Mathematics Education and Subjectivity: Cultures and Cultural Renewal

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Mathematics Education and Subjectivity

Mathematics Education Library VOLUME 51 Managing Editor A.J. Bishop, Monash University, Melbourne, Australia

Editorial Board M.G. Bartolini Bussi, Modena, Italy J.P. Becker, Illinois, U.S.A. M. Borba, Rio Claro, Brazil B. Kaur, Singapore C. Keitel, Berlin, Germany G. Leder, Melbourne, Australia F. Leung, Hong Kong, China K. Ruthven, Cambridge, United Kingdom A. Sfard, Haifa, Israel Y. Shimizu, Tsukuba, Japan O. Skovsmose, Aalborg, Denmark

For further volumes: http://www.springer.com/series/6276

Tony Brown

Mathematics Education and Subjectivity Cultures and Cultural Renewal

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Prof. Tony Brown Manchester Metropolitan University Education and Social Research Institute Didsbury Manchester, M20 2RR, UK [email protected]

Chapter Four results from work with Una Hanley. ISBN 978-94-007-1738-1 e-ISBN 978-94-007-1739-8 DOI 10.1007/978-94-007-1739-8 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011931871 © Springer Science+Business Media B.V. 2011 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)

Acknowledgements

I would firstly like to thank the people who have generously contributed to this book. The book commences with some joint writing with Krista Bradford resulting from a project in Africa, where I worked with British volunteers in schools and teacher colleges. Some material is used by kind permission of flm publishing. Chapter 2 draws on a paper published in Educational Studies in Mathematics. It also includes drawings by Sally Jackson and some writing by James Clegg, along with some very stimulating contributions from their colleagues in the class of 08/09. The work on planets was shared with Dave Heywood and Yvette Solomon. Thanks particularly to our students: Clare Crookall, Claire Shakeshaft and Jenny Taylor. Chapter 3 draws on an earlier paper in Educational Philosophy and Theory. It further includes an image inspired by the work of the mathematician Caroline Series, from the site maintained by Peter Liepa (brainjam.ca/fractals.html). Meanwhile, Chapter 4 results from work with Una Hanley. Chapters 5, 6 and 7 draw on material originally published in Educational Studies in Mathematics. This work provoked an exchange with Luis Radford and Norma Presmeg following my response to the special issue of that journal to which they contributed. Their response also appeared in the same journal. Subsequent conversations have resulted in major shifts in my thinking that I have sought to represent in this book. Bill Kirkby provided the work on Doubling Modulo. In the background, however, there were a number of people who have consistently offered intellectual support over many years. For this book most notably Rob Lapsley helped me to engage with Badiou, having already provided a substantial part of my basic Lacanian education. Dennis Atkinson, Julian Williams and Yvette Solomon have always provided excellent conversations, as does the Woodstock group. And although I have only spent 1 day in conversation with him, Slavoj Žižek is the person from whom I have learnt most. Manchester Metropolitan University continues to be a good supportive base after some 20 years. Thanks, particularly, to Jean Davidson for dealing with very regular backup requests. I am especially indebted to Andonis Zagorianakos who conducted a very close examination of the text and offered numerous insightful suggestions. As always, love to Alison, Elliot and Imogen.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Book Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I

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Some Cultural Contexts of Mathematical Learning

2 The Regulation of Spatial Perception . . . . . . . . . 1 Encircling Subjectivity . . . . . . . . . . . . . . . . 1.1 Extracts from Krista’s Reflective Journal . . . 1.2 Circular Arguments . . . . . . . . . . . . . . 1.3 Rounding Up Reflexivity – Cyclical Research 2 Experiencing Shape and Space . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . 2.2 Shaping Up . . . . . . . . . . . . . . . . . . . 2.3 Calculus . . . . . . . . . . . . . . . . . . . . 2.4 Mancunian “String Theory” . . . . . . . . . . 2.5 Planetary Movements . . . . . . . . . . . . . 3 The Seduction of Language . . . . . . . . . . . . . 3.1 Conceptualising the Past . . . . . . . . . . . . 3.2 Time and the Circle . . . . . . . . . . . . . . 3.3 Conceptualising the Future . . . . . . . . . .

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3 Cultural Mediation of Mathematics . . . . . . . . . . . . . . . . . . 1 Art as Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Public Images of Mathematics . . . . . . . . . . . . . . . . . . . .

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4 Teachers’ Conceptions of Curriculum 1 Introduction . . . . . . . . . . . . . 2 Conceptualising Curriculum Reform 3 Researching Curriculum Change . . 3.1 Mathematics in Context . . . . 3.2 Methods . . . . . . . . . . . . 3.3 Discussion and Context . . . . 3.4 Analysis of Data . . . . . . . . 4 Problems with Consensus . . . . . .

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Contents

5 Subjectivity in Mathematics Education Research . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminary Remarks on Reading Text . . . . . . . . . . . 3 Conceptualising the Distribution of Psychology . . . . . . 4 Positioning of Author, Reader and Mathematics Education Research Community . . . . . . . . . . . . . . . . . . . . 5 “Teachers”, “Students” and “Mathematics” . . . . . . . . . Part II

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Cultural Renewal in Mathematical Learning

6 The Lacanian Subject of Mathematical Learning . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 Cognitive Psychology Versus Psychoanalysis . . . . . 3 Mathematics Education, Psychology and Semiotics . 4 Lacan Piaget Vygotsky . . . . . . . . . . . . . . . . 4.1 The Context of Lacan’s Work . . . . . . . . . . 4.2 Piaget . . . . . . . . . . . . . . . . . . . . . . . 4.3 Vygotsky . . . . . . . . . . . . . . . . . . . . . 5 The Lacanian Subject . . . . . . . . . . . . . . . . . 5.1 The Mirror Phase . . . . . . . . . . . . . . . . 5.2 The Imaginary, the Symbolic and the Real . . . 5.3 The Subject of Mathematics Education Research 6 Restrictive Registers . . . . . . . . . . . . . . . . . . 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . .

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7 The Cultural Renewal of Mathematical Learning . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mathematics and Badiou’s Philosophical Terrain . . . . . . 3 Badiou: Truth and Knowledge . . . . . . . . . . . . . . . . 3.1 Truth, Knowledge and Mathematical Activity . . . . . 3.2 A Radical Education Project . . . . . . . . . . . . . . 3.3 The Four Domains of Truth and Meaning . . . . . . . 4 The Lacanian Premises to the Work of Badiou . . . . . . . 5 Understanding Mathematical Learning . . . . . . . . . . . 5.1 Being and Multiplicity . . . . . . . . . . . . . . . . . 5.2 Counting as One . . . . . . . . . . . . . . . . . . . . 5.3 “That’s It” Encounters “That’s Me” . . . . . . . . . . 6 Examples in Mathematics Education . . . . . . . . . . . . 6.1 Example One: Mathematical Investigations . . . . . . 6.2 Example Two: Gattegno’s Science of Education . . . 6.3 Example Three: A Vygotskian Conception of the Objectification of a Graph . . . . . . . . . . . 6.4 Example Four: The Cultural Theory of Objectification 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

8 The Political Shaping of Mathematical Learning 1 Introduction . . . . . . . . . . . . . . . . . . . 2 The Regulation of School Mathematics . . . . . 2.1 Curriculum and Assessment . . . . . . . . 2.2 Mathematics as a Commodity . . . . . . . 3 Regulation Through Teacher Education . . . . . 4 Regulation Through School Policy . . . . . . .

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9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . .

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About the Author

Originally from London, Tony Brown was educated in Canterbury and Exeter, before moving to Holland Park School in central London where he taught mathematics for 3 years. This was followed by 3 years as a mathematics teacher educator for Volunteer Services Overseas in Dominica in the Caribbean. In 1987 he completed his PhD at Southampton University. His doctoral research focused on language usage in mathematics classrooms, especially where the fluent use of English could not be assumed. After a spell as a mathematics coordinator in a middle school on the Isle of Wight, Tony moved to Manchester Metropolitan University in 1989. Tony has been the leader for doctoral studies in education as well as participating in a range of other courses and became Professor of Mathematics Education in 2000. Projects have included: Economic and Social Research Council funded studies examining teacher education; piloting a distance-teaching programme enabling British volunteers based in Africa to research their own teaching practice within a programme of professional development; and, a General Medical Council funded project on how senior doctors learn. Tony also spent 2 years on leave from Manchester at the University of Waikato where he became the first Professor of Mathematics Education in New Zealand. There he led a project funded by the New Zealand Council for Educational Research on the experiences of Pasifika teachers working in New Zealand schools. Tony has published two other books in the Mathematics Education Library series. Mathematics Education and Language outlines his interest in mathematics in schools. Becoming a Mathematics Teacher (with Olwen McNamara) considers the formation of practices of new primary school teachers. Meanwhile, Action Research and Postmodernism (with Liz Jones) provides a theoretical perspective on teachers carrying out practitioner research within higher degrees. Regulative Discourses in Education (with Dennis Atkinson and Janice England) considers teacher practices through psychoanalytic theory. The Psychology of Mathematics Education, edited by Tony, introduces psychoanalytic theory as an alternative to more cognitive understandings of psychology. Tony has also written extensively in journals such as Educational Studies in Mathematics, for the learning of mathematics and the British Educational Research Journal.

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Chapter 1

Introduction

This book is centrally concerned with how we represent mathematical teaching and learning with a view to changing them to suit new circumstances. It considers teachers, students and researchers. It explores their mathematical thinking and the concepts that this thought produces, concepts that shape subsequent thought. The book examines some of the linguistic and cultural filters that influence mathematical understanding in schools. But above all it is concerned with how we understand ourselves in relation to the school-learning contexts that produce mathematics. Yet representation can never be neutral. “Teachers”, “students”, “researchers” and “mathematics” can be read through a multitude of filters according to our particular interests at any given time. What are we trying to represent with each of these terms? In which form do we represent them? To whom are we trying to represent them? More generally, successive attempts to represent the world and its features have produced markers, symbols, paintings, artistic traditions, books, theories and entire languages that have come to define our relations to the world, a world to which these creations had sometimes sought to alert us. What we experience as reality is not the “thing itself”. Reality is always already symbolised and structured through the filter of these creations. And these creations have meanwhile become part of the world, shaping its everyday business and influencing new creations. Indeed we cannot remember the times when this interference from our own creations did not affect our sense of the world. And through having introduced such features, perspectives and points of access, we cannot help but realise that we ourselves, those we address, and the representational apparatus we use, are also functions of those creative enterprises resulting from past decisions to organise and account for the world in particular ways. We have crafted ourselves, and our relations to the world, using our own apparatus, and have no possible way of understanding ourselves, or of representing ourselves, without that self-referential apparatus. Yet, fully implicated as we are in these accounts of who we are, they always leave something out. The self, seen as a body with thoughts and emotions, is not the only centre for any individual person; it is not the only representational mode. The book introduces some contemporary theories of human subjectivity from the point of view of school mathematical learning. In these theories, subjectivity relates to the ways in which we ourselves make sense of who we are, and how we represent who we are, but T. Brown, Mathematics Education and Subjectivity, Mathematics Education Library 51, DOI 10.1007/978-94-007-1739-8_1, C Springer Science+Business Media B.V. 2011

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Introduction

also how other people understand us according to a variety of alternative frames. In some of these new theories, human subjects are not so much a function of their biological or cognitive selves. Rather, they are implicated in the stories told about them. Our sense of who we are results from how we see ourselves in relation to the various accounts of the world that we encounter. The new theories can trouble more familiar approaches in mathematics education research where standalone “humans” apprehend distinct mathematical “concepts”. The theories can disturb conventional understandings of what mathematics is and how it exists in an “objective” sense. In the proposed orientation, mathematical objects relate to the world through our collective perceptions of them as we go about our lives. The objects can be derived from the social filters that have been applied to the world. And also, they can serve as filters on the world, capable of producing new social entities. The attitude of the proposed reconsideration is to think of the purpose of school mathematics as being to provide filters on life, a grammar through which we tell stories of life. That is, mathematics provides ways of making sense of life, or of modelling life. It also presents frameworks, or analytical apparatus, against which life is constructed. But, in particular, mathematics gives us ways of seeing how we are all a part of that life, with the capacity to change that life, and to be changed by it. The book proposes that mathematics can provoke us to think differently about our environments, whether they are spatial, social, cultural, educational, philosophical or political, as successive chapters examine. This reorientation might enable us to build our world differently. Part I of this book is a portrayal of how learners and teachers in different cultural settings construct mathematical knowledge according to their particular social needs. This raises the question of the extent to which mathematical concepts can be consistent across cultures. Examples of cultures to be touched on in this book include, school mathematical learning in Uganda, mathematics lessons for trainee teachers and teacher development initiatives in England, reform mathematics in the USA, the culture of international mathematics education research itself, the political environment of teacher education. In so doing the book proposes that we rethink culture in the context of mathematics education to embrace more contemporary conceptions of subjectivity. Here, rather than seeing mathematics as being consumed by so many individuals each with a brain, the analysis is shaped by the ways in which both individuals and mathematics itself result from immersion in particular cultures and the collectivised ways in which these cultures make sense of our world. Through such recognitions it may be possible to resist the one-size-fits-all preferences that fuel international comparisons of mathematical performance in schools, the policies that pursue them and the rather limited conceptions of mathematics that go with them. Part II focuses on renewal. There is a common propensity to see scientific activity in general, and mathematics in particular, as being concerned with eternal entities. Such entities, it is often supposed, anchor or limit the social relations that can take place around them. Many practices of mathematics in educational contexts are set in familiar habits that restrict our perceptions of the objects in question. This supposed stability can sometimes serve reactionary agenda. For example, the supposition that

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Book Outline

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there is permanence in mathematical forms can lead to teaching practices in some quarters aiming to fix the way that we see the forms through preferred methods or “standards” or testing mechanisms. In some countries this results in teachers of mathematics being tightly constrained within the formal education system, working to curriculums where mathematics is firmed up as a set of concepts, procedures and preferred modes of apprehension. Such arrangements paint the teacher as a postal worker delivering items in standard brown paper parcels. Here both the pedagogical strategies of teacher and demonstrable skills of the learner are respondents to externally imposed demands. Yet, this book argues that such demands, in their very assertion, can fix mathematical knowledge in defiance of the life that lives around this knowledge. The pedagogical strategies and the demonstrable skills become the indicators of whether mathematics is understood correctly. And in standing in for mathematics, these ultimately sterile indicators replace mathematics. If we are to break out of this reductive account, we cannot see mathematics research as being in the business of finding better methods or indicators to replace those that we have. The book argues that mathematics needs to be centred in an attitude of experimentation and critique, to support ever-fresh approaches to the new challenges that we will surely face. Mathematics itself can be responsive to life and not just serve as a stable point of reference. Yet it may be all too easy to suppose that teachers will champion approaches that lift their students from the quagmire of centralised regulation. But the actuality of teacher supply in many countries can temper some of our more optimistic aspirations. For example, non-specialists who have themselves struggled with mathematics often teach the subject to children below the age of 11. Quite apart from governments trying to impose preferred models of practice to change educational outcomes, the difficulties of recruiting suitably qualified mathematics teachers in many locations can be the very cause of, or rationale for, more prescriptive centralised guidance, and the enforced administration of sober practices. To resist such tendencies, the book argues that mathematics education research must maintain a scope that goes beyond merely supplying technical solutions to current models of practice.

1 Book Outline Part I of the book commences with Chapter 2. The chapter focuses on mathematical learning. It introduces some instances of geometry teaching to explore aspects of how humans connect with mathematical phenomena. The chapter considers how the learning depicted relates to specific culturally and historically bound circumstances. It offers brief examples: of a child being directed to move around precisely in a seemingly haphazard banana plantation in Uganda, an equatorial country where consistent patterns of daylight-assisted orientation; and of teenage students in school in the same country trying to apprehend geometric configurations

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Introduction

derived from Western culture. It then provides some extensive discussion of British trainee teachers engaged in a range of geometric or spatial constructions centred in bodily movement. These learning experiences, which relate to more or less familiar geometrical ideas, are considered against a broader understanding of human spatial awareness. The chapter proposes that such awareness can be understood as being built through on-going immersion and growing familiarity with movement in a spatial environment conceptualised according to geometrically defined facets. It is argued that this subjective awareness is held in place by specific descriptive practices and physical environments constructed in these terms, but motivated by possibilities beyond our immediate grasp. Yet our very selves have been created according to a physical organisation and an analytical heritage consequential to a long history of spatial awareness. We sit on chairs, climb stairs, wash round dishes, swim with fishes, ride on Ferris wheels, remove orange peels, travel on trains and fly in planes. Our bodies have learnt to function and know themselves in physical environments that result from culturally embedded conceptions of space. We fit in to the social/physical world through participation in shared ways of organising, apprehending and constructing. Our perceptions of the physical environment are inevitably processed through aspects of this symbolic heritage. But so too are our perceptions of ourselves. We make sense of who we are through using the same vocabulary. The ensuing discussion considers the broader conceptual challenges of seeing learning in terms of participation in processes of historical change with respect to how humans and how spatial phenomena are conceptualised. Chapter 3 contemplates how mathematics provides analytical apparatus through which we can make sense of the wider social environment and our subjective immersion in it. It re-thinks mathematical objects as potential filters of this world by drawing analogies with artistic objects, towards examining how we as humans relate to such objects. The chapter considers some instances of contemporary art and draws parallels with some examples of mathematics in the public domain. The mathematical examples include a public lecture on advanced mathematics, the publication of a popular book discussing mathematical physics, a public mathematics examination for aspiring primary teachers, a controversy surrounding the publication of some school examination statistics, and the mathematical underpinning of the on-going world financial crisis. The chapter argues that the social and linguistic conditioning of mathematics within such embodiments is a crucial aspect of the discipline being addressed in school. In particular, being proficient with embodiments of mathematics and the social dynamics that attend these is integral to the broader ability of moving between concrete and abstract mathematics. Chapter 4 examines teacher responses to curriculum reform within continuing professional development processes. It considers how wider educational circumstances shape the mathematics that is encountered in schools. Here mathematical thinking is seen as a function of curriculum structures. Teachers are depicted as being subject to particular models of teaching. Yet the chapter tempers any supposition that curriculum changes can easily change practice. It questions a common assumption that curriculum reform can readily produce visible and measurable

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Book Outline

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effects in teaching approaches and pupil progress. A curriculum development initiative and some associated research are described to show how some teachers understand their teaching of mathematics against alternative conceptions of practice. Interview data document four teachers attempting to shift from one teaching paradigm to another. The chapter discusses the ways in which the teachers interpret what the new scheme requires of them and how in turn, engaging with the scheme causes them to reconceptualise their classroom practices. Certain key terms referenced to earlier approaches are shown to variously anchor teacher conceptions during this transition. Here the individual’s teaching practice is depicted as developing over their career as a result of it being understood and enacted through a succession of curriculum rationales, each modifying the cumulative experience of the teacher. Chapter 5 takes the example of mathematics education research itself as a cultural domain where meanings are produced. It examines how certain assumptions are sometimes built into the fabric of research, which set the categories of subsequent analysis. The chapter inspects a special issue of the journal Educational Studies in Mathematics on semiotics in mathematics education, comprising reports written by major authors; the reports variously considered how mathematical objects are produced as signs to convey ideas. By examining how the authors of those reports reflexively situate themselves, the chapter is targeted at deducing how the authors in those reports variously understand the notion of subjectivity. Specifically, it asks: how are concepts of “mathematics”, “students” and “teachers” understood and contextualised within the report narratives? The chapter includes a citation analysis to consider how the authors conceptualise their research domain, and their implied positioning in relation to teachers, teacher educators, researchers and other potential readers. Part II commences with Chapter 6. The chapter explicitly argues that theories of subjectivity and psychoanalysis provide a powerful alternative to psychology in the context of mathematics education research. By engaging with social relations in preference to individualistic learning engagements, alternative filters through which to conceptualise evolving practice become available. This argument is centred in a presentation of some important themes from the work of the psychoanalyst and theorist Jacques Lacan. His work on subjectivity combined Freud’s psychoanalytic theory with processes of signification as developed in the work of Saussure and Peirce. The chapter positions Lacan’s subjectivity initially in relation to the work of Piaget and Vygotsky who have been widely cited within mathematics education research. But more extensively it is shown how Lacan’s conception of subjectivity provides a development of Peircian semiotics as a model for understanding mathematical signification, or how mathematical objects might be recast as pedagogical objects, or more broadly as analytical strategies. Through this route Lacan’s work enables a conception of subjectivity that combines yet transcends Piaget’s psychology and Peirce’s semiotics and in so doing provides a bridge from mathematics education research to contemporary theories of subjectivity more prevalent in the cultural sciences. These theories release analytical opportunities to reshape our practice, introduced here and developed in subsequent chapters.

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Chapter 7 argues that mathematics itself changes as a result of human experience. This has important consequences for teaching and learning directed at seeing mathematics as productive and responsive analytical apparatus for everyday life. Locally, we can become stuck in particular pedagogical formulations that may not necessarily have adjusted as well as they might have done to new conditions. As a result mathematics may be lost in caricatures of it. For this reason it is not good enough to see the goal of education as being solely to bring students into existing practices. We want to cultivate their ability to discover new things, follow new paths, to be critical and able to reconstruct what is given as complete and ideal. More broadly, mathematics has a common image of presenting eternal truths. This anchors some conceptions of learning, such as Piagetian constructivism, to classical logic and analytic philosophy. Yet the chapter argues that mathematical universality is constantly on the march, redefined, as it constantly is, with respect to historical circumstances, social demand and academic advance, as humans approach it through successive encounters. And any subjective apprehension of a mathematical form comprises a symbolic layer that structures this apprehension with respect to contemporary parameters. Some examples are offered of how re-conceptualisations of the philosophical environment might enable us to trigger classroom mathematical activity towards more futuristic possibilities. Here learning is understood as participation in cultural renewal. Chapter 8 focuses on how political structures shape teachers, students and mathematics. The chapter depicts mathematics as a function of central regulation, marking out the mathematical terrain according to a one-size-fits-all attempt to meet the demands of international comparative tests. The chapter argues that mathematical content has been recast as so many commodities that mask the full productive capabilities of mathematics as an analytic tool. The chapter revisits some research on pre-service teacher education in which trainee and new teachers conceptualise their emergent professionalism through the filter of their implementing a governmentdefined mathematics curriculum. Here trainees find themselves immersed in socially acceptable ways of describing their own practice. Teacher identity is predicated on an impossible reconciliation of personal aspirations with a multitude of external demands. The chapter concludes by depicting a head teacher locked into a regulative policy framework that restricts her school’s functioning. An attempt at sedition is rewarded by seduction into the pleasures of following rules. The book concludes in Chapter 9. It gathers some remarks on how we might understand the interface of mathematics with humans. Subjectivity is depicted as embracing both our physical experience of the world and our capacity to symbolise this experience. Ultimately teachers, students and researchers need to be attentive to how they are subject to restrictive encounters with mathematics, and with each other.

Part I


Mathematics is a function of the community that embraces it, and it evolves in relation to the needs expressed and tasks performed. There are grounds to resist moves in which mathematical achievement in schools is read against a register of commodified procedures, in a “one-size-fits-all” model, spanning diverse nations and communities. Teachers, students and mathematics fit, or are required to fit, particular images. Such moves seem symptomatic of the twentieth century that has left a legacy of techno-scientific control governed by the ideology of “real” social forces (Lather, 2003). The field of mathematics education spans science and social science and there is much contestation about the boundaries of each of those domains. Consensus is neither possible nor desirable. We do not want to pin down teachers, students or mathematics in to well-defined entities. Cultural renewal relies on our certainties being troubled and disrupted to allow new ways of thinking and being. Whereas “mathematics” often continues to be conceptualised as a discipline resistant to social discourses, “education” resists conceptual immersion in the broader social sciences. These conceptions can open the door to assertive governance by those ill equipped to evaluate mathematical or educational priorities. We begin to experience a new century in which such rationalistic aspirations have been rerouted in so many areas of social theory. Mathematics education research needs to move away from earlier instrumentalist tendencies concerned with understanding and “improving” mathematical performance against social registers that have not been problematised. Mathematics and people grow together and influence each other in this growth, every day. In this first part of the book I outline four very different locations for mathematics, which each process conceptions of mathematics according to the forms and life that govern each of them. But it is life that is at stake in each of these locations. In Chapter 2, by nailing the continuities of geometrical form in to the discrete categories of algebra, something may be gained in terms of control, but something may be lost in experiencing the shapes. Chapter 3 emphasises that we are not seeking to produce or encounter objects in which meaning is fixed. The meaning resides in the life around. And in the forms that seek to pin that life down. Chapter 4 depicts

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how administrations petrify mathematics in to curriculums that become the reality of mathematics for teachers and children. Finally, Chapter 5 argues that researchers themselves can produce reductive accounts of teachers, students and of mathematics. These accounts can restrict analytical opportunities by missing locations where we might intervene.

Chapter 2

The Regulation of Spatial Perception

1 Encircling Subjectivity (with Krista Bradford) In thinking mathematically, how does mathematics itself become something different? And how do we ourselves change? To unsettle the ground a little for later chapters, I commence with an exploration of these issues through considering extracts from a personal reflective diary work by a former student, Krista Bradford. The diary was created as a part of a research masters’ degree programme. The enquiry attended to the cross-cultural perception of mathematical concepts during some classroom research. Formerly a primary teacher in the United Kingdom, Krista was teaching mathematics to teenagers for Voluntary Services Overseas in Uganda. The research was carried out within a practitioner enquiry frame as part of a distance education course that I had initiated within a charity-funded project. As Krista had not worked on mathematics at this level since the end of her own schooling, she experienced a steep learning curve. This curve was made steeper as she became more aware of how mathematics was constructed in Ugandan schools. Its derivation from Western curricula compounded difficulties for the students she was teaching. Krista’s raised awareness of the cultural issues also brought into question her own agency within this development context. As a white person from the west she faced the challenge of mediating the externally defined demands of the western inspired curriculum and the more immediate educational needs of her students. Krista shifted in her conception of how mathematics could be understood according to the ways in which it was progressively described. The shift was multi-layered. First, she found her focus in teaching needed to change from improving her own explanations to striving to understand how her students constructed their own mathematical knowledge. As Krista’s understanding of the students’ needs evolved, so she shifted and adapted to teaching in a different way. Finally, there was a shift in Krista’s understanding of how words or symbols stand for – and sometimes disguise – mathematical concepts when seen from the perspectives of different cultures.

T. Brown, Mathematics Education and Subjectivity, Mathematics Education Library 51, DOI 10.1007/978-94-007-1739-8_2, C Springer Science+Business Media B.V. 2011

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2 The Regulation of Spatial Perception

What follows is some of Krista’s writing produced as she worked with her students making sense of shape and achieving a broader spatial awareness. The original journal extracts are accompanied by some discussion by Krista of the issues they raise, which can be seen to concern the linguistic filters and socio-cultural factors that influence our understanding of mathematical concepts.

1.1 Extracts from Krista’s Reflective Journal The Runyankore language allows for very accurate description of precise localities. For example, a child can be sent into a large banana plantation with instructions on which bunch to collect. The trees are not in rows or in any “organised” arrangement and the plantations can be vast. However, the instructions as to the location are exact and the child is able to follow them precisely. The language used is descriptive, involving the slope of the ground, the angle and the way the trees are “facing”, a description of a particular group of trees and in relation to the sun. The translation into English can hardly be done and causes quite a lot of frustration to the person asked to try. An equivalent description in an English culture would almost certainly be based around approximate distances using standard units or counting rows with the trees being planned in rows in order to overcome the problem of locating a particular spot. Being an equatorial country, the sun consistently rises and sets at the same time and in exactly the same place each day. This stable consistency in position (and appearance) has allowed for the development of sun-based location and its associated language in a much more profound way than has been possible in a European language. Perhaps this would have made the development of alternative systems unnecessary. The words for East and West are derived from the rising and the setting of the sun. There are no specific words for North and South. They are not necessary. They can be described in relation to the sunrise and the sunset. Runyankore does not include words for any geometrical shapes. This indicates that the Banyankore have either not needed these words, not used the shapes or have not felt the need to explain and classify them as such. There are no words for triangle, rectangle or even for square. There is a word, oriziga, meaning circular or curved but it does not refer to a circle specifically (as I shall illustrate shortly). Once again the language has developed descriptively rather than finding ways to categorise as the English language favours. Where the English language can use the word sphere, a description involving the word for curved and comparison to similarly shaped objects will be needed in Runyankore. K: So if I give you an orange, a football, this small ball, and this stone, what shapes are they? How would you describe the shape in vernacular? Him: They are all circular and shaped like the small ball. K: But in English you would say sphere? Him: Sphere, yes. But that is a muzunga (white) word.

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Encircling Subjectivity

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The absence of these words surely makes a profound statement about the direction of the development of the indigenous ethno-mathematics. At present the western curriculum is being taught in schools using the English language in order to improve the children’s English but also to overcome the problems of awkward translations. However, the concepts of simple regular shapes such as squares and triangles are so basic to the imposed curriculum that they are included in the syllabus for nursery and primary. These children have little or no English so the words sikwera (square) and turyango (triangle) have been incorporated into the language. But the concept is a taught one and it is an unfamiliar one. These basic shapes that surround a western child in their everyday life are only theoretical, abstract and purely academic shapes on the chalkboard to the Banyankore children. There is no allowance for this in the school syllabus. These children progress on to Pythagoras’ Theory just as quickly. In order to promote as much discussion as I could, I chose to start the lesson with an open question. I reworded the question many times in order to choose one that was obviously as open and as un-oppressive as I could. Using a (handmade) chart with four diagrams on a sphere, a spiral, an oval and a regular polygon with 16 sides, the question that I gave was Why are these not circles? At first many students were not involved so I quickly told them that they should discuss for a few moments in small groups and then they could put their ideas out to the class. The discussions went well and there was lot of describing and fairly argumentative discussion. During the second and third lessons I was much less intrusive in the class discussion and allowed the students to debate across the class more. I was surprised at the level of discussion that the question had caused. I had assumed that we would be describing the properties of the sphere in comparison to a circle but I realised that the students were not very confident with the words sphere or oval and even less so with spiral. A majority of students were arguing that the sphere was a circle and it was when they were convincing the non-believers that I came in and steered the discussion by explaining 2D and 3D. The confusion may have been because my representing of 3D on a 2D learning aid was confusing this issue. The oval was more straightforward and seemed to be a language issue only as did the spiral. I became aware of my use of the words circular and circle and tried to explain the difference. But I am surprised at the confusion over the use of the word circle. When I taught this lesson my focus was on student interaction, the purpose of which was twofold. At that time I was measuring the success of my teaching by the quality of the interaction but also I was anxious to promote as much discussion as possible in order to find out as much as I could about the students’ thinking. I hoped that by doing this I could become “closer” to them and teach more effectively. This second extract is a recording of a session using the same activity 6 months later. However, by this time I am more aware that the possible confusions may be attributed to the different cultural background and uses of language and classification systems.

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I presented a small group of students with a diagram of a circle, a sphere, a spiral and an oval. After a few moments, I asked them what they could see and if they knew what the shapes are. St1: They are circles! St2: Well this one is more of a circle. This is a pure circle (Pointing to the circle). K: What about these other ones then? Are they the same? St2: This one is bent and too pointed. (Referring to the oval) Pause K: mmmm? (Prompt to continue) St3: This one is more of a coil. (Referring to the spiral) K: Why is it not a circle? St3: Because it is not closed and it is not as perfect as this one (the circle). It is continuing. St1: But it is a circle sort of. St2: It is a coil. Pause K: What about this one? St4: This is a circle (General agreement over the sphere), but why have you shaded it? I decide that it is a fault on my part to try and represent a three-dimensional sphere as a diagram and use a nearby globe as the sphere. K: How is this (the globe) different from this (the circle)? St2: This one (the globe) has this line the equator around it. (I am really regretting not bringing a plain ball!) K: How is the shape of this (globe) or say the volleyball or the football, different from this circle? Or are they the same? St3: They are both similar. The students then begin to argue in vernacular and from the gestures they seem to be discussing the three dimensions of the globe. When I ask them for a conclusion in English they have decided that they are both circles. So I explain about the three dimensions and that the globe is a circle from all sides, which they are fully aware of and have been discussing. I give them the word sphere, which they all eagerly write down. One student now points out that he has heard the word in Geography when learning about hemispheres and they now see the connection. A connection that is obviously emphasised and perhaps limited to my use of the globe as a learning aid. The above dialogue shows that although there is no discrepancy in describing the differences and the similarities of the various circular shapes, I think there is a mismatch in the use of the word circle between myself and my students. There is

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also a mismatch in the language structure for categorising these shapes. The English word circle is being used in place of the Runyankore word oriziga. But I think that a more accurate translation for oriziga must be the English circular. The students then are correct when they say that “they are all circles” because some students mean that they are all circular. In the vernacular then all these circular shapes are separated by a description incorporating the word for circular. For example the oval is usually described as meaning “circular like a stone” and the spiral is engata meaning circular like a basket. As mentioned before the sphere is circular like a ball. In a western culture specific words have evolved in order to denote exactly each category. Each category has precise requirements. Much of the academic mathematics is based around the requirements of each category and the consequential properties of the individual shapes. Mathematical values and attitudes in a western society emphasise this need to categorise according to precise properties and generally to leave nothing uncategorised of explained. These are characteristics of a western technological society and not of a practical one (Bishop, 1988; Kline, 1962). Thus, I feel that these students have not only to work in a second culture but also become aware of a different structure for categorising. If adjusting from a language with direct translations of all these words, I feel that the adjustment would be comparatively simple. I had originally assumed that the students would use statements such as “this one is a sphere because”. But if I am to align my teaching with the students thinking and to make the subject more accessible, then I should shift my own thinking and provide an activity that begins with discussing the differences and similarities rather than focusing on each category and its label.

1.2 Circular Arguments As these extracts show Krista’s teaching developed into an on-going displacement of the term “circle”. The teaching itself became an enquiry in to how “circle” was understood and shared from a range of perspectives. Something seen by Krista was “circular” but seen by her students was a “circle”. But one might ask what then is a real circle, or even when is a circle, and who decides? How might one best experience a circle? What form of authority would one invoke to adjudicate alternative claims? Or rather, how might this process of adjudication proceed? And what sorts of things would be offered in evidence? Who would be called as expert defence witnesses? What status would “circle” have at each stage of the proceedings? Clearly, this book is not going to sort out the ramifications of attempting to impose the most suitable linguistic frames upon mathematical conceptualisations. Could we ever finally decide how “circle” might be understood in a more “mathematical” way? Husserl’s essay The origin of geometry, which will be referred to throughout this book, sought to speculate on how mathematical ideas have come into being through historical processes:

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2 The Regulation of Spatial Perception It is easy to see that even in [ordinary] human life, and first of all in every individual life from childhood up to maturity, the originally intuitive life which creates its originally selfevident structures through activities on the basis of sense experience very quickly and in increasing measure falls victim to the seduction of language. Greater and greater segments of life lapse into a kind of talking and reading that is dominated purely by association; and often enough, in respect to the validities arrived at in this way, it is disappointed by subsequent experience (Husserl, 1936, p. 165, his emphasis).

The very idea of a clearly defined “signifier” like “circle” locating a precise “signified” object presents a number of difficulties. The term “circle” spans many diverse situations, which each evolve through the passage of time. It seems unlikely that these situations could have an intersection within which a word could have a positive and unconditional meaning. Signifiers, such as the word “circle”, simply cannot be attached to states of being and becoming without some sort of effect. And any such attempts to settle matters would surely result in successive displacements of meaning. To insist on a particular understanding would entail a very insistent brand of teaching. As this book proceeds, it may become clear that we have a limited capacity to constrain the self-generating aspects of language towards settling, once and for all, the meaning of mathematical objects. We cannot finally decide to what extent mathematical objects are products of discourse or not,1 or whether they are a symbolic layer structuring the “reality” beyond,2 or bearers of an invisible cultural veneer that resists access to their more abstract existence. There is not just an issue of what mathematics is. We need to consider how alternative understandings of the teacher, her students and mathematics evolve in the research and teaching processes. Learning can be conceptualised as having the power to disrupt the very state of mathematics. Learning does not just discover things, it produces things. Krista’s journal extracts hint at the linguistic filters that are in operation and the socio-cultural factors which influence both the teacher’s and the students’ understanding of the mathematical concept of a circle. The journal depicts a hermeneutic circle of understandings passing through various explanatory domains, English/ Runyankore, mathematics/ geography, novice/ expert, descriptive/ abstract. In one, “circle” may be seen as a noun, in the other, a type of adjective. And quite apart from such distinctions the term resides in alternative worlds where people move around in diverse ways and understand or experience their movements quite differently. There are words such as circle, sphere, oriziga, engata, spiral, ball or coil that appear in sentences that get spoken in the presence of certain objects or diagrams and the task of education here seems to some extent to be about how to distinguish them. Yet these sentences are being visited from a range of experiences, in a range of linguistic home bases and self-concepts. This can create an illusion of signifiers being isomorphic to signifieds, or even that the signifieds exist in a clearly defined manner.

1 Radford 2 Žižek

(2003). (2007).

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The phenomenological experience of “circle” is different for each of the participants in Krista’s story. The hermeneutically circular path of “circle” takes the word through a broad range of scenery waving to passersby, as they wave back in partial recognition: “we thought we knew you but you seem different now”. But as Krista holds on to her circular calling card she herself changes, in her understanding of what she is trying to do; of how her teaching relates to her students’ learning, of how she understands the students learning, in her way of sharing words, in her ways of getting words to do things, or of letting them do things to her. As the calling card is passed around, it bounces or rotates or swirls or rolls or gyrates or orbits or radiates, and “arrives at its destination”.3 The card (the word “circle”) stays the same, in some sense at least, but people respond to it differently, link it to new experiences, shape themselves around it differently, as do Krista’s written analyses. The signifier, or name, stands in for, even produces, the object it seeks to locate. We ourselves follow our ancestors’ attempt to make sense of the world and find ourselves drawing on the symbolic universe that they created in trying to make sense of who we are today.4

1.3 Rounding Up Reflexivity – Cyclical Research Krista’s analyses took place at different times. She occupied successive modes of immersion in various linguistic domains. She sought to observe, understand, share, resist and transform these domains through her participation. She recorded successive perspectives on successive actions. Yet in the research process it was the writing generated by her that provided anchorage. Yet this anchorage echoes the limited sort of way in which the word “circle” served as an anchor for more mathematically oriented discourse. The words themselves were more stable than the way they held meaning. Similarly, the writings simultaneously sought to explain the past and shape the future. But in the meantime they provided orientation and a conceptual space for examining how the term “circle” was being used. Yet each component of the writing was constantly in the process of having its status amongst its neighbours unsettled. Krista was involved in the production of stories that had a limited shelf-life as “stories in their own right”. The process of research entailed generating perspectives and framings to enable solutions to professional difficulties or perhaps to recast the

3 Lacan

(1988, p. 53).

4 In his later work Lacan suggests that “the displacement of the signifier determines the subjects in

their acts, in their destiny, in their refusal, in their blindnesses, in their end and in their fate, their innate gifts and social acquisitions notwithstanding, without regard for character of sex, and that, willingly or not, everything that might be considered the stuff of psychology, kit and caboodle, will follow the path of the signifier” (1988, pp. 43–44).

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difficulties into a more manageable form. The successive reflective writings5 provided a “trace technology” comprising an archive seen through a “perpetual present”.6 Yet this sense of archive transcended mere associations with the past. In the analysis offered above, the reflexively defined teacher/researcher followed the path that had been set in motion through previous actions and interpretations. The on-going collation of writings was about formatting the future, setting a trajectory for an ongoing journey, in which the term “circle” would be met over and over. And the understanding Krista has of herself, and of her task as a teacher, would be persistently updated. Yet the consistency of such terms is necessarily tainted by their derivation from and evolution through their seduction by language. Our narratives seduce us. They draw us in to their grasp. But this is at some cost to the experiences we have sought to capture. Subsequent experience disappoints us. The narratives never quite fit, deriving as they do from past values or earlier ways of making sense. As Mason (1994, p.176) put it: “Words generate more words in explanation, but often draw us away from the experiences from which they stem”. Terms, including mathematical terms, are historically defined entities, colonised by particular modes of usage, as are the people looking at them. The passage of time, however, can provide the distance necessary to see the previous frame as being outside of oneself. And of how it had encapsulated the teacher, the learner and the mathematical objects that they had sought to share.

2 Experiencing Shape and Space 2.1 Introduction The possibility of mathematics being objective, the way that objectivity might be understood and the different modalities under which ideas can be ideal, will be recurrent themes throughout this book. In geometrical study we are seemingly confronted with ideal mathematical objects that are also a function of their cultural heritage. Yet these supposedly ideal objects typically derive from human constructions, made with respect to configurations observed in the physical world by humans at a particular stage in their own evolution with their given natural perceptual apparatus (eyes, fingers, bodies, movements, senses, etc) and technological supplements (telescopes, cameras, computers, sensors, etc). Badiou (2009a) has argued that these objects are products of particular culturally and historically situated frames of knowledge. Any particular form of knowledge relates to what he calls a world, a

5 I have written extensively about these issues of practitioner research being processed through reflective writing (Brown, 1996b; Brown & Jones, 2001: Brown & England, 2004; Brown & England, 2005; Brown, Atkinson & England, 2006; Brown, 2008d). 6 Luke (2003, p. 336).

