Perspectives on Practice and Meaning in Mathematics and Science Classrooms (Mathematics Education Library)

PERSPECTIVES ON PRACTICE AND MEANING IN MATHEMATICS AND SCIENCE CLASSROOMS

Mathematics Education Library VOLUME 25

Managing Editor A.J. Bishop, Monash University, Melbourne, Australia

Editorial Board H. Bauersfeld, Bielefeld, Germany I.P. Becker, Illinois, U.S.A. G. Leder, Melbourne, Australia A. Sfard, Haifa, Israel O. Stovsmose, Aalborg, Denmark S. Turnau, Krakow, Poland

The titles published in this series are listed at the end of this volume.

PERSPECTIVES ON PRACTICE AND MEANING IN MATHEMATICS AND SCIENCE CLASSROOMS

Edited by

DAVID CLARKE University of Melbourne, Victoria, Australia

KLUWER ACADEMIC PUBLISHERS NEW YORK / BOSTON / DORDRECHT / LONDON / MOSCOW

eBook ISBN: Print ISBN:

0-306-47228-7 0-792-36938-6

©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2001 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

http://kluweronline.com http://ebooks.kluweronline.com

Editor’s Dedication To Sharon

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TABLE OF CONTENTS Preface Acknowledgements Chapter 1: Negotiating Meanings - An Introduction

ix xi 1

David Clarke

Chapter 2: Complementary Accounts Methodology

13

David Clarke

Chapter 3: Untangling Uncertainty, Negotiation, and Intersubjectivity

33

David Clarke

Chapter 4: Accounting for Accounts of Learning Mathematics: Reading the ZPD in Videos and Transcripts

53

Stephen Lerman

Chapter 5: Mathematical Interactions and Their Influence on Learning

75

Derek Holton and Gill Thomas

Chapter 6: Interest in Learning and Classroom Interaction

105

Mary Ainley

Chapter 7: Cognitive Engagement in the Mathematics Classroom

131

Sue Helme and David Clarke

Chapter 8: Measuring Values in Classroom Teaching and Learning

155

Carol Lewis-Shaw

Chapter 9: Opportunities to Learn Science? Multiple Contexts at Work in a Science Classroom

197

Susan Rodrigues

Chapter 10: The Role of Gesture in Co-constructing Mathematical Understanding

231

Robert Reeve and Fiona Reynolds

Chapter 11 : Learning and Teaching from Ignorance to Understanding

255

John Baird

Chapter 12: Teaching/Learning

2 91

David Clarke

Appendices: A. Summary of Lessons Analysed B. Sample Integrated Text Author Index Subject Index

321 331 339 347

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PREFACE This book represents the culmination of an unusual process of interpretation, negotiation and synthesis. It is the realisation of a vision of an integrated description of classroom practice and learning that motivated the research team members, who are also the chapter authors, during their analysis and in the subsequent integration of their results into the kaleidoscope that is this book. It is not complete. We would be the first to assert that any portrayal of a social setting as complex as a classroom is inevitably incomplete. But not, we think, without value. In part, it was our perception of the inadequacies of previous descriptions of classroom teaching and learning that initiated the complex process leading to the book you now hold in your hands. It is not our intention to reveal to you what mathematics and science classrooms are “really like”. Such a goal would be in contradiction of the theoretical and methodological positions on which the project has been founded. A classroom takes on a different aspect according to how you are positioned within it or in relation to it. Any claim to a single, definitive or consensus view of the classroom ignores this. As you will discover, the analyses reported in this book were not undertaken with consensus in mind. Complementarity was our reference point and our goal. This places you, as reader, in an interesting position. You will discover that two (or more) chapters that discuss the same lessons, and even the same events, do not report the same findings, address the same issues, or draw the same conclusions. Consensus was not our goal. Teaching and learning are complex practices. Only a multi-faceted analysis of classrooms can hope to do justice to the complexity we find there. No single person is equipped to carry out the essential diversity of analyses. And so the logic of the multi-perspective research team is framed. A team of researchers, each with particular areas of expertise, contributed their analyses of a shared collection of classroom data. The result is a variegated picture of science and mathematics classrooms that challenges a research tradition that converges on the truth. In this book, we surround you with different images of the classroom. Some will hopefully address issues of interest, some will confirm beliefs you have long held, some will challenge these same beliefs, and some may surprise you. There were certainly enough surprises to sustain our interest, we hope there are enough to sustain yours. David Clarke January, 2001

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ACKNOWLEDGEMENTS The research reported in this book benefited substantially from funding awarded by the Australian Research Council. Several individuals assisted in the data collection process in the course of this project. Particular thanks are due to Cathy Kessel, Alison Ashe, and to Shokria Niazmand. Technical support was provided by Eric North, Tim Mason, Scott Smallacombe, and Simon Clough. Their patient assistance was greatly appreciated. The Department of Psychology, UCLA, and Professor James Stigler, provided facilities essential to the collation of the final manuscript during the Editor’s sabbatical year at UCLA. The formatting of the final manuscript was supported by a grant from the University of Melbourne. A substantial proportion of the editing of this book was only possible due to the hospitality and tolerance of Yvette and the staff of Starbucks Coffee, Beverly Glen Circle, Los Angeles, California, USA. Grateful thanks are extended for coffee, conversation, and a wonderful work environment. Similar thanks are extended to James and his colleagues at Genevieve on Faraday in Carlton, Victoria, Australia, where the proofing of the final manuscript was sustained with coffee and good humour. The Editor would like to extend his thanks and appreciation to all contributing authors (colleagues and friends) for their tolerance of his pedantry and for their patience. Finally, and most importantly, the research team would like to extend their thanks to “Mrs Brown”, “Mr Black” and their students for their willingness to participate in this project, for their time, the thought they gave to their interview responses, and the insights that these provided. This book is very much a legacy of their generosity.

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DAVID CLARKE

CHAPTER ONE Negotiating Meanings - An introduction

Classrooms are complex social settings, and research that seeks to understand the learning that occurs in such settings must reflect and accommodate that complexity. This accommodation can occur if your data collection process generates a sufficiently rich data set. Such a data set can be adequately exploited only to the extent that the research design employs analytical techniques sensitive to the multifaceted and multiply-connected nature of the data. The challenge for this type of classroom research is to portray teaching and learning as these occur in the highly complex social context of the mathematics or the science classroom. In order to meet this challenge, we had to develop a methodology that was in harmony with our theoretical orientation to the classroom. If we, in fact, approach social settings (and the situations they frame) as multiply-constructed and open to multiple construal, then the methodology employed in their study must offer a voice to the several participants in these settings and avoid the identification of authority with any one voice (even that of the Researcher). We must avoid the threat of over-simplification of setting or situation; a threat more likely to be realised if we were to commit to a single interpretation. Instead, we need to acknowledge the multiple potential meanings of the situations we are studying by deliberately giving voice to many of these meanings through accounts both from participants and from a variety of “readers” of those situations. The implementation of this approach requires the rejection of consensus and convergence as options for the synthesis of these accounts, and instead accords the accounts “complementary” status, subject to the requirement that they be consistent with the data from which they are derived, but not necessarily consistent with each other, since no object or situation, when viewed from different perspectives, necessarily appears the same. Complementary Accounts Methodology, as we have developed and applied it (Chapter 2, and Clarke, 1998), is distinguished from other approaches to classroom research by: _ The nature of the data collection procedures, leading to the construction of "integrated data sets" combining videotape and interview data, _ The inclusion of the reflective voice of participant students and teacher in the data set,

1 David Clarke (ed.), Perspectives on Practice and Meaning in Mathematics and Science Classrooms, 1—11. © 2001 KluwerAcademic Publishers. Printed in the Netherlands.

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An analytical approach that utilises a research team with complementary but diverse areas of expertise to carry out a multi-faceted analysis of a common body of classroom data. The object of our research was the integrated documentation of not just the obvious social events that might be recorded on a videotape, but also the participants’ construal of those events, the memories, feelings, and actions invoked, and the mathematical/scientific and social meanings and practices which arose as a consequence. The research procedure recounted here was designed explicitly to achieve this integration. Each member of the research team was encouraged to select from and interpret the data sets from a distinct, carefully articulated, theoretical perspective. The goal of this process was complementarity rather than consensus, and each researcher’s interpretation was subject to the same criteria of coherence, consistency with the videotape data, and plausibility. The synthesis of such accounts is not a matter of unification through consensus into a single account, but rather the interweaving of accounts providing in their combination a richer portrayal of the classroom. Further, an interpretation from one perspective may support, inform, justify and even explain the account constructed from another perspective. While you, the reader, will carry out the most important synthesis of our analyses, one synthesis is provided in Chapter 12. The authors of the chapters in this book report their analyses of a common body of classroom videotape and interview data relating to four mathematics lessons and four science lessons selected for analysis from the substantial data set generated by the Classroom Learning Project, based at the Faculty of Education, University of Melbourne, Australia (see Appendix A). The method of data collection and the rationale of lesson selection are given in chapter two. Each researcher or pair of researchers has subjected the data to an analysis informed by a particular theoretical perspective and consistent with the specific research expertise of that individual or team. While the lessons that constitute the primary data source took place in Australia, the Classroom Learning Project has been an international research effort and the participating researchers come from Australia, New Zealand and the United Kingdom. A key feature of our portrayal of the classroom is the juxtaposition of researchers’ quite different theoretical emphases. Our portrayal (the content of this book) derives its coherence and focus from the shared concern of all researchers with the construction of meaning and practice in the mathematics and science lessons studied and our interest in furthering our understanding of the discourse options and the operative practices within mathematics and science classrooms, how these position participants, and how discourse and practice appear to facilitate or inhibit the learning of mathematics and science. The most obvious source of coherence in our work is provided by the common data base accessed by all researchers, allowing the research team and the readers of this book to consider the compatibility and utility of conceptually independent analyses of the same data. While each analysis has its own message and integrity, what benefit do we, as a community of educators, derive from the combination of -

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analytical perspectives on the same lessons? It is our intention in this book to present independent analyses of a common body of classroom data and to address the implications of these analyses for our understanding of the practices prevalent in these classrooms and their role in the learning of mathematics and science. The authors of each chapter drew from the data set to different degrees. Some chapters analysed only the mathematics lessons (Chapters 4, 5, 7 and 10), others only the science ones (Chapters 6, 8, 9 and 11). The Contents listing identifies the academic content focus of each chapter. Some chapters, such as Chapters 5 and 6, analysed several lessons, others, such as Chapters 4 and 9, focussed on only one lesson. In each case, the data were employed in a manner consistent with the researcher’s analytical approach. The authors of each chapter set out the type of analysis employed and relate this to their use ofthe available data. Our accounts are framed in terms more or less familiar to the reader: negotiation, meaning, intersubjectivity, uncertainty, metacognition, context, interest, engagement; the list is considerable. Some chapters specifically address the need to ground these ill-defined terms empirically and to situate them in relation to each other. Chapter 3, for example, addresses negotiation as a social process related to the practices of the mathematics classroom and associates the need for negotiation with the occurrence of uncertainty. Data are presented to illustrate the role of intersubjectivity as both mediating agent and product of the negotiative process associated with the resolution ofuncertainty. The argument put forward in chapter 3 is as follows: Negotiation depends on language (or at least on some form of communicative process, cf Reeve and Reynolds in Chapter 10 on the use of gesture), and language is constitutively intersubjective (Todarov, 1984, p. 30). Thus, a level of student-student and teacherstudent intersubjectivity is prerequisite to the negotiative processes occurring in classroom settings and, moreover, appears to be characteristically prompted by the occurrence (and recognition, see Baird in Chapter 11) of uncertainty. The demonstration of this pre-existing or previously-accomplished intersubjectivity is a key element of the argument advanced in Chapter 3. Specific examples of classroom data are reported in Chapter 3 in which the relationship between uncertainty, negotiation and intersubjectivity is made explicit. Central to Chapter 3 is an analytic frame by which classroom dialogue is parsed using an hierarchical scheme of which the critical element is the “negotiative event.” Rather than attending analytically to the separate utterances of the individuals participant in the interaction, the negotiative event identifies the interaction itself as the object of interest. This accords primacy to social interaction. It will be argued, however, that an individual’s participation in that interaction is a consequence of multiple interpretive acts associated with the participant’s social affiliations and with the iterative refinement of meaning and practice. The reflexive nature of the relationship between social interaction and individual cognition is a common premise of many of the analyses reported in this book. Since the setting with which we are dealing is fundamentally social, a Vygotskiist perspective on learning and teaching and the social interactions recorded within these settings is evident in many chapters, even where this is not explicitly identified as such by the chapter authors. A Vygotskiist perspective is most explicit

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in the theoretical stance adopted by Lerman in Chapter 4. Lerman’s strategy was to look for learning in the zone of proximal development (ZPD) (Vygotsky, 1978). Chapter 4 provides an account of that theoretical ‘space’, how it is recognized and what it offers to our investigation of the classroom. Issues of justification, validity and coherence are also addressed. The particular extracts of classroom dialogue analysed by Lerman are of sustained conversation about mathematics between two peers. Lerman’s analysis suggests that zones of proximal development (ZPD) were not created in the interactions studied, but it is his assertion that this reveals as much about learning as might be revealed in instances where a ZPD is created. The interpretation of Vygotsky’s ZPD that is used in Lerman’s analysis (see also Lerman, 1994) is close to that of Davydov (1988) and is perhaps less ‘internalist’ than many other interpretations of Vygotsky’s own formulation. In this view, a learning activity proffered by the teacher or a researcher (Newman, Griffin & Cole, 1989; Brown, Ash, Rutherford, Nakagawa, Gordon, & Campione, 1993), or a text, or peer, can set up a ZPD for each of the participants. In relation to classrooms, Brown et al. (1993) have described what arises as “multiple overlapping zones of proximal development”. The interpretation of the ZPD used in Lerman’s analysis, however, emphasises the intersubjective origins of all learning, and therefore is focused on the learning activity. For Lerman, the roots of psychogenesis for each individual are in sociogenesis, which is the focus through which to read the internalisation by each individual. The process of analysis involved searching for events of communication in the videos and the transcripts in which the students “catch each other’s thoughts.” Such instances were used as clues to the sociocultural settings which affect the creation or otherwise of a ZPD. In the particular classroom interaction discussed in this chapter, the students’ different focuses in the mathematics and the nature of their relationship appeared to preclude the creation of a ZPD. Lerman’s analysis reinforces the centrality of social interaction in the classroom in the creation of mathematical meaning. Chapter 5 utilizes the concept of scaffolding to examine teacher-student and student-student interactions. The authors, Holton and Thomas, examine scaffolding and its use by “a more competent other” and then extend this notion to students’ use of questioning in a fashion that resembles scaffolding. They propose that this questioning can be considered an externalisation of metacognition as a group process and characterise this process as “reciprocal scaffolding”. Metacognitive selfinterrogation is referred to as “self-scaffolding.” The chapter provides examples of reciprocal scaffolding in practice. Importantly Holton and Thomas call into question the need for the “scaffolder” to be the more competent other, and extend this proposition to consideration of the alternative scaffolding roles that a teacher might adopt. Holton and Thomas also identify instances of learning occurring in group situations. In particular, they report that the inferred cognitive processes were catalysed by talk. They conclude that much of what was achieved in dyads in the mathematics lessons studied could not have been achieved as quickly, if at all, by whole class teaching or by students working alone. One framework used by Holton and Thomas to analyse the interactions between students was adapted from a scheme developed by Bennett and Dunne (1992). In this framework, student talk was

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AN INTRODUCTION

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progressively partitioned through a series of dichotomous pairs: Non-Task/TaskRelated; Social/cognitive; and Action/Reflection and this classification scheme is illustrated empirically in Chapter 5. A quantitative analysis of all recorded student interactions utilizing this scheme provided significant evidence of worthwhile taskrelated talk in the lessons analysed in this chapter. The nature of the mathematical interactions was the focus of a further “layer” of analysis. A feature of the analysis carried out by Holton and Thomas was the identification of interactions in which students displayed significant mathematical thinking. A case is put that in relation to one of the posed problems, the teacher did not know the answer. This raises the question of whether the teacher can then provide the necessary scaffolding to sustain student learning. It is the researchers’ contention that she can and did. The teacher’s specific competence in this instance was the knowledge of how such problems work and the steps that she herself would go through to solve them. As a consequence, the teacher provided a form of less specific guidance that actually promoted some very useful mathematical interactions among the students. These opening chapters share elements of a common vocabulary that was refined in the many conversations, personal and electronic, that accompanied the genesis of this book. Differences in interpretation and usage among members of the research team were a source of fruitful exchange. The reader will still discern differences of emphasis and interpretation among the multiple accounts voiced in the chapters of this book. Our tolerance of difference in interpretation extends even to the choice of terms we have employed, but not to the point where the meaning ascribed to a term by the author(s) of one chapter is in conflict or contradiction with that employed in another chapter. The reader of this book will hear several voices, which, while not all singing the same note or even the same tune at the same time should be heard as a harmony rather than a discord. In fact, it was one of the revelations of the group interpretive process that our multiple accounts were in such sympathy and accorded significance to such a coherent set of related terms. In Chapter 6, Ainley starts from a common consensus as to what are the indicators of interest and then proceeds to identify and discuss the processes whereby interest emerges from what is not interest. The analytical approach is straightforward: A point at which an individual student showed signs of interest can be identified. It is then possible to sift through what preceded that event to try to identify what it was that aroused interest and also to identify what followed. Ainley’s discussion of the link between uncertainty and engagement resonates with Chapters 3 (Clarke) and 7 (Helme and Clarke), in which uncertainty is identified as a key element in the negotiation of meaning and related to student cognitive engagement. In fact, the significance of uncertainty is raised in several chapters. The pervasiveness of uncertainty throughout this book is one of several structural patterns emerging from the synthesis of the various accounts offered in the different chapters (Chapter 12). Ainley distinguishes interest as a personal trait from interest as a characteristic of the learning environment from interest as a psychological state, and accords particular significance to the third aspect of interest. In Chapter 6, Ainley proposes particular indicators of interest as evident in the data. One example is particularly

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compelling in which a student’s question, unanswered in one lesson, is pursued again by the student in a subsequent lesson some days later. The process whereby “Joanne” resolved her uncertainty over the question “Does paper have fibres in it?” is used by Ainley to illustrate the way in which interest can support learning. Examples of other indicators are presented and the identification of “boredom” is usefully employed as a counter-example to consolidate and clarify our conception of “interest.” Ainley demonstrates powerfully that even those activities intended to arouse student interest may miscarry even while they succeed, in that they arouse an interest that is not pursued within the lesson. It is enlightening, in this regard, to juxtapose Ainley’s analysis of this lesson with that of Rodrigues in Chapter 8. The chapter includes the happy phrase “providing opportunities for students’ interest to be raised in the service of learning.” The distinction, with which Ainley concludes, between “Interest in Learning” and “Actual Learning” focuses the reader’s attention appropriately on the learning consequences of the immediate, but intermediate, goal of fostering and arousing student interest. Helme and Clarke, in Chapter 7, define cognitive engagement as the deliberate task-specific thinking that a student undertakes while participating in a classroom activity. Most studies that have investigated the link between cognitive engagement and learning have relied on imprecise and indirect measures of cognitive engagement such as time on task, or student self-reports. The consistent approach employed in the Classroom Learning Project (and in this book) has been to avoid sole reliance on student self-reports of cognitive behaviour and to seek evidence of constructs such as cognitive engagement within videotapes of classroom behaviour, supplemented by student reconstructive accounts of the incidents and behaviours evident on the videotape. Helme and Clarke examined cognitive engagement in four distinct classroom situations—parallel and collaborative classroom activity, whole class instruction, and student-teacher interaction in a small group—and observed that different patterns of cognitive engagement appeared to characterise each type of activity. From this evidence, Helme and Clarke established that cognitive engagement can be consistently recognised in classroom situations by specific linguistic and behavioural indicators. Student-student interactions appeared to offer more scope for high-level cognitive engagement than teacher-student interactions, both in whole-class instruction and in small groups Among other findings, Helme and Clarke identify some important differences between peer interactions and student-teacher interactions in this classroom. These differences are particularly apparent in the comparison of students working in small groups with and without the teacher present. Helme and Clarke note that the arrival of the teacher triggers a shift away from enquiry and problem solving by the students to explanation and justification of their solution to a “higher authority”. This shift was accompanied by a very different pattern of interpersonal interactions, with the overwhelming majority of student utterances (twelve out of fourteen) being responses to teacher questions. Helme and Clarke argue that in situations dominated by teacher questioning there was little opportunity for students to initiate ideas or spontaneously express and resolve uncertainty. However, it should be noted that the teacher’s use of open-ended questions appeared to provide useful cognitive and metacognitive scaffolding. It is suggested by Helme and Clarke that the closed

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questioning that characterises much of classroom practice is unlikely to provoke much thought on the part of students and consequently is likely to curtail higher level cognitive engagement. Lewis-Shaw opens Chapter 8 with the premise that “the extent to which a teacher can appropriately motivate and help to enhance the capacity of the learner is dependent upon the teacher’s understanding of his or her own value position” and her analysis provides a fascinating dissection of the teacher’s values and the extent to which these are similar to or different from those of a particular group of three students. To achieve this comparison, Lewis-Shaw applies the analytical framework of Hall and Tonna (Hall, 1994) to an 8th grade science lesson involving a heart dissection. The Hall-Tonna framework employs 125 values with standardised definitions to provide a common language by which to describe the values extant in the statements recorded during the particular science lesson. The text analysis applied by Lewis-Shaw is both quantitative and qualitative in approach. The researcher begins with her own interpretation of what the science lesson participants appear to value, the meanings and senses in which language and behaviour are used, and then iteratively systematises the findings. The transcription of the science lesson was first analysed by Lewis-Shaw using the Hall-Tonna definitions to identify encoded values. Through trial and error, analysis became more discerning, tapping into tacit knowledge and value assumptions. Evidence was sought of value patterns that could characterise the speaker regardless of context. Lewis-Shaw’s careful analysis demonstrates that there are differences in the value ranking held by the teacher and the particular three-student group being studied. Each ranking is taken to represent a different world-view. Both teachers and learners continuously engage in a process of determining which values are held in common and the extent to which value difference exists. In fact, negotiation proceeds continuously at the affective level, whether the teacher and learner are conscious of it or not (Lewis-Shaw, chapter 8, page 200).

Lewis-Shaw illustrates the significance of an understanding of values for an adequate understanding of the interactions of the science classroom (or, we would suggest, any other classroom). The point is made that insights emerge both from the values demonstrably present in classroom interactions and those apparently absent. Further, values underlie the power relationships in operation in the classroom and assist our understanding of some of these relationships. The implications for teacherstudent rapport of the match and mismatch of values are also discussed. At the heart of the interactions studied in this one lesson, we find the negotiation of values. The recurrence of negotiation as a central characteristic of the classroom interactions analysed by Lewis-Shaw resonates with the other analyses in this book. We emerge with the impression that classrooms are alive with negotiative activity, but that the negotiations may be simultaneously concerned with mathematical or scientific meanings, social or procedural meanings, practices or values. Rodrigues, in Chapter 9, examines the construct “context” as it relates to one particular science lesson. Rodrigues identifies three facets of context (communication, resources, processes) and argues that the lesson in question “became a student inappropriate context” because aspects of those three facets were

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not addressed. In particular, Rodrigues argues that contemporary enthusiasm for teaching science “in context” has led to confusion among teachers, who find themselves exhorted to “use” contexts to teach science, but who have little guidance with regard to what this entails. Contexts are favoured that are “familiar” to students, however if, as Rogrigues asserts, context is constructed and reconstructed in situated practice and by the individual, the extent to which these everyday contexts become student appropriate contexts from which to learn science concepts is dependent upon both the operations used within a situation and the situation itself. In Chapter 9, the author identifies various potential hazards for the effective construction of meaning (for example, differences in meaning between ‘everyday’ and ‘technical’ uses of common terms). The effects of these hazards are then illustrated by reference to the chosen lesson. Rodrigues describes the process whereby students struggled to make sense of terms that were used and ideas that were suggested. The negotiative process engaged in by the students is contrasted with the communication between teacher and student. While student-student exchanges had more negotiative potential than those between teacher and student, they occurred within a context constrained by a shared subscription on the part of all participants to the tenets of the traditional science classroom and dominated by the need for students to identify and meet the teacher’s expectations. Rodrigues quotes one student’s statement: “Everything has more than one meaning” and illustrates how the potential richness of the students’ interactions remained unrealised under the dominance of the perceived classroom goal to “make the longest possible list.” Rodrigues also raises the problematic legitimacy of interest as an indicator of likely effective learning. Chapter 10, by Reeve and Reynolds, explores the significance of gestures in a collaborative learning setting where two students are trying to make sense of, and solve, a mathematical problem involving speed and time. Of particular interest to the authors is how the students use gesture themselves and in how they interpret the gestural meaning of others in order to co-construct an understanding of mathematical ideas. In particular, the analysis reported examines the claim that gestures serve various signaling functions in collaborative problem solving communication generally, and in mathematics problem solving more specifically (Roschelle & Clancey, 1992). The overall purpose of this chapter is to illustrate the integral role of gesture in dyadic communication and to argue that its theoretical significance in analyzing collaborative problem solving has been overlooked. The general thesis is that gesture is among the first devices employed by the preverbal infant for sustaining joint-attention and that it continues to possess similar communicative significance throughout the life span. This analysis offers evidence that gesture signals both communicative difficulties and shifts in understanding. Reeves and Reynolds suggest that it is highly plausible that gestures may be a particularly important communication medium in individuals who are less verbal or who do not possess an adequate grasp of a particular domain “language” (for example, some aspect of mathematics) that would allow them to engage in meaningful discussions about that domain. Finally, it is proposed that researchers must focus on the interpretive significance of different kinds of gestures in sustaining ongoing communication. In particular, the authors suggest that it is

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important to establish whether there is a lexicon of gestures and whether this lexicon is unique to situations or individuals and whether it can be interpreted by nonparticipant onlookers (for example, teachers; Alibali, Flevares & Goldin-Meadow, 1997) to help understand the confusion of those involved in collaborative problem solving, The chapter provides an outline of a possible framework for analysing the role of gesture in learning settings more generally. Baird’s analysis reported in Chapter 11 illustrates more than any other author/researcher the reflexivity in the teaching/learning dyad (so frequently read as a dichotomy) that typifies our reading of the classroom. Chapter 11 adopts two separate stances and examines both the learner and the teacher through a common lens, Baird postulates a learning model that employs ten hierarchically related constructs. It is the construct “Ignorance” and its role in learning that serves to focus Baird’s analysis of particular incidents in the science classrooms studied. Consideration of the learner’s awareness or non-awareness of ignorance precedes analysis of their consequent actions and Baird employs “volitional control” as a vehicle for discussing how learners might be empowered to facilitate their own learning. In particular, “perceptions of self, task and context were demonstrated to be central to levels of motivation, perseverance, and informed progression through metacognitive and volitional control” (Baird, Chapter 11, p. 326). The characterisation of desirable and undesirable learning situations is informed by Baird’s analysis of situations occurring in the science lessons included in common data base. Whether we look to the contemporary French use of “didactique” or the Russian “obuchenie”, we find that other communities have acknowledged the interdependence of instruction and learning by encompassing both activities within the one process and, most significantly, within the one word. If our framing of “instruction” in language presumes a “learner” and if “learning” is inextricably entwined with an “instructive” setting, then our interpretations of the activities of the classroom are more likely to identify communal practices and the progressive participation in a common discourse as essential features than to fragment the classroom into teaching and learning activities undertaken by individuals. Is it “learning” or “teaching” that creates the zone of proximal development? Vygotsky’s intention, surely, was to signify that it was in social interaction, where learners and teacher converse, and where the more able other will frequently be a classmate, that the zone of proximal development was co-constructed for each participant (rather than by each participant). These issues are addressed from different perspectives in Chapters 3, 4, and 5, and discussed in more detail in Chapter 12. It is a major contention of this project, elaborated in Chapter 12, that classrooms are more effectively understood as sites for bodies of mutually-sustaining practice that in combination characterize a process we might call (in English) “teaching/learning.” Baird’s two-part chapter examines the “teacher” and the “learner” in the same science classroom, participant in the same activities; and examines them through essentially the same lens. But co-construction and joint participation do not connote commonality of purpose. Among other things, teachers and learners are distinguished by the goals they bring to the classroom situation. Baird’s analysis addresses this difference and implicitly poses two different questions with respect to

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the classroom participants. For the learner: How best should s/he act (includes “think”) in order to benefit from participation in the activities of the science classroom? For the teacher: How best should s/he act (and think) in order to maximise the benefit accruing to the student from their participation in the activities of the science classroom? In that sense, both teacher and learner share a common purpose, but they are not positioned identically within that purpose, and their classroom participation will both confirm these positionings and co-construct them. The problems of synthesis

Separately, the chapters of this book offer a variety of perspectives on mathematics and science classrooms, and most particularly on the practices and meanings that characterise such settings. It is our contention that the interpretations offered in each chapter hold significance for the promotion of learning in other settings. These interpretations are our “readings” of the combined accounts offered by the videocamera, and the transcribed reconstructive accounts obtained from classroom participants. These accounts are the evidence on which our various readings are founded. Any number of interpretations, guided by any number of interests, can be built on the same foundation of evidence; but an interpretation ignoring that evidence can never be a defensible one (Vendler, 1997, p. 24).

In this book, we construct different interpretations based on a common body of data. Helen Vendler, quoted above, is introducing her interpretations of Shakespeare’s sonnets. In her introduction, Vendler is critical of the type of literary critic who “leaves it up to the reader to construct the poem” and states her goal explicitly: “I have hoped to help the reader actively to that construction by laying out evidence that no interpretation can ignore” (Vendler, 1997, p. 24). The interpretations offered in each chapter are open to challenge, but not, we hope, for their consistency with the data. To the extent that it is essential to our argument, we have presented the data to you. We have not abdicated the researcher’s responsibility to articulate the meanings we constructed. Like the reader of a sonnet, you are free to construct a different meaning from ours; but, like Vendler, we hope that your interpretation will be informed by the “evidence” we have chosen to highlight and enriched by the perspectives offered in each chapter. There is no doubt that there are other interpretations that could be constructed on the same foundation. Different readers will find different messages of hope, discouragement, insight and challenge in our accounts. Those of us seeking to interpret classrooms do so in the hope of improving them as sites for teaching/learning. It is our hope that the work we have commenced will continue and that the culmination will be a richer understanding of teaching/learning in classroom settings: the forms it takes, the practices through which and in which it is constructed, and the means by which it might be promoted.

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REFERENCES Alibali, M. W., Flevares, L.M., & Goldin-Meadow, S. (1997). Assessing knowledge conveyed in gesture: Do teachers have the upper hand? Journal of Educational Psychology, 89, 183-193. Bennett, N. and Dunne, E., 1992. Managing Classroom Groups. Hemel Hempstead: Simon and Schuster Education. Brown, A., Ash, D., Rutherford, M, Nakagawa, K., Gordon, A,, & Campione, J.C. (1993). ‘Distributed expertise in the classroom’. In G. Salomon (Ed.) Distributed Cognitions: Psychological and Educational Considerations,, pp. 188-228, New York: Cambridge University Press. Clarke, D.J. (1996). Refraction and Reflection: Modelling the Classroom Negotiation of Meaning. RefLecT 2( l), 46 - 51. Clarke, D.J. (1998). Studying the classroom negotiation of meaning: Complementary accounts methodology. Chapter 7 in A. Teppo (Ed.) Qualitative research methods in mathematics education. Monograph Number 9 of the Journal for Research in Mathematics Education. Reston, VA: NCTM, pp. 98-111. Davydov, V. V. (1988) Problems of developmental teaching. Soviet Education, 30, 6-97. Hall, B.P. (1994) Values Shift: Personal and Organizational Development. Rockport, MA: Twin Lights Publishers. Lerman, S. (1994) 'Towards a unified space of theory-and-practice in mathematics teaching: a research perspective. In L. Bazzini (Ed.) Proceedings of the Fifth International Conference on Systematic Cooperation between Theory and Practice in Mathematics Education, Universita degli Studi di Pavia, p. 133-142. Newman, D., Griffin, P. & Cole, M. (1989) The Construction Zone. New York: Cambridge University Press. Roschelle, J., & Clancey, W.J. (1992). Learning as social and neural. Educational Psychologist, 27, 435453. Todarov, T. (1984). Mikhail Bakhtin: The Dialogical Principle. Translated by Wlad Godzich. Minneapolis: University of Minnesota Press. Vendler, H. (1997). The Art of Shakespeare’s Sonnets. Cambridge, Mass: Harvard University Press. Vygotsky, L.S. (1978). Mind in Society. Edited by M. Cole, V. John-Steiner, S. Scribner, & E. Souberman. Cambridge, Mass: Harvard University Press.

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CHAPTER TWO Complementary Accounts Methodology

1. INTRODUCTION The research procedure employed in this project was developed in an attempt to study learning in legitimate classroom settings, while minimizing researcher inference regarding participants’ thought processes and maximizing the richness of the research database. It is unlikely that any individual has the extensive expertise or the time required to adequately analyse such complexity. The formation of multidisciplinary research teams for the purpose of analysing such complex data sets is an obvious step. As a consequence, the interpretive process enacted by such teams has become the subject of methodological discussion (Davidson Wasser & Bresler, 1996). This chapter outlines the method of generation and integration of the multiple accounts reported in the chapters of this book. Included in the multiple accounts of classroom data are the voices of the classroom participants, whose retrospective reconstructions of events provide another form of data for interpretive analysis by the research team. Underlying the research approach outlined in this chapter is the question of whose meaning is the product of such research. Much of classroom research is fundamentally about meaning: The content-related meanings constructed by students during classroom activity; the meanings ascribed by the participants to the social activity by which classroom interaction is sustained; and the meanings constructed by the researcher for the observed practices of a particular classroom. It is a trite but essential observation to assert that a research report can only ever document the researcher’s constructed meanings. When the researcher’s goal is the portrayal of the process whereby children construct mathematical or scientific meaning (for example), the question of whose meanings are the matter of the research report must be addressed with care. When the single researcher is replaced by a multidisciplinary research team, the reconciliation of alternative meanings becomes a methodological issue of significance. This methodology and the Classroom Learning Project in general, are predicated on a conception of the researcher as learner. Given this, the consistency in application of conceptions of negotiation, meaning, and the role of complementary 13 David Clarke (ed.), Perspectives on Practice and Meaning in Mathematics and Science Classrooms, 13—32, © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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accounts to the activities of both learner and researcher, has led to a high level of correspondence between research practice and applied learning theory. The themes explored in this chapter, which guided the development of the project methodology, arose in large part from the project director’s sense of the extent to which the research process mirrors the process being researched, and represent a conscious attempt to subject the methodology to a form of scrutiny and analysis comparable to that being applied to the data. It should be evident from the discussion thus far that the research reported in this book presumes a significant similarity between classroom learning and the activities of an interpretive research group. It is a major premise of this chapter that the practice of any researcher enacts a theory of learning (and of knowledge) that structures the data that is collected and affords certain interpretations of that data while constraining other possible interpretations. The researcher has an obligation to anticipate the possibility that this correspondence between research practice and theories of learning and epistemology might become prescriptive, constraining rather than informing data interpretation. The use of a multi-disciplinary research team has the virtue of subjecting each team member's account to the critical scrutiny of several colleagues. Differences in the theoretical positions adopted by research team members constitute an important safeguard against methodological monism or myopia. 2. COMPLEMENTARY ACCOUNTS METHODOLOGY The focus of this chapter is the specification of a qualitative research approach called Complementary Accounts Methodology (see also Clarke, 1998). Available technology was utilized to combine videotape data with participants' reconstructions of classroom events. This integrated data set then provided the basis for complementary accounts constructed by the research team. As outlined in Chapter 1, Complementary Accounts Methodology is distinguished from other approaches to classroom research by: _ The nature of the data collection procedures, leading to the construction of "integrated data sets'' combining videotape and interview data, _ The inclusion of the reflective voice of participant students and teachers in the data set, _ An analytical approach that utilises a research team with complementary but diverse areas of expertise to carry out a multi-faceted analysis of a common body of classroom data. An interest in an empirical elaboration of the theoretical constructs of meaning and classroom negotiative processes prompted the development of the general interpretive framework employed in this the study, and therefore the method of data collection. Classroom negotiative processes, as conceived in this study, involve communication, interpretation, intersubjectivity, the iterative refinement of meaning, and accommodation. Such processes are most evident in situations characterised by student uncertainty, and in those interactions whereby conjectures and arguments

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arising in classroom discourse are compared and assessed (including the development of social context meanings). With regard to meaning: The presumptions of meaning are community, purpose and situation. It is not the word that has meaning, but the utterance. The emphasis on utterance, derived from Bakhtin (1979), is evident in Chapter 3 in a discussion and illustrative analysis focussing on intersubjectivity. Similarly, the situatedness of meaning is emphasised by Lewis-Shaw in Chapter 8. In Chapter 10, Reeve and Reynolds extend the consideration of communication to include gesture. In every chapter, researchers draw inferences about the meanings of practices and communicative acts. These inferences are grounded in consideration of the situation in which the act occurred, the purposes of the individuals involved, and the social (and sometimes cultural) groups with which each individual might claim affiliation during the interaction in question. The anticipated need to document the interactive constitution of meaning through an iterative process of communication and interpretation within social groups situated in mathematics and classrooms posed particular methodological challenges. One challenge for this type of classroom research was to portray the learning process enacted by individuals and groups participant in a highly complex social context. This learning process was taken to be an integration of not just the obvious social events which might be recorded on a videotape, but also the individual's construal of those events, the memories, beliefs and attitudes invoked, and the meanings and practices which arose as a consequence. The research procedure recounted here was designed specifically to achieve this integration. The data base for the project consisted of videotape records of fifty-five secondary mathematics and science lessons obtained using two cameras, together with over 100 transcribed interviews with students. In constructing the videotape records, one camera was directed solely at the teacher throughout the lesson, while the other camera was focused on a group of about four students. The teacher's utterances were recorded through a radio microphone, and a single microphone was used to record the conversations of all four students. The two video images were mixed on-site in the classroom as the lesson proceeded to produce a composite picture in which the students occupy most of the screen with the teacher image superimposed in a corner of the screen.

Figure 2. 1. Video Image

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This combined image was recorded onto video-8 tape using a compact video recorder attached to a laptop computer. The researcher, seated at the rear of the classroom, was able to listen simultaneously to both student conversations and teacher utterances and to record field notes onto a word processing document on the computer. The field notes were "time-tagged", as they were made, to corresponding events in the video record using CVideo software (Roschelle, 1992). The field notes enabled the researcher to document impressions of classroom episodes and learning events as well as to provide reference markers for the subsequent interviewing of student subjects and the teacher. Student interviews were conducted immediately after the lesson and were intended to produce reconstructive accounts of the students’ thoughts, feelings, motives, actions and perceptions relating to events during the lesson and to the lesson as a whole. For ethical reasons, under an agreement negotiated with those students being videotaped, the teacher reconstructive accounts were generated several months later, when the teacher was no longer responsible for the teaching of those groups of students. A consistent practice during data collection was the use of a familiarisation period of about a week (that is, about four lessons). During this familiarisation period the full data collection procedure was followed - both in terms of videotaping and interviewing. This period allowed students and teacher to develop an attitude of familiarity and ease with regard to the presence of the video cameras in the classroom and with regard to the interview procedure. The familiarisation period also allowed the research team to optimise camera location and microphone deployment to anticipate classroom configuration and teacher and student movement in order to be minimally disruptive of the lesson. The use of students' and teachers' verbal reconstructions of their motivations, feelings and thoughts requires justification. The circumstances under which such verbal accounts may provide legitimate data have been detailed in two seminal papers (Nisbett & Wilson, 1977; Ericsson & Simon, 1980). Brief excerpts from these papers set out the principal points. Though people may not be able to observe directly their cognitive processes, they will sometimes be able to report accurately about them. Accurate reports will occur when influential stimuli are salient and are plausible causes of the responses they produce (Nisbett & Wilson, 1977, abstract).

Videotapes of classroom interactions constitute such salient stimuli, and individuals' verbal reports of their thoughts and feelings during classroom interactions, when prompted by videos of the particular associated events, can provide useful insights into those individuals' learning behaviour. Ericsson and Simon (1980) identify conditions under which the validity of such accounts is questionable. Inconsistent retrospective reports can be produced as a result of probes that are too general to elicit the information actually sought, and as a result of subjects' use of inferential processes to fill out and generalize incomplete or missing memories (Ericsson & Simon, 1980, p. 247).

Videotapes provide a specific and immediate stimulus that optimizes the conditions for effective recall of associated feelings and thoughts. An individual's

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video-stimulated account will be prone to the same potential for unintentional misrepresentation and deliberate distortion that apply in any social situation in which individuals are obliged to explain their actions. A significant part of the power of video-stimulated recall resides in the juxtaposition of the interviewee's account and the video record to which it is related. Any apparent discrepancies revealed by such a comparison warrant particular scrutiny and careful interpretation by the researcher. Having relinquished the positivist commitment to identifying 'what really happened', both correspondence and contradiction can be exploited. On-site production of the linked field notes and split-screen video record allowed the researcher to conduct the student interviews immediately after the lesson. The videotape record was used in the interview to stimulate student recall of classroom events. The use of the CVideo software enabled the researcher to locate within the field notes reference to actions of the student that seemed to be of significance either to the student or to the researcher. Having found a particular item in the word document, the software could be used to find very quickly the corresponding moment on the video record. This was then played back and discussed. Thus students were able to reconstruct their motivations, thoughts and actions, prompted by the video record of the classroom events. The audio record of the interview provided a third source of data. As noted above, interviews with the teacher were conducted several months later. The teacher viewed the videotaped lesson and paused the tape whenever she wished to comment on events in the classroom that showed something about teaching or learning that she considered to be important. The audio record of the teacher's commentary provided a fourth source of data. The student interview conformed to a consistent structure. While the specific wording of the questions varied a little from interview to interview, the structure of each interview followed this basic pattern: Focus prompt one [before viewing the videotape]: Please tell me what that lesson was about.

Focus prompt two: Was there anything in the lesson which you now know, but which you did not know before the lesson? Please explain what it is that you now know and how you came to know it? Is there a point in the lesson by which you are confident you now knew this new thing? I am going to play the videotape from a few minutes before that point, please watch the video and tell me what you remember doing, thinking and feeling during the events shown on the video.

Focus prompt three (repeated as necessary): Was there a moment or an event in the lesson that you think was particularly important (for any reason)? I am going to play the videotape of that event, please watch the video and tell me what you remember doing, thinking and feeling.

Focus prompt four (repeated, as necessary, and if interview time permitted): There was a particular point in the lesson that I was interested in. I am going to play the videotape of that part of the lesson, please watch the video and tell me what you remember doing, thinking and feeling during the events shown on the video.

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Focus prompt one provided a fascinating documentation of student construal of lesson purpose. Focus prompts two and three were responded to variously, and sometimes in the negative. Focus prompt four was most commonly used by the researcher to draw out student reconstructive accounts of observed occasions of student uncertainty, apparent confusion, student-student debate, disagreement, or other social interactions of a negotiative character. The interview structure was designed to move progressively from the learner’s agenda to the researcher’s. In every case, however, it was the learner’s voice that was given priority and focus prompt four was not pursued until the student had responded as fully as they wished to focus prompts one, two and three. The protocol for the teacher interviews involved handing the teacher the remote control device for the video player and asking her/him to replay the lesson while providing a moment-by-moment spoken commentary into an audio recorder. If the teacher felt that a particular event warranted lengthy comment, they were encouraged to pause the videotape and provide that comment. Sometimes this would occur where the interviewer felt additional detail was needed for the teacher’s commentary to be understood. Such prompts from the interviewer might relate to the teacher’s motives in using a particular task or taking a particular action, or the teacher’s interpretation of a student’s response, or the teacher’s comments on a particular student-student interaction. The interviewer’s role was minimally directive or interrogatory, and was more concerned with obtaining sufficient elaborative detail for the teacher’s commentary to be understood. 2. 1 The School, the Teachers and the Students The school that provided the site for all the lessons collected was a co-educational private school in outer suburban Melbourne. Students attending the school were drawn from a variety of communities that shared a level of affluence sufficient to afford the cost of private school education, but which would not be considered “elite” or “wealthy” in the sense in which those terms would be employed locally. The school principal provided written approval for the school’s participation in the study, but did not otherwise have any active involvement in the project. In particular, there was no sign that teachers involved in the study received either specific encouragement or discouragement from the school administration regarding their participation. The mathematics and science teachers whose lessons were videotaped were all volunteers. Participating teachers received no monetary remuneration from the research team for their involvement. Teachers were told by the project director that their participation would contribute to a project intended to inform and improve classroom teaching and learning. As a consequence, teacher participation in the study should be seen as an act of professional goodwill on the part of the individual teachers. The participating teachers agreed to have their lessons videotaped and their students interviewed provided that this could be done with minimal disruption of their normal instruction and with minimal additional teacher workload. Similarly, the students who were videotaped and interviewed over the three years of the project

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had parental approval to participate provided that their studies were not disrupted. Both parents and students were provided with an outline of what participation in the project would involve and the likely extent of disruption of the students’ studies. Such disruption consisted of missing up to 30 minutes of the class immediately following the lesson videotaped. Since different students were videotaped from lesson to lesson, the degree of intrusion on the studies of any one student was minimal. The particular group of four students chosen to be videotaped (and possibly interviewed) on a given day were selected, in consultation with the teacher, more for their stability as a social unit than for any other reason. Since students typically worked in pairs, the stability requirement translated into the identification by the teacher of two pairs of students with a history of working together (as separate pairs) and who happened to be sitting in such a fashion that both pairs could be captured on the “student camera”. Since the genesis of the study derived significantly from an interest in negotiation and student-student interaction, it was felt that the most useful data would be obtained from students with some history of working together. Student ability was not a criterion at any stage, however attempts were made over each year’s data collection to videotape comparable numbers of boys and girls. 2.2 The Lessons

It must be emphasised that the classrooms videotaped were not experimental in any clinical research sense. The intention was to capture “ordinary” lessons in mathematics and science. Fifty-five such lessons were videotaped over the threeyear period of the study. For the specific purpose of carrying out the multidisciplinary analyses leading to the writing of this book, eight “integrated data sets'' were generated from four mathematics and four science lessons (see Figure 2.1, and Appendices A and B). It is the analysis of these eight integrated data sets that constitutes the empirical base on which this book has been constructed. These lessons were selected by the project director on the basis that each lesson included elements of student-student and teacher-student interaction. Since the underlying aim was to study negotiative processes in mathematics and science classrooms, this selection criterion did no more than maximise the visibility of the sort of situations that were of interest to the research team, without anticipating or constraining the nature of those situations. It was unusual for teachers to make structural use of working groups of more than two students. By far, the most common working unit was the student dyad (pair), either with explicit teacher sanction, or as an informal unit of classroom social interaction to which the teacher gave implicit sanction but not explicit encouragement. In the experience of the research team, the videotaped classes contained many elements familiar to educators in Australia, the USA, and countries that share a European heritage (including, for instance, Malaysia or Hong Kong).

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Lesson Date Number 1 4/10/94

Year Subject level 8 Science

Teacher

Lesson theme

Students interviewed Mrs Brown Group chart: what Anna you know and Grace would like to know about the heart Mrs Brown Problem solving Karen using standard Lauren number Mrs Brown Compass bearings Anna Karen Rates and ratio; velocity-time graphs Heart dissection

2

19/10/94

8

Maths

3

18/11/94

8

Maths

4

13/6/95

8

Maths

Mrs Brown

5

19/6/95

8

Science

Mr Black

6

17/6/96

8

Maths

Mrs Brown

7

16/7/96

9

Science

19/7/96

9

Science

8

Rates and ratio; velocity - time graphs Mrs Brown Beginning of Fibres unit; feeling fibres blindfolded Mrs Brown Looking at fibres under a microscope

Patricia Kevin Stephen

Mark Darren Joanne Narelle Joanne Frances

Table 2.1. Lessons analysed in this book It will be clear from the various chapters that some lessons proved more amenable to particular analytical approaches than others. For example, the physical configuration of the classroom did not always allow for the unimpeded videotaping of gesture that Reeve and Reynolds would have preferred for their analysis. To reiterate: the fundamental criterion for selection of the particular lessons chosen for analysis was that the chosen lessons should include instances of various forms of classroom negotiation. In addition, the set of eight lessons offered the research team a large number of possible analytic alternatives. For example, the eight lessons chosen offered the research team the possibility of contrasting mathematics and science teaching by the same teacher (Mrs Brown), the teaching of similar science content by two different teachers (The Heart, Mrs Brown and Mr Black), the teaching of the same content to different classes in successive years (rates and ratio, Lessons 4 and 6), the learning of the same content by different students (all lessons), the learning of mathematics and science by the same student (Lessons 1 and 3, Anna), teaching and learning in consecutive science lessons (Lessons 7 and 8, Mrs Brown and Joanne) and the learning of different mathematics

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content by the same student (Lessons 2 and 3, Karen). Selection was the responsibility of the project director (Clarke) and not all of these potential comparisons were exploited in the analyses reported here. It is quite possible that a different selection might have proved more or less amenable to one or more of the analytical techniques employed by the various members of the research team. However, all research team members found the chosen lessons sufficient to sustain their analyses, as is evidenced in each chapter. Since there was no intention to frame conclusions of a normative or statistical nature, selection could be premised on the occurrence of the phenomenon on which the project was focussed: negotiation. Another selection of different lessons could have been made if, for instance, the research emphasis had been on student response to the teacher’s use of particular instructional techniques. It was a consequence of the selection process outlined above that only two teachers came to be included in the set of eight lessons that provided the data base for the analyses reported in this book. “Mrs Brown” was an experienced teacher in her mid-thirties. Her experience and organizational skills led to her appointment as “Year 8 Coordinator”, which gave her significant administrative, disciplinary and pastoral responsibilities for eighth grade students. Mrs Brown held this position in all three years of the study. “Mr Black” was a younger, but experienced, teacher in his early thirties. Both teachers had been teaching at the school for several years and had previous experience teaching the content covered in the lessons videotaped. A brief example of one section of the text component of an integrated data set is given in Appendix B. In this sample text, the researcher’s field notes, the transcribed classroom dialogue, and the students’ and teacher’s accounts are combined. It was integrated text of this type that, in combination with the corresponding video record, provided the data base for the analyses carried out by the members of the research team. Readers are invited to consider what analyses they might have undertaken had they had access to such a data base. 2.3 The Research Team These data sets and the corresponding videotape and audiotape records were made available to a multidisciplinary team of eleven university academics with expertise and research experience in areas such as mathematics and science, mathematics education and science education, developmental psychology and cognitive science, ethnography, sociology, epistemology, values analysis, motivation, and metacognition. The participation of team members in the project was negotiated after their consideration of the available data sets and the development by each team member of a proposed analytical approach commensurate with the available data and consistent with the overall project goal of studying learning, teaching, and the negotiation of meaning in classroom settings. Individual members of the research team were encouraged to sample and interpret the data sets from a distinct, carefully articulated, theoretical perspective. The goal of this process was complementarity rather than consensus, and each researcher’s interpretation was subject to the same criteria of coherence, consistency with the videotape data, and plausibility.

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DAVID CLARKE 3. GROUP INTERPRETIVE RESEARCH AND MULTIPLE ACCOUNTS

Given that group-conducted studies, rather than lone-researcher studies, are becoming a widespread feature in educational research, it was somewhat surprising to find that there had not been a more thorough discussion of group interpretive processes (Davidson Wasser and Bresler, 1996). For example, group analysis of qualitative data characterised the recent study by Cobb, Bauersfeld and their coworkers (Cobb & Bauersfeld, 1995). The focus of the Cobb and Bauersfeld study was the operation of mathematics classrooms in primary school. Of course, the analysis of large data sets by multi-disciplinary research teams is not restricted to qualitative data. The Third International Mathematics and Science Study (TIMMS) offers a current example of a large quantitative data base that is presently being analysed by research teams all over the world (Lokan, Ford & Greenwood, 1996). Whether the data is qualitative or quantitative, the integration of multiple analyses of data is not a simple matter of aggregation. The integrative process is better characterized as synthesis, and different means may be employed to achieve this synthesis. Alternative approaches to the synthesis of the various research analyses are discussed and implemented in Chapter 12. Many research teams using videotape as their fundamental data source establish group negotiative processes aimed at constructing consensus accounts of the data. As an illustration of how the processes of the classroom can mirror the research process, in discussing the emergence of mathematical meaning in a second-grade classroom, Krummheuer (1995) invoked Goffman's (1959) notion of a "working consensus" as the immediate goal of classroom argumentation. This conception of working consensus, the authority of which derives from its local and immediate viability, can be applied to describe the interpretive processes followed by many research teams engaged in interpretive research into complex social settings such as classrooms. Despite the evident similarities, it is useful to identify the difference between the working consensus of students and teacher and the working consensus achieved by a research team undertaking classroom research. Students in a mathematics classroom and their teacher participate in the construction of a class consensus relating to mathematical meanings which are ultimately subject to criteria of consistency with the tenets of mathematical correctness as endorsed by the mathematics community. In such negotiations, the goal of the consensus process is clear and conforms to established classroom conventions. In researching classrooms, the goal is the generation of knowledge that is entirely new, both to the research team and to the research community. In this case, ultimate appeal is not to an established body of community-endorsed ‘knowledge’, but to the community’s endorsement of the standards of rigor by which the new knowledge was generated. Despite this difference, when the object of argumentation is the interpretation of classroom data, Goffman's conception of a working consensus as a transient convergence on a locally viable interpretation is a particularly apt characterisation of the goal of the consensus process operating in group interpretive research teams. Explicit in Goffman's conception of the working consensus is the need for resolution of the competing claims of different interpretations. Since the goal of the

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interpretive process in classroom research is not the identification of ''what really happened'' but rather the development of viable accounts, care must be taken in specifying the criteria by which the relative merits of competing interpretations might be compared. Scholle (1992) has provided reassurance as to the legitimacy of such comparisons of merit. The question mark that postmodernism puts on the 'truth' of all discourse means only that all knowledge is contextual, not that all knowledge is false, nor that one cannot support the validity of one claim over another in a specific circumstance (Scholle, 1992, p. 276).

Any comparisons made between the researcher’s reading of the videotape data and the accounts of participants cannot be undertaken for purposes of “validating” a particular account (participant’s or researcher’s) but for the insights offered by discrepancies and consistencies between accounts. “Triangulation” in such research is employed because of the intrinsic value of multiple perspectives and not with the aim of determining “what really happened.” The consistency of a particular account with the researcher’s reading of the videotape data adds plausibility to both accounts through their corroboration, but it does not establish either or both accounts as definitive. The account provided by one student, for example, may be in correspondence with the researcher’s reading of the videotape data but may contradict in several details the account provided by a classmate. Such differences may significantly inform our understanding of how particular classroom situations or teacher actions are interpreted by different participants, and our subsequent instructional design may be more effective by its anticipation of these possible multiple interpretations. Inconsistency between a participant’s account and the researcher’s reading of the videotape data suggests the possibility of unlooked for meanings in the situation in question, or of the participant’s need to reconstruct/recount the events in a particular fashion. Such inconsistencies of account are likely to be as useful methodologically, or theoretically, or empirically, as correspondence. In this sense, such multiple interpretive accounts of social situations or events are always complementary, contributing additional detail to the emerging picture of classroom practice. Central to the process by which a research team carries out its interpretive activities is the question of whether the goal of that interpretive activity is consensus or complementarity. In the case of the Classroom Learning Project, consensus was sought within the research team with regard to the consistency of each particular account with the available data; that is, consensus as to 'fit'. There was no requirement of consensus across the team as to the ‘truth’ or ‘correctness’ of the account other than with respect to this consistency with the data. It is to be expected that an interpretation of classroom data emphasizing volitional control of cognition will differ from an interpretation of the same data focussing on the use of gesture, and both will differ from a values analysis. Significant negotiation occurred within the team regarding the meaning of the key terms from which the various interpretive accounts were constructed. Key synonyms of 'complementary' for the purposes of this study were 'interrelated' and 'parallel.' A simple analogy would be the construction of physical

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and intellectual descriptions of an individual, which might be said to be complementary in that they represent distinct aspects of that individual; neither, separately, constituting an adequate portrayal, but each capable of making an important contribution to such a portrayal. While the various accounts in this book refer to a common body of data and may resemble each other in some details, the recruitment of the research team was intended to maximize the variety of perspectives represented and the breadth of research expertise. The resultant interpretive accounts can be read both for their individual validity and for their differences and similarities. Each analysis is thorough, rigorous, insightful and warrants publication in its own right. The premise of this book, however, is that it is through the synthesis of such distinct analyses of the same body of data that we are most likely to approach an understanding of learning in classrooms settings and the means by which that learning might be promoted. Further, any interrelatedness evident in the researchers’ accounts can be exploited in the interest of synthesis and the identification of transcendent structural features of classroom learning. It is in this sense that complementarity is invoked as the distinguishing characteristic of this approach to classroom research. The Classroom Learning Project and its methods raise interesting questions about the interpretive process and in particular the distinction between data collection and data analysis. It has become recognised in qualitative research that there can be no clearly defined point where data collection ends and data analysis begins. Miles and Huberman (1994) refer to this issue in their discussion of qualitative research methods, emphasising the continuous and cyclic nature of this process: Even before the data are actually collected, anticipatory data reduction is occurring as the researcher decides (often without full awareness) which conceptual frameworks, which cases, which research questions and which data collection approaches to use. As data collection proceeds, further episodes of data reduction occur (p. 12).

The students who provided their own interpretations of their actions and motivations through the process of video-stimulated recall were engaging in a form of data analysis, with the videotape providing the primary data. Their analytical tools were not those of the researchers and their accounts were grounded in a theory of classroom practice that was not known to the researcher and may never have been articulated explicitly by the students. Yet these students were the first analysts of the videotape data and their accounts are subject to similar criteria to those of the research team: consistency with other available data, coherence and plausibility. What constitutes a "complete" data set? A data set is only "complete" to the extent that it meets the needs of the researcher. The eight lessons selected as the focus for the analyses reported in this book proved such a rich resource that several members of the research team found themselves compelled to restrict their analyses to only a few key episodes, and further analyses of the same data set are already in train. In addition, there is no doubt that the data collected in the Classroom Learning Project was triangulated extensively, with the integration of many voices providing a resource rich in detail. However, it is also possible that later analysis of the videotape might identify an event, the significance of which was not recognised at

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the time by either student, teacher, or researcher, and which, therefore, was not pursued in interview. As with any clinical interview, the researcher may miss one insight through the pursuit of another. The process of transcription highlights the extent to which these issues are intertwined. The integrity of the integrated data sets, of which Appendix B provides an edited example, derives fundamentally from the conventions adopted during transcription and the manner in which these were applied. How detailed should the transcript be? Should facial expressions, gestures and tone of voice be recorded? Transcription is not (and never will be) a simple mapping of ‘classroom reality’ onto the page, nor can transcripts of classroom events or of interviews capture the social and political milieu within which the data was collected. However, having the data interpreted by a team of people from different theoretical perspectives reduces the likelihood that an important interpretive perspective has been omitted and subjects both the adequacy of the transcripts and the individual interpretations to an exacting scrutiny. In summary: The complementary accounts that constitute the outcomes of the Classroom Learning Project (and the chapters of this book) are subject to the requirement of consistency with the available data (which might be termed 'vertical consistency'). They are not required to be consistent with each other ('lateral consistency') and indeed may be most informative when different accounts offer alternative explanations of classroom actions, practices, or events. However, consistent use of terminology is essential if the insights offered by different accounts are to be meaningfully interpreted and compared. The synthesis of such accounts is not a matter of unification through consensus into a single account, but rather the interweaving of accounts providing in their combination a richer portrayal of the classroom. Further, as will be illustrated throughout this book, an interpretation from one perspective may support, inform, justify and even explain the account constructed from another perspective. 4. ANALYSIS: FINDING STRUCTURE IN DIVERSITY It is not the purpose of this chapter to detail in full the analytic techniques employed in the Classroom Learning Project. Each chapter that follows includes details of the analytic procedure from which its findings were generated. There are many analyses that might be carried out on data such as the sample in Appendix B. The following discussion provides some illustrative examples of analytical approaches employed with the common data set that provided the focus for the analyses reported in this book. 4.1 Complementary Participant Accounts Comparisons can be made of the students' interpretations of a particular exchange. The following sample text includes two students' reconstructed accounts of the same event (italicised text), after viewing the video record of the interaction transcribed below (plain text). The researcher's observational field notes, recorded during

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videotaping, are shown in bold type. These brief and somewhat cryptic notes provided the reference points by which particular classroom events were located on the videotape record during the student interviews. The convention of recording classroom dialogue in plain text and interviews in italics is maintained in all chapters in this book (T = Teacher; K = Karen; L = Lauren). 00:39:27 to 00:41:30 T asks K and L what they've done. K explains, T is dubious, then says I think you're right, L explains, says we're right. T: Where'd you get a hundred and eighty from? K: Width. Equals forty— T: Why did you multiply them together? Why not add? K: To get the area. I know that much. T: [to Lauren] You've been tutoring her? K: Equals forty-five thousand, therefore you'd need—oh, how'd I get that? T: Forty-five thousand? K: Forty-five thousand. That's what we got. T: Forty-five thousand? Can you press that— can you press that— can you do that again? Two hundred and fifty times a hundred and eighty? Oh, hang on, hang on, I think you're right. Hang on. I think they're wrong. K: Yup, they're wrong, we're right. (Lauren holds up calculator). I(Intewiewer with Lauren): Uh huh. So why were you so sure your answer was right? Or were you sure your answer was right? L: Um, because when she asked us what it was, she thought it was right too. I: I'd like you to tell me this last bit. So say that again for me. L: She came and asked us to do the answer that we found and we had a different answer to the one that another group had given her and when she heard our answer it must have clicked that, um, it sounded more right than the other one did, so she went to tell them that they were wrong. I: It must have clicked with her, so that's why she thought it was right. Why did you think it was right? L: 'Cause if Mrs Brown thinks it's right, it probably is. I(Inteviewer with Karen): There she's going over to your group.

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[videotape continues] Yeah, so you seem pretty sure, you got the, you had answers for everything she said. K: Yeah, because, um don't say anything to her, but the girls said that she'd pick on me a bit because I'm new, so if I show her that I know what I'm talking about then she 'll lay off I: Oh, well that makes sense. K: Because I don't want to seem like I'm going, "Well, you're right." I: But you did know, didn't you? K: Yeah, but I just thought it, 'cause yeah. I: Yeah. K: So I just wanted to show her that I knew what I was talking about, because otherwise she'd keep on at me. I: Yeah. K: Admit that actually sort of think that she made a little bit of a mistake. Yeah. I knew she'd lay off if I sort of had an answer for everything, so that's why I just said, straightaway she'd ask me a question, I'd have an answer, and she'd go, um, think about it for a while and then straightaway say "You're right or you're wrong." Yeah, I just wanted to say everything quickly so she didn't have time to think of another question.

, In a study whose major focus was classroom negotiation, Karen s account of her motives and her perception of the interaction offered some insight into what it was that was actually being negotiated in an exchange whose surface content was mathematical. Such examples strongly suggest that without the student's reconstruction, the researcher's account of the interaction would be unlikely to capture the student's motivations and construal of the particular social situation. Lacking this detail, any inferences the researcher might make regarding the student's participation in the classroom and her associated learnings would be extremely restricted (and problematic). 4.2 Analysis by Exemplification The key constructs employed in the articulation of our theories of classroom learning must be empirically well-founded. Chapter 3 illustrates the use of "Analysis by Exemplification" to identify pattern and structure within possible instances of the key construct "intersubjectivity" identified within the data set. In analogous fashion, Lerman examines the nature of the “Zone of Proximal Development” as an empirical entity in the analysis reported in Chapter 4. A major goal of the Classroom Learning Project from its inception was the empirical grounding of some of the theoretical constructs central to current theories of learning.

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4.3 Structural Analysis of Text Another type of analysis of the integrated data sets involved the classification of the text into six stratified levels: the lesson, the activity, the episode, the negotiative event, the turn, and the utterance. In the example discussed in Chapter 3, the levels 'lesson' and 'activity' play no part. Episodes comprise the dialogue and actions that students engage in as they approach, work on and complete a particular classroom task, such as solving a mathematical problem. Thus, each episode is a coherent sequence or combination of behaviours unified by a single encompassing purpose. Each such episode may involve several negotiative events. A negotiative event is defined by a single identifiable intermediate purpose; a purpose whose realization is an intermediate goal within the encompassing episode. As is indicated by the term “negotiative”, such events are characterised by their identification with the attempted resolution through social interaction of a particular issue or uncertainty. Each negotiative event may be composed of several turns, each with its own immediate (typically communicative) purpose. A turn comprises a communicative act by an individual, consisting of one or more utterances, intended in combination to serve a single communicative purpose. An utterance is the simplest meaningful communicative element. An utterance may be a sentence, a phrase, or a single word. As is demonstrated by Reeve and Reynolds in Chapter 10, a gesture may also constitute an utterance. It will be argued in Chapter 3 that the negotiative event, as a unit of analysis, has significant methodological and theoretical significance, since it constitutes a “unit of social interaction”, conceptually distinct from the separate utterances of the individuals participating in the interaction. Its theoretical significance derives in part from the possibility that such events are the social correlates of important cognitive processes. Methodologically, such events set the lens of our analysis at a level of “magnification” corresponding to the least complex social interactive unit above the level of the individual. Negotiative events may be utilised by individuals to achieve social goals (see Karen’s interview, above), to exercise power (see Barnes, 1998) or to advance conceptual understanding or task completion (see Chapter 3). Because the negotiative event presumes the interaction of two or more individuals, the enactment of any of the above purposes by an individual is inevitably situated in a social context that accords significance to the reciprocal negotiative character of interactions among the participants. 4.4 Replicable Analysis of Text Software packages such as NUD•IST (Qualitative Solutions and Research, 1994) can be employed to undertake a wide variety of forms of textual analysis. One of the most simple of these is the determination of the frequency of association of key terms or phrases employed in student classroom discourse or in student or teacher discussion of video-recorded classroom situations. Analysis of this type can reveal not just the frequency of use of such terms or phrases, but their frequency of association either with each other or with aspects of classroom practice or elements

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of the social situation. Further, software tools such as NUD•IST facilitate the identification of chains or networks of association that may reveal structures of theoretical significance, common across diverse bodies of text. Such software is in widespread use in studies in which text analysis is a key feature. The use of an indexing tool such as NUD•IST offers a form of replicability for textual analyses not previously associated with conventional content analysis; that is, a particular sequence of commands applied to one body of text, may reveal a network of associated clusters of text elements with a particular structure, and the robustness of the emergent structure can be tested by applying the same sequence of commands to a different body of text. This approach was not used extensively in this project and contributed specifically only to Chapter 3, where the results of the analysis reported by Clarke and Kessel (1995) are cited to elaborate the argument being proposed. Nonetheless, replicable analysis of text is a noteworthy technique when dealing with a database consisting of significant bodies of largely unstructured text. 4.5 Complementary Researcher Accounts A key design element in this project was the bringing together of a research team of sufficiently diverse expertise to adequately implement the Complementary Accounts Methodology. The research described here utilized a team of university academics with expertise in mathematics education, developmental psychology, sociology, epistemology, values analysis, motivation, mathematics, science, science education, , children s conceptual frameworks, metacognition, gender and a range of qualitative and quantitative research methodologies. As has been stressed already, individual members of the research team, rather than seeking a consensus interpretation of an event, an episode, or an interaction, were encouraged to interpret the documented interaction from a distinct, carefully articulated, theoretical perspective. The goal of such a process is complementarity rather than consensus, and each researcher's interpretation is subject to the same criteria of coherence, consistency with the videotape data, and plausibility. As stated above, group consensus was sought in relation to the consistency of one account with the data, not with regard to the consistency of one account with another. Different perpectives, reflecting different objects of study within the encompassing goal of modeling classroom learning, can be used to analyze the same integrated data sets. For example, Clarke and Helme (1997) made use of the data in Appendix B to discuss the role of negotiation in the resolution of uncertainty in mathematics classrooms. This data is also amenable to analysis with respect to the several instances of student metacognition contained within the text and the function of these self-evaluative reflections within the learning process. These two accounts of the Appendix B data: one from the perspective of the resolution of uncertainty and one from the perspective of student use of metacognition, are not in competition; they represent complementary interpretations of the same integrated data set. Such complementary accounts have the potential to be mutually informing

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and to constitute in combination a richer portrayal of classroom learning than would be possible by the consideration ofeither account separately. In acknowledgement of the singularity of the analytic path followed by each researcher, chapter authors address the process of analysis specifically. For example, in Chapter 4, Lerman discusses the multiple layers of interpretation that preceded his analysis and the interpretive acts by which his account was constituted. One danger of utilising well-established research methods can be the absence of a critical perspective on one , s own research practice. The self-conscious methodological deliberation evident in the chapters in this book is a healthy by-product of working in a novel research paradigm. The researchers' various perspectives employ intersecting sets of constructs and their interpretive accounts provide a form of empirically-grounded 'crossreferencing' for the recurrent components of a model of classroom learning. As a result, one method of synthesising the various analyses involves the utilisation of patterns or structures arising from one analysis as analytical tools or explanatory devices in another analysis. It is in their capacity to be mutually informing that much of the value of complementary accounts resides. This is particularly true where one analysis suggests an explanation for the outcomes of another. For example, it is not surprising that an analysis of the values underlying classroom interactions should inform an analysis of student interest and motivation. Educational researchers have always appropriated, adapted and applied each other's analytical tools. Within a team employing a complementary accounts approach, this process of appropriation and adaptation is facilitated by a common focus on the same complex data set, coupled with the regular opportunity for comparison and negotiation of meanings and accounts afforded by the group interpretive process. 5. CONCLUSION In the research discussed in this book, an attempt was made to optimize the use of currently available technology through the synthesis of classroom videotape and interview data in an integrated video and text data base. An important, possibly essential, perspective on the classroom can be obtained from the students themselves in interview situations, where the significance of classroom events and their associated thought processes can be reconstructed by the participants themselves with the assisting prompt of the classroom video record. This reconstruction does not signify a direct correspondence between the classroom event (and associated teacher/learner thought processes) and the reconstructive account, and precise correspondence is not claimed. However, as can be seen from the interviews with Lauren and Karen, the researcher’s interpretation of the videotape data can be significantly informed by such reconstructive accounts. Other chapters in this book will provide many additional examples of this. The resultant data set is amenable to a variety of forms of analysis and some of these have been illustrated. Recent developments in educational research (and in learning theory) have led to acceptance of the idiosyncratic and legitimate subjectivity of both the research subjects and the researcher, and to the consideration of what can be learned from the

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comparison of the multiple stories compiled from the accounts of the various participants in the social setting, and from the constructed accounts of the research team. Judgements regarding the relative merit of one account over another will relate to the purpose for which the comparison is being made and do not call into question the value of either account with regard to any other purpose. It is through the accumulation of such complementary accounts in relation to a common integrated data set that our portrayal of classroom learning will approach the complexity of the process we seek to model. The meaning that emerges from this research will be our meaning; commensurate with our methods, our theories and the purposes that motivated our research. The interpretive process from which such meanings are derived must be subject to the same scrutiny and conform to the same theoretical considerations as the classroom interpretive processes that are the focus of our research. REFERENCES Bakhtin, M. (1979). The aesthetics of verbal creation. Moscow: Bocharov. Cited in Todarov (1984). Barnes, M. (1998). Towards a framework for analysing power relationships in small group discussions. Draft of a paper submitted for inclusion in the 1998 conference of the Mathematics Education Research Group of Australasia, to be held at Broadbeach, Queensland, July 4 to 7. Clarke, D.J. (1998). Studying the classroom negotiation of meaning: Complementary accounts methodology. Chapter 7 in A. Teppo (Ed.) Qualitative research methods in mathematics education. Monograph Number 9 of the Journal for Research in Mathematics Education. Reston, VA: NCTM, pp. 98–111. Clarke, D.J. & Helme, S. (1997). The resolution of uncertainty in mathematics classrooms. In F. Biddulph and K. Carr (Eds.) People in Mathematics Education. Proceedings of the Twentieth Annual Conference of the Mathematics Education Research Group of Australasia, held at Rotorua, New Zealand, 7-11 July, 1997. Waikato, New Zealand: MERGA, pp. 116-123. Clarke, D.J. & Kessel, C. (1995). To know and to be right: Studying the Classroom Negotiation of Meaning. In B. Atweh & S. Flavel (Eds.). Galtha: MERGA 18 . Proceedings of the 18th annual conference of the Mathematics Education Research Group of Australasia. Darwin, NT: University of the Northern Territory, pp. 170-177. Cobb, P. & Bauersfeld, H. (Eds.) (1995). The emergence of mathematical meaning: Interaction in classroom cultures. Hillsdale, NJ: Lawrence Erlbaum. Davidson Wasser, J. & Bresler, L. (1996). Working in the interpretive zone: Conceptualizing collaboration in qualitative research teams. Educational Researcher 25(5), 5-15. Ericsson, K.A. & Simon, H.A. (1980). Verbal reports as data. Psychological Review 87(3), 215-251. Goffman, E. (1959). The presentation of self in everyday. New York: Doubleday. Cited in Krummheuer (1995). Krummheuer, G. (1995). The ethnography of argumentation. Chapter 7 in P. Cobb and H. Bauersfeld, The emergence of mathematical meaning: Interaction in classroom cultures. Hillsdale, NJ: Lawrence Erlbaum, pp. 229–269. Lokan, J., Ford, P. & Greenwood, L. (1996). Maths and Science on the Line: Australian Junior Secondary Students’ Performance in the Third International Mathematics and Science Study Camberwell, Vic.: Australian Council for Educational Research. Miles, B. B. & Huberman, A. M. (1994). Qualitative Data Analysis: An Expanded Sourcebook. (2nd Ed.). Newberry Park, Calif.: Sage Publications. Nisbett, R.E. & Wilson, T.D. (1977). Telling more than we can know: Verbal reports on mental processes. Psychological Review 84(3), 231-259. Qualitative Solutions and Research (1994). Non-Numerical Unstructured Data • Indexing Searching and Theorizing (NUD•IST). Software package. Melbourne: Qualitative Solutions and Research. Roschelle, J. (1992). C-Video. Software package. San Francisco: Knowledge Revolution.

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Scholle, D. (1992). Authority on the Left: Critical pedagogy, postmodernism and vital strategies. Cultural Studies 6(2), 271-289. Todarov, T. (1984). Mikhail Bakhtin: The Dialogical Principle. Translated by Wlad Godzich. Minneapolis: University of Minnesota Press.

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CHAPTER THREE Untangling Uncertainty, Negotiation and Intersubjectivity

1. INTRODUCTION In this chapter, the goal is to interrelate three separate analyses that share a common purpose. Each analysis had as its focus the empirical grounding of a term of some significance to our theorising about learning. The research from which the Classroom Learning Project evolved arose from the perceived need to clarify the function of negotiation within a model of learning in classroom settings. Usage of the term “negotiation” seemed to slide all too easily from the description of a social process whereby mathematical meanings were exchanged and refined in the course of classroom interaction to an implicit description of an internalised process whereby the social negotiation of meaning was in some way re-enacted as a mental process within the individual. This internal negotiation seemed not dissimilar from the assimilatory/accommodatory processes invoked in Piagetian theory, however use of the term “negotiation” appeared to assume a direct correspondence between a social and a psychological process and even to confound the two as a single process. This is not to identify the confusion of the interpersonal with the intrapersonal with any particular theoretical position, only to stress the need for care in our use of “negotiation” as an element in any theory of learning. Steffe, for example, drew the distinction absolutely by stating that “Individual-environment interactions and interactions within an individual constitute nonintersecting domains” (Steffe, 1995). However, to draw the distinction between social and cognitive processes is not to preclude the influence of one upon the other (in either direction). Cobb (1994) framed the relationship as one of mutual contextuality, where the reflexivity between social and cognitive processes can be located in the implicit presence of each theoretical perspective in the other. Learning as acculturation via guided participation implicitly assumes an actively constructing child . . . . Learning as cognitive self-organization implicitly assumes that the child is participating in cultural practices (Cobb, 1994, p. 17)

Reflexivity between the social and personal domains has been addressed explicitly elsewhere in descriptions of negotiation as the iterative refinement of meaning (Clarke, 1996). One of the original motivations for the use of video33 David Clarke (ed.), Perspectives on Practice and Meaning in Mathematics and Science Classrooms, 33—52. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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stimulated recall with students was the need to investigate whether the social negotiation documented on videotape corresponded in any sense to a mental process (recountable by students) that also possessed negotiative characteristics. Attending to negotiation of meaning inevitably led to consideration of “intersubjectivity” and to the contemporary debate over whether or not intersubjectivity is reconcilable with radical constructivism (see, for example, Lerman, 1996). In developing a research approach with the potential to study negotiation and intersubjectivity within realistic classroom settings, the question arose as to the type of classroom situations in which these constructs might be most evident. For negotiation to occur something must be at issue, unresolved, uncertain. In the classroom, one characterisation of learning as experienced by a student is as a progression of emergent uncertainties and the resolution of those uncertainties. The catalytic role of uncertainty within the learning process has been widely recognised. Rowland, for example, stated that "uncertainty is a productive state, and a necessary precondition for learning" (Rowland, 1995, p. 328). This chapter integrates the results of separate analyses intended to contribute to our understanding of the role of uncertainty, negotiation and intersubjectivity in the learning process as it is enacted in classroom settings (Clarke, 1996; Clarke & Helme, 1997; Clarke & Kessel, 1995; Helme & Clarke, 1996; Helme, Clarke & Kessel, 1996). It is a measure of the degree of interdependence of the three terms: uncertainty, negotiation, and intersubjectivity, that each section of the following discussion makes frequent reference to all three terms, irrespective of the specific (first-named) term that is the nominal focus of the discussion. 2. NEGOTIATION, INTERSUBJECTIVITY AND UNCERTAINTY The phrase "negotiation of meaning" has been used to describe the means by which learning occurs in social settings. Earlier papers (Clarke, 1993, 1994, 1996) examined the legitimacy with which negotiation is employed to characterise a cognitive process associated with learning. The analysis reported here addresses negotiation as a social process related to the practices of mathematics (and science) classrooms and associates the need for negotiation with the occurrence of uncertainty. It must be stressed that negotiation may be triggered by other states besides uncertainty: disagreement, for instance. However, this chapter is focussed on documented instances of uncertainty, the processes by which such uncertainties were resolved, the negotiative nature of such processes and the role of intersubjectivity in such negotiations. In the analyses reported here, the goal of resolving uncertainty is taken as the starting point, and the means by which any resolution is achieved is documented. In this account of classroom learning, intersubjectivity enters as a mediating agency, essential to the negotiative process, whereby uncertainty is resolved, and new knowings are constructed. Whereas Voigt's analysis starts from the learner's subjectivity and models the achievement of intersubjectivity (Voigt, 1995, this account starts from evidence of students' intersubjectivity and documents alternative

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forms of uncertainty and the processes whereby resolution might be achieved; processes in which intersubjectivity has a central role. Steffe and Thompson have argued that “one cannot take intersubjectivity as given _ as an explanatory concept in the construction of knowledge. . . Rather, intersubjectivity must itself be explained” (Steffe & Thompson, 2000, p. 197). The study reported in this book accepts the obligation to minimise the number of concepts in our theorising that are taken as axiomatic and, where possible, to give all key concepts an empirical foundation. The purposes served by our analyses are partly concerned with explanation (and therefore potentially theory-building) and partly with the clarification, through empirical grounding, of terms in widespread use in our educational theorising. The argument on which this chapter is predicated starts with the propositional chain that one pathway to knowing is via the resolution of uncertainty, that the process of resolution is fundamentally negotiative, that negotiation is mediated by language, that language presumes intersubjectivity, and that the matter of intersubjectivity is meaning. It is essential that a theoretical framework be established with regard to each ofthe constituent terms in the preceding sentence. 2. 1 Negotiation Negotiation has been characterised in some detail elsewhere as a cyclic process of refraction (construal), reflection, and representation, the goal of which is consensus (Clarke, 1996 and 1999). Cobb and Bauersfeld defined the negotiation of meaning succinctly as ''the interactive accomplishment of intersubjectivity'' (Cobb & Bauersfeld, 1995, p. 295). This definition certainly reifies the connection between negotiation and intersubjectivity. Use of the term “interactive” suggests the participation of more than one person in the negotiative process (locating the process in the social realm), unless the anticipated interaction is with inanimate aspects of the learner’s environment. This latter interpretation need not be dismissed out of hand if we adopt a view of meaning as a distributed attribute, embodied in such artifacts as a row of desks facing a blackboard or the regular ringing of a bell to signal the end of a lesson. In such an interpretation intersubjectivity becomes an operative correspondence of meaning that enables separate individuals to utilise the artifacts effectively. The development of such correspondent meanings by individuals, however, would be an interpretive process enacted by each individual. Lave and Wenger, in associating learning with participation in practice, assert that ''Participation is always based on situated negotiation and renegotiation of meaning in the world'' (Lave & Wenger, 1991, p. 52). Implicit is the requirement to attend simultaneously to the situated character of meaning and to its progressive refinement by the individual “participant” through negotiation. Since classrooms represent legitimate sites of situated mathematical practice, this perspective supports the potential value of the empirical documentation of negotiative processes in the classroom.

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2.2 Meaning The presumptions of meaning are community, purpose and situation. It is futile to discuss the meaning of a word or term in isolation from the discourse community of which the speaker claims membership, from the purpose of the speaker, or from the specific situation in which the word was spoken. Indeed, it is not the word that has meaning, but the utterance. Our understanding of meaning has been usefully informed by consideration of the writings of Bakhtin and his colleagues and interpreters. In all these cases, we are dealing not with the isolated words as a unit in language, nor with the signification of this word, but with the completed utterance and its concrete meaning, the content of this utterance (Bakhtin, 1979, quoted in Todarov, 1984, p. 53).

Since the data with which we work consist primarily of utterances of one form or another, and since the focus of this chapter is the resolution of uncertainty, Baktin's statement that "Meaning always answers some questions" (Bakhtin, 1979, quoted in Todarov, 1984, p. 54) is particularly useful. It identifies meaning with the need to know and, by implication, with the resolution of uncertainty. 2.3 Negotiative Events As was noted in the previous chapter, it proved useful to parse classroom dialogue into six stratified levels: the lesson, the activity, the episode, the negotiative event, the turn, and the utterance. For our purposes, the most significant level of analysis centres on the negotiative event. A negotiative event is defined by an identifiable purpose; a purpose whose realization is an intermediate goal within the encompassing episode. The significance of this particular form of structural analysis can be found in the consideration of patterns of association of a particular construct or practice with a particular level. The interpretation of the significance of a given episode requires an interpretation of each constituent level of negotiation: the negotiative event, the turn and the utterance. The identification of the negotiative event as the unit of analysis has important theoretical implications. The negotiative event is a “unit of interaction.” Rather than attending analytically to the separate utterances of the individuals participant in the interaction, the negotiative event identifies the interaction itself as the object of interest. Our analysis can then attend to the characteristics and structure of the negotiative event. In Episode 1, a sample of classroom dialogue, selected from Appendix B, has been 'parsed' by partitioning the text according to the occurrence of "negotiative events" within an "episode" (Clarke & Helme, 1997). Karen and Lauren are student interviewees, S20 and S22 are other students, and T is the teacher (Mrs Brown).

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Turn E V E N T 1

Transcript

1

L: [writing] Five hundred sheets. Height equals.

2

K: OK, question two [Find the height of a stack of one million sheets of paper].

3

L: Does everyone understand what we did with number one?

4

K: No, but. Anyway. Five hundred sheets.

5

L: And how many sheets do we need?

6

K: Five hundred sheets of what? Five hundred sheets.

7

L: Their height equals five point eight.

8

S20: We've done that.

9

K: I know. But we've got to do it all together so

10

L: One point oh times ten to the power of six divided by five hundred.

11

K: Oh yeah sure everyday what are you talking about? What are you talking about?

12

L: Finding out how many five hundreds there are in a million.

13

K: How many five hundreds there are in a million. That would make it one thousand. How many thousands are there in a million? That would make a thousand, two thousand. What? [to another S] I have a lot to say.

14

L: [uses calculator] Two thousand. Well done!

15

K: This is called skill. This is what you do. Five hundred into a one hundred which is two. Then you do a hundred [correcting herself] which is a thousand.

16

L: Times five point eight. Shush.

17

K: Which is two. Then you do a thousand into a million which is a thousand, so one thousand times two is two thousand.

18

L: Eleven thousand six hundred.

Figure 3.1a. Episode 1, Negotiative Event 1

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Turn E V E N T 2

Transcript

19

T: With your working out folks I want you to tell me what you are mult— Matthew— what you are multiplying by, and you simply put a little arrow telling me what and why.

20

K: What are we doing? Is it a million sheets of paper though?

21

L: Yeah. One point—

22

K: We're doing a million sheets of paper.

23

L: Yeah, you need—

24

K: Yes we do. We do, shut up

25

L: Therefore,

26

K: Huh? I did that wrong.

27

L: One [inaudible, working]

28

K: Two thousand times five point eight is eleven, six, zero, zero.[ie 11,600]

29

L: [sounding out letters] M-ms?

30

K: Centimetres—which would make it eleven point six metres, right?

Figure 3. 1b Episode 1. Negotiative Event 2

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Transcript

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L: Or eleven—yeah. It'd be eleven point six metres, wouldn't it, 'cause you take off one to get the centimetres, and another one, yeah. [pause]

32

K: [looking up] That's quite high, isn't it?

33

L: All right. And you've got to point out what the units (?) are, right?

34

K: You've got to point out what the what is?

35

L: We have to show what we're multiplying by. [S22 says something to Karen, Karen laughs]

36

S20: That's not how you know, you look like you know what you're doing and you just do it.

37

K: Exactly, you go into a state of total concentration, it lasts about two seconds, that's when you get the answer, and then you don't know what you're doing, so it doesn't matter. Five hundred sheets equals, height equals five point eight centimetres. I don't even understand what I wrote. [pause as Lauren, Karen write]

Figure 3.1c. Episode 1, Negotiative Event 3

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Turn E V E N T 4

Transcript

38

K: But why do we divide a million by five hundred to get that answer?

39

L: Because you know, if you know what the height is-

40

K: So what am I doing. Tell me what I'm doing here, tell me what I've done.

41

L: All right. You know that five hundred sheets equals fiftyeight—

42

L and K: Five point eight centimetres.

43

K: There is a point there, it's up there.

44

L: Oh, 1 can't see it.

45

K: Get some other glasses.

46

L: Now, we need to know-we need to know the height of a million sheets of paper. Therefore you must divide a million by five hundred and times that number by five point eight.

47

K: [writing] Equals two thousand. Sheets of paper.

48

L: OK.

49

K: Two thousand times five point eight centimetres equals eleven thousand six hundred centimetres, equals eleven point six metres of paper. [bell rings]

Figure 3. 1d. Episode 1, Negotiative event 4 As can be seen in the example above, the episode that began when Karen says "OK, question two" and concluded with the statement, "Two thousand times five point eight centimetres equals eleven thousand six hundred centimetres, equals eleven point six metres of paper" can be partitioned into four negotiative events: – Event 1 combines the refinement of intersubjectivity within the dyad with their first solution attempt. – Event 2 revisits the procedure employed in Event 1. – Event 3 involves the negotiation of appropriate units of measurement. – Event 4 reviews the procedure again and links it to the task. Not every episode can be parsed so clearly into discrete negotiative events. Participating sub-groups may pursue overlapping negotiative purposes. Episode 1,

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however, serves to illustrate the theoretical significance of the negotiative event as a key structural element in any parsing of classroom discourse. One structural characteristic, found in several episodes within the data set (including Episode 1, above) and apparently typical of collaborative group problem solving, involves the repeated rehearsal of a successful problem solving attempt; that is, a sequence of negotiative events in which first the problem is solved, then the problem is solved, and then the problem is solved again. Each repetition of the problem solving process appears to serve a distinct purpose: to identify a possible solution, to establish a common understanding of the procedure, or to review the method of solution for viability, validity and/or completeness. These structural elements within the text may reflect parallel structures within the process of learning. Certainly there seems no doubt that each socially-enacted revisiting of the task served a specific cognitive purpose: the resolution of a distinct form of uncertainty. An analysis of the function of 'uncertainty' in the learning process can include the identification of the particular levels within the text with which uncertainty is most commonly associated or expressed. The results of such an analysis will be reported later in this chapter. The transcribed dialogue throughout all four negotiative events provided evidence of both intersubjectivity as a pre-existing accomplishment of the dyad (Lauren/Karen) through the history of their previous interactions, and of the iterative refinement of that intersubjectivity in the course of Episode 1. This dual character of intersubjectivity as agent and product will be examined more carefully in the next section. 3. INTERSUBJECTIVITY, UNCERTAINTY AND NEGOTIATION In this account of classroom learning, intersubjectivity enters as a mediating agency, essential to the negotiative process, whereby uncertainty is resolved, and new knowings are constructed. The contribution of analysis by exemplification (see Chapter 2) lies in the accumulation of empirical instances that conform to the theoretical characteristics of the particular construct. Given this correspondence, the cumulative set of instances can be examined for transcendent elements or relationships by which the nature of the postulated construct can be grounded and elaborated empirically. Theoretical definitions and hypothesised relationships can then be tested for consistency with the empirical data, and patterns within the identified instances can then serve to elaborate the construct. Such an iterative procedure of theorising, empirical exemplification, elaboration of theory, and further empirical exemplification, can appear tautologous. This is not the case. The progressive refinement of theory by reference to observation, iterated until the elaborative potential of the data is exhausted (saturation), has a long history in all branches of science. The intended exemplification does not reside in the particular examples, but in the recurrent characteristics and relationships which transcend any particular instance and thereby exemplify the construct. General statements of principle, which define one construct in terms of another, or relate one construct to another, require

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empirical demonstration of the postulated relationship. This is especially obligatory when the proposed relationship is one of process and product. For the purpose of this discussion, both 'negotiation' and 'intersubjectivity', and the relationship between them require such empirical elaboration, since neither construct is taken as given. Cobb and Bauersfeld define the negotiation of meaning succinctly as "the interactive accomplishment of intersubjectivity", where intersubjectivity is described as "a mutual or taken-as-shared understanding" (Cobb & Bauersfeld, 1995, p. 295). In this view, intersubjectivity is the product of a negotiative process. One contention of the research reported in this chapter is that while one product of negotiation may be 'enhanced intersubjectivity', the nature of the negotiative process itself demands a form of intersubjectivity that enables and sustains negotiation. As was argued earlier, negotiation is only possible because of the prior accomplishment of intersubjectivity by participating individuals. A significant component of this preexisting intersubjectivity is enshrined in shared language. The argument is as follows: Negotiation depends on language (or at least on some form of communicative process), and language is constitutively intersubjective (Todarov, 1984, P. 30). Thus, a level of student-student and student-teacher intersubjectivity is prerequisite to the negotiative processes by which the resolution of uncertainty is attempted. A relationship between the constructs 'negotiation' and 'intersubjectivity' was postulated earlier in the following form: One pathway to knowing is via the resolution of uncertainty; the process of resolution is frequently negotiative; negotiation is mediated by language; language presumes intersubjectivity; and the matter of intersubjectivity is meaning (Clarke & Helme, 1997, p. 117). Equipped with this theoretical association, the researcher is in a position to examine the data set for interactions which might serve as instances of the postulated relationship and thereby of the constructs under investigation. Consider the following transcript (All utterances are by students. Karen and Lauren were subsequent interviewees, S19 and S20 were not):

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Turn

Transcript

1

S19: It says how many sheets of graph paper would you need to show one million one millimetre squares.

2

L: To show one million, you know you don't divide it by one hundred, because there's more than a hundred one millimetre squares. I mean you're going to find the area of this.

3

K: What?

4

L: You've got to find the area of this, there's more than one hundred one millimetres.

5

K: That's right. I was doing length by—oh screw that.

6

L: One hundred one millimetre squares.Take length—

7

K: Um, there's how many down here?

8

L: And along that side there is-

9

K: Ten, twenty, thirty, forty, fifty. How many are there down there?

10

L: There's a hundred one miltimetres there.

11

L: No, there wouldn't be.

12

K: There wouldn't be, that's not right.

13

L: There'd be two hundred and fifty.

14

K: Yeah.

15

L: Yeah, there'd be two hundred and fifty.

16

K: And we just totally screwed it all—

17

L: Length of graph.

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K: OK, so it would be length times width [inaudible]

19

L: And uh, two hundred and fitly millimetres. Width—

20

K: What's width?

21

L: That’s—

22

K: That's ten, twenty, thirty, forty, fifty, etcetera.

23

L: eighteen, one hundred and eighty.

24

K: Times one hundred and eighty. OK here we go. Two hundred and fifty times one hundred and eighty equals forty-five thousand. OK, that's forty-five thousand. We need a million. What's a million divided by forty-five thousand and times it by that?

