From Text to 'Lived' Resources: Mathematics Curriculum Materials and Teacher Development (Mathematics Teacher Education)

From Text to ‘Lived’ Resources MATHEMATICS TEACHER EDUCATION VOLUME 7 SERIES EDITOR Andrea Peter-Koop, University of ...

Author: Ghislaine Gueudet | Birgit Pepin | Luc Trouche

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From Text to ‘Lived’ Resources

MATHEMATICS TEACHER EDUCATION VOLUME 7 SERIES EDITOR Andrea Peter-Koop, University of Oldenburg, Germany Patricia Wilson, University of Georgia, United States EDITORIAL BOARD Andy Begg, Auckland University of Technology, New Zealand Chris Breen, University of Cape Town, South Africa Francis Lopez-Real, University of Hong Kong, China Jarmila Novotna, Charles University, Czechoslovakia Jeppe Skott, Danish University of Education, Copenhagen, Denmark Peter Sullivan, Monash University, Monash, Australia Dina Tirosh, Tel Aviv University, Israel SCOPE The Mathematics Teacher Education book series presents relevant research and innovative international developments with respect to the preparation and professional development of mathematics teachers. A better understanding of teachers’ cognitions as well as knowledge about effective models for preservice and inservice teacher education is fundamental for mathematics education at the primary, secondary and tertiary level in the various contexts and cultures across the world. Therefore, considerable research is needed to understand what facilitates and impedes mathematics teachers’ professional learning. The series aims to provide a significant resource for teachers, teacher educators and graduate students by introducing and critically reflecting new ideas, concepts and findings of research in teacher education.

For other titles published in this series, go to http://www.springer.com /series/6327

Ghislaine Gueudet · Birgit Pepin · Luc Trouche Editors

From Text to ‘Lived’ Resources Mathematics Curriculum Materials and Teacher Development

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Editors Ghislaine Gueudet IUFM Bretagne site de Rennes Rue Saint- Malo 153 35043 Rennes CEDEX France [email protected]

Birgit Pepin Sør-Trøndelag University College 7004 Trondheim Norway [email protected]

Luc Trouche Institut français de l’Education École Normale Supérieure de Lyon 15 parvis René-Descartes, BP 7000 69342 Lyon cedex 07 France [email protected]

ISBN 978-94-007-1965-1 e-ISBN 978-94-007-1966-8 DOI 10.1007/978-94-007-1966-8 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011935535 © Springer Science+Business Media B.V. 2012 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Foreword

‘Mathematics Curriculum Material and Teacher Development’ can be read as the title of a dull book on an old fashioned topic from the era of curriculum development in the 1980s, perhaps ‘jazzed up’ by the catchword ‘teacher development’ from the 1990s. So, one might expect a latecomer to research in Mathematics Education. A simple cursory look over the content of the table of contents of this book shows that this is a false assumption. There are at least three major issues investigated in this book, which make it an up-to-date and fascinating contribution to research in Mathematics Education (or Didactics of Mathematics as I would prefer to call it): – ‘Curriculum material’ has definitely not been perceived in the restricted way it had been discussed two or three decades ago. The fact that the authors use the concept ‘curriculum resources’ highlights that beside the traditional curriculum materials, like textbooks and other curricular documents, a whole range of texts and other resources have been taken into account, including software, electronic resources and the Internet. All these resources seem to become increasingly important in expressing and sharing ideas not only on curriculum materials themselves, but also in terms of curriculum development. They also help in terms of teacher education and everyday practice. The inclusion of more modern resources does not deny the most important teacher resource – the textbook. A main message of this book is to place the artefact ‘mathematics textbook’ in a wider, systematic perspective of material resources available for (mathematics) teachers and students. The book also shows that this broadening of the concept of teacher resources is helpful for understanding practices in various contexts. In selected countries, and communities of mathematics teachers, it is a fact that a wide range of ‘resources’, apart from textbooks and traditional curriculum documents, is present and relevant for teachers’ daily practice. Teachers’ professional knowledge, practical constraints (like money and other classroom arrangements) and cultural resources like language, collegiality, organisation and time, amongst others, have to be analysed to comprehensively understand the processes involved in teacher use of resources. In fact, this book opens a perspective on resources, which is not necessarily material.

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– The book supports recent trends in research on teaching and learning mathematics with the help of artefacts: to fully understand the role of curriculum material, it is not sufficient to simply analyse the artefact as such. A comprehensive content analysis of an artefact used by teachers can help to develop deeper knowledge of its functions in mathematics education. Nevertheless, it is only by analysing the use of the artefact that one may be able to adequately judge upon the affordances and constraints of a given artefact. For example ‘instrumental genesis’ (initiated and introduced to Didactics of Mathematics by Rabardel) analyses how an artefact is turned into an ‘instrument’ via the genesis of individual or social utilisation schemes. The research literature claims that a curriculum resource can only be judged by an analysis of its inherent features in addition to an analysis of the ways in which the different agents of the educational process use these resources. In an instrumental genesis approach, this is condensed in the concept of ‘utilisation scheme’, which is also fundamental to the documentation approach described in this book. As a consequence, the documentation approach conveys the notion of an agent having created the ‘document’ for a specific purpose. – In the book the word artefact is used in a broad sense, leaning on Wartofsky’s (1979) notion (XIII: ‘anything which human beings create by the transformation of nature and of themselves’) which differs from the traditional understanding of curriculum resources. The texts in this book are not only analysing material resources, but pay due attention to immaterial sources available to (mathematics) teachers. Beside material resources, a comprehensive analysis of teachers’ resources must also take into account immaterial resources like colleagues and communities of teaching practices. The book discusses ‘collaborative use’, and selected chapters explore the relations between teacher communities of practice, the documents shared in these communities and the consequences for the professional development of teachers from this collaboration. Here, the individual use of resources is adequately complemented by using resources in an environment shared by a community of teachers. Moreover, the book shows under which conditions such collaboration can empower teachers to become active instructional designers. With the broadening of the view from material to immaterial resources, from individual to collective use of resources, methodologies investigating documentation and professional interaction (sharing of knowledge) of teachers also have to be extended beyond the ‘standard’ features of classroom and school research (often done by video-taping and consecutive case study analysis) or large scale statistical research using questionnaires (maybe complemented by interviews and the like). A reader sensitive to research methodology will find a whole range of research methods to explore the diverse phenomena – with various foci according to the different theoretical stances taken by the authors. As a consequence of the innovative character of the book, no consensus on research methodology has been reached yet – and this heterogeneity seems to be appropriate for a newly developed approach and the explorative character of the investigation of resources used by mathematics teachers.

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Having stated this, one characteristic nevertheless stands out, and for the majority of the book’s chapters: nearly all texts heavily rely on case studies. The empirical results point to the necessity of a mix of research methods to better understand teachers’ use of resources. Although the last paragraph typically puts forward an argument, which shows the value and importance of the book for researchers in Mathematics Education, I would like to highlight that the texts in this book can also be very helpful for practising teachers, who could learn about the wide range of resources available for enhancing their teaching practice. Curriculum developers and policy makers may benefit from the book’s reports of investigations, which show once again that implementing change in education and educational reform is not a straightforward, top-down process. Researchers are reminded that having the best available ideas and concepts for change does not imply factual change of teaching. The book shows that sharing artefacts and collectively developing utilisation schemes in collaborative groups of teachers and researchers can be a more effective means to curriculum change. Cooperation around appropriately designed resources – be they material and/or conceptual – can be a way to develop teaching and learning mathematics. Giessen, Germany

Rudolph Straesser

Reference Wartofsky, M. W. (1979). Models. Representation and the scientific understanding. Dordrecht: Reidel Publishing Company.

Introduction Ghislaine Gueudet, Birgit Pepin, and Luc Trouche

The teachers, in their professional activity, interact with a wide range of resources; these interactions and their consequences hold a central place in teachers’ professional development. The purpose of this book is to develop this perspective and to explore it in the field of mathematics education. We consider on the one hand curriculum material. Traditionally, textbooks remain central resources for the teaching of mathematics in most countries. Nevertheless, other kinds of resources, in particular digital resources, and amongst them resources accessible via the Internet, are increasingly used. Understanding the evolutions brought by digital material is a central motivation of our work. On the other hand, the reason for introducing the term ‘resource’ instead of ‘material’ is to broaden the perspective on the elements available for the teachers’ work, and to include in particular interactions with a variety of agents: – Interactions between the teacher and her students constitute central resources for this teacher. Digitisation creates new forms of students’ productions and new modes of communication between students and teachers; but even an expression on a student’s face in class can constitute a resource for the teacher. – Interactions between the teacher and her colleagues seem to hold an increasing place. Teachers can collectively design curriculum plans, lessons, and once again the digital means convey new forms of communication, networking and association. Teachers collect resources, select, transform, share, implement and revise them. Drawing from the French term ‘ingénierie documentaire’, we call these processes ‘documentation’. The literal English translation is ‘to work with documents’, but the meaning it carries is richer. Documentation refers to the complex and interactive

G. Gueudet (B) CREAD, Université de Bretagne Occidentale, IUFM Bretagne site de Rennes, 35043 Rennes Cedex, France e-mail: [email protected]

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ways that teachers work with resources; in-class and out-of-class, individually, but also collectively. We propose a new perspective, considering teachers not as passive users, but as designers, creative ‘users’ and ‘sharers’ of their own resources, and viewing these resources as ‘lived’ resources. Teachers’ professional knowledge influences this design; at the same time, the documentation work extends existing-and generates new-professional knowledge. Working in 12 different countries, the authors develop a variety of perspectives on teacher resources, on their use and on the associated teachers’ professional development, with different foci and theoretical frameworks. The book is organised in four parts. Each is complemented by a reaction, presenting an expert’s view of the whole section. The first part focuses on the different kinds, and nature of, curriculum resources for mathematics teachers from a practical, methodological and theoretical point of view. It examines what is, or is not, available for teachers’ professional activity. It also introduces the question of what kinds of changes are afforded by digital resources: – Jill Adler introduces a conceptualisation of resources as re-sourcing teachers’ professional activity. She focuses on teacher professional knowledge, and provides evidence of different uses, by teachers in class, of knowledge resources. – Ghislaine Gueudet and Luc Trouche propose what they coin as documentational approach of didactics for the study of the teacher’s documentational work. This new theoretical approach emphasises that geneses, documentational geneses as well as professional geneses, are strongly intertwined. The authors also expose a specific methodology for the study of these geneses: the reflexive investigation of teachers’ documentation work; – Maria Alessandra Mariotti and Mirko Marracci consider the question of semiotic mediation initiated by the development of the available digital resources. They explore the semiotic potential of an artefact for teacher use in their classrooms. – Gérard Sensevy focuses on didactical intentions, for individual teachers and for different kinds of teacher groups. He studies the influence of resources on teachers’ pedagogical intentions. Furthermore, he considers teacher action in class in terms of joint actions, and which include student actions, where the students’ contributions constitute a major resource for teachers. The Reaction to Part I is written by Bill Barton. The second part of the book focuses on the characteristics of curriculum material. The articles raise questions about the design of curriculum materials, and about their integration, appropriation and transformation by teachers in and for their everyday teaching. Is the teacher use of curriculum materials aligned with the use envisioned by curriculum designers? What are the consequences of teacher transactions with resources for teacher professional development? The various factors shaping the nature of the resources, their design and their use, are examined here, with a

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specific focus on sociocultural factors and how these influence the development of curriculum materials. – Kenneth Ruthven investigates the use and integration of technology in mathematics classroom practice. He identifies five structuring features, of different natures, that shape the incorporation of new technologies into teachers’ practice: working environment, resource system, activity format, curriculum script and time economy; – Janine Remillard considers different modes of engagement which teachers develop vis-a-vis curriculum resources and how they develop. She argues that teachers are often positioned, or position themselves, as passive users; and her overarching aim is to reframe the teacher–curriculum relationship such that teachers are positioned as partners and collaborators with curriculum resources. – Birgit Pepin investigates the role of resources, more precisely a task analysis schedule, as catalyst for teacher learning. She explores the different forms of feedback resulting from developing and working with a ‘tool’ designed to analyse mathematical tasks/curriculum materials for instruction. Her results provide deeper insights, at one level, into the processes of teacher learning with the help of analytic tools and the feedback these may afford, and at another, how a tool or artefact may change into a catalytic tool at the interface between task design and enactment. – William Schmidt describes the development of a textbook content metric that can be used in longitudinal studies to map and measure the curricular experience of individual students. Teachers and schools, sometimes districts, choose textbooks, and teachers in turn decide on the ‘coverage’ of those textbooks. This, in turn, has implications for student exposure to these curriculum materials, and the ways of working with them. – Christine Proust proposes a historical perspective on the nature of ‘school documentation’, in terms of design and use of mathematical texts in the scribal schools of Mesopotamia about 4,000 years ago. She observes patterns of this documentation across different schools indicating strong institutional conditioning. The Reaction to Part II is proposed by Malcolm Swan. The third part focuses on the use of resources by teachers and students, in-class and out-of-class, and includes studies that explore the influence of the resources’ characteristics on teacher and student activity. Furthermore, the articles in this part consider the interactions between the various educational agents, and the effects of these interactions on the development and design of resources: – Carolyn Kieran, Denis Tanguay and Armando Solares study the ‘how’ and the ‘why’ of teachers adapting researcher-designed resources, and in the context of integration of computer algebra system (CAS) technologies. They claim that the whole adaptation process, from its beginning (how teacher engage with a resource designed by researchers) to the changes made in class during the implementation, rests on teacher knowledge and beliefs.

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– Using classroom videos, Dominique Forest and Alain Mercier analyse how teachers can organise their pedagogic practice and student interventions drawing on language and gestures. They show how classroom videos can become resources for teacher professional development and research. – Sebastian Rezat focuses on textbooks, considering teachers’ and students’ use of textbooks. He establishes links between teacher’s use of mathematics textbooks effecting students, and vice versa, and argues that students’ use of resources must be considered as an important aspect within teachers’ documentation work. – Maria Trigueros and Maria-Dolores Lozano study documentational geneses of teachers working within Enciclomedia, a national project in Mexico that offers a particular online resource. They identify developments in terms of teacher documentation systems and of teacher pedagogic practice, which includes the use of the digital means offered and supported by traditional textbooks. – Paul Drijvers uses and further develops the concept of orchestration. He argues for a specific focus on what happens in class, the didactical performance, and identifies types of orchestrations. Survey results suggest that teachers’ intentions may differ from their actual teaching. He investigates factors leading teachers to retain a given type, and conditions for evolutions and development. The Reaction to Part III is proposed by Luis Radford. The fourth part of the book focuses on the collaborative aspects of teacher documentation, considering that teachers are in contact, and work, with various groups and communities in their professional lives. In this part concepts are introduced that illuminate the influence of the nature of groups and communities, the particularities of the processes of documentation within groups, and individual–collective relationships. The articles in this part identify various potential roles and interventions of collaborative teacher documentation in mathematics teacher education. – Carl Winsløw proposes a comparative study of two kinds of teacher collectives: lesson studies as a means for professional development of mathematics teachers in Japan; and Danish high-school teachers’ collaboration in the setting of multidisciplinary modules. He introduces, and provides evidence for, the importance of didactic infrastructures, their constraints and affordances in terms of teachers’ collaborative work in preparing, observing and evaluating their teaching. – Ghislaine Gueudet and Luc Trouche extend the documentational approach of mathematics didactics presented in chapter two by emphasising the importance of collective aspects in teachers’ documentation work. Drawing on the notion of ‘communities of practice’, they introduce the notions of community genesis and community documentation genesis, and study the relationships between the different kinds of geneses. – Jana Visnovska, Chrystal Dean and Paul Cobb problematise the rhetoric of teachers as instructional designers. They argue that all teachers engage in documentation work; but the ability of designing coherent instructional sequences requires specific support and appropriate professional development structures.

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The Reaction to Part IV is proposed by Barbara Jaworski. Deborah Ball offers a general view on all contributions; the conclusion section presents a synthesis of the book’s main results. The authors in this book provide different lenses to view the interactions between teachers and teaching resources, and the implications for teacher professional development. These different views come together in the book, resulting in the emergence of a new theorisation of teacher documentation work, and a new perspective on teachers’ resources.

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Contents

Part I

Teacher Resources

1 Knowledge Resources in and for School Mathematics Teaching . . Jill Adler

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2 Teachers’ Work with Resources: Documentational Geneses and Professional Geneses . . . . . . . . . . . . . . . . . . . . . . . Ghislaine Gueudet and Luc Trouche

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3 Patterns of Didactic Intentions, Thought Collective and Documentation Work . . . . . . . . . . . . . . . . . . . . . . . Gérard Sensevy

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4 Resources for the Teacher from a Semiotic Mediation Perspective . Maria Alessandra Mariotti and Mirko Maracci

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Reaction to Part I: Resources Can Be the User’s Core . . . . . . . Bill Barton

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Part II

Text and Curriculum Resources

5 Constituting Digital Tools and Materials as Classroom Resources: The Example of Dynamic Geometry . . . . . . . . . . . Kenneth Ruthven

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6 Modes of Engagement: Understanding Teachers’ Transactions with Mathematics Curriculum Resources . . . . . . . Janine T. Remillard

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7 Task Analysis as “Catalytic Tool” for Feedback and Teacher Learning: Working with Teachers on Mathematics Curriculum Materials . . . . . . . . . . . . . . . . . . . . . . . . . Birgit Pepin 8 Measuring Content Through Textbooks: The Cumulative Effect of Middle-School Tracking . . . . . . . . . . . . . . . . . . . William H. Schmidt

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9 Masters’ Writings and Students’ Writings: School Material in Mesopotamia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christine Proust Reaction to Part II: Some Reactions of a Design Researcher . . . . Malcolm Swan Part III 10

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Researcher-Designed Resources and Their Adaptation Within Classroom Teaching Practice: Shaping Both the Implicit and the Explicit . . . . . . . . . . . . . . . . . . . . . . . . Carolyn Kieran, Denis Tanguay, and Armando Solares

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Classroom Video Data and Resources for Teaching: Some Thoughts on Teacher Education . . . . . . . . . . . . . . . . Dominique Forest and Alain Mercier

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Interactions of Teachers’ and Students’ Use of Mathematics Textbooks . . . . . . . . . . . . . . . . . . . . Sebastian Rezat

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Teachers Teaching Mathematics with Enciclomedia: A Study of Documentational Genesis . . . . . . . . . . . . . . . . . Maria Trigueros and Maria-Dolores Lozano

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Teachers Transforming Resources into Orchestrations . . . . . . . Paul Drijvers

Part IV

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Use of Resources

Reaction to Part III: On the Cognitive, Epistemic, and Ontological Roles of Artifacts . . . . . . . . . . . . . . . . . . . Luis Radford

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Collaborative Use

A Comparative Perspective on Teacher Collaboration: The Cases of Lesson Study in Japan and of Multidisciplinary Teaching in Denmark . . . . . . . . . . . . . . . . . . . . . . . . . . Carl Winsløw

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Communities, Documents and Professional Geneses: Interrelated Stories . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghislaine Gueudet and Luc Trouche

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Mathematics Teachers as Instructional Designers: What Does It Take? . . . . . . . . . . . . . . . . . . . . . . . . . . . Jana Visnovska, Paul Cobb, and Chrystal Dean

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Reaction to Part IV: Teacher Agency: Bringing Personhood and Identity to Teaching Development . . . . . . . . . . . . . . . . Barbara Jaworski

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Afterword: Using and Designing Resources for Practice . . . . . . . . . Deborah Loewenberg Ball

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Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghislaine Gueudet, Birgit Pepin, and Luc Trouche

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contributors

Jill Adler School of Education, University of the Witwatersrand, 2050 Johannesburg, South Africa; King’s College London, London, UK, [email protected] Deborah Loewenberg Ball School of Education, University of Michigan, Ann Arbor, MI 48109-1259, USA, [email protected] Bill Barton Department of Mathematics, University of Auckland, Auckland 1142, New Zealand, [email protected] Paul Cobb Peabody College, Vanderbilt University, Nashville, TN, USA, [email protected] Chrystal Dean Department of Curriculum and Instruction, Appalachian State University, Boone, NC, USA, [email protected] Paul Drijvers Freudenthal Institute, Utrecht University, PO Box 85170, 3508 AD Utrecht, The Netherlands, [email protected] Dominique Forest IUFM de Bretagne, Université de Bretagne Occidentale, Rennes Cedex, France, [email protected] Ghislaine Gueudet CREAD, Université de Bretagne Occidentale, IUFM Bretagne site de Rennes, 35043 Rennes Cedex, France, [email protected] Barbara Jaworski Mathematics Education Centre, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK, [email protected] Carolyn Kieran Département de Mathématiques, Université du Québec à Montréal, Montréal, QC, Canada H3C 3P8, [email protected] Maria-Dolores Lozano Instituto Tecnológico Autónomo de México, CP 1000, México City, Mexico, [email protected] Mirko Maracci Department of Mathematics, University of Pavia, Pavia, Italy, [email protected]

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Maria Alessandra Mariotti Department of Mathematics and Computer Science, University of Siena, Siena, Italy, [email protected] Alain Mercier ADEF: Université de Provence, ENS-Lyon IFE, 32 rue Eugène Cas, 13004 Marseille, France, [email protected] Birgit Pepin Faculty of Teacher and Interpreter Education, Sør-Trøndelag University College, 7004 Trondheim, Norway, [email protected] Christine Proust Laboratoire SPHERE (CNRS & University Paris-Diderot), Paris, France, [email protected] Luis Radford École des sciences de l’éducation, Laurentian University, Sudbury, ON, Canada P3E 2C6, [email protected] Janine T. Remillard University of Pennsylvania, Philadelphia, PA 19146, USA, [email protected] Sebastian Rezat Institut für Didaktik der Mathematik, Justus-Liebig-University Giessen, 35394 Giessen, Germany, [email protected] Kenneth Ruthven Faculty of Education, University of Cambridge, Cambridge CB2 8PQ, UK, [email protected] William H. Schmidt Michigan State University, East Lansing, MI 48824, USA, [email protected] Gérard Sensevy Brittany Institute of Education, University of Western Brittany, France, [email protected] Armando Solares Universidad Pedagógica Nacional, México City, México, [email protected] Malcolm Swan Centre for Research in Mathematics Education, School of Education, University of Nottingham, Nottingham NG8 1BB, UK, [email protected] Denis Tanguay Université du Québec à Montréal, Montréal, QC, Canada, [email protected] Maria Trigueros Instituto Tecnológico Autónomo de México, CP 1000 México City, Mexico, [email protected] Luc Trouche Institut français de l’Education, École Normale Supérieure de Lyon, 15 parvis René-Descartes, BP 7000, 69342 Lyon cedex 07, France, [email protected] Jana Visnovska School of Education, The University of Queensland, St Lucia, QLD 4072, Australia, [email protected] Carl Winsløw Department of Science Education, University of Copenhagen, 1350 København K, Denmark, [email protected]

Part I

Teacher Resources

Chapter 1

Knowledge Resources in and for School Mathematics Teaching Jill Adler

1.1 Introduction This book, and the range of chapters within it, take as its starting point the role of curriculum resources in mathematics teaching and its evolution. Teachers draw on a wide range of resources as they do their work, using and adapting these in various ways for the purposes of teaching and learning. At the same time, this documentation work (as it is referred to by Gueudet and Trouche, Chapter 2) acts back on the teacher and his or her professional knowledge. Documentation work is a function of the characteristics of the material resources, teaching activity, the teachers’ knowledge and beliefs, and the curriculum context. The chapters that follow explore and elaborate this complexity. An underlying assumption across chapters is an increasing range of textual resources for teaching and wide availability of digital resources. The empirical work that informs this chapter took place in mathematics classrooms with limited textual and digital resources, and it is this kind of context that gave rise to a broad conceptualisation of resources in mathematics teaching that included the teacher and her professional knowledge, together with material and cultural resources, like language and time. In Adler (2000) I describe this broad conceptualisation, theorising material and cultural resources in use in practice in mathematics teaching in South Africa. The discourse used is of a teacher ‘re-sourcing’ her practice – a discourse with strong resonances in documentation work. This chapter builds on that work, foregrounding and conceptualising professional knowledge as a resource in school mathematics teaching. I begin by locating our concern with knowledge resources, a discussion that leads on to the methodology

J. Adler (B) School of Education, University of the Witwatersrand, 2050 Johannesburg, South Africa King’s College London, London, UK e-mail: [email protected]

G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_1, C Springer Science+Business Media B.V. 2012

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we have developed in the QUANTUM1 research project to adequately describe their use in mathematics teaching. This current research has as its major question, what and how mathematics comes to be constituted in pedagogic practice? Professional knowledge in use in practice, and how this shapes what is made available for learning, come into focus. The methodology we have developed is then illustrated through recent empirical work in two secondary mathematics classrooms in South Africa. These illustrations add force to the argument for foregrounding knowledges in use in descriptions of classroom practice and teachers’ interactions with resources. Moreover, while the methodological tools offered here emerge in response to a particular context, related data and theoretical gaze, they are, I propose, useful for studying the evolution of knowledge resources in use in teaching across contexts.

1.2 Locating the Study of Knowledge Resources QUANTUM has its research roots in a study of teachers’ ‘take-up’ from an upgrading in-service teacher education programme in mathematics, science and English language teaching in South Africa (Adler & Reed, 2002). By ‘take-up’ we mean what and how teachers appropriated various aspects of the programme, using these in and for their teaching. The notion of ‘take-up’ enabled us to describe the diverse and unexpected ways teachers in the programme engaged with selections from the courses offered and how these selections were recontextualised in their own teaching. We were able to describe teachers’ agency in their selections and use, and illuminate potential effects. Amongst other aspects of teaching, we were interested in resources in use. We problematised these specifically in school mathematics practice (Adler, 2000), where I argued for a broader notion of resources in use that includes additional human resources like teachers’ professional knowledge (as opposed to their mere formal qualifications), additional material resources like geoboards which have been specifically made for school mathematics, everyday resources like money as well as social and cultural resources like language, collegiality and time. I also argued for the verbalisation of resource as ‘re-source’. In line with ‘take-up’, I posited that this discursive move shifts attention off resources per se and refocuses it on teachers working with resources, on teachers re-sourcing their practice. In focus were selected material (e.g. chalkboards) and cultural resources (language, time). Theoretical resources were drawn from social practice theory, leading to an elaborated categorisation of resources, supported by examples of their use in practice in terms of their ‘transparency’ (Lave & Wenger, 1991). These combined

1 QUANTUM is a Research and Development project on mathematical education for teachers in South Africa. Its development arm focused on qualifications for teachers underqualified in mathematics (hence the name) and completed its tasks in 2003. QUANTUM continues as a collaborative research project.

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to illustrate that what matters for teaching and learning is not simply what resources are available and what teachers recruit, but more significantly how various resources can and need to be both visible (seen/available and so possible to use) and invisible (seen through to the mathematical object intended in a particular material or verbal representation), if their use is to enable access to mathematics. Out of focus in this work were human resources: teachers themselves, their professional knowledge base and knowledges in use. The teachers in our study were studying courses in mathematics and mathematics education. We were thus interested in their ‘take-up’ from these courses. However, we had difficulty ‘grasping’2 teachers’ take-up with respect to mathematical content knowledge in particular. Our analysis of interviews, together with observations in teachers’ classrooms over 3 years, suggested correlations between teachers’ articulation of the mathematical purposes of their teaching and the ways in which they made substantive use of ‘new’ material and cultural resources (language in particular). These results are in line with a range of research that has shown how curriculum materials are mediated by the teacher (e.g. Cohen, Raudenbush & Ball, 2003). Remillard (2005) describes the interaction between a teacher and the curriculum materials he or she uses as relational, and thus co-constitutive. A relational orientation to teachers and resources serves as a starting point for a number of chapters in this volume (see Chapters 5 and 7). Our analysis, in addition, pointed to unintentional deepening of inequality. The ‘new’ curriculum texts selected by teachers from their coursework and recontextualised in their classroom practice appeared most problematic when teachers’ professional knowledge base was weak. Typically, this occurred in the poorest schools (Adler, 2001). These claims are necessarily tentative. Our methodology did not enable us to probe teachers’ take-up with respect to mathematics content knowledge over time. Moreover, as we attempted to explore professional knowledge in practice in the study, we appreciated the non-trivial nature of the elaboration of the domains of mathematical knowledge, knowledge about teaching and the didactics of mathematics in the construction of teacher education – a point emphasised recently by Chevallard and Cirade (in Gueudet & Trouche, 2010). In a context where contestation over selections from knowledge domains into mathematics teacher education continues, the importance of pursuing knowledge in use in teaching through systematic study was evident. Mathematical knowledge for and in teaching, what it is and how it might be ‘grasped’ became the focus in the QUANTUM study that followed. The methodology we have developed makes visible the criteria teachers transmit for what counts as mathematics, and through these, the domains of knowledge teachers recruit to ground mathematics in their classroom practice. It is this conceptualisation that has enabled an elaboration of knowledge resources in use in mathematics teaching.

2 I use ‘grasp’ here in a technical sense to convey the message that knowledge in use in practice is not unproblematically ‘visible’, but is made so through the deployment of specific methodological tools and analytic resources.

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1.3 Conceptualising Knowledge Resources In Adler (2000), and as discussed above, I argued for a conceptualisation of ‘resource’ as both a noun and a verb, for thinking about resource as ‘the verb “re-source,” to source again or differently where “source” implies origin, that place from which a thing comes or is acquired’. Here too, ‘resource’ is both noun and verb – ‘knowledge resources’ refers to domains of knowledge – the objects, processes and practices within these – that teachers recruit as they go about the work of teaching. This conceptualisation of knowledge as resource coheres with the orientation to the notion of ‘lived resources’ that underpins this volume. While my focus is domains of knowledge (not curriculum material), I am similarly concerned with what is selected, transformed and used in practice, and what is produced as a result. Selecting from domains of knowledge and transforming these in use for teaching is simultaneously the work of teaching and its outcome, that is that which comes to be legitimated as mathematical knowledge in a particular practice. Teachers recruit (or appeal to) knowledge resources to legitimate what counts as mathematics in a school classroom context. We work with a social epistemology, and thus understand that what comes to count as mathematics in any pedagogical practice (such as in school) is a function of the inner workings of pedagogic discourse (Bernstein, 1996). In other words, mathematical knowledge is shaped by the institutions of schooling and curriculum and by the activity of teaching within these. In this sense, professional knowledge in use in practice needs to be understood as shaped by pedagogic discourse. Consequently, a methodology for ‘seeing’ knowledges in use in teaching requires a theory of pedagogic discourse. An underlying assumption in QUANTUM, following Davis (2001), is that pedagogic discourse (in both teacher education and school) proceeds through the operation of pedagogic judgement. As teachers and learners interact, criteria will be transmitted of what counts as the object of learning (e.g. what an ‘equation’ is in mathematics) and how the solving of problems related to this object is to be demonstrated (what are legitimate ways of knowing, working with and talking about equations). As teachers provide opportunities for learners to engage with the intended object, at every step they make judgements as to how to respond to learners, what to offer next and how long to pursue a particular activity. As Davis argues, all pedagogic judgements transmit criteria for what counts as mathematics. For example, in many South African classrooms, learners can be heard describing the steps in solving a linear equation as follows: to ‘solve for x’ in 3x − 7 = 5x + 11, learners say ‘We transpose or take the xs to one side and the numbers to the other side’. The teacher in this case could judge this expression as adequate, as reflecting shared procedural meaning in the classroom; alternatively, the teacher could judge the description as unclear; the language used does not refer adequately to the objects (algebraic terms) being operated on and also potentially misleading from a mathematical point of view. The teacher could then question the learner as to the specific meaning of ‘transpose’ or ‘take’ as the learner is using it, probing so as to transmit more mathematical criteria for the transformation of the

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equation, and in particular, the operation of adding additive inverses. In this latter case, through responses learners provide, and further questioning, the teacher then negates (even if only implicitly) the first description by legitimating mathematically justified steps offered. In this interaction process, the criteria transmitted are that steps for solving equations require mathematical justification. In QUANTUM we describe these moments of judgements as appeals, arguing that teachers appeal to varying domains of knowledge to legitimate what count as valid knowledge in their classrooms. What comes to count as valid is never neutral (Bernstein, 1996).3 Pedagogic discourse necessarily delocates and relocates knowledges and discourses, and recontextualisation (transformation) creates a gap wherein ideology is always at play. What teachers recruit is thus no simple reflection of what they know. An underlying assumption here is that the demands of teaching in general, and the particular demands following changes in the mathematics curriculum in South Africa, bring a range of domains of knowledge outside of mathematics into use. A range of mathematical orientations are discernable in the new South African National Curriculum, including mathematics as a disciplinary practice, thus including activity such as conjecturing, defining and proof; mathematics as relevant and practical, hence a modelling and problem-solving tool; mathematics as an established body of knowledge and skills, thus requiring mastery of conventions, skills and algorithms; and mathematics as preparation for critical democratic citizenship, and hence a use of mathematics in everyday activity (Graven, 2002; Parker, 2006). What mathematical and other knowledge resources teachers select and use, and how these are shaped in pedagogical discourse, are important to understand. In our case studies of school mathematics teaching, we are studying what and how teachers recruit mathematical and other knowledge resources in their classroom practice so as to be able to describe what comes to function as ground in their practice, how and why. Five case studies of mathematics teaching in a secondary classroom have been completed, each involving a different topic and unit of work.4 We pursued a range of questions, the first of which was, from what domains of knowledge does the teacher recruit knowledge resources in her teaching? I focus here on this question, and its elaboration in two of the five case studies, cognisant that as knowledge in use come into focus, so other resources, as well as details on other aspects of teaching, go out of focus.

3 In this chapter I do not explore the ideological or political in the constitution of mathematics in and for teaching. We have done this elsewhere, particularly in our reporting of the constitution of mathematics for teaching in teacher education (see Adler & Davis, 2011). 4 Studies in school classrooms have been undertaken by master’s students and a postdoctoral fellow at the University of the Witwatersrand, working in QUANTUM. I acknowledge here the significant contribution of Mercy Kazima, Vasen Pillay, Talasi Tatolo, Shiela Naidoo and Sharon Govender and their studies to the overall work in QUANTUM, and specifically to this chapter.

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1.4 Evaluative Events, Criteria at Work and Knowledge Resources in Use As is described in more detail elsewhere (Adler, 2009; Adler & Davis, 2006; Davis, Adler, & Parker, 2007), our methodology is inspired by the theory of pedagogic discourse developed by Basil Bernstein, and its illumination of the ‘inner logic of pedagogic discourse and its practices’ (Bernstein, 1996, p. 18). Any pedagogic practice, either implicitly or explicitly, ‘transmits criteria’; indeed this is its major purpose. What is constituted as mathematics in any practice will be reflected through evaluation, through what and how criteria come to work.5 How then are these criteria to be ‘seen’? The general methodology draws from Davis (2005) and the proposition that in pedagogic practice, in order for something to be learned, to become ‘known’, it has to be announced in some form. Initial orientation to the object, then, is through some (re)presented form. Pedagogic interaction then produces a field of possibilities for the object. Through related judgements made on what is and is not the object, possibilities (potential meanings) are generated (or not) for/with learners. All judgement, hence all evaluation, necessarily appeals to some or other locus of legitimation to ground itself, even if only implicitly. An examination of what is appealed to and how appeals are made (i.e. how ground is functioning) delivers up insights into knowledge resources in use in a particular pedagogic practice.6 Following the linear equation example above, if the teacher probes for or indeed inserts the notion of additive inverses, then he or she is appealing to mathematical discourse and recruiting resources from the mathematical domain. If, however, the teacher proceeds with everyday terms such as move, take over or transpose, then the grounds functioning are non-mathematical. Where appeals to the everyday dominate, and the sensible comes to overshadow the intelligible, potential mathematical meanings for learners might well be constrained (see Davis et al., 2007). Of course, what teachers appeal to is an empirical question. Our analysis to date has revealed four broad domains of knowledge to which the teachers across all cases appealed (though in different ways and with different emphases) in their work: mathematical knowledge, everyday knowledge, professional knowledge7 and curriculum knowledge. Teachers, in interaction with learners, appealed to the domain of mathematics itself, and more particularly school mathematics. We have described, a posteriori, four categories of such mathematical knowledge and/or activity that, in turn, are resonant of the multiple mathematical orientations in the current South African curriculum as discussed above: 5 It is important to note this specific use of ‘evaluation’ in Bernstein’s work. It does not refer to assessment nor to an everyday use of judgement. Rather it is a concept for capturing the workings of criteria for legitimation of knowledge and knowing in pedagogical practice. 6 This set of propositions is elaborated in Davis et al. (2003), as these emerged through collaborative work in QUANTUM. 7 In Adler (2009), everyday knowledge and professional knowledge are collapsed, both viewed as knowledge from practical experience. The separation comes from the development of this chapter.

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• mathematical objects have properties, mathematical activity follows conventions (e.g. in an ordered pair, we write the x co-ordinate first); • mathematical knowledge includes knowledge of (justifiable) procedures, mathematical activity is following rehearsed procedures (e.g. the first step to add two proper fractions is finding a common denominator); • mathematical justification can be empirical (e.g. testing whether a mathematical statement is true by examining an instance – substituting particular numbers or generating a particular visual display); • mathematical argument or justification involves generalising and proving (e.g. examining whether a statement is always true). The second domain of knowledge to which teachers appealed was nonmathematical and is most aptly described as everyday knowledge and/or practice. Across the data, teachers appealed to sensible, that is practical or experiential, knowledge to legitimate or ground the object being attended to.8 For example, the likelihood of events was discussed in relation to the state lottery, or obtaining a ‘6’ when throwing dice; simplifying algebraic expressions (e.g. 2x + 3y − 3x + 2y) was exemplified by grouping similar material objects (two apples, three bananas, etc.); in a task that required students to cut up a fraction wall containing a whole, halves, thirds, quarters, fifths, etc., up to tenths, and then reorganise/mix the fraction pieces and make wholes from different unit fractions, some students pasted pieces that together formed more than a whole. The teacher’s explanation as to why this was inappropriate was grounded in the way bricks are cemented to form walls. Connecting, or attempting to connect, mathematical ideas to everyday knowledge and experience is a topic of considerable interest, indeed concern in mathematics education in South Africa, where the goals of application, modelling and critical citizenship in the curriculum have produced a prevalence of such discourse in many classrooms. What is critical, of course, is that whatever is recruited extra-mathematically needs to connect with learners’ meaning-making while simultaneously holding the integrity of the intended mathematical idea. A third domain is teachers’ own professional knowledge and experience: what they have learned in and from practice. For example, all five teachers called on their knowledge from practice of the kinds of errors learners make and built on these in their teaching. Knowing about student thinking and misconceptions is a central part of what Shulman (1986) termed pedagogic content knowledge (PCK), and its centrality in teachers’ practice is well described in Margolinas (in Gueudet & Trouche, 2010). There are two inter-related sources for practice-based knowledge: the teacher’s own personal experience and the accumulated knowledge from research in mathematics education, that is from research on practice beyond the individual teacher. In this chapter I refer only to the former, which we have called experiential knowledge.

8 In our description of ground, we are not concerned with their mathematical correctness or whether they are appropriate. Our task is to describe what teachers recruit, whatever this is.

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Teachers’ appeals extended beyond the three domains discussed above to include what we still rather loosely call curriculum knowledge. In all our cases, and in some cases this was a significant resource for the teacher, the teacher appealed to the official curriculum, recontexualised in, for example, a textbook or an examination question. In other words, what counted as legitimate was based on exemplification or description in a textbook or what would count for marks in an examination (e.g. the definition of a polygon is that which is found in the textbook; the justification for why it is important to label axes and points on a graph is that these attract marks in an examination). Of interest is whether and how this legitimation is integrated with or isolated from any mathematical rationale. In the remainder of this chapter, I present two of the five cases to illustrate our methodology and to illuminate the knowledge resources in use in mathematics teaching.