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Experiencing Shape and Space

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particular set of circumstances, where any particular analytical apparatus has definite limits of applicability. Newtonian physics, for instance, draws on Euclidean geometry in defining such a world. This analytical apparatus that had earlier seemed to be universally applicable, does not apply to the very large, such as in deep space, or to the very small, as in quantum mechanics. The apparatus would also probably not convince dolphins with their sensual apparatus and their very different ways of moving in space. There is clearly a big outside to the world of Newtonian physics. Yet it may be that our immersion in a world conceived in Newton’s terms mediates our intuitive senses and our capacity to see outside that way of thinking. The geometric objects that Euclid described are “ideal” only within a very specific human apprehension of the world and can only ever be accessed through technology or perceptual filters that are both time and culture specific. Husserl (1936, p. 173) argues that “to understand geometry or any given cultural fact is to be conscious of its historicity, albeit ‘implicitly’”. “Ideal” objects can only ever be accessed through technology or perceptual filters that are both time and culture specific. Those technologies or filters display some historical continuity, revelatory of how they emerged from earlier manifestations. Yet we cannot be geometrically naïve insofar as our very sense of self results from identifications with this heritage. As Derrida (1994, p. 54) puts it: “the being of what we are is first of all inheritance, whether we like it or know it or not”. We may not be fully aware of how this is happening. Our physical experiences are processed through that vocabulary of set moves and analytical strategies.7 We have learnt some of these things in school, or through everyday life experiences, but in a fundamental sense they are also part of us, contributory as they were to our very formation. Gattegno has considered the algebraicisation of geometry, that is, how geometrical experience is transformed, perhaps compromised, by an insistence on it being converted to symbolic form (e.g. 1971, 1988). Schubring (2008, p. 140) has later argued “that the processes of algebraicisation are among the most marked characteristics of the historical evolution of mathematics”.8 At a seminar that I attended in 1979 Gattegno spoke about a baby pointing to a fly walking across the ceiling. Each fly position on a continuous path was associated with a particular (discrete) arm position, which Gattegno saw as an algebraic relationship. But a key concern was that in school, geometrical experience generally gets converted into algebraic

7 Bartolini-Bussi & Boni (2003) describe the circle not as “an abstraction from the perception of round shapes” but as reconstructions, by memory of “a library of trajectories and gestures” (p. 17). (Quoted by deFreitas and Sinclair, forthcoming.) 8 Mathematics is initially experienced intuitively prior to its later encapsulation in symbolic form, where there is also some later evolution of the symbolic forms. For example, Spyrou, MoutsiosRentzos and Triantafyllou (2009) discuss some experimental work with 14-year-old children where “embodied verticality” was linked through gravity with “perpendicularity”, which led “to the conquest of the “first level of objectification” (through numbers) of the Pythagorean Theorem, showing also evidence of appropriate “fore-conceptions” of the second level of objectification’ (through proof) of the theorem” (cf. Radford, 2003).

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experience and that this results in a loss. Whilst not in anyway detracting from the importance of algebra in emergent mathematical understanding Gattegno was keen to educate the “whole brain” where experiences of the continuity of geometry were more often fore-grounded in classroom geometry. In this section we encounter first year undergraduate students training to be teachers of mathematics in British secondary schools (aged from 18 to undeclared middle age). They are depicted in four instances of activities concerned with the apprehension of spatial environments. In each instance the continuity of space is encountered through some sort of discrete mark-up. These examples will be drawn on in the following section where a broader perspective is taken on how historical processes generate mathematical entities. There will also be some preliminary remarks on how humans present on the scene are also classified according to historically generated descriptive categories. Later chapters will return to this material in considering theoretical approaches to how learning might be understood as being in closer alignment with such processes that produce both conceptions of geometry and conceptions of what it is to be human.

2.2 Shaping Up I have a weekly session centred on broadening the students’ perceptions of mathematics and of how mathematics might be taught. For many of these students their recent experiences of mathematics will have been shaped around performance in prescriptive school examinations. In one session we tried out a few activities in which various instructions were followed that resulted in the students walking the loci of certain geometric objects. They later at home drew the figures they had walked, or created computer representations using standard packages: Walk so that you are always equidistant from your partner who is standing still (circle). Walk so that you are at all times equidistant from your stationary partner and a wall. Figure 2.1 depicts a parabola. Some variations are included as Fig. 2.2. Now get in to groups of three where there are two people each standing still at some

Fig. 2.1 Parabola

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Fig. 2.2 Constructions of circle

distance apart: Walk so that you remain equidistant from both partners. Walk so that you remain twice as far from one partner as you do to the other. Walk so that you can still touch a piece of loose string held firmly at each end by your two partners (ellipse – see Fig. 2.3a–j, where a fourth member observes and records the trajectory, and Fig. 2.4 drawn after the event). In setting the task on the first occasion for some time I had some expectations, based on my own hazy memories, of some of the figures that would be generated. But given the zest and determination of this particular group of students, explorations went further than expected with some very familiar geometric figures emerging from unexpected directions. And for the students there appeared to be a

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a

b

c

d

e

f

g

h

i

Fig. 2.3 (a–i) The construction of an ellipse

Fig. 2.4 Ellipse drawing

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very real sense of acting out shapes and feeling them before recognising them as more or less familiar. Yet perhaps now they were being understood differently given the novelty of the approach. The “ideal” nature of any given object could not be apprehended in an instant. Or rather, that “ideal” dimension gave forth its properties in many ways. There comes into being a perceptual architecture that supplements the “ideal” with a necessarily cultural layer. In steering a particular course a student had to stay twice (or three times, four times, etc) the distance from one partner as she was from the other. As I observed I had some vague memory that a hyperbola might be the result. Yet it eventually became clear to those present that there was just one curve and that it seemed to be closed (see Fig. 2.5a–d). Yet the relative imprecision of the body movements resisted anyone achieving complete certainty as to whether it was closed and if so if its regularity suggested a circle or an ellipse. We all experienced glimpses of possibilities but remained unsure if our conjectures could be confirmed without more sustained analysis using drawings or calculations. A conceptual layer, or some sort of proof, was needed to confirm intuitive assessments. But these initial moments provided exciting insights into emergent understandings. These were all the more intense for the person attempting to walk the path of the curve, experiencing the mathematical rules through actual bodily movements. For others, positioned differently, there was the challenge to assume some specific perspective on the emerging locus. For the other partners this was from a fixed point. As all of these activities involved walking on the floor, the shapes constructed were all two-dimensional. Yet one interpretation literally shifted the ground. With an instruction where the moving player was required to be equidistant from two stationary partners I had anticipated a straight line but the moving partner, Sally Jackson, who is also responsible for the drawings included in this section, decided to stand on a chair and then on a table between her two partners (see Figs. 2.6, 2.7, and 2.8). A third dimension was brought in to play where for any given distance a circle in the third dimension could be imagined. This departure led to an unexpected exploration later on for all of the other erstwhile two-dimensional shapes. In another set of exercises, illustrated in Fig. 2.9, students formed a circle and issued instructions to a blindfolded member standing inside the circle – “turn right”, “walk forward three steps”, “right turn 45◦ ”, etc. – to walk in the form of some prescribed shape such as a number 4 or the letter M. This required that participants agreed on the orientation of the figure and a particular form, such as an open or closed top number 4. This entailed detailed discussions between group members, all of whom were oriented differently before the individual could commence walking. In a final example the only instruction allowed was “come to me” where the poor sightless fellow was obliged to walk continuously in the direction of the person last speaking the instruction, whilst people in the circle frantically (but in strict silence) decide who to point at to issue the next “come to me” such that a curved 3-shape or

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Fig. 2.5 Circles

8-shape is produced.9 This time a series of discrete instructions provoked a relatively continuous path to be walked (Fig. 2.9). Finally, a string was held between two people where only one person was allowed to move. There was however a third person in a fixed position closer to the other fixed person than was the mobile person. Here the mobile person experienced

9 Figure 2.9 depicts a different group engaged in this activity. See also Brown (2001, pp. 119–126).

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Fig. 2.6 Equidistance from both partners on floor

Fig. 2.7 Equidistance from both partners on chair

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Fig. 2.8 Equidistance from both partners on table

Fig. 2.9 Come to me


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Fig. 2.10 Changes in curvature

a change in the rate of curvature as the string hit the second stationary person (Fig. 2.10). Together such activities provided the students with experiences of moving in space according to more or less precise instructions, more or less drawing on conventional geometrical terminology, such that continuous movement was associated with a sequence of discrete instructions. Yet the words and sentences that we were using were never fully up to the task but were often successfully suggestive of the “ideal” forms beyond their reach. Back in the regular classroom and later at home subsequent attempts were made to capture the bodily movements in drawings and reflective writing and a new world of geometric figures were generated. Much work was carried out on the two-dimensional shapes. The mathematical objects were generally familiar once encapsulated but the routes to them made them seem somehow new, as though they were being encountered in a fresh way that made them seem different. And following the ascent of the chair and table, later developments considered how the various tasks could be extended in to three dimensions. Ellipses became eggs. Circles became balls. Lines became walls. And various bowls and saddles of infinite

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Fig. 2.11 Ellipsoid

Fig. 2.12 Sphere

dimension and curious orientation also emerged. And in certain circumstances eggs could become balls or even walls. (Figs. 2.11, 2.12 and 2.13). In a separate session several weeks later, as part of an activity inspired by Krista’s focus on circles I asked the members of the same group to each write answers to the question, “What is a circle?” They then read out their thoughts for everyone to hear. Here are some of the results: A circle is a 2D shape, which starts and finishes at some point. It is a continuous curve and has 360 degrees. Clockwise from the centre point to the curve is called the radius and the radius is the same distance to the curve all the way around the circle. We use the radius to calculate the area and the diameter, which is twice the radius gives us the circumference when multiplied by π. A circle is a regular 2D shape, which has no straight sides. Every point on the circle is an equal distance from its centre point. This distance is called its radius. The distance around the outside (circumference) is known from the formula 2πr and the area from πr2 . Lots of coordinates plotted on a graph and when joined with a line it makes a circle shape. It has a centre point, and from the centre point to the edge is called the radius of the circle. Double the radius = diameter. The points can form an equation in the form (x − a)2 + (y − b)2 = c2 , where a, b is the centre point and c is the radius.

The students then speculated on how circles were invented: Circles were first invented by the Aztecs. They are widely regarded as the first astronomers of our time. They saw the shape of the moon and the sun and recreated the same image on the ground with sticks in the mud, which later became marks on walls like Egyptian hieroglyphs.

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Fig. 2.13 “Saddle”

By God when he made the human eye – Ask him! In the days of caveman they decided it was easier than carrying certain objects to put them on a sledge type thing and pull them along. . . . But when they travelled over gravely ground they realised the ground was assisting the movement. This gave them the idea of raising the sledge up off the ground and attaching large bits of gravel to the bottom. Over time they developed the axle helping the stones move and again over time the stones wore down to a circular shape. In a POLO factory (a popular round mint with hole!) After Allah created the moon and sun they were observed by man and copied. Circles were invented when a man cut down a tree and noticed the shape of the stump was of a different shape and the logs it created were a different shape. He also noticed that it rolled easily enough and he realised this may be a good template for a new shape. . .

2.3 Calculus In a later session the same students were invited to talk about their memory of learning calculus in school. They each sought to recover learning that for some had taken place as long ago as nearly 20 years before. The following extract comprises the first 10 min of speech (apart from the brief prompts and murmurs) of a 70-min discussion by about seven first year

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undergraduate students trying to recover their memories of the calculus they had done in school more or less recently. My purpose in offering these extracts is to track the way in which understandings were recovered and developed through the length of the exercise taking place over two sessions a week apart. But my specific interest is in how geometrical or spatial concepts are held in place with linguistic constructions and also how algorithms become placeholders for distant memories of mathematical procedures. Again this provides an example of continuous entities being understood through discrete filters. Dorenda: I remember that, differentiation, you times by the power and then you decrease the power by one and then, integration, you increase the power by one and then divide by the new power. Sophie: Trying to find the gradient of things. Sanyia: I have done calculus and I can remember some rules but, as to where it did come from, I did have a really, really good teacher and that good of a teacher that I’ve forgotten, so [Laughter] I’m not too clear about it. No, no, no, he was really good but, right now, I can’t think; my memory has just gone blank, all I can remember is differentiation and how it’s done, how we take away one power. David: I’ve done some rules on it, I think, didn’t Isaac Newton [pause] or something, as well, something to do with that and I remember that and it was to do with the description of curves, basically. That’s what I thought it was. I just know that you use it in schools and increase the power and find out what the plus value was and stuff like that. Just basically probably find a little method, and that, for when I was doing it. Sally: This is a learning curve for me. I don’t know a lot about it. . .I know it’s changing the powers and it’s to find gradients. That’s the basics I know of it. That’s it, that I know about it, really. James: I don’t remember any of it. The only things I remember are the terms integration and differentiation and something to do with dy × dx, . . . That’s all I remember. David: With respect to dx, whatever that means. I remember seeing that. They used to always make me write that. I don’t know what it meant though. Sophie: Integration is going from the other one back up to what the actual graph is. . . . it’s going from the gradient back up to what the graph was. Dorenda: We found the area under the curve. . . .I remember with integration that there is a big, long formula that you can use to find the area and it was between limits as well so you had like between x = 0 and x = 5, or something, like you imagine 0 and 5 on the x axis, you could find that area under the curve. If you imagine a curve you could find the area between those two. . . Differentiation, thinking about it, I suppose, because you’re decreasing the power you’re

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Sanyia:

David:

Sophie: David: Sanyia:

James:

Sally:

James:

bringing it to a – say it was y2 or an x2 thing then, I don’t know, because it brings it to a straight line because gradient is like the straight line point on a curve. If you’ve got a curve you want the straight line of it so maybe that’s how it. . .. I do remember with area under a curve, that rings bells, but what bells it does ring, I’m not too sure but a bulb did turn on in my head while Dorenda was talking. That trapezoid rule and Simpson’s Rule. . .. They’re describing the area underneath the curve I think. I remember a little – we didn’t use that actual big formula, we just used a little box method that kind of broke down but they – I can just remember like a line and one bit shaded in one colour and one bit shaded in another and then – It’s all these little rectangles and you measure an area board of rectangles and add them together. [Draws a curve on the board] It was like area. It was like lines. It was like shaded in like that. Didn’t each quarter of underneath that curve you split it into quarters, like sections? . . . I don’t know why. That’s coming into my head and I don’t know why but I’m thinking of the words minimum and maximum. . . On the curve and finding out what the minimum is and what the maximum is. . .. One over x? I don’t know. I’m just getting random stuff coming into my head. Nothing that’s flowing. I can’t remember but I remember now, looking at that visual, I can – minimum and maximum comes to my head and how you’ve got to find the minimum and maximum area – that it is possible. I’m trying to think. Is this just trying to find out the area underneath the line? Is that what we’re? Right so if you split it down into four things and get the approximate area of that. If you split it more you get a more accurate. . .. If you split into 16 it’s more accurate. Right? . . .And if you split it into infinite amounts you’d have the exact area but that’s not possible and the same thing as the – Is it when you do the maximum, the U, if your curve goes like that you take the width of the section that you’ve chosen and you take it as the highest point and then you add all of those up and the minimum is if you take your lowest point and you take it underneath the curve and that’s your minimum size of your curve. . .. If you had it like and that’s your curve and you’ve chosen all these sections here, if you worked out all these areas as to that, that would be your maximum, doing those areas, working on those areas and adding them up. You can do them underneath your curve and work that area out and added those up and that would be your minimum size. Well the more blocks you split it up into rather than just having four blocks there.

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Dorenda: Then you get a more accurate account. James: They get closer to the actual value they should be. Dorenda: Close to each other [if you take that to the extreme?] Infinity. . .. Like you do it really, really accurate, like you’ve got four sections there, within those four sections, there’s a lesser gap between them but four sections which means your minimum and maximum is going to be far more accurate, obviously, because you’re using more – because it’s going to be more close. I’m just repeating what everyone’s saying. Similar discussion continued for a further hour where eventually diagrams began to be shared on a white board. It became increasingly difficult to distinguish between memories that were being recovered through the relics of their educational pasts, as opposed to new understandings being created as a result of participation in the discussion. Shortly we shall follow James (on left in Fig. 2.6) into more complex territory, partly since he appeared to have no recollections of calculus at the beginning of the discussion, having left school more than 10 years earlier. I want to provide further evidence of his contribution to this continuing discussion, but to save space here, without providing the full transcript of the class discussion. So I provide here James’ subsequent contributions with annotations to indicate brief contextualisation. Unfortunately I do not have a record of the drawings mentioned. Well, suppose I just sketch you a slightly more accurate version of what it should be where you can break it into more chunks. OK. [Draws] Or can I just edit this one at the bottom there. Rather than having four like that why not bring it up there and there so now, instead of just having the four parts to measure, now you’ve got eight parts, so you can get it more accurate and do it that way so you’ve got like [Draws] and then if you wanted to get it more accurate than that you can break it up again and so on and so on until eventually you would get it right. [How would you do the calculation?] Formulas. [The line that you’ve just drawn is that a tangent to the curve?] That’s just joining up the two points. [What’s the biggest area I can fit under the curve? – When your maximum equals your minimum – and when the strip is?] Infinitely small. Equal value to the lower value multiplied by the height. [How would you work out the gradient of a curve? Would someone like to explain?] Do you not get two points on the curve, work out the gradient of that and, the closer the points are, the more accurate the gradient of the said point becomes until you get where both points are exactly the same which gives you the tangent at that point. . . It’s sort of, as it gets closer and closer, you’re gradient will be changing and changing and changing until, at the exact point where you first point and your second point are the same point, you should have the exact gradient which would be a tangent to the curve.

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A tangent is a line that’s at 90◦ to the perpendicular of a line. I don’t know! On a straight-line graph, technically, the gradient of a straight-line graph is still a tangent to that line. It’s just that the tangent is that line. It’s just that as the line’s changing. Would it be better drawing it like on a massive scale and showing how it would . . . Can I just draw? [Draws] So if you start from a big curve like this, and say this was zoomed really, really far in, and you were trying to find the gradient of that point there when this is – well, it’s just zoomed now in, far in – and you start off with that point and you link those two up and you get the gradient of that. I don’t know if you can tell because it’s almost a straight line really there. Isn’t it? So if you take that as being your first gradient and you want to get it more accurate so you bring it down to here and you get it more accurate. Bring it down to there – more accurate – until, eventually, you’re doing it here and here until that point there is the same and that gives you the actual gradient. It’s not that accurate. But every time you move that point it gets more and more and more accurate. It’s just like the arrow thing. [The impossible story where a soldier runs directly away from an arrow approaching him, but each time the arrow gets to where the soldier had been the soldier has run to a new spot.] You’re just trying to find out the exact point of impact. I just thought it would work better with a bigger scale drawing of it. Is that the dy or the dx that I remember? . . . Is that when both of the points are exactly the same? I eventually asked the students to each independently write a short definition of calculus and then to read it out. This is what they read out: Sanyia: Oh. Don’t start with me. I’ve not written what calculus is. I’ve just written calculus is a form of mathematics so, OK, I’ve just contradicted myself. Calculus is a form of mathematics. Integration is one aspect of it and differentiation is another. Integration is how we get an area of a curve. In order to find the area of the curve, we would need to know the equation of the curve and the formula for integration is the integral with the highest point at the top and the lowest point and the bottom – dx – and that gives you the area of a curve. x is actually the equation of the actual curve. Differentiation gives us the gradient of an equation. It’s not necessary that we know the equation; it’s just we need to know the points that are on the line. Dorenda: Calculus is differentiation and integration. Differentiation is used to find the gradient of a curve at any given point. The function is dy × dx, y with respect to x. The double differentiation d2 y by dx2 is used to determine the maximum or minimum points on a curve. Integration is used to find the area under a curve between two given

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Sophie:

Stuart:

James:

Sally:

boundaries along the x axis. It involves splitting the area under the curve into an infinitely large number of equal-sized strips. If you have a graph and you need to find the area underneath a line to the axes then we need a quick method of doing this. We could do it the long way by splitting the area up into lots of trapeziums and working out the areas of them all and adding them together or you can use integration if you have the equation of the line to find the area. If you want to find the gradient of a line at a certain point you can use differentiation, which is where you use the equations and use the power and times it by the coefficient of the letter and reduce the power by one. Integration is part of calculus. Integration is where you create trapeziums to get as close to the line as possible. The more trapeziums you use, the closer you get and this will give you the area, and also differentiation is used to find the gradients of the line. Calculus is a mathematical way of calculating ever-changing variables within a never changing set of circumstances. Things, like rates of acceleration, gravity, are all worked out with sets of rules from calculus. Integration is a specific area of calculus that deals with areas under curves and other lines that are not straight lines. Differentiation is another specific area linked to integration inversely that is used to calculate gradients of lines that aren’t straight. Part of calculus is differentiation. This is the way of altering the equation of a curve that will enable you to work out the gradient of the curve at any point along it. Another part of calculus is integration. This is the way of changing the equation of a curve that will enable you to work out the area underneath it, between two specific points. There is another way to work out the area under a curve, which is called trapezium rule. This method is calculated by dividing the area up into narrow strips, working out the area of each strip, and then adding them all up. The more strips there are, the more precise the calculation is.

I was intrigued by how such partial memories at the outset could be recovered and combined with new awareness produced within the discussion. Most striking was James who had not done calculus for many years. It is his extension of this activity that is presented in the next section.

2.4 Mancunian “String Theory” In the session the following week we were based in a large church building where a string was suspended diagonally across, spanning some 20 m such that one end was higher than the other (Fig. 2.14).

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Fig. 2.14 Suspended string

Here the students were asked to investigate the string’s properties, specifically those relating to calculus, such as the gradient at different points and the area beneath it. Finally, as an option, students were invited to write up their experiences on these activities as part of their course assessment. The description of the activity will show how continuous geometric phenomena were transformed through attempts to fix them in analytical apparatus, such as words or procedures. The continuous phenomena were talked back into existence in the duration of the session within an attempt to connect with earlier learning but also to establish a dialogue through a shared project with fellow class members on this occasion. It appeared that the students had not in their earlier school learning seen calculus much beyond the technical apparatus. Yet these formulae and algorithms had clearly helped the students to acquire their advanced secondary school qualifications. Or at least that was the core of what had been preserved. The apparatus enabled the tangible link that permitted some sort of continuing grip. Indeed the technology appeared to provide an unreliable handle on earlier learning that was perhaps at the time focused on the knowledge being tested in an examination context, where possibly results in the form of “correct answers” were more easily detected than deeper understanding in testing procedures. The students worked in groups of three to four pooling their diverse experiences. James Clegg provides a report of his participation (highly edited for brevity here):

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2 The Regulation of Spatial Perception To start this lesson, our tutor took a long piece of string and attached it to one wall of the room we were in. He then looped it down across the room, and attached it to a pillar at the opposite end of the room. The premise was that we had to measure the area that was underneath the string as accurately as possible. Using metre sticks and measuring tapes, we set about trying to measure certain things that we thought would aid us to find the information that was asked. The first thing we thought about measuring was the height of the 2 fixed points at either end of the string. These static points were measured as 425 cm at the top end, and 114 cm at the bottom end. Some people set about trying to measure the length of the piece of string that was used; however, as people were trying to measure other information, the string was moving constantly and was impossible to measure accurately. The next piece of information that I thought would be necessary was the horizontal distance from one end to the other, but with the stage in the way, this was relatively difficult to get an exact reading. We took the reading at approximately 1,600 cm (rounded to 5 cm due to accuracy issues and ease of calculation). A strategy was then formulated to try to better measure both the length of the string, and the area underneath it. What if we took sections of the string, measured heights at set intervals and worked out how a polygon shape would fit under this area. The result would be an inaccurate version; however, it would also be a fairly good estimation of the area that we were trying to measure. From here we measured every 100 cm across the floor, and took a reading as accurate as possible, for the height of the string at that point (Figs. 2.15 and 2.16). This gave a table of results as follows:

Fig. 2.15 Measuring a height

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Fig. 2.16 Ascending the y axis

Distance from wall

Vertical height

Distance from wall

Vertical height

0 100 200 300 400 500 600 700 800

425 362 308 237 188 149 114 87 67

900 1, 000 1, 100 1, 200 1, 300 1, 400 1, 500 1, 600

53 44 41 44 53 67 88 114

From this table, you can clearly see the start and end points of the string, and that the string dips down below the lower end, before it is connected to the wall at the bottom. From here we used prior knowledge that we could create a simplified version of this curve, by plotting these points onto graph paper, and connecting each point up with the next. The curve wouldn’t be very accurate with straight lines connecting each point up, but it would give a good estimation of it. . . From here, to work out the area under the curve, we used what is called The Trapezium Rule. This involves taking the curve as a whole, and building an amount of polygons (in

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2 The Regulation of Spatial Perception this case trapeziums) underneath the curve, to estimate the area that is under it. The sensible thing to start off with here was making 16 separate areas at each 100 cm intersection, and working out the area of each one, then adding them all together.

This diagram shows the building up of the trapeziums under the curve. The lines connecting each point on the curve are straight, so the accuracy of it will not be exact. Working out the area of each trapezium, you add the length of each side together and divide by 2, to get an average height, then multiply by the base length. Doing this for each section and adding them together gave us a value of 217,150 cm. This translates as 21.725 m2 , which using our own judgement seemed to be a good approximation. After doing this first trapezium, it was quickly realised that if we just took more readings, every 50 cm perhaps, then the trapezium rule would give us a more accurate version of the area we were trying to find. This could be expanded further to taking measurements every 25 cm, or even every centimetre; however, this approach isn’t the most practical way of measuring the area more accurately. We fast realised that to be able to measure the area with an incredible amount of accuracy, we would need to know an exact equation for the curve that is being created by the piece of string. Looking at the table of results, and just using our own judgement, we decided that the string looked like a fairly simple quadratic curve, so we set about trying to find an equation for it. The way I came up with of trying to do this was to pick the first three points that I had in my table, and solve three simultaneous equations of the form y = ax2 + bx + c (the general equation for a quadratic curve). Using the points (0, 425) (100, 362) and (200, 308) I set about solving this. I substituted the coordinates into the x and y values of the general equation, and came up with three new equations; 425 = 02 a + 0b + c 362 = 1002 a + 100b + c 308 = 2002 a + 200b + c Straight away you can work out the value of c from the first equation. Which is where the curve crosses the y-axis, so c = 425. This can then be substituted into the second two

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equations, leaving only two unknown variables. By expanding the last two equations, and rewriting one of them so that both values of b had the same coefficient, I could solve the value of a, and substitute back into the equations to get a value for a, b and c, thus giving me an exact equation for the curve of the piece of string! . . . (After some extensive calculation three equations were thus produced – see graph below.) All three of these equations are slightly different, although the second two are relatively close to each other. From this I decided to use the three most accurately measured points to create and equation to use. I assumed that the points we’d measured most accurately would have been the minimum point, and the two points on the wall. Using these points, I got a fourth equation. I plotted these four lines on a graph so I could compare the accuracy of what I’d workedout.

From this you can clearly see that one of the lines that I’d worked out is completely out, while the other three are relatively accurate. This line is the equation that was worked out by only using the first three points as reference. From the general overview you can see now that the first three points that were measured appear to not be where they should be expected, and that they all seem to have been measured about 20 cm too high. This inaccuracy will be due to the fact that the highest point of the string was atop a stage at that end of the room, making accurate measurements quite hard to follow. The green line in my diagram, however, appears to be extremely accurate for almost every other point (except the three measured from on top of the stage), and from here I decided to use this line as the one that most closely represents the string that we had in the room. This equation was y= ···

66x 6x2 − + 404. 20000 100

To now work out the actual area under the curve, I can use integration to work out the area under this curve with an infinite amount of accuracy to that specific curve. The only thing that limits this being related to the area under the string was the accuracy of the initial measurements, which with the green curve on my graph seem to have been taken with a

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2 The Regulation of Spatial Perception good degree of accuracy, all things considered. The rules of integration are that you have to increase the power of the term by 1, and divide the new number by the number of the new power, for each term in the equation. . .this gives a final area of 21.12 m2 . This value isn’t that far away from the first area I worked out with the trapezium rule of 21.715 m2 , and taking into account the adjustments that I made, when it was realised that the highest 3 measurements were taken inaccurately, I think this is the most accurate result that I can come up with, without getting all the measurements to a far more accurate scale. After having not done integration for the last 10 years, I had no recollection of what it was about, or how it was even used, but this exercise has managed to get my level of understanding very high. I realise now why it is used, and what the theory behind it is. The visual effect of the string, and the measuring part of the lesson at first seemed to be relatively simple and getting a good approximation of an answer wasn’t too difficult. Getting an answer to a very accurate degree of accuracy was a lot more difficult, and required a lot of thought and relearning to be able to do it.

James’ route was more or less typical of the various attempts around the room, although there was much sharing of ideas following the various group discussions. The process might be characterised in terms of a continuous geometric construction being subdivided into discrete elements for more detailed scrutiny. This scrutiny takes the students away from the continuous object in an attempt to regain it subsequently with alternative understandings. Armed with recovered but fragmentary memories of strategies such as the trapezium method, Simpson’s rule, integration and various measuring implements, the students initially sought to investigate the area under the curve. The task was quickly understood in terms of subdividing the space beneath the curve into areas that would be more readily quantified. A few of the groups used more than one method to double check results. All of these approaches comprised familiar approaches to analysing this particular sort of situation that entailed seeing the continuity as being partitioned so that a story could be told. The various stories produced linked the cultural technology of handling areas under curves.

2.5 Planetary Movements (with Dave Heywood and Yvette Solomon) I later worked with an equivalent group and some colleagues on building the students’ capacity to relate mathematical models of the solar system to their own empirical experiences of space on a grander scale.10 Students were asked to explore in discussion, by drawing, and then by acting out, how they imagined night and day, or the seasons, being explained by planetary movements. Three students working

10 This work had been first developed elsewhere: Parker and Heywood (1998); Heywood and Parker

(2010); Brown and Heywood (2010).

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together considered how the earth moved in relation to the sun, using a globe to represent the earth.11 Fingers pointed to where the sun was imagined to be. The earth was spinning on its axis and rotating about the sun. England was rotating around the earth’s axis. Such words pointed to the continuity of experience. Spinning and rotation do not stop. The continuous cycle of yearly lived experience on both a daily and seasonal basis is couched in terms of circular (or elliptical) and cyclical movement, spin and orbit. In explaining the phenomena however, the students introduced discrete elements where the continuous was interrupted. With the task of explaining the seasons, words such as winter and summer appeared, as in “this side is in winter, and that side is in summer”. This cut introduced phases in to words like spin and rotation, such that some parts of the spin, were night or day, or some parts of the rotation were summer or winter. Yet such pairings, night/day, winter/summer, dark/light soon proved to be inadequate when the task moved on to explaining how some periods of daylight were longer than others, or how some days were colder. There was also some discussion as to whether the earth was rotating clockwise or anticlockwise, backwards or forwards, and how that was related to the order of seasons, length of day, time of day, etc. Was a discrete ordering of winter, spring, summer and autumn linked with a continuous clockwise or anticlockwise rotation? And did this relate to where one was positioned? The students moved through a range of perspectives. They imagined themselves to be positioned on England on the globe tracking where the sun moved. Here they made hand gestures around the globe, spun the globe, lifted the globe in to different positions, moved around a stationary globe in different ways, or adopted the perspective of a space ship in a fixed location watching the earth spin before it (Figs. 2.17 and 2.18). The yearly cycle, broken down into winter and summer, was also explained in terms of specific mathematical shapes circular, elliptical, eccentric and oval orbits (“Does the earth go round the sun in a perfect circle?” “I know Pluto goes round in an ellipse. The rest are closer. It just depends on what the tilt is”. “It depends on how close to the sun”. “It depends on the eccentricity of its orbit. I know that Pluto has the most eccentric orbit. Therefore it’s the most elliptical”.). This was captured by drawing sun and earth (in a number of positions), in a fixed plane, on a piece of paper. Subsequent explanations included increasing incidence of mathematical framing such as orbital plane being the same in each of the discrete phases (“I think I saw something where all the planets are in the same plane”.). Having produced the drawing of planetary positions they then went through a phase of gesturing with their hands around the drawing, looking at the drawing from different perspectives (Fig. 2.19) and returning to the globe to check various things.

11 Aspects

of this activity were inspired by the opening scene of a Hungarian film called Werckmeister harmonies, in which a bar of drunken men are encouraged to perform planetary movements. This can be viewed at http://www.youtube.com/watch?v=VFmu7BYbthY

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Fig. 2.17 They made hand gestures around the globe, spun the globe, lifted the globe in to different positions and moved around a stationary globe in different ways

Fig. 2.18 The perspective of a space ship in a fixed location watching the earth spin before it

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Fig. 2.19 Gesturing with their hands around the drawing

Fig. 2.20 A fixed human sun with a human earth rotating around it

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Fig. 2.21 Rotating in a circle (or ellipse) around the sun

Fig. 2.22 But also spinning and leaning whilst rotating

You get less daylight in winter because the axis is pointing away. Imagine there’s an arrow at the top of the axis. Yeah? Yeah. In the winter the axis is pointing away from the sun, so therefore the southern part is pointing more towards the sun. So because it’s close to the top of the earth. . . This is the sun. This is winter. So this bit is getting more sun. So imagine it’s

3

The Seduction of Language

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starting to get sun now. It’s dawn, middle of the day, and it’s sunset. And round the middle it’s constantly getting sun. Getting very little darkness. Imagine in winter it’s light for short period of time. In summer start getting daylight here – it’s pointing closer. . . Whereas in autumn it’s in the middle, in the autumn you’re getting daylight starts from here. . . until it’s got to that line. That’s about half the time.

Finally they played the part of planets, featuring a fixed human sun with a human earth rotating around it, but not only rotating in a circle (or ellipse) around the sun but also spinning and leaning whilst rotating (Figs. 2.20, 2.21, and 2.22).

3 The Seduction of Language The classroom situations enabled the students to encounter mathematical phenomena in novel ways. The phenomena were “known” in advance, but they were coming to be known in new ways through the activities. How do these physical activities signify more widely known mathematical phenomena? And do the phenomena themselves change as a result of the activities? That is, do these localised encounters provide participation in evolving conceptions of mathematics across populations that ultimately change what mathematics is? This book’s answer to the last question is in the affirmative. And for this reason the book will conceptualise learning as active participation in this evolution. In short, I will be arguing that mathematical objects do change as pedagogical objects since their cultural housing changes. And given that I have already claimed that ideal objects are culturally specific, the ontological status of such objects is further troubled. This section commences by considering how evolving historical perspectives affect our apprehension of mathematical objects. Circle is taken as an example towards considering how the existence of that object relates to cultural apparatus. I discuss how sensory and physical experiences provide so much cultural cladding of the notional ideal objects. Perceptions of planetary movement are used to consider how cultural supplements locate mathematical phenomena in modelling exercises. This cultural supplementation is then considered as a paradigm for wider advances in the physical sciences, seen as a province of applied mathematics. The chapter concludes by emphasising how the crafting of mathematics and of people in the language we use to describe them enables this advance. Yet this language simultaneously polices these advances in line with older conceptualisations.

3.1 Conceptualising the Past How could we today engage with Husserl’s quest to understand the evolution of geometrical configurations? Where and when could we possibly start? We could envisage extending the search to other mathematical objects, or indeed any empirically derived scientific object. Such an attempt would alert us to the cultural nature of each and every mathematical idea encountered in our educational quest, and of

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the cultural derivation of the frameworks that produce those ideas. Or do we, in any sense, encounter situations in which some mathematicians suppose that they can identify mathematical objectivity beyond culture and its history? And if we do encounter such situations how would they impact on our understandings of how humans apprehend mathematical phenomena? Could one possibly suppose a clear historical perspective on such concerns? History itself and our collective understandings of time are both linguistic constructions. History is not singular. There are many ways of rewriting history to produce new accounts of who we are now. Or more radically, according to a thinker who will greatly influence the pages that follow: “History does not exist. There are only disparate presents whose radiance is measured by their power to unfold a past worthy of them” (Badiou, 2009a, p. 509). We sometimes too readily assume a particular version of the past to explain and justify aspects of the present. The telling of history can normalise arrangements that favour some people more than others. Yet, time itself, Ricoeur (2006) argues, is a function of the stories we tell about it. But those stories relate to how we experience life unfolding. Yet the processes of history cannot be fully captured in the stories about them. Ricoeur (1984) argues that we cannot agree on the existence of key characters, places or events, let alone the relationships between them. And people in earlier times did not understand history better than we do today. This greatly troubles our capacity to make sense of the world from any sort of firm ground. In turn this absence of grounding disrupts the objects that we conjure. During a visit to Venice, Tony’s then 7-year-old daughter Imogen was rather taken aback by Tintoretto’s sixteenth century painting Creation of the animals: “Where are the dinosaurs?” Her youthful awareness of cultural history could detect the limits of Tintoretto’s worldview. Dinosaurs, a twentieth-century human construction, were unknown to our earlier ancestors. Her brother Elliot, meanwhile, had not realised that God was a man. Cultural narratives have been revised since the painting was created and altered how individuals understand themselves fitting in to the world we inhabit. But we need to enquire in to the explanations that we currently draw on to make sense of ourselves now. For how long will they remain valid, if they have not already lost their validity? Similarly, Galileo’s provisional account of the universe being heliocentric preceded contemporary conceptions. But the coexistence of his account with other contemporary astronomy redefines the relationship his ideas have to the entirety of human knowledge, and how we understand his ideas fitting in, and how we ourselves relate to them. Black holes or black stars12 may have been a shock to Galileo. Yet Galileo in his time was surely formalising, through telescopic observations and deductions, ideas that had previously been known intuitively, or more informally. He could not have been the first person to notice the phenomena that he described, but perhaps his encapsulation enabled alternative modes of noticing, that shaped

12 Barceló,

Liberati, Sonego, and Visser (2009). See also Susskind (2008); Smolin (2007).

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later thought. Any supposed universality of earlier conceptions would be disrupted, or localised, by later developments. Our narratives define who we are and hold us in place. Similarly, Husserl’s enquiry into how geometry came into being concluded that without the anchorage of words, or other culturally specific technology, it was quite difficult to conceptualise. Such narratives mythologise certain expressions or points of reference which contribute to socially constructed phenomenologies which serve as anchorages or frameworks for given communities.13 That is, we collectively share arbitrary ways of constructing the world as a set of objects and assign relationships between them. At any stage the signifier and signified of a supposed object can get jarred into a fixed relation to produce the object within the specific mode of common sense social construction that is currently being lived. The potential meanings and actual usage of such expressions change through time for the individual but not necessarily in the way that the person immediately detects or monitors. My common sense may not keep up with the vicissitudes of the world. Words may start off as placeholders for a particular conception and then go through a phase of being a useful working definition. Later, however, the term may be discarded as it becomes too much of a cliché without functionality. But in this fluid existence, the use of the word collides with other words being used at the time. Words are combined in sentences and impact on each other’s meaning. The introduction of any new word activates strains and stresses throughout the whole discursive framework and results in the meaning of all words and symbols being challenged in some sense.14 But in many important respects the words and symbols that had predicated objects were all that had held the objects in place. The term mathematics locates nothing but the stories of what mathematics is.15 Somehow in his calculus work described above James had recovered lost memories as he made sense of a significant challenge with great speed. He proved well up to the challenge of speaking for himself in this recovery process, attaching firm labels to what had become for him mere intuition or very hazy relics of memories, perhaps attached to specific words. How might we understand this recovery in terms of James reconnecting with aspects of his past, with stories constitutive of him perhaps linking to his earlier educational experiences? Here this recovery results from both mathematics, and James himself, being functions of cultural categories and tradition. The processes of subdividing the curve in particular ways give some access to the precise qualities of the curve but now being captured in language so that we can control it through a variety of culturally derived apparatus. This very formality can rediscover and revive previously known strategies. Part of James’ make up was a result of earlier immersion in cultural ways of making sense. Memories rely on tangible markers, yet those markers reconfigure the events that they mark. Yet that earlier immersion is part of

13 Barthes

(1972); Gabriel and Žižek (2009, pp. 50–81). (1984). 15 To be discussed in Chapter 8. See also Brown and McNamara (2011). 14 Ricoeur

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James, formative as it was of his very self. Images from such earlier experiences define who we are.16 Geometry is often seen as a field comprising ideal objects held in place in the collective memory through the technologies that have been developed to access it. In the school context those technologies are often used to formally assess understanding of it. Yet the reliance on such markers is at a cost to the profundity of the understanding that may have been achieved. In the case of calculus the students could recover the technical approach defined under the banner of the trapezium rule, a visually intuitive way of apprehending the area under a curve. In some cases they could also recover the technique of raising the power and dividing by that power as a proxy for integration. The symbolic configuration perhaps marked memories of a conceptual manoeuvre. But the more profound aspects of area being the limit as the strips or trapeziums reached in infinitesimally small width was the hardest to retrieve. The technology always has its limits of functionality, both as an aid to memory, but also in terms of its analytical reach. Husserl saw geometrical understanding as being linked to an implicit awareness of its historicity. We understand who we are through the narratives we use to explain spatial connections. The sum total of cultural knowledge about geometry remains incomplete, but “the infinite totality of possible experiences in space in general” (Derrida, 1989, p. 52) could never be completed. Yet this “infinite totality”, insofar as it is imagined or sensed, is processed through geometrical knowledge as a field of ideas held in place by the forms that it has taken. A perceptual architecture supplements any supposed ideal objects with a necessarily cultural layer. This provides access for those learning the subject. For example, in our calculus example, integration is the limit case of Simpson’s rule, such that when the exact answer was secured the perceptual technology that got him there was abandoned for final access to a conceptually different domain. As the strips get narrower integration produces a more accurate area, or differentiation produces a more accurate gradient, but only at the point it becomes purely abstract and no longer grounded in perceivable geometric space, explainable within a socially derived discourse. As the empirical world disappears abstract mathematics becomes possible but at that point the social supports for making certain choices as regards how we should understand mathematics also disappear. In integration the sense of the strips narrowing can provide a powerful insight into the object at the limit (perhaps understood as a symbolically defined area), where the object is tainted by the journey to it in the students’ conception. Sometimes mathematical abstraction requires that we see the object as a thing in itself, as though it is now separate to the journey. Derrida (2005, p. 127) characterises Husserl as saying that “objectivist 16 Elsewhere

I have written about instances of depleted pedagogical technology being all that remained of past learning experiences. A student could remember the phrase Silly Old Harry Caught A Herring Trawling Off America (SOHCAHTOA) having something to do with trigonometry. Yet the student was unable to recover the associated trigonometrical relations (Brown & McNamara, 2011). I was arguing that our mathematical selves are built through caricatures of earlier experiences.