25

L: Hang on, hang on, hang on, hang on. Don't go too fast. OK. Therefore there are forty-five thousand million mm squares.

26

S20: Forty-five thousand million?

27

L: Yeah.

28

S20: Forty-five thousand.

29

K: Twenty-two point two.

30

L: On one piece. Of graph paper.

Figure 3.2. Episode 2 Our goal in this case is the empirical grounding of intersubjectivity through the analysis of an accumulation of such examples. As Lerman has noted, "Intersubjectivity is a function of the time and place and the goals of the activity and the actors" (Lerman, 1996, p. 137). Bruner, among others, has noted that the situatedness of intersubjectivity is ultimately cultural (Bruner, 1996, p. 11). Intersubjectivity, if it is to be understood at all, must be understood in context and in action. In the preceding transcript, there are several indicators suggestive of intersubjectivity. First, much of the recorded dialogue is incoherent as written text: That is, sentences are ungrammatical or incomplete; pronouns are used without textual clues as to their referents; single word utterances are frequent. Communication in this form is only sustainable because the participants share understandings of the referents of the pronouns or key words, and of the processes,

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actions, or relationships suggested (but not stated) by the sentence fragments and the participants' gestures. Secondly, evidence of intersubjectivity can be seen in the manner in which one speaker will complete the sentence of the previous speaker, as occurs in 'turns' 8 and 9. The overt text in the form of the literal transcription is here being distinguished from the implicit text being co-constructed by the participants. The existence of an implicit text is inferred on the basis that the interaction appears to have been both purposeful and successful. The preceding episode and others like it, constitute examples of intersubjectivity in practice in a classroom setting, conforming to theoretical conceptions of the construct. In particular, the various instances of intersubjectivity within this episode and others support the postulated function of intersubjectivity as enabling and sustaining negotiation. For such interactions to occur, the negotiative accomplishment of intersubjectivity must have occurred already (before the episode in question). Pre-existing intersubjectivity within the dyad with regard to mathematical meanings, social meanings, and the dyad’s “ways of working” enables the operation of the dyad as a social and co-cognitive unit and is itself iteratively refined in the course of the same interactions that it enables. This evidence of the enactment of intersubjectivity locates intersubjectivity within the learning process as both agency and as outcome. At issue is whether acceptance of intersubjectivity as an operative feature of the classroom negotiation of meaning challenges the viability of a constructivist model of learning. A pragmatic reading of the data reported throughout this book suggests (i) that individuals interact in both mathematics and science classrooms as though they held many meanings in common; but (ii) can, on occasion, construe the actions of others highly idiosyncratically (discussed at greater length in other chapters). The imprecision of social exchange absolves us from the obligation to postulate any absolute correspondence of meaning and allows us to accept Episodes 1 and 2 as providing evidence of “operative intersubjectivity.” The successful anticipation of another’s statements or actions, so evident in Episode 2 , confers “second order viability” (von Glasersfeld, 1995) on the knowledge held to be common and confirms the status of that knowledge as operatively intersubjective at that time and specific to that situation (Bruner, 1996). We can thus confirm, on empirical grounds, the feasibility of intersubjectivity as a socially-situated phenomenon, while also identifying a role for this intersubjectivity within the process of cognitive construction of the participating individuals (in Episode 2, say) since the successful resolution of the problem in hand demonstrates co-construction of knowledge (the solution to the problem) novel to both individuals. 4. UNCERTAINTY, INTERSUBJECTIVITY AND NEGOTIATION 4.1 Identifying Uncertainty In the argument being developed in this chapter, Uncertainty has the status of a triggering condition stimulating the initiation of the learning process. "Uncertainty" is taken to refer to a behavioural state: socially prompted, but cognitively

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experienced. Examples can be found in the data set of several forms of student uncertainty with regard to academic content encountered in the lessons analysed in this book. Uncertainty regarding a fact or correct term: —

["Why is it V, what does V have to do with speed?"] —

Uncertainty regarding the correct form of an action or procedure; [I: Can you tell me how to work that one out? M: You get eight four oh which is the number of students, divide five, times two.]

—

Uncertainty as to the meaning of a term or an activity;

—

Uncertainty regarding a form of representation;

["How can the graph go below zero? You can't have a minus ten speed"]

-

["It's the velocity, not the way it bounces" in reference to a graph]

Uncertainty as to the correctness ofthe student's opinion, understanding, method, or outcome. ["Tell me if you think this looks sort of logical"] Uncertainty is not ignorance in any absolute sense. In order to be uncertain the learner must have some awareness or primitive knowledge of what they are uncertain about. This immediately situates the learner at the periphery of their existing knowledge base. Viewed in this light, Uncertainty in a learning situation may constitute a prerequisite condition for the formation of Vygotsky’s Zone of Proximal Development (Vygotsky, 1978). In the next chapter, Lerman examines in some detail the creation and non-creation of Zones of Proximal Development in classroom settings. This is not the purpose of this chapter. Rather, it is enough for my immediate purpose to identify examples of uncertainty likely to initiate the sort of negotiative interactions associated with learning. The preceding categorization of Uncertainty is unlikely to be exhaustive with regard to all possible classroom settings or situations. Rather, the categories provide an adequate classification of this particular data set, and enough distinct instances to sustain the postulated associative chain that connects the condition of uncertainty with negotiation, intersubjectivity, and construction of meaning. The practical role of Uncertainty in mathematics and science classrooms is revisited in Chapter 12. The identification of Uncertainty as a “productive state” (Rowland, 1995) rests with the learner’s interest in the resolution of that uncertainty. A learner might recognise uncertainty with regard to some aspect of academic content, but choose to acquiesce to that uncertainty; that is, not to pursue resolution. In this view, Uncertainty alone is not a sufficient condition for the initiation of learning. The felt need to resolve perceived uncertainty, however, commits the learner to the sort of social and/or cognitive activities that are the focus of this chapter and much of this book. —

4.2 Resolution Resolution is yet another term open to interpretation as either process or product. The categorization system, outlined below, is based on the dominant authority appealed to in the process of Resolution. For the purposes of this discussion, Resolution is taken to involve one of five forms, but these are not mutually

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exclusive, and a particular episode may involve student appeal to several authorities in seeking resolution of the same uncertainty. — Resolution via an appeal to empirical evidence; — Resolution via an appeal to a person (a knowledgeable other, such as the teacher or another student); — Resolution via an appeal to a text; — Resolution via an appeal to prior experience; — Resolution via an appeal to logical argument. Since, theoretically, each of the categories of Uncertainty may be resolved by any of the above means of Resolution, it is possible to consider the interaction of Uncertainty and Resolution. For example, does the knowing which is prompted by student uncertainty as to a procedure and which is resolved by appeal to a peer differ in a significant manner from the knowing which is prompted by student uncertainty regarding a representation and resolved by reference to a text (for example)? These differences might well be profound. Examination of the lessons (both science and mathematics) that provided the database for this book revealed over thirty distinct classroom "episodes" in which one or more students expressed uncertainty. The episodes comprised the dialogue and activities that students engaged in as they approached, worked on and reached completion of a particular classroom activity, such as a problem solving task. Thus, each episode was a coherent unit of activity unified by a single purpose. Within each episode there were identifiable negotiative events consisting of a number of utterances — questions, statements, answers — where a particular form of uncertainty was expressed and sometimes resolved. In Episode 1, the students had been asked to find the height of a stack of one million sheets of paper. Uncertainty about procedures and outcomes was resolved through peer interaction. Appeal to peers was in frequent evidence in the data and Episode 1 is a particularly rich example of the features of this form of classroom negotiation. There were a number of instances where students were not able to resolve their uncertainty within the time allowed by the lesson or in the interview following that lesson. Not surprisingly, more superficial forms of uncertainty (such as procedural uncertainty) were usually cleared up within the lesson. Uncertainty about more complex and difficult material was less readily resolved. The classification scheme used in this chapter for the parsing of classroom discourse suggests a structure to classroom interactions in which the negotiative event holds a pivotal place. In a separate study, Barnes made use of the scheme in an analysis of power relations within student collaborative groups and concluded: Negotiative events appear to be a key unit for analysis, because the transitions from one to the next mark the progress of a group's work on an activity. A student who enacts closure of one negotiative event and initiates the next does to a very large extent control the discourse (Barnes, 1998).

This is not to underestimate the practical difficulties of parsing classroom dialogue into negotiative events. By comparison, parsing by both Episode and Utterance is relatively straightforward. Parsing by Turn is more problematic, since individuals do not avail themselves of all proffered “Turns” and a single individual

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may take consecutive turns. Further, what constitutes a Turn may be highly culturally specific. One can imagine an analysis in which utilised and non-utilised Turns provided significant insight into power differentials and conventions of behaviour within social groups. Such an analysis was not part of this study. What has been argued and demonstrated empirically is the utility of “negotiative events” as a conceptual category, correspondent to specific instances of uncertainty and associated negotiation. The analysis of the eight lessons reported in this book confirmed the significance of uncertainty as a catalyst of the negotiative exchanges commonly seen to be a precursor to learning. The relative infrequency of the resolution of identified uncertainty within the lessons studied may be an artifact of the particular lessons chosen for analysis. Alternatively, it is possible that while uncertainty can be identified as a key element in classroom interactions and specific uncertainties matched to particular negotiative events or episodes, the resolution of those uncertainties may sometimes only be documented, studied and understood over a time frame longer than the single lessons that were the focus ofthis analysis. A related point concerns the authority to which students appeal in resolving uncertainty, particularly in the sanctioning of a “claim to know”. In a study that preceded those reported in this book, Clarke and Kessel (l995) suggested that when asked to reflect on the consequences of a lesson, students in the sample classrooms did not tend to employ terms such as "learn" or "know." Students in these classrooms, when reflecting on the consequences of their participation in a lesson, or in the course of classroom conversation, made frequent use of the term “right” (as in, “I got it right” or “The teacher said I was right”). Such a finding surely prompts us to conjecture as to what view of knowledge and classroom learning is held by these students, such that their success in a mathematics or science lesson can be adequately (and consistently) described as being “right.” The restriction of the goals of mathematics and science instruction to only those things about which one can be ''right'' suggests an impoverished model of mathematical and scientific knowledge and a likely misconstrual on the part of the students of the teacher's classroom goals. Such a finding has clear implications for practical action to develop classroom situations that promote more sophisticated learning outcomes. Specifically, if uncertainty is a preparatory condition for student activity in the zone of proximal development, and if that activity is directed towards the resolution of that uncertainty, then it becomes a matter of some significance as to how a learner judges resolution to have been achieved and the nature of the authority invoked in enacting closure on a negotiative event. It is, to some extent, a validation of the model of learning proposed in this chapter that consideration of the associative connections between constituent constructs prompts such potentially useful speculation. In this particular instance, a too ready acquiescence to an external authority such as teacher or text could restrict the learner’s knowledge construction. Reference to instances of such premature closure of the knowledge construction process can be found in the early writings of Baird (Baird & Mitchell, 1986) and in the identification by Clarke, Frid and Barnett of a three-tiered hierarchy in the authorities invoked by teachers and learners, not dissimilar from that found in Kohlberg’s model of moral development (Clarke, Frid & Barnett, 1993).

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Acknowledgement by teachers of the significance of uncertainty in the learning process must be accompanied by recognition of the importance of distinguishing between the various means by which that uncertainty might be resolved and the promotion of negotiative processes that achieve resolution by appeal to internalised cognitive frameworks rather than by acquiescence to the arbitrary dictate of authorities external to the learner: To the authority of “that makes sense” rather than the authority of “well, that’s the answer in the back ofthe textbook.” 5. CONCLUSIONS The data reported in this chapter are intended to contribute to our understanding of intersubjectivity as both an outcome of classroom negotiation and as a mediating agency in the resolution of student uncertainty in mathematics classrooms. This chapter reports evidence for the relationship between negotiation and intersubjectivity as associative, with intersubjectivity identified as prerequisite for the classroom negotiation of meaning. The argument has been made in this chapter that intersubjectivity should be seen as the means by which the negotiative process is sustained and social closure of classroom activity signalled through group acknowledgement of the resolution of uncertainty. In the examples discussed in this chapter, learners appear to refine an intersubjectivity previously accomplished and in which their on-going interaction is grounded. Starting from the intersubjective constitution of language, the preceding discussion has elaborated the situated character of intersubjectivity, specific to the proclivities of the participants in an interaction and their personal history of interaction, and specific to the particular situation of the interaction. The data cited suggest a reflexive relationship between social interactions and individual cognitive activity. It is argued that acceptance of the situated character of intersubjectivity provides us with a rationale for the compatibility of operative intersubjectivity as a socially-mediated phenomenon with a Piagetian or constructivist theory of subjective cognitive construction of knowledge. It is clear that the resolution of uncertainty in mathematics classrooms can have a negotiative character. Student classroom activity within a single lesson can be interpreted as occurring in a hierarchy of levels of purpose: the lesson, the episode, the negotiative event, the turn, and the utterance, where each level can be associated with a specific purpose. Other data, not reported here, documents the purposeful resolution of student uncertainty over several lessons, and even over several weeks. Resolution over such periods of time goes beyond the confines of classroom activity and is not considered in this chapter. The resolution of student uncertainty within the classroom can provide the pretext for purposeful activity within at least two of the proposed levels of classroom discourse: the episode, and the negotiative event, and this has been demonstrated empirically here. The practical implications of attending to the resolution of uncertainty at these two levels centre on the form of authority by which learners judge that resolution has been achieved. Attending to the negotiative event as a unit of interaction appears to offer an opportunity to associate social and cognitive phenomena. Both Episodes 1 and 2

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provide useful illustration of the pre-existence within the Lauren/Karen dyad not just of intersubjectivity regarding the meanings of the mathematical terms they exchange during their interaction, but also the pre-existence of established ways of working as a dyad. Lauren’s contributions to a negotiative event are frequently interpretive or explanatory with respect to the task or a particular procedure. Karen, on the other hand, advances the discussion by asking many questions of Lauren (and, implicitly, of herself), and by verbalising calculations (Episode 1: Turns 15, 17, 28, 37, and 49 and Episode 2: Turn 24, provide good examples of this). The co-construction of these patterns of dyadic practice involves the iterative refinement of intersubjectivity, just as is the case for mathematical meanings. However, in this case, the matter of the meanings relates to the individuals’ actions and statements, and the product of this process of iterative refinement is a body of dyadic practice, specific to the Lauren/Karen dyad. It will be argued in Chapter 12 that Classroom Practices are similarly co-constructed by teacher and learners. The focus on negotiative events and on evolving practice places an emphasis on interaction (dyad and/or whole class) appropriate to the fundamentally social nature of negotiation and intersubjectivity. This chapter has addressed the significance of uncertainty as a constructive catalyst for reflection, negotiation of meaning, the development and refinement of intersubjectivity, and for the learning of both mathematics and science in classroom contexts. Educators seeking to exploit uncertainty in the interests of promoting learning may find their efforts undermined if the resolution of the students’ contentrelated uncertainties rests with authorities that trivialise the nature of mathematical and scientific knowledge. This proviso does not call into question the significance and educational potential of the structured utilisation by teachers of student uncertainty, but does, however, distinguish between the negotiative process documented empirically in this chapter and the optimal utilisation of that process for the promotion of learning. Most importantly, this chapter contributes to the empirical grounding of key elements of the learning process such as negotiation and intersubjectivity, and of their interrelationship. In the preceding argument, conceptual or theoretical primacy has not been accorded to any one of negotiation, intersubjectivity, interaction, uncertainty, resolution, or meaning; rather, each term participates in a network of association that collectively characterises learning in classroom settings. Elaboration of each term inevitably invokes all other terms, and no term is taken as fundamental. This reflexivity of relationship between the key constructs addressed in this chapter, whereby constructs mutually elaborate each other, is an essential characteristic of the model of learning developed in this chapter. The interpretive nature of each individual’s participation in the classroom is progressively elaborated in several of the following chapters. Integration of this interpretive process into a theory of learning is not undertaken until Chapter 12. It is sufficient for the purposes of this chapter to have characterised the nature of the relationships between Uncertainty, Negotiation and Intersubjectivity.

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ACKNOWLEDGEMENTS The author would like to thank Sue Helme and Cathy Kessel for their assistance with data collection, for their meticulous transcription of many hours of classroom videotape and interview data, and for their input in many conversations regarding data interpretation related to the content of this chapter. Comments from Les Steffe also influenced the final form taken by this chapter, and thanks are extended to Les for his input. REFERENCES Baird, J.R., and Mitchell. I.J. (Eds) (1986). Improving the quality of teaching and learning: An Australian case study - the PEEL project. Melbourne: Monash University Printery Services. Bakhtin, M. (1979). The aesthetics of verbal creation. Moscow: Bocharov. Cited in Todarov (1984). Barnes, M. (1998). Towards a framework for analysing power relationships in small group discussions. Draft of a paper submitted for inclusion in the 1998 conference ofthe Mathematics Education Research Group of Australasia, to be held at Broadbeach, Queensland, July 4 to 7. Bruner. J. ( 1996). Celebrating Divergence: Piaget and Vygotsky. Keynote address given in Geneva, September 15, in honour ofJean Piaget's birth. Clarke, D.J. (1993). Unguarded metaphors in educational theorizing: Metaphorically modeling the social-personal interface (Abstract only). In J.R. Becker and B.J. Pence (Eds.) Proceedings of the Fifteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 2. San Jose, CA: San Jose University, 251. Clarke, D.J. (1994). The metaphorical modelling of "coming to know". In J. P. da Ponte and J. F. Matos (Eds.) Proceedings of the Eighteenth International Conference for the Psychology of Mathematics Education. Volume I. University of Lisbon, Portugal, July 29 to August 3, pp, 193 – 200. Clarke, D.J. (1 996). Refraction and Reflection: Modelling the Classroom Negotiation of Meaning. RefLecT 2( l), 46-51. Clarke, D.J. (1999). Refraction and Reflection: Modelling the Negotiation of Meaning in Science and Mathematics Classrooms. In J. Baird (Ed.) Reflecting Teaching Learning: Perspectives on Educational Improvement. Melbourne: Hawker Brownlow Education, pp. 167-180. Clarke, D.J., Frid, S., & Barnett, C. (1993). Triadic systems in education: Categorical, cultural or coincidence. In B. Atweh, C. Kanes, M. Carss, & G. Booker (Eds.) Contexts in Mathematics Education. Kelvin Grove, Qld: MERGA, pp. 153–160. Clarke, D.J. & Helme, S. (1997). The resolution ofuncertainty in mathematics classrooms. In F. Biddulph and K. Carr (Eds.) People in Mathematics Education. Proceedings of the Twentieth Annual Conference of the Mathematics Education Research Group ofAustralasia, held at Rotorua, New Zealand. 7-11 July, 1997. Waikato, New Zealand: MERGA, pp. 116-123. Clarke. D.J. & Kessel, C. (1995). To know and to be right: Studying the Classroom Negotiation of Meaning. In B. Atweh & S. Flavel (Eds.). Galtha: MERGA 18. Proceedings ofthe 18th annual conference ofthe Mathematics Education Research Group ofAustralasia. Darwin, NT: University of the Northern Territory, pp. 170 -177. Cobb, P. (1994). Where is the Mind? Constructivist and Sociocultural Perspectives on Mathematical Development. Educational Researcher, 23(7), 13-20. Cobb, P. & Bauersfeld, H. (Eds.) (1995). The emergence of mathematical meaning: lnteraction in classroom cultures. Hillsdale, NJ: Lawrence Erlbaum. Helme, S. & Clarke, D. J. (1996). Moments in the process ofcoming to know. In P. Clarkson (Ed.) Technology in Mathematics Education. Proceedings ofthe 19th annual conference of the Mathematics Education Research Group ofAustralasia. Melbourne, Victoria: University of Melbourne, pp. 269 - 276. Helme, S. & Clarke, D. J. (1998). Capturing complexity: The use of complementary accounts in the analysis of classroom learning. Paper presented at the 1998 Deakin Research Methodology Conference, Deakin University, Burwood, Victoria, Australia.

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Helme, S . , Clarke, D.J., & Kessel, C. (1996). Moments in the process of coming to know. In P.C. Clarkson (Ed.). Technology in mathematics education. Proceedings of the 19th annual conference of the Mathematics Education Research Group of Australasia (MERGA). June 30 - July 3, 1996, at the University of Melbourne. Melbourne: MERGA. pp. 269 – 276. ( 1996) Lave. J. & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge University Press. Lerman, S. (1996). Intersubjectivity in mathematics learning: A challenge to the radical constructivist paradigm? Journal for Research in Mathematics Education. 27(2). 133-150. Rowland, T. ( 1995). Hedges in mathematical talk: Linguistic pointers to uncertainty. Educational Studies in Mathematics, vol. 29, 327-353. Steffe. L.P. (1995). Alternative epistemologies: An educator’s perspective. In L.P. Steffe & J. Gale (Eds.), Constructivism in Education (pp. 489 - 523). Hillsdale: Lawrence Erlbaum Associates. Steffe, L.P. & Thompson, P.W. (2000). Interaction or Intersubjectivity? A Reply to Lerman. Journal for Research in Mathematics Education. 31(2), 191–209. Todarov. T. (1984). Mikhail Bakhtin: The Dialogical Principle. Translated by Wlad Godzich. Minneapolis: University of Minnesota Press. Voigt, J. (1995). Thematic patterns of interaction and sociomathematical norms. Chapter 5 in P. Cobb & H. Bauersfeld (Eds) The Emergence of Mathematical Meaning. Hillsdale, NJ: Lawrence Erlbaum, 163-202. Von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Washington. D.C.: The Falmer Press. Vygotsky, L. S. (1978) Mind in Society Cambridge, Mass.: Harvard University Press.

STEPHEN LERMAN

CHAPTER FOUR Accounting for Accounts of Learning Mathematics: Reading the ZPD in Videos and Transcripts

1. INTRODUCTION In this chapter I aim to give an account of the communication taking place in a particular classroom between the particular children chosen by the research team for the video camera’s focus, and also to account for that account. In that this chapter seeks to treat both ‘levels’ of accounts as social research, and in its attempt to be reflexive, it is ethnographic (in its literal sense of writing culture (Atkinson, 1992)). Somewhat questionably, the warrant of ethnography is often given as ‘having been there’ (Atkinson, op cit.). I was not in the classroom, but I was given the opportunity to sit and look at videos of the classroom as set up and recorded by the research team and read the transcripts prepared by them. This account, then, will need to be as much about the latter event in which I did take part as about the recorded classroom events. At one level my intention is that this account will offer some insights into how answers are arrived at by students in a mathematics classroom, and how these students make mathematical meanings from their activities. I might hope that those insights could assist teachers, including me, in deciding how to manage peer working, when and which kinds of interventions to make, and so on. This is intended to be a piece of research in mathematics education. At another level, given my situation in this project as a researcher and given the nature of the research event, that of someone visiting1 the data, what can I legitimately say about the data? How is my purpose in the reporting reflected in my choice of the manner of its presentation? This chapter will intend also to address the issue of the role of video analysis in educational research and, therefore, the validity of the account I will give. In selecting extracts of the data set to study I was looking for periods in the mathematics lessons where there was sustained conversation, student-student or teacher-student, either of which would have served my purposes for this analysis. We (I looked at, and worked on, the videos with another invited researcher Clive Kanes) felt it necessary to study the videos, going over them many, many times. This detailed work enabled us to make some changes to the transcripts, clearing up 53 David Clarke (ed.), Perspectives on Practice and Meaning in Mathematics and Science Classrooms, 53—74, © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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some of the initially incoherent utterances and altering a number of utterances which we found to have been differently transcribed. Such a process is never-ending; other researchers might well make further changes to the same sections of the tape, and perhaps change some of our changes. Of course it must be said that the aspects of the lesson which are captured by the video are, inevitably, already part of the layers of interpretation which are being written, in the sense that the choices made by the camera operator, the particular students being filmed, the number and direction of microphone, are just some of the factors that are present at that stage. Although an event took place on the date, at the time, and in the place recorded, all accounts of it, including the original video-tape recordings, reports by the students, the teacher, the researchers present, the researchers visiting later, are necessarily interpretations (Hammersley & Atkinson, 1983). Every account after the event will be set in a different context, with different intentions. The intention of my account was to describe the mathematics learning of two students engaged in a task set by the teacher. Below I will set out what I take to be an event of learning and how I claim to recognise it in the data-set. Indeed I must do so, as I am creating the object of my research and must justify the selection to whatever extent possible. I want first to consider the form of the account. I was tempted to write two stories, each being as if written by one of the students (Lewis, 1961, in Atkinson, op cit.). This would have provided a particularly interesting perspective, given that the students could have told their own stories in the postlesson interview, had it been set up that way by the interviewers. In writing those stories at least it would be overt that I would not be making claims to any incontrovertible facts of those students’ learning. Such a device would recognise the affinity of ethnographic accounts with fiction rather than with a scientific paradigm. Some historical books have been written in a fully narrative style, although with the intention that the stories told could well have been good descriptions of actual people’s lives. Were I to have chosen such a method of writing this would have been my intention. However, for reasons of personal preference as well as for reasons of familiarity with particular writing genres in the mathematics education community I have used a descriptive/analytical style. It is widely argued that research from this perspective must be reflexive, and self-conscious of the relationship between the data and the account (e.g. Atkinson, op cit.; Usher, 1993; Scott, 1996; Hitchcock & Hughes, 1995). The choice of extracts, for instance, is guided by the argument one wishes to present, although it is usually justified as I have above, as ‘convenient’ for the research. Using a system based on periods of time, or number of words, is an alternative approach that is intended to remove at least that bias. That system is taken up in others’ analyses. A key element in research is the interaction of theory-building and data collection. As a visitor to the data, I was in the position of not being able to affect its design or collection at source, that is, in the classroom, in the other interactions of the researchers with the teacher and students, or in the final form of the data-set, except that I contributed to the researcher’s questions to the teacher in her interview. However, in the choice of extracts, in the choice to alter the transcript through repeated replay of the video, in the choices of reconstructions of the text where the sound was not clear, and crucially in the choice of object of study in my analysis, I

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was constructing the text for research. With this goal I have collected some data and I will attempt to construct a “logic-in-use” (Hammersley & Atkinson, 1983, p. 2) for the account written of it. I am bringing a theory to the data and intending to develop that theory through the analysis. Naturalistic ethnographers might be unhappy with this but I would suggest that it more honestly reflects the process of research. Like most contemporary researchers, I (and my colleagues in the Classroom Learning Project) find it impossible to conceive of the notion of approaching any research without a theory. The choice of extracts for analysis was made, as mentioned above, on the basis that I wanted to look at sustained, uninterrupted communication, although I was not concerned whether this was between students or with the teacher. I was looking for how learning takes place, or fails to take place, in zones of proximal development. In the next section I will attempt to justify this through a critical review of some of the literature. 2. THEORETICAL FRAMEWORK 2.1 The Socio-Cultural Perspective What has become largely accepted in recent years in the mathematics education research community is that a focus on the constructions of individual children is inadequate unless it takes account of the socio-cultural setting of those activities (Lerman, 1996a, 1996b, 1997). The ways in which researchers view the role and significance of the socio-cultural setting varies widely of course. To employ some terms of Smith (1993) that I have found useful in this context, the socio-cultural setting can be seen as causative of learning or constitutive of learning. The former perspective, he argues, was that adopted by Piaget, and developed today by the radical constructivists: social interactions, along with physical and textual interactions, can cause disequilibrium in the individual, leading to conceptual reorganisation. This is the meaning of learning for Piaget: “Intelligence organizes the world by organizing itself” (1937, in von Glasersfeld, 1989, p. 136). Piaget cites four factors that lead to the development of one set of structures into the other: maturation; experience of the physical world; social transmission; and equilibration, the latter being the fundamental one (Piaget, 1964, p. 13). Most (all?) constructivists emphasise that social interactions are the major element in causing disequilibrium. Von Glasersfeld claims that social interactions have a privileged role in conceptual development in that “the most frequent source of perturbations for the developing cognitive subject is interaction with others” (von Glasersfeld, 1989, p. 136), but Perret-Clermont makes it clear that the causation is not manifested in the outcome: Social-cognitive conflict may be figuratively likened to the catalyst in a chemical reaction: it is not present at all in the final product, but it is nevertheless indispensable if the reaction is to take place (1980, p. 178).

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Smith’s other perspective, that socio-cultural factors are constitutive of learning, argues that social factors are “the substance of operational knowledge” (Smith, 1993, p. 128) and are therefore constitutive of consciousness. It is in discourses, subjectivities, significations and positionings that psychological phenomena actually exist (Evans & Tsatsaroni, 1994). Subjectivity is constituted through social practices, not independently arrived at by a decontextualised individual (Walkerdine 1988). Development takes place in social settings in which meanings are carried, rather than meanings being freely and "rationally" constructed by the individual; a shift away from a view of the cognizing subject constructing her/his subjectivity and knowing, to one of the construction of human consciousness in and through communication. It is in this sense that Vygotsky writes of the social plane as primary both in time and in fact. This is not to see individuality and difference as non-existent, nor is it to argue that the individual mind is reducible to the social plane. It is to argue that the gaze of the researcher must be on the social and cultural practices in which meanings and purposes function such that people engage with life. The study of the mind is a way of understanding the phenomena that arise when different sociocultural discourses are integrated within an identifiable human individual situated in relation to those discourses (Harré & Gillett, 1994, p.22).

Social settings, such as the mathematics classroom, are determined by all the actors, both present and absent, and so the search for common understanding (Edwards & Mercer, 1987) is a function of the time and place and the goals of the activity and the actors. The term 'absent' here is intended to convey the image of those who are part of making the setting what it is, such as textbook writers, school administrators, parents, and others, but who are not physically present in the classroom. A visual metaphor that I find useful is that of looking through a zoom lens (Lerman, 1998b), starting from a focus on an individual student, and as the zoom progressively draws outwards one sees another student, some materials including texts, other students, a teacher, a classroom, a school and so on. There is also the reverse procedure, of course, starting from the wide angle lens and zooming in towards the individual. Wherever one chooses, as a researcher, to pause in the zooming process one cannot ignore the other pictures as they came into, and then went out of, focus. A psychologist may choose to focus on an individual or a small group, as I do in this chapter, but the task I have set myself is to be aware of the classroom, the framing of the activity, and the other students, and to integrate them into the account. The metaphor sets limits, though. One might wish to look at the lives of these students around other parts and arenas of school life, and in their free time, as well, but the zoom lens is linear, and a-historical. As I mentioned above, a fictional narrative might attempt to tell a broader story (Winbourne & Watson, 1998). 2.2 Zone of Proximal Development The assumption of this chapter, regarding learning, is that it takes place in the zone of proximal development. As is well known, Vygotsky defined the zone as follows:

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... the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers (1978, p. 86).

Vygotsky introduced the ZPD in a lecture given in March 1933 (Van de Veer & Valsiner, 1991, p. 329), although he pointed out that the idea was not originally his own. He died only fifteen months later and clearly had not been able fully to elaborate his thoughts on the ZPD. Three moments can be identified in his development of this concept (van der Veer & Valsiner, op cit): (i) a criticism of traditional IQ testing and an attempt to compare differential scores between individual and assisted performance in problem solving; (ii) an extension of the first phase, with emphasis on the social aspects of assistance and guidance rather than on scoring performance; and (iii) further developments of the zone of proximal development, taking it out of the immediate socio-interactional situation, and focusing on the “object-mediated world” in such activities as play. The interpretation of Vygotsky’s ZPD which I use and develop in this study (see also Lerman, 1994) is close to that Davydov (1988, 1990) and is perhaps less ‘internalist’ than many interpretations of Vygotsky’s own formulation. The ZPD is not the individual’s, it is not a physical space, or a region of potential that the individual brings with her/him to any situation. What the individual brings is a previous network of experiences, and the limitation of specific genetic propensities, largely undeterminable as separate from environment. In relation to school, the learning activity can set up the ZPD. The learning activity is certainly the task set by the teacher but is also a function of the style of classroom interaction, the texts, the ethos of the school, the possibilities arising out of the particular mix of the actors that day, even that moment, and so on. It does not exist, for the class, for the teacher, or for the individual students, before the interaction. This is the sense I take from the following: We propose that an essential feature of learning is that it creates the zone of proximal development; that is, learning awakens a variety of developmental processes that are able to interact only when the child is interacting with people in his environment and in collaboration with his peers (Vygotsky, 1978, p. 90).

A learning activity, then, proffered by the teacher or a researcher (Newman, Griffin & Cole, 1989; Brown, Ash, Rutherford, Nakagawa, Gordon, & Campione, 1993), or a text, or peer, can set up a ZPD for the participants. In a classroom, or amongst a group of students, what arises has been described as “multiple, overlapping zones of proximal development” (Brown et al, 1993). The interpretation of the ZPD which I use here, however, emphasises the intersubjective origins of all learning, and therefore is focused on the learning activity. The roots of psychogenesis for each individual are in sociogenesis, which is the focus of study through which to read the internalisation by each individual. Thus, in a successful learning activity, the child’s response to a question will originate from her/his previous network of experiences but may well pull a second child into her/his ZPD and a second response might well pull both pupils into their ZPDs. It is the intersubjectivity which sets up the ZPD, that leads to internalisation, although quite often a ZPD is not created, of course. Describing this as overlapping zones might

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suggest that these zones are brought to the activity by the students, like force fields, in which, if they overlap, children learn. In my interpretation, from Davydov (1988, 1990; also Meira & Lerman, forthcoming) there is a shared ZPD, created in the learning activity, and the process of internalisation occurs in each child’s ZPD. These latter are dependent on the former. Vygotsky viewed intersubjectivity as preceding intrasubjectivity. Some formulations of intersubjectivity seem to work from the other direction. Thus Tudge (1992) writes: ‘The second concept that, from a Vygotskian perspective, is essential for understanding the processes and consequences is intersubjectivity ... As Rommetveit (1979) states it, “Communication aims at transcendence of the ‘private’ worlds of the participants. It sets up what we might call ‘states of intersubjectivity”’ (Rommetveit, 1979, p. 94). In other words, individuals may arrive at some mutually agreed-upon, or intersubjective, understanding.

I have discussed the problems of such a formulation elsewhere (1996a). Briefly, where intersubjectivity is conceived as mutually agreed-upon understandings originating in ‘private’ worlds one encounters the boot-strapping problem of learning, namely, unless the new ideas are already present in the learners, how do they recognise them? In Vygotsky’s version, the internal plane is formed through internalisation, which is mediated by and originates in the intersubjective plane. All learning is a consequence and outcome of communication. I want to argue that one can take the zone of proximal development to be the explanatory framework for learning as a whole, both in intentional settings, such as schooling, and in informal settings; in other words all socio-cultural milieus. It recognises the fundamental asymmetry of the teacher-student(s) relationship, and of peer interaction, an asymmetry often denied or underplayed by more individualistic approaches. It provides the framework, in the form of a symbolic space (Meira & Lerman, forthcoming), for the realisation of Vygotsky’s central principle of development: In our conception, the true direction of the development of thinking is not from the individual to the social, but from the social to the individual (Vygotsky,1986, p. 32).

Such a definition opens a space for a unit of analysis of consciousness which incorporates affect and cognition. When we approach the problem of the interrelation between thought and language and other aspects of mind, the first question that arises is that of intellect and affect. Their separation as subjects of study is a major weakness of traditional psychology... (Vygotsky, 1986, p. 10).

Recent studies in mathematics education have taken the integration of cognition and affect as axiomatic (Evans 1993; Matos, 1995; Santos, 1995; Brodie, 1995). Teachers use a variety of strategies to try to create a zone of proximal development, including reminders of past lessons, events and language (Edwards & Mercer, 1987), references to outside school objects or meanings, and so on. Much can be learned by the researcher, focusing on these strategies of the teacher. Similarly, pairs of students can create their own zone of proximal development if they are motivated, taught how to share ways of working (Cobb, Wood & Yackel,

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1991), have an appropriate personal relationship, and/or other factors. Indeed students can be, and very often are, pulled into their ZPD by imposition. This may occur for reasons of desire to become like another person, or to please another person, to be accepted into a group, or achieve other such goals people will copy/emulate another, and subsequently that behaviour may become part of that person. However, it is certainly not the case that learning always takes place. In both teacher-student interactions and student-student interactions the participants may not engage together in the activity. They may act separately or one and/or the other may not act at all. In Meira and Lerman (forthcoming) we give some instances of a zone of proximal development being created in an activity and instances where, despite the teacher’s best efforts, a zone of proximal development is not created in the activity. In the two examples chosen for analysis below the students do not create a zone of proximal development, but I would argue that as much may be learned from incidents where a zone of proximal development is not created as when it is. 2.3 Peer Collaboration Studies of peer collaboration have a fairly recent history. Forman and Cazden (1985) refer to the work, drawing on a Piagetian theoretical framework, of lnagaki and Hatano (1977), Kol’tsova (1978), and Perret-Clermont (1980) as some of the earliest, their work being aimed at identifying cognitive conflicts and examining the outcomes. Forman’s (1981) approach was to focus on pairs of students collaborating, without the insistence on one being more capable. She studied outcomes of their work as well as transcripts of their collaboration whilst working together. Forman identified three different forms of peer interaction: parallel, in which the students pursue separate directions; associative, in which students communicate their work but do not develop their ideas together; and cooperative, where the work is the product of both students. The students discussed in this chapter are mostly interacting in a parallel way, and at times in an associative way. In Forman and Cazden (1985) they compare the work of individuals and pairs, pointing out in their data that the pairs achieve considerably more advanced levels of work even though a post-test of the individuals in the pair do not reveal the same success. They argue that this makes sense from the point of view of the zone of proximal development, whereby the students achieve things together today that they will achieve separately at a later date. Tudge (1990) emphasises the need for more studies of peer collaboration, both because the great emphasis on teacher-student activity, from the perspective of Vygotsky’s theories, may lead readers to suspect a transmission-style orientation, and because so much classroom activity is actually between students, albeit framed by and mediated through the teacher’s talk, the textbook and other elements. Forman and McPhail (1993) point out the elements of analysis that researchers should draw upon in analysing peer interactions in the ZPD, beyond that of achievement, including motivational and epistemological developments as indicated in discursive features. Recognition of the extent and significance of interactions between students is a key issue for the Clasroom

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Learning Project as a whole and this chapter is intended to develop Forman and McPhail’s elements to include cases where a ZPD is not created and to attempt to take into account other elements such as the teacher’s construction of the students’ abilities, the teacher’s interventions and her specific teaching strategies. 2.4 Logic-in-use Although the study reported in this chapter is a small one, I want to identify in large part with the aims of Edwards and Mercer (1987): We see our analysis as having implications ... for understanding the relations between discourse and learning and between knowledge and action (p. 168).

Many researchers have worked with the notion of the zone of proximal development. Newman and Holzman (1993) give a critical review of much of this work (including that of Tharp & Gallimore, Newman, Griffin & Cole, and Bruner) in terms of the authors’ adoption of Vygotsky’s work into their existing cognitive paradigms rather than recognising its revolutionary character. In proposing their own interpretation and development they write: It is helpful to think of the ZPD ... as a language game ...we play a language game, though the game has no existence or significance independent of its playing ... (p . 90).

I looked at the interactions between the subjects of the video, trying to identify when they were communicating their ideas and reasoning to each other. Acts of communication, as objects of study, are, I am arguing here, the signs of sociogenesis, the social origins of psychogenesis and internalisation. I looked for clues from all the elements of the data-set to help in drawing inferences about the nature of that communication. I have referred above to other research on videos which uses the notion of ZPD (Meira & Lerman, forthcoming). That work is with much younger children and there we look for the children’s attention being captured by the teacher, or vice versa, and their utterances and actions being directed and modified by her. Here I was looking for instances where the utterances or actions of one student affected the other, resulting in a modification of what they then said and did. Elsewhere (Vile and Lerman, 1996), drawing on Peirce (in Buchler, 1955), we have called this ‘catching other’s ideas’. Immediately one encounters a constraint: whilst this chapter can include verbatim extracts from transcripts it cannot include the video, although I have used it in drawing these inferences. The technology to include video extracts in a ‘paper’ already exists, of course, but there are ethical constraints on the presentation of data. Names can be changed but changing pictures to give anonymity is more complicated. I will attempt to indicate clearly where seeing the video has influenced the inferences I have made. There are many possible interpretations of the videos and transcripts and, as I have mentioned above, others will be given elsewhere in this book. It could be argued that some of these other interpretations would supplement mine. For instance, I will indicate below that the behaviour of teacher and the students with regard to their mathematical activity is framed by a discourse of ability. This constitutes one of the student, Mark, as more able and therefore more powerful in

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the interactions between them. The teacher read the video and the interactions of the students as confirming her evaluation in terms of ability. Thus power relations could be used to provide an account from the data-set. In order to give an account which is adequately framed in a theory, and which also offers a description of the objects about which one, as researcher, is making statements, one has to delimit one’s text. A Piagetian model would argue that power, in the form of authority, inhibits equilibration. Vygotsky’s ZPD, working from a sociogenetic perspective, assumes imbalances in social relations as part of being human and communicating, and therefore these will be identified as part of my account. To summarise, I searched for events of communication from the videos and the transcripts in which the students catch each other’s thoughts. This does not explain, however, how I analysed the transcripts of the interviews with the students and the teacher. In fact I was using these for clues to the socio-cultural setting which affects the creation or otherwise of a ZPD, to the imbalances in relationships and knowledge, to the ways in which the working together and the task are framed by the teacher, the students, and many other factors of which only some can be captured. Clearly this is much less specified than interactions in the ZPD and the use of the term ‘clues’ is to indicate that the interviews were used for rather general observations. 3. DATA AND ANALYSIS 3.1 Background In my reading of the videos I felt that the two boys, Darren and Mark, did not seem to work well together and I was therefore particularly interested to know how and why they were chosen for the recording. The researcher who interviewed the teacher asked the following questions on my behalf: “How was the pair chosen? Their choice or the teacher’s? Have they a history of working together? Did they get on well at the time the video was made? Did they work together afterwards?” In the interview the teacher replied that she had chosen Mark and Darren for several reasons: they were friends and usually worked together; they were, in her view, of different abilites and this would make their interactions more interesting for the researchers; for reasons of convenience of the video recording that pair of subjects was the most suitable. She added, Yes they were [getting on well when the video was made]. I think that Darren, um Darren said that he wasn’t feeling well during the tape. I also think that it was compounded by the fact that really he was well out ofhis depth as far as the ratio work went. Yes [they worked together afterwards], and they’re still friends, even one or two years later, or one year later. They weren’t the best behaved students um, you know. usually they were just pretty average typical kids. Mark tends to be fairly bright, Darren tends to not understand a great deal and I thought that some of that might come out in the dialogue and give you a good insight as to where students can have misconceptions and whether other students can help them and things like that.

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The teacher’s orientation throughout her lessons is to encourage student interaction as a route to learning, a view which I support, as this chapter demonstrates. The class of students, who were about 13 years old, had been set an exercise which included a question containing parts (a) to (j), on ratios. Parts (a) to (e) were given to the whole class to solve, and the teacher had instructed the students to cross out the letters (f) to (j). In what appeared on the video to be an afterthought, but from the interview with the teacher was probably planned, she called several students to the front of the class, including one of the two boys in the focus pair, Mark. She referred to those students as ‘those who like working ahead’. When two named students turned down the invitation to come to the front, the teacher asked if anyone else wanted to join the group instead, and the second student in the pair, Darren, volunteered. In her interview, the teacher explained that she likes to give people the chance to volunteer. She gave them some extra instructions on cancelling fractions, which she called an ‘algebra trick’, so that they could also answer the crossed-out parts ofthe first question. Question 1 consisted of the following: 1. Reduce to simplest term (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

Figure 4. 1 Question 1

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Extract 1 The first extract below is work which Darren and Mark undertook when they sat down after the extra instruction. There were a few interchanges in which they confirmed what they were supposed to do. The analysis below begins immediately after those interchanges, when they began work on part (f).

Table 4.1. Extract I Turn

Transcript

1

M: What? Equals ab? [pause, D looks on M’s page] Equals ab?

2

D: Yeah.

3

M: No, it equals one

4

D: Wait a second..

5

M: 'Cause one, [punching calculator buttons] twelve times tw... no. One, look, look, look. One times two, divide one times two... it shouldn't equal four. [M appears to be substituting the values one and two for a and b]

6

D: [laughs]

7

M: Um, yeah, it's, 'cause I'm doing [punching buttons] one times two, divide one times two, equals one.

8

D: So that's cancelled. The two b's are cancelled out.

9

M: Equals one.

10

D: Right? The two b’s are cancelled out.

11

M: Hey, where'd my pen go? No come on, look, look, look, look. You've got to do BODMAS. Watch, watch, watch, watch. [punching buttons] One times two, divide one... come on, one times two. That's stuffed up. [with emphasis] One.

12

D: ... I'm going to ... this is ... better ....

13

M: Look, look, look, look at this one, look at this one.

14

D: ... Hang on ...

15

M: Divide.

16

D: ... I'm going to do these, this one first.

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M: Equals 1, it does equal 1. I've got to do this first.

18

D: Takes [inaudible]. [pause] Two.

19

M: Oh, reduce to the simplest terms. Oh, OK, um. [pause, punches calculator] All right. Shit. [pause, both work]

20

D: [inaudible] here and speed.

21

M: One point 1? [inaudible]

22

[T talks with nearby student]

23

M: Mrs Brown? Mrs Brown, do we simplify that as well? Question one?

24

T: Yes, you do.

25

M: OK.

26

T: And you're right.

27

M: OK.

28

T: [looking at M’s work] So you got one right and you got two right.

29

M: I know.

30

T: Yep. Ah no because it’s a ratio. Oh, I guess you could say yes, I'll accept that. Yep.

31

M: Thank you. [T goes to Anna and Lauren's desks]

32

D: We don't go on to that yet.

33

M: Yes, we do. [with emphasis]

34

D: Are we supposed to do that?

35

M: Yes. [with emphasis]

36

D: Yeah, that [points with pencil on M’s paper], that's not, that equals ab, doesn't it?

On video-stimulated recall of this section, the teacher said the following: With what Mark’s talking about um quite often when we’re working with algebra symbols I’ll go ahead and have said to the class, OK well if you’re not certain about something, substitute in some simple numbers which is what he’s doing here and uh I was really pleased to see that he’s actually followed what I’ve been trying to say, um, to check with the algebra numbers by substituting in simple, simple numbers and he obviously found, hey that didn’t work so let’s try something else. And the fact that he’s

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helping Darren with this is fantastic because it’ll help Mark in the process. Mark went on to um virtually, he’s now in year 10 and he’s doing year 11 maths which is what I thought he’d end up doing. So he’s actually a year ahead with his maths. So he’ll do extra maths in his VCE. And Darren I think is just keeping it there at the same pace .... So we accelerate over those two years 7 and 8 to get up to year 9 maths and uh, you know, for someone like Mark it’d be quite easy for him to do that. Someone like Darren wouldn’t have a hope. But uh it’s good to see them working together helping each other out.

Mark often spoke with a pencil in his mouth, as a consequence of which there are a number of utterances that are inaudible. Mark sat on the left of the picture, on Darren’s right hand side. Mark made almost continuous use of a calculator, with his left hand, placed between the two, although he wrote with his right hand. Darren did not use a calculator until towards the end of the lesson; indeed, I was not aware that he had one until then. The teacher pointed out, in reply to one of my questions, Calculators are an integral part. In fact even on our assessment report, on our maths report, we have a particular section for ‘able to use technology‘ so we make a deliberate move to incorporate both calculator skills and computer lesson skills as part of technology. Sometimes we even introduce the graphic calculators to year 8 at the various parts, so its part of our um helping kids to feel familiar with and confident with various technologies.

Mark and Darren began with part (f). On Turn 1 (henceforth written as italic 1) in Extract 1 Mark quietly proposed the answer ‘ab’ with an interrogative lift to his voice and repeated the same answer. He then changed his mind, although there is no evidence of why this should have been so. Darren appeared not to know why (4) and had, at first, agreed with the answer ‘ab’ (2). At 5 Mark justifies his answer by apparently substituting numbers for ‘a’ and ‘b’, 1 for a and 2 for b. Initially he entered 12 into his calculator, perhaps forgetting that ‘ab’ is the product of ‘a’ and ‘b’ but quickly realised his error. He then seemed to have trouble with the order of operations, and when he arrived at the answer 4 he knew it was incorrect. At 8 Darren seemed to be trying to answer the question in his own way, beginning by using the teacher’s language from when they were at the front of the class for the extra lesson, and cancelling the letter b. In her interview, the teacher did not seem to notice that Darren had done this, noticing only Mark’s substitution moves. Darren attempted to draw Mark’s attention to what he was doing (10), but Mark persisted in showing Darren, with very rapid presses on the calculator buttons, his rationale. He continued to make mistakes with the order of operations but remained sure that the answer should be 1. Most importantly, at no time did he explain to Darren why he was working with the numbers 1 and 2 instead of ‘a’ and ‘b’. Later in the lesson the teacher used this substitution technique in an explanation to another student and the teacher makes it clear in her interview that he had seen this done before. Perhaps Darren had not, or had forgotten. In any case, Darren apparently wanted to pursue his own method (12), preferring it over Mark’s. This pattern of relationship dominated throughout their exchanges. Darren continually attempted to engage Mark’s attention but Mark ignored the attempts and persisted in just describing what he was doing in a fairly domineering and insistent voice, offering no explanations.

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At 17 Mark appeared to have managed to get 1 through his calculator actions. At 19 he seemed to focus on the fact that the question contains a colon, a ratio sign, rather than some other form. He called the attention of the teacher, who was just nearby their desks (23), although we do not know to which part of question 1 he was referring. At 30 the teacher noticed Mark’s part (f) answer, which he had written as just the number 1, not the ratio ‘1: l’. The teacher accepted the answer, then retracted, and finally accepted his answer. Darren, throughout Mark’s exchange with the teacher, did not appear to be listening at all. He had turned away, sideways, and appeared to be writing on his worksheet. Finally, at 36, after the teacher had moved away, Darren leaned over Mark’s book, and apparently pointing at Mark’s answer of 1, asked whether the answer shouldn’t be ‘ab’. Again, Mark ignored Darren’s intervention. Mark’s answer sheet has the single answer ‘1’ for (f) whereas Darren’s has ‘ab’ crossed out, followed by ‘ 1’. It seems to me clear that they are not engaging each other in their zones of proximal development. Put another way, they used no strategies to try to establish joint understandings (Edwards & Mercer, 1987). For this to be happening we might expect to see them listening to each other and considering each other’s utterances, finding common terms or common intentions for task-solving activity. There seem to be sufficient indications that they could have worked fruitfully in their zones of proximal development since Mark was using substitution of numbers to justify his answer, which was apparently new to Darren, and Darren wanted to cancel letters, the method that the teacher had shown them a few minutes earlier but which Mark was not intending to use. The nature of their relationship appears to have precluded that possibility, as manifested in Mark’s impatient insistence on demonstrating, not communicating, his justification, and Darren’s lack of assertiveness in insisting on Mark listening to him. Their utterances and mathematical activities are in parallel here (Forman, 1981), despite plenty of talk between them. One wonders whether the assumption of Mark’s superior ability by both students had an unspoken structuring effect on the way they reacted to each other, in spite of their being friends who often worked together. Extract 2 The second extract began a short time later. The two boys continued working on question 1. The beginning line of this extract has been chosen because the utterance at Turn 1 draws their work together, where in the lines before it was too disparate usefully to be analysed. On video-stimulated recall the teacher said the following, just before this extract: Darren really is struggling with the abstractness of this. This is what I thought, I was surprised when he came out the front, um, and wanted to be in the fast group but that’s OK. At least hopefully he’ll be able to cope with it later on.

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Transcript

1

D: Um, two-forty, yeah, three hundred divided by two-forty.

2

M: Hang on, twelve, so we'd have twelve ... [pause]

3

D: No, that ... it's divided by it, right? That's one ...

4

M: ... over four [interrupting], Four divided by ...

5

D: ... or one third.

6

M: ... it's going to be three over one.

7

D: Why?

8

M: 'Cause, it's just going to be the opposite. Look.

9

D: Yes.

10

M: Times twelve.

11

D: 'Cause that's bigger number. Yep.

12

M: Um. Um. Two forty over three hundred. Um, two four ‘o’ divide... no.

13

D: Three hundred.

14

M: Three hundred, divide two four ‘o’ ... no. No, um. [Presses 300/4 = 75] Seventy-five.

15

D: Did you get twelve over. [pause]

16

M: [Press 240/6 = 40] ... six ... forty. And that was one.

17

D: Is it ... yeah, it's eight over.

18

M: Seventy-five divide eight doesn't work, no, can't be eight.

19

D: It's um... try ten.

20

M: Five, No, seventy-five, it's going to be five there.

21

D: Yeah, five over... no, try five.

22

M: By five, is fifteen.

23

D: It's eight over fifteen.

24

M: I think, hang on.

25

D: It is right, that's right.

26

M: Yeah, it's right.

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STEPHEN LERMAN At this stage during video-stimulated recall the teacher remarked: I’m surprised that Darren did as well as he did. It’s easy to sort of pigeonhole students into an area and think that they may struggle with or they shouldn’t try but with Mark beside him he’s got a a lot further than I thought he would

As will already be clear from my analysis, my impression is that Mark was not helping Darren at all, nor is it clear that he always needed help. At 1 Darren signalled that he wanted the two of them to begin part (d). Mark was working on (b) and wanted to finish it. From 2 to 6 they talked over each other but Mark insisted that the answer was 3/1 whereas part (a) was 1/3. Darren wanted to write the same answer for (b) as for (a) and on 7 he asked Mark why it is not correct. Mark’s “Look” at 8 called Darren’s attention to the calculator, where Turn 1 0 appears to be showing that the “opposite” leads to multiplication by 12, returning 3/1 to 36/12. The word “opposite” may also have referred to the answer to (b) being the reciprocal of (a). Darren’s response at 11 might be unrelated to the justification or the question, or it might indicate that he saw the numerator ‘36’ in (b) as bigger than the denominator ‘12’ and realised that the answer should be top-heavy. However Darren’s answer paper has 1/3 for both (a) and (b), whereas Mark has 1/3 for (a) and 3/1 for (b), suggesting the former. From Turn 12 to the end they worked together on part (d). Darren’s Turn 1 5 seems unrelated to the question, and may just have been triggered by the discussion over the number 12 which preceded. After a great deal of rapid button pressing on his calculator Mark, at 16, divides the numerator by 6, having divided the denominator by 4. Darren’s statement at I7 again appears unrelated to the question and certainly to the work which Mark has been doing. Nevertheless Mark picks it up, not as a numerator but as a possible cancelling factor. At 18 he pointed out that it would not work and Darren, perhaps randomly, suggests 10 as a numerator (19). By this time Mark had seen that the denominator which he had written down, 75, would divide by 5(20), and Darren picked this up to propose 5 as the numerator (21). At 22 Mark calculates that the denominator should be 15 and at 23 Darren proposes firmly that 8/15 is the correct answer. This is a very interesting interchange because they are listening to each other for the first time but they arrive at the incorrect answer, arising from Mark’s error in dividing the numerator by 6 and the denominator by 4. This is easily done, as 6 and 4 are the two factors whose product is 24, a number on which Mark may well have been focusing. Furthermore, Darren’s offerings appear to be quite haphazard, a combination of repetitions from earlier utterances and numbers mentioned by Mark in the middle of his working out, with the continual assistance of the calculator. Here again I would not want to say that their interchange engages either of them in their zone of proximal development. The apparent communication in fact gives Mark some numbers he can attempt to work with but does not appear to give Darren any meaning from the work. It is a deceptive extract in that on first seeing the video I thought they were communicating, in the sense in which I am looking for it here, and clearly the teacher did as well. In fact she saw it as Mark helping Darren whereas my reading is that Darren’s possibly random suggestions are being picked up by Mark.

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Extracts of interviews with Mark and Darren Below are the sections of the interviews with Darren and Mark which relate to the extracts I have discussed above. D’s interview: I: OK, So all right. [plays video] There you are trudging back to your seats, there's Mrs Brown. [comment from the teacher, who was passing through the interview area, that she still hasn't played the video tapes from previous years.] [to Darren] Oh, you have to rush off, don't you? D: Oh, we're working out the algebra things, on the ratio pep test then. I: Right. D: Yeah. So. And we were just swapping answers and discussing, so, yeah. I: Yeah. D: Because if one of us got it wrong we just checked each other's answers, to see. I: Right. D: lf they were right, then we, if they were wrong then he, he would tell me if um, how to do it, and I would tell him how to do it. I: Uh huh. D: But in this case he was telling me how to do the first of them. I: Yeah, yeah. And here he's explaining something to you. D: Yeah. I: So what's that about? D: Just one of the sums I guess, like one that I didn't get right. I: And it's sort of funny 'cause you don't look like you 're really that, that impressed. D: Oh, some of them like I didn't really I didn't think that they were right, and I checked them and checked them again. I: Um hm. Well, do you, so you don't remember which this one was? D: No.

The teacher briefly came into the room during the interview. It appears that Darren does not give responses about the mathematical activity to which the

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interviewer is referring. Rather, Darren emphasises the symmetry of their working relationship most of the time, the particular incident being by chance one where Mark was doing the explaining, although he then picks up the hint from the interviewer about not being impressed with what Mark has said and Darren says that he wasn’t sure about Mark’s answer. This is a clear example of the way in which an interview about an event, even using the video, is a different event, other issues and reactions are elicited by the framing of this other environment. Mark’s interview: I:

[pause] [plays tape] Can’t see too much of that. I’ll jump this forward a bit. You’re all back in your seats, Darren is asking some stuff [Mark on tape: divide 1 times 2.] So do you know what that’s about?

M: Yeah, um, it’s. [sound of paper] It’s about trying to reduce um, algebra ratios. I: Oh, right. So it’s not the, it’s not the thing that Mrs Brown was showing you up there at all? M: Yeah, that’s sort of what it was. I: Oh, it is? OK. M: ‘Cause its about cancelling each other out. I: Yeah. M: Which is what she was explaining. I: Yeah, yeah. Oh, so was it, was she explaining this bit, was she explaining the stuff involving this sheet? M: Um, sort of, like um, she had x values and y values on the board and she was explaining if you had um, five x divided by x you ’d just have five left. I: Right. M: It’s about how to cancel things out. I: Yeah, yeah, yeah. So it was sort of so, and what was, what was the thing that Darren was asking about? M: Um, this one. I: Uh huh. M: ’Cause of what they were saying. I: Oh, right. And that’s where you say, that’s where you say ‘cause, yeah, that it equals one.

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M: Mm. I: OK, yeah, all right. And then, and then, and so, did that, what was his response to that? He didn’t sound like he was all that keen on that answer. M: He didn’t agree. Um, yeah, he thought it was just working out that [??] I forgot to simplify, there’s no way of simplifying it because there’s no numbers. I: Um hm, um hm. M: So wouldn’t it just be zero um, to zero. I: Oh. M: Which would have been. I: Oh. Something else entirely. M: Mm. I:

Yeah. [plays tape: On tape Mark says: doesn’t seem to do a lot for him.

1 times 2] [laughs] That, that, that

M: It did. It worked once, and then it didn’t work for the next couple of times. I: What! What didn’t it work? M: I just pushed the wrong button.

Mark also does not engage with the details of the interactions. The nature of the interview as an activity has framed his responses, as it did with Darren. The students want to represent themselves to the interviewer in particular ways, to tell stories which are inevitably about the interview rather than the classroom interactions. A different analysis, in terms of the powerfulness and powerlessness of multiple subjectivities, would develop this theme (Lerman, 1998a). Here, it is sufficient to have the issue of the forming of identity in the classroom (Wenger, 1998; Winbourne & Watson, 1998) brought to our attention by the responses of the students to the video-stimulated interviews. 4. CONCLUDING REMARKS In this final section I will address two questions that were posed at the beginning: _ Is the account of the nature of the interactions between Mark and Darren helpful to (me and) other researchers and teachers? _ Is the account which I have constructed from the data set of the interactions between Mark and Darren valid? My account could have drawn on other theories, such as astrology, and still have been valid in those terms. I conjecture that such an account might not have been

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particularly helpful to teachers and/or researchers. Thus helpfulness, fruitfulness or some other similar notion is an important feature. In that my account of how Mark and Darren worked together differs considerably from the teacher’s, I hope that it is at least helpful since it is an alternative one. My first viewing of the video suggested to me too that these two boys were collaborating in building a common understanding of what one must do to objects like “ab:ab” and “240/300” in the mathematics classroom, but my subsequent detailed analysis suggests otherwise. I also suggested that there were other elements which frame the situation, including: the teacher’s assumptions of their ability and the likelihood that such a perception had been unintentionally conveyed to the boys by the teacher’s and other pupils’ behaviour; some aspect of their relationship which results in Mark’s impatience and dominance; the function of the calculator in Mark’s reasoning; and the differential effect of the extra instruction on each of the boys, in the way that the tasks became framed. There is a great desire to know more, to see if other information about the boys, the class, the teacher/pupil relationships might extend the picture of the relationship between the boys at that moment, in that place, with those tasks, and so on. I want to emphasise the importance, for me, of the particularities of the activity, in trying to develop an account of learning. I am not looking for decontextualised learning; rather the localisation in the context, in as much richness as is possible. For the second question, one might first consider what could constitute an answer. The validity of one person’s account of videos and transcripts collected by others will rest on whether the account is coherent as a story, whether the judgements of the interpretations of events have been justified, and whether the ‘community’ of readers of the account find it of some use. I therefore rephrase the second question as: _ Does the account which I have constructed from the data set of the interactions between Mark and Darren suggest that this kind of analysis of this kind of data-set is coherent and justifiable? In that I have tried to make this account a socio-cultural one, that is, as responsive as possible to the consequences of the zoom lens image I offered above, it must always be unsatisfactory. The data-set will always be a snapshot, however extensive the variety of elements in the data-set. As a reader of the videos and transcripts I will necessarily bring my own network of experiences (Vile & Lerman, 1996) to the interpretation. Furthermore I have based my analysis on looking for zones of proximal development, and the coherence must also rest on how that theoretical framework has been developed. I have argued for an interpretation of the ZPD as being created, or not, in the learning activity and dependent on peers catching each other’s thoughts. In the account I have given of the interactions between Mark and Darren I have attempted to show that their different focuses in

the mathematics and the nature of their relationship appeared to preclude the creation of a ZPD. It has been my intention, through a Vygotskian analysis, to reinforce the centrality of the social relationships in the classroom in the creation of mathematical meaning, a theme common to the Project team and the chapters in this book.

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Notes I. On television or radio news reporters speak of “We are now going over to ...” as if they are virtual visitors. In this sense I too am a visitor to the classroom. REFERENCES Atkinson, P. (1992) Understanding Ethnographic Texts London: Sage. Brodie, K. (1995) ‘Peer Interaction and the Development of Mathematical Knowledge’. In D. Carraher & L. Meira (Eds.) Proceedings of Nineteenth International Meeting of the Group for the Psychology of Mathematics Education, Recife, Brazil, Vol. 3, 216-223. Brown, A,, Ash, D., Rutherford, M, Nakagawa, K., Gordon, A., & Campione, J.C. (1993). ‘Distributed expertise in the classroom’. In G. Salomon (Ed.) Distributed Cognitions: Psychological and Educational Considerations, , pp. 188-228, New York: Cambridge University Press. Buchler, J. (Ed.) (1955) Philosophical Writings of Peirce New York: Dover. Cobb, P., Wood, T. & Yackel, E. (1991) ‘A constructivist approach to second grade mathematics’. In E. von Glasersfeld (Ed.), Radical Constructivism in Mathematics Education, (pp. 57-176). Dordrecht: Kluwer. Davydov, V. V. (1988) Problems of developmental teaching. Soviet Education, 30, 6-97. Davydov, V. V. (1 990) Soviet Studies in Mathematics Education: Volume 2. Types of Generalization in instruction (J. Kilpatrick (Ed.), trans. J. Teller), Reston VA: National Council of Teachers of Mathematics. Edwards, D. & Mercer, N. (1987) Common Knowledge: The Development of Understanding in the Classroom, London: Methuen. Evans, J. (1993) Adults and Numeracy Unpublished PhD Thesis, University of London Library. Evans, J. & Tsatsaroni, A. (1994) ‘Language and “subjectivity” in the mathematics classroom’. In S. Lerman (Ed.), Cultural Perspectives on the Mathematics Classroom, Dordrecht: Kluwer, 163-182. Forman, E. A. (1981) The Role of Collaboration in Problem-Solving in Children Doctoral Dissertation, Harvard University. Forman, E. A. & Cazden, C. B. (1985) ‘Exploring Vygotskian Perspectives in Education: The Cognitive Value of Peer Collaboration’. In J. V. Wertsch (Ed.) Culture, Communication and Cognition: Vygotskian Perspectives Cambridge: Cambridge University Press, 323-347. Forman, E. & McPhail, J. (1993) Vygotskian Perspective on Children’s Collaborative Problem-Solving Activities. In E. A Forman, N. Minick & C. A. Stone (Eds.) Contexts for Learning: Sociocultural Dynamics in Children’s Development (pp. 213-229), New York: Oxford University Press. Harré, R. & Gillett, G. (1994) The Discursive Mind London: Sage. Hammersley, M. & Atkinson, P. (1983/92) Ethnography: Principles in Practice London: Routledge. Hitchcock, G. & Hughes, D. (1995) Research and the Teacher 2nd edition London: Routledge. Inagaki, K. & Hatano, G. (1977) Amplification of Cognitive Motivation and its Effects on Epistemic Observation American Educational Research Journal, 14, 485-491. Kol’tsova, V. A. (1978) Experimental Study of Cognitive Activity in Communication (with Specific Reference to Concept Formation) Soviet Psychology, 17, 23-38. Lerman, S. (1994) ‘Towards a unified space of theory-and-practice in mathematics teaching: a research perspective. In L. Bazzini (Ed.) Proceedings of the Fifth International Conference on Systematic Cooperation between Theory and Practice in Mathematics Education, Universita degli Studi di Pavia p. 133_142. Lerman, S. (1996a) Intersubjectivity in Mathematics Learning: A Challenge to the Radical Constructivist Paradigm? Journal for Research in Mathematics Education, 27(2), 133-150. Lerman, S. (1996b) Editorial Educational Studies in Mathematics Special Issue, Socio-Cultural Approaches to Mathematics Teaching and Learning, 31(1/2), 1--9. Lerman, S. (1997) ‘Research on Socio-Cultural Perspectives of Mathematics Teaching and Learning’. In J. Kilpatrick & A. Sierpinska (Eds.) Mathematics Education as a Research Domain: A Search for Identity Dordrecht: Kluwer. Lerman, S. (1998a) The Intension/lntention of Teaching Mathematics. In C. Kanes (Ed.) Proceedings of Mathematics Education Research Group OfAustralasia, Griffith University at the Gold Coast, Australia, Vol. 1, 29- 44.

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Lerman, S. (1998b) A Moment in the Zoom of a Lens: Towards a Discursive Psychology of Mathematics Teaching and Learning. In A. Olivier & K. Newstead (Eds.), Proceedings of the Twenty-second Annual Meeting of the International Group for the Psychology of Mathematics Education, Stellenbosch, South Africa, Vol. 1, 66 -81. Matos, J. F. (1995) ‘Ethnographic Research Methodology and Mathematical Activity in the Classroom’. In D. Carraher & L. Meira (Eds.) Proceedings of Nineteenth International Meeting of the Group for the Psychology of Mathematics Education, Recife, Brazil, Vol. 1, 211. Meira, L. & Lerman, S. (forthcoming) ‘The Zone of Proximal Development as a Symbolic Space’. Newman, D., Griffin, P. & Cole, M. (1989) The Construction Zone New York: Cambridge University Press. Newman, F. & Holzman, L. (1993) Lev Vygotsky: Revolutionary Scientist London: Routledge. Perret-Clermont, A. N. (1980) Social lnteraction and Cognitive Development in Children New York: Academic Press. Piaget, J. (1964) ‘Development and learning’. In R. E. Ripple & V. N. Rockcastle (Eds.) Piaget Rediscovered, Ithaca: Cornell University Press, 7 - 19. Santos, M. (1995) ‘Mathematics Learning as Situated Learning’. In D. Carraher & L. Meira (Eds.) Proceedings of Nineteenth International Meeting of the Group for the Psychology of Mathematics Education, Recife, Brazil, Vol. 1, 222. Scott, D. (1996) ‘Ethnography and Education’. In D. Scott & R. Usher (Eds.) Understanding Educational Research (pp. 143-158) London: Routledge. Smith, L. (1993) Necessary Knowledge: Piagetian Perspectives on Constructivism Hove: Lawrence Erlbaum Associates. Tudge, J. (1990) ‘Vygotsky, the Zone of Proximal Development, and Peer Collaboration: Implications for Classroom Practice’. In L. Moll (Ed.) Vygotsky and Education, Cambridge: Cambridge University Press, 155-172. Tudge, J. R. H. (1992) Processes and Consequences of Peer Collaboration: A Vygotskian Analysis Child Development, 63, 1364-1379. Usher, R. (1993) Reflexivity, Occasional Papers in Education as Interdisciplinary Studies, 3, Southampton: University of Southampton. van der Veer, R. & Valsiner, J. (1991) Understanding Vygotsky: A Quest for Synthesis Oxford: Blackwell. Vile, A. & Lerman, S. (1996) ’Semiotics as a descriptive framework in mathematical domains’. In L. Puig & A. Gutiérrez (Eds.) Proceedings of the Twentieth Meeting of the International Group for the Psychology of Mathematics Education, Vol. 4, 395-402. von Glasersfeld, E. (1989) ‘Cognition, Construction of Knowledge, and Teaching’, Synthese, 80, 121140. Vygotsky, L. S. (1978) Mind in Society Cambridge, Mass.: Harvard University Press. Vygotsky, L. S. (1986) Thought and Language (Revised Edition, A. Kozulin (Ed.)), Cambridge, Mass.: MIT Press. Walkerdine, V. (1988) The Mastery of Reason London: Routledge. Wenger, E. (1998) Communities of Practice Cambridge, UK Cambridge University Press. Winbourne, P. & Watson, A. (1998) ‘Learning Mathematics in Local Communities of Practice’. In A. Olivier & K. Newstead (Eds.) Proceedings of the Twenty-Second Annual Meeting of the International Group for the Psychology of Mathematics Education, Stellenbosch, South Africa, Vol. 4.

DEREK HOLTON AND GILL THOMAS

CHAPTER FIVE Mathematical Interactions and Their Influence on Learning

1. INTRODUCTION In this chapter we are concerned with the mathematical interactions that took place between the participants in the various classrooms of the Classroom Learning Project. We first outline the underlying theory for our work and follow this with approaches to analysing the value of mathematical interactions. Various lessons are then analysed. Here we are solely concerned with the interactions that took place in mathematics lessons. Our aim is to try to understand what aspects of these interactions enhance learning and understanding. As a result we have considered the transcripts of the four mathematics lessons: Problem solving using standard form.

Lesson 2 (19th October 1994): Lesson 3 (18th November 1994):

Compass bearings.

Lesson 4 (13th June 1995):

Rate and ratio: velocity-time graphs.

Lesson 6 (17th June 1996):

Rate and ratio: velocity-time graphs.

From these transcripts we have been particularly concerned with interactions of the following types: — teacher - class — teacher - group — student - student A qualitative analysis of all of these types of interactions has been made as well as a quantitative analysis of some aspects of the student-student interactions. The purpose of these analyses is to try to understand how mathematical learning and understanding might take place through talk and how this might be influenced by the task and other classroom variables. We conclude by discussing some issues that arose as the result of our analysis of the lesson transcripts. First we discuss exactly what is meant by the term "scaffolding" (see Wood, Bruner and Ross, 1976) and who can provide scaffolding for someone else's learning. Does scaffolding have to be provided by the teacher or can it be provided by a peer? Is there such a thing as reciprocal scaffolding? Must scaffolding be intentional or can it be accidental? 75 David Clarke (ed.), Perspectives on Practice and Meaning in Mathematics and Science Classrooms, 75—104. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Then we look at learning and whether or not it can be aided by talk in group interactions. Certainly the talk analysed shows a high degree of on-task cognitive and even reflective talk. There are certainly examples of talk which provide learning and understanding. However, talk between peers is not a necessary and sufficient condition for learning. What then does a teacher have to do to improve the chances of learning taking place? 2. THEORETICAL FRAMEWORK As the work of this chapter is based on the transcripts of lessons, the mathematical interactions that we have observed have been exclusively based on the reported conversations. The topic of talk and learning is a complex one. The aim of this section is to capture the essence of the links between talk and learning by identifying the core issues, and by considering the ideas of different theorists in the discussion. What children learn in the classroom depends to a large extent on what they already know. This seemingly simple statement underlies constructivism and the debate that surrounds it. The book by Steffe and Gale (1995) considers the various aspects of constructivism from social constructivism to radical constructivism (von Glasersfeld, 1995) with Ernest (1995) describing a range of constructivist approaches. The constructivist influence has led to there being widespread agreement that it is the child who makes sense of her experiences by constructing links with her prior knowledge. This has led to constructivism appearing implicitly as the underpinning of mathematics curricula in Australia and New Zealand (see Australian Education Council, 1991, and Ministry of Education, 1992). For instance in Ministry of Education (1992, p. 12) we find As new experiences cause students to refine their existing knowledge and ideas, so they construct new knowledge.

While there is little argument among theorists that learning involves the construction of knowledge through experience, there is debate in relation to the conditions under which such learning is optimised. For instance, is learning best promoted through individual activity or by the occurrence of particular types of social interaction? Can different things be learnt best in different ways by different people? Bruner and Haste (1987) illustrate this argument when they make the contrast between the child as a social being or a lone scientist. We had fallen into the habit of thinking of the child as an “active scientist”, constructing hypotheses about the world, reflecting upon experience, interacting with the physical environment and formulating increasingly complex structures of thought. But this active, constructing child had been conceived as a rather isolated being, working alone at her problem-solving. Increasingly we see now that, given an appropriate, shared social context, the child seems more competent as an intelligent social operator than she is as a “lone scientist” coping with a world of unknown (Bruner & Haste, 1987, p. 1).

Bruner (1986) observes that most learning in most settings is a communal activity and this leads him to emphasise the role of learning as a shared and social experience in the classroom. Bruner has been largely influenced by the work of Vygotsky who assigned social interaction a central role in facilitating learning.

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While much learning is based on social activity it is not clear that all learning is so based. Gifted children especially, may need independence (Maker, 1982, p. 216). 3.

STUDENT-STUDENT INTERACTION

The interest in peer-based learning has arisen from research within psychology and education which has established that children can have a powerful influence upon one another’s intellectual development (Damon, 1984). Most of us have an intuitive feeling for the cognitive benefits of discussion. We recognise in our own discussions that process of clarifying ideas as we communicate them to others. It seems, then, reasonable to assume that the same process may occur when children talk to each other. As well as our own intuitive beliefs about discussion there exist a number of theoretical traditions which underpin this assumption that children benefit from talking to one another. Piaget and Vygotsky, two quite distinct and often opposing theorists, are in agreement on this issue, albeit for different reasons (Thomas, 1994, p. 1).

Cognitive-developmental psychologists in the tradition of Piaget have taken the view that peer interaction facilitates development by posing cognitive conflicts. Psychologists in the tradition of Vygotsky value the thought processes generated by peer communication. Piaget made numerous references to collaboration between children particularly with his theory of socio-cognitive conflict. In Piagetian theory, peer interaction works as a stimulus for change but it does not provide the substance of the change. Piaget believed that the disequilibrating effect of peer feedback initiates a process of intellectual reconstruction in the child. But once this process has begun, according to Piaget’s theory, the main work of formulating new knowledge is undertaken by the individual and the ideas themselves remain the product of the child’s internal processes (Damon, 1984). Piaget’s theory is most helpful in explaining those situations where conflict is overtly expressed. However, in situations where overt conflict is not apparent and where mutual guidance and support are evident, his theory provides few clues concerning the role of social factors in the development of learning. Vygotsky, however, focused closely upon “talk as a medium for sharing knowledge and potentially transforming understanding” (Mercer, 1994, p. 95). In Vygotskian theory, peer feedback not only initiates change, it also shapes the nature of the change itself (Damon, 1984). According to this view, children benefit from one another by internalising the cognitive processes implicit in their interactions and communications. In other words, children are introduced to new patterns of thought when they engage in dialogue with peers. This is because peer dialogue is by nature a cooperative exchange of ideas between equals and therefore emulates several critical features of rational thinking. Damon (1984) suggests that there are a number of reasons why a child’s peers act as an effective source of cognitive conflict. First, children speak to one another using a vocabulary that they both understand. This has also been noted in Holton, Anderson and Thomas (1997, p. 45) where it is observed that children sometimes understand other children’s explanations better than they understand a teacher’s explanation. Second, children tend to take the feedback of another child seriously

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and are motivated to reconcile contradictions although this appears to depend on the confidence with which the opposing viewpoints are held (Tudge, 1990). Third, they tend to speak directly and openly to one another. Finally, informational communications with other children are often less emotionally threatening than corrective feedback from an adult. A number of studies conclude that students’ learning is enhanced by their active interaction. For instance, Webb (1989) found that the giving of elaborate explanations by children was positively related to their individual achievement. On the other hand, the receiving of elaborate explanations had few significant positive relationships with achievement. Russell and Kelley (1991) found that requiring students to explain aspects of their work led to increased understanding of their course material. Dole, Valencia, Greer and Wardrop (1991) showed that the intellectual stimulation of children is enhanced when their parents read to them, if the parent engages the child in discussions of the stories being read. Chi, Bassok, Lewis, Reimann and Glaser (1989) also showed the benefit of self-explanation. Of the students in their study, the ones who were subsequently more successful at problem solving, were the ones who had spontaneously generated the most selfexplanations while studying sparsely stated, worked-out examples. Chi, de Leeuw, Chiu and La Vancher (1994) produced similar results. Presumably learning involves the integration of new information into existing knowledge. Generating explanations seems to facilitate that integration process. Talking to oneself, an adult or a peer seems to facilitate the generating and clarifying of explanations. 4.

TEACHER-STUDENT INTERACTION

Vygotsky (1962) noticed that children who by themselves were able to perform tasks at a particular cognitive level, in cooperation with others and with adults were able to perform at a higher level. The difference between these two levels, or the potential for a child’s development, Vygotsky called the zone of proximal development. Vygotsky (1962) then says In the child’s development, ... imitation and instruction play a major role. They bring out the specifically human qualities of the mind and lead the child to new developmental levels. In learning to speak, as in learning school subjects, imitation is indispensable. What the child can do in cooperation today he can do alone tomorrow. Therefore the only good kind of instruction is that which marches ahead of development and leads it; it must be aimed not so much at the ripe as at the ripening functions. It remains necessary to determine the lowest threshold at which instruction in, say arithmetic may begin since a certain minimal ripeness of functions is required. But we must consider the upper threshold as well; instruction must be oriented toward the future, not the past (p. 104).

We assume then that the zone of proximal development is both a function of the individual child and a function of time. It is a construct that the teacher needs to be aware of in order to provide an optimal learning situation. However, it is difficult, if not impossible to define it precisely. (Note the different interpretation of the concept in Chapter 4.)

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The zone of proximal development provided the foundation for the notion of scaffolding that was introduced in Wood, Bruner and Ross (1976). Bruner (1985) talks about these two concepts in the following terms. (The emphasis in this quote is ours.) I have puzzled about this matter (zone of proximal development) for many years, and I think I understand what Vygotsky may have meant. Or at least I understand the matter somewhat as follows ... If the child is enabled to advance by being under the tutelage of an adult or a more competent peer, then the tutor or the aiding peer serves the learner as a vicarious form of consciousness until such a time as the learner is able to master his own action through his own consciousness and control. When the child achieves that conscious control over a new function or conceptual system, it is then that he is able to use it as a tool. Up to that point, the tutor in effect performs the critical function of ‘scaffolding’ the learning task to make it possible for the child, in Vygotsky’s word, to internalize external knowledge and convert it into a tool for conscious control (p. 215).

Greenfield (1984) describes scaffolding in more detail as follows, where the worker represents the learner and the building represents what is to be learned. The scaffold is a metaphor to describe the ideal role of the teacher. The scaffold, as it is known in building construction, has five characteristics: it provides a support; it functions as a tool; it extends the range of the worker; it allows the worker to accomplish a task not otherwise possible; and it is used selectively to aid the worker where needed. To illustrate this last point, a scaffold would not be used, for example, when a carpenter is working five feet from the ground.

Scaffolding is a concept discussed in much of the current literature on children and learning (see, for example, Berk and Winsler, 1995) and refers to the “guidance and interactional support given by a tutor in the zone of proximal development” (Smith, 1992, p. 20). Cambourne (1988) identifies “focusing - on a gap which the learner needs, extending - challenging or raising the ante, refocussing - encouraging clarification and redirecting - offering new information” (p. 117) as the most common interactions used in scaffolding. Vygotsky, because he viewed learning as a profoundly social process, emphasised dialogue and the varied roles that language plays in instruction. The mere exposure of children to new materials neither allows for adult guidance nor for collaboration with peers. In Thought and Language, Vygotsky (1962) presented a sophisticated argument demonstrating that language, the very means by which reflection and elaboration of experience takes place, is a highly personal and at the same time a profoundly social human process. As Smith (1992) explains: Language is central to children’s development. Language first occurs externally between people and is then converted into internal processes that can be used to solve problems independently. Children first use language to communicate with other people, but their language is then converted into internal speech or thought which is used for self-regulation (Smith, 1992, p. 19).

In Vygotsky’s writing, the social relationship referred to as teaching is primarily the one-to-one relationship between adult and child. When Vygotskian perspectives are explored for education, the classroom setting must be considered. Forman and Cazden (1985) see two separate but related issues concerning the group presence in the classroom. First, there are the problems posed for the teacher in carrying out instruction to a group of students who have varying zones of proximal development.

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The second issue concerns the potential of social interactions among the children themselves. According to Forman and Cazden, interactions among peers focused on intellectual content can be placed on a continuum depending on the distribution of knowledge and therefore on the roles they take toward one another. At one extreme, one child knows more than the other and is expected to act as a tutor. At the other extreme the knowledge is equal and collaboration is expected. The other important concept here is that of metacognition. Basically metacognition is “thinking about your own thinking”. According to Schoenfeld (1987), metacognition encompasses the three areas of (i) self-regulation, or monitoring and control; (ii) knowledge of our own thought processes; and (iii) beliefs and intuitions. We will show a link between metacognition and scaffolding. Schoenfeld (1992) underlines the value of metacognition in what he calls “ real problem solving” (as opposed to undertaking problems which repeat problem types already solved by a teacher). The use of metacognition appears to be an important area where novice and expert problem solvers differ. Schoenfeld suggests For the most part, students are unaware of or fail to use the executive steps demonstrated by the expert. However, it is the case that such steps can be learned as the result of explicit instruction that focuses on metacognitive aspects of mathematical thinking. That instruction takes the form of “coaching” with active interactions as students work on problems (p. 316).

This “coaching” can be thought of as scaffolding and so the route to metacognition for the student is through scaffolding. Pólya (1973) says this in a slightly different way which includes the notion of zone of proximal development. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking (p. 2).

The train of thought initiated by the quotes of Smith (1992), Schoenfield (1987) and Pólya (1973) lead us to suggest that scaffolding and metacognition are intimately related. In fact we suggest that metacognition could be referred to in part as self-scaffolding and that this is, in fact, one of the main goals of education. In other words we want to produce citizens who can critically reflect on their environment and their functioning within that environment. To see this in perspective let us consider what we hope to achieve by the school experience. Surely one goal is to enable children to cope with problems of a wide variety of types in their daily lives. These problems are unlikely always to be mathematical problems. However, the general approaches we consider below (see, for example, Pólya, 1973) apply generally to problem solving of any kind. Now, in school, teachers are unable to present in advance all of the problems that an individual will face in life. The best that a teacher can hope to do in this area then is to prepare the student to be able to tackle problems, to provide general strategies that will help in many situations. Hence when a teacher is scaffolding in a classroom situation, there are at least two reasons for this scaffolding. The first reason is immediate. The aim is to take the student past the current content barrier or explore the problem more effectively. But another aim is to establish long term good practice in the child. Effectively the

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teacher is saying “look, when I’m having difficulty with a situation like this, I try this or that strategy. You might find it useful here but you will almost certainly find it useful in many similar situations in the future”. Hence this second aim is to overcome process barriers which may occur at a later time. By repetition, if necessary, it is hoped that the child will internalise this scaffolding so that, perhaps gradually, the child will need less scaffolding, and the scaffolding will become part of the metacognitive capability of the child. Hence when the child meets a similar problem in the future, the teacher’s scaffolding will have become the child’s scaffolding and so the child will be able to self-scaffold around or through the difficulty. 5. TASK-STUDENT INTERACTION In solving problems in the classroom there has to be a task that needs to be accomplished. The learning that can be achieved through the task will depend on the match between the task and the student’s zone of proximal development. (Many authors have highlighted the “match” between the task and the child, see for example, Bennett and Desforges, 1988, Stacey, 1994, and Holton and Thomas, 1999.) Because we are interested in this chapter in analysing the talk of students, we will concentrate here on the relationship we see between the cognitive demand of the task and the resulting talk between peers which leads to cognitive gain. In Figure 5.1, we summarize graphically the discussion which follows.

cognitive demand of task Figure 5.1. A model for the link between productive talk and the cognitive demand of a task. The basic proposition here is exemplified by the three regions A, B, C of the graph in Figure 5.1. In region A, the cognitive demand of the task is low. The problem set is easy for the children involved, perhaps because they have

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successfully tackled many similar problems in the past. Whatever the reason, the task presents no great difficulties for them. Hence they have no need to engage in meaningful conversation to aid them to complete the task. Region A then, demonstrates that if the cognitive demand of the problem is low, then so will be the productive talk between peers engaged in solving the problem. At the other end of the scale, we have region C. Here the cognitive demand of the task is high. It is at the edge, or even outside, the zone of proximal development of the children involved in the task. In such a case the children will find the problem hard and are unlikely to be able to talk to each other in any meaningful way. They will not be in a situation to provide any scaffolding for each other. Most probably they will go off-task until the teacher provides some appropriate scaffolding. The net result is then, that if the cognitive demands of the task are too high, the talk for cognitive gain will again be low. Instances of this occurred in Lesson 3 (Compass Bearings) where the instructions given were so unclear that the students were unable to make any progress. The optimum region for productive talk is region B. In this case the cognitive demand of the task is sufficiently high that the children need to discuss how they will proceed to its completion but not so high that it is beyond their ability to make progress. The problem regarding one million sheets of paper discussed later in this chapter, is an example of a problem in the optimum region. The model above is part of one proposed in Holton and Thomas (1999). While we have represented the model as approximately parabolic, we have no empirical evidence to support this. We do, however, believe that the talk for cognitive gain does increase as the cognitive demand of the task increases. It is our conjecture that the level of talk then flattens off as the task becomes more difficult and then goes into decline from there on. Naturally the model proposed above recognises the individuality of the children involved. For some students a particular task will have low cognitive demand while for others it will be high. It is the teacher’s job to ensure that the task set is appropriate to the students’ zones of proximal development. 6. QUANTITATIVE ANALYSIS OF STUDENT-STUDENT INTERACTIONS In this section we consider the methodology used to quantitatively analyse the student-student interactions. We then provide the results obtained from this analysis. 6.1 Methodology In order to determine to what extent children help one another learn through their talk, it is first necessary to ascertain the type of talk in which they engage. The framework used here to analyse the interactions between students was adapted from one developed by Bennett and Dunne (1992). Their model involved an initial distinction between task-related and non task-related talk. They then distinguished between talk that was not directly relevant, for example the management of

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materials for the task, and relevant talk. Finally a distinction was made between task-related talk which was socially oriented (group management) and talk which was cognitively oriented. Thomas (1994) adapted the framework with a further distinction of the cognitively oriented talk into action and reflection talk. The delineation of cognitive talk into action and reflection is supported in the literature by researchers who argue that the value of talk lies in the speaker’s clarification of ideas, which most often occurs in the form of explanations (see for example, Barg & Schul, 1980, Gooding & Stacey, 1993, Hoyles, 1989, Webb, 1991, as well as the research cited earlier). The framework used in this analysis for classifying the talk between children is shown in Figure 5.2.

Figure 5.2. Classification system for group talk The talk is first classified as task- or non-task-related. Non-task-related talk is talk related to activities or events other than the task at hand. There were few instances of this talk and most of these were brief. Examples of talk classified as non task-related include: Karen:

My hair’s all bobbly? (non-task)

Karen:

I don’t want PE tomorrow. (non-task)

Lauren: A ball thrown in the air, (action, reading) Do you think Frances missed the bus? (non-task)

Task-related talk is then sub-divided, depending on its relevance to the social or cognitive demands of the task. Social talk, associated with the management of the group or materials, was further classified according to the sub-categories shown in

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Figure 5.2. Some categories were derived initially from the research of Thomas (1994), however others were derived empirically as the framework was applied to the current learning project data. Examples of talk classified as social include: M: Hey, where’d my pen go? (social, materials) D: I’m going to do these, this one first. (social, progress) D: Are we supposed to do that? (social, instructions) D: That’s what I’ve done. (social, dispute)

Cognitively oriented talk relates to the learning objectives of the task. A distinction is made between cognitively oriented talk, which was in nature reflective or active. Action talk is talk related to the activity of the moment. The child talks about their activity or comments on the activity of another child. Examples of action talk and the sub-category designation include: M: You’re not supposed to work out the ratio, you’re supposed to simplify it. (action, comment) M: Look. Forty divide forty, equals one. Two hundred divide forty equals five. See. (action, doing) F: We didn’t do this? We didn’t do this last year. (action, comment) P: A bouncing ball. I think a bouncing ball would be like D. (action, answer)

Reflection is talk no longer directly connected with the activity of the moment but is concerned with the child reflecting on their understanding. Reflective talk is talk associated with the clarification of ideas and is predominantly associated with the giving or receiving of explanations. The sub-categories of reflection were essentially derived from the research literature that supports the link between learning and the giving and receiving of explanations (Gooding and Stacey, 1993, Webb, 1991). The inclusion of the thinking aloud category came primarily from instances in the data where the children appeared to be verbalising their thinking. The thinking aloud sub-category closely resembles the notion of metacognition. Examples of reflective talk include: Frances: Yeah. No I reckon it would gain speed, and then it would sort of go down this constant pace, so I reckon it could be C. Once it hits the water it’ll go down at a constant pace. (reflection, explanation) Patricia: I think it’s probably more, Frances, ‘cause it’s not going to be totally constant pace? (reflection, explanation) Patricia: Frances, let me ask you one question, this is just one thing that is bugging me. Why is it V, what is V got to do with speed? (reflection, request for help)

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SHAPA (James and Sanderson, 1990), an interactive software tool which helps encode and identify patterns and sequences in verbal protocol data was used to apply the talk framework to the transcripts of the four mathematics lessons. The protocol data, which in this study comprised the transcripts of the talk between children, was encoded using the categories and subcategories previously described. After the text is coded, the program prepares reports on the number of instances in each category. The program can also report on patterns and cycles of talk by the whole group or by individual children. 6.2 The pattern of talk between students In considering the four lessons cited in the introduction to this chapter, it was observed that there were very few examples of student-student mathematical interactions in the lesson of 18th November 1994 (compass bearings). Consequently no attempt was made to analyse student-student talk for this lesson. The talk that occurs between children gives an indication of their task involvement and allows us to examine the extent to which talk contributes to their mathematical learning and understanding. Figure 5.3 illustrates the proportion of talk in each of the categories for the three lessons which contained significant talk between students. In each box, the numbers refer to the following lessons in that order: (a) Problem solving using standard format; (b) Rate and ratio, 1995; (c) Rate and ratio, 1996. Hence in the box marked “Cognitive”, 65% of the student-student talk was of a cognitive nature for the lesson of 13 June, 1995.

Figure 5.3. Analysis of student/student talk

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As a result of the above analysis, we can see that there is significant evidence of worthwhile task-related talk occurring in the three lessons that have been analysed. This is in contrast to the classes analysed by Thomas (1994). The major difference between the two situations analysed would appear to be the closer match between task and the zone of proximal development of the students involved. Hence the tasks provided in the three lesson transcripts from the Classroom Learning Project are much closer to region B in the model of Figure 5.1 then were the tasks in Thomas’ project. While contributing to our overall perception of what took place between the students, the quantitative analysis above does not help us to see the precise nature of the mathematical interactions within groups. We take this up later in this chapter. 7. ANALYSIS OF TEACHER-STUDENT INTERACTIONS This section parallels the outline of the last section but considers the analysis of teacher/student interactions. We first note that there seem to be three types of teacher/student interaction. These are (i) interactions with the whole class; (ii) interactions with a group of students; and (iii) interactions with a single student. The second point to note is that there seem to be two different approaches which influence the teaching of mathematics. These are the traditional approach and the constructivist approach. According to Freire (1972) the former: Involves a narrating Subject (the teacher) and patient, interviewing objects (the students). The contents, whether values or empirical dimensions of reality, tend in the process of being narrated to become lifeless and petrified (p. 45).