1.5 Knowledge Resources in Use in School Mathematics Teaching The five case studies noted above have been described in detail elsewhere (Adler & Pillay, 2007; Kazima, Pillay, & Adler, 2008). The two selected for discussion here are telling: they present different approaches to learning and teaching mathematics, together with similar and different knowledge resources in use. In so doing, and akin to material resources, they problematise notions of professional knowledge that are divorced from practice and context, opening up questions for mathematics teacher education.

1.5.1 Case 1. Procedural Mathematics, Justified Empirically, Sensibly and Officially9 Nash,10 is an experienced and qualified mathematics teacher. He teaches across Grades 8–12 in a public school where learners come from a range of socioeconomic backgrounds. He has access to and uses curriculum documents issued by the National Department of Education (DoE), a selection of mathematics textbooks, a chalkboard and an overhead projector. He collaborates with other mathematics teachers in the school, particularly for planning teaching and assessment. He is well respected and regarded as a successful teacher in his school and in the district. In this case study, Nash was observed teaching linear functions to a Grade 10 class. His approach to teaching can be typically described: he gave explanations from the chalkboard; learners were then required to complete an exercise sheet he

9

For a detailed account of this study, see Pillay (2006) and Adler and Pillay (2007). This is a pseudonym.

10

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prepared. He did not use a textbook nor did he refer his learners to any textbook during the lessons observed. A six-page handout containing notes (e.g. parallel lines have equal gradients), methods (steps to follow in solving a problem) and questions (resembling that of a typical textbook) formed the support materials used. This handout was developed by Nash in collaboration with his Grade 10 teaching colleagues.11 In the eight lessons observed, Nash dealt with the notion of dependent and independent variables, the gradient and y-intercept method for sketching a line, the dual intercept method, parallel and perpendicular lines, determining equations of straight lines when information about the line is given in words and also in the form of a graph and solving simultaneous linear equations graphically. He completed the unit with a class test. The overall pass rate was 94%, class average was 65% and 34% obtained over 80%. Of course, success is relative to the nature of the test and the pedagogy of which it forms part. The test questions were a replica of questions in the handout given to learners and so a reproduction of what had been dealt with in class. In the first two lessons, Nash dealt with drawing the graph of a linear equation first from a table of values, and then using the gradient and y-intercept method. In Lesson 3, he moved on to demonstrate how to draw the graph of the function 3x – 2y = 6, using the dual intercept method. The extract below is from the discussion that followed. It illustrates an evaluative event, the operation of pedagogic judgement in this practice and the kinds of knowledge resources Nash recruited to ground, and as grounds for, the dual-intercept method for graphing a linear function. The beginning of the event – the (re)presentation of the equation 3x − 2y = 6 – is not included here. Extract 1 picks up from where Nash is demonstrating what to do. The appeals – moments of judgement – are underlined, and related grounds described. Judgments in this extract emerge in the interactions between Nash and four learners who ask questions of clarification, thus requiring Nash to recruit resources to ground and legitimate what counts as mathematical activity and so mathematical knowledge in this class. Learners’ questions were of clarification on what to do, suggesting they too were working with procedural grounds. There were possibilities for mathematical justification and engagement, for example why only two points are needed to draw the graph and how the direction of the graph is determined. However, these are not taken up and the grounds offered remain empirical – in what can be ‘seen’. Here the dual-intercept method is the simplest because it is accurate. It avoids errors that come with changing the equation into ‘standard form, that is y = mx + c. To ‘do’ the dual-intercept method, you use the intercepts on the axes, that is when x = 0 and when y = 0. You need only these two points. They determine the shape of the graph.

11

This documentation practice, unfortunately in the light of this book, was not in focus in our research.

You don’t need all the other parts? You don’t have to put down the other parts . . . its useless having −6 on the top there (points to the y axis) what does the −6 tell us about the graph? It doesn’t tell us much about the graph. What’s important features of this graph . . . we can work out . . . from here (points to the graph drawn) we can see what the gradient is . . . is this graph a positive or a negative?

(chorus) positive. it’s a positive gradient . . .we can see there’s our y-intercept, there’s our x-intercept (points to the points (0;−3) and (2;0) respectively)

Lr 1: Nash:

Lrs: Nash:

(in the next minute, a learner asks about labelling of points, and Nash responds with emphasis on the marks such labelling attracts in examinations

. . . first make your x equal to zero . . . that gives me my y-intercept. Then the y equal to zero gives me my x-intercept. Put down the two points . . . we only need two points to draw the graph

Nash:

Extract 1. Lesson 3, Case 1. (Lr = learner)

Mathematical conventions are official – those expected in the examination

Grounds: curriculum knowledge.

Mathematical activity is procedural and properties justified empirically

Grounds: empirical Important features of a graph are what can be ‘seen’

There is no justification for only needing two points. Nash might understand the geometry theory here, but this is simply asserted The assertion is questioned by L1, and the theory not followed. Rather, an empirical explanation is given

Grounds: procedural. Steps to follow are described, and justified mathematically.

Knowledge resources in use

12 J. Adler

Nash: Learner 4: Nash:

Lr 2:

Nash: Lr 2: Nash:

The simplest method and the most accurate . . . Compared to which one? Compared to that one (points to the calculation of the previous question where the gradient and y-intercept method was used) because here if you make an error trying to write it in y form . . . that means it now affects your graph . . . whereas here (points to the calculations he has just done on the dual intercept method) you can go and check again . . . you can substitute . . . if I substitute for 2 in there (points to the x in 3x – 2y = 6) I should end up with 0

Mathematical activity demands accuracy and is error free

Grounds: avoiding error

Mathematics is procedural

Explanation focuses on how you get the correct gradient by following the steps. Grounds: procedural

Grounds: procedural

Further procedural question

Lr 2: Lr 3:

Sir, is this the simplest method sir? How do you identify which side must it go, whether it’s the right hand side (Nash interrupts) (response to Lr 2) You just join the two dots That’s it? Yeah . . . the dots will automatically . . . if it was a positive gradient it will automatically . . . if this was (refers to the line just drawn) negative . . . that means this dot (points the x-intercept) will be on that side (points to the negative x axis) . . . because if the gradient was negative, how could it cut on that side? (points to the positive x axis). Is this the simplest method sir?


Extract 1. Lesson 3, Case 1. (Lr = learner)

1 Knowledge Resources in and for School Mathematics Teaching 13

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In this event, Nash’s responses were about what to do. Legitimation was provided by steps to follow or what could be ‘seen’. Appeals were to procedural knowledge, to some empirical feature of the object being discussed or to curriculum knowledge (what counts in the examination). This event, and the operation of pedagogic judgement, is typical of how Nash conducted his teaching of this particular set of lessons. Table 1.1 summarises the full set of 65 events across the eight lessons, and the knowledge resources Nash recruited. As indicated above and in the numbers in the table, more than one kind of knowledge resource could be called on within one event. Nash’s appeals to everyday knowledge and his professional experience were not evidenced in this event. Briefly, his recruiting of everyday knowledge, which were to add meaning for learners, was often problematic from a mathematical point of view. For example, he attempted to explain independent and dependent variables by referring to a marriage, husband and wife and expressed amusement and concern when discussing this in his postlesson interview! Table 1.1 Case 1, linear functions, grade 10 Total occurrences

% Occurred

Events

65

Appeals/knowledge resources Mathematics Empirical Procedures/conventions

24 43

37 66

18 14

28 22

6 7

9 11

Experience

Professional Everyday

Curriculum

Examinations/tests Text book

In overview, mathematical ground in this set of lessons was procedural, with justification empirical, sensible and official. Nash recruited from the domains of mathematical, professional and curriculum knowledge. That these latter are key in Nash’s practice were reflected in his post-lesson interview. Nash talked at length about how he plans his teaching, key to which is a practice he calls ‘backwards chaining’. First and foremost when you look[ing] at the topic/my preferred method is . . . backwards chaining. [which] means the end product. What type of questions do I see in the exam, how does this relate to the [Gr 12] exams, similar questions that relate to further exams and then work backwards from there . . . what leads up to completing a complicated question or solving a particular problem and then breaking it down till you come to the most elementary skills that are involved; and then you begin with these particular skills for a period of time till you come to a stage where you’re able to incorporate all these skills to solve a problem or the final goal that you had.

He also illuminated how his experience factors into his planning and teaching, and his attention to error-free mathematics. Learners’ misconceptions and errors are a teaching device – and in the context of the perspective of this book – a resource in his teaching. They are not a feature of what it means to be mathematical.

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You see in a classroom situation . . . you actually learn more from misconceptions and errors . . . than by actually doing the right thing. If you put a sum on the board and everybody gets it right, you realise after a while the sum itself doesn’t have any meaning to it, but once they make errors and you make them aware of their errors or . . . misconceptions – you realise that your lessons progress much more effectively . . . correcting these deficiencies . . . these errors and misconceptions.

1.5.2 Case 2. Mathematical Activity as Conjecture, Counterexample and Proof12 Ken13 is also an experienced and qualified mathematics teacher. He has a 4-year higher diploma in education majoring in mathematics, an honours degree in Mathematics Education and at the time of the data collection was studying for his master’s degree. He has thus had opportunity to learn from the field of mathematics education research. He has 11 years’ secondary teaching experience across Grades 8–12. The conditions in his school are similar to those in Nash’s school, and gradelevel teachers similarly prepare support materials and assessments for units of work. Ken too is well respected and successful in his school. Ken prepared and presented a week’s work focused on polygons; the relationship between its sides, vertices and diagonals; generalisation and proof to his Grade 10 class. He described his plans for the lessons as a set of ‘different’ activities to ‘revise’ and enable learners to reflect more deeply on geometry. The five lessons were organised around two complex, extended tasks. The first involved the relationship between the number of sides of a polygon and its diagonals. The second was an applied problem requiring learners to interpret a situation and recognise the need for using knowledge of equal areas of parallelograms on the same base and with the same height to solve the problem. The extract below is from the first of the five lessons and the initial work on the first task: learners were to find the number of diagonals in a 700-sided polygon, a sufficiently large number to require reasoning and generalising activity. The extract captures an evaluative event, with the presentation of the task marking the beginning of the event. It continues for 14 min as the teacher and learners interact on what and how they could make a conjecture towards the solution to the problem. Some progress is made, as learners are pushed to reflect on specific empirical cases. As with extract 1, the underlined utterances illustrate the kinds of appeals and so knowledge resources Ken recruits in his practice. All judgements towards the object – a justified account of the relationship between the number of sides and diagonals in a polygon – emerge from utterances of either or both learners and the teacher.

12 13

For detailed account of this study, see Naidoo (2008). This is a pseudonym.

Ok! Guys, time’s up. Five minutes is over. Who of you thinks they solved the problem? . . . . I just divided 700 by 2. You just divided 700 by 2. Sir, one of the side’s have, like a corner. Yes . . . (inaudible), because of the diagonals. Therefore two of the sides makes like a corner. So I just divided by two . . . (Inaudible). So you just divide the 700 by 2. And what do you base that on? So what do you base that on because there’s 700 sides. So how many corners will there be if there’s, 700 sides?

Lr2:

Ken:

Ken: Lr2:

Let’s hear somebody else opinion Sir what I’ve done sir is . . .First 700 is too many sides to draw. So if there is four sides how will I do that sir? Then I figure that the four sides must be divided by two. Four divided by two equals two diagonals. So take 700, divide by two will give you the answer. So that’s the answer . . . So you say that, there’s too many sides to draw. If I can just hear you clearly; . . . that 700 sides are too many sides, too big a polygon to draw. Let me get it clear. So you took a smaller polygon of four sides and drew the diagonals in there. So how many diagonals you get? In a four sided shape sir, I got two

[. . .] there is discussion about 700 sides and corners, whether there are 350 or 175 diagonals

Ken:

Lr 1: Ken: Lr 1:

Ken:

The class begins with Ken (standing in the front of the class), placing the following problem onto the Overhead Projector: How many diagonals are there in a 700-sided polygon? The students are asked to work on it for 5 min. After 7 min, Ken calls the class’ attention, and the interaction below follows:

Extract 2. Lesson 1, Case 2

L2 grounds his conjecture empirically, pragmatically and procedurally

L1s response is procedural. Following a challenge from the teacher, the grounds extend to include perceived properties of the mathematical object. Again this is challenged by the teacher


16 J. Adler

Yes, I don’t want to confuse myself So you don’t want to confuse yourself. So you’re happy with that solution, having tested only one polygon? Inaudible response . . . What about you Lr4? You said you agree. He makes sense. (referring to Lr1). . .He proved it. . . . He used a square. He used a square? Are you convinced by using a square that he is right? But sir, here on my page I also did the same thing. I made a six-sided shape and saw the same thing. Because a six thing has six corners and has three diagonals.

Lr2: Ken:

Interaction continues. Ken intervenes as he hears some confusion between polygon and pentagon, and turns the class’ attention to definitions of various polygons having learners look up meanings in their mathematics dictionaries

So what about a five-sided shape? Then sir

What about a five-sided shape? You think it would have five corners? How many diagonals?

Lr1:

Ken:

Lr2: Ken: Lr4: Ken: Lr5:

These grounds are again challenged by Ken

Two. So you deduced from that one example that you should divide the 700 by two as well? So you only went as far as a four sided shape? You didn’t test anything else?

Ken:

Mathematical activity involves reasoning; providing examples and counterexamples Mathematical objects have properties and are defined

Grounds functioning in this interaction remain empirical and include counterexamples

Learners first confirm with an additional example – six sides, then ask about five sides, and Ken picks up on this additional empirical case and counterexample

Learners ground responses in the empirical and sensible Challenge to the empirical ground and single case

Ken challenges the empirical ground and single case


Extract 2. Lesson 1, Case 2

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The discussion and clarification of different polygons continued for some time, after which Ken brought the focus back on to the problem of finding the number of diagonals in a 700-sided figure, and work on this continues through the rest of this lesson and the next two lessons. It is interesting to note that in all the discussion on the 700-sided figure, the empirical instances discussed, and the diagrams made public, a polygon is assumed to be regular and convex. Properties discussed focus on the number of sides and related number of angles in a polygon (again regular and convex), and a diagonal is defined as a line connecting two non-consecutive corners. One route to solving the problem – noticing a relationship between the number of corners and the number of diagonals from each corner – and so the possibility of a general formula becomes dominant. It is interesting too that the term ‘vertex’ is not used, and the everyday word ‘corner’ persists in the discussion. Ken’s focus throughout the two lessons is on conjecture, justification, counterexample and proof as mathematical processes. A shared understanding of the mathematical object itself – a polygon and its diagonals as defined geometrically – through which these processes are to be learned is assumed. Judgements in this extract flow in interaction between Learners 1, 2, 4, 5 and the teacher. The knowledge resources called in fit within the broad category of mathematics. In particular, the ground for the teacher is reflected in his insistence on mathematical justification. However, these grounds are distinctive. The first appeal (Lr1) is to the empirical, a particular case that can be ‘seen’ (two of the sides makes like a corner) and a related procedure (I just divided by 2), followed by Ken’s challenge through an appeal to properties of a 700-sided polygon. The appeal of Lr2 is also to the empirical, to a special case (four sides), and this is supported by Lr4, and then by Lr5 (who did ‘the same thing’ with six sides). It is interesting to reflect here on what possible notion of diagonal is being used by Lr5. While there has been discussion on diagonals as connecting non-consecutive corners, it is possible Lr5 is considering only those that pass through the centre of the polygon. Ken does not probe this response, rather picking up on Lr1’s suggestion of a counterexample (what about a five-sided shape?), which is also an empirical case. The appeals by the teacher, as he interacts with, revoices and responds to learner suggestions, are to the meta-mathematical domain, and so providing the criterion that the justifications provided are not yet mathematically adequate – they do not go beyond specific cases. The grounds that came to function over the five lessons are summarised in Table 1.2. In sum, a range of mathematical grounds (with empirical dominant, and including appeals to mathematics as generalising activity) overshadowed curriculum knowledge, with everyday knowledge barely present. In the pre-observation interview, Ken explained that his intention with the lessons he had planned was to focus on the understanding of proofs. He wanted them to see proof as ‘a way of doing maths, getting a deeper understanding and communicating that maths to others’. In the postlessons interview, interestingly, Ken explained that these lessons were not part of his normal teaching. He used the research project to do what he thought was important, but otherwise did not have time for. He nevertheless justified this inclusion in terms of the new curriculum, which had a strong emphasis on proof, on ‘how to prove and

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Table 1.2 Case 2, geometric thinking, grade 10 Total occurrences

% Occurred

Events

37

Appeals/knowledge resources Mathematics Empirical General Procedures/conventions

23 14 8

64 36 23

0 2

0 5

11 0

32 0

Experience

Professional Everyday

Curriculum

Examinations/tests Text book

what makes a proof’. When probed as to why he did not do this kind of lesson in his ‘normal’ teaching, he explained that there was shared preparation for each grade, and ‘because of time constraints and assessments, you follow the prep and do it, even if you don’t agree’.

1.6 The Significance of Knowledge Resources in Use in Practice In the introductory sections of this chapter, I argued that the knowledge resources teachers recruit in their practice are important. Earlier research has suggested that teachers’ professional knowledge was a significant factor in the relationship between teachers and curriculum materials, and particularly so in contexts of poverty. Where curriculum resources are minimal, the insertion of new texts critically depends on what and how teachers are able to use mathematics and other knowledge domains appropriately for their teaching. By implication, a study of curriculum text as ‘lived’ needs to foreground knowledge resources in use. This chapter has offered a methodology – structured by evaluative events and criteria in use to ground objects of learning and teaching – for illuminating knowledges in use. It contributes to the overall perspective offered in this book – a perspective that problematises the interactions between teachers and the resources drawn on in their professional activity. The methodology was put to work in two classrooms, enabling a description of the knowledge resources two teachers who were teaching different topics recruited to ground the mathematics they were teaching. Together with the mathematical domain, and particularly procedural mathematical knowledge, Nash drew on extra-mathematical domains of knowledge, particularly curriculum knowledge and everyday knowledge. Ken drew largely from the meta-mathematical domain. The knowledge resources that sourced the work of these two teachers were substantively different, and so too was the mathematics that came to be legitimated in these classrooms.

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As he explained, Nash backward chained from valued school knowledge reflected in national examinations and built in teaching strategies to elicit errors from learners that he could then correct, and he did this by focusing on procedural knowledge and what is empirically verifiable. This practice produces student ‘success’, though, in Ruthven’s terms, he could be described as following a mathematically constrained script and activity format (see Chapter 5). Ken, on the other hand, uses mathematics in extended ways to engage learners in reasoning practices like conjecturing leading to proof. What is not available here, of course, is the knowledge resources Ken might recruit if he were teaching linear functions, and similarly whether the script in Nash’s class is uniform across topics. We could surmise from Ken’s interview and his ‘confession’ that the observed lessons were done outside of his normal teaching, that grounds different to what we have seen in this episode might well function in his ‘normal’ classes. These teachers’ intentions, and what else they might do, are not at issue here. The object of QUANTUM’s research is not on what a particular teacher does or does not do, in some decontextualised sense, but rather on what comes to be used, and thus how mathematics is constituted in specific practices. Through the cases in this chapter, we see that observing teachers in practice is a window into the varying knowledge resources in use within a particular curriculum practice and set of institutional constraints. These insights were ‘revealed’ through the notion of ‘ground’ as that which is recruited to legitimate what counts as mathematics in teaching. The methodological tools developed in the QUANTUM project probe beneath surface features of pedagogic practice to reveal substantive differences in the way teachers recruit and ground knowledge objects as they go about their mathematical work, and so into how knowledges become ‘lived’ resources.

1.7 In Conclusion: Some Questions for Professional Development Activity In this chapter we have described two teachers’ practices in their mathematics classrooms. Nash and Ken teach in similarly resourced schools, and in a similar policy context. They recruited different knowledge resources, and thus different opportunities for learning mathematics were opened up in their classrooms. The methodology we have used enables us to understand and think about what might support expansion of the potential meanings these two teachers open up in their classrooms. Nash’s practice and his talk about this in his interview reveal the value he places on the high status official curriculum. This suggests possibilities for productive work and reflection with Nash on his privileging of the official curriculum, and how this shapes the ground functioning in his classroom in his teaching reported here. Ken, on the other hand, might benefit more from an investigation of the integration of meta-mathematical knowledge into his teaching more generally.14

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This challenge for teacher education is explored more directly elsewhere (see Adler, 2010) where I problematise the teaching of mathematical reasoning, and its implications for teacher education.

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In QUANTUM, our overall goal has been to ‘see’ across sites of practice (teacher education and school). We have studied pedagogic discourse and the constitution of mathematics for teaching in teacher education sites as well as the school classrooms illuminated in this chapter. For, if we are to improve mathematics teacher education, we need to understand what potential meanings are opened and closed in and across these sites, and how those emerging in teacher education relate to those emerging as dominant school practices. In the introductory section of this chapter, I asserted that the methodology described would be useful for studying the evolution of knowledge resources in use across contexts, and that this was particularly important in contexts of limited material resources. It is certainly useful in our current work where we are studying teachers’ practices over time, with an interest in whether and how professional development interventions focused on aspects of content knowledge in and for teaching relate to knowledges and other resources in use in practice. Acknowledgements This chapter emerges from the QUANTUM research project on Mathematics for Teaching, directed by Jill Adler, at the University of the Witwatersrand (Wits) with Dr Zain Davis, University of Cape Town, as co-investigator. The methodology described here was developed through joint work in mathematics teacher education. The elaboration into classroom teaching was enabled by the work of master’s students at Wits. This material is based upon work supported by the National Research Foundation (NRF), Grant number FA2006031800003. Any opinion, findings, conclusions or recommendations expressed here are those of the author and do not necessarily reflect the views of the NRF.

References Adler, J. (2000). Conceptualising resources as a theme for teacher education. Journal of Mathematics Teacher Education, 3, 205–224. Adler, J. (2001). Re-sourcing practice and equity: A dual challenge for mathematics education. In B. Atweh, H. Forgasz, & B. Nebres (Eds.), Sociocultural research in mathematics education: An international perspective (pp. 185–200). Mahwah, NJ: Lawrence Erlbaum Associates. Adler, J. (2009). A methodology for studying mathematics for teaching. Researchers en Didactique des Mathematiques, 29(1), 33–57. Adler, J. (2010). Mathematics for teaching matters. Education as Change, 14(2), 123–135. Adler, J., & Davis, Z. (2006). Opening another black box: Researching mathematics for teaching in mathematics teacher education. Journal for Research in Mathematics Education, 37, 270–296. Adler, J., & Davis, Z. (2011). Modelling teaching in mathematics teacher education and the constitution of mathematics for teaching. In K. Ruthven, & T. Rowland (Eds.), Mathematical knowledge in teaching (pp. 139–160). Dordrecht: Springer. Adler, J., & Pillay, V. (2007). An investigation into mathematics for teaching: Insights from a case. African Journal of Research in SMT Education, 11(2), 87–108. Adler, J., & Reed, Y. (Eds.). (2002). Challenges of teacher development: An investigation of takeup in South Africa. Pretoria: Van Schaik. Bernstein, B. (1996). Pedagogy, symbolic control and identity: Theory, research and critique. London: Taylor and Francis. Cohen, D. K., Raudenbush, S. W., & Ball, D. L. (2003). Resources, instruction, and research. Educational Evaluation and Policy Analysis, 25, 119–142. Davis, Z. (2001). Measure for measure: Evaluative judgement in school mathematics pedagogic texts. Pythagoras, 56, 2–11.

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Davis, Z. (2005). Pleasure and pedagogic discourse in school mathematics: A case study of a problem-centred pedagogic modality, Unpublished PhD dissertation, University of Cape Town, Cape Town. Davis, Z., Adler, J., Parker, D., & Long, C. (2003). Elements of the language of description for the production of data. QUANTUM Research Project, Working paper #2, Johannesburg: University of the Witwatersrand. Davis, Z., Adler, J., & Parker, D. (2007). Identification with images of the teacher and teaching in formalized in-service mathematics teacher education and the constitution of mathematics for teaching. Journal of Education, 42, 33–60. Graven, M. (2002). Coping with new mathematics teacher roles in a contradictory context of curriculum change. Mathematics Educator, 12(2), 21–27. Gueudet, G., & Trouche, L. (Eds.). (2010). Ressources vives. Le travail documentaire des professeurs en mathématiques. Lyon: Presses Universitaires de Rennes & INRP. Kazima, M., Pillay, V., & Adler, J. (2008). Mathematics for teaching: Observations from two case studies. South African Journal of Education, 28(2), 283–299. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press. Naidoo, S. (2008). Mathematical knowledge for teaching geometry to Grade 10 learners. Unpublished Masters Research Report. Johannesburg: University of the Witwatersrand. Parker, D. (2006). Grade 10–12 mathematics curriculum reform in South Africa: A textual analysis of new national curriculum Statements. African Journal of Research in SMT Education, 10(2), 59–73. Pillay, V. (2006). An investigation into mathematics for teaching: The kind of mathematical problem-solving teachers do as they go about their work. Unpublished Masters Research Report. Johannesburg: University of the Witwatersrand. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.

Chapter 2

Teachers’ Work with Resources: Documentational Geneses and Professional Geneses Ghislaine Gueudet and Luc Trouche

Chapters 2 and 16 constitute two connected components of the presentation of a theoretical approach focusing on phenomena central in this book: the interactions between mathematics teachers and resources, and their consequences for professional growth. We name it documentational approach of didactics (Gueudet & Trouche, 2009). We begin (Section 2.1) with a discussion of the elementary concepts of this approach: documentation work, teachers’ documentation, resource/document dialectics and documentational genesis. We then elaborate (Section 2.2) the methodology we use for studying teachers’ documentation and the data we have collected using this methodology over 2 years. In Section 2.3, we detail a case study, extracted from these data. Finally, we present the perspective on teachers’ professional growth yielded by this approach (Section 2.4). We pay particular attention to digital resources, the constituting factors of major evolutions.

2.1 The Documentational Approach of Didactics We begin our discussion of the elementary concepts of the approach we develop by explaining its theoretical roots.

2.1.1 Teachers’ Professional Activity and Professional Growth The approach we present here has a specific orientation to studying mathematics teachers’ activity and development. Informed by activity theory, firstly introduced by Vygotski (1978) and developed by Leont’ev (1979), we consider that the teacher’s activity is oriented by goals (the object of the activity). Moreover, it

G. Gueudet (B) CREAD, Université de Bretagne Occidentale, IUFM Bretagne site de Rennes, 35043 Rennes Cedex, France e-mail: [email protected]


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must be studied as a social activity, which leads us to pay attention to its context: institution and different social groups. We are interested in the evolution, and factors of evolution, of the teachers’ professional activity. The studies about practising mathematics teachers sometimes separate their practice, their knowledge and their beliefs (Perrin-Glorian, DeBlois, & Robert, 2008). We consider here teachers’ professional growth as a joint evolution of these three aspects. Conceptualising the way the practice articulates with knowledge and beliefs is one of the aims of the theory we expose here. We do not separate knowledge and beliefs, because the boundary between both is often unclear; we use the expression of professional knowledge to refer to both and focus particularly on knowledge related to mathematical content. The reference to activity theory is also directly connected with our interest in mediation and mediating artefacts. However, we refer to resources rather than artefacts and discuss the reasons for this choice in the next section.

2.1.2 Resources and Documentation Work Adler (2000, Chapter 1) proposes a conceptualisation of resources, emphasising the variety and the broadness of the range of resources intervening in teachers’ professional activity. We retain here a similar conceptualisation and perspective: ‘It is possible to think about resource as the verb re-source, to source again or differently’ (Adler, 2000, p. 205). With this perspective, a resource can be an artefact, i.e. an outcome of human activity, elaborated for a human activity with a precise aim (Rabardel, 1995, Chapter 4). But resources exceed artefacts: For a teacher who draws on them in her activity, the reaction of a student, a wooden stick on the floor can also constitute resources. The teacher interacts with resources, selects them and works on them (adapting, revising, reorganising, etc.) within processes where design and enacting are intertwined. The expression documentation work encompasses all these interactions. We consider that documentation work is central in teachers’ professional activity. It pertains to all the facets of this activity: all the places, all the groups teachers are involved in. We also use the word documentation, which means, for us, both this work and its outcomes. Retaining a wide perspective on resources does not mean ignoring the specificities of different kinds of resources. The work presented here originates in an interest in digital resources and their consequences for the teaching of mathematics. Teachers download lesson plans and exercises texts on websites; they modify, combine several files, elaborate their own texts, share them, etc. Digital resources evidence the documentation work. More generally, material resources have a particular status, at least for the researcher, from a methodological point of view. The interaction between teachers and material resources can indeed be visible in some cases: written notes on a book, an answer to an email, modifications in a file, etc. Non-material resources are more difficult to capture; some of them are nevertheless determining, like interactions in class with students. Several chapters in this book emphasise the importance of these interactions: chapters about the use of

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resources (Chapter 11 evidences the importance of non-verbal interactions) as well as chapters considering the joint action of teachers and students (Chapters 3 and 11), the teacher–student interactions in the use of textbooks in particular (Chapter 12). These interactions constitute a specific kind of resource, in particular because they are likely to modify other resources, or the relation between the teacher and a given resource (an exercise text can be modified, because of a student’s reaction, for example). They are both resources and a part of the social dimension of the teachers’ professional activity.

2.1.3 The Resource/Document Dialectics and the Documentational Geneses The documentational approach draws on the instrumental approach, developed by Rabardel (1995) in cognitive ergonomics and then integrated into mathematics didactics (Guin, Ruthven, & Trouche, 2005). Rabardel distinguishes between an artefact, available for a given user, and an instrument, which is developed by the user, starting from this artefact, in the course of his/her situated action. These development processes, the instrumental geneses, are grounded, for a given subject, in the appropriation and the transformation of the artefact, to solve a given problem, through a variety of usage contexts. Through this variety of contexts, utilisation schemes of the artefact are constituted. A scheme (Vergnaud, 1998) is an invariant organisation of the targeted activity, which is structured by operational invariants, developed in various contexts met for the same class of situations. This approach also distinguishes, within the instrumental geneses, two intertwined processes: instrumentation (constitution of the schemes of utilisation of the artefacts) and instrumentalisation (by which the subject shapes the artefacts). The instrumental approach has mostly been used to study the consequences of technology-rich environment for the student learning, despite a growing interest for teachers in the educational research about technology in mathematics (Hoyles & Lagrange, 2010). We propose here a theoretical approach extending the scope of the instrumental approach. Moreover, it borrows from other research studies, about document management (Pédauque, 2006), which enlightens the evolutions brought by digital resources, and about curriculum material (Remillard, 2005, Chapter 6). The teacher, in her documentation work, for a given class of situations, draws on a set of resources of various nature. Introducing a new vocabulary, we consider that this set of resources bears, for this class of situations, a document, within a documentational genesis (Fig. 2.1). The documentational genesis jointly develops a new resource (made up of a set of resources selected, modified and recombined) and a scheme of utilisation of this resource. We can represent this process, in a very simplified way, by the following equation: Document = Resources + Scheme of utilisation The document can be considered as an outcome of the subject’s activity. This static equation must nevertheless not obscure the fact that geneses are dynamic;

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Fig. 2.1 Schematic representation of a documentational genesis

they are ongoing processes: a document comprises resources, which can be associated with others and involved in the development of other documents. A scheme of utilisation is an invariant organisation of the activity to achieve a type of task; however, it can evolve in the course of the documentation work. It can be adapted to take into account new features of the context; several schemes can be associated, etc. We illustrate our model with a first short example, drawing on a previous study. Sarah has taught mathematics from grade 6 to grade 9, in France, for 10 years. An important objective assigned by the official curriculum is to introduce students to rigorous proofs in the context of geometry. For the class of situations ‘designing and setting up the introduction to proof in geometry’, Sarah selects exercises in the textbook where the figures are coded (equality of lengths and right angles). She uses dynamic geometry software and with it elaborates coded figures. Her students write in their workbook ‘a property of a figure cannot be claimed from mere observation, if there is no coding symbolising this property’. She declares, in an interview, that her long experience in grade 9 classes has led her to pay attention to the difficulties raised by proof in geometry, especially difficulties linked with the use of figures. In this case, we consider that the teacher, in the course of her work and over several years, developed a document, comprising recombined resources: extract of the textbook, dynamic geometry software, etc. This document also entails a scheme of utilisation of these resources, with operational invariants like ‘the proof of a result in geometry must be associated with a coded figure’ and ‘a coded figure helps to identify the relevant properties for the proof’. We share with other authors in this book [in particular Adler (Chapter 1), Remillard (Chapter 6), and Pepin (Chapter 7)] a perspective considering that teachers ‘learn’ when choosing, transforming resources, implementing them, revising them, etc. The documentational approach proposes a specific conceptualisation of this learning, in terms of genesis. Documentation being present in all aspects of the teacher’s work, it yields a perspective on teachers’ professional growth as a complex set of documentational geneses. Understanding this growth requires a holistic view on these geneses, by considering all the documents developed by the teacher: her documentation system.

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2.1.4 Resource System and Documentation System Studying teachers’ documentational geneses evidences articulations between different documents. Naturally, the class of situations ‘designing and setting up the assessment corresponding to the cosine lesson’ is connected with ‘designing and setting up the introduction of cosine’: the objective of the teacher’s activity is different, but the mathematical content being the same, the same operational invariants are likely to intervene in the documents developed for each class. However, the mathematical dimension is not the only element accounting for articulation between documents. ‘Designing and setting up an assessment’ can also be considered as a class of situations, corresponding to a general aim of the teacher’s activity, connected with ‘managing the class and following the students’, another general aim. Within the framework of the instrumental approach, Rabardel and Bourmaud (2003) consider systems of instruments, whose structure depends on the structure of the subject’s professional activity. The classes of situations are articulated and organised, because the various aims can be more or less similar or linked. Drawing on this conceptualisation, we consider that the documents of a teacher are articulated in a structured documentation system. The resource system of the teacher constitutes the ‘resource’ part of her documentation system (i.e. without the scheme part of the documents). Ruthven (Chapter 5) also introduces a concept of resource system, belonging to the five key structuring features of classroom practice he identifies. What we consider here as resource system does not fully coincide with Ruthven’s definition, because of the broader meaning of resources we retained. The resource system comprises material elements, but also other elements that are more difficult to collect, like conversations between teachers. We have presented here the theoretical construct framing our research. The complex object we study also requires a specific methodology, connected with this theory.

2.2 Studying Documentation Work: Reflective Investigation We elaborated the theory and an associated methodology simultaneously in the development process of the documentational approach. We briefly present here this methodology that we named the reflective investigation of the teacher’s documentation work.

2.2.1 Methodological Principles The main principles grounding this methodology are as follows: – A principle of long-term follow-up. Geneses are ongoing processes and schemes develop over long periods of time. This indicates the need for long-term observation, within practical constraints.

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– A principle of in- and out-of-class follow-up. The classroom is an important place where the teaching elaborated is implemented. As mentioned above, these direct interactions with students are crucial resources for the teacher. They bring adaptations, revisions and improvisations, as Drijvers (Chapter 14) emphasises, distinguishing between an exploitation mode of a didactical configuration, planned by the teacher, and the didactical performance she realises in class. However, an important part of teachers’ work takes place beyond the students’ presence – at school, at home, in teacher development programs, etc. We pay attention to all these different locations. – A principle of broad collection of the material resources used and produced in the documentation work, throughout the follow-up. – A principle of reflective follow-up of the documentation work. We closely involve the teacher in the collection of data, with the pragmatic aim of broad collection and in-class and out-of-class follow-up previously discussed. The active involvement of the teacher yields a reflective stance (Schön, 1983). We built a data collection device, presented in the following section, corresponding to these principles.

2.2.2 Data Collection Tools The data collection we propose is planned to last several years; a teacher is followed at least 3 weeks each year. We detail here the schedule and the tools used. Figure 2.2 presents the overall agenda of the yearly follow-up. During the first year, the teacher fills in a logbook over at least 3 weeks, describing her activity relative to one of the classes she teaches. The researcher visits the teacher three times at home for interviews and collection of resources. He/she asks (during the first interview) the teacher to draw a schematic representation of the structure of the resources she uses. We call it a schematic representation of the resource system (SRRS). An example of an SRRS is displayed in Fig. 2.4. During the following years, the teacher is still followed in a class of the same level, for the same mathematical content. The overall structure remains the same (Fig. 2.2), but the focus is much more on developments: the teacher is asked to bring the necessary modifications, to explain the changes, compared to the previous year about the questionnaire, the SRRS and during the first interview. We focus in this book on two teachers whom we followed for 2 years: 2008–2009 and 2009–2010.

2.2.3 Choice of Two Teachers The two teachers we followed teach in middle school (from grade 6 to 9). They have been selected with very different profiles, according to several dimensions that we assume as crucial for the topic of our study: they differ in terms of ICT

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First encounter:

First visit:

Second visit:

- presentation of the methodology, its spirit and its tools.

- about the resources in general;

- about the lesson observed.

Classroom observation

Third visit: - About the lesson observed; - Complements about the chapter and resources.

- about the chapter followed.

Tools:

Tools:

Tools:

Tools:

Tools:

- Schedule; - Questionnaire;

- Interview guidelines;

- Interview guidelines

- Observation guidelines

- Interview guidelines;

- Logbook.

- SRRS

- SRRS; - Collection of resources.

Week 1

Week 2

Week 3

Filling the logbook

Fig. 2.2 Agenda of the follow-up

degree of integration (Assude, 2007), of participation in collectives and of institutional responsibilities (Gueudet & Trouche, 2009). These teachers were neither at the beginning nor at the end of their career. Myriam (50 years in 2009) has a strong degree of ICT integration; she regularly takes part in in-service training; she took part in several IREM1 groups. Pierre (35 years in 2009) has a strong degree of ICT integration; he is responsible for ICT in his school and member of Sésamath2 association. This association gathers mathematics teachers; we describe it in more detail in Chapter 16. We mention here only several of the resources designed by the association which are accessible on its website, in particular the Sésamath textbook (which exists in digital and paper versions). As explained above, we followed each teacher in only one class. Moreover, we chose that class to capture phases of reorganisation of the documentation work. We thus followed Pierre in one grade 6 class with a data-processing speciality which brings an opening towards new forms of work (better equipped students, motivated 1 2

IREM, Institute for Research on Mathematics Teaching. http://www.sesamath.net/

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for out-of-class interactions). We followed Myriam in grade 9 on a chapter (functions) corresponding to a change in the official curriculum. In 2008–2009, it was the first time that she taught that topic. Myriam and Pierre are not chosen as ‘exemplar’ teachers. We do not aim at describing ‘good’ documentation work. Myriam and Pierre share a strong professional involvement; they spend probably more time on their out-of-class work than do the average mathematics teacher in France. This longer time is likely to evidence better the phenomena that we want to capture; however, we consider that these phenomena take place for every teacher. In this chapter, we detail the case of Myriam; the case of Pierre is studied in Chapter 16.

2.2.4 Analysing the Data Collected This data collection is followed by a data exploitation device, which comprises various aspects. We carry out a quantitative treatment of the logbook: length of the out-of-class and in-class work, places for this work, number of occurrences of a given activity, length, number of uses of a given resource, length, nature and number of collective work moments, list of implied participants, etc. For the interviews, we note in the same way the occurrences of the types of activity, resources and persons mentioned. The questionnaire provides us with concrete information about the teacher’s career and her current working environment. We also gain access, through the questionnaire, to elements of her professional and personal history, in particular in terms of family environment and collective involvements. We complement these first treatments with the SRRS to evidence elements of structure of the teacher’s activity and of her resources, systematically identifying moreover collective dimensions (Chapter 16). We identify, in the logbook and the interviews, all the elements relating to the lesson observed in class. We observe in the lesson’s transcript how the interactions between teacher, students and knowledge lead to adaptations of resources, during or after the lesson. We conduct a systematic comparison of the first-year and second-year data, quantitatively and qualitatively. In the next section, we present a case where we applied this data collection and analysis device.

2.3 A Case Study In this section we study the case of Myriam and of her teaching about functions in grade 9 to illustrate the concepts presented in Section 2.1.

2.3.1 Synthetic Description of Myriam’s Activity The synthetic description of Myriam’s activity during the follow-up (Fig. 2.3) corresponds to the year 2009–2010.