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naïveté . . . is produced by the very progress of the sciences and by the production of ideal objects, which, . . . cover over or consign to forgetting their historical and subjective origin”. That is, the objective reality of knowledge conceals its own history – its non-objectivity. Someone has created the object after his or her own intellectual journey. The objects derive from our narratives within a reality frame that is not as complete as we had sometimes thought.

3.2 Time and the Circle How might we understand the circle’s formation as a mathematical object? How have apprehensions of circles evolved? Circles have acquired so much baggage as they have been progressively used in building our stories. Some curious perspectives are apparent in the student definitions of “circle” above. They use words or ideas derivative of circles. Indeed the examples both use the word “circle” in their definitions. How might we imagine circles without this linguistic apparatus that is seemingly consequential to the supposed existence of circles? Inevitably contemporary conceptions get in the way of any such attempt. School experiences of “circle” were often in the past centred in the construction with a pair of compasses. Things could go wrong by the pencil slipping. The circle came to be understood through the control one had in generating it. The experience of circle would be different in LOGO or Cabri Geometre, or through the exercises described above. Mathematical terms are situated in shared and in individual histories, and the terms’ meanings derive from their relations with other terms. The terms do not have meanings in themselves. The uptake of geometrical terms would be different across people according to how the terms intervened in everyday living or were included in intellectual life. The natural environment of our rural Ugandan would not provide many instances of squares or triangles to which Western educations refer, and on which they rely in so many instances of orientation. As different aggregations of such objects shape our wider apprehensions of life, the formative impact of “circle” continues to evolve and operate in diverse ways. Increasingly, such usage conceals its original historical contingency as a construction, motivated by empirical observations, at some point in the past. Residues of previous eras, and earlier conceptions of those eras, remain locked in to the later formulations of the spatial environment. Circles are a function of contemporary thinking (and vice versa). We have also changed as humans, such that those earlier humans could not have known circles in contemporary terms, and those earlier humans and their apprehensions could not be processed in contemporary terms. So many other mathematical constructs would have histories and meanings rooted in different, more or less recent, intellectual circumstances. But, most people can immediately apprehend a circle. It is a widely recognised cultural object. Yet there could be a considerable variety of meanings brought to it as indicated. Other mathematical entities would also have been generated, signified or encountered through physical embodiment and embellished in similar ways (relation, straightness, counting, iteration). Many entities, however, require rather more specialist training to even apprehend their existence, let alone their finer qualities.

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Circles also structure our everyday conceptions of time. Observed cyclical events, such as night and day, the phases of the moon, seasonal variation and planetary motion provide alternative spatial perspectives on following a cyclical or circular path. Our senses of self are shaped in relation to repeated cycles, marked events, rites of passage, the working week, annual school plays, harvest festival, Christmas, birthdays, the taking of exams, starting university, entering a profession and so on. And at a micro level these time phases are experienced through the beat of a heart, the tick of a clock, the beat of music or a sequence of TV programmes. The self is mapped and linked to cyclical changes, which are part of lived experience. The models of explanation abstracted from these observed events inevitably impact on the way in which we read and internalise the experiences we have of them, whether that be following a closed loop in space as on a Merry-go-round, or following the numbers on a clock in modulo 12. These experiences not only relate to personal experience, they also underpin wider contemporary scientific revolutions (Kuhn, 1985). From Copernicus through to Newton and beyond, understandings of the physical world have impacted on the spatial awareness of successive generations and hence the evolution of the mathematical and geometrical constructs that are conjured. The circle, or cycle, is a multi-faceted tool used in modelling many such situations. In relation to planetary movements we as humans encounter the phenomenon from many diverse perspectives that are not easy to assimilate into the organisation of our empirical experience (such as being on a fixed point on an imaginary circle on a very real earth that rotates every day, observing a moon following a closed circuit too big for us to grasp for most of the day, travelling on a route around the sun completed every 365 and a quarter days.) Qualitative features of varying intensity (Badiou, 2009a) such as light, warmth, growth of living things, length of day, time of day, position of sun as seen from earth and length of shadow, dressed and structured the students’ apprehensions of those cycles/circles. Those apprehensions were linked to personal experience, but more fundamentally to that person’s constitution in a specific location on earth’s surface. Continuous understanding or experience (of spin, rotation, of changing temperatures or degrees of light) is mapped into discrete categories, “Things are getting light so they have enough energy to grow”. Cultural apparatus was introduced to hold on to and orient the supposed ideal objects. But that immersion and mark-up that comprises so much cultural cladding now makes the objects what they are.

3.3 Conceptualising the Future How then might we conceptualise our future and the ways in which learning might be seen as participation in cultural renewal? Michael Green17 has speculated on how

17 Michael

Green is a string theorist now occupying the chair previously held by Newton and Hawking. Woit (2006) provides a contrary view.

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new frontiers of mathematics and science might be presently understood against past discoveries that had unified earlier work: The whole history of physics, for centuries, has been one of unifying things. In the 19th century electricity and magnetism were considered to be two completely disconnected phenomena and then it was realised that they were different aspects of the same thing. And that was a great breakthrough in understanding. And of course more recently, with Einstein, there was an understanding of the implication of ideas about space–time and gravity. I guess in the biological sciences things work completely differently – although, of course, a great model, the most glamorous thing you can possibly imagine happening, is the work of Crick and Watson, who had no right to believe that there was a simple elegant solution for how animals, how entire biological systems procreate – but they understood the structure of DNA, and with that understanding came along the understanding of how it all worked. And so in a completely different context, and obviously in a very different way, that’s the kind of thing that we are looking for.18

The future is being conceptualised as bringing things together in new ways. For example, on a grand scale, can theories relating to the very big be linked to theories explaining the very small? Mathematics leads and follows such frontier pursuits. On a smaller scale one might seek to make a mathematical generalisation, such as a formula that could be seen as the production of a unified object. The very act of unifying a domain, broad or local, redefines the world, as we understand it, since the world as a whole and, each of the elements within it, are understood differently within the new configuration.19 Mathematics is an infinite realm that is still expanding. It would be difficult to achieve consensus on how any limits could be drawn at any point in time. Even mathematicians refuse to reach consensus on the philosophical and social terrain, or the ontological status of mathematical objects: “Realists cannot explain how mathematical perception works, formalists cannot explain why meaningless mathematical statements apply so conveniently to physical reality, and intuitionists cannot explain why so much of classical mathematics seems reliable and coherent” (Hallward, 2003, p. 74). The meaning brought by someone to a given mathematical configuration is necessarily a function of its axioms, and it is a function of the world in which those axioms are supposed to reside.20 Any assumption of a wider terrain for mathematics provokes an adjustment to any supposed meaning of its constituent terms, new and old. Meaning is a function of how mathematics is apprehended. Indeed, the term “meaning” might be strategically avoided to emphasise that the meaning of any expression is no more than its use in language.21 People are diverse in character and any individual can be understood through a variety of social filters

18 Quoted

in Edemariam (2009, p. 34). attempt to find a common ground for all mathematics by using the set theory collapsed when Russell provided his famous barber paradox, and when Gödel proved that such an attempt is impossible. 20 Badiou (2009a). 21 Wittgenstein (1983/1958); Williams (1983). 19 Frege’s

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to produce alternative subjective modes (e.g., cognitions, subjects, bodies, reflective practitioners). They can identify or be identified with different ways of making sense of the world. It is not just a case of what you see, but from where you see it, and who you are. How might mathematical objects be understood in pedagogical situations given their changing relations with the social apparatus that locates them? Geometry and spatial perception exercises carried out by students have provided examples in this chapter of mathematical activity centred on the apprehension or production of curves. I have speculated on how notions of “circle”, as an example of a mathematical concept, are developed, transmitted and transformed through the need to traverse cultural and historical, or teacher and research, perspectives. The concept was shown to be far from stable, from place to place and time to time. Apprehensions of circles have evolved as circles have acquired so much baggage as they have been progressively used more as elements in building constructions of the world around us. And individuals access and utilise such histories differently. The students were bringing together their past knowledge from school with new observations centred in their current task of training to teach to produce new understandings. Their conceptions of mathematics are being shaped to new circumstances. Individually they are moving from being school students to trainee teachers. But in doing this they are also adapting to new collective understandings of what it is to teach mathematics in schools. In school, as pupils, they were sharing that generation’s absorption and construction of school mathematics. In college as trainee teachers they are participating in a cultural renewal of mathematics since the schools in which they will teach will collectively influence what mathematics is for the next generation. It is important to re-search, re-connect, re-learn, re-integrate, and re-cast insights in the discourse of the times. I see working on education not in terms of an edifice of knowledge, adding new theorems to old, but rather as a journey of discovery and development in which what others have learned has to be re-learned, re-integrated and re-expressed in each generation (Mason, 1994, p. 177).

There are however costs and benefits to this participation in cultural renewal. Geometry sometimes appears as a field comprising ideal objects held in place by the technologies that have been developed to provide access. And so often this area of mathematics has been converted into particular linguistic forms for accountancy purposes or formal recognition, such as in tests or examinations. This can compromise aspects of geometrical learning, in the way that Gattegno highlighted, such as where continuous experience of certain geometric forms is prematurely seen in terms of discrete categorisation. This may obscure or close off potential apprehensions of spatial phenomena, and thereby limit learning. But in another sense geometry relies on its objects for its existence. As Derrida (1994, p. 54, his emphasis) points out: “Inheritance is never a given, it is always a task . . . there is no backward looking fervour in this reminder, no traditionalist flavour, reaction, reactionary or reactive are but interpretations of the structure of inheritance”. That is, the formation of an object is a choice to see things in a particular way. “That we are heirs does not mean that we have or that we receive this or that, some inheritance

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that enriches us 1 day with this or that” (ibid.). But we are not all equally powerful, or attentive, in deciding how things should be seen. The renewal of mathematics is often in the hands of those controlling the curriculums, and wishing to exercise that control in the name of “raising standards”. The constriction of mathematical forms does not only happen in geometry of course. It is endemic in school mathematics that curriculum constraints shape mathematics to fit descriptive registers. These registers (as represented in TIMSS22 test items, school exams, or curriculum frameworks, such as those to be discussed in Chapter 4, etc.) are so often fixed to prescribe particular (reproductive or reactive) perspectives on chosen objects. The preceding pages have not been so much concerned as to whether particular teaching strategies were achieving good pedagogical results. Such a call would depend on the specific educational regime and the way in which it framed mathematical objects. The crucial issue has been to do with how mathematical objects were located in the activities described, and how those locations suggested interfaces with humans more generally. Mathematical objects are necessarily functions of symbolic or axiomatic or discursive regimes. These regimes make mathematics what it is.

22 Trends

in Mathematics and Science Study.


x

Chapter 3

Cultural Mediation of Mathematics

1 Art as Object The last chapter considered examples of mathematical objects embodied in the spatial environment. This chapter contemplates how the cultural environment confers particular images on mathematics that further shape how it is perceived, and how it evolves. To begin this discussion I want to consider the notion of objects from a different perspective towards setting the scene for a preliminary consideration of how specifically mathematical objects derive from cultural contexts in everyday life. In the first instance I take, as an example, an art object, not so much as thing in itself, but to consider its provenance as an object. Could an analogy with the apprehension of artistic objects help us to think a little differently about the apprehension and provenance of mathematical objects? How do humans relate variously to mathematical or artistic objects? How are objects situated or created in relations? I shall argue that the objectivity of the piece is a function of its social location. Damien Hirst’s piece, For the love of God, is a small, delicate object, a life-size human skull cast in platinum and encased entirely in diamonds, some 8,500 in all. It became the most expensive work of art ever created, put on the market initially for £50 m. Hirst, who was recently ranked as the most powerful individual in the contemporary art world by Art Review magazine, is reputedly worth in the region of £100 m. Such quantitative appraisal appeared widely in the press. But what might we make of the title and possible meanings of Hirst’s piece? Love is the ultimate in relationality. God is the most profound yet elusive of objects. Our relation with God, seen as money, produces death, it would seem; a skull encrusted in diamonds. And the object, as art, is fully implicated in the circulation of money as a result of its intrinsic value in those terms. Also Hirst’s own persona is a function of his financial value. The veteran art critic Robert Hughes condemned Hirst and the piece itself. Hughes saw it as being “a letdown unless you believe the unverifiable claims about its cash value”. Yet in Hirst’s terms this comment would be further stimulus to market forces and public image that sustain the operation of the work in our human psyches. We are all implicated in the same flow of money with our identities regressively crafted against our own exchange values, whilst supposing that we can humanistically transcend capitalism. Yet engagement with Hirst’s piece can make us sickeningly aware of our incapacity to do that in a meaningful way. The piece T. Brown, Mathematics Education and Subjectivity, Mathematics Education Library 51, DOI 10.1007/978-94-007-1739-8_3, C Springer Science+Business Media B.V. 2011

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3 Cultural Mediation of Mathematics

provokes us to acknowledge that our own designations, subjectivations and selfdefinitions are inescapably quantified as a function of our exchange values and of our positioning in an economically defined world. Hirst’s ultimate assertion is that the parameters of art have changed. The meaning is not so much in the object, as in the symbolic networks that assign meanings. Why did the piece receive so much attention? What can we say about the way in which this work activated public reaction? How did it transgress public conceptions of how art functions? This very attitude, asserting its own awkwardness of fit into what counts as art, activated the response that became part of the piece’s specific evolution in public conception and of art generally in its now expanded form. Through provoking public outrage, the piece secured media attention that in turn activated debate on the theme of the meaning of art, and on how we might understand ourselves differently through reflection on that art. Meaning then is produced as much through identification with the debate as with the art object itself. The objectivity of the piece results from specific modes of identification, within the supposed social parameters. Such events are merely the tip of the iceberg of wider artistic activity. The events successfully spearhead art’s impact on public debate, and hence the reception of art produced subsequently. The thrust of contemporary art has long since moved on from notions of art objects being admired by independent observers. French artist Nicolas Bourriaud’s (e.g. 1998) work on relational aesthetics provides a current example of cutting edge activity connected to contemporary theory. In 2009 he curated the Triennial exhibition at the Tate Modern in London where the art was described in the press in the following terms: open-ended, formless, and refers to something outside itself. Walking through the show is like spending a few hours aimlessly surfing the net. A seemingly endless stream of politics, porn, science fiction, history, culture and science flows past you so fast that when you leave it is hard to say where you’ve been (Dorment, 2009).

Within such contemporary art, objects are no longer the point in themselves. Rather, meaning is provoked through the interplay of events/participants, things against other things and so forth. Hitherto, comparisons of art and mathematics have often been centred in the beauty of mathematics, and in how various forms of mathematics might be seen as being aesthetically pleasing. I attended a conference session focusing on how mathematics could be seen in terms of its aesthetic qualities and how such qualities might become more prominent in our teaching of students (Sinclair, 2003). The session appeared like an art-appreciation class. Various people commented on the qualities of different mathematical proofs presented using words such as “economic”, “neat” and “elegant”. I found myself adopting a fairly guarded attitude as, so often, when mathematics is presented as being pleasing aesthetically, it draws attention to qualities missed by so many people. We hear countless stories of students struggling with mathematics where perceptions of beauty are not always readily attained. In this instance, however, the paper was responding to an earlier paper where aesthetic quality in mathematical form had been seen as an objective quality, where students

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Public Images of Mathematics

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either had aesthetic sense or they did not, where instances of their work were either elegant aesthetically or not.1 Sinclair’s paper was arguing that aesthetic awareness could be distributed differently to provide motivation for all learners.2 Such alternative modes of distribution promise an alternative to objects being seen as having intrinsic qualities. Yet, as in the examples of Hirst’s piece, or Bourriaud’s exhibition, the prominence of beauty as an attribute is now less prevalent in contemplations of contemporary art. Rather, art has a much broader scope, perhaps with a greater emphasis on providing apparatus through which we might think more imaginatively about the world towards better understanding who we are. That is not always beautiful, or even nice! Mathematics, like art, can teach us about ourselves, but not necessarily through clear and direct messages. It is met through all manner of social packaging. The next section identifies examples of mathematics in the public domain with view to pinpointing further aspects of how mathematics is located in public perception.

2 Public Images of Mathematics The image of mathematics as a private activity grappling with symbols has been disrupted in many contexts. In England, for example, mathematics, as practised in schools, has often become a regulated activity defined more by government agencies than by teachers or mathematicians. That is, mathematics is defined by its image within the regulative environment. This is especially apparent in the processing of prospective teachers for employment. My colleague Olwen McNamara captured a scene at a large teacher training college where student teachers were sitting for a government-mandated Numeracy Skills Test, required for all those entering the primary classroom.3 Quite apart from the standardised ways of portraying mathematics in the test items, the whole event could be presented as a performance that went well beyond mere mathematical concerns. There was a cast of 830 trainee teachers, a stage comprising 34 classrooms where the exam was to take place and a script provided by the government agency responsible. The directors had worked solidly for days prior to the event enlisting back stage support, planning, producing room lists, counting out scripts; preparing individualised instruction packs for the stage managers/runners; and stopping builders from building, gardeners from cutting lawns, and beer lorries from delivering. The stage-managers comprised 40 invigilators and 20 runners drawn from amongst the academic and administrative staff. A team of stagehands worked tirelessly for days setting out the props, including requisite amount of chairs and tables and ensuring security and access. The performance was very much about 1 Dreyfus

and Eisenberg (1996). also Sinclair (2006) and de Freitas (2009). 3 Brown and McNamara (2011). 2 See

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orchestrating mathematics in the eyes of the students, the government setting the test, but in the background the public audience keen to ensure that standards in schools were maintained. The event tells us a lot about the country, its mode of governance, its sense of how things get done. Underlying this social nexus is a very specific interpretation of mathematics and its pedagogy linked to some specific assumptions about the mathematical knowledge teachers must demonstrate before they can begin earning a salary as a primary teacher. Mathematics in this instance was a function of a complex social architecture, displayed through ritualised performance. School mathematics is often thus. What follows are brief accounts of four events that have a mathematical dimension: a public lecture on advanced mathematics, the publication of a popular book discussing mathematical physics, a controversy surrounding the publication of some school examination statistics, and finally an item dominating the news as I write, the on-going world financial crisis. These accounts each situate mathematics in a public domain and are chosen to exemplify different cultural manifestations of mathematics. The depictions perhaps offer alternative points of entry to better understand the public image and social situation of mathematics as a wider backdrop to school learning. This provides a route towards understanding how mathematics, inspected more locally, might enable us to learn a little about ourselves through our wider complicity in mathematics shaping the world. A public lecture in mathematics: The mathematician Caroline Series gave a number of funded public lectures in New Zealand. The publicity for the events featured intricate and colourful images reminiscent, to a novice at least, of the widely known Mandelbroit images of fractals (Fig. 3.1). The lecture, which focused mainly on the limit sets of Kleinian groups,4 was widely attended. It would be unusual for many mathematicians to command such attention. The mathematics presented was probably inaccessible to many of the people present, including myself. Yet the images were sufficient for me to warm to the associated symbols. It was as if the beauty of the images invested the symbols with a meaning that could not be deduced through the symbols themselves. The images were seemingly to be understood as a profound association between the symbolic presentation and the hidden meaning. Yet for so many, certainly for me, it appeared to be in the surface, attractive images that stood in the way of access to any depth behind. In some sense this is familiar territory as regards assumed links between mathematics and art. Geometric images such as Islamic patterns are common in classrooms.5 Here art and mathematics meet in images or objects for the physically independent human to behold. Stephen Hawking’s “The Brief History of Time”: Hawking’s (1988) book achieved remarkable public exposure despite the complexity of its themes. It 4 Mumford, 5 For

Series, and Wright (2002). example, Sharp (2002).

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Fig. 3.1 A Schottky group with 4 generating circles shown

documented the successive post-Euclidean ways in which we have apprehended the physical space around us. Along with a simplified version (2005) it made a bold attempt to convey the issues at the frontiers of mathematics and physics to a mass audience. It has “sold about one copy for every 750 men, women and children on earth” (op. cit., p. 1). This entailed a profound processing of the ideas concerned to make them comprehensible to a wider public. Yet, despite Hawking’s considerable efforts much of the material would remain inaccessible to many of these readers. To many people in his audience the book functioned on an iconic level. Hawking successfully created an image of a complex world, but one very much associated with him, through pictures in the book and the associated publicity. The cultural theorist Žižek (1997, p. 173) offered a somewhat unsympathetic reading of Hawking’s success, contending that his iconic status resulted from his disability, indicative as he saw it of the general state of subjectivity today (p. 135), where countless people for much of the time mediate their social encounters through electronic media.6 Such media sets the terms and form of communication in so many instances and mere humans are obliged to shape themselves with respect to its demands. As a phenomenon it is not the meaning of what Hawking says but the mode of performance and how the performative aspects of his delivery speak for mathematics and physics and the relationship they are perceived as having with the world by a broader public. Žižek (2001b, p. 213) argues 6 If these finger movements on a keyboard and mouse were to be seen as “gestures” this popular area of mathematics education research would extend to a rather different conception of the human subject, less centred in a physical body.

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that Hawking is a “new type of public intellectual . . . who, in the eyes of the wider public, stands more and more for the one ‘supposed to know’, trusted to reveal the keys to the great secrets which concern us all”. The (im)possibility of perfection in children’s achievement: An education initiative for secondary pupils in New Zealand was based around an assessment regime designed to include all pupils, such that everyone could supposedly achieve acceptable results. Nevertheless, some controversy led to the dismissal of a prominent school principal after 100% success had been declared for pupils in her school. It transpired that children had been assigned a wide variety of occasionally trivial tasks to enable them to make this perfect score an achievable and defensible reality. Some people felt that the principal’s tactics had been a blatant attempt to massage life to create statistics. The story was headline news in New Zealand for a number of weeks, focusing on the assessment strategies used, the dispute that led to the principal’s dismissal, and the subsequent student protests seeking to reinstate the principal. This public performance of statistical arguments and their underlying premises exposed how such arguments are used and abused to shape our world. The mathematical dimension could not have been extracted to become independent of this highly interpretive environment. The credit crunch: The “credit crunch” locates a phenomenon resulting partly from the term itself being coined, to label and hence intensify a response to particular economic circumstances. It comprises mathematics at play on a grand scale, a mathematical event par excellence, where at short notice we can learn that a major British bank has accrued debts of over 10 billion pounds, major US banks have collapsed and that entire countries have been bankrupted. There seemed to be a widespread public disbelief of the circumstances whereby such a loss of control could arise. An impression was given of an infinite number of intricate and precise calculations made meaningless by seismic sliding. Capitalist economics shapes all of our lives and rests on specific modes of accountability and quantification but within a terrain without limits or anchorage.7 A massive failure of confidence has undermined the production of mathematical certainty at a more local level. The media explanation seemed to be that people who were good at mathematics had used their skill to calculate their own bonuses, whilst hoping for chance events that were altogether too chancy. Confidence had confirmed its intangibility, and confidence intervals had delivered their outliers, refusing to set clear boundaries on certainty, where “elaborate mathematical models [had] proved to be false gods” (Ferguson, 2009, p. 14). Bootle (2009) argues that economic movements are primarily psychological, driving any mathematical models that might be introduced. Seemingly this performance affects us all, shaping our lives and our futures. The description of the mathematics test paints a picture of a country where former aspirations in which teachers and children create mathematics in classrooms are replaced. School mathematics in England at least has come to be defined by 7 Harvey

(2006, 2010).

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a government seeking to ensure that its specific conceptions of education and of mathematics are embraced by all, partly through its quest to hold its own in international comparisons of school performance.8 The signifier of test performance has become at least as important as the signified mathematical capabilities. Yet a high profile major study documents the failure of British primary schools resulting from earlier government-backed initiatives to focus on the basics of mathematics, writing and reading, to the cost of subjects like geography, history, music, art, but also of mathematics itself, located now in a more barren academic environment (Alexander, 2009). Mathematics appears to have become impoverished through such prescription. Meanwhile, there is much benefit to be gained from understanding the discipline’s aesthetic qualities and in finding ways to enable our students to share these pleasures. There are risks, however, in underwriting those pleasures as though they provide access to a more beautiful world beyond. The works of mathematicians like Series and physicists like Hawking paint a complex and distant picture of mathematics that offers few markers or tools for school children across the world looking for ways to tackle real life problems. As “they are clarifying the ultimate enigmas . . . (they) silently pass over the burning questions which actually occupy centre stage in current politico-ideological debates”.9 Mathematics enables us to inspect the world for its beauty and for its deformities. It is possibly unhelpful to demand that school mathematics be underwritten by its supposed connections to a pristinely defined advanced mathematics. School mathematics is at least in part about enabling children to learn about themselves and to express themselves, where also they begin to understand how the mathematics they learn is shaped around life and how life is shaped around mathematics. For the New Zealand principal, her very attempts to deliver the ideal held up by the government as aspirational, made the system itself, to which she was subscribing, all too faithfully, appear to lack credibility. Žižek (e.g. 1989, 2008b) argues that regimes require a cynical distance among their populace for their policies to work, a point to be discussed more in Chapter 8. Statistics no longer remain a measure of life if life is forced to fit the required image. The mathematical measure demonstrating perfection or the guidelines advising how this could be achieved showed that the conception of perfection was not altogether perfect. This dilemma was graphically mediatised in an episode of the American television series The Wire. A new teacher having finally secured the attention of a difficult class with some imaginative mathematics teaching approaches was instructed by his principal to work in a different way. The principal had in turn been instructed by her local educational authority to spend 2 months preparing for the standard tests with dreary fixed format questions. This change of strategy was supposedly to ensure the statistics that would secure the school’s continued accreditation. This echoes a problem made real in a very recent research. High school mathematics in England has

8 Department 9 Žižek

for Education (2010). (2001b, pp. 215–216).

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become so shaped around test items that its functionality in supporting university mathematics and its success in motivating children’s enjoyment of the subject, have been brought into question.10 With the financial crisis: “We did our sums wrong. We did the wrong sums” (Stronach, in press). Models crack and the world cracks. We can, however, learn from these events, but not necessarily by holding them against notions of perfection or universality, which generally have had a poor track record. Walkerdine (1988) argued that school mathematics was built on a fantasy of control. Yet, the achievement of this control also requires life to behave itself in particular ways.11 The image of mathematics and its predicative capabilities have surely been tarnished by such events. As I shall argue more fully later, mathematics in schools is shaped according to very specific understandings of “teachers”, “learners” and of “mathematics” itself. Each of the terms needs to be cropped according to specific filters for them to control their supposed content. Yet the events we now encounter are beyond the reach of the ordinary use of those terms. Indeed, in the new order, perhaps those terms no longer assume the centrality that they have until now. Insofar as mathematics is seen as equating to control technology, the credit crunch defies the certainty of those calculations. The crisis undermines the concepts of the world within which those calculations are seen to hold true. Science cannot explain everything. There is much in the world that is irrational. And whilst mathematicians may be tempted to argue that the objectivity of mathematics transcends such mundane concerns they could be challenged from two directions. Firstly, the kudos of mathematics in schools often derives from its supposed utility and functionality in control of our mundane lives. Secondly, mathematics itself is a social construction whose historically derived objects now depend on their meaning being built relationally both to other objects and to those who apprehend them. In short, no matter how objective mathematics may be, and no 10 Pampaka,

Williams, Hutcheson, Wake, Black, Davis, and Hernanadez-Martinez (in press). (2010) has discussed the role of mathematics in confronting issues of climate change. He argues that we only really know anything about climate change because of mathematics. The changes, he suggests, are at too large a scale for humans to discern them individually for the most part. We need mathematics to understand and (maybe) respond to a huge threat. I see a number of analogies between climate change and the financial crisis in terms of the demands placed on mathematics. The certainty of mathematics is always with respect to some motivated choice of axiomatic field, and often, false analogies are drawn between axiomatic fields and real life. The economy is governed by psychology and by exact calculations, and in the case of climate change there is also a psychological dimension to the reading of mathematical models that feed directly in to any mediatised account, such as the ones that Barwell presents. Populations respond to various stimuli. We are perhaps destroying our world and our children’s world, and if we really knew that we might act differently. But the stories that govern our actions are not yet quite enough to convince us that we could change. Or rather they do not convince us that we might have the capacity to act collectively in a different way. There is not an objective picture because it is only possible to understand the world in terms of how it might be possible to change it, and that requires people to change – and media plays a part in how they change. People do not typically obey mathematical models. There is a psychological dimension that needs to be integrated. For mathematics to be true we have to select the correct story in which it appears.

11 Barwell

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matter what one means by “objective”, mathematics has had a long history of people making sense of it and using it as a filter in building and making sense of their lives. This activity itself confers supplementary meaning to any mathematical form. On frequent occasions familiar strategies (to apprehend physical space, to control the economy, etc.) fail as a result of the supposed model being too limited. Mathematics, like art, has reflective and reflexive functions that help humans to affirm images of themselves. Perhaps, more importantly, those functions can disrupt these images to enable growth. These functions are encountered through objects, rituals and other events. Learning mathematics is intricately tied up with the architecture of emotional and intellectual space. But societal relations define the very contents of individual brains and perhaps individuals cannot see themselves outside of those parameters. Yet those are the parameters that govern and explain the individual’s actions. Mathematics can be part of the kit bag of resources that might enable us to better understand how those parameters work in a terrain that is rather unpredictable.


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Chapter 4

Teachers’ Conceptions of Curriculum

1 Introduction In Chapter 2 we considered mathematics in the spatial environment. In Chapter 3 we have seen some instances of mathematics in cultural media. This began our discussion of how mathematics is a function of its location, how it is part of us, and how it is built into the spaces we occupy. We now turn to the context of schools where curriculum frameworks filter teacher perceptions of their task of teaching mathematics. Mathematical learning in schools cannot be understood fully in terms of individual students encountering mathematical objects. Those objects are formed across a much broader context, and can be understood in many different ways. And “the meanings circulating in the classroom cannot be confined to the interactive dimension that takes place in the class itself; rather they have to be conceptualised according to the context of the historical–cultural dimension”.1 Mathematical objects in a school context are typically defined in relation to a curriculum that prescribes roles for students and teachers. The actions of teachers and students are so often designed, recognised and assessed according to how they conform to those definitions. This pedagogical housing of mathematics influences the objects that are studied.2 The housing sets the conditions for learning and the resulting apprehension of mathematics. In raising such issues, I find myself moving to a very different author voice to the one introduced in the previous chapters. I am also drawn to very different sources of literature. Local administrations often handle school mathematics in very different ways to conceptualisations more commonly found in international mathematics education research. Yet, these administrative and pedagogical layers have many facets, which reshape mathematics to facilitate learning in particular settings where:

(with Una Hanley) 1 Radford 2 Brown

(2006b, p. 23). and McNamara (2011).


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• assessment regimes insist that student performance is recorded on a pass/fail basis; • compulsory education fixes choices, such as class size, resource allocation, styles of teaching; • access to mathematics is differentiated according to social group (e.g. Uganda/Sweden, private/state schools, political preferences); • primary education is handled by generalist teachers who may be insecure or unknowledgeable about mathematics, which affects how students encounter the subject, perhaps through overly didactic approaches with reduced scope for exploration3 ; • international curriculum criterion is applied in specific local contexts (I visited a school in a rural tea plantation in Sri Lanka where education was specified not in terms of local need but in such a way that successful students left the area for work in the capital city); • the resistance of adolescents to adult guidance results in teaching styles shaped primarily by the needs of classroom management; • the teacher’s version of events is valued more than the student’s. More generally, teacher capabilities are dependent on a broad range of factors. Setting policy to bring about widespread adjustment to teacher practices towards raising “standards”, or national test scores, is a persistent aspiration, so often disappointed.4 In England, for example, policy makers5 do not work to a consistent agenda in governing school mathematics, and other stakeholders, such as, advisory groups,6 regulators,7 trainers, research and development funding agencies8 and potential employers/universities/colleges, work according to a variety of perspectives and priorities. Curriculum decisions are thus split and shared between these various groups, which do not necessarily see eye to eye resulting in potential disjunctions between policy setting, implementation by teachers and the conceptualisations made of such implementations by researchers.9 At the same time there are significant difficulties in influencing the professional learning of educational researchers themselves towards changing the practices of educational research.10 And so much research effort is dissipated across countless small studies that defy a coherent picture being

3 Brown

and McNamara (2011). Day, Kington, Gu, Stobart, and Smees (2007). 5 For example, Department for Education and Skills, Teacher Development Agency, Qualifications and Curriculum Development Agency. 6 For example, Advisory Committee on Mathematics Education, National Centre for Excellence in the Teaching of Mathematics. 7 For example, Office for Standards in Education, examination boards. 8 For example, Economic and Social Research Council, Leverhulme, Gatsby. 9 Saunders (2007); Whitty (2007). 10 Rees, Baron, Boyask, and Taylor (2007). 4 Sammons,

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created more generally. As a consequence the theoretical underpinning of such processes has been somewhat fragmentary, sometimes switching between cognitive psychologies at the level of the individual student learning mathematics, to an array of policy sciences and budgetary led political expediency at the macro level. These various areas of work each have their own specialists who rarely meet with specialists from other areas to swop notes. How then do such states of affairs set the scene for developing mathematics education, theoretically and practically? The complexities just described can be variously conceptualised, perhaps as part of a wider attempt to depict “how things are” towards considering “how things could be” following a research-informed development initiative. Classroom interaction, policy dynamics and curriculum reform can each be shaped through particular lenses. These lenses can enable or restrict analytical and strategic opportunities. In turn these lenses produce conceptions of teacher development and quality that can or cannot be sustained with the actual or potential supply of teachers and other resources. Much research in the field of mathematics education is targeted at individual teachers or teacher educators, from the perspective of how they might adjust their individual practices with students, whilst an array of policy interventions split between diverse stakeholders operates in the wider domain. Might alternative perspectives, or points of leverage, offer more effective models of curriculum change? How might we conceptualise mathematics education research impacting on populations of teachers through affecting policy decisions? Research is often predicated on identifying deficiencies in current practices as part of a rationale for implementing a new approach. Hargreaves (1996, p. 5) has suggested that educational research must demonstrate “conclusively that if teachers change their practice from x to y there will be a significant and enduring improvement in teaching and learning”. Hence, a history of research might be characterised as a series of projects, papers and books, with many arguing the case for some sort of improvement against various priorities. Yet, looking back at any one time, it is not easy to argue how we might assess retrospectively the nature of this cumulative improvement over any given period of time. It is quite difficult to provide evidence of improvement except in narrow terms. With the introduction of any new initiative there comes an implicit assumption that it will bring improvement over the previous regime. Priorities, however, are not always consensual and evaluation strategies change over time. Alternative versions of history craft their heroes, objects and time phases differently. The term “improvement” can be understood in many different ways and resists stability across time, space and circumstances. Meanwhile, teacher biographies are typically characterised by engagements with a number of teaching approaches throughout any one career. Each shift from one to another entails mathematics being framed in a slightly different way that perhaps results in a different teaching style and, perhaps also, in a different conception of mathematics. Elements derived from each phase feed into composite experience and contribute to that teacher’s modes of practice and emergent, and perhaps convergent, professional identity. These elements might be attributed variously to fashions in school practices, learning theories, assessment preferences, career phase of the

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individual teacher, etc. The shifts in teaching approach would normally be locally negotiated on the basis of some supposed improvement on the previous model. There are, however, a number of studies11 that describe curricular innovations and the (modest) benefits that have followed. More specifically, teachers’ orientations to teaching mathematics are identified as influential factors in mediating the strength of any initiative’s effect. Evaluations of initiatives, for example, have referred to teachers’ deep-seated beliefs, which are left largely unchallenged12 such that structural changes can be “bolted on” to existing practice.13 Similarly, any such development will have multiple interpretations made of its impact.14 Local context also appears to influence the nature of classroom implementation. Specifically, this could be teachers’ perceptions of their school’s priorities within the wider education system,15 or their beliefs about pupils’ needs beyond mathematics.16 Others provide models that illuminate shifts in teachers’ thinking and practice as a recursive interchange between beliefs and how they talk about teaching and experimentation.17 One specific aspect that has drawn attention is the quality and variation of interactions observed in whole class teaching.18 Still other pertinent and diluting factors include teachers’ lack of detailed awareness of how they operate19 and for how long they have been teaching. Inexperienced teachers are more likely to engage fully with new curricula,20 whilst experienced teachers can find it very difficult to alter practice,21 as personal beliefs about practice are based on what they see as compelling evidence derived from daily classroom experience.22 Asking teachers to move from one teaching approach to another can, it seems, never be regarded as a straightforward substitution.23 Nevertheless, for those charged with setting policy, there is often a perceived obligation to do something. Often this involves doing something big. In the United Kingdom, New Zealand and Australia, for example, governments have prescribed detailed curricula for students and teachers alike, along with associated industries concerned with preparing materials.24 In terms of research literature, however, support for these moves was slight.

11 Swan

(2000); Hickey, Allison, and Pellegrino (2001). Askew, Millett, and Rhodes (2003). 13 Galton (2002). 14 Brown, Millet, Bibby, and Johnson (2000). 15 Kynigos and Argyris (2004); Ng, Lopez-Real, and Rao (2003). 16 Sztajn (2003). 17 Senger (1999); Skott (2001). 18 Brown et al. (2003); Burns and Myhill (2004). 19 Sahin, Bullock, and Stables (2002); Torrance and Pryor (2001). 20 Remillard and Bryans (2004). 21 Romberg (1997). 22 Handal and Herrington (2003). 23 cf. Fullan (2001). 24 Brown and McNamara ( 2001). 12 Brown,

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Conceptualising Curriculum Reform

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Considerably more information is readily available about the effect of major curriculum reform in the United States, where there is also a considerable emphasis on the widespread adoption of new curriculum materials as a primary strategy for improving mathematical education.25 And, this research has influenced the parameters through which we understand curriculum reform and related research more generally. This chapter considers how conceptualisations of mathematical learning emerge through alternative curriculum models and development initiatives. It offers examples of teachers making sense of their practices adjusted in line with new descriptive lenses. It explores how teachers identify with successive curriculum models and how these identifications frame mathematical learning. The chapter argues that within any curriculum implementation, both the teacher’s sense of what they are doing and the curriculum itself is reconstituted through the encounter, thwarting any supposed convergence to an endpoint. This argument has implications for how we think about initiatives designed to work at creating consensus in teaching approaches. In particular, we need to question how research agenda encourage teachers to align with a particular model or philosophy of practice conceptualised in advance. By way of introduction, the chapter briefly considers how curriculums might be understood as providing ideological bases to teacher practice. Aspects of the American and British systems and related research are compared. A much more extensive account of the British system has been provided elsewhere.26 Secondly, the chapter focuses on a group of secondary mathematics teachers participating in a small British teacher development initiative that sought to understand better how teachers, schooled in differing sets of philosophies and practices, respond to a new, alternative model. Data presented are analysed as evidence of the teachers’ “identifications” with new paradigms, and how these identifications are made through specific traits of that paradigm.