This traditional approach is one which treats students as “vessels to be filled” or as “tabula rasa”. Constructivism on the other hand has been defined by Confrey (1990) as: A theory about the limits of human knowledge, a belief that all knowledge is necessarily a product of our own cognitive acts. We can have no direct or unmediated knowledge of any external or objective reality. We construct our understanding through our experiences, and the character of our experiences is influenced profoundly by our cognitive levels (p. 108).

Consequently a constructivist teacher would not treat students as empty vessels but as thinking beings. And the role of the teacher changes from being ‘the sage on the stage’ to ‘the guide by the side’. The third major point that needs to be made is that mathematics teaching revolves around students tackling problems. Hence teacher/student interactions may take place (i) during the posing of a problem (Getting Started, see the next section); (ii) during the solving of a problem as a check to progress or because the student has reached a barrier to progress (While Students Are Working On The Problem) or (iii) when a problem has been solved to determine the correctness of the solution, other possible solutions, possible extensions, and so on (After Students Think They Are Finished). Holton, Spicer and Thomas (1999) suggest that there are three types of barriers encountered by students. These are strategic, when students are not sure which

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strategy or heuristic to apply or how to apply it, mathematical, when students require a piece of mathematical knowledge, or affective, which clearly covers a range of possibilities in the affective domain. In all stages of a problem, but perhaps especially when students have met with some difficulty, the teacher needs to employ scaffolding. Holton, Spicer and Thomas (1999) propose therefore that there are three types of scaffolding: cognitive, metacognitive and affective. Cognitive scaffolding is given to a student who requires support with the cognitive demands of a task. Similarly metacognitive scaffolding is related to the metacognitive demands of the task. The final affective category, again relates to issues such as transferring the task labour from the teacher to the student or refocussing on new targets to be reached. The analysis of the teacher/student interactions was undertaken by first breaking them up into categories of whole class, group or single student interactions. It was then initially planned to consider the interactions in each of these three categories according to where in the problem the interaction took place and according to the nature of the scaffolding. This approach was abandoned because the teacher was either transmitting information or because the nature of the problems posed was such that virtually all interactions were instigated because of mathematical problems experienced by the student and the scaffolding given was of a cognitive nature. We illustrate this by the following transcripts (T = Teacher; L = Lauren; K = Karen). Lesson 2

Standard Form.

1.

T:

Good. A shorthand way of writing this is 10 by itself 8 times, -one, two- yes? and 10 by itself eight times, multiplied by 3, and if I wanted to write it in our normal standard form the answer’s that. Right? [Works 3 x 108]

2.

L:

Umhm.

3.

T:

Now that has the three parts of a standard number. I have my number between one and ten [points to 3], ‘cause the decimal point’s there, and I have multiplied by 10 to the power 8.

4.

K:

5.

T:

Some of you I saw, you were doing this like this. You got your number right, and you got it between 1 and 10 right. And you thought “Great, there are 8 hops”, so you do that up here. OK, this is actually 3 by itself 8 times. Now could you please work that out. And the quickest, listen first, listen, ears [puts hand to ears]

6.

T:

[Getting Karen’s attention] Karen, ears.

7.

T:

The quickest way is to put 3 into your memory. So you put 3, so do this now, 3 and then go, put it into M in, which is put into your memory, good. Now you go times MR, recall memory, you do that seven times. So it’s 3 by, memory recall, which give you 3 again, times 3, get memory recall, times MR, times MR, times MR and do that 7

[laughs, taps Lauren’s hand]

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times so you’ve actually got 3 by itself [writes on board: 3 × 3 × 3 × 3 × 3 × 3 × 3 ×3 = 6561]. And Ilsa do you think it’s what? 8.

S4: 6, 5; 6, 5.

9.

T: OK, anyone check that or verify that for me?

10.

S5: Um.

11.

T:

12.

L: [calls out] That’s right.

13.

T: What’s 3 by 3 by 3? What’s 3 by 3 by 3? Right, what’s this 3 by 3 by 3? What’s this? Good, could you check it please by going 27 times 27 times 9? See if you come up with the same answer. Right, who agrees? [Lauren raises hand.] OK, can you see the difference? This is 300 million, yet this way of saying it is only 6 point or six thousand five hundred and sixty-one. Can you see the difference? This is, this number here multiplied by itself 8 times. It’s not 3 to the power 8, it’s 3 by itself 8 times. Do you remember me saying you’ve got to watch your language otherwise you’ll make mistakes along the way.

Check it.

The above is the first few minutes of a section of the lesson which continues for over 14 minutes. During that time the teacher provides information and assesses whether the students are following what she says by asking a series of small problems based on straightforward arithmetic. This approach can also be found in Lesson 4 (P = Patricia; F = Frances). Lesson 4: Velocity-Time Graphs 1.

T: [Speaking to the whole class] OK. OK, I think I’m - I’ll tell you what we’ll do, I’ll move on and I’ll come down - well I’ll ask you to come and see if I can check your work today. I’ll move on to page 19 and some of you may have [inaudible] 19 and 18 when you were away. OK, now let’s read page 19. You’ve got eight different graphs here. OK? In many circumstances you draw the graph of speed versus time. Now this isn’t distance versus time, this is speed versus time, this is how fast you’re going. The speed of a car, how far you go in a certain time. So speed is how fast you’re going, distance divided by time. It’s how fast you go. OK, I want you to underline, circle the word speed on the first line. [of page 19 of booklet][pause] You all know that there’s a meeting at recess don’t you? There is a meeting at recess time, the very beginning of recess time, yeah. In center space. OK, so circle the word speed. OK, let’s read it again. [reads] In many circumstances drawing the graph of speed versus time gives us a very clear indication of how fast or slow or if the object is moving at all. In these cases we put the speed variable on the dependent, the vertical axis [draws set of axes and labels vertical axis] speed, and time goes down there [labels horizontal axis], OK? These graphs if read correctly give a very accurate story of what is going on. OK, let’s have a look at graph A. Oh, no, we’ll start with graph B. OK, what do you notice? Where’s your book Ryan? [talks with student in undertone] OK, can you see how that graph’s going down all the time? OK, so we’ve got a situation

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where we actually have a graph going down like that. [draws graph B on axes] Now, the speed, have a look at it there. And have a look at it just a little bit later. What’s the speed, is it higher or lower? 2.

P and F: Lower.

3.

T: So what’s the car doing? Slowing down? OK, now watch this very carefully. At this particular time [points to where line intersects x-axis] what’s happening?

4.

F: It’s stopped.

5.

T: Has it?

6.

F[?] Um.

7

T: OK. Your car is going down and slowing and slowing and slowing. Is it going down nice and evenly?

8.

F: Yes. [P nods head]

9.

T: OK. So, all right. Let’s say that this is our car [P laughs] and this is the house [draws car and house].

10.

F: Doesn’t look like a house.

11

T: OK, I reckon you could break that into two halves. Do you agree? Do you agree?

12.

P: [nods head]

Again this is the initial section of a lesson segment which continues for over 14 minutes, during which time the teacher is speaking to the entire class, providing information and asking relatively simple, closed questions such as those on lines 1, 7 and 9 of Lesson 2 and on lines 1, 3, 7 and 11 of Lesson 4. All of these questions just require direct answers which do no provoke much thought on the part of the students. There is also little opportunity provided for the students to offer any explanations. Because much of all four mathematics lessons seemed to be given over to whole class activity we analysed the percentage of the lessons spent this way. The results are given in Table 5.1. Table 1. Percentage of time spent in whole class and group activities

Lesson 2 Lesson 3 Lesson 4 Lesson 6

Whole class 45% 100% 58% 75%

Group work 55% 0% 42% 25%

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The table suggests that the teacher does not value group work. Given the productive talk that we have shown to occur during group work time (see the following quantitative and qualitative analyses), it does seem a pity that more time has not been productively spent in class this way. However, the teacher clearly enjoys lessons of the type of Lesson 2 that had the most student interaction of all four classes. In responding to the videotape of Lesson 2 she has the following to say. I love this lesson. I still do it years later. It ‘ll be very interesting for me to see how effectively it comes across to the kids. I’ve often had them work for the whole lesson on just one or two problems and I feel that they get a lot out of it, and uh sometimes the trivia in mathematics and the finer points can quite often be useful. I think they like the idea of working out the height of a thousand sheets of paper and the estimation and the methodology I see as very important and far too often in maths we tend to go ahead and just, you know, the tendency is to give them textbooks and nothing to really work out and talk and negotiate and problem solve. So I particularly chose this because it’s a wonderful problem solving exercise and um it’s fun, and I love doing it as well.

This raises the question of why she does not give more of these lessons. First it should be said that we have only taken a sample of four lessons here and it is possible for one reason or another that they were not given in the teacher’s usual style. Second we should point out that the Compass Bearings Lesson (Lesson 3) was a different style of lesson. But it is also worth noting the teacher’s comments in response to the videotape of Lesson 6. There she says the following. One of the criticisms of the maths course, particularly by, say some science and maths teachers who are coming into the course after not having taught maths for a couple of years is that the maths course we’re teaching at the moment is very teacher-directed and as such you don’t get a chance to spend a lot of time going around and uh checking the student’s work. And for a student to go on with work without having the teacher standing up the front and explaining is very difficult. And that is one of the criticisms of this particular maths course. So um I know as a practising teacher that it does make it hard for me to _ I have to work extraordinarily hard to allow time in the syllabus to go round and check the students’ work. It’s something that just doesn’t fit in very easily into uh into the maths syllabus. So um whereas before, you know, um, not so much in our school, but previously you know teachers would teach a little bit and then the kids would have to go and do a whole lot of rote practice. and the teacher had plenty of time to go round but then it was pretty boring for the students. So one of the debates within the faculty is are we allowing enough repetition, work, for the students to really understand the concept before we move on. So they’re all philosophical questions that are coming up time and time again.

So the teacher was aware of the conflict but she seems to be in a situation imposed by her school mathematics department. It is not clear from her tapes though, whether the mathematics course is forced to be that way by time or other external constraints or whether it is designed that way to assist being “pretty boring for the students”. It does seem unfortunate. The students clearly gained from the approach of Lesson 6 and the material of the speed-time graphs (Lessons 4 and 6) could have easily been presented in a problem solving format. Perhaps it was the uncertainty on

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the part of the Mathematics Department over this topic which made her teach it the way she did. Amongst the staff it was quite a difficult lesson to teach and there was still some controversy about the exact accurate interpretation of the graphs.

It is perhaps worth noting the following comment by Dewey (1910). Many, if not most, instructors would be surprised if informed at the end of the day of the amount of time they have talked as compared with any pupil. (p. 185)

This suggests that many of us spend more time talking in class than we realise. 2. SCAFFOLDING – THEORY AND PRACTICE Before investigating specific episodes of scaffolding, we need to consider scaffolding itself in more depth. 2.1 Scaffolding revisited The metaphor of a scaffold relates to a building. In our opinion, what is being built is the knowledge and understanding of the learner. When constructing a building the scaffold is used to put the worker in a position to add the next bricks, insert a window, etc. A learner is facilitated in this activity by assistance from a “more competent other” (either adult or peer), who puts them in a position to construct knowledge and understanding that they would be unable to achieve on their own. There are many ways in which this scaffolding can be effected. During a mathematics lesson, where the emphasis is frequently on solving a problem of some description, scaffolding is often presented by way of questions. Almost always these questions are open questions which lead the learner to think about the situation. Because thinking is valued in the scaffolded experience, closed questions, which require only a low level of cognitive response, are almost universally avoided. Hence open questions such as the following have been suggested by Schoenfeld (1992), Holton et al (1997) and others. What are the important ideas here? _ Can you rephrase the problem in your own words? Why are you doing this? Are you convinced that bit is correct? Holton et al (1997) have divided a set of generic open questions into three parts, each part relating to a problem solving stage. These parts are Getting Started, While Students are Working on the Problem, After Students Think They are Finished. The latter part consists of the following questions, examples of which appear in an episode from Lesson 2 which we will comment on soon. —

—

—

After Students Think They Are Finished Have you answered the problem? Have you considered all the cases? —

—

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– – – – – – –

Have you checked your solution? Does it look reasonable? Is there another solution? Could you explain your answer to the class? (Orally or in written form.) Is there another way to solve the problem? Could you generalise the problem? Can you extend the problem to cover different situations? Can you make up another similar problem?

2.2 A teacher-student interaction We now consider one of the teacher-student interactions and then discuss the scaffolding in that episode (L = Lauren; K = Karen). Lesson 2 – Standard Number 1.

T: Which one are you up to guys?

2.

L: Three.

3.

T: Kate, where are you up to?

4.

K: Three.

5.

T: Yes, fine, what have you done so far?

6.

S20: It’s hard to explain.

7.

K: Which bits do you want to know?

8.

T: Uh, question 1. Explain your working.

9.

K: Question 1. We did length times width.

10.

T: Of?

11.

K: Of the sheet of paper.

12.

T: And what’s your length?

13.

K: Length was two hundred and fif - two hundred and fifty what?

14.

T: Yeah, two hundred and fifty whats?

15.

L: [quietly] millimetres.

—

—

–

–

MATHEMATICAL INTERACTIONS AND THEIR INFLUENCE ON LEARNING 16.

K: Um. Yeah, millimetres.

17.

T: She hasn’t had the pure joy of me with units yet, has she? Hm, Hm. Times?

18.

K: Yeah, yeah. Minus the outlay, border thing, [makes page-shaped gesture in air] that didn’t have graphs on it, the graph on it, times one hundred and eighty millimetres —

19.

T: Where’d you get one hundred and eighty from?

20.

K: Width. Equals forty

21.

T: Why did you multiply them together? Why not add?

22.

K: To get the area. I know that much.

23.

T: [to Lauren] You’ve been tutoring her?

24.

K: Equals forty five thousand, therefore you’d need. Oh, how’d I get that?

25.

T: Forty five thousand?

26

K: Forty-five thousand. That’s what we got.

27.

T: Forty-five thousand? Can you press that — can you press that — can you do that again? Two hundred and fifty times a hundred and eighty? Oh, hang on, hang on, I think you’re right. Hang on. I think they’re wrong.

28.

K: Yup, they’re wrong, we’re right.

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[L holds up calculator.] 29.

T: What is it? Oh dear, And what’s the answer?

30.

L: Forty-five thousand millimetres on one piece.

31.

T:Yes.

32.

L: – and then you divide, you divide a million by that, and it gives you twenty two point two, so therefore you need twenty two and a fifth sheets.

33.

T: Excuse me [goes to group at front of room.]

The style of the episode above is in stark contrast to all of the other teacherstudent interactions in the transcripts. The episode above lasts for over two minutes. During that time the teacher asks several open questions. These are Line 5

... what have you done so far?

Line 19 Where’d you get a hundred and eighty from?

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Why not add?

Line 29 Then what’s the answer?

In addition a comment made in line 8 is the equivalent of a question. Line 8

Explain the working.

All of the questions and the comment, in the context in which they were presented, required a more thoughtful response than the closed questions of the two previously cited transcripts. The question from line 29, in particular, elicited a complete explanation of the solution of the problem. The five lines quoted above, all represent scaffolding from the “After Students Think They Are Finished” subset of questions listed earlier in this section. It would seem then, that we have at least five instances of scaffolding here that the teacher was using to ensure that the students had successfully completed the problem and understood what they had done. It has to be noted, however, that the teacher probably did not know the answer to the problem that the students had just solved. The evidence for this comes from the fact that Lauren in a stimulated recall interview says: L: Cause, um, she ’d already got the answer off some other kids, and we got a different answer that was right, and so she told them that they were wrong.

Further, in line 29 the teacher says “Oh dear. And what’s the answer?’ Then as soon as the teacher has finished with the group of Karen and Lauren, she immediately “goes to group at front of room”. This raises the question that if the teacher did not know the answer, could she in fact be scaffolding in the transcript? We consider this in the context of Bruner’s (1985) comment cited in Section 2 that “the tutelage” is by “an adult or a more competent peer”. Can it always be assumed that an adult is always more competent? In the current example, there is a sense in which we believe the teacher is less competent than the students. We have advanced a case that the teacher did not know the answer to the problem. Can she then scaffold the student’s problem solving activity? Our feeling is that she can and that she did. The competence that she has that the students don’t necessarily have is the knowledge of how such problems work and the steps that she herself would go through to solve them. The teacher’s scaffolding then, follows her internal line of reasoning in response to the answers that the students give. This is an important feature of solving problems. In order to get students to solve problems successfully, the teacher does not have to know everything about all problems that she may want to pose in the classroom. At times it is enough to have a general knowledge of how solving problems works and to know the kinds of questions to ask, that is the scaffolding to give, that will lead the students forward and consolidate their learning and understanding. Teachers may not be able to correctly solve all the problems that their students are working on more quickly than their students or even more accurately than their students. Of course, they should be “more competent” than their students in this sense more often than not. However, they should have the ability to elicit the

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correct response from their students by appropriate scaffolding. This ability comes from their experience of mathematics and it is in this sense that they should always be “more competent” than their students and hence able to produce appropriate scaffolding when required. 2.3 A student-student interaction In the previous section, we carried out a qualitative analysis of the nature of talk between students. Here we want to look at an episode of student-student interaction in some depth. This episode is from Lesson 2 and it involves two students working on the following problem (L = Lauren; K = Karen). Find the height of a stack of one million sheets of paper. To assist with solving this problem they are given a pack of 500 sheets of paper. They have measured the depth of this pack and have found it is 5.8 centimetres thick. 1.

L: [writing] 500 sheets. Height equals.

2.

K: OK. Question 2.

3.

L: Does everyone understand what we did with number one?

4.

K: No, but, anyway. 500 sheets.

5.

L: And how many sheets do we need?

6.

K: 500 sheets of what? 500 sheets.

7.

L: Their height equals 5.8.

8.

S20: We’ve done that.

9.

K: I know. But we’ve got to do it all together so.

10.

L: One point oh times ten to the power of 6 divided by 500.

11

K: Oh year sure everyday what are you talking about? What are you talking about?

12.

L: I’m finding out how many five hundreds there are in a million.

13.

K: How many five hundreds there are in a million? That would make it one thousand. How many thousands are there in a million?

14.

K: That would make a thousand, two thousand. What? [to another S]. I have a lot to say.

DEREK HOLTON AND G ILL THOMAS

96 15.

L: [checks result on calculator] Two thousand, well done!

16.

K: This is called skill. This is what you do. Five hundred into 100 which is 2, then you do a hundred [correcting herself] which is a thousand.

17.

L: Times 5.8. Shush.

18.

K: Which is 2. Then you do a thousand into a million, which is 1000, so a thousand times two is 2000.

19.

*L: Eleven thousand six hundred.

20.

T: With your working out folks I want you to tell me what you are mult Matthew - what you are multiplying by, and you simply put a little arrow telling me what and why.

21.

K: What are we doing? Is it a million sheets of paper though?

22.

L: Yeah. One point -

23

K: We’re doing a million sheets of paper.

24.

L: Yea, you need -

25.

K: Yes we do. We do, shut up.

26.

L: Therefore, I did that wrong.

27.

K: 2000 times 5.8 centimetres is eleven, six, zero, zero. [i.e. 11,600]

28.

L: Mms?

29.

**K: Centimetres - which would make it 11 point 6 metres right?

30.

L: Or eleven - yeah. It’d be 11 point 6 metres wouldn’t it, ‘cause you take off one to get to centimetres, and another one, yeah. [pause]

31.

K: [looking up] That’s quite high, isn’t it?

32.

L: All right. And you’ve got to point out what the [inaudible] are right?

33.

K: You’ve got to point out the what is?

34.

L: We have to show what we’re multiplying by. [S22 says something to Karen, Karen laughs]

35.

S20: That’s not how you know, you look like you know what you’re doing and you just do it.

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36.

K: Exactly, you go through a total state of concentration, it lasts about 2 seconds, that’s when you get the answer, and then you don’t know what you’re doing, so it doesn’t matter. 500 sheets equals, height equals 5 point 8 centimetres. I don’t even understand what I wrote. [pause as Lauren, Karen write, sound of other students discussing problem]

37.

K: But why do we divide a million by 500 to get that answer?

38.

L: Because you know, if you know what the height is -

39.

K: So what am I doing? Tell me what I’m doing here, tell me what I’ve done.

40.

L: All right. You know that 500 sheets equals 58 -

41,

L and K: 5.8 centimetres.

42.

K: There is a point there, it’s up there.

43.

L: Oh, I can’t see it.

44.

K: Get some other glasses.

45.

L: OK. Now, we need to know - we need to know the height of a million sheets of paper. Therefore you must divide a million by 500 and times that number by 5.8.

46.

K: [writing] Equals 2 thousand. Sheets of paper.

47.

L: OK.

48.

* * *K: That would make it 2000 times 5.8 centimetres equals 11,600 centimetres, equals 11.6 metres of paper. [bell rings]

*The asterisks partition the transcript into distinct attempts to solve the problem. These are discussed in the next section.

We now want to analyse this from a scaffolding viewpoint. To do this we will examine the questions that are asked in this episode and indicate the possibility that they are of a scaffolding nature. We relate all questions back to the list in the first subsection of this section. To simplify the discussion, we present the results in table form. Hence in Table 5.2 we list the questions raised and where appropriate, indicate a parallel question from our scaffolding list.

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Table 2. Questions and Scaffolding Question

Scaffolding list

Comment

Line 3. L: Does every one understand what we did with Number 1?

Could you explain your answer to the class?

Scaffolding not taken up by K.

Line 5. L: And how many sheets do we need?

What information is given?

Part of understanding the problem.

Line 6. K: 500 sheets of what?

What infomation is given?

Probably a rhetorical question.

Line 11. K: What are you talking about?

Tell me what you are doing here?

Seeking explanation.

Line 13. K: How many five hundreds are there in a million? ... How many thousands are there in a million?

Calculation.

Line 21. K: What are we doing?

Tell me what you are doing here?

Seeking explanation.

Line 21. K: Is it a million sheets of paper though?

What information is given?

Understanding problem.

Line 28. L: [Millimetres]

Have you considered all the cases?

Trying to determine the units of the answer.

Are you convinced that bit is correct?

Checking calculation.

Mms?

Line 29. K: Which wou1d make 11 point 6 metres right? Line 31. K: That’s quite high, isn’t it?

This may simply be a rhetorical question relating to the magnitude of the answer.

Line 32.L: You’ve got to point out what the [inaudible] are right?

Relates to line 20 and teacher’s requirement for explanations.

Line 33. K: You’ve got to point out the what is?

Clarification of question from Line 33.

Line 37. K: But why do we divide a million by 500 to get that answer?

Tell me what you are doing there?

Seeking clarification.

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Clearly not all the questions asked are of a scaffolding nature. However, it is not always clear which are and which aren’t scaffolding questions. The first point that should be made here is that many of the questions that are asked in this episode have obvious parallels in the scaffolding list presented earlier. There are a number of questions that are clearly questions that would be considered as scaffolding questions if they had been asked by the teacher. If they are asked by a student, who is at first sight not necessarily “a more competent other”, do they still represent scaffo l ding ? The aim of scaffolding is to help the learning of the participants. In most cases, the students make progress on the problem following these questions. Hence we are inclined to conclude that each such question does constitute scaffolding for at least one of the students in the dyad. Since both students in the dyad asked questions of a scaffolding nature, the form of scaffolding displayed here can be termed ‘reciprocal scaffolding.’ On the competency issue, the students involved in asking the question have sufficient competence and experience, and the problem is sufficiently within their zones of proximal development, for them to be able to ask appropriate questions of themselves and of each other. The ‘internal’ voicing of such questions constitutes a form of metacognitive activity or, in the terms of this discussion, ‘ self-scaffolding.’ The voicing of such questions as a component of the collaborative problem solving activity of the student dyad is then termed ‘reciprocal scaffolding.’ It would seem that metacognition here is not simply represented by questions. In line 37, Karen tries to explain how her brain functioned to produce an answer. Line 40 also appears to be metacognitive as Karen attempts to understand the solution to the problem, and in so doing seeks assistance from Lauren. This broadening of the sphere of application of ‘scaffolding’ to include the reciprocal character evident in the dyadic interactions reported here challenges the asymmetry previously assumed to be a characteristic of the process of scaffolding. Further, it is proposed that questions of the type documented have metacognitive characteristics and can be usefully thought of as ‘self-scaffolding’ in situations when a student poses such questions to themselves in the course of solving a problem or attempting to learn new content. 2.4 Discussion This section has analysed two episodes, one involving a teacher-student interaction and one involving a student-student interaction. In the former episode there is evidence of scaffolding and in the latter similar questions are probably being asked metacognitively as self-scaffolding for at least one of the dyad, or interactively as reciprocal scaffolding. It appears that scaffolding was initially conceived as a deliberate action taken by a more competent other with regard to activity in the cognitive domain. However, the relative competence of the scaffolder and scaffoldee may be in the metacognitive domain as we saw in the teacher-student interaction.

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When similar questions were asked in a group setting it is not at all clear that the person asking the question was more competent. This was most clearly evident when both members of the dyad asked such questions. The questions did seem to move the dyad forward. They were never of the type “Just tell me the answer”. Hence we conclude that they were largely metacognitive in nature and, given the symmetry of their social occurrence, we feel justified in referring to them as reciprocal scaffolding. One final point needs to be made concerning the student-student episode. Although the questions being asked had the character of scaffolding, they were being asked by a student. We doubt that teachers often engage in an interaction with student(s) where a student is undertaking most of the scaffolding. Possibly this is something that should be encouraged. The important thing for us is that the students were engaged in reciprocal scaffolding and that this should be recognised as a significant component of group work. In such situations, students should feel free to ask such questions and know that they will be positively received (see Damon, 1984). Learning to ask such questions is at least as important as being able to answer them. Groups may be the best place to practice asking such scaffolding questions, and such practice should be nurtured by the teacher. 3. SOMETHING OLD, SOMETHING NEW In this section we want to revisit the episode regarding the million sheets of paper problem. The episode reported in the last section is an interesting one for reasons other than the reciprocal scaffolding that took place. Having considered the episode line by line, we now consider it holistically. First we note that the problem is essentially solved three times. On the transcript they are marked with one, two and three asterisks. The single asterisk is at line 19, the double asterisk is at line 31 and the triple asterisk at line 50. It is worth noting that the first answer is 11,600, the second is 11.6 metres and the third is also 11.6 metres. (It should perhaps be pointed out that there is a small error here and the actual answer should be 116 metres.) The first answer is found by Lauren almost without Karen’s help. In fact, Karen has been more of a hindrance in lines 16 and 18, with Lauren telling her to “shush” in line 17. So in the first phase of the episode, Lauren begins to come to grips with the problem. The burden of working towards the answer the second time is on Karen’s shoulders. In fact it is not clear what Lauren is doing in this phase of the episode. However, through what she has heard from Lauren and through her own efforts in the second phase, Karen produces an answer to the problem. Furthermore Lauren agrees with this answer, even though it is incorrect. Units have appeared here probably because the students remember that the teacher stresses the importance of correct units. Lauren brings this up in one of her stimulated recall interviews. 51.

L: Mrs Brown is very big on units.

52:

I: Oh.

MATHEMATICAL INTERACTIONS AND THEIR INFLUENCE ON LEARNING 53:

L: if you leave off units, you get marks off.

54.

I: So do you leave off the units?

55.

L: I used to, I don’t any more.

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The final phase appears to be more of an interaction between the two students, with Lauren finally stating the method of solution in line 46 and Karen carrying out the computation in line 49. This phase was undoubtedly motivated by the teacher’s interjection on line 19, where she insists that they show their working. Perhaps the thrust of this final section is to understand the problem sufficiently in order to write it up in accordance with the teacher’s specification. We believe that this three-phase process towards a solution is not uncommon both in problem solving exercises and in mathematical research. Commonly in the first phase the originator of the solution is not totally comfortable with the solution obtained, even though the method is correct. We do not have any direct evidence from this episode or from stimulated recall interviews, that Lauren was less than happy with her solution at line 18. However, it is not until line 46 that she articulates the method of solution. She could have stated the method at any point after line 15 if she had been confident that it was correct or fully understood what she was doing. Hence this first phase might be considered to be an intuitive phase. In the next phase, while listening to Karen and working on her own, Lauren becomes convinced of the answer. When pressed because of the teacher’s requirements, Lauren is then able to articulate the solution method. At this stage it seems fairly clear that Karen has also understood the problem too. Perhaps the statement at line 18 could be considered as a conjecture. This conjecture was finally justified in lines 46 to 49. The sequence that we have described above may be quite important to the learning and understanding of mathematics. What is more, it is unlikely that such a sequence could occur so naturally except in a group situation. The ‘way of working’ of the dyad has allowed the students to express their ideas. Through the discussion of the episode both students have achieved learning and understanding. Being allowed to talk was essential to the successful conclusion of the episode. While Lauren may have completely understood the problem in a reasonably short time if she had been working by herself, it is by no means clear that this would have been the case for Karen. In both cases though, it is likely that understanding came more quickly, and at a greater depth, as a result of the interactions between the two students. 4. CONCLUSIONS In this chapter we have explored some of the mathematical interactions which have taken place in some classrooms involved in the Classroom Learning Project. We have used these interactions to try to determine how learning and understanding might take place and to test current theories.

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In particular, we have looked at the concept of scaffolding and seen how it can be used by “a more competent other”. However, we have also seen how students used questioning similar to that involved in ‘conventional’ scaffolding. We see this questioning as partly metacognitive in character and are inclined to call it selfscaffolding when practised by an individual and reciprocal scaffolding when practised by a group or student-student dyad. We have given examples of these forms of scaffolding in practice. Using both qualitative and quantitative data we have observed learning and understanding take place in group situations. It is clear that these cognitive processes are catalysed by talk. Much of what has been achieved in dyads in the classrooms studied could not have been achieved as quickly, if at all, by whole class teaching or by students working alone. Talk is not a necessary and sufficient condition for learning and understanding however. In the fraction problem cited in Chapter 4, the dyad showed very little understanding of the processes involved in manipulating fractions. Even in the one million sheets problem, although the group there showed understanding they did not obtain the correct answer. Hence it is clear that a very important role for teachers during group work, is to monitor the progress of the groups. Finally, we note that the mathematical interactions that appear to have been of the highest quality took place between students in situations where they were presented with problems of a demanding nature. It would seem that the teacher in the lessons considered in this chapter could have improved the chances of learning by the students in three ways. First by providing problems that would extend the students in the way that, for instance, the one million sheets of paper problem did. Second, by providing appropriate scaffolding. That is, by asking questions which stimulated discussion and required some explanation. And third, by providing more time for students to interact in groups and less time in whole class, teacher-driven situations, where the students were not stimulated to make any significant contribution. ACKNOWLEDGEMENTS The authors would like to acknowledge the many helpful discussions on the topics of this chapter that we have had with Paul Ngee Kiong Lau and Chris Linsell.

REFERENCES Australian Education Council (1991). A national statement on mathematics for Australian schools. Carlton, Vic: Curriculum Corporation. Barg, J. and Schul, J. (1980). On the cognitive benefits of teaching. Journal of Educational Psychology, 72 (5), pp 593-604 Bennett, N. and Desforges, C. (1988). Matching classroom tasks to student’s attainment. The Elementary School Journal, 88 (3), pp 221-234. Bennett, N. and Dunne, E. (1992). Managing CIassroom Groups. Hemel Hempstead: Simon and Schuster Education. Berk, L.E. and Winsler, A. (1995). Scaffolding Children’s Learning: Vygotsky and Early Childhood Education. Washington, DC: National Association for the Education of Young Children.

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Bruner, J. (1985). Vygotsky: A historical and conceptual perspective. In Wertsch (Ed.), Culture, Communication and Cognition: Vygotskian Perspectives, pp 21-34. Cambridge: Cambridge University Press. Bruner, J. (1986). Actual Minds, Possible Worlds. Harvard, MA: Harvard University Press. Bruner, J. and Haste, H., (Eds.) (1987). Making Sense: The Child’s Construction of the World. Bath: Methuen & Co. Ltd. Cambourne, B. (1988). The Whole Story: Natural Learning and the Acquisition of Literacy in the Classroom. Auckland, N.Z.: Ashton Scholastic. Chi, M.T.H., Bassok, M., Lewis M., Reimann, P. and Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13, pp 145-182. Chi, M.T.H., de Leeuw, N., Chiu, M-H. and La Vancher, C. (1994). Elicting Self-Explanations Improves Understanding. Cognitive Science, 18, pp 439-477. Confrey, J. (1990). What constructivism implies for teaching. In Davis, R.B., Maher, C.A. and Noddings, N. (Eds.), Construcitvist Views on Teaching and Learning Mathematics. Journal for Research in Mathematics Education, Monograph No. 4, pp 107-124. Hillsdale, NJ: Lawrence Erlbaum. Damon, W. (1984). Peer education: The untapped potential. Journal of Applied Developmental Psychology, 5, 331-343. Dole, J.A., Valencia, S.W., Greer, E.A. and Wardrop, J.L. (1991). Effects of two types of prereading instruction on the comprehension of narrative and expository text. Reading Research Quarterly, 26, pp 142-159. Ernest, P. (1995). The One and the Many. In L Straffe and J. Gale (Eds), Constructivism in Education. Hillsdale, N.J.: Lawrence Erlbaum. Forman, E. and Cazden, C. (1985). Exploring Vygotskian Perspectives in Education: The cognitive value of peer interaction. In Wertsch, J. (Ed.), Culture, Communication and Cognition: Vygotskian Perspectives, pp 323-347. Cambridge: Cambridge University Press. Freire, P. (1972). Pedagogy of the Oppressed. Ringwood: Penguin Books. Gooding, A. and Stacey, K. (1993). Characteristics of small group discussion reducing misconceptions. Mathematics Education Research Journal, 5 (1), pp 60-73. Greenfield, P.M. (1984). A Theory of the Teacher in the Learning Activities of Everyday Life. In Rogoff, B. and Lave J., (Eds.), Everyday Cognition: Its Development in Social Context, pp 117-138. Cambridge, MA: Harvard University Press. Holton, D.A., Spicer, T, Thomas, G. and Young, S. (1996). The Benefits of Problem Solving in the Learning of Mathematics. Report No. 546 to the Ministry of Education, Wellington. Holton, D.A., Anderson, J. and Thomas, B. (1997). OPE-N Plan for Teaching Mathematical Problem Solving.: Report No. 587 to the Ministry of Education, Wellington. Holton, D.A., Spicer, T. and Thomas, G. (1999). Scaffolding and Mathematical Problem Solving, submitted. Holton, D.A. and Thomas, G. (1999). The Nature of Task, submitted. Hoyles, C. (1989). What is the point of group discussion in mathematics? In Murphy, P. and Moon, B. (Eds.), Developments in Learning and Assessment, pp 121-129. London: Open University Press. James, J. and Sanderson, P. (1990). SHAPA 2.0: Software to heuristically assist protocol analysis. Department of Mechanical and Industrial Engineering. University of Illinois at Urbana - Champaign. Maker, C. (1982). Curriculum Develoment for the Gifted, Rockville, Maryland: Aspen Systems Corporation. Mercer, N. (1994). Neo-Vygotskian theory and classroom education. In Stierer, B. and Maybin, J. (Eds.), Language, Literacy and Learning in Educational Practice. Milton Keynes: Open University Press. Ministry of Education (1992). Mathematics in the New Zealand Curriculum. Wellington, N.Z.: Learning Media. Russell, D.M. and Kelley, L (1991). Using IDE in instructional design through automated design tools. Paper presented at Annual Conference of American Education Research Association. Schoenfeld, A.H. (1987). What’s all the fuss about metacognition. In Schoenfeld, A.H. (Ed.), Cognitive Science and Mathematics Education. Hillsdale, N.J.: Lawrence Erlbaum. Schoenfeld, A.J. (1992). Learning to Think Mathematically: Problem Solving, Metacognition and Sense-Making in Mathematics. In Grouws, D.A. (Ed.) Handbook of Research in Mathematics Teaching and Learning, pp 334-370. New York: Macmillan. Smith, A. (1992). Understanding Children’s Development. Wellington, N.Z.: Bridget Williams.

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Stacey, K. (1994). Challenges and constraints for constructing curriculum. In G. Bell, R. Wright, N. Lessen and J. Geake (Eds.), Challenges in Mathematics Education: Constraints on Construction. Proceedings of MERGA 17. Steffe, L.P. and Gale, J. (1995). Constructivism in Education. Hillsdale, NJ: Lawrence Erlbaum. Thomas, G. (1994). Discussion in Junior Mathematics: Helping one another learn? Unpublished Ph.D. Thesis. University of Otago, Dunedin, New Zealand. Tudge, J. (1990). Vygotsky, the zone of proximal development and peer collaboration: Implications for classroom practice. In Moll, L. (Ed.), Vygotsky and Education: Instructional Implications and Applications of Sociohistorical Psychology, pp 155- 172. Cambridge: Cambridge University Press, von Glasersfeld, E. (1995). A Constructivist Approach To Teaching. L.P. Steffe and J. Gale, (Eds.) Constructivism in Education. Hillsdale, NJ: Lawrence Erlbaum. Vygotsky, L. S. (1962). Thought and Language. Cambridge, MA: MIT Press and Wiley. Webb, N.M. (1989). Peer interaction and learning in small groups. International Journal of Education Research, 13, pp 21-39. Webb, N. (1991). Task-related verbal interaction and mathematics learning in small groups. Journal for Research in Mathematics Education, 22 (5), pp 366-389. Wood, D., Bruner, J.S. and Ross, G. (1976). The Role of Tutoring in Problem Solving. Journal of Child Psychology and Psychiatry, 17, pp 89-100.

MARY D. AINLEY

CHAPTER SIX Interest in Learning and Classroom Interactions

“Interest”, then, is not in the object, nor in the mind of the child, but it emerges as a result of processes that link the two in irreversible time. Once we have been able to “diagnose” its presence, we have already missed the opportunity to study its emergence (Valsiner, 1992, p.33).

1. INTRODUCTION Any attempt to model the learning process as it occurs in classroom settings must include some components which represent the motivation which directs and energises the learner’s behaviour. A large number of descriptors have been used to refer to different aspects of the motivational structure of classroom learning: curiosity, intrinsic and extrinsic motives, goals, interest, the list could go on. There are important points of shared meaning among these terms and some of these will be explored as the present study focuses specifically on the construct of student interest in learning. At a time when there is increasing significance being placed upon issues of ‘selfregulated learning’ and policy commitments to the encouragement of ‘lifelong learning’, an analysis of the motivational dynamics of learning as seen in the data of the Classroom Learning Project may provide some useful insights into the psychological process variables which relate to student interest in learning. Valsiner’s contribution to the collection of papers The Role of Interest in Learning and Development (Renninger, Hidi, & Krapp, 1992), highlights something of the difficulty inherent in studying student interest. He draws attention to the difference between individual personal meanings and general or shared meanings of the term interest. For example, in the claim “I am interested in aliens and extraterrestrial beings”, the state of interest referred to consists of a set of meanings and associations unique to the individual, acquired through their own special history of interaction with such things as literature, movies and associated phenomena. When the researcher describes a group of people as being “interested in aliens and extraterrestrial beings” the state of interest carries a more restricted meaning. Its meaning is constrained to an agreed set of psychological processes. Armed with knowledge of that set, an external observer can then, in theory, reliably infer that the 105 David Clarke (ed.), Perspectives on Practice and Meaning in Mathematics and Science Classrooms, 105—130. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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members of the group share a common psychological state in relation to aliens and extraterrestrial beings. The general or shared meaning is essential for making inference and generalisation. At the same time, appreciation of the unique and personal reminds us that the basic referent of the concept ‘interest’ is an individual’s experience in relation to the object of their interest. Thus. if N persons state that “I am interested in X” (statements at the level of personal sense), we might be tempted to claim that “interest in X” (statement at level of meaning) is widely present among people, hence general. Of course by doing this we have fused the levels of the individual (and unique) use of the word in everyday life. and the psychologist’s generalization (level of shared meaning or special term) that supposedly is applicable to many persons in similar ways most of the time (Valsiner, 1992, p.29).

The present analysis of student interest in learning as seen in the Classroom Learning Project, gives due consideration to the personal and unique aspects of interest as well as the general shared meanings. It gives special emphasis to the student’s own voice as a way of understanding what interest in learning is about. The strategy which will be adopted starts with the shared meaning, the consensus among researchers on what are the appropriate indicators of interest. A number of instances of behaviour matching these criteria for identification of ‘interest’ were then located within the transcripts from the Classroom Learning Project. The detail of the data set was used to explore these examples of ‘interest’ from the personal experience perspective, the level of personal meaning. This approach addresses what Valsiner claimed is an important agenda in the psychological understanding of ‘interest’: to distinguish the processes whereby interest emerges from what is not interest. The Classroom Learning Project transcripts provide a useful data source for addressing this issue. A point at which an individual student showed signs of being interested in some aspect of their learning environment can be identified. It is then possible to sift through what preceded that event to try and identify what went before and aroused interest as well as to identify what followed. A range of perspectives are available in the transcripts. This is an important strength of the Classroom Learning Project methodology. It is possible to examine the character of a specific event, to turn back time to observe possible foreshadowing events in previous lessons. The student’s reflective judgements about the event can be used to further test those interpretations and the teacher’s reflective judgements can be added. Together these multiple perspectives on the one event offer a rich insight into the character of student interest. 2. RESEARCH PERSPECTIVES ON INTEREST IN LEARNING When applied to classroom behaviour, constructs such as interest, curiosity, engagement, and intrinsic motivation all carry the implication that the student has become actively involved with the content, or some aspect of the content, of the learning activity. The different constructs which have been used in this area put different shades of meaning on ‘active involvement’ but all propose that the person acts to maintain or extend their contact with the object in order to increase their knowledge of it. A look at some of the ways in which the construct of curiosity has

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been framed by researchers over the years highlights some important aspects of the relationship between learner and object which are the focus of current research on the role of interest in learning and development. Berlyne’s (1960) classic work on curiosity described this relation between person and object in terms of exploratory behaviour. When curiosity is aroused, the learner seeks further information to resolve the perceived ambiguity, complexity, or uncertainty. Grounded in a drive theory framework, Berlyne’s main focus was on the characteristics of the situation. He was concerned to identify those stimulus properties which aroused what he referred to as the curiosity motive which in turn prompted exploratory behaviour. According to this view the character of the learning situation is primarily responsible for arousal of the learner’s interest. The outcome of Berlyne’s research was a very detailed description of a number of stimulus properties which across any group would predictably generate further investigation of those stimuli. Together this set of stimulus properties (for example, incongruity and ambiguity) were referred to as collative variability. In one well known experiment (Berlyne, 1957) the effect of incongruity on the arousal of curiosity was investigated using specially constructed stimulus patterns. Each stimulus consisted of animal parts which the viewer would, through past experience have come to regard as incompatible; for example, the head of an elephant on a lion’s body. Building on Berlyne’s work, Hunt (1971) coined the phrase ‘motivation inherent in information processing’ to refer to the uncertainty generated by puzzling and novel experience and the information seeking behaviour which followed. This view, like Berlyne’s, gave a prominent role to the arousal properties of the situation (or stimulus). However, for Hunt the situation was seen as deriving its arousal properties from the cognitive status of the observer. The puzzling and novel assume their puzzling and novel character by reference to the individual who has encountered them. It is the tension of uncertainty generated when an incoming stimulus does not quite fit the known and familiar cognitive categories, which prompts further exploration and information seeking. The additional information in turn brings about changes in the learner’s cognitive categories or structures. Adopting a trait or dispositional approach, Beswick (1971) described curiosity as behaviour which is directed towards seeking, maintaining and resolving conceptual conflicts. In doing so, Beswick and other curiosity trait theorists moved the focus of attention to the individual differences aspect of the curiosity phenomenon and emphasised the role of the learner’s disposition in the arousal of interest. Not everyone confronted with the same novel event seeks further knowledge and information, resolving uncertainty and expanding their cognitive structure. Still following the individual differences path, more recently concepts of ‘flow’ and ‘undivided interest’ (Csikszentmihalyi & Rathunde, 1992) have been used to refer to active involvement between individual and the object of interest. These terms refer to the personal experience which occurs when the learner appears to be totally absorbed in their task. They become “completely involved in something to the point of forgetting time, fatigue, and everything else but the activity itself” (Csikszentmihalyi & Rathunde, 1992, p.59).

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Despite obvious differences in emphasis all of these approaches imply an active information or knowledge seeking character to the relation between person and object. It is the active engagement between person and object, or, person and task, which is central for learning. As indicated by Valsiner interest (or curiosity, or engagement, or intrinsic motivation) does not adhere exclusively in the object or in the person but refers to a relation between person and object. The different perspectives which can be distinguished in the early views on curiosity are reflected in recent theories of interest. One group of researchers have sought the origins of, or explanation for, this person-object relation in relatively stable personal characteristics or dispositions and have studied individual interest (see Krapp, Hidi, & Renninger, 1992; Keller, Schneider, & Henderson, 1994). A second group (see Hidi, 1990; Hidi & Baird, 1986) look to explain the person-object relation that is interest through an examination of the special characteristics of the situation and refer to situational interest. Krapp et al.(1992) have summarised these perspectives adding a third: Three major points of view are reflected in interest research: (1) interest as a characteristic of the person (interest as a personal trait or disposition), (2) interest as a characteristic of the learning environment (interestingness), and (3) interest as a psychological state (active interest, aroused interest). Both individual interest, in the sense of relatively stable preferences, and interestingness can bring about experiences and psychological states in an individual that are generally referred to as interest. Typical characteristics of this state might include increased attention, greater concentration, pleasant feelings of applied effort, and increased willingness to learn . . . (p.9).

It is the third of these the actualised state of interest, sometimes called ‘active’ interest, which is of special significance for the present analysis. The aim is therefore to identify and elaborate some of the psychological processes which are active when a specific student or group of students demonstrate behaviour which suggests that they are interested in a classroom activity, or some aspect of a classroom activity. To adopt this vantage point directs the focus to the experience of the student. When the same classroom activity is presented to a number of students there will be diverse reactions. What are the characteristic behaviours which indicate that a student is interested in the task, or, finds the task interesting? Krapp et al., (1992) suggest increased attention, greater concentration, pleasant feelings of applied effort, and increased willingness to learn as typical features of the state of active interest. Sansone and Morgan (1992) suggest that features such as engaging in an activity freely, with persistence, energy and intensity are the important hallmarks. Data from The Classroom Learning Project will be used to examine the experience of active interest. Because this study is attempting to look at the personal meaning which characterises interest, the basic data will consist of what the students and teacher are saying about what they are doing in the classroom. This is supported by some gestural information from the videotape record. The section of the data set which will be used consists of Lessons 7 and 8, which recorded the interaction of the same group of girls, and this analysis will focus on one of those girls. The two lessons are

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science lessons dealing with an investigation of fibres: one on the Tuesday, the second on the Friday of the same week. The analysis will look at the three forms of data available - the record of interaction, the student’s reflections on her own actions and the teacher comments recorded while viewing the lesson tapes 14 months later. A number of episodes have been isolated for detailed examination. Some deal with question asking and others look at the expression of positive emotion. In each case the behaviour of the student is explored for the personal meaning to uncover what it has to say about the student’s experience of active interest and in so doing to examine the character of the processes which link person and object. 3. “DOES PAPER HAVE FIBRES IN IT?” QUESTIONS AS INDICATORS OF INTEREST One of the tasks set by the teacher in the second of these lessons was for each group of students in the class to make a list of as many things as they could think of that were made out of fibres. In her group Joanne gave the response ‘paper!’ The transcriber marked the tone of this utterance with an exclamation mark indicating that the tone was different to other utterances between group members. Examination of the videotape confirmed the documented emphasis. Observation of the record of their actions showed that the three girls were talking quietly coming up with items which were on task as well as other off task comments. Joanne was looking down at the books and pens on the desk, quickly lifted her head, looked at Narelle and Frances and in a louder voice exclaimed “Paper!”. This was met with disagreement and uncertainty. J:

Hankies.

F:

Yeah, hankies.

J:

Towels.

N: Oh, is it to do in a group? J:

Yeah.

N: Oh, I’ve got blankets, sheets, doonas, towels, tablecloths, furniture, upholstery, curtains. F:

Pencil cases and bags.

J:

Yep, bags.

N: Tents. S10: Mrs B.? Would a bullet-proof vest be clothing? T: Um, put it down. It's on the borderline.

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What else did you have? [looks at Narelle's page] Tablecloth.

F:

Upholstery.

T: No, I don't want you to look in your book. I want things that you use them for. J:

Tablecloths. I have tents. Um. Carpet.

F:

Yeah. carpet.

J:

Um.

F:

Lab coats. [laughs]

J:

Paper!

F:

Do we?

J:

'Cause fibre is in paper, isn't it?

F:

No fibre.

N: [shakes head] I'm not sure.

About four minutes later when the teacher was getting answers from all of the groups Joanne raised her hand and asked “Does paper have fibres in it?” This became the prompt for a suggestion from the teacher about how to examine the fibres in cardboard. N:

Playground equipment.

T:

Right. Playground equipment. What, with fibreglass? [Joanne nods] All right, I guess I'll have to accept that because it does have certain fibres there.

J:

[raises hand] Does paper have fibres in it?

T: Does paper have fibre, folks? Ss: No. No. Yes. Yes. F:

It might if it was given a chance.

T: How many of you have actually made uh, made recycled paper? N: Yeah. T: Right. Who's seen the fibres in it when you mash it up? S: No.

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T: I think paper does have fibres, particularly obvious in home-made paper. We’re going to be looking in a microscope today, and if some of you have got some cardboard, particularly if you’ve got a folder that’s uh, splitting a little bit, like Kevin has, you might take a tiny piece of this cardboard, have a look at it underneath the microscope and see if you can see various fibres in it. And we know that you can get a lot of fibres from wood products as well. OK, today we’re going to do activity two...

On what basis is this interest in learning on the part of Joanne? On what grounds is it reasonable to interpret this question as an indicator of interest, that is, an involvement with the object which is directed to acquiring further information or knowledge about the phenomenon? Did Joanne really want to know whether paper is made of fibres? This example clearly illustrates Valsiner’s point that “once we have been able to ‘diagnose’ its presence, we have already missed the opportunity to study its emergence” (Valsiner, 1992, p.33). The initial “Paper!” was a simple exclamation. In the context of the set of answers being given (textiles and carpet) it clearly represents a divergent line of thinking about things that were made out of fibres. This was followed up by “'Cause fibre is in paper, isn't it?” The first part of this statement came as an elaboration, but it tailed into a question. The suggestion was met with a rebuttal and uncertainty. At this point there was no real evidence that Joanne was interested in the answer to the question posed by her friends’ responses to her ‘paper’ suggestion. However, when Joanne breaks into the flow of the reporting activity four minutes later to ask her question it is clear that she was keen to find out whether paper is in fact made of fibres. The certainty which went with the exclamation had been challenged generating uncertainty. The experience of uncertainty appears to have prompted an information seeking question to an ‘authoritative’ source and the uncertainty was resolved with Joanne now knowing that paper did in fact have fibres in it. This more formal account is similar to how Joanne described the experience when she was shown this section of the videotape. I.

Here ... it’s the bit about where you’re asking about is fibre in paper?

J:

Oh. Uh, huh, I remember that.

I:

Yeah, I remember that too now.

J:

I knew, like I thought it was in that, because um, I just remember, you know, you see little things like you see in material. All those little lines like in that recycled paper.

I:

Uh, huh.

J:

I didn’t know, like I thought fibres were, I don’t know. Yeah, I asked, ‘cause Naomi said they weren’t, and I was thinking they were.

I:

Um hm.

J:

And so, but the only way I could see is ask the teacher.

I:

Um, hm.

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MARY D. AINLEY J: So, I did that. I:

... why did you feel like, it would be it, it would work to ask the teacher?

J: I don’t know, it’d just confirm, um, if I like, I wasn‘t, you know saying, “Oh, who was right?” I: Yeah. J:

It was just, I was wanting io see, ’cause otherwise I’d keep on going, but I’d either think that they weren’t or I wouldn’t know, ‘cause I usually just listen to other people, what they think, and I was thinking, “Oh, it might be in there” and I thought ‘‘ Well, she might know” so I asked her. Yeah.

The fact of asking a question is not of itself a sufficient indicator that the student is interested in expanding their knowledge or understanding. Most classroom questions are teacher initiated and are a check on knowledge rather than seeking to expand knowledge. Joanne refers to two purposes of questions when she explains the reasons for her action of asking the teacher - to resolve her uncertainty (“I was wanting to see” - information seeking), or, to point score against the other members of the group (“I wasn’t, you know saying, “Oh, who was right?”). The difference here is important because if the question was prompted by uncertainty it is within the domain of ‘interest’. However, if the question’s main purpose was to prove that Joanne was right and the others wrong, the motivation would have involved an external goal, wanting to show superiority over the other students, and so would not meet the criterion for classification as ‘interest’. This episode concerning the fibre in paper has a history which supports an interpretation of the question as information seeking, or interest. In the preceding lesson. The teacher has presented the group task as: T: What do I know about fibres? Write that down please. And I’m going to give you five minutes for you to write down all the things that you people know about fibres.

Some minutes later the teacher enquired of the four girls what they each had contributed to the group response (N = Narelle; J = Joanne; L = Leonie; F = Frances): N: Um, they’re matter - like it’s made, makes up materials and fabrics and things. T:

Great, what’s your contribution?

J:

I said they were particles and they were in fabric.

T:

OK. What’s your contribution?

L: I said they were small. T:

Sorry?

L: I said they were small.

INTEREST IN LEARNING AND CLASSROOM INTERACTIONS T: So you said the same thing. You have to come up with something different. L: I didn’t, no, I didn’t. N: No. she was talking about the Omo ads [Her comment was about the reference to microscopic fibres in the laundry detergent advertisements]. J:

That (points to paper).

L: I was not. I said they were little. T: Oh alright. OK that’s fine .... N: [writing] More than one meaning, cereal. J:

No, just food. ‘cause fibre’s in all kinds of things.

L:

Uuh, stop it [inaudible]

F: No. L: Frances. are fibres in everything? F: Huh? L: Frances, are fibres in everything? J:

Yeah, in paper

.... T: Anything else about fibres, ladies? J:

They’re in everything.

T:

In everything?

F:

No, they’re not.

J:

Nearly everything.

L: [to J] where did you get an idea like that? [laughs] ... F:

They’re made into wool.

T: Hang on.

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MARY D. AINLEY F: They’re twisted to make wool. T: We’ve got two things here. J:

I think like. ‘cause there’s fibres in there, and fibres in there, like in that paper stuff, and there’s fibres in there [taps things on table].

L:

I think Narelle’s right.

T: Ah, are there fibres in the plastic? J:

No. ‘Cause it’s like, it’s solid.

T: But you think there’s fibre in the [points to table]. J:

In the paper.

T: OK, do you think there’s fibres in the paper? [Leaves table]

Sandwiched in the discussion of the microscopic fibres in the Omo advertisement, and unnoticed by the other group members was Joanne’s reference to fibre in paper. Joanne came back to it several times, but it was not taken up by any of the other members of the group. A few seconds later it was the content of an exchange with the teacher. However, the teacher’s contribution ended as a query. Joanne did not have her assertion that there are fibres in paper confirmed, but rather has been left feeling uncertain. A state of uncertainty was generated and in terms of personal meaning this is some of the background to the question put to the teacher in the next lesson several days later. The substance of the question was only one small element in two very ‘busy’ classes. However, a state of wanting to know the answer, has persisted over a number of days. Knowledge of the conditions through which this question arose; that is, a state of uncertainty being generated and active questioning about the object, supports the interpretation that it was information seeking and indicative of an interest in knowing about the content of paper. As Valsiner pointed out, the study of interest and its role in learning is constrained by the difficulty of identifying interest. The same question (“Does paper have fibres in it?”) can be directed to different ends. The character of the question needs more than its surface elements for the observer to interpret the personal meaning of the questioner. In this example, the range of data available from three vantage points (previous encounter with topic, the context in which the student’s question was posed, and the personal reflection on what was happening at the time) have provided a stronger interpretive base than any one on its own. The persistence, increased attention, and heightened concentration which characterised the way in which Joanne focussed on finding an answer to her question makes it clear that interest was aroused, that she really did want to resolve her uncertainty. Resolution of the uncertainty was achieved through a change in the student’s understanding. In terms of the substance of the question this is a relatively simple example but it does demonstrate the way in which interest can support learning.

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4. “WHAT DO YOU RECKON’S IN THESE THINGS?” QUESTIONS AS INDICATORS OF BOREDOM Another form of question involves the situation where the person is asking for information about an object or activity but is asking as a form of self stimulation or stimulus seeking, rather than because they are intent on finding the answer to their question. Berlyne (1960) referred to this type of behaviour as diversive curiosity. Using an optimal arousal model of motivation, Berlyne proposed that diversive curiosity arises when the level of stimulation drops below a certain threshold level. Tension rises and the person seeks stimulation to reduce that tension, or, to restore it to a more acceptable level. The individual differences or dispositional aspects of this model have been taken up in sensation seeking and risk taking theories (eg., Zuckerman, 1979). Consider the following example. Joanne was standing waiting for a turn at the microscope (J = Joanne; N = Narelle; F = Frances): N: Is this the fibre? F:

Yes, that red thing. Can I have a go?

N: Yep. Here I just want to put this in. I can’t even see it—oh, there it is, it’s red. F: Yep. N: Why isn't it staying in there? Oh. It’s not there. F:

Yeah, it’s 'cause it’s out of focus. No, you don’t use that one, you have to—this one.

J:

I used that one, it’s heaps easier to see. What do you reckon’s in these things? [looking at wall vents] “Compressed air can cause serious injury.” Is that compressed air in there? Is that air coming out? It stinks.

N: It smells like gas. J:

I can smell gas [sniffing]. How do you tell if gas is on?

N: You can smell it. T: Right, there shouldn’t be people standing around with their hands in their pockets. N: How many have we done? F: One. J:

Let's go make some more.

The arousal of uncertainty here did not develop into any further information seeking, at least within the segment of Joanne’s life captured in these transcripts and

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tapes. As far as we can ascertain from the data set, a series of questions were asked (“What do you reckon’s in these things?”, “Is that compressed air in there?”, “Is that air coming out?”, and “How do you tell if gas is on?”). But there was no real interest in finding out about how the air vent operated or what it was releasing into the room. The questions were posed and then dropped. Within the space of a few minutes Joanne asked about the air vents, tested the effect of dripping water from the pipette onto her pencil (“Oh look Narelle, this is waterproof, look, see it’ll just drip off’), and asked Frances about other apparatus near the microscope bench. A look at the broader context in which these questions occurred shows Joanne had just ‘discovered’ an exciting new perspective on the holes in the Band-aid on her finger when it was put under the microscope (“like you looked in it and there’s all these little fibres in it, like in my Band-aid, in that soft part inside”). Then Narelle had taken a turn at the microscope. In these circumstances questions such as the one about the air vents, questions which arise out of nowhere and then are just as quickly dropped can be considered to be seeking stimulation, or a response to boredom. This is what Berlyne referred to as diversive curiosity. Joanne had just had a turn at the microscope which she found to be highly stimulating (“Yuck, Oh, wow you can see all the ink”). Her questions (“What do you reckon’s in these things?”, “Is that compressed air in there?”, “Is that air coming out?”, and “How do you tell if gas is on?”) filled the gap while she waited for another turn at the microscope. Analysis of the detail of the context in which these questions occurred supports this interpretation. Three students (Joanne, Narelle and Frances) were sharing the one microscope for their science activity. In order to show the context for these questions the same section of the transcript has been quoted but the videotape record has been used to supplement the record of their utterances with an indication of their movements (in bold). N: Is this the fibre? F: Yes, that red thing. Can I have a go? N: Yep. Here I just want to put this in. [Narelle fiddling with adjustments on microscope] I can’t even see it—oh, there it is, it's red. F: Yep. N: Why isn't it staying in there? Oh. It's not there. F: Yeah, it’s 'cause it’s out of focus. No, you don’t use that one, you have to-this one. [Frances moves behind Narelle to other side of microscope where Joanne is watching something across the other side of the room. Frances bumps Joanne and Joanne then moves to the side of the microscope vacated by Frances] J: I used that one. it‘s heaps easier to see. [Joanne has come to the left of the microscope, picks up tweezers] What do you reckons in these things? [Joanne stretches across bench to look into vent just above the bench, looks at wall vents, reads small notice on the wall] ‘Compressed air can cause serious injury’. Is that compressed air in there? Is that air coming out? It stinks.

&

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[Frances at the microscope] N: It smells like gas. J:

I can smell gas [sniffing]. How do you tell if gas is on?

N: You can smell it. [Narelle next to Joanne and looks in vent] T: Right, there shouldn’t be people standing around with their hands in their pockets. N: How many have we done? F: One. J:

Let’s go make some more.

In the time they were working at the microscope Joanne spent more time with the microscope than either of the other two girls. The strong impression from the record of their actions and movements is that when Joanne was not actually looking down the microscope she was looking around for something else to do. The questions about the wall vents and being able to tell whether gas is on, all suggest that Joanne was seeking something to fill the lull between turns at the microscope. They suggest restless activity between the more engaging episodes at the microscope rather than arousal of a state of uncertainty seeking new information. The two forms of question, seeking information and seeking stimulation, described in the preceding sections represent different levels of interest in actually finding out the answer to the question. The ‘fibres in paper’ question was characterised by persistence. The student was seeking knowledge which would resolve the uncertainty generated by the conflicting information from her past experience with the paper question. Her view of the phenomenon had been rejected by her classmates, and received an ambiguous response from the teacher. A state of interest had been aroused and the question was seeking information. On the other hand the ‘air vent’ question was quickly dropped when new direction for the microscope task was given. The questions were a means of maintaining arousal between turns at the microscope. The question form itself is not necessarily a pointer to the state of interest. Persistence, increased attention, and concentration seeking further knowledge or understanding are important characteristics of questions that indicate the state of interest that serves learning. In the examples explored here, the context in which the question was embedded allowed identification of features of the relation between the student and the subject of the question. The context provided useful pointers for distinguishing interest questions from time-filling questions. 5. “OOH, MY GOSH. THAT IS SO COOL” POSITIVE AFFECT AND THE EXPERIENCE OF INTEREST Another important element in the experience of interest is the affective component. The cognitive emphasis of theories of curiosity and intrinsic motivation in the 1970s

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(eg., Beswick, 1971; Hunt, 1971; Day, 1971) commonly accorded positive affect a place in the behaviour which was being described. They saw its role as a ‘byproduct’ of the information seeking which brought about resolution of cognitive conflict and uncertainty. More recent theories dealing with constructs of interest (eg. Krapp, 1994; Krapp & Fink, 1992) treat positive affect as an essential part of the psychological state of interest. The same is true for the related constructs of intrinsic motivation (eg. Deci, 1992; Sansone & Morgan, 1992) and flow (Csikszentmihalyi & Rathunde, 1992). Emotions that have been shown to be associated with a general interest in learning are feelings of surprise, excitement and enjoyment (Ainley, 1998). The Classroom Learning Project offers two sources of data for an analysis of affect as it relates to interest in learning. Facial expression and gesture from the video records, and, emotional expression through language both during the lesson and in the subsequent reconstructive interviews provide important information on student affect. The project was not set up to record all of the facial expressions of the participating students and so the record here is fragmentary. However, the quality of the language in the transcripts from the recording does support a close analysis of qualitative changes in language which signal changing affect. The task that had been set required the students to look at fibres under the microscope and to describe the weave in terms of its porosity. Completion of this task required making up microscope slides of a range of fabrics and fibres, inspecting them, and describing them. The group of girls had not been long at the microscope when their language underwent a dramatic change with a lot of very expressive positive exclamations being heard (“Oo, my gosh. That is so cool”, “It’s so amazing”). J:

[coming back to microscope] I want to have a look at my finger. It’s not on, is it?

F:

Yes, it is. There you go that’s a fibre.

J:

[looking at her finger in microscope] Yuck. Oh, wow you can see all the ink, that’s disgusting.

N: Where do you get the water from? F:

The tap.

J:

My gosh, have a look at the holes in it. Have a look at that, look at the holes in the Band-Aid, have a look. [Narelle looks] You have to adjust it properly. But have a look at the holes in the Band-Aid. Those holes right, those kind of, in there. Isn’t that disgusting?

N: I can’t see it. Stop moving your hand. [pause] Oo! [stops looking] J:

[looking] Isn’t that disgusting? I want to look at my cut.

N: Oh, J don’t be disgusting. J:

Yuck!

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The language has changed dramatically. It conveys surprise, excitement and enjoyment, the three key positive emotions which have been found to be associated with scores on a dispositional measure of interest in learning (see Ainley, 1998). The data set allows some cross checking of the meaning of these exclamations to validate this interpretation. For example: In Class: J:

Oh my god. [nudges Narelle]

N:

Um.

T: . ... J:

I was looking up and I looked at the camera there. [makes a face]

In Interview: J:

When I walked in I just thought “Oh, I saw all the equipment”, and I was just sitting there, I was so secure, and I turned around, and I saw this big thing.

I:

Yeah.

J:

I went “Oh, my gosh”, it‘s like ’cause it was just such a big bulky thing kind of

I:

It is.

J:

Oh, my gosh. [laughs]

For these students, especially Joanne, “Oh, my gosh” and variants of it (“Oh my god”; Oh, gosh”; “my gosh”), are clearly exclamations of surprise and delight. In what sense is this behaviour indicative of interest? It appears that Joanne was surprised and fascinated by the new perspective on both her finger and the Band-aid. The language used indicated that she was arrested by the grotesque and hideous in the familiar, it was after all her own finger. In this situation it appears to have been the character of the object, the interestingness of the object, that took her by surprise and prompted her state of intense involvement. Here the object of interest was what could be seen through the microscope eyepiece (a new perspective on herself, the ‘interestingness’ of her finger) rather than just the novelty of the microscope itself. This interpretation is confirmed in the comments made by Joanne after the lesson. The same positive affect is evident in the language used to reflect upon what was happening. I:

All right, so that’s that bit, now I’m going to jump way back, way over to the part where you’re looking at the microscope.

J:

I love looking through them at your finger, and like, ‘cause such a little cut on your finger looks so disgusting.

I:

[laughs]

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You can see all the lines on it, and it looks like a big bit of dead skin, it’s disgusting.

I:

[laughs] Have you done - is that the first time you’ve done that?

J:

No. We, we’ve done um, microscopes in um, in other classes, like in science this year, I think a bit earlier on and also last year, and we’ve got one at home which I just look in, its not as if it’s a big - “oh, my gosh, it’s a microscope”, we don ,t have a look, it’s just a microscope, cool. Put your finger under.

I:

Uh, huh, uh, huh.

J:

No, I looked a my Band-aid, um, and the holes in it, you know, like you looked in it and there’s all these little fibres in it, like in my Bandaid, in that soft part inside.

I:

Yeah.

J:

Oh, wow, ‘cause there was all lines in it, looked very interesting. [laughs]

In her work on situational interest in text, or interestingness of text, Hidi (Hidi & Baird, 1986) made an important distinction between the arousal of interest through uncertainty generated by the structural components of the text and the interest aroused through associations between elements of the task and aspects of the person’s values, preferences or goals. The first of these is bound up with uncertainty generated by the particular relation pertaining between the information being presented in the text and the prior knowledge of that content which the subject brings to the task. The second is bound up with processes activated through some personal significance of the information. Hidi argued that within this second area there are themes of universal personal significance which arouse interest as well as elements of significance to more limited groups of individual learners. Encountering a new perspective on her finger with its cuts and Band-aid is an example of this second form of situational interest, and with the intensity and energy of the response aroused has the potential for supporting effective learning. Joanne’s interest in her finger was an accidental and surprising discovery. However, after observing the videotape and recognising the intense interest aroused by the band-aid on Joanne’s finger, the teacher was moved to consider the possibility of using a different approach to this unit of work, an approach which would cover the same concepts but would also draw on themes closer to Hidi’s “universal personal significance”. I guess for extension material what I could have done was to go and have say four or five slides made up and if people were working faster maybe even have two unknown slides made up. and then go ahead and say to them ‘okay now can you tell, ’ and come at it from the point of view of being a forensic scientist and that would have been a different approach, and then say okay and we set up a mock situation and see if they can identify the fibre that was taken off this dead ‘victim’ that I’d [inaudible] make up. And that may have been a better way of instigating it. I mean had this topic been done through forensic science for example, it would have been a totally different angle on the whole thing. I think we approach the topic wrongly. We could have done a lot more with the topic had we come at it not as fibres, but we had given it a different name. You know it’s like part of forensic science and mixed it in with a couple of other things.

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But obviously our thinking, or the thinking of the people who put the unit together was ... Okay lets go and get the information on fibres, do a couple of experiments, and do an assessment task and that’s the unit, but not—I don’t think there was a lot of evidence of time and preparation and creativity to capture kids’ imaginations, so an interesting point, forensic science and have some mystery slides and saying okay, what do you think, this came from the outer garment of a person. Were they wearing a woollen jumper or not? So it would have been good to do it like that.

The teacher’s reflections direct this discussion back to the critical issue for educators of the ‘interestingness’ of the situation and how steps might be taken to maximise this potential for learning. Attention to the quality of the affect observed here is crucial. Central to Joanne’s engagement was the emotion of surprise. The bandaid under the microscope was not at all part of the teacher’s script for the lesson. When it did occur the resulting surprise linked student and object (bandaid on the finger under the microscope) in a way that maintained the student’s attention sufficient for her to explore it further. Restructuring of the lesson to introduce mystery (uncertainty) with its attendant emotion of surprise offers a way of developing this science content in a form which is likely to arouse and maintain interest in the service of learning the central scientific concepts. The teacher hinted at some awareness of the part these positive emotions play in learning when she was viewing the videotape of Joanne’s reaction to seeing her finger and the bandaid under the microscope: The girls hadn’t even basically started. ... but it is good that she is at least enjoying her Band-Aid finger. But it just shows that there is so much basic background that we could go through, you know it is almost like we have gone into the technical stuff before she has had time to enjoy it and play with it and sort of go through that stage to then go on to the next stage. It is like we are almost too academic, and we haven’t sort of allowed the child to come through, you know that sort of a child part of the person when you come through and enjoy and feel comfortable and really enjoy the piece of apparatus and what she can see. And I don’t think we have ever. you know, we have never really gone that particular way in science. Maybe we just need to allow structured play time even if it’s only for 3 or 4 minutes, but it doesn’t seem like she has had that opportunity to do that.

The teacher here has used the language of curiosity, excitement, and playful enjoyment to describe the student behaviour in front of her. Just those experiences which educational writers across time have seen as the educational potential of the young child (eg., Rousseau, 1762/1963; Dewey, 1933), a potential which they suggest many appear to lose through their years of formal schooling: We destroy the disinterested (I do not mean uninterested) love of learning in children, which is so strong when they are small, by encouraging and compelling them to work for contemptible rewards ... for the ignoble satisfaction that they are better than someone else ... (Holt, 1969, pp.165-166).

These instances of student behaviour have been closely inspected in order to discern the personal meaning of the events for the participant, Joanne, and to determine what insight they give into her interest in learning science. In terms of the teacher’s lesson plan and objectives for student learning these incidents do not add up to a successful lesson. They do however indicate that the student was engaged by,

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or showed interest in some aspects of the lesson activities. In the course of the two lessons certain aspects of the task did arouse her interest. She experienced uncertainty and pursued it through questioning to find the answer. A novel perspective on her finger generated surprise and uncertainty which sustained a quite extended period of microscope work. Alongside this, however, it must be acknowledged that these episodes were not substantial if measured against the teacher’s curriculum plan. The structure of the lesson is crucial. To the extent it confronts students with scientific phenomena in a way that prompts surprise and excitement, and to the extent it allows means of seeking answers, it is providing opportunities for students’ interest to be aroused in the service of learning. 6. DISPOSITION AND SITUATION The integrated data set provides a record of classroom interaction and reflections by participants on certain sections of that interaction. As Rodrigues, Helme and Clarke (1997) point out the framework of the researchers who collect the data influences their selection within the complex data of a classroom. It is inevitable that this will not provide all that is essential to the complementary perspectives which might to be brought to bear on the data set. For the purposes of an analysis of interest, an important missing element is direct information on student orientations or dispositions toward learning. Such information is usually assessed using some form of self-report questionnaire. The advantage of this type of data is that it allows assessment of the relative standing of any given student against the general norm of students of similar age and background. However, it must be acknowledged that such measures assume people are relatively consistent in their behaviour, that individuals have a general way of responding in similar circumstances. Whether measured as curiosity, intrinsic motivation, or, individual interest the critical assumption is made that the score on the questionnaire represents the likelihood of students responding with that trait. Early curiosity trait theorists such as Beswick (1971) described the dispositional aspect of curiosity as the degree to which the individual seeks, maintains and resolves conceptual conflicts. A more recent trait measure such as the Two Factor Curiosity Scale (see Ainley, 1994), distinguished two separate styles of approach to novelty. A depth-of-interest curiosity trait, the tendency to approach novel and puzzling phenomena in order to understand, is distinguished from a breadth-ofinterest curiosity trait, the tendency to approach novel and thrilling experiences in order to experience what they are like. The first of these two dimensions has been used in a number of contexts as a measure of students’ general interest in learning (see Ainley, 1998; Ainley, Batten, Collins & Withers, 1998). Another self-report measure which has been used widely in Australian research is the Learning Process Questionnaire (Biggs, 1987) which assesses the student’s orientation to learning in terms of deep, achieving and surface approaches. The pattern of scores across deep, achieving and surface approaches has been used to characterise students’ general style of engagement with learning, for example, committed, disaffected, and disengaged styles. These general styles have been shown to be significantly related

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to the types of strategies students adopted on specific tasks. For example, the ‘committed’ students were more likely to employ stategies of trying to understand the meaning of a task, whereas the ‘disengaged’ students were more likely to depend on strategies such as learning work off-by-heart (see Ainley, 1993). Student self-report via standardised questionnaires provides a window on the disposition from one perspective. Within the integrated data of the Classroom Learning Project there is also a window on the student’s disposition toward learning. This is in the form of students’ reflections on their own learning as these occurred spontaneously in the reconstuctive interviews. There are a number of points in these interviews where Joanne used the language of disposition to explain or comment on what she could see herself doing on the videotape. The following excerpt from the reconstructive interview after the first of the two lessons on fibres presents a number of statements which are offered as generalisations about her preferences, or orientation to learning. J:

I usually stay confused, and it, it doesn’t bother me, I just ask Narelle “What are we supposed to be doing?”

I:

Um hm, uh, huh.

J:

Or Leonie, next to me.

I:

Um huh, uh, huh.

J:

And then I get it and I just write it down. I don’t listen to [T.] very much though.

I:

Yeah, I know later on when she's reading out it doesn’t look like you’re listening too much.

J:

Yeah, I’m, I was - like I can hear her, but it seems as though, I don’t know, I can’t concentrate very much on this just being talked at.

I:

Yeah.

J:

I need to be um, either involved or um, like I hate just listening.

I:

Yeah.

J:

To the one person just telling you all the facts, just straight like that. I like it, you know, um. Talking to my friends helps because we like, when we had to do those activities.

I:

Uh, huh.

J:

You find out who is the [inaudible], you hear them talking and then you can, you know, kind of stay around the one topic for just more than a couple of seconds.

In these claims there is a restlessness about her response to classroom learning. Most of her statements were about what she does not like doing: “don’t generally listen”, “can’t concentrate ... just being talked at”, “hate listening ... to one person”.

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Among these negatives were a few positives: “talking to my friends helps” and “when we had to do those activities”. Then in the interview following the second lesson on fibres she presented a different view. There she claimed to really be listening despite appearances to the contrary. When confronted with the observation of her own classroom behaviour on tape Joanne was clearly aware of the disparity between what she was doing and how her actions appear to an observer, especially her teacher. J:

It’s funny ‘cause when I was um, like I was just looking there and it looks like um, and it looks as though I’m not listening at all but I, I was listening to everything that she’d say.

I:

Oh!

J:

And I was thinking, “Gee, that’s why she probably gets annoyed ‘cause she thinks I’m not listening” but I find that I can’t listen when I’m sitting there doing nothing just listening, I have to sit there going like this [imitates position on videotape, sitting over paper, pen in hand looking down at desk], ‘cause I’m not listening.

I:

[laughs]

J:

So I have to.

I:

Well, that’s what ...

J:

I remember I was listening to everything she was saying and I was just looking then.

I:

Um hm.

J:

Yeah. Oh, well, I’ll have to make sure I look as though I’m listening. [laughs]

I:

Oh, well, everyone - you’ll be able to see, but sometimes you’ll notice it doesn’t look like anybody in your group -

J:

Is really listening.

I don’t listen but I do listen: the juxtaposition of reflections from two separate lessons. It would be useful to have been able to cross the boundaries of the two lessons and see what sense she made of these apparent inconsistencies. Other sections of the transcript convey a sense of confusion. Joanne presented herself as someone who received inconsistent feedback about her own learning: at times she felt dumb, she found the homework difficult, was not able always to understand the teacher’s explanations of the information or of what was required in a task. Inconsistency is itself part of the personal disposition towards learning shown by Joanne, The material examined throughout this chapter suggests a student who was easily distracted but who when she encountered something surprising or puzzling was persistent in her attempts to sort it out to her own satisfaction. Distractibility and involvement occur side by side and this highlights an essential characteristic of interest. It is not in the person or in the situation but in the relation between them.

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A second perspective on the student’s disposition toward learning is provided by the teacher through her retrospective comments on the videotapes. From the teacher’s point of view Joanne was generally not engaged with her learning. Analysis of the full transcript of the comments made after both of the science lessons on ‘fibres’ indicated that more comments were made about Joanne by the teacher than about any other of the four girls who were videotaped during these two lessons. Joanne was mentioned by name 4 times in the teacher commentary on the first ‘fibres’ lesson, and 10 times in the commentary on the second ‘fibres lesson’. Across both lessons Frances and Narelle were mentioned each a total of six times. The majority of these comments suggest that the teacher saw Joanne as being uninvolved with the science classes: “off with the fairies”(three times), “off with the birdies”, “doesn’t stay on task”, “always behind the eight ball”, “is never really involved in what is going on”, “basically seems to be off target a fair bit of the time”. These comments have a broader reference than just the immediate incident in which they occurred. It is clear from the commentary that these are dispositional statements representing the teacher’s view of Joanne’s general style of engagement with learning, her typical approach to her schoolwork. T:

Now Kevin, where have you done your eight questions?

S8:

I haven't done them.

T:

You haven't done them. Is there a reason for it?

S8:

[inaudible] [Narelle plays with rolled up sticker, pretending it is a cigarette, then hands it back to Joanne]

T:

Does anyone know where JJ is? [Student answers.] I can't hear. He goes what? [Student: Riding.] Riding. On a Friday. So school doesn't exist for him on those days? Hm. OK, this is a question. Write this down.

T: Besides clothing, besides clothing. Sean, got your sheet ready? Besides clothing, come on, what else. J:

[to Narelle] Whereabouts? How many do you need for the questions? [Joanne is apparently referring to the number of pages that should be allocated for the homework questions she has not done][Narelle shakes head] About two? [Narelle nods]

T: Do we use fibres for? T: [Laughing] See I’m not the least bit surprised that Joanne is like this. I know I can’t see things in minute detail, but the broad brush stroke ... impression that Joanne gave me was that she was just not with it. There was always a lag. So in the group that I

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MARY D. AINLEY chose, you have a wide variety. Joanne who can sometimes do well but is often vague, Narelle who tries hard but tends to go ahead and misunderstand some of the concepts and gets terribly anxious if I don’t dot every ‘i’ and cross every ‘t’. Frances who basically flows along with it, and then there was Leonie who tended to be fairly sharp both in tongue and in intelligence as well, and can be quite negative. So I really did give you quite a nice variety to work with. So from up the front you can actually get a fair idea about how kids are going.

Bringing together both student and teacher perspectives on this student’s approach to learning is informative for the way it highlights their different starting points. The teacher’s general characterisation of Joanne as ‘away with the fairies’ and not concentrating, sums up the degree to which Joanne mastered the basic learning purposes which the teacher had for these lessons. Joanne had given very little attention to the homework requirements for the unit and in class she did not produce a significant number of slides in the microscope activity. However, the episode of extended concentration on her finger and her Band-aid under the microscope show that when her attention was arrested by something novel she needed no prompting to give it considerable time and concentration. Ironically, this incident seen by the teacher as peripheral to the lesson purpose did allow her to inspect ‘fibres’ under the microscope as she describes in the post-lesson interview: I:

[laughs] Have you done - is that the first time you’ve done that?

J:

No. We, we’ve done um, microscopes in um, in other classes, like in science this year, I think a bit earlier on and also last year, and we’ve got one at home which I just look in, it’s not as if it’s a big - “oh, my gosh, it’s a microscope”, we don’t have a look, it’s just a microscope, cool. Put your finger under.

I:

Uh, huh, uh, huh.

J:

No, I looked a my band-aid, um, and the holes in it, you know, like you looked in it and there’s all these little fibres in it, like in my band-aid, in that soft part inside.

I:

Yeah.

J:

Oh, wow, ‘cause there was all lines in it, looked very interesting. [laughs]

The arousal of interest links together person and object. The object might be a complex scientific principle or something as simple as the holes in a Band-aid viewed under the lens of a microscope. The likelihood of interest being aroused depends in part on what the student brings to the lesson, their attitudes and preferences for specific scientific content as well as their likes and dislikes for listening to the teacher, working alone, or working with friends. In Joanne we have seen a student who was easily distracted but who could also experience intense surprise and persistent enquiry when confronted with something novel and puzzling. Both what the student brings to the classroom and the structure of the experience they encounter are essential to the arousal of the sort of interest that prompts seeking further knowledge and understanding.

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And what about evidence of any learning associated with the points at which Joanne has shown some interest in the class activities? The most intense concentration and attention by Joanne throughout the second lesson on fibres occurred when she was looking through the microscope. The objects of interest, the finger and the bandaid, were generally not those which matched the teacher’s lesson objectives. The term ‘porosity’ was one of the critical concepts in the second lesson on fibres. Joanne had been distracted and missed some of the teacher’s explanation. To pick up what she had missed she asked questions about the meaning of ‘porosity’ of her classmates and then the teacher. As with many of the issues in these two lessons Joanne has shown she wants to find out about the meaning of this specific term. It was however, interspersed with distracted episodes. What has she learned? The transcript shows a lot of confused statements. In the reconstructive interview when Joanne tried to explain the meaning of porosity, her confused comments indicated considerable uncertainty. At the same time some effort was being made to pull together and make sense of the fragmentary information she had. I: J:

But you’re asking, you’re asking her again. Do you know why? Yeah, um, from, I just understood, um I didn’t know whether the um, porosity was the holes or the gap around it.

I:

Oh!

J:

Um. ‘cause she drew it like that, and then if had all gaps.

I:

Um, hm. um, hm.

J:

Around if, I didn’t know whether you had to say how many of them or how much space there was or?

I:

Um, hm.

J:

I probably just forgot and asked her again.

To appreciate the extent of this confusion over the meaning of ‘porosity’ it is essential to go back to the actual tape to see what had transpired. The teacher has defined porosity as the “surface area percentage of the holes in the fabric” and drew a diagram on the board - a rectangle with a series of circles inside the rectangle. In her explanation of the diagram she pointed to the circles and said “Now these holes [pointing to the circles] are the holes. And if I asked you for the porosity of the fabric what would you estimate? It looks like these gaps in between are about the same or a little bit more.” (p.7) What the teacher has referred to as the ‘gaps’ between the holes as shown on the blackboard diagram was actually the fabric surrounding the holes. While the teacher explained about porosity Joanne sniffed and appeared to be looking for a tissue or handkerchief. She looked down and noticed a rolled up sticker on her sleeve. She pointed out the sticker to Narelle.

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noticed a rolled up sticker on her sleeve. She pointed out the sticker to Narelle. sitting next to her. She looked up at the board just as the teacher was pointing to and referring to the ‘gaps’. She had not been attending to the whole explanation and had only fragmentary pieces of information to put together to make sense of the term ‘porosity’. Tuning in to the diagram on the board at this late stage made it highly likely that she would misinterpret ‘gaps’. The confusion in her understanding of the meaning of the term ‘porosity’ as she related it to the interviewer would appear to stem from her misinterpretation of what the teacher was referring to as the ‘gaps’. There was however a serious attempt to understand the meaning of this term. I:

Do you think you understand what it is now?

J:

I’m j- I don’t know the, like I’m sure there’s a technical meaning to it, but I just need, a low one, is it’s very close together, there’s no little holes.

I:

Um hm.

J:

And high is where it’s got heaps of holes. So I suppose that it’s, um.

I:

So where did, and where did you find that out?

J:

Um, when she drew it up on the board.

I:

Aha!

J:

Actually it’s a bit confusing having it as um, like. [transcript refers to some equipment difficulties]

J:

It’s confusing having it, low porosity (sic) is um, where its got heaps of holes, I would have thought it wouldn’t have had many ‘cause, low as in less holes.

I:

Yeah. Aha, yeah.

J:

Low as in no holes, like hold on, um, the low um, hold on, I’m thinking about it.

I:

OK, that’s OK.

J:

The low - can I have a look in my book?

I:

Oh, yeah! Absolutely.

J:

It’s written but I just can’t describe it. OK, um [sound ofpaper rustling], I’ve got it there, so the low porosity, um, I would have thought that that was - actually it does make sense ‘cause low poros - porosity or whatever, um, if doesn’t have any holes in it. so it’s very low as in no holes in it.

Despite the inability to master the term porosity Joanne had clearly grasped that an important feature of fabrics relates to differences in the openness of the weave their porosity. This episode provides a good example of how something experienced

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as being ‘not quite right’ generated tension or uncertainty. Joanne has grappled with it until she worked it out. The same pattern of behaviour was seen with the ‘fibres in paper’ question. In this case it was not until she worked it through again in the reconstructive interview that she achieved some understanding of the issue. In both cases the process was the same: uncertainty, exploratory behaviour and the achievement of some new knowledge. Joanne came to a basic understanding of the concept ‘porosity’. Although there is evidence that her knowledge had increased she had not mastered the terminology and this failure to master the language created difficulties when she tried to explain or apply her knowledge. 8. THE PROCESSES OF INTEREST In summary, research on arousal of students’ interest in their learning from a number of perspectives suggests that persistence, attention, concentration and feelings of surprise, excitement and enjoyment are important processes which occur together when the state of interest is aroused. As the analyses presented here have shown the arousal of active interest is a complex interaction between the dispositions and past experiences which a student brings to classroom and the structure of the situation they encounter. Interest adheres in a particular relationship between learner and the object of the learning whereby the learner seeks to extend their exposure to the object to find out more about it. There are ways in which learning tasks can be structured in order to increase the likelihood that students will experience interest. Coming face-to-face with information which challenges the known categories, being surprised by a novel perspective, having the opportunity to investigate the points of uncertainty, to examine the surprising event, allow these interest processes to support change in the learner’s knowledge and understanding. Extending understanding of the types of classroom experiences which will arouse the processes characteristic of active interest in science remains a challenge. REFERENCES Ainley, J., Batten, M., Collins, C., & Withers, G. (1998). Schools and the social development of young Australians. Camberwell, Vic: ACER. Ainley, M.D. (1998). Interest in learning and the disposition of curiosity in secondary students: Investigating process and context. In L. Hoffmann, A. Krapp, K.A. Renninger, & J. Baumert (Eds.), Interest and learning. Proceedings of the Seeon-Conference on interest and gender. IPNSchriftenreihe Kiel: IPN. Ainley, M.D. (1994). Engagement with learning: Adolescent perceptions of self and school. The Australian Educational and Developmental Psychologist, 11, 15-23. Berlyne, D.E. (1960). Conflict, arousal and curiosity. New York: McGraw-Hill. Berlyne, D.E. (1957). Conflict and information-theory variables as determinants of human perceptual curiosity. Journal of Experimental Psychology, 53, 399-404. Beswick, D.G. (1971). Cognitive process theory of individual differences in curiosity. In H.I. Day, D.E. Berlyne, & D.E. Hunt (Eds.). Instrinsic motivation: A new direction in education. Toronto: Holt, Rinehart and Winston of Canada. Csikszentmihalyi, M., & Rathunde, K. (1992). The measurement of flow in everyday life: Toward a theory of emergent motivation. Nebraska Symposium on Motivation, 57-98.

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Day, H.I. (1971). The measurement of specific curiosity. In H.I. Day, D.E. Berlyne, & D.E. Hunt (Eds.). Intrinsic motivation: A new direction in education. Toronto: Holt, Rinehart and Winston of Canada. Deci, E. L. (1992). The relation of interest to the motivation of behavior: A self-determination theory perspective. In A. Renninger, S. Hidi, & A. Krapp (Eds.). The role of interest in learning and development. Hillsdale, NJ: Lawrence Erlbaum. Dewey, J (1933). How we think Boston: Heath. Hidi, S. (1990). Interest and its contribution as a mental resource for learning. Review of Educational Research, 60 549-571. Hidi, S. & Baird, W. (1986). Interestingness -A neglected variable in discourse processing. Cognitive Science, 10, 179-194. Holt, J. (1964). How children fail. Middlesex: Penguin. Hunt, J McV. (1971). Towards a history of intrinsic motivation. In H.I. Day, D.E. Berlyne, & D.E. Hunt (Eds.). Intrinsic motivation: A new direction in education. Toronto: Holt, Rinehart and Winston of Canada. Keller, H., Schneider, K., & Henderson, B. (1994) In H. Keller, K. Schneider, & B. Henderson (Eds.) Curiosity and exploration. Berlin: Springer-Verlag. Krapp, A. (1994). Interest and curiosity. The role of interest in a theory of exploratory action. In H. Keller, K. Schneider, & B. Henderson (Eds.) Curiosity and exploration. Berlin: Springer-Verlag. Krapp, A. & Fink, B. (1992). The development and function of interests during the critical transition from home to preschool. In A. Renninger, S. Hidi, & A. Krapp, (Eds.). The role of interest in learning and development. Hillsdale, NJ: Lawrence Erlbaum. Krapp, A,, Hidi, S., & Renninger, A. (1992). Interest, learning and development. In A. Renninger, S. Hidi, & A. Krapp, (Eds.). The role of interest in learning and development. Hillsdale, NJ: Lawrence Erl baum. Rodrigues, S., Helme, S., & Clarke, D. (1997). What we see and what we know depends on where we stand: Issues in studying classroom learning from multiple perspectives. Paper presented at the 1997 Deakin Research Methodology Conference, Deakin University, Burwood, Victoria, Australia. Rousseau, J.J. (1963). Emile. London: Dent. (Originally published, 1762). Sansone, C., & Morgan, C. (1992). Intrinsic motivation and education: Competence in context. Motivation and Emotion, 16, 249-270. Valsiner, J. (1992). Interest: A metatheoretical perspective. In A. Renninger, S. Hidi, & A. Krapp, (Eds.). The role of interest in learning and development. Hillsdale, NJ: Lawrence Erlbaum. Zuckerman, M. (1979). Sensation seeking: Beyond the optimal level of arousal. Hillsdale, NJ: Lawrence Erlbaum Associates.

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CHAPTER SEVEN Cognitive engagement in the Mathematics Classroom

Interviewer: So can you tell me what made it, what was good about it, what worked for you? Hilary: Uh, I think, um, it was just that we all really put in together, we really put our minds to it, thought about it.