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Myriam introducing functions in grade 9 Myriam has one grade 9 class, with 20 students. The theme of functions was introduced into the grade 9 curriculum in 2008–2009. The official curriculum is divided between ‘core content of knowledge’, which every student should learn, and other contents. Functions do not belong to the core content. The students must obtain a diploma, ‘diplôme de brevet des collèges’, at the end of the year. This diploma comprises a computer certification ‘brevet informatique et internet’ (shortened as B2i*); Myriam is responsible for ensuring some of the corresponding skills.

Preparing the lesson about functions For the preparation of her lesson, Myriam uses several websites: Sésamath, and institutional websites. She also uses the classroom textbook and her personal notebook from the previous year. She finds on an institutional website an activity ‘the box’, where the students are asked to compute (with their calculator and then with a spreadsheet) the volume of a rectangular box for several values of the side x of squares, withdrawn on each corner to build the box.

She retains this activity and takes rough paper to propose to the students to build their own boxes. Introduction of functions and graphics She implements the ‘box’ problem in class and uses it to introduce the vocabulary and notations: function, image, antecedent, f(1)= 8, f: 1

8. The whole activity, with the spreadsheet, and the course synthesis last 3 h

(H1–H3). Myriam has observed, in 2008–2009, that many students failed to place a point given by its coordinates in a Cartesian coordinate system. Thus she presents during the fourth hour a mere placement activity, before introducing the notion of the graph of a function, during the fifth hour (H5), and presenting exercises about graphs (H6).

Fig. 2.3 Synthetic description of Myriam’s activity, introducing functions in grade 9, 2009–2010. ∗ http://www.educnet.education.fr/formation/certifications/b2i

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Work on exercises After the introduction of the different notions, vocabulary and representations, she presents her students a sheet with five exercises, coming from several sheets downloaded on the Sésamath website (H7). One exercise concerns rectangle areas; all the others are situated within extra-mathematical contexts. The students are organised in homogeneous groups. They have to write their solutions on a slide (this session is filmed and observed). Myriam expects everybody to succeed the two first exercises, which actually happens; she observes only some difficulties in the notation and vocabulary in the students’ productions, which are discussed and corrected the following day (H8). Other exercises are presented in H9 with e-exercises video projected and solved by the whole class, and in H10 with the calculator. Snow, email and spreadsheet The eleventh hour of the lesson was planned for Wednesday, January 6. But the snow begins to fall, and Myriam is blocked at home. On January 7, she comes to school, but only three students managed to reach it. She starts an exercise with them of the textbook about graphs, which requires the use of a spreadsheet. She is concerned about the following days, because school transport is cancelled. She decides to send the exercise by email to the students and asks them to solve the exercise and to send back the graph, drawn with the spreadsheet. Fifteen students send it, and it is later corrected in class. In February, Myriam gives a short test on functions. She is not very satisfied with the results: some students still use incorrect notations or are unable to properly read a table of values. She presents them with additional work with the spreadsheet that the students have to send to her by email.

Fig. 2.3 (continued)

In this description for the class of situations ‘designing and setting up the introduction of functions’, we observe that Myriam uses many material resources of various kinds; digital and non-digital resources are strongly intertwined. We give below examples of geneses which occurred in the course of this activity.

2.3.2 Resources, Documents and Geneses: The Case of Myriam We have selected, amongst all the geneses we can infer from our data, examples involving knowledge and resources which seem to be of particular importance for Myriam (mentioned on several occasions in the logbook, during the interviews, etc.). We also retained the example of a genesis corresponding to a new class of situations recently introduced in Myriam’s system of activity for material reasons.

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2.3.2.1 Mathematics as a Tool for Other Topics: Influence of an Operational Invariant Myriam’s professional knowledge and beliefs strongly influence her documentation work: her choice of resources, the way she associates them, etc. For example, Myriam is convinced that mathematics is a tool useful for other scientific topics: biology, physics, etc. An important factor for this belief is that Myriam is married to a physics teacher. The discussions with her husband are resources coming from a specific community, which re-source her practice. In the Sésamath exercises, as in the class textbook, she chooses many exercises related to biology or physics: this is an instrumentalisation process; her knowledge and beliefs guide the choice of resources and drive the teacher’s agency. In the exercises she presents, many different letters are used to symbolise functions and variables: not only f(x) but also h(t), d(v), etc. We consider that she has developed an operational invariant like ‘the students must be able to manipulate functions with different names, because they will be asked to do so in physics and biology’. 2.3.2.2 Official Texts as Resources Myriam cares a lot about official recommendations (national, in France). Every Friday, she reads the ‘official publication of national education’3 which presents the official curriculum, announces the dates of the exams, etc. She intervenes as an in-service teacher trainer,4 discusses with the regional inspectors, etc. She has read a lot of texts about the ‘core-content’ reform. The organisation she chooses for the exercise sessions we observed, with a homogeneous group, is directly related to this reform. One of the official texts that Myriam often uses describes such an organisation. This text is an important resource for Myriam (coming from the ‘official’ institution); it frames her choices in an instrumentation process. We separated here, somewhat artificially, the associated processes of instrumentation and instrumentalisation for the sake of clarity; both processes are nevertheless strongly intertwined, as illustrated in the following examples. 2.3.2.3 Students’ Productions as Resources Students’ productions constitute essential resources for teachers. Many of Myriam’s choices are grounded in observations of difficulties encountered by students in 2008–2009. She changes her introductory activity, as she tells us in the third interview: ‘I think that my starting activity this year was easier, more concrete [. . .] Choosing a good starting activity is very important, it determines the mood for the whole lesson [. . .] Last year the students rejected the notion of function, they said 3

http://www.education.gouv.fr/pid285/le-bulletin-officiel.html In France every teacher can propose in-service training, on a given subject, usually for 1 or 2 days. The regional inspectors select some of the propositions to constitute the yearly ‘training offer’. 4

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it was useless’. The starting activity of 2008–2009 was situated within the frame of geometry (perimeter of a parallelogram inscribed in a right-angled triangle) and required delicate modelling work to determine a rather simple formula. Moreover, all the students conjectured the formula without calculation using a GeoGebra dynamic figure, which contributed to their reluctance towards a formal proof. It led Myriam to change her introductory activity; we consider that it also produced an evolution of her operational invariants (instrumentation). 2.3.2.4 Distant Work with the Students and Development of a Document In 2008–2009, Myriam participated for the first time in the assessment of the computer certification (B2i, Fig. 2.3). For this reason, she had to ask the students to send her emails with attached files. She created a special email address (we can consider this process as instrumentation). In 2009–2010, the heavy snowfalls prevented the students from coming to school for almost 1 week. Myriam used a students’ mailing list to present homework in an instrumentalisation movement. She is only starting with such requests, so she does not give much attention to being precise about the name of the file to send back or its format. She did not yet develop a stable orchestration (Trouche, 2004; see also Chapter 14) for such situations. The students sent back files with non-significant names; some sent spreadsheet files, while others copied their graphs in a word-processing file – in this case, Myriam cannot see how they built their graphs. We consider that Myriam is developing a document for the class of situations ‘designing and setting up distant work about the graphs of functions’. The document has a ‘resource’ part, associating in particular the classroom textbook, a spreadsheet, email addresses for the teacher and the students, amongst others. Our observation took place at a moment of important evolutions, for this class of situations, linked with new digital means. We hypothesise that Myriam starts to develop operational invariants like ‘when asking the students to send spreadsheets productions, it is necessary to precise that the spreadsheet file itself must be sent, and not copied into a word processing document’ and ‘correcting spreadsheet productions requires access to the formula written in the spreadsheet’. With the data we gathered, we cannot claim that Myriam actually developed these operational invariants; further observations are necessary to confirm this hypothesis. We consider it nevertheless as consistent, being connected with a more general operational invariant: ‘correcting the students’ mathematical exercises requires access to their procedures’, which seems to intervene in many documents developed by Myriam. Beyond these examples, we are interested in capturing more generally Myriam’s (and other teachers’) professional growth; for this purpose, we need to consider the documentation system as a whole.

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2.4 Documentation Systems and Professional Geneses We discussed in Section 2.3 several examples of documentational geneses of instrumentation and instrumentalisation processes in the case of Myriam. In this section we first go back to this case, trying to capture the structure of Myriam’s documents, their evolutions and factors of evolution. Then we address more generally the issue of documentation systems and professional geneses, focusing especially on the evolutions brought by digital resources.

2.4.1 Myriam’s Documentation System and Its Evolutions Figure 2.4 presents Myriam’s representation of her resource system, drawn in 2009–2010. Myriam represents herself, and specific aspects of her documentation work (in particular her teaching project), in the centre of a space organised in four zones: – On the left and in the middle, her work at home. This does not mean individual work: for example she is inscribed in mailing lists and receives at home information from these lists. She also works at home with resources given by colleagues. – Up right, her work at school without students. Meeting with colleagues, entering the students’ marks in the special software (Pronote5 ), which builds the school report at the end of each term. – In the middle of the right side, her work at school in the classroom with her students. Myriam has her own classroom equipped with a computer, a video projector and an overhead projector (which she intends to replace soon with a webcam). – Down right, in-service training collectives. Myriam is involved in two such groups. One group gathers some mathematics teachers of nearby schools (five teachers); they exchange exercises and discuss changes in the curriculum. They meet once a month. This group is not officially recognised by the institution. Myriam is also a member of an ‘official’ group, where a regional inspector participates. This group works on problems and investigations in mathematics. These articulated zones correspond to a structure of Myriam’s professional activity. This confirms the relevance of our ‘global’ positioning: the teacher’s resources are structured according to her activity. We emphasise here central features of Myriam’s professional activity and related characteristics, and evolutions of her documentation system: – Myriam develops real agency in the elaboration of her courses linked with the available digital means: she collects exercises from different downloaded files 5

http://www.index-education.com/fr/telecharger-profnote.php, ‘Pronote’ means ‘professional marks’.

36 Binder with the courses of previous years

G. Gueudet and L. Trouche Official curriculum

Entering students’ marks (Pronote− Profnote)

Websites Class textbook (Transmath grade 9)

Eduscol

Special meetings for some students

Official publication of national education

Photocopies for students

Vade mecum Problems basis Other textbooks Exam texts

ME

Overhead projector

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Video projector with the classroom computer

Reflection time

MatouMatheux

Discussions with other teachers: physics, biology

Mathenpoche

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GeoGebra

Evaluation

Spreadsheet

Websites (Institutional − Mathenpoche − MatouMatheux* − Sésamath grade 9)

Documents of training sessions followed or given by colleagues

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Notebook with

Training

USB key

planning each day

Local group (St Basil, St Anton)

Folders Report sheet of grade 9 before each holiday (summarising the main notions to know)

Class notebook

Notebook written as a student

Reflection group

Fig. 2.4 Schematic representation of the resource system in 2009–2010, Myriam. The original SRRS was handmade and naturally in French. We translated this one and typed it. ∗ Online exercises, http://matoumatheux.ac-rennes.fr/accueilniveaux/accueilFrance.htm

to build one exercise sheet; she integrates new software that she does not fully master, etc. Her preparation work at home is represented on more than half of the SRRS. This corresponds perhaps more to the importance of this work in her opinion than to the time actually spent. She filled in the logbook for over 27 days (about the work with her grade 9 class; this includes Saturdays and Sundays). She mentions about 14 h of work in class with the students, 10 h at school for other purposes (several kinds of meetings) and 12 h at home. She can certainly be considered as an expert teacher. We consider that this characteristic acts as a lens, evidencing phenomena that happen for all teachers. – She is very concerned about official instructions. She follows them and even anticipates further institutional requirements. Her involvement in the assessment of the ‘B2i’ leads her to send work by email and to develop professional knowledge linked with these email exchanges with the students.

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– Myriam is involved in many collectives, which strongly influence her documentation (we detail such processes in Chapter 16). As discussed above, the collective she forms with her students (Section 2.3) and her interactions with students are central resources. She also discusses with colleagues at school and within the in-service training groups. These groups provide her with new ideas of exercises and develop her interest in problems and in investigation situations. This is probably one of the reasons accounting for her choice of introductory activity (‘the box’, Fig. 2.3). We have also mentioned the discussions with her husband. Myriam evokes in fact many discussions with her family, which intervene in her documentation work: not only her husband, who teaches physics, but also her sister, who teaches maths, and her daughter, who is now in grade 11. In H10 (see Fig. 2.3), she devotes a whole hour to the work around the calculator, much more than what was done in 2008–2009, because she realised the importance of the calculator at high school by observing the work of her daughter. – She considers that mathematics needs to provide tools for other scientific topics, physics and biology in particular. This feature is linked with the issue of collectives: her husband is a physics teacher, and at school her discussions are especially with the physics and biology colleagues. It strongly influences her choice of exercises and problems, which are often connected to these scientific topics. The whole of Fig. 2.4 could be interpreted as a description of Myriam’s documentation work: on the left, the resources she draws on; in the centre, her own ‘creations’; on the right, the implementation which supplies new resources – back to the left (kept in the ‘binder with the lessons of previous years’). The presence, in the centre, of ‘ME’ written in capital letters emphasises that this work and, according to our perspective, the associated geneses deeply influence the teacher.

2.4.2 Professional Geneses and Integration of Technology In this section, we address more generally the question of documentational geneses for teachers and the consequences of the use of digital resources, drawing on observations realised with several teachers (including Myriam and Pierre). 2.4.2.1 A Specific Perspective on Professional Growth The documentational approach offers a specific perspective on teachers’ professional growth. The documentation systems articulate professional knowledge and the teacher’s resource system. Therefore, considering teacher’s documentation systems leads in particular to identify structuring elements in their professional knowledge. These elements include what Ruthven (Chapter 5) names the curriculum script (model of goals and actions guiding the teaching of a particular topic). We consider nevertheless

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that this script is mostly adapted to enlighten the decisions that the teacher takes in class; the documentational approach aims at presenting a more holistic view of the teachers’ activity. It can naturally be used to study technology integration phenomena and more generally to understand the professional evolutions resulting from the generalised availability of digital resources. 2.4.2.2 Using Digital Resources: Consequences We notice here different types of such evolutions: – The balance evolves between what is limited to the group formed by the professor and her students, and what is more largely accessible. In particular, Myriam and Pierre use the Pronote software, which was retained by the administration in each of their schools. It confers a public dimension to the marking, making the marks immediately accessible to the other teachers from the same class and to the administrative staff. – The spatial organisation in class includes new forms of display. The two teachers observed use a video projector; Pierre, moreover, has an interactive whiteboard that he combines with his traditional whiteboard (Chapter 16). Myriam also uses an overhead projector to exploit two forms of display as well. This leads to raise the question of new forms of ostension (the teacher showing the content to be learned, Salin, 1999) associated with these new displays, which would require a complementary study. – Using digital files allows an immediate modification of these files as soon as the teacher observes a problem during the implementation in class. The impact of the interactions with the pupils thus seems increasingly important for the teacher’s resources. – The email allows fast and flexible exchanges of files between the teachers and permits out-of-class exchanges between students and teachers.

2.5 Conclusion The stake of a documentational approach of didactics is not limited to the analysis, in terms of professional genesis, of the consequences for teachers of their interactions with resources (Cohen, Raudenbush, & Ball, 2003). It constitutes a change of perspective and an invitation to see documentation work as central in the teachers’ activity and documentational geneses as the components of a complex professional genesis. The expression documentational approach of didactics aims at emphasising that the objective is to not only propose a didactical analysis of the teacher’s documents but also consider the documents as central within the didactic phenomena, and in particular within teachers ongoing professional development. This perspective has already been discussed in Gueudet and Trouche (2009), where we introduced the resource/document dialectics and the concept of documentational genesis. The specific methodology that we have implemented here enabled

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us to investigate further, taking advantage of the long-term evolution, over several years. Complementary studies are needed; the theoretical aspects of the approach have to be refined, in particular by confronting the documentational approach and other theories used for the study of teachers’ growth, of professional activity and of mediations in/for this activity. The consequences of this approach for the design and use of resources, in particular with an objective of professional development (Chapters 7 and 17), require a specific attention. It is also an aspect of our work in progress, in particular about innovative teacher training programs grounded in collaborative documentation work (Gueudet, Soury-Lavergne, & Trouche, 2009). Acknowledgements The authors warmly thank Jill Adler for her comments, and in particular for her revision of the English language in this chapter.

5.00–5.15 (P.M.)

The bursar

The class (grade 9)

Other persons involved

Source: Extract of a filled logbook (Myriam, January 4, 2010)

Home

Report of the day’s lesson

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10.00–10.55 (A.M.)

Classroom

Dining hall

Time

Precise place

Discussion with the bursar

Activity type (and math. theme, if relevant) Lesson about functions

Myriam, Additional Data

Appendix

Materials produced

Videoprojector

Method sheet Students sheet Online about reading exercises images (MatouMatheux) and antecedents

Resources used

Personal notebook

Ring binder grade 9

Archiving (what? where?)

She informs me that my webcam has been delivered

Comments

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References Adler, J. (2000). Conceptualising resources as a theme for teacher education. Journal of Mathematics Teacher Education, 3, 205–224. Assude, T. (2007). ‘Teachers’ practices and degree of ICT integration’. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (pp. 1339–1348). Larnaca, Cyprus: CERME 5. Retrieved May 2011, from http://ermeweb.free.fr/CERME5b/ Cohen, D. K., Raudenbush, S. W., & Ball, D. L. (2003). Resources, instruction and research. Educational Evaluation and Policy Analysis, 25(2), 119–142. Gueudet, G., Soury-Lavergne, S., & Trouche, L. (2009). Soutenir l intégration des TICE: quels assistants méthodologiques pour le développement de la documentation collective des professeurs? Exemples du SFoDEM et du dispositif Pairform@nce. In C. Ouvrier-Buffet & M.-J. Perrin-Glorian (Eds.), Approches plurielles en didactique des mathématiques (pp. 161–173). Paris: Laboratoire de didactique André Revuz, Université Paris Diderot. Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? Educational Studies in Mathematics, 71, 199–218. Guin, D., Ruthven, K., & Trouche, L. (Eds.). (2005). The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument. New York: Springer. Hoyles, C., & Lagrange, J.-B. (Eds.). (2010). Mathematics education and technology – Rethinking the Terrain. The 17th ICMI Study. New York: Springer. Leont’ev, A. N. (1979). The problem of activity in psychology. In J. V. Wertsch (Ed.), The concept of activity in Soviet psychology (pp. 37–71). New York: M.E. Sharpe. Pédauque, R. T. (Coll.) (2006). Le document à la lumière du numérique. Caen: C & F éditions. Perrin-Glorian, M.-J., DeBlois, L., & Robert, A. (2008). Individual practising mathematics teachers. Studies on their professional growth. In K. Krainer & T. Wood (Eds.), Participants in mathematics teacher education (pp. 35–59). Rotterdam, The Netherlands: Sense Publishers. Rabardel, P. (1995). Les hommes et les technologies, approche cognitive des instruments contemporains. Paris: Armand Colin (English version at http://ergoserv.psy.univ-paris8.fr/Site/default. asp?Act_group=1). Rabardel, P., & Bourmaud, G. (2003). From computer to instrument system: A developmental perspective. Interacting with Computers, 15(5), 665–691. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. Salin, M.-H. (1999). Pratiques ostensives des enseignants. In G. Lemoyne & F. Conne (Dir.) (Eds.), Le cognitif en didactique des mathématiques (pp. 327–352). Montréal: Les presses de l’Université de Montréal. Schön, D. (1983). The reflective practitioner: How professionals think in action. London: Temple Smith. Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9, 281–307. Vergnaud, G. (1998). Toward a cognitive theory of practice. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity (pp. 227–241). Dordrecht, The Netherlands: Kluwer. Vygotski, L. (1978). Mind in society. Cambridge, MA: Harvard University Press.

Chapter 3

Patterns of Didactic Intentions, Thought Collective and Documentation Work Gérard Sensevy

Understanding someone’s action requires, in particular, understanding his intention. Understanding an intention, or a system of intentions, does not necessarily mean understanding the whole action, but at least an essential part of it. This statement, however, could be understood as a form of solipsism (the intentions are specific to the individual) and mentalism (intentions are in the head). On the contrary, this text defends and illustrates an alternative conception, in which intentions are regarded as more or less shared and more or less external to the individual (Duranti, 2006). This chapter participates in the general project of this book in underlining the essential dialectics between the documentation work and the shaping of intentions. In this perspective, it aims to demonstrate how intentions are formed in a system of resources (Chapter 2). Intentions are therefore understood, through the documentational genesis process as resulting largely from a documentation work (Chapter 2) performed by the teacher. This contribution falls within the scope of the Joint Action Theory in Didactics (JATD) (Amade-Escot & Venturini, 2009; Ligozat, 2008; Sensevy, in press; Sensevy & Mercier, 2007; Schubauer-Leoni, Leutenegger, Ligozat, & Flückiger, 2007), a theory situated in the general paradigm of joint action (Blumer, 2004; Clark, 1996; Eilan, Hoert, Mc Cormack, & Roessler, 2005; Mead, 1934; Sebanz, Bekkering, & Knoblich, 2006; Tomasello, 2008). In this framework, human activity is seen as grounded in the recognition of signs founded in others’ behaviors. It is viewed as a social game (Bourdieu, 1990, 1992; Bourdieu & Wacquant, 1992). The didactic activity is modelized as occurring in a didactic game that can be described, in particular, with the concepts of didactic contract and milieu (Brousseau, 1997; Sensevy, in press; Sensevy & Mercier, 2007; Sensevy, Mercier, Schubauer-Leoni, Ligozat, & Perrot, 2005) and their relation. According to this theory, the joint work of teacher and students can be seen under the description of two articulated moments: (1) didactic activity in situ, in which the

G. Sensevy (B) Brittany Institute of Education, University of Western Brittany, France e-mail: [email protected]


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teacher makes the students play the didactic game; (2) the preparation of this activity when the teacher builds the game he will implement. We argue that the teacher’s intentions are shaped in his documentation work. The morphogenesis of intentions, in the documentation work, is thus the link between the building of the game and the actual play that the teacher institutes. In this chapter, we rely on practical descriptions of teachers’ and students’ practices, but our first objective is theoretical. We propose conceptual elements with a three-fold purpose. We try to achieve a better understanding of (1) the relations between intentions and didactical action; (2) the relations between classroom preparation and the actual implementation; and (3) how these relations unfold in a collective that can in some cases produce a specific thought style (Fleck, 1979, p. 99), a system of categories shared in this collective, that ultimately produces ‘the readiness for directed perception and appropriate perception of what has been perceived.’ We then propose a description of the elaboration process of the game, supported by three related assertions we work out in this chapter. First, the resources system that the teacher mobilizes (in the process conceptualized by Gueudet and Trouche, Chapters 2 and 16) is a key source of his action. Second, the teacher’s prior didactic intentions do not have to be found ‘in his head’ or ‘in the situation,’ but in the dialectical relationship between resources or documents, and the way he anticipates the progress of the game in situ. The didactic intentions in action stem from the dialectical relationship between prior intentions and the game as it is enacted in didactic transactions. Third, the process that connects documents, prior intentions, and intentions in action is rooted in the inclusion of the action of individuals in a collective structure. In the first part of this chapter, elaborating on Baxandall’s Patterns of Intention (1985), we develop a framework to understand intentions from a generic viewpoint. We argue that prior intentions function as strategic rules that drive the teacher’s game. The second part is devoted to the study of two empirical examples, which may illustrate the above framework. In particular, we show how prior intentions, as strategic rules (Hintikka and Sandhu, 2006), are drawn from the documentation work, and how the strategies they enact depend on the structure of the milieus suitable for the didactic action in situ. In the third part of the chapter, we briefly summarize our findings.

3.1 Patterns of (Didactic) Intentions In his book (1985), Baxandall formulates a system of descriptions of the intentions of certain artists (e.g., Picasso) in relation to specific paintings (e.g., the Portrait of Kahnweiler). For this purpose, he first built a generic framework for studying how an English engineer, Benjamin Baker, built in the east of Scotland a bridge over the Forth River. To summarize Baxandall’s conceptions, we can look at the following quotation:

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The intention to which I am committed is not an actual, particular psychological state or even a historical set of mental events inside the heads of Benjamin Baker or Picasso, in the light of which – if I knew them – I would interpret the Forth Bridge or the Portrait of Kahnweiler. Rather, it is primarily a general condition of rational human action which I posit in the course of arranging my circumstantial facts or moving about the triangle of re-enactment. This can be referred to as ‘intentionality’, no doubt. One assumes purposefulness – or intent or, as it were, ‘intentiveness’ – in the historical actor but even more in the historical objects themselves. Intentionality in this sense is taken to be characteristic of both. Intention is the forward-leaning look of things. It is not a reconstituted historical state of mind, then, but a relation between the object and its circumstances (Baxandall, 1985, pp. 41–42).

One can notice the importance of the so-called ‘triangle of re-enactment’: there is a situation (first term), a problem arising from this situation (second term) and the solution-object (third term). Let us see how Baxandall summarizes his investigation into the Forth Bridge: One came first to the general Charge that the agent, Benjamin Baker, would be responding to, and noted that while it could be terse – ‘Bridge!’ – it was a rubric for performance that contained within it various general terms of the problem – spanning, providing a way, not falling down. From this one moved on to specific terms of the problem, which I called the Brief, though the name does not matter . . . Together Charge and Brief seemed to constitute a problem to which we might see the bridge as a solution (Baxandall, 1985, p. 35).

To finish summarizing the framework provided by Baxandall, I will address the issue raised by the ‘relationship between object and its circumstances.’ Baxandall argues in the following way: Some of the voluntary causes I adduce may have been implicit in institutions to which the actor unreflectively acquiesced: others may have been dispositions acquired through a history of behavior in which reflection once but no longer has a part. Genres are often a case of the first and skills are often a case of the second (Baxandall, 1985, p. 42).

We can now put forth a first formal framework for the description of intentions we will project on the description of didactic intentions. 1. The objects (and actions) can be described as solutions to a particular problem. To understand an object or action, it is worth asking the question of the problem they are supposed to respond to, and, in some way, which shaped them. One can see a close relationship between this way of conceiving things and the background epistemology in Dewey’s (1922) and Brousseau’s (1997) works, both of which focused on the notion of a situation. 2. Intentions are inherent to physical objects and environments in which these objects (and actions) are located. This view is obviously opposed to mentalistic or psychologizing conceptions of intentions. To understand the intent of an agent in a situation, even before questioning him or eliciting his rationale, we have to understand how the symbolic and physical milieus within which he is acting will lead to such or such intention. In this perspective, material objects themselves (e.g., tools) are purveyors of intentions and plans (Suchman, 1987),

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for the use of people playing the appropriate social game. The concept of ‘affordance’ enables us to understand how objects may be viewed as purveyors of intentions: ‘what we perceive when we look at objects are their affordances, not their qualities. We can discriminate the dimensions of difference if required to do so in an experiment, but what the object affords is what we normally pay attention to’ (Gibson, 1979, p. 134). More broadly, it is the symbolic milieu (e.g., the meanings associated to a specific genre), and therefore the identification of the games that the agents are expected to play in specific situations that may give access to the intentions. One of the fundamental aspects of this milieu is that symbolism is not confined to action in situ and the here and now. Most of our actions are prepared. 3. It is useful and relevant to consider these intentions at various levels of granularity (specificity). In this respect, Baxandall distinguishes the ‘Charge’ that can ‘summarize’ the general intention specific to a particular action and the ‘Briefs’ that characterize these intentions locally. It is interesting to notice that these scale levels call for a differential description of the action. In this perspective, one may usefully appeal to Searle’s (1983) distinction between ‘prior intentions’ and ‘intentions in action’ to figure out how the prior intentions are redesigned as intentions in action in the current action at stake.1 4. The intentions have to be thought about in a broader framework than that fixed by the common epistemology. We saw in particular how Baxandall seeks to extend the meaning of the word ‘intent’ to both institutional practices (including genres) and skills. One can therefore read the intentions in the categories of perception and action that are provided by the institutions, and in skills inherent in the ‘handling’ of a particular object. The four dimensions of the framework presented above can and should be specified in didactic action, and more specifically to the situation of the teacher who ‘prepares the classroom.’ We must be aware of the specificity of this situation. In the intentional part of the documentation work, the teacher uses the resources of a given milieu to organize them into a document. Following Gueudet and Trouche (Chapter 2), we can consider such devices as artefacts monitored by a scheme of use. We have to acknowledge the intentional structure specific to the documentation work. The teacher, related more or less to a group, selects resources according to certain intentions. The arrangement he produces from these resources in turn redefines the system of intentions, which will be further reorganized in the effective course of action. As a ‘historical object’ (Baxandall, 1985, p. 42), a document embeds purposefulness. Intentional structure and actional structure codetermine the other in the document. In some ways, this is both the condition and the effect of this codetermination.

1

In this respect, Pacherie’s recent work (2008) may be also of some help.

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The second part of this chapter will be devoted to the empirical study of some elements of this process.

3.2 Didactic Intentions: An Empirical Study The short empirical study that follows will allow us a first use of the theoretical framework above. For this, we will refer an example in mathematics in elementary school, in which we try to understand what is going on when the teaching intentions are designed in a specific collective, within an implementation process of a new version of a given instructional design. The collective we talk about is a group of teacher educators and researchers. The teacher educators are half the time in their primary school classroom and so able to implement the instructional sequences designed by the group. The collective work we describe has been conducted for 2 years and is inscribed in an ongoing process. The instructional sequence is called Treasures Game. It is a months-long didactical sequence, designed for Kindergarten by Brousseau and his team at the beginning of the 1980s, as documented in Pérès (1984). Brousseau (2004) has presented strong theorization of this research design, which he considered as a fundamental situation for the notion of a representation. The Treasures Game consists of producing a list of objects to be remembered and communicated. The didactic device takes place over a long period (about 45 sessions, of variable duration), which thus becomes a ritual time, but one where the rules change as the game progresses. There are four main stages in the game (Schubauer-Leoni, Leutenegger, Ligozat, Flückiger and Thevenaz-Christens, 20102 ): In Stage 1, the teacher presents two or three small new objects belonging to the world of children to all the children. The objects are passed from hand to hand, the teachers ask the students to name them and then she puts them in a gray box (the treasure chest). She then asks: what’s in my box? A student then calls out the name of an object, the teacher pulls it out of the box and places it in full view of everyone. ‘Is my box empty?’ she asks, and if not, the game continues, and so on. The game takes place every morning. Every two or three days, new objects (two or three objects) appear and are added to the previous ones. By the end of 1 month, the whole class has emptied a box of 40 objects seemingly disparate but carefully chosen. This stage is played out with the entire group of students and focuses on the creation of a verbal system of reference for the objects in the treasure chest. Stage 2 starts the individual memory game, as each pupil must remember the two objects that are hidden daily in the treasure chest. All the children understand the game and are able to succeed, which means, memorize short lists of items from the morning to the evening. 2

I would like to thank Florence Ligozat for sharing this text with me.

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Stage 3 is an individual game aimed at making lists whose production is driven by an important change in the rule of the game (jump from 2 to 10 or 12 hidden objects): the informational leap that finishes Stage 2. Stage 4 is aimed at collectively developing a common code and is driven by a communication game between pupils. The purpose is to offer children new to school life, an opportunity to experience the necessity to rely on a graphical code (drawn or written) to remember a set of objects and to communicate about them with others. It is the very basis of the representation process that is triggered through this game.

3.2.1 Designing New Versions of the Treasures Game: The Building of Intentions as a Collective Strategic System We now focus on Stage 3 of the instructional sequence at the end of Stage 2. It is an important moment, in that the students are confronted with what Brousseau coined as an ‘informational leap.’ The epistemic strategies that enable the students to memorize the right objects (by relying on their ‘internal’ memory) are invalidated by the large number of objects they have to retrieve. For students, it is impossible to memorize 10 or 12 objects without an external (public) representation of these objects. Therefore, introducing this informational gap aims to foster the students’ passage to an external representation, or inscription system. In the following, we compare three ways of thinking about this crucial moment, in the two studies we mentioned in the previous sections, and within our collective. The first way of thinking is presented by Brousseau (2004, p. 256): The passage from 3 to 10 represents a considerable complexification of the situation. The unruffled teacher notices failures, but remains encouraging. ‘Think, we’ll get it’ . . . No child of that age can invent or even conceive the answer all of a sudden, by making a list of objects designed with small drawings of these objects, because the process can succeed only if one controls together all the components. On the other hand, the project can be meaningful only if the children consider, at the outset, specific means to carry it out. The situation appears to be blocked, which causes teachers’ anguish. Yet, we observed that each year, drawings and lists appear. For Brousseau, the adoption of a list of written codes, even if it ‘causes teacher’s anguish,’ is not really problematic. ‘Elements of solution appear and spread in the classroom’ and ‘The method of making lists of drawings is quickly adopted.’ Schubauer-Leoni et al. (2010), in the ‘second generation’ of the implementation of the Treasures Game, consider this issue as follows: ‘This is the trickiest moment and one should not expect the pupils to put in place the relevant strategies straight away . . . The problem faced by the pupils is that they must feel empowered to shift to a remembering process based on inscriptions. T cannot suggest drawing as this would be too strong a command for the pupils and

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it would prevent them from feeling the need for a list. It is in the discussion between the pupils that this idea can come to light.’ We can acknowledge a similar conception of the way the teacher and the students have to deal with the informational leap and the necessity to adopt a ‘remembering process based on inscriptions.’ As ‘Brousseau’s teacher’ must stay ‘unruffled’ and trust the students’ invention, ‘Schubauer-Leoni and coworkers’ teacher’ ‘cannot suggest drawing,’and the necessity of designing writing codes (the inscription process) has to stem from ‘discussion between the pupils.’ In the same light, in the two texts, one can find that the teacher’s role is to encourage the students, in particular by assuring them there is a way to win the game. If we now look at the way our collective dealt with this issue, we have to keep in mind the following points: First, Brousseau’s and Schubauer-Leoni and coworkers’ conceptions were well-known by the collective, given that their papers have been studied before implementing the teaching sequence, and discussed throughout the implementation process. From this viewpoint, the collective documentational genesis (Chapter 16) encompasses the elaboration of these texts, in relation to the actual implementation. The collective was thus sure that the teacher had to stand to the side, and leave the students to figure out how to solve the informational leap problem. One can notice that such a perspective is consistent with the roots of the theory of didactic situations (Brousseau, 1997) as it is usually understood. According to this theory, an essential purpose of the didactical process consists of enabling students to build a first-hand relationship to a given piece of knowledge. To reach that goal, the teacher has to monitor this process by making sure that the students experience the mathematical necessity (in this case, the power of public representations). Nevertheless, at the end of the Stage 2, when the informational leap had to be realized in the classroom, a discussion unfolded in the collective about this issue, initiated by the teacher who has the responsibility to carry out the lesson. Indeed, the collective habit of this group was to anticipate as precisely as possible students’ actual participation and the range of didactic behaviors that students might produce in the didactical situations. In doing so, the collective tried to identify a link between the milieu and the teacher’s action, and the students’ behaviors. When trying to fulfil this a priori analysis pattern, in the case of the session in which the informational leap was presented, the collective was not able to anticipate by what concrete means students would be able to figure out the necessity of using inscriptions. In this respect, it was the teacher’s responsibility to manage the situation by improvising on the basis of the conceptual background that was at the root of the collective’s work. One can thus consider how the collective work on available resources (from Brousseau’s team and Schubauer-Leoni’s team) provides a specific strategic system that one can describe as follows: confronted with the inevitable failure of his students, the teacher had to let the students know that the game can be played with success (encourage the students by giving them the assurance they can win); she had to stand to the side to allow the students to experience the necessity of the inscription system (let them find they can make a list); she knew that she was going

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to face uncertainty stemming from her ignorance of the students’ possible moves to find a ‘solution’ (she is prepared to use any opportunity to guide the students’ learning trajectory). Let us now consider the actual implementation of this part of the situation, which means, according to our theoretical framework, how the system of these strategic rules is enacted in actual strategies.

3.2.2 Implementing Instructional Sequences Within a Collective: A Teacher’s Rational Improvisation This part of the instructional sequence has been videotaped and transcribed.3 Several months after the sequence was carried out, an auto-analysis interview was conducted between the teacher and another member of the research team. The studied episode took place at the end of Stage 2. This session occurred in a workshop gathering five students. It was the first day of a two-day process, in three phases (Day 1: morning; Day 1: afternoon; Day 2: morning). In the following, we focus on one of the crucial moments of the Treasures Game, in describing how the joint action of the teacher and the students fosters the emergence of ‘making a list.’ 3.2.2.1 Day 1: Morning The teacher presented the 10 objects that had to be remembered for the afternoon. She handed the items in the bag (the equivalent of the ‘treasure chest’), and stressed the goal of the game: the students had to remember, and ‘each child will be on his own to remember all things this afternoon.’ At the end of this episode, a significant dialogue takes place between a student (Ima) and the teacher: You have to write down. . . I have to write down what, my dear (inaudible) I have to write down all the children who did the Treasures Game? I have to write down all the objects? What do I have to write? Ima: You have to write down (inaudible) of Treasures Game.

Ima: T: Ima: T:

In fact, at this moment, Ima wanted the teacher to write down the list of the students who played the game on that day. This is a generic classroom habit, specified to the Treasures Game situations. In the classroom, it is important to write down who has done the activity, to know who hasn’t. 3 This part of the chapter has been written on the basis of data collected by two members of the collective, Dominique Forest and Anne Le Roux-Garrec. I would like to thank them. I am grateful to Dominique Forest for the fruitful discussions we had about the interpretation of these data.

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So the teacher takes the list from a table near the students and points to the students’ names: T: Ok, I know that’s you, the other ones will do it afterwards . . . It is interesting to notice a kind of uncertainty in the teacher’s behavior. Ima referred to a list. Even though after the teacher understood that the list Ima was talking about was not the type of list she was waiting for, this behavior could be considered as an opportunity for the teacher to give the students an incentive to think about making a list, for instance, by revoicing Ima’ s proposal. The teacher comments on this event as follows: I know I’ll have trouble, finally I am afraid I find it hard to make them think about the written record. Oh, I know that, I know that because we talked a lot about it in the group, and I know that at a point the written record must appear, and I do not see how it will appear. I do not know what I was thinking then, but when Ima said ‘write down,’ then I said to myself there is something, something that I must keep under my sleeve, because the idea of writing record, if it does not emerge after, at this moment there are traces, which emerge now, traces I will be able to rely on. We can understand how the teacher was able to reenact her intentions in the dialogue focused on her videotaped practice. It is possible to recognize the strategic system we mentioned above. In particular, she knew she must let the students ‘find by themselves’ that they can make a list. This is one of the core constituents of the strategic system elaborated within the collective (‘we talked a lot about in the group’). One may say that this strategic rule stems from the ‘thought style’ (Fleck, 1979) inherent in the work of this collective.4 There are some fundamental relations and properties that are impossible to challenge in a thought style, a kind of ‘bedrock’ (Wittgenstein, 1997), which turns ‘individual thought over to an automatic pilot’ (Douglas, 1987, p. 63). We argue that the ‘let the students find by themselves they can make a list’ strategic rule is such a core principle in the collective thought style. Nevertheless, it is worth noticing that this strategic rule is not easily converted to an actual strategy. As we put it above, it is in some ways contradictory to the habit of thought, elaborated in the collective, which consists of drawing a precise a priori analysis to anticipate the students’ learning behavior. This uncertainty is obvious in the teacher’s comments (‘I know that at a point the written record must appear, and I do not see how it will appear’), which seems to mirror Brousseau’s and Schubauer-Leoni and coworkers’ statements we quoted above. Thus, it is perfectly understandable that the teacher be tempted to use all the

4 We argue that one can consider the educational process as the slow elaboration of a thought style (Sensevy et al., 2008).