2 Conceptualising Curriculum Reform Let us think about the common assumption that research is about encouraging movement towards some improved conception of teaching. Let us begin by very briefly considering, as an example, some of the debates taking place around the US reform in relation to how this discussion influences curriculum change and research literature, sometimes beyond the boundaries of the United States. The US reform has shaped research and practice domains in particular ways with commensurate conceptions of improvement and gained prominence in the highest official quarters.27 I know very little about this subject except as a casual observer of its impact on the international research literature, where it has been widely reported. I am interested 25 Remillard

(2005). and McNamara (2011). 27 US Department of Education (2008). 26 Brown

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in it primarily from the point of view of how it has leaked into international discussion, exporting from the United States’ particular culturally specific conceptions of how issues in mathematics teaching and learning might be understood. This “reform” is defined in relation to the guidelines of the US National Council of Teachers of Mathematics and is, for many teachers, seen as the transition from a transmission to a constructivist pedagogical approach.28 Constructivism, which dominated international mathematics education research for some two decades,29 has been characterised as “genuine mathematical problems for students to solve”,30 with a focus on “conceptual understanding”.31 Research in the area has sometimes been conceptualised as tracking progress towards some improved state of affairs.32 Many other studies focusing on how teachers respond to curriculum changes centre their analysis on individuals shaping their practice in response to the perceived reform agenda.33 Many of the authors positively identify and readily subscribe to this agenda. Some, however, note some resistance in some quarters, such as “veteran” or “traditional” teachers who are unable to shift so fundamentally in terms of their beliefs as to what it is to be a teacher.34 Such reform, however, does not offer a trajectory with universal appeal or applicability. There are widespread disputes within the United States itself, centred on debates that have come to be known as the “math wars”. The “inquiry” methods associated with constructivist reform, characterised by greater learner and teacher autonomy directed at conceptual understanding, have been resisted by more traditional teachers who prefer an emphasis on computational skills. The inquiry methods would also be less acceptable in many Eastern or Pacific cultures where curricula, teacher/student roles and the collective good are defined differently.35 Further, the alleged autonomy understood within the “reform” agenda conflicts with the reality teachers have come to accept in many countries, assessed as they are through legislative documentation and recognised through the filter of their compliance with this. In England, for example, student-centred pedagogies emphasising problem solving, investigations and project work dominated curriculum reform agendas some 30 years ago. A later backlash resulted in prescribed curricula for both teachers and students in which student-centred approaches became tightly structured.36 Reasons cited for this backlash included right wing politicians37 claiming that given

28 Fennema

and Nelson (1997). and Kieran (1994); Brown (2001, pp. 7–21). 30 Lloyd (1999, p. 228). 31 Wilson and Goldenberg (1998, p. 269). 32 Simon and Tzur (1999); Tzur, Simon, Heinz, and Kinsel (2001). 33 Remillard and Geist (2002); van Zoest and Bohl (2002). 34 Cohen (1990); Wilson and Goldenberg (1998); Lloyd (1999). 35 Brown, Devine, Leslie, Paiti, Sila’ila’i, Umaki and Williams (2006). 36 Brown and McNamara (2011). 37 Such as Kenneth Clarke, a Conservative Minister of Education speaking in 1996. 29 Steffe

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difficulties with teacher supply the average teacher could not teach to such highminded ideals. Left wing commentators, meanwhile, argued that aspirations to child-centred approaches merely replaced overt regulation with a form of covert regulation.38 Conceptions of improvement can be very localised. It is this concern with the specificities of local circumstances that underpins this present chapter’s focus on adjustments to new paradigms. In particular, it is unhelpful to suppose that we could identify trajectories of improvement that would apply across all people and all phases of development. “Improvement” is not a universal term. There is always a cost in the form of local preferences being suppressed resulting from one-sizefits-all suppositions. Further, such consensual aspirations and alignment with them are not always so easy to grasp through the immediacy of everyday practice. Consequently, there is a need to build a theoretical frame that permits alternatives to consensual aspiration. Recent neo-Marxist theory has questioned notions of human progress being shaped by ideals relevant across all communities.39 This is hardly a radical proposal. Mathematics education, for example, would be seen as culturally dependent with each country predicating alternative conceptions of mathematics. For example, international comparative testing has resulted in many countries aiming at forms of mathematics and the styles of teaching commensurate with those forms of assessment. So, in the name of international conformity, tea pickers in Sri Lanka do not get an education suited to their local needs and the United Kingdom sacrifices its relative facility with problem-solving approaches.40 Laclau (2005) has rejected the notion of the “people” as a collective actor, and, by extension, the possibility of a research “community”, or a set of governments, being able to define a common interest with regard to the purposes of school mathematics. For example, to what extent is it possible for the mathematics education research community to assume some consensus in its purposes? Exam results, facility with mathematics and enjoyment of mathematics do not always pull in the same direction.41 Instead, Laclau has examined the nature and logics of the formation of collective identities and suggested that such collectives can be seen as being held together through identifications with specific populist aspirations (raising standards, making children happier, supporting the economy, building richer mathematical experiences, etc.). Althusser (1971) meanwhile has focused on how the individual understands herself through ideology. Here an ideology is understood as a specific conception of life, a particular version of common sense.42 We always occupy an ideologically 38 Walkerdine

(1984). (2005). 40 England succeeded in moving from 18th to 7th position on TIMSS in 2007, whilst dropping in its rankings from 8th to 25th on the more problem focused PISA in 2006. 41 Pampaka, Williams, Hutcheson, Wake, Black, Davis, and Hernanadez-Martinez (in press). 42 Here ideology is seen as common sense as experienced by someone within that ideology. One can only inspect an ideology from the perspective of another ideology, “we are ‘naturally’ in ideology, our natural sight is ideological” (Žižek, 2008c, p. xiii). And generally people, including this author 39 Mouffe

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derived position. We never have the luxury of speaking from outside of an ideology. Althusser described schools as an instrument within the “ideological state apparatus”. Here schools are seen as a device through which the preferred ways of the state are disseminated with general consent. For many pupils and their parents progression through school is an ideological movement to which they are readily mobilised. Althusser is not persuaded by consensual aspirations where difficulties are ironed out. He sees the supposition that you could get to a consensual ideal beyond conflicting ideologies as the biggest ideology of all. The individual may recognise herself in some ideologies but not others. But, there is always a gap in this identification, a distance between the person and the story in which she sees herself. This gap stays there. For example, some American teachers may truly believe that they are subscribing to Reform agenda and following such approaches in their practice, whether or not others see it this way.43 But, at the same time, some other American teachers may be sceptical about reform projecting them higher up the international league tables, and they may not always agree with the content of that ambition. Yet, both groups find their working practices defined and evaluated in that way securing compliance at a practical level. In such circumstances, we may ask whether it is appropriate to celebrate any supposed “improvements” in the quality of mathematical learning. Perhaps, instead, such “improvements” may simply be indicative of success in the administration’s project of convincing the public that the administration’s understanding of mathematics is the correct one, and that the content of standardised tests define what mathematics is. The administrative authorities are measuring a version of mathematics that lends itself to being measured in a particular way, rather than according to universally accepted criteria. Standards have not been raised as such. Rather, the criteria through which standards are understood have been changed. Certain approaches to teaching work better in given circumstances. Each successive model rests on teacher memories of the ones that preceded them. The models cannot be gauged according to a supposed underlying truth of practice. The individual’s immersion in successive ideologies of practice might be understood as a task of taking bits from a range of models. Individuals are unlikely to be fully aware of how they draw on a range of alternatives, since their own identities as teachers derive from a variety of sources. The various ideologies are crafted together into a functional whole in some more or less personal way. Given a history of influences it seems unlikely that individuals could distance themselves and detect the experience of being immersed in one distinct ideology rather than in another. The US “reform” functions as an ideology, in Althusser’s sense, a specific version of common sense, insofar as it has set key parameters shaping discussion relating to

and other authors in the field of mathematics education, are unable to detect the precise limits of their own ideology (e.g. Žižek, 2008b, pp. 181–193) even if ideologies could lend themselves to more compact definition. 43 Cohen (1990).

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curriculum innovation. It defines a professional space governed by certain assumptions as to how improvement might be achieved. The limitations of its validity remain peripheral to this definition. In so many instances of mathematics education research, “reform” functions as a supposed consensual aspiration. Further, even within that culture: “Based on their concepts of students’ needs, teachers select which parts of the reform documents are appropriate for their students”, which translates as “children from upper socioeconomic backgrounds get problem solving, those from lower socioeconomic backgrounds undergo rote learning”.44 Viewed internationally, however, even such variation might be regarded as modest, framed as it is within the parameters of national boundaries, compulsory schooling infrastructure, economic status and a host of other societal assumptions. So there is a need to be cautious in observing how such assumptions have slipped into the apparatus for curriculum reform research more generally. Yet, affinity with an ideology does not necessarily fix the mode of association, or how that is viewed. Remillard (2005, pp. 215–223) examined alternative ways in which teacher/curriculum interfaces are understood within the research literature. She contrasted “following or subverting” a curriculum text with “drawing on” a curriculum text or “interpreting” a curriculum text. In these three alternatives, the text is present in some form and teachers respond to it. Finally, however, Remillard considered how curricula might be understood as teachers participating with the text. For a teacher “enacting” a curriculum in this mode, she suggested that teacher and curriculum might be seen as mutually constitutive. Here, curriculum use is understood as participation with the text (pp. 221–223). She identifies this with “Vygotskian notions of tool use and mediation, wherein all human activity involves mediated action or the use of tools by human agents to interact with one another and the world”.45 Such an approach is familiar within mathematics education research.46 But how might we unfold the features of this mutual constitution of teachers and curricula? Understood in terms of Foucault’s (1989) notion of “discursive formation”, both teacher and curriculum would be functions of how they are implicated in the stories that unite them. Both change as a result of curriculum development activity. Remillard (ibid.) identifies some studies where teachers change or learn from their use of resources.47 Yet, teacher change can also be understood as being the result of increased compliance with respect to a curriculum initiative. A core assumption of this book is that aspirations to consensus can suppress the specificities of alternative needs, responses, etc, and thereby serve those who are already the most powerful. Elsewhere, I provided a book length account of how trainee and new teachers in the United Kingdom begin to include official curriculum descriptors into accounts of

44 Sztajn

(2003, p. 53). Cole (1996). 46 Lerman (2000); Blanton, Westbrook, and Carter (2005); Goos (2005). 47 Lloyd (1999); Remillard (2000); van Zoest and Bohl (2002). 45 Cf.

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their own practices as they move through the accreditation process.48 This presented an account of a national policy initiative in England in which all primary level teachers were obliged to work according to a centralised curriculum with high degrees of specification. The study, premised on the perspectives of the trainee/new teachers themselves, sought to better understand how the policy framework was articulated through the teachers’ accounts of their own practices and how such regulation was internalised.49 Here it was the government who determined the constitution of mathematics within a legislated curriculum, thus creating a very different form of regulation centred on discursive compliance. In this model, individuals were defined relationally (e.g. teacher/student) through the discursive order, rather than by their supposed individualistic qualities. In this way, the study sought to show how teachers mediated the policy framework, with a view to examining how policies might be adjusted towards achieving alternative effects. That is, the study demonstrated how teachers were subject to the policy framework and the terminology it employed. Their validity, professionalism and identities as teachers were understood through the filter of their compliance with this regime. Indeed, in Althusser’s terminology, there are no identities, just identifications with particular ideological ways of making sense of the world. The study sought to understand how mathematics was shaped and regulated by this policy initiative. It also sought to show how teachers and their students identified with its apparatus, and shaped their actions, and moulded their sense of self as a consequence. Conceptions of self are governed by a tussle between personal aspirations and meeting external demands. The task of socialisation often entails the teacher gradually introducing social/official language into her self-descriptions. She becomes increasingly implicated in official accounts of her practice as she begins to recognise herself in such accounts and to describe herself in those terms.50 And, in so doing, she loses aspects of her earlier conception of self, perhaps developed through school and family relations. Teachers interviewed as part of the above study saw this as necessary from the point of view of their accreditation as teachers. Yet, they found the discourse highly prescriptive, albeit a form of prescription that released them from the need to make so many content decisions in a curriculum area where often they had in the past lacked confidence in their own capabilities. The research perspective offered in this chapter similarly attempts to weigh up the relative advantages of achieving personal aspirations or fulfilling external demands, rather than supposing external demand is to be favoured. Although individuals may, correctly or incorrectly, see their actions as being supportive of

48 Brown

and McNamara (2011). (1997). 50 A detailed example of a senior teacher becoming absorbed in policy discourses will be provided in the final part of Chapter 8. 49 Butler

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some collective enterprise as defined externally, for example, conforming to the US reform, or following English National Numeracy Strategy51 correctly. The curriculum is not ready constituted. The constitution is contingent on teacher participation. The curriculum is not a thing in itself, since the teacher can fix how it works. Curriculum development is not just about teachers aligning themselves or not with an overarching rationale or model such as reform. Alignment by a teacher with a new curriculum is not in itself necessarily to be viewed as success, since improvements are a function of the ideological stance implicit to the curriculum. Yet, both teacher and curriculum change through any curriculum initiative, as do the parameters through which those changes are understood.52 In the next section a curriculum initiative and associated research projects are described in an attempt to give some modest empirical flesh to these issues. Some accounts of teachers reporting on their experience of participating in the initiative are described. The purpose here is not to establish the specifics of how the curriculum has been implemented but rather to better understand how both the teachers and curriculum have been transformed as a result of the initiative. It is suggested that the curriculum is understood and enacted differently according to the teachers’ previous experience.

3 Researching Curriculum Change 3.1 Mathematics in Context Since the advent of international comparisons governments have been jockeying for a better position in the resulting league tables. Within Europe, the Netherlands achieved notable results and this country’s approach was seen in some quarters as providing an achievable model upon which other Western countries could base their aspirations.53 For a number of years some schools in the country have pursued practices based on Freudenthal’s54 Realistic Mathematics Education (RME). In his view, when constructing a school mathematics curriculum, educators need to be aware of two basic principles. Firstly, mathematics needs to be seen as connected to the learners’ world. Realistic refers to problems, which are meaningful or imaginable to the learner. The contexts are there to help them make sense of the work, and to keep it close to their reality, though contexts are chosen initially for mathematical rather than social reasons. Secondly, mathematics needs to be seen as a human activity and central to this is the idea of guided reinvention.55 This approach privileges the 51 A

major curriculum initiative introduced in all British state schools, later called the National Numeracy Framework. 52 Schön (2003). 53 de Lange (1996). 54 Freudenthal (e.g. 1991). 55 Gravemeijer (1994).

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informal strategies that students bring and learners make small steps in a context which guides learning toward more formal perspectives. The Freudenthal Institute and the University of Wisconsin collaborated in producing materials in English, based on RME principles. The resulting scheme, “Mathematics in Context” (MiC), was created as a series of textbooks to support the introduction of RME principles into a pilot curriculum in Wisconsin. A major initiative based at Manchester Metropolitan University trialled these materials in British schools.56 Una Hanley was involved in an associated project57 that tracked the shifting perceptions of the teachers in the larger study. It asked how the teachers experienced their exposure to a new paradigm within their existing conceptions of practice. It followed the teachers through their first year of participation, and documented changes to the ways in which they accounted for their practice with reference to old and new paradigms. Prior to the introduction of the new scheme, a significant number of teachers in the sample had been introduced to a “Whole Class Interactive” approach to teaching.58 Here the teacher sought to maximise opportunities for students to contribute to an evolving group account of the mathematical situation being considered. Having posed a question or presented a situation to the class the teacher then sought to ensure that a significant number of children were included as she sought to help the children to craft a composite story out of the diverse comments that they offered, perhaps moving into areas not anticipated by the teacher. This discursive generation provided children and teacher with a frame within which they could begin to share ways of talking in relation to mathematical stimuli. Here, rich discussion was seen as evidence of shared mathematical construction and so a common objective for teachers working within this approach was to work toward extended conversations. Whilst familiar mathematical concepts would be touched on within such discussion, the teacher sought to promote the students’ own mathematical constructions. That is, the students were encouraged to encapsulate the mathematical phenomena in their own terms. This provided the basis for later translation into more conventional terminology. Such an approach had been influential for some of the teachers involved and shaped their practice in significant ways. Yet, other teachers involved were unfamiliar with this approach. Further, in the period that the study covers, a government-sponsored National Numeracy Strategy that had earlier been introduced in to primary schools was being extended into the secondary curriculum. This framework featured overtly programmatic aspects that entailed crafting all lessons to a prescribed format. Whilst discussion featured as a key component of these lessons, this discussion was contained within particular parts of the lesson and was targeted at specific elements of the wider curriculum. There was some potential conflict for teachers pursuing Whole Class

56 Eade

and Dickinson (2004–2007). project was funded in the following year by the UK government’s Economic and Social Research Council (Torrance & Hanley, 2005–2006). 58 Harrington (1998). 57 This

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Interactive Teaching objectives as they sought to participate within this emerging frame. But, more generally, the implementation of this framework had not been embraced with consistent enthusiasm across the teaching force. This resulted in uneven early implementation. This need to reconcile alternative paradigms was intensified as schools in the study began their participation in the current project. At least three paradigmatic understandings of curriculum innovation were represented: the National Numeracy Strategy, an administrative framework designed to enable teachers to introduce mathematical concepts as defined in the National Curriculum59 ; Whole Class Interactive Teaching, a discursive teaching style derived from the diagnostic teaching approach60 ; Realistic Mathematics Education, a philosophy of learning. These understandings were not readily commensurate, although there was some productivity in thinking them against each other.

3.2 Methods Teachers, moving from one paradigm to another, are considered here through looking at how certain key terms oriented their conceptions of the schemes during this transition. The teachers were drawn from the sample of six secondary schools using MiC materials with at least two classes of children aged 11–12. Each of the four teachers was from a different school, these varying in terms of pupils being either of mixed or of single sex, and in their ethnic, social and religious background. In addition, the teachers attended regular training days related to the materials at 6 weekly intervals. Colleagues from both the Freudenthal Institute and the University of Wisconsin participated on two of these days and visited project schools. The intention was that all participating teachers should be interviewed at the beginning and at the end of the first year of the project, 2004/2005. Eight teachers attended both interviews. Each interview lasted approximately half an hour. The interviews were semi-structured in form, and were designed to ascertain teacher beliefs with regard to teaching but, more particularly, how these beliefs evolved as a result of participation in the initiative. They focused on how the teachers understood the key components of their own teaching, the skills that they thought were important and how these shifted as MiC61 was introduced. The resulting transcripts were analysed by two researchers and two issues were found to arise consistently as themes that marked shifts in practice. As will become clear in the following excerpts, the ways in which the two particular words, discussion and context, were included provided good illustrations of teachers re-positioning themselves. These words were pivotal in each of the old

59 Department

for Education and Employment (2001). (1993). 61 In the discussion that follows, the term MiC will be used to represent both the RME philosophy and the associated classroom materials. 60 Bell

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and new teaching approaches being considered. Teacher objectives could be understood in relation to how such discussion was crafted. Yet, such crafting needed to be adjusted when MiC objectives were introduced, where specific features of discussion came to be seen as more significant. Meanwhile, the prominence of “context” as a key word in MiC resulted in mathematical ideas being processed through alternative pedagogical devices. Whereas, for teachers trained in other paradigms, both words became prominent tools in re-describing practices. Specific quotes were selected that seemed to capture most clearly the essence of what the teachers were trying to say; overall their transcripts showed high internal consistency. More particularly, however, the analysis here is directed at showing how the two key terms were used in distinguishing aspects of shifting practices. The intention is to examine how the teachers understood their broader practices in relation to their deployment of alternative paradigms through their use of these terms and surrounding material. It was assumed that there would be some gap between the teachers’ practices and how they described them as well as some gap between each of these and alignment with MiC practices. Accordingly, we sought to ascertain how much we could say about these gaps as a result of interrogating the teacher accounts of their practice and of the alternative paradigms.

3.3 Discussion and Context “Discussion”. Teachers’ perception of “discussion” shifted as they participated in the study and sought to be governed more by their understanding of MiC principles. Jennifer worked in a school where the predominant pedagogy was centred in a Whole Class Interactive style of teaching, with the government’s National Numeracy Strategy more in the background. Looking back, she described her teaching before her involvement in MiC: “Before I was quite happy to discuss all lessons. . .. There was just one question in the class and we worked towards that one. . .. We tried to develop skills in prolonging discussion and some of us got quite good at it”. (Interview 1) Jennifer began to realise that her discursive style was not a great fit with the newly introduced MiC scheme. As she tried to accommodate aspects of this new approach, she experienced some discomfort: I’m torn with: Are we now rushing through the materials instead of discussing? . . . What’s hard now is stopping the discussion. Or deciding what’s worth sharing as a class and what’s ok just to leave to discuss in pairs. . . Last lesson I started wondering who was the discussion for. Is it just that I need to be involved? Is it for the few who know what’s going on? Is it to give ideas to others who are stuck? And do we need a long discussion all the time? (Interview 1)

This discomfort continued throughout the year, which seemed to imply difficulties in being able to process her practice through the two paradigms simultaneously: “I found myself wondering: What are we discussing? Are we just discussing the answer or are we discussing the strategies. . . Before I was quite clear what I was discussing and now I’m doing this. I’m not sure any more”. (Interview 1)

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Jonathan had also been involved with the local development of Whole Class Interactive Teaching before embarking on the project: I think what we had been doing was we’d been getting a good question, getting the kids to work on it for a while and then having a long discussion and hearing lots of different viewpoints and then trying to come to some sort of conclusion about which is the most efficient method. . . (This) would take the lesson, . . . maybe lesson and homework. (Interview 1)

Equally, Jonathan has found himself having to make similar adjustments to his teaching, so as to accommodate the MiC scheme, though his inner tension in doing so appears to be less evident than it was for Jennifer: I’ve started to try and make it snappier and pacier and you know cut off a discussion when I wouldn’t normally cut it off. . . I’m kind of thinking that a lot of the important and most useful comments come out at the beginning anyway and sometimes when a discussion drags on. It’s other people repeating what’s already been said or making refinements that actually don’t improve the overall understanding. (Interview 1)

Later on, Jonathan seems to have become comfortable with this change: I’ve stopped feeling the need for closure on topics. I think. . . at the very early stages it was like walking through every question and . . . having to have everyone happy with the answer before you’ve moved on and I’m very much less bothered by that now and I think the class are more used to doing that. (Interview 2)

Lucy entered the project from a rather different starting point. In her background “normal classrooms” had assumed a framework where discussion seemed to fit around the edges of more individually focused activity triggered by some teacher input: In normal classrooms you have your introduction where you kind of bounce ideas and then the kids kind of get on with it, but I don’t encourage them to be working with each other as much because obviously I don’t think the focus is on too much paired working and group working. . . I have got to see how that individual is doing so she has to work on her own. (Interview 2)

Clearly for Lucy the journey has been in a different direction. Discussion had become a vehicle for pupils’ learning and in a way that both surprised and delighted her: MiC . . . is all the time (asking), “how?” or “why?” or “where did that come from?” So my questioning has changed and my taking answers, leaving them there for a few minutes . . . allowing them to get it wrong . . . They bounce so many ideas off each other so much and they help each other so much. (Interview 2)

Michael, in describing his teaching before the project, made no reference at all to discussion as such: “I have questions that I’m going to model and I expect two or three to be done. Then we’ll go through the answers . . . It’s just . . . is the word didactic, where I would say “this is the way, and this is what I want you to do”. (Interview 1) However, once he has started using the MiC materials, discussion has clearly become a significant feature in these classes and in a way that is very comfortable:

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I’ll have moved around the room when they’re doing their questions and I’ll see particular things that I want to talk about and I will ask those kids particularly to give me their answers . . . I’d say just take what kids say and leave them hanging there. And if no one comes up with a contradiction or a problem with what is offered, then just leave it and move on and don’t bother too much about being the font of all knowledge. (Interview 2)

Thus, we have four teachers providing thoughts on how the role of discussion changed as a result of encountering MiC; two who were seeking to regulate the extensive discussion that they had previously promoted, and two who were interpreting MiC as an approach where discussion would figure more prominently than before. “Context”. The four teachers provide comments that reveal how the notion of “context” changed in the same period. Jennifer was rather less forthcoming about context, since her base point in Whole Class Interactive Teaching resulted in her evaluating her own lessons primarily from the point of view of how discussion was generated. Yet, in the second interview, she made the following comment: “I’m trying to take them back to the context when it feels you know that they are not accessing what’s happening. . . Developing mathematical ideas through contexts that are real to kids, not necessarily real life but something they can relate to so that that is an entrance to a problem”. (Interview 2)

Jonathan, looking back, thinks that before the project: The talk that would have been going on in a non-MiC classroom might be more related to “Do you just do that to the top and multiply by this and divide by that? And that’s how you do them”. So even if . . . you had actually started the topic with a context, the discussion would be a discussion of the maths out of the context. (Interview 2)

Whereas, after several months using the MiC materials, Jonathan can see that context could have a different function: The context thing is the thing that stands out most; that is they learn more with maths in a context that they can get their teeth into it. They can get a grip on it, rather than just working with pure numbers . . . We’re still working with maths when we’re doing MiC but I think . . . that the context and the relevance of it to them makes it easier for them to engage in it. (Interview 2)

During the project Lucy became aware how much of her previous teaching had centred on algorithms and how these perhaps did not lead to enduring learning. Here she reflects on how she taught the area of a rectangle: Right well you can see that there’s so many squares on that row, and there’s so many rows, and therefore the area is such and such . . . Then you’d clarify it. Then you’d write the formula on the board and then we’d practice various different problems using that formula. . . It kind of just brought me back to thinking they’ve heard this, but they’ve not understood it, and therefore now they’re using it incorrectly, which then made me think, maybe we shouldn’t be giving these algorithms. (Interview 1)

Equally, pupils did not find the algorithms useful: “They’d come in ‘but how does this relate to, you know, what’s this got to do with my life?’” (Interview 1) In the second interview, Lucy saw that MiC was offering an alternative: The context – if the kids can grab that context they can understand what it is starting to talk about. They can then begin to understand how to solve it . . . Then once it got to the stacking

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cups, even though it got more difficult, the kids could still access it and begin to solve it in their own way. (Interview 2)

Michael expressed a degree of dissatisfaction with his usual teaching from very early in the project. This centred on his attempts to work in the abstract: I usually start with a context and work within one but then unfortunately the resources that I have to hand such as texts that are produced commercially have quite poor context or they move into the abstract very quickly and you then end up running around the length of the classroom patching things up. (Interview 2)

The MiC scheme represented a panacea to this situation: I really like it because . . . instead of being a load of abstract, stand-alone skills, you can see that people might want to see . . . how much carpet they’d need to cover a room. . . There’s a natural interest in it. . . I found it very interesting that thinking from Utrecht was that very few individuals will ever need to or will have the ability to move into fully abstracted generalised mathematics, which is fair enough. But our education system predominantly tries to push people there. Which seems very bizarre. (Interview 2)

3.4 Analysis of Data The presentation of data has attempted to pinpoint how the teachers demonstrated their attachment to, or identification with, a particular way of comprehending their own teacher practice. Yet, there is a need to exercise care in supposing that a shift in the teachers’ terminology indicates an adjustment to their underlying practice. Indeed, the implied distinction between surface and depth, or between evidence and supposed truth, needs to be avoided. Such couples are specific to particular and alternative discursive domains rather than universal. For example, Hammersley (1997) argues that the “evidence-based practice” movement privileges research discourse over the discourse of professional experience. The data in these two domains, he suggests, would be constructed rather differently and would be linked to a broader reality through alternative procedures. Thus, teacher speech and the selfconceptions implied in this need to be considered in relation to the alternative practice/research/policy paradigms. A teaching scheme is a function of the history into which it is being inserted. Yet, the experience of this history will be different for each individual according to how the individuals have accessed this history through a variety of alternative cultural forms. For Hammersley, alternative discursive forms are alternative forms of life and cannot readily be compared side by side. Specifically above, the data are centred on the teachers’ use of the terms “discussion” and “context” and how this use defined modes of allegiance to alternative paradigms. Yet these words derive their meaning through a play of difference with other words.62 The terms are shaped more by particular ways of making sense of the world, rather than by some notion of underlying practice. Practical engagement 62 Derrida

(1978).

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is never precisely anchored by these ways of making sense. There is also a gap between the discipline of following a particular ideology and the agency of more personal action. More particularly, however, there is a gap between the teacher’s rationale and practices offered in respect of this. This split identity is discussed more in later chapters.63 At the outset of the curriculum initiative, the teachers did not have an overall sense of what it might be to be a teacher following RME principles. Having spent a period of time between the two interviews, they grasped some understanding through particular traits that they had begun to identify as significant. “Discussion” and “context” emerged as two key words that acted as both variables in assessing practice and levers in developing it. Consequently, these words provided a useful research instrument in grasping how the teachers revised the map of their teaching domain. With respect to the teachers’ conception of the term “discussion”, however, we find teachers coming from very different directions. For two teachers (Jennifer and Jonathan), the word represents an aspect of their practice that is already well developed, yet needs to be reconceived in relation to the objectives of the MiC scheme. For the other two teachers (Lucy and Michael), it represents, at least in hindsight, a deficiency in their previous practice. So, in their attempts to connect with MiC, “discussion” emerges as a buzzword that commands a key filtering role in the register that shapes the teachers’ new practice. They identify with it as a component of teaching as they now understand it. All four teachers can subscribe to it as an important ingredient in this new conception of teaching. There is some evidence that for all four teachers the re-conception of “discussion” also reconfigures the composition of activities that together comprise their “teaching” and that they appear to see these changes as improvement. Jennifer questioned the target of her earlier faith in discussion and now wanted to be clearer about what discussion was getting at and how it might be better shaped to meet those objectives. Jonathan wanted to be “snappier” and “pacier” rather than let the discussion “drag on”. He also began to see less need to include everyone’s perspective – perhaps a result of being governed more by a clearer advance sense of what he was trying to achieve as teacher – perhaps now specified in terms of content defined by the teacher rather in the amount of discussion shaped around maximum inclusion and student participation. Meanwhile, the other two appeared to be converts who had previously marginalised “discussion”. Lucy, in sharp contrast to Jonathan, was now delighted that children could “bounce ideas” and had the opportunity to “get things wrong”. Michael, meanwhile, similarly seems to have become more relaxed about not achieving closure rather like the earlier stance adopted by Jennifer. So MiC functioned differently as a motivator for the two pairs of teachers: it seemed to provoke their seeing future improvements as movement in opposite directions to each other. While the teachers’ shifting conceptions were not sharing a trajectory to a common position, each of them was in the process of some pedagogical identity shift.

63 Brown,

Atkinson, and England (2006) provide detailed discussions of this.

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Their descriptions of themselves engaging with MiC were of a conception of teaching defined differentially to their previous practices. Personal improvement was seen alternatively as “less but more managed” discussion, as against “more” discussion. Their sense of their own teaching was in terms of the next step for their own improvement in relation to where they perceived themselves being before. The mathematical activity produced in this new teaching configuration was not pinned down or seen equivalently by the four teachers, though on the face of it all were apparently convinced by their MiC-inspired approaches. The notion of “context” functioned rather differently. For the two teachers already schooled in discursive approaches (Jennifer and Jonathan), “context” was seen as making mathematics more engaging by providing entrances accessible to pupils through employing more informal strategies. It provided a clear alternative to promoting discussion for its own sake as in Whole Class Interactive Teaching. And perhaps this new concern for context prompted these teachers to question if discussion for its own sake really was a good thing. In this way, context was now seen as a vehicle for mathematical understanding not an additional layer of difficulty. For the two other teachers (Lucy and Michael), the new centrality of context was seen as a direct substitution for earlier emphases on formulae and algorithms that were now viewed as being unnecessarily confusing and abstract. For all the teachers, there was a sense in which context was seen as enhancing accessibility to the mathematics which resulted in more convergence than with discussion although for different reasons. For Jennifer and Jonathan, it provided an approach to shaping discussion and thus making it more purposeful and aligned to learning objectives, whereas for Lucy and Michael there was a much greater sense that context functioned as a substitute for abstraction. Context provided something to discuss. The discussions were centred on a context. The data suggest that the implementation of the MiC scheme has had an effect on the practice of all four teachers in tangible ways. Certainly this has been achieved with respect to how they create the register through which they describe and evaluate their own practice, even though the scheme activated very different sorts of movements and outcomes between the two pairs of teachers. Within a project that is admittedly small scale the data show that whilst the teachers all made pedagogical shifts, and in a way that they would view as improvement, one would hesitate to claim that they have reached a pre-determined point. Indeed the “point” they have reached is not singular and could be understood differently according to a variety of possible interpretive schemes. Or indeed, Žižek (2006a) remarks that such a point is achieved through reading paradigmatic filters against each other. On a much larger scale, it could be inferred that any move to set curriculum guidance according to a specific agenda will not meet everyone’s aspirations or sense of practice in a consistent way, although it might nevertheless be seen positively as providing the teachers with an alternative frame through which they might inspect and stimulate their evolving practice. The benefit was in stimulating change rather than in the supposition of a better end point.

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4 Problems with Consensus I have sought in this chapter to unsettle the ways in which curriculum reform is often seen as shaping practice. I have argued that both teacher and curriculum change in processes of policy implementation. Teachers did not converge to a particular conception of teaching but rather evaluated their overall and cumulative teaching identity through a new filter. This filter could not be apprehended in its entirety from the outset. The emphasis was on how the ways of describing practice evolved, rather than on supposing that these descriptions can be checked out in relation to some supposed empirical truth of the teacher’s everyday practice. Teacher responsiveness to policy is shaped through successive paradigmatic adjustments. In many ways Whole Class Interactive Teaching could be seen to be supportive of the philosophy of Mathematics in Context. Yet there were also clear differences and it would not be possible to adjudicate between which was best without fixing evaluative criteria that would favour one teaching philosophy or another. Teachers learnt about the MiC approach to teaching mathematics and “identified” with this approach. They saw themselves in it, as it were, and began to describe their own practice in such terms, and through the filter of the new paradigm. The teachers engaged with and explored the two terms “discussion” and “context” as key instruments in better understanding how successive approaches contrasted with each other with respect to their practice. The terms provided conduits to more holistic yet still partial conceptions of the successive approaches. In successive adjustments, the mathematics that teachers teach and the mathematics that children experience are each processed differently. Human identity might be understood as an amalgam of partial identifications with co-existing ideologies.64 Who I am, or my teacher identity, is a function of how I draw on elements from the alternative discourses in which I am immersed, a split identity, as it were. But links between immediate tools and broader conceptions may be transitory or unreliable. Governments and researchers may be motivated by the long march to improve standards through major holistic change in practices. Such progress, however, might better be understood as a succession of ideological changes that resist a unified conception of what improvement might be, or of what mathematics might be in the school context. There is no consensually preferred conception of teaching, or an ultimate configuration of mathematics. In the next chapter, I suggest that the very anchorage of curriculum reform referenced to keywords, such as students, teachers and mathematics itself, needs to be questioned.

64 Laclau

and Mouffe (2001).

Chapter 5

Subjectivity in Mathematics Education Research

1 Introduction Research discourses create the analytical frames that we use in observing classroom activities, which in turn create the objects that we research, including “students”, “teachers” and “mathematics”. Such notions cannot be seen independently of the analytical lens brought to them by researchers. And in doing this the researchers themselves reveal something of who they are, and how they see themselves fitting in. The research discourses, and the lenses chosen from within them, however, are set against the complex backdrop of motivations, preferences and trends built into our use of language, of which we are not always aware. This chapter continues with the theme of mathematical cultures that characterises the first part of this book. Yet on this occasion the culture relates to the community of people who make it their business to observe the cultures of mathematics education. That is, the chapter focuses on mathematics education researchers themselves and the realities they construct in relation to their chosen area of work. Through inspecting research reports and the author perspectives revealed, some insight into those realities is achieved, and a place for mathematics is suggested. Specific examples are drawn from a Special Issue of the journal Educational Studies in Mathematics, comprising papers centred on the semiotic work of the American philosopher C. S. Peirce.1 The various authors provided discussion of how mathematical ideas are signified. Specifically, through discursive analysis of the reports, the chapter seeks to examine the productivity of reading the reports from alternative perspectives. The Special Issue authors will be understood through what they say, drawing as they do on the discursive resources that they have available. These resources link them to various communities. For example, the journal Educational Studies in Mathematics in which their work appears has certain conventions of style, an acceptable range of topics, familiar themes, etc. To be included in the journal an author needs to demonstrate that his or her work fits those criteria. Similarly each author would have a professional location where he or she would talk to colleagues 1 Saenz-Ludlow

and Presmeg (2006).


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in particular ways and engage in practices commensurate with that location. In this chapter, the authors are seen as subjects considered with respect to certain discursive perspectives. Their reports reflexively situate who they are in relation to the discursive perspectives that they assume. That is, these perspectives, in their depiction by the author, reveal the author. The author becomes known through the way in which he or she makes sense of the world. Subjectivity locates the individual, in relation to the stories they tell of themselves, the stories they tell of the world, and the stories people tell about them. The individual is not so much seen as a cognitive or biological entity. The reports are also considered in terms of how they locate the teachers and students, as subjects,2 in particular accounts of the world. In their writing, the authors declare the perspectives that they have on the teachers and students, and in so doing depict a world with codes and practices that shape the actions of the teachers and students. To illustrate this positioning, a study already mentioned discussed a national policy initiative in which all primary level teachers were obliged to work according to a centralised curriculum with high degrees of specification.3 The study, premised on the perspectives of the teachers themselves, sought to better understand how the policy framework was articulated through the teachers’ accounts of their own practices. A key element of this study was that it was the government who determined the constitution of mathematics within a legislated curriculum rather than mathematicians or teachers. The teachers were examined speaking through the filter of government policy apparatus, not to get at the intention of what the teachers were saying, but rather to understand how the policy was operating through the teachers. The authors, however, were not advocates of this regime but sought to examine how the teachers processed the demands of the regime. That is, the authors sought to understand how mathematics, students and teachers were shaped by this policy initiative and how they were included in the world that the policy apparatus depicted.4 This chapter has similar ambitions in its attempt to understand how teachers and students are subject to the discursive filters that the Special Issue authors have employed in depicting them. Teachers and students appear in the stories that the authors tell. These stories inevitably reflect the assumptions that the authors have made. Yet these assumptions may not be the central plank of the story that the authors are telling. The assumptions may be in the background to the author’s own pressing concern. Previous chapters have sought to locate embodiments or social constructions of mathematics, in spatial and cultural environments, or in curriculum frameworks. I have argued that mathematics is a function of those environments. This chapter is focused more on the configuration of teachers and students

2 Foucault’s notion of discursive formations implicates subjects. Althusser sees individuals as being interpellated by ideology. 3 Brown and McNamara (2011). 4 Brown, Atkinson, and England (2006) provide further examples of subjectivity in education.

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positioned around mathematics. The central aim is to examine how “students”, “teachers” and “mathematics” are variously understood or created in the realities of those authors, and perhaps in mathematics education research more generally. It is supposed that this environment shapes perceptions of mathematics and how people work with it. I shall, firstly, examine how the writers variously conceptualised psychology or subjectivity, in the context of mathematical learning activity, by examining the psychologically oriented material that they have cited. The authors, however, were more concerned themselves with issues of signification, and only talk about psychology and subjectivity obliquely. Nevertheless, this oblique reference can normalise a particular conception of psychology, which gets built into the fabric of mathematics education research. Secondly, I examine the assumptions made with regard to how the reports conceptualise the processes through which their recommendations might be achieved. I consider how the research audience was understood as suggested through implied positioning of teachers, teacher educators, researchers and other potential readers. This includes examining the choices of literary resources that have been made and how those choices shape the phenomena being researched. A citation analysis provides an indicator of the authors’ perceived research domain. Thirdly the chapter asks: How do the reports position “mathematics”, “student”, “teacher” and conceptions of the research domain in these processes? That is, how are teachers, students and mathematics understood and implicated as subjects in the versions of the world that the authors portray?

2 Preliminary Remarks on Reading Text My interest in this chapter is directed towards how subjectivity is produced in some examples of mathematics education research. I am not so much concerned with encapsulating any final meaning that the authors may have intended. The authors produce particular ways of making sense of the world, which result in particular configurations of “teachers”, “students” and “mathematics”. This subjectivation makes them what they are. Foucault has been very influential in shaping contemporary understandings of subjectivity. His work was typically referenced to particular historical situations, as evidenced through texts from the period concerned. For example, Foucault (2001/1961) took documents from the eighteenth century to decide how madness was variously understood at the time, even though the documents he consulted were not always centrally about “madness”, a term whose supposed meaning has shifted in relation to symptoms, and understood against evolving frames of reference. Foucault’s discourse analysis was concerned with how madness was positioned in the text and derived its meaning from the words around and the world they portrayed. In such enquiry the meaning of any text is located in later interpretations,

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perhaps with respect to other agenda, rather than in any original intended meaning.5 Similarly, I am taking the Special Issue as an example of a contemporary historical document and asking how the subjectivity of the teachers and students has been variously understood or implied by the authors.6 The teachers and students are seen as a function of the stories that they appear within. They are subjects within these stories. I am also considering how the authors reflexively imply themselves in the stories they tell. How do they imply the role they see themselves performing as researchers and their own specific interventions in the social domains of which they are part? This analysis is directed towards asking if we can better understand how conceptions of subjectivity are intentionally or unintentionally produced through the way in which the authors depict research objects such as “students”, “teachers” and “mathematics”. The chapter provides a limited focus that does not seek to encapsulate the Special Issue authors themselves as intentional beings. The authors cannot be fully attentive to all dimensions of the productivity of their work. They cannot predict all of the interpretations that will be brought to their work. Yet in using the term “intention” I am cognisant of how this term has faltered as a result of hermeneutic interrogation. To illustrate this, Gallagher (1992) has drawn a useful analogy between teacher– student and author–reader relationships to define four alternative conceptions of hermeneutics (interpretation) that between them rather dissipate the notion of intention.7 In conservative hermeneutics, the student, or reader, seeks to understand the intention of the teacher, or author, on the latter’s terms. In moderate hermeneutics (e.g. Ricoeur, 1981), meaning is a co-construction, such that the author’s or teacher’s original intention is modified in the encounter. Critical hermeneutics (e.g. Habermas, 1972) is concerned with the “emancipation” from the ideological structures that govern our actions, such as the administrative and pedagogical layers introduced at the beginning of the previous chapter. In radical hermeneutics (post-structuralism, e.g. Derrida, Foucault), new frameworks are produced as a

5 In a contemporary re-enactment of such discursive analysis one of my own PhD students, who is a senior medic in a hospital Emergency Department, observed the production and operation of knowledge economies in three areas of his professional work; everyday medical practice, a curriculum development for emergency medicine doctors that he was initiating, and the Emergency Medicine Journal, where he is a regular author (Kilroy, 2010). Again fluidity in the use of language was highly apparent as a result of structural changes to the profession, such as in; the government seeking to control medical practices through regulation; Emergency Medicine as a specific professional domain seeking to be recognised as a bona-fide medical area with its own Royal Charter; doctors themselves undergoing a major demographic overhaul in which women and ethnic minorities now comprise the majority of new doctors but in a domain shaped by their white male predecessors who had sometimes displayed rather more conservative attitudes, etc. Categories of scientific knowledge were shown to be unstable as were the practices that derived from them. 6 I have provided extensive discussion of this interpretive approach in relation to children’s mathematical learning (Brown, 2001). 7 These categories resonate with the discussion of curriculum by Remillard in the last chapter.