1. INTRODUCTION In this chapter, we undertake an analysis of videotape and interview data related to the four mathematics lessons from the perspective of student engagement. Our focus on the nature, role and significance of engagement stems from a consistent valuing of engagement within the literature on learning that has not been accompanied, in our view, by a satisfactory empirical demonstration of the role played by engagement in the learning process. The term "engagement" usually refers to the extent to which a student is actively involved with the content, or some aspect of the content of a learning activity, where "active involvement" suggests that the person acts to maintain or extend their contact with the object in order to increase their knowledge of it (Ainley, Chapter 6). The "quality" or "level" of this involvement is generally believed to have a profound effect on learning outcomes, in that students who "really put their minds to it" are much more likely to learn successfully than students whose engagement with the subject matter is low. The term "engagement" has been used by researchers to encompass both the motivational and cognitive aspects of the construct, encompassing students' initiation of action, effort and persistence on academic tasks, as well as their ambient emotional states during learning activities (Skinner, Wellborn and Connell, 1990; Skinner and Belmont, 1993). There seems to be general agreement in the literature about this definition. In a recent review of the literature, Stipek (1996) describes actively engaged students as approaching challenging tasks eagerly, exerting intense effort using active (that is, deliberate) problem-solving strategies, and persisting in the face of difficulty. 131 David Clarke (ed.), Perspectives on Practice and Meaning in Mathematics and Science Classrooms, 131—153. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Various studies have found that motivational and cognitive components of student engagement do not operate in isolation from each other, but rather support and complement one another in a synergistic manner. Pintrich (1989) explored the interactive relationships between students' motivation and cognition and found that students who were more mastery and challenge oriented used more cognitive strategies and engaged in more metacognitive activities than students who were less intrinsically oriented. He concluded that motivation and cognition were interdependent and proposed a "multivariate contextual model of student learning" which stressed the importance of the pattern of relationships among the various cognitive, metacognitive, and motivational components of learning, particularly emphasising "the dynamic interplay between motivation and cognition" (Pintrich, 1989, P. 153). The interdependence of cognition and motivation is a key feature of notions of self-regulated learning (Corno and Mandinach, 1983; Zimmerman, 1990). Zimmerman described such learners as "metacognitively, motivationally, and behaviourally active participants in their own learning" who "set goals, organize, self-monitor and self-evaluate at various points during the process of acquisition" (p. 4). According to Zimmerman, self-regulated learners have a greater sense of control over their learning processes and outcomes than their more passive counterparts. The notion of perceived control was explored in more detail by Skinner, Wellborn and Connell (1990). They found that children's perceived control influenced academic outcomes by promoting or undermining the effort children exerted in performing cognitive tasks. A number of studies have attempted to clarify the relationship between the motivational, cognitive and behavioural aspects of engagement. Fullarton (1996) studied students' belief structures over the primary - secondary school transition with the aim of identifying students whose belief structures put them at risk by undermining their engagement with the subject. She identified an underlying construct of control and confidence and found that a substantial proportion of the students showed a decline in both their perceived control and their engagement (as rated by teachers) over the transition from primary to secondary school. The concept of "flow" which has been theorised and researched in many different settings (Csikszentmihalyi & Csikszentmihalyi, 1988, Csikszentmihalyi & Rathunde, 1992) also underscores the dynamic interplay between cognition and motivation. The notion of "flow" refers to an individual's experience of an activity in which they are intensely involved. The flow experience is usually associated with a high degree of control, enjoyment and cognitive engagement with activities which are just within the limits of an individual's "performance envelope", and where there is a dynamic balance between the challenge of the task and the individual's skill level. Ainley's (1993) study examined the interaction between motivation, learning strategies and achievement. Three "styles of engagement with learning" were identified from questionnaire data, which were associated with either a "deep", "achieving" or "surface" approach to learning. When these styles were combined with information on exam preparation strategies and achievement outcomes, six style-of-engagement clusters were obtained. Engagement itself was associated by

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Ainley with "students' general orientation to learning" and with "student construction of the task" to influence the learning strategies they adopt, with the proviso that any relationship between engagement and achievement may also be a function of features of the specific learning domain (Ainley, 1993, P. 404). These associations suggest possible indicators by which engagement might be recognised within classroom videotape and interview data such as that collected in this study. 2. COGNITIVE ENGAGEMENT Our research into engagement has focused on what we have called "cognitive engagement", in part to distinguish active mental involvement from the motivational and emotional aspects of engagement. The term "cognitive engagement" appears to have sufficient currency in the literature however for some authors to see no need to define the construct, either theoretically or operationally. For example, Pintrich and De Groot (1990) make repeated reference to cognitive engagement, in a paper dealing with motivation and self-regulated learning, without definition. Nolen (1995) similarly employs the construct in relation to self-efficacy, but without definition. Researchers generally agree though that cognitive engagement involves the thinking that students do while engaged in academic learning tasks. For example, Meece, Blumenfeld and Hoyle (1988) defined active cognitive engagement by "students' reported use of metacognitive and self-regulation strategies rather than by help-seeking or effort-avoidant strategies" (p. 515). Similarly, Blumenfield, Mergendoller and Puro (1992) defined cognitive engagement as the use of "thinking, metacognitive, and self-regulatory strategies to approach learning thoughtfully" (p.207). Corno and Mandinach (1983) proposed a model of cognitive engagement which distinguishes four forms of cognitive engagement: self-regulated learning (where students' cognitive processing is driven by higher-order or "metacognitive" components); task focus (where students use task-specific planning and selfmonitoring, for tasks where information transformation rather than acquisition is required), resource management (in which students garner help from external sources); and recipience (in which students respond passively with little mental investment, often to instruction which has short circuited their self-regulatory cognitive processes). They note that students need to be able to adapt their style of cognitive engagement to varying task situations, and to avoid dependence on a particular strategy. For example, success with resource management may create a preference for this style of engagement and restrict future use of higher forms of cognitive engagement, such as self-regulation and task focus. A number of studies have explored the relationship between the motivational and cognitive aspects of engagement. Meece, Blumenfeld and Hoyle (1988) investigated the relationship between student goal orientations and cognitive engagement. Student cognitive engagement was assessed using questionnaire items that tapped cognitive strategies and dimensions of self-regulated learning (active engagement) as well as help-seeking and effort-avoiding strategies (superficial engagement).

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Students who placed greater emphasis on task-mastery goals reported more active cognitive engagement than students oriented toward gaining social recognition. Blumenfeld et al (1992) found that high levels of motivation are not necessarily associated with high levels of cognitive engagement. They compared two science classes taught by two different teachers and found that, although both classes reported high levels of motivation, one class reported significantly higher levels of cognitive engagement. Students' cognitive engagement was assessed using a questionnaire which asked them to report their use of cognitive and metacognitive strategies. Although teacher practices were monitored via classroom observation there was unfortunately no such detailed observation and analysis of students' actual classroom behaviour. The findings of these and other studies highlights the importance of motivational factors in explaining variations in students' levels of cognitive engagement in the classroom, and by implication, their learning outcomes. We define cognitive engagement as the deliberate task-specific thinking that a student undertakes while participating in a classroom activity. Our consistent approach in the Classroom Learning Project has been to avoid sole reliance on student self-reports of cognitive behaviour and to seek evidence of constructs such as cognitive engagement within videotapes of classroom behaviour. From this evidence, we hope to contribute to a viable model of classroom learning, in which student cognitive engagement is usefully situated. 3. COGNITIVE ENGAGEMENT AND LEARNING Students need to have both the "will" and the "skill" to be successful in classrooms. It is the experience of teachers that students who are motivated to learn and who think carefully about what they are learning develop deeper understanding of the material being covered, and there is a growing body of research indicating the important role that self-regulated learning strategies (as well as key motivational factors) play in students' academic achievement (Pintrich, 1989; Pintrich and De Groot, 1990; Zimmerman, 1990). Swing, Stoiber & Peterson (1988) undertook a large and complex study which, among other things, explored the effects of increasing the quality of cognitive engagement in classroom tasks. The study compared two classroom-based interventions on students' mathematics achievement: increasing students' academic learning time ("engaged time") versus instruction in using specific cognitive strategies. The study did not directly study students' actual skill use in the classroom, but used video-stimulated recall to facilitate students' reporting of these strategies, and presented a problem to students in interview which was solved using a "think aloud" protocol. As would be expected, students in the thinking skills group reported significantly more use of specific cognitive strategies than students in the learning time group. Comparison of individuals' verbal protocols during the "think aloud" problem solving task supported the conclusion that a more thoughtful approach to problem solving was taken by students in the thinking skills group. Shortcomings in defining and operationalising student engagement limit the usefulness of the results of many studies in this area. Skinner et al (1990), for

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example, found that scores on an engagement scale completed by teachers correlated significantly with grades and measures of student achievement. However information on the actual behaviours associated with engagement was not collected. Teacher reports of student engagement, although an accurate measure of active involvement, did not provide a detailed enough description of what students actually did while engaged with classroom activities. Studies employing the process-product approach, such as the work of Good and Brophy (1984) in the USA and, in Australia, Bourke (1984), sought to identify specific behaviours (such as frequency of student-initiated questions) associated with desired outcomes (such as test performance or attitude to the subject). The inability of such studies to distinguish association from influence, cause and consequence limited their explanatory power, while nonetheless drawing attention to those practices that most characterised certain classroom types and offering results that promoted productive speculation about causal relationships. The persistent tension in the study of classroom practices and outcomes has been between the drive to generalise and the need to understand process. The complexity of the behaviours being studied challenges any generalisable categorisation scheme, and conclusions as to causation are undermined by the multiplicity of possible interactive effects. For example, Webb (1989) reviewed nineteen published studies linking peer interactions and achievement in small groups learning mathematics and computer science, and highlighted the difficulties of forming meaningful conclusions on the basis of correlational studies. Some research on small group learning in mathematics has been fine-grained enough to be able to specify the forms of verbal interaction (indicative of cognitive engagement) which appear to promote learning. For instance, Gooding and Stacey (1991) did a detailed analysis of knowledge development in small groups learning division. They found that groups that were more effective (showed a net gain on a post-test) were more likely than ineffective groups to show evidence within their interactions of particular forms of cognitive and metacognitive activity. These included proposing ideas; responding to questions; giving explanations with evidence; and refocussing the discussion. Increasingly, the complexity of the practices and the outcomes being studied is leading to interpretive studies in which support for influence and causation is sought directly in the data collection process (from informants), rather than being concluded by inference on statistical grounds. In such studies, convergence of informant accounts and the theoretical coherence of any documented relationships are taken as more compelling authorities for causation. This is the approach employed in the research reported in this chapter. 4. INFLUENCES ON COGNITIVE ENGAGEMENT Cognitive engagement is located within a complex and cyclic interplay of influences that include the classroom situation, the task, and the individual. As Ainley (1993) points out, "Characteristics that the individual learner brings to the learning context shape and combine with the learner's construction of the task and its demands, to

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influence the learning strategies adopted and the learning outcomes" (Ainley, 1993, p. 395). With this proviso in mind, we discuss three factors that impact on cognitive engagement. 4.1 The individual Individual students bring to the learning situation numerous characteristics that influence their cognitive engagement. These include skills, knowledge, dispositions, aspirations, expectations, perceptions, needs, values and goals (Corno and Mandinach, 1983; Meece, Blumenfield & Hoyle, 1988; Ainley, 1993). Students' reported cognitive and emotional engagement has been found to be influenced by their perceived control over academic outcomes (Fullarton, 1994). As discussed earlier, self-regulated learners actively seek out opportunities to learn and systematically use metacognitive, motivational and behavioural strategies to achieve desired learning outcomes. 4.2 The learning environment Within any classroom, a 'classroom culture' emerges creating conditions that either constrain or promote particular teaching and learning strategies and particular styles of interaction with the teacher and other students (Edwards & Mercer, 1987). Certain teaching practices have been shown to enhance thoughtfulness and selfregulatory behaviour, while others tend to restrict cognitive engagement (Corno and Mandinach, 1983; Blumenfield, Puro & Mergendoller, 1992). Current analyses of data collected in the Classroom Learning Project suggest that teaching style and the nature of peer interactions have a powerful influence on student cognition and metacognition (see the chapters by Lerman, Holton and Thomas, and Baird, in this book). 4.3 Tasks Classroom learning tasks and activities provide the vehicle for a student's cognitive engagement. Recent research into task characteristics such as complexity (Williams & Clarke, 1997), challenge (Csikszentmihalyi & Csikszentmihalyi, 1988), familiarity (Helme, 1994), intrinsic interest (Ainley, Chapter 6), and personal meaningfulness (cf. Clarke, 1996) suggests a possible relationship between such characteristics and forms of cognitive engagement. What is 'at stake' for the student (e.g. class assessment, peer status, or personal satisfaction) is a further possible influence on cognitive engagement. In our analysis we have endeavoured to locate and explain any evidence suggestive of interdependence between these factors and different forms of cognitive engagement.

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Many early studies of student engagement appear to have operationalised the construct in terms of time-on-task (for example, Peterson & Fennema, 1985, and Hart, 1989), although the usefulness of this measure has been challenged over a significant period of time (for example, Peterson & Swing, 1982). As we have argued above, cognitive engagement is qualitatively different from time-on-task or student participation. The fine-grained nature of our data collection has provided us with examples of occasions when a high level of student percentage time-on-task was not accompanied by a high level of student cognitive engagement, as we would define the construct. Very few studies have used direct observation of student classroom behaviour to assess levels of cognitive engagement, and have instead relied on indirect measures such as student questionnaires or teacher reports. Meece, Blumenfeld and Hoyle (1988), for instance, used a questionnaire to assess the level of students' cognitive engagement, which included items which tapped cognitive and metacognitive strategies and dimensions of self-regulated learning such as "I asked myself some questions as I went along to make sure the work made sense to me" and "I went back over things I didn't understand". Self-reports of cognitive processes are, however, subject to bias (Corno and Mandinach, 1983) and point to the need for more direct measures. Wilson (1999) has explored ways in which self-reporting can be made more reliable, and has developed a technique by which students' self-reports of cognitive and metacognitive activity can be combined with other data sources such as observation and video recording. Other studies have made use of direct observation of student behaviour, such as Gooding and Stacey, 1991; 1993) and Leder & Forgasz (1992). Leder and Forgasz attempted to describe simultaneously students' cognitive and affective engagement in mathematical tasks, and identified three forms of cognitive engagement, providing operational definitions for each. These were: independent thinking; task/work autonomy; and persistence. Cognitive behaviours were categorised as high or low level. The purpose of the present analysis was to identify instances of cognitive engagement using classroom videotape data as a primary source. Thus we have foccussed primarily on linguistic indicators of cognitive and metacognitve activity as our primary data source, such as verbalisation of thinking, questions, explanations and other forms of communication. The reflective comments students made in interview provided a second source of data. As the previous discussion has indicated, the interdependence of cognitive and motivational aspects of engagement requires that we take account of a range of behaviours in our analysis of the video record. We have therefore noted motivational correlates of cognitive engagement such as active participation, persistence in completing tasks, and resistance to interruptions. Non-verbal correlates of cognitive engagement such as gestures, eye contact and body orientation have also been taken into account, as well as indicators of emotional involvement such as expressions of enthusiasm, enjoyment, and satisfaction. Data from the present study has cautioned us about making inferences on the basis of one source of information only. A

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student in a science class who observed herself in video-stimulated recall commented that her own behaviours could be easily misinterpreted by the teacher as lack of attention, when in fact she was listening to what the teacher was saying. This chapter is primarily the report of an investigation of the cognitive dimension of engagement, but we have attempted not to lose sight of the bigger engagement picture which, as well as cognition, includes behaviour, motivation and affect. By developing a comprehensive and rich understanding of cognitive engagement and its relationship to classroom practices we hope to be able to deploy our resources in a more informed fashion to enhance cognitive engagement in the classroom, and possibly the quality of student learning. 6. RESULTS AND DISCUSSION We identified a number of student behaviours in the videotaped lessons which we believe can be associated with cognitive engagement. These are listed in Table 7.1. The observed behaviours occurred in clusters which could be associated with distinct classroom situations. It is worth noting that the five situations in Table 7.1 can be divided into those situations in which the teacher is present and those in which the teacher is absent. The clustering of these behaviours and their consistent association with qualitatively different social situations provided retrospective support for their validity as indicators of the single phenomenon of cognitive engagement.

Table 7.1. Indicators of cognitive engagement Classroom situation Individuals working in parallel

Behaviour Verbalising thinking; self-monitoring; concentration (resisting distractions and/or interruptions); gestures (interpreted as externalising thought processes); seeking information and feedback. Collaborative small group Questioning; completing peer utterances; exchanging ideas, directions, explanations, or activity information; justifying an argument; particular gestures. Small group interactions with Answering teacher’s questions; giving teacher information; explaining procedures and reasoning; questions addressed to teacher; reflective self-questioning. Whole class interactions with Asking and answering questions; making teacher evaluative comments; contributing ideas; completing teacher utterances

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Interactions between teacher and There were no episodes of sustained interaction individual student in the data set currently available which could be subjected to analysis From the interview records we were able to identify a number of additional possible indicators of cognitive engagement. These included student claims to have made a genuine attempt to learn something, to resolve uncertainty or to have learned something in the lesson; student discussion, communication or recall of details of lesson content; and the claim to have been engaged during the lesson (e.g. ''we really put or minds to it”). An example from four of the classroom situations listed above will be discussed in detail below. Numbers in brackets refer to line numbers in transcript excerpts. Unidentified students are referred to as S1, S2 and so on. 6.1 Individuals working in parallel Karen and Lauren were working alongside each other on a task which required them to calculate how many blood cells they have lost in their lifetime up until today. They worked mostly in parallel, probably because, being slightly different ages, they were working with different numbers (T= Teacher; K = Karen; L = Lauren): 1.

T: I want you to tell me how many blood cells to the day you have lost.

2.

K: [to Lauren] Does that mean how many up to now? [Overlapping talk between the Teacher and Lauren for next three lines]

3.

T:

So you’ve got to multiply by the years, by the months-

4.

L: And the months, and the days.

5.

T: By the days. OK?

6.

L: Well!

7.

T: That’s when you’ve finished that.

8.

K: Seven hundred and thirty million a day, no, per year, times fourteen years. Shit.

9.

S 1: In your lifetime?

10.

T: Yes, how many blood cells have you lost in a lifetime. If it's two million a day.

11,

S2: I’m not dead yet. We’ve got to do so many months, so many days. Oh oh.

12.

T: I’ll go round and check the rest [Teacher goes around room and checks work].

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13.

L:

14.

K: [working aloud] You times that by fourteen, equals one point oh two two to the power of ten. Oh yeah, I understand that. One oh two two. One, two, three, four, five, six, seven, eight, nine, ten. Would that be right? [looks towards Lauren] How many days in a month approximately?

15.

L: [working aloud] Times nineteen. [looks towards Karen]

16.

K: [to Lauren] Would you do approximately thirty days for the month? It would be thirty point five days. No it’d be twenty-nine point nine days or something.

17.

L: Hang on. [goes back to own work] Times, times [inaudible].

18.

K: Would it be this? Do you reckon, Lauren? Um, Lauren, can I borrow your calculator? Lauren.

19.

L: Yeah [inaudible, working aloud].

20.

K: [talking as she works, L does not appear to be listening] That equals one point oh, oh, oh, no, one point oh two two to the power of ten, make it—

21.

L: [to herself] That’s wrong.

22.

K: [to herself]—ten billion, two hundred and twenty million. [Looks toward L] Ten billion, two hundred and twenty million. Is that right? [no response, L is bent over work] Don’t worry.

23.

L: I hate you Mrs Brown! [possible emotive response related to frustration of challenging task]

[counts on fingers, talks to self] May, June, July, August, September, October. That’s six. And how many days have we had in October? [looks at diary] The nineteenth.

Indicators of cognitive engagement Lauren thinks aloud as she works (13, 15,17, 19), and her intense concentration is evidenced by her resistance to Karen’s interruptions (17, 19). She actively uses available resources to help her resolve uncertainty, such as repeating the teacher’s instructions in order to clarify the task requirements (4), and referring to her diary to work out the number of days (13). She makes use of gesture to externalise her thought processes (13), and shows evidence of self-monitoring (21). The emotive dimension of her engagement is apparent in line 23, in which she expresses a certain amount of frustration about the task. Karen’s engagement style takes a different form to that of Lauren. She showed less evidence of self-monitoring and relied more on external assistance. Apart from one instance of self-monitoring (8), she asked Lauren for assistance with basic information (14), to clarify the task requirements (2), and to give her feedback about her ideas (16), and her progress with the task (18, 22). Her cognitive style in this situation was more aligned with Corno and

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Mandinach's (1983) conception of "resource management" whereas Lauren's style was more task focussed, with greater evidence of self-regulated learning. Lauren’s interview record contains further insights into the quality of her engagement with mathematics. Reflecting on the novelty of "real-life" situations in mathematics, she claimed that they made things clear to her, and more real: I:

Was there anything new in that lesson for you?

L: Not really, it was just um, the only new thing was using it in sort of real-life situations. I:

Um hm.

L: Where, because, usually the numbers don’t make any sense 'cause you're not putting them with anything that you’d think of, but when you’re using stuff like rice and paper and dollars and so on it makes it clear. I:

Yeah, do you feel that you learned anything new from that?

L: Um. Yeah, it just makes it seem more real, if you use it with something you know about. OK?

This excerpt reinforces our contention that task characteristics afford and constrain student participation and consequent engagement. Although the data available to us cannot confirm that cognitive engagement facilitates learning in this situation, both the video record and the teacher’s interview suggest that Lauren is a successful learner: She seems to be fairly competent at her work and she’s like a little mini teacher in the room.

The teacher considered the task to be a challenging extension to a task that the students were working on (It’s quite a tricky task) and her opinion of Lauren’s competence is confirmed by the fact that she addressed Lauren personally when she initially set it (Although she immediately asked the rest of the class to do it too). 6.2 Peer interactions Frances (F) and Patricia (P) have a set of graphs (Figure 7.1) from which they are to select the one that best fits the v-t graph of a ball thrown into the air. It is an extremely challenging task, mainly because of the risk of confusing the trajectory of the ball with the graph of its velocity (which in their final selection the girls appear to have done). It is important to note that none of the graphs provided correctly depicted the situation described. The students had also been asked to do a sketch of the correct graph.

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Figure 7.1 Velocity-time graphs presented to students

COGNITIVE ENGAGEMENT IN THE MATHEMATICS CLASSROOM 1.

P:

[To F] Ready, what do we do with the ball thrown into the air?

2.

F:

This [points to material on table].

3.

P:

Which one?

4.

F:

I reckon [selects an alternative by tapping her page with pencil].

5.

P: Uh uh, I don't reckon. It would go up—it wouldn't go up fast, and it would come down real fast, so.

6.

F:

But it doesn't come down real fast.

7.

P:

It does [nods head].

8.

F:

[shakes head]

9.

P:

It comes down faster than it goes up.

10.

F: No. What happens, is it goes really fast [with pencil traces what appears to be trajectory of ball thrown into air] and then it slows down once it gets to the top, and comes — it comes up and then slows down [repeats first gesture], and stops when it turns around. And then it comes up again [pencil moving up] it goes down quite fast and then it slows down when it gets to the bottom [sketching second half of parabola] because it has to—

11.

P:

[looking at book] Yep, this one. Because it doesn't, but it doesn't, that's like it goes up and then [speaking slowly] kind of moves real [speaking slowly, gesture with hand], you can see how it. But that one just kind of goes like that.

12.

F:

But it doesn't go Oo-op [moves pencil upwards from desk in straight diagonal line] slow down suddenly, it sort of gradually slows. [another student briefly interrupts about another matter] I reckon—

13.

P: D.

14.

F: D. 'Cause it doesn't really just go uomp [moves ruler up from desk in steep curve with a jerk] and then slow down straightaway [moves ruler slowly in arc] and then—

15.

P: All right, do D. [makes side comment about pen]

16.

F:

Then we have to draw it.

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Indicators of cognitive engagement Patricia and Frances employ various resources to help them with the task. Both students use gestures to create a ‘slow motion’ version of the path of the ball and to help them work out how the velocity changes during its flight (10, 11, 12, 14). Gesture here seems also to enable them to create a 'shared representation' (Clarke, 1996) which they can both 'see' and modify until they reach agreement. (For a comprehensive discussion of the role of gesture in collaborative problem solving, and a detailed analysis of the interaction cited above, see Chapter 10 by Reeves and Reynolds). Patricia uses her voice to enhance this process. As she describes the changes in the ball's velocity, her speech slows down and speeds up accordingly (11). The episode is primarily characterised by a ''negotiative event" (as discussed in Chapter 3), in which the students initially disagree and use argument and explanation to eventually reach consensus (4 - 15). There is evidence of monitoring of task requirements in line 16, in which Frances reminds Patricia that they have to draw the graph. The data available to us does not enable us to demonstrate a connection between cognitive engagement and learning outcomes. We would argue though that the challenging nature of the task pushed these students to make full and creative use of their cognitive resources to help them make progress on what was for them a very very difficult problem. 6.3 Small group interactions with teacher Karen and Lauren worked on a problem in which they had to calculate how many sheets of graph paper they would need to show one million millimetre squares. The following interaction occurred when the teacher approached the group, having previously accepted a different answer from another group, and was trying to decide which of the two was correct. The change in social dynamics-a shift away from enquiry and problem solving to explanation and justification of their solution to a "higher authority"—was accompanied by a very different pattern of interaction, with the overwhelming majority of student utterances (twelve out of fourteen) being responses to teacher questions. 1.

T:

Uh, question 1. Explain your working.

2.

K: Question 1. We did length times width.

3.

T:

Of?

4.

K:

Of the sheet of paper.

5.

T:

And what was your length?

6.

K: Length was two hundred and fif—two hundred and fifty what?

7.

T:

Yeah, two hundred and fifty whats?

COGNITIVE ENGAGEMENT IN THE MATHEMATICS CLASSROOM 8.

L:

[quietly] millimetres.

9.

K: Um. Yeah, millimetres.

10.

T:

11,

K: Yeah, yeah. Minus the outlay, border thing [makes page-shaped gesture in air], that didn't have graphs on it, the graph on it, times a hundred and eighty millimetres —

12.

T:

13.

K: Width. Equals forty—

She hasn't had the pure joy of me with units yet, has she? Hm Hm. Times?

Where'd you get a hundred and eighty from?

14.

T: Why did you multiply them together? Why not add?

15.

K: To get the area. I know that much.

16.

T:

17.

K: Equals forty-five thousand, therefore you'd need—oh how'd I get that?

18.

T:

19.

K: Forty-five thousand. That's what we got.

20.

T:

21,

K: Yep, they're wrong, we're right.

22.

[to Lauren] You’ve been tutoring her?

Forty-five thousand?

Forty-five thousand? Can you press that- can you press that- can you do that again? Two hundred and fifty times a hundred and eighty? Oh, hang on, hang on, I think you're right. Hang on. I think they're wrong.

[Lauren holds up calculator.]

23.

T:

What is it? Oh dear. And what's the answer?

24.

L:

25.

T: Yes.

26.

L: -and then you divide, you divide a million by that, and it gives you twentytwo point two, so therefore you need twenty-two and a fifth sheets.

27.

T: Excuse me. [goes to group at front of room.]

28.

L:

Forty-five thousand millimetres on one piece-

Aha! We’re right.

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146

Indicators of cognitive engagement As stated above, the overwhelming majority of student utterances in this episode (twelve out of fourteen) were made in response to questions and requests initiated by the teacher. These included responses to requests for information (2, 4, 6, 8, 13, 19), and giving an explanation which was either procedural (2) or some combination of procedure and reasoning (11, 17, 24, 26). Thus the social dynamics of this episode were very different to those in the instances of peer interaction discussed previously, where questions and responses were more equitably balanced. But some aspects of self-regulation were present, such as reflective self-questioning (6, 17). Karen used gesture to enhance her explanation of how she worked out the area of the page (11). Line 17 captured three exemplars of cognitive engagement: explanation of procedure, verbalising reasoning, and reflective self-questioning. The emotive dimension of engagement is evident in line 28, in which Lauren clearly expresses pleasure and satisfaction in being right. This episode illustrates some important differences between peer interactions and student-teacher interactions in this classroom. With the teacher asking virtually all the questions, there was little opportunity for students to initiate ideas or spontaneously express and resolve uncertainty. However, the teacher's use of openended questions may have provided useful cognitive and metacognitive scaffolding (see Holton and Thomas, Chapter 5), in contrast with her usual style of straightforward closed questioning which tended not to provoke much thought on the part of students and, according to Corno and Mandenach (1983), could contribute to a pattern of "recipience" by "short circuiting" higher level cognitive engagement. 6.4 Whole class interactions with teacher Mark and Darren participated in a teacher-led class discussion about the v-t graph of a bouncing ball which is initially thrown into the air. The teacher drew a set of axes and a diagonal line from a point on the y-axis representing the initial velocity to a point on the x-axis. This was a very challenging task (for teacher as well as students), and both the teacher and students appeared to be confused about whether they were discussing speed, velocity, displacement, or the actual motion of the ball itself. 1.

T: There are two different ways it can go. It can go here to zero so I've, I've started it off. Right? It's slowing, slowing, slowing, slowing [gestures upward movement which stops] and then at z—something is zero. Well I've got the zero and then it's going to change direction, and get faster and faster and faster and faster. Now basically because it changes direction, it-you could argue that it [referring to graph] will go down [gestures below x-axis]. And then, and then, somehow or other-

2.

M: [whispers] It can't go down.

3.

T: —It has to hit the ground. When it hits the ground what happens then?

4.

M: Can't.

COGNITIVE ENGAGEMENT IN THE MATHEMATICS CLASSROOM 5.

Ss: It bounces.

6.

T: OK, it bounces so it's going to start going ...

7.

D: Up again.

8.

T: Back up again. So, is the graph like that [hand gesture up from point on xaxis].

9.

S28: It is.

10.

T: Or is the graph ...

11.

D: Yeah, it's because the speed [points].

12.

M: Yes.

13.

S10: It's the velocity, not the way it bounces.

14.

T:

15.

D: It's the speed.

147

... like that? [hand gesture going up from point below x-axis]

Indicators of cognitive engagement Because of the manner in which the teacher was controlling the flow of the discourse, there was little opportunity for students to ask questions and express and develop their ideas. Within these constraints, however, we were able to identify several indicators of cognitive engagement, and thus of the strategies students were using to make sense of the teacher's argument. These included: responding aloud to teacher's questions (12), completion of teacher utterances (7), and spontaneous reflective comments (2, 4, 13, 15). Both Mark and Darren discussed this episode at length in their interviews, lending further support to our contention that both students were both significantly cognitively engaged with the problem in the lesson. It could also be argued that their engagement with the problem had extended beyond the immediate classroom situation in that they continued to express uncertainty and elaborate and justify their ideas in the interview itself (Interview with Darren) I:

Do you remember, remember the stuff about the graph, about Mrs B—drew the thing about the bouncing ball, and she said did it go like this or did it go below the axis like this?

D: Oh,yeah. I:

Um hm.

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SUE HELME AND DAVID CLARKE D: I, I think it’s the top one. I:

OK.

D: Because you can't go under the graph because that means that it would just be floating or.. . I:

How do you know it would be floating? Can you tell me that quickly?

D: It-hang on.I've got to think about this one. Um, the ball goes, so she’s throwing the ball up in this case, right? I:

Yeah.

D: With this graph it would have to be thrown, no dropping the ball. yeah, dropping the ball, so it goes up, hits the um, ground so, and then it goes back down again. I:

Yeah.

D: And then it’s up in the air and it stops, then it goes back up again. I:

Yeah.

D: So it wouldn’t go straight through. I:

OK.

D: Because that would mean it would be through the ground virtually.

(Interview with Mark) I:

OK, why can’t it go below the graph?

M: Because it’s speed and you can’t get below zero speed. I:

Right, but actually.

M: That’s what I thought. 1:

Yeah, yeah, so what do you think now?

M: Um. Well, it’s the bit I didn’t understand, I didn’t understand how it could go below. I:

Um hm. OK, so you still don’t know, you still don’t know on that one?

M: Yeah.

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149

OK. [plays tape: teacher says ''slowing, slowing, slowing''] Does that make any sense to you?

M: No. I:

OK. [plays tape] So which one are you saying no about?

M: The bottom one where it goes below the line. I

Yeah, that’s what I thought, that’s kind of what I thought.

M: ‘Cause she’s sort of saying that balls, the way the ball bounces. I:

Yeah.

M: Goes up and down. I:

Yeah.

M: It not the way it goes, it’s the speed going up and then it slows down, it’s going back down.

Understandably, when the students were given the opportunity in the interview to reflect on the lesson and elaborate their ideas, they revealed far more information about their uncertainties and difficulties than was evident in the excerpt above. The episode convincingly illustrates the constraints that whole-class instruction can place on the form, level and expression of students' cognitive engagement. Our data suggests that student-student interactions offered greater potential for high-level cognitive engagement for these students than did interactions with their teacher. 7. THE CASE OF HILARY: “WE REALLY PUT OUR MINDS TO IT” Hilary made this statement in her interview after a lesson in which students worked in groups of four to name some rates that affect their life. The lesson was part of the lesson sequence that included Lesson 6. Members of the group were Darren, Mark, Hilary and Briony. This was the only occasion in over 100 interviews relating to 55 lessons in which any student made a spontaneous positive claim about their cognitive engagement. It is the uniqueness of this event that justifies our inclusion in this chapter of data outside the eight lessons that provide the focus of the rest of this book. Our immediate interest was in the nature of the classroom episode that prompted this apparent heightened cognitive engagement. We believe that the distinctive features of the episode for the student were its novelty and the emergent connection with her personal experience. Below are two short excerpts from the group discussion: 1.

D: What are we writing?

2.

B: Name some rates that affect your life.

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150 3.

H: Name some?

4.

B: Rates that affect your life.

5.

H: Rates?

6.

M: Housing rates, um.

7.

H: That affects me personally?

8.

B: Affects your life.

. . . . . 9.

H: Bank rates. [Darren writes, the Teacher approaches then moves off] Child ‘cause my dad has to pay it. maintenance rates. Affects me, '

10.

B: What?

11.

H: Child maintenance. [to Darren] Child maintenance rates. [Darren writes] Stop laughing Mark, it’s true.

12.

B: School rates.

13.

H: [nods] School.

14.

B:

15.

H: Write school fees.[Darren writes]

16.

D: Pay TV rates.

17.

H: Does anyone here have pay TV?

18.

B: Doesn’t affect me.

19.

H: Doesn’t affect me.

20.

M: Can’t get it yet.

[to Darren] Up there.[Darren writes]

Hilary participated actively in the group discussion, and her contribution of child maintenance rates was clearly about something quite personal and important in her life. She appeared to value the opportunity to work in a group: H: I think it was good the way we went into groups and kind of thought about it, ‘cause it was easier to do than just to give Mrs Brown feedback, just from sitting in our seats by ourselves. I:

Yeah. So which bit about the groups part was good?

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H: I thought it was good that we didn’t get into friendship groups ... 'cause if we get into friendship groups it’s just like you talk all the time and you don’t actually do your work, but. So we were with people, Briony and I were with Darren and Mark, and we kind of really put in together, and that worked really well. I:

So can you tell me what made it, what was good about it, what worked for you?

H: Uh, I think, um, it was just that we all really put in together, we really put our minds to it, thought about it.

This last statement of Hilary's reflects the "dynamic interplay" between cognition and motivation discussed earlier in the chapter, and supports Csikszentmihalyi and Rathunde's (1992) contention that cognitive engagement can be intrinsically satisfying and motivating. Our conjecture is that it was the novelty of the task and its connectedness with Hilary's personal experience that promoted the high level of cognitive engagement. The following interview excerpt appears to support this interpretation. H: 'Cause we hadn’t done anything like that before, in maths. I:

Uh huh.

H: In maths, yeah. I:

Uh huh, so what at home do you know, what was that, what were you thinking about then?

H: Well, water rates, 'cause I remember my brother did an assignment on that, um, child maintenance rates because um, my father has to pay them for my brother and I, um. I:

Right.

H: All, all the rates that we pay, bank rates, home loans, we just bought a new home and stuff like that. So yeah.

8. CONCLUSIONS Cognitive engagement is valued by the education community and widely held to facilitate learning. Most studies that have investigated this link tend to rely on imprecise and indirect measures of cognitive engagement such as time on task, or student self-reports. The present study attempted to contribute to our understanding of this construct by locating empirical evidence for its occurrence within the classroom, and exploring the relationship between forms of cognitive engagement and different classroom situations and activities. On the basis of the data we have examined, we are prepared to assert that cognitive engagement, as we have defined it, is observable in classroom situations, and can be consistently recognised by specific linguistic and behavioural indicators. We examined cognitive engagement in four distinct classroom situations—parallel

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and collaborative classroom activity, whole class instruction, and student-teacher interaction in a small group—and observed that different patterns of cognitive engagement appeared to characterise each type of activity. We would argue that student-student interactions appeared to offer more scope for high-level cognitive engagement than teacher-student interactions, both in whole-class instruction and in interactions with small groups. The social rules governing these different activities appeared to play a pivotal role in the form and expression of cognitive engagement. We have also found evidence to support the view that task characteristics influence cognitive engagement, as do individual factors, such as personal meaning. Emotional and motivational aspects of engagement are also evident in our data. The data we have examined thus support the view that cognitive engagement is located within a complex interplay of particular aspects of the classroom situation, of the task, and of the individual. REFERENCES Ainley, M. (1993). Styles of engagement with learning: Multidimensional assessment of their relationship with strategy use and school achievement. Journal of Educational Psychology, 85 (3), 395-405. Blumenfield,P.C., Puro, P. & Mergendoller, J.R. (1992). Translating motivation into thoughtfulness. In H. H. Marshall (Ed.). Redefining student learning: Roots of educational change. Norwood, N J.: Ablex Publishing Corporation. Clarke, D.J. (1996). Refraction and Reflection: Modelling the Classroom Negotiation of Meaning. RefLecT 2(1), 46 - 51. Corno, L. and Mandinach, E. (1983). The role of cognitive engagement in classroom learning and motivation. Educational Psychologist, 18 (2), 88-108. Csikszentmihalyi, M. & Csikszentmihalyi, I. S. (Eds.) (1988). Optimal experience: Psychological studies of flow in consciousness. New York: Cambridge U. P. Csikszentmihalyi, M. & Rathunde, K. (1992). The measurement of flow in everyday life: Toward a theory of emergent motivation. In J.E. Jacobs (Ed.). Developmentalperspectives on motivation. Nebraska Symposium on Motivation, 40, 57-97. Edwards, D. & Mercer, N. (1987) Common Knowledge: The Development of Understanding in the Classroom. London: Methuen. Fullarton, S. (1996). Transition to secondary school: A help or hindrance to engagement in mathematics? Paper presented at the 1996 joint conference of the Australian Association for Research in Education and the Educational Research Association of Singapore, Singapore, November 25th to 29th, 1996. Gooding, A. & Stacey, K. (1991). Aspects of knowledge development through interaction in small groups. Procedings of the Fifth Conference on the Theory of Mathematics Education. Paderno, Italy. Hart, L.E. (1989). Classroom processes, sex of student, and confidence in learning mathematics. Journal for Research in Mathematics Education, 20 (3), 242-260. Helme, S. (1994). Mathematics embedded in context: The role of context in task perceptions, performance and the solution methods of adult women students. Unpublished Master of Education Thesis, Australian Catholic University. Leder, G.C. and Forgasz, H.J. Inside the mathematics classroom. Paper presented at AARE/NZARE Joint Conference, 1992. Meece, J.L., Blumenfeld, P.C. & Hoyle, R. (1988). Students' goal orientations and cognitive engagement in classroom activities. Journal of Educational Psychology, 80 (4), 514-523. Nolen, S.B. (1995). Effects of a visible author in statistical texts. Journal of Educational Psychology , 87(1), 47-65. Peterson, P.L. & Fennema, E. (1985). Efective teaching, student engagement in classroom activities, and sex-related differences in learning mathematics. American Educational research Journal, 22 (3), 309-335. Peterson, P.L. & Swing, S.R. (1982). Beyond time on task: Students' reports of their thought processes during classroom instruction. The Elementary School Journal, 82 (5), 481-491.

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Pintrich, P.R. (1989). The dynamic interplay of student motivation and cognition in the college classroom. In M.C. Maehr & C. Ames (Eds.) Advances in motivation and achievement, Volume 6: Motivation enhancing environments. Greenwich, Connecticut: JAI Press Inc. Pintrich, P.R. & De Groot, E.V. (1990). Motivational and self-regulated learning components of classroom academic performance. Journal of Educational Psychology, 82(l), 33-40. Skinner, E.A. & Belmont, M.J. (1993). Motivation in the classroom: Reciprocal effects of teacher behaviour and student engagement acreoss the school year.Journal of Educational Psychology, 85 (4), 571-581. Skinner, E.A., Wellborn, J.G. & Connell, J.P. (1990). What it takes to do well in school and whether I've got it: A process model of perceived control and children’s engagement and achievement in school. Journal of Educational Psychology, 82(l), 22-32. Stipek, D. J. (1996). Motivation and Instruction. In D. C. Berliner and R. C. Calfee (Eds) Handbook of Educational Psychology. New York: Simon & Schuster Macmillan. Webb, N.M. (1989). Peer interaction and learning in small groups. International Journal of Educational Research, 13, 21 - 39. Swing, S.R., Stoiber, K.C. & Peterson, P.L. (1988). Thinking skills versus learning time: Effects of Alternative classroom-based interventions on students’ mathematics problem solving. Cognition and Instruction, 5(2), 123-191. Williams, G. & Clarke, D.J. (1997). Mathematical task complexity and task selection. In D.M. Clarke, P. Clarkson, D. Gronn, M. Home, L Lowe, M. Mackinlay & A. McDonough (Eds.). Mathematics: lmagine the Possibilities. Brunswick, Vic: MAV. Wilson, J. (1999). Methodologically Speaking: “I do that well most of the time.” Published in the Proceedings of the 22nd Annual Conference of the Mathematics Education Research Group of Australasia, The University of South Australia, Adelaide, July 4-7, 1999. Zimmerman, B. J. (1990). Self-regulated learning and academic achievement: An overview. Educational Psychologist, 25(l), 3-17.

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CHAPTER EIGHT Measuring Values in Classroom Teaching and Learning

1. INTRODUCTION What values underpin the teacher-learner relationship and what might be their significance for more effective teaching and learning? This research attempts to define and examine the values dimension and to identify and discuss particular sets of value priorities negotiated between the teacher and students in a Year 8 science lesson. In reality, teachers and learners are constantly communicating their value positions through language and behaviour. The influence of these, often tacitly held, values is far more significant than is often acknowledged. One reason for this lack of recognition may be the complexity of value patterning that occurs during classroom interaction. The significance embodied within such complexity is often daunting to interpret in the absence of some systematic approach. A coherent map, and language for values, is first required by which to survey the overall values landscape before more detailed analytical interpretation can commence. The data set analysed in this chapter consists of a videotape and transcription of a teacher and a three-student group during a Year 8 science lesson (Lesson 5). It also includes transcripts of individual post-lesson interviews with two students and with the teacher. The aim was to investigate the value patterns being communicated and to consider their implications for teaching and learning. The research sought to address the following questions: What are the value patterns expressed by the teacher and three-student group? — — What values, both espoused and in practice, appear to be educationally important to the teacher? — What is educationally significant about the relationship between the value ranking of teacher and students? — What are the educational implications of instructional leadership styles reflected by teacher and students? The chapter is set out in four broad sections. The first introduces the classroom context within a theoretical framework in which the tasks and achievements of teaching and learning can be seen as both value-laden and developmental. It is 155 David Clarke (ed.), Perspectives on Practice and Meaning in Mathematics and Science Classrooms, 155—196. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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argued that teachers and students reflect, communicate and internalise values through their transactions with one another in ways that are intimately connected with processes of learning. The second section describes the research methodology that combines quantitative and qualitative aspects of value analysis in an interpretive approach. The approach is based on the assumptions that values are indeed identifiable and measurable and that there are differing degrees of subjectivity and rigour that can be brought to any work that addresses the complex affective dimension of human performance. The central method of the analysis, using the Hall-Tonna value definitions and framework, is briefly outlined. The third section illustrates how values are identified in language, attitude and behaviour. The text analysis yields a value profile for both teacher and three-student group that forms a map of interrelating patterns for more detailed analysis. Evidence was sought from the post-lesson interviews that might confirm, corroborate or reveal anomalies in the value analyses. Not least, findings and conclusions are discussed along with the implications. 2. LEARNING THEORY, VALUE THEORY AND THE SCIENCE CLASSROOM Although the research is empirically grounded, it is undertaken with the awareness that different value perspectives can be brought to bear on the conceptual framing of the problem and interpretation of the evidence. While there may not be one right interpretation, some interpretations may be more rigorous and useful than others, and more suited to their intended purposes. The intention here is to provide a basis for commencing future dialogue about values as these arise in a situation-specific context. Through some shared understanding of how values might be systematically identified and discussed, recognition of their importance is made more explicit. To this end, the educational values expressed by the science teacher guided the choice of learning theory that contributes to the conceptual framework of the current research. Whether the science teacher would recognise this framework as one that aligns with his own educational position, or whether the position he consistently communicates is primarily a reflection of his students’ value positions or the school’s ethos and organisational imperatives, remains the topic of future discussion. Peters (1976: 1-22) proposes a task-achievement conception of education. Education does not refer to any one type of task or achievement, nor to a particular process. It encompasses criteria to which any one of a number of processes must conform. Tasks are relative to achievement and when someone learns he or she “achieves the standard or attains the end which is internal to the activity and which gives it point” (p.2). To learn something is to come up to some standard, to succeed in some respect. Peters also refers to an attitudinal aspect of the sort of knowledge that is built into the concept of being educated. Education implies that an individual’s outlook is transformed by what he or she knows, that it is insufficient merely to be knowledgeable. Knowledge must not be inert in the sense that:

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It must involve the kind of commitment which comes through being on the inside of a form of thought and awareness. . . . All forms of thought have their internal standards of appraisal. To be on the inside of them is both to understand this and to care (p.8).

The tasks of the teacher consist in the employment of various methods, such as teaching, instructing and training, to get learning processes going; but in order to count teaching as successful, the achievement must be that of the learner in the end. Not only must the achievement of learning be considered a worthwhile end in itself but the associated tasks and processes must also be regarded as morally unobjectionable. Peters states that the moral requirements intrinsic in the concept of education, that help to distinguish it as different from ordinary cases of tasks and achievements, is that “it is inseparable from judgements of value” (p.3). Since, under this general heading of desirability, education does not refer to any one type of task or achievement, it is this breadth of possibility that accounts for two major problems: people differ in their estimates of what is desirable and, even if they agree on this, they often differ on how it might be achieved. Teachers develop their own philosophical positions about education as part of their broader “appreciation systems” (Schon, 1987: 36-37). Blunden (1996) describes how a well-reasoned educational philosophy might look: (It) would be a general conception which included a sound theory of knowledge pertinent to the relevant teaching domain and including a defensible ethical basis and advanced with systematic rational argument. Having said this ... it would be common enough to find good quality educational philosophy which was rather indeterminate.

It is not difficult to detect that each teacher’s position towards his or her subject matter and students rests upon personally internalised value priorities, usually including a belief system and ethical position relevant to education. These personal positions may have developed through experience and understanding of the subject or discipline being taught, standards and norms of professional practice, needs and interests of students and contexts in which learning occurs. Over time, individuals may develop toward what Bloom (Krathwohl et al., 1964) refers to in the affective domain as "characterisation by value complex". Here the individual has "controlled his behaviour for a sufficiently long time for him to have developed a characteristic 'life style'. Thus the behaviour is pervasive, consistent and predictable". For example, some teachers will predictably prefer to teach by first building rapport with the student, and aim to develop the students’ self worth as a foundation for learning. Others may focus on teaching the subject, believing that its intrinsic worth will engage the student’s interest. Teachers reflect values from numerous sources. They respond in predictable ways to the values expressed through the attitudes and behaviour of students, for example, by designing activities suited to their interests and abilities. They conform to, or call upon, externally imposed value priorities, such as those reflected by the larger educational organisation and community. How a teacher personally ranks these often competing value priorities influences what it is to which he or she will pay more attention in class. A certain authority is conferred upon the classroom teacher, whether it is sought or not. Teachers are often looked to as a model in their attitudes and behaviour. Even though this may not constitute a formal part of the curriculum, teachers are

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well aware that much “incidental learning” (Peters, 1976) occurs through observing others. How do teachers sort out the values that they inevitably communicate to students through their goals, behaviour, lifestyles, moral codes, speech patterns and attitudes to work, learning, people and so on? Many of us often display contradictory behaviour and inconsistency between espoused values and practice, while acknowledging that modelling consistency between these is more likely to have a stronger influence. Kirschenbaum (1977: 8) states that it is not enough to achieve the goal of being clear about one's own values. There needs to be a concern for consequences of one's position - both personal and social. To this end, the extent to which a teacher can appropriately motivate and help to enhance the capacity of the learner is dependent upon the teacher's understanding and strategic use of his or her own value position and subsequent instructional leadership style. In the present case, the forms of thought immanent in the biological sciences are communicated to Year 8 boys and girls in the science lab. The topic of the lesson is the heart and the method is the dissection of a sheep’s heart. The students assemble to watch the teacher’s demonstration and then work in small groups on their dissections. The teacher’s emphasis is on experience-based learning, his stated objective is for students to learn to “enjoy science.” He communicates his own enjoyment and sense of appreciation of the subject; and he assesses the lesson in terms of the students’ enjoyment and extent to which they have “gained something from learning about a sheep’s heart and relating it to (their) own bodies”. This is early evidence of educational values reflected by the science teacher. The teacher has chosen an experience-based approach to the lesson and anticipates the students’ enjoyment of this ‘hands on’ experience. In a review of the theory, Andresen, Boud and Cohen (Foley, 1995:208) describe three defining characteristics of an experience-based approach to learning: (i) it involves the whole person – intellect, feelings and senses; (ii) it involves recognition and active use of the learner’s own previous experience; and (iii) it involves continuing reflection upon earlier experiences in order to add to and transform them into deeper understanding. Kolb’s (1984) cyclical approach to learning, through concrete experience, reflective observation, abstract conceptualisation and active experimentation, exemplifies the widely held view of the importance of combining experience and reflection. Kolb’s “transformation of experience into deeper understanding” echoes Dewey’s (1910) earlier view of constructing knowledge based on the development of conceptions: Education in its intellectual aspect and getting an idea from what is experienced are synonymous. What does having an experience amount to unless, as it ceases to exist, it leaves behind an increment of meaning, a better understanding of something, a clearer plan and purpose of action; in short, an idea? (p.154)

Dewey indicates how educationally important it is to arrive at conceptions. He describes conceptions as meanings that are general because they apply in a great variety of different instances, in spite of their difference; they are constant, uniform or self-identical in what they refer to; and they are standardised, known points of

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reference. It is by this means that we get our bearings when we are plunged into the strange and unknown. So, for example, throughout the lesson, the teacher continually uses correct scientific terms in order to establish a connection between those elements of the dissection discovered by students and the larger scientific classification system. This paves the way towards learning the language and comprehending the more complex, abstract conceptions that characterise the discipline. Notably, such conceptions also inherently contain value assumptions of some significance in their meaning. This conceptual development occurs within a social context in which value assumptions and priorities form part of the often implicit subtext of social transaction and negotiation. Vygotsky’s (1997: xxv) work on the development of consciousness provides insight into the study of concept formation through his distinction between “lower” and “higher” mental functions. The lower natural, or biological, mental functions are characterised as elementary perception, memory, attention and will; and the higher, or cultural, mental functions of consciousness are the development of logical memory, selective attention, decision making and comprehension in language. He argues that socially meaningful activity is the generator of consciousness and that higher mental functions must be viewed as products of mediated activity. Although the higher and lower mental functions are interrelated, lower functions do not disappear as higher functions develop, but are structured and organised according to specifically human social goals and means of conduct. Vygotsky argues that the role of mediator is played by psychological tools, for example, gestures, language and sign systems, mnemonic techniques and decision-making systems, and by means of interpersonal communication. Language and speech are psychological tools that help to form other mental functions and are themselves one of these functions, which means they also undergo a cultural development. Kozulin (op.cit.) describes how Vygotsky’s experiments indicate a long and complex developmental process in an individual’s concept formation “leading from classification based on unorganised congeries of physical characteristics of objects, through the stages of 'complex' and 'pseudoconceptual' thinking, to mature forms of classification based on conceptual thinking" (p.xxxii). Pseudoconceptual thinking is "a form of child’s reasoning that phenotypically coincides with reasoning in the adult and yet has a different, preconceptual nature" (p.xxxiii). Preconceptual thinking also forms the basis of the everyday, normal reasoning of adults, such as that of the teacher. Of relevance to classroom practice, Vygotsky distinguishes two basic forms of childhood experience that give rise to two interrelated groups of concepts: “scientific” and “spontaneous.” "Scientific concepts originate in the highly structured and specialised activity of classroom instruction and impose on a child logically defined concepts; spontaneous concepts emerge from the child’s own reflections on everyday experience" (p.xxxiii). He argued that scientific concepts “far from being assimilated in a ready-made form, actually undergo substantial development, which essentially depends on the existing level of a child’s general ability to comprehend concepts. This level of comprehension, in its turn, is connected with the development of spontaneous concepts” (p.xxxiv). Thus, the

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dialogical character of learning becomes critical as progress in concept formation by a child is achieved in cooperation with an adult. Vygotsky terms this the “zone of proximal development” which represents: The place at which a child’s empirically rich but disorganised spontaneous concepts “meet” with systematicity and logic of adult reasoning. The depth of zone varies, reflecting children’s relative abilities to appropriate adult structures. The final product of this child-adult cooperation is a solution, which, being internalised, becomes an integral part of a child’s own reasoning (p.xxxv).

Thus, the nature of the teacher-student relationship is seen as highly significant, as is the teacher’s ability to stimulate and manage student receptivity to learning. Andresen et al. (Foley, 1995) also note that the design of experience-based learning “often assumes relatively equal relationships between facilitator and learner, involves the possibility of negotiation, and gives the learner considerable control and autonomy” (p.209). These features warrant consideration in any teacher-student relationship where the development of personal responsibility, collaborative skill, independent thinking and creativity are seen as desirable educational objectives and means to learning. 2.1

The concept of values

The success of the teacher-student relationship is also based on the importance of understanding what motivates the learner, that is, which values move an individual to engage in the tasks and achievements of learning. Vygotsky’s notion of language and speech as psychological tools that undergo cultural development lies at the heart of the argument for examining the pervasiveness of the affective dimension in learning. Human values are the basic elements that stand behind all behaviour. Hall (1994: 24) explains that when we talk about human values, we are assuming something about the quality of a person’s life, and about human excellence. He defines values as: The ideals that shape and give significance to our lives, that are reflected through the priorities that we choose, and that we act on consistently and repeatedly (p.21).

Rokeach (1973:5) defines a value system as "an enduring organisation of beliefs concerning preferable modes of conduct or end-states of existence along a continuum of relative importance "(p.5). Each culture has its own conception of the desirable expressed in terms of value hierarchies, thus educational values may be considered a subset, or a diversity of subsets, of the broader array. While the concept of human values and consideration of the virtues came from philosophers and religious thinkers (Singer, 1993), recent social psychologists and educators have contributed much toward value theory and theory of human development. Combs and Snygg (1959: 102-164) are representative in their explanation of how individuals form their earliest values. They describe the individual in terms of the development of the phenomenal self which is distinguished, on the one hand, from the phenomenal environment (the not self) and, on the other, from the deeper more consistent core of the phenomenal self, or “self

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concept”. What we think and how we behave is primarily influenced by our concepts of self and our abilities. Behaviours in any given situation will depend upon how we perceive ourselves, and how we perceive the situations in which we are involved. To illustrate the importance of the self concept, Combs and Snygg cite Lecky (1945) who postulates a need for self consistency as a basic need of prime importance: Immersed in an environment which he does not and cannot understand, the individual is forced to create a substitute world which he can understand and in which he puts his faith. He acts in consistency with the conception, derives his standards of value from it, and undertakes to alter it only when convinced by further experience that it fails to serve the goal of unity. Since this self-made scheme of life is his only guarantee of security, its preservation soon becomes a goal in itself. He seeks the type of experience which confirms and supports the unified attitude (p.129).

Thus, the preservation of the self concept, expressed as the desire for "self consistency", becomes a personal need or goal that is actively valued through the satisfaction of needs and motives that compel decisions and actions in daily living. This ability to observe the difference between the phenomenal self and the self that is required to respond to a given situation requires an ability to see ourselves as others do. The individual not only behaves in a certain way but also is able to interpret this behaviour objectively, that is, in terms of the values of the culture in which he or she operates. The concept of self is differentiated with greater and greater clarity throughout life as we continue to discover who we are and what is valuable to us. This development is inextricably linked with the external environment. We learn the most significant and fundamental facts about ourselves from what Sullivan (1947) calls "reflected appraisals" or "inferences about ourselves made as a consequence of the ways we perceive others behaving toward us" (cited in Combs and Snygg, 1959: 134). Hofstede (1991 : 8) states that values form the core of culture and that they are among the earliest things that are learned - not consciously, but implicitly. Many theorists have observed that individuals construct and reconstruct the meaning or significance of their experiences throughout life. Rokeach (1 973:6) suggests that as a child matures and deals with greater complexity he or she is increasingly likely to encounter social situations in which several values compete and require a weighing of one against another. Gradually, through experience and a process of maturation, we learn to integrate the isolated, absolute values we have been taught in this or that context into a hierarchically organised value system, wherein each value is ordered in importance relative to the others. Of enduring concern is the extent to which values are considered to be personal subjective choices, or visible and objective standards imposed by an external system of authority. The values clarification work of the 1960s (Raths, Harmin and Simon, 1966; Kirschenbaum, 1977) emphasises their subjective nature, while for Kohlberg (1981, 1976), whose work centred on moral development, the critical issue is not the making of subjective choices but the ability to recognise and exercise such external moral values as justice and truth. For Kohlberg and others (Maslow, 1968; Erikson, 1978; Fowler, 1981) this can only emerge in human behaviour developmentally, in much the same way that cognitive skills are learned and develop over time. Hall

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(1994: 30) concurs with Maslow and Rokeach in recognising that human beings both choose their lives and also need exposure to a larger list of possibilities for individual growth. Hall (1994) argues that meaning is derived from language and actions and that values act as the brokers between the inner world of thoughts and feelings and the outer world of language and actions. He (op.cit.: 35-39; Hall, 1987) terms this process the Genesis Effect where behaviour is imbued with significance derived from the unique combinations of values that each individual holds. Since the relationship between values and consciousness is bound up with the development and use of language, the development of language vastly accelerates the process of differentiation of perceptions as it provides a means by which experience can be "symbolised, manipulated and understood with tremendous efficiency" (Combs and Snygg, 1959: 133-4). Through the development and use of language people are able to see their reality differently and conscious awareness of this can provide steps towards greater possibilities for learning and human development. For Friere (1979), working with illiterate adults, this altering of human consciousness through dialogue and praxis leads to personal transformation and liberation. Friere listened for particular “emotionally laden words” that he considered the central factor of conscientisation in spoken language. Hall (1994:3) later identified these emotionally laden words, inherently containing dynamic clusters of energy, as values. He subsequently came to believe that the words representing this energy were common in all language and not limited to a particular population. In summary, behaviour is imbued with significance derived from the unique combinations of values that each individual holds. When the individual's phenomenal self is understood, the various and diverse behaviours of people become consistent and predictable. Encoded within language and behaviour lie the cues to the needs, motives and values of the individual. It follows that values are indeed identifiable and measurable, from the simple activities of observing changes in voice tone and animation, or sampling a person's conversation to get some idea of motive patterns, to the application of instruments and techniques of values analysis. 2.2

The Hall-Tonna framework of values

Without becoming reductive in approach, research methods are few for systematically identifying and assessing the impact of values on teaching and learning in the classroom. And yet, values identification is a necessary precursor to any comprehensive effort at the values clarification that is needed for the further development, alignment and strategic employment of values in the interests of making teaching and learning more effective. Such value clarification and subsequent development may include discussion about, for example: – The suitability, relevance or priority of particular values within the learning environment; – The coherence or consistency with which they are espoused and practised by both teacher and learners; and

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How values may best be aligned or used strategically to challenge and motivate learners as part of the process of negotiating meaning. Through the use of the Hall-Tonna framework of values, the researcher investigates the value positions being communicated throughout the science lesson. Hall and Tonna"s (Hall, 1994) work on value theory represents a synthesis of numerous theories into a comprehensive conceptual framework and associated language. They have identified a list of 125 values with universally standardised definitions which provide a common language and which “appear to be stable and evident across all samples and all populations” (p.3). These definitions have been standardised through reliability and validity studies that take into account cultural, linguistic and gender differences (Hall, op.cit: vii; Hall, Harari, Ledig and Tondow, 1986: 23-41). Hall and Tonna have also constructed a classification system, a framework called the Hall-Tonna Values Map. Together with the definitions, this technology can be used to systematically identify and represent the infinite variety of unique and complex value patterns of individuals, groups and organisations. Table 8.1 represents the Values Map which is organised into four phases and eight stages of human development. The four phases each show a different world view by which an individual conceives and relates to the world. Each phase has two stages, the left stage represents internal values associated with the self concept, and the right stage represents external values by which the phenomenal self relates to the external world. Each of the 125 values shown on the Values Map is classified as a means value or goal value, incorporating Rokeach’s (1973) notion of instrumental and terminal values. Goal values represent states of "being" that individuals may desire to achieve. Means values relate instrumentally to goal values. For example, a teacher may aim for the Stage 4 goal of being competent/confident and use the Stage 4 means value of communication/information as one means by which to achieve this goal. Holding a particular means value is not the same thing as having the skills required to realise that value but, in motivational terms, its a good start. Once individuals understand their value relationships, or dialectics, they are able to make decisions and direct their actions more purposefully. In the example of the teacher, the means and goal value are located in the same stage of the Values Map. This represents a positive means-goal dialectic. To say that a dialectic is positive or negative is not to judge it as right or wrong, but to assess it in terms of possible consequences for behaviour. The criterion is whether or not the behaviour may be useful for the individual to realise his or her value expectations. Positive and negative dialectics occur for an individual within and across the eight stages on the Map. The Hall-Tonna Values Inventory questionnaire (Values Technology, 1995) is one instrument by which to obtain an individual value profile and have it graphically represented on the Values Map. The distribution pattern of a value profile forms three value clusters known as foundation, focus and vision values. The foundation values provide the basic support for daily living and, if not fully integrated, may eventually take priority over higher ideals. This notion reflects Maslow’s (1968) earlier Hierarchy of Needs theory but is more developed. Vision values are those

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that give motivation and meaning. They form part of an individual's vision for the future and require translation into concrete actions and needed skills in order to help reach higher goals. However, it is the focus values that act as a bridge between foundation and vision values. Focus values represent the highest priorities in daily life, the ideals into which the most energy is put and the focus of an individual’s developmental efforts. From these three value clusters a particular point is obtained, called the “central dynamic”, at which the individual may focus much of his or her interest and energy at any given time. The central dynamic may vary somewhat according to changes in the individual and context. Most individuals hold values from across all phases at any one time in various stages of integration and with varying degrees of skill for actualising them. Finally, the central dynamic corresponds with a leadership style represented on a linear scale on the Values Map. One particular leadership style is not necessarily superior to another by virtue of its place on the scale. It is an existential rather than a developmental perspective that is being sought here. A particular leadership style is valued for its appropriate match to the situation. Hall indicates that the leadership style of someone in authority can potentially motivate or lower the performance of those in close proximity. Thus, a student’s motivation and capacity to learn can be influenced by the instructional leadership style that prevails in the classroom setting. In the present research, the generic quality of the Hall-Tonna 125 definitions helped to put names to values and to classify instances of their occurrence in the language and behaviour of the teacher and three-student group in the science class. By making values explicit and establishing a broad overview of their interrelating patterns, a context is provided in which to examine selected value clusters in greater detail for their educational relevance. Because of the nature of the data set received, the present research does not employ the Hall-Tonna questionnaire but a method of text analysis that is situation-based in approach and so more suited to the research aims. 3. THE METHODOLOGY The data set, gathered independently of the researcher, consists of a video recording that focuses simultaneously upon the dialogue of the teacher and one group of boys, Kevin, Stephen and James, throughout the science lesson. The transcription of the dialogue is the central focus of the value analysis. Kevin and Stephen and the teacher are also interviewed individually as they watch themselves on the video replay of the lesson. The post-lesson interviews were examined for additional evidence that might confirm, corroborate or reveal anomalies in the values analysis.

Physical Safety/ Survival

Shelter Function/

MEANS VALUES Food/Warmth/

Self Preservation Wonder/Awe/Fate

Self Interest/ Control

GOAL VALUES

Territory/Security Wonder/ Curiosity

MEANS VALUES Affection/ Physical Economics/profit Property/Control Sensory/Pleasure

GOAL VALUES Physical Delight Security

PHASE 1 SURVIVING The world is a mystery over which I have no control. 2.SECURITY I.SAFETY

Rights/Respect Social Affirmation Support/Peer Tradition

MEANS VALUES Being Liked Care/Nuture Control/Order/ Dis. Courtesy/ Hospitality Dexterity/ Coordination Endurance/ Patience Equilibrium Friendship/ Belong Obedience/ Duty Prestige/Image

GOAL VALUES Family/ Belonging Fantasy/Play Self Worth

3.FAMILY

Duty/Obligation Economics/ Success Education/ Certification Efficiency/ Planning Hierarchy/Order Honor Law/Rule Loyalty/Fidelity Management Membership/ Institution

MEANS VALUES Achieve/ Success Administration / Control Communicat/Info Competition Design/Patt/Order

GOAL VALUES Beli ef/ Philosophy Competence/ Confidence Play/ Recreation Work/Labor

4.INSTITUTION

The world is a problem with which I must cope.

PHASE II BELONGING

Generosity/ Compassion Health/Healing Independence Law/Guide Limitation/ Acceptance Mutual Obedience Quality/

Joy

MEANS VALUES Adaptability/ Flexibility Authority/ Honesty Congruence Decision/ Initiation Empathy Equity/Rights Expressivness/

Liberation Integration/ Wholenes Self Actualization Service/Vocation

Equal ity /

GOAL VALUES

New Order Creativity Detachment/ Solitude Discernment Educat/ Knowledge Growth/ Expansion Intimacy

Complimentarity Corporat./

MEANS VALUES Accountability/ Ethics Collaboration Comm./ Supportive

Construct/New Order Contemplation Faith/Risk/ Vision Human Dignity Knowledge/ Insight Presence

GOAL VALUES Art/Bcauty Being Self

PHASE III SELF-INITIATING The world is a project in which I want to participate. 5.VOCATION 6.NEW ORDER

Solitude

Minessence Prophet/Vision Synergy Transcendence/

Interdependence

Commun./ Personalist

MEANS VALUES

GOAL VALUES Intimacy/Sol itude Truth/Wisdom

Macroeconomics

Human Rights

Global Justice

MEANS VALUES Convivial Technology

GOAL VALUES Ecority Global Harmony word

PHASE IV INTERDEPENDENCE The world is a mystery for which we care on a global scale. 7. WISDOM 8. WORLD ORDER

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Table 8.1a. The Hall-Tonna Values Map 3.0 (Part One)

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PATERNALIST Benevolent paternalist with followers who are dependent and obedient.

3.FAMILY

4.INSTITUTION Ownership Patriotism/ Esteem Productivity Reason Responsibility Rule/ Accountability Technology/ Science Unity/Uniformity Workman/Art/ Craft

MANAGER Efficient manager with followers who are loyally devoted to the organization.

5.VOCATION Evaluation Relaxation Search/Mean/ Hope Self Assertion Share/Listening/Trust

FACILITATOR Listener, clarifier and supporter with followers also listeners. clarifiers and supporters

6.NEWORDER Justice/Social Order Leisure Limitation/ Celebratio Mission/ Objectives Mutual Accountability Pioneerism/ Innovation Research Ritual/ Communication Simplicity/PIay Unity/Diversity

COLLABORATOR Facilitator, producer and creator with active peer participation.

7.WISDOM

SERVANT Interdependent administrator with collegial participation.

8.WORLDORDER

VISIONARY Liberator with a global network of peer visionaries.

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AUTHORITARIAN Authoritarian with followers whoare totally dependent.

2.SECURITY

Table 8.1b. The Hall-Tonna Values Map 3.0 (Part Two)

1.SAFETY

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The text analysis is both qualitative and quantitative in approach. Geertz (1973) notes that ethnographic enterprise is "not as an experimental science in search of law but as an interpretive one in search of meaning." (p.5). It consists not only of observing, recording and analysing but of the kind of intellectual effort that he cites as Ryle's notion of "thick description" (p.6). The difference between thick and thin description, Geertz suggests, is the difference between a camera-like, phenomenalistic observation of the surface behaviour compared with the analysis and interpretation of the layers of meaning that underlie the appearance of any given set of transactions. Its strength lies in "guessing at meanings, assessing the guesses, and drawing explanatory conclusions from the better guesses" (p.20). Thus, the researcher begins with her own interpretation of what the science lesson participants appear to value, the meanings and senses in which language and behaviour are used, and attempts to systematise the findings. Of particular relevance to unravelling complexity of meaning is Vygotsky's (1997) identification of the problems of inner speech, a realm which cannot be directly observed. Inner speech has two important interwoven processes: the transition from external communication to inner dialogue and the expression of intimate thoughts in linguistic form, thus making them communicative. The problem consists in the generalisation versus contextualisation of word meaning. Kozulin (op.cit) cites Vygotsky's distinction between word meaning, which reflects a generalised concept, and word sense, which depends on the context of speech. Kozulin explains: The sense of the word is the sum of all the psychological events aroused in a person's consciousness by the word. It is a dynamic, complex, fluid whole, which has several zones of unequal stability. Meaning is only one of the zones of sense, the most stable and precise zone. A word acquires its sense from the context in which it appears: in different contexts, it changes its sense (p.xxxvii).

The necessary movement back and forth, from generalisation in order to be understood by others to contextualisation in order to make experience personally meaningful, also involves a shifting and re-organising of value relationships as part of the mediation process. The transcription of the science lesson was first analysed using the Hall-Tonna definitions to identify encoded values. This commenced with trying to interpret values embedded in multitudes of words, not just the “emotionally charged code words”. For example, at times the pronoun 'I' was used in a structural sense to distinguish self from not self, but elsewhere it embodied the speaker's valuing of personal authority and honesty. Through trial and error analysis became more discerning, tapping into tacit knowledge and value assumptions (Schon, 1987) embedded in the speaker's technical language, subject expertise, familiar ways of speaking and awareness of group norms. Not least, evidence was sought of value patterns that could characterise the speaker regardless of context. Although the procedures adopted ensured a systematic and rigorous approach, they could not ensure total objectivity in any strong scientific sense. The researcher's own subjectivity, in selecting words and attributing values, could bear consequences for the objectivity of research outcomes. This was dealt with in two ways. The first was the use of a computer software program, Text Analyser (Broughton, 1996),

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devised to assist with recording quantitative tabulations of identified values. Figure 8.1 shows how Text Analyser records each underlined word or phrase from the transcript and the values elicited. The program allows all underlined occurrences of the same word to be manually confirmed as containing the same values, or modified according to the sense in which the word is used. Evidence of every value occurrence is recorded so the source of any single instance of interpretation can be traced back and has some empirical basis for discussion of alternative interpretation. The second method of dealing with the difficulty of subjective interpretation was to be aware of possible interpretive alternatives and, if necessary, take these into account during analysis. In some instances, meaning in the transcript was unclear. The only conclusions that could be drawn here were that more information was required or that ambiguity and confusion are internal to meaning and thus values remain obscured. The overall aim was for logical coherence within the parameters of the research, with appreciation for the fact that there are differing degrees of subjectivity and rigour that can be brought to any work addressing the affective dimension. Not least, it was important not to assess the "rightness" or otherwise of an individual’s choices, but to identify what the values are and try to understand and illuminate something of the relationships formed by them. Once the text analysis was complete for the teacher and three-student group, the accumulated values were then quantitatively distributed across the stages of the Hall-Tonna Values Map and the central dynamic calculated for each value profile. This formed the basis for the next level of analysis: examination of the internal dialectics and interrelating value patterns held by each and their implications for teaching and learning. 4. ANALYSIS AND FINDINGS The analysis is set out in four parts. Figures 8.1, 8.2 and 8.3 are representative of the values expressed by the teacher. They illustrate how the Hall-Tonna value definitions are applied and identified values are quantitatively distributed across the Values Map. Figure 8.4 discusses an excerpt of conversation from the three-student group during its initial stages of dissection. Next, examples are discussed of teacherstudent interactions that illustrate the reflecting of values as a means to maintain rapport and an exercising of control by utilising implicit norms. The resulting Tables 8.2 and 8.3 show the value patterns of the teacher and three-student group and Table 8.4 shows their interrelationship and central dynamic for each. Figure 8.5 provides a summary and discussion of findings, represented as value ranking, world view, central dynamic and corresponding leadership style of each. (a) Values expressed by the teacher In Figure 8.1, shown in italics, the teacher introduces the lesson and establishes the tone of the class. He commences with reference to the “brilliant” new science lab, its prestige qualified by the fact that a lot of money has been spent on it. He imparts to the students a sense of belonging (“this is where you’ll only be from now on”) and accessibility (“so you’ll get half a year’s use out of it”). He communicates a sense of

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responsible ownership (“keep it clean and respect it”); considers that the new lab will contribute to the students’ enjoyment of science (“you’ll even enjoy science because its such a good lab”); and concludes by greeting the students courteously and denoting their membership within the larger institution (''good afternoon, Year eights”). In summary, the evident values are about the importance of science, a desire for students to enjoy learning about science, establishing a sense of belonging and respect for the students and facilities.

Excerpt 1: Teacher’s introduction to the lesson T: So this is the new science lab, a lot of money has been spent on it, this is where you’ll only be from now on, until the end of the year. In Year 9 I don’t think you’ll use this lab so you’ll get half a year’s use out of it. Brilliant lab, keep it clean and respect it, and you’ll even enjoy science because its such a good lab. OK good afternoon Year eights.

Values Analysis new (3Prestige/Image) keep it clean (4Responsibility) science lab (4Technology/Science) respect (3Rights/Resp) enjoy (5Expressiveness/Joy) a lot of money has been spent (4Economics/Success) science (4Technology/Science) this is where you’ll only be from such a good lab (4Technology/Science, now on (2Security, 5Expressiveness/Joy) 3Care/Nurture) good afternoon Year eights I don’t think (3 Courtesy/Hospitality, (4Reason,5Authority/Honesty) 4Hierarchy/Order, Brilliant lab 4Membership/Institution) (4Technology/Science, 5Expressiveness/Joy) Total number of expressions of value for each stage on the Hall-Tonna Values Map Phase I II III SELFIV SURVIVING BELONGING INITIATING INTERDEPEND. 3 5 6 4 1 2 8 7 Stage No. time, expressed 0 0 0 Figure 8.1: Value analysis of teacher's introduction to the lesson The middle section of Figure 8.1 shows the underlined words and phrases and identified values from the text. Each value is numbered according to its stage on the Values Map. The generic definition for each value is used to help classify instances of its occurrence. (Note: some abbreviated definitions are used in the present chapter, where the meaning of a generic value may not be so evident in the context; see Hall, 1994 for the full definition). For example, a key value embodied in the word “science” is the value of technology/science which Hall and Tonna define as “Systematic knowledge of the physical or natural world and practical applications of

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the knowledge through inventions and tools.” A second example is the teacher’s statement, “this is where you’ll only be from now on” which communicates a sense of belonging. It embodies the values of security and care/nurture. These values help to establish a conducive learning environment. The teacher also shows, in his attitude towards students and facilities, the values of responsibility and right/respect. The bottom section of Figure 8.1 shows the quantitative distribution, of all the values identified so far, across Phase I, II and III of the Values Map. The aim was to gather evidence of these, and other values, that accumulate to form any significant patterns throughout the lesson duration. Figure 8.2 shows the teacher commencing organisational arrangements and stating the topic of the lesson. There is a shift towards student control (“One, two, three, next person who talks. OK you girls listen, please”) and management of the task (“There’s going to be four groups. Four groups will get plucks ”). The teacher asserts his authority through the use of such directions as “We’re going to ...” and “There’s going to be...”, although this control is courteously exercised through continued use of “please” and inclusive use of “we”. The key values are about control/order/discipline (Providing restraint and direction to achieve methodological arrangements of persons or things according to the prescribed rules) and administration/control (Exercising given authority to complete specific management tasks). Amidst these arrangements, the teacher commences to teach, evidenced in his explanation “Plucks are hearts, lungs and liver”. This statement indicates three possible values: – Communication/information (Transmitting ideas and factual data between people and components of an organisation); – Technology/science (Systematic knowledge of the physical or natural world and practical applications of the knowledge through inventions and tools); and – Education/knowledge (The experience of ongoing learning as a means of gaining new facts, truths, and principles. One is motivated by the occasional reward of new understanding that is gained intuitively). The teacher implicitly demonstrates the value of communication/information in addressing the class, a process central to the transmission of knowledge. The information is scientific in content, hence the value of technology/science; and since it also represents the teacher’s own knowledge of science, it is an intrinsic indication of the teacher’s valuing of education/knowledge. If the statement “Plucks are hearts, lungs and livers,” is to count as an instance of communication/information or technology/science, then it can be classified as a Stage 4 value. Alternatively, if the statement is classified as expressing the value of education/knowledge, then as a Stage 6 value it has different implications for the overall result. In the event, communication/information was chosen because there is no additional evidence to suggest that education/knowledge is driving the teacher to a greater extent than his intent to introduce basic information to his students at this early stage, even though it alerts us to the possibility that he may place greater

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personal value on education/knowledge. Although this interpretive decision is relatively straightforward, in this example, it illustrates the complexity of interpretive analysis. The problem, in the present case, was resolved by calculating results using two sets of readings, one with the greater numbers of Stage 4 and the other with Stage 6 values, and then checking for any significant difference. Excerpt 2: The teacher's shift into classroom management and introduction of the topic T: Everyone up here please (students move to front of class, Kevin stands on teacher's left; Stephen at back of group in centre) One. two three, next person who talks. OK you girls listen. please. OK we were going to do the kidney today and the heart on Wednesday but since both sets of organs came. we’re going to freeze the kidneys and do the hearts today. (00:01:50 to 00:09:05. A summary in bold by transcriber ofwhat happens next) Now I didn’t expect plucks to come. Plucks are hearts, lungs and livers. We only thought hearts so there’s going to be four groups. Four groups will get plucks though the main prac is the heart.

Values Analysis , Now I didn t expect plucks Everyone (6Community/Supportive) 5Authority/Honesty) up here please . . . One two three lucks are hearts lungs and livers (3Control/Order/Discipline, (4Communication/Information) 3Courtesy/Hospitality) We (6Community/Supportive) talks (4Communication/Infomation thought (4Reason) listen please (3Courtesy/Hospitality, here’s going to be four groups. Four 5Sharing/Listening/Trust) groups will get we were going to do the kidney today plucks (3Support/Peer, and the heart on 4Administration/Control) Wednesday but since both sets of the main prac is the heart organs came we’re going to freeze the kidneys and do the hearts today (4Competence/Confidence, (4 Administration/Control) 4Work/Labour, 6Mission/Objectives) Total number of expressions of value for each stage on the Hall- Tonna Values Map IV II III SELFI Phase BEL ONGING INITIATING INTERDEPEND SURVIVING Stage 1 2 4 5 6 7 8 3 0 0 No. time 0 0 expressed Figure 8.2: Value analysis of teacher's shift into classroom management and introduction of the topic Compared with Figure 8.1, Figure 8.2 shows the distribution of the teacher's values increasing in Phases II and III, with focus on the instrumental values of classroom management. As the teacher moves into the body of the lesson, Figure 8.3 shows how he communicates his knowledge and experience. He demonstrates and , instructs on how to dissect by cutting thin cross-sectional slices of a sheep s heart and describing the procedure. He names the different parts of the dissected organ,

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demonstrates through gesture the pulsating action of the chambers of the heart and commends several sources of information to the students. Through these activities he is implicitly valuing the importance of communication/information, competence/confidence and education/knowledge in an attempt to engage the students’ interest. His use of language, including such phrases as “the model shows the chambers brilliantly” and “if we had a lovely video,” expresses an enthusiasm that derives from his own experience, appreciation and knowledge of the topic. He is also eager to make the subject accessible through other sources of authority to guide the students (“I’ve got a drawing of the heart to give you, which can guide you” and “it’s all in the book, it’s all in the book, it’s all there”). We begin to sense that what he is now communicating transcends the more instrumental value of communication/information. He models the value of competence/confidence as he demonstrates how to dissect. The values of expressiveness/joy and education/knowledge are now providing greater impetus to his teaching. Expressiveness/joy consists here in spontaneous expression through emotionally encoded words like "brilliantly" and "lovely"; herein lies an engaging and motivational quality that may appeal to the students. At the same time, the teacher is conscious of a continuing need for control/order/discipline (“listening, listening, take it in please”) as he maintains the attention of the class. His repeated reference to the word listening, here and elsewhere in the lesson, invokes the value of sharing/listening/trust. However, throughout the lesson this value remains clustered with such values as administration/control and efficiency/planning which are demonstrated through his awareness of the need to manage the time available (“just quickly and briefly”). A conflict begins to emerge for the teacher between value priorities associated with the need for time management, classroom control and desire for students to engage in learning. Figure 8.3 shows another shift in the distribution of values. There is a large number of instrumental values accumulating in Stage 4 and the teacher’s enthusiasm for science now reveals more Phase III values that he may consider to be personally motivating. These represent movement towards a different world view, one that encompasses educational values towards which the teacher might personally aspire and, perhaps, towards which he implicitly aims on his students’ behalf.

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Excerpt 3: Teacher demonstrating, instructing and communicating T:. . . Now the heart is what I’m going to talk about now. Now for those doing the heart dissection, the way the heart is cut - o f course you’ll have to cut the tissue away here ‘cause its all joined - is to cut, is to cut thin slices, now it’s all in the book, it’s all in the book, it's all there, and vou cut the first thin slice and you put it on a piece of tissue paper, I hope there’s some around, or on your board. Then there’s another little slice , and you keep cutting till you get a whole series of slices. Now I’ve got a drawing of the heart to give you, which can guide you. Listening, listening. listening. I've also got a model of the heart. Now that model is available for people to look at at any time. The model’s a bit unrealistic because it’s not quite as uh clear as that. But people you may be lucky to see different sorts of blood vessels. Just quickly and briefly there’s fatty tissue, the heart itself is made of muscle tissue, there’s - listening, listening. take it in please - there’s some blood vessels here, and inside the heart are chambers, and of course the model shows the chambers brilliantly. The chambers in your heart right now are pulsating (moves fist towards and away from chest) and if we had a lovely video, they’re going up and they’re going - they’re pulsating all of the time.

Values Analysis a model of the heart going to talk about now (4Design/Pattern/Order) (4Communication/Information, model (4Design/Pattern/Order) 6Mission/Objectives) available for people for those doing the heart dissection (6Community/Supportive) (4Competence/Confidence) models a bit unrealistic because the way the heart is cut of course , (4Design/Pattern/Order, you ll have to cut the 4Workmanship/Art/Craft, 4Reason) tissue away here cause its all people (6Community/Supportive) joined is to cut is to cut thin slices lucky (1Wonder/Awe/Fate) (4Competence/Confidence, to see different sorts of blood vessels 4Communication/Information, (4Communication/Information) 4Design/Pattern/Order,4Reason,) , Just quickly and briefly it s all in the book (4Efficiency/Planning) (4Communication/Information) , , There s fatty tissue the heart itself is made it s all in the book of muscle (4Communication/Information) , tissue, there ,s - listening, listening, take it s all there (5Law/Guide) , it in please -there s some blood vessels you cut the first thin slice and you here, and inside the heart are chambers put it on a piece of (3Courtesy/Hospitality, tissue paper 4Communication/Information, (4Competence/Confidence, 5 Sharing/Listening/Trust) 4Communication/Information) model (4Design/Pattern/Order) I hope (5Authority/Honesty, shows (4Communication/Information) 5Search/Meaning/Hope) brilliantly (5Expressiveness/Joy, Then there’s another little slice and 6Art/Beauty) you keep cutting till The chambers in your heart right now are you get a whole series of slices pulsating (1Function/Physical, (4Competence/Confidence,

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4Design/Paternt/Order) 4Commmunication/Information, I’ve got (4Ownership) 4Ownership,6Education/Knowledge) a drawing of the heart if we had a lovely video (4Design/Pattern/Order) (4Communication/Information, to give you 4Technology/Science, (6Community/Supportive) 5Expressiveness/Joy, guide (5Law/Guide) 6Community/Supportive) Listening, listening, listening pulsating all the time (3Control/Order/Discipline, (1 Function/Physical) 5Sharing/Listening/Trust) I’ve also got (4Ownership) Total number of expressions of value for each stage on the Hall-Tonna Values Map Phase I II III SELFIV SURVIVING BELONGING INITIATING INTERDEPEND. 2 7 Stage 3 4 6 8 5 1 No..times 0 0 0 expressed Figure 8.3: Values analysis of teacher demonstrating, instructing and communicating

In summary, Figures 8.1-3 establish an initial picture of the teacher's values regarding his learners, subject matter and the tasks at hand. In this manner, the teacher’s values were identified for the duration of the lesson and the total results distributed across the Values Map shown in Table 8.2. The values analysis for the three students, Kevin, James and Stephen, was undertaken in an identical manner, by following the thread of conversation and behaviours for each throughout the lesson. However, the boys’ values are not expressed as individual profiles but as a collective profile which is shown in Table 8.3. (b) Values expressed by the three-student group

The transcript of the lesson shows that Kevin is the first of the three students to actively participate in the lesson. During the teacher’s demonstration and prior to the commencement of group work, Kevin is recorded as following attentively. For example, a student points to the pluck and asks, “What’s that thing here?” When the teacher replies, Kevin expresses the value of wonder/curiosity by indicating more specifically, “No, that”. Next, he is attentive to the teacher’s description of the adams apple which includes the teacher touching his throat and indicating its strong muscle tissue. Both Kevin and Stephen reflect the teacher’s gesture in feeling their own throats. This sensate experience embodies the values of physical delight and wonder/curiosity. A third example of engagement occurs when the teacher asks groups whether they prefer to dissect a heart rather than a full pluck. Kevin volunteers, “Me and Stephen will”, (values of self assertion, friendship/belonging). Kevin also shows awareness of existing procedures (law/rule) in the science lab in

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asking, “Do we have any gloves”, and, of the established teacher-student relationship, by getting the gloves as directed, and seeking permission at each stage of preparation, “Can we take it (the heart) over to our table?” (obedience/duty). On return to the three-student group with a heart, Kevin calls, “Stephen, Stephen, here’s your heart” (friendship/belonging). Next, he grabs Stephen’s glove by mistake. In response to Stephen’s, “No, they’re mine”, he replies, “Are they yours? Oh, OK” (rights/respect), and gets another pair. Within the group, Kevin shows awareness of the teacher’s expectations (mission/objectives, responsibility) as evidenced in his reminders to Stephen, “You don’t need to blow the heart you only need to blow the lungs” and “We’re expected to do discs.” As group activity commences, Figure 8.4 shows conversation between Kevin, Stephen and James that reveals the friendly, cooperative and supportive nature of the group. Kevin shares in the playfulness (fantasy/play) of the dissection activity. For example, while watching James’ cutting procedure he calls out, “Squeeze it!” and moves his clenched fist across his body as if to squirt. He then smiles, picks up the heart and repeats the squirting action. Further on he supports and approves James’ dissection efforts with, “Oh, good on you” (social affirmation). In the same conversation, Stephen also expresses the values of support/peer (“They’re in my pencil case. Use the scissors”), expressiveness/joy (“You’re joking, aren’t ya?’), function/physical (fingers a slice of heart and comments ‘‘It stinks”) and social affirmation (sought from peers through comments like “It stinks, doesn’t it? ”). Stephen also expresses the values of design/pattern/order (connecting his discoveries with patterns and shapes from previous experience, “It’s like a cheese, isn’t it!”) and fantasy/play (holds up skewered heart announcing “spit roast”). Not least, James provides numerous expressions of reason (“On my calculations, if you put it right through the heart, it will come out the other end” and “But you won‘t be able to see the chambers, will you?”) and wonder/curiosity (“I’ll see if that hole leads anywhere”).

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Table 8.2 Quantitive distribution of values expressed by the teachers PHASE I 1. SAFETY GOALS 10 Self Int/C Self Pres 05W/Awe/F

2. SECURITY GOALS Phys Del. 01 Security

MEANS MEANS Food/W/S Affect/Phy 08Func/Phv 01 Econ/Prof 04Safe/Surv Prop/Cont Sens/P/Sex Territ/Sec 03 Wond/Cur

PHASE II

PHASE III

PHASE IV

3. FAMILY

4. INSTIT.

5. VOCAT.

6.NEW ORDER

7. WISDOM

GOALS Fam/Belon Fant/Play Self Worth

GOALS Bel/Phil 18 C/Conf Play/Rec 03 W/Lab

GOALS Equal/Lib Integrat/W Self Act Serv/Voc

GOALS 01Art/Beaut Being self Const/N/O Contempl Faith/R/V Human Di 04 Know/Ins Presence MEANS 01 Acc/Ethic Collab 29 Co/Supp Complem Corp/N/O Creativ Detach/Sol Discern 16 Edu/Knw 02Grow/Exp Intimacy Just/SocO Leisure Lim/ Celeb 07 Miss/Obi M/Account Pio/Innov 07 Research Ritual/Com Simp/Play 01Uni/Div

GOALS Intim/Soli Truth/Wis

MEANS Be Liked 04 Care/Nur 12 Cont/O/D 08Court/Hos 03 Dext/Co 04Endur/Pat 02 Equilib 03Friend/Be 02Ob/Dutv 01Pre/Imag 05Ri’ts/Res 13 SocAffir 05 Sup/Peer 02 Tradition

MEANS MEANS Achie/Suc Adap/Flex 16 Admin/C 09Auth/Hon 53Com/Info Con,Dec/Init Compet 01 Empathy 20Design/P Equity/R 01Dut/Oblig 10Expr/Joy 02Eco/Succ Gener/C 01Educ/Cert Health/H 05Effic /Plan Independ 06Hi/Order 02Law/Guid Honor Limit/Acc 01Law/Rule MutObed Loyal/Fid Qua/Eval Manage Relax 01Mem/Inst 02Sear/M/H 06 Owner Self Assert Patriot/Est 07Share/L/T 04 Productiv 20 Reason 04 Resoons Rule/Acc 15 Tech/Sci Unity/Unif 01 W/A/C

8. W/ORDE R GOALS Ecority Global Har Word

MEANS MEANS Comm/Pers Conviv/Tech Interdepen Glob/Justice Miness Human Prophet/V Right Synergy Macroecon Trans/Soli

From the full text analysis of Kevin, Stephen and James’ participation during the lesson, it is evident that there are three unique value sets, as well as numerous values held in common by the three students. It must be noted here that only one group of three male students is being studied in this composite class of male and female students. Although not all student conversations could be audio taped, a collective voice emerges in the transcript that provides an element of expressiveness not unlike the chorus of a Greek tragedy. The students' utterances, individually and as a class, range from “Ugh, yuk!” when initially confronted with the sight and smell of dissected organs, to expressions of delight, horror, interest and humour as they proceed through the experience. Although this data is impressionistic, it reiterates a

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lively sense of curiosity, expressiveness, reason, friendship, peer support, communication and respect that constitutes the tone of the classroom environment. (Kevin looks at diagram, sits down) J: See that? Cut it down the middle. K: No you cut it (makes slicing movement with hand). J: But you wn't be able io see the chambers, will you? Oh yeah (starts to cut a slice off). K: 'Cause we're expected to do discs. (Stephen returns, Stephen and Kevin watch). S: They're in my pencil case. Use the scissors. (Kevin takes a scalpel) (Stephen pointing to the diagram) Doesn't look much like that, does it? K: My cat used to eat these (Cuts a slice). S: You're joking, aren't ya? K: Nup. (Kevin hands scabel to Stephen) J: What's that baby food they eat, brains? K: Squeeze it! (Kevin moves clenched fist left to right across body as if to squirt. Smiles) J: Hold it up here and squeeze it! Whoosh! Gee that's lumpy. It looks like mince. (Kevin picks up heart and repeats squirting action. Stephen leans back in response.) (James cuts a slice. Stephen and Kevin watch closely). S: It stinks, doesn't it? K: Yep. (00:15:46 to 00:20:07. A summary inserted by transcriber of what happens next. Transcriber uses term "general hilarity”). S: Oh, cool. Are they supposed to be as thin as that? (short inaudible conversation). J: Now. squeeze it, squeeze it. You stick the straw through there and you S: (Holding slice on finger) It's like a cheezel, isn't it? J: Where's the straw? K: Blow. J: It’ll come out the other end, I bet. OK, turn it round. S: (Pokes finger at Kevin) It's like a cheezel, it's like a cheezel, isn't it? J: OK, turn it round. K: Oh, get if off, Stephen. J: On my calculations, if you put it right through the heart, it will come out the other end. K: Oh, good on you. S: (Holds up heart as fit it is on a spit) Spit roast. K: Spit roast. (Kevin takes heart, removes straw and places it on table). S: No! Not on my pencil case! (Kevin cuts another slice, passes heart to Stephen)

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(James holds slice up to Kevin’s face) Don't scare people like that James. (Stephen cuts away) Is this what we're supposed to do then? (inaudible) No, no, I didn't get the whole thing off J: I’ll see if that hole leads anywhere. K: No James, we don’t need to do that. J: No, no (inaudible) it's a chamber. (Stephen pokes straw into heart, Kevin glances up at camera) K: Oh, yuk!