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opportunities she could find in the students’ utterances, even though there is a risk of misunderstanding. The following part of the teacher’s auto-analysis sheds light on this topic: I think I try to bring out small things, because it can be reused, maybe there are seeds that are sowed there. On the excerpt in which the teacher shows students the list of children: This is the list of children, but it’s true, I do not present it without purpose. This is a sample of a list, which . . . So there is an emergence of something, I thought, if it is difficult for them to achieve the written record, perhaps I will be able to build on it. The interviewer asks: But you do not go any further at this time? No, because it is not the right time, it’s not the game. And here it is about a list, which describes the children who played the game, it is not at all the idea of a written trace, which keeps a permanent memory for later. In the list of children Ima refers to, we deal with a written trace, which allows us to validate: has each child played? So it’s not at all the same approach. In this excerpt, we can understand how the teacher’s action (the teacher’s game on the student’s game, in the theoretical sense of the JATD) surrounding the ‘question of the list’ consists of reducing uncertainty by ‘sowing seeds,’ that is, by paying attention to the student’s mention of the list of children, to be able to reuse this meaning later on. But it is interesting to note that this behavior does not entail a Jourdain effect5 (Brousseau, 1997). Indeed, the teacher explains that the strong conceptual difference between the two types of lists (the ‘list of children’ and the ‘Treasures Game list’) prevents her from relying too firmly on the student’s designation of the list of children. One can acknowledge from the teacher’s declaration how two fundamental aspects of the didactic game are at stake. First, the chronogenesis (the genesis of time) constraint (No, because it is not the right time, it’s not the game) that explains that the teacher has to wait for the right time, the kairos, as the ancient Greeks said, to engage the classroom discussion on the issue of the list. Second, the mesogenesis constraint, which is closely linked to the chronogenetic one. In this episode, the mesogenesis (the genesis of milieu) necessity refers to the need, for the teacher, to introduce some specific meanings to create common ground and upon which she will be able to elaborate in order to help the students figure out how to produce a remembering process. 3.2.2.2 Day 1: Afternoon The teacher asked each student to recall the names of objects, without success. Although some of them recalled more than others, nobody was able to recall the ten objects (carefully chosen, in kind and number.) The teacher asked the students to name and count the objects, and she made them acknowledge their failure. At a point in this session, the teacher insisted: ‘I would like us to succeed, because we did

5 A Jourdain Effect occurs when the teacher pretends to acknowledge a specific piece of knowledge in an ordinary student’s behavior.

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not win’. It is important to note how the teacher emphasizes the students’ failure, as an impossibility to retrieve the ten objects. But one has to identify, in the teacher’s speech, the use of the pronoun ‘we,’ which means the teacher includes herself in the failure and the necessary subsequent research. For her, it is a way to deal with the difficult uncertainty students could feel. In doing so, she tries to enact the two main strategic rules in this part of the Treasures Game: (1) the students have to experience the limits of internal memory; (2) this failure must not alter their commitment in the enquiry. It is interesting to focus on a slight move, in the teacher’s game, which occurred at the end of the session. Out of the blue, the teacher first intended to show the ten objects to the students, then she changed her mind: ‘Well, I am going to show you the objects [the teacher takes the bag]. I intended to let you see, but before that, I would like us to succeed, cause we did not win.’ To understand this point, the interviewer asks the teacher a question: I: You said ‘I am going to show you the objects’ and you didn’t do that . . . T: No, because the problem is elsewhere, I would like to get it, I would like it to emerge, and I am afraid that, I tell myself that, by showing them the objects, they think ‘oh, it’s easy, I could have done it,’ they could be in trouble. In fact, I want to leave them feeling the failure, I want to leave them telling themselves ‘the hidden objects that I can’t see yet, what is the representation I could give them.’ In my opinion, it’s the point. During the following interactions, the teacher went on by underlining several times the reality of the failure, and its inescapability: ‘you will not succeed, it’s too difficult.’ In the same time, she diffuses the idea of a possible solution: ‘we should find a means.’ She characterized this means as a ‘little means,’ thus signifying that every student had the possibility to find a solution. Even though the teacher tries to help the students, they do not provide a solution. One could identify a kind of fatigue among the students. So the teacher introduces in the milieu the ‘meaning of writing’ in the following way: S1: T: S2: T:

And if you tell us? Oh, me, I won’t say anything. Ah, ah, ah, she tells us nothing because it’s the Treasures Game. Yes indeed you are playing, but Ima, what did she say she wanted to do in order to remember in the evening? S3: Write!

The teacher gave a clear incentive, by focusing student’s attention on Ima’s word about the possibility of writing, and the students acknowledge this reminder. It is interesting to consider the teacher’s analysis of these moves. Here I am cheating, I am cheating, because what Ima wanted to do in the morning was keep a trace of the students’ participation in the game to be sure they have participated. On my side, it meant keep a memory to remember . . .

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The teacher critiques her own behavior, but after having emphasized again her recognition of the difference between the student’s viewpoint and her own, she reconsiders her previous analysis: After all, it’s sure that the idea of a list, the idea of ticking the students’ participation to be sure all the students have played, this idea is in the same spirit to keep a memory, it’s what I reactivate here. Even though they do not have this anticipatory idea of keeping a trace as a representation to use it later on . . . So, it’s not really cheating, it’s, umm, bridging the gap from my behavior to a behavior that they can adopt in their personal approach. One can see this latter assertion as witnessing the complexity of the didactical practice. If one follows the teacher’s justification, one can say that to reach her goal (enabling the students to refer to writing), the teacher admits a kind of minor misunderstanding of the nature of the remembering process. One can raise the hypothesis that to the extent to which the failure of the internal memory has been acknowledged by the students, the production of the ‘solution’ (writing) is not a major stake. The crucial point is that the students commit themselves to the writing process, given that the teacher’s monitoring of this process will enable them to understand the very nature of the remembering process, and thus to correct the initial minor misunderstanding that will have allowed the joint process to proceed. 3.2.2.3 Day Two: Morning The day after, after having reminded the students of their difficulty to retrieve the ten objects, the teacher reactivated the writing solution: T: And I, I still wanted 10 objects, even though it was a lot. So, this evening, will it be easy to remember them? S1: No! T: No, so what could we do to remember this evening? Ima: We write. T: You, you would like to write, so you need a sheet of paper. So, go ahead (the teacher gives Ima a sheet of paper and a pencil), for me, it’s alright. T: But she does that for herself, OK. To remember on her own. You, if you want to remember you have to do something too? S2: Yes, me, I want to write, too (the teacher gives him a sheet of paper and a pencil). S3: Me too. It’s that, we all are going to write down (the teacher gives a sheet of paper and a pencil to every student). Eventually, the instructional sequence continued. The students started to produce some inscriptions as a means to remember the objects, and the Treasures Game proceeded.

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3.3 Conclusion In this chapter, we first focused on the issue of intentions, on the basis of Baxandall’s work. Within the framework of the JATD, we consider human practices as social games. In this respect, we argue that to understand people’s actions, we have to identify what we modelize as the game they play. Thus, people’s intentions are to be drawn from these games, and we consider intentional systems as strategic systems. In doing so, we highlighted a conception of intentions in which intentions are public, found in the milieu of the action. In that sense, an intention is more or less always collective, not necessarily in the sense that it stems from a collective, but in that it has to be viewed as the expression of an institutional thought style that stems from the social game at play. This thought style plays a prominent role in the orchestration process (Chapter 14) that teachers enact. In the empirical study we outlined in this chapter, the teacher’s intentions were collective, in the first sense of the term that we acknowledge below. The teachers work in a particular institution, broadly speaking, a didactic institution, which brings them to a specific thought style. For example, a teacher has to enable the students to establish more or less a first-hand relationship to a given piece of knowledge, and one who wants to understand the dynamics of the teaching--learning process has to take this general feature into account when identifying the teacher’s intentions. In this chapter, the case study allows us to understand how the didactic intentions lie in the documents designed by the teachers and in the relationship the teacher has built with these documents. The case at stake is interesting in that it shows the nature of the teacher’s intention, about the necessity, for the students, to experience the failure of internal memory, and the consequent adoption of a writing strategy. This system of intentions is not an individual’s system, but the result of a collective documentation work, which is based on the study of the previous versions of the Treasures Game. But in the texts presenting these previous versions, as we saw, not enough was said about the way of dealing with the necessity of the list, even though the researchers present this necessity as critical in the teaching process. In this respect, we have shown how the teacher’s strategic system, as a system of prior intentions, is designed to achieve her two-fold purpose (failure of internal memory, necessity of a writing strategy), by standing to the side. We argue that it is impossible to understand the joint action of the teacher and the students, in this classroom, without acknowledging this two-fold purpose, which is purpose of the collective. But taking into account this collective purpose is not sufficient. We try to show that it is necessary to document the way the teacher, against this common ground, puts in place actual strategies that concretize the strategic rules that monitor his behavior. To understand the concrete action of a teacher, even though it has been designed in a collective documentation work, one has to acknowledge the teacher’s ‘feel for the game’ (Bourdieu, 1990) that enables her to rationally improvise, and to reach the collective goals beyond the collective preparation. In this respect, teachers could be seen as ‘instructional designers’ (Chapter 17) to the extent to the results of their improvisation modify the research design.

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In this perspective, a thought style, conceived of ‘the readiness for directed perception and appropriate perception of what has been perceived’ is a precious support for people’s practices, but it does not provide people with all the ‘solutions’ of the practice. In this study, the classroom concretization of prior intentions, as a strategic system, rests on the teacher’s capacities to enact a particular way of ‘standing to the side’ within the joint action. In the Treasures Game situation, as it was implemented here, we have to acknowledge that this enactment is not easy. It seems that a major reason for this difficulty could be the ‘lack of intentiveness,’ to use Baxandall’s neologism, of the scientific texts the collective was using. The resources and documents embed purposefulness, but in some cases, not enough.

References Amade-Escot, C., & Venturini, P. (2009). Analyse de situations didactiques: Perspectives comparatistes. Dossiers des Sciences de L’éducation. Numéro Spécial, 20. Baxandall, M. (1985). Patterns of intention: On the historical explanation of pictures. New Haven, CT: Yale University Press. Blumer, H. (2004). George Herbert Mead and human conduct. Walnut Creek, CA: AltaMira Press. Bourdieu, P. (1990). The logic of practice. Cambridge: Polity Press. Bourdieu, P. (1992). Language and symbolic power. Cambridge: Polity Press. Bourdieu, P., & Wacquant, L. (1992). An invitation to reflexive sociology. Cambridge: Polity Press. Brousseau, G. (1997). The theory of didactic situations in mathematics. Dordrecht, The Netherlands: Kluwer. Brousseau, G. (2004). Les représentations: étude en théorie des situations didactiques. Revue des sciences de l’éducation, 30(2), 241–277. Clark, H. (1996). Using language. Cambridge: Cambridge University Press. Dewey, J. (1922). Human nature and conduct. New York: Modern Library. Douglas, M. (1987). How institutions think. London: Routledge. Duranti, A. (2006). The social ontology of intentions. Discourse Studies, 8(1), 31–40. Eilan, N., Hoert, C., Mc Cormack, T., & Roessler, J. (2005). Joint attention: Communication and other minds: Issues in philosophy and psychology. Oxford: Oxford University Press. Fleck L. (1979). Genesis and development of a scientific fact. Chicago: The University of Chicago Press. Gibson, J. J. (1979). The ecological approach to visual perception. Hillsdale, NJ: Lawrence Erlbaum Associates. Hintikka, J., & Sandu, G. (2006). What is logic? In D. M. Gabbay, P. Thagard, & J. Woods (Eds.), Handbook of the Philosophy of Science. Volume 5, 20: Philosophy of Logic. London: Elsevier. Ligozat, F. (2008). Un point de vue de didactique comparée sur la classe de mathématiques. Etude de l’action conjointe du professeur et des élèves à propos de l’enseignement/apprentissage de la mesure des grandeurs dans des classes françaises et suisses romandes. Thèse de Sciences de l’Education, Université de Genève et Université d’Aix-Marseille. Mead, H. G. (1934). Mind, self, and society. Chicago: University of Chicago Press. Pacherie, E. (2008). The phenomenology of action: A conceptual framework. Cognition, 107, 179–217. Pérès, J. (1984). Use of the theory of situations with a view to identify didactic phenomena during a period of school learning. PhD thesis, University of Bordeaux II. Schubauer-Leoni, M.-L., Leutenegger, F., Ligozat, F., & Flückiger, A. (2007). Un modèle de l’action conjointe professeur-élèves: les phénomènes qu’il peut/doit traiter. In D. G. Sensevy &

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A. Mercier (Eds.), Agir Ensemble. L’action didactique conjointe du professeur et des élèves dans la classe (pp. 52–91). Rennes, France: PUR. Schubauer-Leoni, M.-L., Leutenegger, F., Ligozat, F., Flückiger, A., & Thevenaz-Christens, Th. (2010). Producing lists of objects to be remembered and communicated. The « treasure game » with 4 and 5 year old children. Fapse Genève University, Translated from French by N. Letzelter & F. Ligozat Searle, J. R. (1983). Intentionality. Cambridge: Cambridge University Press. Sebanz, N., Bekkering, H., & Knoblich, G. (2006). Joint action: Bodies and minds moving together. Trends in Cognitive Science, 10(2), 70–76. Sensevy, G. (in press). Overcoming fragmentation: Towards a joint action theory in didactics. In B. Hudson & M. A. Meyer (Eds.), Beyond fragmentation: Didactics, learning, and teaching. Leverkusen, Germany: Barbara Budrich Publishers. Sensevy, G., & Mercier, A. (2007). Agir ensemble. L’action didactique conjointe du professeur et des élèves. Rennes, France: PUR. Sensevy, G., Mercier, A., Schubauer-Leoni, M.-L., Ligozat, F., & Perrot, G. (2005). An attempt to model the teacher’s action in mathematics. Educational Studies in mathematics, 59(1), 153–181. Sensevy, G., Tiberghien, A., Santini, J., Laubé, S., & Griggs, P. (2008). Modelling, an epistemological approach: Cases studies and implications for science teaching. Science Education, 92, 424–446. Suchman, L. A. (1987). Plans and situated actions: The problem of human-machine communication. New York: Cambridge University Press. Tomasello, M. (2008). Origins of human communication. Cambridge: MIT Press. Wittgenstein, L. (1997). Philosophical investigations. Oxford: Blackwell.

Chapter 4

Resources for the Teacher from a Semiotic Mediation Perspective Maria Alessandra Mariotti and Mirko Maracci

4.1 Introduction The potentialities of ICT tools for learning have been extensively studied with a main focus on the their possible use by the students and the consequent benefits for them, but there has been the tendency to underestimate the complexity of the teacher’s role in exploiting these potentialities. In this chapter, assuming a semiotic mediation perspective (Bartolini Bussi & Mariotti, 2008), we will discuss different kinds of artefacts that are offered to the teachers to enhance the teaching–learning activity in the classroom. Thus, as Adler (Chapter 1) suggests, we shift “attention off resources per se, and refocus(es) it on teachers working with resources; on teachers re-sourcing their practice. Teachers’ “re-sourcing practice” can be viewed in at least two ways: firstly, it may be interpreted as exploiting resources and developing professional activity (e.g. professional growth), as explained and discussed by Gueudet & Trouche in the theoretical frame of documentational approach (Chapter 2). Secondly, it may be interpreted as exploiting resources in the classroom to achieve a specific educational goal. This latter sense opens the research direction that we followed in our study, assuming the specific frame of the Theory of Semiotic Mediation (TSM) (Bartolini Bussi & Mariotti, 2008). Such theoretical approach explicitly considers the role of the teacher and describes how she can exploit the use of an artefact, managing different didactical situations to make the expected semiotic process happen. Following Bartolini Bussi (1998), we describe the teacher’s action making use of the metaphor of orchestration. As argued in Mariotti & Maracci (2010), the term orchestration here can be related to what is labelled the didactical performance component of the instrumental orchestration within an instrumental approach (Chapter 14), but the objectives are different. In fact, the objective of the teacher’s orchestration within a semiotic mediation approach is not that of guiding students’ instrumental

M.A. Mariotti (B) Department of Mathematics and Computer Science, University of Siena, Siena, Italy e-mail: [email protected]


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geneses, but that of developing shared meanings, having an explicit formulation, de-contextualized from the artefact use, recognizable and acceptable by the mathematicians’ community. In the following, we want to go further in the description of the “use of an artefact”, discussing the use of different kinds of resources related to its functioning as a tool of semiotic mediation (Bartolini Bussi & Mariotti, 2008, p. 754).

4.2 Mediation and Teaching–Learning According to a Semiotic Approach The TSM is centred around the seminal idea of semiotic mediation introduced by Vygotsky (1978) and it aims to describe and explain the process that starts with the student’s use of an artefact and leads to the student’s appropriation of a particular mathematical content. The TSM addresses this issue combining a semiotic and an educational perspective, and elaborating on the notion of mediation while considering the crucial role of human mediation (Kozulin, 2003, p. 19) in the teaching–learning process. Taking a semiotic perspective means to acknowledge the central role of signs in the teaching–learning activity. The use of the term “sign” is inspired by Pierce. We assume an indissoluble relationship between signified and signifier. In the stream of other researchers (Arzarello, 2006; Radford, 2003) we developed the idea of meaning that originates in the intricate interplay of signs (Bartolini Bussi & Mariotti, 2008). Consequently, specific attention is paid to the processes of production of signs and of their transformation, which in turn is considered as evidence of learning. Fostering or guiding this process is a crucial issue and a demanding task for the teacher. In the following sections, we outline how one can organize a teaching–learning sequence by integrating the use of an artefact. Such description is developed around the key notions of semiotic potential of an artefact and of didactic cycle. Within this frame we describe different resources1 which can support the teacher in exploiting the semiotic potential of a given artefact.

4.3 The Semiotic Potential of an Artefact and the Didactical Cycle Following Hoyles (1993), one can speak about the relationship between artefact and knowledge as evoked knowledge. For experts the artefact may evoke specific knowledge corresponding to what is mobilized to solve specific problems.

1 Assuming a semiotic mediation perspective, Mariotti and Maracci (2010) address the issue of how an ICT tool can be a resource for the teacher.

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Fig. 4.1 Copy of a Roman abacus (1st century AD)

For example, the positional notation of numbers may be evoked by an abacus (Fig. 4.1). Similarly, a Dynamic Geometry System may evoke the classic “rule and compass” geometry. However, there is the need to distinguish between meanings emerging from the practice based on the use of the artifact and the mathematics knowledge evoked in the expert’s mind. The notion of semiotic potential of an artifact is meant to capture that distinction and to make it explicit. By semiotic potential of an artefact we mean the double semiotic link which may occur between an artefact and the personal meanings emerging from its use to accomplish a task and at the same time the mathematical meanings evoked by its use and recognizable as mathematics by an expert.2 Thus, taking a semiotic perspective we will focus on the semiotic processes occurring in the classroom when the teacher manages the use of an artefact according to specific didactical goals. According to the TSM, the teaching–learning process starts with the emergence of students’ personal meanings in relation to the use of the artefact to the accomplishment of a task. The emergence is witnessed by the appearance of specific personal signs – the unfolding of the semiotic potential. The process of semiotic mediation develops in the collective construction of shared signs, related to both the use of the artefact and to the mathematics to be learnt (Fig. 4.2). The evolution of signs can be promoted through the iteration of didactic cycles (Fig. 4.3) where different categories of activities take place, each of them contributing differently but complementarily to develop the complex process of semiotic mediation: (a) activities with the artefact on the basis of the tasks purposefully designed for promoting the emergence of signs referred to artefact-use; (b) activities

2 The distinction between personal meanings and mathematical meanings may remind of Brousseau’s distinction between knowing (in French: connaissance) and knowledge (in French: savoir) (Brousseau, 1997). Even if they are not in antithesis, the two perspectives cannot be reduced to one another: the former stresses the semiotic dimension of the teaching--learning processes, which is in the shadow for the latter.

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Personal meanings Artefact signs

Use of an artefact for accomplishing a task

Mathematical meanings Mathematical signs

Teacher’s mediation

Fig. 4.2 Semiotic potential of an artefact and teacher’s mediation

of individual writing involving students in semiotic activities concerning written productions. For instance, students might be asked to write individual reports on the previous activity with the artefact, reflecting on their own experience, and raising possible doubts or questions; (c) classroom discussions which constitute the core of the semiotic process. According to the idea of mathematical discussion (Bartolini Bussi, 1998), the teacher’s main objective is to exploit the semiotic potentialities of individual contributions that move towards mathematical meanings. In other words, assuming a semiotic mediation perspective calls for the establishment of a specific activity format (Chapter 5) which consists of the iteration of didactical cycles. Fig. 4.3 The didactical cycle Activities with the artefact

Individual production of signs

Collective production of signs

Though the ICT tool can be considered the key resource, the use of which is not limited to the initial phase (Mariotti & Maracci, 2010), other resources may support the teacher’s actions throughout all the didactical cycle. In this contribution we will consider the specific resource provided by written texts. According to their different potentials, the teachers may exploit different types of texts – either internal or external to the mathematics class community: the texts produced by the teacher to describe the task, those produced by students in the different moments of the teaching–learning sequence, or the texts provided by a historical source. To show the potentialities of such texts as resources for the teacher, we will focus on how the teacher can effectively use them in the classroom for triggering and

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sustaining semiotic mediation, and hence on the semiotic processes activated by the students and prompted by the use of those texts.

4.4 Texts as Resources for the Semiotic Mediation Process We consider the term “text” in a broad sense including any kind of organized set of signs, also belonging to different semiotic systems, although in the following we will limit ourselves to considering mostly written verbal texts. A text provides a number of signs organized in a stable structure that may become object of reflection and discussion, and for this very reason the text has the potential of triggering the production of new signs. To understand the specific types of resource we intend to discuss, we can explicitly refer to Wartofsky’s (1979) classification of cultural artefacts in primary, secondary and tertiary artefacts. As Wartofsky explains: What constitutes a distinctively human form of action is the creation and use of artefacts, as tools, in the production of the means of existence and in the reproduction of the species. [. . .] Primary artefacts are those directly used in this production; secondary artefacts are those used in the preservation and transmission of the acquired skills or modes of action or praxis by which this production is carried out (1979, p. 202).

Secondary artefacts are therefore representations of modes of actions with artefacts. When representations of modes of actions are drawn on systems of signs, the element of convention comes to play a large role generating new semiotic systems. The relationship between the abacus and the positional system of number representation is a paradigmatic example. There is also another class of artefacts that Wartofsky calls tertiary artefacts and may be often an evolution of secondary artefacts. [. . .] which can come to constitute a relatively autonomous ‘world’, in which the rules, conventions and outcomes no longer appear directly practical, or which, indeed, seem to constitute an arena of non-practical, or ‘free’ play or game activity (1979, p. 202).

Examples of tertiary artefacts are the mathematical theories which organize the mathematical models constructed as secondary artefacts. Assuming such a perspective, Bartolini Bussi, Mariotti, & Ferri (2005) discuss the semiotic potential emerging from combining the use of a primary artefact, a perspectograph, and secondary artifacts, texts drawn from ancient treatises of painting, to form together the base of the development of tertiary artefacts. The semiotic potential of such combination of intertwined elements was based on the potential of different artefacts of evoking each other. The polysemy or multi-voicedness of cultural artefacts make them useful as teacher resources to foster mathematical discussions in the classroom (Bartolini Bussi, 1998): because of its evocative potential, a text may be used to fuel the evolution of signs, objective of a semiotic mediation process. A condition for a text for being potentially useful is that of being interpretable by the students in terms of their experience with the primary artefact in play as well as with the mathematics. In our teaching experiments, the teacher utilized texts of different types:

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(a) written texts produced by a pair of students and (b) written text drawn from a historic source. Each type of text has different potentialities with respect to the semiotic mediation process. Consider the case (a). A text written by a classmate may assume the status of the simulation of a pair interaction, where one of the interlocutors expresses herself through a written text. In the other cases (b), an asymmetry appears immediately between the reader and the voice expressed by the text whose authority may come from the well-known reputation of the author or from official reference to the community of mathematicians. Generally speaking, reading an original source is a specific activity on the basis of an hermeneutic effort referring to the tension between the meaning of the text in the perspective of the author and the meaning for the reader in her personal perspective (Jahnke, Arcavi, Barbin, Bekken, Furinghetti, Idrissi, da Silva, & Weeks, 2000). In the following sections, we will show how all the resources described above can be used in synergy by the teacher. We start by illustrating the key role of the text that describes the task (the formulation of the task) in fostering the unfolding of the semiotic potential; then we discuss examples concerning the teacher’s utilization of written texts: texts produced by students and a text drawn from a historic source. The examples are drawn from the same teaching experiment centred on the use of Cabri (Laborde & Bellemain, 1995), which involved Italian and French 10th grade classes (for details, see Falcade, 2006; Falcade, Laborde, & Mariotti, 2007).

4.5 The Teaching Experiment The educational goal was to use Cabri for introducing students to the idea of function as co-variation. The design of the sequence of activities was consistent with the structure of the didactic cycle. Students’ productions and audio-recordings of classroom activities were collected and analysed. The idea of function was introduced within a geometrical setting, as a relation between points of the plane which are linked through a geometrical construction. One can recognize the possibility of establishing a rich system of connections between certain components of Cabri and their use – such as basic points and points obtained through a construction, the dragging tool and its effect on the different kinds of points, the trace tool and the macro tool – and the mathematical notion of function and all the related notions – such as that of independent and dependent variables, parameter, domain, image and graph. The first activity proposed to the students concerned the exploration of the effect of a macro-construction. They had to explore systematically the effect of dragging a point, experiencing both the free and the conditioned movement. The aim was to introduce the notion of variation and co-variation as a base for a definition of function. The trace tool was extensively used, and the study of the trajectories of the different points contributed to the appropriation of rich meanings embedding a

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Fig. 4.4 What appears on the screen activating the trace tool

dynamic component, for the notion of domain and image of a function (Fig. 4.4). Collective discussions were orchestrated by the teacher with the aim of formulating shared mathematical definitions of the notions of function, domain and image. Later on, after the introduction of numerical functions, the students were assigned the problem of providing a geometric representation of a numerical function. Once obtained the solutions to this problem, the students were asked to interpret an excerpt of a text by Euler addressing the same question, and to compare their own answer with the method described by Euler. The ensuing collective discussion had the aim of sharing the individual interpretations and promoting the evolution of personal meanings towards the mathematical meaning of graph of a function.

4.6 The Text Formulating the Task The design of the starting activity was intended to foster the students’ production of personal signs related to the use of the dragging tool that could subsequently evolve towards the desired mathematical signs. Specific attention is put in the formulation of the task, specifically in the choice of the words referring to different aspects to be focussed on. Task. Displace all the points you can. Observe what moves and what does not. Explore systematically, that is, displace one point at time and note which points

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move and which do not. Summarize the results of your exploration in the table below.

Points which can be dragged Points which move Points which do not move

The expressions “displace”, “move” and “drag” are present, and are used with different meanings. “Displace” and “drag” are used as nearly synonymous to refer to the direct action made by the user upon the points. There is a slight difference between the two words, as the first is a word of “natural language”, while the second is a word of “Cabri language”: “move” is used to refer to the movement of a point as a result of direct or indirect action upon it. This difference is not made explicit. It is left to the pupils to make sense of this difference through their exploration in Cabri. The transcripts of students’ conversations show interchangeable use of the expressions “move” and “displace”, until the students realize and express the distinction between what moves and what can be displaced. We can therefore say that as an effect of working on the task, the students produced and shared two distinct signs, “displace” and “move”. These two signs directly refer to the activity with Cabri, but they have the potential of being related to the mathematical signs of independent variable (point that can be moved) and dependent variable (points that move but cannot be moved). The following exchange between two students is a good example of what can be expected during the solution of this task. 30. Egi: I wanted to ask . . . points that can be displaced, in what sense . . . that every time move. 31. Mar: Can be displaced . . . I told you its hard . . . all of them move but you can displace only three of them. H moves under the action of A, B and P. A semiotic perspective introduces a specific dimension in the design of the task: the production of certain signs can be considered the effect of the specific task, but what is crucial is not only what is requested to be done, but also how such request is worded.

4.7 Written Texts Produced by Students The second example concerns an episode occurred during a collective discussion designed with the twofold aim of clarifying and systematizing the ideas that emerged during the first phase of activities with the artefact, and finally, expressing these ideas in a “mathematical statement”: the definition of function. This kind of

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activity was not new for the students, who were accustomed to engaging in defining processes. The discussion developed over three lessons (approximately 5 h). The first phase started when the teacher recalled the recent activities with Cabri: both the pupils and the teacher referred to Cabri tools and phenomena experienced during those activities. Different elements in play were highlighted by the students and explicitly related to the corresponding mathematical ideas of (independent and dependent) variable, domain and image. Then a crucial point arose: the students realized that characterizing a function implies determining common features and this corresponds to determining when two functions can be said to be “equal”. Grasping the opportunity, the teacher shifted the focus of the discussion and asked the students, working in pairs, to try and formulate a “definition of equal functions”. Cabri was available, and students were prompted to check different examples to test their conjectured definitions. Finally, students were asked to express through a written text a “definition of equal function” that took into account the ideas from the activity in Cabri. The following definitions were proposed. And–Ale: “Two functions are equal if they have the same domain and the same image for all the domains subsets of the original domain which defines the functions.” Gio–Fed: “Two functions are equal if they have the same number of variables, the same domain, and the same procedure (in the construction of the macro).” Mar–Gab: “Two functions are equal when they have the same image and (when) the same domain is fixed (for both).” Tiz–Seb: “In our opinion two functions are equal if having the same domain and the same definition procedure they have the same image. If either the domain, or the definition procedure, or the image are not equal, neither the functions are not equal.” Almost all the definitions mentioned the main elements in play: the domain, the procedure and the image. The first definition presents a characterization in which the domain is conceived in terms of subsets and uses a quantifier (“for all”). This is a static definition in terms of sets: no reference to variation is made. Though it may appear quite strange, nevertheless this characterization originated from the pupils’ previous experience, specifically from the relation built between the idea of image of a function and that of trajectory coming from the use of the trace tool. This will emerge from the collective discussion.

4.7.1 Comparison of Texts for Sharing a Definition All the produced texts above highlight clear potentialities with respect to developing a relationship between meanings emerging from the Cabri experience and the mathematical meanings at stake. Hence, these texts may be considered artefacts, secondary artefacts with respect to the primary artefact Cabri. The teacher

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decided to exploit the semiotic potential of these secondary artefacts and organized a collective discussion centred on the comparison between the four produced texts. She provided the students with a copy of the produced texts, left a few minutes for reading them, and opened the discussion clarifying the aim: to formulate a shared definition of “equal functions” starting from the four given produced texts. In the following, we report some excerpts from the transcript of the discussion. Excerpt 1 1. T (teacher): [. . .] we must find an agreement on a definition, which can be one of these, or an improvement of one of these, or the fusion of these . . . We must decide. 2. And: According to me, Gab’s and Mar’s definition is wrong. 3. T: So, And, according to you, Gab’s and Mar’s definition is wrong. Let’s read it again (she reads again) “two functions are equal when they have the same image and (when) the same domain is fixed for both”. 4. And: Because to get to the same image, someone could pass through . . . we could have several journeys; in fact, if there were a subset of the domain . . . we can’t say that the functions are. . . 5. T: . . . Tiz, could you try to explain it? 6. Tiz: Yesterday, we saw that we can, by doing the same domain, we can create the same image and this, with different functions (procedures). After declaring the main goals of the activity, the teacher moderated the interventions focussing on one of the produced texts. The intervention of And (4) made explicit the origin of his and Ale’s definition: the equality of function is related to the coincidence of the trajectories for each subset of the domain. Still, the reference to the use of the trace tool was not explicit; rather it was introduced by the metaphor of journeys (4). Realizing that perhaps some students could not share And’s way of reinvesting the experience with the trace tool, the teacher prompted an explanation (5). Tiz intervenes referring to previous work in Cabri and raised the issue of considering explicitly the procedure that realizes a function. In the following, other interventions focussed on this same issue until the teacher redirected the discussion to the comparison between the definitions and asked to go back to And and Ale’s text. The re-formulation of this sentence in terms of procedure was collectively achieved. Excerpt 2 44. T: Let’s read the text. You say that if they have the same domain and the same image for each subset of the domain. . . 45. Tiz: But, here it’s like to have the same procedure. 46. T: Hummm, and why it’s like to have the same procedure? 47. Several voices: . . . Because . . .

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48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.


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Gab: . . . As we go further, the subsets of the domain and vice versa . . . T: Do you agree, And? Gio: The domain is the plane, then you have the straight line, then a segment . . . T: What are these? And: The domain can be whatever. Gio: They are subsets. T: And then, the procedure, what does it do? That is to say, I . . .. Where does it start from? And: The domain can be one point too . . . if we want! T: The subset of the domain can be one point too. Oh! And: For whatever point, we get the same point of the image. T: And this gives the idea to say that . . . Gio: I’m doing the same procedure. And (together with Gio): I’m doing the same procedure. T: I’m doing the same procedure. Therefore, for whatever point of what? And: For each point of the domain we have the same . . . as the result of the function, the same point of the image. T: Do you agree? (referring to Tiz) Perplexed silences. The teacher writes on the blackboard and reads: “For each point of the domain, we have as the result of the function, the same point as the image”.

At the beginning, students seemed to accept that “to have the same domain and the same image for each subset of the domain” it’s like “to have the same domain and the same procedure” (45, 48 and 50). The agreement with And was based on previous experiences in Cabri, when students’ actions with the tools generated different phenomena according to which And and Ale’s definition appeared sensible. Nevertheless, when the teacher asked for an explicit agreed-upon statement, students remained silent and perplexed (64). In fact, the conclusion that was written on the blackboard by the teacher does not explicitly recognize the key role of the procedure in the identification of a function. In addition, it requires a conceptual move from an experience-based definition, tightly tied to Cabri activities, to a purely mathematical definition, where any reference to moving points and procedures disappears. Further discussion was needed to reach the acceptance of comparing functions point by point. In summary, relying on the potential of students’ produced texts for developing a relationship between experience-based meanings and mathematical meanings, the teacher decided to exploit these produced texts launching a collective discussion based exactly on their comparison. During the discussion she guides the semiotic process towards the inter-subjective construction of a specific mathematical meaning which may be quite different from the students’ personal meaning. We claim that texts produced by the students provide a powerful resource for the teacher.

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4.8 Texts Drawn from History In this section, we discuss the potentialities of a text drawn from a historical source: an excerpt of a text by Euler dealing with the problem of providing a geometric representation of a numerical function: Introduction in Analysis Infinitorum, Tomus secundus, Theoriam Linearum curvarum (1748).3 The students were presented with the excerpt4 and were asked to interpret it. The text analysis shows the potential of evoking the mathematical meanings of graph of a function, and also actions and meanings related to the use of Cabri. For this reason, we consider it as a resource to be exploited to develop a semiotic mediation process. As mentioned earlier, because of its belonging to the shared cultural background, a text drawn from a historical source brings the voice of mathematics to the classroom, through the voice of a famous mathematician. In addition, to the extent to which it can evoke to the students their experience with Cabri, it can contribute to establishing a connection between the students’ personal meanings and the mathematical meanings. The text presented to the students contained Euler’s description of the main steps of the graph construction. For the reader’s convenience, we organize the text into a sequence of steps and mark the missing paragraphs. First Step: Representing the independent variable x as a variable segment AP on a straight line RS: 1. A variable quantity is a magnitude considered in general, and for this reason, it contains all determined quantities. Likewise in geometry a variable quantity is most conveniently represented by a straight line RS of indefinite length [. . .] Since in a line of indefinite length we can cut off any determined magnitude, the line can be associated in the mind with the variable quantity. First we choose a point A in the line RS, and associate with any determined quantity an interval of that magnitude which begins at A. Thus a determined portion of the line, AP, represents the determined value contained in the variable quantity. 2. Let x be a variable quantity which is represented by the line RS, then it is clear that any determined value of x which is real can be represented by an interval of the line RS. For instance, if P is identical with the point A then the interval AP vanishes and represents the value x=0. The farther removed from A the point P is, the greater the definite value of x represented by the interval AP. The interval AP is called the abscissa. The abscissas manifest the determined values of x.

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Introduction to Analysis of the Infinite, Book II. In the text presented to the students some parts were omitted and diagrams and graphs were removed. 4

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Second Step: Representing the dependent variable as a segment PM on the line perpendicular to RS and passing through P. 3. [. . .] 4. Since the indefinite straight line represents the variable x, we would like to see how a function of x can be most conveniently represented. Let y be any function of x, so that y takes on a determined value when a determined value is assigned to x. After having taken a straight line RAS to denote the values of x, for any determined value of x we take the corresponding interval AP and erect a perpendicular interval PM corresponding to the value of y [. . .] Third Step: Associating functions with curves. 5. [. . .] 6. For all determined values x of the line RS, at the point P we erect the perpendicular PM corresponding to the value of y, with different M’s for different P’s [. . .]. All the extremities, M, of the perpendiculars form a line that may be straight or curved. Thus, any function of x is translated into geometry and determines a line, either straight or curved, whose nature is dependent on the nature of the function. 7. In this way, the curve which results from the function y is completely known, since each of its points is determined by the function y. At each point P, the perpendicular PM is determined, and the point M lies on the curve. [. . .] Although Euler certainly had no idea of such a technical drawing device as Cabri, the dynamic description of the graph that he provides is highly consistent with what could be obtained using the Cabri tools. The variation of the independent variable x is represented by the variation of the segment AP (1 and 2), and the variation of the dependent variable PM is implied by the variation of the independent variable. Co-variation is made explicit by the direct link between the two segments. The reference to a point P, moving on a line in Cabri is immediate: in its motion, P “drags” the segment PM, whose length changes in function of the position of P. Thus, the metaphors used in the description of the graph may be directly related to Cabri tools (line, point on a line, dragging . . .). Finally, the potential reference to the trace tool is also very clear (all the extremities, M, of the perpendiculars form a line which may be straight or curved).

4.8.1 The Activities Centred on Euler’s Text Before the activity with Euler’s text the students were introduced through a lab activity to the geometric representation of a function, that is the representation of the variation of a numerical variable and the co-variation of two numerical variables

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linked through a given function. This is the same problem that is at stake in the text by Euler that shortly after the students were asked to interpret. In the following session, the students were asked to read and try to make sense of Euler’s text. Though the activity took place in the computer lab, the students were not explicitly asked to use Cabri. As homework, the students had to explain what they had understood about the method proposed by Euler and to compare it with the method they elaborated for representing a numerical function geometrically through Cabri. A final discussion aimed at sharing a definition of graph of a function – as a geometric representation of a numerical function – and a method for realizing such representation.

4.8.2 Unfolding of the Polysemy of the Text The analysis of the students’ written texts reveals that almost all of them accomplished the interpretation task by providing a paraphrase of the text. The paraphrases are characterized by the use of the mathematical terms previously introduced, such as function, independent or dependent variable. It seems that, the use of these terms fostered the interpretation of the text, that is using these terms helped the students to penetrate the text. Egi: [. . .] the values that are enclosed by AP represent the independent variable x, called abscissa. Thus, to represent a function of x [Euler] decided to take a line perpendicular to x, let’s call it y. To represent any value of the line RS, so that according to the variation of AP (independent variable) the numbers of PM vary too. Besides the use of the specific terminology, many students referred, more or less directly, to the possible use of Cabri. Fed: “Euler describes the function y with independent variable x, whose domain is an unlimited straight line. [. . .] The magnitude of AP, which varies on the straight line according to the movement of P, is called abscissa.” Mar: “The distance of P from A varies as x varies, that [P], as a consequence drags with itself the line MP, perpendicular to RS. This line has been called y; in this way one can say that when x varies y varies, so that a function is created.” The use of Cabri is evoked through metaphors linked to the idea of movement as variation, which is at the core of any DGE. Below is an example where the trace tool is explicitly referred to: Fil:

“[. . .] Let us consider the function y(x), the dependent variable will change according to the variation of the values given to x; this function can be represented geometrically, drawing a straight line RAS which represents the

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values of x, for each value of the variable x assigned to AP, one will draw a perpendicular PM to RAS, such that PM is equal to y. Now if we apply the trace tool at the point M, we find all the points of the function of x [. . .]” Finally, it is worthwhile noticing that some students autonomously decided to use Cabri to make sense of some parts of the text of Euler (we remind the reader that no diagrams were available in the text): Gab: “[. . .] I have no perplexities, though I met some difficulties in understanding the paragraphs 4 and 6, until I used Cabri and reproduce . . ..” Because of its evocative power, Euler’s text works as a secondary artefact related to the primary artefact Cabri: the dynamic description of variables, function and graph provided is consistent with what could be experienced in Cabri. At the same time, realizing Euler’s construction within Cabri may help making sense of the text itself. Hence, the two artefacts have the potential of evoking each other. The articulation of the world of Cabri and the mathematical world evoked by the text can offer the teacher a resource to be exploited in the collective discussion: a multiple perspective relating the activities in Cabri to the mathematical meanings of graph. Indeed, all this happened in the subsequent classroom discussion that was orchestrated by the teacher, with the aim of sharing a definition of graph of a function and a method for producing it. In that discussion, after a first phase in which the students shared their understanding about the method described in Euler’s text, the teacher decided to ask for a drawing illustrating Euler’s method with the temporarily-notdeclared aim of soliciting the reference to Cabri (40. T: “Would anybody be able to make a drawing [. . .] there was not even a drawing [in Euler’s text] [. . .]”). The need of a direct experience of executing the operations to understand the text was also recognized by students: at first the students and the teacher alternated at the blackboard, trying to produce together a suitable drawing, then the class agreed to use Cabri to illustrate Euler’s method: “now we have Cabri which can help us a little. As a matter of fact, what we are going to do is to try to construct all this stuff within Cabri” (182). Thus, Cabri is given the role of contributing to clarify the text. The intertwinement between the text (secondary artefact) and Cabri (primary artefact) is evident.