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result of the interaction between teacher/student or author/reader.8 In this latter perspective the original intention may be completely displaced or neglected in the actions that follow, consequential to the ground on which they stand being reconfigured in later analysis. In earlier work, I used this frame to contrast intention of the teacher with significance to the student in some school mathematical examples through moderate and critical perspectives (Brown, 1996b, 2001). In short, I argued that the meaning of a mathematical encounter does not have to remain on the teacher’s terms. This relates to recent work carried out by some other mathematics education researchers.9 Meanwhile, contemporary literary theory10 and political science11 respectively argue that authors and our political masters produce effects that are not necessarily within the remit of their original intentions. Such concerns are of central importance in the field of qualitative inquiry.12 Likewise, the authors of the Special Issue through pursuing certain intentions also, perhaps unintentionally, support other moves. Here I shall argue that one of those moves, which I pick up as a reader, is unwitting support for potentially restrictive conceptions of subjectivity. For example, as will be seen, cognitive perspectives are sometimes normalised at the expense of more socialised conceptions of the human subject. A reviewer of an earlier draft argued that in responding to the Special Issue authors on my agenda rather than theirs I am committing an “ethical violation” in distorting their original meanings. I do not agree. I would respond by arguing that some accounts of contemporary ethics13 are centred on subjects being obliged to operate within oppressive discursive domains rather than on individuals doing their own thing. And such obligations bring with them support for the agenda built into those domains. This can result in symbolic violence to those disadvantaged in those modes of depiction.14 For example, in mathematics education, wouldbe problem solvers would not be served in a landscape depicted according to a basic competency ethos. The mathematical thinking of primitive tribes would not be commensurate with mathematics in “Western” nations.15 Or those of a collectivist persuasion (e.g. a social practice theorist) would not feel comfortable in a world depicted in individualistic constructivist terms, or vice versa. I see my analysis as an interrogation of the discursive domains that shape the Special Issue authors’ work and my own, as well as school practices, rather than

8I

have summarised this model elsewhere (Brown & Heggs, 2011). recent work on teacher–student positioning has been carried out (Herbel-Eisenmann, B. & Wagner, D., 2010; Herbel-Eisenmann, B., Wagner, D. & Cortes, V., 2010; Wagner & HerbelEisenmann, 2008; Wagner, forthcoming) 10 Barthes (1977); Eagleton (2006); Colebrook (2007). 11 Laclau (2005); Mouffe (2005); Harvey (2006); Nancy (2007). 12 Lather (2007). 13 Badiou (2001); Butler (2005). 14 Žižek (2008). 15 Bellos (2010). 9 Extensive

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of the individual authors and their intentions.16 My target is the practices of mathematics teaching in school where ethical violations are held in place by the custom and practice of discursive operations. The authors are pitching their work in to a domain with specific architectonic preferences that define the acceptability of work, a consensus that drives the “context of expectation”17 that generates meanings. Accordingly, I am seeking to comment on how this community within the wider mathematics education research community is reflexively defined, and how the objects of research are shaped by these decisions. This chapter extends this focus to consider how these formulations also signify teachers and students, and the processes through which they change.

3 Conceptualising the Distribution of Psychology The notions located by the terms “mind”, “psychology” and “subjectivity” have been treated in various ways in the Special Issue. The work of Piaget and Vygotsky, created during the middle of the last century, has had considerable impact on how the field of mathematics education understands the psychology of learning. Their work has been normalised in many instances of mathematics education research and it is evident throughout the Special Issue reports. The normalisation of these psychological paradigms, together with the certainties that have been attached to mathematics, have anchored the field, making it resistant to more contemporary conceptions of psychology or subjectivity. I shall argue in the next chapter that the Piagetian emphasis in mathematics education research echoes a very specific and controversial reading of Freud’s theories. This reading saw the task of psychology in terms of nudging individuals towards conventional, that is, state-sanctioned modes of behaviour, or specific models of what it is to be human. I shall also argue that Vygotsky’s perspective assumes a benevolent community drawing in new members that do not always square with the student’s experience of learning mathematics. What alternatives might there be to understanding psychology in these terms? In many contemporary theoretical debates preferences for different psychological theories can be explained by the demands of the specific task in hand.18 Nordtug, for example, argues that post-structuralist accounts of subjectivity19 so often only 16 This

response to the Special Issue is offered in the spirit of academic debate as an attempt to engage critically and productively with aspects of the Special Issue, a research publication that I admire greatly. The authors are major contributors in the field thus avoiding the charge that I am inspecting inferior research. I do not see my remarks as a criticism of the individual authors. This debate has been pursued following the publication of earlier versions of Chapters 5 and 6 being published (Brown, 2008a, 2008b), leading to a response from some of the authors concerned (Presmeg & Radford, 2008). 17 Colebrook (2007, p. 194). 18 This point is made by Nordtug (2004) in her comparison of Lacan and Peirce. 19 She cites Malson (1988) and Lovlie (1992).

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apply to groups, or how individuals relate to social trends, which is not so useful to her since she works as a medic with individual patients. She insists that theoretical models need to be fit for purpose. Mathematics education research deals with individuals, social groups and social systems. Consequently it needs a variety of apparatus that enables analysis to span variously conceived domains. The choice of apparatus depends on the task being addressed, whether that is trying to support individual teachers or pupils, or alternatively trying to design and implement a policy. Mathematics education research has a choice of positions for itself and its supposed readership, and how it imagines its dissemination might operate. For example, the policy level task of improving particular mathematical capabilities for specific populations of students requires very different apparatus to an individual teacher assessing her own personal capabilities for work with particular individual children. And both of those alternatives are very different to the psychological models that might be needed in assessing localised situations that might be uploaded to a wider domain. The configuration of student, teacher and mathematics can be understood from many perspectives and the constitution of each is a function of the perspective assumed. Self and subjectivity are often seen as much the same sort of thing. As various alternative perspectives find a more prominent voice in contemporary thought, however, the notion of psychology has been challenged by emergent conceptions of subjectivity. A key shift in contemporary social theory has been towards seeing the individual caught up in more or less committed participation in a multitude of discursive activity.20 That is, individuals partake in social languages that more or less fit what they are trying to say. The individual is obliged to use these languages if they are to be included in social exchanges. In this book, their subjectivity will be understood as a function of this participation. This new conception of subjectivity attends to the discourse-specificity of any assignation and the distribution of the psychological. Subjectivity is a key analytic term understood variously across fields as diverse as critical psychology, postcolonial studies, film theory, gender studies, social theory, geography, anthropology and cultural studies. Subjectivity, in this chapter and more generally across the fields mentioned above, relates to individuals whose psychological existence is distributed across a multitude of linguistic filters. Subjects are centred in the linguistic/narrative accounts provided of them by the subject herself, or by others. Indeed to be noticed their very presence may need to be asserted in a preferred linguistic mode. For example, within mathematical learning there are many alternative approaches to demonstrating achievement, yet these may count for little if they do not align themselves with the specific assessment regime in force (e.g. basic skills versus problem solving). An individual is subject to the specific discursive framework presently being applied and is recognised according to the degree of compliance. The individual is obliged to express herself and be understood through externally imposed linguistic filters.

20 Butler

(2005); Žižek (2006a).

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Self, meanwhile, has often been understood as the biological entity held together by a cognitive unity.21 But as Lemke (1995, p. 82) argues, from a “post-modern view [the assumption of a unified self] was a massive sleight of hand. Even within the natural sciences there is no guarantee that physical, chemical and biological definitions of an organism coincide for all purposes”. The self is not the only centre of coherence. Ten papers make up the Special Issue.22 Issues of psychology or subjectivity are handled variously across the papers. The papers of Radford and Steinbring are premised on Vygotsky’s notion of cultural tools. Radford’s formulation is attentive to potential links between Vygotsky and Peirce in relation to the production of, and learning about, cultural objects.23 Otte’s work centres on mathematical epistemologies as understood through semiotic apparatus that places psychological or student concerns on the margins. For example, he reduces people, in almost behavioural terms, to those able to perceive and respond “appropriately” (p. 13) according to the teacher’s transformation of symbolic material. In each of these papers, to varying degrees, the conception of pupil is somewhat cropped. That is, they are read against registers centred on the interface of student, teacher and mathematics in the classroom setting, as though that were the natural focus. Meanwhile, Duval nudges towards Piaget’s cognitive psychology in positioning students. His use of semiotic structures, as applied to mathematics, is very much in focus, sidelining a more nuanced conception of the student engaging with them. Students fill their assigned space and supposed mode of apprehension in the appropriate construction of child, seen as being at this or that stage, within the framework presented by the research author. This privileges a specific view of students that perhaps marginalises other attributes that would be noticed on a different register. In this instance, there appears to be little scope for contestation of places assigned to participants. They are reduced to “types”,24 responsive in predictable ways, according to prevailing ways of making sense. If the teacher controls the environment, particular responses will be achieved. The research is conceptualised as adopting a relatively objective eye, rather than subjective “I”, in positioning teachers and students in roles from which they cannot readily escape. This tendency to create “types” promotes an instrumental rationality in our research ambitions. This perspective suggests assessment as a control technology to bring about tangible change. Such orientations risk influencing populations seen as homogenous to be more homogenous rather than promoting differences. Such 21 For

example, this is true of Colapietro’s (1989) book “Peirce’s approach to the self ”, cited by Presmeg, which is centred on an individualistic conception of consciousness. 22 The papers of the Special Issue (Saenz-Ludlow & Presmeg, 2006) that were all published in 2006 are written by Radford, Steinbring, Otte, Duval, Ernest, Saenz-Ludlow, Presmeg, Morgan, Ongstad and Hoffman. For brevity the date will not be repeated each time the paper is discussed in this chapter. 23 This paper is discussed in Chapter 8. 24 Berger and Luckmann (1972).

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ambitions are sometimes centred on a quest to find the best method. These orientations are about management and response rather than about learning. There are alternatives to such a research perspective in which individuals might be allowed to be creative, unpredictable, not “appropriate” and human. Semiotic formulations favoured by Ernest and Saenz-Ludlow combine aspects from both Piaget and Vygotsky. This work is centred on individuals defined according to normal behaviour. The work marginalises more socially conceived notions of subjects, where social diversity might open up alternative relations to mathematical objects. Morgan and Presmeg each offer alternatives in favouring the anti-Piagetian discursive stance of Walkerdine (e.g. 1988). Presmeg links this to the mid-career semiotic work of Lacan but without mentioning his conception of subjectivity for which the apparatus was designed. In each case however, analysis is centred on interactions in the individual mathematics classroom. It is focused on how the teacher’s role might be adjusted to get a better effect. There is little concern with wider policies such as curriculum construction, teacher education, social definition of mathematics, etc. Morgan, for example, opens the door to some other analytical approaches from social semiotics25 that “can provide a systematic means of gaining insight into the dynamics of classroom interactions and the roles of individuals within these” (p. 238). Nevertheless, she centres her conclusion on how such tools “may help mathematics teachers and students to develop more purposeful and hence more effective use of language” (p. 239, my emphasis). That is, she provides “some flavour of the extent of the contexts of situation and of culture taken into account in the analyses and of their use in forming interpretations” (p. 239). The analysis, however, privileges relevance to classroom actions at the level of interaction and effectiveness according to some agenda, rather than the wider social parameters that govern those actions and define the actuality of classrooms. Ongstad is centred elsewhere. He provides a more fully social model centred on Habermas’ opposition to Peirce with links to Bakhtin’s conception of dialogical dynamics. Some of the individual reports in the Special Issue are predicated on small-scale research understood from the perspective of an individual teacher, teacher educator or researcher. How might a teacher improve his or her teaching so that a child might improve his or her learning? In a Piagetian frame there are some essential human characteristics that can guide the ambitions of the teacher to meet the child’s readiness at certain stages of their development. Piaget portrays a child who naturally passes through developmental stages. An influential book by Skemp (1971) conceptualised school learning through Piaget’s analytical categories, shaping and classifying school mathematics according to this image of a human’s development. Suitably equipped with knowledge of the child’s development, the teacher can decide how best to introduce mathematical ideas. But, similarly, this approach conceals a very specific version of mathematics, derived from the assessment of what a

25 Hodge

and Kress (1988); Fairclough (1995).

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child is, and what this child is supposed to be able to achieve. The child is a known cognitive entity. Mathematics is similarly fixed as a standardised set of knowledge. In Piaget’s perspective, it may indeed be best to equip the teacher to attend to the child’s natural mathematical development. In the past this explained many teacher practices being centred on free play and moving bricks, guided as they were by notions of young children not being ready for formal mathematics. Curriculums, meanwhile, may seek to serve that supposed image of a naturally developing child. By focusing on the level of classroom interaction the Special Issue reports lack instances of a large-scale policy implementation in which semiotic structures would be an equally valid analytical tool. The choice of emphasising interactive analysis in the classroom is not wrong. The choice, however, does have analytical consequences in terms of how a wider map is created. It points to particular normative understandings of how mathematics education should be framed for analysis. Action is deemed to take place at the level of the teacher working with students, gauged according to their developmental needs. The demands, in a sense, are non-negotiable since children, at particular stages of their development, are seen as needing mathematics of a particular sort, although if we could hasten that process so much the better. Yet, as discussed in the previous chapter the teaching of mathematics always takes place within the context of some curriculum or wider social structure, where factors outside of immediate mathematical concerns intervene. Semiotic analysis, drawing on the work of Peirce, lends itself to the investigation of many such contexts. A semiotic reading would be quite feasible in analysing policy initiatives for instance.26 Yet this potential has not been explored extensively within the reports as a result of emphasising the level of classroom interaction. The restrictive field of analysis within the Special Issue comprises a choice, which I argue pervades mathematics education research more widely. The choice emphasises individual teacher action as the key variable for change, rather than more structural adjustments, for example. That is, the preponderance of mathematics education research reports focus on the interactive level and this emphasis naturalises this focus, as though it were necessarily the preferred point of entry for mathematics education research analysis. This choice defines the field and the architectonic parameters that govern collective points of reference. In particular, conceptions of mathematics are relatively fixed, as are the communities or structures that house them. This results in the child’s challenge being directed at a stable object, where his or her teacher is guided by what the child ought to know at a particular point in his or her development. A background theme of this present chapter is to consider how mathematics education issues might be thought otherwise, towards identifying the alternative levels at which we might intervene to meet different needs. A child may not want to

26 As an example, although not explicitly semiotic in its remit, a study mentioned above (Brown &

McNamara, 2011) was premised on understanding how the curriculum (the sign) represented mathematical activity (the object) to the population of teachers in a certain way (the interpretant) (cf. Peirce, Collected papers, vol. 2, p. 274).

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develop in a prescribed fashion nor be welcomed into the community as a compliant citizen. Communities do not always define participation on an equitable basis. To focus on the interactive level in our research is to obscure difficulties that may reside in the wider structure. Further, our capacity to influence individual teachers across a diverse world is surely limited and moves towards homogenisation as in comparative testing can suppress the needs of some groups.

4 Positioning of Author, Reader and Mathematics Education Research Community The last section considered aspects of the voices being assumed by the researchers, and how these voices defined their research objects, such as teachers and students, according to alternative conceptions of psychology. I now turn to better understanding how the research audience is understood, as implied by the apparent address of the reports. Each researcher has a particular image of how his or her work addresses the community and a sense of the responses that might be achieved in terms of changed teacher practices, or from other researchers. And these images relate to particular configurations of the social world and the people within it. For example, the researcher depicts teachers and students as being and acting in particular ways within particular settings and supposes that it might be possible to influence them to act otherwise to achieve certain results. I am suggesting that these social configurations produce situated versions of mathematics that more or less draw on wider discursive parameters. Mathematics education research is shaped across a complex network of discourses. Authors in the domain are reflected and produced in these multiple participations. The traditional realm of mathematics education researchers has been anchored theoretically by educational psychology and the philosophy of mathematics, with work targeted on improving teaching techniques. The field is not especially experienced in responding to alternative social paradigms. A survey by Lerman, Xu, and Tsatsaroni (2002) of articles in Educational Studies in Mathematics considered their engagement with other theoretical fields. The survey depicted a situation in which only a few alternative theories have sustained interest. The chief exception in the period reviewed was constructivism in its many guises but in forms primarily exclusive to mathematics education.27 Attention to this theme was initially shaped around the supposed debate between Piagetian and Vygotskian perspectives.28 The latter extended to the cultural psychology of Cole,29 the activity theory of Engestrom30 and the situated cognition of Lave and

27 von

Glasersfeld (1991, 1995); Ernest (1998). (1991). 29 Roth and Lee (2004). 30 Williams and Wake (2007). 28 Confrey

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Wenger.31 There have also been some studies building on the sociology of Bourdieu and Bernstein, including some specifically examining school mathematics texts as cultural products.32 Such moves have characterised the major challenges to cognitive perspectives. There has, meanwhile, been a light sprinkling of reference to other contemporary theory such as post-structuralism and hermeneutics33 and feminism.34 Yet the influence of such alternative models is less evident than in the broader field of education. Bartolini Bussi and Bazzini (2003) provide a proposal for how mathematics education research might reach out to other social scientific fields. I provide below what must be at best an idiosyncratic taxonomy of citations made by the Special Issue authors. My purpose in compiling this was to build some sense of how the authors are situating themselves, to establish to whom they are listening and to whom they might be addressing their work and thus how the literary domain, research community and audience are being conceptualised. I classify and distinguish between some of categories of material that the authors do cite whilst considering these together with the wider domain invoked in my oblique reading. The categories I have chosen are as follows: the author’s own previous work, work by other mathematics education authors in the area of semiotics, other mathematics education research (including the philosophy of mathematics), other philosophical material, non-educational social scientific material, non-mathematics-specific education research, policy-oriented material, teacher education research and books or journals aimed directly at teachers. The materials could be classified differently given that each citation is put in just one category. Yet, the resulting tabulation does provide some evidence of how the authors are positioning themselves in relation to their conceptions of their community and academic domain. In doing this, I hope to indicate how the authors variously conceive implementation in respect of their own research, which is not always made explicit in their reports, and how that implementation prescribes roles for students, teachers, teacher educators, etc. Consequently, I consider how those roles restrict or produce conceptions of mathematics. I seek to understand how the reports could be positioned within a bigger picture, where the processes of dissemination are included. In this approach, I endeavour to pinpoint how the authors understand the building of control technology that would enable the consequences of their research to be realised. The citations primarily emphasise the domain of mathematics education research, centred on the authors’ own work, with some reference to other semiotic specialists, and others within mathematics education research, to more mathematically oriented philosophical/theoretical material. Radford, Otte, Steinbring, Ernest

31 Lave

and Wenger (1991). See also Cobb and Bowers (1999); Graven (2004). (1998); Cooper and Dunne (1999); Morgan (2002). 33 Walkerdine (1988); Brown (2001); Walshaw (2004, 2007); de Freitas and Nolan (2008); Walls (2009). 34 Burton (1995); Walshaw (2001). 32 Dowling

8 45 21 14 4 6 11 20

10

0 5 7 0 6 10 5 13

8

MER/philmath

Semiotic MER

Self

Duval 15 Ernest 9 Morgan 7 Otte 7 Ongstad 13 Presmeg 5 Radford 7 Saenz 4 Ludlow Steinbring 6

3.

2.

1.

3

7 5 0 17 15 5 27 19

Other phil

4.

1

0 6 13 1 15 4 5 1

Soc-sci-non-educ

5.

0

0 0 0 1 4 0 1 5

Education-non math

6.

0

3 0 0 0 0 2 0 1

Policy

7.

9.

0

0 0 0 0 0 0 0 0

0

0 4 2 0 0 0 0 0

Math teacher Teacher educ book/journal

8.

4 Positioning of Author, Reader and Mathematics Education Research Community 95

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and Ongstad are primarily addressing specialists in analytical apparatus and do not appear to be talking directly to teachers or even teacher educators, or, at least, not in ways that would suggest an immediate impact on educational practice. The theoreticians they cite are generally no longer with us. Their analysis is centred on structures rather than on people, except when those people have a well-defined role to perform within a structure. Radford’s paper is ambivalent as regards its audience and supposed route of impact. Briefly, it pitches itself at a mathematically oriented audience protective of mathematical meaning being objective. With this audience in mind, he argues the case against transcendental objectivity and truth in mathematics. This philosophical argument is seemingly crafted to persuade some more traditional mathematicians not to hold on to their purist aspirations in the school context. Such aspirations, Radford argues, are philosophically unsustainable and opposed to mathematics adjusting to new circumstances. This contrasts with the audience Radford normally supposes. His papers,35 which typically occupy a Vygotskian perspective, are addressed to teachers or teacher educators, governed primarily by educational concerns and centred on changing their own practices in line with cultural norms. Chapter 7 discusses Radford’s paper and associated work in more detail from the perspective of how objectivity might be understood. It will be argued that the two audiences supposed are necessarily split and that reconciliation is not always achievable. Presmeg and Saenz-Ludlow each more readily address the integration of their theoretical input into the practices with teachers and children yet issues of dissemination remain understated. For example, Saenz-Ludlow (p. 213) promotes “interpreting games” as potential “teaching–learning tools” and remarks: For this to happen, it is necessary to build up classroom environments in which a communicative relationship between teacher and students naturally emerges with immediate and mediated intellectual results. That is, teacher and students should be equally committed to the establishment of a communicative relationship in which sign use, sign interpretation, and inquiry become a continuous state of affairs.

This statement comprises advice on how teachers and students should conduct themselves. Such changes in practice are left to the individual teacher or teacher educator reading this report. This assumes that individual teachers would read this research paper, understand it, and then be free to adjust their practices accordingly. It is not policy talk. It assumes some teacher autonomy with respect to the curriculum. Routes to implementation are not outlined in great detail: Because the curriculum in elementary schools tends to be less intense and children tend to be more open to dialogue due to their age, classroom environments in which interpreting games tend to flourish tend to be easier to establish in these schools. It might also be possible to establish this type of classroom environment in secondary schools but more research is needed in this direction (p. 215).

35 E.g.

Radford, Bardini, and Sabena (2007).

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It is not made explicit as to whom this author is talking: a lone teacher, a teacher educator, a school principal choosing staff or a mathematics teaching scheme, a curriculum writer able to include “interpretation games” into the curriculum, or fellow researchers considering alternative angles? The author makes assumptions about the character of elementary schools and the reader’s professional capacity to adjust practices within them. Paths to implementation are not conceptualised as part of the research. And the assumptions made about implementation are unsupported. The complete statement is surely culturally specific. Images of schools and children are being provided that may not be lived up to in real life. This portrays a conception of mathematical learning in which mathematics emerges through interpretive approaches and seemingly not so much through external demand. Presmeg (p. 180) works within a frame spanning the National Council of Teachers of Mathematics (NCTM) and Realistic Mathematics Education. This confers a particular image of mathematics in the school setting in line with those influences. The paper is not specific about implementation by way of teachers: “It seems clear that these theoretical lenses have the potential to cast light on some of the processes involved in attempts to construct connected knowledge in mathematics classrooms”. Presmeg is keen to support policies that emphasise how connections can be made between everyday practices and mathematical concepts, where meaning is negotiated. Yet, such negotiation encounters multiple interests in any given setting. In his assessment of the Special Issue, Hoffmann (p. 284) argues that Presmeg assumes a teacher could intentionally define learning routes through specific semiotic chains, as though they were necessarily the most obvious to follow. He suggests: “Learning might be much more complex, and different paths could be taken by different students”. I agree with this point, but unlike Hoffman (ibid.) I find Presmeg’s combination of Walkerdine and Lacan, with links to linguistics, epistemology and psychoanalysis, as entirely feasible within mathematics education. Presmeg (pp. 172–173) acknowledges how mathematical objects are “mutually constituted in the classroom by the participants involved”. This far Presmeg covers implementation strategies for her research. But she does this without explicitly hitching the process to the subjective constitution of those participants themselves. For Lacan, who she cites, participation in discourse equates to that person’s self-formation. Also the fluidity of mathematical objects that Presmeg espouses contrasts with her depiction of students acquiring “some mathematical concept that is desirable for the students to learn” (p. 166). She envisages pursuing a specified chain of thinking to a “connection” conceived in advance by the teacher. Such a specification suppresses the conceptions of subjectivity that Walkerdine or Lacan portray, beyond their usage of Saussure’s linguistic apparatus, to which Presmeg does refer.36

36 Walkerdine’s (1984, 1988) neo-Marxist stance paints the process of learning as being about social regulation and student compliance. Whilst for Lacan (e.g. 2006, pp. 6–48) the human subject is often a regressive affair consequential to a multi-layered linguistic making sense of the world.

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But what other general points can be made? All of the authors refer extensively to their own work.37 This tendency emphasises that they are each engaged in a continuing personal project, continuing a chain of thinking maybe, rather than as part of some shared or coordinated initiative. In the citation count this personal aspect is in most cases stressed more strongly than their associations with the community of mathematics education writers in the immediate area. Group identity, in the Special Issue papers, is defined primarily through identification38 with Peirce and the theoretical domain he symbolises. The main areas where other citations occur are in relation to mathematics education research generally, the philosophy of mathematics and philosophy, areas not always easy to sub-classify. A key area of absence is more generalist education material, as if to suggest such material would be marginal to the theme being addressed. Thus, in relation to the discipline of “mathematics education”, the stress appeared to be more on a particular framing of mathematics than on education more broadly. A more broad based discussion of educational issues would bring a concern for greater cultural diversity or human interaction with it. Similarly, material addressing mathematics teacher education or teacher education more generally is poorly represented. Some conception of teacher education processes would necessarily underpin any proposed dissemination strategy, if teacher practices are to be adjusted. Teacher mediation cannot be regarded as a neutral layer in dissemination, as any shift in policy would have major ramifications for how teachers understand and adjust their practices. It is thus not always clear how the authors envisage the suggestions of their work being integrated into the work of classroom teachers, as though their work can be developed without that being a major concern at this stage. The emphasis is on the point of contact between mathematics and students rather than on how this might be transformed through the active participation of teachers, or communities of teachers through the adjustments of broader parameters. There is also a general absence of articulated links to policy-oriented audiences. Yet, in many countries it is not teachers who decide the content of mathematics that they teach, or even the style in which they do it. So to summarise, according to the citation count, in order of descent, the authors are listening to themselves, dead philosophers, other specialists in semiotic work in mathematics education research, other mathematics education research researchers and then, just occasionally, to social scientists, but almost never to other education researchers, including mathematics teacher education researchers, school teachers and teacher educators. The engagement with Peirce is being understood primarily through personal engagements with the original material rather than as a result of working through the filters of history, including those evidenced within mathematics education research reports in the immediate area. Through this depiction of the terrain, the Special Issue authors emphasise a relatively closed world in which teachers encounter students learning particular

37 A

vice I share! (2005).

38 Laclau

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conceptions of mathematics. The reports, and the hierarchy of power relations implicit in them, marginalise links to fields such as education, teacher education, policy implementation or the broader social sciences. This allows teachers, students and the mathematics they share to be largely untroubled by the outside world and the persistent challenges it presents to the frameworks that govern our teaching. Finally, the formation of mathematics education research depends on how authors in that domain understand their field and their audience. Generally, writing is shaped as a result of authors jockeying for acceptance within that domain. That is, those who feel a need to be part of that domain, shape the domain, partly in their own image. Their writings might be seen as requests to be included. Laclau (2005, p. 53) insists that group formations derive from libidinal motivations. They result from excitements for individuals to do with feelings of being a part of or apart from current trends, fashions, innovations, transgressions or taboos. Individuals find themselves identifying or not with a range of alternative modes of life, according to the particular social needs that they feel, the form of alignment or non-alignment they wish to pursue, or the particular ambitions that they set themselves. In a field like mathematics education research there are tightly stratified arrangements for contributions to be received and disseminated.39 Through peer review processes, a few major journals exert significant influence on the themes to be explored within the domain, what counts as important, what is seen as interesting and what needs to be cited for effective positioning to have been achieved. Meanwhile, local professional circumstances, funding provision and personal preferences all have an impact on the types of research carried out. And it is against these parameters that individual authors offer their contributions and become who they are with respect to the domain. The activities cannot be seen independently of the analytical lens brought to them by the researchers.

5 “Teachers”, “Students” and “Mathematics” The Special Issue organises its papers into three categories: (i) semiotic and epistemology (Otte, Radford and Ernest), (ii) implications for teachers of mathematics (Duval, Steinbring, Presmeg and Saenz-Ludlow) and (iii) classroom communication (Morgan and Ongstad). The authors inevitably stress some features rather than others and this results in some specific shaping of the research objects (mathematics, students, teachers, etc). Those choices have analytical consequences. By attending to semiotic systems and the place of mathematical entities within them, the first set of authors restricts conceptualisations of mathematical students. In these papers, improvements to mathematics teaching are understood in terms of the teacher providing better definition of, and access to, the existing symbolic networks of mathematics. The individual is understood with respect to an externally conceived frame and is given little opportunity to contribute an alternative conception of their 39 Valero

and Zevenbergen (2004).

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role. Little mention is made, for example, of how teachers, or populations of teachers, might be assisted in acquiring these new frameworks of understanding their task or of making these changes to their practices. That is, the layer of teacher mediation implied is not developed as a teacher education task. Similarly, little or no mention is made of the policy implementation structures that would govern curriculum definitions or scope for teacher agency in defining the content and style of their lessons. The diversity of students seems not to be addressed. These restrictive conceptions of learners and teachers are also true of the second set of authors insofar as the roles of teacher and student are specified in ways that assume particular social arrangements, with pre-supposed learning objectives. Duval’s (p. 107) strict distinction of “the represented object from the semiotic representation” is an analytical choice that downplays the cultural/semiotic dimensions of mathematical formation, or teacher or student participation in this formation. What sort of teacher is being addressed for example? How would she change her practice? How is the teacher positioned in writers’ intentions? Teachers could be hero/leaders innovating new approaches to teaching in response to powerfully argued research, or be mere administrators of centralised policy, doing what the school principal or government demands of them. Members of the latter group probably need more activation. Yet such teachers, or those managing their work, are not conceived as part of the research audience. Few proposals are made as regards modifications to teacher practice or policy parameters. The third set of authors focus their interest on communication. Morgan stresses interpersonal and social discourses without a specific structuring of the mathematical content. This provides a flip side to the world offered by Otte, Duval and Steinbring, but no explicit link is made. In Morgan’s formulation, it is mathematics that is left to fit in around the social world that she depicts. Ongstad offers a more holistic alternative that I have discussed elsewhere (Brown, 2008b). Briefly, Ongstad supposes that a consensus could be achieved through rational discussion. Such a possibility it seems would motivate particular conceptions of “teachers”, “students” and of “mathematics” anchored in absolute notions supportive of rationality, such as truth, fairness, usefulness, or we may add mathematical objectivity, or supposed “best practice” in mathematics teaching. Such notions, however, require a degree of consensus that may not be achievable in many formulations of life. The neo-Marxist writers,40 mentioned in the last chapter, specifically argue that it is not possible to remove the antagonisms that resist a happy ending. Consensus ultimately promotes the will of the most powerful. For example, conceptions of mathematics and the centralised apparatus designed to promote styles of teaching differ between countries. Such conceptions are time and culture dependent. Ultimately, notions such as teachers, students and mathematics are functions of particular ways of seeing the world. You cannot step outside of time and culture to offer a point of view on what they should be.

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(1971); Rancière (1998); Laclau (2005); Mouffe (2005); Žižek (2008a).

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In summary, without doubt mathematical activity lends itself to being considered as signifying activity spanning multiple discursive domains (mathematical, social, philosophical, etc). Any specific focus within this, however, can exclude some discursive aspects. Particular reductions considered have related to how students, teachers and mathematics are variously implied by the report authors. The foci of papers have predicated specific social worlds, variously delineating the domain of mathematical activity. For example, the individual student/teacher is reduced in formulations centred on mathematical structure. The positioning of subjects more generally can assume somewhat restrictive possibilities within such work. This restriction may be characterised by suppositions that all subjects would witness equivalent events in given circumstances. This applies to all people implicated in research processes, whether they are the teachers and students being researched, or the researchers themselves, as well as the audience predicated within the research design. In concluding her paper, Morgan (p. 239) suggests: “a fuller articulation of social theory is needed in order to characterise the context more systematically”. Yet, context can be understood in many ways. It is not amenable to singular encapsulation, or to being systematised without cost. Any attempt to produce a system will always result in exclusions. And by focusing on the case of a teacher working with a pupil on mathematics, the heterogeneity of the parameters that define such relations may be lost.


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Part II


[E]ach age seems to draw on its leading scientific innovation in order to reinvent understanding of the ‘social’ in that image. From Boyle’s Law to the Double Helix and then on to current evolutionary biology, a scientific discovery is hailed as a new metaphor of the social, a new determinism, and latest candidate for a reductionist ‘explanation’ (Stronach, forthcoming).

How are conceptions of science or mathematics produced? And how do these constructions of science imply specific models of human subjectivity? How might we understand the encounter of mathematics and humans in educational contexts? Part I has sought to examine these questions in a variety of locations. Mathematics and science are reflections of the humans that create them. Our apprehensions of mathematical and scientific phenomena also say something about us, and the way we understand ourselves. To compound, but also distort, this circularity, mathematics and science can be used to model human action, such as in psychology and the wider quantitative social sciences, or in the quantitative assessments of our mathematical performances. But one consequence of this modelling can be that these models are then turned around to police our actions more generally. Distorted as they may be, the models come to define humans, mathematics and their mutual encounters. What implications does this reflexive conception of mathematics and science have for education in these areas? Historical and cultural conceptions of mathematics and science are passed from one generation to the next. These necessarily reflexive conceptions can shape the pedagogical objects through which this inheritance may be secured in schools. The teaching devices deployed to deliver school mathematics can also deliver themselves. The devices present themselves as seemingly implicit or neutral components of the mathematical ideas our students encounter. The pedagogical objects they conjure, like Trojan horses, contain shorterterm attributes that more explicitly carry the ideology, mode of governance, common sense, fashions or other cultural baggage of the day. In this way, suppositions that there are preferred ways of apprehending mathematics can standardise pedagogical strategies for diverse populations across a multitude of locations. Scientific knowledge can guide but need not restrict human governance. Yet the image of infallibility

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that science sometimes confers can give science more licence than it warrants in everyday life. The actions of teachers and students are typically subject to specific curriculums and to local demands. Teachers are not things in themselves. The term “teacher” is set according to a particular social construction of that term. The expectations that go with it differ across schools and countries. As an individual teacher, I may have all sorts of personal optimistic aspirations. But, if I want a teaching job in a school, I have to fit in with the regulative structures, and understand myself through the terms of that regulation. Likewise, students are subject to specific assessment regimes that map out mathematics in particular ways and shape student learning accordingly. It is through these processes that mathematics leaves its mark on students, to make them what they are. The politics of mathematics education are negotiated in a world that is forever changing. These wider transformations are not always noted within the locality of any individual’s participation in mathematics in schools. The task of mathematics teaching can get to be framed in relation to a temporarily supposed stability of state of knowledge. Here, education is predicated on students achieving functionality within existing situations. This can lead to delivery and receipt models of education, since the mathematics to be learnt is held as fixed. There are, however, distinct limitations to education being understood in this way. There is clearly a potential role for teachers to re-craft mathematics in their classrooms. Similarly, students are not only recipients of culture but also creators of it. Fresh perspectives on mathematical situations can reform and reframe mathematics itself as part of a wider attitude shift. How might education be rethought towards supporting perpetual adjustment to new circumstances? Such adjustments implicate changes in teachers, students and mathematics. It is possible to see education as the assertion of new ways of seeing things, if only rigidities in the construction of “teachers”, “students” and “mathematics” could be eased. This part of the book focuses attention on processes of cultural renewal. It considers how renewal reshapes both mathematics and the people that work with it. Towards achieving this, the next chapter provides a contemporary account of human subjectivity. Teachers and students, understood as human subjects, are presented as fluid entities responsive to ever changing social demands. The chapter provides a substantial introduction to Lacan whose work provides a radical, highly productive and profoundly mathematical reconfiguration of how we understand human subjectivity in relation to mathematical learning. Lacan is not noted for his simplicity. For this reason the chapter adopts a need-to-know rationale so as to support the subsequent chapters where Lacanian perspectives are pursued through contemporary authors. Chapter 7 considers processes of renewal in relation to the formation of mathematics within these changing social circumstances. Chapter 8 theorises the contingent political circumstances that shape humans and mathematics in educational contexts. It considers the mechanisms that might enable the dissemination of operational adjustments to practices across a population of teachers. Chapter 9 concludes the book by drawing these themes together.

Chapter 6

The Lacanian Subject of Mathematical Learning

1 Introduction In Chapters 2 and 3 I sought to question the past primacy of notions of psychology centred on individuals apprehending mathematical objects. Objects, whether mathematical or artistic, develop meaning relationally to other objects, and in relation to the persons apprehending them. Objects do not have meaning in themselves. They are accessed through stories told about them. These stories structure the reality in which the objects appear. The objects are, in a sense, identified with images outside of themselves, built through the linguistic or symbolic apparatus at our disposal. And further, the persons apprehending such objects can be themselves understood through a variety of analytical lenses. Individuals know themselves through the stories of who they are, told by themselves, or by others. My stand-alone self, understood as some sort of cognitive or biological core, is peripheral to my everyday perception of life. In short, the logic of the psyche does not need to be understood in cognitive or individualistic terms. Meanings can be a function of socially defined filters, such as the curriculums described in Chapter 4, that in turn shape individuals and how they understand themselves. This chapter proposes to de-throne cognitively oriented psychology from its prominent location in mathematics education research. Instead the chapter builds on the conception of subjectivity introduced in the last chapter. In this move the psychological is redistributed. Individuals are defined differentially by their relations to other people, rather than as entities in themselves. The individual is seen as being caught up in more or less committed participation in a variety of social activity. That is, individuals partake in social languages that more or less fit what they are trying to say, where the individual is obliged to use these languages if they are to be included in social exchanges. In this way the human subject identifies with something outside of himself. They see themselves in the social languages, but the languages never quite fit. And through these identifications they craft their subjectivity. This could apply to all people implicated in research processes, whether they are the teachers and children being researched, the researchers themselves, or the audience predicated within the research design. This chapter seeks to introduce and better understand the relationships that link and define the social distribution of school mathematical understanding. This is T. Brown, Mathematics Education and Subjectivity, Mathematics Education Library 51, DOI 10.1007/978-94-007-1739-8_6, C Springer Science+Business Media B.V. 2011

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shaped around a conception of psychology distributed across a complex variety of discursive fields in a contemporary model of subjectivity centred in the work of Jacques Lacan. I will argue that psychology as a discipline filtered through the work Vygotsky and particularly Piaget has lost some of its poignancy in fuelling contemporary social debates. Despite its continuing success in providing widely used control technology, psychology as a wider discipline has been subject to much recent criticism. Some of this criticism emanates from within its own ranks, as a result of its perceived restrictions that cast humans as caricatures of what it is to be human.1 By surveying some contemporary work in social theory and philosophy, we would encounter what might at first seem a surprising re-conception of psychology impacting on how we understand social relations. Such work to be discussed in the next two chapters2 draws on contemporary psychoanalysis in providing an account of how humans interact. Here humans are seen as relational beings implicated in symbolic networks. I commence with a discussion of how psychology and psychoanalytical theory emerged from the work of Sigmund Freud. This will support later discussion of subjectivity and in particular the work of Lacan in conceptualising a mathematical learner.

2 Cognitive Psychology Versus Psychoanalysis An adult’s sense of self cannot have been the same from the beginning. It must have undergone a process of development. . . . Pathology acquaints us with a great many conditions in which the boundary between the ego and the external world becomes uncertain or the borderlines are actually wrongly drawn. There are cases in which parts of a person’s body, indeed parts of his mental life – perceptions, thoughts, feelings – seem alien, divorced from the ego, and others in which he attributes to the external world what has clearly arisen in the ego and ought to be recognised by it. Hence, even the sense of self is subject to disturbances, and the limits of the self are not constant. (Freud, 2002/1929, p. 5)

How might we renew our conceptions of psychology in the context of mathematics education? Let us begin by thinking about the provenance of psychology and its roots in the work of Sigmund Freud. Freud’s work was produced over a very long professional life. It was set against rapidly changing social and intellectual circumstances. The work was centred on how human beings made sense of who they are. The work was therapeutically motivated, with the theory being built around encounters with Freud’s own patients. Whilst originally motivated by activating neurological shifts in his patients, Freud’s legacy might be better understood, in retrospect, in terms of enabling patients to reassess their pasts with view to opening up and making visible alternative paths for the future.3 By understanding the

1 Parker

(2007). (2001, 2006, 2007, 2009a, 2011); Žižek (2006a, 2008a, 2008b). 3 Giddens (1999). 2 Badiou

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present and past differently alternative futures could be imagined. At root psychoanalytical thinking is predicated on a reality centred on two people talking in a doctor–client relation, for the benefit of the client. This benefit, however, can be understood in various ways. Initially, at least, Freud saw psychoanalytic consultancies as being about achieving a cure by helping the subject to overcome distortions resulting from earlier repressions. The psychologist purported to know what to do to achieve this result. But how would the analyst know? The analyst would need to be speaking from a more secure location, from a version of life that has placated its disquietude. Throughout Freud’s work a central concept was that of ego. The term developed and transformed over some 50 years and is crucial in orienting conceptions of mind, or alternatives to mind. A variety of interpretations have been provoked, leading to some especially contentious debate. Without doubt, Freud shifted in his understanding over time, between his academic papers and professional locations. Even some of his later work left his precise meaning unresolved. In his earlier work, Freud (e.g. 1991/1923) understood the ego as a biological entity and his papers established a cartographic representation of the human mind. This comprised ego, id and super ego: “we suppose that the ego is the layer of the mental apparatus (of the id) which has been modified by the influence of the external world (of reality)”.4 The super ego meanwhile is a sort of internal policeman, “an ethical-moral critical agency, . . . which develops on the basis of identifications” (Freud, A. 1991, p. 3). In this conception of the ego, psychoanalytical treatment was understood in terms of developing the ego to increasingly occupy the territory governed by the id. This was announced by the slogan: Where the id was the ego shall be. “Yes, all will be well if the ego is in possession of its whole organisation and efficiency, if it has access to all parts of the id and can exercise its influence on them”.5 The US ego psychology school from the middle of the last century, led by émigré analysts,6 embraced this version of the ego. This approach has gained an image of seeing psychoanalytic therapy in terms of calming the ego to be more conformist. The ego was understood as a biological entity to be strengthened in line with a supposed model of good citizenship, to placate any tendencies to disrupt smooth running according to the governing ideology. In this first model of the ego we would find a fairly familiar depiction of an individual human subject. The individual exists as a standalone biological entity who casts his or her attention over mathematical or artistic objects that also standalone. The psychological basis of much early mathematics education research would fall under this banner, as encapsulated in the tradition and name of the Psychology of Mathematics Education organisation, centred, as much of this work was, on individual teachers interacting with individual classes or individual pupils.