Figure 8.4 Excerpt of Kevin, Stephen and James working as a group

(c) Interactions between teacher and students The following examples highlight the interrelated nature of values expressed through teacher-student interaction. In particular, they illustrate the reflection of values as a means to maintain rapport and the exercise of control based upon the utilisation of implicit norms. The first examples show evidence of students reflecting values in their adoption of language. When the teacher drops a piece of the dissection on the floor, a student humorously pipes up, “Put that in the right way!'' mimicking an adult’s speech pattern for order and control. The three-student group also unselfconsciously emulates the teacher's form of address, ''Now people”, within the privacy of its characteristic group activity. As the lesson proceeds and interest is engaged the students begin to replace words like “that thing” and “those little stringy bits” with correct scientific terms. They become less self-conscious. For example, Stephen reaches a point where he is prepared to try and pronounce more difficult terms (Stephen: "Tendyus -'' ) with the teacher guiding his pronunciation (“ Tendinous cord” ). Conversely, the teacher seeks to establish rapport as part of the classroom environment by using language and behaviour approved of by the students. As well as consistently using scientific terms, he also adopts less formal language reflecting student vernacular, for example, words like “brilliant” and instructions such as “Get your lab coats on fellas, lab coats. ... Put ‘em on, put ‘em on”.

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Table 8.3 Quantitative distribution of values expressed by three-student group PHASE I

PHASE II

6. NEW ORD. GOALS GOALS GOALS GOALS 15 Phys Del. Fam/Belon Belief/Phil Equal/Lib Art/Beauty 05 Self Int/C 20 Fant/Play 17 C/Confid Integrat/W Being self 01 Self Pres Security Const/N/O 04Sel.Worth 03 Play/Rec Self Act 05W/Awe/F Contempl Work/Lab Serv/Voc Faith/R/N HumanDi 03Know/Ins Presence I. SAFETY

2. SECURIT Y

MEANS Food/W/S 07 Func/Phy Safe/Surv

MEANS Affect/Phy Econ/Prof Prop/Cont Sens/P/Sex 02Territ/Sec 17Wond/Cur

3. FAMILY

MEANS Be Liked Care/Nur Cont/O/D 01Court/Hos 14 Dext/Co Endur/Pat Equilib 11Friend/Be 10 Ob/Duty Prest/Imag 05Rts/Res 5SocAffirm 31Sup/Peer Tradition

4. INSTIT.

PHASE IV

PHASE III 5. VOCAT.

04Ach/Suc Adap/Flex 04Admin/C 04Auth/Hon 16Com/Info Congru 01Decis/Init Compet 05Des/PO Empathy Dut/Oblig 01 Equity/R Econ/Succ 07Exp/Joy Gener/C Educ/Cert Health/H 02Eff/Plan Hied/Order Independ 01 Honor Law/Guid 05Law/Rule Limit/Acc MutObed Loyal/Fid Manage 02 Qua/Eval Relax Sear/M/H Memb/Ins 04 Owner 04SelfAssert Patriot/Est Share/L/T 01Productiv 34 Reason 04Respons. Rule/Acc Tech/Sci Unity/Unif Work/A/C

7.

WISDOM

GOALS Intim/Soli Truth/Wis

8 W/ORDE R GOALS Ecority GlobalHar Word

MEANS MEANS Acc/Ethics Comm/Pers Conviv/Tech 06 Collab Interdepen Glob/Justice Miness Human Right 04 Co/Supp Complem Prophet/V Macroecon Corp/N/O Synergy 04 Creativ Trans/Soli Detach/Sol Discern 01Ed/Know Grow/Exp Intimacy Just/SocO Leisure Lim/Celeb 03 Miss/Ob M/Account Pio/Innov 02 Research Ritual/Com Simp/Play Uni/Divers

As the lesson progresses there is increasing engagement through student-teacher interaction. Two examples characterise the extent of rapport maintained by Stephen and the teacher through humour and sharing the value of expressiveness/joy. First, Stephen questions the teacher's omission of a procedure: S: Shouldn't you put some gloves on? T: No, I just washed my hands, with that uh, there it is (points) See that alcohol stuff there?

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In this example, the teacher's sense of cameraderie is evident in his willingness to explain and use of less formal language. At the same time he introduces new information during the transaction (“See that alcohol stuff there”). In a second example, the three boys struggle to dissect while the teacher offers cheerful encouragement with regard to the accumulating slices of sheep's heart, “Nice. Keep them in order guys. Lovely, lovely.” Much to the amusement of everyone, Stephen quickly responds, “Nice? God, you sick man!” Despite his enjoyment of rapport with the students, it is still based on a tacit acceptance that it is the teacher who decides when and how to exercise control. His skill in maintaining interest and cooperation reduces the need for strong measures, but there is one incident that demonstrates the influence of control conflicting with his own educational value priorities. Early in the lesson the teacher refers to an ethical matter, “These dissections can't be compulsory so I need to sort out who wants to do it.” He acts in a responsible manner, whether or not he personally agrees with a school policy that allows students to be excused from animal dissection on ethical grounds. A later incident alerts us to possible implications of the teacher’s value position, although the position itself is not made explicit. The incident occurs while the teacher describes the mechanical nature of the heart. A student suddenly realises what is happening: T:. . . It's (the heart) a most remarkable pump. This is doing the same thing in a sheep. S12: Is that a sheep? T: Yep. And the sheep was killed S13: What’s that over here? S14: That's so cruel. (The teacher examines the pluck in response to the question by student 13. Together, he and students I5 and I6 confirm that it is a bit of the kidney. Student 17 perseveres with the line of inquiry about killing the sheep). S17: Are they killed for them or are they T: They're killed people, for those having emotional problems -;(student 18 inaudible) People, they are killed S19: Don't you feel guilty though? (student laughter) T:Kevin. S20: Put the vacuum cleaner down it.

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T: Next person talking doesn’t do this and will copy out of a book. The sheep are killed for their meat, it’s just that when schools ring up and want these things they provide them. Now of course the liver is provided to the butchers, and so is the heart. So they actually are collected and sold.

This brief excerpt may be interpreted as students exaggerating their horror, perhaps as a means to deal with what confronts them for the first time. However, it also contains legitimate curiosity about the killing of sheep and reference to the morality of doing so. The teacher’s response, "for those having emotional problems - ”, perhaps can be construed as humorous chiding to help students get over their initial reaction. However, the comment may also send another message: anyone who feels squeamish (or even legitimate curiosity?) risks being labelled as having "emotional problems". For students, strongly focused on values of belonging at this stage in their development, this represents a risky line of thought to pursue. The fact that control is exercised so suddenly by the teacher (“Next person talking doesn’t do this and will copy out of a book”) closes off any further possibility. The students laugh, social cohesion prevails and an opportunity is overlooked to engage learners in a discussion initiated from their own interest and curiosity. Instead, the teacher provides a rational explanation about the killing and selling of sheep and sidesteps any possibility of having to overtly engage within the affective domain of learning. Whether intended or not, what incidental learning are students picking up here? Science must always be seen as rational and objective, rather than also including the affective and subjective? Is the moral issue of killing animals beside the point? Must curiosity conform to acceptable lines of exploration? Are some issues just too difficult and time consuming to discuss? Not least, knowledge obtained from a book is being associated with control and discipline. By implication, is it less desirable than experience-based learning? The answer to these questions partially resides in the teacher’s own value position. Although there is no further evidence within the transcript to explain the ambiguity of this incident, it is significant in that the teacher contradicts his own intention, stated in the post-lesson interview, to follow the students' line of questioning in a “spontaneous” manner. The incident is valuable because it exemplifies the kind of conflict in value priorities with which teachers are often faced: In this case, trying to achieve a balance between classroom control and receptivity to learning under pressure of insufficient time. 5. VALUE PRIORITIES OF THE TEACHER AND THREE-STUDENT GROUP From the initial text analysis, Table 8.2 shows that the teacher's value profile contains values in Phases I, II and III, with the majority in Phase II. Table 8.3 shows a similar span of distribution for the three-student group. Table 8.4 combines Tables 8.2 and 8.3 to show the total distribution of values expressed by teacher and threestudent group and the central dynamic calculated for both value sets. The central dynamic is calculated on the quantitative distribution of means and goal values overall and represents the current point of energy focus. As previously discussed, a second central dynamic has been calculated for the teacher based on the possibility of greater quantitative emphasis being placed on the Stage 6 value of

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education/knowledge. For the students, the high numbers of values in Stage 2, of Phase I, and Stage 3, of Phase II, account for the central dynamic being in Phase I, even though a majority of values appear in Phase II. The first finding, from Table 8.4, shows that there are many more means than goal values expressed by the teacher and three-student group. This corresponds with existing empirical research (Rokeach, 1973) about the value sets of most individuals. The proliferation of means values without clear means-goal dialectics can contribute to lack of direction that may lead individuals to treat means values as ends in themselves. Hall (Values Technology, 1995, 1996) indicates that while the possession of numerous values may be the case, individuals are likely to operate with only a small set of value priorities, perhaps as few as half a dozen, at any given time. This leads to consideration of the value priorities that are in currency in the science classroom setting. The next findings, distilled from Table 8.4 and presented as a summary in Figure 8.5, show that there is a different value ranking held by the teacher and three-student group. Each ranking reflects a different world view, central dynamic and leadership focus. It is evident that each is operating from a particular rationale or mental schema that encompasses a recurring value set. This world view represents a generalised form of predictability of behaviour. The students' world view is from Phase I Surviving, “the world is seen as a mystery over which I have no control”. Their perspective consists in relating to the world with less awareness of the nature and extent of their personal authority or control. Consequently, they strive for a sense of belonging, count on the teacher for some level of security and authority and regard him as the holder of knowledge and expertise. In contrast, the teacher’s world view is from Phase II Belonging, “the world is a problem with which I must cope”. His view consists in relating to the world from a problem-oriented perspective. His language and behaviour are embedded with Phase II values. He communicates a sense of responsibility for the management of the teaching task. In the post-lesson interview, he is intent on the problems of time management and supervision. He explains his dislike of having the practical lesson shortened that day and the additional workload required to accommodate "difficult" students. He deals with these problems from within a Phase II framework. These differences in world view may be expected, given the teacher's greater experience, maturation and the likelihood that many of his foundational values are already integrated. However, Hall (1994) reminds us that foundation values, in combination with vision and focus values, remain with us throughout adult life and that their maintenance also remains important.

PHASEIII SELF-INITIATING The world is a project in which I want to

control

1.SAFETY GOAL VALUES Self Interest/ Control Self Preservation

2. SECURITY GOAL VALUES Physical Delight

Wonder/Awe/Fate

MEANS VALUES Affection/ Physical Economics/Profit

MEANS VALUES Food/Warm/ Shelter Function/ Physical Safety/Survival KEY:

Property/Control Sensory/Pleasure Territory/ Security Wonder/ Curiosity

3.FAMILY GOAL VALUES Family/ Belonging Fantasy/Play Self Worth MEANS VALUE Being liked Care/Nuture Control/Order/ Dis. Courtesy/ Hospitalit Dexterity/ Coordinat Endurance/ Patience Equilibrium Friendship/ Belong Obedience/ Duty Prestige/Image Rights/Respect SocialAffirmation Support/Peer Tradition

4.INSTITUTION GOAL VALUES Belief/ Philosophy Competence/ Confid. Play/ Recreation

5. VOCATION GOAL VALUES Equality/ Liberation Integration/ WholenesSelf Actualization Service/Vocation

Work/Labor MEANS VALUES Achieve/ Success Administra./ Control Communicat/ Info. Competition Design/Patt/ Order Duty/Obligation Economics/ Success Education/ Certificat. Efficiency/ Planning Hierarchy/Order Honor Law/Rule

MEANS VALUES Adaptabil./ Flexibility Authority/ Honesty Congrue n ce Decision/ Initiation Empathy

6.NEW ORDER GOAL VALUES Art/Beauty Being Self Construct/New Order Contemplation Faith/Risk/ Vision Human Dignity Knowledge/Insight Presence MEANS VALUES Accountability/ Ethics Collaboration Commun/ Supportiv

Equity/Rights Expressiveness/Joy Generos/ Compassion Health/Healing lndependence Law/Guide

Complimentarity Corporat./ New Order Creativity Detachment/ Solitude Discernment

Limitation/ Acceptanc Mutual Obedience Quality/ Evaluation

Educat/ Knowledge Growth/ Expansion Intimacy

The world is a mystery for which we care on a global scale 8.WORLD ORDER 7.WISDOM GOAL VALUES GOAL VALUES Ecority Intimacy/Solitude Global Harmony Truth/Wisdom Word MEANS VALUES MEANS VALUES Commun./ Convivial Personalist Technology Interdependence Global Justice Minessence Human Rights Prophet/Vision Macroeconomics synergy Transcenden./ Solitud

183

Underlined =Teacher’s values Bold =Students' values * =Student’s central dynamic ** =Teacher’s central dynamic *** = teacher’ s central dynamic , if greater no. Commun/Information interpreted as Educat/ Knowledge

Security

participate

PHASE IV INTERDEPENDENCE

MEASURING VALUES IN CLASSROOM TEACHING AND LEARNING

This world is a mystery over which I have no

PHASE II BELONGING The world is a problem with which I must cope.

Table 8.4a. Distribution of values expressed by teacher and three-student group across Hall-Tonna Values Map (Part One)

PHRASE 1 SURVIVING

3.FAMILY

4.INSTITUTION

5.VOCATION

Loyalty/Fidelity Management Membership/ Instit

Relaxation

Owernership Patriotism/ Esteem Productivity Reason Responsibility Rule/

Search/Mean/ Hope Self Assertion Share/Listen/ Trust

Accountability

*

PATERNALIST Benevolent paternalist with followers who are dependent and obedient.

8.WORLD ORDER

Limitation/ Celebratio Mission/ Objectives Mutal Accountability Pioneerism/ Innovatio Research Ritual/ Communicat Simplicity/Play Unity/Diversity

*

* **

*

B AUTHORITARIAN Authoritarian with followers who are totally dependent.

7.WISDOM

C MANAGER Efficient managerwith followers who are loyally devoted to the organization.

FACILITATOR Listener, clarifierand supporter with followers also listeners, clarifiers and supporters.

COLLABORATOR Facilitator, producer and creator with active peer participation.

SERVANT Interdependent administrator with collegial participation.

VISIONARY Liberator with a global network of peer visionaries.

CAROL P. LEWIS-SHAW

Technology/ Science Unity/Uniformity Work/Art/Craft A

6 NEW ORDER

Justice/Social Order Leisure

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2. SECURITY

Table 8.4b. Distribution of values expressed by teachers and three-student group across Hall-Tonna Values Map (Part Two)

1. SAFETY

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Rank order of teacher ,s values Value Descriptor ValueType Quantity

Rank order of three-student group'svalues Value Descriptor Value Type Quantity

1.Communication/Information (and/or Technology/Science) means

1.Reason

means

34

2.Support/Peer

means

31

3.Social Affirmation

means

25

4.Fantasy/Play

goal

53

2.Community/Supportive means 29 3. (a)Reason means 20 (b)Design/Pattern/Order means 20 4.Competence/Confidence goal 18 5.(a)Administration/Control means 16 (b)Education/Knowledge means 16 , Teacher s world view: Phase II Belonging, “ the world is a problem with which I must cope". Teacher , s central dynamic is situated in Phase II. Leadership focus: (a) lower point and (b) mid-point of Managerial style, defined as "efficient manager with followers who are loyally devoted to the organization".

5. (a) Competence/Confidence goal (b)Wonder/Curiosity means

20 17 17

, Students world view: Phase I Surviving,“the world is a mystery over which I have no control ,., , Student s central dynamic is situated in Phase I. Leadership focus: mid-point of Paternalist style, defined as “ benevolent paternalist with followers who are dependent and obedient ,,.

Figure 8.5 Rank order of value priorities expressed by the teacher and three-student group From the generalised world view to the more particular level of individual value priorities, it can be observed that all of the values in the teacher's ranking could be regarded as educationally significant. Specifically, his hierarchically organised value system shows communication/information (or science/technology), community/supportive and reason ranked as first, second and third respectively. Communication/information reflects the teacher's desire to communicate to others his knowledge and experience of science. It is supported with evidence from such emotionally encoded phrases as "brilliant lab" and "you'll even enjoy science". However, when he models the importance of communication/information, he is more concerned with getting his message across rather than encouraging students to improve communication skills as a means to develop their own thinking and ideas. The value of communication/information is expressed as pedagogic (Knowles, 1984) behaviour of the teacher as transmitter of knowledge and expertise.

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The second highest ranked value is community/supportive which is reflected in interest and concern for students. There is qualitative evidence in the teacher's cheerful encouragement and rapport with the three-student group. There is also supportive evidence, in Table 8.2, in the numerous Stage 3 and Stage 5 relational values held by the teacher. These form useful antecedents to community/supportive. These two highest ranked values, communication/information and community/supportive, form a complementary balance between the teacher's respect for the subject and his interest in the students. However, under pressure the teacher predictably leans towards the more task-oriented communication of subject content. He is aware of the tension between these sometimes competing priorities. For example, he is apologetic to his students about the regrettable lack of time, which is both respectful of them and shows a desire to maintain a sense of connection with them. In watching the post-lesson video replay, he reflects intently on his own time management and is anxious to see whether students are getting the right idea in their group activity. He is delighted to see the three-student group "getting right into it". There is only one dialectic in the teacher's ranking. Dialectics are calculated by taking the highest ranked means value and relating it to the highest ranked goal value, the second highest means value is related to the second highest goal value and so on (Values Technology, 1995; 1996). In the present research, this is done with only the most frequently expressed values because the extent of the data set is insufficient to interpret relations between infrequently expressed value indicators). In the teacher's ranking the means value of communication/information forms a positive dialectic with the goal value of competence/confidence. It is positive because both values are in the same stage and thus form a potentially useful relationship for the teacher's actions. This positive dialectic is modelled through the teacher's confidence and competence in demonstrating how to dissect. This dialectic is educationally important because the goal of competence/confidence is an essential prerequisite for achieving further educational goals. There are a few indirect references to competence, for example, action-oriented phrases like “we're doing the heart”and “you’ve got to cut it this way”, but competence/confidence is never articulated as a goal. In the present context, does this goal need to be made more explicit, in order to guide the students' sense of direction beyond the immediacy of classroom activity, or is it sufficient for them to see it modelled in practice? One clue is that competence/confidence is also as a high priority in the students' ranking which helps to explain their receptivity to the task at hand. They are enjoying their engagement in the dissection activity and there is some evidence of purposefulness. For example, Kevin states, "We're expected to do discs", and James expresses a sense of progress in statements like, “I think we're getting somewhere” and "We're going very well". However, there is also evidence of lack of direction in Stephen's "What exactly are we looking for Mr Black? We're cutting it up into small strips, but what are we looking for?" This particular Phase II value of competence/confidence is the closest the teacher and three-student group get to a shared view of an educational goal, and it may be quite sufficient at this point. The teacher’s ranking also shows the values of reason and design/pattern/order. He demonstrates both of these constantly, for example, in how he classifies elements

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of knowledge, structures learning through his questions and acknowledges student replies. His replies to students’ questions consist in acknowledging that they have understood a fact or provided a correct response in terms of recognising formal scientific concepts. Not least, the teacher's ranking includes the equally ranked values of education/knowledge and administration/control. Education/knowledge by definition is of central importance in that it represents valuing the means to realise the goal of knowledge/insight. Education/knowledge has been consistently related with communication/information and technology/science throughout the lesson to the extent that a second central dynamic has been calculated taking account of the possibility of its greater frequency of expression. The educational relevance of administration/control raises the issues of accountability, ethics and the place of power in the teacher-student relationship. From the analysis, this value is expressed in two aspects of the teacher's behaviour, that is, classroom control and management of the learning task. For example, administration/control is illustrated in his techniques for orderly classroom management, through maintaining interest and rapport. It is also expressed in the incident where he exerts authority to curb discussion when the students express curiosity about the killing of sheep. In comparison with the teacher's ranking, the students' highest ranked value is reason followed by support/peer and social affirmation. The students share the Stage 4 means value of reason with the teacher, and their high ranking of it corresponds with the teacher also having a highly ranked Stage 4 means value, communication/information. Both teacher and students also rank the goal value of competence/confidence among their priorities. The students’ second and third ranking, support/peer and social affirmation, are two relational values. While not prioritised to the same extent by the teacher, they form useful antecedents to the teacher's second highest priority, community/supportive. It is significant that these people-oriented values are ranked second in importance by both. The implication of holding such similar and related values is that if the teacher makes certain decisions, for example, a task-oriented over people-oriented decision, then these students are more likely to be accepting of the teacher's decisions and actions. For learning purposes, this means they are more likely to be receptive to the teacher's emphasis on content and his approach to it because it corresponds so well with some of their own priorities. The three-student group shows two dialectics in its ranking. The means value of reason relates negatively to the goal value of fantasy/play. The students exemplify this dialect in using reason for the goal of fantasy/play in itself. This is considered negative because the relationship tracks back from a Stage 4 means to a Stage 3 goal value on the Values Map which is problematic for development. This is not to say that fantasy/play is not an important value in education. In the second dialectic, the means value of support/peer positively supports the goal value of competence/confidence. While support/peer also does not lie in the same stage as competence/confidence, it is regarded as positive because it tracks forward from a Stage 3 means to a Stage 4 goal value and offers developmental potential. This positive dialectic is exemplified by James' reasoned, "On my calculations, if you put

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it right through the heart, it will come out the other end," and Kevin’s supportive response, "Oh, good on you", which encourages James’ efforts towards the goal of competence/confidence. The next findings in Figure 8.5 relate to the central dynamic and corresponding leadership style. The teacher’s central dynamics correspond with the lower and midpoints of the Managerial leadership style, defined as "an efficient manager with followers who are loyally devoted to the organization". This may be interpreted in terms of the teacher's classroom management and instructional leadership style. As for his world view, it shows a predominance of Phase II values reflected in language and behaviour. Not unexpectedly, the three-student group's central dynamic corresponds with the mid-point of the Paternalist leadership style. The style of leadership to which the boys are presently most responsive is defined as "the benevolent paternalist with followers who are dependent and obedient". A key finding, having significant implications for what goes on in the classroom, lies in the two point proximity shown on the Map's leadership continuum between the teacher’s managerial and students' paternalist leadership styles. The central question is whether the teacher’s instructional leadership style is sufficiently motivating for all the class, or is it merely a reflection of the students’ values in order to maintain rapport and control? By definition, leadership consists in initiating, guiding and envisioning alternative possibilities for the future. In fact, the teacher achieves a good motivational relationship with the three-student group, as evidenced by the two step proximity. Hall and Kroening (Values Technology, 1995) indicate that if the person in authority operates in a style that is one or two steps ahead it is more likely to motivate others to work towards their capability. If the focus is too many steps distant either way, leadership may appear disconnected from the follower's perception of reality. If the teacher operates developmentally below the students' point of focus they are likely to perform below capacity which, in turn, can also affect the teacher's own performance. If the teacher holds previously integrated values that reflect the current focus of his students, even though he may actually hold a different current focus of his own, he is better able to empathise as a means to appeal to them. Thus there may be value in a teacher holding a leadership focus that is flexible enough to reflect through language and behaviour a leadership style that is more likely to connect with students’ current interests, and at the same time, is strategically capable of motivating them to move beyond their current focus. The teacher's second dynamic, influenced by the value of education/knowledge points to motivational capability. 6. DISCUSSION OF FINDINGS The research commenced from the assertion that when values are systematically identified and relationships between value priorities are understood, this provides greater insight into the process of negotiating meaning and has the potential to lead to improved teaching and learning. The argument is based on assumptions about the nature of conceptual development and the influence of values within that development. An individual’s concept formation moves through several stages of

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development from basic preconceptual forms of thinking to "mature forms of classification based on conceptual thinking" (Kozulin cites Vygotsky, 1997:xxxii). Along with the individual development of thought and language through inner speech, the dialogical character of learning is critical to progress in concept formation. The outcome of cooperation in the teacher-student relationship is seen in terms of a solution, which, being internalised, becomes an integral part of a student's own reasoning. Simultaneously, behaviour is imbued with significance derived from the unique combinations of values that each individual holds. Encoded within language and behaviour, these are brought to bear on all social discourse. The teacher and students reflect, communicate, internalise and integrate values through their transactions with one another in ways that are intimately connected with learning that takes place. How the teacher and students perceive and relate to their world is governed by their prevailing world view. How they implicitly rank their respective value priorities influences what it is to which they will pay more attention, and these choices influence the terms and conditions for the negotiability of meaning. The teacher has chosen an experience-based approach to science in the practical dissection activity. As already noted, Andresen et al. (1995) identify negotiation between relative equals as a key feature of the design of experience-based learning; a negotiative aspect may be viewed as an integral part of the learning experience. The teacher exhorts students to listen and "take it all in"; but to assimilate as one’s own meaning that has been constructed by others requires adaptation and adjustment. To assimilate says nothing of the process (Sargent, Nilan & Winter, 1997, pp.231-232) by which this might occur. Negotiation, on the other hand, implies some mutual form of engagement between participants, some arrangement of the terms and conditions under which learning may be transacted. Negotiation takes account of the dialogical and social nature of learning. By its very nature a mutual form of engagement must be based on first establishing points of shared meaning. Both teacher and learners continuously engage in a process of determining which values are held in common and the extent to which value difference exists. In fact, negotiation proceeds continuously at the affective level, whether teacher and learner are conscious of it or not. They are both constantly checking, testing and calibrating, for example, through reflected appraisals, their respective affective states and its likely influence on what is taking place. Through a tacit form of affective negotiation participants proceed, often in a reiterative manner over time, through the process of framing and reframing constructs of meaning as they deal both individually and socially with difference, uncertainty and the unknown. The present analysis provides evidence of reflected values and values held in common. Even though differences in value priorities may exist between the teacher and students, the need to seek and maintain some common ground helps to account for the social and cultural elements of meaning-making. Reflecting values is a give and take affair. For example, the teacher concedes the students’ interest in play/recreation in giving permission to “enjoy” the lesson. He is prepared, to a certain extent, to go where the students’ wonder/curiosity takes them during this new experience of animal dissection. Thus, it is the nature of student interest, as well

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as the teacher's valuing of this interest, that partly contributes to how subject content is approached, although students may not be so aware of this. Reflecting the values of others, something that both teacher and students are constantly doing, may serve an immediate social purpose and, at the same time, in many instances leads to eventual internalisation of these same values that are initially merely reflected. If this internalisation occurs it is done in combination with an individual’s existing unique value patterns. The newly internalised value perspective impacts the existing patterns and it is, in turn, interpreted through an existing set of value priorities. In an educational sense, such increments of internalised values over time accumulate to represent new insights and changes in the perspective by which an individual conceives and relates to the world. It is the subtle shifts in the quality of appreciation, often through processes of communicating and negotiating meaning that characterise increments of new understanding or learning. Andresen, Boud and Cohen (Foley, 1995) state that “Where new learning can be related to personal experiences, the meaning thus derived is likely to be more effectively integrated into the learner’s values and understanding” (p.208). During the dissection activity the students' previous experience is expressed through such spontaneously expressed ideas as "it's like a cheezel, isn't it?" and "spit roast". How might this connecting with prior experience move students towards increasing their conceptual understanding? Boud and Walker (Foley, 1995) emphasise three stages of reflection for structured activities based on an experiential approach that rely upon reflective skills: (a) preparation for experiential events, where it is important to focus on the learner, the learning milieu and the skills and strategies employed in reflection; (b) reflection during an experiential activity, with its phases of noticing and intervening; and (c) reflection after the event, involving the individual in returning to the experience, attending to feelings, and re-evaluating the experience (pp.214-215).

The teacher is aware of the less formal aspect of experience-based learning, as shown in his intention to “spontaneously” respond to student questions rather than to follow a set plan and give a prescribed amount of content. However, the evidence shows that he does not fully recognise the validity of the students' own feelings, perceptions and spontaneous conceptions. Their ability to link previous experience to new experience is not developed nor is there active encouragement towards the acquisition and practice of questioning attitudes and reasoning skills that will move the enjoyment of the experience, and students' imaginative responses to it, into the realms of reflection and conceptualisation. For example, the teacher acknowledges, in the post-lesson interview, that no follow-up discussion occurred for this practical lesson and that the next lesson moved on to another practical dissection activity. The findings reveal evidence of consistency between the teacher’s espoused values and practice. Consistency is observed when speech is accompanied with congruent body language, such as gesture, facial expression, a synchronistic leading and following the cues of others and so forth. Positive dialectics give meaning to the teacher’s own language and behaviour, and consistency reinforces self-concept which provides a more confident basis from which to negotiate and collaborate with others. From this position, the teacher persuades, that is, he negotiates with the

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students the worth of engaging in the present learning task. Modelling consistency lends credibility to the position he advocates. It is also useful in reinforcing positive dialectics, or trying to redirect negative dialectics, held by students. In the present case, the students value reason but in a negative dialectic with the goal value of fantasy/play, and they value support/peer in a positive dialectic with the goal of competence/confidence. Without some clear guidance, the students risk perceiving the science lesson in terms of fantasy/play for its own sake rather than as a step towards achieving a particular learning objective. The teacher can help students to align their efforts more purposefully, especially if they may be unsure what to expect of themselves. Whether acknowledged or not, the teacher's work includes development and alignment of the students' value priorities, not merely to his own way of thinking, but in line with broader educational objectives. This is why it is so important to understand his own and his students' value priorities. Congruence is also observed in the three-student when it is working well together; as the teacher observes, "they're really getting into it!" Once a social dynamic is established and maintained, value difference may be more easily tolerated. Instead of difference, it may be conceived in terms of interesting diversity or useful complementarity. Difference then becomes less of a threat or cause for anxiety and more a source of surprise, interest, stimulation or challenge. This responsiveness to change and difference is closely related to the ability to learn (Dunphy, 1981: 94 -194). Instructional leadership rests upon the teacher having an awareness of his own philosophy of teaching and learning and a clear sense of direction by which he can navigate and guide the students. What might be some other values, in evidence in the value profiles of teacher and students, that could be useful for, say, increasing interest in reflecting on experience or enhancing teacher-student reciprocity entailed in negotiating meaning? One such value in evidence in the teacher’s profile is the Stage 5 relational value of sharing/listening/trust, "The capacity to hear another's thoughts and feelings actively and accurately and to express personal thoughts and feelings in a climate of mutual confidence in one another’s integrity". The teacher expresses this value mostly in the context of classroom control where he is appealing to students to listen to what he has to say, and also when he wishes to share with them his concern about the pressure of time constraints. The defining feature of this value is its element of reciprocity. It implies a mutual need for give and take. Within the teacher’s own frame of reference, this relational value could be further developed as a useful means to achieving his highly ranked value of community/supportive, if he were to consider it important enough. Means values also need to be viewed dialectically with the goal value they are intended to serve. So, for example, both sharing/listening/trust and community/supportive could be directed toward the goal value of knowledge/insight. This particular goal value is also one which has occasionally been expressed by both teacher and three-student group. The three-student group value profile also shows the useful Stage 6 means value of collaboration (The ability of an organisational leader to cooperate interdependently with all levels of management to ensure full and appropriate delegation of responsibility). Collaboration is practiced within the privacy of their

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own group dynamic and so the teacher does not observe it. The group highly values support/peer and social affirmation which form useful antecedents to the more conceptually complex collaboration. Similar to the teacher's highly ranked community/supportive, collaboration lies in the same Stage 6. Does it require the teacher to be aware of the students' collaborative attitude in order to reinforce its value by drawing attention to and by helping to develop the skills required for its more systematic implementation? If the teacher is clear about his own educational vision and values within the larger framework in which he and the students operate, then he is better able to understand the educational relevance of students developing and integrating particular value choices over others. For example, the students' values of wonder/curiosity and fantasy/play are useful antecedents to such future values as creativity, research and knowledge/insight, so it is appropriate to encourage their development and integration. Similarly, the students' values and skills of working together cooperatively, which are currently focused on such goals as self worth and belonging, are useful antecedents for more complex forms of cooperation, such as collaboration, complementarity and community supportiveness. All of these values represent important aspects of an educational track that encourages growth and development and leads toward realising higher goal values. The teacher may not articulate Stage 6 values at length but, as with the shared Stage 5 value of expressiveness/joy and the potentially useful value of sharing/listening/trust, he can begin modelling them, as he models competence/confidence, and as the three boys model aspects of collaboration independently of the teacher. The teacher can also select intermediate dialectics to motivate and provide developmental focus for students beyond the immediacy of the present. This ability to be aware of future possibility has considerable impact on value choices that govern behaviour in the present. Understanding this is a key to personal empowerment in the face of dealing with complexity.

6.1 Values not in evidence The Hall-Tonna framework helps in the identification of values that are present. It is useful for such purposes as identifying value priorities and examining internal dialectics, central dynamics and relationships between value sets. The understanding that is obtained provides a foundation for discussion of the deeper and more problematic issues associated with the influence of values. The Hall-Tonna framework can say little about what values ought to be present in a given situation. A number of points can be made about the absence of particular values from the present empirical study. The first is about values that are embodied within the science discipline. Values are as critical to facts as facts are to values (see Bottery, 1990:46-52) and the teacher models the importance of such values as reason, design/pattern/order, technology/science and education/knowledge. Although it is beyond the scope of the present research to engage in debate about the subjective-objective nature of science (see also Greenfield, 1992, and Hodgkinson, 1991), its importance must be

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acknowledged. There is evidence that an experience-based approach to learning about science appeals to the senses and emotions as well as the intellect, and there is evidence that the value choices by the teacher are being modelled in the very selection of which facts are to be taught and how they are to be approached. Second, values are central to the position of power within the teacher-student relationship and this raises issues about values associated with moral and ethical behaviour. Implicit in this classroom setting are the norms of respect for authority, both the teacher's for institutional priorities and the students' for the teacher's authority. Andresen et al. (Foley, 1995) note that experience-based learning generates dilemmas, "two of which are the ethics of working with deeply-felt experience, and the difficulty of operating within organisational contexts in which there are expectations of learner participation which undermine the learning which is intended" (p.219). Conflict of values may exist for the teacher who must negotiate his way between the expectations of the larger organisation and the interests of his students. For example, when is it appropriate to discuss the moral and ethical issues of animal dissection and to respect those who may hold differing or, as yet, undeveloped views? With respect to abstaining from the dissection activity (which a number of female students did), we can only wonder whether the teacher complies with school policy or genuinely respects the right of students to make their own choice. The post-lesson evidence shows the teacher questioning the sincerity of some students who abstain from the dissection activity on conscientious grounds, and there may be good reason for his concern. Who should be responsible for addressing this issue if students use this as an excuse to avoid a particular subject, lesson, or teacher? The constraints of the school environment, the norms of pedagogoic practice, these can often induce "shallow" rather than "deep" approaches to learning (Ramsden, 1992). A teacher's responsibility lies in employing methods intended to achieve satisfaction both in what is learned and how it is learned, as well as to establish habits and skills associated with moving both teacher and students towards a conception of greater personal autonomy and awareness of the interdependent nature of learning. Instructional leadership partly consists in recognising and initiating the actions required to mediate and negotiate between conflicts of interest that inevitably exist and create barriers to deeper approaches to learning. Third, while we know something of the value ranking of the three-student group, we know nothing of other groups in class, including the group of girls who abstained from the dissection activity and were excused to work in the library. Since the research is based upon the values analysis of three male students and a male teacher, the excused group presents a potentially different situation in terms of studentteacher relationship. There is a brief teacher-student interaction, in evidence on the video tape of the lesson but not included in the transcript, in which the girls negotiate their way out of the classroom. In the post-lesson interview, the teacher provides corroborative evidence that there is much less rapport between himself and the excused group who are perceived as "difficult students" and "silly girls" who represent the additional workload of preparing an alternative lesson to the dissection.

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From these brief comments, we can only surmise that while the teacher values his subject, his own efforts to communicate it and an experiential approach to teaching it, these things might not be valued to the same extent by the excused students. For example, if these students were to hold a simple reversal of the teacher’s first and second value ranking, say relationship precedes task, then this could have significant implications for behaviour in the teacher-student relationship. Would the kind of rapport that the teacher maintains with the all male three-student group work as well for these female students? Might difference in value priorities between the teacher and the excused students reduce potential interest in the subject or contribute towards the marginalisation of these students over time? Conversely, have the excused students tacitly recognised value difference and exercised sufficient power to successfully negotiate their way out of a particular teacherstudent relationship that may not match so well with their own priorities - and is this in their developmental interests or not? We begin to appreciate the extent to which the teacher's value priorities influence what goes on in the classroom, and that he has some choices within his control over the means of negotiating meaning in this teacher-student relationship. His choices may be governed by the extent to which he perceives there are possible alternatives open to him and that these possible options do not conflict with his existing value priorities. These choices may also be modified to the extent of his awareness of the value priorities of others. Similarly, if the students were made aware of the purpose of more dialogical approaches to learning, and they were prepared with appropriate questioning and discussion skills, based on respect for the values of sharing/listening/trust and collaboration, they might also participate in the negotiation of learning in a deeper and more purposeful manner. 7. CONCLUSION In conclusion, using a values analysis perspective to study what takes place in a Year 8 science lesson, the research provides empirical evidence that affirms much that is already known to be the case from previous research and theory about the nature and influence of values on teaching and learning. If we already sense the research outcomes to be familiar this is not unexpected inasmuch as all individuals constantly live with and take account, both tacitly and explicitly, of the influence of values in daily living. The significance of the research is that it allows researchers and educators to observe and consider in more systematic detail the significance of values in the relationship between teacher and students in the classroom. Hall (1973:9) notes that value ranking, like the valuing process itself, as espoused by Raths, Harmin and Simon (1966:28-30), has to be chosen, prized and acted upon. The more aware teachers are of the discrepancies that may exist between their espoused values and practice, the more likely they are able to focus upon and prioritise their own professional development needs and to work towards greater personal congruence between their beliefs and philosophy about education and the praxis that informs their decisions and actions in the classroom. For students, if the direction is clear they can also proceed within a shared framework,

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as rational and purposeful individuals better able to contribute to the identification of their own learning needs and interests and better able to negotiate and achieve understanding. The importance of the Hall-Tonna technology lies in its overall conceptual framework and comprehensive set of 125 value definitions. Its strength lies in its representation of immense systemic complexity. The framework provides a starting point to understand and address more complex value-laden issues. It represents one means to systematically name values, and to move beyond an often fragmented awareness of individual instances of their occurrence in language and behaviour. An understanding of individual value priorities and how they interrelate with the value priorities of others, provides a basis from which to navigate through more difficult value-related issues and dilemmas. An increased appreciation of value complexity may also enable us to move from a positivist, empiricist approach towards the development of shared language values and conceptual frames in order to address important ethical and philosophical issues and the development of new discourses that can qualitatively enhance our knowing and understanding. REFERENCES Andresen, L., Boud, D. and Cohen, R. (1995) Experience-based Learning, in Understanding Adult Education and Training, G. Foley (Ed.), Allen & Unwin: Australia. Bloom, B.S. (Ed.) (1964) Taxonomy of Educational Objectives: the classification on educational goals. Handbook II: The Affective Domain, Krathwohl, D.R., Bloom, B.S. and Masia, B.M., University of Chicago: Longmans, Green & Co. Blunden, R. (1996) Email discussion with C.P. Lewis-Shaw, October 1996. Department of Vocational Education and Training, The University of Melbourne, Melbourne, Australia. Bottery, M. (1990) Facts, values and the educational enterprise, in The Morality of the School: The Theory and Practice of Values in Education, London: Cassell Educational Ltd. Broughton, A.P. (1996) Text Analyser. Computer program, Version 1.0, Hobart, Tasmania: Digital Kinetics. Combs, A.W. and Snygg, D. (1959) The Development of the Phenomenal Self, IndividualBehavior: A Perceptual Approach to Behavior, New York: Harper & Rowe Publishers. Dewey, J. (1910) How We Think, Boston: Heath and Co. Dunphy, DC. (1981) Organizational Change by Choice, Australia: McGraw-Hill. Erikson, E.H. (1978) (Ed) Adulthood, New York: W.W. Norton. Foley, G. (Ed.) (1995) Understanding Adult Education and Training, Australia: Allen & Unwin. Fowler, J.W. (1981) Stages of Faith, San Francisco: Harper & Row. Friere, P. (1979) Education for Critical Consciousness, London: Sheed & Ward. Geertz, C. (1975) The Interpretation of Cultures, London: Hutchinson & Co. Greenfield, T. (1992) Science and service: the making of the profession of educational administration, Educational Leadership: Challenge and Change, Miklos, E. & Ratsoy, E. (Eds) Department of Educational Administration, University of Alberta, Edmonton, Canada, pp. 137-166. Hall, B.P. (1973) Value Clarification as Learning Process, New York: Paulist Press. Hall, B.P. (1976) The Development of Consciousness: A Confluent Theory of Values. New York: Paulist Press. Hall, B.P. (1987) The Genesis Effect: Human and Organizational Transformations, New York: Paulist Press. Hall, B.P. (1994) Values Shift: Personal and Organizational Development. Rockport, MA: Twin Lights Publishers. Hall, B.P., Harari, O., Ledig, B.D. and Tondow, M. (1986) Manual for the Hall-Tonna Inventory of Values, New Jersey: Paulist Press.

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Hodgkinson, C. (1991) Educational Leadership: The Moral Art, Albany, USA: State University of New York Press. Hofstede, G. (1991) Cultures and Organizations: Intercultural Cooperation and its Importance to Survival, Glasgow: Harper Collins. Kirschenbaum, H. (1977) Advanced Values Clarification, California: University Associates Publishers. Knowles, M. (1984) Andragogy in Action, San Francisco: Jossey-Bass Publishers. Kolb, D. (1984) Experiential Learning: Experience as the Source of Learning and Development, Englewood Cliffs: N.J: Prentice Hall. Kohlberg, L. (1981) The Philosophy of Moral Development, San Francisco: Harper & Row. Kohlberg, L (1976) Moralization: the cognitive-developmental approach. Morality: Theory, Research and Social Issues, T.Likona (Ed), New York: Holt, Rinehart & Winston. Maslow, A. (1968) Toward a Psychology of Being, New York: D. Van Nostrand. Peters, R.S. (1976) What is an educational process? in The Concept of Education, R.S.Peters (Ed), London and Henley: Routledge & Kegan Paul. Ramsden, P. (1992) Approaches to learning, Learning to Teach in Higher Education, London and New York: Routledge. Raths, L.E., Harmin M. and Simon, S.B. (1966) Values and Teaching: Working with Values in the Classroom, Columbus, Ohio: Charles E. Merrill Publishing. Rokeach, M. (Ed.) (1979) Understanding Human Values: Individual and Societal, New York: Free Press. Rokeach, M. (1973) The Nature of Human Values, New York: The Free Press, Macmillan Publishing. Sargent, M., Nilan, P. and Winter, G. (1997) The New sociology for Australians, 4th edition, South Melbourne: Addison Wesley longman Australia Pty Ltd. Schon, D.A. (1987) Educating the Reflective Practitioner, San Francisco: Jossey-Bass. Singer, P. (Ed.) (1993) A Companion to Ethics, Oxford, UK, Blackwell Publishers. The Random House Dictionary of the English Language (1967) Eds. Stein, J. and Urdang, L., The unabridged edition, New York, Random House. Values Technology (1996) Values-Based Organizational Alignment - Training Manual, Santa Cruz, CA: Values Technology Inc. Values Technology (1995) Certification Level 1 - Training Manual, Santa Cruz, CA: Values Technology Inc. Vygotsky, L.S. (1997) Thought and Language, Cambridge, MA: The MIT Press.

SUSAN RODRIGUES

CHAPTER NINE Opportunities to Learn Science? Multiple Contexts at Work in a Science Classroom

1. INTRODUCTION ‘The negative tron oxigises the baryon cell.’ Many students would believe that the previous sentence describes something scientific. The sentence involved jargon, sounded strange and they didn’t understand it. The fact that the sentence really is nonsensical would be irrelevant because, for many students, science doesn’t make sense. Not surprisingly, to help students make better sense of science concepts, and to take them beyond the remit of assessment, the 1990s science classroom has been encouraged to use everyday familiar scenarios from which to teach and learn school science. This chapter provides a critical rationale which is subsequently employed to review a lesson involving ‘fibres’ as an appropriate context. However, before I review the lesson, I would like to review the evolution of the use of “context” as a means of making science more attractive to students and follow this with a description of what I consider to be three important aspects (communication, resources, processes) for a context blueprint. I will then use these three aspects to demonstrate how the lesson under observation became a student-inappropriate context, because elements of the three aspects were not addressed. The word context has been widely used to mean a variety of things. In order to differentiate between these meanings, I have used “System Context” to represent the traditional classroom setting in which science is taught. ''Socio-cultural Contexts'' is used to represent the move to include common themes and topics from which to teach science and “Internalised context'' is used to represent the view of context as a process as much as a situation, a view I present in this chapter. Teaching and learning cannot occur without a context and in most schools science has been taught in the System Context. A System Context employs traditional classroom practices, typically transmissive in character. More recently, themes such as clothing, transport, or global balance have been used to teach and learn about concepts related to materials, forces and energy. This approach has been equated with teaching science in context. However, for me, this is only one 197 David Clarke (ed.), Perspectives on Practice and Meaning in Mathematics and Science Classrooms, 197—230. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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component necessary when “teaching in context”. Appropriate contexts for learning science concepts and processes consist of more than System or Socio-cultural Contexts, both of which are external to the individual. Context, in the sense in which I employ it, is not external to the cognitive being. Establishing and using contexts to teach and learn particular concepts requires more than employing a familiar, but external, milieu. Such external contexts (System and/or Socio-cultural) do not automatically constitute an appropriate context for learning because the construct identified and developed by an individual may not be that originally intended. An Internalised Context is constituted in the operation by which someone comes to have a shared understanding of a particular skill/concept in a given situation. Context is not the (external) situation in any absolute sense. Context comes to life for each participant as it is enacted. An investigation of this enactment requires consideration with respect to the nature of communication, resources, and processes, if we wish to identify the learning opportunities made available to students. Viewing contexts in this way focuses our attention on critical philosophical and social questions regarding what is taught and how. In this chapter, I review the idea of familiar situations providing more “studentappropriate” contexts in terms of learning science. In what sense are such Sociocultural Contexts more student-appropriate? What does the use of a familiar theme mean in terms of changes in classroom practice? Is it possible to engage in ‘typical’ classroom discourse, use traditional science classroom practices and codes, employ familiar resources in a traditional science classroom manner and consider it to be a context-based approach, in the sense in which this appears in the reform literature? 2. THE USE OF ‘CONTEXT’ TO TEACH AND LEARN SCIENCE Everything occurs in some form of context and the goal of generating Internalised Contexts is not new. Many cultures have institutionalised this process of learning in situ, and many apprenticeships are served under the auspices of gaining skills while immersed in the work place. Some identify such practices as “learning in context”. I believe that all learning occurs in some form of context because context encompasses both the operation by which we make sense of a situation and the situation itself. More on this later. The importance of context in learning science has been signalled for many years. For example, the Assessment Performance Unit science assessment framework suggested that contexts, concepts and process skills were crucial aspects in learning school science (Department of Education and Science, 1979). But school science has focused on concepts and process skills and paid scant attention to the importance of context in fostering learning of science concepts. Consequently school science lessons tend to be thought of as conduits that convey particular patterns and views of science. Clearly this type of school science context is failing to engage students, given repeatedly voiced concerns with respect to the numbers of students opting out of science (Kahle, 1985).

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Groups concerned about the public understanding of science and those investigating the notion of scientific literacy began suggesting that context-based approaches to science would appeal to students and provide them with more relevant experiences. There was a growing belief in the value of real world situations as potential Internalised Contexts that built on industrial, technological, economic and social implications. This valuing of real world situations was underpinned by the belief that they would lend some degree of purpose to learning what are often perceived as abstract concepts. The push to use Internalised Contexts to teach and learn science grew further in the early eighties due in part to the research literature on women and science. Research of that era indicated that women, when given a choice opted not to study the physical sciences, (see for example, Kahle, 1985), because many felt that science was too masculine. Fuelled by the research on gender (Belenky et al., 1986; Johnson, 1987), the research on situated cognition (Lave and Wenger, 1991), the research on scientific literacy (Jenkins, 1990) and the Science-Technology-Society (STS) movement, contexts have become a password to encourage change in the way in which science is taught in classrooms. The call to teach science “in context” has seen many science curriculum documents advocate contexts as front-endapplications, themes or topical issues. Therefore in many classes science is introduced through a common theme or topic (a Socio-cultural Context) before teaching or learning the science concepts and skills in the traditional System Context. In the past, these themes and topics were addressed after the concepts and skills were taught in the System Context. These applications, themes and topics are advocated as more likely to interest students and induce learning. As a consequence, what used to be considered project work or an application task usually undertaken after a concept was taught is now considered the vehicle through which to address the concept. Girl-friendly science, involving context-based approaches, supported by research indicating a concern for the public understanding of science (Belenky et al., 1986; Kahle, 1985; Johnson, 1987; Rodrigues, 1993), saw the onset of several new courses such as the UK SATIS (Science and Technology in Society, 1986) and Salters Science (University of York Science Education Group, 1990),and PLON (Dutch Physics Curriculum Development Project, Eijkelhof & Lijnse, 1988). Science educators came to the belief that science should not be taught simply to prepare students for science careers or further study but instead needed to focus on empowering students to become citizens of a society dependent upon technological and scientific advances. As a consequence of the STS movement, the science classroom has seen a greater use of resources and issues pertinent to the realm of the everyday world beyond the classroom, whereby everyday themes are employed to generate situations from which students are supposed to learn science. Toward the late eighties, SATIS (Science and Technology in Society, 1986) and Salters Science (University of York Science Education Group, 1990) in the United Kingdom, CEPUP (Chemical Education for Public Understanding Programme, 1991) in the United States, and PLON (Eijkelhof & Lijnse, 1988) developed context based materials or courses.

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The push to use context in classroom teaching has been followed by a spate of research investigating the role and effect of Internalised Contexts in the classroom. In recent times, cognitive psychologists have explored the role of context in general human reasoning (eg Light & Butterworth, 1992). Science educators have explored students’ preferences for different contexts (eg Choi & Song, 1996; Rodrigues, 1993). In this view, Context is a Socio-cultural artifact - simply a common theme involving common objects and common settings. As such, contexts may be investigated in terms of students' preferences for System Contexts and Socio-cultural Contexts. Socio-cultural Contexts are an attempt to embody the scientific activity in a common setting, through which the teacher attempts to relate the lesson’s content. Yager (1983) has shown that students who participated in lesson sequences involving science-technology-society (STS) outperformed (in terms of conceptions of the nature of science, attitudes to science and interest in science) students who were engaged in traditional science lessons. In traditional science lessons, students usually encounter the science devoid of holistic relationships to the world beyond the classroom walls. As a consequence, the relevance of the science is often relegated to an afterthought. Rodrigues (1993) and Ramsden (1998) suggested that Socio-cultural Context approaches were as effective as traditional teaching approaches in developing understanding of chemical concepts, but that students thought context-based approaches were more worthwhile experiences. Likewise, Byrne and Johnstone (1988) and Rodrigues and Bell (1995) suggested that students found chemistry more appealing when they thought the subject matter was useful and relevant. However the argument to consider more “appropriate” contexts when planning science lessons includes more than a view to promoting student motivation, where a context is deemed more appropriate if it appeals to students in terms of relevance and usefulness. There is an argument that school science exemplifies pseudo science practice; pseudo because it is attempting to mimic the workplace of the scientist, while having to take into account the needs of a prescriptive school system. It is pseudo science because working scientists are not simply regurgitating history, they are exploring new ideas or analysing and critiquing old ideas. Most school science curricula have to balance preselected information (often justified by the scientific community) with historical explanations that can be readily taught and understood (Gilbert et al, 1998) within the education system’s established organisation and culture. They have to balance the need to convey contemporary science paradigms with given school practices and behaviours. The result in science classrooms is often a piecemeal approach to science. Most scientists follow a particular scientific field and they report to a specific community for endorsement of their ideas. Students do not commonly report to their peers for endorsement and they tend to rediscover scientists’ thinking and models rather than carry out legitimate science. Recent times have also shown that there is limited interest in research for its own sake. Instead, much scientific investigation is sponsored by industry and driven by economics. As a consequence scientists’ goals commonly involve more than the pursuit of a qualification or “knowledge for its own sake”.

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School science has a different character entirely. School science has an emphasis on anticipating and meeting the requirements of assessment, typically for credentialling purposes, rather than on the pursuit of an answer or of framing further questions for investigation. In many schools, science investigation is primarily undertaken for the purpose of ‘re-discovering’ existing ideas or for learning about the process of investigation. This process of investigation is manifested through the rhetoric of practical work in schools. Yet, amongst others, Hodson (1992, 1996), has shown that the value of practical work is problematic because it does not reflect the reality of practical work in the scientific field and because it does not develop sound understanding of the science concepts. Hodson (1992, 1996) has also indicated that practical work in school science has several objectives: to keep students busy; to keep students interested, and to teach them science concepts or skills. Quite clearly, therefore, a scientist and a student have different agendas in undertaking practical work. These different goals are evident in the science lessons analysed in this study. For example, in the following student exchange, which occurred during a practical task requiring blindfolded students to determine what fabrics they were handling, two students said: F: If you guess polyester for all of them, you might get one right. N: Cotton-polyester.

In school science, students’ investigations are reported in a particular manner, (sometimes, ironically, referred to as “the scientific method”) where observations are predetermined and conclusions are drawn by ignoring any inadvertent mistakes or unwanted findings. Such approaches misrepresent scientific research, while also stifling the legitimate curiosity and interests of students. The documented failure to lure the majority of students to study science (Jones, Porter and Young, 1996) has seen the challenge of teaching in more appropriate contexts being taken up by some schools. Distinguishing between System and Socio-cultural Contexts is in part an attempt to make science more relevant to the student. It is also an attempt to help the students accept and accommodate scientists’ accounts or theories. For these reasons, the use of everyday contexts has gradually made its way into the 1990s science classroom. Recent authors of resource material (for example Rodrigues, 1993; Stinner, 1994; SATIS, 1986) demonstrate how this notion of Socio-cultural context can be realised through starting points involving real life or topical materials, events or phenomena. Many textbooks have also modified their content to incorporate and emphasise applications and the use of Socio-cultural Contexts; although, sadly, many continue to initiate a chapter with an interest catching introduction only to lapse into pared down abstract science a few paragraphs later. Some teachers have interpreted the use of context as signalling a change from the concept-driven approach to the process-based approach. The concept-driven approach views classroom science as the learning of key scientific concepts. The process-based approach promotes the skills and processes common to science rather than simply emphasising key concepts. The inadequacies of the process-based approach have been described and an authoritative review can be found in Miller and Driver (1987). Not surprisingly, confusion has reigned, leaving some teachers

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bemused about “teaching in context”. They are faced with curriculum documents stipulating the use of contexts to teach science but they have little guidance with regard to what this entails beyond the presence of some application-type resource materials and a few textbooks. The significance of Internalised Contexts in science education has not been fully explored by teachers or by science educators . However, perceived directives in curriculum documents and the motivation associated with student preferences for out-of-school related topics has seen some teaching of science through real life themes of clothing, transport, and so on. The lessons analysed in this chapter should be seen as deriving their rationale from these curricular roots. 3. THE LEARNERS’ GOALS, THE LEARNERS’ ENVIRONMENT AND THE PURPOSE OF CONTEXT The purpose of context-based approaches is to provide a more familiar milieu, through which to motivate students to learn and accept socially-accepted propositions commonly called scientific knowledge. In this approach, the learner’s goals are taken to be central to the learning process. What motivates the learner determines what they pay attention to, what they build on, and how they use it in the future. In a science classroom, the learner’s goal, their interpretation of the situation in which they find themselves and their determination of the adequacy of their existing understandings in meeting the demands of that situation are critical factors in their acceptance of new scientific propositions or in supporting their development of ideas which take into account accepted science understandings (Driver & Easely, 1978). It is argued that providing familiar situations from which to investigate science ideas may help students to accept and use current scientific propositions (Qualter, 1993; Rodrigues and Bell, 1995). These familiar situations are deemed to be more appropriate classroom science teaching contexts in comparison with traditional science classrooms because they make the science more accessible to the students. However, given that context is constructed and reconstructed in situated practice by each individual, the extent to which these everyday contexts become studentappropriate contexts from which to learn science concepts is dependent upon both the language and behaviour employed within a situation and the physical environment of the situation itself. What opportunities do everyday themes provide for the purpose of reconstruction or reflection that would not be otherwise present in the classroom? To what extent does the shift from school science to everyday themes make the science more student-appropriate and palatable ?

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3. 1 Appropriate contexts? The extent to which a context becomes appropriate depends largely upon the realm in which communication, resources, and processes are experienced. This is equivalent to saying that the operations used within a situation are as important as the situation when considering the notion of context. All contexts can be represented by the three aspects: communication, processes and resources. What renders a context appropriate in terms of making the science accessible to students, will depend on the contextual realm in which the three aspects operate and the extent to which they help establish links between existing constructs and new experiences. In this chapter I use a lesson on the theme of “fibres” to illustrate how important these aspects of communication, processes and resources are in the construction of student-appropriate contexts. – Communication is the language and association required to make a context shared and meaningful. It is route toward shared understanding or what Gadamer (1990) refers to as the fusing of horizons . We use language to communicate and make links between frameworks held by those present during the communication. It includes verbal and written discourse as well as gesture. – Resources are artefacts manifested by or explored within the situation as encountered. They include people and things. Resources include things because context involves understanding various artefacts and using these artefacts to reach a shared understanding. – Processes are skills and practices employed within and by specific situations. Processes include both the cognitive (eg memory, thinking, planning) as well as the physical nature of the task (ie the “doing”). Scientific processes are the very acts in which we engage when we are learning or doing science. All science classrooms involve learning in context, albeit, sometimes inappropriate contexts. Not just contexts that fail to replicate the work setting of scientists or science practitioner practice but, more importantly for my purpose, contexts that are inappropriate in terms of fostering students’ understanding of science concepts, processes and skills. In most cases, traditional school science employs forms of communication, resources, and processes that are predefined and constrained by a static perception of science and by accepted codes of school requirements. In addition, within this System Context, the language, resources and processes do not always sit comfortably with the individual’s experiences (Rodrigues, 1992). The equipment and processes employed are not familiar and the language used is either new or the words mean something different in science classrooms to their meaning in other situations that are more familiar to the student. The Language of the Science Classroom Sociolinguistics, the study of language in social contexts, involving a study of who is speaking and for what purpose lends itself to the examination of the language of classrooms. In science lessons, much of the language used is initially foreign to the students. Words such as “oxidation” and

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“reduction” are used in school science and students come to accept the existence of these words, but they have little personal meaning for the student. In such a situation, the words will be stored in lexicon memory rather than semantic memory. That is to say, word meaning has not been stored; the student knows the words exist, but has no real understanding of their scientific meaning. To further exacerbate the problem, many words encountered in school science have dual meanings, one for the scientific world, and one for everyday use. Consider words such as ‘reduction’, ‘basic’, and ‘cells’. In some cases science meanings are at odds with the already established understanding for the term. For example, our everyday use of the word “reduction” would suggest shrinkage, yet a chemical definition of the word reduction reads: “reduction is the gain ....” Not surprisingly, confusion stems from this use of common words which have alternative meanings, especially as we devote so little time to clarifying these confusions or even recognising their existence. To make matters worse, in some cases words have different meanings within the various scientific fields. For example a cell in biology means something quite different to a cell in chemistry or a cell in physics. Many students feel alienated by the language employed in science lessons. Another ramification of this is that traditional science classroom discourse becomes unidirectional, with the science teacher dictating much of the conversation as well as the direction of the conversation (Barnes and Todd, 1977). Instructive discourse in the classroom is both uni-directional and bi-directional. It is uni-directional when a teacher is setting a scene, or explaining a topic to a class, and it is bi-directional when a teacher invites discussion about a topic. Within a traditional classroom environment, science is underpinned by particular processes and sanctioned ways of knowing science are dictated by these processes. Due to their lack of correspondence to the personal constructs of the individual, many of these practices and processes are counter-intuitive. The processes employed in the classroom also strip science as taught of any association with beliefs, attitudes or values. There is an assumption that orthodox science does not have these appendages, it is after all ‘objective truth’. Objectivism suggests that objects can be accurately described by scientific concepts, and thinking is simply the manipulation of these concepts. As a consequence teaching is then viewed as a means of transferring objective knowledge from an expert to a less knowledgeable individual (Roth and Roychoudhury 1993) because the expert has the ability to describe and manipulate objects scientifically. In recent times this objectivist position has been subject to criticism (see von Glaserfeld, 1987). This criticism and the constructivist philosophy of learning, (Kelly, 1955) has resulted in a growing conviction that knowledge and meaning is constructed by the individual and not simply transferred from the teacher’s head to the student’s head. Despite differences in what is considered science and how experiences inform views, there is agreement that scientists are influenced by existing personal views when they make or interpret observations. Accepting this, scientists’ theories are therefore influenced by relatively subjective views of reality. Consistent with this view of science, constructivist learning environments emphasise interaction between learners and the environment in which they find themselves. This in turn suggests a change in the nature of processes found in classrooms.

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Besides processes, resources, in terms of equipment or people, are also distinguishing characteristics of different approaches to science, and particular resources (as is the case for particular forms of language use or particular practices) support and even promote a particular way of knowing (Belenky, Clinchy, Goldberger and Tarule 1986). The recent development of units of work (for example Rodrigues, 1993; Stinner, 1994; Salters, 1996) focussed on identifying topics and applications that were familiar to students, through which concepts specified by curriculum documents could be addressed. In such units, the discourse, resources, and practices invoke (or make reference to) the wider community. Issues of drink driving, global warming, genetic engineering and so on become vehicles through which we teach oxidation and reduction, the carbon cycle and genetics. To what extent do these curriculum initiatives involve a shift in resources? The artefacts used in those classrooms implementing the new curriculum included more than beakers and testtubes, but more is required to provide more student-appropriate contexts. The issues associated with the provision of student-appropriate contexts are much more evident when examined in situ, through the analysis of data drawn from an actual science lesson. 4. AN ILLUSTRATIVE LESSON TO REVIEW This chapter focuses upon only one of the eight lessons analysed in the chapters of this book. I use the lesson to portray what I consider to be multiple, though not necessarily appropriate contexts at work in science lessons. Four female students and their teacher provide a focus for my analysis with respect to the three aspects of context indicated in preceding paragraphs. My analysis centres upon the role of context with respect to the students. The classroom videotape data was rich in detail and the interviews provided key elements of triangulation. The data analysed in this chapter included the combined video image (see Chapter 2), the transcripts of the tapes, the interviews, the interview transcripts and field notes by the researcher for that particular lesson. Perhaps some readers of this chapter would consider my analysis problematic as it involves only one lesson. However, I should indicate that I am intending to use the lesson to illustrate the nature of the three aspects of context within the lesson, hence the fullness of the data for that lesson renders it both rich and time-consuming in terms of analysis. There is nothing to prevent others from conducting similar analyses. Obviously my choice of excerpts is influenced by my quest to illustrate my theoretical position vis a vis the role of context in the science classroom. Whenever possible I justify the choice of these extracts either in terms of them being exemplary of the lesson on the whole, or in terms of highlighting a particular issue I wish to address in more detail. Therefore my choice of extracts is guided by the case I wish to illustrate and the argument I wish to present. In the transcript excerpts, an initial letter represents each of the four focus students, or if the speaker was one of the wider collective of students, they are represented with ‘st’ or ‘sts’. The teacher is represented with ‘T’ and any

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researcher/interviewer comment is represented with an ‘I’. The lesson is the first on the topic of "Fibres". T : All right, this unit's on fibres. If you have a look at the front cover we'll talk quickly about where the unit's going. So let's turn to the objectives on page two. ... [reading]

I use data stemming from this lesson to show how use of a familiar topic does not render it automatically an appropriate context for the students, even though the topic had the potential to be an appropriate context for the students. The theme is indeed an alternative context to that normally employed in science classrooms in most schools. I use the data to illustrate why the topic of Fibres was summarily a partially appropriate student context. 5. COMMUNICATION IN THE ‘FIBRES’ LESSON In traditional science classrooms, there is a recognised pattern of teacher orchestrated turn taking (Barnes and Todd 1977, Cazden 1988, Edwards and Westgate 1987, Gilbert 1990), with the teacher initiating and evaluating student comment. This pattern has been described as the basic unit of interaction in an hierarchical model of verbal interaction in most lessons (Sinclair & Coulthard 1975). These initiate-respond-follow up patterns have signalled and typified the power structure that exists in most classrooms. A different pattern of talk, “the cascade effect” (Rodrigues & Bell, 1995), was observed in science lessons in which ‘internalised contexts’ empowered students to test the adequacy of their concepts and establish links between new and existing understandings (Rodrigues and Bell, 1995). In the Fibres lesson, the pattern of talk between the students and the teacher was similar to the basic unit of interaction seen in traditional science lessons. Teacher orchestrated talk was responsible for most of the discourse in the lesson. The following excerpt illustrates what I have called unidirectional talk. T: Understand the difference between natural and synthetic, so could you please underline "difference between natural and synthetic fibres"? And underline the last word "sources"; you need to find out the various sources of each. OK, number three, Danny [S9], what—Danielle, what do you reckon is the most important? There're two words there that could really summarise the wholeS9: Um, compare. T: That's one word. And the other word, Jeff? [S10] S10: Uh. T:

In the beginning of the sentence.

S10: Know. T: Aah, a little bit more than that, know what though?

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S10: Know the property. T: Yeah, so therefore underline the word "properties" and the word “compare.

The previous transcript illustrates the initiating and evaluating role adopted by the teacher and the respond role allocated to students. The orchestrating of turn taking to reach a predetermined outcome is a common pattern of talk employed in many classrooms. This Fibres lesson was no exception. Teacher orchestrated talk is not without purpose. On video-stimulated recall, the teacher provided reasons for the task she had set the students: The underlining is also to try and highlight to the students what is going to be important for them to learn as they go through the unit, and the terms of “properties” and “compares” are used so much in testing and you know the language that we use to test and evaluate students that the sooner that we can try and get them io understand the meaning of it the better. But it is really sad if we say ‘evaluate’ or ‘compare’ in a test and the students don’t have a very good understanding of it then they can’t do the test question particularly well because they don’t understand our language. So it is part of introducing the language associated with science. And um also again trying to model it, so if they don’t understand something they can write the word above, or they can highlight it so it just doesn’t become a meaningless sea of words that are quite often too difficult for kids to understand.

However, the video and transcript would suggest that the students did not comprehend the purpose of the underlining. Consider the following excerpt (T = Teacher; J = Joanne): T: Yep, I’d underline “clothing industry” and the word “use.” And the other bit might be demands of different clothes. OK, the next one, Joanne, should be pretty easy for number five. J:

Define?

T: Sorry? J: Define. T: Yes, but define what? J:

Um, polymer.

T: Yeah, polymer and? J:

And monomy.

T: Monomer. J:

Monomer.

T: Yeah, monomer. OK, number six please, Stewart?

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The students were confused. They were not sure what was expected even though the teacher considered it to "be pretty easy ". As we shall see in the following quote, Joanne explains why she thought “define” was more important than monomer, Nevertheless, through teacher maintenance of the talk pattern, the task is accomplished for the teacher but not the student. The student bemusement was further signalled in the video-stimulated interview between the interviewer and the student concerned. After the lesson, Joanne watched the video record of the excerpt quoted above. She explained: J:

No, I would have thought it was define ‘cause I, it was just underlining monomer or whatever—it didn’t mean anything .... Monomer meant nothing.

I:

So which did you underline?

J:

I underlined the define. ...

J:

And then, but when we do do that, she’s done that a couple times, but then we have to say it, um and she goes through it, saying “OK, you should have underlined, this, this, this, and this” and most of the time I haven’t underlined the things that she wants me to, but when I do I don’t understand what.

Yet according to the teacher, the underlining task would assist Joanne, as she explained during her video stimulated interview: And you can see by Joanne’s comments that she was off with the fairies, she really, you know, doesn’t really listen particularly well and get through the task, so um, maybe the underlining can help her.

Although the teacher’s goal was to help the students focus on vital elements of the topic, the students did not perceive this focus and the pattern of talk did not allow them to voice or clarify their confusion. T: And number seven, what do you reckon, S13, is going to be the most important one, or two? S13: Um. Recognise. T: Nope. S13: Principles. [Joanne laughs] T: Nope. S13: [inaudible] T: Yes, well done, Sam. [laughter among the focus group] The dyeing process, OK, 1 think that’ll do.

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As Duschl (1994) suggests the answer provided by the student is used by the teacher to emphasise ‘final form science’, the type of science that would be assessed or could be presented to an external audience. T: I think book work is very important and the way you set up the book is important, it sort of gives them a sense that they are achieving and they can look back

In the Fibres lesson, for the most part the conversation between student and teacher was unidirectional. Even when there was opportunity to work with the expertise and experience of the students, the teacher doggedly pursued her agenda (T = Teacher; F = Frances). T:

Sorry, can't hear.

S29: Mrs Carlton did this [inaudible]. T: Great. How many of you have done the textiles unit? F: We all-we're in the same class. [comments from other students in class] T: Have you guys done it? S30: Yeah. T:

You have, right. OK. Other seed hairs include kapok, used for pillow stuffing. Fibres taken from the plant leaf are called hard, or cordage, fibres because they are principally used to make rope. Most leaf fibres come from tropical areas. Why?

The students had investigated a topic of textiles which would have enabled them to contribute more to this lesson on fibres, if the teacher had recognised the worth of their previous experiences. In contrast, peer group communication involved a cascade pattern of talk. The pattern of talk between students during their group discussion was similar to that described by Rodrigues and Bell (1995). The students appeared to be using strategies and processes to establish shared understandings. They considered peer comments and used familiar examples to try and clarify their standpoint. Consider this discussion of “fibres” (J = Joanne, F = Frances; L = Leonie): J:

There’s a weft and a warp.

F: [laughs] J: A weft and a. L: [patting Joanne on shoulder] We’re not in textile class today. J:

Yeah, that’s what a fibre is, isn’t it?

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Due to a state of confusion established earlier between the students and the teacher regarding fibres and fabrics, the students are now trying to clarify the difference between fibres and fabrics. In fact the students’ confusion is never resolved. The teacher appears to use the terms interchangeably, without clarifying or acknowledging any difference. During student peer group talk the students have a typical out of classroom conversation pattern, they interrupt, they suggest, they add, they modify. The parity of conversational status accorded to each student allows them to interject and clarify ideas. This in turn led to changes in some students’ understanding. It also afforded the opportunity to embed new information. Unfortunately due to previous classroom tasks involving fibres and fabrics, the terms “fibres” and “fabrics” are confused. This is discussed in more detail in Baird's chapter. However it is interesting to note that the teacher's introduction to the unit included the following: T: So by the end of this unit the student should aim to understand the process by which fibre is used to form a fabric.

Yet given the manner in which the materials were used and the tasks set, there is no indication that students “understand the process by which fibre is used to form a fabric.” Overall, when bi-directional conversations occurred in the classroom, they involved initiation, turn taking, topic management, clarification, and the cognitive functions of attention and memory. These tended to be more pronounced during small group work between the students. In contrast, conversations between teacher and student were unidirectional, which followed the teacher’s content agenda rather than, for example, identifying the students' intuitive science based on their everyday experiences. 6. PROCESSES USED IN THE FIBRES LESSON In tandem with communication and resources, processes employed within the classroom help determine goals, the situation and the making of meaning. In science, the term processes refers to both thinking practices and practical skills. As a consequence, many curriculum documents discuss science concepts and processes. In this paper, we will consider processes, to be skills and practices (mental and practical), employed by the students. In the Fibres lesson these skills and practices differed when students worked within their peer group and when they worked for the teacher. T: The question is-you writing this down? What do I know about fibres? Write that down please. And I'm going to give you five minutes for you to write down all the things that you people know about fibres. So it’s what do I know about fibres? In a group.

By setting this task the teacher appears to be seeking the students' intuitive science. However, as the lesson progresses, it soon becomes apparent that much of the insight that could have been made public was lost.

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T: Ah, are there fibres in the plastic? J:

No. ‘Cause it's like, it's solid.

T: But you think there's fibre in the [points to table] J:

In the paper.

T: OK, do you think there‘s fibres in the paper? [leaves table] [to class] OK, folks, one more minute. Are the ideas drying up already, are they?

As the teacher left Joanne's group, the following dialogue occurred (J = Joanne; N = Narelle, F = Frances; L = Leonie). N: Yeah, things with air in it. 'Cause like if you've got like a puff pastry, let's take this as an example. And you know how like, like it puffs, it puffs air in between. That would have fibre in it. F:

Maybe.

N: But. L:

Like this wouldn't.

F: OK, it's twisted to make wool. L: Yes. We did how they're twisted to make wool, we did fabrics, sort of, and so just-. F:

But, she's not writing it down.

L:

Oh. It's not important.

Indeed when the teacher did return her attention to the group it was to render the task she'd set as trivial because her focus became one of creating the longest list: T: Joanne, how many has yours got? J:

[counting] One, two, three, four, five, six, seven, eight, nine.

T: OK how many has yours got, Neil? N: Um, how many? Nine. T:

Susan?

S:

Seven.

It appears that what could have been a useful exercise that explored students intuitive conceptions and the processes they used in their everyday experiences to make sense of scientific concepts was lost. Indeed, there were several instances

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when students made public their intuitive concepts which were not pursued for further clarification by the teacher. For example: L: Yeah, aren't there fibres in animal skin? Like 'cause that's what they used to wear when there were cave men. T: Well. That's, that's a point. OK, something else please, Neil? N: They've got all—there's lots of different fibres. T: Such as? N: Lots of types. T: I'll leave that, I'll leave that one.

It is hard to see how this exchange could have benefited either the contributing students or the progress of the lesson, and the development of the class's understanding of the concepts associated with fibres. In their working group, the students were engaged in discussion and in verification. They sought clarification of ideas, and challenged each other's thinking. During a retrospective interview, one student explained how she was led to reflect on her own ideas: N: Yeah, I wasn 't sure what polyester was... Yeah, I knew like it was a man-made fibre but, didn't know that it could be like, had different forms, like it could feel like silky or rough or (pause). I: Yeah. So where did you find out that it could have different forms? N: Um, when Joanne didn't know what it was, that sort of made me think, "Well, I'm not really sure either."

Such confusion remained largely unresolved and stemmed from a very early teacher question asking the students to list fibres, in which the students actually listed fabrics (such as: satin, chiffon, etc) and some fibres (such as: cotton, wool, etc) and no attempt was made by the teacher or the students to distinguish between fibres and fabrics. Even when the opportunity arose to question students’ thinking, the opportunity was missed due to a focus on a task-driven curriculum (L = Leonie; F = Frances; N = Narelle; J = Joanne; T = Teacher): L:

No, they’re microscopic. Everything got. Am I just making this up ‘cause don't they say [laughs] in the Omo ads, they say microscopic fi-[laughs].

F:

Yeah, they can be microscopic ‘cause of the Omo ads.

N: So they are microscopic. L:

[gesturing] The enzymes.

N: They um. They wash in cold water.

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L: And they get fat. [Frances and Leonie laugh] J:

And seriously they’re particles.

N: Oh. F:

I thought they had [some overlapping inaudible talk].

N: Material. F:

Aren’t they just one little bit, when you have a piece of wool, just one little strandy bit?

T:

[comes over to study group, looks at Leonie’s and Joanne’s papers] Good. This is not a very big talking group guys. [to Leonie] How about you stop your colouring in-

F:

Well, we’re just talking about Omo.

L:

I’ve been suggesting.

J:

She has, we’ve all suggested, she said they were small and microscopic.

T: OK, well keep colouring in as long as you’re suggesting then.

Such pondering, stemming from reflecting on other students’ ideas, was common in peer group discussions. Students working in groups used everyday processes and knowledge to make sense of the activity they were set. Not all of these were successful in terms of reaching or accepting current scientific theory, because to a certain extent much of the science cannot be ‘discovered’ in the classroom without appropriate facilitation and this was not provided in the lesson studied. The previous student transcripts also illustrate the significance in such conversations of everyday connections and ties. Even though these constructions do not appear sophisticated, the students were establishing links; for example, the reference to the OMO [laundry detergent] advertisement to try to clarify their understanding of fibres. Searle (1969) suggested that meaning rests on implicit but shared empowering and constitutive rules that exist only in their application. During the group conversation, the students were simultaneously negotiating and developing these rules. Through collaboration and elaboration they tried to make sense of terms that were used and ideas that were suggested. Once they discerned consequences for a particular term or idea they re-presented this to the group for confirmation. In contrast, communication between the teacher and the students and consequently the processes being advocated, were constrained by predetermined goals, most of which were known to the teacher but not the students. Sequences of talk were shaped by the power relationship between the students and teacher as well as the subject matter. In the case of the Fibres lesson, the teacher invoked a form of

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classroom management discourse to manoeuvre the students and to signal goals required (T = Teacher; S27 and S28 were students not in the group being studied). T: OK. Yours. [pause] Come on folks, quickly. S26: Cotton, wool, chiffon, lycra, silk, polyester, satin, velvet, [inaudible], canvas, [inaudible]. T: OK, and your group, please. S27: Wool, um polyester, cotton, denim, polyester, cotton, satin, and rayon. T: OK, all right, folks. S28: Fibreglass. T:

And what?

S28: Fibreglass. T: And fibreglass.

Students identified these goals very early in the lesson and they appeared accustomed to identifying the process required and adjusting their ensuing activity accordingly. Approving the colouring in of the title page and counting the number of listed textiles set the tone for the lesson. Enforcing the initiate-respond-follow-up pattern of speech maintained the power structure in favour of the teacher, but it also dictated the process being valued. In this case it signalled an exercise to complete a routine task rather than engage in a process of clarifying concepts and terminology (N = Narelle): T: I’m going to time this one. When I say go, I want you to write down all the fabrics and fibres you have heard of. OK? N: In our group? Does our? T: In your group, all the fabric and fibres you have heard of. Go. See which group can come up with the biggest list.

After the allocated time, the teacher asked each group to read out their list of fabric and fibres. In reading out the list, there was no discussion, it was a recounting exercise. It was also notable that the students conformed to the goal to make the longest list regardless of whether their ideas were scientifically correct (T = Teacher; N = Narelle; L = Leonie; F = Frances; S25 = non-interviewee): T: OK, your group? Guys, I asked you to wrap it up a moment ago. How many are there, folks? How many for your group? S23: Sixteen. N: Twenty-eight.

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S24: Twenty-seven. T: Put twenty-three down in a circle. S25: Sixteen. T:

Sixteen in a circle.

N:

Twenty-eight.

T:

Twenty-eight, right.

L: Which are probably like half of them wrong. [F laughs] F:

Probably wrong.

N: You made them up. L: Yeah.

By the time the students have read out their lists, the lesson has been going for 37 minutes. The students have a title page and have constructed and read out a list of fabrics and fibres. As yet, there has been no real elaborating of meaning at a whole class level. This elaboration only occurred during the group process of constructing the list. Without formal acknowledgment of the value and legitimacy of the small group process of negotiating meaning, traditional science classroom practices and values are not challenged and continue to flourish. The subject framework is subtly changed but the approaches and processes employed within the classroom continue to replicate those found in most traditional science classes. Huibregste, Korthagen, and Wubbels (1994) suggest that even teachers who consider themselves to be constructivists and favour an active role for learners, still heavily direct student activity and maintain strong control over the type of student learning activities deployed. Hence, even when using everyday student familiar contexts, there can be a tendency to retain traditional classroom practices and processes. The processes seen in this lesson on fibres are akin to the processes commonly found in science lessons using traditional school contexts, where the students have a closed prescriptive routine to follow. T: Right, equipment. There are ten samples of clothes to be identified. And there should have been some blindfolds here. And I can't see the blindfolds so I'm not sure how we're going to work round that one. OK, what I think we need to do. S37: Is close your eyes. T: Close your eyes and don't cheat, if you don't mind. There may be two or more samples of the same fibre. You have to, number one, copy the heading onto a new page, underline that. Circle number two, copy the table below, and you've got to allow two lines per um, sample description. Number three, underline "in pairs."

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Likewise the hands-on approach as implemented in this classroom, does not appear to constitute a challenge to those processes employed in traditional science lessons. The students have a closed routine to follow rather than an investigation (T = Teacher, J = Joanne). T: [Joanne raises hand, Teacher comes over] OK, come on folks, come and get some samples please. J:

Do we do the other person's as well? Like, do we write down theirs?

T: Who's your partner? J:

Narelle.

T: Then you'll be blindfolded. J:

Um.

T: She will take your book. J:

Um.

T: And she will write down your results, how you described it. J:

So l'll describe it to her, she'll quickly write it down and I'll say "I reckon it's."

T: Yeah, yeah, neatly.

Their ideas are not central to the lesson's development of scientific meaning and their preoccupation with guessing the teacher's intention continues to take precedent. They completed tasks as prescribed, but have little if any understanding of the purpose of the task as was evident when the teacher asked Joanne to underline what the teacher considered important . They followed a recipe of instructions, undertaking a practical activity, but still have no personal sense of purpose for the task. Once students had discerned the rules of ‘final form science’ for this lesson, there was very little incentive to seek clarification of ideas or make meaning of new ideas. T: And underline "blindfolded." So you're going to work in pairs. One person should be blindfolded, the other person should hand one sample at a time to the blindfolded partner who must describe the feel of the material and make a guess. So could you please underline the word "describe" and "make a guess"?

Given that the blindfolds were not available, what was the purpose of underlining that word. Yet students undertook that task. T: The person who's not guessing, in other words the one with their eyes open, they should be the one to record the results in your book, and number four says each sample should be identified before moving to the next one or they may get mixed up. OK? So you've got to try and identify them before you go. They say five samples, I'm going to ask for three to start with. So after five, after three samples I'd like you to swap and I'd

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like you to change over so the other person has their eyes shut and you have to try and choose a different fabric to the one that they had. And if you can remember what it felt like as well, or you remember the name or whatever, then that should be fine. Your conclusion, how many fabrics could you identify, were there any fabrics that felt different than you, than you expected, and in what way, and could some fibres be made to feel similar to others and if so how. So, go to question two.

The focus of this extended teacher statement was on requirements in terms of performance or meeting tangible outcomes (J = Joanne). J: Oh, well, I’ll write down Narelle’s results. If we don't do it now she’ll want us to do it later. [to Teacher] Do we write down the other people's results? I hate doing. We’re just going to have to ask, like for this, we’re just going to have to ask, I’m going to have to ask my mum and dad, ‘cause I wouldn't have a clue. Like what are we learning, just getting them to do it.

Hence opportunities to build on, challenge, modify or discard ideas were unlikely to arise as students sought to conform to expected practice and demonstrate required consequences. These expected practices were signalled by the teacher; T:I'd like you to circle those seven questions, very easy, very quick to answer. At the bottom of page 5 I'd like you to put question 8, question 8 and could you say PTO, for turn over. If you turn over to page 8, there's a table that you will need to copy and—oh, sorry, page 6, question 8, um, you'll need to copy and do. Now, throughout this book, there are a series-you virtually have to have together, a number of research questions of this-of which this forms the first eight. Fairly easy and there's really not much that you should have any trouble with at all. OK? There is just one, question 3, could you underline the word "husks" H-U-S-K-S? Underline that. That's about the only thing that might, question 3, underline the word "husks." [inaudible student question] T: No. OK. The other thing is that they do ask you to write up a full practical report for a number of pracs. I'd be quite happy for you to write up the first one in full detail which means you've got to write the method out and if you do a really good job of that, I won't ask you to write up the next prac. Those of you who do not write the first one up properly, I will be asking you to write it up, every prac up fully until you actually give me a really good prac. So if you do the first one really thoroughly and well, then l'll just ask for the short method during lesson time.

The abrupt alternation between trivial directions to underline particular words and the emphasis on a well-written practical report, suggests that students are given little guidance as to the relative importance of these very different instructions. Any student confusion over lesson goals or priorities seems quite understandable. While peer group collaboration afforded opportunity to voice personal ideas, it did not appear to lead to either movement toward the scientific community’s ideas or clarification of personal ideas. In several instances, confusion persisted as the peer group had personal ideas that were in various stages of development themselves (J = Joanne; F = Frances). J: You don’t say that it’s um that fibres are in food because we don’t know that she’s (the teacher) talking about clothes fibres, she could be talking aboutF: No, she isn’t talking about fibres. J:

How do you know?

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The processes used in this lesson derived from the priority attached by students to the interpretation of the teacher’s intentions and expectations, and from the students’ acquiescence to these and their confusion over both the purpose of their activities and the terminology that the topic inevitably required. 7. RESOURCES AVAILABLE IN THE FIBRES LESSON In this section I consider how context is influenced by the various components of resources. I consider both artefacts and people as resources. I discuss the interactions between the resources, students and teachers and also consider the role of peoples' attitudes, values and beliefs in the development of appropriate contexts. von Glasersfeld (1993) suggested that learning is a personal activity and teaching is a social activity, the first operating within the cognitive domain of the learner, the second relates to the interrelationship between the learner and other available resources. In effect teachers are the intermediate agents between learners and curriculum science (Watts & Jofili, 1998). As expected the teacher determined the focus of the situation when setting the goals for the Fibres lesson. But a chasm existed between setting and achieving these goals, because of the manner in which the inanimate resources were used. Though the inanimate resources involved familiar materials, the manner in which the students were being instructed to use them reflected processes that were similar to those encountered in traditional science lessons. The task was 'closed' and formally structured. There was no real investigation involved. The teacher was to later comment during video stimulated recall that she could have tried another approach: T: See, you know I just wonder what would happen if we got some various types of fibres and soaked them in water, plant fibres, and tried to get the kids to make ropes and see who could make the strongest rope out of various things, or even had some silk worms set up around the room so you could actually see them. There is so much that we could do, but anyway.

However, for the most part, the artefacts made available were used in a manner that promoted a System Context rather than either the Socio-cultural Context of scientific practice or Internalised Contexts of students’ personal science. In one practical activity, students were encouraged to work in pairs, with one student blindfolded. The blindfolded student had to guess the fabric. The following is taken from a discussion held soon after a pair had undertaken the activity (L = Leonie; J = Joanne; F = Frances): L: And did you, and did you, put wool or you guessed wool? But it didn’t feel like when you have a wool jumper, you know, like your mum knits for you. J:

No. Yeah.

L: It felt really different. F: Yeah, that’s what I’m saying, I thought it’d be more. L: Woolly!

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F: A closer weave than I thought.

Once again students are establishing links to existing stores of data. These links were seldom pursued by the teacher in whole class discussion or in greater detail by the students. Artefacts can have value in terms of helping reconstruct science concepts, if used appropriately. The critical issue concerns the extent to which this activity was helping to establish links between the students’ knowledge and the knowledge of the informed (scientific) community. Even though the materials used in the task were everyday, the students did not think they were using everyday understanding when working on the fabrics task (I = Interviewer; N = Narelle). N So um, like we’re learning about fabrics, but say in Science we’ll learn about um, how the fabrics are um, like the science in the fabrics, and in Textiles we’ll learn about like materials and things like that ... Yeah, so it was. I:

What about outside of school?

N: Um, no, not really, I didn’t-oh, no I didn‘t really learn what I knew today from outside of school.... Probably learned from inside of school.

Although the lesson uses the theme of Fibres to learn science, the mechanism employed within the theme rendered it ineffective. Everyday materials were available as resources during the lesson. Students used them in a recognition activity. However it is difficult to discern to what extent this activity was developed as either a motivating prelude to another cognitive or skill-based task, or how it was intended to encourage scientific thinking. It is difficult to determine the purpose of the blindfold activity in terms of the science being taught. If the purpose of the topic is to use a relevant familiar theme, then one has to encourage students to draw on their out of school experiences when they are engaged in practical work. Primarily the students were distracted by the inferred goal of the task. In one task the goal was understood to be “to make a list”. Once each group read their lists that particular task was considered to be complete. The goal had become to obtain the longest list rather than to negotiate meanings and clarify uncertainties such as the confusion between the notions of fibre and fabric, and the students identified the task requirement from the outset of the task (T = Teacher; N = Narelle). T: I'm going to time this one. When I say go, I want you to write down all the fabrics and fibres you have heard of. OK? N: In our group? Does our? T: In your group, all the fabric and fibres you have heard of. Go. See which group can come up with the biggest list.

Apart from the artefacts used, the use of resources in terms of people is also a significant facet in determining the nature of the resulting context, be it System, Socio-cultural or Internalised. There was also a significant difference between the

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roles of the teacher as a resource and students as resources and, as we shall see, both had associated dilemmas and strengths. From previous and the following transcripts we should be able to see how the teacher and peer group attend and accord importance to different aspects of the task and hence adopt different roles within the discussion. The teacher had the capacity to help students remake sense of accepted science ideas but did not use the opportunities to clarify ideas in order to come to a shared understanding class-wide (T = Teacher; J = Joanne). T: Have you got answers for them in your mind? J:

Yes. How many fabrics could I correctly got.

T: That's easy. Next one? J:

One. Um, were there any fabrics that felt different. Yes, 'cause I thought that the cotton felt like wool. Yeah.

T: Um hm. Good, so you can answer that. And the third one? J:

And, feels similar because they were all kind of, they were fellow [?] material, you just.

T: Hm. What else though? J:

Well, 'cause the polyester can be silky stuff. Soft.

T: I gave you that hint that I asked you to underline. [to class] To answer number three, I gave you a hint at the top of the page, remember where they said a hint? Sts: No. T: Well, they said—I asked you to underline something near the top three lines. Try and use that in some of your answers if you could, but you might find it's easier to put it in words if you use those words there.

The student working groups provided the process for ideas to be voiced and challenged but did not successfully remake sense of accepted science ideas (J = Joanne; N = Narelle; F = Frances; L = Leonie). J:

They’re clothes [That is, “fibres” are clothes].

N: Huh? J:

They’re clothes.

F: No they’re in clothes. They’re not clothes. L: Like you make stuff, they’re not.

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N: [writing] In fabric. F: Yeah, in fabric.

The following transcript tracks the conversation between students who worked in a group to create a student list of fabrics and fibres. They used their peer group members as resources to explore and negotiate meaning: L: Everything has more than one meaning. [nods] N: Yes. [writes] L:

Like Fibre Plus.

J:

No, like Nutrigrain and that.

L:

Yes. [laughing] Like fibre.

F:

No, that’s different fibre.

J:

How do you know, they could be talking about fibre in your body.

L: Except for the fact that she isn’t. N: Hey, do you realise, right, to find it like we’ve used information from other classes, in that one and we are. F:

Right here?

L:

[makes weird noise]

N: You just said textiles, and I said fibre from home etc.

Many of the student insights were not publicly shared with the teacher or students in other groups. Opportunities to tap into the students’ existing knowledge and establish links to science and the Fibres theme were missed by the teacher but stumbled upon by the students. In one task the teacher had asked for a list of fabrics and fibres, and after a period had elapsed she asked the student to provide a list of fibres they knew (T = Teacher; J = Joanne; F = Frances; S24 = non-focus student): T: All the fibres you've ever heard of. S24: Polyester, vinyl, cordoroy. T: Can’t hear you, it's not clear. S24: Polyester, vinyl, corduroy, cotton, tencel, velvet, nylon, silk, satin, denim, leather, rayon, wool, lycra, corduroy [?], linen, plastic, suede, um, flax, cotton, synthetic. T: Great. All right, [to study group] your group. Read out your list please.

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Cotton, nylon.

T: No, loudly. Speak to the people at the back of the class. J:

Cotton, nylon, polyester, silk, flax, wool, lycra, canvas, satin, felt, Velcro, chiffon, tencel, denim, poplin, dacron, down, feathers [bell rings], tulle, lace ribbon, elastic, angora, leopard skin [laughter from group], mink, hair, and that rat fur stuff, and Mason, those brushes with pig hair on top of them.

F: Mason bristles.

The list provided by the students included fibres and fabrics, but there was no move on the part of the teacher either at this point or at any point during this lesson and the following lesson to clarify the difference between fabric and fibre. This is rather alarming given that the teacher had initially stated that the purpose of the unit was to “understand the process by which fibre is used to form a fabric”. In fact, during the course of the lesson the teacher interchanged the terms frequently. For example, at one point she says: So after five, after three samples I'd like you to swap and I'd like you to change over so the other person has their eyes shut and you have to try and choose a different fabric to the one that they had.

Toward the completion of the task she says: So I think for the last five minutes, I'd like all the fibres to be put back here on the table and I'll sort them out.

This lack of attention to the differences between the terms was commonplace throughout the lesson. As stated early in this chapter, when considering resources we have to take into account people as resources. These people, be they the teacher or students bring with them beliefs, attitudes, interests and other values. In the following paragraphs I describe how these aspects of the resources influence the structure of the context. One of the common processes employed in System Contexts involves practical work. Practical work is often used in school science to motivate and interest students. Interest has been defined as the extent to which someone will willingly spend their time on an activity (Kleine, 1976). Using interest to determine whether students are engaged in meaningful learning is questionable. To infer interest on the part of students because they spend time on an activity is problematic. Ainley addresses this issue directly in Chapter 6. As a consequence, to infer that learning has occurred just because students spend time on task is problematic for several reasons. First there could be an attitude behaviour inconsistency, hence expressed interests and manifested interests may contradict each other. Second, according to Gardner (1985) there are at least three components to interest: subject matter, extent of commitment and the purpose of the activity in question. The presence or absence of these components can be inferred, to some extent, from the students’ classroom dialogue (J = Joanne; L = Leonie).

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This is, there isn't an aim.