4.9 Conclusions Within the frame offered by the TSM the use of an artefact has a twofold nature: on the one hand, it is directly used by the students as a means to accomplish a task; on the other hand, it is indirectly used by the teacher as a means to achieve specific educational goals. In this sense, a specific ICT tool can be considered a fundamental resource for the teacher. Nevertheless, according to the model of the teaching action

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provided by the TSM, other types of resources meant to foster and enhance the semiotic mediation process, can be outlined. Asking students to work in pairs at the computer is expected to foster social exchange, accompanied by production of signs related to the use of the artefact, words, sketches, gestures and the like. In this respect not only the specificity of the task but also in particular its formulation constitutes basic resources to trigger the unfolding of the semiotic potential provided by the artefact. Moreover, students may be involved individually in different semiotic activities concerning written productions. All these activities are centred on semiotic processes leading to the production and elaboration of signs, related to the previous activities with tools. Wartofsky’s (1979) classification into primary and secondary artefacts helped us make explicit the synergy between Cabri (primary artefact) and different kinds of written texts related to it (secondary artefacts). Such synergy made these artefacts resources for the teacher to exploit according to her didactic goals. In summary, the realization of the evocative potential of the primary and the secondary artefacts may feed the semiotic mediation process and, thus, foster the evolution towards the mathematical meanings at stake (that is the “tertiary artefacts” which can frame and organize what has been constructed in relation to the use of – primary and secondary – artefacts). As a final remark it seems important to stress a particular contribution offered by this study with respect to teachers’ education. Beside the theoretical contribution given by this study for the development of the TSM, there are interesting implications concerning teachers’ classroom practice and teacher education in general. The awareness of the semiotic potential of written texts and the capacity of selecting and exploiting them in the classroom could become an educational aim for teacher education. The functioning of a text as a resource for developing a semiotic mediation process depends on the possibility of triggering an interpretative process. This may happen either through an explicit or through an implicit request, such as a request of comparison or elaboration. Interpreting concerns both meaning making and expressing, and consequently producing and elaborating signs. In exploiting the polysemy of a text the teacher intentionally articulates meanings coming from the experience with a primary artefact and meanings emerging from a secondary artefact.

References Arzarello, F. (2006). Semiosis as a multimodal process. Relime V1, Especial, 267–299. Bartolini Bussi, M. G. (1998). Verbal interaction in mathematics classroom: A Vygotskian analysis. In H. Steinbring, M. G. Bartolini Bussi, & A. Sierpinska (Eds.), Language and communication in mathematics classroom (pp. 65–84). Reston, VA: NCTM. Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 750–787). Mahwah, NJ: LEA. Bartolini Bussi, M. G., Mariotti, M. A., & Ferri, F. (2005). Semiotic mediation in the primary school: Dürer glass. In M. H. G. Hoffmann, J. Lenhard, & F. Seeger (Eds.), Activity and

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sign – Grounding mathematics education. Festschrift for Michael Otte (pp. 77–90). New York: Springer. Brousseau, G. (1997). Theory of didactical, situations in mathematics. Dordrecht, The Netherlands: Kluwer. Falcade, R. (2006). Théorie des Situations, médiation sémiotique et discussions collective, dans des séquences d’enseignement avec Cabri-Géomètre pour la construction des notions de fonction et graphe de fonction. Grenoble: Université J. Fourier, unpublished doctoral dissertation. Falcade, R., Laborde, C., & Mariotti, M. A. (2007). Approaching functions: Cabri tools as instruments of semiotic mediation. Educational Studies in Mathematics, 66(3), 317–333. Hoyles, C. (1993). Microworlds/schoolworlds: The transformation of an innovation. In C. Keitel & K. Ruthven (Eds.), Learning from computers: Mathematics education and technology (pp. 1–17). NATO ASI Series. Berlin: Springer. Jahnke, H. N., Arcavi, A., Barbin, E., Bekken, O., Furinghetti, F., Idrissi, A., et al. (2000). The use of original sources in mathematics classroom. In J. Fauvel & J. van Maanen (Eds.), History in mathematics education. The ICMI study (pp. 291–328). Dordrecht, The Netherlands: Kluwer. Kozulin, A. (2003). Psychological tools and mediated learning. In A. Kozulin, B. Gindis, V. S. Ageyev, & S. M. Miller (Eds.), Vygotsky’s educational theory in cultural context (pp. 15–38). Cambridge: Cambridge University Press. Laborde J.-M., & Bellemain, F. (1995). Cabri-géomètre II and Cabri-géomètre II plus [computer program]. Dallas, TX: Texas Instruments and Grenoble/France: Cabrilog. Mariotti, M. A., & Maracci, M. (2010). Un artefact comme outil de médiation sémiotique: une ressource pour le professeur. In G. Gueudet & L. Trouche (Eds.), Ressources vives. Le travail documentaire des professeurs en mathématiques (pp. 91–107). Rennes, France: Presses Universitaires de Rennes et INRP. Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37–70. Vygotsky, L. S. (1978). Mind in society. The development of higher psychological processes. Cambridge: Harvard University Press. Wartofsky, M. (1979). Perception, representation, and the forms of action: Towards an historical epistemology. In M. Wartofsky (Ed.), Models, representation and the scientific understanding (pp. 188–209). Dordrecht: D. Reidel Publishing Company.

Reaction to Part I Resources Can Be the User’s Core Bill Barton

How does a resource become “lived”? If we may play on the etymology for a moment, becoming “lived” means enlivened, to get full of life, to become, to be born. The four chapters of this part tell us how a resource enters the world “mewling and puking in the nurse’s arms” (Shakespeare, As You Like It, Act II, Scene 7). The bard gets it right, again. The carer of the resource is responsible for nurturing and shaping its potential, helping it to grow, ignoring the unseemly squeaking, and clearing up the spilt milk. On the one hand, Adler shows us how the nurse invests herself into the new life: how does teacher knowledge emerge during events in the classroom? Gueudet and Trouche want us to focus on the nurse’s actions in caring for the baby: how do resources become transformed in a particular teacher’s hands? Sensevy, on the other hand, wants us to look at the nurse’s aims: how are the teacher’s actions driven by developing intentions? Mariotti and Maracci ask us to watch the baby itself as it interacts with the nurse and others in the world: how can resources change the way people think and act? The common stance is one of mediation, the transformation of resources by teachers as they are reborn from a prior, relatively fixed state to a new dynamic existence in action in the classroom. I am tempted to play with etymology yet again. Mediation does not derive from media, but in this section we are being asked to pay attention to media. Famously, “the medium is the message” (McLuhan, 1964), or, more appropriately for this context, The Medium is the Massage (McLuhan and Fiore, 1967). Not only the resource but also the form of the resource alter the way it can be used and transformed by a teacher. We are presented with four different ways to conceive of resource mediation. The authors draw heavily on established theory, modifying it for their purpose, and we are left with a strongly grounded feeling. What do the four perspectives offer us? Adler draws on social practice theory to present us with an integrated view. Teacher’s knowledge, their access to texts, the classroom environment, the language

B. Barton (B) Department of Mathematics, University of Auckland, Auckland 1142, New Zealand e-mail: [email protected]

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resources available, and the pre-defined curriculum merge through the act of teaching to legitimise a particular view of mathematics. Teachers, whether they like it or not, whether they are prepared for it or not, are central in this process and bear its responsibility. I understand immediately the research-based wisdom that teachers are the most important factor in learning. As mathematics educators we are asked to pay attention to how we, in teacher education, open or close opportunities for particular mathematical perceptions. Gueudet and Trouche introduce documentational genesis, the evolution of teaching materials in the hands of a teacher drawing on several resources for a particular classroom outcome. On the basis of activity theory and an instrumental approach, the focus on documents changes the way we look at teaching. Tracing documentary evolution enables us to see, physically, the teacher’s moves in the game of instruction, and also the development of a teacher’s ideas, intentions and pedagogic orientation over a long time. For me, the importance of this perspective is the way it highlights continual change. I believe that many teachers would regard their practice as relatively stable – and many developers and education researchers comment on teachers’ resistance to change. A documentational genesis is likely to prove the lie to such statements, and thereby challenges us all to think again about the way development can be influenced. For example, it will reveal constant but gradual change – the antithesis of many programmes of teacher development. Sensevy also relies on documentation, and follows Bourdieu’s idea of a social game and Brousseau’s didactic contract. He asks us to pay attention to the way a teacher sets up the game (or contract), embedding explicit pedagogic intentions in both the resources and the elaboration of the game. The research data forces me to consider the ways the process goes wrong: during classroom interaction the response to the resources can diverge from the intention. This creates a didactic moment, a decision point, a phenomenon investigated by many researchers. Mason (1999, 2010) also focuses on teaching moments, and Schoenfeld (1987, 2008) persists in his analysis of classroom decisions. Schoenfeld’s KOG analysis (knowledge, orientations and goals) of teacher behaviour resonates with Sensevy’s work. My response to this perspective is to wonder anew how to prepare for such moments. The very act of Sensevy’s research sensitised his teachers to their predicament. They knew that they would be questioned on their actions at the critical moments, and it was almost as if that knowledge altered the decision they made. Can heightened awareness be a mode of professional development? How could we bring this about? Mariotti and Maracci turn our attention to the learner to learn about the teacher’s mediation of resources. Semiotic mediation of artefacts require us to investigate the meaning given to a resource, and how that meaning changes (or can be changed) with teacher action. A key word I take from the chapter is “invoking”. Meaning is invoked; learning does not reside in the resource, it is invoked by it. My reaction, then, is to think about the invoking power of a resource. This gives us, for example, a way of investigating technology: does modern technology have a greater power to invoke, perhaps because it is interactive and dynamic compared to texts. Are recorded lessons to be seen in the same way? Mariotti and Maracci note in their last paragraph that written texts have the advantage (over spoken words and gestures)

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of permanence and reproducibility. No longer! Video recordings, where the richness of gesture and articulation are preserved, may have more invoking power than conventional texts. In several places Mariotti and Maracci themselves invoke the multiple roles of resources: the double semiotic link, Wartofsky’s triple classification of artefacts and Winsløw’s pragmatic and didactic roles. Artigue (2002) distinguishes between three “values” when discussing the role of technology in mathematics education. The pragmatic value or productivity of the technology: how it helps us in the mathematical action we are currently undertaking. The epistemic value: how technology helps students understand the mathematical objects they are dealing with. The heuristic value: how technology contributes to understanding future or more advanced concepts. Hence, not only is the mediation of the resource transforming a generalised object into an object-in-action, but the mediation occurs on several levels simultaneously. To what extent are teachers aware of this in general, and in the moment? Taken collectively, the four chapters raise the issue of teacher awareness of their mediation role with respect to resources. Assuming the analyses are well-founded (and I have argued that indeed they are), we must ask ourselves how teachers come to know to transform as well as how to transform. We must also ask about developing both the confidence (to undertake mediation of resources) and the habit (to do so). If nothing else, these chapters emphasise the importance of such tasks. But, seated as they are in well-tried theoretical frames, we might expect that the four chapters will illuminate familiar phenomena within classroom experiences. Do they do this? The familiar phenomenon of teacher resistance to change has already been mentioned, and we are asked to re-evaluate this perception in the light of evidence that teacher change evolves over long time periods. University lecturers’ reluctant weaning from blackboards and slow adoption of technology can be better understood as being wedded not to the practice of chalk dust and dusters, but to the particular construction of mathematics that blackboards activate. Similarly, the apparently wasteful teacher habit of writing and rewriting mathematical notes that are readily available in neatly formatted and triple-checked textbook form can be explained by documentational genesis, and the need to personally transform ones’ pedagogical intentions through the resource. What about the research-verified phenomenon of teachers excitedly engaging in mathematical content whether or not it relates to their teaching? Their enthusiasm is not just a product of an inordinate love of their subject. Teachers’ mathematical knowledge also goes out of date, and they are aware of it. They are also aware that subject knowledge is one of the most powerful sources of effective teaching – and it needs to be re-sourced. We would think twice about using a 30-year-old text, but many teachers are still using their 30-year-old mathematics. No wonder that a judgement-free opportunity to re-source is welcomed with open arms. And finally, what does the analysis tell us about the powerful effect of communities of professional teachers? Why is teacher development so much better when done in a community? Because resource mediation is a social practice. Explicitly, in

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Sensevy’s thought collective but also implicitly. An unmediated resource is someone else’s voice; the teacher, through mediation, has a conversation with its originator. How much more powerful it is when the dialogue is a discussion. Teaching resources, like their fuel namesakes, must be mined. Often the extraction is an expensive business, requiring an investment of time and money (e.g. software or textbook production). Sometimes there is significant pollution and waste (e.g. travesties of repetitive exercises masquerading as mathematics), corruption (e.g. false claims for technology) and a carbon footprint that needs to be compensated (wasted teacher time in top-down workshops). Nevertheless, when their energy is released by the internal combustion of a teacher in action, the results can be explosive and, as the anagram of the title to this reaction suggests, become the core of the teacher’s task.

References Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245–274. Mason, J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and structure of attention. Journal of Mathematics Teachers Education, I, 243–267. Mason, J. (2010). Attention and intention in learning about teaching through teaching. In R. Leikin & R. Zazkis (Eds.), Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice (pp. 23–47). New York: Springer. McLuhan, M. (1964). Understanding media: The extensions of man. New York: Mentor. McLuhan, M., & Fiore, Q. (1967). The medium is the massage. New York: Random House. Schoenfeld, A. (1987). What’s all the fuss about metacognition. In A. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189–215). Hillsdale, NJ: Lawrence Earlbaum. Schoenfeld, A. (2008). On modelling teachers’ in-the-moment decision-making. In A. Schoenfeld (Ed.), A study of teaching: Multiple lenses, multiple views (pp. 45–96). Reston, VA: National Council of Teachers of Mathematics.

Part II

Text and Curriculum Resources

Chapter 5

Constituting Digital Tools and Materials as Classroom Resources: The Example of Dynamic Geometry Kenneth Ruthven

5.1 Introduction This chapter examines the often unrecognised challenges that teachers face in seeking to make effective use of new mathematical tools and representational media in the classroom, highlighting several key facets of professional learning associated with overcoming these challenges. It focuses on the appropriation of digital tools and media as resources for the mainstream practice of secondary-school mathematics teaching, taking the particular example of dynamic geometry to illustrate this process. First, the chapter demonstrates the interpretative flexibility surrounding a resource and the way in which wider educational orientations influence conceptions of its use. It does so by showing how pedagogical conceptions of dynamic geometry have shifted between pioneering advocates and mainstream adopters; and how such conceptions vary across adopters according to their wider approaches to teaching mathematics. Second, the chapter outlines a conceptual framework intended to make visible and analysable the way in which certain structuring features shape the incorporation of new technologies into classroom practice. This conceptual framework is then used to examine the case of a teacher leading what – for him – is an innovative lesson involving dynamic geometry, and specifically to identify how his professional knowledge is being adapted and extended. This shows how the effective integration of new technologies into everyday teaching depends on a more fundamental and wide-ranging adaptation and extension of teachers’ professional knowledge than has generally been appreciated.

K. Ruthven (B) Faculty of Education, University of Cambridge, Cambridge CB2 8PQ, UK e-mail: [email protected]


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5.2 The Interpretative Flexibility of Educational Resources Studies of the social shaping of technology have drawn attention to the ‘interpretative flexibility’ through which the function and operation of a tool remain open to adaptation (MacKenzie & Wajcman, 1999). In particular, conceptions of a technology influence its non-adoption by potential users, or its appropriation by them in the light of their interests and circumstances; indeed, technologies may be taken up in ways which, in terms of the speculative intentions of their designers, appear as something of a misappropriation. The concept of ‘innofusion’, then, blurs the conventional technocratic model of development in proposing that innovation carries on throughout the process of diffusion, as a technology and its modalities of use become aligned with user concerns and adapted to use settings (Williams & Edge, 1996). Contemporary educational studies adopt a similar perspective on curriculum materials and pedagogical guidance. Such resources have long provided a staple approach to influencing classroom practice. However, attempts to ‘teacher proof’ them, and the recurring failure of these efforts even more so, testify that teachers act as interpreters and mediators of them. This reflects a broader pattern in which the unfolding of innovation in education is shaped by the sense-making of the agents involved (Spillane, Reiser, & Reimer, 2002). Teachers typically select, combine and adapt resources, and they necessarily incorporate them into wider systems of classroom practice (Ball & Cohen, 1996). Accordingly, conceptualisations of how resources are used have developed from rather limited views of teachers simply following or subverting them, to more sophisticated perspectives encompassing teacher interpretation of, and participation with, them (Remillard, 2005). Interpretative flexibility became very apparent during the early development of geometry software. Originally intended to provide computer-supported analogues to established manual processes for the construction of figures, geometric software underwent a significant evolution with the recognition that, on a computer screen, such figures could be made dynamic, changing shape in response to the dragging of points or segments, but preserving their defining properties (Scher, 2000). Although the dragging operation rapidly became a defining feature of dynamic geometry software, its functional versatility and corresponding complexity were not anticipated, and are still in the process of being established (Arzarello, Olivero, Paola, & Robutti, 2002; Laborde, 2001). Equally, although dynamic geometry systems were developed with educational purposes in view, they were not initially devised with a particular pedagogical approach in mind (Scher, 2000). However, pioneering work quickly associated dynamic geometry with a pedagogical orientation in which such software served ‘to create experimental environments where collaborative learning and student exploration are encouraged’ (Chazan & Yerushalmy, 1998, p. 8), so that ‘mathematics becomes an investigation of interesting phenomena, and the role of the mathematics student becomes that of the scientist’ (Olive, 2002, p. 17). Nevertheless, evidence about how dynamic geometry has actually been taken up in schools offers an enigmatic picture. For example a national survey conducted in the United States found

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an association between teachers nominating dynamic geometry as their most valued software and reporting skill-development as their main objective for computer use (Becker, Ravitz & Wong, 1999).

5.3 An English Study of Teacher Constructions of Dynamic Geometry A recent English study has thrown further light on the use of dynamic geometry in mainstream practice (Ruthven, Hennessy, & Deaney, 2008). Much of the pioneering development of dynamic geometry systems has taken place in countries – notably France and the United States – which comparative studies show to have retained a strongly Euclidean spirit within their school geometry curriculum, resulting in greater attention to formalisation and systematisation, including an emphasis on proof (Hoyles, Foxman, & Küchemann, 2001). The Euclidean lineage of dynamic geometry might be expected to fit poorly with a national curriculum which refers – as does the English one framing the practice studied – not to ‘geometry’ but to ‘shape, space and measures’. However, the scope to employ the software as a means of supporting observation, measurement and calculation resonates with the empirical style of English school mathematics, and such modalities of reasoning were found to be prevalent when dynamic geometry was used. The study found echoes of the exploratory rhetoric of the software’s advocates in teachers’ suggestions that dynamic figures helped students to ‘find out how it works without us telling them’, or ‘tell you the rule instead of you having to tell them’, so that students were ‘more or less discovering for themselves’ and could ‘feel that they’ve got ownership of what’s going on’, even if teachers might have to ‘structure’, ‘hint’, ‘guide’ or ‘steer’ students towards the intended mathematical conclusion. Case studies identified a range of practical expressions of this idea. One case involved a strongly teacher-led, whole-class approach, in which dynamic presentation by the teacher was used to make it easier for students to ‘spot the rule’ so that ‘you’re not just telling them a fact, you’re allowing them to sort of deduce it and interact with what’s going on’. In the other cases, the classroom approaches involved more devolution to students, through investigations structured towards similarly preconceived mathematical results, with the teacher ‘drawing attention to’, ‘flagging up’ and ‘prompting’ them. On the issue of students themselves making use of the software, classroom approaches were found to be based variously on avoiding, minimising or capitalising on the demands of using dynamic geometry. In the first case referred to above, the software was used only for teacher presentation on the grounds that ‘it would take a long time. . . for [students] to master the package’ and ‘the return from the time investment. . . would be fairly small’, so that ‘the cost benefit doesn’t pay’. In two further cases, the normal pattern was ‘to structure the work so [students] just have to move points [on a prepared figure]’, so that ‘they don’t have to be complicated by that, they really can just focus on what’s happening mathematically’. In the final

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case, getting students to construct their own dynamic figures was seen as a vehicle for developing and disciplining their geometrical thinking; using dynamic geometry was introduced to them in terms of: ‘It’s not just drawing, it’s drawing using mathematical rules’. Thus, the degree to which students were expected to make use of dynamic geometry was influenced by the extent to which this was conceived as promoting mathematically productive activity. A related issue concerned handling the apparent mathematical anomalies which arise when dynamic figures are dragged to positions where an angle becomes reflex (with the associated problem of measurement), or where rounded values obscure an arithmetical relationship between measures (as featured in Fig. 5.1). The potential for such situations to arise was considerable in the type of topic most widely reported as suited to dynamic geometry: the study of angle properties. For example two of the case studies included a lesson on the angle sum of polygons (both employing a figure of the type shown in Fig. 5.1). In the first case, the teacher took great care to avoid exposing students to apparent anomalies of these types, through vigilant dragging to avoid ‘possibilities where students may become confused, or things that might cloud the issue’. In the other case, the teacher actively wanted students to encounter such difficulties so as to learn ‘that you can’t assume that what you’ve got in front of you is actually what you want, and you have to look at it . . . and question it’; equally, resolving such situations was seen as serving ‘to draw attention to . . . how the software measures the smaller angle, thus reinforcing that there are two angles at a point and [that students] needed to work out the other’. Thus, approaches to handling these apparent mathematical anomalies were influenced by whether they were seen as providing opportunities to develop students’ mathematical understanding, in line with a more fundamental pedagogical orientation that saw analysis of discrepancies as supporting learning. This study, then, highlights several noteworthy aspects of the interpretative flexibility of dynamic geometry. First it shows that the forms of guided discovery that dynamic geometry is typically used to support in English classroom practice, as well as the empirical and arithmetical modes of reasoning associated with them, are very different from the types of mathematical enquiry and modes of mathematical reasoning envisaged by the original proponents of the software. Equally, it shows

72.0°

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Fig. 5.1 Dynamic geometry figure for establishing the angle sum of a pentagon

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Angle sum = 394.1°

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how differing approaches to staging guided discovery, and organising the associated software use, reflect varied interpretations of the functionality for students of dynamic geometry, shaped by contrasting conceptions of what it means for students to learn mathematics. These case studies were carried out in mathematics departments that were professionally well regarded for their use of digital technologies. Even in these departments, the exposure of any one class to dynamic geometry was of the order of a handful of lessons each year. Moreover, when the software was used, teachers largely sought to minimise disruption to customary patterns of classroom activity. Indeed, research on how teachers make use of the interactive whiteboards now widely available in English classrooms reports that software such as dynamic geometry is generally rejected as over-complex or used only in limited ways (Miller & Glover, 2006). Such observations suggest that it is not just the way in which teachers conceptualise dynamic geometry as a teaching resource that influences their response to it, but more basic concerns about how to realise its incorporation within a viable classroom practice.

5.4 Structuring Features of Classroom Practice Such concerns are often overlooked in educational reform, and with them the craft knowledge that underpins everyday classroom practice (Brown & McIntyre, 1993; Leinhardt, 1988). In particular, much proposed innovation entails modification of the largely reflex system of powerful schemes, routines and heuristics that teachers bring to their classroom work, often tailored to their particular circumstances. The conceptual framework that I will now develop focuses, then, on the functional organisation of a system of (often tacit) pedagogical craft knowledge required to accomplish concrete professional tasks (consequently this framework does not directly consider the subject disciplinary knowledge required of the teacher, although this too plays a part). This section will introduce five key structuring features of classroom practice and show how they relate to the constitution of digital tools and materials as classroom resources: working environment, resource system, activity format, curriculum script and time economy.

5.4.1 Working Environment Making use of computer-based tools and materials in teaching often involves changes in the working environment in which lessons are conducted; namely, the physical surroundings where lessons take place, their general technical infrastructure, and the social organisation associated with them. In many schools, lessons have to be relocated from the normal classroom to a dedicated computer suite so as to make machines available in sufficient numbers for students to work with them. Such use entails disruption to normal working practices

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and makes additional organisational demands on the teacher (Jenson & Rose, 2006; Ruthven, Hennessy, & Deaney, 2005). Well-established routines which help lessons to start, proceed and close in a timely and purposeful manner in the regular classroom (Leinhardt, Weidman, & Hammond, 1987) have to be adapted to the computer suite. The alternative of providing sets of handheld devices or laptop computers in the ordinary classroom raises similar organisational issues. For example teachers report having to develop classroom layouts that assist them to monitor students’ computer screens, as well as classroom routines to forestall distraction, such as having students push down the screens of their laptops during whole-class lesson segments (Zucker & McGhee, 2005). More recently, there has been a trend towards provision of digital projection facilities or interactive whiteboards in ordinary classrooms. Their attraction to many teachers is that they require fewer modifications to the customary working environment of lessons (Jewitt, Moss, & Cardini, 2007; Miller & Glover, 2006). Such facilities can be treated as a convenient enhancement of a range of earlier display and projection devices, and allow a single classroom computer to be managed by teachers on behalf of the whole class.

5.4.2 Resource System New technologies have broadened the types of subject- and topic-specific resources available to support school mathematics. Educational suppliers now market textbook schemes alongside exercise and revision courseware, concrete apparatus alongside computer micro-worlds and environments, manual instruments alongside digital tools. The collection of mathematical tools and materials in classroom use constitutes a resource system which depends for its successful functioning on their being used in a co-ordinated way aligned with educational goals (Amarel, 1983). Studies of the classroom use of computer-assisted instructional packages have attributed strong take-up of particular materials to their close fit with the regular curriculum and their flexibility of usage (Morgan, 1990). Equally, teachers report that they would be much more likely to use technology if ready-to-use resources were readily available to them and clearly mapped to their scheme of work (Crisan, Lerman, & Winbourne, 2007). An important factor here is the limited scope that many digital materials offer for the teacher customisation characteristic of the use of other resources, and recognition of this has encouraged developers to offer greater flexibility to teachers. However, whatever the medium employed, teachers need to acquire knowledge in depth of materials so as to make effective use of them and to integrate them successfully with other classroom activity (Abboud-Blanchard, Cazes, & Vandebrouck, 2007; Bueno-Ravel & Gueudet, 2007). Something close to the textbook – even if taking a digitised form – remains at the heart of the resource system in many classrooms, valued for establishing a complete and coherent framework within which material is introduced in an organised and controlled way, appropriate to the intended audience. Indeed, one common use of

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interactive whiteboards in classrooms is to project and annotate textbook pages or similar presentations (Miller & Glover, 2006). More broadly, educational publishers are seeking increasingly to bundle digital materials with printed textbooks, often in the form of presentations and exercises linked to each section of the text, or applets providing demonstrations and interactivities. Such materials are attractive to many teachers because they promise a relatively straightforward and immediately productive integration of old and new technologies. Textbook treatments of mathematical topics necessarily make assumptions about what kinds of tools will be available in the classroom. Nowadays, it is increasingly assumed that some kind of calculator will be available to students. If well designed, textbooks explicitly develop the calculator techniques required and establish some form of mathematical framing for them. However, it is rare to find them taking account of other digital mathematical tools. Here, textbook developers face the same problems as classroom teachers. In the face of a proliferation of available tools, which should be prioritised? And given the currently fragmentary knowledge about bringing these tools to bear on curricular topics, how can a coherent use and development be achieved? Such issues are exacerbated when tools are imported into education from the commercial and technical world. Often, their intended functions, operating procedures, and representational conventions are not well matched to the needs of the school curriculum.

5.4.3 Activity Format The processes of classroom teaching and learning are played out within recurring patterns of teacher and student activity. Classroom lessons can be segmented according to recognisable activity formats: generic templates for action and interaction which frame the contributions of teacher and students to particular types of lesson segment (Burns & Anderson, 1987; Burns & Lash, 1986). The crafting of lessons around a succession of familiar activity formats and their supporting classroom routines helps to make them flow smoothly in a focused, predictable and fluid way (Leinhardt, Weidman, & Hammond, 1987), permitting the creation of prototypical activity structures or activity cycles for lessons as a whole. Monaghan (2004) studied secondary teachers who had made a commitment to move from making little use of ICT in their mathematics classes to making significant use. For each participating teacher, a ‘non-technology’ lesson was observed at the start of the project, and further ‘technology’ lessons over the course of the year. Monaghan found that technology lessons tended to have a quite different activity structure. In all the observed non-technology lessons, teacher-led exposition including the working-through of examples was followed by student work on related textbook exercises. Of the observed technology lessons, only those which took place in the regular classroom using graphic calculators displayed this type of structure. Most of the technology lessons focused on more open tasks, often in the form of investigations. These featured an activity structure consisting typically of a short introduction to the task by the teacher, followed by student work at computers

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over most of the session. Both types of technology lesson observed by Monaghan appear, then, to have adapted an existing form of activity structure: less commonly that of the exposition-and-practice lesson; more commonly that of the investigation lesson. Other studies describe classroom uses of new technologies that involve more radical change in activity formats, and call for new classroom routines. For example to provide an efficient mechanism through which the teacher can shape and regulate methods of tool use, Trouche (2005) introduces the role of ‘sherpa student’, taken on by a different student in each lesson. The sherpa student becomes responsible for managing the calculator or computer that is being publicly projected during wholeclass activity; what is distinctive about this activity format is the way in which it is organised around the teacher guiding the actions of the sherpa student, or opening them up for comment and discussion by the remainder of the class; the particular function it serves is in providing a mechanism by which the teacher can manage the collective development of techniques for using the tool. A new activity format of this type calls, then, for the establishment of new classroom norms for participation and the adaptation of existing classroom routines to support its smooth functioning.

5.4.4 Curriculum Script In planning to teach a particular topic, and in conducting lessons on it, teachers draw on (evolving) knowledge gained in the course of their own experience of learning and teaching that topic, or gleaned from available curriculum materials. Such knowledge is organised as a curriculum script, where ‘script’ is used in the psychological sense of a form of event-structured organisation: a loosely ordered model of relevant goals and actions that guides teachers’ handling of the topic, and includes variant expectancies of a situation and alternative courses of action (Leinhardt, Putnam, Stein, & Baxter, 1991). A curriculum script interweaves ideas to be developed, tasks to be undertaken, representations to be employed and difficulties to be anticipated in the course of teaching that topic, and links these to relevant aspects of working environment, resource system and activity structure. Teachers frequently talk about the use of new technologies in terms which appear to involve the adaptation and extension of established curriculum scripts (Ruthven & Hennessy, 2002). For example they talk about a new technology as a means of improving existing approaches to a topic, suggesting that it serves as a more convenient and efficient tool for supporting specific mathematical processes, or provides a more vivid and dynamic presentation of particular mathematical properties. Nevertheless, it is easy to underestimate the host of small but nuanced refinements which existing curriculum scripts require so as even to assimilate a new technology, let alone adapt the approach taken to a mathematical topic in the light of fresh insights gained from using the technology to mediate it. When teachers participate in development projects, they experience pressure (often self-administered) to use technology more innovatively. Monaghan (2004) reports, for example, that teachers had difficulty in finding resources to help them

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devise and conduct technology lessons on an investigative model. Consequently, they were obliged to plan such lessons at length and in detail, and then found themselves teaching rather inflexibly. The extent and complexity of such adoption is still greater when ‘imported’ technologies need to be aligned with the school curriculum. Monaghan compares, for example, the relative ease with which new lessons could be devised around the use of graphware specifically devised for educational use, with the much greater demands of appropriating ‘imported’ computer algebra systems to curricular purposes. These challenges become particularly severe in an educational culture, such as the French one, which emphasises a rigorous articulation of mathematical ideas and arguments (Artigue, 2002; Ruthven, 2002).

5.4.5 Time Economy The concept of time economy (Assude, 2005) focuses on how teachers seek to manage the ‘rate’ at which the physical time available for classroom activity is converted into a ‘didactic time’ measured in terms of the advance of knowledge. Although new tools and materials are sometimes represented as displacing old to generate a time bonus, it is more common to find a double instrumentation in operation, in which old technologies remain in use alongside new. In particular, old technologies may make an epistemic, knowledge-building contribution as much as a pragmatic, task-effecting one (Artigue, 2002). This double instrumentation means that new technologies often give rise to cost additions rather than to cost substitutions with respect to time. Thus, a critical concern of teachers is to fine-tune resource systems, activity structures and curriculum scripts to optimise the rate of didactic return on the time investment (Bauer & Kenton, 2005; Crisan et al., 2007; Smerdon, Cronen, Lanahan, Anderson, Iannotti, Angeles, & Greene, 2000). A critical issue is what teachers perceive as the mathematical learning that results from students using new tools. As noted in the earlier discussion of dynamic geometry, teachers are cautious about new tools which require substantial investment, and alert for modes of use which reduce such investment or increase rates of return. These concerns to maximise the time explicitly devoted to recognised mathematical learning are further evidenced in the trend to equip classrooms with interactive whiteboards, popularised as a technology for increasing the pace and efficiency of lesson delivery, as well as harnessing multimodal resources and enhancing classroom interaction (Jewitt et al., 2007). Evaluating the developing use of interactive whiteboards in secondary mathematics classrooms, Miller & Glover (2006) found that teachers progressed from initial teaching approaches in which the board was used only as a visual support for the lesson, to approaches where it was used more deliberately to demonstrate concepts and stimulate responses from pupils. In the course of this development, there was a marked shift away from pupils copying down material from the board towards use ‘at a lively pace to support stimulating lessons which minimise pupil behaviour problems’ (p. 4). However, in terms of the type of mathematical resource used with the board, there was little progression beyond textbook type sources and prepared presentation files; more generic

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mathematical resources such as spreadsheet, graphing and geometry programs were rejected by teachers as over-complex or used by them only in limited ways.

5.5 Practitioner Thinking and Professional Learning in an Innovative Lesson The conceptual framework sketched in the last section will now be used to analyse the practitioner thinking and professional learning surrounding one of the lessons from the earlier study of classroom practice incorporating dynamic geometry use (Ruthven et al., 2008). In the original study, this specific nomination was followedup not only because the teacher concerned had talked lucidly about his experience of teaching such a lesson for the first time, but also because he displayed particular awareness of the potential of dynamic geometry for developing visuospatial and linguistic aspects of students’ geometrical thinking. Thus, this case was chosen for investigation as a prospectively interesting outlier where a teacher appeared to be developing a form of classroom practice more consonant with the style of dynamic geometry use envisaged by its protagonists. Because the teacher was unusually expansive in interview, touching on a range of aspects of practitioner thinking and professional learning, this case was also particularly suited to further analysis in terms of the structuring features identified in the conceptual framework outlined in the previous section. Nevertheless, it should be borne in mind that the original study was not designed or conducted with this conceptual framework in mind; rather, it has provided a subsequently convenient means of exploring application of the framework to a concrete example.

5.5.1 Orientation to the Lesson When initially nominating a recent lesson as an example of successful practice, the teacher explained that it had been developed in response to improved technology provision in the mathematics department, notably the installation of interactive whiteboards in ordinary classrooms. He reported that the lesson (with a class in the early stages of secondary education) had started with him explicitly constructing a triangle, and then the perpendicular bisectors of its edges. The focus of the investigation which ensued had been on employing dragging to examine the idea that this construction might identify the ‘centre’ of a triangle (Fig. 5.2). According to the teacher, one particularly successful aspect of the lesson had been the extent to which students actively participated in the investigation. Indeed, because of the interest and engagement shown by students, the teacher had decided to extend the lesson into a second session, held in a computer room to allow students to work individually at a computer. For the teacher, the ready recall by students in this second session of ideas from the earlier session was another striking aspect of the lesson’s success. In terms of the specific contribution of dynamic geometry to

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Fig. 5.2 The basic dynamic figure employed in the investigative lesson

this success, the teacher noted how the software supported exploration of different cases, and overcame the manipulative difficulties which students encountered in using classical tools to attempt such an investigation by hand. But the teacher saw the contribution of the software as going beyond ease and accuracy; using it required properties to be formulated precisely in geometrical terms. These, then, were the terms in which this earlier lesson was nominated as an example of successful practice. We followed up this nomination by studying a lesson along similar lines, conducted over two 45-minute sessions on consecutive days with a Year 7 class of students (aged 11–12) in their first year of secondary education.

5.5.2 Working Environment Each session of the observed lesson started in the normal classroom and then moved to a nearby computer suite where it was possible for students to work individually at a machine. This movement between rooms allowed the teacher to follow an activity cycle in which working environment was shifted to match changing activity format. Even though the computer suite was, like the teacher’s own classroom, equipped with a projectable computer, starting sessions in the classroom was expedient as doing so avoided disruption to the established routines underpinning the smooth launch of lessons. Moreover, the classroom provided an environment more conducive to sustaining effective communication during whole-class activity and to maintaining the attention of students. Whereas in the computer suite each student was seated behind a sizeable monitor, blocking lines of sight and placing diversion at students’ fingertips, in the classroom the teacher could introduce the lesson ‘without the distraction of computers in front of each of them’. It was only recently that the classroom had been refurbished and equipped, and a neighbouring computer suite established for the exclusive use of the mathematics department. The teacher contrasted this new arrangement favourably in terms of the easier and more regular access to technology that it afforded, and the consequent

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increase in the fluency of students’ use. New routines were being established for students opening a workstation, logging on to the school network, using shortcuts to access resources and maximising the document window. Likewise, routines were being developed for closing computer sessions. Towards the end of each session, the teacher prompted students to plan to save their files and print out their work, advising them that he’d ‘rather have a small amount that you understand well than loads and loads of pages printed out that you haven’t even read’. He asked students to avoid rushing to print their work at the end of the lesson, and explained how they could adjust their output to try to fit it onto a single page; he reminded them to give their file a name that indicated its contents, and to put their name on their document to make it easy to identify amongst all the output from the single shared printer.