4 Freud

(1991/1926, p. 17). (1991/1926, p. 22). 6 The group included Heinz Hartmann, Ernst Kris and Rudolph Lowenstein. Freud’s daughter Anna aligned herself with this group after her father’s death. 5 Freud

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At various other points, however, including some of Freud’s very latest work, the ego was understood by him as a relational entity produced through the subject’s identification with other people and the world around.7 Lacan (e.g. 2006/1966) was the most famous of those who followed this path from Freud. He promoted a shift from bio-scientific to narrative emphases in interpreting Freud’s work. Lacan’s approach contrasted markedly with the ego psychology school developments. There was much hostility between the contesting schools. The psychoanalytic task as conceived by Lacan was not to remove supposed distortions in speech in the style of Freud. Rather, the analyst’s task was to learn from speech to see what it revealed. Such speech was scanned for symptoms of, what Lacan called, “the truth of desire”. Lacan (1990) once famously declared, “I always speak the truth”. By this he meant that whatever he, or anyone else, says reveals things about the speaker about which the speaker is not necessarily aware. By understanding how emotional flows of a patient were activated Lacan could, as an analyst, better understand how these shaped the patient’s actions in the outer world. In this way, actions were explained. These actions, however, were not corrected against a model that was supposed to be correct in advance. Badiou (2001) has recently argued that ethics and the rights accorded to Man are very much a function of how we understand “Man” and, more particularly, whom we include in “Man”. Lacan similarly resisted the countenance of specific images to which we should aspire. For Lacan the ego is an inauthentic agency derived from a delusional stance in which the human subject has an image of his or her self. I don’t know who I am, I can only conjecture. Since it is delusional, the ego is something to be challenged. The ego is both formative and informative but not something that one would want to strengthen in line with a good model. Thus, Lacan rejected a self-contained ego “that is a biological result of the interaction of psychical and social relations” that can be objectively described. The ego he envisaged “depends on the subject’s relations with others” and “is governed by fantasy, and modes of identification, and introjection” (Grosz, 1989, p. 31). Lacan’s relational, or narcissistic ego, is a result of fantasy. Lacan portrays our understanding of reality as being structured through such fantasies, or filters. How could it be otherwise? This reality cannot be perceived directly, and in some other senses is not there at all, except through its manifestation in the fantasies of individuals. In this arrangement, we would not have a standalone biological human confronting an independent object. For analytical purposes, the space would be carved up differently with no overarching perspective from an independent arbiter. That is, an expert position would not be feasible as there are so many different ways of reading the situation. The analysis would be centred on the human subject’s supposed relation to the object and the world he or she crafts around it. The insistence on

7 The following statement late in his career points to a responsive ego, rather than it being a thing in itself: “the harshness of the super-ego, is thus identical with the severity of the conscience; it is the ego’s perception of being supervised in this way, its assessment of the tensions between its own strivings and the claims of the super-ego”. (Freud, 2002/1929, p. 73)

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any image as fixed brings with it a violence to ensure conformity. Lacan, as psychoanalyst, did not purport to know the best outcome. As such the ego was something to be challenged. Treatment is seen in terms of understanding how the unconscious functions in conditioning the patient’s expressed demands. Therapy in this instance is targeted at locating the “truth” of the patient’s desire. Lacan suggests that when the analysand says “I”, the analyst should be mistrustful. That is, the image of self that is portrayed needs to be inspected to discover how it is a distortion of the desires that are being activated. The notion of the unconscious, central to the work of both Freud and Lacan, is rather absent from mainstream psychology. According to Freud, the unconscious was like an iceberg making only a small part of itself visible: “an unconscious conception is one of which we are not aware, but the existence of which we are nevertheless ready to admit on account of other proofs or signs”.8 How could we understand the ontological status of the unconscious? Would it stand up to empirical enquiry? If the unconscious does indeed exist, how does it? But empiricism itself, as commonly understood within research enquiry, defines a particular way of looking, associated with specific processes of validation. A science needs to have an object.9 For Freudian and Lacanian psychoanalysis the object is the unconscious. For Lacan the unconscious is to the human subject what, in psychology, the mind is to the brain. But do we believe in the unconscious or not? To engage with Freudian or Lacanian psychoanalysis that assumption is necessary. We might similarly ask questions as to psychology’s status as a science given that that too rests on the assumption of an object, namely the mind. The mind is a notion assumed by the apparatus of psychology and held in place by that very apparatus. Yet, mind is less prominent as a notion in some cultures, and not a tangible entity in any clear way. It can be approached from many directions. This resistance to immediate encapsulation puts it in a similar boat to the unconscious as regards its ontological status. The two aspirant sciences then, cognitive psychology and psychoanalysis, each require ontological assumptions relating to their objects. That is, there is a need to believe in each them one way or another. They each operate from their own particular form of illusion.

3 Mathematics Education, Psychology and Semiotics Piaget and Vygotsky, foundational figures in the formation of mathematics education research, still have considerable impact on how we understand the psychology of learning mathematics. Indeed they are so much a part of our language that it can be difficult to spot their functioning within us. Debate continues as to whether we should privilege the individual cognitive psychology of Piaget, or more the socially 8 Freud

(1991/1912, pp. 135–136). (1971).

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oriented model of Vygotsky. That is, do we conceptualise the task of mathematics teaching to activate and transform the minds of children, which are assumed to be responsive to such external agitation? Or do we suppose that individuals can only be understood as integral to more collective conceptions of who humans are and how they develop? Surely, such oppositions are irresolvable, though Vygotskian theory remains on the ascendance in fuelling vibrant contemporary debate. So much of this debate,10 however, is arguably somewhat distanced from contemporary cultural theory as understood within a broader social scientific domain. Psychoanalytic theory has made a few appearances in the field of education. Some writers have taken the work of Lacan to explore issues of pedagogy and learning.11 The authors in a book edited by Todd (1997) have discussed the place of desire and fantasy in teaching and learning. Britzman (1998, 2003) has used the work of Anna Freud and Melanie Klein to investigate problematical and ambivalent aspects of teaching.12 Bibby (2010) has used a psychoanalytic lens in examining children’s discomfort in the school setting, and in particular the mathematics classroom. Meanwhile, as seen, a growing number of authors in mathematics education research have become interested in the semiotic theory of Peirce, and draw on his theory in conceptualising signification within mathematical activity. This work has provided alternative conceptions of how mathematical phenomena are processed as ideas.13 Peirce, as a semiotician, however, was less interested in issues of psychology and his work predates Piaget and Vygotsky, as well as contemporary discursive conceptions of subjectivity in which subject and object relate differently. Through attention to wider discursive networks and how subjectivities are produced, Derrida and Lacan in different ways, each extend the scope of semiotic activity beyond the territory that Peirce’s work explicitly covers. Derrida and Foucault have been used extensively in educational contexts.14 For Derrida each word and by extension each text contains layers of meanings that have grown up through cultural and historical processes. As argued in the last chapter, a writer may not know it, but what he puts on paper has all kinds of other significance than the obvious and such content can be “deconstructed”. Subjectivities are effects of discursive networks and mathematical objects like all symbolic objects are produced

10 E.g.

Confrey (1991). Hollway, Urwin, Venn, and Walkerdine (1984); Felman (1987). 12 Other authors to have considered psychoanalysis in the context of education include: Appel (1996); Jagodzinski (1996, 2001); Robertson (1997); Pitt (1998); England and Brown (2001); Atkinson (2002, 2004); Brown and England (2004, 2005); Brown, Atkinson, and England (2006); Brown (2008a); Cho (2009). In mathematics education there has been a few examples (e.g. Baldino & Cabral, 2006, 2008; Breen, 1993; Brown, Hardy, & Wilson, 1993; Brown & McNamara, 2005; Tahta, 1993a, 1993b, 2007; Brown, 2008c). 13 This activity has led to recent Special Issues of Educational Studies in Mathematics (SaenzLudlow & Presmeg, 2006) as discussed in the last chapter and Revista Latinoamericana de Investigación en Matemática Educativa (Radford & D’Amore, 2006). 14 Walls (2009) and Walshaw (2007) use Foucault’s work within mathematics education contexts. Peters and Biesta (2008) discuss Derrida in relation to pedagogy. 11 Henriques,

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through their situation in such evolving networks. In Foucault’s terms for example I am a subject of discourse. I am defined by my mode of participation in discursive activity. It is Lacan’s system of thought,15 however, that provides the analytical filter that I shall pursue now and throughout the remainder of this book.16 He combined a theory of mind with theories of linguistics and semiotics to provide a sophisticated model of subjectivity that spans issues of mathematical signification. I commence with a brief discussion of how Lacanian theories of subjectivity transcend the supposed debate between Piaget and Vygotsky. I then provide a more detailed account of Lacan’s conception of subjectivity against some contemporary concerns in mathematics education. I follow this with an indication of how Lacan is positioned in relation to Peirce. I suggest that the latter writer’s undeveloped conception of human subjectivity restricts the scope of his theory in tackling broader social concerns.

4 Lacan Piaget Vygotsky 4.1 The Context of Lacan’s Work Lacan’s professional career spanned the middle decades of the twentieth century.17 He was influenced primarily by Freud but also by Saussure’s linguistics, the structuralism of Levi-Strauss, Hegel’s philosophy, and arguably the semiotics of Peirce.18 Lacan’s work also engages with the discursively oriented theory of his contemporaries in France. But, unlike the post-structuralist writers in that group, Lacan more explicitly supplements an attachment to the Real, to be discussed shortly. Saussure’s (1966) influence is explicitly and extensively cited.19 Saussure had famously argued that linguistic signs connect sound images to concepts, instead of names to things. This distinction underpinned a key shift in Lacan’s work relating to how we might encapsulate humans. That is, we apply language to the concept of

15 See

Lacan (2006/1966). Homer (2005) provides a more general introduction to Lacan. & England, 2004, 2005; Brown & Jones, 2001). 17 Lacan became famous through giving a sequence of twenty annual seminars. His main collection of writing appeared complete in translation for the first time fairly recently (Lacan, 2006/1966). 18 Peirce’s semiotics, which integrated the notion of an interpretant impacting on the meaning of any signifying system, influenced Lacan’s work more obliquely (e.g. Lacan, 2002/1961) Peirce’s influence on Lacan has been identified by Lacan’s son in law and intellectual custodian, JacquesAlain Miller, who claimed in an introduction to an earlier edition of Lacan’s “Ecrits” that Lacan’s understanding of subjectivity was inspired by Peirce’s semiotic triad of sign, object and interpretant, and by his ideas of signifying production as an unlimited semiosis. Wiley (1994) however, insists that Lacan and Peirce represent different understandings of subjectivity. Since Lacan often defied academic conventions in referencing sources, the complexity of the territory resists an easy resolution to this question. 19 Lacan (2006/1966, pp. 412–441). 16 This builds on my work in the area of teacher education (e.g. Brown, 2008d; Brown

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a human (rather than to an essentialised individual biological entity). The concept of the human can shift as a result of this application, as can that to which it refers. For Lacan (2002/1961),20 the signifier does not mark a thing. It marks a point of pure difference or movement in a discursive chain. It thus triggers a way of thinking, a system of differences, marking out the territory according to a certain schema, or points of orientation, or supposed binary divides (individual/social, mind/brain, function/relationship, regular/non-regular, sense/non-sense). In a psychoanalytic encounter it activates a particular approach to telling the story of who I am. Through developing his psychoanalytic theory into the domain of linguistics and semiotics, Lacan provided an important extension to Freud’s understanding of the workings of the human mind. Lacan’s theory produced a radical conception of subjectivity modelled on how clients presented (signified) themselves to an analyst. When I describe myself to an analyst as a client I reveal a lot about how I understand myself, and the things in the world that I see as significant. I also shed light on how I (consciously and unconsciously) understand my social relationships as defined through the sorts of conversations I link in to. I might, for example, tell you the story of who I am through what I do: I read the Guardian on Saturday, teach mathematics to trainee teachers, listen to Brian Eno with my son, discuss household matters with my wife, drink beer with friends, ignore football, write books for charming people, etc. Lacan’s theory of the subject is centred on this sort of self-definition (influenced by Hegel’s philosophy21 ). But in this self-definition I can never fully complete the picture, and the picture I present would change from one instance to another. Žižek (2000) offers Hegel’s example of a plant being akin to a human with intestines on the outside. A plant draws nourishment through its roots. A human draws nourishment through her intestines to meet her dietary needs. Yet the human also draws nourishment through her multiple connections to symbolic networks and in a sense becomes understood through the filter of her participation or implication in these networks, which are external to her.22 To understand me just 20 See

also Derrida (e.g. 1978). model draws on conceptions of reflection derived from Hegel (1977/1807). 22 Žižek (e.g. 1989, 1993, 1999) has extensively outlined Hegel’s influence on Lacan. Hegel was especially influential with regard to Lacan’s key concept of the mirror phase to be discussed shortly. In Hegel’s philosophy objects are apprehended in relation to what the cognition brings to them. But the mind itself is then conceived of as being constituted out of these apprehensions. The act of cognition results in an aspect of the object being partitioned off according to how the human apprehends it. The “in-itself” of the object becomes the “in-itself only for consciousness” (Hegel, 1977/1807, p. 55). That is, Hegel argues that the object “in being known, is altered for consciousness” (ibid.). And this aspect in the object corresponds to an aspect of the human mind, “the pure apprehension”. That is, “the pure apprehension” mirrors the “in-itself only for consciousness” of the object. Thus, in Hegel’s formulation: “Consciousness simultaneously distinguishes itself from something, and at the same time relates itself to it, or, as it is said, this something exists for consciousness: and the determinate aspect of this relating, or of the being of something for a consciousness, is knowing” (p. 52. Hegel’s emphasis). Lacan’s (2006/1966, 75–81) conception of the mirror phase echoes Hegel’s couple of the “in-itself only for consciousness” and “the pure apprehension” with regard to how a human develops an understanding of who she is. However, having taken this Hegelian step in constituting the human subject, the picture as regards how the 21 Lacan’s

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through these participations, however, does not tell the full story of who I am. The very notion of self is problematised, or fragmented in Lacan’s work. This seemingly curious move privileges humans being understood as being implicated in particular ways of talking about the world. Their cognitive and biological status is peripheral to this perspective. Such perspectives have led to re-evaluations of the definition and tasks of the discipline of psychology itself.23 Lacan reconceived Freud’s psychoanalysis as narrative-based rather than scientifically based. In so doing he provided a powerful link to contemporary discursive conceptions of subjectivity. Contemporary take up of Lacan’s work, however, has not so much been concerned with individual minds engaged in a psychoanalytic relation. Lacan’s work has more typically been taken up in cultural studies, political theory24 and philosophy.25 The reference to Lacan in these fields has been indicative of a refusal to treat individuals as analytic units outside of a social frame, since any individual characteristic would be a function of the social frame supposed. Lacan’s work bypassed the respective psychologies of his contemporaries Piaget and Vygotsky. I commence by offering a brief account of Lacan’s attitude to Piaget, before attempting to join dots in calculating his view of Vygotsky. This brief passing reference to psychology, however, serves to emphasise that Lacan set the provenance of his theory rather differently. His concerns transcend psychology as understood by most psychologists. Lacan certainly did not regard himself as a psychologist, in so far as psychology is about individual minds. He privileged the notion of the subject produced through symbolic engagements over conceptions of individual cognitive entities. Through this route humans are seen more as players in social organisations. In the context of mathematics education, encounters between teachers and students are primarily shaped by the social arrangements that prevail rather than by their specific conduct as individuals. Accordingly, this chapter is directed more at understanding how those social organisations function, rather than with how individual teachers’ might adjust their classroom practice.

4.2 Piaget Lacan is scathing about Piaget, precisely because the latter assumed that a separation between individual and social was a useful analytic strategy in understanding human apprehends objects becomes rather more convoluted since Hegel’s second object, “the pure apprehension”, becomes a function of a fantasy self. That is, for both Hegel and Lacan, all objects apprehended are tainted according to the human’s conception of who she is and, specifically, her conception of how she fits in to the social network. 23 Parker (2007). 24 Žižek (2006a); Laclau (2005). 25 Badiou (2001); Butler (1997, 2005). Others seek to explore the ramifications of combining semiotics and subjectivity within the social sciences in a more accessible way (Fairclough, 1995; Gee, 2005; Hodge & Kress, 1988; Lemke, 1995).

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learning. Piaget’s assertion of an individual child passing through successive stages denied the full cultural dependency of the child’s constitution asserted by Lacan. For Lacan, it would be hopeless to suppose that biological, physical and chemical aspects of the body could be held in by the same limits. Any attempt to synthesise a supposed individual’s activity in a multitude of discursive networks seems unlikely to succeed. Lacan castigated Piaget’s emphasis on an ego-centred analysis of learning. He preferred to interpret children’s actions as responding to what the child perceives to be demands to fit in. In short, the child, and the adult he or she grows in to, is formed around a desire to please someone, whether that be parents, friends, teachers, or later, one’s boss, husband, the tax man, etc. The Piagetic error . . . lies in the notion of what is called the egocentric discourse of the child, defined as the stage at which he lacks what this Alpine psychology calls reciprocity. Reciprocity is very far from the horizon of what we mean at that particular moment, and the notion of egocentric discourse is a misunderstanding. The child, in this discourse, which may be tape-recorded, does not speak for himself, as one says. No doubt, he does not address the other, if one uses here the theoretical distinction derived from the function of the I and the you. But there must be others – it is while all these little fellows are there, indulging all together, for example, in the little games of operations, as they are provided with in certain methods of so-called active education, it is there that they speak – they don’t speak to a particular person, they just speak to nobody in particular. (Lacan, 1986, p. 208)

For Lacan, the individual’s understanding of who she is, is encapsulated in their response to the symbolic network. This symbolic network directs and controls their acts. The network comprises the discourses that I inhabit, try out for size, explore myself through, in which I see myself reflected, etc. Lacan calls this network the Symbolic, or big Other, to be discussed in the next section. This idea first appeared in mathematics education research when Walkerdine (1988) showed mathematics educators how the concrete objects of Piaget’s analysis were necessarily implicated in the child’s conception of social relations. Whilst Piaget (e.g. 1965) centred his approach on individual cognition, Walkerdine contrarily posited both “student” and “mathematics” as being constructed in discourse. That is, “student”, “mathematics” and “teachers” are understood through specific constructions of the world. They are not things in themselves. For example, there are many ways in which we can pay attention to the presence of a student once she has arrived in a lesson described as “mathematics” by her teacher or school. We may admire: her capacity to follow some prescribed mathematical steps; her capacity to relate mathematical apparatus to real life problems; her performance in certain forms of examination; her alignment with other pupils in terms of her physical presence in the classroom (clothing, behaviour, mode of engagement, etc); her residential or fee paying status (or even gender in some schools or countries) that may have some bearing on her admission to the room; her capacity to speak the local dialect; her physical make up (body, brain, eyes, ears, hands, etc); and so forth.

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Similarly, “mathematics” or “teachers” can be processed through a wide variety of social filters that undermine any supposed unity to those terms.26 My attention always goes beyond the mere objects in my immediate apprehension, shaping my talk and thus reflexively revealing my conception of the social network and how I fit into it. The objects I see are always built into the stories I could tell about them and the worlds that those stories reveal. The individual human is in no way selfcontained in this view of the world. My very sense of how I shape my own words, objects and gestures and how I interpret the words of others is built through shared social experience.27 Lacan (2006/1966) claims that we all want to be looked at by a particular other in a particular way, and that this is part of who we are. This starts with the parents naming the child and having in mind some fantasy of who their child might be. The child’s actions are gauged against such fantasies, such as responding in particular ways, connecting with family members, fitting in at nursery/school, etc. Later this desire to fit in translates as not appearing to be out of sorts with broader social expectations, such as, behaving in a civil fashion, dressing in particular ways, forming relationships, developing practical skills, reaching certain academic standards, etc. So many of our actions are governed by norms. I am sure that if I went into my doctoral research methods class or my mathematics class and started washing cabbages, my students and colleagues would have a thing or two to say. In general I am a conformist and do what is expected of me. What matters, is who is looking at whom and how that shapes the space we share. Lacan’s notion of the human subject is shaped by guessing the answer to its own question (What is it you want from me?) addressed to the supposed demands of the symbolic network (Žižek, 2006b, p. 8). The human subject is not an entity in itself but a relational entity built through social interaction in the discursive environment where the relations have many subtle or concealed features. All mathematical activity, in Lacan’s account, would be predicated on being noticed. Actions would be shaped around that supposed noticing. The objects of mathematical activity would be implicated in and affected through the social processes that produce the demands to which the individual sees his or herself responding. For any mathematical gesture made by the student there is an attendant effect produced through this discursive embroilment. This effect transforms the gesture and the objects contained or created therein. For these reasons Piaget’s notion of each child passing through developmental stages, whatever their social location and associated demands, points to an essentialist conception of the human that Lacan rejects. 26 For example, Sammons, Day, Kington, Gu, Stobart, and Smees (2007) have carried out a major UK government funded study in to how differences in teachers’ lives, such as personal pressures or career stage, impact variously on student performance. 27 Elsewhere I discuss this in relation to the co-dependency of early language and mathematics (Brown, 1996a, 2001).

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4.3 Vygotsky Vygotsky28 shared some of Lacan’s objections to Piaget, such as, how minds are social from the outset and how children are differentiated from adults. Lacan, however, was responding to a much older Piaget than was Vygotsky. Vygotsky’s concise work under the banner of psychology preceded Lacan’s more intensive analysis over some 50 years. Vygotsky’s encapsulation of the child posits an intentional being with essentialist characteristics: the child, with the help of speech, creates a time field that is just as perceptible and real to him as the visual one. The speaking child has the ability to direct his attention in a dynamic way. He can view changes in his immediate situation from the point of view of past activities, and he can act in the present from the viewpoint of the future (1978, p. 36)

This contrasts with Lacan’s notion of the human subject seen as an effect of social discourses. Vygotsky’s work, however, has had a longer-term influence on a significant band of major thinkers motivated by a desire to create a more culturally oriented conception of psychology. Michael Cole, a student of Vygotsky’s colleague Luria, is a key figure in the area. In Cole’s conception, mind is “co-constructed and distributed” (Cole, 1996, p. 104), an agenda compatible with Lacan’s.29 Cole argues that the “central thesis of the Russian cultural–historical school is that the structure and development of human psychological processes emerge through culturally mediated, historically developing, practical activity” (p. 108). The objects created in this structure reflect and define the human’s sense of self and her relation to the world. Key elements in this construction included: “mediation through artifacts” (p. 109) where the term “artifacts” encompasses Vygotsky’s notion of tools, including linguistic tools; “historical development” in which already-created tools are arranged for rediscovery in each succeeding generation; and “practical activity” (p. 110) where it is assumed that psychological processes are grounded in humans’ everyday activity. Radford (2006a, p. 7) provides an extended contemporary analysis of how this might be understood in mathematics education. He analyses a classroom example in which a wooden ruler, a number line and mathematical signs on a piece of paper are all seen as artefacts, which “mediate and materialise thinking”.30 Meanwhile, artefacts such as works of art, scientific models, architectural designs, food recipes and other such cultural forms provide filters for understanding for successive generations as part of historical development. Further, my sense of the world is a function of what I do. Lacan and Vygotsky would agree on much of this. Both would claim that humans feed off the linguistic apparatus that surrounds them. They would differ, however, in their understanding of how humans relate to this symbolic mediation. Vygotsky’s 28 E.g.

Vygotsky (1986, pp. 12–57). the longevity of these enterprises there has been little communication between them, with a few exceptions (e.g. Bruss, 1976). 30 See also Blanton, Westbrook, and Carter (2005) and Goos (2005). 29 Despite

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psychological notion of Zone of Proximal Development31 attends to the localised case of children trying to learn from adults, through which children are brought into the social world. This notion contrasts sharply with Lacan’s assertion that humans’ alienation from language is built into their very constitution as subjects. The language we use to describe people never quite fits with their own sense of reality. And they can be alienated from the very apparatus we use to include them. For Vygotsky, the child’s realization of his separateness from society is not a crisis; after all, his environment provides both the form and content of his personality. From the start, dialogue reinforces the child’s grasp on reality, as evidenced by the predominantly social and extraverted nature of his earliest egocentric speech.32 For Lacan, on the contrary, dialogue seems to function as the alienating experience, the stade du miroir phase of a child’s development. (Emerson, 1983, p. 256)

The next section shows how Lacan’s model of child development pivots on the notional point at which the child identifies with an image outside of herself (such as a mirror image) and says “That’s me”. And the opposition this creates between the “me” and the “I” results in a “permanent hunger” (ibid.) to close this gap. As Emerson continues: “The child is released from this alienating image only through discovering himself as subject, which occurs with language: but this language will inevitably come to him from the Other” (ibid.). That is, the only way out of the restrictive caricature of self is to accept the turbulence of participation in discursive activity. Yet any attempted identification with specific discourses or ideologies is tainted by the individual’s desire to please. For Lacan, the difficulty in fit, the alienation, can be experienced as a positive condition, releasing an individual who has grown out of the linguistic clothing bestowed upon her. Yet further for Lacan, the alienation results from the human baby being born too early, committing her to a long dependency on her parents and a susceptibility to the structures that govern her fellow humans. He suggests that the act of her parents naming her throws her into a complex social network where a place for her has been prescribed in so many ways. The growing human’s assertions of self will always be a response to this initial set of expectations. In turn her responses reflexively create through time an evolving account of the external world to which she continues to respond. Thus, “Lacan argues that the child’s first experience of society is that of alienation because ‘the subject depends on the signifier’. . . the child must make a choice: either be alienated by the signifier and enter society or reject it and remain unrecognizable in social terms”.33 Yet as we shall see later “social terms” do not last forever and the individual’s growth is linked to social growth and changes in the terms of reference through which individuals and their worlds are conceptualised. For this reason Vygotsky’s Zone of Proximal Development is not the neutral place one might imagine. Bibby (2008, p. 38) argues:

31 Vygotsky

(1986, pp. 174–208). is discussed by Gade (2009). 33 Cho (2009, pp. 39–40), quoting Lacan from Seminar 11. 32 This

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The seductive imagery conjured by Vygotsky’s metaphor of the “zone of proximal development” leaves hanging the nature of the zone and obscures the space it occupies, it allows us to ignore the difficulties and resistances which the learner will encounter and develop. Indeed, it demonises them – any resistance must be wilful and destructive: why would anyone want to resist benevolence and kind intentions? In doing this, the metaphor encourages us to ignore any differences between the learner and the teacher and seems to suggest that the learner’s differences will be unimportant and willingly subjugated to the teacher’s benevolent intentions. Similarly, the metaphor locates the teacher in a place of idealised omnipotence – an impossible place from which to teach or relate, a place from which the teacher’s own difficulties and resistances, perhaps difficulties with particular students, become intolerable and unspeakable.

Teachers and students variously act according to perceived demands. Teachers may or may not identify with particular aspects of the curriculum. Children may or may not connect with the account of the world that the teacher seeks to present. For a teacher working according to a specific curriculum there may be forms with particular meanings that she feels she needs to communicate. Yet, the child’s apprehension of the form may be out of line with the teacher’s intended meaning. Tahta (1989) draws attention to the problematic nature of “meaning” and related words like “understanding” or “concept”: Very often, these words are used with a single, unique sense: the meaning, this understanding, that concept. This, then, inevitably leads to metaphors of ownership and control: obtaining the meaning, having the understanding, getting the concept. And, consequently, of course, there will be the mathematical descaminados, the shirtless who have not understood, who never get the concept.

I recall an incident in a mathematics lesson where a young boy was moving matchsticks around a table like his fellows in a seemingly focused way. Yet, these movements bore no relation to the numerical intention as outlined by the teacher on the blackboard. He was merely imitating the physical actions of his fellows who were acting in a seemingly focused way. The mathematics education research literature is full of examples of students not quite understanding the meaning that their teacher assigns to particular mathematical configurations.34 Psychologically, these might be regarded as localised problems resulting from the pedagogical definition of the task and the child’s inexperience with making the required sort of intellectual connection. Educationally, the learning situations might be seen as reproductive. But education does not have to be about children learning specific ideas from a teacher.35 There is the potential for processes of learning as distributed across populations to produce cultural renewal, transcending the parameters that may have been envisaged. The next section seeks to delve deeper into these issues through a fuller account of Lacan’s terminology.

34 E.g.

Radford (2003). and Passeron (1977).

35 Bourdieu

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5 The Lacanian Subject 5.1 The Mirror Phase In Lacan’s model, there is a cost attached to conceptions of psychology that process and understand the human through prescribed registers. In these models students get judged through an idealised and restrictive account of what they should be, as though a broader truth could be captured in those local circumstances. Piaget paints an idealised version of the child, against which the individual child will be gauged in teacher practices. Vygotsky paints an idealised version of society, of which a child would surely want to be a member. Psychology as a discipline has commonly worked in the name of the supposed normality of the status quo.36 The shaping of the individual through inherited or imposed artifacts, tools or words, Lacan would argue, can begin to misrepresent the human’s sense of self. This demands or supposes compliance with a false caricature. This results in the individual being spoken about through the filter of particular ideologies, which serve some people better than others. Lacan insists that there is always a failure of fit between the psyche and the discursive tool kit. That is, there is a gap. This gap prevents the individual having a completed sense of his or her self. And for Lacan this gap locates and activates desire. This desire is brought about by a promise of perfection, or new exciting territories. This desire, however, often mistakes its object. For example, the child misunderstands what would be achieved in getting the mathematics correct and this misunderstanding affects the nature of the child’s motivation. We may well have fantasies of who we are and fantasies of the world that we occupy, fantasies emanating from different aspects of our fragmented selves. But for Lacan, there is always something beyond these fantasises and this supplement interferes with the operation of our fantasies. These fantasies structure our reality but never fully account for this reality. Freud claimed that the human does not have a sense of self from the outset. He argued that some new “psychical action” must take place to constitute the ego. He did not say what this psychical action was. As indicated, in the earlier Freud quote, much of a baby’s sense of where her body ends and where the world starts is clearly undecided. What happens to create a more mature account of these limits? Even adults have their fair share of delusions about who they are, their personal limits and how the world impacts on their actions. Freud’s own patients were often seen as having developed unhelpful accounts of their histories that piloted them through both the real and imagined obstacles of their lives. Lacan proposed a more tangible “psychical action” to fill the gap in Freud’s account. He saw his action relating to the child building an understanding of her self. This understanding was encapsulated in the child assuming an image of who she was. Lacan linked this assumed image to the child seeing herself through her 36 Parker

(2007).

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own mirror image. That is, the child conceives of herself as being reflected in a mirror. This conception, however, results in a transformation of that image to contain just those bits that the child supposes it to be. The child, in seeing “herself” in the mirror, is identifying with an image outside of herself, in characterising the image as being “me”. This brings to her body a unity that she had not previously conceived. Here the notion of the mirror need not be taken too literally; it could be that the child recognises “herself” in another child. This marks the stage at which the child becomes able to conceptualise herself as complete with more clearly defined limits. Lacan, however, argues that this recognition is deluded, symptomatic of an order that he calls the Imaginary, to be elaborated shortly. The assertion of “that’s me” comprises the assertion of the ego. Lacan,37 however, claims that the ego is “an inauthentic agency functioning to conceal a disturbing lack of unity”. The mirror image is most certainly a caricature, at best a holding device, for an ego that would never be fully complete. And this promotes a deluded sense of self. This is why Lacan had no interest in strengthening the ego through analysis. The importance of the mirror is in what it misses as well as in what it succeeds in reflecting. If the process of analysis enables the patient to generate a lot of stories, it may be possible to detect a pattern of territory that gets avoided. The analyst then may ask why that territory is avoided, and examine if it is linked to repression of stuff the client would rather avoid. More generally, Lacan sees the human subject as having a conception of self located in a fantasy of that self. Žižek (2000, p. ix) offers an example to illustrate this. He suggested that the director Woody Allen, in the wake of his separation from Mia Farrow, appeared in a number of broadcasts to be like one of his own neurotic and insecure film characters. Žižek asked whether we could assume that Allen had put his own character in to the films. Žižek answered his own question in the negative, preferring to suggest that Allen was in fact copying a certain model that he had elaborated in his movies. He argued that real life was imitating symbolic patterns expressed at their purest in art. That is, human subjects do not have access to their true selves. They are decentred. Rather they play according to a fantasy of who they are or of who they think they should be. Elsewhere,38 and in Chapter 8, I provide examples of new teachers seeking to craft their actions according to the strictures of school and government requirements.39 The image of who they should be is specified in great detail. In due course, new teachers identify with such specifications. They see themselves in those terms. And, like their employers and regulators, begin to assess their own performance (find pleasure even) in those terms.40 Ultimately, their practices are only noticed to the extent that they conform to the official image.

37 Quoted

by Leader and Groves (1995, p. 24). and McNamara (2011). 39 See also Valli and Buese (2007). 40 See also Hanley (2007); Nolan (2007). 38 Brown

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As seen earlier, many US mathematics teachers may see themselves working according to a specific conception of the Reform agenda,41 an agenda sometimes embraced by mathematics education researchers outside of the USA.42 They conform to an image of a “teacher” outside of themselves as it were, as a result of this image seeming to provide a version of life to which they can subscribe to, or identify with. Perhaps such identification43 with a movement can provide individual teachers with a sense of collective purpose, and with a kitbag of resources. Reform functions as an ideology that can provide an effective point of reference for teachers and researchers evaluating the social connectivity of their respective work. Teachers can aspire to evaluating their own practices in those terms. Researchers can judge their work according to how it supports the cause of Reform objectives. That is, the vocabulary of the ideology provides the apparatus or technology through which one is recognised, and through which one learns to recognise oneself.44 Learning or teaching effectively in those terms only guarantees subscription to that ideology. It only guarantees successful mathematics learning teaching if one happens to subscribe to that ideology and the infrastructure and values that supports it. In the second chapter, I provided an account of some Ugandan students being asked to describe a variety of shapes in terms of their circularity. This was not a straightforward task for them as words for basic geometric shapes were not part of the students’ indigenous vocabulary. In their language, words for objects such as square, triangle (sikwera and turyango) were only recently introduced and based on their Western equivalents. The students were caught between using their own indigenous language and the language of the Western-oriented curriculum, taught by a Western teacher (or encountered through exposure to wider media), which defined their mathematical lessons. Their own language was shaped around descriptions of everyday activity, rather than around the classification of objects, which is more common in English. For example, the closest approximation to the word circle was the word oriziga which, roughly translated, meant circular. As a result, objects such as spiral, globe, oval were all oriziga. Words such as “circle” and “sphere”, however, were muzunga (white) words and entailed labelling objects in a mode of abstraction less common in the students’ indigenous language (Teacher: So if I give you an orange, a football, this small ball, and this stone, what shapes are they? How would you describe the shape in vernacular? Student: They are all a circular and shaped like the small ball.) In this classroom situation the students’ conceptions of mathematics were recognised, with respect to the curriculum, only to the extent that they could express themselves in a Western register of mathematical terms. A student’s mathematical self was only acknowledged to the extent that it was expressed though a Western filter. In this regard, correctness in mathematics is culture-dependent according to

41 van

Zoest and Bohl (2002); Remillard and Bryans (2004); Remillard (2005). (2001). 43 Laclau (2005). 44 Žižek (2006a); Brown (2008d). 42 Skott

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how mathematical phenomena are framed.45 This point will be discussed further in the next section. As seen, Husserl (1936) had sought to enquire how geometry came into being. He concluded that without the anchorage of words (that is, culturally specific constructs) it was quite difficult to conceptualise.46 The cross-cultural example, however, provides an excellent analogy in respect of students in more or less cross-cultural situations, around the world and in any country. There is a common requirement that students translate their experiential involvement in the world through unfamiliar linguistic or symbolic registers. And as in Žižek’s assessment of Woody Allen, they get accustomed (or not) to occupying a particular mode of being defined by a language outside of themselves. As Žižek (2001a, p. 75) puts it in describing a similar example, “he does not immediately display his innermost stance; it is rather that, in a reflective attitude, he “plays himself”. Even Piaget (1972, pp. 20–23) questioned the possibility of a clear distinction between mathematical and empirical knowledge. And so long as empirical experience impacts on mathematical knowledge the cultural or intersubjective dimension of human perception will be implicated even in the most abstract manifestations of mathematical knowing and the most generalised accounts of mathematical knowledge. Psychoanalytic theory emphasises relational conceptions of the human subject. In Lacan’s formulation, teacher and student cannot be understood in isolation from each other. Rather like the relationship between analyst and analysand, they are co-formative, each seeking something from the other. Their specific relationship is symptomatic but also generative of the culture in which they reside.

5.2 The Imaginary, the Symbolic and the Real Lacan’s notion of subject is based on three orders: the Imaginary, the Symbolic and the Real. I have mentioned these in passing but they might now be gathered together in a more systematic way. In short, the human subject can only start with his own illusions, or those that we share (the Imaginary), which he makes sense of through the Symbolic (which traps him in to a borrowed language that confers inclusion and constraint), which never quite captures the Real. I will now try to unfold that a little. The Imaginary might be seen as self-identification, or rather, the creation of images of oneself.47 The notion of a young child looking into a mirror is Lacan’s iconic example. Seeing a whole self, an image of completeness, gives the child a sense of mastery. It produces an inside and outside to oneself, a “me” and “not me”. But this has some cost since the child is identifying with an image outside of himself. The crucial point here is that the individual, looking in on himself, sees an image

45 I

have discussed this issue extensively elsewhere (Brown, 2001). (2002/1961) has provided an extensive account of the evolution of numerical counting, although his efforts exceeded this reader’s patience. 47 This term perhaps has some affinity with Vygotsky’s notion of egocentric speech. 46 Lacan

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(a fantasy) of his self. Not the “real me” as it were. This identification, however, lays a foundation for a more symbolic engagement with the world. Bhabha (1994, p. 77) pinpoints this: “The Imaginary is the transformation that takes place in the subject at the formative mirror phase, when it assumes a discrete image, which allows it to postulate a series of equivalences, samenesses, identities, between the objects of the surrounding world”. The image of self, as characterised by a name, fixes an egocentric image of the world shaped around that image of self. The Imaginary then can be summarised as the individual looking in on a fantasy self to create a sense of unity. But, more broadly, the entire human race occupies fantasies of itself, which the whole of academia does not succeed in fully displacing. The Symbolic, meanwhile, relates in some respects to Althusser’s notion of “interpellation”48 discussed earlier. The Symbolic encapsulates this individual looking out to a fantasy world, where his vision is mediated through ideological framings. The identification with this fantasy is alienating, however. I cannot fully connect with it since it operates within a “previously formed language”.49 I can swim in the Symbolic but cannot quite grasp it. The Real might be seen as the space in which the Imaginary and Symbolic are enacted. The fantasies built within the Imaginary and the Symbolic fail to capture the self or the world.50 This brings into play a space for desire motivated by the supposed possibility of closing the gaps between the Imaginary and Symbolic and the Real that hosts these fantasies. The Real, by definition, resists symbolisation. The resources of language cannot mop up the whole of experience. Otte (2006, p. 16) gives a flavour of the Lacanian Real in the world of mathematics education where he remarks “It is impossible . . . that everything means something. Not everything in the world is reasonable and intelligible. There exist pure feelings or brute facts that seem to escape any reasonable explanation. We therefore cannot describe or explain everything”. Any account of science has an outside. The world of science and “reasonable” explanations can always be expanded or revised.51 Yet science and “reasonable” explanations are not the only ways of making sense of the world and the outside can sometimes be more interesting. Lacan’s system of thought combines Imaginary, Symbolic and Real in a Borromean knot of mutual dependency, where no element is privileged and each has a contingent status. Each order impacts on the other two orders.