L: No, just write the heading. J:

OK.

L: And then the table.

As a further example, the extent of commitment may be dependent on the student’s intention to behave. L: Frances, she said we're doing three, then just swapping, so does that mean we only need to rule up three for each person? F: I'd rule up more in case you get through more maybe. J:

Do we want to get through more?

This suggests that Joanne, at least, is not guided by an inherent interest in the task, instead she is trying to discern the tangible outcomes as required by the teacher. The extent of the students’ commitment and purpose seems to be determined by their perception of the teacher-required outcomes. This in itself is not an issue, providing it does not become the sole focus of a learning activity. If it does, it will limit the value of the situation as an appropriate situation (by the criteria that I am applying in this chapter) and render the operations within the situation purposeless for the individual student. Furthermore, the extent of the students’ commitment does not necessarily allow them to display their interest. In analysing the data it was difficult to identify student interest because opportunity and the classroom social practice constrained student participation. Similarly, purpose may result in a variance in student manifested and purported interest because of the manner in which they wish to use the skills and knowledge they gain. Therefore student interest as a measure of an appropriate context or indeed as a means of generating student interest is open to discussion and contention. In promoting a more appropriate context from which to learn, many assume that generating student interest or fostering a change in attitude will increase the appeal of science. Often in the past, attitudes and values were thought to have no place in traditional school science and issues were related to the aftermath of a science lesson. For example, after learning about fission and fusion reactions, students may then have had an opportunity to briefly voice their opinions of nuclear energy. More recently, some socio-cultural context-based strategies have been proposed on the grounds of spawning student interest in the topic by considering their (the students’) issues, attitudes and values. Many of the ‘teach science in context’ activities involve practical tasks. The Fibres lesson followed the same principle. The teacher had identified students’ interest in practical work. But she also indicated that this interest was likely to result in mayhem because students did not take adequate notice or sufficient time to

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discern the exact task requirements. This again signals the closed approach adopted and the lack of consideration of students' ideas. As the teacher explained in the interview, her strategy to focus their attention and hence discern their task was to ask students to underline key terms to direct their activity: The big problem I face with students doing pracs, is that they are so keen to get into the prac, they never ever, ever read the prac. So you give them time to read the prac, and they still don’t know what to do. So I guess my way of doing it is, particularly with kids with short term memory problems etc., is to go ahead and just get them to underline it as I said, because otherwise it becomes a huge sea of words and the kids can’t sort of focus on the main parts.

But the underlining task did not seem to make explicit the desired goal for the students. Yet from student behaviour in response to the underlining activity, the teacher believed she was successful in keeping Joanne interested. You see Joanne was just looking around the ceiling etc., like that and I am sure she was off with the birdies then, but to someone like Joanne, it can actually tend to keep her back on task.

This is problematic, as we have already indicated, time on task is not necessarily an indication of interest. Neither does spending hands-on time on an activity equate with students being engaged in minds-on meaningful learning. Earlier in this chapter I suggested that cognising beings use perceptions of relevance, beliefs, values and attitudes, I would also suggest that these are closely related to levels of interest. The student may be interested because of the perceived relevance or value of the task in which they are engaged. However we cannot assume that because they are spending time on task they are interested, nor can we assume that busy hands-on work leads to minds-on development of the science concepts being taught. The students appeared to be engaged and interested in the practical work. However, though the materials, such as the fabrics made available, were thought to be familiar to the students, the nature of the task resulted in few links being established between students’ out of school experiences and within school experiences. From the data examined it is possible to infer that this was because the operations required for practical work were not conducive to helping students establish links between the System and Socio-cultural Contexts. There is a body of research to indicate that much practical work undertaken in science classrooms is conducted for the purpose of busy work rather than cognitive or skill development. If one considers people to be resources, then it becomes clear that their attitudes and their interests have a bearing on the development of an appropriate context. Students and teachers bring with them a set of attitudes. “Attitude” is an abstract concept that involves both a cognitive and affective dimension. For Shaw and Wright, (1968) attitudes involve affective reactions towards an object or situation based on the belief system of the individual. In 1975, Gardner’s review of the term “attitudes” in science education highlighted a distinction between scientific attitudes and attitudes toward science. For the most part, people have advocated the use of contexts to learn science, in order to promote positive attitudes toward science, rather than changing scientific attitudes. For example, many studies (as evidenced in

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Gardner, 1975 and in Jones, Porter and Young, 1996) show that students have negative attitudes to science. Negative attitudes to science are a problem because attitudes affect interest and in turn, as reported by Gardner (1975) and Jones, Porter and Young, (1996), interest is determined by commitment to task and purpose. If students see no purpose, or have no commitment to the task then they are unlikely to develop an understanding that is consistent with the scientists’ view of science (I = Interviewer; J = Joanne). I:

And was there any, was there anything new in that lesson? For you.

J:

I can’t remember it, really, um.

I:

[laughs]

J:

No, 1 suppose I go into class and sit through the class, then come out, and then it's just all forgotten.

I:

Um hm.

J:

And then I go back into class and I just take, I don’t um, really. I listen and I remember it for next class but I don’t think about it in between all the classes.

This exchange suggests a dichotomisation, in the student’s mind, of in class and out of class activity that carries unfortunate implications for the perceived purpose of class work. Issues related to attitude surfaced during the interview, and provide some understanding as to why the lesson unfolded as it did. For example, the teacher said: I never really wanted to teach this unit. I know it is on the CSV and I can see the importance of it, but really there are other things in science I think that would capture kids' minds, and they could get a lot of this say through Textiles. I really don't know the importance of teaching it in a science unit. I think it is just another way that kids can get turned off science. The choice of pracs, if they are fairly directed, and the students are not given the choice I think is one thing that turns the students off A lot of writing turns them off. Some of the projects we have given them in the STA [Science, Technology and Art] area in Year 7 were really boring, so they are not corning in with that, you know, really looking forward to the science class and I just wonder what they are going to do to our science numbers [enrolments] later on.

The teacher’s negative attitude regarding the theme may have been communicated to students during the Fibres lesson. As a consequence, for both the teacher and the students, the lesson became one of meeting externally prescribed requirements. A few unshared issues underpinned the tasks set, the resources provided and the communication engaged in. Given that the teacher is one of the resources in this classroom, her role in making the context more appropriate for students is crucial. However, in starting the lesson, some time was spent asking the students to create a title page. The teacher’s reason: So it is just part of the productivity that I am trying to get them to understand, and it is also important part of teaching I think that the parents can have a look at the students’ books and see that there has been care taken and that this [inaudible]. Otherwise kids

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SUSAN RODRIGUES tend to just scrawl down and they really don’t have a very, some people it comes naturally but for a number of students it doesn’t so um, I think book work is very important and the way you set up the book is important, it sort of gives them a sense that they are achieving and they can look back [inaudible].

Therefore clearly her values were more closely associated with producing a wellpresented record of work. But the students may have had difficulty discerning this value. Interview comments also show that students thought that the lesson on the theme of fibres was similar to other science lessons they had encountered (J = Joanne). J:

So I just. I switch into Science when I’m in Science.

I:

Um hm.

J:

And then I switch out of it as soon as it finishes.

I:

Um hm.

J:

As soon as it finishes.

I:

Um hm, um hm.

J:

When the bell goes.

I:

Um hm. OK. Um, let’s see, so you felt there wasn’t anything new in that class for you.

J: Um, Not really.

The students made links between their existing ideas and new information, and this was visible in the transcripts involving students working with their peers, but in interview the students did not think they had made any new links or enhanced their learning. It is quite possible that clarifying an existing meaning is not perceived by students as “new knowledge”. The manner in which resources, be they people or artefacts, are used is vital in terms of creating student appropriate contexts. The lesson on fibres, though using everyday resources, did not become a student appropriate context, because the operations involving the artefacts and people were limited and remained largely in the System Context, even though the actual materials were drawn from a broader context.

8. WHERE TO NOW? In contrast with the students’ relatively successful negotiation of meaning when working in small groups, many opportunities to establish links to the students’ existing ideas regarding fabrics and fibres were largely missed by the teacher. These disparities illustrated by the various quotes throughout this chapter help to identify the emphasis placed on specific elements of the operations used within the lesson. With hindsight, during video stimulated recall, the teacher commented: They’ve got quite a wide range of knowledge about different fabrics, wider than I thought they may have. I thought they may have been confined to mainly basic raw

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fibres. They seem to know quite a lot of them like poplin and chiffon so maybe they have picked that up through the textiles.

There is no doubt that the notion of teaching in context has gathered momentum. Curricula, textbooks and research papers advocate the use of everyday relevant experiences and real issues to teach science. However this itself will not result in the utilisation of more appropriate contexts at the chalk face. In this chapter, I have suggested that three aspects of context must be considered when analysing the use of every day familiar situations to teach science in context. These aspects of communication, resources, and processes can exist within three contexts: an Internalised Context, a System Context and a Socio-cultural Context. Appropriate Contexts are constituted not only in the situation that is explored but in the links made by the individual between existing and new knowledge. General acceptance of the importance of promoting this link between the situation experienced and individuals’ knowledge would require science to be taught using Internalised Contexts. By simply adopting a theme, and providing the necessary material artefacts, the initial shift is toward a Socio-cultural Context. However, if the artefacts are used in processes that are similar to those currently employed in a System Context, then the Socio-cultural Context becomes constrained (and constraining) and it becomes difficult for the individual to construct an Internalised Context. The lesson on the theme of fibres failed as an appropriate context for the students (and the teacher). It simply shifted some elements of the communication, resources and processes from the System Context to the Socio-cultural Context, but overall, the majority of the elements remained within the System Context . For example, the lesson used resources that were familiar to students, but did not employ operations that allowed for open investigation or a sharing of understanding between the teacher and student, although it did allow for sharing of meaning between students. The majority of the processes employed were school practice determined. Teacher and students needed to be seen to be meeting the requirements enshrined in the curriculum. Highlighting and underlining, for example, became meaningless and purposeless since the students had no concept of the importance or understanding of what they were underlining and highlighting. But they engaged in the task, performed the task, demonstrated the expected processes and remained, largely, none the wiser. The task involving familiar resources has to be purposeful for the student. If it isn’t, then it is unlikely to provide an opportunity to access an appropriate student context. Trying to see the science through the window of society events and episodes because they involve society will not work unless the window is made transparent by those sharing the ideas or kept open by those interested in the ideas. This requires a change in the nature of communication, (between students and teachers), a more informed use of resources and a change in the type of processes to allow for more open communication. The use of Fibres as an Internalised Context did not work in this lesson because the students (and the teacher) did not identify with the purpose of the “theme” and consequently did not move beyond the routine completion of the tasks mechanically. For example, instead of simply identifying which group had

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constructed the largest list of fabrics and fibres, the teacher could have used the list to explore and clarify the students’ understanding of the terms fabrics and fibres and to analyse which materials listed fell into which category. In the time it took to ask each group to read out very similar lists, the teacher could have managed a discussion about classifying the materials. What sorts of categories would the students use? What sorts of categories would scientists use? This would have meant that the 'listing' task was more than busy work. There is evidence of the recent development of units of work or modules that focus on topics and applications that are familiar to students, through which the concepts specified by curriculum documents could be taught. This is a start, but it should not be viewed as necessarily teaching in student-appropriate contexts, because this will only be possible if the following are addressed: – A change in communication patterns between students and teachers as both seek to determine each other’s understandings. The teacher must be sensitive to the students’ perspective and knowledge of the topic and work with this perspective. This is possible when the conversation between students and teacher becomes bi-directional rather than unidirectional. – Using hands-on and minds-on processes that are common to the Sociocultural Context, the System Context and the Internalised Context. This would involve planning investigations that are more open rather than recipe or didactic in character. Such investigations would enable students to mimic more closely the behaviour of scientists while helping them make more public their understanding of particular science concepts and processes. – Meaningful sharing and engagement ofthe resources and, as a consequence, a heightened awareness of the beliefs and value systems associated with the concepts being promoted in the classroom. Issues such as drink driving, fibres, global warming and genetic engineering could be powerful vehicles through which to learn science, but only if communication, resources, and processes are encountered on more than just the Socio-cultural Context but are engaged in by students within an Internalised Context. The nature and mechanism for engagement will affect the students’ ability to access the science. If learning is to be enhanced, the contexts need to be more than themes or Socio-cultural Contexts through which we continue to use didactic practices and System Contexts. The operations that are used while working with these themes will create more internalised contexts. Topics can provide the blueprints for appropriate learning contexts, whereby we expect students to explore and relive their everyday world in the light of science thinking scaffolded by the teacher and in the light of the students’ own science thinking. But these blueprints will only be realised if we put in place operations that allow both the teacher and student opportunities to explore each other’s thinking. If use of a topic or theme does not create a situation in which classroom participants are able to engage in meaningful negotiation of each other’s meanings, the context will not be “appropriate”, for either the student or the teacher.

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If we accept that cognition is a process of organisation and reorganisation of personal subjective experience, involving a reinterpretation of previous, current and possible actions and conceptions, then appropriate contexts are more than the situation ‘out there’ (ie System or Socio-cultural Contexts). Appropriate contexts for learning are an interwoven fabric, of which the individual is an integral part. Indeed, if we continue to perceive context as residing beyond the individual, then we will continue to promote the notion of knowledge as some form of corporeal object . This does not fit in with the constructivist view of the dynamism of knowledge nor with the notion that contexts include the individual. Perhaps in moving our context rhetoric into “reality” we have simply only heeded teaching in context, but not considered learning in context by moving from the System Context to a Sociocultural Context without due attention to the Internalised Context within which learning occurs. As can be seen from the fibres lesson, simply immersing students in everyday situations in which they encounter science concepts, even where they discuss these in groups, does not necessarily lead to the students establishing or modifying links between their existing knowledge and their new situation. This suggests that “teaching through immersion” does not necessarily lead to a shared understanding of the science concepts advocated by the scientific community. Using themes is simply the first step; providing an instructional blue print. For the blueprint to produce constructions that are more akin to those accepted by the scientific community and more relevant to the student, the nature of communication, resources and processes need to be carefully reconsidered and employed so as to work more effectively with the student. Context must be viewed as a construction by the individual. Most importantly, teachers must be assisted to promote the connections between contexts and the students’ evolving scientific knowledge. The analysis reported in this chapter suggests that if teaching in context is to facilitate learning in context then the nature of communication, resources and processes need to be employed carefully to engage students in more meaningful ways. This will only be possible when we afford equal status to the System Context, the Sociocultural Context and the Internalised Context. REFERENCES Aotearoa New Zealand Ministry of Education, (1983) Science in the New Zealand Curriculum, Wellington, NZ: Ministry of Education. Barnes, D., and Todd, F. (1977). Communication and learning in small groups. London, Routledge and Kegan Paul. Belenky, M. F., Clinchy, B.M, Goldberger, N.R and Tarule, J.M. (1986). Women’s ways of knowing: the development of self, voice, and mind. NY Basic Books. Byrne, M and Johnstone, A. (1988) How to make science relevant. School Science Review, 70,43-45. Cazden, C. (1988). Classroom discourse. Portsmouth, NH: Heinemann. Choi, J-S., and Song, J. (1996). Students’ preferences for different contexts for learning science. Research in Science Education, 26 (3), 341-353. Department of Education and Science (1979). Science Progress report 1977-1978. London, Department of Education and Science. Edwards, A. D., and Westgate, D. P. G. (1987). Investigating classroom talk. Lewes, Falmer Press.

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Eijkelhof, H.& Lijnse, P. (1988). The role of research and development to improve STS education: experiences from the PLON-project. International Journal of Science Education, 10, 4, 464-474. Gadamer, H.-G. (1990). Truth and method. New York: Continuum. Gardner, P. L.(1975) Attitudes to science: A review . Studies in Science Education, 2, 1-41 Gilbert, J. (1990). Secondary school students talking about science: Language functions, gender and interaction in small group discussions. Unpublished Masters thesis, Victoria University of Wellington, Wellington. Gilbert, J., Boulter, C and Rutherford, M. (1998) Models in explanations, part 2. International Journal of Science Education 20(2) 187- 204. Hodson, D (1996). Laboratory work as scientific method: three decades of confusion and distortion’ J.Curr Studies, 28, (2), 115- 135. Hodson, D. (1992). Assessment of practical work; some considerations in philosophy of science. Science and Education, 1, 115-44. Huibregste, I., Korthagen, F. and Wubbels, T. (1994) Physics teachers’ conceptions of learning, teaching and professional development. International Journal of Science Education, 16 (5), 539-561. , Jenkins, E. (1990). Scientific literacy and school science education. School Science Review, 71 (256), 4351. Johnson, S. (1987). Gender differences in science: parallels in interest, experience and performance. International Journal of Science Education, 9 (4), 467-481. Jones, J., Porter, A., and Young, D.J (1996) Perceptions of the relevance of mathematics and science: Further analysis of an Australian Longitudinal Study. Research in Science Education, 26 (4) 481-494. Kahle, J. (1985) Women in science: a report from the field. Falmer Press. Kleine, P. (1976) Psychological Testing: The Measurement of intelligence, ability, and personality. London: Malaby Press. Lave, J. and Wenger, E. (1991). Situated learning: Legitimate peripheral participation. New York: Cambridge University Press. Light, P., and Butterworth, G. (1992). Context and cognition: Ways of learning and knowing. London Harvester Wheatsheaf. Miller, R and Driver, R (1987) Beyond processes. Studies in Science Education, 14,33-62. Qualter, A. (1993) I would like to know more about that: a study of the interest shown by girls and boys in scientific topics. lnternational Journal of Science Education, 15 (3), 307-318. Ramsden, J (1997) How does a context based approach influence the understanding of key chemical ideas at 16+? International Journal of Science Education, 19 (6) 697-710. Rodrigues, S. (1992). Confusion in Learning Oxidation and Reduction Concepts. New Zealand Science Teachers Association Journal 69 (Autumn issue), 30 -33. Rodrigues, S. (1993). Oxidation and Reduction: Some form six units. Centre for Science and Mathematics Education Research, University of Waikato, Hamilton, New Zealand. Rodrigues, S. and Bell, B. (1995) Chemically speaking: Students’ talk during chemistry lessons. International Journal of Science Education, 17 (6) 797 -809. Roth, W. M and Roychoudhury, A. (1993). The nature of scientific knowledge, knowing and learning: The perspectives of four physics students. International Journal of Science Education, 15 (1), 27-44. Salters (1996) Salters Science Key Stage 4. Oxford: Heinemann. Salters’ Science Project. (1990). Science: Double Award (MEG Syllabus no 1764). A brief summary of the course and the supporting materials available.York, University of York Science Education Group. Association for Science Education (1986-1991) SATIS 14-16 Units. Herts., UK: ASE. Shaw, M. E. and Wright, J. M. (1968) Scales for the Measurement of Attitude. New York: McGraw Hill. Sinclair, J. McH., and Coulthard, M. (1975). Towards an Analysis of Discourse. London, Oxford: University Press. Stinner, A. (1994). Providing a contextual base and a theoretical structure to guide the teaching of high school physics. Physics Education, 29, 375–381. Von Glaserfeld, E. (1987). The Construction of Knowledge. Seaside, Intersystems Publication. The Systems inquiry series. Watts, M and Jolifi, Z. (1998). Towards critical constructivist teaching. International Journal of Science Education, 20 (2) 173- 186. Yager, R. E. (1983). Science Technology Society as reform, School Science and Mathematics, 93, 145151.

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CHAPTER TEN The Role of Gesture in Co-Constructing Mathematical Understanding

In mathematics, as in other forms of inquiry, imprecise, inadequate, or incomplete information impedes effective communication and, ipso facto, effective understanding. Lacking an understanding of the formal language of mathematics, students in collaborative learning settings often attempt to dynamically co-construct an understanding of problems in order to solve them. The question of how coconstructions occur in opaque learning contexts, or, where problems are ill-defined from the participants' perspective, is of theoretical interest to those studying learning and development. Indeed, a pragmatic characterisation of development is the process of constructing ways of understanding in order to solve a never ending succession of unfamiliar and increasingly more difficult problems (A. Brown & Reeve, 1987). Among other issues, it is important to identify transition events in collaborative learning that reliably signal participants' changes in understanding, independent of whether that understanding is sufficient to solve problems correctly (A. Brown, Campione, Reeve & Ferrara, 1992; Siegler, 1996). Needless to say, transition events are notoriously difficult to identify and are likely to differ as a function of, among other factors, participants' domain knowledge, interest, task engagement, metacognitive abilities and motivation, all of which are likely to covary as problem solving collaboration proceeds. Here, we explore the modest claim that gesture serves a communicative function in collaborative problem solving settings by signaling the quality of individuals' ongoing understanding. We will also attempt to assess the more contentious claim made by Goldin-Meadow and her colleagues that some forms of gesture signal transitions in the growth of understanding. The focus of our analyses is on the way that two high school students collaborate to solve a "speed-time" problem and their attempts to develop an understanding of graphical representations of speed as a function of time. In the first part of this chapter, we review perspectives on collaborative solving in order to make explicit our meta-theoretical orientation and to illustrate why we believe that gesture is an important feature of dyadic problem solving interactions. Second, we review research on the role of gesture in communication and consider the limited work that has explored its significance in learning settings. Third, we offer evidence in support of the claim that gesture signals both communicative difficulties and shifts in understanding by analysing students' interactions in their attempt to solve a 231 David Clarke (ed.), Perspectives on Practice and Meaning in Mathematics and Science Classrooms 231—253. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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particular “speed-time’’ problem. Fourth, we conclude by outlining the beginning of a possible framework for analysing the role of gesture in learning settings more generally. 1. COLLABORATIVE PROBLEM SOLVING The claim that working with others (peers, friends, siblings, parents, teachers, or other adults etc.) collaboratively to solve problems may affect the understanding of the individual collaborators is uncontroversial. However, answers to the question of how and in what ways problem solving collaboration affects changes in understanding tend to be more controversial in that responses tend to reflect differences in theoretical frameworks as well as values about what constitutes learning (Damon, 1991; Strayer & Moss, 1989); in other words, answers reflect differences in metatheoretical perspectives. For example, in a well-cited analysis of Situated Cognition and the Culture of Learning, J. Brown, Collins and Duguid (1989) claimed that situated cognitive acts tend to reflect inventiveness and expediency in a way that abstracted cognition cannot. In support of their argument, the authors cite Lave’s (1988) example of an individual on a diet plan who managed to solve a fraction problem (2/3 of 3/4 of a cup of cottage cheese) in an inventive fashion. However, in her commentary on J. Brown et al.'s (1989) paper, Palincsar (1989) rather courageously noted that it is unclear what the dieter had learned or whether it would be useful in solving other kinds of rational number problems. Palincsar’s query is relevant for researchers interested in learning for at least five intimately interrelated reasons. First, it raises the general question of whether a particular act of learning has contributed to the learner’s intellect in a functional manner (see similar comments by some developmentalists who have conducted research on the mathematics skills of unschooled children: e.g., Guberman, 1996; Nunes & Bryant, 1996; Saxe, 1991). Second, it leads to the question of how best to describe and assess the fruits of learning. Third, it raises questions about what kind of tasks, contexts, settings and, indeed, talk should be used to study learning and development. Fourth, for those interested in socially shared and distributed cognition, it raises interesting questions about the unit of analysis: should it be the individual, the dyad, or some larger ensemble (Bruner & Bornstein, 1989; Fischer & Granott, 1995; Valsiner, 1988; Winegar, 1997)? Fifth, how should we describe the changing dynamic in collaborative learning interactions (Karmiloff-Smith, 1992)? The problem of how best to answer the matrix of questions raised in the previous paragraph has plagued and has been furiously contested by those interested in how social relationships contribute to cognitive growth. Much of the research conducted over the last twenty years on peer collaboration and its effects on cognitive development has been nominally conducted within either a Piagetian or a Vygotskian framework (for reviews see: Azmitia & Perlmutter, 1989; Garton, 1992; Tudge & Rogoff, 1989; Tudge & Winterhoff, 1993; Tudge, Winterhoff & Hogan, 1996). However, other frameworks are emerging (cf. Bronfenbrenner, 1993; Bidell & Fischer, 1994). Researchers working within a Piagetian framework tend to argue that cognitive growth is likely to occur when participants have different perspectives

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or abilities and engage in sociocognitive conflict (Doise & Mugny, 1984; Doise & Palmonari, 1984). Moreover, according to Piaget (1965), sociocognitive conflict is more likely to occur when children interact with peers than if they interact with adults because of status differences (DeVries, 1997). Sociocognitive conflict is somewhat difficult to define, however. In contrast to the sociocognitive conflict hypothesis, Vygotskian-oriented researchers have suggested that change in understanding occurs in collaborative problem solving context because participants develop a shared understanding of the meaning of problems (A. Brown & Reeve, 1987; Tudge & Rogoff, 1989). However, the research that has been conducted on the effects of dyadic collaboration on learning has not provided strong support for any particular hypothesis. We suggest that part of the problem is methodological in nature in that inappropriate analytic research methods, dependent on linear model assumptions, have been employed to study variability and change in problem solving competence. In other words, the kind of stories that can be told about the role of social interaction in cognitive change have been highly constrained by the analytic methods employed to study the fruits of collaboration. Recently, a number of developmental psychologists (e.g., Fischer & Granott, 1995; Karmiloff-Smith, 1992; Siegler, 1996) have used so-called microgenetic methods to study individual differences in changes in understanding. Fischer and Granott's (1995) work is of particular interest because it demonstrates both the theoretical and methodological difficulties and the advantages inherent in a microgenetic analysis of dyadic problem solving. In Fischer and Granott's task, two students were video-taped working together (for approximately four hours) trying to understand the operations of an electronic toy robot. The authors were interested in analysing the way the students developed an understanding of the device and the language they employed to describe their activities. Fischer's skill theory (i.e., description of the transitions between actions, representations and abstractions) was used to characterise transitions in the development of the ensemble's understanding. Three aspects of Fischer and Granott's analyses are worthy of comment. First, the skill level of the ensemble differed in terms of whether they were communicating with each other more generally or trying to understand the operations of the robot. Second, initial attempts to understand and communicate about the operations of the robot appear to be largely gestural in nature. (It should be noted that even though some of their transcripts illustrate the role of gesture in problem solving communication, Fischer and Granott do not comment upon this fact.) Third, in terms of Fischer's skill theory, the ensemble's understanding of the operations of the robot changed from an action-level (largely gestural) to a representation-level understanding over the course of the interaction sequence. In our view, Fischer and Granott's work illustrates well how people work collaboratively to solve problems. It also illustrates how the quality of the interaction changes over time, and the role played by gesture in problem solving interactions.

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Gesture is thought to be an inherent aspect of communication that appears to be present from birth. The pointing gesture has been identified as the first definite gestured reference to objects, and the possible link between pointing and language acquisition has been emphasised (Werner & Kaplan, 1963). Pointing has been observed in children as young as three months, occurring frequently in conjunction with speech-like vocalisations (syllabic sounds) but seldom with other vocal sounds (Masataka, 1995). Several categories of verbal behaviour has been observed in infants during the period 6-months to 2-years old (Rome-Flanders & Cronk, 1995). Of particular relevance is the category of prelexical comment, which comprises the non-lexical utterance of standard phonemes accompanied by strong visual or gestural indicators of attentional focus, such as pointing, touching, or intent gaze. Prelexical comments emerge at 12 months of age, remaining stable until the age of 18 months, and decreasing to zero at 2 years, with girls making more prelexical comments than boys. Prelexical comment may be an early manifestation of identifying and commenting on a topic within a single process. Its disappearance by the age of 2 may be because of the availability of multiple word utterances. Tomasello and Camaioni (1997) noted that human neonates engage in dyadic interactions with adults, although it is not until the second six months that human infants begin to engage in triadic interactions with adults and objects, such as engaging in social referencing, coordinating attention with adults to objects and events, monitoring adult gaze, and displaying emergent imitative behaviour. Deictic gestures such as pointing, showing, offering and ritualized requests also appear in the later part of this period (but see Masataka, 1995 for evidence of earlier emergence of pointing). Tomasello and Camaioni (1997) argue that human infants’ gaze monitoring indicates an awareness of the effect their signals will have on the recipient, while infants’ early deictic gestures are often declarative and have a genuine communicative intention. It also appears that pointing plays a part in the achievement and maintenance of joint-attention with respect to the infant’ response to mother’s pointing (Morissette, Ricard & Decarie, 1995). From the age of 12 months, infants will follow their mother’s gaze to a target, but this behaviour becomes reliable only at 18 months. By contrast, infants will follow their mother’s gaze and pointing reliably at 15 months. Prelexical infants also use gestures symbolically; for example, forming a cup with their hand to indicate the desire for a drink, making a swooping movement with the hand to indicate a slide (Acredolo & Goodwyn, 1997; Goodwyn & Acredolo, 1993). The evidence, then, is that gesture is an important precursor to spoken language, having at least two functions: first, as a means of achieving and maintaining jointattention, and second, as a way of communicating symbolically. But what is the significance of gesture after the prelinguistic period? While the link between gesture and cognition has been recognized for a considerable period (Efron, 1941; Werner & Kaplan, 1963; Wundt, 1921/1973), the function of gesture once humans attain spoken language has only recently become the focus of research attention. It was not until the 1970’s that Kendon proposed an essential unity between gesture

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and speech (Kendon, 1972, 1980). Kendon (1972, 1980) contended that the connection between gesture and language lies on a continuum, from (spontaneous) gesticulation through emblem (V for victory, crossed fingers etc.) and pantomime to formal languages, such as American Sign Language. The nature of spontaneous gesticulation and its relation to speech has been further investigated by McNeill (1985, 1992) in the context of narrative discourse. McNeill proposed a five-level categorization of spontaneous gestures, each of which plays a different part in discourse. First, iconic gestures are closely related to the semantic content of speech, enacting the same act referred to in speech. Second, metaphoric gestures, like iconics, are illustrative, but they enact an abstract image rather than a concrete event or object. Third, beats (or batons: Efron, 1941; Ekman & Friesen, 1969), which are typically finger or hand flicks, serve as markers in discourse. Thus they occur with the introduction of new themes, information and structure. Fourth, cohesives, which may be iconic, metaphoric, or pointing gestures operate to connect related but temporally separated parts of the discourse. Finally, deictic gestures, or pointing, are used to indicate of concrete events and objects as well as abstract concepts. Gesture also seems to be implicated in the way that understanding of the world develops. The classic Piagetian position is that understanding of the world is based upon action: direct actions upon the surrounding environment gradually become interiorised through a process of reflective abstraction. It is possible that gesture, seen as action on the world, may be central to cognitive development. Indeed, several researchers into children’s early mathematical understanding have emphasised the essential role that gesture, including pointing and finger patterns (e.g. raising and lowering fingers), plays as children construct and give meaning to the number sequence (Steffe, 1988; Steffe, 1991; Steffe, 1992a; Steffe & Cobb, 1988; Steffe, von Glasersfeld, Richards, & Cobb, 1983), create composite units (Steffe, 1992b, 1994), and begin to develop fraction understanding (Steffe & Olive, 1996). For Steffe and his colleagues, the gestures used by children in the process of acquiring mathematical understanding have at least three functions for the children themselves: (1) as a means of monitoring counting and partitioning activities, (2) as a representation of countable items, and (3) as an enactment of an abstracted mathematical entity. Moreover, Steffe (1988) has observed that: The mathematics of children can be known by an adult only through interpretative experiences of the child’s language and actions (P. 137).

The interpretation of children’s language and gestural actions has received further consideration in the context of children’s problem-solving performance. In a well-cited series of studies of children’s solutions and explanations of conservation and mathematical equivalence problems, Goldin-Meadow and her colleagues observed that children who were in the process of acquiring a novel concept sometimes gestured when explaining their understanding of that concept (Alibali & Goldin-Meadow, 1993; Church & Goldin-Meadow, 1986; Goldin-Meadow, Alibali & Church, 1993; Goldin-Meadow, Nusbaum, Garber & Church, 1993; Perry, Church & Goldin-Meadow, 1988, 1992). It was noted that children’s spoken

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explanations were often wrong while their gestures indicated a different, sometimes fuller, conceptual understanding (“gesture-speech mismatch”). Goldin-Meadow and colleagues suggested that such children were experiencing cognitive dissonance, a state in which several ways of solving a problem are simultaneously activated. Goldin-Meadow and colleagues also found that such children were more likely to benefit from instruction in the particular concept, that is, they were ready to learn more about that concept. Goldin-Meadow and colleagues further proposed that the occurrence of cognitive dissonance as evidenced by gesture-speech mismatch is an index of impending cognitive change. The effect of gesture-speech mismatch on recipients was further explored in a series of studies in which adults (both teachers and non-teachers) and children (Alibali, Flevares & Goldin-Meadow, 1997; Goldin-Meadow, Wein & Chang, 1992) viewed video vignettes of children solving problems and exhibiting both gesturespeech match and gesture-speech mismatch. Goldin-Meadow and colleagues have claimed that the results of these studies demonstrate that the occurrence of gesturespeech mismatch does indeed have an influence on the way in which both adults (whether or not they have teaching experience) and children interpret children’s conceptual understanding. However, these studies did not involve face-to-face situations; thus there was no opportunity to explore whether the dynamics of collaborative interaction would result in different findings. Few formal analyses of the significance of gesture in classroom learning or other settings have been conducted. While some mathematics researchers have included gesture as part of their protocol analysis of problem solving, they have typically regarded gesture as optional extra information rather than as possessing significance in its own right in the ongoing discourse. In general, references to gesture have been limited to noting their occurrence in protocol analysis (Goos, 1994), transcribing gestures that indicate order of operations (Gooding & Stacey, 1993), and reporting iconic gestures used by participants to enhance their verbal explanations (Forman, 1989; Hall, 1995). However, most researchers have not given any attention to the possible role of gesture in the construction of meaning. The work of Roschelle (1992), Roschelle and Clancey (1992), and Moschkovich (1996) goes some way to redressing to this neglect of gesture. All three studies were concerned with collaborative learning by peers using computer-generated examples. Moschkovich (1996) examined three pairs of 9th and 10th grade students who were exploring slope and intercept using a computer graphing program. Students were given an equation and its graph and were asked to predict what effect changing the equation would have on the graph. Only one pair of students made consistent use of gestures in their conversation. Most of the gestures involved pointing to the line on the computer screen, or using coordinated iconic gestures and conversation to clarify the meaning of, for example, “steeper”. Moschkovich observed that: The dialogue ... exemplifies the use of gestures and talk interactively to disambiguate the meaning of a description. Through repeated gestures.. . and coordinated talk.. . each student elaborated their understanding of the situations in which the description steeper would apply. This . . . highlights the importance of gestures in general.. . and specifically when describing graphical objects. Gestures were an integral part of these students’

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descriptions of graphs, and their language use might have been interpreted differently without the videotape as a source of data (p.254).

However, she did not consider the implications of these claims about gesture. Roschelle (1992) reported an investigation of two students engaged in discovery learning about velocity and acceleration in a computer microworld. Roschelle noted that the students’ language was ambiguous, quite unlike written scientific language, nor did their behaviour exhibit anything resembling the hypothetico-deductive methodology of scientific investigation. However, the students managed to converge on a deep and novel understanding of velocity and acceleration. Roschelle claimed that the gestures made by the students, that comprised pointing to the computer display, using the mouse to make gestures on the computer screen, tracing the display with fingers or with the mouse were essential to the interpretation of the accompanying speech. He suggested that the role of gesture in this collaborative situation was that of “metaphors-in-action”, and were clearly understood by the recipient. The combined effect of speech and gesture was thus to produce for these students a concept similar to a physicist’s concept of acceleration. Rochelle and Clancey (1992) also studied the interaction of two students learning about acceleration using a microworld. As in the two studies already mentioned, the students pointed to objects on the computer display in order to clarify their references, and used iconic gestures to demonstrate their understanding of the concepts under consideration. Roschelle and Clancey therefore claimed a twofold role for gesture: to coordinate the students’ conversation and to draw each others’ attention to important features of the computer display, and to construct a shared understanding of the meaning of the concepts of velocity and acceleration. Roschelle and Clancey emphasised the importance of gesture in understanding the students’ interaction. Without gesture their verbal conversation was almost devoid of meaning: At the level of the significance of notations in activity, we draw attention to the richness of modalities available to students for perceiving and talking about meaning: In the specific examples we considered, one key aspect that students sought to understand was the relation between the arrow notation and changes of direction. To do so, they used gestures, metaphors and experimentation with the simulation. As we pointed out earlier, these resources are essential - without them, [students’] talk was gibberish. Indeed, in actual scientific practice and in everyday talk, simultaneous use of multiple modalities of experience in close synchrony is a prominent fact (p.450).

While these three studies, particularly the latter two, constitute a useful beginning of an analysis of the role of gesture in collaborative learning, it is arguable that they do not address several important theoretical and methodological issues. First, there is no consideration of the function of gesture in the complex processes of transitions in understanding. In particular, no attempt is made to link the use of gesture to the underlying mechanisms of cognitive change. Second, the description of the gestures made by the students is coarse-grained. A more precise description and classification of gesture (see for example Alibali & Goldin-Meadow, 1993; Bavelas, 1995; Bavelas, Chovil, Coates & Roe, 1995; Crowder, 1996; GoldinMeadow, Alibali & Church, 1993; Levy & McNeill, 1992; McNeill, 1992; McNeill,

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Cassell & McCullough, 1994) may have allowed insight into the nature of the interplay of speech and gesture, that may in turn have made possible a clearer understanding of the nature of collaborative cognitive change. Third, it is possible that the claims made about gesture in these three studies may be restricted to collaborative learning situations involving computer simulations. While the importance of physical or computer-based manipulatives in conceptual change is undoubted (Roschelle, 1996), the role of body movements in creating shared representations has been claimed to be crucial in achieving an understanding of relative motion (Hall, 1996). Whether gestures in the absence of concrete manipulatives have a similarly pivotal representational role in the creation of shared understanding is yet to be addressed. 3. INTERPRETING SPEED-TIME GRAPHS WITH GESTURES On the basis of our analyses of the collaborative problem-solving episodes reviewed, we contend that gesture serves several distinct but equally important functions in sustaining attention and elaborating meaning. Furthermore, it is evident that the tempo of the discourse (speed of speech and pauses) and its synchrony with gesture appear to signal phase changes in the problem-solving episode. (We do not mean to imply that the problem-solving sequences are linear in any sense, because in the episodes reviewed no definitive solution was reached. Indeed, an appropriate geometric metaphor needs to be developed to characterize the movement of discourse in collaborative problem-solving.) Given the relative absence of work on the role of gesture in collaborative problem-solving, we are interested in three general issues. First, we are interested in whether gesture acts as a cognitive amplifier. In particular, we are interested in the degree to which gestures signal shifts in attention to different aspects of the task. Second, we are interested in the degree to which gesture elaborates meaning conveyed in speech. As noted above, Roschelle has observed that interpreting collaborative problem-solving without reference to gesture is often quite difficult. Third, we are interested in whether the occurrence of changes in speech tempo and pauses, together with gestures, indicates some alteration in conceptual understanding. We regard this possible connection as having a functional similarity to Goldin-Meadow and colleagues’ claims concerning gesture-speech mismatch. As noted above, Goldin-Meadow and her colleagues have argued that such discrepancies may well signal impending changes in understanding. Indeed, the overarching concern of our analysis is the identification of transitional stages. Such changes would clearly be of importance in the acquisition of problem-solving competence. The topic of the mathematics lesson that we use as the basis of our exploration of the role of gesture in the co-construction of understanding was the graphical representation of speed as a rate. The lesson began with a teacher-led discussion of graph interpretation tasks in the school-produced mathematics booklet (see Figure l), with particular reference to a car slowing down and reversing, the speed of a ball

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thrown into the air at different points of its journey, a stone dropped from a great height, and a bouncing ball.

Figure 10.1 Speed-time graphs presented to students She then discussed a student’s bus trip to school, drew a schematic diagram of the journey, and then constructed a speed-time graph on the blackboard. The teacher then led a class discussion interpreting this graph, pointing out different aspects of the motion depicted, including constant acceleration from zero velocity, constant velocity, and deceleration to zero velocity. Students were then instructed to work in pairs. The first task was to assign appropriate speed-time graphs to five different

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situations. The second task was to consider a speed-time graph of a school bus travelling on its morning run (see Figure 2), and from that graph describe the bus’s journey. In our analysis, we review and present protocols of three brief exchanges between two students, Fran (“F”) and Patricia (“P”). The protocols include the students' speech and gestures, and, where appropriate, a suggested purpose for those gestures. A characteristic of collaborative problem-solving episodes is their brevity, with rapid, almost overlapping, turn-taking between participants. As we were interested in speech tempo and the occurrence of pauses, those are also recorded in the protocols. Speech that was slower than normal is underlined, while pauses of or greater than one second are notated ''***''. Example 1. Discussion of speed-time graph for ball thrown in the air

The first problem-solving episode we analysed involved deciding which of the graphs in Figure 1 best represented a ball thrown in the air. TURN

TIME

NO. AI

(secs) 0:00

SPEECH

A2

0:04

P: Ready, what do we do with ball thrown into the air? F: This

A3 A4

0:06

P: Which one?

0:07

F: I reckon

A5

0:09

A6

0: 17

A7 A8

0:20 0:22

P: Uh uh, I don’t reckon. It would go up *** it wouldn’t go up fast, and it would come down real fast, so. F: But it doesn't come down real fast. P: It does F:

nods head shakes head

A9

0:25

A10

0:27

P: It comes down faster than it goes up. F: No. What happens, is it goes really fast

moves pencil along desk

0:29

and then it slows down once it gets to the top, and comes *** it comes up and then slows down

0:37

and stops when it turns around. And then it comes up again

GESTURE

PURPOSE

points to material on table.

establishes joint-attention

tapping her page with pencil

selects appropriate graphs

repeats first gesture

hand and pencil move up into the air

represents acceleration of ball thrown into air represents acceleration of ball thrown into air represents acceleration of ball thrown into

THE ROLE OF GESTURE IN CO-CONSTRUCTING MATHEMATICAL UNDERSTANDING

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air 0:40

it goes down quite fast and then it slows down when it gets to the bottom

0:45

because it has to— P: [looking at book] Yep, this one. Because it doesn't, but it doesn't,

A11 0:46

hand and pencil describe a curve

sketches second half of parabola

051

that's like it goes up and

raises left hand and arm

represents acceleration as shown on graph

055

then kind of moves real

059

you can see how it

pauses at top, extends index and middle fingers, hand moves clockwise

represents acceleration as shown on graph focuses jointattention on specific part of graph

1:00

But that one just kind of goes like that.

moves straight down to point at graph

1:02

F: But it doesn't go *** Oo-op

moves pencil upwards from desk in straight diagonal line

1:06

slow down suddenly, it sort of gradually slows

further attempt to represent acceleration of ball

1:09

[another student briefly interrupts about another matter]

1:11

I reckon ***

A12

A13

1:13

Al4

1:14

F: D. ***

1:17

'Cause it doesn't really just go uomp

moves ruler up from desk in steep curve with a jerk

sketches way in which ball would not move

1:19

and then slow down straightaway

moves ruler slowly in arc

sketches way in which ball would not move

1:21 1:22

and then— P: All right, do D.

A15

P: D.

We identified several examples of gestures used to maintain joint-attention on the problem. For example, at A2 Fran focused the dyad's attention on one graph, then at A4 she shifted attention to another graph. The complex gesture made by Patricia at Al 1 ended with an emphatic point at yet another graph.

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There was also extensive use of gesture to expand upon speech. Fran's four gestures at A10 were particularly interesting. What Fran seemed to be doing was developing a notion of acceleration of a ball to maximum velocity. Indeed, her gestures seemed to describe the correct mathematical function. The problem then became to identify the appropriate graph. Patricia's gesture (at Al1) in response to this verbal/gestural statement was considerably more stylized. Taken in conjunction with the tempo of Patricia's speech, which slowed markedly at that point, we interpreted Patricia as trying to enact the acceleration described in another graph and to contrast her gestured description with Fran's. Fran then responded (at A12) with an enactment of acceleration described in yet another graph as if to compare the mapping between the two possibilities. The girls then settled on graph D as the correct solution, with a final demonstration by Fran (at A14) of the type of acceleration described in (possibly) graph F. This episode was also notable for the occurrence of slow speech and pauses occurring in conjunction with representational gestures. The co-occurrence of gesture, pause and slow speech in Fran's turn taking (A10) was a particularly clear example of a possible convergence of pragmatics and gestures in the conveying of understanding. While we were unable to identify any mismatches between gesture and speech in this episode, we considered that it provided a good illustration of the way in which gesture functions in dyadic problem-solving with respect to maintaining jointattention and elaborating meaning. In particular, we interpreted this episode as demonstrating considerable understanding on the part of both students of the correct mathematical function of acceleration, somewhat impeded by their uncertainty of how to map that understanding onto graphical representations. Example 2. Discussion of speed-time graph for bouncing ball The second episode of collaborative problem involved determining which graph best represented a bouncing ball. The students' exchange was as follows. TURN TIME NO. (secs)

SPEECH

B1

0:00

P: A bouncing ball. I think a bouncing ball would be like D.

B2

0:07

F: ** Or E.

B3

C:08

P: D. No, it wouldn't. Course it wouldn't. Stupid. Wouldn't be like E, it's not, a ball's not gonna, like—

B4

0:16

F: Go backwards. It's going to go woo-woo, woo-woo

0:20

It's not going to do that. It's going to go

GESTURE

moving hand buck and forth horizontally two and a half times

PURPOSE

Sketches ball's movement in horizontal plane Sketches ball's movement in horizontal plane

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moves pencil in hand back and forth once fast, both laugh.

0:25

sl.. .cchh!

0:27

It's gonna go

0:29

speed up

moves pencil up

sketches ball's movement in vertical plane

0:30

and slow down, it gets to the top and then.

moves pencil down

B5

0:34

P: [leaning back in seat] It's gonna go like this, it's gonna go like this

tapping page pencil

sketches ball's movement in vertical plane refocuses jointattention

B6

0:37

F: I reckon it would go like that

moves pencil up, then taps page with pencil

0:39

It would be like B again.

represents ball's acceleration THEN jointattention

B7

0:41

P: Like B? It's not like B.

B8

0:43

F: D.

moves pencil in hand up and down 5 times, hitting desk

represents ball's acceleration to maximum velocity when bouncing

B9

B10

0:44

P: Yeah, that's what I reckon. It'll go

0:48

bo-ing (sound only continues for first part of gesture)

054

except it'll bounce a little lower each time *** 1 reckon.

1:02

So I reckon it'll be like D.

1:03

F: Yeah. know.

Mm.

with

I don't

This very short episode also provided instances of gesture being used to achieve joint-attention. First, Patricia (B5) refocused the dyad's attention on her initial choice of graph D after the earlier discussion about graph E. Fran immediately responded (B6) to this by pointing to her own choice of graph B The first, and possibly most interesting, use of gesture to elaborate meaning was Fran's explanation of the type of acceleration represented by graph E (B4). Fran apparently misinterpreted the graph as depicting horizontal motion. Her gestures clearly indicated a ball moving in a horizontal plane, and involved a considerable elaboration beyond her speech. Fran used gesture again (B6) to enact her understanding of acceleration as shown in graph B. Finally, Patricia's gestures (B9)

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244

appeared to demonstrate her actual understanding of acceleration. As in the first example, we considered that her gestures were mathematically correct. As in the first episode, there was some evidence of synchrony between slow speech, pauses and explanatory gestures. Patricia’s sequence of gestures at B9 accompanied speech that was notably slower than the remainder of the episode, and was followed by a three second pause. We interpreted this to indicate some uncertainty on Patricia’s part, possibly because the collaborative problem-solving episode was converging upon the same graph as that selected in the first, thrown ball, example. Example 3. Discussion of graph representing the "bus trip" The final episode we reviewed involved examination of a graph (see Figure 2) purporting to illustrate the journey of a school bus.

Figure 10.2. The journey of a school bus The students’ attempts to describe the journey appear in the following protocol. TURN NO. C1

TIME (secs) 0 :00

SPEECH P: [reads] Consider th e following graph, the speed versus time of a school bus travelling to school on its morning run [sings do, do, do]

GESTURE

PURPOSE

THE ROLE OF GESTURE IN CO-CONSTRUCTING MATHEMATICAL UNDERSTANDING

C2 C3

0:12

*** Bus.

0:20

F: The journey *** the journey of a school bus.

0:26

P: A school bus, bus, bus.

0:34

[reads] (a) break up the graph into its various parts and write a story describing the journey of the school bus. Hm. [sings]

245

both writing in workbooks

0:47

F: Break up the graph into its various parts *** Peculiar.

052

P: Metres per second, this is in.

C6

0:57

F: The bus, wouldn't be that—

C7

0:59

P: It's going, it's going broom, broom, baaah

imitates someone at steering wheel of bus, uses both hands

enacts driving a bus

1:04

It picks someone up, right? *** Then it goes innuh

draws approximately 45 degree line in air

attempts to describe acceleration

1:10

then it goes uuuh

more or horizontal in air

attempts to describe constant velocity

1:12

Then it gets like out onto the highway or freeway or something

left hand moves left in open scoop

continues bus journey

1:15

It goes riiy, riiy

draws steep line in air

attempts to demonstrate acceleration

1:17

and then it gets to school and it goes uuun

moves down

attempts to describe deceleration

1:21

blap

hits desk

imitates stop

C8

1:23

F: No it goes uuh

draws on page

horizontal line

C9

1:27

P: No, you have to look at

tapping page

establishes joint-attention

1:30

this graph. So, first of all it's just picked someone up so it's accelerating to the speed limit there

points at graph

emphasises need for jointattention

C4

C5

Iess line

pencil

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ROBERT A. REEVE AND FIONA J. REYNOLDS

C11

1:37

C12

1:43

P

points to page

F: No, I reckon what it does—you left hand and know how they have those little side pencil move bits, it's sort of driving a bit along desk

focuses jointattention on specific part of graph sketches actual motion of bus

1:47

along there just to see where it can abrupt move right, hand and pull out, and then it pulls out pencil move forward faster

1:50

and then it goes really fast for a bit and then it goes "Oops, there's the bus stop."

hand and pencil move left), slow down and stop

1:56

P: Just when you're driving along

both hands sketch holding steering wheel

1:57

waiting for all the traffic

right hand makes 2 downward flicks

1:58

to go past, you're not going 10 metres per second ***

hands move up and down in contrary motion

2:04

Oh, I didn't really want it to do that.

gets out calculator, opens bag, pencils fall out

C14

2:13

F: So, it can be like the bus that comes out of, down ***

C15

P: [sings]

C16

2:18 (to 2:22) 2:19

C17

2:31

P: It's going 60, right *** I should have known that, that was dumb

putting calculator away

2:39

Kilometres per second, going 60 kilometres per hour, climb out to go 60, right? So it's like pulling out

both hands as if turning steering wheel

enacts driving bus

2:47

And you know how you pull out

left hand tilts, moving from left

as if pulling into middle of road

C13

imitates driving bus

F: It can be like the um, [leaning back, yawning] the bus is all [inaudible] on top of the tape [?] *** 'Cause it comes around like this and it comes down *** like you said.

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2:49

and you go unnn

hands angled, right up to left, left pointing diagonally right

attempts to represent acceleration

251

and you get faster and faster and faster. And then he's just goin' along, goin' 60, cool

both hands flat, fingers wiggling

attempts to represent constant velocity

255

and then he pulls out like onto the highway or something

left hand makes downward point

focuses jointattention on specific part of graph

C18

257

F: Yeah.

C19

258

P: And so he speeds right up to

left hand scoops down, then up

attempts to represent acceleration

3:01

* * * 20 metres

hand makes "stop ” signal at top

represents maximum velocity

C20

3:02

F: 120 *** 120, yeah.

C21

3:05

P: *** I didn't think school buses would go that fast

hand clenches then flattens

uncertainty: refocuses jointattention

3:08

I've never been in one that can, except for when Neil drove

both hands describe small circles

abstract demonstrat'n of driving bus

3:13

and then he just goes, p–ouw

left hand slides down diagonally

represents deceleration

3:15

and then he stops and stops and stops and stops and stops

left hand moves slowly down to desk at 45 degree angle

represents deceleration to stop

C22

3:17

F: Then he goes "Oh, a bus.

C23

3:20

P: [bell rings] Yeah, he could be stopping at red lights.

C24

3:25

F: Yeah.

C25

3:26

P: It's a constant time that he's going down, so he's going unnn

imitating someone at steering wheel

enacts driving bus

3:33

All right.

3:35

F: School bus.

C26

ROBERT A. REEVE

248

AND

FIONA J. REYNOLDS

3:36

P: Oh, no, look. This is time in Seconds ***

3:42

So he only goes 60 for 2 seconds. So, this is a very short trip ***

C28

3:52

F: Through a lot of red lights.

C29

3:55

P: *** Huh?

C30

3:57

F: Through a red light. *** Pretend, get me some [inaudible].

C31

4:04

P: Here.

C27

C32

4:05

Points at x axis

joint - attention on specific part of graph

F: 'Cause I mean, I can't think of another story.

In this episode, Patricia was clearly the dominant partner. She used gesture regularly to direct joint-attention to aspects of the graph. First (C9), she emphasised the need to focus on the graph. In her next turn taking (C11) she indicated the part of the graph showing acceleration. Again, at C17 she drew attention to the part of the graph marking a change from constant velocity to acceleration. Patricia then seemed to experience a discrepancy between her understanding of the velocity that she expected to find in a bus journey and the velocity depicted on the graph. Her gesture at C21 was interpreted as drawing Fran’s attention to that uncertainty. Finally, at C27 she accentuated that uncertainty by pointing out the scale of the xaxis. Both girls made extensive use of gesture to illustrate their understanding of how a school bus would proceed. Again, Patricia appeared to be the dominant partner, first demonstrating driving at C7, those gestures being reflected by Fran (C10). Fran then took the initiative (C12), offering another verbal and gestured explanation of the journey. Patricia briefly echoed and extended that explanation at C13. We regarded Patricia’s contributions at C17, C19 and C21 as effectively continuous and significant. In the course of them, we interpreted Patricia as demonstrating her accurate understanding of how the bus would speed up and slow down, and as suddenly realising an apparent discrepancy between that understanding and the graph. This episode differs from the two previously reviewed because, while pauses can be found throughout the discourse, many of them occurred for reasons other than the need for dyadic communication, such as writing, putting calculators away, and yawning. It was only towards the end of the episode, when Patricia began to realise the conflict between her understanding and the graph, that co-occurrence of slow speech, pauses and descriptive gestures can be identified. We maintain that Patricia’s contributions at C21 and C27 can be interpreted as indicating first, a sudden discovery of discrepancy, and second, a possible reason for the existence of that confusion. Because the lesson ended shortly thereafter, however, that conflict remained unresolved. Our analysis of the three protocols goes some way to answering our three questions. First, it is clear that gesture is used by problem-solving dyads to achieve,

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249

maintain and refocus joint-attention on various aspects of the problems under consideration. Second, gesture obviously provides an explanatory context without which the words of the two students would be difficult to interpret. Third, there seems to be some support for our contention that there is a relationship between the occurrence of gesture, slow speech and pauses, and the existence of an element of cognitive uncertainty. 4. CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH The significance of gesture in communication has received surprisingly little empirical attention, although it is acknowledged to play an important role in many different aspects of discourse (Armstrong, Stokoe & Wilcox, 1996). It has been suggested, for example, that when communication is problematic or when concepts are difficult to explain verbally (as is common in many areas of science and mathematics), gesture is frequently used to signal difficulty or to help elaborate meanings. Perhaps more importantly, it has also been claimed that individuals are particularly sensitive to the meaning of gestures that signal the way in which discourse is proceeding. In other words, it appears that gesture may serve cognitive, attentional as well as monitoring and regulating functions. We have been particularly interested in evaluating the generality of the indexical significance of gesture-speech activity in collaborative problem solving, both as a support for communication more generally and as an index of change in understanding more specifically. The use of gesture is particularly useful in a representational sense because the young possess a relatively poor linguistic competence (constructs, concepts, and symbols). The role of gesture as a functional means of adding to discourse is important. The point here is to build a notion that gesture is important in contexts where words are not freely available. In other words, there is ample evidence to suggest that gestural signs are used to convey desires/meaning. The connection here is that humans are well practiced at using gestures in difficult contexts. The question of how well humans are able to interpret the gestures of others is of interest and less well understood than one might expect. In our analysis of the literature on the role of gesture in collaborative problem solving specifically and in communication more generally, we have been deeply puzzled by the tendency of many researchers to treat, without comment, gesture as if it were equivalent to speech. We suspect that some researchers have conflated speech acts with communication and ignored the possibility that different aspects of communication (speech and gesture, for example) may serve different cognitive functions on different occasions in different contexts and settings. Although some researchers have noted that gestures are often used to disambiguate the “gibberish” of speech (Roschelle & Clancey, 1992) in collaborative problem solving, it seem to us that it would be theoretically useful to more fully explore the relative functions of gestures in communication.

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250

In the present paper, we have shown that gestures serve several distinct purposes in maintaining problem solving interactions in a setting where two students are discussing “time-speed” problems. We chose these problems because of their obvious spatial properties and the concomitant likelihood that they would elicit gestures in the individuals discussing them. It seems to us that it is equally important for researchers to examine the use of gesture in problems that possess less obvious spatial properties. Part of our rationale for studying the role of gesture in communication is that gestures are acknowledged to be one of the earliest forms of preverbal communication. However, very little is known about the developmental implications of gestures, especially in the way that different aged students employ them in collaborative learning settings. It is highly plausible that gestures may be a particularly important communication medium in individuals who are less verbal or who do not possess an adequate grasp of a particular domain “language” (e.g., some aspect of mathematics) that would allow them to engage in meaningful discussions about that domain. Finally, researchers must focus on the interpretive significance of different kinds of gestures in sustaining ongoing communication. In particular, it seems important to establish whether there is a lexicon of gestures and whether this lexicon is unique to situations or individuals and whether it can be interpreted by non-participant onlookers (e.g., teachers) to help understand the confusion of those involved in collaborative problem solving. We would like to end with a telling extract from an interview that took place immediately after the lesson. Patricia was questioned by the investigator (“I”) about Fran's gestured interpretation ofgraph E: I:

But she's, Fran's doing this thing with her hands

P:

Yeah, 'cause we kept on sort of just going on with the ball like this, with the ball and then she was kind of going like that [inaudible] [laughs] [pause]

I:

And that, that must be the end. Oh, I know what that is. Did you find yourself doing a lot of that stuff?

P:

Yeah.

I:

Why?

P: It's because, well that's just sort of how we explain stuff, 'cause we know what we're on about, but we can’t explain it to the other person. I:

So there's a lot of physical movement, so you're moving your hand up and down or?

P: Yeah, 'cause that's how we thought the ball would move. I:

I’m trying to work out what Fran was moving her hand across from left to right. Which example was she talking about when she was doing that?

P: Um, when we were doing the bouncing ball, she said it would go up and not across and I was saying [laughing] "I don’t think so".) (emphasis added)

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We consider that this provides further support, not only for our interpretation of this gesture as a shared misunderstanding of graph E, but also for our position that gesture can act as an amplifier of speech, and that, furthermore, the students concerned were aware of this important function of gesture. ACKNOWLEDGEMENTS We would like to extend our heartfelt thanks to David Clarke and Sue Helme for providing the opportunity to review tapes and transcripts. We also thank David, Sue and other participants in the Classroom Learning Project for their insightful comments on our work. We extend particular thanks to Rodney Reynolds for the graphics. Finally, we thank Les Steffe for his extremely insightful and helpful comments on an earlier version of this chapter that was presented at the annual meeting of the American Educational Research Association conference in April, 2000. REFERENCES Acredolo, L. & Goodwyn, S. (1997). Furthering our understanding of what humans understand. Human Development, 40, 25-31. Alibali, M.W., Flevares, L.M., & Goldin-Meadow, S. (1997). Assessing knowledge conveyed in gesture: Do teachers have the upper hand? Journal of Educational Psychology, 89, 183-193. Alibali, M.W. & Goldin-Meadow, S. (1993). Gesture-speech mismatch and mechanisms of learning: What the hands reveal about a child’s state of mind. Cognitive Psychology, 25, 468-523. Armstrong, D.F., Stokoe, W.C., & Wilcox, S.E. (1995). Gesture and the nature of language. Cambridge, UK: Cambridge University Press. Azmitia, M. & Perlmutter, M. (1989). Social influences on children’s cognition: State of the art and future directions. In H. W. Reese (Ed.), Advances in child development and behavior (Vol. 22, pp. 89-144). New York: Academic Press. Bavelas, J.B., Chovil, N., Coates, L., & Roe, L. (1995). Gestures specialized for dialogue. Personality and Social Psychology Bulletin, 21, 394-405. Bavelas, J.B. (1995). Gestures as part of speech: Methodological implications. Research on Language and Social Interaction, 27, 201-221. Bidell, T. R., & Fischer, K. W. (1994). Developmental transitions in children’s early on-line planning. In M. Haith, J. Benson, R. Roberts, & B. Pennington (Eds.), Intellectual development (pp. 141-176). Chicago: University of Chicago Press. Bronfenbrenner, U. (1993). The ecology of human development. In R. H. Wozniak & K. W. Fischer (Eds.), Development in context: Acting and thinking is specific environments (pp. 3-44). Hillsdale, NJ: Erlbaum. Brown, A. L., & Reeve, R. A. (1987). Bandwidths of competence: The role of supportive contexts in learning and development. In L. Liben (Ed.), Learning and development; Conflict or congruence? (pp. 173-216). Hillsdale, NJ: Erlbaum. Brown, A. L., Campione, J. C., Reeve, R. A., & Ferrara, R. A. (1992). Interactive learning and individual understanding: The case of reading and mathematics. In L. T.Landsmann (Ed.), Culture, schooling and psychological development (pp. 124-178). Hillsdale, NJ: Erlbaum. Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, l8(1), 32-42. Bruner, J. S., & Bornstein, M. H. (1989). On interaction. In M. Bornstein & J. Bruner (Eds.), Interaction in human development (pp. 1-16). Hillsdale, NJ: Erlbaum. Church, R.B., & Goldin-Meadow, S. (1986). The mismatch between gesture and speech as an index of transitional knowledge. Cognition, 23,43-71.

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Crowder, E.M. (1996). Gestures at work in sense-making science talk. Journal of the Learning Sciences, 5, 173-208. Damon, W. (1991). Problems in direction in socially shared cognition. In L. Resnick, J. Levine, & S. Teasley (Eds.), Perspectives on socially shared cognition (pp.384-397). Washington, DC: American Psychological Association. , DeVries, R. (1997). Piaget s social theory. Educational Researcher, 26 (2), 4-17. Doise, W., & Mugny, G. (1984). The social development of intellect. Oxford: Pergamon Press. Doise, W., & Palmonari, A. (1984). Social interaction in individual development. Cambridge: Cambridge University Press. Efron, D. (1941). Gesture and environment. Morningside Heights, NY: King’s Crown Press. Fischer, K. W., & Grannott, N. (1995). Beyond one-dimensional change: Parallel, concurrent, socially distributed processes in learning and development. Human Development, 38, 301-321, Forman, E. (1989). The role of peer interaction in the social construction of mathematical knowledge. lnternational Journal of Educational Research, 13, 55-70. Garton, A. F. (1992). Social interaction and the development of language and cognition. Hillsdale, NJ: Erlbaum. Goldin-Meadow, S., Alibali, M.W., & Church, R.B. (1993). Transitions in concept acquisition: Using the hand to read the mind. Psychological Review, 100, 279-297. Goldin-Meadow, S., Nusbaum, H., Garber, P. & Church, R.B. (1993). Transitions in learning: Evidence for simultaneously activated strategies. Journal of Experimental Psychology: Human Perception and Performance, 19, 92-107. Goldin-Meadow, S., Wein, D., & Chang, C. (1992). Assessing knowledge through gesture: Using children's hands to read their minds. Cognition and Instruction, 9, 201-219. Gooding, A,, & Stacey, K.( 1993). Characteristics of small group discussion: Reducing misconceptions. Mathematics Education Research Journal, 5, 60-73 Goodwyn, S.W., & Acredolo, L.P. (1993). Symbolic gesture versus word: Is there a modality advantage for onset of symbol use? Child Development, 64, 688-701. Goos, M. (1994). Metacognitive decision making and social interactions during paired problem solving. Mathematics Education Research Journal, 6, 144- 165. Guberman, S. R. (1996). The development of everyday mathematics in Brazilian children with limited formal education. Child Development, 67, 1609-1623. Hall, R. (1995). Representation as shared activity: Situated cognition and Dewey's cartography of experience. Journal of the Learning Sciences, 5, 209-238. Karmiloff-Smith, A. (1992). Beyond modularity: A developmental perspective on cognitive science. Cambridge, Mass: MIT Press. Kendon, A. (1972). Some relationships between body motion and speech. In A. Siegman and B. Pope (Eds.), Studies in dyadic communication (pp. 177-210). New York: Pergamon Press. Kendon, A. (1980). Gesticulation and speech: Two aspects of the process of utterance. In M.R. Key (Ed.), The relation between verbal and nonverbal communication (pp. 207-227). The Hague: Mouton. Lave, J. (1988). Cognition in practice. Cambridge: Cambridge University Press. Levy, E.T., & McNeill, D. (1992). Speech, gesture, and discourse. Discourse Processes, 15, 277-301. Masataka, N. (1995). The relation between index-finger extension and the acoustic quality of cooing in three-month-old infants. Journal of Child Language, 22, 247-257. McNeill, D. (1985). So you think gestures are nonverbal? Psychological Review, 92 , 350-371. McNeill, D. (1992). Hand and mind: What gestures reveal about thought. Chicago: University of Chicago Press. McNeill, D., Cassell, J., & McCullough, K.E. (1994). Communicative effects of speech-mismatched gestures. Research on Language and Social Interaction, 27, 223-237. Morissette, P., Ricard, M., & Decarie, T.G. (1995). Joint visual attention and pointing in infancy: A longitudinal study of comprehension. British Journal of Developmental Psychology, 13, 163-175. Moschkovich, J.N. (1996). Moving up and getting steeper: Negotiating shared descriptions of linear graphs. Journal of the Learning Sciences, 5, 239-277. Nunes, T., & Bryant, P. (1996). Children doing mathematics. Oxford: Blackwells. Palincsar, A. S. (1989). Less charted waters. Educational Researcher, 18(4), 5-7. Perry, M., Church, R.B., & Goldin-Meadow, S. (1988). Transitional knowledge in the acquisition of concepts. Cognitive Development, 3, 359-400.

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Perry, M., Church, R.B., & Goldin-Meadow, S. (1992). Is gesture-speech mismatch a general index of transitional knowledge? Cognitive Development, 7, 109-122. Piaget, J. (1965). The moral judgment of the child. London: Free press. Rome-Flanders, T., & Cronk, C. (1995). A longitudinal study of infant vocalizations during motherinfant games. Journal of Child Language, 22, 259-274. Roschelle, J. (1992). Learning by collaborating: Convergent conceptual change. Journal of the Learning Sciences, 2, 235-276. Roschelle, J. (1996). Guest editor's introduction. Journal of the Learning Sciences, 5, 169-171. Roschelle, J., & Clancey, W.J. (1992). Learning as social and neural. Educational Psychologist, 27, 435453. Saxe, G. B. (1991). Culture and cognitive development: Studies in mathematical understanding. Hillsdale, NJ: Erlbaum. Siegler, R. S. (1996). Emerging minds: The process of change in children’s thinking. New York: Oxford University Press. Steffe, L.P. (1988). Children’s construction of number sequences and multiplying schemes. In J. Hieberi & M. Behr (Eds.). Number concepts and operations in the middle grades (pp. 119-140). Hillsdale, NJ: Erlbaum. Steffe, L.P. (1991). The learning paradox: A plausible counterexample. In L. Steffe (Ed.), Epistemological foundations of mathematical experience (pp.26-44). New York: Springer-Verlag. Steffe, L.P. (1992a). Learning stages in the construction of the number sequence. In J. Bideaud, C. Meljac, & J.-P. Fischer (Eds.), Pathways to number: Children’s developing numerical activities (pp.83-98). Hillsdale, NJ: Erlbaum. Steffe, L.P. (1992b). Schemes of action and operation involving composite units. Learning and Individual Differences, 4, 259-309. Steffe, L.P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp.3-39). Albany, NY: SUNY Press. Steffe, L.P., & Cobb, P. (1988). Construction of arithmetical meanings and strategies. New York: Springer-Verlag. Steffe, L.P., von Glasersfeld, E., Richards, J., & Cobb, P. (1983). Children’s counting types: Philosophy, theory, and application. New York: Praeger. Steffe, L.P., & Olive, J., (1996). Symbolizing as a constructive activity in a computer microworld. Journal of Educational Computing Research, 14, 1113-138. Strayer, F. F., & Moss, E. (1989). The co-construction of representational activity during social interaction. in M. Bornstein & J. Bruner (Eds.), Interaction in human development (pp. 173-196). Hillsdale, NJ: Erlbaum. Tomasello, M., & Camaioni, L. (1997). A comparison of the gestural communication of apes and human infants. Human Development, 40, 7-24. Tudge, J. R., & Rogoff, B. (1989). Peer influences on cognitive development: Piagetian and Vygotskian perspectives. In M. Bornstein & J. Bruner (Eds.), Interaction in human development (pp. 17-25). Hillsdale, NJ: Erlbaum. Tudge, J. R., Winterhoff, P. A. (1993). Can young children benefit from collaborative problem solving? Tracing the effects of partner competence and feedback. Social Development, 2, 242-259. Tudge, J. R., Winterhoff, P. A,, & Hogan, D. M. (1996). The cognitive consequences of collaborative problem solving with and without feedback, Child Development, 67, 2892-2909. Valsiner, J. (1988). Ontogeny of co-construction of culture within socially organized environmental settings. In J. Valsiner (Ed. ), Child development within culturally structured environments (Vol. 2. pp. 283-297). Norwood, NJ: Ablex. Werner, H., & Kaplan, B. (1963). Symbol formation. New York: Wiley & Sons. Winegar, L. T. (in press). Developmental research and comparative perspectives: Applications to developmental science. To appear in J. Tudge, M. Shanahan, & J. Valsiner (Eds.), Developmental Science: Vol. 2. Comparative approaches in human development. Wundt, W. (1921/1973). The language of gesture. The Hague: Mouton

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JOHN R. BAIRD

CHAPTER ELEVEN Learning and Teaching: From Ignorance to Understanding

INTRODUCTION In this chapter, interpretation and analysis of the integrated data sets are guided by a perspective on learning that emphasises integration of thinking (cognition), feeling (affect), and acting (conation and associated behaviour). Equipped with a model of learning and informed by that model as to how best to promote learning and empower the learner, it should be possible to develop a model of quality teaching. In the second part of this chapter, the focus is on the teacher, and how teachers can orchestrate conditions for converting ignorance into understanding in learning, through personal attention to their thinking, feeling and acting. I have chosen to examine the Science classroom from the perspective of both the learners and the teacher. Among the several reasons for adopting this approach, one of the most compelling is the inappropriateness of presenting any representation of the classroom that does not give explicit acknowledgment to the centrality of the teacher-student dyad and to the significance of each of these roles in the coconstruction of the practices of the science classroom. PART A. FROM IGNORANCE TO UNDERSTANDING IN LEARNING: THINKING, FEELING AND ACTING Desirable learning proceeds with the learner both willing and able to take responsibility and control over personal learning practices. Particular themes are developed to structure data interpretation in terms of the learning constructs discussed below. These themes centre initially upon aspects of thinking then extend to the affective and conative domains. 1.1 A perspective on learning Ten terms that structure discussion to follow are shown in Figure 11.1. These terms form the basis ofthe learning model to be considered later. 255 David Clarke (ed.), Perspectives on Practice and Meaning in Mathematics and Science Classrooms, 255—289. © 2001 Kluwer Academic Publishers. Printed in the NetherIands.

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Figure 11. 1. Key Terms in a Model of Learning In Figure 11.1 the terms, which cover cognitive, affective, and conative aspects of learning, are arranged somewhat sequentially in terms of learning approach, progress and outcomes (Baird, 1998). I now briefly review these aspects, argue for their effective integration in attempts to enhance learning quality, and explain the place of ignorance in the figure. Excluding ignorance, the constructs start with perceptions, motivation, and attitudes/emotions. These terms are clustered together in order to emphasise the importance of (cognitive) perceptions for all that occurs next, and the influence of these perceptions (whether accurate or inaccurate) on various feelings. The nature of perceptions and associated affective elements will determine the extent to which the learner is positively, productively challenged by what is to be done. I have defined and discussed perceived challenge elsewhere (Baird, 1994; Baird and Penna, 1997) as not simply a measure of extent of thinking (cognitive demand), but as such thinking coupled with extent of interest and motivation (affect). If interest and motivation are high, cognitive demand will be viewed positively and the learner will be disposed to engage in active reflection and associated action to set worthwhile learning goals and to achieve these goals. These processes involve metacognition and volition. I define metacognition as having three components: metacognitive knowledge (knowledge of the nature of learning, effective learning techniques, and personal learning characteristics); metacognitive awareness (of task nature and progress - essentially knowing what you are doing and why you are doing it) and metacognitive control (making productive decisions about approach, progress and outcomes) (e.g. Baird, 1991). Volition connotes strength of will and a sense of striving; it is associated with such attributes as diligence, perseverance and resourcefulness (e.g. Corno, 1993). Volition has been distinguished from motivation

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in terms of the timing of decisions made by the learner regarding learning goals. Where motivation is important for establishing a sense of the extent of productive challenge and thereby influences the nature of decisions made (and is thus “predecisional”, e.g. Kuhl, 1985), volition involves determination to carry out decisions made by controlling unhelpful emotions and resisting distractions. It is thus “postdecisional”. Volitional control, a meta-emotional and meta-motivational executive control mechanism, complements metacognitive control with its cognitive focus. Finally, as indicated in Figure 11.1, the desirable ultimate outcomes of learning are again a balance of cognition and affect - understanding and enjoyment. The arrangement of terms in Figure 11.1 mirrors recent attempts by others to view learning in a way that emphasises its purposive character, where such things as perceptions, aspirations, feelings about self (e.g of competence and confidence), decisions, learning goals, and behaviours are brought together to inform and illuminate learning actions and the meanings that underlie such actions. Snow, Corno and Jackson (1996) present a provisional taxonomy of individual difference constructs that links cognition (procedural and declarative knowledge), affect (or affection: temperament and emotion) and conation (motivation and volition). Research into the increasingly popular areas of “learning to learn” (e.g. Candy, 1990) and of self-regulated learning (e.g. Boekaerts, 1997, 1998; Zimmerman, 1998; Zimmerman and Schunk, 1989) emphasise interactions between perceptions, learning goals, metacognition and volition. For instance, Zimmerman (1 998, P. 4) presents learning as a “self-fulfilling cycle” that encompasses forethought (e.g. goal setting, strategic planning, intrinsic interest), performance/volitional control (e.g. attention focussing, self-monitoring) and self-reflection (e.g. self-evaluation, attribution, adaptivity). Boekaerts (1997) distinguishes between cognitive selfregulation and motivational self-regulation in a six-component model of selfregulated learning (P, 164). From a different perspective, Boekaerts (1998) considers self-regulated learning according to an action hierarchy involving multiple cognitively- affectively- and volitionally-related goals at various levels of abstraction (from the most general level, self-concept, through to more specific levels of principle control, action programs, and scripts - P. 17) 1.2 The role of ignorance in learning It may seem strange to the reader that ignorance is the first term listed in Figure 11.1. It is placed there because ignorance is both a vitally important and a common feature of all learning. Ignorance - not knowing something or how to do something is a common condition in daily life and is the basis for new learning. As important as ignorance, however, is whether or not you are aware of it. Sometimes you realise that you don’t know something, sometimes you don’t. Knowing that you don’t know or can’t do something is prerequisite to effective learning. Indeed, this situation means that the learner has achieved metacognitive awareness, which is necessary for adequate metacognitive control over learning. Alternatively, you may not know that you don’t know or can’t do something. In this

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situation, you are metacognitively unaware - a term that has been entitled secondary ignorance (e.g. Brown, 1978). An unfortunate aspect of the first situation above is that many metacognitively aware students may be disinclined to act to convert their ignorance into understanding. In this case, the problem is affective rather than cognitive, and the outcome is inaction. A reason a learner may not act on recognised ignorance is because he or she mistakenly relates ignorance with stupidity, and consequently makes the decision to not reveal ignorance for fear of appearing foolish. Not taking the risk to betray personal ignorance would be a tactic especially favoured by those students who lack confidence in their ability as learners (e.g. Miserandino, 1996) and exhibit ego-involvement rather than task-involvement in their achievement motivation (e.g. Nicholls, 1983; Spaulding, 1992). The other undesirable learning situation, where the learner is metacognitively unaware of the nature and extent of personal cognitive ignorance, has all of cognitive, affective, and volitional aspects that act together to directly diminish a person’s willingness and ability to achieve desirable learning outcomes. One type of metacognitive unawareness is when you don’t realise that you don’t know or can’t do something. This unawareness may arise from inaccuracy - for instance, when you think you know what you are doing when in fact you don’t. This situation often involves the setting of unproductive learning goals. A similar but different type of metacognitive unawareness is if you don’t know whether you do or don’t know something (you can’t call the information to mind, even if only temporarily). Here, motivation would be expected to be low and emotions negative, because you don’t know what to do next. In Figure 11.2, the ten constructs in Figure 11.1 are subsumed within an overall model of learning that forms the frame for interpretation of data arising from the integrated data sets.

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Figure 11.2. Learning Model Two terms from Figure 11.1 are not labelled in Figure 11 .2 and their place in the model requires clarification. One aspect of metacognition, metacognitive knowledge, is not shown explicitly, but is considered a component of stored memory. Thus, it is located together with other knowledge, beliefs, attributes and competencies within the Self Conditions (Traits) element of Figure 11.2. Next, reflection is considered to be a part of a process of purposeful enquiry that comprises thinking (reflection) and associated action (behaviours). Reflection thus will be considered in terms of particular learning behaviours, as described below. All arrows shown in Figure 11.2 and in subsequent figures signify proposed sequential or causal relationships between different constructs. Most attention will be paid to the interactions in learning between the constructs shown in the large triangles at the centre of Figure 11.2. Perceptions (cognition) and motivation and emotions/feelings (affect) influence the level of metacognitive awareness of self, task and context conditions. In turn, rnetacognitive awareness influences learning decisions and associated behaviours that include metacognitive and volitional control. Throughout, desirable behaviours comprise actions taken in concert with focussed reflection, evidenced by the learner asking evaluative questions. As will now be discussed, two types of learning can be distinguished on the basis of the extent to which the learner habituates this process of questionasking.

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In an undesirable type of learning that I entitle Perceptions-Driven Learning (PDL - Figure 11 .3) the learner approaches the learning task reactively, and initiates few reflective behaviours. In PDL, task, context, and self conditions - that include nature and level of cognitive ignorance - are inadequately or imperfectly diagnosed, with the results indicated as dotted lines in the figure. Lack of active reflection results in deficient perceptions, perhaps associated with negative emotions, all of which contribute to a low level of perceived personal challenge. As metacognitive awareness is low, decisions regarding learning goals may be unproductive, and metacognitive control, volitional control and learning outcomes limited.

Figure 11.3 Perceptions-Driven Learning

By comparison, in Reflection-Driven Learning (RDL - Figure 11.4) the learner has habituated a process of purposeful enquiry, involving active reflection and associated action. The learner’s routine manner of engagement in the learning task is to actively interrogate its nature, characteristics and purposes. and to determine personal attributes for successful completion. Accordingly, the learner determines what needs to be done. and what he or she does and does not now know - thus achieving metacognitive awareness. A key desired outcome in the learning model shown in Figure 11 .4 is understanding. Understanding is achieved by appropriate reflection upon aspects of cognitive ignorance and action to remedy such ignorance. In Australia. the Project for Enhancing Effective Learning (PEEL - Baird and

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Mitchell, 1986; Baird and Northfield. 1992) continues as a significant school-based project to help students to become more willing and able to engage in such reflection and associated action. to take greater responsibility for, and control over, their own learning and thus, to convert ignorance into understanding.

Figure 11.4. Rejection-Driven Learning

In the next section, I present data that illuminate the assertions made above. Particularly, by presenting the data below, I explore the influence of metacognitive awareness and unawareness on decision-making related to setting appropriate learning goals, on learning behaviours, and inferentially on learning success. 1.3 Data organisation and analysis Data sources for the current analysis were Lessons 1, 7, and 8 — the three Science lessons taught by Mrs Brown. As set out in Chapter 2. the integrated data set, constructed for each lesson, comprised four components (verbatim lesson transcripts; classroom researcher annotations/field notes; student post-hoc comments at interview: teacher post-hoc comments at interview). I then annotated this data set by highlighting interesting statements or sequences and assigning to them codes based upon some of the key constructs shown in Figs 11.1-4, The main coding categories, and their nature based on my interpretation ofdata in the sequence, were:

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Cognitive ignorance or knowledge– explicit or inferred Perceptions of task – accurate or faulty Decisions related to learning goals – productive or unproductive Metacognitive awareness or metacognitive unawareness Motivation -positive or negative Behaviours -positive/reflective or negative/reflective Metacognitive control - presence of or lack of Volitional control - presence of or lack of Learningoutcomes (understanding, enjoyment, successful completion) presence of or lack of I accompanied each code with a short statement to support my interpretation of the data.

— — —

— — — — — —

—

1.4 Findings from the data Findings that inform the theoretical perspectives summarised above will be considered according to two related themes: — Perceptions, Motivation and Learning Cognitive Ignorance And Metacognitive Awareness And Control Of Learning — Another perspective on learning approach, progress and outcomes concerns the role and influence of the teacher. This perspective is not developed here, but is considered in detail in Part 3 of this chapter. —

1.5 Perceptions, Motivation and Learning Data relevant to this section will be considered by concentrating upon the thoughts, feelings and actions of some of Mrs Brown’s Year 9 students. 1.5.1 Perceptions Of Task and Motivation Joanne had motivational and volitional limitations to her learning, brought about by her perceptions that the work lacked interest and challenge. There were various instances of Joanne implementing undesirable learning strategies (T = Teacher, J = Joanne, N = Narelle, and F = Frances). T: [to Joanne] Your questions eight—oh, one to eight you haven't done. OK, so when will they be done? J:

On Monday.

T: OK. All right. J:

Just um, I left my bag at the doctor's. And we didn't know where it was, and so. And I haven't [inaudible]...

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In comparison with such instances of lack of commitment, however, were many episodes where she adopted a focussed, reflective approach to her learning, especially when interest or perceived challenge were high. For instance, she was intrigued by the microscopic appearance of a waterproof object (her pencil) when viewed under the microscope: J:

You've got to focus it with this thing here .... Hey, look, it's waterproof. [dripping water from pipette on to her pencil]

N: Can I have a look Frances? F:

Yeah.

J:

Oh look Narelle, this is waterproof, look see it'll just drip off. See? there's water in there?

See how

In the following interview transcript, Joanne makes a conscientious effort to resolve the relationship between porosity of a fabric and its structure. Even though , she doesn t resolve the situation accurately, she demonstrates the ability for protracted, reasoned enquiry: J:

It’s confusing having it, low poroxsity (sic) is um, where it's got heaps of holes, I would have thought it wouldn't have had many 'cause, low as in less holes.

I:

Yeah. Aah, yeah.

J:

Low as in no holes, like hold on, um. the low um, hold on, I’m thinking about it.

I:

OK, that’s OK.

J:

The low—can I have a look in my book?

I:

Oh, yeah! Absolutely.

J:

It's written but I just can’t describe it, OK, um [sound of paper rustling], I’ve got it there, so the low porosity, um, I would have thought that that was-actually it does make sense ‘ause low poros-porosity or whatever, um, it doesn't have any holes in it, so it's very low as in no holes in it.

I:

Uh huh.

J:

But, um. [pause] Yeah, I would thought it was high because it just meant it was all tight together so low, ‘cause I kind of think porosity would mean that it’s very um, close together. I just think of that.

I:

Uh huh.

J:

So when it’s low porosity you think it’s very far apart .... Most words usually say, you know, porosity is how thick something is, or you know, um, how dense.

I:

Uh huh.

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That’s how I think of it.

I:

Yeah, yeah.

J:

, So it, if it’s very high porosity it s very close together, but if it’s low it means it’s very open apart.

I:

Um hm, um hm.

J:

, That s how I-think it’s just from other words mostly high, you know, high is up.

I:

Um hm.

J:

And all that.

I:

Yeah.

J:

But, yeah, porosity is close together.

At interview, there were many occasions when she demonstrated lack of metacognitive awareness of the learning task. Such metacognitive unawareness seemed to result from low motivation and commitment, both associated with perceived lack of challenge in the teaching/learning situation: I:

So you feel, um, how do you feel like you learn then?

J:

Copying of my friends.

I:

[laughs]

J:

No, I find that I ask, if I ask the teacher usually, they give me a response that they just told the class which I don’t understand still.

I:

Um hm.

J:

And, 1, I suppose it’s not that I don’t understand, it’s just I’m a bit confused about what she’s talking about.

I:

So, how do you get unconfused? Or do you?

J:

, I usually just stay confused, and it, it doesn t bother me, I just ask Narelle “ What are we supposed to be doing?"

I:

Um hm, uh huh.

J:

Or Laura, next to me.

I:

Uh huh, uh huh.

J:

And then 1 get it and I just write it down. I don’t listen to Mrs B. very much though.

LEARNING AND TEACHING: FROM IGNORANCE TO UNDERSTANDING I:

Yeah, I know later on when she’s reading out it doesn’t look like you’re listening too much.

J:

Yeah, I’m, I was-like I can hear her, but it seems as though, I don’t know. I can’t concentrate very much on this just being talked at.

I:

Yeah.

J:

I need to be um, either involved or um, like I hate just listening.

I:

Yeah.

J:

To the one person just telling you all the facts. just straight like that. I like it, you know, um. Talking to my friends helps because like we, when we had to do those activities.

I:

Uh huh.

J:

You find out who is the [inaudible], you hear them talking and then you can, you know, kind of stay around the one topic for just more than a couple of seconds ...

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Again, in the following short interview transcript, she perceives the lesson as like a lecture, where she is asked to absorb material that she is not particularly interested in: In this class it’s just a matter of coming in here, getting, you know, a lecture, about getting facts poured out at you really... Answering a couple of questions and then coming out.

Joanne was experiencing conflict between her essentially positive perception of , herself as a learner and her negative feelings associated with Mrs Brown s style of teaching and the nature of the work: I found last year I had Mrs Brown as well and I, I think the reason that I don’t concentrate as much in science or do as well is just ...I don’t know that method of teaching or something, because I had her for maths and last year was horrible, for maths, and then this year I came into maths and I knew everything, and I just did my work and I get good marks and I was thinking how, like, if I, 'cause um, I thought I was dumb, like last year, because of maths, and like this year I feel dumb in science and, I think “Oh, I'm not ” because last year I did fine in science um, the year before and I ve , always liked science and this year ,s just writing and I don t learn anything from just writing.

1.5.2 Perceptions of role as learner Students’ perceptions of their role as learner directly influenced their learning decisions and behaviours. For example, the following two interview transcripts indicate that these students perceive their role in learning as reactive rather than active, and largely involving recording information supplied by others G = Grace, A = Anna): G: Oh, um, the blood travels through to the left of the heart... That’s what Mrs. Brown said.

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JOHN R. BAIRD I:

That’s what she’d said earlier, hadn’t she?

G:

So, I just repeated it.

I:

You really believe it’s true though?

G:

Yeah, I just believe what teachers say. Is it true?

A:

Yes, there was lots of things, like I didn’t really know that the blood in your body was yellow until one of the other groups said that ... And, um I hadn’t heard the lymphatic system before, that s’ what Lauren suggested. And there's ’lots of-you knew some words, but you didn't know what they were, you just knew them and wrote them down. 'Cause of the other groups were saying things and Mrs. Brown explaining things.

1.6 Cognitive ignorance and metacognitive awareness and control of learning Throughout the three lessons, there were many instances where students were metacognitively aware of their cognitive ignorance. For instance, in this short sequence related to understanding of concepts: T: So you know all about the heart? Do they know all that? A: No, we just know some words.

This recognition of limits to concept understanding was reiterated at interview, for example by Grace on two occasions: I:

As a word? What about as a "what it is" thing?

G: I knew it as a word. G: That thympatic (sic) system again, I never even thought of it as a word, I never knew it at all, and um, there was something else, I, oh, no, that's about it, but I kind of didn’t know which side of the heart -the left or the right-was the thickest or the strongest, I didn’t know which one it was, and it's the left.

As will now be discussed, methods of dealing with such ignorance varied considerably and involved different levels of metacognitive control. The following interview sequence with Grace demonstrates the powerful influence of acknowledged cognitive ignorance (and the confusion with which it was associated) on learning decisions made and on opportunities to effect metacognitive control over learning. In the sequence, the student clearly is metacognitively aware that she doesn't know something. This ignorance centres upon confusion associated with a disparity that she identifies between class discussion of heart function and her knowledge of a textbook diagram of the human heart (in which the diagram presents a mirror image reversal of left and right sides): G: 'Cause, um. majority of the class said it was, but we don 't know it for a fact yet, we 're going to ask the teacher but we haven 't yet ....

LEARNING AND TEACHING: FROM IGNORANCE TO UNDERSTANDING I:

Right, so that group’s saying it, and you think most of the class?

G:

Yeah, well, we kind of ended up thinking that it was on the left as well. ‘Cause, yeah, ‘cause the left ‘Cause, we—I kind of remember that it being— but I still think it’s the right, because 1 remember my PE thing [from her Physical Education studies], but it could be the other side.

I:

But you still think it’s the right?

G:

Yeah, ‘cause, um, but I could be wrong, but.

I:

Sure, but what makes you think it’s

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G: ‘Cause on my PE book, it’s got a picture of a heart just there. I:

OK, so you've got a picture in there.

G: Yeah, and this side is wider than the left, and I thought— I:

Oh, I see it, right, OK, yes.

G: But it could be actually the other way, the other way, it could be put down on paper the wrong side. I:

Oh, like seen from the back, or? (pause)

G:

I'll ask Mrs. Brown (pause) it’s up to the teacher.

The last statement in this sequence reveals her decision about how best to resolve the difficulty - to leave it “up to the teacher” rather than attempting to resolve it herself. Sometimes, a decision by the learner to remedy ignorance by asking other students or the teacher led to an undesirable outcome. On these occasions, the learner failed to question the accuracy of the information provided (L = Lauren, G = Grace, and A = Anna): L: What are those great big veins? Cob? Cosh? G:

Capillaries.

L: Yeah, that's the one. Capillaries. [Anna writes] G: And there's some other ones.

On other occasions, however, when students demonstrated awareness of their cognitive ignorance and then acted reflectively to seek knowledge from another person, the result was positive. In the transcript below, for instance, once Anna had realised that she could not resolve whether veins travel through the body or blood travels through the body, she asked the appropriate diagnostic question of Lauren and her ignorance was converted into understanding.

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JOHN R. BAIRD A: [writing] The veins travel throughout the body. L: Yeah. A: Isn't that the blood travels throughout the body? L: No, the veins travel through our body and the blood travels through the veins. The veins connect the body. A: Veins travel throughout body and blood travels through veins [writes]. L: Yeah, veins.

Alternatively, there were instances of students dealing more actively with acknowledged ignorance, through application of Good Learning Behaviours (GLBs - Baird, 1986) to pursue useful learning goals. For instance, in the following interview sequence, Anna demonstrates how she dealt with acknowledged ignorance of heart structure by actively invoking episodic and visual memory: I:

Aorta, very good. What's that one, let's pause [on] that one.

A: It's the biggest vein in your body and it comes out of the top of your heart. I remember a little picture of it coming out. I:

Right, well done. So you had an image in your head.

A: Yeah. I:

What made you think of it at that point then?

A: Because, um, when I was on holiday, I looked at my PE book, and I've got to study for this test, I don't want to do badly. I looked at my book and I opened it up. I looked at this picture and you 've got to label it and I thought “Oh, no” and then I thought “There's a picture back here and I'll label it.” And there was this little thing sticking out and I go "What in the world is that?” and then I looked back and it's the aorta and I go “Oh, yeah, I remember my sister telling me the aorta is the biggest vein in the body”. And so that's how I remembered it. 1:

Right, OK, 'cause yeah, you've got this whole collection of memories associated just with that particular word.

A: I knew, as soon as I looked at it, I knew the aorta first. That's because, my sister's in year ten, she'd been through all this before, and she said "Did you know the aorta's the biggest vein in your body? ” and I said “Where is it?” and she goes “Oh, it comes out of the heart.” And that's how I remembered it.

It is unfortunate for this student that, in pursuing this potentially productive approach to learning, she has been misinformed by her sister's factually incorrect comment.