5.5.3 Resource System The department had its own ‘schemes of work’ (a term used in English schools for a written schedule of topics to be taught to particular year-groups, that usually includes suggestions for resources to be used) with teachers encouraged to explore new possibilities and report to colleagues. This meant that teachers were accustomed to integrating material from different sources into a common scheme of work. However, so wide was the range of computer-based resources currently being trialled that our informant (who was head of department) expressed concern about incorporating them effectively into departmental schemes, and about the demands of familiarising staff and students with such a variety of tools. In terms of coordinating use of old and new technologies, work with dynamic geometry was seen as complementing established work on construction with classical manual tools, by strengthening attention to the related geometric properties. Nevertheless, the teacher felt that old and new tools lacked congruence, because certain manual techniques appeared to lack computer counterparts. Accordingly, old and new were viewed as involving different methods and having distinct functions. While ruler and compass were seen as tools for classical constructions, dynamic software was ‘a way of exploring the geometry’. Equally, some features of computer tools were not wholly welcome. For example the teacher noted that students could be deflected from the mathematical focus of a task, spending too much time on cosmetic aspects of presentation. During the lesson the teacher had tried out a new technique for managing this, by briefly projecting a prepared example to show students the kind of report that they were expected to produce, and to illustrate appropriate use of colour coding. In effect, by showing students to what degree, and for what purpose, he regarded it as legitimate for them to ‘slightly adjust the font and change the colours a little bit, to emphasise the maths, not to make it just look pretty’, the teacher was developing sociomathematical norms (Yackel & Cobb, 1996) for using the new technology, and developing a classroom strategy for establishing these norms.

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5.5.4 Activity Format Each session of the observed lesson followed a similar activity cycle, starting with teacher-led activity in the normal classroom, followed by student activity at individual computers in the nearby computer suite, with this change of rooms during sessions serving to match working environment to activity format. Indeed, when the teacher had first nominated this lesson, he had remarked on how it combined a range of activity formats – ‘a bit of whole class, a bit of individual work and some exploration’ – to create a promising lesson structure; one that he would ‘like to pursue because it was the first time [he]’d done something that involved quite all those different aspects’. In discussing the observed lesson, however, the teacher highlighted one aspect of the model which had not functioned as well as he would have liked: the fostering of discussion during individual student activity. He identified a need for further consideration of the balance between opportunities for individual exploration and for productive discussion, through exploring having students work in pairs. At the same time, the teacher noted a number of ways in which the computer environment helped to support his own interactions with students within an activity format of individual working. Such opportunities arose from helping students to identify and resolve bugs in their dynamic geometry constructions. Equally, the teacher was developing ideas about the pedagogical affordances of text-boxes, realising that they created conditions under which students might be more willing to consider revising their written comments because this could be done with ease and without his interventions being seen by students as ‘ruin[ing] their work’ by spoiling its presentation. This was helping him to achieve his goal of developing students’ capacity to express themselves clearly and precisely in geometrical terms through refining their statements of properties.

5.5.5 Curriculum Script The observed lesson followed on from earlier ones in which the class had undertaken simple classical constructions with manual tools: in particular, using compasses to construct the perpendicular bisector of a line segment. Further evidence that the teacher’s curriculum script for this topic originated prior to the availability of dynamic geometry was his reference to the practical difficulties which students encountered in working by hand to accurately construct the perpendicular bisectors of a triangle. His evolving script now included not only the knowledge of ‘unusual’ and ‘awkward’ aspects of software operation liable to ‘cause[] a bit of confusion’ amongst students, but also of how such difficulties might be turned to advantage in reinforcing the mathematical focus of the task so that ‘sometimes the mistakes actually helped’. Equally, the teacher’s curriculum script anticipated that students might not appreciate the geometrical significance of the concurrence of perpendicular bisectors, and incorporated strategies for addressing this, such as trying ‘to get them to see that

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. . . three random lines, what was the chance of them all meeting at a point’. This initial line of argument was one already applicable in a pencil and paper environment. Later in the interview, however, the teacher made reference to another strategy which brought the distinctive affordances of dragging the dynamic figure to bear on this issue: ‘When I talked about meeting at a point, they were able to move it around’. Likewise, his extended curriculum script depended on exploiting the distinctive affordance of the dynamic tool to explore how dragging the triangle affected the position of this ‘centre’. This suggests that the teacher’s curriculum script was evolving through experience of teaching the lesson with dynamic geometry, incorporating new mathematical knowledge specifically linked to mediation by the software. Indeed, he drew attention to a striking example of this which had arisen from his question to the class about the position of the ‘centre’ when the triangle was dragged to become right angled. The lesson transcript recorded: Teacher: What’s happening to the [centre] point as I drag towards 90 degrees? What do you think is going to happen to the point when it’s at 90?. . . Student: The centre’s going to be on the same point as the midpoint of the line. Teacher [with surprise]: Does it always have to be at the midpoint? [Dragging the figure] Yes, it is! Look at that! It’s always going to be on the midpoint of that side. . . . Brilliant! Reviewing the lesson, the teacher commented that this property hadn’t occurred to him; he ‘was just expecting them to say it was on the line’. Reacting to the student response he reported that he looked at the figure and ‘saw it was exactly on that centre point’, and then ‘moved it and thought . . . of course it is!’. What we witness here, then, is an episode of reflection-in-action through which the teacher’s curriculum script for this topic has been elaborated.

5.5.6 Time Economy In respect of the time economy, a very basic consideration of physical time for the teacher in this study was related to the proximity of the new computer suite to his normal classroom. However, a more fundamental feature of this case was the degree to which the teacher measured didactic time in terms of progression towards securing student learning rather than pace in covering a curriculum. At the end of the first session, he linked his management of time to what he considered to be key stages of the investigation: ‘the process of exploring something, then discussing it in a quite focused way as a group, and then writing it up’, in which students moved from being ‘vaguely aware of different properties’ to being able to ‘actually write down what they think they’ve learned’.

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A further crucial consideration within the time economy is investment in developing students’ capacity to make use of a tool. As noted in the larger study from which this case derives, teachers were willing to invest time in developing students’ knowledge of dynamic geometry only to the extent that they saw this as promoting their mathematical learning. This teacher was unusual in the degree to which he saw working with the software as engaging students in disciplined interaction with a geometric system. Consequently, he was willing to spend time to make them aware of the construction process underlying the dynamic figures used in lessons, by ‘actually put[ting] it together in front of the students so they can see where it’s coming from’. Equally, this perspective underpinned his willingness to invest time in familiarising students with the software, recognising that it was possible to capitalise on earlier investment in using classical tools in which ‘doing the constructions by hand first’ was a way of ‘getting all the key words out of the way’. As this recognition of a productive interaction between learning to use old and new technologies indicates, this teacher took an integrative perspective on the double instrumentation entailed. Indeed, this was demonstrated earlier in his concern with the complementarity of old and new as components of a coherent resource system.

5.6 Discussion Although only employing a dataset conveniently available from earlier research, the case study presented in this chapter starts to illuminate the professional adaptation on which the constitution of digital tools and materials as classroom resources depends. While the status of the conceptual framework that has been used to identify structuring features of classroom practice must remain tentative, it prioritises and organises previously disparate constructs developed in earlier research, and has proved a useful tool for analysis of already available case-records. It has the potential to be employed not just in relation to secondary mathematics teaching, but also to other school phases and curricular areas, and to other types of resource; indeed, much of the earlier research from which the various central concepts have been drawn has such a range. At the same time, however, the differing provenance of the five central constructs raises some issues of coherence. The original construct of curriculum script, for example, is very clearly psychologically based, focusing on individual knowledge schemes. One might also add that the term ‘script’ (originating in a psychological metaphor for memory structures) risks failing to convey the sense intended here of an organised repertoire of potential actions and interactions for teaching a topic as opposed to a specific sequence. By contrast, the construct of working environment may appear to refer to a material situation independent of the teacher. However (as suggested by Adler in Chapter 1), a more adequate theorisation takes a structuring feature as being constituted not just by an existing system of contextual constraints but by teachers’ interpretation of these and adaptation to them. Moreover, this co-constitution takes place on the social plane as well as the individual; indeed,

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these planes interact inasmuch as individual adaptation to such constraints is subject to a degree of socialisation, while the corresponding social norms evolve by virtue of a wider cultural appropriation of what originated as innovative micro-genetic adaptations on a very local scale. Thus, while each of these structuring features of working environment, resource system, activity format, curriculum script, and time economy are anchored in a particular form of constraint under which the work of teaching takes place, these constraints do not wholly condition practice, but interact and afford some degree of adaptation. For example in the case study detailed in this chapter, the proximity of the teacher’s normal classroom to the computer classroom afforded him the option to move between them as the location for the lesson. Moreover, the way in which he exercised this option was guided by his assessment of the suitability of the two locations for different activity formats. This, in turn, permitted the teacher to develop a new type of activity structure covering each session as a whole, efficient in terms of time economy, and providing what he considered a promising structure for an investigative lesson to capitalise on student use of digital resources. In terms of the specific digital resource in play, dynamic geometry, the teacher established a resource system in which this software fulfilled complementary functions to classical tools, each supporting particular aspects of students’ learning of mathematics, and so justifiable in terms of time economy. Finally, the teacher’s curriculum script for the topic was evolving, through adaptation and extension of an investigative task previously carried out without digital tools, the associated activity formats and corresponding refinement of his knowledge about supporting the interactive development of mathematical ideas. Acknowledging the concern of some chapters of this book with the collaborative use of resources, the collective role of the school department in fostering teachers’ professional learning was not a focus of this case study. In this department, however, it was clear that the internally developed schemes of work provided a key means of prompting the spread between teachers of new teaching ideas, often supported by self-devised materials. Nevertheless, the teacher had not yet reached the point at which this particular teaching sequence could be incorporated in the relevant departmental scheme. Indeed this case illustrates the bricolage which typifies the process of appropriating a new tool in the absence of well-established professional practice; a bricolage which, in the English educational system at least, is often left to the individual teacher rather than organised collectively. Likewise, the teaching sequence studied in this chapter was far from being captured in documented form. Although he had prepared a worksheet to remind his students of certain pieces of advice for their work, the teacher was generally rather sceptical of the value of such aids: ‘I don’t like pre-prepared worksheets’; ‘Normally I don’t use worksheets very much at all’. This arose from his strong valorisation of the explicit collective (re)construction of mathematical situations: ‘I always like to start with a blank page and actually put it together in front of the students so they can see where it’s coming from’. For him, it was this interactivation of a teaching sequence (guided by his curriculum script) that lay at the centre of his teaching. Under these conditions, then, this new teaching sequence might be expected to eventually be shared with

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colleagues more through observation or simulation of a lesson than by the reading or following of a documentary reification of this professional knowledge. This prompts comparison of the ideas developed in this chapter with those of other chapters in this book, notably those chapters that focus on the integration of digital tools and materials into everyday mathematics teaching. In terms of the core idea of ‘resource’ itself, following the concrete sense in which that term is widely used within the teaching profession, the focus of this chapter is on material ‘resources’ in classroom use, whereas, as Gueudet and Trouche note in Chapter 2, they use the term more loosely to cover any teacher resource, material and nonmaterial. Another significant contrast between the conceptual framework used in this chapter and that of Gueudet and Trouche lies in the central metaphor employed to capture the organisation, retrieval and exchange of professional knowledge. For Gueudet and Trouche, this is the ‘document’; in the conceptual framework employed in this chapter it is the ‘script’. Although neither Gueudet and Trouche nor myself are entirely happy with our respective metaphors, they do point to an important contrast in modalities of memory and thought, similar to that discussed by Proust in Chapter 9. This may well reflect divergences of professional practice and values between educational systems, notably as these bear on the planning of lessons. Such divergences might be linked, for example, to differing types of evidence used for professional accountability (lesson planning, for example, as against student progression) and models of lesson process (establishing disciplinary narrative, for example, as against ensuring curricular coverage), as well as intensity of work (with contrasting expectations as regards lesson preparation reflecting very different volumes of teaching and other duties required of teachers). Relatedly, although Gueudet and Trouche note in Chapter 2 how a teacher’s curriculum script serves particularly to guide the decisions that the teacher takes in class, it is important to emphasise that this script also plays a crucial part in preactive planning of a lesson agenda, and in post-active reflection on (and learning from) a lesson (Leinhardt et al., 1991). Indeed, I would hypothesise that every ‘document’ expresses elements of some underlying ‘script’. Nevertheless, it is important to acknowledge the part that the use and adaptation of documentary materials may play in supporting and developing the personal curriculum scripts of teachers, particularly those whose subject knowledge is modest (as noted by Pepin in Chapter 7). In Chapter 14, Drijvers raises the question of how the conceptual framework used in this chapter relates to the construct of instrumental orchestration. In terms of the concrete instrumental orchestrations that Drijvers describes, the answer is simple: each corresponds to a particular type of activity format centred on a specific use of one or more tools. More broadly, as described by Drijvers, didactical configuration and exploitation mode are features of what is commonly referred to within research on teacher thinking and planning as pre-active teaching, and didactical performance is likewise an aspect of interactive teaching (Clark & Peterson, 1986). In terms of the structuring features of classroom practice identified by the conceptual framework employed in this chapter, didactical configuration concerns organisation

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of the working environment as well as some more generic aspects of the functioning of the resource system; exploitation mode relates to more topic-specific aspects of the functioning of the resource system as well as to the tool mediation of processes within the curriculum script; and didactical performance relates to the way in which the curriculum script guides interactive teaching. Drijvers notes that the conceptual framework presented in this chapter is a more generic one, not specifically tied to the integration of technological resources in the way that the orchestration framework is. Arguably, these qualities are complementary. Indeed, an important conceptual weakness, both of advocacy for technology integration and research into it, has been lack of attention to the broader situation in which ordinary teachers find themselves (Lagrange, 2008; Ruthven & Hennessy, 2002). It is in this spirit that the conceptual framework used in this chapter has been developed by synthesising observations from recent studies of technology use, particularly in school mathematics, in the light of earlier conceptualisations of classroom teaching and situated teacher expertise. Turning to future development of the conceptual framework presented in this chapter, other insights have already been gained through a parallel analysis of mathematics teachers’ appropriation of graphing software (Ruthven, Deaney, & Hennessy, 2009). However, further studies are now required in which both data collection and analysis are guided by the conceptual framework, so that it can be subjected to fuller testing and corresponding elaboration and refinement. If they are to adequately address issues of professional learning, such studies need to be longitudinal as well as cross-sectional, and to focus on teachers’ work outside as well as inside the classroom. Likewise, the current reach of this conceptual framework is deliberately modest; it simply seeks to make visible and analysable certain crucial aspects of the incorporation of new technologies into classroom practice which other conceptual frameworks largely overlook. By providing a system of constructs closer to the lived world of teacher experience and classroom practice, it may prove able to fulfil an important mediating function, allowing insights from more decontextualised theories to be translated into classroom action, and serving to draw attention to practical issues neglected in such theories. Acknowledgements Particular thanks are due to the teacher colleague featured in the case study; to Rosemary Deaney who carried out the fieldwork for it; and to the UK Economic and Social Research Council which funded the associated research project. This chapter draws on and develops ideas and material from two earlier publications (Ruthven, 2009, 2010). These publications drew, in turn, on papers discussed at the CERME conferences in 2007 and 2009, in the RME and TACTL SIGs at the AERA conference in 2009, and at the CAL conference in 2009.

References Abboud-Blanchard, M., Cazes, C., & Vandebrouck, F. (2007). Teachers’ activity in exercisesbased lessons. Some case studies. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (pp. 1826– 1836). Larnaca, Cyprus: CERME-5. Retrieved 31, January, 2011, from http://ermeweb.free. fr/CERME5b/WG9.pdf

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Amarel, M. (1983). Classrooms and computers as instructional settings. Theory into Practice, 22, 260–266. Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245–274. Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. Zentralblatt fur Didaktik der Mathematik [now known as ZDM: The International Journal on Mathematics Education], 34(3), 66–72. Assude, T. (2005). Time management in the work economy of a class. A case study: Integration of Cabri in primary school mathematics teaching. Educational Studies in Mathematics, 59(2), 183–203. Ball, D. L., & Cohen, D. K. (1996). Reform by the book: What is –or might be– the role of curriculum materials in teacher learning and instructional reform. Educational Researcher, 25(9), 6–8 & 14. Bauer, J., & Kenton, J. (2005). Towards technology integration in schools: Why it isn’t happening. Journal of Technology and Teacher Education, 13(4), 519–546. Becker, H., Ravitz, J., & Wong, Y. (1999). Teacher and teacher-directed student use of computers, teaching, learning, and computing: 1998 National Survey Report #3. Centre for Research on Information Technology and Organizations. California: University of California Irvine. Retrieved 31, January, 2011, from http://www.crito.uci.edu/TLC/FINDINGS/.../REPORT_3_ PDF_REV.PDF Brown, S., & McIntyre, D. (1993). Making sense of teaching. Buckingham: Open University Press. Bueno-Ravel, L., & Gueudet, G. (2007). Online resources in mathematics: Teachers’ genesis of use. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (pp. 1369–1378). Larnaca, Cyprus: CERME-5. Retrieved 31, January, 2011, from http://ermeweb.free.fr/CERME5b/WG9.pdf Burns, R. B., & Anderson, L. W. (1987). The activity structure of lesson segments. Curriculum Inquiry, 17(1), 31–53. Burns, R. B., & Lash, A. A. (1986). A comparison of activity structures during basic skills and problem-solving instruction in seventh-grade mathematics. American Educational Research Journal, 23(3), 393–414. Chazan, D., & Yerushalmy, M. (1998). Charting a course for secondary geometry. In R. Lehrer & D. Chazan (Eds.), New directions in the teaching and learning of geometry. (pp. 67–90). Hillsdale, NJ: Lawrence Erlbaum. Clark, C. M., & Peterson, P. L. (1986). Teachers’ thought processes. In M. Wittrock (Ed.), Handbook of research on teaching (3rd ed.) (pp. 255–296). New York: Macmillan. Crisan, C., Lerman, S., & Winbourne, P. (2007). Mathematics and ICT: A framework for conceptualising secondary school teachers’ classroom practices. Technology, Pedagogy and Education, 16(2), 21–39. Hoyles, C., Foxman, D., & Küchemann, D. (2001). A comparative study of geometry curricula. London: Institute of Education. Jenson, J., & Rose, C. B. (2006). Finding space for technology: Pedagogical observations on the organization of computers in school environments. Canadian Journal of Learning and Technology, 32(1). Retrieved 31, January, 2011, from http://www.cjlt.ca/index.php/cjlt/article/ view/59/56 Jewitt, C., Moss, G., & Cardini, A. (2007). Pace, interactivity and multimodality in teachers’ design of texts for interactive whiteboards in the secondary school classroom. Learning, Media and Technology, 32(3), 303–317. Laborde, C. (2001). Integration of technology in the design of geometry tasks with CabriGeometry. International Journal of Computers for Mathematical Learning, 6(3), 283–317. Lagrange, J.-B. (2008). ‘Ordinary’ teachers using technology: Concerns, theoretical approaches, case studies. Regular lecture at ICME-11. Abstract accessed on 31, January, 2011, at http:// icme11.org/node/1441

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Leinhardt, G. (1988). Situated knowledge and expertise in teaching. In J. Calderhead (Ed.), Teachers’ professional learning (pp. 146–168). London: Falmer. Leinhardt, G., Putnam T., Stein, M. K., & Baxter, J. (1991). Where subject knowledge matters. Advances in Research in Teaching, 2, 87–113. Leinhardt, G., Weidman, C., & Hammond, K. M. (1987). Introduction and integration of classroom routines by expert teachers. Curriculum Inquiry, 17(2), 135–176. MacKenzie, D., & Wajcman, J. (Eds.). (1999). The social shaping of technology (2nd ed.). Buckingham: Open University Press. Miller, D., & Glover, D. (2006). Interactive whiteboard evaluation for the Secondary National Strategy: Developing the use of interactive whiteboards in mathematics: Final report. Keele: Keele University. Retrieved 31, January, 2011, from http://nationalstrategies.standards.dcsf. gov.uk/node/96272 Monaghan, J. (2004). Teachers’ activities in technology-based mathematics lessons. International Journal of Computers for Mathematical Learning, 9, 327–357. Morgan, C. (1990). Carry on SMILEing. Micromath, 6(3), 14–15. Olive, J. (2002). Implications of using dynamic geometry technology for teaching and learning. In M. Saraiv, J. Matos, & I. Coelho (Eds.), Ensino e Aprendizagem de Geometria. Lisbon, Portugal: SPCE. Retrieved 31, January, 2011, from http://www.spce.org.pt/sem/JO.pdf Remillard, J. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75, 211–246. Ruthven, K. (2002). Instrumenting mathematical activity: Reflections on key studies of the educational use of computer algebra systems. International Journal of Computers for Mathematical Learning, 7(3), 275–291. Ruthven, K. (2009). Towards a naturalistic conceptualisation of technology integration in classroom practice: The example of school mathematics. Education & Didactique, 3(1), 131–149. Ruthven, K. (2010). Constituer les outils et les supports numériques en ressources pour la classe. In G. Gueudet & L. Trouche (Eds.), Ressources vives, le travail documentaire des professeurs en mathématiques (pp. 183–199). Rennes, France: Presses Universitaires de Rennes/Institut National de Recherche Pédagogique. Ruthven, K., Deaney, R., & Hennessy, S. (2009). Using graphing software to teach about algebraic forms: A study of technology-supported practice in secondary-school mathematics. Educational Studies in Mathematics, 71(3), 279–297. Ruthven, K., & Hennessy, S. (2002). A practitioner model of the use of computer-based tools and resources to support mathematics teaching and learning. Educational Studies in Mathematics, 49(1), 47–88. Ruthven, K., Hennessy, S., & Deaney, R. (2005). Incorporating Internet resources into classroom practice: Pedagogical perspectives and strategies of secondary-school subject teachers. Computers and Education, 44(1), 1–34. Ruthven, K., Hennessy, S., & Deaney, R. (2008). Constructions of dynamic geometry: A study of the interpretative flexibility of educational software in classroom practice. Computers and Education, 51(1), 297–317. Scher, D. (2000). Lifting the curtain: The evolution of the Geometer’s Sketchpad. The Mathematics Educator, 10(1), 42–48. Smerdon, B., Cronen, S., Lanahan, L., Anderson, J., Iannotti, N., Angeles, J., et al. (2000). Teachers’ tools for the 21st century: A report on teachers’ use of technology. Washington, DC: US Department of Education, National Center for Education Statistics. Retrieved 31, January, 2011, from nces.ed.gov/pubs2000/2000102.pdf Spillane, J. P., Reiser, B. J., & Reimer, T. (2002). Policy implementation and cognition: Reframing and refocusing implementation research. Review of Educational Research, 72(3), 387–431. Trouche, L. (2005). Instrumental genesis, individual and social aspects. In D. Guin, K. Ruthven, & L. Trouche (Eds.), The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument (pp. 197–230). New York: Springer.

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Williams, R., & Edge, D. (1996). The social shaping of information and communications technologies. Research Policy, 25(6), 856–899. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477. Zucker, A., & McGhee, R. (2005). A study of one-to-one computer use in mathematics and science instruction at the secondary level in Henrico County public schools. Menlo Park, CA: SRI International. Retrieved 31, January, 2011, from ubiqcomputing.org/FinalReport.pdf

Chapter 6

Modes of Engagement: Understanding Teachers’ Transactions with Mathematics Curriculum Resources Janine T. Remillard

6.1 Introduction The last decade has seen considerable progress in theory building related to teachers’ use of mathematics curriculum resources (Adler, 2000; Brown, 2009; Gueudet & Trouche, 2009; Remillard, 2005). Scholars agree that the process of using a curriculum resource is not one of straightforward implementation; rather curriculum use involves an interaction between the teacher and the resource. A number of studies, including many in this volume, have documented a variety of personal, professional, and classroom-based results from teachers using curriculum resources as tools (Chapters 7, 10, and 14; Gueudet & Trouche, 2010; Remillard, Herbel-Eisenmann, & Lloyd, 2009). In this chapter, I explore and theorize the relationships that teachers develop with curriculum resources as they use them. I focus, in particular, on mathematics curriculum texts produced to guide teachers in the design of daily instruction. In the United States, these resources tend to be published in print format most commonly by commercial companies. The chapter is informed by research on elementary and middle school teachers in the United States, where the nature of mathematics curriculum resources have undergone substantial change since the publication of the NCTM Standards in 1989. Traditionally, the primary focus of mathematics textbooks was student exercises and practice problems with minimal attention paid to pedagogy. Because the Standards targeted both the kinds of mathematical tasks students are asked to do and the nature of instruction around these tasks, new curriculum materials place a great deal of emphasis on pedagogical guidance for the teacher. As a result, teacher’s guides now provide teachers with new kinds of information to read and suggest different kinds of teaching practices to enact. Researchers are finding that using these changed resources, often referred to as Standards-based curriculum materials, presents challenges for many teachers and requires considerable reorientation (Drake & Sherin, 2009; Lloyd, 1999; Remillard, 2000); moreover, many teachers use them in ways J.T. Remillard (B) University of Pennsylvania, Philadelphia, PA 19146, USA e-mail: [email protected]


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not intended by the designers (Collopy, 2003; Remillard & Bryans, 2004). My aim in this chapter is to examine these teacher–curriculum interactions from a conceptual perspective and offer a framework for characterizing them. I argue that teachers are positioned by and through their encounters with curriculum materials as particular kinds of users of them. I explore how this positioning happens, including how teachers participate in it, as well as its implications for teachers and for the prospect of reform in mathematics teaching and learning. In her book, entitled Teaching Positions, Ellsworth (1997) draws on film studies to argue that whenever two or more people engage in an interaction, be it spoken, textual, in film, or pedagogical, the speaker always makes assumptions about the audience. “Films, like letters, books, or television commercials are for someone . . . Most decisions about a film’s narrative structure, ‘look’, and packaging are in light of conscious and unconscious assumptions about ‘who’ its audience ‘is’, what they want, how they read films” (p. 23, emphasis in original). This concept, known in film studies as mode of address, Ellsworth argues, involves positioning the audience in particular ways that are the necessary starting place for interaction. This starting place is where the viewer (or hearer, or reader) enters a relationship with the story or ideas in the text. However, this positioning is also problematic in its shaping of the relationship around power and authority in the interaction. Using Ellsworth’s perspective as an analytical lens, I examine the relationships teachers enter into when they use curriculum resources, how they are positioned by the materials, and how they position themselves as readers and users of texts. In doing so, I build on the idea of mode of address and Rosenblatt’s (1980, 1982) theory of transactions with text to offer a model of how these interactions are shaped and how they shape the role of curriculum materials in teaching. I argue that, in addition to having a mode of address, curriculum materials have forms of address, particular “looks” or formats that reflect and reinforce the mode of address. Moreover, teachers interact with curriculum resources through an identifiable stance or mode of engagement. Like the modes of address, modes of engagement have particular forms. To interrogate current patterns in how teachers use curriculum resources or to imagine alternatives, it is necessary to understand these constructs and their interrelationships.

6.2 Modes of Address and Curriculum Resources In film studies, a mode of address captures who the film’s designers think the audience is, what they want, and how they read. All films (or texts) have an intended audience and are written to capture and appeal to, to speak to that audience. Ellsworth uses the metaphor of seats in a theater to explain. There is one seat or “position” from which the film does its best work or looks its best. Building on this idea of position, she argues: There is a “position” within power relations and interests, within gender and racial constructions, within knowledge, to which the film’s story and visual pleasure is addressed. It’s from that “subject position” that the film’s assumptions about who the audience is work with the least effort, contractions, or slippage (p. 24).

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In other words, to work as imagined, films, and I would add texts, need the intended audience to be who the mode of address assumes they are. Thus, part of the work of a mode of address is to affirm that position, to keep them wanting what they should want, to enlist the viewer in reinforcing her positioning or epistemological stance. It is in this way that modes of address do not merely speak to an intended audience, but actually seek to assert control over that audience or to enlist a particular kind of participation. Modes of address are not neutral. Moreover, the drive behind the mode of address is not merely literary or artistic in nature (i.e., like the relationship between author and reader); it is commercial. Films and texts need the intended audience to be who they (the designers) think they (the audience) are to sell. In systems with strong commercial publishing industries, the need for a text to sell is particularly important, but even in school systems that use a single, state sponsored textbook, the need for it to appeal to and be usable by teachers remains present. The same is true for materials developed noncommercially. Even though the designers may not be interested in “selling” the curriculum, per se, they need the teacher to “buy in” to the orientation set out in the materials. Curriculum materials, then, are written with particular teachers (readers) in mind – teachers who exist in the minds of the writers. They are written to both appeal to those readers’ needs and desires and to affirm them to keep the text–teacher relationship intact. Later, I discuss how this affirmation occurs through the specific forms of address used most frequently in curriculum materials in the United States. Naturally, texts have multiple modes of address or, as Ellsworth (1997) suggests, multiple entry points. A film might use different characters to draw in audience members outside of the primary intended audience. A film written with white, adolescent, males as the primary audience, Ellsworth explains, uses other characters, such as a strong, intelligent woman, to capture other audience members. The concept of multiple entry points applies to curriculum materials as well. For example, many curriculum designers assume their audience desires or needs day-by-day guidance for teaching mathematics lessons and they design their materials attentive to this desire. However, they may be cognizant that some teachers seek challenging activities that they can pick up and use without making a substantial investment in the structure of the curriculum. These designers make such activities accessible and visible while addressing the needs of their primary audience. The concept of multiple entry points as it applies to mathematics texts and their use is illustrated in findings from a qualitative study I undertook in the early 1990s (Remillard, 1996). The study included a document analysis of a commercially developed elementary mathematics textbook published in the wake of the Standards and a concurrent year-long study of two fourth-grade teachers using this text for the first time. Through regular classroom observations (18 distributed across the year), I wrote field notes that described the classroom interactions and use of the textbook for each teacher and then used audiotaped interviews to triangulate observations and uncover the teachers’ approach to textbook use. My analysis of the text revealed two distinct entry points in the text. The basic, core program consisted of 13 traditionally titled chapters that emphasized the mastery of computational procedures.

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At the same time, the text included language and work practices associated with the NCTM Standards, such as exploration, group work, and manipulatives, along with a number of highly visible but auxiliary options, including a daily problem to solve and exploratory activities. From this presentation, one might infer that the intended audience was the typical elementary teacher at the time, someone seeking a mainstream, procedurally focused mathematics curriculum, but who was open to intermittently incorporating problem solving, manipulative work, and partner activities into lessons. These Standards-aligned activities themselves, however, did offer a second entry point. The possibility of these two entry points is further illustrated by the two teachers in the study. One of the teachers, Ms. McKeen, fits this description of the intended audience. Her use of the textbook focused on the routine practice problems it provided for students to complete. She also used some of the optional, reform-oriented activities available in the text when time permitted. The other teacher in the study, Ms. Yarnell, was attracted by another entry point into the text. She focused on the supplemental exploration activities found in the teacher’s guide (but not the student text), the manipulative-based instruction described in the margins of the teacher’s guide that surrounded the picture of the student’s page, and the pages of daily problems available at the back of the teacher’s guide. In her use of the textbook, Yarnell drew primarily from these resources and used the student practice pages infrequently (see Remillard, 1999, for more details.) I return to this example when discussing teachers’ modes of engagement.

6.3 Forms of Address and Curriculum Materials In my analyses of teachers using curriculum materials, I have come to appreciate the significance of form and look. When speaking of film, Ellsworth (1997) argues that narrative structure, look, and packaging all represent the mode of address – who the film makers think the audience is. I argue that when it comes to curriculum materials, form takes on its own significance and meaning. And while it is a critical component of the mode of address, form deserves a particular analytic focus. The form of address of a curriculum resource refers to the physical, visual, and substantive forms it takes up, the nature and presentation of its contents, the means through which it addresses teachers. The form of address is what teachers actually see, examine, and interact with when using a curriculum resource. The form of address is akin to what Otte (1986) referred to as a text’s objectively given structure. In his exploration of the concept of text and textbook, Otte argued that one must consider the text as both an “objectively given structure of information” (the physical form that the text takes), and a “subjective scheme” (how it is understood or perceived). Drawing on Otte’s writing, Love and Pimm (1996) referred to objectively given structures as, “what can be seen when looking at such materials” (p. 379). As the following discussion suggests, the form of a curriculum resource includes, but goes beyond, what is seen. Naturally, what is

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seen and encountered in a resource is inextricably linked to the subjective schemes that surround it, the tradition, meanings, and expectations that mediate the reader’s interpretation of the objective structure. The form of address is multi-faceted and includes all aspects of a resource. Not all forms of address have been examined in studies of teachers and curriculum materials, however. I use the following descriptions to illustrate the multifaceted nature of these forms. In the 1980s, elementary mathematics textbooks used in the United States tended to look alike. Developed and marketed by commercial publishers, the typical text consisted of 12 or 13 chapters that placed primary emphasis on computational processes, such as addition, subtraction or two-digit multiplication. These volumes also devoted chapters to several noncomputational topics, including measurement and geometry. The chapters were arranged into two-page modules; each on a different skill, around which each day’s lesson was shaped. The pages were glossy, colorful, and included photographs of children happily engaged in activities that employed math skills, like cooking, selling, or constructing. There might be an example of the particular skill at the top of the first page, followed by a series of similar exercises. One would likely find a set of story problems at the bottom of the second page. The large, spiral-bound teacher’s guide contained pictures of the student pages surrounded by wide margins that provided instructional suggestions, including questions to ask or even scripts to follow, and answers to all the exercises and questions on the students’ page. Teachers could teach a lesson from this two-page spread with minimal preparation. This example illustrates how a particular form can become a cultural convention, an accepted and expected package. In the mid-1990s, a number of curriculum developers began to challenge that cultural convention. Working outside of the commercial publishing market, they designed new materials intended to reflect the vision put forth by the NCTM Standards (NCTM, 1989). These Standards-based materials offered an instructional approach and set of mathematical goals and activities that differed substantially from the typical textbook described above. At the same time, curriculum authors (sometimes steered by pressure from publishers1 ) made different choices about the form of address their materials took. Some curriculum developers adopted forms familiar to teachers and presented their unfamiliar curriculum designs within the familiar curriculum package described above. At first glance, Everyday Mathematics, an elementary school program first published in the early 1990s by the University of Chicago School Mathematics project, looked very much like conventional textbooks. The teacher’s guide was a large, spiral-bound book with glossy pages packed with images of the student’s pages and teaching suggestions along the margins. Other designers opted for forms that appeared radically foreign to the textbook market. The elementary program, Investigations in Number, Data, and Space, developed by TERC and first published in 1998, for example, presented its program in a set of modules, each bound separately. The pages were printed with blank ink on

1 Even though they were not designed by commercial publishers, the Standards-based curriculum programs are published and marketed by commercial publishers.

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matte paper and contained descriptions of the teacher’s role in setting up and directing the lesson in addition to a good bit of blank space. Small pictures of the students’ work page, if applicable, were placed off to the side. The periodic pictures were pen and ink sketches of children engaged in the activities described in the lesson.2 These brief descriptions illustrate some of the various forms of address that print curriculum resources take. There are unlimited possibilities and variations. Increasingly, designers are making curriculum resources available in the form of webpages with links to different kinds of support and guidance that do not stop at mathematical and pedagogical suggestions. Some resources include video clips of classrooms using suggested activities and live discussion forums where teachers may seek insights from others. (See Chapters 2 and 5 for examples of discussions of how electronic curriculum resources influence teachers’ use of them.) In my examinations of mathematics curriculum resources and my studies of teachers using them, I have found that the large number of characteristics that make-up the form of address can be loosely classified into five interrelated categories: structure, look, voice, medium, and genre. Some of these characteristics tend to be given more attention in discussions of curriculum resources than others. I contend that each category is relevant to how teachers engage and utilize resources. Moreover, each category represents a set of design considerations and decisions that are not always made explicitly.

6.3.1 Structure Structure is the feature most commonly examined in curriculum resources. It refers to how the resource is organized and what it contains. Some refer to structure as the nature and organization of the content of the curriculum, the particular mathematical concepts and goals, and the underlying pedagogical assumptions. The components of structure can be parsed in a variety of ways. On the basis of his analysis of science curriculum resources, Brown (2009) identified three basic facets of curriculum resources that comprise their structure: (a) representations of concepts specific to the domain, (b) representations of tasks or procedures that students are expected to undertake, and (c) physical objects and representations of physical objects that are intended to support students’ work on the tasks and understanding of the concepts. These three facets, Brown argues, “encompass the most fundamental aspects of the curriculum’s content and structure: its core ideas, the activities undertaken in their explorations, and the objects that support such activities” (p. 27). Brown’s use of the term “representations of” signals the notion that most curriculum resources represent concepts (like the Pythagorean theorem) and tasks (like “Find the area of the triangle”). In some cases, the resource provides physical objects (like rulers) to be

2 It is worth noting that the second edition of the Investigations (TERC, 2008) has a physical appearance more in line with conventional mathematics teacher’s guides than the first edition, although it continues to be organized in modules.

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used by students. Often, the materials represent these and other objects pictorially (like a picture of a bridge in which triangles are used). As I discuss later, these representations makingup the structure of the curriculum are then read and interpreted by teachers. Analyses of the structure of curriculum resources tend to consider how the various components are organized, the mathematics content included or excluded through the representations, and the valence or emphasis of the content, including how the content is represented. Organization refers to how the features in the curriculum resource are packaged. Earlier, I described some structural elements of the typical commercial textbook published in the United States before the mid 1990s. These curriculum resources generally contained work pages for the student, answers to the student problems, guidance or even scripts to use during instruction, auxiliary activities, orienting resources (such as the table of contents, scope and sequence chart, and other resources that might help teacher structure the curriculum). In my analysis of Standards-based curriculum resources, I have found that many of these organizational elements are present, however, they may be packaged differently. For example, when it was first published, Investigations in Numbers, Data, and Space (TERC, 1998) consists of individual lessons or multi-lesson sessions (grouped according to a larger idea or investigation), but each session is not organized around the students’ work pages. Rather, the sessions are typically organized around a number of activities that are intended to occur in the class, some of which have associated student pages.3

6.3.2 Look Look refers to the purely visual appearance of the resource – what teachers see when they look at it. In the United States, cultural and institutional traditions exist, which influence the designed look of curriculum resources, even those designed by different publishers. Many of the commercially designed curriculum resources, for example, have a decidedly commercial look. They are printed on glossy pages, contain colorful photographs of smiling children, and include pages that read like advertisements for the materials. Colors and fonts are used in such a way that particular words seem to jump out at the reader. A number of the noncommercially published materials I have reviewed have a look that appears subdued when compared to those just described. Look is the result of a number of design choices, and is also influenced by the structure of the program. For example, a resource that represents reasoning and problem solving as central components of mathematics will have a different look than a resource that places primary emphasis on mastery of discrete skills.

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The Investigations curriculum is described in more detail in Section 6.6.

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6.3.3 Voice Voice refers to how the authors or designers are represented and how they communicate with the teacher (Love & Pimm, 1996). In the case of most curriculum resources I have examined, the authors are invisible and little information is provided about who they are or what their experience is. The invisibility of the author may be a device to depersonalize the text and increase its authority. It may well be a tradition that has evolved over time. Despite the invisibility of the authors, curriculum resources have a voice that is manifested through the way they communicate with the teacher. Most curriculum resources place primary emphasis on what the teacher should do. I think of this as talking through teachers (Remillard, 2000). That is, the authors communicate their intent through the actions they suggest the teacher takes. Few resources speak to the teacher by communicating with teachers about the central ideas in the curriculum. Some researchers have argued, however, that speaking to teachers is one way that curriculum resources can be designed to be educative for teachers (Davis & Krajcik, 2005; Schneider & Krajcik, 2002). Davis and Krajcik identified a set of design heuristics that curriculum designers might follow to make their resources explicit for teacher learning. Offering transparent and direct guidance related to reasons and purposes underlying task selections or anticipating students’ responses to tasks are two such examples. In their analysis of two elementary mathematics programs, Stein and Kim (2009) found differences in how designers communicated with teachers. The teacher’s guide of one program spoke primarily through the teacher. It offered pedagogical guidance, but few explanations. The other program included a number of efforts to speak to the teacher, including elaborations of reasons underlying pedagogical recommendations, notes to the teacher about common student errors or developmental learning trajectories, and example student dialogue. Another curriculum resource I examined, which was designed for teacher educators, included a journal written by a fictitious facilitator of the program. The journal was intended to provide facilitators using the resource with insights into the decision-making processes a facilitator might go through when using it with a group of teachers. In this sense, voice is related to structure because it is the inclusion or exclusion and placement of particular structural elements that shape the resource’s voice. The voice of curriculum resources is also evident in the language used. HerbelEisenmann used discourse analysis tools drawn from Morgan (1996) to analyze the voice of the student text of a Standards-based middle school curriculum, focusing on how the authoritative structures in the writing constructed the author, the reader, and mathematical reasoning. She noted an absence of first person pronouns-a common approach taken in student texts–and suggested that this tendency concealed the presence of human beings in the design of the text. She also suggested that the authors’ frequent use of second person pronouns in conjunction with objects in statements such as “the graph shows you,” obscures the authority of the authors and gives inanimate objects power to perform animate activities.