48 Althusser

(1971, p. 174). (1971, p. 213). 50 The self and the world are an assumed binary divide, but they are a couple that imply each other and structure the space in a specific way (self/rest of world). 51 “The subject is dependent on the articulated chain represented by science’s acquired knowledge. The subject has to take his place there, situate himself as best he can in the implications of that chain. He constantly has to revise all the little intuitive representations he has come up with, and which becomes part of the world, and even the so-called intuitive categories. He’s always having to make some improvements to the apparatus, just to find somewhere to live. It’s a wonder he hasn’t been kicked out of the system by now. And that is in fact the goal of the system. In other words, the system fails. That is why the subject lasts.” (Lacan, 2008/1967, p. 81). 49 Althusser

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At the Imaginary level we have signifiers in the social space of mathematical learning activity each shaped and characterised by its name: “student”, “teacher”, “circle”, “functional relation between x and y”, “3×4”, “teacher accreditation criterion”. Each such “mathematical” object has a name and sense attached to that name (Imaginary). That sense, however, is different for each person, teacher, pupil, at different times, in different circumstances, within alternative curriculum documents, in different countries, etc. Any role such terms have in a wider symbolic network is filtered through the Imaginary perspective on them, even though that symbolic network is at least partly generative of the individual human subject. Each individual person has a different sense of how any object is positioned in relation to the discursive network. Such terms relating to perceived objects can be suggestive of character and function but sensual aspects can shield alternative or more generalistic aspects.52 At the Symbolic level these words are articulated in particular ways according to the multiple sets of cultural rules that prevail, such as a teacher giving a task to a student and expecting a specific style of response. Different discourses (e.g., Reform mathematics in the United States, cognitive psychology, constructivism) make use of different sets of words or use similar words differently. Other examples might include, grammatical rules followed blindly, cultural rules that allow partners to communicate, unconscious prohibitions that stand in the way of certain paths of action (e.g., not wanting to be seen as being too clever in class, not wanting to reveal sensitivities, keeping up proper appearances, etc.). The Symbolic provides a yardstick against which I can measure myself and understand myself in relation to the social frame. That is, the self is not egocentric but defined in response to social expectations. This dimension may obscure access to clear meanings yet such suppositions of clear or static meanings require reductive accounts of the life we are seeking to capture and serve through mathematical activity. In Lacan’s account, the meaning of words is constantly in flux refusing to settle for the purposes of unambiguous communication shaped around such clear or static meanings. The rules and conventions (the Symbolic order) that govern mathematical activity are context dependent insofar as in different mathematical domains (school, university, shopping, engineering, economics), different questions are asked, different things are emphasised, different assessment instruments are applied, different fashions prevail, and different objects are invoked. Yet, the understanding of the social space as the enactment of these rules does not mop up everything. There are other factors governing mathematical activity: emotion, intelligence, mode of compliance/resistance of pupil, school context, affective dimensions of mathematics in particular learning sites, conceptions of learning relation; the quest to please the teacher; the satisfaction derived from connecting with a teacher; the perverse pleasures achieved by those who paint themselves as mathematical illiterates, etc. My sense of self, that is, my ego, however, is necessarily a function of how I see the world. The dual fantasies, of myself, of the world, are

52 cf.

D’Amore (2006) provides an example in mathematics.

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inseparable in Lacan’s model. These fantasies are processed respectively through the Imaginary and the Symbolic. My fantasy of self, my Imaginary identification, is the delusory mirror image I have of myself. Meanwhile, my fantasy of the world is processed though the Symbolic, the ideological apparatus that surrounds and engulfs me. Desire is present in both of the fantasies as the gap (or the surplus) that separates the fantasy from the reality it seeks to capture. In Lacan’s terminology, I never attain the object of my desire. Indeed, my desire mistakes its object, ever caught between a language that does not quite fit. The notion of desire, perhaps better translated as “wish”, explains my motivation in terms of something that I want to acquire, even if I am not quite sure what this thing is exactly. In human sexuality, for instance we may know desire as a promise for the future, an unexplainable and seemingly youthful sparkle that draws us towards someone for a satisfaction not quite specified in advance. Indeed the mystery element is part of the appeal. Desire might also be expressed in relation to how I wish my future life to unfold more generally. Yet this desire and the way it shapes my progress into the future can never quite be captured. There is something beyond my reach that excites me, a surplus beyond that which I can express in words. Or, perhaps alternatively, if I take the desired outcome to be the yardstick, this surplus might be seen as a lack. That is, there is something that prevents me reaching the desired outcome. My explanation of how I could get what I want always misses something. This quandary, of course, applies to my developing sense of self and any work I might do on understanding what that is. I may have a sense of who I am striving to be. Perhaps, however, this is a positive reading of myself that I am trying to actualise in my current actions. I have a sense of the world and how my idealised self fits into this. My sense of where I am going is pleasurably tainted by the promise it holds. And such promise can enable individuals to act, not as types, beyond culturally defined parameters. Lacan sees both fantasy and gap (surplus) as positive elements. The fantasy structures the reality that the individual perceives and lives. The gap serves as the motivation that gives the fantasy meaning for the individual. For Lacan, there is a world “outside” of these two fantasies, what he calls the Real, but this is an “outside” beyond the scope of the individual’s grasp. The Real, as seen, resists any symbolic account. It is “that which remains the same in all possible (symbolic) universes” (Žižek, 2006a, p. 26). The mutual formation of Imaginary and Symbolic shapes itself around the resistance of this Real. We can never quite capture ourselves in language but it is this “surplus of the Real over every symbolization that functions as the object-cause of desire” (Žižek, 1989, p. 3). Such is the domain of the Real.

5.3 The Subject of Mathematics Education Research A key argument of this chapter is that mathematics education research as a discipline often restricts its perspectives to familiar pathways. Firstly, one often finds an assumed naturalness of names at the Imaginary level (e.g. “students”, “teachers” and

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“mathematics”) such that certain subjective dimensions are privileged over others. Secondly, it can entertain narrow conceptions of the Symbolic interaction (required performance of given mathematical procedures, tight specification of social roles). Thirdly, it can neglect the Real except as understood through external factors (e.g., affect, curriculum power relations, intelligence seen merely as better performance on a given register). The Real cannot be ignored. Otherwise, the research would apply to “types” rather than the diversity of childhood.53 Lacan refuses the possibility of critical distance, or of an objective standpoint from which we can view the truth of the world. And for Lacan the only truth is the truth of desire. By that he means the emotional flows activated by engagement with symbolic structures and the attempts to close the gaps between those structures and the lives they seek to capture. Mathematics and the structures that guide its conduct in schools are not neutral activities. They provoke diverse responses where emotion supplements any compliant action. Such desire never settles and is never satisfied. And as a consequence conceptions of mathematical objects, the people working with them and the social arrangements that host such conceptions and such people continue to evolve and shape themselves without the comfort of an imminent ideal outcome. The triad of Imaginary, Symbolic and Real, resists convergence to supposed idealisms (e.g., idealist geometric objects co-existing in a Euclidean framework, a standardised conception of a pupil learning mathematics in school governed by a robustly conceived curriculum, or a systematically conceived conception of mathematics operating in a harmoniously functioning society). Or more prosaically, to take some examples from the Special Issue discussed in the last chapter, Lacan’s system of thought would resist idealisms such as “cognitive systems that . . . give access to mathematical objects” (Duval, 2006, p. 103) or “real mathematical activity (that provides) representations of mathematical entities” (Otte, 2006, p. 11). Such preferences are culture dependent and time dependent and may well be symptoms of blockages to certain alternative ways forward. They suppose that cognitive and mathematical entities are things in themselves that can be seen from an objective standpoint.

6 Restrictive Registers In the last section, and in Chapter 2, the work of some Ugandan students could only be recognised to the degree it could be read through westernised registers. That is, only some of their mathematics would be assessed. Only certain aspects of the students themselves would be noticed. I linked that to the more generally

53 Such reductionism is captured in Lacan’s definition of the signifier: “The signifier . . .functions as a signifier only to reduce the subject in question to being no more than a signifier, to petrify the subject in the same movement in which it calls the subject to function, to speak, as subject (Lacan, 1986, p. 207).

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experienced requirement that students translate their experiential involvement in the world through unfamiliar linguistic or symbolic registers. It is this difficulty of translation that underpins this section’s more complex theoretical treatment of signification. Lacan’s construction of subjectivity builds on Peircian semiotics, and in particular Peirce’s notion of the sign.54 Lacan’s work reaches more extensively towards contemporary debate on subjectivity. He states in his characteristically slippery way, that The whole ambiguity of the sign derives from the fact that it represents something for someone. This someone may be many things, it may be the entire universe, in as much as we have known for sometime that information circulates in it . . . Any node in which signs are concentrated, in so far as they represent something, may be taken for a some-one. What must be stressed at the outset is that a signifier is that which represents a subject for another signifier (1986/1973).55

Žižek (1998, p. 74) argues that Lacan’s famous last sentence might be understood through an example of a chart at the end of a hospital bed. The old style hospital bed has at its feet, out of the patient’s sight, a small display board on which different charts and documents are stuck specifying the patient’s temperature, blood pressure, medicaments, and so on. This display represents the patient – for whom? Not simply and directly for other subjects (say, for the nurses and doctors who regularly check this panel), but primarily for other signifiers, for the symbolic network of medical knowledge in which the data on the panel have to be inserted in order to obtain their meaning. One can easily imagine a computerised system where the reading of the data on the panel proceeds automatically, so that what the doctor obtains and reads are not these data but directly the conclusions that, according to the system of medical knowledge, follow from these and other data.

The signifier, a graph maybe, represents the subject, a patient in the bed, for another signifier, a doctor or nurse reading the graph with view to it impacting on a specific dimension of their subsequent actions. That is, we are not attending to patient or medic as holistic subjects. Rather we are considering the patient through the restricted registers of the patient, with particular symptoms, and a medic only interested in those symptoms (perhaps with view to setting a correct dosage), according to the wider system of medical knowledge. To take some other examples: Some of my Manchester medical colleagues designed a virtual environment to train doctors for major incidents in which trainees had to cut through horrific surface appearances to get to the key “numbers” (blood 54 This is most apparent in Lacan’s (2002/1961) notoriously difficult Seminar Nine on Identification, referred to above. 55 The first and last sentences may be read as mapping closely into Peirce’s famous definition. “A sign, or representamen, is something which stands to somebody for something in some respect or capacity. It addresses somebody, that is, creates in the mind of that person an equivalent sign, or perhaps a more developed sign. That sign which it creates I call the interpretant of the first sign. The sign stands for something, its object. It stands for that object, not in all respects, but in reference to a sort of idea, which I have sometimes called the ground of the representamen”. (Peirce, Collected Papers, vol. 2, p. 228).

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pressure, heart rate, temperature, etc.) they needed to prioritise patients. As another medical example, a doctor was concerned that patients presenting with headaches may or may not have something seriously wrong with them. All he had to go on was the way in which the patient, unaccustomed to making sense of their bodies in medical terminology, expressed their experience of pain into a set of more familiar words.56 The doctor found himself slipping into all sorts of stereotypical assumptions about the patient’s lifestyle and associated risk factors from the way in which they spoke. Meanwhile, in a New Zealand government study that I directed, Pacific Island teachers sometimes felt that they were only noticed within school contexts “insofar as they fitted the conventional image of a New Zealand teacher”.57 Finally, I was on a training course for recruiting staff where a case of alleged unfairness was being considered. Unfairness itself is not necessarily against the law and so the victim in the case was obliged to identify racial elements of the unfairness to create a legally viable complaint, since racial discrimination is against the law. He was obliged to express his complaint within the available legal discourse of racial discrimination even though this did not match the exact character of his central complaint. Similarly, a child in a mathematics lesson can only express their understanding, however that term is understood, through the symbolic apparatus that they have available. What they say is often only heard to the extent it aligns itself with the assessment regime. And for all the emotion and insight they experience, teachers may only be accredited if they demonstrate externally defined competencies. For instance, mathematics teachers might be understood and recognised by their employers only insofar as they fulfil the remit of a government policy directive. That is, they are seen as no more than a statistic. In this regard, Stephens (2007, p. 32) reported on his involvement as an education authority manager in a Neighbourhood Renewal Strategy where his work was prescribed by a central government directive: “Overall the targets set in 2000 are aimed at . . .ensuring that at least 25% of pupils in every school and 38% in every local education authority can achieve five or more GCSEs a grades A∗ -C (British 16+ examination)”. Such targets were supplemented by policy apparatus specific to mathematics: For schools; National Curriculum, National Numeracy Strategy, Standardised Attainment Tests, Standardised training programme for teachers administering the Strategy, Government Inspections; For training colleges; National Curriculum for Initial Training, Numeracy Skills Tests for teachers, Government Inspections for training colleges. Within such a frame children’s mathematical work may only be appreciated to the extent that it fits within the teacher’s immediate objectives as defined within such apparatus. Common to all these examples, a specific sign relation only works or applies within a particular discursive register.58 The ambiguity is located on how we understand the subjectivity or identities of such actors when filtered through such limited

56 This

is reminiscent of Wittgenstein’s (1983/1958) discussion of pain. Devine, Leslie, Paiti, Sila’ila’i, Umaki, and Williams (2007, p. 115). 58 In a specific “ground”, to use Peirce’s terminology. 57 Brown,

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registers. The “someone” or “subject” are in a sense, in any instance, discourse specific. They only register as entities in certain modes of discourse (or ground). And in the context of mathematics education we might wonder how this reduction functions when understood in the context of teachers, learners and mathematics. The last chapter focused on the reductions of teacher, learner and mathematics might result from semiotic analysis of mathematical activity. What do we hold still in seeking to understand how those words signify? And what cost does that suspension have? The convenience of holding a supposed meaning of a word or symbol still for the purposes of communication privileges some interests over others. For example, Duval’s (2006) Piagetian treatment of Peirce’ theory asserts that mathematical objects are only known through signs yet underplays the productive role of the situated individual who gets to know or produce these signs. Semiotic systems are culture dependent and subjectivity is entwined in each dimension of this dependency and what you see results from this entanglement. The ambiguity for Lacan I believe is centred on how the “someone” is predicated in semiotic activity. What aspect of the whole person is activated (or brought into being) in any given semiotic configuration? That is: How are they created as subjects? Which discursive aspect responds, or appears, and why? Yet Lacan’s work on subjectivity provides a crucial albeit less travelled link that connects Peirce’s semiotic insights to the philosophy of the later twentieth century, where the production and analysis of subjectivity have assumed centre stage in many important debates. It is this more extensive engagement with discursive networks and their production of subjectivity that fuels Lacan’s concerns (and in turn Žižek’s, 2006a account of these in a broader social arena, to be discussed in the next chapter). This connection is hinted at in Lacan’s curious suggestion that the “someone” could be the “entire universe”. This term is made yet more obscure by the clause “in as much as we have known for sometime that information circulates in it”. Here I am drawn back to an example I discussed earlier in which Žižek saw Stephen Hawking as a metaphor for contemporary subjectivity. Connectivity to the Internet re-centres our sense of self, our sense of reach and our scope of receptivity. Being wired into this network echoes Hegel’s suggestion that we have intestines (or perhaps tentacles) on the outside. Our connectivity to the network affects how we process information, make gestures, impact on others, etc. Contemporary understandings of subjectivity centred on human immersion in discursive and signifying activity provide a backdrop to Lacan’s pre-Internet assertion that someone might provide access to the entire network of discursive activity. Everyone is implicated in the discursive construction of society and everyone draws on that construction. And thus: “Any node in which signs are concentrated, in so far as they represent something, may be taken for a some-one”. Yet between the “entire universe” (which I take to mean the universe of the discursive domain as defined by participation in it) and the example I have offered of a medic with a specific brief there are many possibilities, each defined by their specific mode(s) of engagements with the discursively created world. It is important to maintain this range of possibilities in analysing mathematical phenomena and the teachers and learners working with these. If we were to suppose

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that research were in the business of providing formulas for action, there would be a need to take great care in ensuring that the diverse entities of learner, teacher and mathematics are not reshaped or reduced for the purposes of fitting such formulas. Or perhaps, rather, we need to be attentive to how such reshaping and reductions, inevitable as they may be, transform our understandings of what we are trying to achieve with groups of people or with individuals.

7 Conclusion Theories of subjectivity are crucially important to mathematics education research. Lacan’s three orders, the Imaginary, the Symbolic and the Real that make up his conception of the subject provide a pliable analytical framework for inspecting the wider ramifications of subjectivity in mathematics education research and beyond. We must be sensitive to the alternative needs of learners, their teachers and the communities with which they associate. Whilst science can support the creation of control technology, we must not let control technology blind us to the wider potentialities of science. Mathematics is a function of the community that embraces it and evolves in relation to the needs expressed and tasks performed. But so often, as communities, we reflect on ourselves in commodified form, packaged for certain types of scrutiny. This chapter has sought to illustrate how psychological and semiotic apparatus utilised within mathematics education research can fix our understandings of both teacher functionality and the dimensions of children’s mathematical learning into time- or culture-dependent configurations. The very definition of psychology as a field of study has over-emphasised “American individualist ideology”,59 resulting in a partisan and sometimes dysfunctional control technology. I have provided examples of common psychological theories and accounts of signification, which produce overly reductive conceptions of the student and teacher within the fabric of mathematics education research output. Piaget supposes progression through a sequence of predetermined and seemingly innate stages. Neo-Vygotskian theory supposes unproblematised engagement with the tools of society. Peircian semiotics, as utilised in some mathematics education research, separates mathematical signification from subjectivity. In our everyday activity in the name of mathematics education we are guided by terms that can petrify the phenomena to which we give our attention. Phenomena can become to be understood only as a function of outmoded control technology. Pedagogical devices can begin to shield or replace the mathematical concepts they were designed to reveal.60 Lacan’s conception of subjectivity, whilst complex, provides a way of thinking differently. “Teachers”, “students”, “mathematics” and the

59 Wertsch, 60 Brown,

quoted by Newman and Holzman (1993, p. 31). Eade, and Wilson (1999).

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frameworks that define them (curriculums, policy initiatives, research frames, learning theories, public expectations, employer demands, international comparisons, mathematics education journals, etc.) can be conceptualised as mutually evolving entities resulting from the play of discursive activity. Yet, for Lacan, discursive activity is more than mere words and symbols. Rather, the operation of discursive networks shapes us all, along with the very world in which we live and the objects contained therein. And in so doing this weaves us into these multiple realities with varying degrees of comfort in positions that confer many and sometimes conflicting perspectives. We need to live with reductionism that results from attaching labels to life but we also need to learn how specific forms of reductionism serve particular priorities and interests. Mathematics education research has had a tendency to be targeted at the everyday practices of teachers, teacher educators and researchers. This activity itself masks us from the limited impact that it has. Such research can only ever reach a small proportion of such individuals, restricting any process of dissemination. Even the capacity such individuals have for impact on broader states of affairs in mathematical learning must be questioned. Politicians and government administrators can often have more influence on the shape of mathematical learning in school through dealing with populations rather than individuals, social organisation through policy directives, rather than face-to-face encounters. Hard-edged language is often an instrument of such governance and research is often commissioned in such terms. That language can have real impact on who people are. Mathematics education research needs to be attentive to how such handling of populations impacts on broader conceptions of subjectivity and how we might impact on the factors that shape this subjectivity. And for this reason our engagement with theory requires apparatus that reaches out to what may at first seem more distant concerns. At first sight there are real issues of transfer for Lacan’s theory that is predicated on analyst and client in a room discussing the life context of the client. This is hardly a promising starting point for a social model spanning broader populations, or for considering mathematics education. Lacan, however, was a contemporary of those major thinkers of the late twentieth century who fore-grounded issues of subjectivity. His own work was a major part of those developments and this chapter has sort to show how that work offers a more refined alternative to mainstream psychological perspectives in mathematics education research. More recently, Slavoj Žižek and Alain Badiou, two of the world’s leading philosophers still writing, have developed Lacan’s work in other domains. Their work has had much impact in defining the opening of the new century, and speaking to large audiences regularly today.61 It is these writers that will help me to take my discussion forward from here. Their work brings Lacanian theory firmly in to the present in support of social and philosophical projects that have much more extensive reach, but with very specific local interpretations. My use of Lacan’s theory

61 Žižek and Milbank (2009); Žižek (2009a, 2009b, 2010); Badiou and Žižek (2009); Badiou (2009a, 2009b, 2011).

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in earlier chapters and elsewhere has drawn extensively on more recent moves that have proliferated his work. Indeed, these contemporary writers have offered useful pathways to Lacan’s work and its convoluted style. In the next two chapters I discuss the relevance of their work in building theoretical frameworks for understanding mathematics education. I cannot provide anything other than a brief glimpse, as both Badiou and Žižek are highly prolific authors, whose ideas are subject to on-going revision. In the next chapter Badiou’s theory provides a model for considering how cultural knowledge evolves. I show how this can apply to mathematical learning, which can be seen productively as active participation in cultural renewal. Meanwhile in the subsequent chapter Žižek provides a framework for examining how political structures shape changing social relations and hence educational and mathematical practices.

Chapter 7

The Cultural Renewal of Mathematical Learning

1 Introduction The last chapter sought to introduce a contemporary understanding of subjectivity in support of the theme of this second part of the book: cultural renewal. A key point of the chapter was to resist essentialist or fixed conceptions of the human subject, since they might fix the conceptions of mathematics that we feel able to entertain, or renewal trajectories specified around objects reconfigured in accounts of renewal. A particular target was Piaget with his reliance on an image of a human passing through successive stages of development common to all. Lacan’s conception of the subject on the other hand is endlessly responsive to new conditions. Lacan argued that Piaget’s conception of the child’s formation avoided the centrality of social demand, initially from parents and later from the wider social sphere. In Lacan’s formulation, the changing symbolic environment and the Real to which it uneasily relates unsettle the individual’s already mistaken understanding of who they are. If that chapter focused on the changing status of the human individual, this chapter offers a theoretical treatment of how we understand mathematics being shaped by its changing cultural parameters. I introduce Badiou, who draws on Lacan in providing a model of how notions of truth underpin cultural conceptions of knowledge. The account of knowledge and mathematics adjusting to changing cultural circumstances provides analytical apparatus crucially relevant to mathematics education research. It revisits some of the mathematical learning themes of Chapter 2 towards introducing sturdier apparatus to account for the mathematics/human interface. I commence by situating Badiou in the contemporary philosophical terrain in relation to mathematics. This is followed by a discussion of how Badiou’s distinction of truth and knowledge might be understood in the context of mathematical learning within a radical education agenda. A more theoretical account is then provided of how Badiou draws on Lacan. The subsequent section uses this theory in support of an account of how mathematical knowledge evolves. I then consider these themes in relation to more detailed examples of mathematical learning and associated research.


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7 The Cultural Renewal of Mathematical Learning

2 Mathematics and Badiou’s Philosophical Terrain Philosophies of mathematics, often centred in positivistic notions of mathematical truth, objectivity and stable meaning are not especially disposed to the predominant philosophies of the day centred on what has been called a “linguistic turn”. And as mathematics educators we are obliged to examine the interface of mathematics and humans where we encounter historical and cultural issues relating to the ontological status of both mathematics and humans, where each is implicated in the other. Badiou (2007) argues that the linguistic orientation is privileged in all three mainstream contemporary philosophical traditions of the twentieth century: hermeneutics (Heidegger, Gadamer), analytic philosophy (Wittgenstein, Carnap) and postmodernism (Lyotard, Derrida, Foucault). In each of these three cases truth, insofar as it is entertained, is processed through language to produce knowledge. And knowledge is not always in the business of portraying a world defined by consensual harmony or where final answers might be available. The traditions each position mathematics in different ways but perhaps analytic philosophy comes closest to reaching out to mathematics as a logical system centred as the tradition is on logical and grammatical analysis to demarcate which utterances have meaning and those that do not. Yet even within that tradition the notion of meaning underwent substantial revision when Wittgenstein (1983/1958, p. 20) equated the meaning of a word with its use in language. Attention to usage of language and by implication social practices necessarily brings with it interpretations that project us beyond the strict categories of logical analysis. Further, this tradition, with its emphasis on logical statements, has distinct limitations when it comes to discussing the human subject. It is this sort of projection that muddies the water between logical frameworks such as mathematics and how those frameworks are encountered by human minds and the meanings they bring to them. The world comprises diverse people who do not all use language in the same way. Usage varies from group to group, country to country, time to time. As a result, within many contemporary perspectives, “the universe is no longer comprehended as a whole regulated by the universal conditions of language: all that remains are lateral connections between partial domains” (Žižek, 1998, p. 83). In recognising this complexity Badiou does not follow or reject any of the three orientations mentioned. Rather he incorporates them into a wider model. Addressing these themes we are necessarily confronted by some profound questions: In which sense can the logical or reality frameworks exist independently of the communities that created them? How much can mathematics have an existence outside of discursive practices and the audiences and participants that these practices provide? How might we understand mathematical truth in its encounter with humans? The supposition that mathematics has an objective existence has been prevalent in many contexts. What could this mean in an educative context? This mathematics itself has already been susceptible to cultural and historical turbulence in its very formation. There is on-going controversy as to the mathematical models that explain our connection to the physical world. Disputes within philosophy sometimes focus on whether there is a greater truth beneath our many descriptions

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world. If there is some greater truth how might we understand that, and what sort of apparatus might be available to access it? Badiou’s philosophy offers an approach to tackling this issue against the backdrop of the wider questions introduced.

3 Badiou: Truth and Knowledge 3.1 Truth, Knowledge and Mathematical Activity Badiou’s canvas extends into the territory of potential futures. His work has begun to occupy the space vacated by the coterie of French philosophers who dominated intellectual life in the second half of the twentieth century (Derrida, Foucault, Lacan, Deleuze, Levi Strauss, Lyotard).1 Badiou’s lineage can also be traced through Bachelard, Lakatos and Althusser who each saw science as a practice marked by the production of new objects of knowledge.2 Badiou was slightly handicapped internationally by his work having had to wait quite some years for translation into English. His most recent declaration, however, that there are bodies and there are languages except that there are truths (2009a, p. 4)3 has been translated fairly punctually, coinciding with the release of much of his back catalogue and his rapidly growing popularity and newsworthiness (e.g. 2009b). This assertion builds on his concern that linguistically oriented twentieth century philosophy is excessively centred on what he calls democratic materialism, where there are only bodies and languages. Here the world is understood and created through reference to contemporary conceptions of the unit of the human, organised according to language-centred analyses. In short, these contemporary philosophies, according to Badiou, neglect truth. For Badiou truth is to be distinguished from knowledge. It has nothing to do with correctness, since correctness relates to a particular view of the circumstances we face. In his conception truth cannot be substantiated or represented in culturally derived media. Social practices necessarily portray particular ways of making sense of the world, more or less held in place by those exerting power within any given situation. These practices use familiar and specific styles of language to depict particular conceptions of bodies. Piaget’s account of childhood would be an ideology in this sense, associated with specific teaching practices centred on supposed modes of mathematical apprehension by humans conceived in a particular way where formal mathematics is beyond reach until the child has advanced to a certain stage. Truths are exceptional to stories restricted to bodies and languages. Badiou’s quest is to

1 Badiou is now widely regarded as France’s greatest living philosopher, who in turn regards Lacan as “the greatest of our dead” (quoted by Hallward, 2003, p. 11). Badiou’s work is extremely complex yet he writes in a clear style and is well served by quality translations and secondary texts (e.g. Hallward, 2003; Feltham, 2008). 2 Feltham (2008, pp. 20–21). 3 Badiou (2011) provides a more recent outline summary of this major work.

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understand how knowledge4 is shaped and how it evolves in relation to a truth that is never finally secured in a culturally tangible form. In science, his conception of truth concerns the invention of theoretical parameters. “Truth can only be reached only through a process that breaks decisively with all established criteria for judging (or interpreting) the validity (or profundity) of opinions (or understandings) . . . access to truth can be achieved only by going against the grain of the world and against the current of history”.5 As Badiou puts it very succinctly and prominently at head of his new book “Truths have no substantial existence”.6 Any attempt to pinpoint truth ultimately disappoints us. This attitude to truth and knowledge can be illustrated through a preliminary classroom example to touch base with our purpose for introducing Badiou’s abstract theory. Modular doubling entails number bracelets where successive numbers are doubled but with varying upper limits. Modulo 11 and modulo 13 produce simple bracelets 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1. . . 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1. . . Having introduced these examples students are invited to explore other modulo, which can produce rather more complex results (Fig. 7.1). Subsequent work initiated by a student included investigation of length of bracelets for odd modulo, leading to statements of generalisation such as: “All odd number modulo produce simple bracelets” or “For any n, that is prime and in the form n = 2k − 1/m (where k, m ∈ N): No. of cycles = n − 1/log2 (nm + 1). Size of cycles = log2 (nm + 1)”.

Fig. 7.1 Bracelets for 18, 20 and 22

4 “Knowledge” in Badiou’s terminology has some similarity to Althusser’s notion of “ideology” discussed earlier. It might also be seen as being a truth pertaining to a “discursive regime” as depicted in Foucault (1991, p. 55). See also Foucault (1989). 5 Hallward (2003, pp. xxiii–xxiv). See also pp. 209–221. 6 (2009a, p. 5), his emphasis.

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The activity is characterised by experimentation and critique. The task for the student is not so much about checking the correctness of particular configurations as being about deciding upon productive avenues to follow. For example, it may be discovered that modulo of the form 2n generate trees. The student here was guided more by some sense of aesthetic completeness or qualitative unity7 than by some pre-ordained model of correctness foreseen by her teacher. He was motivated by encountering new ways of seeing, both within and beyond the situation in question. The purpose was not to learn about doubling modulo as such. Rather, the task was to learn about the production of generalisations, or to become more adept at predicting interesting lines of enquiry, or organising the space according to the degrees of certainty we have about different aspects, perhaps managing the complexity of multiple results. The task for the students was about researching how mathematical forms can function as analytical apparatus to produce insights, more than supposing that there are specific existing mathematical ideas to be learnt. In Badiou’s terms, the results can be cashed in as knowledge.8 The quest however is motivated by truth. This attitude to investigation provides an example of why Badiou insists that truth is to be distinguished from knowledge. For Badiou (2009a), knowledge relates to a diverse range of domains, or “worlds”. “If what we call the world or the universe is some kind of totality, then we must agree it is primarily a totality made up of subsets, of domains of objects” (Gabriel & Žižek, 2009, p. 15). Badiou includes in his examples subsets, such as “words, gestures, violence, silences, expressions, groupings, corpuscles, stars, etc”.9 Each such domain provokes a form or system of knowledge, within which we can say that statements are verifiable, but we are unable to say that they are true. Importantly, in Badiou’s account, these alternative forms of knowledge cannot be harmonised into one overarching frame. Truth always exceeds the sum of all the systems of knowledge. Whilst there have been many attempts at different times to capture truth, successive generations invariably rewrite these attempts. So many failed attempts, but perhaps we learn to fail better. The idea of truth, as commonly understood in mathematics, has proved to be inadequate in some areas of mathematical thought. Euclidean geometry, as presented in Euclid’s Elements, was considered an embodiment of truth. It was believed that the Elements contained truths about space. The creation of Non-Euclidean geometry showed that it was not the case and that it was inadequate to continue speaking about truth in mathematics.10 In Badiou’s formulation truth stimulates the generation of knowledge, but truth itself cannot be captured as knowledge. This can help us to understand the onward march of mathematical discovery that results in earlier work being assigned new meanings alongside the mathematics of

7 Sinclair

(2006). (2009a, p. 75) sees this cashing in as “an atonic exposition of the sciences, whose real norm, in the final analysis, cannot be any other than that of the profits which are expected of them (their lucrative ‘applications’”. 9 Quoted by Hallward (2003, p. 94). 10 I thank an anonymous referee for this example. 8 Badiou

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today. And in mathematics education, Badiou’s quest can provide a framework for situating the many knowledges that we have produced to describe human encounters with mathematical truth. Badiou’s work would suggest an orientation to teaching centred on experimentation and critique rather than on the delivery of existing knowledge.

3.2 A Radical Education Project Alternative philosophical traditions handle human relationships to language in many different ways. Gallagher’s survey of some of these traditions, mentioned earlier, shows how the various options support alternative conceptions of education. For educators, a key question presents itself: Do we conceptualise our task in terms of initiating our students into existing knowledge? More radically, our task can be seen as troubling the certainties of those knowledge. It may explore the limits of those certainties, to keep open the prospect of our students accessing a truth that transcends the parameters of our own teaching. Gallagher’s radical options (e.g. Foucault) are defined in terms of students reaching beyond the frameworks that their teachers offer to produce a new future governed by structures unavailable or inconceivable in the present. All my books . . . are if you like, little tool boxes. If people want to open them, or to use this sentence or that idea as a screwdriver or spanner to short-circuit, discredit or smash systems of power, including eventually those from which my books have emerged . . . so much the better (Foucault, quoted in Patton & Meaghan, 1979, p. 115).

Without doubt Badiou emphasises an educational attitude centred on renewal. Yet, unlike Foucault who considers truths in discursive formations, Badiou holds onto a notion of an eternal truth. What could this look like within mathematics education? The doubling modulo activity provided an example of an investigational attitude to learning mathematics. This suggested a way of conceptualising learning governed by experimentation and critique, rather than by a predetermined checklist of concepts or procedures. Whole Class Interactive Teaching,11 as described in Chapter 4, provides a possible pedagogical framework for presenting this style of work. As seen the teacher’s task is to facilitate an evolving group account of a mathematical situation. The end point is not necessarily agreement, or an outcome that the teacher had in mind in advance. The challenge is to build an understanding of the analytical apparatus that mathematics provides rather than to see this analysis as being shaped around a set of pre-existing ideas that suggest particular outcomes to the discussion. This discussion provides students and teacher with a frame within which they can begin to share ways of talking in relation to mathematical stimuli. This enables students to conjure their own mathematical objects from encapsulations of certain aspects. The activity is anchored by mathematical correctness, whilst some of the

11 Harrington

(2002).

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pathways more familiar in school are loosened to allow the mathematics to function as a filter on truth. It is explored as a filter, or as an approach to structuring the way we see things. It is not just about learning the “content” of school mathematical knowledge. Mathematics is seen more as a sort of grammar, guiding the path of thinking, rather than as a vocabulary comprising words with fixed meanings. The emphasis of the activity is more on process and structure rather than on objects or mathematical content. As another brief example from my own teaching, I ask students to imagine a path, “any path at all that you like”, connecting two specified but random points on a 4×4 grid lattice. Having collected multiple suggestions from the class that normally reveal the infinity of possibilities I then ask for suggestions for rules that future paths must obey. Sometimes this produces an interesting problem to pursue, perhaps where students work independently or in groups to classify the different sorts of possibility. Or I set challenges such as “Can we fix a set of rules such that there are between 15 and 20 possible paths?” This enables students to explore how rules might be variously set to produce alternative results. Here, the pedagogical emphasis is as much on the adjustment of parameters as on the production of results. Such options require that teachers forego a comprehensive understanding of what their students should be able to achieve. Student achievements, which can offer fresh perspectives or nuances, may not be in the teacher’s register, or be in a terrain deemed too risky. The achievements may also be in the future, recognised and framed only in hindsight, and not be foreseen from within the activity. For the lessons of school can take many years to settle and take us by surprise when we least expect it.12 The territory of mathematics can be marked out in many different ways. The familiar ways in schools have often become familiar as a result of their tangibility. And pressures to compare educational locations can result in mathematical ideas being framed to enable such comparisons. This can result in certain pathways through mathematics becoming known as mathematics. In a rare reference to mathematical learning Badiou (2009c, p. 39) suggests that “we must consider taught mathematics to be inexact, giving us no idea at all of what mathematics is as a subjective and historical process. What is taught is not mathematics but only its locus”. He contrasts the “invented mathematics” of the mathematician with the “taught mathematics” of the university professor of mathematics. For the former, he proposes an attitude “out of place with respect to” pedagogy, “that provokes a thorough reshuffling”. Pedagogy does have a necessarily reproductive dimension since, as Badiou continues: “it is pointless to try to live without repetition”. Sometimes we do merely want to understand earlier formulations. For example, examination formats in English schools present mathematics as a wholly predictable enterprise, where mathematics gets to be known through certain sorts of questions that reoccur year after year. This can result in teachers teaching to the test, shaping their account of mathematics accordingly. This establishes particular states of representation that become normalised. This normalisation of

12 Britzman

(2003).

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particular states of representation, as in exam-maths, portrays a reduction of mathematics for a particular mode of cultural consumption. The normalisation fixes a mode of existence for any given mathematical object (as an item in exam-maths, as a mathematical fact, as comprehended by a child at a particular developmental stage, etc.). It also normalises modes of apprehension and intuition. As another example, in the history of physics, the prevalence of Newtonian physics made Einstein’s work seem counter-intuitive, but then in turn the wider acceptance of Einstein’s work made the subsequent work of Bohr and Heisenberg seem counter-intuitive. Resistance to such normalisation may open alternative states of representation, or different modes of existence, that deliver alternative results. By extending the range of mathematics beyond this reduced state we may enhance our analytical range and depth. Badiou’s philosophy is centred on our capacity to open alternative states of representation and to benefit from the opportunities that they allow. Earlier, I argued that geometric objects and the analysis that go with them change through time in their relations to human perceptions. The reality of geometry is a function of how humans perceive it. Analyses of these objects can change, as these analyses are included in new ways of making sense of the world. Perceptions of objects and their provenance may change as a result of new technology. This is true of much of the physical sciences, especially in relation to the realisation of radically new macro-space conditions, where the earthly ones are but a localised “exception”. Mathematics education needs to be responsive to these changing parameters. Atkinson (2007, see also 2011), who discusses Badiou’s work in relation to education, introduces a notion of “pedagogy against the state” where “immanent to pedagogy is a movement against itself” to “embrace new ontological possibilities for the learner and teacher beyond established states of representation”. Atkinson sees Badiou’s philosophy as opening the door to such possibilities.

3.3 The Four Domains of Truth and Meaning For Badiou truths are created with respect to four domains, science, art, politics, and love, each of which host eternal truths.13 Yet truth (singular) relies on the co-operation of these four domains. In discussing scientific truth Badiou (2009a, pp. 10–16) provides the example of number theory derived from the ancient Greeks in which conceptions of infinity are fully present but where the statement “There are as many prime numbers as numbers” is for an ancient Greek, even for a mathematician, to speak an entirely unintelligible jargon (p. 11). Badiou continues: 13 Badiou’s

Maoist sentiments are wholly secular with no space for religion allocated in his four domains of truth. The individual is understood as being very much part of a collective, where the collective is privileged as a unit of analysis over the individual. Humans are a function of a collective way of seeing, as reflected in their discursive productions. In this perspective students and their teachers more or less behave themselves according to the norms that prevail and this sets a frame for mathematics to be enacted in generally unsurprising ways.

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First of all, no infinite set can exist for such a Greek, because everything that is really thinkable is finite. There exist only sequences that continue. Secondly, what is contained in some thing is less than the thing. This can be stated axiomatically (it is one of the explicit formal principles in Euclid’s Elements): “The whole is greater than the part”. Now prime numbers are part of numbers. Therefore there are less prime numbers than numbers (ibid).

Yet after Cantor’s radical re-depiction of infinity in the twentieth century the statement is permissible. So “‘prime numbers’ does not have the same meaning in Euclid’s Greek language as it does in ours, since an ancient Greek could not even comprehend what is said about prime numbers in the modern language” (p. 12). Similarly, with regard to the domain of art Badiou compares two artistic depictions of horses: the prehistoric cave paintings in Chauvet, and a Picasso piece produced 30,000 years later, but before the Chauvet paintings were discovered in the last century. Badiou argues that “horse” cannot have the same meaning in both. Yet Badiou (2009a, p. 18) contends there is “an eternal truth, which is at work, between the Master of the Chauvet cave and Picasso”. In each of these four domains truth is always new yet eternal. Knowledge, however, constantly needs to adjust to new circumstances.

4 The Lacanian Premises to the Work of Badiou ‘truth is always new’, and if it is to be true, it has to be new. So you have to believe that what truth says is not said in quite the same way when everyday discourse repeats it (Lacan 2008/1967, p. 17). I always speak the truth. Not the whole truth, because there’s no way, to say it all. Saying is literally impossible: words fail. Yet it’s through this very impossibility that the truth holds onto the real (Lacan, 1990, p. 3).

It is Lacan’s revitalisation of the term “truth” that motivates Badiou. The first quote might be best understood in relation to Lacan’s psychoanalytic therapy. Therapeutic sessions entail interrogating the stories the client has constructed about who they are. Such stories may have become fixed in unhelpful ways. This rigidity can prevent movement to a new story that might suit the client better in new circumstances. Here “truth” is not served by the ways in which these stories have settled in to common sense. We can all sometimes ignore the fact that stories that we tell of ourselves in the world are failing us. Perhaps, we have not yet learnt other stories with which to replace them. Psychoanalysis is premised on rewriting the storyline of our lives. We can organise our pasts differently. We can mark out events in different ways. But it is not about replacing an old story with a new story but more to keep the story writing alive. These strategies can open up different ways of understanding possible futures. Some writers extend this in to our capacities to collectively rewrite history to highlight alternative historical trajectories.14 Earlier accounts could restrict the avenues 14 Pavon

(2010).