LEARNING AND TEACHING: FROM IGNORANCE TO UNDERSTANDING 1.7 Discussion: Learning through purposeful enquiry In Table 11.1, I propose four types of active learning. These types have emerged from my observations of students’ classroom learning over many years. The types, which form a taxonomy of increasing reflection-driven inquiry by the learner, serve to highlight the boundaried nature of the learning exhibited regularly in Mrs Brown’s lessons, and illustrated by much of the data considered above.

Table 11.1. Proposed Four Types of Active Learning TYPES OF ACTIVE LEARNING 1

Nature of “Active” in Active Learning Receptive; Compliant

Major Strategy

Answering questions

d al

Reflection

Metacognition

Uniformly low

Limited at best to metacognitive awareness of whether you know or don’t know specific content. Essentially no metacognitive monitoring or control

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Type 1 is clearly distinguished from the higher types. For Type 1 Active Learning, the term “active” connotes compliant attention and application to the work, but such application is limited to answering questions or completing tasks set by the teacher. For this type, to be active is to be conscientious and busily engaged, but not, in Bruner’s term, “to go further than the information given”. In contrast, Types 2 and 3 learning centre upon proactive, purposeful enquiry that is initiated by question-asking. Increasingly in these levels, activeness connotes curiosity - a striving to make sense - that fosters personal responsibility and control over personal practice. Learning of Types 2 and 3 demonstrate the type of Reflection-Driven Learning shown diagrammatically in Figure 11.4 above. Many of the transcripts considered above, however, demonstrate PerceptionsDriven Learning (Figure 11.3). Here, students are question-answerers rather than question-askers. Often, students’ learning was limited to reacting to externallycentred demand; in such cases, perceived challenge was low, perceptions of task sometimes deficient or faulty and perceptions of self unproductive. At best, the type of Active Learning would be characterised as Type 1 (Table 11.l) where reflection is low and metacognition limited. In episodes of learning in the three lessons, there were some instances of Level 2a learning, little if any evidence for Type 2b, and none for Type 3. Only occasionally did students initiate a process of purposeful enquiry by reflecting on task or context to generate productive questions, and acting to find answers to these questions. The Reflection-Driven Learning evidenced as Types 2 and 3 in Table 11.1 is, however, the most useful for replacing cognitive ignorance into understanding because it permits (and requires) effective metacognitive and volitional control. As successful implementation of both of these executive control mechanisms is associated with positive affective feelings, there is productive integration of thinking, feeling and acting. Students are both competent and confident to take responsibility and control over learning practice. These conditions involve learning content and physical and interpersonal context. Some such conditions were illustrated in the data presented in this chapter. Classroom data are complex and multiple. Worthwhile, trustworthy ethnographic research needs to present selected data in a way that invites interpretation of observed actions in terms of possible underlying reasons or purposes. Well-grounded interpretive research requires some credible and relevant referential bases that allow isolated episodes or events to be organised, considered and disseminated coherently. Many conditions influence students’ willingness and

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ability to take such responsibility and control over learning. The data above show many different aspects of personal and interpersonal actions and learning conditions that either help or hinder conversion of cognitive ignorance to understanding. Above all, perceptions of self, task and context were demonstrated to be central to levels of motivation, perseverance, and informed progression through metacognitive and volitional control. Teaching and learning are social activities. Consistent with my prior research in this area, it seems that both teacher and students will be stimulated to think, feel, and act to transform personal ignorance into understanding when each person perceives personal productive challenge in the learning tasks, and also feels that the classroom climate allows for positive interpersonal interactions within a supportive atmosphere. Crucial to the generation of such an atmosphere is first the classroom condition that “it is O.K. not to know” and second that lesson activities stimulate students to feel that “I know that I don’t know, and I want to find out”. PART B. QUALITY CLASSROOM TEACHING: THINKING, FEELING AND ACTING This section complements the preceding section: From Ignorance To Understanding In Learning: Thinking, Feeling and Acting, that took a student learning focus. In this section, the focus is on the teacher, and how teachers can orchestrate conditions for converting ignorance into understanding in learning, through personal attention to their thinking, feeling and acting. 1.1 A first look at quality teaching: Observable “Good Teaching Behaviours” In this section, I summarise some of my earlier research on classroom teaching in order to establish a basis for interpretation, analysis, and discussion of the current data. This research focussed upon “Good Teaching Behaviours” - those behaviours perceived as actions that foster worthwhile learning. When considering how to teach in a way that will enhance learning quality, it is tempting for teachers to focus upon teaching behaviours, and to search for those behaviours that seem to foster productive learning. In this research (Baird, 1998), I invited more than 200 practising teachers to construct short descriptors of what they considered to be “good teaching”. Subsequently, I combined their responses with various descriptors reported in the literature (e.g. Larrivee and Algina, 1983; Wright and Nuthall, 1970) to construct a list of 72 desirable teaching behaviours. A questionnaire based upon this list was then completed by 393 preservice teachers-intraining and experienced practising primary, secondary and tertiary teachers. It required each teacher to prioritise each of these 72 behaviours by grading it as “Crucial”, “Desirable but not Essential”, or “Not significant or relevant” to good teaching. This research illuminated the nature and range of behaviours that teachers consider to be important operational indicators of quality teaching. Each of the

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teacher respondent groups above assigned a mean of approximately 50 of these 72 behaviours as being crucial to good teaching. This high figure emphasises the complex and multidimensional nature of teaching. As an example of this multidimensional character, each of the behaviours considered crucial could be characterised according to the main contribution of one or more of three dimensions or elements: cognitive/ metacognitive (thinking); affective (feeling); conative/ volitional (acting). As will be elaborated next, behaviours chosen as crucial were a mix of all three types of elements. When responding to the questionnaire, some teachers were asked to complete an additional task. These teachers were members of the Project for Enhancing Effective Learning (PEEL), an Australian teacher-led, school-based action research project to improve classroom teaching and learning that has been in progress for sixteen years (e.g. Baird and Mitchell, 1985; Baird and Northfield, 1992). These respondents were required to select, from the approximately 50 behaviours they considered to be crucial, those they considered overall to be the ten— most important descriptors of quality teaching. The most commonly-selected “top ten” behaviours selected, in decreasing rank order, were: — Takes risks in teaching in order to trial better techniques (V) Employs a range of teaching strategies (V) — — Promotes a reflective attitude by students to themselves and their work (C) — Promotes linking of ideas in learning (e.g. through concept maps) (C) — Actively promotes conditions where students can ask questions (C) — Caters for individual differences among students; tries to extend their learning and understanding (C) — Is flexible - changes teaching approach/strategies as required (V) — Encourages and supports students (A) — Shows respect for students and their needs and concerns (A) — Demonstrates enthusiasm for subject and the work done (A) — Uses strategies that foster students’ self-esteem and confidence (A) This list of behaviours demonstrates clearly the multi-dimensional nature of good teaching, and how cognitive/metacognitive (C), affective/prosocial (A), and volitional (V) elements all contribute to desirable practice. Subsequently, a slightly modified questionnaire (shortened to give a total of 62 teaching behaviours) was completed by 116 practising teachers, none of whom knew about PEEL. Again, these teachers responded by providing the “top ten” behaviours they considered crucial to good teaching. Here, the most commonly-selected behaviours in decreasing rank order were: — Encourages and supports students (A) — Provides clear purposes/ instructions for work to be done (C) — Demonstrates enthusiasm for subject and the work done (A) — Maintains class discipline (V) — Doesn't "put down" any student (A) — Uses language that students can understand (C)

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Regularly monitors students understanding and gives regular, appropriate and prompt feedback on progress (V), (C) Employs a range of teaching strategies (V) — Has high (but potentially achievable) expectations of students (C) — Applies discipline fairly (A) — Demonstrates concern for individual students and their needs (A) — Moves around classroom, actively interacting and participating (V) — Demonstrates personal mastery of subject (C) — While only some behaviours are common to both lists, there is again a clear mix of behaviours that reflect significant cognitive/ metacognitive (C), affective/ prosocial (A), and volitional (V) elements. In the first list, there appears to be a greater emphasis on the teacher trialling approaches and behaviours that encourage students’ competence and confidence to enact reflective question-asking. This emphasis is consistent with teacher aspirations and practices in PEEL. In the second list, there appears to be greater emphasis on teaching for student and classroom organisation and management. Finally, a student version of the questionnaire was completed by almost 1,700 Year 7 - 11 students in thirteen Victorian government and independent schools. I provide one finding only for this questionnaire: the items chosen most commonly by students as important for good teaching. These items, grouped again in decreasing rank order, were: Encourages me in what I try to do — Makes me feel it is always O.K. to ask questions in class — — Helps me to understand what I am supposed to do and why — Speaks in a way that I understand — Shows a thorough understanding of the topics he/she teaches — Encourages me to believe that Ican do the work — Encourages me to think carefully about the work I am doing — Expects me to achieve as much as I can — Doesn’t put me down — Gives us tasks to do that make us think in a variety of ways — Pays attention to what I say or do in class From the students’ viewpoint, therefore, direct actions taken by the teacher to assist them cognitively and affectively in their learning are the ones that they rate most highly. Expressing it another way, they value being valued, as individuals and as learners. These findings emphasise the multiple, and strongly purpose- and contextdependent nature of good teaching expressed as observable actions. In the next section, I consider the implications of these features of good teaching for the analysis and interpretation of integrated data sets in the current research. —

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1.2 Observable teaching behaviours as data The data sources that formed the focus of this chapter were Lessons 1, 7, and 8 _ three science lessons taught by Mrs Brown. An integrated data set for each lesson, comprising verbatim lesson transcripts, classroom researcher annotations/field notes, student post-hoc comments at interview, and teacher post-hoc comments at interview was prepared as discussed Part A of this chapter: From Ignorance To Understanding: Thinking, Feeling And Acting. The videos and associated integrated data sets provide detailed information on observable teaching (and learning) behaviours. It is tempting, therefore, to examine this video evidence to identify whether and, if so, to what extent, identified examples of behaviours such as those considered above can be found in the teacher’s practice. Because of the complex nature of good teaching this approach is suspect, however, if the meanings that underlie teacher actions are not known. The need to explore meanings together with actions reflects the increasing research interest over recent decades in ethnographic, interpretive approaches to the study of classroom practices. The post-lesson interviews by teacher and students form essential parts of the integrated data sets, as they supplement behavioural evidence with self-report on intention, purpose and perspective. In order to pursue this nexus between intention and action, I shall frame inferences and interpretations of the integrated data sets in terms of the taxonomy of teaching practice shown in Figure 11.5. The figure represents a taxonomy of influence, with each level influencing the level below it. The superordinate level is “Teacher Values and Beliefs ”. Conscious or tacit values and beliefs held by the teacher influence teaching intentions and purposes. These intentions and purposes, in turn, influence observable teaching approaches and behaviours (the focus of the preceding discussion) that, again, influence the teachers’ selection of methods and procedures for student learning.

•

• • •

Teacher Values/Beliefs Teacher Intentions/Purposes Approaches/Behaviours exhibited by teacher Methods/Procedures selected by teacher to foster student learning Figure 11.5. Taxonomy of influence on teaching practice

More generally, I shall argue that quality in teaching is fostered by the teacher establishing and sustaining productive personal integration of each of the levels in the figure. This integration in an individual’s thinking, feeling and acting does not come easily, especially if it is not guided and structured by purposeful reflection. In this chapter, I shall advocate self-regulation in teaching as a means of improving teaching quality - a situation analogous to similar advocacy by others for the efficacy of self-regulation for improving quality in learning (e.g Boekaerts, 1997; Zimmerman, 1998). This self-regulation is guided by reflection and evidenced by

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enhanced metacognitive and volitional control over personal practice - for most teachers, a necessary condition for quality. It follows from the perspective above that, in order for successful integration of thinking, feeling and acting at the levels shown in Figure 11 .5, a teacher must reflect upon personal teaching practices as they influence students’ learning. This reflection can, and should, occur before, during and after teaching. I have discussed elsewhere (e.g. Baird, 1990) the nature and importance of all of anticipatory, contemporaneous, and retrospective reflection in teaching. Engaging in these three types of reflection can illuminate, for the teacher and others, the reasons that underlie teaching actions. In so doing, the quest for quality in teaching extends beyond simply a desire to exhibit particular behaviours to efforts to have these behaviours serve desirable teaching values, beliefs, intentions, and purposes. In the next section, I move to consider some classroom data. First, however, I shall summarise a perspective on teaching that emerged from the data that bears directly on what has come before. I mention this perspective now because it is central to presentation of the data below. The perspective relates particularly to the teaching of one of the study teachers, Mrs Brown. Some of Mrs Brown’s teaching indicated lack of integration of the levels of teaching in Figure 11.5. Both her teaching behaviours and her post-hoc comments suggest that she engages to only a limited extent in anticipatory, retrospective and, particularly, contemporaneous reflection about her teaching. As a result, Levels 3 and 4 are inadequately integrated with Levels 1 and 2 of Figure 11.5. In her teaching behaviours and in the learning methods and procedures she selects, she seems to be enacting teaching “scripts”. As such, she applies approaches and behaviours somewhat mechanistically, and pays insufficient attention to particulars of student feedback or learning outcomes. Her teaching lacks the volitional element that, as discussed above, both teachers and students perceive as contributing to quality. 1.3 Fostering effective learning: Teacher beliefs, purposes and actions In this section, as later, the relationships between teacher beliefs, purposes and actions will be considered in ways that attempt to illuminate the nature of quality in teaching. There was some evidence of Mrs Brown grounding her teaching practice in educational theory or productive prior personal experience. For example, in the following transcript, she starts by helping the students centre upon task objectives, and checks their existing knowledge (F = Frances). T: All right, this unit’s on fibres. If you have a look at the front cover we’ll talk quickly about where the unit’s going. So let’s turn to the objectives on page two. Now whenever you’re learning about a unit it’s good for you to know basically what we expect you to know. If I'm doing any reading for a uni essay, then what I need to know is what is the topic of the essay because then I can get information as I go along, otherwise I’m just sort of learning blindly. [reading] So “by the end of this unit the student should aim to: understand the process by which fibre is used to form a fabric”. OK. So underline the word ''process'' for me please, put a number one beside that term, easier if you number them. Put a number one beside the ..

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JOHN R. BAIRD dot .... Objective number one. OK, put number two down. OK, number two says comprehend. Who knows what that means? F:

Understand.

T: OK, above that, put understand if you're not certain what comprehend means,

Then, in her post-lesson interview, she provides justification for her teaching approach and a recognition of different student approaches to learning: [Teacher Interview]: It amazes me how teachers will continually prepare material with this wonderful teacher jargon which says, process, or comprehend and we're putting into student books and it amazes me because a lot of students don't understand those terms and I really wonder what we are achieving by going along those lines. Sometimes in the units we prepare the language is even very difficult for reachers to understand so it is pretty stupid io teach it The other thing too is that there are big picture thinkers who like to know where the topic is going and then they will have an idea and they can put it into context, and then there are those people who like to build up in chunks an idea where the unit is going so what I try and do is to, if there are one or two big picture thinkers in the class, is to show them basically where we are going and what we are going to do, so at least they have an understanding. Those people for whom it is not important I am pretty sure they’lljust ignore that and they willjust work on building the topic up piece by piece.

Her comments indicate her reasons for requiring students to underline and number key words. She considers such procedures to be helpful in aiding comprehension of text. In another interview, she elaborates her reasons for the underlining strategy: [Teacher interview]: The big problem I face with students doing pracs, is that they are so keen to get into the prac, they never ever, ever read the prac. So you give them time to read the prac, and they still don’t know what to do. So I guess my way of doing it is, particularly with kids with short term memory problems etc., is to go ahead and just get them to underline it as I said, because otherwise it becomes a huge sea of words and the kids can’t sort of focus on the main parts. So, yes I am into underlining in a big way, and it seems to work. For those kids that are organised, it may make it quicker for them to eye ball the right part of the page, and those kids who just haven’t got a clue, it gives them a bit of a structure So, where you can, you underline it. I usually start off by showing them what I want and then Iprogressively let go and ask them to say well what do you think it would be, so at least they learn and model on what I am doing.

As will be seen later, however, these reasons may not be similarly appreciated by some of her students, On other occasions, she indicated her recognition of different learning requirements of students, and how she approached attending to such differences: [Teacher Interview]: Often when we give instructions to kids and we give data, it’s pretty meaningless for a kid to say, you know, understand how big ... a heart is, if they give it in centimetres so often if you make a fist or get the kids to do something, I'm trying to tap into the kinesthetic learners I guess who tend to, you know, like to see things much more physically than just simply hearing things. The other thing is that I also think it’s very important that kids know something exactly like the location of the heart. That it is slightly off centre, and we find that because first aid and CPR are very important components. I’m trying to link into that, to make cross links with other types of first aid and subjects.

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important components, I'm trying to link into that, to make cross links with other types of first aid and subjects.

As also indicated above, she made various attempts to enhance students' interest and understanding by relating work to episodes from her life, and to students' lived experiences: [Teacher Interview]: I like to keep the kids understanding some of the basic technology, so that some of their parents or grandparents might have had a pacemaker, and my father had a pacemaker so I guess that I was always interested in that. So where possible try and make the links with things that might be occurring in their family.

She indicated further that she valued students' questions that seek to clarify the nature of her science lessons: [Teacher Interview]: Because the kids had quite a substantial amount of knowledge about the heart from PE lessons, I also thought that there might be some questions that they wanted to have answered, and I see that as a fairly important part of teaching, is to try and meet the students' expectations of what a subject is. And uh if they've got questions they'd like answered, then that involves them in the process. And I find that the student questions can be really exciting. They add a lot of stimulus and they involve the students a great deal ...

As will again be mentioned later, however, this belief seemed not to translate clearly into her classroom practices. Other values and beliefs directed her thinking and, in the next episode, her acting regarding her teaching approach: T: Right, folks, a quick revision. Right, don't open your books. OK, what's the name of the topic? Ss: Fibres. [Teacher interview]: I remember being impressed by someone who was doing some research, and they were saying that they took a group of students who were basically quite good B students, and they wanted to be A students, and amongst other things he worked with them on study techniques. But he said four key questions that you need to improve your knowledge. One is when you come into a class check where you are up to the last lesson. So immediately you are right on target. During the lesson the teacher is really only tying to teach you two or three things, or three or four things, so ty and work out what you think the teacher is tying to do. The third thing is at the end of the lesson, revise in your mind what were those two or three things, or three or four things the teacher was trying to get across during this lesson, then when you go home, go ahead and revise it again. So I think this was my attempt to go ahead and say, 'Okay let's get you back into the lesson, see what you know,’ so that people are sort of starting to think along the same lines and techniques. T: Fibres, good. Fibres, when did we first use fibres?

The episodes and interviews above demonstrate beliefs and purposes that Mrs Brown values, and upon which she wishes to base her teaching. As will be discussed next, however, other beliefs, purposes and actions are less helpful to effective learning.

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1.4 Constraining effective learning: Teacher beliefs, purposes and actions There were instances where students’ approach to, progress through, and outcomes of learning were influenced negatively by teacher attitudes and behaviours. Five types of teacher influence on learning are considered here: Vagueness or Ambiguity in Classroom Instructions — Requiring Student Compliance — Inadequate Feedback — Inappropriate Feedback — Inaccurate Perceptions of Students’ Learning — Vagueness or Ambiguity in Classroom Instructions The first type of behaviour concerned vagueness or ambiguity in classroom instructions. This problem was common; there were at least fourteen episodes related to it for the three lessons. In the following extracts, for instance, the topic of the lesson is the construction of fabrics and the fibres of which they are made. She did not make this topic particularly clear, however: T: So in other words, in this unit we’re going to look at the properties, we’re going to look at wear and fabric and stuff like that ...

Later, the teacher asks students to volunteer information about “all of the fibres you’ve ever heard of”. She does not distinguish in her question, or in her acceptance of student answers, between fabrics and the fibres from which they are made. This ambiguity led to considerable confusion in the minds of students. T: All the fibres you’ve ever heard of. ... S24: Polyester, vinyl, corduroy, cotton, tencel, velvet, nylon, silk, satin, denim, leather, rayon, wool, lycra, corduroy ‚ linen, plastic, suede, um, flax, cotton, synthetic. T: Great. All right, [to study group] your group. Read out your list please.

At interview, the teacher still was unclear about the distinction, as is evident in her comment: They’ve got quite a wide range of knowledge about different fabrics. wider than I thought they may have. I thought they may have been confined to mainly basic raw fibres. They seem to know quite a lot of them like poplin and chiffon, so maybe they have picked that up through the textiles. It is amazing what you do learn when you give kids open questions rather than just going through closed questions. [pause]. Doing an open-ended thing like this, if you tried to get through the topic, you often wonder at what point is it becoming just busy time, and at what point is it really stretching the imagination or moving their knowledge forward, so that is why you try and reduce it to 2- or 3 minutes to see the variety.

The classroom transcript from this stage of the lesson indicated that her “open questions” may have drawn diverse responses from students because they were, in fact, ambiguous rather than open, and generated some student confusion. In the third lesson, her instructions again did not serve to clarify some important concepts, for example:

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T: OK, page eight. [reads from page 8 of booklet] In the previous activity you tried to identify fibres from the feel of the woven fabric. You will now have the opportunity to look at fibres for themselves with the aid of a microscope. When looking at the fabric it is important to take note of the weave and then the yam. So can you underline that please? Third line. It's important to take note of the weave and then the yarn. OK.

For students, it seems that clear distinction between fibre, weave and yarn would have assisted them in their learning, but the teacher did not pursue such a distinction. Shortly afterwards, Mrs Brown developed the notion of porosity of fabrics but, again, somewhat ambiguously (T = Mrs Brown; J = Joanne): T: And underline the word "the weave." So you'll be required to describe the weave which you should do in terms of its porosity. So you'd better underline that ... Porosity can be defined as the surface area percentage of the holes in the fabric. OK, so if you have a look at the surface area and you work out the holes in the fabric and then you try and work out roughly the percentage. So [walks over to board, draws rectangle with circles in it]. Here's a piece of cloth. All right, if we looked under the microscope and we saw that these were all holes, all the way around, and if you did this with the whole of the fabric, if you had to estimate how many holes there were, what would you estimate? ... Now these holes are the holes. And if I asked you for the porosity of the fabric, what would you estimate? It looks like these gaps in between are about the same or a little bit more. J:

Um.

T: Than the holes. What do you reckon?

Here, ambiguity between her use of the terms percentage and “how many holes there were” does not assist students’ understanding. Other instructions by Mrs Brown similarly lacked clarity, and would be expected to lead to student uncertainty. A further example: T: I'd like you to circle those seven questions, very easy, very quick to answer. At the bottom of page 5 I'd like you to put question 8, question 8 and could you say PTO, for turn over. If you turn over to page 8, there's a table that you will need to copy and—oh, sorry, page 6, question 8, um, you'll need to copy and do. Now, throughout this book, there are a series—you virtually have to have together, a number of research questions of this—of which this forms the first eight, fairly easy and there's really not much that you should have any trouble with at all. OK? There is just one, question 3, could you underline the word "husks" H-U-S-K-S. Underline that, that's about the only thing that might, question 3, underline the word "husks."

Requiring Student Compliance A second teacher behaviour that limited systematic and informed progression by students in their learning related to requiring them to “guess what's in the teacher's mind”. It is clear in the following transcript, for instance, that both she and the student involved understood that she knew the “correct answer” and his job was to pick it: T: And number seven, what do you reckon Sam [SI3] is going to be the most important one, or two?

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JOHN R. BAIRD S 13: Um. “Recognise”. T: Nope. S 13: “Principles”. [Joanne laughs] T: Nope. S 13: “Dyeing” T: Yes, well done, Sam. [laughter at the focus students’ table] The dyeing process, OK, I think that'll do.

Inadequate Feedback The third type of behaviour that negatively affected purposeful progression in learning related to shortcomings in the nature and extent of feedback given to the students by the teacher. On many occasions, students had difficulty resolving aspects either understood or not understood because the teacher did not distinguish key aspects of the content material. For instance, the following transcript is a portion of a protracted questioning sequence on the heart and the circulatory system. Through this whole sequence, there were repeated occasions when the teacher’s lack of directed, focussed feedback precluded understanding (“G’’ stands for Grace, “A” for Anna): S30: You need the heart to keep pumping all the blood. T:

Yes, indeed.

G: It’s got the lym— T:

Loudly, please.

G: Lymphatic system. T: It [the heart?] has got a lymphatic system? G: Yeah. T: Right. What’s that? Who’s ever heard of a lymphatic — who's heard of the word? Lymphatic system. [Lauren and Anna raise hands] G: Not me. T:

Who’s heard of the lymph nodes? [Lauren and Anna don’t raise hands.] Anyone tell me where they are?

A: [to group] In your heart.

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S3 I: In your [inaudible], like you can get cancer from them. T: Yes, you can. And lymph nodes are under here, press underneath your jaws, until it almost feels like you'll choke. They're right there, and if there're [inaudible] they get quite swollen, so if you've got glandular fever, the lymph nodes are the ones that get infected. And, yes, it is a part of the blood system, a very important part which I'll explain quickly later. Any other comments, please? Your group. S32: The heart is part of the circulatory system. T: Circulatory system. Who knew that? [Show of hands.] Good. Anything else? Right, anyone-your group, finish any points you've got please.

In order that students could have made sense of the discussion of which this is a part, they would have had to already realise that the heart was a part of the circulatory system, yet the teacher did not ask this question until this late stage. In the above episode, it is possible that the teacher did not hear Anna's incorrect response regarding the location of lymph nodes, but other transcript data reveal situations in which she failed to correct inadequate student understanding. The following sequence, for instance, shows evidence of inadequate teacher feedback: T:

What about capillaries?

G: They're veins, part of the vein. T: So they're part of the system. G: Yup. T:

OK, this group?

S23: When the blood's inside the body it's yellow and when it comes out it's red. T: When the blood's inside, it's? S23: Yellow. T: And when it's outside, it's? S23: Red. T: OK. OK, and that group here?

In this episode, the teacher appears to misconstrue an incorrect student response regarding the relationship between veins and capillaries, and she accepts without comment another clearly incorrect response. As will be discussed again later, she repeatedly appears to inadequately listen to and reflect upon the substance of students' input, but limits herself more superficially to question-answer recitation. In the last episode above, Mrs Brown seemed either to lack knowledge of the content area or had prepared inadequately for the lesson. On at least one occasion,

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she admitted to limited content knowledge, as in the following class discussion about plant material samples that she had provided for the students to examine (“F” stands for Frances): T:

Don't know. I wasn't sure about that one.

F: That's made out of a tree, isn't it? T: Don't know. I wasn't sure about that one. Yes, and that wood.

Perhaps as one result of such behaviours as those above, the teacher occasionally acknowledged that she had only a limited idea of lesson learning outcomes: [Teacher interview]: So you sit back and you think I wonder what the kids got out of the double lesson, and I wonder what they learnt that was new. Uhm and really was it more or less busy time, really. You know, they knew that there were different types of fibres, it was a bit of fun to feel the different textures of them but so what? That’s my feeling, so what. The lesson went fairly smoothly, we started a new unit. We handed out the books from the old unit, reinforced liitle bit about how to write up a book. Kept them busy and on task, but they come out of it and they think, “oh yeah so what.” So I really don’t know what they came out knowing on that one, I’m afraid. So it was just another lesson, another day Nothing really exciting or different or momentous.

In various ways, lack of timely and adequate teacher feedback had significant impact upon students’ approach to, and success in, learning. In the following sequence, lack of appropriate teacher action persuaded a student to accept as correct a fallacious student comment (“A” stands for Anna): A:

Yeah. Like there was a girl in a group, in the group at the table next to us, um, they said something about how the blood in your body’s yellow and when it comes out it’s red, and I didn’t know that I didn’t really know that at all...

I:

Do you think it’s true?

A:

Yeah, I do actually.

I:

What makes you think it’s true?

A:

I don’t know. It’s just—I heard something about its different colour, I didn’t understand what they meant, so I don’t know, I always heard that um, it was blue or something, I didn’t believe that

I:

Yeah, right.

A:

But, the, I know that it would be true but I don’t know why...

I:

You’re saying you believe it, I’m just trying to work out—do you believe it because the kid said it or because you saw Mrs. Brown say “Yeah, that’s right”?

A:

Yeah, well she would have said no actually, because she likes to get things clear, that’s what Mrs. Brown would have said, and that’s, and I suppose that’s why I really thought it was true otherwise ‘cause she would have said “Oh, no, actually it’s not, but don’t worry about it.’’

LEARNING AND TEACHING: FROM IGNORANCE TO UNDERSTANDING I:

So you think she would have probably said, if a kid said something that was wrong

A:

Yeah, she would have corrected.

I:

She would have said no, I don’t think, that’s, that’s the case.

A:

Yeah. She just would have made sure that it clear to everyone so that they don’t get the wrong impression.

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This strategy of depending upon the teacher to intervene appropriately breaks down when the teacher fails to take appropriate action. This episode demonstrates the subtlety of classroom exchanges, and the potentially powerful effects of minimal teacher action. Inappropriate Feedback Learning problems arise when there is mismatch between values and goals regarding learning held by teacher and student. An example of this situation is presented below. It involved an interaction between Mrs Brown and “Joanne”. As mentioned above, the teacher had advocated an underlining strategy for focussing upon important terms in written information. The difficulty here was that the student was unclear about the basis of the strategy and particularly the teacher’s criteria for determining which words should be underlined and which not. In this sequence, Joanne believed that the term “define”, a term in the text, was especially important to underline because of its implications for her learning, but she felt less strongly about the word “monomer”, even though she did not know its meaning: J:

She should say that “Go ahead and underline everything that you think is [important].”

I:

Uh huh.

J:

And then, but when we do do that, she’s done that a couple of times, but then we have to say it, um and she goes through it, saying “OK, you should have underlined, this, this, this, and this” and most of the time I haven’t underlined the things that she wants me to, but when I do I don’t understand, what ....

I:

And you were saying that the [word] monomer didn’t, you know, didn’t mean anything to you.

J:

No ... Because it was just a word there but define kind of said you have to understand .... and if it had define, it means that I know that I have to understand the following stuff.

I:

Um hm, um hm.

J: Which. And I have. I: Yeah, yeah.

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JOHN R. BAIRD J:

So she was just telling me to underline that [monomer] but that didn’t make sense to me.

I:

Um hm.

J:

When I haven’t underlined it she says “Oh, right, come on, hurry up, you’re keeping the class waiting”.

Other types of teacher feedback on students’ uncertainties were more obviously unhelpful, for example: J:

Me and Narelle just sit there going “What are we doing?’ like ‘cause we don’t know, we don’t have a clue what we’re doing and.

I:

So how do you find out? Yeah, you were telling me, you ask other [students], yeah, let’s see, what do you do?

J:

Well, if, if we ask Mrs Brown. she tells us, she kind of tells us as though “why don’t you already know this?” in, in that kind of tone. As though, you know, you do this, you do this, and you do that, why don’t you know it?

The episodes above illustrate the subtle and complex nature of the social interactions that are central to teaching and learning. Another dimension of these interactions is discussed in the final type of teacher attitudes and behaviours that act to constrain desirable learning, considered next. Inaccurate Perceptions of Students’ Learning An important aspect of the influence of social interaction for nature and quality of learning centres on teacher perceptions of students, and how such perceptions influence both teacher and student attitudes and behaviours. Again, interactions between Mrs Brown and Joanne illustrate this effect. At interview, the teacher repeatedly denigrated Joanne and her learning. These negative attitudes and beliefs seemed to limit the teacher’s ability to dispassionately evaluate Joanne’s learning behaviours. In the classroom interaction considered next, the teacher makes a disparaging comment about Joanne’s response, but does not highlight similar responses by other students: T: Sally? You don’t know? Did you do your homework? S5: Yep. T:

Michelle [S7]?

S7: I don’t know. T:

Oh, dear. Sally?

N: [to T] What was the question?

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T: I'm sorry, you said you didn't know. Kate [S8]. S8: Um. T: Oh dear, oh dear, oh dear. Joanne? J:

What was the question?

[Teacher interview]: On yer Joanne, oh dear, off with the fairies.

In the following episode, Mrs Brown’s comment about Joanne seems inappropriate, given that the student is making an effort to pursue the learning task (“J” stands for Joanne, “F” for Frances). J:

What else did you have? [looks at Narelle’s page] Tablecloth.

F: Upholstery. T: No, I don't want you to look in your book. I want things that you use them for. J:

Tablecloths. I have tents. Um. Carpet.

F: Yeah, carpet. J: Um. F: Lab coats. [laughs] [Teacher interview]: Yeah, see I don ’t think Joanne is ever really involved in what is going on. You know l just think that luckily she is not a behavioural problem but she basically seems to be off target a fair bit of the time. She will do a little bit of work and then go and rely on those around her, to sort of supplement her work. And I don’t think at any stage I’ve really engaged her except to maybe feel the materials. Hm, interesting. J:

Paper!

F:

Do we?

J:

‘Cause fibre is in paper, isn't it?

Again, in the next sequence, Joanne is attempting to understand the meaning of porosity, and she asks the teacher a clarification question. Here, Joanne has recognised what she doesn’t know, she enacts a reasonable behaviour to determine the answer, and she tries to embed the teacher’s response within her existing knowledge of the structure of wool. The teacher, when viewing the videotape, does not perceive the interaction positively, however: J: Is that how big the holes are? T: High number of holes. High number of holes.

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JOHN R. BAIRD J:

[turns away from T] So that has very low porosity. For wool.

T: Like high percentage of holes, high porosity. J:

So that has very low porosity. For wool.

T: So very few holes. [Joanne shakes head] Good. Right. J:

Yes, there were hardly any holes.

T:

[moves to next table]

[Teacher interview]:So there really is quite a reluctance for Joanne to be an independent learner. I think that’s what I really picked up. That as long as you hold her by the hand and take her through, she’ll do it, but if it means she has to go back to a book and check or whatever, she’s always asking. And I don’t think its because she’s an oral learner rather than a visual learner, I just think she lacks a lot of the skills of an independent learner, and is quite willing to ask other people for an answer so she can get a quick answer and go on rather than follow something for herself:

It is interesting that, when Joanne viewed the classroom videotape, she came to a realisation about how her own behaviours could be misinterpreted by the teacher as showing lack of attention. Perceptively, she decided to try to amend such behaviours in the future: J:

It’s funny ‘cause when I was um, like I was just looking there and it looks like um, and it looks as though I’m not listening at all but I, I was listening to everything that she’d say.

I:

Oh!

J:

And I was thinking, “Gee, that’s why she gets probably gets annoyed ‘cause she thinks I’m not listening” but I find that I can’t listen when I’m sitting there doing nothing just listening, I have to sit there going like this [imitates position on videotape, sitting over paper, pen in hand looking down at desk], 'cause I’m not listening. ..I remember, I was listening to everything she was saying and I was just looking then.

I:

Um hm.

J:

Yeah. Oh, well, I’ll have to make sure I look as though I’m listening. [laughs]

1.5 Enhancing Teaching Reflection: A role for stimulated recall Before concluding the presentation of data from the integrated data sets, I shall emphasise a methodological feature of stimulated recall using videotape. As will be discussed in the final section, many of the teaching shortcomings evidenced above are related to a lack of on-going reflection by the teacher. By stimulating retrospective reflection, videotape review can assist teachers to learn more about

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their teaching and their students’ learning, as evidenced by such comments as the following, made by the teacher at interview. First, an insight about the efficacy of her strategy of probing student knowledge: I’m most amazed to hear the lymphatic system. I wouldn’t have expected them to know that. As well as vena cava. So I’m really glad that I chose the general introduction of finding out what the kids know. Because for Lauren, I think I would have bored the socks off her if I hadn’t um at least got a fair idea as to what they know and what they don’t.

Second, an insight about student understanding: I’ve never really considered that kids just think that the blood would go round the body and not through tubes. Their distinction between the veins going round the body and the blood going round the body more or less gives me a different insight into how kids think.

Next, an insight about students’ learning: I am very impressed with the amount of knowledge sharing that’s going on between the four students. I often try and find out for myself what the kids know but the amount of peer teaching is quite considerable. So it really reinforces this as a constructive approach at the right times as apart of student learning. And I also think it gets the kids to be involved and active so I’m most impressed with the quality of the conversation that’s going on.

1.6 Discussion: Integrating levels of teaching for enhanced teaching quality As described earlier, quality teaching is perceived by both teachers and students as an inextricable blend of thinking, feeling and acting. Teacher cognition/metacognition, affect and behaviours all contribute to desirable, productive professional practice. A perspective that arises from the data above relates to how mismatch between espoused beliefs and beliefs in action limits quality in teaching, and how focussed thinking, feeling and acting to integrate the four levels of teacher practice shown in Figure 11.5 can generate enhanced teaching quality. Particularly, the data above indicate that: — It is possible for a teacher to espouse certain values and beliefs, and intentions and purposes, but not succeed in translating these features into teaching practice; — Lack of reflection may lead to a lack of informed metacognitive and volitional control over the specifics of particular teaching and learning episodes. This lack of reflection may be exhibited as enactment of teaching “scripts”, where the teacher enacts known and valued approaches or behaviours somewhat mechanistically, without taking adequate action on student feedback shown either by questions or learning behaviours; — The gap between what is espoused and what is enacted may arise because of limited teacher reflection, particularly reflection that is framed in terms of establishing functional integration between each of the four levels in Figure 11.5;

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Lack of integration among the four levels may diminish the scope of behaviours, valued by both teachers and students as characterising quality teaching, that a teacher exhibits; — Through lack of reflection and routine enactment of a teaching “script”, the teacher may limit the range of active informed student learning, such as that described in Part B of this chapter: From Ignorance To Understanding In Learning: Thinking, Feeling And Acting. According to the types of active learning discussed there, the teacher is not promoting Active Learning Types 2 or 3 that centre on student initiation of evaluative questions. Rather, the type of Active Learning that she fosters is at best Active Learning Type 1, characterised by student compliance and busyness and limited in student metacognitive and managerial control over learning practice. —

REFERENCES Baud, J.R. (1990). Individual and group reflection as a basis for teacher development. In P. Hughes (Ed.), Teachers' professional development. Hawthorn, Victoria: Australian Council for Educational Research. Baird, J. R. (1994). A framework for improving educational practice: Individual challenge; Shared Adventure. In J. Edwards (Ed.), Thinking: International and interdisciplinary perspectives. Melbourne, Hawker Brownlow Education. Baird, J.R. (1998). Promoting willingness and ability to learn: A focus on ignorance. Reflect - The Journal of Refection in Learning and Teaching, 4, (1), 21-26. Baud, J.R., and Mitchell, I.J. (Eds) (1986). Improving the quality of teaching and learning: An Australian case study - the PEEL project. Melbourne: Monash University Printery Services. Baird, J. R., and Penna, C. (1997). Perceptions of challenge in science learning. International Journal of Science Education 19, (10), 1195-1209. Boekaerts, M. (1997). Self-regulated learning: A new concept embraced by researchers, policy makers, educators, teachers, and students. Learning and Instruction, 7, (2), 161-186. Boekaerts, M. (1998). Boosting students’ capacity to promote their own learning: A goal theory perspective. Research Dialogue in Learning and Instruction, 1, (1), 13-22. Brown, A. L. (1978). Knowing when, where and how to remember: A problem of metacognition. In R. Glaser (Ed.), Advances in instructional psychology. Hillsdale, NJ: Erlbaum. Candy, P. C. (1990). How people learn to learn. In R. M. Smith (Ed.), Learning to learn across the life span. San Francisco: Jossey-Bass. Corno, L. (1993). The best-laid plans. Modern conceptions of volition and educational research. Educational Researcher, 22, (2) 14-22. Kuhl, J. (1985). Volitional mediators of cognition-behavior consistency: Self-regulatory processes and action versus state orientation.. In J. Kuhl and J. Beckmann (Eds.), Action control: From cognition to behavior. New York: Springer-Verlag. Larrivee, B., & Algina, J. (1983, April). Identification of teaching behaviors which predict success for mainstreamed students. Paper presented at the annual meeting of the American Educational Research Association. Miserandino, M. (1996). Children who do well in school: Individual differences in perceived competence and autonomy in above-average children. Journal of Educational Psychology, 88, (2), 203-214. Nicholls, J. (1983).. Conceptions of ability and achievement motivation: A theory and its implications for education. In S. Paris, G. Olsen, and H. Stevenson (Eds.), Learning and motivation in the classroom. Hillsdale, NJ: Erlbaum. Spaulding, C. L. (1992). Motivation in the classroom. New York: McGraw Hill. Snow, R. E., Corno, L., and Jackson III, D. (1996). Individual differences in affective and conative functions. In D. C. Berliner and R. C. Calfee (Eds.), Handbook of Educational Psychology. New York: Macmillan.

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Wright, C., & Nuthall, G . (1970). Relationships between teacher behaviors and pupil achievement in three experimental elementary science lessons. American Educational Research Journal, 7, 477-491. Zimmerman, B.J. (1998). Developing self-fulfilling cycles of academic regulation: An analysis of exemplary instructional methods. In D.H. Schunk and B.J. Zimmerman (Eds.), Self-regulated learning: From teaching to self-reflective practice. NY: The Guilford Press. Zimmerman, B. J., and Schunk, D. H. (1989). Self-regulated learning and academic achievement: Theory, research, and practice. New York: Springer.

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DAVID CLARKE

CHAPTER TWELVE Teaching/Learning

1. SYNTHESIS The synthesis of the findings of a study such as this constitutes a significant challenge. The aggregative approaches of quantitative research do not apply. Each of the preceding chapters has its own message, its own legitimacy, and its own significance. The premise on which this book is predicated, however, is that the combination of accounts should represent far more than merely the sum of its constitutive chapters. Of course, the synthesis of the results of such qualitative research analyses must be interpretive rather than aggregative (Jensen & Allen, 1996). Sandelowski (1997) has asserted that the purpose of an interpretive synthesis of qualitative research is not to generate predictive theories, but to facilitate understanding of the phenomenon, context or culture under consideration. In the case of a project such as this, where the methodological and contextual coherence of accounts is high, the enhancement of our understanding of learning and teaching in classroom settings is our most basic aspiration. Since each contributing author has utilised a specific theoretical framework in their analysis of the common body of data, we would hope to have advanced the reader’s appreciation of the multiplicity of theories that might inform our understanding of classroom learning and teaching. The question of whether theory itself has been advanced by our analyses is left for specific discussion later in this chapter. In a study that is both ethnographic and interpretive in character, this concluding chapter is arguably phenomenographic in its multi-perspective examination of mathematics and science classrooms and its pursuit of patterns across the various interpretive accounts represented by the chapters. Noblit and Hare (1988) list some alternative approaches to such a qualitative synthesis: — Reciprocal Translational Synthesis, where key metaphors or themes evident in individual analyses are identified and reapplied to the findings arising from the other reports. The goal (as identified in Chapter 2) is parsimony of construct, and those terms or findings are selected that can most succinctly describe the findings of all reports in the study. In this book, the various usages of uncertainty and negotiation in several chapters have served to illuminate the reading of other chapters and to confirm Uncertainty and 291 David Clarke (ed.), Perspectives on Practice and Meaning in Mathematics and Science Classrooms, 291—320. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Negotiation as key elements of any synthesised account of Classroom Practice. — Refutational Synthesis, in which key findings are compared in order to identify contradictions (explicit or implicit). Through an interpretive approach, implicit contradictions are made explicit. The goal is to develop new metaphors to explain the contradictions between accounts. The tensions addressed in this book reside more in advocacy of the need to abandon the prescriptive identification of Intersubjectivity or Negotiation, for example, as either exclusively process or product. The challenge of constructing a suitably integrative metaphor is addressed later in this chapter. Other tensions addressed include the various forms of asymmetry identified in teacherstudent and student-student interactions, and the form (and optimal utilisation) of teacher knowledge in scaffolding valued learner practices. The conjectured fluidity in the relative positioning of classroom participants within such interactions constitutes a form of metaphor intended to sustain the applicability of Vygotskiist learning theory to teaching/learning situations that bear little resemblance to those with which Vygotsky was familiar. Lines-of-argument Synthesis, where key categories that are more powerful in — representing the combined set of accounts are identified. The goal is to link these key categories interpretively to create a holistic account of the whole phenomenon (classroom teaching and learning). Running through all the chapters of this book are conceptions of interactivity and co-construction of practice. The objects of this co-construction may be ‘values’ or ‘social norms’ or ‘mathematical meaning’, but the pervasive argument to emerge from consideration of the preceding chapters is the importance of considering each such construct as both the product of co-construction and as the vehicle by which the co-constructive process is advanced. In the discussion that follows all three of the above approaches to synthesis are evident to some extent. One aspect of this interpretive synthesis must be emphasised: It is ultimately one person’s synthesis (mine). Other readers of the preceding chapters might place the emphasis differently, notice other similarities or differences, and draw different conclusions. My co-authors would all have framed this chapter somewhat differently. Having made that point, I would argue that the synthesis that follows is consistent with the research we have conducted and with the various complementary accounts reported. In a summative chapter such as this, the organisation of the content poses several challenges. The most obvious is the choice of a structural organization for the chapter that appropriately and clearly sets out some of the main features of the synthesised account. For example, Holton and Thomas report that there were few instances of non-task-related talk in the four mathematics lessons they analysed, and they situate this observation in an analysis that reveals how classroom talk was structured. How might we benefit from their analysis? Is it most usefully interpreted as a reflection on the motivation of the students, related to Ainley’s analysis and the analysis of Helme and Clarke, and our attention best focussed upon the optimisation of that motivation? Is it, rather, a positive reflection on the teacher’s management or

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organisational skills that provided the appropriate combination of affordances and constraints to minimise non-task-related talk and maximise task-specific discussion and discussed in relation to Lewis-Shaw’s discussion of the teacher’s ability to model and foster the development of particular values leading to productive classroom participation? If so, then we might usefully reflect on the teacher actions by which this was achieved. Alternatively, perhaps our attention might be most usefully directed to the tasks themselves, their characteristics and their function within the lesson to engage students. In which case, we might relate the Holton/Thomas analysis to Rodrigues’ discussion of the key features of internalised contexts and the manner in which tasks and classroom situations combine to constitute appropriate contexts for students. Essential to this chapter and this book is the commitment to an integrated view. In Chapter 11, Baird commented on “the inappropriateness of presenting any representation of the classroom that does not give explicit acknowledgement to the centrality of the student-teacher dyad.” Of all contributing authors, Baird is most consistent in his reference to “the teaching/learning situation.” We can interpret Baird’s advocacy as encompassing all interactions between classroom participants as constitutive elements in the co-construction of the practices of the science or mathematics classroom and, implicitly, of the social situatedness of these practices and the co-constructive process. Any attempt to synthesise the analyses reported in this book must reflect this integrated and situated perspective. This chapter takes its structure from the most obvious features of mathematics and science classrooms: the teacher, the learners, the lessons, the methods by which these might be studied, the theories that inform our study, and the collective practices by which (and of which) such settings are constituted. 2. METHODOLOGY Some comment should be made regarding the manner in which the goals of complementary accounts methodology have been realised in practice. The details of the methodology were introduced in Chapter 2. The implications of the methodology are best explored here, where they can be related to the results of the previous chapters and evaluated in terms of their demonstrated effectiveness. Also, the way in which the researcher is positioned in relation to the data can be better understood after reading the various authors’ accounts. In some chapters, this issue of positioning has been addressed very directly. In his opening remarks, Lerman makes the point “This account, then, will need to be as much about the latter event [the analysis of the classroom video and the classroom and interview transcripts] in which I did take part as about the recorded classroom events.” The distinction that Lerman draws between his role as interpretive analyst and other possible roles he did not undertake, such as interpretive data collector, is a useful reminder that the complementary accounts from which the methodology derives its title include the videotape account, the students reconstructive accounts, the teachers’ retrospective commentaries, and the research team’s interpretive analyses. Differences exist between each type of

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account. For example, Lerman states, “An interview about an event, even using the video, is a different event, other issues and reactions are elicited by the framing of this other environment.” Given this, each account should be read as fundamentally situated, reflective of its author’s position with respect to the project data and the project’s purpose. The nature of the data available to the research team provided both new challenges and new opportunities. In carrying out her analysis, Lewis-Shaw could not utilise the standard Hall-Tonna questionnaire and had to adapt the Hall-Tonna framework to the analysis of text (in this case, transcripts of classroom and interview dialogue). Baird (Chapter 11) utilised an existing theoretical model of learning to generate a coding scheme for elements of the data set that exemplified the key constructs of his model (see also, Baird, 1998). In Chapter 6, Ainley identified the range of perspectives offered by the classroom participants’ complementary accounts as a significant methodological strength, facilitating the exploration of a complex construct such as ‘interest’ by contrasting participants’ perspectives on a specific event. Reeve and Reynolds (in Chapter 10) assert that the inability of research to resolve the differences between Piagetian and Vygotskiist accounts of changed understanding in collaborative problem solving situations is, at least in part, a methodological problem. They suggest that a major constraint on “the kind of stories that can be told about the role of social interaction in cognitive change” has been the analytic method(s) employed. In citing the work of Fischer and Grannott (1995), Reeve and Reynolds draw our attention to the need for a methodology that would reveal features of the development of the ensemble skill level or functional competence as students worked collaboratively in pairs to solve problems. The relative scarcity of studies of gesture in classroom settings further suggests a lack of data amenable to that form of analysis. Portions of the videotape data available in this study provided the fine-grained record of student-student gestural interaction needed to sustain such an analysis. Given the demonstrable value of an analysis of gesture, future studies utilising classroom videotapes should ensure that the physical configuration of camera(s) and microphone(s) maximises the opportunity to integrate both speech and gesture in the documentation of classroom interactions. Of all the analyses reported in this book, that by Reeve and Reynolds provides possibly the clearest indication of the potential of videotape data for fine-grained interpretive analysis of social interaction. Fran’s four gestures at A10 were particularly interesting. What Fran seemed to be doing was developing a notion of acceleration of a ball to maximum velocity. Indeed, her gestures seemed to describe the correct mathematical function. The problem then became to identify the appropriate graph. Patricia’s gesture (at A11) in response to this verbal/gestural statement was considerably more stylised. Taken in conjunction with the tempo of Patricia’s speech, which slowed markedly at that point, we interpreted Patricia as trying to enact the acceleration described in another graph and to contrast her gestured description with Fran’s (from Chapter 10, P. 291).

Reeve and Reynolds’ suggest the possible existence of a lexicon of gestures that might help researchers (and, ultimately, teachers) decode aspects of the process of collaborative problem solving. They conclude that “gesture obviously provides an

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explanatory context without which the words of the students would be difficult to interpret.” Lerman, for example, and others in the research team, also interrogated the video record in substantial detail, seeking non-verbal cues as to the referents and significance of student statements and actions. As stated above, the need for this careful analysis of gesture and oral inflection places a methodological obligation on researchers studying the co-construction of meaning in classrooms to employ data collection techniques that provide a sustained, continuous record of student-student interactions, both verbal and non-verbal. Acceptance of the mutuality of teaching and learning, as will be argued below, places an obligation upon researchers to collect data that offers voice to all participants in a social setting. Participants’ voices must be heard both in terms of documenting the substance of social interaction and through the provision of the opportunity for participants to interpret and comment upon their own actions in order to provide data on the meanings that triggered, accompanied and arose from those actions. Within the pragmatic contingencies of budget, time, and available personnel, and the constraints of ethics and school context, the database for this project went some way towards meeting these requirements. 3.

THEORETICAL CONSIDERATIONS

Every effort has been made by the authors of the chapters in this book to establish the theoretical position from which each analysis is undertaken. Lerman, for instance, does this by explicitly acknowledging that “the assumption of this chapter, regarding learning, is that it takes place in the zone of proximal development”. The remainder of Lerman’s chapter is predicated on this assumption. In citing Smith (1993), Lerman endorses the view that socio-cultural factors are constitutive of learning. In this view, primacy is accorded to the social practices through which subjectivity is constituted and the “social settings in which meanings are carried”. The challenge is to any theory that postulates learning and meaning construction as the unfettered activity of a decontextualised individual. While various chapters refer to learners as ‘cognising agents’ and several references are made to the construction of meaning, none of the researchers represented in this book fail to acknowledge the importance of social context in the learning process. Equally, while explicitly embedding their accounts in identifiable social settings, chapter authors acknowledge the subjectivities of individuals acting within those social settings. This dialectic echoes Confrey’s succinct dictum that “The self is both autonomous and communal” (Confrey, 1995), but (as will be argued below) demands that individual autonomy be seen as socially constrained and individual participation in communal activity as mediated by personal proclivities and capabilities. In Chapter 3, meaning itself is specifically identified as a ‘distributed attribute’, residing to a significant extent in the social practices and artifacts of the classroom. The reader will have noticed some differences in various authors’ usage of specific theoretical terms. In Chapter 5, Holton and Thomas employ the zone of proximal development as a learner potentiality realised or not according to the constraints and affordances of the classroom environment. Lerman, by contrast,

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asserts that “The ZPD is not the individual’s, it is not a physical space, or a region of potential that the individual brings with her/him to any situation” (from Chapter 4). These differences in theoretical position both informed and fueled discussion within the research team. As has been stated already, it was the independent coherence of each author’s theoretical position that was the principle criterion for the acceptability of a researcher’s account. The obligation, accepted by chapter authors, was to set out clearly and coherently the theoretical basis for their analyses and for their use of any constituent constructs. One challenge for theorists has always been to account for the demonstrable diversity of individuals’ knowings within the evident commonalities of action associated with participation in a common social setting. A focus on learning as a form of incrementally increasing participation in an existing body of social practice provides one useful lens. A valuable perspective is also offered by consideration of the extent to which features of the social setting constrain or afford particular practices associated with learning and thereby constrain or afford the learning itself. In a book such as this, so dependent on the recounting of various texts, it becomes essential to consider how the individuals in the classroom are positioned by the discourses in which they participate. The discourse of educational research also acts to position participants in ways that afford and restrict certain interpretations. Analyses that attribute characteristics such as interest, motivation or values to individuals require a theory of psychology of the individual, albeit a sociallysituated individual, that recognises personal histories and perceptions. The recurrent use (either explicit or implicit) made by the authors of the chapters in this book of concepts associated with Vygotsky can be interpreted as an indication of the viability of a Vygotskiist view of classroom learning, and also of the theoretical alignment (as distinct from correspondence) of the separate analyses reported in this book, despite their initially independent generation. While Lewis-Shaw invokes Kolb’s highly individualistic model of learning (Kolb, 1984), she constructs a bridge between the picture of knowledge construction by an individual and the Vygotskiist position that socially meaningful activity is the generator of consciousness, citing Vygotsky’s own conception that “The final product of this child-adult cooperation is a solution, which, being internalised, becomes an integral part of a child’s own reasoning” (Vygotsky, 1997, P. xxxv). The need to relate individual knowledge construction to learner participation in social interactions was, inevitably, a common component of the chapters in this book. Lewis-Shaw cites Hall (1994) in asserting that “values are the brokers between the inner world of thoughts and feelings and the outer world of language and actions.” By way of contrast, Rodrigues in Chapter 9 states, “Context, in the sense in which I employ it, is not external to the cognitive being.” Rodrigues identifies Internalised Context as constituted through interaction “in a given situation”; a conception that accords an external character to the situation but stresses the unique character of each individual’s experience of that situation and postulates the emergence of an internalised context as “enacted” through participation in the social situations we find in science and mathematics classrooms. It is a characteristic of each analysis that the construct central to the concerns of that chapter finds a place at the interface

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between the personal and the social or, depending on the perspective, within their union. The concern with negotiation that provided the major motivation for this study is echoed throughout the chapters of this book. Lewis-Shaw endorses the use of “negotiation” as a metaphor for one aspect of the learning process on the grounds that “Negotiation takes account of the dialogical and social nature of learning.” From the perspective of a values analysis, she further asserts that “negotiation proceeds continuously at the affective level, whether teacher and learner are conscious of it or not.” The various accounts provided in this book amply illustrate the many forms of negotiation through which learning is constituted. The social character of negotiation as metaphor and as process is entirely apposite to the interactions that constitute the basic data for this study. The question is less “Are our classrooms sites for negotiation?” but rather “What aspect of our classrooms is not the subject of negotiation?” Clearly, the subject of classroom negotiation (explicit or implicit) might be mathematical or scientific meaning, but it might also be social norms or values. One reading of the analyses reported here is that such negotiation should become, in every case, much more public and explicit; partly, in order that the contribution of learners to that negotiation might be both recognised and valued, but also in order that the skills required for negotiation become a visible component of the curriculum and a valued feature of everyone’s classroom practice. Reeve and Reynolds commented on the inability of research to distinguish conclusively between the viability of Piagetian and Vygotskiian accounts of learning in classroom settings. Viability is very much the criterion to which the research team’s various analyses were held accountable. The combined account of classroom learning documented here is grounded in social interaction but mediated by individual interpretation and construction. Classroom learning, as it is portrayed in the chapters of this book, originates in the social interactions of the classroom and in the practices in which the participants engage. However, as will be argued below, these practices and interactions are only to be understood in terms of each individual’s construal ofsituation and history of affiliation. Our research suggests that any account of learning in classroom settings that sought to locate learning solely within social practice or solely within the cognising individual would be seriously mistaken. Cobb’s suggestion that each of learning as acculturation and learning as cognitive construction presumes the other has merit (Cobb, 1994). The distinction lies in the primacy accorded to the social or the individual aspect of learning. Because we have studied learning in classrooms, our accounts have social interactions as their genesis. But these same accounts have demonstrated repeatedly the significance of individual interpretive acts on the part of all participants and with respect to all aspects of their classroom experience. Further, the idiosyncratic character of an individual’s learning demonstrated here and in many other studies; notably in Erlwanger’s celebrated and seminal account of Benny’s conceptions of mathematics (Erlwanger, 1975) challenges any attempt to subordinate individuality to conformity to communal practice. If we are to move forward, we must conceive of socio-cultural and constructivist theories of learning not as competing but as complementary. That they can be constructed so as to be in competition is evident. It is equally evident that our pursuit of yet another dichotomy

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(as with cognition and affect) is counter-productive. Each theoretical frame provides coherent accounts and explanations for particular forms of learning in particular settings. Any conception of either theory that precludes the other is arguably inadequate. The identification (construction) of a theory of learning compatible with a given situation may take the social or the individual as its starting point but ultimately will be obliged to make appeal to the other if a coherent account is to be constructed. 4. THE TEACHER It must be acknowledged from the outset that no-one’s teaching would survive such detailed scrutiny without criticism. The issue to be addressed and the focus of our analyses was not the competence of the teacher but the viability and the appropriateness of particular teacher acts, situated in time and in the socio-cultural context of the classroom being studied. This context included the topic being studied. It is of interest that three of the science lessons and all of the mathematics lessons were taught by the same teacher. This provided an opportunity for comparison not just of one teacher’s practices in the two domains of mathematics and science, but of differences in the analyses carried out on that teacher’s practice and the criteria by which that practice was challenged or affirmed by each member of the research team. That said, several authors commented on the manner in which the teacher’s expectations of a student coloured both her interaction with particular students and her interpretations of various students’ actions or statements. Lewis-Shaw argues that these interpretations are made on the basis of a system of values that “stand behind all behaviour”. Certainly, the various analyses draw our attention to differences in participants’ interpretations of the same events. Differences in underlying values are one plausible reason for such differences in interpretation. Lerman drew our attention to the apparent discrepancy between the teacher’s interpretations of the tutor-tutee relationship between Darren and Mark in the lesson on ratio, and the students’ actual interaction as documented in the videotape and reported in student interviews. In analysing the Fibres lessons (Lessons 7 and 8), both Baird and Rodrigues discussed the teacher’s characterisation of Joanne as “off with the fairies” and suggested that Joanne’s curiosity and reflective nature went largely unrecognised. It is of particular interest that Joanne herself, on viewing the videotape, commented that it did not look as though she was paying attention at all. While acknowledging the dangers of teachers’ mistaken typifications of students (see, for example, Clarke, 1992), the data reported here also suggest that those aspects of a student’s classroom practice that a teacher might observe can be a misleading indicator of the student’s engagement and that other strategies should be employed to supplement the data available to a teacher through observation (for example, Clarke, 1987, or Clarke, Waywood & Stephens, 1994). Baird devotes a significant portion of his discussion of “Quality Classroom Teaching” to identifying instances of lack of clarity in the teacher’s instructions and in the feedback (or lack of feedback) that the teacher provided to students. This

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discussion in Chapter 11 is set against the context of previous research by Baird in which teacher and student groups were asked via questionnaire to identify the “ten most important descriptors of quality teaching.” Baird draws on the results of this survey to highlight the “multi-dimensional nature of good teaching” and the importance attached by all respondent groups to cognitive/metacognitive, affective/prosocial, and volitional elements of teaching. Quality in teaching is identified by Baird with self-regulation in teaching in correspondence to his similar advocacy of self-regulation as a characteristic of good learning practice. While identifying within the teacher interview data several instances of Mrs Brown grounding her practice in educational theory, Baird also identifies several instances in which the teacher’s espoused theories did not seem to be borne out in practice. Particular counter-productive teacher practices are identified by Baird and their negative consequences illustrated from the data. One of the most crucial of these was a tendency to “inadequately listen to and reflect on the substance of students’ input.” Among the useful perspectives offered in Chapter 8, Lewis-Shaw suggests that teachers (like students) are continually confronted with competing value priorities. From her analysis, Lewis-Shaw concludes that the teacher is more concerned with “getting his message across” than with developing students’ thinking skills. This can be seen as a tension between cognitive and metacognitive curricula. It is a theme of several of the chapters in this book that teachers should accept responsibility for scaffolding students’ metacognitive activity as well as their cognitive activity. Baird’s very specific advocacy of teacher promotion of Active Learning equates to the teacher fostering “student metacognitive and managerial control over learning practice.” Constructing a classroom practice that accords due weight to both cognition and metacognition requires a teacher to make precisely the type of value-based decisions that Lewis-Shaw suggests. The other way in which a values analysis informs our picture of teaching practice lies in the suggestion that teacher pedagogical decisions are more likely to be accepted by students where such decisions are based on a similar ranking of values to the one held by the student/s. Lewis-Shaw provides the example that if the particular teacher she studied were to make a task-oriented over people-oriented decision, the particular students in question would be more likely to be accepting of the teacher's decisions and actions. As can be seen from the framing of the example, such a situation is highly sensitive to the correspondence between teacher and student value systems, which suggests that a particular instructional decision may be accepted by one student group and rejected by another. The conclusion to be drawn from this is not that teacher values should be highly malleable and adapted to the student group in question but, rather, that the negotiation of values should be acknowledged as a significant aspect of classroom practice, particularly where these are in conflict. It is more than likely that anyone reading the various analyses recounted in this book, who also happened to be considering teaching as a career, would decide very quickly that the occupation of teacher is far too complex and difficult to be contemplated. Anyone already a member of the teaching profession might conclude that our (collective) expectations of the profession are unreasonable and unrealistic.

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It is not our intention to reconstruct good teacher practice in a fashion that renders it an impossible aspiration. In suggesting that teachers model problem solving practice, metacognitive activity, interest, engagement, and values, we are documenting activities that are already inevitable components of science and mathematics classrooms. These are not new obligations that teachers must accept, they are aspects of teacher practice that must be recognised and optimised. By documenting the role played by each of these aspects of teacher practice, it is our hope that teachers will gain increasing respect for the complexity of their professional activity and be given increasing support to carry out that activity as effectively as possible. Similarly, the analyses reported here suggest that there are aspects of learner practice to which teachers might usefully attend, and various chapters advocate consequential actions that might follow teacher observation of particular student actions.

5. THE LEARNER/S We find, in contemporary education, an enthusiasm for frequent student-student interaction as a key attribute of effective classrooms. There are many justifications proffered for the effectiveness of student-student interactions as facilitators of learning. Whatever the rationale, in a variety of forms, from peer-tutoring to communities of learners, it is being advocated that the learning of one student can be advanced through interaction with other students. The contributors to this book also subscribe on the whole to this view. The separate chapters display, however, useful shades of difference in their advocacy or interest in student-student interaction. Lerman postulates “the fundamental asymmetry of the teacher-student(s) relationship and of peer interaction” in contrasting socio-cultural theories of learning with “more individualistic approaches”. Other authors asserting the importance of student-student interactions for the promotion of learning apparently see no need to insist on asymmetry as a characteristic. Lewis-Shaw, for example, cites Andresen et al. in suggesting that experienced-based learning (as found in Lessons 5, 7 and 8, in particular) often assumes relatively equal relationships between facilitator and learner (Foley, 1995). Certainly, even the imbalance invoked by Lerman need not be a static imbalance between individuals, but could be considered as a fluid relativity of position highly dependent on context and shifting in character from moment to moment during a lesson. Lerman, among others, links student-student interaction to individual learning by paraphrasing Vygotsky (1962, p. 104) and asserting that “students achieve things together today that they will achieve separately tomorrow”. This elegant statement nicely affirms the primacy of social interaction in the learning process. However, if this is to be interpreted as a model of learning, then what is being postulated is the emergence in interaction of a co-constructed practice synonymous with learning but not yet internalised separately by the collaborating students. The status of that (implicitly novel) co-constructed practice preparatory to internalisation remains undetermined. Holton and Thomas make a related observation regarding student capacity to “perform at a higher level” when working cooperatively than

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individually. In making this assertion, Holton and Thomas suggest that one reason for this heightened capability is the reciprocal scaffolding of group members, offering individuals a support for their cognitive activity that would not be available were they acting individually. Central to such reciprocal scaffolding is the ‘operative intersubjectivity’ invoked by Clarke in Chapter 3. This chapter identified intersubjectivity as both a product of social interaction and as a mediating agent “essential to the negotiative process, whereby uncertainty is resolved and new knowings are constructed.” Holton and Thomas propose a three-phase structure to student problem solving (also noted in Chapter 3) consisting of an initial solution, the review and “understanding” of that solution, and, finally, the full articulation of the solution. It is their suggestion, as research mathematicians as well as mathematics educators, that this sequence may be a characteristic of both classroom problem solving and mathematical research. They further postulate (i) that the cyclic process of review and refinement may be “quite important to the learning and understanding of mathematics” and (ii) that such a process is more likely to occur when students work in small collaborative groups. This suggests that teachers should watch for and encourage the occurrence of all three phases in their students’ problem solving activity. Ainley focused her attention on “interest as a psychological state” and sought to examine “the personal meaning which characterises interest” and in so doing “to examine the character of the processes which link person and object”. Writing from a perspective informed by the traditions of individual psychology, Ainley’s learner is autonomous and self-directing. The character of student interest is illustrated very clearly in one student’s persistent pursuit over two lessons, several days apart, of the question as to whether there are fibres in paper. The learner, as portrayed by Ainley, is an amalgam of attributes, some transient, some enduring: notably, curiosity, interest and persistence, but in another context, boredom. Ainley’s analysis addresses the affective component of interest, in contrast to the cognitive emphasis of early theories that identified interest predominantly with the resolution of cognitive conflict. As she elaborates her portrayal of the interested learner, Ainley anticipates the analytic perspective of Lewis-Shaw by linking individual values, preferences and goals to the process by which interest is aroused. Baird, by contrast, links motivation (or the lack of it) to metacognitive awareness of ignorance and the willingness, interest or motivation required to act on the recognition of that ignorance. Helme and Clarke (Chapter 7) distinguish cognitive engagement from other, less specific, forms of interest, attention, or participation, and, in particular, distinguish active mental involvement from the motivational and emotional aspects of engagement. Their definition of cognitive engagement as, “the deliberate taskspecific thinking that a student undertakes while participating in a classroom activity” identifies cognitive engagement as perhaps the most critical mediating psychological state by which the classroom situation, as experienced by participant learners, is translated into new learnings. The emphasis placed on the term “deliberate” resonates with the importance attached by Baird (Chapter 11) to volitional control. At issue, of course, is whether cognitive engagement is most

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usefully seen as a desirable psychological state or as a set of behaviours or learner practices that all learners should be assisted to adopt or develop. Like Ainley and Lewis-Shaw, Helme and Clarke postulate a learner whose participation in classroom practice is a product of pre-existing personal inclinations and attributes that are variously realised in interaction with others and within the constraints and affordances offered by the particular classroom in which they find themselves. If cognitive engagement is accepted as an essential precursor to learning, then its promotion should be a priority of our instruction and teachers should be assisted in identifying and promoting cognitive engagement in their students and in their classrooms. Similarly, the role of gesture identified by Reeve and Reynolds in learner conceptual advance suggests that, in line with the recommendations of other analyses, the opportunity for task-related student-student interaction should be maximised in order that students might have every opportunity to supplement the verbalisation of their new understanding with gestural amplifiers. It was a significant finding of Reeve and Reynolds’ analysis that the students themselves recognised the important role played by gesture. In situating the learner within the classroom, each analysis did so in a manner reflective of the author’s analytical focus. Lewis-Shaw, for example, characterised the classroom as the site of “social situations in which several values compete and require a weighing of one against another”. Optimal participation in the classroom, from this perspective, requires acknowledgement and accommodation within classroom practice of the values held by all participants and the utilisation of those values for the promotion of student interest and cognitive engagement in the lesson’s focus. Learners must be assisted to make personally useful decisions about the relative and immediate significance of competing values in classroom situations. Also, an awareness of existing student value systems may enable teachers to exploit these in supporting the development of more sophisticated values. Lewis-Shaw gives the specific example of student subscription to “support/peer” and “social affirmation” values from which the more conceptually complex “collaboration” value might be constructed with appropriate scaffolding by the teacher. Of all the analyses reported in this book, Baird’s is most explicit in advocating a model of “effective” or “desirable” learning. In contrasting Perceptions-Driven Learning (PDL) with Reflection-Driven Learning (RDL), Baird asserts the importance of learners habituating a “process of purposeful enquiry, involving active reflection and associated action.” Baird’s RDL learner routinely interrogates the nature, characteristics and purposes of a learning task and relates these to the personal attributes required for successful completion. It is fair to say that such learners were not evident in the data collected for this study. Scattered through the data, however, we do find instances of aspects of the practices which Baird values (see Table 11.1 “Proposed Four Types of Active Learning”) and, where evident, these practices were universally associated with improved learning. Thus, Baird’s advocacy of the RDL learner receives the tacit endorsement of his colleagues and co-authors on the research team.

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6. UNCERTAINTY AS A CONNECTING LINK The integration of the complementary accounts reported in the chapters of this book into a rich portrayal of classroom learning can occur by a variety of means. As noted in Chapter 2, one approach, employed by the research team, was to identify particular constructs employed in several accounts and to contrast the function of the particular construct within each account. One such construct was 'uncertainty', which occurs in several complementary accounts and has been consistently associated with learning. As noted in Chapter 3, Rowland stated that "uncertainty is a productive state, and a necessary precondition for learning" (Rowland, 1995, P. 328). Several members of the research team employed 'uncertainty' in interpreting the integrated data sets. Ainley, for example, employed the student's response to uncertainty as an indicator of student interest. In discussing student interest in the context of the science lessons on ''fibres", Ainley stated, "The certainty which went with the exclamation had been challenged, generating uncertainty. The experience of uncertainty prompted an information-seeking question to an 'authoritative' source and the uncertainty was resolved with Joanne now knowing that paper did in fact have fibres in it" (Chapter 6, pages 149-150). Baird's analysis centred on cognitive ignorance and metacognitive awareness. Baird utilised the classroom data to demonstrate "the powerful influence of acknowledged cognitive ignorance (and the confusion [or uncertainty] with which it was associated) on decisions made and opportunities to effect metacognitive control over learning" (Chapter 11, page 322). Clarke, in Chapter 3, located uncertainty in relation to other key constructs within one model of classroom learning. Reynolds and Reeve, in exploring the functional contribution of gesture to classroom discourse, have suggested that gesture can serve to compensate for inadequacy or uncertainty with regard to appropriate language. Rodrigues, in Chapter 9, contrasted the productive "pondering" arising from one student's uncertainty prompted by the uncertainty of a classmate with the less problematic and arguably less productive communications of the teacher. Pursuing a related point, Holton and Thomas argued that the most productive teacher questioning occurred in a situation where the teacher, herself, was uncertain of her knowledge ofthe mathematical content. In combination, we have the following multi-faceted documentation of the diverse role of uncertainty in mathematics and science classrooms. — The experience of uncertainty prompted an information-seeking question (Ainley, -interest). — Uncertainty acts to trigger cognitive engagement, subject to student interest and values (Helme & Clarke — cognitive engagement). — Acknowledged cognitive ignorance (and the confusion [or uncertainty] with which it was associated) exerted a powerful influence on decisions made and opportunities to effect metacognitive control over learning (Baird – metacognition). — One pathway to knowing is via the resolution of uncertainty; the process of resolution is frequently negotiative; negotiation is mediated by

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language; language presumes intersubjectivity; and, the matter of intersubjectivity is meaning (Clarke - negotiation). — Gesture can serve to compensate for inadequacy or uncertainty with regard to appropriate language (Reynolds & Reeve - the role of gesture). — Contrast the productive ''pondering'' arising from one student's uncertainty prompted by the uncertainty of a classmate with the less problematic and arguably less productive communications of the teacher (Rodrigues – communicative and miscommunicative acts) — The most productive teacher questioning occurred when the teacher, herself, was uncertain of her knowledge of the mathematical content (Holton and Thomas – mathematical interactions). It is proposed that the resolution of uncertainty in relation to academic content constitutes one form of learning occurring in classroom settings. While the resolution of uncertainty is only one of many perspectives that may inform our analyses of classroom learning, it not only provides insight into the negotiative process in operation in the classroom, but suggests at least two areas for further investigation. First, the nature of the cognitive objects which are the focus of student uncertainty, and, second, the forms of knowing that can be associated with particular uncertainty-resolution combinations. In Chapter 3 it is suggested that teacher awareness of the significance of uncertainty and resolution as aspects of learning must be accompanied by the development of strategies to promote modes of resolution that emphasise sense-making. A further area for investigation is the tracking of the resolution of uncertainty concerning more complex and difficult academic content over longer periods than those addressed in this book.

7. THE LESSONS The point has already been made that no-one’s teaching could survive this level of scrutiny without criticism. Similarly, it is probably less useful to evaluate the lessons themselves than to identify those elements that appeared to act as constraints or affordances of learning. For example, in analysing the four mathematics lessons, Holton and Thomas reported that their analysis of student-student interactions “shows a high degree of on-task cognitive and even reflective talk”. This is a universally valued characteristic and one that we might seek to promote in lessons in any content domain. However, both the analysis by Holton and Thomas and that by Rodrigues concluded that the occurrence of such task-related peer interaction was a necessary but not a sufficient condition for learning. Lerman’s analysis could be interpreted as drawing the same conclusion. This is not to underestimate the importance of meeting the prerequisite conditions for learning. Several of Mrs Brown’s lessons seem to have acted to stimulate useful student-student interaction and we would do well to consider how this was done. Lessons 1, 6, 7 and 8, for example, included specific invitations to the students to pool their collective understandings and experience of the topic under discussion. In other lessons, 2 and 4 for instance, student-student interaction was not a structural component of the lesson but was more a matter of students

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spontaneously collaborating in their efforts to understand new content or complete class tasks, Holton and Thomas endorse the use of the “million sheets of paper” problem (Lesson 2) as being optimally effective in providing the opportunity for productive student-student interaction. It is noteworthy that the teacher gave these interactions her implicit sanction by not discouraging such student talk. Holton and Thomas address the issue of how a teacher might act to exploit these student interactions and scaffold the learning occurring in such interactions by supporting the process they call “reciprocal scaffolding.” Central to the analysis of Holton and Thomas (and Lerman) is a belief that students can “perform at a higher level” when working in cooperation than individually. Holton and Thomas attribute this, in part, to the scaffolding function served by one student for another in the course of attempting a new problem. Holton and Thomas, however, accord a potential reciprocity to this scaffolding that distinguishes their conception of the functioning of student groups from that of Lerman who, as noted above, postulates a fundamental asymmetry. In terms of informing teacher practice, Holton and Thomas identify the challenge for teachers as being to “ensure that the task set is appropriate to the students’ zones of proximal development”. This advocacy is particularly interesting because it presumes the existence of the students’ zones of proximal development (ZPD) prior to the initiation of task completion, at least to the extent that the teacher can anticipate its likely form. This distinction between the ZPD as either pre-existing or able to be anticipated (Holton and Thomas) and the ZPD as entirely emergent in social interaction (Lerman, for example) can be examined on pragmatic grounds. If the zone of proximal development of the student cannot be anticipated with regard to a particular content domain, then teachers are unable to make optimally effective task selection decisions. There is a substantial body of contemporary literature that assumes such task selection decisions can be made in such a way as to optimise constructs such as “challenge” and facilitate valued phenomena such as “flow” and “undivided interest” (for example, Csikszentmihalyi, M., 1992, and Csikszentmihalyi & Rathunde, 1992). In this regard, Steffe, von Glasersfeld, Richard and Cobb (1993) distinguish between first- and second-order models, where first-order models refer to the knowledge an individual may construct, while secondorder models refer to the knowledge one may construct of another’s knowledge. Clearly, it is the second-order models that a teacher might construct that would inform any task selection. Steffe et al. go so far as to assert that the first order models that constitute children’s mathematics are essentially inaccessible to the teacher. Whether or not their conception of the ZPD as a student potentiality matches Vygotsky ’s original conception, Holton and Thomas align themselves with those who see teacher task selection as being undertaken: (i) on the basis of a (second-order) model of student knowledge and learning potential; (ii) with real hope of pitching task demand at a level that optimises (rather than maximises) challenge; and (iii) with the intention of maximising productive student-student on-task interaction.

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Lerman makes the point that Vygotsky proposed the zone of proximal development shortly before his death and that the elaboration of this concept has been variously and productively pursued since. The essential criterion by which we might judge the various authors’ use of the zone of proximal development to advance their analyses should not be one of “truth”, or even of consistency with Vygotsky’s usage, but should address its immediate viability for the realisation of the authors’ various purposes and for the constructive informing of the actions of the various participants in our classrooms. In terms of informing our development of effective lessons, these arguments suggest that we should pursue research into task characteristics and their role in promoting useful classroom interactions (such as Henningsen & Stein, 1997, or Williams & Clarke, 1997) and into organisational structures that afford the sort of classroom interactions encouraged by such tasks (see, for example, Cohen, 1994; Slavin, 1990; Webb, 1991). In Chapter 6, Ainley examined the teacher’s reaction to the videotape record of a group of students obviously stimulated by the experience of using a microscope. The evident interest that the activity engendered in the students prompted the teacher to reflect “I don’t think there was a lot of evidence of time and preparation and creativity to capture the kids’ imaginations.” This discussion connects interest as a desirable psychological state of the learner with interest as a consequence of “interesting tasks”. In Ainley’s view, the teacher’s subsequent enthusiasm for an emphasis on forensic science in future versions of that teaching unit is an appropriate recognition of the need to create particular conditions antecedent to interest; in particular, mystery/uncertainty. Where Holton and Thomas and Lerman direct our attention to task selection on the basis of the optimisation of challenge and the promotion of student-student interaction, Ainley exhorts us to select tasks that exploit uncertainty to arouse student interest. The culminating link in Ainley’s associative chain connecting curiosity, uncertainty, interest, and persistence, is the link to learning. Ainley makes the telling observation, having convinced us of Joanne’s interest, that the objects of interest did not relate to the teacher’s objectives for that lesson. This observation suggests that the generation of desirable learning outcomes requires that the ‘objects of attention’ invoke the concepts, principles or procedures which constitute the lesson’s objectives. In the same manner that task-related student-student interaction has been identified as a necessary but not sufficient condition for learning, the arousal of interest per se should not be construed as the appropriate instructional goal. Rather, the arousal of interest, the connection between person and object as Ainley frames it, must connect the learner with the objects (including and especially the cognitive objects) that constitute the goals of the lesson. This is, of course, just as applicable to the teaching of mathematics as to the teaching of science. Lewis-Shaw puts a related point slightly differently when she distinguishes “means values” from “goal values” and suggests that the domination of the former over the latter can contribute to a lack of direction in the lesson. Helme and Clarke distinguish particular configurations of social interaction (classroom situations) and list indicators of learner cognitive engagement corresponding to each configuration. At one level, teachers are potentially

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empowered by being sensitised to the occurrence of these indicators in their classrooms. Our goal of improving teaching and learning will only be achieved, however, if teachers are able to go beyond recognition to active promotion of cognitive engagement in as many students in as many classroom situations as possible. This can be achieved if we conceive of the listed indicators as specifying outward forms of cognitive engagement afforded by each social interaction configuration. Videotapes of classroom practices offer one means by which teachers can be sensitised to issues such as these. The use of classroom video material for teacher professional development is the subject of significant current debate (Clarke & Hollingsworth, 2000). Prerequisite to the effective promotion of cognitive engagement is a recognition of the significant differences in the social dynamics of the different configurations. From the data cited by Helme and Clarke, the teacher has a major role, a wide range of options, and important pedagogical decisions to make with regard to each social configuration. In choosing to employ a particular social configuration (small group work, for example), a teacher must then act to ensure that the affordances of the situation with respect to cognitive engagement are optimally realised. Some detail is provided as to how this might be done. Chapter 7, however, reads like a documentation of constraints rather than affordances, and the teacher’s effectiveness will reside as much in the minimisation of these constraints as in the maximisation of affordance. Perhaps the most telling and compelling empirical result of Chapter 7 is the observation that, of a total of 100 interviews analysed, on only one occasion did a student make spontaneous positive reference to their cognitive engagement. This suggests that the valuing of cognitive engagement must become an explicit, structured, and public component of our lessons; as much an aspect of learner practice as of teacher practice. Helme and Clarke’s analysis of the mathematics lessons also drew attention to the role served by the instructional use of reference to student everyday experience as an agent for the heightening of cognitive engagement. This theme is explored at greater length by Rodrigues. In Chapter 9, Rodrigues focuses our attention on the contemporary enthusiasm for the introduction into science instruction of “everyday” or Socio-cultural Contexts, and the importance of the subsequent construction by the student of Internalised Contexts if conceptual advances are to occur. The significance of the everyday in mediating “between the conceptual system [of scientific concepts] and the world of objects” was identified by Vygotsky (1987, p. 180). One viable theoretical description of the process of internalisation by which Rodrigues’ Appropriate Contexts promote the construction of Internalised Contexts can probably be found in Vygotsky’s conception of internalisation as “internal reconstruction” (see Vygotsky, 1978, p. 56). The enthusiasm for “teaching in context” that Rodrigues has identified in science education is also evident in recent mathematics curricular initiatives (AEC, 1994; de Lange, 1992; NCTM, 1989). Among others, Clarke and Helme (1998) have drawn attention to the problematic aspects of mathematics instruction that utilises ‘everyday contexts’ as though each learner construed and engaged with such contexts in an identical (or even similar) fashion. Rodrigues’ analysis elaborates our emergent picture of the science (or mathematics) classroom and contributes to our conception of effective lessons by

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identifying structural elements of context in instructional settings and the criteria for their optimisation. Commencing with the premise that “Context encompasses both the operation by which we make sense of a situation and the situation itself”, Rodrigues identified the key components: Communication, Resources and Processes, and used her analysis of Lesson 7 to illustrate the function of each component in the development of an Internalised Context and to discuss how this development might be facilitated. Of all the analyses reported in this book, Rodrigues analysis makes the most frequent reference to “the lesson” as a conceptual unit of instruction. As we accumulate information on teaching in different cultures, we may discover that the unit of instruction is a culturally-specific artifact. Certainly evidence exists to suggest that in Japan teachers’ professional activity is focussed significantly on the refinement of single lessons. In other cultures it may be that creative professional activity is invested in the development and refinement of particular activities from which lessons are constituted (to some extent this was true of mathematics curriculum development in Australia and the USA during the 1980s; see, for example, Lovitt and Clarke, 1989, and NCTM, 1989). Elsewhere, such professional effort may be directed at the development of thematic sequences of several lessons (an approach employed over an extended period by the Freudenthal Institute in the Netherlands, by the Australian Science Education Project in the 1970s and, most recently, by the “Maths in Context” team at the University of Madison — Wisconsin). The fine-grained nature of the analyses reported in this book are not conducive to recommendations at the level of optimal lesson structure, but the findings summarised in the preceding discussion should inform instructional design and teacher practice at the level of activity, lesson, or thematic unit. In particular, Rodrigues’ chapter offers specific guidance regarding the creation and use of Appropriate Contexts in the teaching of science. Rodrigues singles out lack of shared purpose as a critical reason why “Fibres” was unsuccessful in promoting the development of an Internalised Context for any of the participating students. A consequence of this observation is the recommendation of “a change in communication patterns between students and teachers as both seek to determine each other’s understandings.” This recommendation repeats the emphasis placed in almost every chapter on the need to make classroom negotiative processes more visible as a component of classroom practice. The advocacy of more open investigations aligns Rodrigues with the empirically grounded recommendations of Ainley, Helme and Clarke, and Holton and Thomas. Similarly, Rodrigues’ support for “meaningful sharing and engagement” reiterates the importance attached by other researchers to student-student interactions and student engagement as distinct from mere participation. In combination, such recommendations should contribute to the development of more effective lessons in both mathematics and science.

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8. CLASSROOM INTERACTIONS It is a strength of this research project that the data collected allowed the voices of both teacher and student to be heard. The availability of multiple perspectives on the various classroom interactions analysed in this book reduces the likelihood of simplistic mis/representations of these interactions. The importance of the teacher-student relationship and the dialogical character of learning are repeatedly stressed in the various analyses. This is particularly the case in a chapter such as that by Lewis-Shaw dealing with the interaction of individuals whose actions were motivated by different systems of values. The apparent intangibility of a construct such as “values” is overcome by the meticulous inferential analysis carried out by Lewis-Shaw and by the practicality of the conclusions she draws from that analysis. Empirical examples of teacher-student interaction provided by Lewis-Shaw illustrate “the reflection of values as a means to maintain rapport and the exercise of control based upon the utilisation of implicit norms”. This reading of teacher-student interactions has a plausible subtlety to it that invokes the notion of a didactic contract (Brousseau, 1986) through its reference to implicit norms, while suggesting “values” to be a constitutive element of such didactic contracts. Rodrigues’ discussion of the mutual misconstrual by teacher and students of the intentions of the other offers a related perspective on negotiation in science (and mathematics) classrooms. Teacher-student interactions are the medium for multiple interpretive and negotiative processes and Rodrigues provides clear illustration of occasions when these processes acted to lead one or other of the participants to misconstrue the actions of another and consequently to restrict the capacity of the classroom to function as an effective learning environment. Rodrigues also contrasted unfavourably the uni-directional character of teacher-student interactions with the bi-directional character of student-student interactions, arguing that a richer diversity of cognitive, metacognitive and social processes was evident in the bidirectional interactions. In discussing teacher-student interactions, Lerman identified particular elements that framed the situation, including: the teacher’s perceptions/assumptions of student ability, the likelihood that such a perception had been unintentionally conveyed to the students by the teacher’s and other pupils’ behaviour; some aspect of the students’ relationship on which their actions in concert are predicated; the function of the calculator in student reasoning; and the differential effect of instruction on each student. Holton and Thomas asserted that “mathematics teaching revolves around students tackling problems” and thereby simultaneously invoke and interrelate teacher, students, and tasks. They go on to relate teacher-student interactions to the process of classroom problem solving: during the posing of a problem; during the solving of a problem; and in the ensuing period when a problem has been solved. Both analyses address the significance of teacher-student interactions, but the second takes as its focus the mathematical activity in which the students are engaged, while the first is at least as much concerned with the students’ construction of themselves as mathematicians. These approaches are not in conflict and in combination they offer us a more complete documentation of the complexity

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of teacher-student interactions. The discussion of scaffolding in Chapter 5 arises naturally from Holton’s and Thomas’ focus on mathematical activity, while Lerman’s explanation of why a zone of proximal development was not created between Mark and Darren draws directly on his consideration of the social interactions and underlying perceptions of classroom participants. Each analysis identifies a different aspect of teacher-student interactions as worthy of our attention. Holton and Thomas explored in some detail the teacher’s role in scaffolding student learning. Their analysis calls into question the nature of the competence differential on which scaffolding is usually predicated; suggesting that the teacher’s competence is more complex than simply mathematical (or scientific) knowledge and also resides in a more generic competence as a problem solver. When considering student-student interactions, Holton and Thomas directly confronted assumptions about the relative competence of “scaffolder and scaffoldee” and put forward the notion of reciprocal scaffolding as a social enactment of metacognition. This proposal hinges on the recognition that a question (from whatever source) can provide effective scaffolding for the advancement of student learning. It seems a plausible consequence of this line of reasoning to accept that such productive questions could be posed by a colleague or classmate who, in other respects, might not be characterised as more competent. In Episode 1, discussed in Chapter 3, a feature of Karen’s role in the Lauren/Karen dyad was to pose questions of Lauren and of herself. Some evidence can be found to suggest that Lauren was the more mathematically capable student. Nonetheless, the successful culmination of the dyad’s problem solving efforts must be attributed, in part, to Karen’s persistent framing of task-related questions. The effectiveness of such reciprocal scaffolding as a component of dyadic problem solving will derive significantly from the appropriateness of such questions and the extent to which one learner attends to the questions (and other contributions) of the other. If students can be sensitised to the potential value of each other’s contributions, it then becomes desirable that students should learn to ask such questions. This adds a further complexity to teacher-student interactions, since the teacher’s actions as scaffolder of student learning now include the modelling and promotion (scaffolding) of the students’ reciprocal scaffolding. We would argue that student-student interactions appeared to offer more scope for highlevel cognitive engagement than teacher-student interactions, both in whole-class instruction and in interactions with small groups (Helme and Clarke, Chapter 7, p. 191).

What are the implications of this statement? What role does it assign to the teacher? The statement by Helme and Clarke should be seen as entirely consistent with the analysis reported by Rodrigues and with the advocacy by Holton and Thomas of teacher-student interactions that model and promote particular forms of student-student interactions, We are familiar with the image of the teacher modelling the practices of the scientist or the mathematician. Less commonly proposed is teacher modelling of good learning practices. Whatever the professional history or current involvement of the teacher in their discipline, the practices into which the students are being initiated are primarily the practices of the classroom; a setting

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whose pretext is not mathematical or scientific research but the learning of mathematics or science. One of the most common themes pervading the chapters in this book is the desirability of the facilitation of metacognitive activity by students. Acceptance of this as a major instructional goal requires a reconception of the nature of teacher-student interactions. If the goal of instruction is cognitive, the teacher models, promotes and scaffolds cognitive activity: techniques for solving particular problems, perspectives on valued concepts, or important principles or facts. In focussing upon cognitive goals, the teacher can inadvertently preempt the students’ metacognitive activity, by appropriating activities such as procedural decision-making and questioning. If the goal of instruction is metacognitive, then such processes are not appropriated by the teacher, but rather are explicitly valued, occasionally modelled, and always encouraged as important aspects of learner practice. In Chapter 11, Baird offers a carefully articulated theory of teacher and learner practice in this regard. Reeve and Reynolds suggest another dimension to the teacher’s practice. Their analysis sensitises us to the significance of student gesture and calls teachers to attend to the occurrence of frequent gesturing by students when interacting in small groups. The occurrence of “gestures, slow speech, and pauses’’ are likely to signify a state of cognitive uncertainty. It is suggested by the analyses in several chapters (particularly Chapters 3, 5, 6, 7 and 10) that the occurrence of this state indicates a state of readiness on the part of the learner for conceptual advance and an important opportunity for discerning scaffolding by the teacher. Reeve and Reynolds discuss the communicative role of gesture and cite the description by Roschelle and Clancy (1992) of student talk as ‘gibberish’ without insight into the gestures that accompanied the students’ conversation. A related point was made by Clarke (Chapter 3) in discussing evidence for the existence of intersubjectivity between individuals based on the apparent incoherence, to an outside observer, of the transcript of student classroom dialogue during a problem solving exercise. The extended nature of the dialogue and the success of the problem solving activity provide compelling evidence that the participants in the dialogue shared a common knowledge of the referents of key terms and, specifically, of pronouns. It is likely that this shared knowledge was supported by the students’ use of gesture, particularly during the initial establishment of the new referents in the dialogue. Reeve and Reynolds draw to our attention the very important role that gesture may play as a mediating device during the students’ development of concepts for which they lack the appropriate terms. Vygotsky (1978) has discussed at some length the evolution of gesture from a purposeful physical action to a symbolic communicative action. The link with language, particularly with the development of facility in the correct use of mathematical terms, is evident in both chapters 3 and 10. It may well be that in attempting to understand the process of internalisation we should study much more closely the transformation of student gesture from direct pictorial portrayal of a phenomenon under discussion to more symbolic gestural representations of less physical (more conceptual) aspects of the phenomenon to the representation of these concepts in language.