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6.3.4 Medium Medium refers to the form of delivery of the resource and has particular relevance with the increased availability of electronic instructional resources. Currently, the majority of curriculum resources are print based, a medium familiar to most teachers. However, as the use of electronic media and access to computer and networking technologies are becoming common, more teachers are using electronic and webbased resources (Gueudet & Trouche, 2009; Chapters 2 and 5). This evolution brings to the fore the need to consider medium in the examination of teachers’ interactions with these resources. Unlike those that are text-based, electronic resources allow for and often assume a nonlinear path through their offerings, giving the user a degree of navigational decision-making control not as apparent with print medium. On the other hand, as Gueudet and Trouche (2010) argue that the notion of author and authorship is often less transparent in online sources than in printed texts.

6.3.5 Genre The final category of form is genre. Unlike the other four categories, which reflect authors’ or publishers’ decisions, genre reflects what a curriculum guide is within a larger classification of written material for teachers. The curriculum guide is designed to offer a package that will aid in the construction of curriculum. In essence, it is meant to guide action and in this sense, it is more like a cookbook or manual than a novel. For this reason, there are elements of form that curriculum resources cannot completely transcend, despite designers’ efforts. The notion that a curriculum resource is a particular kind of artifact connects to Otte’s (1986) suggestion that texts have both objectively given structures (what can be seen) and subjective schemes (ways of being understood or expectations upheld about them). The curriculum-text genre signals particular subjective schemes among teachers who are familiar with them. Genre is important because it has implications for the expectations teachers bring to a curriculum resource that influence the way they engage it. I take this discussion up in the section that follows. The role of genre in meaning making is elaborated by Ongstad (2006) in his semiotic analysis of communication in mathematics and mathematics education. “Genre,” Ongstad explains, “precisely presupposes much of what can be expected in the kind of communication in question” (p. 262). Its familiarity conjures a “zone of expectation” and aids in how one makes sense of any form of communication, textual, or discursive. Naturally, any form of communication is likely to contain unfamiliar elements as well. Ongstad uses the term “rheme” to identify the unfamiliar or new. Learning or making meaning necessarily involves an interaction between the familiar (the theme) and the new (the rheme) in which the theme contextualizes and aids in the interpretation of the rheme. “Particular genres such as textbooks,

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definitions, explanations, and proofs for instance, will often have an implicit regime for balancing theme-rheme (or given and new) that we learn to use and recognise” (p. 263). An examination of the recent expansion in the development of curriculum resources, in response to both curriculum reform efforts and new digital technologies, raises questions about whether there exists a single curriculum-text genre. Even though all curriculum resources share a purpose or theme that distinguishes them from other forms of text, categories of resources have emerged within this genre. Standards-based curriculum materials, for example, which are researcher-developed as opposed to commercially developed, for many teachers, have become a genre within this broader class of resources. In other words, they provoke a particular theme that is distinct from conventional resources. That said, the examples described in Section 6.5, suggest that the broad curriculum-text genre can be powerful in the interpretive process for many teachers.

6.3.6 Why Forms Matter Forms of address are powerful mediators of teachers’ engagement with a particular curriculum resource. My position is influenced by sociocultural theory, which explains how artifacts mediate human activity (e.g., Cole & Engeström, 1993; Pea, 1993; Vygotsky, 1978; Wartofsky, 1973). From this perspective, curriculum resources are artifacts or tools that are part of the material world made and used by humans to accomplish goal-directed activity. They have material dimensions, but as constructions of culture, they also have social and cultural meaning. Indeed, they are “products of sociocultural evolution” (Wertsch, 1998) and are both shaped by and have the power to shape human action through their affordances and constraints. From this perspective, curriculum resources have the potential to enable, extend, or constrain human activity. My understanding of the power of forms of address has also been influenced by Rosenblatt’s (1980, 1982) work in literacy theory on transactions with text. In her writing about children’s interactions with literature, Rosenblatt argued that reading involves a transactional process between the text and the reader in a particular context. By framing this transactional relationship between the reader and the text, she makes an important distinction that is easily missed: the reader’s relationship is with the text and not the author. This is not to say that the author’s presence in the text cannot be detected in its design nor that subjective schemes or the genre is not at play in this transaction. But the reader engages and interacts with the designed artifact, not the author. When speaking of literature, Rosenblatt (1982) explains that although the author may have a particular plan for a book, he or she cannot predict what the reader will make of it. Similarly, when teachers engage with curriculum resources, they interact with the designed artifact and not the author’s intentions. This is why form matters. Form denotes certain meanings to the reader, a zone of expectation (Ongstad, 2006), which influence how teachers engage the resource.

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6.4 Modes of Engagement I use the term modes of engagement to refer to how a teacher interacts with the forms of address of the text. According to Rosenblatt (1980), readers enter into a transaction with the text they are reading and “there ensues the adoption, either consciously or unconsciously, of a predominant attitude or stance” (p. 388). This stance focuses the reader’s attention. Similarly, mode of engagement refers to what a teacher does in her transactions with a particular curriculum resource, how she engages, infuses meaning, and makes sense of its offerings. A teacher’s mode of engagement reflects her beliefs and epistemological stance. The notion of mode of engagement is also related to positioning theory (Harré & Langenhove, 1999). People use narrative, communication, or what Harré and Langenhove call “storylines” to position or locate themselves with respect to the scripts or roles made available in a particular context. As discussed earlier, these roles emerge from the teacher’s engagement with the forms of address. And as Ongstad (2006) suggests, the zone of expectation is influential in this interaction. When examining teachers’ use of curriculum resources, it helps to account for modes of engagement because the act of reading curriculum guides is different from the act of reading many other texts. This is where forms of address, and particularly genre, come into play. The forms used to address the reader signal to the reader what to expect, offering possible scripts. When a reader sees a poem written on a page, she sees many elements of its physical form – the blank page surrounding it, the uneven margins, the placement of the author’s name – all of which indicate to her that it is a poem and this indication prompts a particular mode of engagement that is unique to that reader in relation to the poem. As I described earlier, curriculum guides represent a particular genre of text that contain predictable elements or themes and these elements engender a response or mode of engagement from the teacher.

6.5 Forms of Engagement Just as the mode of address of a resource can be seen in its forms, a teacher’s mode of engagement can be understood through the forms that engagement takes up. In my research, I have found that a teacher’s mode of engaging a curriculum resource includes four primary forms or kinds of reading: what she reads for; which parts she reads; when she reads; and who she is as a reader. These forms of engagement overlap with several kinds of readings described by Sherin and Drake (2009) in their research. Earlier, I contrasted two teachers, Ms. McKeen and Ms. Yarnell, who read the same textbook in different ways. Specifically, they read different parts of the teacher’s guide and they read looking for different kinds of guidance. McKeen tended to read the routine lessons and focused on what the text had designed for students to do. Yarnell, in contrast, tended to read the auxiliary exploratory components of the text and read for the big mathematical understandings students were intended to develop (see Remillard, 1999, for details). These examples illustrate the first two

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forms of engagement. Sherin and Drake referred to distinction in what teachers read for as reading for activities or reading for big ideas. Few researchers have examined the different parts of the textbook read by teachers. This form of engagement seems particularly important in the United States where curriculum programs at all levels are becoming increasingly more laden with supplementary and alternative offerings, increasing the number of entry points, to invoke Ellsworth’s term. A third form of engagement is when a teacher reads the text. In their study of 10 elementary teachers’ curriculum strategies, Sherin and Drake (2009) found that teachers read their curriculum guides differently and at different times relative to instruction – before, during, and after. When teachers’ read is related to what they are reading for and their particular stance toward curriculum materials in teaching, discussed below. The fourth primary form of engagement is who a teacher is as a reader. In other work (Remillard, 2005; Remillard & Bryans, 2004), I refer to this positioning as stance or orientation. Teachers generally have a stance toward curriculum materials that is influenced by their views about mathematics, teaching, and the role that curriculum resources can and should play in the process of teaching mathematics. It is also influenced by their view of the particular resource. In my research, I have found teachers’ orientation toward curriculum materials in general to be strikingly influential in what they read, what they read for, and when they read their particular teacher’s guide. Rosenblatt (1982) used the term “reader’s stance” in a similar way: The reader may be seeking information, as in a textbook; he may want direction for action, as in a driver’s manual . . . In all such readings he will narrow his attention to building up the meanings, the ideas, the directions, to be retained; attention focuses on accumulating what is to be carried away at the end of the reading (1982, p. 269).

Clearly, forms of engagement are open to revision as a result of further reading and experience. Moreover, the claims made earlier about the role of the forms of address in signaling to the reader what to expect, might lead one to conclude that different forms of address, would prompt a change in how teachers initially engage new curriculum resources. Findings from a number of studies, however, suggest otherwise. At least initially, teachers are inclined to engage a new curriculum resource in similar ways to their interactions with previously used resources (Collopy, 2003; Lloyd, 1999; Remillard, 1991; Remillard & Bryans, 2004; Sherin & Drake, 2009). I posit that this tendency illustrates the two-way transaction between the teacher and the resources described by Rosenblatt (1982). Even though the forms of the resource contribute significantly to the reading, the reader does as well. Moreover, the genre figures substantially in the transaction: The words in their particular pattern stir up elements of memory, activate areas of consciousness. The reader, bringing past experience of language and of the world to the task, sets up tentative notions of a subject, of some framework into which to fit the ideas as the words unfurl (Rosenblatt, 1982, p. 268).

I believe that the genre of a curriculum guide – what it is and what it represents – provokes a mode of engagement that is particular to the teacher and, consequently, shapes the teacher–curriculum transaction. Thus, despite other elements of form, the

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genre, for many teachers, seems to trigger a “zone of expectation” (Ongstad, 2006) and assumptions early in the encounter.

6.6 Examples of Modes of Engagement What follows are several brief illustrative examples of what I mean by modes of engagement. They are all drawn from the first two years of a qualitative study of 14 elementary teachers from two different schools. Both schools had recently adopted the first edition of Investigations in Numbers, Data, and Space (TERC, 1998), an elementary program published in the United States. A brief description follows. Although it fit the genre of a curriculum guide, the Investigations curriculum was different from conventional materials in structure, voice, and look. It was designed to reflect the vision of NCTM Standards (NCTM, 1989, 2000), which was evident in its mathematical and pedagogical structures. The program emphasized conceptual understanding and reasoning about mathematical ideas. The majority of the recommended student activities involved collaborative exploration and problem solving, followed by class discussions. The program included worksheets that students were to complete, but they were minimal and generally intended to be integrated into the interactive part of the lesson. The program also included features designed to be educative for teachers (Davis & Krajcik, 2005). These features included information for the teacher in the form of mathematical explanations, rationales for the mathematical decisions made, examples of student work and talk, summaries of relevant research, and suggestions for assessment. The authors included the following statement in the introduction to each unit guide: ‘Because we believe strongly that a new curriculum must help teachers think in new ways about mathematics and about their students’ thinking processes, we have included a great deal of materials to help you learn more about both’ (p. 6). In this way, the program spoke in a different voice than was typically the case. Finally, the layout of the program offered an unfamiliar look. A great deal of text was devoted to describing the teacher’s role and what she might do and look for during the lesson’s activities. The following descriptions illustrate different modes teachers took when engaging this resource. They also illustrate how these modes interacted with the various forms of address, including the multiple entry points.

6.6.1 Mr. Jackson: Reading for Worksheets Mr. Jackson, a fourth grade teacher, had been teaching for 30 years when his school adopted the Investigations program. He was familiar with and a dedicated user of commercially published mathematics textbooks. In fact, although the school had officially adopted Investigations as its primary mathematics program, he had a set of older textbooks he distributed to his students and used along with Investigations. He used both programs in strikingly similar ways.

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During our observations of Jackson’s interactions with the curriculum, we noticed that he went first to the student pages to see what written work was part of the lesson. Even though these worksheets were not necessarily designed to be central to the lesson, he gave them central focus. Next, he would get a general sense of the structure of the lesson – what students were to do in what order. We have no evidence that he read the description of the teacher’s role provided with each lesson. In the lessons we observed, he moved fairly quickly to assigning independent student work and seemed to be most comfortable with this sort of teaching format. He orchestrated these lessons using the same teaching practices he used when using the commercial textbook. He asked questions and accepted answers, rarely asking for an explanation. When students worked at their desks on the assigned task, he sat at his desk and graded papers rather than interacting with them about the work. He often reminded students to “do your own work” and infrequently followed individual work with whole-class discussions. Often he ended class by asking students to hand in their papers after being sure their names were on them. When asked how math was going mid-November, he said: “Well we’ve followed along in the math books that we’re using, and the um, with Investigations, we’re basically right where we’re supposed to be, according to the schedule that we set up at the beginning of the year.” We identified Mr. Jackson as reading for worksheets because that is what he looked for and read in the curriculum. Although many aspects of the form of address of Investigations differed from the familiar text, he found the worksheets familiar and they dominated his reading of the curriculum. Mr. Jackson’s focus on the worksheets illustrates one way that multiple entry points work in a curriculum resource.

6.6.2 Ms. Hatcher: Reading for the Script Ms. Hatcher, a second grade teacher, had been teaching for 20 years when her school started using Investigations. Like Mr. Jackson, she had a long history using curriculum resources faithfully, but she avoided commercially published textbooks, opting for alternative resources that focused on problem solving and conceptual understanding. She was a careful reader of these curriculum guides and tried to follow them as best she could. For her, following the Investigations curriculum meant doing exactly what the authors suggested. Sometimes, this presented a challenge for her because, the teacher’s guide did not always tell her exactly what to do. As this quote suggests, Ms. Hatcher used the curriculum to create a script for herself. When asked how she used the curriculum to plan, she said: I reread the curriculum, reread whatever it is in the Investigations book we’re using . . . I am really following the teacher’s guide. If I’m having trouble understanding it, I will sometimes script it out; otherwise I highlight or maybe mark what I want to make sure I touch on.

When reading the teacher’s guide she focused on the lesson description, making careful notes in the margins. She also consulted the book frequently during each

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lesson. Indeed, we observed that Ms. Hatcher had her class follow each step of the lesson provided in the guide. She too tried to fulfill her role in the script by asking students to explain their answers and interacting with and challenging students during small group work. Ms. Hatcher’s stance on curriculum resources was that they provided the teacher with a script. When reading curriculum materials, which she did with great care, she read as much as she could, looking for a script. Even though she did not always find as much detail as she would have liked, she used what was available to create her own script.

6.6.3 Ms. Jordan: Reading for Big Ideas Ms. Jordan taught third grade and had been teaching for 4 years when she began using Investigations. She had limited experience with curriculum resources and clear ideas about the kind of mathematical understanding she wanted to foster in her teaching. She was attracted to Investigations because of its structure – the mathematical ideas it offered. She engaged the resource through these ideas. When reading the text, rather than wanting to know what to do, she wanted to find the important mathematical ideas. She then used them to shape her use of the lesson descriptions. When she first looked at the book, she went to the section that described the mathematical emphasis. She described her planning this way: I look at the mathematical emphasis first to see, first of all, what it is that I’m trying to get from them by the end. I’ll look at the teacher notes as they come up within the actual script of the lesson. I read the lessons a lot of times over because there are certain components I want to say, but I don’t want it to be scripted. . . . It requires rereading to make sure I have the mathematical emphasis down and I know what I’m trying to get. So even if the lesson leads a different way, I know the math aim I’m going for that day and I try to stick to that even if we have to veer off somewhere.

Ms. Jordan was the only teacher we studied who talked about using the mathematical emphasis and she was one of two who read the support pages at the beginning of each unit. We observed many instances of Ms. Jordan veering off her plans during a lesson. This happened most often when she felt her students were not getting the important ideas. When this happened, she often inserted an improvized review session of what she saw as the important ideas or made explicit connection to the previous day’s activity. We identified Ms. Jordan as reading for ideas, because it was the mathematical idea that most guided her decisions when using the curriculum. Like the others, her mode of engagement illustrates Rosenblatt’s (1980) assertion that when entering into a transaction with a text, readers adopt a predominant attitude or stance. Ms. Jordan’s stance was shaped by her goals in teaching mathematics and her view of how a curriculum guide could support those goals. Ms. Jordan was one of two teachers in the study (n = 14) who engaged the Investigations materials in ways that aligned with their dominant mode of address.

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They were the only teachers focused on the mathematical goals and emphases and who read the additional information for the teacher, using it to inform their teaching decisions and understanding of student learning.

6.7 Possibilities of Shifting Modes of Engagement The examples above illustrate how past experience with and assumptions about curriculum resources in teaching are drawn into the teachers’ transactions with new resources. Returning to Ellsworth’s notion of (1997) positioning, we see that curriculum materials have particular modes of address – ways of communicating with teachers – and these modes of address prescribe roles for teachers that position them as certain kinds of readers. But teachers enter this transaction with their own expectations, beliefs, and routines that shape their modes of engaging. Most often, these modes of engagement are formed in response to past experience with curriculum resources. In this way, teachers are positioned by their own encounters with curriculum materials as well as by the materials themselves. Margolinas and Wozniak (2010) used the term “generating documents” to refer to encounters with curriculum documents early in the career that have a forming or generative role in teachers’ future practice and, I argue here, future modes of engagement. My interest in how teachers’ modes of engagement are shaped and their tendency to be resilient over time is influenced by the design of nonconventional materials that potentially offer a new genre of resource and anticipate a different kind of teacher use. Many Standards-based programs seek to alter conventional teaching and curriculum-material practices. Like all materials, they are designed with a target audience in mind. These materials work best when the audience is who the designers intend and behaves as the designers expect it to. To be successful, the materials must enlist the teacher in being part of that target audience. Doing so necessarily involves contributing to a shift in the way teachers use these resources. As the evidence in this chapter and others (e.g., Chapter 10) suggests, achieving this shift requires more than a simple change in modes of address. The possibility that these modes can develop over time gives research an important focus to consider. Gueudet and Trouche (2009) found that the process of constructing and reconstructing resources leads to substantial change in teachers’ approaches. Moreover, Drake and Sherin (2009) found that, over time, teachers using a Standards-based program developed greater trust in the curriculum along with a clearer vision of its purpose. I see the domain of modes of engagement, how teachers engage and read curriculum materials, to be a fruitful place for learning to take place and an ideal focus for professional development of teachers. These learning opportunities would also offer valuable sites for inquiry. At this point, we understand little about the processes through which teachers might learn to engage with curriculum resources in substantially new ways and position themselves as partners with them.

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Acknowledgements The author is grateful to the insightful and substantive feedback provided by Carolyn Kieran and the three editors of this volume. This chapter develops concepts and analyzes presented in several earlier publications (Remillard, 1999, 2010; Remillard & Bryans, 2004). This research described within was funded by the National Science Foundation (Grant nos. REC-9875739; ESI-9153834) and the Pew Charitable Trust (Grant no. 91-0434-000). The views expressed within are those of the authors and are not necessarily shared by the grantors.

References Adler, J. (2000). Conceptualising resources as a theme for teacher education. Journal of Mathematics Teacher Education, 3, 205–224. Brown, M. W. (2009). The teacher-tool relationship: Theorizing the design and use of curriculum materials. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction, (pp. 17–36). New York: Routledge. Cole, M., & Engeström, Y. (1993). A cultural-historical approach to distributed cognition. In G. Salomon (Ed.), Distributed cognitions: Psychological and educational considerations (pp. 1–46). Cambridge: Cambridge University Press. Collopy, R. (2003). Curriculum materials as a professional development tool: How a mathematics textbook affected two teachers’ learning. Elementary School Journal, 103(3), 287. Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3–14. Drake, C., & Sherin, M.G. (2009). Developing curriculum vision and trust: Changes in teachers’ curriculum strategies. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 321–337). New York: Routledge. Ellsworth, E. (1997). Teaching positions: Difference, pedagogy, and the power of address. New York: Teachers College. Gueudet, G., & Trouche, L. (2009).Towards new documentation systems for mathematics teachers. Educational Studies in Mathematics, 71(3), 199–218. Gueudet, G., & Trouche, L. (2010). Des ressources aux documents, travail du professeur et genèses documentaires. (From resources to documents, teacher’s work and documentational geneses). In G. Gueudet & L. Trouche (Eds.), Ressources vives. Le travail documentaire des professeurs en mathématiques (pp. 57–74). Rennes, France: Presses Universitaires de Rennes et INRP. Harré, R., & van Langenhove, L. (1999). Positioning theory. Oxford, MA: Blackwell. Lloyd, G. M. (1999). Two teachers’ conceptions of a reform-oriented curriculum: Implications for mathematics teacher development. Journal of Mathematics Teacher Education, 2(3), 227–252. Love, E., & Pimm, D. (1996). ‘this is so’: A text on texts. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics, part 1 (pp. 371–409). Boston: Kluwer. Margolinas, C., & Wozniak, F. (2010). Rôle de la documentation scolaire dans la situation du professeur: le cas de l’enseignement des mathématiques à l’école élémentaire (Role of teaching documentation inthe teacher’s situation: the case of mathematics at elementary school). In G. Gueudet & L. Trouche (Eds.), Ressources vives. Le travail documentaire des professeurs en mathématiques (pp. 183–199). Rennes, France: Presses Universitaires de Rennes et INRP. National Council of Teachers of Mathematics (NCTM). (1989). The curriculum and evaluation standards for school mathematics. Reston, VA: Author. NCTM. (2000). The principles and standards for school mathematics. Reston, VA: Author. Ongstad, S. (2006). Mathematics and mathematics education as triadic communication? A semiotic framework exemplified. Educational Studies in Mathematics, 61, 247–277. Otte, M. (1986). What is a text? In B. Christiansen, A. G. Howson, M. Otte (Eds.), Perspectives on math education (pp. 173–202). Dordrecht, The Netherlands: Kluwer.

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Pea, R. (1993). Practices of distributed intelligence and designs for education. In G. Salomon (Ed.), Distributed cognition: Psychological and educational considerations (pp. 47–87). Cambridge, MA: Cambridge University Press. Remillard, J. T. (1991). Abdicating authority for knowing: A teacher’s use of an innovative mathematics curriculum. (Elementary Subjects Center Series No. 42). East Lansing, MI: Michigan State University, Institute for Research on Teaching, Center for the Learning and Teaching of Elementary Subjects. Remillard, J. T. (1996). Changing texts, teachers, and teaching: The role of textbooks in reform in mathematics education. Unpublished doctoral dissertation, Michigan State University, East Lansing, MI. Remillard, J. T. (1999). Curriculum materials in mathematics education reform: A framework for examining teachers’ curriculum development. Curriculum Inquiry, 29(3), 315–342. Remillard, J. T. (2000). Can curriculum materials support teachers’ learning? Two fourth grade teachers’ use of new mathematics text. Elementary School Journal, 100, 331–350. Remillard, J. T. (2005). Examining key concepts of research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. Remillard, J. T. (2010). Modes d’engagement: Comprendre les transactions des professeurs avec les ressources curriculaires en mathématiques (Modes of engagement: Understanding teachers’ transactions with mathematics curriculum resources). In G. Gueudet & L. Trouche (Eds.), Ressources vives, le travail documentaire des professeurs, le cas des mathématiques (pp. 201–216). Rennes, France: Presses Universitaires de Rennes et INRP. Remillard, J. T., & Bryans, M. B. (2004). Teachers’ orientations toward mathematics curriculum materials: Implications for teacher learning. Journal for Research in Mathematics Education, 35, 352–388. Remillard, J. T., Herbel-Eisenmann, B. A., & Lloyd, G. M. (Eds.). (2009). Mathematics teachers at work: Connecting curriculum materials and classroom instruction. New York: Routledge. Rosenblatt, L. M. (1980). What facts does this poem teach you? Language Arts, 57(4), 386–394. Rosenblatt, L. M. (1982). The literary transaction: Evocation and response. Theory into Practice, 21(4), 268–277. Schneider, R., & Krajcik, J. (2002). Supporting science teacher learning: The role of educative curriculum materials. Journal of Science Teacher Education, 13(3), 221–245. Sherin, M. G., & Drake, C. (2009). Curriculum strategy framework: Investigating patterns in teachers’ use of a reform-based elementary mathematics curriculum. Journal of Curriculum Studies, 41(4), 467–500. Stein, M. K., & Kim, G. (2009). The role of mathematics curriculum materials in large-scale urban reform: An analysis of demands and opportunities for teacher learning. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 37–55). New York: Routledge. TERC. (1998). Investigations in numbers, data, and space. Menlo Park, CA: Dale Seymour. Vygotsky, L. S. (1978). Mind in society. Cambridge, MA: Harvard University Press. Wartofsky, M. (1973). Models. Dordrecht, The Netherlands: D. Reidel. Wertsch, J. V. (1998). Mind as action. New York: Oxford University Press.

Chapter 7

Task Analysis as “Catalytic Tool” for Feedback and Teacher Learning: Working with Teachers on Mathematics Curriculum Materials Birgit Pepin

7.1 Background 7.1.1 Mathematics Education and Curriculum Materials The field of teachers and mathematics curriculum materials is rapidly growing. Much of its growth, particularly in the United States, is due to the current explosion of curriculum development projects in response to particular standards (e.g. NCTM, 1989), in addition to an increasingly widespread practice of mandating the use of a single curriculum to regulate mathematics teaching. In educational research, more broadly, some earlier work has focussed on teachers’ use of texts. Ben-Peretz (1984) distinguished between the curriculum “proposed” by materials and the curriculum enacted by the teacher, and she argues that teachers draw on their professional experience (and beliefs) to “assign meaning to the curriculum materials they use daily in their classrooms” (p. 71). She used the term “curriculum development” and argued for two phases of curriculum development, the second being when the teachers work with the materials to make them suitable for their students, and she describes these actions as “uncovering the potential of curriculum materials” for use in the classroom. More recent research in mathematics education views curriculum use as “participation with the text” (Remillard, 2005) indicating the dynamic interrelationship between teachers and curriculum materials. These studies (e.g. Chapter 6) not only develop insights into the use of curriculum materials, but also how teachers learn from their use. This view of dynamic interaction between teacher and curriculum, agent and tool, is also reflected in Lloyd’s study (1999) arguing that “curriculum implementation consists of a dynamic relation between teachers and particular curricular features.” (p. 244). Remillard’s (2000) study of teachers’ use of textbook as a response to Standards-based curricula highlights the importance of considering different perspectives in the field. B. Pepin (B) Faculty of Teacher and Interpreter Education, Sør-Trøndelag University College, 7004 Trondheim, Norway e-mail: [email protected] G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_7, C Springer Science+Business Media B.V. 2012

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Furthermore, and innovatively, Gueudet and Trouche (2009) coined the term “documentation work” indicating teachers’ work with materials: “looking for resources, selecting/designing mathematical tasks, planning their succession, managing artefacts, etc.” (p. 199). This paved the way for a “new” and creative thinking about mathematics curriculum material in connection with teacher “use” of those materials. This also provides a new perspective, viewing teachers not as passive users, but as designers of their own resources, and there are interesting and interrelational dependencies between teachers’ professional knowledge and curriculum design, each influencing each other in the process (Chapter 3). Interestingly, and somewhat in contrast, Ruthven (Chapter 5) developed a conceptual framework on the basis of five constructs, and amongst them the “curriculum script”, as compared to Gueudet & Trouche’s “document” (Chapters 2 and 16). Thus, it is evident that in mathematics education there is a growing body of scholarship and research that places teachers at the centre of the “teaching enterprise” raising questions about the effects of curriculum materials on classroom instruction and pupil learning. What happens when teachers use particular curriculum programmes (e.g. reform programmes), and why? An underlying assumption is that teachers are central players in the process of transforming curriculum ideals, captured in the form of mathematical tasks, lesson plans and pedagogical recommendations, into real classroom events. What they do with curriculum resources matters (Lloyd, Remillard, & Herbel-Eisenmann, 2009).

Thus, what teachers do with mathematics curriculum materials, how they “mediate” them (Chapter 4) and why, how they choose particular mathematical tasks, and how this complex net of choices influences classroom activity, is crucial for understanding not only the “implementation” of curricular programmes, but also for informing the work on the development of new programmes. Moreover, it is important for understanding how students interact and work with the curriculum materials (Chapter 12), and how they may learn in turn (Chapter 8). In line with researchers working in this field (e.g. Davis & Krajcik, 2005; Remillard, 2005), I use the term “curriculum materials”, and sometimes “textbook” materials, to refer to printed and often published materials designed to be used by mathematics teachers and pupils during classroom instruction.

7.1.2 Curriculum Materials, Teacher Use of Materials and Teacher Knowledge Underpinning the study reported in this chapter is the assumption that teacher learning involves teacher autonomy and agency when analysing, choosing, changing and transforming materials, devising alternatives, and “enacting” the materials (Ben-Peretz, 1984). Paris (1993) emphasises that teacher agency in curriculum matters involves “the creation or critique of curriculum, an awareness of alternatives to establish curriculum practices, the autonomy to make informed curriculum choices, an investment in self, and ongoing interaction with others.” (p. 16)

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Moreover, seminal work by Ball and Cohen (1999) discusses the role of curriculum materials, in particular textbooks, with respect to teacher learning. They assert that Curriculum materials could only become central to teacher learning, if the traditional boundaries between texts’ presentation of content and teachers’ teaching were redrawn to make central the work of enacting curriculum. (p. 7)

In terms of improving instruction, materials are often seen to offer resources for teachers’ work with their students, and not designed to encourage teachers’ investigations of and work with the material. Sadly, it is claimed, teachers must often learn alone “with few resources to assist them”. Thus, they call for the creation of curricula that would help teachers to better enact curriculum in practice. If the boundaries of curriculum design and development were reconsidered and redrawn, curriculum materials could offer teachers more opportunities to learn in and from their work. (p. 8)

Furthermore, in the United Kingdom, the recent Williams Review (Williams, 2008) drew attention to the need, and challenge, of strengthening mathematical knowledge for teaching system-wide, and recognised the scale of professional development initiatives required to secure that knowledge. As new mathematics curriculum materials are being and have been developed in many countries (e.g. US: NCTM, 1989; UK: DfES, Standards Unit, 2005), teacher learning is considered an important aspect and part of the “enactment” of these materials: the design and teaching of new materials as a potential place for teacher learning (e.g. Remillard & Bryans, 2004). For example, recent research in science education advocates educative curricular resources as both a tool for teacher learning and as a support for teachers to become curriculum designers (e.g. Davis & Krajcik, 2005). Ruthven, Laborde, Leach, and Tiberghien (2009) argue that “the availability of design tools capable of identifying and addressing specific aspects of the situation under design can support both the initial formulation of a design and its subsequent refinement in the light of implementation” (p. 329). It appears that there is large potential for curricular and pedagogical resources to be designed, or existing materials to be amended and enriched, so as to fulfil an educative function for the teacher (Ruthven, 2008). The rationale for the work with teachers was based on findings from TIMSS (Hiebert, Gallimore, Garnier, Givvin, Hollingsworth, Jacobs, et al., 2003) which report that high achieving countries engage students more frequently in rich mathematical activities, (and more rigorous reasoning) than lower achieving countries. In particular, it is claimed, students (in high achieving countries) are presented with more rich and open mathematical problems that require them to make connections between mathematical ideas. There are likely to be several reasons that may explain “poorer results”, amongst them the following: (a) mathematics teachers do not possess a deep understanding of the mathematics they are asked to teach and (b) teaching mathematics with attention to conceptual underpinnings is difficult, and time consuming, and unless it is given priority, seems not consistent with the tradition of school mathematics. Thus, although most policy makers would probably

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agree that making “learning mathematics with understanding” has become a shared objective, teachers seldom have opportunities, and time, to develop mathematical tasks and teaching sequences, where “richness of tasks” and “learning mathematics with understanding” are emphasised. With these two hypotheses in mind, I embarked on a professional development programme to include opportunities for teachers to deepen their own understanding of selected key concepts of curriculum they were likely to teach, to improve their knowledge of ways students may understand the content, and learn about analyzing, selecting and enriching mathematical activities, and subsequently learn about how to teach these in their classrooms. The approach to teacher learning centres around the analysis and enrichment/amendment of mathematical tasks, that is curriculum materials (work sheets, textbooks, etc.) that teachers typically use for their teaching. Teaching is seen as a dynamic process that goes beyond what happens in the classroom to include analysis and selection of curriculum materials as part of planning and reflection (Chapter 2). The literature claims that improvements in planning and reflection have great potential for improving teaching (Ball & Cohen, 1999; Fernandez, 2002; Hiebert et al., 2003). Moreover, it is in agreement with research on teacher learning which claims that effective professional development must provide opportunities for teachers to work together, analyse and discuss curriculum materials in connection with classroom practice (e.g. Whitehurst, 2002; Chapters 15–17). Teacher learning is seen here in the widest sense, as teachers work together or on their own and with materials that help them to develop their knowledge in and for teaching.

7.2 Tools for Reflection and Feedback In this study, I explore the role and nature of feedback resulting from the development and use of a tool designed to help teachers develop further understandings of characteristics of mathematical tasks, their selection, amendment, enrichment and potential use with their students. This fits largely within the studies on feedback in professional learning (e.g. Hargreaves, 2000) and that teachers can learn from feedback (student feedback in the case of Hargreaves, 2000). There is also a large body of research of teachers’ experiences of learning through enquiry and collaborative projects (e.g. Fennema, Carpenter, & Franke, 1996; Greeno & Goldman, 1998; Chapters 15 and 17), amongst them those that highlight the importance of tools (e.g. Baumfield, Hall, Higgins, & Wall, 2009; Chapter 16). A “tool” can be viewed in different ways. Boydston (1986) claimed A tool is a mode of language, for it says something to those that understand it, about the operations of use and their consequences . . . in the present cultural settings, these objects are so intimately bound up with intentions, occupations and purposes that they have an eloquent voice. (p. 98)

Thus, whilst a tool may have different forms, using a tool in the context of pedagogic practice, it is likely to re-frame teachers’ experiences (this is also

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acknowledged by the instrumental approach, introducing the tool/instrument dialectics, Chapter 2). For example, a new tool is likely to add something to the repertoire; equally it may disrupt participants’ practice and take something away (Baumfield, 2006). This reflects the tool’s catalytic quality: it may change participants’ practice and environment. The individual agency of the teacher rests with the decisions s/he takes as a result of feedback from the use (of the tool), thus her/his reactions to the feedback. Literature on learning through inquiry and feedback in professional learning (e.g. Hargreaves, 2000) emphasises how teachers can learn from student feedback (to bring about learning and/or change in classrooms). There is also a large amount of literature linking student achievement and feedback (e.g. Butler & Winne, 1995) where feedback is conceptualised as “information with which a learner can confirm, add to, overwrite, tune, or restructure information” (p. 275). In their study on the importance of feedback Hattie and Timperley (2007) claim that “feedback is one of the most powerful influences on learning and achievement” (p. 81). They conceptualise feedback as information provided by an agent, may it be a teacher, peer, book, self, experience or curriculum materials, regarding aspects of participant’s performance or understanding. To understand the interaction of enquiry and feedback in teacher learning, Baumfield et al. (2009) investigated how a tool designed for student awareness of their learning also supported teacher professional development. There is ample evidence (e.g. Hattie & Jaeger, 1998) that the presence of feedback (in whichever form it may be) increases the likelihood that learning will occur. In this study I conceptualise feedback not as feedback from students, but from other sources, such as the analysis schedule, curriculum materials, or peers, for example. Moreover, and leaning on research by Hattie and Timperley (2007), I distinguish between four levels of feedback: the task level (how well the tasks are understood/performed); the process level (the main process(es) needed to understand/perform tasks); the self monitoring level (directing and regulating actions); and the personal evaluation level (personal evaluation and affect) (p. 87). Winne and Butler (1994) also claim that feedback can have different sources: external (e.g. provided by contexts or other participants); and internal (e.g. self-generated such as monitoring their actions). Using this theoretical frame I seek to develop deeper understandings, and theorise, “feedback” in connection with “tools” – and where tools are perceived in different ways than previously done – and investigate and relate their connected power to teacher learning.

7.3 The Study The study built on previous work with teachers, textbooks and other curriculum materials (e.g. Haggarty & Pepin, 2002; Pepin, 2008, 2009). Supported by a grant from the National Centre for Excellence in the Teaching of Mathematics (NCETM1 ) 1

https://www.ncetm.org.uk/

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and whilst based in the United Kingdom, I worked with teachers over a period of 9 months. The work with teachers consisted of two phases: 1. Curriculum exploration and transformation (deepening teacher knowledge of mathematics concepts and teaching mathematics “with understanding”, development of analysis schedule for curriculum materials, analysing mathematical tasks, selection of tasks and activities for lessons); 2. Link to practice and lesson analysis (“enacting” the curriculum materials, reflection on enactment-videoed lesson, sharing of work and discussions). Teachers participated on a voluntary basis. Originally, five teachers of the school’s mathematics department intended to participate, but due to illness and private circumstances this number was reduced to two: John and Paul. John was a relatively experienced teacher (who had been educated at the university department), whereas Paul was a trainee teacher on a “Graduate teacher programme” (i.e. learning to teach “on-the-job” whilst working in and being employed by a school). In addition, two other newly qualified teachers joined the group for a few months at the beginning of the project. In this chapter, I will mainly refer to John’s and Paul’s participation. It is important to distinguish between two levels of study here: the professional development work with teachers; and the research study that drew on the work with teachers. The main aim of the professional development project was to work with these teachers to develop their mathematical knowledge in/for teaching, that is (1) to assist teachers to analyse, select and amend mathematically rich problems; and (2) to assist teachers to “implement” their analyzed/amended activities into their pedagogical practice. Here the author acted as “trainer” and “critical friend” to guide and help teachers in their endeavour to develop their pedagogic practice. The main aim of the research study was to develop a deeper understanding of how curriculum materials, and working with teachers on curriculum materials, can support teacher learning and professional knowledge (Chapter 17). After funding was obtained, six meetings were scheduled over the course of 9 months. In terms of professional development, during those meetings the focus was on different activities (see phases and activities, Table 7.1). In terms of research and data collection the following data were collected over the 9 months period: • Observation reports (descriptions) of what went on during each of the six sessions. • Written evaluation/feedback after each of three whole day university sessions. • Lesson observations (Paul and John observing each other’s lesson, also the teacher educator). • Group interview after the lessons (and based on the observed lessons). • Group interview as evaluation of the whole project.

Reading, discussing and presenting literature on “learning mathematics with understanding”

Development/amendment/ re-shaping of mathematical task analysis tool for task analysis (on the basis of Pepin, 2008). Analysis of “different” mathematical tasks (e.g. LEMA). Auditing/collection of resources available in the school’s mathematics department. Use of “tool” to analyse/amend/ enrich mathematical tasks.

Use of “tool” for analysis of assessment tasks/test questions. Linking assessment, mathematical tasks and National Curriculum. Preparing tasks for teacher use in lessons. Learning Walk (lesson observation: teacher educator and each teacher observing the other’s lesson). Observing each other’s lessons. Discussion of lessons (play back) and feedback to each other.

Literature review

Development of task analysis schedule

Assessment, task analysis & National Curriculum

Learning walks

Task analysis

Activity

Phase

Reflective (feedback on practice: suggest alternative strategies & trial out different tasks and practices)

Diagnostic

Diagnostic (focused on learning and skills).

Diagnostic, reflective

Reflective (focussed on knowledge/learning)

Types of feedback

Table 7.1 Types of feedback with respect to activity

“Creative” enactment of materials. Making learning more explicit. Opening to critical enquiry helps to build autonomy.

Creative questioning with respect to the purpose and value of mathematical tasks. Alternative strategies of devising a mathematical task. Selection of appropriate tasks for instruction. Confidence in amending and enriching materials for particular purposes. Linking context (assessment and National Curriculum) and content.