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9. RECONCEIVING CLASSROOM PRACTICE Previous research, and much of our theorizing, has tended to dichotomise teaching and learning as discrete activities sharing a common context. It will be argued below that this dichotomization is a particularly insidious consequence of the constraints that the language (and the English language, in particular) imposes on our theorizing. It is a major premise of this project that such dichotomisation misrepresents both teaching and learning and the classroom settings in which these most frequently occur. It is appropriate to make some specific points in support of this position. There is no intention to challenge the separate integrity of “teacher” and “learner” as labels for individuals engaged in particular practices or discourse modes. Rather, data collection in this project was predicated on the possibility that classrooms are more effectively understood as sites for bodies of mutuallysustaining practice that in combination characterize a process we might call (in English) “teaching/learning”. Any discussion of learning as a social activity calls inevitably upon the writings of Vygotsky. Vygotsky is invoked repeatedly by various authors in the preceding chapters. In this discussion, however, I want to examine the manner in which any understanding of Vygotsky’s theorising must be grounded in an understanding of the language available to him, and the limitations (constraints and affordances) that language places on our theorising about classrooms. In particular, it appears that readers of Vygotsky in English have been denied a richness of meaning present in the original Russian text. Consider the key differences in two published translations of the same passage: From this point of view, instruction cannot be identified as development, but properly organized instruction will result in the child‘s intellectual development, will bring into being an entire series of such developmental processes, which were not at all possible without instruction (Vygotsky, 1982, p. 121, as quoted in Hedegaard, 1990, p. 350).

The translation used by Hedegaard is referenced: “Vygotsky, L.S. (1982) Om barnets psykiske udvikling [On the child’s psychic development]. Copenhagen: Nyt Nordisk.” In this translation, the juxtaposition of “instruction” and “development” is an uneasy conjunction, although the suggestion that “organized instruction will result in the child’s intellectual development” seems transparently appropriate. The passage in this form can be read as a celebration of the role of instruction in facilitating intellectual development in directions not otherwise possible. However, in another translation of the same passage, but this time in the widelycited 1978 translation published as Mind and Society, we find: From this point of view, learning is not development; however, properly organized learning results in mental development and sets in motion a variety of developmental processes that would be impossible apart from learning (Vygotsky, 1978, p. 90).

In this translation, the distinction that is being drawn between learning and development takes on special significance, and it appears to the reader that Vygotsky’s point is to distinguish intellectual development through maturation from intellectual development in response to the environment (which would include instruction). However, in this translation, the phrase “properly organized learning”

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need not connote instruction as this is conventionally understood, since “properly organized learning” could conceivably consist of structured occasions for the child to interact with their environment without the mediation of an adult or more competent other but just by immersion in particular environments — as might be the case on a trip to the zoo or a visit to an art gallery. The pivotal assertion that must be understood is whether Vygotsky was asserting the impossibility of certain forms of intellectual development “without instruction” (which presumes an actively interactive more competent other) or “apart from learning” (which on one level seems a tautology, but which could also be interpreted as equivalent to the assertion that properly organized interaction with the environment is essential for certain forms of development to occur). This distinction is non-trivial, since it calls into question the significance of the mediation of another more able individual (the teacher/instructor). Given what we know of the significance Vygotsky attached to the role of the teacher, it would appear that the most appropriate reading of the major premise is “a variety of developmental processes would be impossible without instruction.” This accords with the significance attached, in the passage quoted below, to the child’s interaction with “people in his environment” rather than just with all aspects of that environment, with or without the mediation of others. The ‘conflicting’ translations arise because of a duality of meaning in the original term employed by Vygotsky. This duality has been noted previously, but its significance seems to have been given scant consideration in the interpretation and application of Vygotsky’s work. The theoretical framework of Vygotsky entails specific understanding of learning, development, and the goal(s) of development. In Vygotsky’s usage, the term obuchenie, frequently translated as learning, more accurately indicates the interaction of teacher and student (Wertsch & Sohmer, 1995, p. 332).

As we have seen, the same term (“obuchenie”) is also translated as “instruction” and clearly shares with corresponding terms in other languages the capacity to invoke both teaching and learning, as these are named in English. Once this duality of meaning is recognized our reading of Vygotsky and our theorizing about the teaching/learning process are greatly enriched. For example, in one of the most famous passages from the translated Vygotsky, the word “learning” can be replaced by the word “teaching” and the resultant text is still meaningful — but, perhaps, with a different meaning. We propose that an essential feature of learning [teaching] is that it creates the zone of proximal development; that is, learning [teaching] awakens a variety of developmental processes that are able to interact only when the child is interacting with people in his environment and in collaboration with his peers (Vygotsky, 1978, p. 90).

If our framing of “instruction” in language presumes a ‘‘learner’’ and if ‘‘learning’’ is inextricably entwined with an “instructive” setting, then our interpretations of the activities of the classroom are more likely to identify communal practices and the progressive participation in a common discourse as essential features, than to fragment the classroom into teaching and learning activities undertaken by individuals. Is it “learning” or “teaching” that creates the

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zone of proximal development? Vygotsky’s intention, surely, was to signify that it was in social interaction, where learners and teacher converse, and where the more able other will frequently be a classmate, that the zone of proximal development was co-constructed for each participant (rather than by each participant). On the one hand, it is comforting to feel that we have some understanding of what Vygotsky intended, despite inconsistencies in translation. It is more to my immediate purpose, however, to reflect on the significance of the translators’ difficulties in interpreting a term that simultaneously invokes both teaching and learning, and the implications that this has for the cultural-specificity of teaching and learning. Speakers of Russian are not alone in their use of a term that combines both teaching and learning. In Dutch there is one term that means both learning and teaching: "leren". To distinguish between the practices of teaching and learning, the Dutch say "leren van" to signify “learning” and "leren aan" to signify “teaching”. A teacher is a "leraar" and a student is a "leerling" (I am indebted to Martin Van Reeuwijk for explaining all this to me). In French, the term “didactique”, and particularly Brousseau’s use of that term (Brousseau, 1996), invokes a mutuality of responsibility and participation not always found in American or Australian interpretations of the classroom. This project has addressed that mutuality very explicitly. It is this explicit acknowledgement of the mutuality of teaching and learning that holds the promise of a significant contribution to current efforts to better connect the disjoint fields of research on learning and research on teachers and teaching. This theoretical position is most clearly in evidence in the chapter by Baird in which consistent reference is made to “the teaching/learning situation.” Lerman (Chapter 4) asserts that “so much classroom activity is actually between students” and this recognition pervades all the analyses. This is, perhaps, not surprising, given that the lessons to be analysed were selected for the negotiative character of their constitutive interactions. Nonetheless, those of us who have spent our recent years viewing videotapes of mathematics and science lessons find a persistent inclination on the part of students to interact with each other, with or without the teacher’s sanction. To attribute this inclination to interact to a general adolescent gregariousness is to trivialise the creativity and resilience of a community (learners) whose imaginative persistence in subverting the teacher’s constraints on their interaction may ultimately be shown to be acting in the teacher’s greater interest by facilitating the learning to which they and their teacher are held accountable. It is the height of platonic sophistry to argue that individuals unfailingly act for their own good, but Holton and Thomas draw our attention to the quality of student-student mathematical interactions when these are minimally fettered by the teacher. The drive to understand seems frequently to be realised in the data as a drive to discuss, to articulate, to elaborate, to question, to explain and to interrogate. Whatever mechanism might mediate between this activity and its consequences as learning product, we appear to have support for the desirability of the activity as such. Some acknowledgement must be made at this point of the possible cultural specificity of the data and of the researchers’ interpretations. Other school systems in other cultures may constrain student-student interactions much more severely than

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was the case in this study, and students’ inclinations to interact may be highly culturally determined. The significance attached in the findings of this study to student-student interactions is strongly grounded in our data, however conclusions related to the function and desirability of such interactions in mathematics and science classrooms must been seen as culturally-specific. The reader will no doubt have noticed the echoing from chapter to chapter of particular practices (both of the teacher and the learners) identified with effective learning, and the highlighting of particular aspects of the learning process. The cumulative images of learning, learner, teaching, teacher, and of the practices of the classroom emerging from the preceding chapters will have been progressively constructed by the reader as elements of each image were integrated into the reader’s framework and the centrality or significance of particular elements became consolidated or amplified by their recurrence in several of the analyses. Each of these emergent images can guide our research and our reform efforts. Advocacy of a particular model of the effective learner will necessarily require the identification of teacher practices and classroom characteristics conducive to the development and sustenance of that particular model learner. The identification of a body of valued classroom practices will position the learner and the teacher within that body of practice. The commitment to an integrated perspective on mathematics and science classrooms obliges us to look for the reflections of each emergent image in the other images through which the classroom is constituted. Every facet of the teacher’s practice will have its echo in some learner practice that it either constrains or affords. Every aspect of the social situation (physical setting, organisational arrangement, cultural values, ethnic alignment, social affiliation) will similarly constrain or afford the various practices of all participants. It is this sense of mutual dependence and reciprocal constraint and affordance that prompts the use of ecological metaphor in recent discussions of classroom learning and teaching (Clarke, 1996; Greeno, Collins & Resnick, 1996). Such metaphorical borrowings are a happy product of the developments in every discipline. In the middle of the last century, the biologist von Uexhull put forward the proposition that the spider’s web is the spider’s model of the fly. This whimsical imagery conceals a powerful reasoning technique similar to reverse engineering. From the structure of a spider’s web: the spacing and strength of the strands, the location and size of the web, and from other characteristics of the spider’s web, we can deduce much about the fly. Classrooms are a little like the spider’s web, From the way in which a teacher structures the classroom (and the practices for which it is the setting), we can infer much about that teacher’s (and that society’s) model of the student. The types of resources provided, the type and duration of the various activities, the forms of interaction that are encouraged and discouraged, all offer insight into the teacher’s conception of what Steffe called “the epistemic student”, the student as constructor of knowledge. Within the confines of accepted practice and available resources, teachers attempt to construct classrooms to afford and constrain particular learner activities. What Brousseau (1986) has brought to our attention is the reciprocity of the construction of classroom practice. Learners (that is, students) engage in practices that afford and constrain teacher actions, and the actions of their classmates. Social interaction by an individual within the classroom

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presumes that the individual has a model of the other classroom participants and can, to some extent, anticipate their capabilities, their needs, their expectations and their responses. Through video-stimulated reconstructive interviews some of these “models of the other” can be elicited. In rare cases, as for Joanne, the individual can even gain insight into the origins of others’ models of oneself. What is clear is the extent to which classroom practice is a jointly constituted body of negotiative social interactions that is best investigated and understood in terms of the mutuality and reciprocity of its constituent activities and of its co-construction as Teaching/Learning.

10.

ADVANCING THEORIES OF LEARNING

A theory of learning must accommodate and relate both individual subjectivities and participation in social practices. Attempts to avoid invoking some form of internalisation process by focussing analysis on participation in social practice remains problematic to the extent that it is necessary to answer the question, “Whose social practice?” That is, one participates in social practice as a member of a social group, but this membership is a matter of interpretive affiliation by the participating individual. The body of practice in which one participates as a high achieving 14 year old Asian female in an eighth grade Australian mathematics classroom may be qualitatively different from the body of practice subscribed to by a less mathematically able 14 year old male of European descent participating in the same classroom. By invoking participation in practice as an alternative to the individual construction of meaning, socio-cultural theorists trade the need for an individually constructed cognitive knowing for an individually constructed socio-cultural affiliation by which the activities and, more broadly, the body of practice in which the individual progressively engages can only be identified to the extent that the individual’s affiliations are known. Very simply, individuals in social settings invoke affiliation every time they employ the first person plural “we” or through their use of “they” dissociate themselves from affiliation with some other social or cultural group. It is an oversimplification to discuss classroom practice as though it were constituted the same for each participating individual. The nature of an individual’s participation is itself an interpretive act. Rogoff (1990) has also drawn to our attention the importance of identifying the meaning that participants ascribe to the activities in which they participate. The interpretive aspect of participation is particularly evident in Joanne’s interactions with Mrs Brown in Lessons 7 and 8. The interpretations of meaning, value, and intention, each of the other, are more than the social mechanisms by which Joanne progressively refines her practice as a learner of science, they are, in fact, the substantive base of that practice. While both Mrs Brown’s and Joanne’s actions in that setting are constrained and afforded by socio-cultural norms, they are also shaped from one minute to the next by the reciprocal interpretation of the meaning of the other’s actions. The progressive refinement of the practices of each is the result of an iterative process of negotiation that is fundamentally social in substance but which requires individual acts of interpretation at each step. This interpretive

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process is most evident when the participating individuals comment on videotapes of their social interaction. The opportunities for re-interpretation of the actions of another and for consideration of the social signals offered by one’s own actions invoke a level of meta-interpretation that is not normally available in the hurly-burly of classroom interaction. The significance of this meta-interpretation is clearly evident in Chapter 11. In such situations, the video-stimulated post-lesson interviews document not just the individual’s interpretations of their own actions and the actions of others, but their interpretations of how their actions might be interpreted by others. By becoming, momentarily, the audience for their own actions, both teacher and learners have the opportunity to “see ourselves as others see us” and reflect on the implications. It was a major goal of this research project to elaborate a theory of learning relevant to classroom settings. Each of the preceding chapters has contributed to that elaboration. A recurrent theme has been the reflexive relationship between social interactions and individual cognitive activity. Each chapter has contributed a novel perspective to our understanding of learning in classrooms: The reflexive connectedness of uncertainty, negotiation and intersubjectivity proposed by Clarke; Lerman’s discussion of what can be learned from the non-creation of a zone of proximal development; Holton and Thomas’ conception of reciprocal scaffolding; Helme and Clarke’s identification of cognitive engagement as a necessary precursor to learning; Ainley’s insistence that the arousal of interest must connect the learner with the objects (including cognitive objects) that constitute the goals of the lesson; Lewis-Shaw’s identification of values as both a critical determinant of student engagement and as the subject of classroom negotiation; Rodrigues’ characterisation of Appropriate Contexts as facilitating an essential form of internalisation; Reeve and Reynolds’ powerful elaboration of Vygotsky’s original conception of the role of gesture in the process of internalisation; and Baird’s location of metacognition and volitional control within Learning/Teaching. The theory of learning that emerges is one that starts from the social situation of the individual in interaction with others, but which accords a significant role to the individual’s interpretive activity. Particular significance is attached to social interaction, and learning proceeds by the iterative refinement of intersubjective understandings that include social, scientific and mathematical meanings, as well as values and modes of collaborative practice. These understandings are enacted as progressive increase in valued practice, including the appropriate utilisation of technical language. Every account of learning provided in these chapters invokes a negotiative process that presumes interaction with others. These interactions are predicated on an interpretive affiliation that situates the learner with respect to the values and goals of others in the learning environment (the classroom) and an interpretive characterisation of the other, by which the capabilities, motivations, values and actions of other participants in the classroom are inferred and this characterisation is then iteratively refined through on-going social interaction. Context is also a matter of interpretation and internalisation. Essential to the emergent picture of learning in mathematics and science classrooms is the coconstructed nature of the practices of these classrooms, and the role of negotiation

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not as a subordinate activity through which classroom practice is constructed but as an essential activity from which classroom practice is constituted. 11. TEACHING LEARNING AND LEARNING TEACHING: OPTIMISING PRACTICE Classroom Practice is a form of communal collaborative activity constituted as it is constructed through the participation of both teachers and learners and only understood (and optimised) through research that accords value and voice to all participants. Teaching and Learning are not simply distinct but interdependent activities that share a common setting, rather they should be conceived as aspects of a common body of situated practice and studied as such. It is ironic that recognition of this fundamental unity is enshrined in several languages other than English and that the dichotomisation of Teaching and Learning may be, in part, an artifact of our use of English as the lingua franca of the international Education community. This project and the analyses reported in this book provide evidence of the mutuality of teaching and learning and support their interpretation as components of a single body of communally constituted practice. Negotiation, the construct that provided the original motivation for this study, is now seen to be a constitutive element of classroom practice. Classroom Practice as a form of communal collaborative and negotiative activity is constituted as it is constructed through the participation of both teachers and learners and only understood (and optimised) through research that accords value and voice to all participants. The implicit confusion in English of process and product (already noted in Chapter 3 as enshrined in language through the existence of “gerunds” as linguistic elements) offers an important perspective on both negotiation and intersubjectivity. As has been argued, intersubjectivity is not to be seen exclusively as the product of classroom negotiative activity but also as a prerequisite condition for the viability of that activity. Similarly, classroom practice (Teaching/Learning) is not to be seen solely as a consequence of negotiation, rather negotiative activity is the essence of classroom practice. The matter of that negotiation can be “values”, for example, or it can be “mathematical meaning”, but critical to any understanding of classrooms is recognition of the organic nature of practice and the iterative and reciprocal refinement of the constitutive elements of that practice through its enactment. Every chapter contains explicit messages for the optimisation of classroom practice. The optimisation of the outcomes of a process is facilitated by the understanding of that process. It is to be hoped that this book has contributed to the understanding of the process we have called Teaching/Learning. ACKNOWLEDGEMENTS The contribution of Harsh Suri to the introductory discussion of research synthesis is gratefully acknowledged.

TEACHING/LEARNING

319

REFERENCES Andresen, L., Boud, D. and Cohen, R. (1995) Experience-based Learning, in Understanding Adult Education and Training, G. Foley (Ed.), Allen & Unwin: Australia. Australian Education Council (AEC): 1994, Mathematics—A Curriculum Profile for Australian Schools, Curriculum Corporation, Carlton, Victoria. Baird, J.R. (1998). Promoting willingness and ability to learn: A focus on ignorance. Reflect - The Journal of Reflection in Learning and Teaching, 4, (1), 21-26. Brousseau, G.: 1986, ‘Fondements et methodes de la didactique des mathematiques’, Recherches en didactique des mathematiques 7 (2), 33-115. Clarke, D.J. (1987). The Interactive Monitoring of Children’s Learning of Mathematics. For the learning of mathemutics 7(1), 2 - 6. Clarke, D.J. (1992). The role of assessment in determining mathematics performance. Chapter 7 in G. Leder (Ed) Assessment and learning of mathematics. Hawthorn: Australian Council for Educational Research, 145 - 168. Clarke, D.J. (1996a). Learning research. In P.C. Clarkson (Ed.). Technology in mathematics education. Proceedings of the 19th annual conference of the Mathematics Education Research Group of Australasia (MERGA), June 30-July 3, 1996, at the University of Melbourne. Melbourne: MERGA, pp. 1 - 9. Clarke, D.J. (1996b) Refraction and Reflection: Modelling the Classroom Negotiation of Meaning. RefLecT 2( l), 46 - 51. Clarke, D.J. (1998). Studying the classroom negotiation of meaning: Complementary accounts methodology. Chapter 7 in A. Teppo (Ed.) Qualitative research methods in mathematics education. Monograph Number 9 of the Journal for Research in Mathematics Education. Reston, VA: NCTM, pp. 98-111. Clarke, D.J. (1999) What can we learn from international comparative studies of mathematics classrooms? Invited Brief Review in Vinculum, November issue, p.24. Clarke, D.J. & Helme, S. (1998). Context as construction. In O. Bjorkqvist (Ed.) Mathematics Teaching from a Constructivist Point of View. Vasa, Finland: Faculty of Education, Abo Akademi University. Clarke, D.J., & Hollingsworth, H. (2000). Seeing is understanding: Examining the merits of video and narrative cases. Journal of Staff Development 21(4), 40-43. Clarke, D.J., Waywood, A,, & Stephens, M. (1994). Probing the structure of mathematical writing. Educational Studies in Mathematics.25 (3), 235 – 250. Cobb, P. (1994). Where is the Mind? Constructivist and Sociocultural Perspectives on Mathematical Development. Educational Researcher, 23(7), 13-20. Cohen, E. (1994). Restructuring the classroom: conditions for productive small groups. Review of Educational Research, 64(1), 1-35. Confrey, J. (1995). A theory of intellectual development: Part III. For the Learning of Mathematics, 15(2), 36-45. Csikszentmihalyi, M. (1992). The flow experience and its significance for human psychology. In M. Csikszentmihalyi & I. S. Csikszentmihalyi (Eds.), Optimal experience: Psychological studies of flow in consciousness (pp. 15-35). Cambridge: Press Syndicate of the University of Cambridge. Csikszentmihalyi, M. & Rathunde, K. (1992). The measurement of flow in everyday life: Toward a theory of emergent motivation. Nebraska Symposium on Motivation, 57-98. de Lange, J.: 1992, ‘Critical Factors for Real Changes in Mathematics Learning’, in G. Leder (ed.), Assessment and Learning of Mathernatics, Australian Council for Educational Research, Hawthorn, Victoria, 1992, 305-329. Erlwanger, S.H. (1975) Case Studies of Children’s Conceptions of Mathematics - Part 1. Journal of Children’s Mathematical Behaviour 1 (3), 157 283. Fischer, K. W., & Grannott, N. (1995). Beyond one-dimensional change: Parallel, concurrent, socially distributed processes in learning and development. Human Development. 38, 301-321. Foley, G. (Ed.) (1995) Understanding Adult Education and Training, Australia: Allen & Unwin. Greeno, J.G., Collins, A.M., & Resnick, L. (1996). Cognition and Learning. Chapter 2 in D.C. Berliner & R.C. Calfree (Eds.) Handbook of Educational Psychology, New York: Simon & Schuster Macmillan. Hedegaard, M. (1990). The zone of proximal development as basis for instruction. Chapter 15 in L.C. Moll (Ed.) Vygotsky and Education. Cambridge: Cambridge University Press.

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Henningsen, M., & Stein, M., K. (1997). Mathematical Tasks and Student Cognition: Classroom-Based Factors that Support and Inhibit High-Level Mathematical Thinking and Reasoning. Journal for Research in Mathematics Education, 28 (5), 524-549. Jensen, L.A. & Allen, M.N. (1996). Meta-synthesis of qualitative findings. Qualitative Health Research. 6(4), 553-560. Kolb, D. (1 984) Experiential Learning: Experience as the Source of Learning and Development, Englewood Cliffs: N.J: Prentice Hall. Lovitt, C. & Clarke, D.M. (1989). Mathematics Curriculum and Teaching Program: Activity Bank. Volumes I and 2. Melbourne: Curriculum Corporation. National Council of Teachers of Mathematics (NCTM): 1989, Curriculum and Evaluation Standards for School Mathematics, NCTM, Reston, Virginia. Noblit, G. W. & Hare, R.D. (1988). Meta-ethnography: Synthesising qualitative studies. Newbury Park: Sage Publications. Rogoff, B. (1990). Apprenticeship in thinking: Cognitive development in social context. Oxford: Oxford University Press. Rowland, T. (1995). Hedges in mathematical talk: Linguistic pointers to uncertainty. Educational Studies in Mathematics, vol. 29, 327-353. Sandelowski, M. (1997). “To be of use”: enhancing the utility of qualitative research. Nursing Outlook, 45(3), 125-132. Slavin, R. E. (1990). Ability grouping, cooperative learning and the gifted. Journal for the Education of the Gifted, 14(l), 3-8. Smith, L. (1993) Necessary Knowledge: Piagetian Perspectives on Constructivism Hove: Lawrence Erlbaum Associates. Steffe, L.P., von Glasersfeld, E., Richards, J., & Cobb, P. (1993). Children’s counting types: Philosophy, theory and application. New York: Praeger. Vygotsky, L.S. (1978). Mind in Society. Edited by M. Cole, V. John-Steiner, S. Scribner, & E. Souberman. Cambridge, Mass: Harvard University Press. Vygotsky, L. S. (1986) Thought and Language (Revised Edition, A. Kozulin (Ed.)), Cambridge, Mass.: MIT Press [reference cited by Lerman was to the 1962 edition, and by Lewis-Shaw to the 1997 edition]. Vygotsky, L.S. (1987). The collected works of L.S Vygotsky (Vol. I), Cambridge, MA: MIT Press, cited in C.P. Panofsky, V. John-Steiner, & P.J. Blackwell (1990) Scientific concepts and discourse. In L.C. Moll (Ed.) Vygotsky and Education, Cambridge, UK: Cambridge University Press. Webb, N. M. (1991). Task-Related Verbal Interaction and Mathematics Learning in Small Groups. Journal for Research in Mathematics Education, 22 (5), 366-389. Wertsch, J. V., & Sohmer, R. (1995). Vygotsky on learning and development. Human Development, 38, 332-337. Williams, G., & Clarke, D. J. (1997). The complexity of mathematics tasks. In N. Scott & H. Hollingsworth (Eds.), Mathematics: creating the future (pp. 451- 457). Victoria: Australian Association of Mathematics Teachers.

APPENDIX

A:

SUMMARY

OF

LESSONS

ANALYSED

1. SUMMARY OF DATA COLLECTED 1994 - 1996

Year 1994

Science 7 (12)

Maths 6 (13)

Total 13 (25)

1995

11 (24)

10 (17)

21 (41)

1996

7 (15)

13 (28)

20 (43)

Total

25 (51)

29 (58)

54 (109)

Notes: 1. Number of lessons shown in plain text, number of student interviews in parentheses 2. All interviews fully transcribed

1994 Breakdown: Year 7 Year 8 Total: 1995 Breakdown: Year 8 Year 9 Total:

Maths 3 (6) 3 (71 6 (13)

Science 2 (3) 5 (9 7(12)

Total 5 (9) 8 (16) 13 (25)

Maths Science 5 (8) 7(16) 5 (9) 4 (8) 10 (17) 11 (24)

Total 12(24) 9 (17) 21 (41)

1996 Breakdown: Year 8 Maths: Year 9 Science: Total:

13 (28) 7 (15) 20 (43)

Note on teachers: Seven (7) teachers participated, all but “Mrs Brown” were male.

32 1

i

322

APPENDIX A

2. LESSONS ANALYSED IN THIS BOOK*

1994 Data Summary Lesson Number 1

2

3

Date 4/10/94

Year level 8

19/10/94

18/11/94

8

Subject

Teacher

Lesson theme

Science

Mrs Brown

Group chart: what you know and

Maths

Mrs Brown

Maths

8

Mrs Brown I

know about the heart Problem solving using standard number Compass bearing

Students interviewed Anna Grace

Karen Lauren

Karen

1995 Data Summary Lesson Number 4

5

Date

Subject

13/6/95

Year level 8

19/6/95

8

Science

Maths

Teacher Mrs

Brown

Mr Black

Lesson theme Rates and ratio; velocitytime graphs Heart dissection

Students interviewed Leonie

Kevin Stephen

1996 Data Summary Lesson Number 6

7

8

Date

Subject

Teacher

Lesson theme

17/6/96

Year level 8

Maths

Mrs Brown

16/7/96

9

Science

Mrs Brown

9

Science

Mrs Brown

Rates and ratio: Velocity - time graphs Beginning of Fibres unit; feeling fibres blindfolded Looking at fibres under a microscope

19/7/96

*As per Figure 2.1 - repeated here for simplicity of reference

Students interviewed Mark Darren Joanne Narelle

Joanne Frances

SUMMARY OF LESSONS ANALYSED

323

3. LESSON OUTLINES

Lesson I.

Subject: Science Topic:

Year level: 8

The heart and circulatory system

Date:4 October, 1994

Students interviewed: Anna and Grace

Teacher: Mrs Brown Lesson structure: Teacher-led discussion followed by group activity and reporting back.

Lesson outline: — Teacher presented historical material about the treatment of disease, including bloodletting using cupping and leeches three hundred years ago. — Teacher explained position of heart in body and its size and mass, equating these with half a can of baked beans. — Students in groups drew a diagram of a heart on a piece of butcher’s paper and on one side listed everything they knew about the blood system and the heart. — A spokesperson from each group reported findings back to the class. — Students in groups wrote down on the same diagram anything about the blood system they would like to know more about. Student work collected: — Diagram of heart with lists of what the students in each group know and what they would like to know.

APPENDIX A

324 Lesson 2. Subject: Mathematics

Year level: 8

Topic: Problem solving using standard form Date: 19 October, 1994 Students interviewed: Karen and Lauren Teacher: Mrs Brown Lesson structure: Teacher-led discussion followed by group activity Lesson outline: Teacher summarised standard form and gave students practice in writing large numbers in standard form (such as the number of blood cells lost in a lifetime) — Teacher explained the difference between a number in index form and a number in standard form (that is, the difference btween 3 to the power 8 and 3 times 10 to the power eight). Teacher used a visualisation exercise to illustrate the difference between a — number raised to a power and a number multiplied by another number (eg 2 to the power 4 as compared with 2 times 4). — Class practice in working out powers of 2, from 2 to the power 3 to 2 to the power 11. — Group problem solving activities, with most students working in pairs. —

The problems: 1. How many sheets of graph paper would you need to show one million squares? 2. Find the height of a stack of one million sheets of paper [to assist with solving this problem a pack of 500 sheets of paper was provided]. 3. Measure each of the following notes: $5 and $10. What would be the length of one million dollars of each when laid end to end? 4. Find the weight of one million grains of rice [Students were provided with the information that 100 grains of rice wieigh 1.523 grams]. Student work collected: Written record of work in class.

—

SUMMARY OF LESSONS ANALYSED

325

Lesson 3. Subject: Mathematics Topic:

Year level: 8

Using a compass

Date:

18 November, 1994

Students interviewed: Karen and Anna Teacher: Mrs Brown Lesson structure:

Teacher-led discussion followed by activity in pairs

Lesson outline: Teacher covered a number of points related to reading maps and using a compass, including the difference between magnetic north and true north, the term ”magnetic declination” and the effect of metals on compass readings Students calculated the change in magnetic declination for a five year period at — 2' per year. Teacher outlined a method for subtracting one angle from another (to calculate a — magnetic bearing) when degrees and minutes are involved, first using money as an example. — Teacher demonstrated the use of clockwise movement in reading a bearing. — Teacher led discussion of definition of acute and obtuse angles — Class exercise in which students estimated compass bearing of a given point J (diagram on board), followed by discussion of methods of obtaining compass bearing of J — Teacher explained difference between magnetic bearings and compass bearings and used an example of converting a bearing of 135° to a magnetic bearing by subtracting 12° 45’. She then introduced the mnemonic GMS (Grandma Sucks) Teacher explained a method for calculating the bearing of a point E on the map — in their booklet. Students then worked in pairs with compasses to calculate the bearing of point E in their booklet (Point E was a point on a map that had a bearing of about N 85°E) Teacher helped K and A find the bearing of E, and then they tried it themselves. — Teacher asked students to convert this compass bearing to a magnetic bearing by — subtracting 13°

—

Student work collected: None

326

APPENDIX A

Lesson 4. Subject:Mathematics

Year level: 8

Topic: Rates and ratio: Velocity-time graphs Date: 13 June, 1995 Students interviewed: Leonie Teacher: Mrs Brown Lesson structure: Teacher-led discussion and work in pairs from the course booklet (P. 19).

.

Lessonoutline: Teacher-led discussion about graph interpretation tasks on page 19 of the mathematics booklet produced by the school. In particular, graph B, of a car slowing down and reversing; the speed of a ball being thrown into the air at different points in its journey; a stone dropped from a great height (for this one she drew a velocity-time graph; a bouncing ball. — Teacher discussed a student’s bus trip to school, drew a schematic diagram of the journey, then constructed a velocity-time graph on the board. She then led a class discussion interpreting this graph, pointing out different aspects ofthe motion depicted, including constant acceleration from zero velocity, constant velocity, and deceleration to zero velocity. — Students worked in pairs on tasks from page 19 of the booklet.

—

Student work collected: Page 19 of course booklet — Leonie’s class notes for 13/6/’95 —

SUMMARY OF LESSONS ANALYSED

Lesson 5. Subject: Science

Year level: 8

Topic: Heart dissection

Date:

19 June, 1995

Students interviewed: Kevin and Stephen Teacher: Mr Black Lesson structure: Teacher-led discussion followed by group dissection activity Lesson outline: — — —

Teacher introduced heart dissection experiment Students in groups dissected heart, cutting it into thin slices, or discs. Teacher discussed dissection, and gave instructions on writing up the prac.

Student work collected: —

Diagram of heart with written answers to questions (for Stephen only)

327

328

APPENDIX A

Lesson 6. Subject: Mathematics

Year level: 8

Topic: Rates and ratio: velocity-time graphs

Date: 17 June, 1996

Students interviewed: Mark and Darren Teacher: Mrs Brown Lesson structure: Teacher-led discussion, short student reports based on an individual class activity and individual seatwork from the course booklet (pages 24 and 25) Lesson outline: Teacher presented material about the Earth’s rotation around the sun, gravity, air resistance, terminal velocity and the difference between speed and velocity. Teacher drew a velocity-time graph on the board and led a class discussion — interpreting this graph, pointing out different aspects of the motion depicted, including constant acceleration from zero velocity, constant velocity, and deceleration to zero velocity. — Teacher-led discussion and construction of a possible velocity-time graph for a bouncing ball. Students wrote a short story to describe a given generic velocity-time graph — (attached). Contexts chosen included skiing, parachuting, driving, drag racing, bicycling and jogging. — Teacher-led instruction in algebra for small group so they could attempt the more challenging questions in the “Ratio Pep Test” (page 25, question 1, parts f to j). — Students worked on page 25 of booklet (“Ratio Pep Test”). — Teacher-led discussion of bouncing ball problem from graph matching exercise on page 24 of booklet. —

Student work collected: Generic velocity-time graph (stimulus material for small group activity) — Maths course booklet, pages 24 and 25. — Students’ class notes for 17/6/’96 — Diagram that Darren drew during his interview —

(The content of this lesson is similar to that covered in Lesson 4)

SUMMARY

OF

LESSONS ANALYSED

329

Lesson 7. Subject: Science Topic: Fibres

Year level: 9 Date: 16 July, 1996

Students interviewed: Joanne and Narelle Teacher:

Mrs Brown

Lesson structure: Teacher-led discussion, group activities followed by reporting back, and practical activity in pairs. Lesson outline: Teacher outlined unit objectives on page 2 of course booklet, requesting students to underline key words. — Group activity in which students were asked to “Write down all the things you know about fibres”. — Groups reported back their findings (Narelle is spokesperson for study group). — Group activity in which students were asked to “Write down all the fabrics and fibres you have heard of’. — Groups report back their findings (Joanne is spokesperson for study group). — Teacher reads out background material from course booklet (pages 3 and 4), “research questions” (pages 5 and 6) and gives instructions for completing assignment on recycling of plastics (page 21). — Students (in pairs) undertook Activity 1 (page 7): Guessing the identity of fabrics by feeling them with closed eyes. Teacher outlined homework requirements: — Complete questions based on Activity 1; Answer questions 1 to 8 on pages 5 and 6; Start major assignment [Note: At next class (19/7/96) Joanne had not done the two sets of questions]. —

-

Student work collected: List of all the things study group knew about fibres: They’re in clothes, they’re in fabric, there’s a weft and a warp, they’re microscopic, they’re particles, they have more than one meaning (there are fibres in food), they’re twisted to make wool, they’re in paper. List of all the fibres suggested by study group was: cotton, nylon, wool, flax, satin, silk, lycra, canvas, felt, Velcro, wool blend, polyester, paper, chiffon, poplin, tinsel, denim, dacron, down, tulle, feathers, ribbing, ribbon, elastic, angora, mohair, leopard skin, leather, hair, cow hair, mink, rat hair, ox hair, bristles. — Narelle’s classwork (Questions based on Activity 1 and questions 1 to 8)

—

—

330

APPENDIX A

Lesson 8. Subject: Science

Year level: 9

Topic: Fibres

Date: 16 July, 1996

Students interviewed: Joanne and Frances Teacher:

Mrs Brown

Lesson structure: Teacher-led discussion, group activities followed by reporting back, and practical activity in groups. Lesson outline: Brief teacher-led discussion revising material covered in previous lesson (Lesson 7 above) — Group activity in which students were asked the following question: “Besides clothing, what else to people use fibres for?’ Groups reported their findings (Joanne is spokesperson for study group). — Teacher explained the concept of “porosity”. — Teacher outlined Activity 2 (page 8) and explained the use of the microscope. — Students undertook Activity 2: Structure of Fibres (page 8). This activity — involved: observing and describing the weave of certain fabric in terms of its porosity — observing fibres under a microscope. — While students were doing the microscope activity, teacher checked homework — questions from previous class (Lesson 7, above) and assisted the study group to focus their microscope. — Write up of Activity 2. —

Student work collected: Response from group to brainstorming question: hankies, towels, blankets, sheets, doonas, tablecloths, furniture, upholstery, curtains, pencil cases, bags, tents, carpet, paper, rope, fibreglass waterslides. Write up of Activity 2. —

—

APPENDIX

B: SAMPLE INTEGRATED TEXT

Year 8 Mathematics Class (videotaped 19/10/94 – classroom dialogue indicated by times plain font, researcher’s field notes indicated by times bold font) with student comments (interviewed 19/10/94 – indicated by times italicised font), and teacher’s comments (account recorded 22/9/97 – indicated by helvetica plain font) The transcription of classroom dialogue from videotape: All audible utterances by focus students are transcribed. Utterances by other students audible to focus students are transcribed. Teacher utterances audible to focus students are transcribed. Additional annotations are also included indicating emphasis, inflexion, or gestures that appeared significant for an understanding of the particular exchange. All transcribed classroom dialogue is in plain font. The inclusion of researcher field notes: Researcher field notes are included in bold font exactly as they were recorded in the course of videotaping the lesson. The field notes accompany the time-tagging markers generated by the CVideo software, linking the text to the videotape record. The transcription of student comments: Inserted in the transcript at the point in the videotape transcription to which they refer. Italicised font. The transcription of teacher comments: Inserted in the transcript at the point at which T paused the videotape to make comments. They are typed in helvetica. General transcription conventions 1. Emphasis denoted by underlining: I was very concerned 2. Inaudible words or utterances less than one second denoted thus: [inaudible] 3. Inaudible utterances longer than one second denoted thus: [next 5 sec inaudible] 4. Where a word is unclear, suggested word is followed by (?) 5. A slight pause in speech denoted thus: It was ... a good idea 6. Self-interruptions denoted thus: She was-I thought she was 7. Interviewer’s comments or questions denoted thus: [I: Year 8 last year?] 8. Researcher’s comments to clarify meaning denoted thus: [referring to previous lesson] Additional Explanatory Notes Study students [Lauren and Karen] are sitting together. The following is an edited selection of the integrated text. 331

332

APPENDIX B

Teacher recorded account at commencement of viewing the videotape (note: the teacher had access to the transcript of the classroom dialogue, but not to the student interview transcripts). I love this lesson. I still do it years later. It'll be very interesting for me to see how effectively it comes across to the kids. I've often have them work for the whole lesson on just one or two problems and I feel that they get a lot out of it, and uh sometimes the trivia in mathematics and the finer points can quite often be useful. I think they like the idea of working out the height of a thousand sheets of paper and the estimation and the methodology I see as very important and far too often in maths we tend to go ahead and just, you know, the tendency is to give them textbooks and nothing to really work out and talk and negotiate and problem solve. So I particularly chose this because it's a wonderful problem solving exercise and um it's fun, and I love doing it as well. 00:29:36 to 00:30:07 L does something metacognitive S 19: It says how many sheets of graph paper would you need to show one million one millimetre squares. L: To show one million, you don't divide it by a hundred, because there's more than a hundred one millimetre squares. I mean you're going to find the area of this. K: What? L: You've got to find the area of this, there's more than one hundred one millimetres. K: That's right. I was doing length by—oh screw that. L: One hundred one millimetre squares. Take lengthK: Um, there's how many down here? L: And along that side there isK: Ten, twenty, thirty, forty, fifty. How many are there down there? L: There's a hundred one millimetres there. I really wonder how Karen would have coped with the task if Lauren wasn't there. I mean she seems to be fairly competent at her work and she's like a little mini teacher in the room. [I: Lauren you mean?] Yes Lauren. And um, yeah, so I just wondered how much Karen really would have understood by herself. At least, you know, she tried a couple of ways and then said, "oh no that's not going to work"' so that was part of the aim, what I was onto. But if the class was busy and Lauren wasn't there to help Karen I wonder how far she would have gone ahead.

00:30:12 to 00:30:51 that's not right, it'd be 250, length times width L: No, there wouldn't be.

SAMPLE K: L: K: L: K: L: K: L: K:

INTEGRATED TEXT

333

There wouldn’t be, that’s not right. There’d be two-fifty. Yeah. Yeah, there’d be two-fifty, And we just totally screwed it allLength of graph. OK, so it would be length times width [inaudible] And uh, two-fifty millimetres. WidthWhat’s width?

It’s interesting that Karen says, “What’s width?” Is she just relying on Lauren to give her the data or does she really know what’s going on? L: K: L: K:

That’sThat’s ten, twenty, thirty, forty, fifty, etcetera Eighteen, one hundred and eighty. Times a hundred and eighty. OK here we go. Two hundred and fifty times one hundred and eighty equals forty-five thousand.

00:30:58 to 00:31:23 45,000 meant K: [L answers question from another student]. OK, that’s 45 thousand. We need a million. What’s a million divided by 45 thousand and times it by that? 00:31:24 to 00:31:33 L says hang on. L: Hang on hang on hang on hang on. Don’t go too fast. OK. Therefore there are forty-five thousand million mm squares. S20: Forty-five thousand million? L: Yeah.. S20: Forty-five thousand. K: Twenty-two point two. L: On one piece. Of graph paper. K: OK, question two. L: Does everyone understand what we did with number one? K: No, but anyway. Yes Karen. I’m not surprised [inaudible] that she doesn’t understand what’s going on, ‘but anyway’. It’s interesting how her main contribution was [3 sec inaudible] so maybe that was her way of saying [2 sec inaudible]. So yes I always find that the difficulty with a lesson like this is to go over it clearly so that people like Karen can understand but I don’t bore the socks off Lauren. If I go through and um say, “those people who got the question right go onto the next thing”, how do I go back and um make certain that they really do understand it, so I always find that’s the hardest thing, how do I help the really weak people, allow the faster

334

APPENDIX B

ones to go on and then you get that grey area where you're not quite sure who understands and who doesn't. So uh, you know one day hopefully 1'11 get a bit closer than I do at other times. K: Five hundred sheets. L: And how many sheets do we need? K: Five hundred sheets of what? Five hundred sheets. L: Their height equals five point eight. S20: We've done that. K: I know. But we've got to do it all together so. L: One point oh times ten to the power of six divided by five hundred. K: Oh yeah sure everyday what are you talking about? What are you talking about? L: I'm finding out how many five hundreds there are in a million. K: How many five hundreds there are in a million. That would make it one thousand. How many thousands are there in a million?

(Interviewer with Karen) I: What's that? K: Um, what were we up to? Did she say borrow? We were doing sheets of paper, um, that's right, Lauren was just saying, I think she was saying something really fast: Five hundred sheets equals five point eight centimetres. I think it was when she did 'cause we needed the height of a million sheets of paper. I think she said, “A million divided by five hundred", something like that, and I didn't know what she meant. I: Uh huh. And? K: Then I figured it out. I: So how did you figure it out? K: I just heard her and I'm trying to do my own work, and she's talking, and it's like I'm -yeah, I agree with you . . . . I didn't understand what she was talking about so I just did it on my own, I think. I: Yeah, so then when you went through it on your own. K: I work better when there's no-one else around anyway, but. 00:36:45 to 00:37:07 K argues with L? K: That would make a thousand, two thousand. What? [to another S]. 1 have a lot to say. L: [checks result on calculator] Two thousand, well done! K: This is called skill. This is what you do. Five hundred into a hundred which is two. Then you do a hundred [correcting herself] which is a thousand. L: Times five point eight. Shush. K: Which is two. Then you do a thousand into a million, which is a thousand, so a thousand times two is two thousand. L: Eleven thousand six hundred.

SAMPLE

INTEGRATED TEXT

335

Yes I was thinking to myself how would I help people like Karen learn and maybe what I need to do is to actually prioritise those tasks into those that may be more concrete for some kids, and leave like the graph paper one, which is quite tricky, leave that one maybe for later on. I might even say, ‘look if you’re having a little bit of trouble this might be the easier place to start, and you might like to talk with some friends more about this particular question, rather than that particular question.’ But once Karen gets it, then suddenly, ‘Oh yes, I can actually handle some of this’. But she sounded very discouraged about the graph paper one. So at least this one here seems to [inaudible]. I guess even if I get-it’s very hard when you’re teaching. You often have to think about where you start from and maybe it’s not where you finish it‘s where you start from, it’s where you move to. That maybe a little bit of success-she may not understand the whole five but even if she understands one of them. Sometimes problem solving helps(?) and the techniques will actually come, be of use for you, a little bit later down the track. And maybe the next time she sees a stack of graph paper she might think, ’now how did I solve that?’ and, ‘oh yes, I know that there is a technique that I can work out how many sheets of graph paper to make a million.’ So if it doesn’t cotton on then, it may actually, hopefully stay in their mind and they may actually use it later. Because one of the beauties of this sort of problem-I’m hoping that may occur for some students. 00:37:08 to 00:37:19 T asks question T: With your working out folks I want you to tell me what you are mult— Matthew-what you are multiplying by, and you simply put a little arrow telling me what and why. K: What are we doing? Is it a million sheets of paper though? L: Yeah. One pointK: We’re doing a million sheets of paper. L: Yeah, you need00:37:20 to 00:37:33 K says I did that wrong K: Yes we do. We do, shut up. L: Therefore. K: Huh? I did that wrong. L: One [inaudible, working] 00:37:36 to 00:38:04 S asks L something 00:38:06 to 00:38:46 K says something to neighbour about not understanding K: Two thousand times five point eight centimetres is eleven, six, zero, zero.[i.e. eleven thousand six hundred] L: Mms?

336

APPENDIX B

K: Centimetres—which would make it eleven point six metres, right? L: Or eleven—yeah. It'd be eleven point six metres wouldn't it, 'cause you take off one to get to centimetres, and another one, yeah.[pause] K: [looking up] That's quite high, isn't it?

That's what I'm after. That's what I'm after. For them to actually visualise it and see, say, 'Hey, that's really quite high.' That makes the whole activity worthwhile in my eyes, from my point of view. L: All right. And you've got to point out what the [inaudible] are right? K: You've got to point out the what is? L: We have to show what we're multiplying by. [S22 says something to Karen, Karen laughs] S20: That's not how you know, you look like you know what you're doing and you just do it. K: Exactly, you go through a total state of concentration, it lasts about two seconds, that's when you get the answer, and then you don't know what you're doing, so it doesn't matter. five hundred sheets equals, height equals five point eight centimetres. I don't even understand what I wrote.[pause as Lauren, Karen write, sound of other students discussing problem]

'Exactly' says Karen. 'You go through a total state of concentration, it lasts about two seconds, that's when you get the answer and then you don't know what you're doing. So it doesn't matter' (laughs) Aha! That's her idea of learning. 00:38:48 to 00:39:25 K asks L to explain, L's explanation* K: But why do we divide a million by five hundred to get that answer? L: Because you know, if you know what the height isK: So what am I doing. Tell me what I'm doing here, tell me what I've done. L: All right. You know that five hundred sheets equals fifty-eight — L and K: Five point eight centimetres. K: There is a point there, it's up there. L: Oh, I can't see it. K: Get some other glasses. L: OK. Now, we need to know — we need to know the height of a million sheets of paper. Therefore you must divide a million by five hundred and times that number by five point eight. K: [writing] Equals two thousand. Sheets of paper. L: OK. K: That would make it two thousand times five point eight centimetres equals eleven thousand six hundred centimetres, equals eleven point six metres of paper.[bell rings]

SAMPLE

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(Interviewer with Lauren) L: She didn't understand why you would times, this is what I think, she didn't understand why you'd times the two thousand by five point eight, because you have to find the height of a million sheets of paper. I: Yeah, L: And we knew that the height offive hundred sheets of paper was five point eight. I think she didn't understand what the two thousand was from. I: Um hm. L: And that was because you divide a million by five hundred sheets, and that gave the two thousand and then you had to times that by five point eight to give you, 'cause there's two thousand lots offive point eight sheets and she didn't understand where the two thousand came from, I think 00:39:27 to 00:41:30 T asks students K and L what they've done. K explains, T is dubious, then says I think you're right, L explains, says we're right. T: Where'd you get a hundred and eighty from? K: Width. Equals fortyT: Why did you multiply them together? Why not add? K: To get the area. I know that much. T: [to Lauren] You've been tutoring her? K: Equals forty-five thousand, therefore you'd need—oh, how'd I get that? T: Forty-five thousand? K: Forty-five thousand. That's what we got. T: Forty-five thousand? Can you press that-can you press that-can you do that again? Two hundred and fifty times a hundred and eighty? Oh, hang on, hang on, I think you're right. Hang on. I think they're wrong. K: Yup, they're wrong, we're right. (L holds up calculator) (Interviewer with Lauren) I: Uh huh. So why were you so sure your answer was right? Or were you sure your answer was right? L: Um, because when she asked us what it was, she thought it was right too. I: I'd like you to tell me this last bit. So say that again for me. L: She came and asked us to do the answer that we found and we had a different answer to the one that another group had given her and when she heard our answer it must have clicked that, um, it sounded more right than the other one did, so she went to tell them that they were wrong. I: It must have clicked with her, so that's why she thought it was right. Why did you think it was right? L: 'Cause if Mrs Brown thinks it's right, it probably is.

337

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APPENDIX B

(Interviewer with Karen) I: There she's going over to your group. [videotape continues] Yeah, so you seem pretty sure, you got the, you had answers for everything she said. K: Yeah, because, um don't say anything to her, but the girls said that she'd pick on me a bit because I'm new, so if I show her that I know what I'm talking about then she'll lay off. I: Oh, well that makes sense. K: Because I don't want to seem like I'm going, "Well, you're right." I: But you did know, didn't you? K: Yeah, but I just thought it, 'cause yeah. I: Yeah. K: So I just wanted to show her that I know what I was talking about, because otherwise she'd keep on at me. I: Yeah. K: Admit that actually sort of think that she made a little bit of a mistake. Yeah. I knew she'd lay off if I sort of had an answer for everything, so that's why I just said, straightaway she'd ask me a question, I'd have an answer, and she'd go, um, think about it for a while and then straightaway say "You're right or you're wrong." Yeah, I just wanted to say everything quickly so she didn't have time to think of another question. [It is noteworthy that the teacher did not comment on this exchange at all.] [I: That was quite an interesting class] Mm, yes. Science is so much more straightforward. But the understanding of maths concepts when you've got so much dependent on previous years and kids have splinter gaps in their knowledge, or wide gaps in their knowledge, it's (?) quite difficult. It's one of the reasons I like teaching maths but it's also my biggest frustration. Is that really, I feel like I'm painting over wood that needs to be replaced at times, because the wood is damaged and all I'm doing is putting a glossy paint over the top.

AUTHOR INDEX A Acredolo, L., 234, 251 Ainley, J., 122, 129 Ainley, M. D., 5, 6, 118, 119, 122, 123, 129, 131-133, 136, 152, 222, 292, 294, 301-3,306, 308, 317 Algina, J., 271,288 Alibali, M.W., 9, 11, 235-7,251, 252 Allen, M.N., 291, 320 Anderson, J., 77, 103 Andresen, L., 158, 160, 189, 190, 193, 195, 300, 319 Armstrong, D.F., 249, 251 Ash, D., 4, 11, 57, 73 Atkinson, P., 53-55, 73 Azmitia, M., 232, 251 B Baird, J., 3, 9, 48, 51, 136, 210, 256, 260,261, 268, 271, 272, 275, 288, 293, 294, 298, 299, 301-3, 311, 314, 317, 319 Baird, W., 108, 120, 130 Bakhtin, M., 15, 31, 36, 51 Barg, J., 83, 102 Barnes, M., 28, 31, 47, 51, 204, 206, 229 Barnett, C., 48,51 Bassok, M., 78, 102 Batten, M., 122, 129 Bauersfeld, H., 22, 31, 35, 42, 51, 52 Bavelas, J.B., 237, 251 Bazzini, L., 73 Belenky, M. F., 199, 205, 229 Belmont, M.J., 131, 153 Bell, B., 200, 202, 206, 209, 230 Bennett,N., 4, 11, 81, 82, 102 Berk, L.E., 79, 102 Berlyne, D.E., 107, 115, 116, 129, 130 Beswick, D.G., 107, 118, 122, 129

Bidell, T. R., 232, 251 Bloom, B.S., 157, 195 Blumenfeld, P.C., 133, 136, 137, 152, 153 Blunden, R., 157, 195 Boekaerts, M., 257, 274, 288 Bomstein, M. H., 232, 251 Bottery, M., 192,195 Boud, D., 158,160, 189,190,195,320 Boulter, C., 230 Bresler, L., 13,22, 31 Brodie, K., 58, 73 Bronfenbrenner, U., 232, 251 Broughton, A.P., 167, 195 Brousseau, G., 309, 314, 315, 319 Brown, A.L., 4, 11, 57, 73, 231, 233, 251, 258, 288 Brown, J. S., 232, 251 Bruner, J. S., 44, 45, 51, 75, 76, 79, 94, 102, 104, 232, 251 Bryant, P., 232, 252 Buchler, J., 60, 73 Butterworth, G., 200, 230 Byrne, M., 200, 229 C Camboume, B., 79, 102 Camaioni, L., 234, 253 Campione, J.C., 4, 11, 57, 73, 231, 251 Candy, P. C., 257, 288 Cassell, J., 238, 252 Cazden, C. B., 59, 73, 79, 80, 102, 206, 229 Chang, C., 236, 252 Chi, M.T.H., 78, 102 Chiu, M-H, 78, 102 Choi, J-S., 200, 229 Chovil, N., 237, 251 Church, R.B., 235, 237, 251-3 Clancey, W.J., 8, 11, 236, 237, 249, 253

339

340

Clarke, D.J., 1, 5,6, 11, 14, 21, 29, 31, 33-36, 42, 48, 51, 122, 130, 136, 144, 152, 153, 292, 298, 301-4, 3068, 310, 311, 316, 317, 319 Clarke, D.M., 153, 308, 319 Clinchy, B.M, 205, 229 Coates, L., 237, 251 Cobb, P., 33, 35, 42, 51, 52, 58, 73, 235, 253, 297, 305, 319, 320 Cohen, E., 306, 319 Cohen, R., 158,160, 189, 190, 195, 319 Cole, M., 4, 11, 57, 60, 74 Collins, A.M., 232, 251, 315, 319 Collins, C., 122, 129 Combs, A.W., 160-162, 195 Confrey, J., 86, 102, 295, 319 Connell, J.P., 131, 132, 135, 153 Corno, L., 133, 136, 137, 141, 146, 152, 256, 257, 288 Coulthard, M., 206, 230 Cronk, C., 234, 253 Crowder, E.M., 237, 252 Csikszentmihalyi, I. S., 132, 136, 152 Csikszentmihalyi, M., 107, 118, 129, 132, 136, 151, 152, 305, 319 D Damon, W., 77, 99, 102, 232, 252 Dole, J.A., 78, 102 Davidson Wasser, J., 13,22, 31 Davydov, V. V., 4, 11, 57,58, 73 Day, H.I., 118, 129, 130 Decarie, T.G., 234,252 Deci, E. L., 118,130 De Groot, E.V., 133, 134, 153 De Lange, J., 307, 319 De Leeuw, N., 78,102 Desforges, C., 81, 102 De Vries, R., 233, 252 Dewey, J., 121, 130, 158, 195 Doise, W., 233, 252

AUTHOR INDEX

Driver, R, 202, 203, 230 Duguid, P., 232, 251 Dunne, E., 4, 11, 82, 102 Dunphy, DC., 191,195 E Edwards, D., 56, 58, 60, 66, 73, 136, 152 Edwards, A. D., 206, 230 Efron, D., 234, 235, 252 Eijkelhof, H., 199, 230 Ericsson, K.A., 16, 31 Erikson, E.H., 161, 195 Erlwanger, S.H., 297, 319 Ernest, P., 76, 102 Evans, J., 56, 58, 73 F Fennema, E., 137,153 Ferrara, R. A. ,23 1,251 Fink, B., 118, 130 Fischer, K. W., 232, 233, 251-3, 294, 319 Flevares, L.M., 9, 11, 236,251 Foley, G., 158, 160, 190, 193, 195,300, 319 Ford, P., 22, 32 Forgasz, H.J., 137, 152 Forman, E. A., 59, 60, 66, 73, 79, 80, 102, 236, 252 Fowler, J.W., 161, 195 Frid, S., 48, 51 Friere, P., 86, 102, 162, 195 Fullarton, S., 132, 136, 152 G Gadamer, H.-G., 203, 230

AUTHOR INDEX Gale, J., 76, 103 Garber, P., 235, 252 Gardner, P. L, 222, 224, 225, 230 Garton, A. F., 232, 252 Geertz, C., 167,195 Gilbert, J., 200, 206, 230 Gillett, G., 56, 73 Glaser, R., 78, 102 Goffman, E., 22, 23, 31 Goldberger, N.R, 205, 229 Goldin-Meadow, S., 9, 11, 231, 235-38, 251, 252 Gooding, A., 83, 84, 103, 135, 137, 152, 236,252 Goodwyn, S., 234, 251 GOOS, M., 236, 252 Gordon, A., 4, 11, 57, 73 Grannott, N., 232, 233, 252, 294, 319 Greenfield, P.M., 79, I03 Greenfield, T., 192, 195 Greeno, J.G., 315, 319 Greenwood, L., 22, 32 Greer, E.A, 78, 102 Griffin, P., 4, 11, 57, 60, 74 Guberman, S. R., 232, 252

Henderson, B., 108, 130 Henningsen, M., 306, 320 Hidi, S., 105, 108, 120, 130 Hitchcock, G., 54, 73 Hodgkinson, C., 192, I96 Hodson, D., 201, 230 Hofstede, G., 161, 196 Hogan, D. M., 232, 253 Hollingsworth, H., 307, 319 Holt, J.. 121, 129 Holton, D., 4, 5, 77, 81, 82, 86, 87, 91, 103, 136, 146, 292, 293, 295, 300, 301, 303-4, 308-10, 314, 317 Holzman, L., 60, 74 Hoyle, R., 137, 153 Hoyles, C., 83, 103 Huberman, A. M., 24, 32 Hughes, D., 54, 73 Huibregste, I., 215, 230 Hunt, D.E., 129, 130 Hunt, J McV., 107, 118, 129, 130 I Inagaki, K., 59, 73

H J Hall, B.P., 7, 11, 156, 160-149, 171, 174,182-4,188,192,194,195 Hall, R., 236, 238, 252 Hammersley, M., 54, 55, 73 Harari, O., 163, 195 Hare, R.D., 291, 320 Harmin M., 194, 196 Harre, R., 56, 73 Haste, H., 76, 102 Hatano, G., 59, 73, Hedegaard, M., 312, 320 Helme, S., 5, 6, 29, 31, 34, 36, 51, 122, 130, 136, 152, 292, 301-4, 306-8, 310, 319

341

Jackson III, D., 257, 288 James, J., 85, 103 Jenkins, E., 199, 230 Jensen, LA., 291, 320 Johnson, S., 199, 230 Johnstone, A., 200, 229 Jolifi, Z., 218, 230 Jones, J., 201, 225, 230 K Kahle, J., 198, 199, 230

342 Kaplan, B., 234, 253 Karmiloff-Smith, A., 23 1, 233, 252 Keller, H., 108, 130 Kelley, L, 78, 103 Kendon, A., 234, 235, 252 Kessel, C., 34, 48, 51 Kirschenbaum, H., 158, 161, 196 Kleine, P., 222, 230 Knowles, M., 185, 196 Kolb, D.,158, 196, 296,320 Kol’tsova, V. A., 59, 73 Korthagen, F., 215, 230 Krapp,A., 105, 108, 118, 129, 130 Krummheuer,G., 22, 31 Kuhl, J., 257, 288

L Larrivee, B., 271, 288 La Vancher, C., 78, 102 Lave, J., 35, 52, 103, 199, 230, 232, 252 Leder, G.C., 137, 152 Ledig, B.D., 163, 195 Lerman, S., 4, 11, 28, 30, 34, 44, 46, 52, 55-60, 71, 72, 73, 74, 136, 293-5, 298, 300, 304-6,309, 310, 314, 317 Levy, E.T., 237, 252 Lewis M., 78, 102 Lewis-Shaw, C., 7, 15, 293, 294, 296302,306,309,318 Light, P., 200, 230 Lijnse, P., 199, 230 Lokan, J., 22, 32 Lovitt, C., 308, 319

AUTHOR INDEX Maslow, A., 161, 163, 196 Matos, J. F., 58, 74 McCullough, K.E., 238, 252 McNeill, D., 235, 237, 251, 252 McPhail, J., 59, 60, 73 Meece, J.L., 137, 153 Meira, L., 58-60, 73, 74 Mercer, N., 56, 58, 60, 66, 73, 77, 103, 136, 152 Mergendoller, J.R., 133, 136, 152 Miles, B. B., 24, 32 Miller, R., 202, 230 Miserandino, M., 258, 288 Mitchell, I.J., 48, 51, 261, 272, 288 Morgan, C., 108, 118, 130 Morissette, P., 234, 252 Moschkovich, J.N., 236, 252 Moss, E., 232, 253 Mugny, G., 233,252 N Nakagawa, K., 4, 11, 57, 73 Newman, D., 4, 11, 57, 60, 74 Newman, F., 60, 74 Nilan, P., 189, 196 Nisbett,R.E., 16, 32 Noblit, G.W., 291, 320 Nolen, S.B., 133, 153 Nunes, T., 232, 252 Nusbaum, H., 235, 252 Nuthall, G., 271, 289 O

M Maker, C., 77, 103 Mandinach,E., 133, 136, 137, 141, 146, 152 Masataka, N., 234,252

Olive, J., 235, 253 P Palincsar,A. S., 232, 252

AUTHOR INDEX

Palmonari, A., 233, 252 Penna, C., 256, 288 Perlmutter, M., 232, 251 Perret-Clermont, A. N., 55, 59, 74 Perry, M., 235, 252 Peters, R.S., 156-8, 196 Peterson, P.L., 134, 137, 153 Piaget, J., 55, 59, 61,74, 232, 233, 235, 253 Pintrich, P.R., 132-4, 153 Porter, A., 201, 225, 230 Puro, P., 133, 136, 152

Roschelle, J., 8, 11, 16, 32, 234, 237, 238, 249, 253 Ross, G., 75, 79, 103 Roth, W. M., 204, 230 Rousseau, J.J., 121, 130 Rowland, T., 34, 46, 52, 303, 320 Roychoudhury, A., 204, 230 Russell, D.M., 78, 103 Rutherford, M., 4, 11, 57, 73, 230 S

Sandelowski, M., 291, 320 Sanderson, P., 85, 103 Sansone, C., 108,118,130 Santos, M., 58, 74 Qualter, A., 202, 230 Sargent, M., 189, 196 Saxe, G. B., 232, 253 R Schneider, K., 108, 130 Schoenfeld, A.H., 80, 91,103 Ramsden, J., 193,196, 200, 230 Scholle, D., 23, 32 Schon, D.A., 157,167,196 Raths, L.E., 194, 196 Schul, J., 83, 102 Rathunde, K., 107, 118, 129, 132, 151, 152, 305, 319 Schunk,D. H., 257, 289 Reeve, R.A., 3, 8, 15, 20, 28, 231, 233, Scott, D., 54, 74 Shaw, M. E., 224, 230 251, 294, 297, 302-4, 311, 317 Siegler, R. S., 231, 233, 253 Reimann, P., 78, 102 Simon, H.A., 16, 31 Renninger, A., 105, 108, 129, 130 Simon, S.B., 194, 196 Resnick, L., 315, 319 Reynolds, F., 3, 8, 15, 20, 28, 294, 297, Sinclair, J. McH., 206, 230 302-4, 311, 317 Singer,P., 160, 196 Skinner,E.A., 131, 132, 135, 153 Ricard, M., 234, 252 Slavin, R. E., 306, 320 Richards, J., 235, 253, 305, 320 Smith, A., 79, 80, 103 Rokeach, M., 160, 161, 163, 182, 196 Smith, L., 55, 56, 74, 295, 320 Rodrigues, S., 6, 7, 8, 122, 130, 199203, 205, 206, 209, 230, 293, 296, Snow, R. E., 257, 288 298, 303, 304, 307-10, 317 Snygg, D., 160-162, 195 Sohmer, R., 313, 320 Roe, L., 237, 251 Rogoff, B., 103, 234, 233, 253, 316, 320 Song, J., 200, 229 Spaulding, C. L., 258, 288 Rome-Flanders, T., 234, 253 Spicer, T, 86, 87, 103 Q

343

344

AUTHOR INDEX

Stacey, K., 81, 83, 84, 103, 135, 137, 152, 236, 252 Steffe, L.P., 33, 35, 51, 52, 76, 103, 235, 251, 253, 305, 315, 320 Stein, M., K., 306, 320 Stephens, M., 298, 319 Stinner, A., 201, 205, 230 Stipek, D. J., 131, 153 Stoiber, K.C., 134, 153 Stokoe, W.C., 249, 251 Strayer, F. F., 232, 253 Swing, S.R., 134, 137, 153

Values Technology (Hall-Tonna), 163, 182, 186,188,196 van der Veer, R., 57, 74 Vendler, H., 10, I1 Vile, A., 60, 72, 74 Voigt, J., 34, 52 von Glasersfeld, E., 55, 73, 74, 76, 103, 204, 230, 235, 253, 305, 320 Vygotsky, L.S., 4, 9, 11, 45,46, 5 1, 52, 56-61, 74,76-79, 103, 159, 160, 167, 189, 196, 292, 296, 300, 305-7, 3114,317,320

T

W

Tarule, J.M., 205, 229 Thomas, B., 77, 103 Thomas, G., 4, 5, 81, 82, 86, 87, 103, 136, 146, 292, 293, 295, 300, 301, 303-6, 308-10, 314, 317 Thompson, P.W., 35, 52 Todarov, T., 3, 11, 31, 32, 36, 42, 51, 52 Todd, F., 204, 206, 229 Tomasello, M., 234, 253 Tondow, M., 163, 195 Tonna, 7 Tsatsaroni, A., 56, 73 Tudge, J., 78, 103, 58, 59, 74, 232, 233, 253

Walkerdine, V., 56, 74 Wardrop, J.L, 78, 102 Watson, A., 56, 71, 74 Watts, M., 218, 230 Waywood, A., 298, 319 Webb, N.M., 78, 83, 84, 103, 135, 153, 306, 320 Wein, D., 236, 252 Wellborn, J.G., 131, 132, 135, 153 Wenger, E., 35, 52, 71, 74, 199, 230 Werner, H., 234, 253 Wertsch, J., 102, 313, 320 Westgate, D. P. G., 206, 230 Wilcox, S.E., 249, 251 Williams, G., 136, 153, 306, 320 Wilson, J., 137, 153 Wilson, T.D., 16, 32 Winbourne, P., 56, 71, 74 Winegar, L. T., 232, 253 Winsler, A., 79, 102 Winter, G., 189, 196 Winterhoff, P. A., 232, 253 Withers, G., 122, 129 Wood, D., 75,79,103 Wood, T., 58, 73 Wright, J. M., 224, 230 Wright, C., 271, 289

U Usher, R., 54, 74

V Valencia, S.W., 78, 102 Valsiner, J., 57, 74, 105, 106, 108, 111, 114, 130, 232, 253

AUTHOR INDEX

Wubbels, T., 213, 228 Wundt, W., 234, 253

Y Yackel, E., 58, 73 Yager, R. E., 200, 230 Young, D.J., 201, 225, 230 Young, S., 86, 87, 103 Z Zimmerman, B. J., 132, 134, 153, 257, 274, 289 Zuckerman, M., 115, 130

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SUBJECT INDEX INTRODUCTION (Ways to use this index) It may seem a little unusual to offer suggestions on how to use a familiar tool such as an index, but I offer these comments nonetheless. Because of the nature of the Complementary Accounts methodology employed in this research project, the index offers an unusual insight into the dynamics of structure and complexity that characterise this and other contemporary classroom research studies. Constructing the subject index for a book such as this is a much more interpretive activity than might be expected. A Subject Index is a surprisingly subjective construction — and in this case it is the Editor’s subjectivity that is on display. For example, a passage that I judged to be primarily concerned with “Negotiation as a Social Process” may not appear so to you. Some occurrences of the term “engage” were judged to be discursive rather than substantive and did not generate an index entry. The alternative — to list indiscriminately all occurrences of “negotiation” or all occurrences of “engage” — would render the index turgid, lengthy, and uninformative. Instead, the index has been constructed in anticipation of several purposes for which it might be used. Superficially, the index catalogues the occurrence of those “key terms” employed in our research. In a study deliberately embracing several distinct theoretical positions, however, it would be naÏve to expect that a term such as “process” would be accorded the same meaning on page 120 as it is on page 210. And this suggests one way in which this index might be used: Identify a term of particular interest (“Context” for example) and use the index to chart its various incarnations in the different chapters of this book. Since each chapter appeals to a literature that is more or less idiosyncratic to the chapter authors’ theoretical approach, a term such as “context” can be revisited in relation to each of these various literatures. Pervasive terms, such as “learning”, offer a specific challenge. If the index were to document every occurrence of the term “learning”, the result would be a lengthy and uninformative list of undifferentiated page numbers. The alternative employed here is to catalogue sub-categories (such as “learning activity”, “collaborative learning” and “learning environment”) of the more embracing term, while also indexing more generic, less differentiated usage of the general term. Not unexpectedly, some chapters devote a major proportion of their content to the explication of just one or two such sub-categories. The index charts these differences in emphasis. It may be that, in addition to its more conventional applications, the index offers some readers a novel way to construct a personal path through our research. I hope this is the case.

347

348

SUBJECT INDEX

A

B

Accounts, 1-6, 10, 13, 14, 16-18, 21-25, 27, 29-31, 34, 41, 53-56, 60, 61, 71, 72, 135, 291-98, 303, 317

Beliefs, 15, 77, 80, 86, 132, 157, 160, 194, 204, 218, 222, 224, 228, 259, 274, 275, 277, 278, 284, 287

Activity, 107, 115, 117, 132, 190, 237, 295, 300, 308 C Classroom, 56, 84, 89, 108, 13436, 151, 152, 157, 159, 172, 186, Case studies, 149 213, 301, 314 Cognitive, 99, 135, 137, 299, Classroom learning — see Learning 311, 317 Learning, 57-59, 72, 76, 91, 131, Co-construction, 231, 238, 255, 292, 218, 223 293, 295, 316 Metacognitive, 99, 132, 135, 137, 299, 311 Cognition, 3-7, 16, 21, 23, 28, 33, 34, Mathematical, 60, 69, 99, 309, 41, 45, 46, 49, 55, 58-60, 76-78, 310 81-87, 91, 99, 102, 107, 117, Practical, 2 16, 2 18, 223 118, 131-41, 144, 146, 147, 149, Scientific, 200, 214 151, 152, 161, 198-200, 203, 210, 218, 219, 224, 229, 232-38, Social, 77, 159, 218, 295, 296, 312 249, 255-60, 262, 266, 267, 27073, 287, 294, 296-99, 301-304, Student, 215, 306, 314 306, 307, 309,311,316 Affiliation, 3, 15, 297, 315-17 Collaboration, 41, 57, 59, 72, 138, 144, 152, 160, 190-92, 194, 213, 217, Appropriation, 30, 311 237, 238, 294, 300-302, 305, Argumentation, 22 317, 318 Artifacts, 35, 200, 295,308,318 Authority, 46-9, 61, 157, 161, 172, 193, 303 Mathematical/Scientific, 6, 22 Methodological, 1 Personal, 167, 182 Research, 22

Communication, 4, 7, 8, 14, 15, 44, 53, 55, 56, 58-61, 66, 68, 77-79, 137, 139, 155, 156, 163, 167, 170, 172, 185-7, 189, 197, 198, 203, 206, 209, 213, 225, 227-29, 231, 234, 248-50, 303,304,308,311 Complementarity, 2, 21, 23, 24, 29, 191, 192 Complementary, 1, 2, 13, 14, 23-25, 2931, 122, 186, 292-94, 297, 303

SUBJECT INDEX

Conation, 255-57, 272 Concepts, 106, 107, 120, 121, 127, 129, 160, 197-99, 201-206, 210-12, 214, 219, 224, 228, 229, 235, 237, 249, 266, 278, 296, 306, 311 Conceptual Conflict, 107, 122 Development, 55, 158, 159, 188, 311,349 Framework, 24, 29, 35, 48, 156, 163, 195 Mathematical/Scientific, 187 Reorganisation, 55 System, 79, 307 Understanding, 28, 159, 190, 236 Consensus, 21-3, 25, 29, 35, 106, 144

349

Context, 1, 3, 7-9, 15, 23, 28, 44, 50, 56, 72, 76, 114, 116, 117, 135, 155, 159, 161, 164, 167, 197-203, 205, 206, 215, 218, 222-229, 231, 249, 259, 260, 270, 271, 273, 291, 296, 300, 308, 317 Appropriate context, 197, 198, 200-203, 205, 206, 218, 223-29, 293, 308, 317 Internalised context, 197-200, 202, 206, 218, 219, 22729, 293, 296, 307, 308 Socio-cultural context, 197-201, 203, 218, 219, 223, 224, 227-29, 295, 298, 307 System context, 197-201, 203, 218, 222, 224, 226-29 Culture, 53, 136, 160, 161, 198, 200, 291, 308, 314-16

Constructs Curriculum, 122, 157, 199, 200, 202, Cognitive constructs, 189, 198, 205, 210, 212, 218, 227, 228, 203, 204 297, 299,307,308 Theoretical constructs, 14, 27, 30, 34, 36 , 41, 42, 45, 78, 105, 106, 118, 131, 133, 134, D 151, 255-59, 261, 291, 292, 294, 296, 303, 305, Discourse, 2, 9, 15, 23, 28, 29, 36, 41, 309, 318 47, 49, 56, 60, 147, 189, 195, 198, 203-206, 214, 235, 236, 238, 248, 249, 296, 303, 312, 313 Constructivism, 45, 55, 76, 86, 158, 161, 204, 215, 229, 295, 297, 301, 315, 316 Discussion, Radical Constructivism, 34, 55, Class, 77, 81, 121, 146, 187, 204, 214, 219, 228, 238, 239, 76 Social Contructivism, 76, 297 266, 281, 282 Group, 114, 139, 149, 209, 212, 213, 220 Disposition, 107, 108, 115, 119, 122-25, 129, 136, 304

350

SUBJECT INDEX

Dyad, 4, 8, 9, 19, 40, 41, 45, 50, 99-102, 231-33, 242, 248, 255, 293, 310

Goals

E Emergent theory, 317 Engagement, 131, 136, 137, 138, 140, 141, 174, 179, 189, 198, 200, 212, 222, 224, 228, 229, 231, 260, 293, 298, 300, 301, 307-309 Cognitive Engagement, 13 1, 136-41, 144, 146, 147, 149, 151, 152, 301-303, 306, 307, 310, 317 Group

Activity/Task, 6, 49, 89, 90, 100, 102, 112,138 Consensus, 29 Cultural groups, 15 Focus student group, 15, 19 Interpretive research, 5, 22, 30 Social groups, 15, 105-7 Talk, 83, 85 Working groups, 6, 19, 41, 57, 101, 108, 109, 135, 150, 209, 210, 213, 215, 217, 299, 301,306, 307

Ethnography, 21, 53-55, 167, 270, 274, 291

F Flow, 107, 118, 132, 305 Framing, 9, 56, 70, 156, 189, 258, 274, 287, 294, 298, 299, 306, 309, 313, G

Educational, 80, 105, 158, 161, 198, 201, 210, 217, 257, 268, 306, 311,317 Learned/student, 9, 48, 112, 120, 132-134, 136, 202, 214, 219, 224, 256 5- 8, 260, 261, 283,301,317 Researcher, 35, 44, 291-93, 317 Social, 28, 56, 59 Teacher, 9, 48, 208, 213, 214, 218, 283, 317 Values, 163, 176, 179, 181, 182, 186-8, 191, 192

I

Gesture, 3, 8, 9, 15, 20, 23, 25, 28, 45, Ignorance, 9, 46, 255-58, 260-62, 266118, 137, 138, 140, 144, 146, 68, 270-72, 288, 301, 303 159, 174, 190, 203, 233-49, 294, 295, 302-304, 311, 317 Image, 15, 16, 56, 72, 169, 235, 268, 315

SUBJECT INDEX

351

Individual, 55-59, 105, 122, 136, 198, Interaction Student-student, 6, 8, 18, 19, 25, 273, 274, 295-98, 300, 301, 305, 27, 47, 58-62, 71, 72, 77, 309, 311, 314-16, 317 78, 80, 82, 85, 100, 136, Activity, 76, 138, 317 141, 149, 233, 292, 294, Decontextualised, 56, 295 Differences, 107, 115, 233, 257, 295, 300, 302, 304-306, 272 308-310, 314, 315, 317 Interpretive activity, 15-17, 25, Student-teacher/teacher-student, 4-6, 8, 19, 22, 59, 78, 86, 45 Learning behaviour, 16, 35, 49, 87, 100, 101, 149, 152, 168, 178, 193, 273, 298, 77,78, 204, 227 309-311,316 Participation, 3, 15, 28, 35, 36, 42, 50, 316 Task-student, 81 Unit of interaction, 36, 49, 206 Practice, 8, 9, 28, 47, 50, 85, 315, 316 Psychology, 3,4, 33, 55 Interest, 105-9, 111, 112, 114, 116-122, Purposes, 15, 28, 223 124, 126, 127, 129, 136, 157, Value sets, 160, 162, 163, 182, 164, 172, 176, 178, 180, 181, 186, 189-91, 195, 199-201, 222185, 190 25, 256, 257, 262, 263, 265, 277 Information Processing, 107 Internalisation, 4, 33, 49, 57, 58, 60, 77, 79, 81, 156, 157, 160, 189, 190, Interaction Classroom, 3, 4, 7, 13-16, 26-30, 197-200, 202, 206, 218, 219, 33, 47, 50, 57, 71, 105, 227-29, 300, 307, 311, 316, 317 122, 155, 271, 293, 294, 306,309,314, 317 Interpretation Group Interpretive Processes, 13Collaborative, 236 15, 22-4, 30, 31, 35, 50, Dyadic interaction, 50, 99, 231 218, 236, 239, 309 Individual-environment, 33, 55, 204, 313 Individual Interpretive Processes, Intra-individual, 33 35, 161, 190, 202, 229, Mathematical/Scientific, 5, 4 5, 297, 316, 317 75, 76, 86, 102, 304, 314 Interpretive Research, 17, 21-3, 25, 29-31, 35, 36, 51, 54, Negotiative interaction, 46 60, 72, 106, 135, 156, 167, Patterns, 3, 6, 42, 50, 75, 144 168, 171, 186, 235, 237, Social, 9, 18, 19, 28, 35, 49, 55, 249, 250, 255, 258, 261, 57, 76, 80, 233, 284, 294262, 270, 271, 274, 29197, 300, 301, 305-307, 310, 314-17 94, 296, 313, 314, 318 Intersubjectivity, 3, 14, 15, 27, 34, 35, 40-42, 44-46, 49, 50, 57, 58, 292, 301, 304, 311, 317, 318

352

SUBJECT INDEX

Interview, 1, 2, 14, 25, 30, 70, 71, 205, Learning, 4, 6-10, 17, 20, 21, 24, 34, 46, 294 48, 50, 54-57, 59, 60, 62, 72, 75Student, 15-20, 26, 54, 225, 226, 79, 81, 94, 102, 108, 120, 127, 261 129, 132, 160, 190, 194, 197, Teacher, 16-18, 29, 54, 261, 299 201, 218, 222, 228, 229, 232, Transcripts, 25-26, 61, 205, 293 255-57, 260, 273, 274, 276, 277, Video-stimulated recall/ 279, 280, 283, 284, 291, 295-97, Reconstructive interviews, 17, 299, 303 -305, 307, 310, 312-18 30, 118, 123, 134, 149, Activity/Task, 57, 58, 72, 106, 208, 274, 316, 317 131, 133, 136, 187, 215, 274, 285, 302, 315 Intrasubjectivity, 58 Behaviour, 16,27,29-31, 105, 259, 261, 284 Intuition, 80, 210- 12 Classroom, 13, 14, 18, 24, 27, 30, 31, 33, 34, 41,48, 50, 105, 134, 162, 236, 291, 292,296, K 297, 303, 304, 315 Collaborative, 231,236-38,250 Knowledge, 5, 7, 14, 22, 23, 35, 45, 46, Decontextualised, 72 48-50, 56, 60, 61, 76-80, 86, 87, Dialogical, 189,297,309 Environment (Settings and 91, 94, 106-8, 111, 112, 117, 120, 126, 129, 131, 136, 156-58, 167, Situations), 5, 9, 34, 46, 169, 170, 181, 185, 187, 188, 106, 136, 170, 204, 231, 200, 202, 204, 205, 213, 219, 232, 238, 250, 309 226-29, 231, 256, 257, 259, 262, Experience-based, 158, 160, 189, 267, 275, 281, 282, 285, 287, 190, 193, 300 292, 296, 303-305, 310, 311, 315 Mathematical/Scientific, 2, 3, 21 , 50, 75, 85, 101, 301, 311 Objectives, 84 L Opportunities, 5, 198 Outcomes, 48, 131, 134, 136, 144, 275, 282, 306 Language, 3, 7-9, 35, 36, 42, 49, 58, 60, Perceptions-Driven, 260, 270, 302 65, 79, 118, 119, 121, 123, 129, 155, 156, 159, 160, 162, 164, Process, 15, 29-31, 34, 41, 45, 167, 172, 178, 189, 190, 195, 49, 50, 105, 131, 157, 198, 202-205, 231, 233-35, 237, 250, 295, 300, 315 272, 276,296, 303, 304, 3 11-3 13, Reflection-Driven, 260, 261, 317, 318 269, 270, 302 Self-regulated, 132-34, 136, 137, Learner, 107, 129, 132, 141, 160, 190, 141,257 202, 215, 232, 257, 258, 273, Situated, 3, 35 291, 293, 297, 300-302, 306, Theory, 4, 14, 28, 33, 50, 156, 307, 310 -18 193, 204, 292, 294, 297, 298, 300, 316

SUBJECT INDEX

M

353

N

Negotiation, 33-35, 40, 42, 49, 189, 190, 193, 228, 297, 301, 303, 304, 318 Internal, 33,34 Meaning, 1-3, 5, 7, 8, 10, 13-15, 23, 24, MathematicaI/Scientific, 29, 309 35, 36, 42, 45, 46, 58, 68, 127, 128, 158, 161, 162, 167, 190, Research process, 23,24, 30 Social process, 3, 5,7, 19, 34, 35, 204, 210, 224, 236-38, 243, 249, 42, 40, 159, 189, 213, 215, 221, 257, 274, 285, 295, 301, 304, 316 226, 304, 308, 309, 316,317 Alternative meanings, 13, 203, Values, 155, 193, 194, 299 204, 313 Mathematical/Scientific, 2, 4, 7, 13, 22,45, 50, 53, 72, 216, Negotiative Event, 3, 28, 36-41, 47-50, 292, 297, 317 144 Negotiation of Meaning, 5, 14, 21, 29, 30, 34, 35, 42, 45, Notation, 237 49, 50, 160, 163, 188-190, 191, 194, 215, 219, 221, 226, 228, 318 O Researcher’s meanings, 13, 23, Objects, 72, 86, 127, 131, 159, 200, 204, 31, 167 234, 235, 237, 292, 301, 304, Shared Meaning (see also Taken306, 307, 317 as-shared meaning), 105, 106, 108,189, 227, 237 Of study, 29,60,61 Social, 2, 7, 13, 15, 45, 56, 317

Mathematical Authority

—

see Authority

Metacognition, 3, 4, 21, 29, 80, 81, 84, 87, 99, 100, 102, 136, 257, 25962, 264, 266, 269-73, 275, 287, 288, 299-301, 303, 309-11, 317 Metaphor, 56, 79, 91, 235, 237, 238, 291, 292, 297, 315 Motivation, 105-8, 112, 115, 117, 118, 122, 131-34, 136, 137, 138, 151, 152, 163, 172, 200, 202, 219, 222, 231, 256-59, 262, 264, 271, 292, 296,301,309, 317 Mutual Contextuality, 33

P Patterns, 5-7, 30, 36, 41, 50, 65, 77, 85, 122,129, 132, 144, 146, 152, 155, 156, 162-64, 167, 168, 190, 198, 206-210, 214, 228, 235, 291,308 Piagetian, 33, 49, 55, 59, 61, 77, 232, 233, 235, 294, 297 Power relationships, 7, 28, 47, 48, 61, 187, 194, 206, 213, 214

354

SUBJECT INDEX

Practice, 15, 36, 312, 316, 317 Classroom, 2, 3, 6, 7, 10, 13, 15, 23-25, 28, 34, 135, 138, 159, 197, 198, 215, 255, 277, 292, 297, 299, 302, 307, 308, 312, 315-18 Communal, 9, 297,313 Cultural, 33, 56 Dyadic, 50 Mathematical/Scientific, 203, 237, 293,310 Mutually-sustaining, 9 Research, 14, 16, 30 Situated, 8, 35, 202 Social, 56,223,293,295-97,316 Student/Learner, 4, 80, 100, 210, 270, 288, 292, 298-300, 302, 307, 310, 311, 31416 Teacher, 21-23, 100, 134, 136, 190, 193, 217, 273-75, 287, 298-300, 305, 307, 308, 311,314, 315 Problem solving, 6, 8, 9, 41, 47, 57, 75, 78, 80, 85, 90, 91, 94, 99, 101, 134, 144, 231-34, 236, 238, 249, 250, 294, 300, 301, 310, 311 Processes, 105, 106, 108, 109, 120, 129, 132, 137, 156, 157, 167, 170, 189, 190, 194, 197, 198, 203, 204, 210-218, 222, 227-29, 234, 235, 237, 256, 259, 260, 270, 300-305, 307-309, 311-13, 31518 Psychology, 21, 29, 56, 58, 77, 296, 301, 302

R Radical constructivism — see Constructivism Rationality, 56, 77, 181 Reality, 25, 86, 162, 188, 201, 204, 229 Reasoning, 60, 72, 94, 99, 146, 159, 160, 189, 200, 296, 309 Reciprocal scaffolding — see Scaffolding Reciprocity, 191, 305, 315, 316 Reconstructive accounts, 16, 18, 25, 30 Reflexive relationships, 49, 50, 317 Reflexivity, 3,9,33, 50, 53, 54 Representation, 17, 35, 46, 144, 233, 235, 238, 242, 249, 293, 309, 311 Resources, 7, 133, 138, 140, 141, 144, 197, 198, 203, 205, 210, 218-21, 224, 226-29, 237, 308, 315

S Scaffolding, 4-6, 75, 79, 80-82, 87, 91, 94, 95, 97-99, 100, 102, 146, 228, 292, 299, 302, 305, 310, 311 Reciprocal scaffolding, 4, 100, 102, 301, 305, 310,317 Self-scaffolding, 80, 99, 100, 102 Self-concept, 161, 190, 257 Social authority

—

see Authority

SUBJECT INDEX

355

Understanding Joint/Shared, 66, 72, 198, 203, Socio-cultural norms, 167, 168, 193, 209, 220, 227, 229, 233, 292, 297, 309, 316 237, 317 Learner’s, 8, 28, 41, 46, 84, 85, 91, 94, 114, 123, 128,202, T 203, 210, 228, 242-244, 248, 267, 273, 277, 281, Taken-as-shared, 42, 44, 105 287, 301, 304, 305, 314 Researcher’s, 34, 36, 49, 56, 138, 151, 237, 238, 250, 291, Teacher, 1, 4-10, 14-23, 25, 48-50, 54302, 311, 312, 314, 317, 62, 64-66, 68, 71, 72, 76, 79, 82, 318 86, 87, 90, 94, 100, 121, 126, Taken- as- shared, 42, 44 146, 147, 155-60, 168-74, 180Teacher’s, 7 94, 200, 204, 206-208, 210, 218, 220, 228, 262, 266, 270-72, 274, 275, 277-87, 292-94, 297-315, 317, 318 V Social interaction — see Interaction

Teacher-Student Interactions — see Interactions Teacher-Student Intersubjectivity — see Intersubjectivity Teaching, 1, 3, 4, 8, 9, 17, 18, 20, 21, 79, 80, 86, 102, 157, 162, 197, 198, 200, 218, 227, 229, 271-75, 284, 287, 288, 291, 298, 299, 306-308, 312-15, 318

Values, 7, 155-95, 204, 215, 218, 22224, 226, 232, 273-75, 277, 283, 287, 288, 292, 293, 296-303, 306, 309, 315 -18 Volition, 262, 270, 272, 273, 275, 287, 299, 301, 317 Vygotsky, 3, 46, 56-61, 72, 76-79, 159, 160, 167, 189, 232, 233, 292, 294, 296, 297, 300, 305-307, 311-14, 317

Teaching/Learning, 9, 10, 264, 291, 292, 295, 312-14, 316-18 W U Understanding, 23, 24, 101, 102, 157, 190, 192, 195, 233, 237, 238, 260-62, 270-72, 317

Working consensus, 22 58, 75-78, 86, 158, 161, 170, 200, 204, 231, 255, 257, 258, 280, 294, 308,

Z Zone of Proximal Development (ZPD), 4, 9, 27, 46, 48, 55-61, 66, 72, 78-82, 86, 99, 160, 295, 305, 306, 310, 313, 314, 317

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Mathematics Education Library Managing Editor:

A.J. Bishop, Melbourne, Australia

1.

H. Freudenthal: Didactical Phenomenology of Mathematical Structures. 1983 ISBN 90-277-1535-1; Pb 90-277-2261-7

2.

B. Christiansen, A. G. Howson and M. Otte (eds.): Perspectives on Mathematics Education. Papers submitted by Members of the Bacomet Group. 1986. ISBN 90-277-1929-2; Pb 90-277-2118- 1

3.

A. Treffers: Three Dimensions. A Model of Goal and Theory Description in MathISBN 90-277-2165-3 ematics Instruction The Wiskobas Project. 1987

4.

S. Mellin-Olsen: The Politics of Mathematics Education. 1987 ISBN 90-277-2350-8

5.

E. Fischbein: Intuition in Science and Mathematics. An Educational Approach. 1987 ISBN 90-277-2506-3

6.

A.J. Bishop: Mathematical Enculturation. A Cultural Perspective on Mathematics ISBN 90-277-2646-9; Pb (1991) 0-7923-1270-8 Education. 1988

7.

E. von Glasersfeld (ed.): Radical Constructivism in Mathematics Education. 1991 ISBN 0-7923-1257-0

8.

L. Streefland: Fractions in Realistic Mathematics Education. A Paradigm of Developmental Research. 1991 ISBN 0-7923-1282-1

9.

H. Freudenthal: Revisiting Mathematics Education. China Lectures. 199 1 ISBN 0-7923-1299-6

10.

A.J. Bishop, S. Mellin-Olsen and J. van Dormolen (eds.): Mathematical Knowledge: ISBN 0-7923-1344-5 Its Growth Through Teaching. 1991

11.

D. Tall (ed.): Advanced Mathematical Thinking. 1991

12.

R. Kapadia and M. Borovcnik (eds.): Chance Encounters: Probability in Education. 1991 ISBN 0-7923-1474-3

13.

R. Biehler, R.W. Scholz, R. Sträßer and B. Winkelmann (eds.): Didactics of MathISBN 0-7923-2613-X ematics as a Scientific Discipline. 1994

14.

S. Lerman (ed.): Cultural Perspectives on the Mathematics Classroom. 1994 ISBN 0-7923-2931-7

15.

O. Skovsmose: Towards a Philosophy of Critical Mathematics Education. 1994 ISBN 0-7923-2932-5

16.

H. Mansfield, N.A. Pateman and N. Bednarz (eds.): Mathematics for Tomorrow’s Young Children. International Perspectives on Curriculum. 1996 ISBN 0-7923-3998-3

17.

R. Noss and C. Hoyles: Windows on Mathematical Meanings. Learning Cultures and Computers. 1996 ISBN 0-7923-4073-6; Pb 0-7923-4074-4

ISBN 0-7923-1456-5

18.

N. Bednarz, C. Kieran and L. Lee (eds.): Approaches to Algebra. Perspectives for Research and Teaching. 1996 ISBN 0-7923-41 45-7; Pb ISBN 0-7923-41 68-6

19.

G. Brousseau: Theory of Didactical Situations in Mathematics. Didactique des Mathématiques 19701990. Edited and translated by N. Balacheff, M. Cooper, R. Sutherland and V. Warfield. 1997 ISBN 0-7923-4526-6

20.

T. Brown: Mathematics Education and Language. Interpreting Hermeneutics and Post-Structuralism. 1997 ISBN 0-7923-4554-1

21.

D. Coben, J. O’Donoghue and G.E. FitzSimons (eds.): Perspectives on Adults Learning Mathematics. Research and Practice. 2000 ISBN 0-7923-6415-5

22.

R. Sutherland, T. Rojano, A. Bell and R. Lins (eds.): Perspectives on School Algebra. 2000 ISBN 0-7923-6462-7

23.

J.-L. Dorier (ed.): On the Teaching of Linear Algebra. 2000 ISBN 0-7923-6539-9

24.

A. Bessot and J. Ridgway (eds.): Education for Mathematics in the Workplace. 2000 ISBN 0-7923-6663-8

25.

D. Clarke (ed.): Perspectives on Practice and Meaning in Mathematics and Science ISBN 0-7923-6938-6 Pb ISBN 0-7923-6939-4 Classrooms. 200 1

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