Develop insights beyond the immediate context. View mathematics/learning mathematics in different ways How to analyze mathematical tasks. Clarify ideas on the purpose of tasks and on what mathematical tasks may inherently possess.

Teacher knowledge/learning

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In terms of analysis, a procedure involving the analysis of themes similar to that described by Woods (1986) and by Burgess (1984) was adopted and using the “constant comparative approach”. Moreover, I tried at one level to maintain the coherence of each teacher’s responses over the different sessions, by analyzing the data with respect to observations and with respect to the different types of responses (e.g. different interviews, evaluation feedback, etc.); at another level I analyzed across the two cases and using the different concepts of “tools” and testing the hypotheses offered by the literature, and building explanations and theorizations anchored in the data. On a third level, I looked for similarities and differences of teacher responses with respect to what the literature claims about feedback, tools and teacher learning. In theoretical terms the analysis focuses on the types and role of feedback, stimulated through the use of the “tool”, in teacher’s learning, and I use the constructs outlined in the previous section to develop a deeper understanding the interaction of enquiry and feedback in teacher learning. However, due to the small number of cases, it was important to address the potential difficulties with respect to validity of the findings. In terms of validity checks, both teachers were invited to comment on the observation reports, also in the final evaluation interview at the end of the project. It was also important to locate and understand teachers’ classroom practices and classroom cultures in context, that is in the particular school environment in which they were working: an inner city comprehensive school in the North-West of England. Here, it was useful to draw on knowledge gained from previous work with at least one of the two teachers and the partnership relationship with the school. In addition, National Curriculum documents, and the school’s guidelines and curricular texts (including textbooks) were analyzed to study the contextual background of mathematics instruction and the potential influences of these texts on teachers’ perceptions and pedagogic practice.

7.4 Findings In this section I discuss the different types of activities undertaken with respect to information gained by and feedback provided to teachers, and in turn its potential for teacher learning. During the 9 months research and development period different kinds of “activities” were undertaken which can be categories under five different “phases” (see Table 7.1): Phase 1: reading, discussion and presentation of the literature on “learning mathematics with understanding” Phase 2: development/amendment/re-shaping of mathematical task analysis tool for task analysis; Phase 3: use of “tool” to analyse/amend/enrich selected mathematical tasks; Phase 4: preparation of tasks for classroom instruction; Phase 5: “learning walks”. During these phases, and in the different activities, the “task analysis schedule (the ‘tool’)” was developed and used, inside and outside class and in very

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different ways, which in turn influenced its nature. This is the focus of discussion, and illustrated in the following. (1) In the first phase reading, discussion and presentation of the literature on “learning mathematics with understanding” helped teachers to view mathematics in a different way. Lively discussions centred around the issue of what it may mean to learn mathematics with understanding, and one of the key issues identified here was related to “making connections”, and in different ways (e.g. to familiar situations, or to previously taught mathematics). Understanding in general is linked to “something”, . . . experiences perhaps. To understand mathematics we must “connect” it to something relevant or “of meaning” to an individual [pupil]. . . . In my experience too many people are concerned with “how to get the answer”. My perception of learning maths with understanding looks at “why” the answer works. (John, Session 1 evaluation)

The emerging discussion on the literature was also seen as a valuable activity to enhance teacher learning in terms of bringing together theory and practice. . . . the discussions have promoted the deep links between the literature and those aspects [identified earlier as individual aspects of teaching mathematics] . . . the overall process of creating a dialectical fusion between theory and practice has become clearer. I feel more able to “read”. (Paul, Session 1 evaluation)

Thus, this activity appeared to help teachers to develop insights beyond the immediate context and “next-day-lesson” and view mathematics learning in different ways (e.g. “to link something to something else”, connecting theory and practice). The type of feedback likely to be “produced” by the activity was reflective and focussed on knowledge and learning. (2) In the subsequent phase, and in subsequent sessions, these notions helped to identify what kinds of characteristics a mathematical task may/should have so that learning with understanding is more likely to happen. On the basis of a “skeleton” task analysis schedule (e.g. Pepin, 2008) teachers developed, amended and re-shaped this for their own analysis of activities and according to their own understandings. This meant that teachers added or changed categories according to what they regarded important in mathematical tasks. For example, and linking to the task analysis schedule (see Appendix), the category of “connections through mathematics” was further re-defined, to differentiate between and include “connections within mathematics” as well as “connections across other subjects”. As another example, several categories under “processes” were further developed, in particular the category of “analysing” in order to differentiate between “reasoning” and “procedural”. As a third example, the whole category of “familiarity” was introduced, and it was apparent that teachers drew on their repertoire of practical experiences here, in order to identify what was important for them and their teaching (evidence: observation reports). One of my reasons for participation on this project was to extend pupils’ knowledge and experience of mathematics beyond the procedural and technical. I believe that the tasks/activities carried out by students play a major part in this experience. The analysis tool which we are developing exposes aspects of the mathematical process, language demands

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and symbolism held within the structure of a task. This exposure has formalised, for me, necessary parameters which I can address individually within my teaching. (Paul, Session 1 evaluation)

In terms of characteristics of a task, particular notions were highlighted, amongst them the following: • “the need for the task to have a purpose – what are we hoping to develop with the task”; • “tasks that incorporate a range of concepts and cognitive demands [are] ‘richer, and therefore of more benefit”; • “communicating mathematically in a range of ways is very important in cementing a pupil’s understanding”; In the light of the discussions and re-designing the task analysis tool, teachers saw the need to re-think their pedagogic practice, and they also realised the difficulties and efforts connected with this. I would like to increase pupils’ exposure to mathematical language and symbolism to allow the possibility of rich dialogue. I would like to use this opportunity to improve the connections within mathematics and between mathematics and (1) other subjects, (2) “real world”. I would like to raise the level of thinking required by pupils. (Paul, Session 1 evaluation)

Thus, in short the development/amendment of the task analysis schedule appeared to help teachers to clarify ideas on the purpose of tasks and what a mathematical task may inherently possess (e.g. in terms of “aspects of mathematical process”, “language demands” and “symbolism”). There is evidence from the data that the type of feedback likely to be “produced” by the activity was diagnostic and reflective. (3) In the third phase, the task analysis itself helped teachers to see alternative strategies of devising a task and what this may mean for classroom practices/processes. After auditing their own school mathematics department resources, teachers brought to the sessions a range of tasks they used in their lessons (e.g. worksheet “House & Garden” that included mathematical tasks on designing a house and garden, decorating the house’s surfaces, designing a pool in the garden, etc.). The teacher educator also provided selected tasks, for example modelling/“open” tasks from an EU project (LEMA, 2009) and selected textbook tasks (e.g. from German and French school textbooks). These were then analysed on the basis of the re-designed task analysis schedule. For example, it was suggested to enrich the “area & perimeter” part of the question to include ideas from the Standards Unit (DfES, 2005), to give pupils more opportunities to address potential misconceptions (e.g. “same perimeter- same area”; evidence: observation report 3; Paul, Session 3 evaluation). On the same topic it was also suggested to use more practical material for the task, such as string or straw to show that the same perimeter can hold different areas. Thus, this activity appeared to help teachers to develop alternative strategies of devising a mathematical task with respect to potential misconceptions, select and

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amend appropriate tasks with respect to their developing ideas and intended instruction, and use creative questioning with respect to the purpose and value of tasks. There is evidence from the data that the type of feedback likely to be “produced” by the activity was diagnostic and focussed on learning and skills for their classroom instruction. (4) In the fourth phase teachers identified the need to look at assessment (and tests) in connection with task analysis, and with respect to the National Curriculum (in England). Thus, as a first step the group decided to use the previously analysed “House & Garden” tasks for “assessing pupils working on tasks”. Sensitized by the previous sessions in terms of “what we may assess”, that is considering the categories relating to task features (e.g. content; connections; contexts; familiarity; representations; etc.), these were mapped against how these could be assessed by a teacher (e.g. what kinds of questions may a teacher ask him/herself?), and the different levels and sublevels of the National Curriculum. As a second step, and in terms of linking assessment and tests to task analysis, departmental grade 8 tests (on number, three different achievement levels) were analyzed. Interestingly, some supposedly “lower level” questions were actually considered (by teachers) to have more potential in terms of “openness” and richness than some of the higher level questions that aimed at procedural understanding, and were described by teachers as “numbing”. Linking this to Assessment for Learning (AfL) teachers developed ideas, in particular with respect to peer- and self-assessment, and developing pupils’ awareness of skills they are using during a series of lessons. Reflecting on assessing tasks and what pupils have produced and learnt is as important as analysing the tasks. (Suzanne, Session 3 evaluation) . . . I will think about a mathematical task and evaluate what areas of the NC levels it addresses rather than the converse, i.e. teach to the NC levels. (Bill, Session 3 evaluation) There is scope to begin with open investigative tasks, . . . Reducing the number of tasks, whilst allowing more time on fewer tasks allows for greater detail in what pupils discover/learn. (Paul, Session 3 evaluation)

Thus, the link to assessment and the National Curriculum appeared to have given teachers another view point and they appeared to have become more secure in their knowledge about tasks, what they can afford and how they can be taught and assessed. The assessment activities helped teachers to gain confidence in amending/enriching materials for particular purposes, for example for “creative” assessment. In addition, this activity helped to link assessment tasks/tests to the National Curriculum, and to see them in a different light (e.g. as formative curriculum materials rather than evaluative). There is evidence from the data that the type of feedback likely to be “produced” by the activity was diagnostic and focussed on assessment and instruction. (5) In phase 5 and during the later parts of the project (Learning Walk – lesson observations) selected lessons of John and Paul (one morning with 1 and 2 lessons each) were videoed. For these both teachers had prepared and discussed, amongst each other, what and how they planned to do things. The two videoed lessons (one each) were played back in an afternoon session at university, and the subsequent

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discussion (on the basis of these two lessons) centred around teachers’ use of their developed/prepared curriculum materials. One of the foci of “modification” of tasks, and pedagogic practice, was to instil more discussion into their lessons (see also worksheet). . . . the lesson I did was modified and adapted from a lesson that I had previously done . . . but the fact that we modified it is a result of this [project] . . . (Paul, video recall) . . . we had looked through the activities that we had already and . . . we used our awareness of this tool to modify the wording . . . we both wanted the discussion to take place . . . both in pairs and in groups . . . to modify the tasks so it was more explicit that it was discussion that we went for. (John, Video recall)

It appeared that the thinking about tasks helped to make the processes involved in doing the tasks more evident. I agree, I think the . . . process of going from coming up with the initial idea, estimating, making a guess, throwing the ideas out and then honing into, to improve mathematical symbols. The idea of mathematical language in order to revisit the problem . . . I think that’s . . . quite a powerful . . . thing that came out of this whole process. . . . I think that the tool has enabled us to, to mediate the tasks. [my italics] . . . Both on paper and then, because of our awareness of what we want, or a greater awareness through out classroom . . . communication. (Paul, video recall)

The final discussions centred around their collaboration, working with each other as “sparring partners” in this project. Teachers emphasised the importance to work with someone “to bounce off ideas” and go beyond what one may develop when working alone. Because I, I would say I have quite a lot of lessons like that, that kind of thing, but they’ve never been developed beyond what I thought of myself . . . I’ve never, in a lot of my lessons, I’ve never bounced my ideas off anyone else. . . . Like, [John] and I did with this one. . . . that part of it was enriching . . . (Paul, video recall)

Thus, the “learning walks” provided opportunities for peers to suggest alternative strategies (for classroom practice), and for encouragement to trial out different things and work together in a team. There is evidence from the data that the type of feedback likely to be “produced” by the activity was reflective and focussed on practice. In summary, it can be said that teachers developed their ideas, whilst going through the different stages: from reviewing the literature; to tasks analysis and task enrichment; to “creative” applications and considerations of task analysis; to enactment in the classroom. At each stage (and these are not seen as hierarchical) they carried “residuals” from previous sessions, and appeared to become more confident in terms of how to proceed, what to do next, why this may be useful, what they may have learnt, for example as the project went on. The “analysis” and “enrichment” of mathematical tasks appeared to have become an analysis and enrichment of their pedagogic practice.

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7.5 Discussion and Conclusions There is general agreement that curriculum materials can support teacher learning and professional development (e.g. Ball & Cohen, 1999; Schneider & Krajcik, 2002). However, it is less clear what kinds of materials (e.g. textbooks) should be used, what characteristics the materials should have, whether educative or otherwise, and moreover what kinds of processes and “use” of materials help teacher learning. In terms of teacher learning, the literature suggests that effective professional development should have three crucial elements: it should be linked to teachers’ classroom context (e.g. Borko & Putnam, 1996); teachers should be supported longer term (e.g. Marx, Blumenfeld, & Krajcik, 1998); and teachers need to be given opportunities to build new knowledge (e.g. Borko & Putnam, 1996). Thus, there is the general view that, whilst curriculum materials have educative potential, they may not be effective without additional professional support. Feedback from others is generally seen as promoting reflection on and inquiry into practice (Fenstermacher & Richardson, 1993; Schön, 1983). However, in Collopy’s (2003) study one teacher seems to have developed through the use of textbook and other curriculum materials, and without additional professional support, whereas the other did not. In this study the mathematical tasks teachers worked on with a university teacher educator were integral to their daily work, both in terms of where the materials originated from (e.g. tasks chosen from their departmental resources) as well as their use in their daily instruction. However, the following seemed crucially important: (1) the situation created by the professional development activity and (2) the tools and processes surrounding the tasks, for example analysing the tasks, and providing and developing a tool for analysis. It appeared that the focus of developing reflection and thinking with teachers was less afforded by the tasks themselves, and the tasks/curriculum materials themselves could be educative or otherwise. Teachers appeared to need the necessary “tools” (e.g. task analysis tool, knowledge of how to enrich a task) to stimulate their thinking, and in turn (re-)shape the mathematical tasks for their teaching. Thus, it is legitimate to ask what may count as a “tool”, and what a tool should afford. Considering Table 7.1, and with reference to the findings outlined in the previous section, I thus conceptualize feedback as information not only provided by a person (e.g. the teacher educator, or peer teacher), but by other “agents”, such as curriculum materials, or more particularly in this case a task analysis schedule. Leaning on research by Winne & Butler (1994) feedback can here be re-conceptualized as information – whether it be mathematics domain-knowledge, meta-cognitive knowledge about processes involved in working with the mathematical tasks, beliefs about self and tasks or cognitive tactics and strategies – which teachers can use to confirm, add to or re-structure their knowledge, in short as feedback for learning. They also claim that feedback can have “external” and “internal” sources (Bangert-Drowns, Kulik, Kulik, & Morgan, 1991): external sources can be incidentally or intentionally provided by others (e.g. peer teacher, interaction with the environment, written

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comments) – in this case such sources are most likely provided by the social situation created by the professional development activity; internal sources for feedback are self-generated (e.g. teachers monitoring their activities and engagement with the learning task). The main message from these studies is that the learning context to which feedback is addressed needs to be considered: in this case teacher learning with curriculum materials (as compared to pupils learning in classrooms). Considering the different types of feedback (see Table 7.1), it appears that the development and use of the task analysis schedule (the “tool”) was crucial in teachers’ awareness raising/developing understandings of task characteristics and potential of particular mathematical tasks for teaching, hence in terms of support for teachers’ learning. There is evidence that this tool was the pivotal point around which most other activities centred, or were linked to, and which was mentioned in all discussions and evaluations (see earlier quote by Paul): as it developed, when it was used for the analysis and enrichment of mathematical tasks, and in the “enactment” of the amended tasks during instruction. Considering its perceived importance, and in terms of the associated “documentational genesis” (Chapter 2), the feedback the tool provided can be perceived at the four different levels (outlined by Hattie & Timperley, 2007). At the “task level”, the analysis tool provided feedback to teachers about the characteristics of the actual tasks, how well these were understood. At the process level, the analysis tool provided feedback in terms of what these tasks may, or may not, afford in terms of pupil learning and skills, the processes needed to understand the tasks. At the self-monitoring level, the tool provided feedback in terms of confidence of working with such an analysis tool. Finally, at the personal evaluation level, it provided feedback in terms of confidence to engage in further enquiries of such type. Thinking in terms of internal and external sources for feedback, it can be argued that the “tool” (analysis schedule) was an external source at the outset, but became an internal source of feedback. Whilst provided (by the university teacher educator) for teacher use in “skeleton” format, and as further developed from the literature with teachers, they shaped the task analysis tool and made it “their own” (see also “instrumentalisation” in Chapter 2) according to what they regarded as important characteristics for a mathematical task for their teaching. This process started with reading and discussing the relevant literature, and subsequently it was amended and then used on their chosen tasks. This in turn triggered ideas for amendment/enrichment of tasks and for comparison of characteristics with National Curriculum “features” and with national/departmental “test tasks”. At different stages of development and use of the analysis tool, different kinds of feedback resulting from the tool became apparent. Moreover, it is argued that by participating in the practice of enquiry (Greeno & Goldman, 1998) to analyze/work with mathematical tasks and the task analysis tool, teachers gained access to feedback that stimulated their professional learning and enabled them to become reflective (Schön, 1983). Particular attention is given here to the role of the tool for enquiry. The tool for task analysis became a tool for enquiry in activated feedback loops between (1) the two teachers amongst each

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other, (2) between them and the “tool” and (3) between them and the teacher educator and the “tool”. This, in turn, provided support for teacher learning. In this way, the level of teacher engagement and learning lifted the “tool” beyond its level of artefact, to become an “epistemic object” (Rheinberger, 1997): a knowledge object that is developmental in nature and depends on the place it occupies in teachers’ collaborative practices. An epistemic object is an object that is beyond the agents’ knowledge and understanding, at the time of first use, and at the edge of the epistemic horizon (see also Miettinen & Virkkunen, 2005). At the same time epistemic objects are grounded in historically developed practices. They function as generators of novel understandings, conceptualization and perhaps innovative solutions, as they are not yet known with certainty. The creative nature of the work with epistemic object appears to be characterized by working “at the edge of the unknown”: working with them produces developing conceptualizations and understandings. This view of use of tools is anchored in socio cultural theory, in particular Vygotsky’s (1978) notion of “tool” and mediation of tools. Cultural Historical Activity Theory (CHAT) developed this further (e.g. Engeström, 2001) claiming that learning can be mediated by a range of tools and instruments. At its inception the “tool” (analysis task schedule) was defined and meant to analyze mathematical tasks, a kind of technical object grounded in familiar pedagogic practices. However, over time and with different activities the insights gained (through the work with the “tool”) triggered communication between teachers (and teachers and teacher educator) and feedback at different levels; and provided access to a developing depth of perspective which encouraged teachers to explore further. In short, the original tool developed into something else: it became an epistemic object which challenged previous perceptions (e.g. creative questioning of value and purpose of mathematical tasks, and what they can afford); it produced “novel situations” (e.g. confidence in amending and enriching materials for particular purposes); and generated novel understandings of pedagogic practice (e.g. viewing mathematics/mathematics learning in different ways). It can be argued that the “tool” has developed catalytic potential, in the sense that it helped teachers, and gave them opportunities, to engage in a re-framed experience. Using the tool had aspects of familiarity (since it is grounded in the “territory” of mathematical tasks and learning), and at the same time of novelty as some of their perceptions are likely to be challenged and something being added to their repertoire. This combination of familiarity and novelty is likely to create “positive conditions” and for the teachers to experience “positive dissonance” (Baumfield, 2006) whereby routines and expectations are likely to be challenged, or disrupted, without the teachers feeling vulnerable, and more importantly for new ways of feedback to be opened up. This is claimed to be the tool’s catalytic quality: it can open up new avenues (e.g. of feedback), whilst maintaining stability by not being changed itself. Thus, the tool’s catalytic potential is provided by its intrinsic features, its usefulness in teacher everyday professional lives, and its potential for empowerment in terms of teacher learning.

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The crucial process element of the catalytic tool is the kinds of and the nature of feedback “produced”. The feedback from the task analysis tool is developmental, context-specific and highly relevant to teachers’ professional needs: be they reflective; diagnostic; focussed on knowledge/learning, or on skills. In Table 7.1, an overview is provided to show which kinds of activities (related to the “tool”) afforded which kinds of feedback, and in turn are likely to enhance which kinds of teacher learning. In summary, there is evidence that the project has had positive benefits to: • Teacher knowledge with respect to “pupil learning mathematics with understanding”. • Teacher selection and analysis of mathematics curriculum material. • Teacher confidence of amending and enriching material. • Teacher reflectivity with respect to the enactment of curriculum material. At a practical level results show that this project has helped teachers to spend time on developing their knowledge for/in teaching, by thinking about and analyzing curriculum material (some of it educative), developing the material further, and by “enacting” the material and reflecting on the processes. The project has succeeded in raising teachers’ awareness, and knowledge, of the educative nature of curriculum material, and what that may mean for their pedagogic practice. It is suggested that we need to help teachers learn from and work with all types of curricular materials – whether they are educative and well-designed, or otherwise – as they prepare for their teaching. This goes beyond “curriculum delivery”, and involves developing strategies to use the support offered by the school environment and uncovering “creative” ways to support their learning with the help of available “resources” and “tools”. Teachers benefit from opportunities to analyse, examine, enrich or amend new curriculum materials with their colleagues. This involves a process of “mutual transformation” – transforming the curriculum material, as well as potentially transforming the teachers’ notions of what can be done in the classroom, their pedagogic thinking. Adding to this, new resources, such as digital resources, the web of interaction becomes even more complex (Chapter 16). Further research is needed that takes us away from the dualistic thinking of “teachers and texts”, to more sophisticated processes and forms of analysis that include the working environment, the resource system, the activity format, and the curriculum script (Chapter 5). At the theoretical level it is evident that the process of interacting with “material” is complex, and it is often neither explicit nor public. There is evidence from this study that curriculum materials, more precisely a task analysis schedule, can act as catalyst for teacher learning. As the task analysis “tool” developed, it became a catalytic tool providing feedback which in turn helped teachers to develop deeper understandings. In the process it afforded feedback loops and changed its character, from “tool” as artefact to epistemic object at the interface between task design and enactment (see Fig. 7.1). Different forms of feedback emerged from the work with the “tool”, at four different levels. The results provide deeper insights into the processes of teacher learning with the help of analytic tools and the feedback these may afford.

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Task level (e.g. Mathematical tasks and their characteristics)

Process level (e.g. What do the tasks afford?)

Self monitoring level (e.g. confidence about working with tool and mathematical tasks)

139 Personal evaluation level (e.g. confidence to engage in further enquiries)

Provides feedback at four levels

Task analysis tool - a catalytic tool

As epistemic object at the interface between Design

Enactment

Fig. 7.1 Catalytic tool in relation to feedback levels, task design and enactment

Acknowledgements Particular thanks to the teachers featuring in the study; my colleague Dr Linda Haggarty who helped to develop some of the thinking behind the analysis schedule and to the NCETM who funded this project.

Appendix Task analysis schedule Text source: Content

Grade/year: Domain

Conn. through maths Processes

Proced. fluency

Number Algebra Geometry Measures Statistics and probability Within Across other subjects Representing Analysing – reasoning Analysing – procedural Interpreting Oral communication – implicit/none Oral communication – explicit 1 Step to be carried out 2 Steps to be carried out 3 Steps to be carried out

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Task analysis schedule Text source: Task type

Grade/year: Familiarity

Typically met in programme Some novel aspects Situation not met before

Context

Pure Artificial/contrived Authentic

Conceptual understanding

Implicit Explicit Subordinated

Cognitive demand

Knowledge (write, list, name) Comprehension (describe, summarise) Application (use, solve, apply) Analysis (compare/contrast, analyse) Synthesis (design, invent, develop) Evaluation (critique, justify)

Mathematical repres.

Analogy Pictorial (e.g. charts) Symbolic Numerical

Tools

Calculator Computer Geometric tools (compass, protractor)

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Chapter 8

Measuring Content Through Textbooks: The Cumulative Effect of Middle-School Tracking William H. Schmidt

8.1 Introduction Textbooks are ubiquitous in schooling worldwide. While other chapters in this part examine the interaction between textbooks and teachers, we focus on one particular inherent characteristic of textbooks, their potential role in providing opportunities for learning mathematics. How textbooks are designed provides a window into the nature of the mathematics that students are expected to learn. They characterise not only the content but also advocate what students are to be able to do with that content – what mathematical behaviours are to be encouraged. In this way they serve as a bridge between the teacher and the students, translating abstractions into reality. They mediate between instruction and the actual behaviours that the students undertake as a part of learning. As a result, such a characteristic of textbooks can constrain opportunity. Using textbook data from a U.S. nationally representative sample of students, we demonstrate a methodology that characterises textbooks related to the content itself but also to the nature of how it is presented especially with respect to the expected behaviours. We do this for different groups of students (those found in different tracks – courses of study) to illustrate how differences in textbooks and their use can constrain opportunity to learn (OTL). Other chapters in this book deal with the interplay between teacher and textbook. For example, Rezat (Chapter 12) shows the linkage between teacher’s and students’ usage of the textbook, while Remillard (Chapter 6) argues that teachers are themselves passive users of curriculum materials. These are studies at the microclassroom level. The emphasis in this chapter is to describe the cumulative effect of textbook usage at the macro-level, across grades, over a student’s high-school career.

W.H. Schmidt (B) Michigan State University, East Lansing, MI 48824, USA e-mail: [email protected]


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8.2 Background The mathematics achievement of U.S. middle- and high-school students is not considered strong by international standards (see, e.g., TIMSS, 2003, 2007; PISA, 2003, 2006) (http://timss.bc.edu/, http://www.pisa.oecd.org). This has prompted an emerging policy focus centred on two key aspects of the educational system – the curriculum and teacher quality. We focus on the curriculum, characterising mathematics opportunities as represented in textbook coverage across grades 7 through 12 and relating that to the common practice in the United States of tracking begun in the middle school. Studies have shown that curriculum is related to student achievement (see Floden, Porter, Schmidt, Freeman, & Schwille, 1981; Schmidt, 2003; Suter, 2000 for a review of this work). Some of this work has focused on the amount of mathematics covered (Schmidt, 1983, 2003; Stevenson, Schiller, & Schneider, 1994). Other studies have examined the role that a particular course such as algebra plays, not only in terms of what they know but also in terms of future career opportunities. Still other studies have focused on characterising the actual content students have studied and relating those specific opportunities to student achievement. In fact, this has been a traditional emphasis of international studies such as the Third International Mathematics and Science Study (TIMSS). The concept of OTL was designed to capture the type of topics studied and then to relate this to cross-national differences in achievement (McKnight, Crosswhite, Dossey, Kifer, Swafford, Travers, & Cooney, 1987). In Finland, Törnroos (2005) characterising OTL with textbooks reported a significant relationship between student achievement and textbook content coverage. Törnroos analysed Finish mathematics textbooks from grades 5, 6 and 7 and correlated the coverage with the seventh-grade student performance on the TIMSS 1999 test. He found that a strong positive relationship between student performance and the amount of cumulative coverage in the textbooks at the content topic level. In order words, the more the topics were covered in the textbooks, the better the students’ performance. The common element in both international and national studies is that the curriculum is a significant factor in explaining student achievement. The fact that these relationships have been established at the student, classroom and country level only strengthens the central role of this relationship to why schools matter (Schmidt, McKnight, Houang, Wang, Wiley, Cogan, & Wolfe, 2001). One of the factors related to what content students are exposed to in the United States is the practice of tracking. Although not typically practiced in other countries, at least among those studied in TIMSS, it is commonly practiced in the United States and begins in middle school. One estimate suggests that only about 25% of eighth-grade students attend schools that are not tracked (Cogan, Schmidt, & Wiley, 2001). We define tracking as the practice of having different students at the same grade take different courses that have different content. This is distinct from ability tracking where students in different classes (usually sorted by ability) cover the same

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topics but to different depths and for different amounts of time. Tracking results in different content exposures. Cogan et al. (2001) describes the number and nature of the different tracks typically found in the United States. Many times three to six different courses are offered in middle schools at eighth grade, most often including general mathematics, prealgebra and algebra. Each course presents a substantively different curriculum, and in turn affects students’ achievement differently. Prior studies have highlighted two important ways that a student’s eighth-grade course affects their subsequent mathematics achievement: positional advantages and differential achievement growth (Adelman, 1999; Atanda, 1999; Hoffer, 1992; McFarland, 2006; Schneider et al., 1997; Stevenson et al., 1994). Using data from the Longitudinal Study of U.S. Youth (LSAY), we developed a textbook-based methodology resulting in measures of the amount of demanding or complex mathematics content taken by a student and used them to obtain national estimates of what is typically taken by students in each of grades 7 through 12. These measures not only refer to the content itself, but also to the nature of what student behaviours are expected with respect to that content. These estimates can then be cumulated to reflect total exposure over middle and high schools to the more demanding aspects of mathematics (gauged by a combination of content difficulty and expected behaviours) given their starting point in seventh grade, that is the track into which they were placed in middle school. In that way we explore the cumulative content exposure for different tracks. In the analyses presented here the measures of curriculum are based on the textbooks used by each student in each of the mathematics courses taken.

8.3 Textbook-Based Estimates of Curriculum It is broadly accepted that textbooks are a good reflection of the implemented curriculum in most countries, and that textbooks are a particularly accurate reflection of the implemented curriculum in the United States (Fuson, Stigler, & Bartsch, 1988; Li, 2000; Mayer, Sims, & Tajika, 1995; Nicely, Fiber, & Bobango, 1986; Schmidt, McKnight, & Raizen, 1997a; Stigler, Fuson, Ham, & Kim, 1986). The growing emphasis on national standards and achievement testing are likely to increase teacher reliance on textbooks as the best available reflection of national standards and the intended curriculum (Crawford & Snider, 2000). Despite the centrality of the textbook to the implemented curriculum and classroom practices, there have been relatively few attempts to quantify the content coverage of textbooks (Porter, Floden, Freeman, Schmidt, & Schwille, 1986) and to measure the level of actual student exposure since few teachers cover the entire textbook during a school term or year. The TIMSS recognised the importance of obtaining reliable cross-national measures of the implemented curriculum and devoted substantial time and resources to the development of a content classification system for use with mathematics and science textbooks (Schmidt et al., 1997b; Valverde, Bianchi, Wolfe, Schmidt, &

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Houang, 2002). The application of this system has been described in numerous reports concerning the TIMSS results (Schmidt, McKnight, & Raizen, 1997a; Schmidt et al., 2001), but the initial translation of this classification system into quantitative measures has been limited to the work at the US TIMSS National Research Center at Michigan State University (Schmidt et al., 2001). Although the initial work by Schmidt and his colleagues was designed for international comparisons (Schmidt et al., 2001), the classification system has the potential to provide useful measures of the implemented curriculum at the classroom level and in the estimation of the influence of the implemented curriculum on student achievement. The idea of comparing the content of mathematics textbooks to the expectations and demands of mathematics problems is not new. Nicely (1985) and others have studied the content and form of problems in U.S. mathematics books. An extensive amount of comparative textbook analysis has been undertaken to understand differences in student performance in the United States and other countries (Fuson et al., 1988; Li, 2000; Mayer et al., 1995; Schmidt, McKnight, & Raizen, 1997a; Schmidt et al., 2001; Stigler et al., 1986). In virtually all of these studies, the content of small segments or specific problems has been analysed and classified, but comparisons between mathematics textbooks have not been made on the basis of the full book. Miller and Mercer (1997) argue that some students have difficulty because many mathematics topics are introduced too quickly by teachers who are trying to “get through the book”. No previous study has attempted to take into account the proportion of each mathematics book that is actually covered by various teachers, especially as an indicator of the scope of material actually covered. Porter (2002) has made a strong argument of the need to provide a metric or language to measure the content of the curriculum. Using a survey approach, Porter proposed a two-dimensional measurement technique that would take into account both the content of the implemented curriculum and the level of performance or understanding expected of the students in a given classroom. This general approach is similar to the textbook measurement technique described here.

8.4 TIMSS Mathematics Textbook Classification System The TIMSS textbook classification system is built on a set of content categories developed by groups of mathematicians and mathematics teachers representing numerous countries (Schmidt et al., 1997b). The classification system included 44 mathematics content topics that are commonly found in elementary and lower secondary schools among the participating countries. In TIMSS, mathematics textbooks that were used by the majority of the fourth grade and separately by the eighth grade students were included from each participating country for analysis. Each book was divided into units (often sections within chapters that corresponded to one to three days of instruction) and each unit was divided into blocks that are similar in content (usually a few paragraphs). The

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content of each block was coded using as many as eight content codes and as many as five performance expectation codes to characterise each block. The resulting data set provided thousands of block level codes for content. The challenge was to develop summary variables that translate the thousands of block codes into variables that provide useful information about the content of each mathematics text book. For purposes of cross-national comparison, Schmidt and his colleagues used a measure of the proportion of each book allocated to each of 44 mathematics content categories and compared this measure to the amount of time that each teacher reported that he or she devoted to the teaching of each of these topics and to student achievement outcomes (Schmidt et al., 2001). Another application of the TIMSS content codes is reflected in the International Grade Placement (IGP) index. The IGP is based on the curriculum data collected using the General Topic Trace Mappings (GTTM) from over 40 countries for the TIMSS (Schmidt et al., 1997b). It is a weighted average of the typical grade level at which countries first include a topic in their mathematics curriculum and the typical grade level at which countries focus instruction on that topic. For example, an IGP of 6.6 for the topic integers and their properties indicates that across TIMSS countries, the average of the grade level of typical introduction and the grade level of typical instructional focus is a little more than half way through grade six. We interpret this as indicating that, from an international perspective, this topic is typically covered in countries’ mathematics curriculum in grades six and/or seven. Similarly, an IGP of 9.0 for the topic patterns, relations and functions indicates that the average of countries’ typical introduction and typical focus is a little more than two grades later than the previously mentioned topic. This means that this topic is typically covered in countries’ mathematics curriculum around grade nine (Cogan et al., 2001; Schmidt, 2003). The IGP assumes that, given the hierarchical nature of mathematics, topics focused on in later grades are likely more complex or difficult, building on the topics covered in earlier grades. It is important to recognise that the TIMSS is a set of three cross-sectional studies designed to provide cross-national comparisons. These international comparisons are important and provide useful insights into the commonalities and differences in mathematics instruction and learning throughout the world. Nonetheless, they cannot be used to provide cumulative measurements on the same students.

8.5 Longitudinal Study of U.S. Youth An important application of TIMSS classification and coding approach would be to a longitudinal data set with curriculum measurements at each grade level. The LSAY1 provides a national longitudinal data set with extensive student,

1 This work was supported by NSF grant RED-9909569. All conclusions and findings reflect the views of the principal investigator and co-investigators and do not necessarily reflect the views of the National Science Foundation or its staff.

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parent, teacher and classroom variables and mathematics and science textbook information, including the percentage of the book that the teacher plans to cover during the semester or school year. Initiated in the Fall of 1987, the LSAY selected and followed two cohorts of public school students from 50 public high schools. Cohort Two was a group of 3,116 seventh grade students and served as the basis for the analyses reported here. The group was from 50 public middle schools of the feeder schools to the sampled high schools and was followed for 7 years (see Table 8.1). During each of the six school years, each student was asked to complete a mathematics achievement test and a science achievement test (usually in October), and two extensive attitudinal and activity questionnaires (in October and April). For every mathematics and science course that included one or more LSAY students, the teacher was asked to complete a course questionnaire that collected – among other variables – the name, publisher and year of the textbook used in the course and the percentage of the textbook that the teacher expected to cover during that course. Because each student attitudinal and activity questionnaire requested a full course schedule, including the name of the teacher and the class period in which the course occurred, it was possible to match each student to specific teachers and courses, allowing the linking of teacher-reported course variables to each student’s record. We focus on the coding and classification of mathematics text books here. One of the advantages of using the LSAY data set for this purpose is that it allows the measurement of the cumulative level of textbook (and presumably, curriculum) exposure to the full range of mathematics topics for students in different tracks throughout their middle- and high-school years. To the extent that textbooks can be coded to reflect the implemented curriculum, it will be possible to map the cumulative impact of tracking over a period of years and examine the influence of differential curricular exposure on individual student achievement.

Table 8.1 Longitudinal Study of U.S. Youth (LSAY) cohort two participation rates Grade

Same school

New school

Early graduate

Dropout

Lost

Quit

N

7 8 9 10 11 12

3,116 2,718 2,267 2,038 1,907 1,743 Percent 100 87 73 65 61 56

0 270 649 736 724 672

0 0 0 0 2 27

0 9 49 134 216 334

0 89 48 53 76 110

0 30 104 155 190 230

3,116 3,116 3,116 3,116 3,116 3,116

0 9 21 24 23 22

0 0 0 0

From Text to 'Lived' Resources: Mathematics Curriculum Materials and Teacher Development (Mathematics Teacher Education)

Becoming a Mathematics Teacher: Identity and Identifications (Mathematics Education Library)

Calculus: Readings from the Mathematics teacher

Curriculum Development in Mathematics

Tasks in Primary Mathematics Teacher Education: Purpose, Use and Exemplars

Teacher, Teacher

Becoming a Mathematics Teacher: Identity and Identifications

Embracing mathematics: on becoming a teacher and changing with mathematics

Developing Inclusive teacher Education

Transforming Teacher Education: Lessons in Professional Development

Teacher Pay and Teacher Quality

Mathematics Teacher Education: Critical International Perspectives (Studies in Mathematics Education Series)

Teacher-Centered Professional Development

Teacher-Centered Professional Development

Sound Foundations Teacher Development

Becoming a Successful Teacher of Mathematics

Teacher

Learning Teaching from Teachers (Developing Teacher Education)

Learning Teaching from Teachers (Developing Teacher Education)

Learning from Teacher Research

Mathematics & Mathematics Education

Rethinking the Mathematics Curriculum (Studies in Mathematics Education Series, 10)

Elephant Odyssey Teacher Resources and Activities

Curriculum Development in Education

Teacher Learning and Development: The Mirror Maze (Self Study of Teaching and Teacher Education Practices)

Cambridge Guide to Second Language Teacher Education

Cambridge Guide to Second Language Teacher Education

Bridging Cultures: Teacher Education Module

Rethinking Teacher Education: Collaborative Responses to Uncertainty

Conceptualising Reflection In Teacher Development

Using Assessment To Reshape Mathematics Teaching: A Casebook for Teachers and Teacher Educators, Curriculum and Staff Development Specialists

From Text to 'Lived' Resources: Mathematics Curriculum Materials and Teacher Development (Mathematics Teacher Education)

Becoming a Mathematics Teacher: Identity and Identifications (Mathematics Education Library)

Calculus: Readings from the Mathematics teacher

Curriculum Development in Mathematics

Tasks in Primary Mathematics Teacher Education: Purpose, Use and Exemplars

Teacher, Teacher

Becoming a Mathematics Teacher: Identity and Identifications

Embracing mathematics: on becoming a teacher and changing with mathematics

Developing Inclusive teacher Education

Transforming Teacher Education: Lessons in Professional Development

Teacher Pay and Teacher Quality

Mathematics Teacher Education: Critical International Perspectives (Studies in Mathematics Education Series)

Teacher-Centered Professional Development

Teacher-Centered Professional Development

Sound Foundations Teacher Development

Becoming a Successful Teacher of Mathematics

Teacher

Learning Teaching from Teachers (Developing Teacher Education)

Learning Teaching from Teachers (Developing Teacher Education)

Learning from Teacher Research

Mathematics & Mathematics Education

Rethinking the Mathematics Curriculum (Studies in Mathematics Education Series, 10)

Elephant Odyssey Teacher Resources and Activities

Curriculum Development in Education

Teacher Learning and Development: The Mirror Maze (Self Study of Teaching and Teacher Education Practices)

Cambridge Guide to Second Language Teacher Education

Cambridge Guide to Second Language Teacher Education

Bridging Cultures: Teacher Education Module

Rethinking Teacher Education: Collaborative Responses to Uncertainty

Conceptualising Reflection In Teacher Development

Using Assessment To Reshape Mathematics Teaching: A Casebook for Teachers and Teacher Educators, Curriculum and Staff Development Specialists

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