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open to us. We may now experience an imperative to replace these accounts. But, initially at least, this imperative provokes a more vital or experimental approach to the generation of accounts of life. This experimentation can reveal the operation of language itself and the realities this operation provokes prior to a new settling as a commonsensical construct. But, the client’s attempt to tell a better story also always ultimately fails as any sort of final version. Yet, these failures are informative, since the attempts provide greater insight into the truth that guides our knowledge. In Lacan’s model, truth always slips away as we try to grasp it. He is persuaded that there is a truth that keeps us alive beyond the reach of settled forms of knowledge. This analytical approach can provide a paradigm for the onward march of mathematical discovery. For example, Einstein’s work on relativity enabled us to see in a new light Newton’s work on gravity. Truth emerged differently from within the older model. Likewise, human engagement with mathematics is necessarily processed through specific cultural manifestations that can distort access to any notional mathematical truth beyond. In schools, curriculum definitions of mathematics might be shaped more by the supposed needs of good employees in the current economic model, rather than according to more humanist aspirations shaped by intellectual endeavour. For the student depicted above working on doubling modulo, an excessive concern with curriculum demands may hamper his investigation since he would be less in control of deciding which aspects of his work would be regarded as important. Badiou’s philosophical and political ambitions are modelled on such psychoanalytic processes. Lacan (2008/1967, p. 27) characterises these processes as being about the detection of “holes in discourse”. And it is the hole that Lacan’s antiphilosophy punctures in philosophy and culture more generally that motivates Badiou’s own more systematically philosophical pursuit. For Badiou, yesterday’s stories never quite live up to the truth. He sees the task of philosophy as being to challenge the consensual status quo. We must see the limits of our current story and move on. Any linguistic or symbolic form will, after a while, settle in to a particular way of making sense that implies a world around, which will serve some people better than others. Euclid’s model does not represent deep space. But, it might be in the interests of curriculum authorities to reduce geometrical thinking to Euclidean principles to set school mathematics in a way that can be assessed more easily. That way teachers and students can be told what to do in precise terms within a given regulative structure. Yet in regulating mathematical thinking it is being compromised according to a partisan cultural agenda. Novelty does not lend itself to assessment procedures since assessment is usually about repeating past forms. In Badiou’s view, an important task of philosophy is to locate holes in the functioning of everyday language to challenge this imbalance and the attendant disquiet from within. Euclidean geometry is not a complete picture and the specific assertion of Euclid’s geometrical world distorts mathematics as a whole. Any designated subset of geometry is knowledge, not truth. In this understanding philosophy is necessarily subversive, out of synch with the discourses that fuel cultural life. The detection of a hole results in attempts to reshape our engagement with life, perhaps through a more experimental attitude to language that achieves alternative

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experiences of truth. That is, senses of “how things are” are disrupted forcing adaptation to a new understanding of reality governed by radically different parameters, invisible to the eye of someone immersed in the previous reality. Philosophy is thus the on-going attempt to fit a new language that never finally settles. But truth is what it seeks. This truth is always new. It is the never-ending attempt to close the gap between truth and knowledge that drives change, and brings about cultural renewal. For Lacan, and for Badiou in his wake, our attempts to encapsulate truth in knowledge never work. But we keep trying. And for as long as we keep trying, for as long as we remain open to new possibilities, we are alive. Truth always slips off our new stories in unexpected, surprising and enlightening ways: “eternal truth is enveloped by different conceptual and linguistic contexts” (2009a, p. 12). The task of philosophy is to challenge the consensual status quo in which particular linguistic forms have settled in to particular ways of making sense that pinpoint an ideology that serves some better than others. The resistance that Einstein experienced to his idea of relativity locates such an instance of adjustment to a new order. In Lacan’s sense, Einstein spotted holes in the previous ways of making sense of the world. Yet this did not stop Einstein strenuously resisting the next generation and their quantum mechanics. The example of the math wars in the United States is another instance of cultural adjustment where those who favoured an emphasis on computational methods resisted the emergence of constructivism. Badiou’s philosophical quest is to locate holes in the functioning of such common sense or ideological formulations that trap us into habitual modes no longer serving us as well as they might.15 In this understanding, philosophy is necessarily subversive, out of synch with the discourses that fuel cultural life, and govern cultural action. The location of a hole results in attempts to reshape our engagement with life, perhaps through a more poetic experimentation with language to achieve alternative experiences of truth. Meanwhile, Badiou (2005, p. 9) sees the task of education as being “to arrange the forms of knowledge in such a way that some truth may come to pierce a hole in them”. In Badiou’s terminology, the location of such a hole, and the consequential activation of a new pursuit of truth, is called an “event”. “Event”, however, is understood on a grand scale with Badiou’s preferred examples being St Paul’s revelations on the road to Damascus, the events in Paris of May 1968, Galileo’s construction of mathematical physics, or a life changing experience in love. That is, perceived overarching senses of “how things are” are disrupted, forcing adaptation to a new understanding of reality governed by radically different parameters, invisible to the eye of someone immersed in the previous reality. Atkinson (2011) however convincingly focuses on event on a smaller scale, linking the term to a piece of learning by a child or teacher that changes their frame of reference.

15 Mason (2002) has done extensive work on how habitual thinking patterns prevent us from seeing

creative solutions in problem solving situations.

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For this part of the book, with its focus on cultural renewal, the intention is to challenge ways of thinking that trap us into past ways of making sense. Enabling students to think creatively against the grain is an essential ingredient of that. Cultural renewal needs to be distributed across the population as a wider attitude to teaching and learning. Badiou’s notion of the subject is defined by, or comes into being through, an encounter with a way of being. The individual participates in historical formation rather than apprehending something fully formed. Badiou’s notion of subject has nothing to do with biological bodies. Rather, he links subjectivity to breaking with tradition, to being alive, to being more fully a facet of structures guiding all of our actions, but actively transforming those structures to guide us all. Understood in this way, a learner of mathematics would be seeing and experiencing mathematics as coming into being. The learner would be experiencing mathematics as part of herself, a self that is also evolving in the process. The encounter with mathematics is a formative experience for the individual. But her participation in the collective enterprise that is mathematics also forms mathematics itself. New awarenesses cannot be attributed to the intentional mathematical quests of individuals taking a lead. One might think alternatively of cultural renewal being consequential to a more widespread innovation being introduced in to a community with more or less unpredictable results. Mathematics, for example, might be seen as an attitude to curriculum policy, and the attempts made to influence practice across populations of teachers and their students. It is such innovations that activate new modes of mathematical engagement or educative encounters across that community. I have provided a book length account of such an attempt in England.16

5 Understanding Mathematical Learning 5.1 Being and Multiplicity How might we utilise the technology within Badiou’s (2007, 2009a, 2011) philosophy to help us make sense of these processes of mathematical acquisition? Badiou’s approach is highly complex. He uses mathematics to develop a systematic conception of how things are, shaped on Cantorian set theory. Yet the basic structure of his philosophy is quite accessible and the technology of his approach provides some highly novel and productive instruments for understanding learning. Badiou does not discuss learning as such but his philosophy promotes a radical attitude to learning. This attitude to learning is readily applicable to mathematical learning, in the manner of the examples provided above. I request that the reader be patient during this brief subsection and the next for they outline the core ideas of two lengthy and complex books. This short discussion opens the door to some major analytical apparatus.

16 Brown

and McNamara (2011).

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For Badiou, “being” “is the sheer multiplicity of the world, a plurality of stuff (facts, states of affairs, etc.) that cannot be reduced to any single organising principle” (Critchley, 2008). Badiou’s platonically oriented theory is entirely formal, rather than a specific determination of action, or concept of life. Mathematics, within his theory, is seen as language reduced to its bare minimum function as a structure. Mathematics is a model for all linguistic construction where “being is to be ultimately explained by mathematics” (ibid). Mathematics is ontology, the very state of being. Yet, somewhat radically, there are no transcendental mathematical objects (2007, p. 7), since he regards mathematics as a grammar organising how we are. Mathematics can only become a body of knowledge within its cultural manifestations. There are no mathematical objects outside of culturally defined knowledge. There is no mathematical knowledge outside of specific cultural forms in culturally defined locations and the social determinations those locations prescribe. Rather, mathematics is a way of structuring life that can be used to produce entities (or generalisations) in a variety of ways within alternative knowledge frameworks. I am labouring these points as they are highly significant for mathematical learning in schools. There are no mathematical entities as such. There are only entities that cultures have put there. It is always a cultural decision to produce objects. Objects do not exist without this decision. To use more familiar educational or sociological terminology, all mathematical objects in schools are social constructions.17 Mathematical objects in schools are a product of curriculums and of the cultural life that generated those curriculums. And that cultural life brings baggage with it. The objects are only there because someone has chosen for them to be there. Objects can only exist in worlds understood as tangible entities. To assert an object also results in the outside of the object being asserted. The assertion of an object also asserts a conception of a world that contains it.18 This idea needs further unfolding.

5.2 Counting as One Badiou’s (2009a) most recent major project is encapsulated in the title of the book Logics of Worlds.19 Worlds are multiple rather than singular. There are multiple ways of introducing logics into any given world. Badiou (2007) commenced with a sheer multiplicity of elements in a pure state of being. In this state the elements are not anywhere. These elements can be combined in subsets of that multiplicity to create or define unities. Badiou’s assertion is that any such unity, or object, derives from an operation of “counting as one”. “Unity is the effect of 17 Berger

and Luckmann (1972). (2009c). 19 This section draws on two translations of books by Badiou (2007, 2009a), although the French originals were published nearly 20 years apart. The terms used by Badiou, and the frameworks to which they referred changed between the two books. I have sought to simplify this text by avoiding any discussion of Badiou’s shifting frameworks and associated terminology. More recently, Badiou (2011) has provided a brief and remarkably clear outline of his philosophical project. 18 Badiou

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structuration – and not a ground, origin, or end”.20 That is, an object is produced by counting a set of elements, within a supposed world, as one object. This operation brings the object into existence within a world. And in a sense it also brings the world into being. The assertion of an object asserts the world that is the outside of that object. The world is itself a result of a wider “counting as one” (of the elements of that world). Everything in a world, so defined, is identifiable within existing forms of knowledge.21 An assertion of a new object comprises an assertion of a new configuration. This configuration entails “counting as one” a set of elements within the multiplicity. This can be achieved through defining a novel combination of elements. Any element can itself be a set and a potential member of other sets. And within any assertion of a set, yet further possibilities are created, resulting from the construction of subsets producing yet more new entities. This very proliferation itself defies any final stability in the universe. For this reason, there can be no settling or convergence in the meaning of the constituent terms. Crudely, objects can always be recontextualised. Their meaning is a function of the context. Badiou contemplates a structured multi-dimensional infinity comprising sets of infinite sets. Yet forms of knowledge are predicated on a world, comprising specific sets of terms within this world. Such forms of knowledge might be disrupted as they readjust around the ever-expanding set of sets being counted as one. This expansion reveals objects not previously identified within that overarching multiplicity. Yet this world, as described so far, is ultimately a closed system where all objects are identifiable within existing forms of knowledge. But how might such an abstract theoretical perspective support the examination of mathematical learning, or more generally the human apprehension of mathematical forms? For a student in school, and probably mathematical learning more widely, mathematics can generally be understood through the pursuit of noticing or asserting generality, a notion resonant with “counting as one”. Much mathematics education research has been predicated on enabling students to experience generalisation to emphasise that mathematics has power beyond mere particularities.22 The noticing of a generality results from an operation that apprehends, or perhaps creates, a set of mathematical elements (e.g. points, numbers, shapes) as a unity. This can be geometric as in seeing a circle as a type of ellipse, or algebraic such as assigning a formula to a specific numerical sequence. These objects cannot necessarily be apprehended in an instant. Indeed the apprehension may result from a gradual assimilation of the object’s components and qualities and how these are combined in forming the object. I may compare new sets with a selection of previously known sets. For example, I may contrast the operation of a newly located function with more familiar functions. The progressive apprehension of the supposed object becomes part of

20 Clemens

and Feltham, introduction to Badiou (2006, p. 8).

21 Likewise Badiou’s model also relies on the idea of a void, or of an empty set, defined in relation

to some potential set members, a space upon which a situation could be built. example, Mason and Johnston-Wilder (2004).

22 For

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the story of my life, a part of getting to understand who I am and how I fit in to a supposed world or how I might make that world otherwise. That is, this progressive apprehension builds culturally specific layers around the more abstract dimensions being located, layers in which any learner is fully implicated since the apprehension was integral to their very own constitution. Whether we are considering students encountering socially known mathematical ideas for the first time, or new innovations by frontier mathematicians, Badiou’s notion of “counting as one” provides a technology. A “counting as one” seen as the acquisition of a new generalisation could be understood in either of these two situations in relation to a newly extended situation. Yet, there is a not a clear dividing line between these two situations. A frontier mathematician may already have an image of being creative. And there is also space for some novelty in the students’ constructions that need not be fully subservient to, or derivative of, the teacher’s cultural agency. The object space in both situations can be understood as a world formed of objects seen as multiplicities in any given space but “counted as one” within particular operational moves. Mathematics can be approached in many ways: Earlier I speculated on how notions of the circle are developed, transmitted and transformed through the need to traverse alternative perspectives. The objectivity of the concept was shown to be far from stable, although it would be difficult to achieve clear consensus on how mathematical objectivity is understood.23 We all know what a circle is but some people may not know that x2 + y2 = 5 defines a circle. We could also experience a circle by using a pencil and compasses, by drawing around a coin, by running whilst holding a rope tied to a flag post, by generating one on a computer, etc. Or perhaps, as seen in Chapter 2, it could be experienced in a new way such as by walking in a path defined on a distance ratio 1:3 between two partners. Similarly, all

23 The

depictions of bodily movement and the production of mathematical objects in Chapter Two occupy similar territory to a variety of mutually incommensurable work on gestures and embodiment. Lakoff and Núñez (2000) aspire to a scientific understanding of mathematics grounded in processes common to all human cognition. Radford (2004, p. 18) suggests that we “consider mathematical objects as fixed patterns of activity in the always changing realm of reflective and mediated social practice”. Nemirovsky and Ferrara (2008, p. 4) frame their analysis in terms of “perceptuo-motor-imaginary activity” that is “fully embedded in the body”. Lappas and Spyrou (2006, p. 12) follow Husserl in proposing a “genetic” conception of embodied mathematics where “man builds his mental representation of the world, through a progressive reorganization of his prior active manipulation of the environment”. Meanwhile Roth and Thom (2008, p. 2) suggest: “Both Kant and Piaget . . . conceive of mathematics generally and of geometry particularly as paradigmatic examples of knowledge that is independent of sensual experience, though always given in the form of representations that can be related to the things that we come to know through sensory experiences”. They contrast this constructivist epistemology with the model of van Hiele: “In the Piagetian model, the human mind necessarily develops to specific endpoints given by classical logic, whereas in the van Hiele model, emphasis is placed on the learning processes that – mediated by language – are specific to the historical period”. Consequently, they argue: “A conception always pertains to the activation of the traces previous experiences have left in the body, and therefore, reflexively, is always an embodied conception” (p. 13). See also Roth (2010).

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mathematical concepts can be understood from a multitude of perspectives and indeed the concept can often be uniquely a function of that perspective. In Badiou’s framework, the term circle entails an operation to “count as one” the objects of a given set. For example, the set of points on the rim of a bowl may be “counted as one” and given a name, circle. Or the moon and the sun might be seen as displaying a “shape” also occurring in naturally occurring objects, such as, berries, oranges, eyes, etc. The group of objects so classified may be given a name, such as “circular shapes”, or “spherical shapes”. But thereafter the term can become a member of other sets of objects such as “regular two-dimensional shapes” (pentagons, ellipses, squares, circles, etc.) seen as making up a world and utilised in organising our apprehension of the world. Algebraization comprises a similar operation of “counting as one” (e.g. identifying the set of points obeying the relation x2 + y2 = 1). The objects get to be there, in a world, as a result of the operation. But they need that prior (or simultaneous) construction, of a world (in this instance two-dimensional space, structured according to some rules), to be there. In Badiou’s account, the existence of an object requires a place for it to exist. In this perspective, any mathematical object is a function of its perceived world, in contradistinction to so many instances where mathematical objects have been understood in a more ideal sense. In Badiou’s formulation a world is merely any presented multiplicity, whether that is an assertion of a mathematical object as a generality, or any cultural configuration such as a social structure. In this sense learning can be seen as putting things there. In Badiou’s terminology, elements are drawn from an undifferentiated multiplicity of pure being to produce objects that exist in a world. Learning comprises the placing of an object in a world. This requires the assertion of an object, and an assertion of a world. Object and world are contingent, and neither can settle. They imply each other. With regard to the students moving around according to geometric loci, the task is to apprehend continuous movement as a sequence of points. These points are then aggregated to “count as one” object, understood in terms of this mode of aggregation. Retroactively, the students can recognise the shape they have walked against a new register and declare: “that’s it”.24 This opens the door to a highly significant theoretical observation.

5.3 “That’s It” Encounters “That’s Me” Much mathematics in schools and elsewhere exists as pedagogical material crafted for supposed modes of apprehension. But, of course, such apprehension depends on social understandings of objects and subjects. Specifically, much mathematics

24 Andonis

Zagorianakos advises me that the notion of geometrical proof started in this way: the Greek word for proof “apodiksi” is related to the action of pointing at something, displaying, showing, making something apparent.

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education rests on supposed cognitive models (e.g. Piaget/Vygotsky) in which the human being is understood in a particular way (as an individual, in a certain conception of the social world, following certain social codes and expectations, etc). Pedagogical apparatus is shaped accordingly. Badiou’s model is centred on a Lacanian conception of subjectivity. Lacan’s notion of subjectivity is a much more fluid affair. It pays little attention to contemporary conceptions of the human form. Lacan’s subject is a function of a broader symbolic universe that survives the culturally specific dimensions of Piaget’s or Vygotsky’s conceptions of humans: Piaget denies the societal demand on child development; Vygotsky paints a rather too compliant social assimilation. In a Lacanian perspective, learning would be understood more as being about an experience through time, rather than apprehending an object located in a fixed conception of space. The task is to locate education in the formation of objects/events in time/space rather than to see it as an encounter with ready-made objects.25 As we have seen, in Lacan’s concept of human formation, a transformation takes place when a young child assumes a discrete image of herself. This allows her to postulate a series of equivalences, samenesses, identities, (and “antitheses”, or rather the uneasiness that stems from the sense of misplacement) between herself and the objects of the surrounding world (the equivalence of my movement on the floor, to the drawing on paper, to the image in my mind, seen as continuous movement, or as a configuration of points). The image of self, as characterised by a name, fixes an egocentric image of the world shaped around that image of self. That is, the assumption of a self (a “that’s me”) results in a supposed relation to the world and a partial fixing of the entities she perceives to be within the world, that the “me” has been gauged against. In due course these relations become implicated in more overtly mathematical phenomena that underpin the child’s formal mathematical education. Unlike Gattegno’s baby, the older student can become aware of symbolised mathematical relationships, such as how specific bodily positioning responds to a coded spatial environment. And notions of humans and of geometrical objects become relatively fixed in such images with consequential restrictions on how relations between people and geometry can be understood. Lacan cautions that we should be wary of this image, since it is illusory. In Badiou’s terminology, this assumption of a self in an assertion of saying “that’s me” comprises a collation of a set of characteristics, attributes, organs, etc., that make up “me”. This set of characteristics is “counted as one” person. We thus have a situation where an individual (that has been counted as one) encounters an object (that has been counted as one). A “that’s me” encounters a “that’s it” and a relation between these two (petrified) entities may be asserted. Our real self is not fully visible to us. The image is crafted retroactively within the limits of the apparatus we have available. And this apparatus has a track record of

25 Cf.

Sinclair (2004).

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being changed on a frequent basis. The operation of “count as one” can always be performed differently according to new circumstances.26

6 Examples in Mathematics Education To illustrate how this analytical technology could work as a model of mathematical learning, I offer some re-readings of educational models in relation to Badiou’s analytical apparatus. I commence by developing the example of mathematical investigations to further illustrate how this alternative orientation to mathematical learning, centred in experimentation and critique, might be understood. This is followed by a brief outline of how the educationalist Gattegno sought to create a systematic science of education. Gattegno’s model provides a rather different illustration of how Badiou sees knowledge being accumulated. The next two examples are more theoretical and centred in instances of mathematics education research. In both I turn to the work of Luis Radford who has done much to develop theoretical approaches in the field. Firstly, his Vygotskian account of a classroom situation is read through the filter of Badiou’s terminology. Then elements of Badiou’s theory are used to interrogate Radford’s paper from the Special Issue described in Chapter 5. In the paper Radford attempts to provide a model of objectivity commensurate with his cultural theory of objectification. I suggest that Badiou’s approach provides a better solution.

6.1 Example One: Mathematical Investigations Mathematical curriculums are typically conceptualised as comprising familiar concepts and procedures, in the service of students being initiated into cultural knowledge, for participation in wider social practices. Through immersion in such curriculums, students encounter a variety of mathematical generalisations familiar to their elders, supplemented by material resulting from more recent trends or current fashions. These curriculums suppose particular orientations in defining the purposes of mathematics as a subject to be learnt. Such orientations, however, are not the only way of conceptualising mathematical learning. There are many possible motivations to learning mathematics and perceived purposes can shape priorities. Mathematics as a discipline can be seen alternatively as being primarily centred on understanding structures, modelling, finding patterns and making generalisations. In Badiou’s terms, making a generalisation is equivalent to performing an operation of “counting as one”. Some generalisations will be familiar results. Yet, if mathematics is to be used by students in more analytical ways, they will need to 26 “Lacan

also notes that scientific truth is only attained at the price of completely forsaking perceptual information, and therefore everything that would connect the world to the organs of the body” (Badiou, 2009a, p. 477).

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develop a capacity to make generalisations of their own. In the manner of research mathematicians the task would be to make sense of a seemingly complex situation, with view to having greater control. This is perhaps about organising and marking out the space to better understand how it fits together, to delineate which bits are known and which bits are not. This may lead to the introduction of mathematical models. Existing apparatus can only help us so much with this sort of enterprise. There is also a need to cultivate experimentation, critique and risk taking, to produce novel encapsulations. What could this look like in the context of mathematical learning? Whole Class Interactive Teaching perhaps provided one conception of teaching in which outcomes were not defined in terms of content knowledge acquisition. The Doubling modulo activity outlined above provided an alternative attitude to generating and framing mathematical ideas. Mathematical investigations such as this gained prominence in a number of locations some years ago.27 Their subsequent wider impact in England has declined since a more content-oriented curriculum was legislated. Investigational work was squeezed into ever more remote corners of successive curriculums. The reduced space resulted in a more constrained conception missing many of the exploratory aspects of such work. This style of work could not be pinned down according to either clear mathematical content objectives or organisational parameters. This made regulative assessment of both teachers and children more difficult in the new climate. Mathematics became ever more policed by prescriptive curriculum defining checklists of skills and procedures that in an important sense asserted a particular and powerful social conception of mathematics.28 In their earlier appearances, investigations had been more or less related to specific mathematical areas and could be interpreted on a number of conceptual levels. They centred on mathematical exploration where the teacher did not necessarily have particular strategies or outcomes in mind. Unlike much work with the sociocultural tradition, the student, working alone or in groups, was encouraged to take the task in their own direction, to mathematise in new ways, rather than to rediscover existing mathematical forms, or the ones that the teacher had in her mind. In setting the Doubling modulo task to a new group recently, only a small minority took it in the direction that I had foreseen as the most productive. I was quite unsettled by the sheer diversity of directions that were pursued. My task as teacher was to find out about these alternative responses and the perspectives and discussions they afforded. That is, they were being encouraged to reach beyond more familiar patterns of reflexive human activity. The limited direction by the teacher was further reduced as students gained more experience with such tasks. This promoted confidence in students posing and addressing their own questions. The initial task specification does not reveal the traditional mathematical area being addressed. The area of known mathematics is not the point. The principal purpose is for the students to introduce mathematical structure in their own way and to make generalisations. In

27 Banwell, 28 Brown

Saunders, and Tahta (1972); Association of Teachers of Mathematics (1977). and McNamara (2011).

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this respect it is not akin to problem-solving education since outcomes, or even the basic shape of the activity, are not specified or conceptualised in advance. Through such learning activities students gradually developed new approaches to orientation in mathematical environments. This activity was centred on getting to know how the environment worked and in due course making some generalisations about how it worked.29 In developing educational models supportive of future life it is necessary to equip students with the work of their ancestors and to work with the students in enabling them to build the capacity to generate their own capabilities in ways that transcend the bounds of existing knowledge. The former risks being reproductive if the original challenges are not re-entered. Objects are housed in familiar territory. The latter, insofar as it can be achieved, lays the ground for new knowledge to be generated. Here new objects and new territories are produced. In Badiou’s terminology, the possibility of promoting new orientations to truth is activated, since the relation to established knowledge and its preferred points of reference is weakened. Mathematical investigation work can potentially provide an effective strategy for the later ambition and be an alternative to overly reproductive models. That is, this sort of work provides a strategy for enabling students to structure mathematics that they have produced and to articulate their own findings around generalisations that they have conjured for themselves. In terms of meeting future challenges students might be better equipped to develop analytical strategies that are fit for purpose, rather than supposing that old problems can always be identified in new ones. I shall offer another activity here, accompanied by a student’s work to illustrate my point, although examples abound in other locations. The example is one that I have used many times with groups of school children and also with trainee and practising teachers. It lends itself to interpretation at many levels whilst providing an effective structure for extensive investigation.30 It comprises imagining a snooker or pool table with pockets just at each corner where a ball is projected at an angle of 45◦ from the bottom left hand corner. The task is to predict by which pocket the ball will leave the table and to understand other aspects of the route it follows for tables of different dimensions. I demonstrate this to the students by drawing a “table” four squares by three and drawing the trajectory of the ball in that case. I then ask them to investigate what happens when the dimensions of the table are adjusted. At this point I absent myself or circulate to observe. For those in the early stages of the task the work entails careful drawing of the routes followed on different sized tables. Very quickly patterns emerge that can lead to analysis of structure with the introduction of algebraic relationships of varying complexity. As is the case in many investigations, the initial task specification does not reveal the traditional mathematical area being addressed. In an important sense

29 See

Brown (2001, pp. 88–100) where I provide an extended discussion of 10-year old children working on an investigation. 30 Cf. Ollerton (2009).

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the area of mathematics is not the point. The principal purpose is for the students to introduce mathematical structure in their own way and to make generalisations. The figures show extracts of work from the same group of student teachers encountered in Chapter 2, where the investigation has been developed further than might be expected with school students. Yet, school students with extended familiarity with this style of work can achieve similar results. My purpose here, however, is more to demonstrate the way in which mathematics is shaped around the student’s own structure and the generalisations that they secure in their explorations. For example, in one student’s work the following statements were made: • All the odd numbers have the same pattern, so we can say for a fact that any table that has an odd numbered width or length the results are top right, bottom right recurring. • All the even numbers have the same pattern, except those in the 4 times table. • We then noticed that all numbers in the 4 times table, but not in the 8 times table, have the same pattern. • As we can see, another pattern is happening here, and without the need of drawing more snooker tables, the pattern will be all numbers in the 8 times table and not in the 16 times table will have all the same pattern. • I feel we can safely say that this will continue with all the numbers in the 16 times table but not in the 32 times table; they have the same pattern and this will continue. In Figs. 7.2 and 7.3 sophisticated work is achieved without anchoring the work in typical mathematical topics and the pedagogical forms so often associated with them. The student, with fellow group members, decided the work sequence, the

Fig. 7.2 Exit points for tables of different dimensions

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Fig. 7.3 Number of bounces for tables of different dimensions

layout and the modes of classification, such as colour coding, abbreviations, etc. For example, Fig. 7.3 results from the following student observation: The odd numbers, except 9, have the same increasing pattern. They begin with 1 less than the number and then the nth number in the line have a separate count starting from zero. There must be a reason why the numbers 8 and 9 don’t fit in with the other patterns so I am going to get a better picture by drawing a bigger table and follow the existing patterns to see if a connection appears.

This leads to the following conclusions: • The diagonals of the even numbers, except 8, 14 and 20, follow the same pattern. The pattern being all the numbers are the same – n minus 1. (marked light blue) • The odd number lines also follow the same pattern; they begin with n−1, with 1 being added each time up to the nth number which has its own sequence starting at 0 and adding 1 each nth number. These patterns are the same except 9 and 15. (marked red) • The central line on the negative diagonal is all zeros, and is also a line of reflection. (marked pink) This work also provides many opportunities for the teacher to engage with the students’ thinking, to expand its scope and perhaps reach out to more powerful realms, more or less relational to culturally familiar territory or objects. For example, the student could be asked to prove her result rigorously, rather than merely “safely

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say”. Or, she could be required to check the consistency of her work with other students, or simply convince other people that she was correct. In facilitating such work the objective is for students to become better able to approach complex or unfamiliar situations with an intelligent attitude. The approach provides an alternative to “real life” problems constructed largely according to particular pedagogical fashions. But more importantly it loosens suppositions that mathematics is principally defined in terms of its history and well-known forms associated with that history. It is not just about learning the content of a curriculum. Rather school mathematics can be seen alternatively as a generative discipline concerned with the activity of defining new generalisations, with counting new things as one, and with mapping out the domain of mathematics in novel and unexpected ways. The purpose is to enable the student to become better equipped to confront complexity and to introduce structure as part of an emerging analytical attitude. Mathematical knowledge may be learnt, and need to be, but the point here is to situate that knowledge amidst a more fluid landscape, in which the student is seeking to live. This sort of approach may foster a more creative attitude to defining and solving problems with a fresher attitude to using mathematics as an approach to filtering and engaging with life.

6.2 Example Two: Gattegno’s Science of Education Caleb Gattegno conceived of an overarching science of education. His conception of knowledge accumulation has some similarity with Badiou’s set theoretic approach to knowledge. The object of Gattegno’s science was awareness. In this science, mathematics education was understood as being built around the acquisition of awareness achieved through securing elements of learning.31 His “unit of learning” was known as the “ogden”. For instance, to own the “structuration of the energy” or “mental quantum” for each of the numbers “one” to “twelve” would cost the student twelve ogdens. “Thirteen” was recast as “three-teen”, where a further ogden was required for “teen”. “Three” had already been purchased. And hence “four-teen”, “five-teen”, “six-teen”, “seven-teen”, “eight-teen” and “nine-teen” all came for no additional cost. A further ogden (to purchase “ty”) enabled “two-ty”, “three-ty” up to “nine-ty”. Then remarkable value could be achieved as a result of three ogden purchases permitting “hundred”, “thousand” and “million”. Thus, one could count up as far as one wanted to for a mere 17 ogdens. Such technology prevailed throughout Gattegno’s formulation of all arithmetic, seen as a field of knowledge to be acquired at minimum cost, as quantified in ogdens. Gattegno used a similar rationale in his teaching of languages. I attended one of his Spanish classes. Gattegno had created a one-hundred-word passage, which, I was given to understand, included all or most possible configurations of Spanish letter sounds. This apparently was not a major challenge given that Spanish is largely 31 Gattegno

(1973, pp. 113–121). See also Gattegno (1987, 1988).

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phonetic with just one sound for each letter. The passage slowly appeared onto a TV screen, but in a curious fashion. Commencing with a blank screen the “a”s from the passage appeared, each in its correct position, one at a time, “a” being the most common letter. The class were required to pronounce the letter each time a new letter appeared, i.e., “a”, “a”, “a” . . . until all the “a”s had appeared. Next, I believe, “o”s appeared so that the class had to pronounce a sequence such as “a”, “ a’’ “o”, “a”, “o”, “o”, “a”, etc. In turn other common letters appeared. Eventually each letter appeared in turn, according to its frequency, until finally, when the last letter had been included, complete words were formed with the class pronouncing successive letters quickly to make the sound of the word, e.g., “b”, “u”, “e”, “n”, “o”. In this fashion I witnessed a group of some 20 non-Spanish speaking people read out a 100 word Spanish passage with passably correct pronunciation just 40 min after their first lesson had begun, at a bargain cost of 26 ogdens. Thus I have depicted an emergent set of numbers, the elements of the Spanish alphabet, or an approach to Spanish pronunciation each progressively “counted as one” as the student’s world was progressively expanded. Gattegno saw the detection of the minimal structure as a potential strategy in all areas of learning. That is, he sought to secure maximum learning for minimum cost. This, however, was most extensively developed in the areas of arithmetic and algebra, where he saw algebra (such as in the example of the relationship of the baby’s arm to the fly on the ceiling offered earlier) as being more primitive than arithmetic. A general case of how something is structured (or an operation) can be more primitive than an exact quantification in standard units.32 For example, if I surround a rectangle made of metre square black tiles with metre square white tiles to form a path I can think of the total of white tiles as “the ones on top”, plus “the ones below”, plus “two lots on the sides” plus the “four corners”, or more generally 2x + 2y + 4. It is familiar territory in mathematics education to identify the generalised or simplified core of a messier statement. Standard units are not necessarily an advance but just an alternative mode of structuring. Similarly, the description of a banana plantation in the second chapter depicted a spatial world structured in tune with local practices. Within Badiou’s model, Gattegno’s numerical system would be centred on the “world” of the infinite set of natural numbers and various subsets thereof, progressively acquired by students and integrated with each other. That is, the concept of natural numbers is the set, or complete sequence, generated thus, “counted as one”. Subsets of the natural numbers could be selected and combined through different procedures (or structurations, or operations) to produce new concepts, or unities. There would be a set of 17 sounds that would enable the pronunciation of all of these numbers, albeit within Gattegno’s slight adjustment to the sounds included in the counting sequence. And each of those 17 sounds would be associated with 1 of 17 concepts that together enable a form of comprehension with respect to that 32 Bellos (2010, pp. 13–42) provides many examples of how relationship is more primitive than numerical counting; sizing up the relative scale of warring factions, choosing the tree with most fruit, etc. He suggests that numbers are less than 10,000 years old, a relatively brief episode in the human history of the world.

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sequence. From that basic situation we could then proceed to learn mathematics by generating new conceptions of sets and operations on sets: primes, multiples of three, square numbers and so forth. Each of these concepts would require the purchase of one new ogden in a wholly additive fashion.33 And in due course we could be in the business of managing concepts into sets of their own to form particular branches of mathematics. All this is a far cry from the concrete real life experiences that characterise so much school mathematics. Gattegno’s pedagogy seems to involve highly structured, imposed, game-like activities that are metonymic in the sense that they play with symbols, transforming them with the barest of “explanations”, with very little metaphor. . . . In directing various communal game-like activities, the teacher looks after the symbols. The sense looks after itself. (Tahta, 1989)

Meanwhile our emergent Spanish speakers would be able to learn new letter combinations to eventually save up for a sizeable and functional set of words to produce a set to be called their “vocabulary”, and, of course, subsets thereof. And as this develops they would combine that with another emergent set comprising grammatical rules. Through this process there will eventually be styles of producing sentences such as those classified as for example, “mathematical”, “arithmetical”, “analytical”, “sarcastic”, “of a left wing persuasion”, “typical man”, etc. Such an educational process would be reproductive so long as each new ogden was derived from the existing knowledge of the community concerned. The security of Gattegno’s models based on additive knowledge would, however, be troubled by what Badiou calls an event. Gattegno depicted a number system based on successive elements being added. Yet, a new regime of truth would be introduced, for example, as a result of the event that was produced by Cantor’s set theory reconceptualising infinity. The context of the number system itself will have been transformed by this work and hence the context and meaning of each of its constituent elements.

6.3 Example Three: A Vygotskian Conception of the Objectification of a Graph This part of the book has been centred in Lacan’s conception of subjectivity. This approach has provided an understanding of the human subject that enables cultural renewal – a reconfiguration of reality that reshapes everything within it (people, mathematics, object definitions, etc.). The analysis above has provided specific criticism of Piaget. Objections have also been raised to a common interpretation of Vygotsky’s work in which the student is drawn in to a pre-formed benevolent culture. This subsection examines, and raises questions in relation to, a piece of 33 LOGO

would provide another example of knowledge being added in this way. Two arcs might make a petal. Eight petals may make a flower. Twenty flowers may make a garden. Papert (1980, p. 171) spoke of knowledge being “broken up in to ‘mind-sized bites’”.

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mathematics education research that draws on Vygotsky, towards developing this concern. The research report depicted children working with Cartesian graphs in the context of a problem of relative motion (Radford, Miranda & Guzman, 2008).34 A “successful creation of a zone of proximal development” results from “the teacher’s ability to find a common ground for the evolution of the students’ meanings” (p. 167, my emphasis). Here the teacher is understood as the representative of the culture that now finds space to accommodate the student’s participation. The provenance of the culture is in question here, however. Care is needed in the metaphors we use. It is possible to conceive the “common ground” as being within the culture, seen as the current state of affairs. In Badiou’s terms, a form of knowledge defines this current state of affairs. That is, the student aligns her actions with current forms of practice as exemplified in the teacher’s classroom behaviour. Or, in line with a suggestion above, culture can be seen as being in a constant state of renewal, where it needs to cast off its encapsulation in successive knowledge and the forms of language that go with these. In this latter formulation, the “common ground” is in a new domain that has now been incorporated into a newly expanded conception of culture. The research process depicted focuses on how the learning is mediated over time by use of voice, gestures, movement and mathematical signs. The paper is centred on students “unpacking” (p. 162) “very dense meanings” (ibid) through their personal meaning and deeds. The authors argue that the graph is an artefact that “bears the imprint and sediments of the cognitive activity of previous generations” (ibid). This unpacking is defined as “the socially and culturally subjective situated encounter of a unique and specific student with a historical conceptual object” (ibid, my emphasis). It is this process that is defined by the authors as “objectification”. Here, objectification is further understood as “an active, creative, imaginative and interpretive social process of gradually becoming aware of something and oneself” (ibid). This awareness that leads to objectification is achieved through “recourse to body (through kinaesthetic actions, gestures), signs (e.g. mathematical symbols, graphs, written and spoken words), and artefacts of different sorts (rulers, calculators and so on)” (ibid). Once again this terminology is open to alternative interpretations. It is possible to understand the term historical conceptual object as an element of “common ground” in both of the above senses. Similarly, the authors’ conception of the term “objectification” could go either way. The analysis could either be about the teacher and students sharing an object that already exists in the knowledge of the culture. Or alternatively, their encounter could define a new object, not previously conceived in quite the same way. The provenance and hence possible meanings of the object are in question. Changing cultural circumstances may provoke such an encounter. For example, the widespread response to international comparative testing may sensitise teachers to instances of mathematics more likely to be objectified and tested in the new regime. That is, objectification follows trends and fashions rather than objects.

34 See

also, Radford (2008b).

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In the specific case being discussed the object sought is a relationship embodied in a graph. The relationship defines the relative distance between two children walking in different directions and at different speeds, as measured with an electronic device. In the analysis (p. 161) graphs are variously understood as: • a complex mathematical sign; • an artefact for dealing with and thinking of cultural realities in a mathematical manner; • something that supposes a selection of elements and depicts relationships between them; • something that rests on a sophisticated syntax and a complex manner of conveying meanings; • not a copy of the phenomena they depict. Through proposing these alternative ways of understanding a graph the authors highlight how different facets of learning mutually support the construction of a mathematical object, an object that ultimately retains some residue of these facets. The research process carried out is centred on how such processing of the “cultural logic” is physically evident in the students’ discourse, gestures and symbols. For example, the authors examine how students move their pens variously to indicate distance or time and suggest that such bodily movements supplement any mental activity in producing the object. The learning task is a layered exercise encompassing initial engagement with the case of only one child moving, finding ways of representing such absolute movement on a Cartesian graph, to eventually creating a similar graph to depict the relative distance between two moving children. Here the graph is seen as representing the relationship between speed, time and distance. The mathematical object, in the process of being objectified, is the relationship itself. The graph comprises a cultural object that mediates thinking along with the movements, gestures and material produced in the learning process. The teacher remained the active agent in bringing out “the students’ meaning from behind, as it were, and helped them push their meanings beyond their initial locations” (p. 167). The relationship cannot be seen entirely as a thing in itself. It is necessarily a function of the prevailing symbolic environment. It is a historical product of the cultural artefacts, such as the graph, through which it is accessed. In being a cultural object, the meaning of the graph cannot reside solely in its observation by an individual student. The meaning relates to the way in which it is explained. This explanation will be drawn from particular resources from the symbolic environment according to the prevalent cultural preferences. Artefacts, such as graphs, cannot be culturally neutral. The object, in this case the graph, can be read differentially against its ground, whether that be a particular social context or a supposition of the mathematical field. There will be alternative modes of engagement across diverse student bodies. For example, graphs in British examination contexts often relate to some practical “real life”, application, whereas schools in many countries emphasise more on abstract relationships.

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The relationship is also a function of the mode of apprehension brought to it. This depends on the conception of psychology or subjectivity that defines such apprehension. That is, the prevailing symbolic environment needs to be extended to incorporate conceptions of the human subject, positioned and understood, as this subject is, in relation to the mathematical relationship in question. In the first account of the “common ground” described above, the teacher recognises a piece of familiar mathematics in the students’ constructions. Or the teacher delineates a pedagogical entity, a pre-defined area of mathematical knowledge, now to be shared with the students. Children’s mathematical re-productions of such entities are evaluated through filters created from the cultural apparatus. This style of teaching is reproductive of culture, in that it either offers existing culture, or recognises student work only insofar as it is aligned with such culture. Moreover, the culture is mediated by the subjective filters of the teacher. Nunes (2010, p. 106) has argued: “The frames and analogies used by teachers help them observe students, rendering some things more visible, but others invisible”. In the alternative conception of “common ground” there remains space for the child to challenge existing conceptions of culture. Education is understood as being about enabling students to transcend this space through experimentation and critique. Students are not only recipients of culture but also creators of it insofar as their fresh perspectives on mathematical situations can be voiced, rather than being merely evaluated with respect to an existing register (Although there is also a pedagogical job to be done of enabling students to recognise how mathematical conceptualisation can be linked to cultural forms so that they can engage with and share culturally preferred approaches to tackling everyday problems.) In his discussion of Badiou, Atkinson remarks: “In pedagogic relations it is not uncommon to have experiences in which what happens cannot be understood within established frameworks when we assume common ground but which in fact is not secure” (2008, my emphasis). A mathematical generalisation is not a “thing” in itself but something understood with respect to a particular discursive frame, or a world. Such a frame derives from a specifically ideological way of making sense (As seen earlier we cannot speak from outside of an ideology.) This ideology is manifest at the level of the pedagogical layer and the materials that support that, such as curriculum specification or favoured ways of setting algorithms. But discursive frames have a limited shelf-life. In Badiou’s philosophy, truth unsettles these discursive frames and the states of knowledge that they hold in place. The symbolic universe has a momentum to it that prevents specific manifestations from staying around for too long. A generalisation comprises a multiple entity “counted as one”. It is a function of the subjective or cultural entities that produce it at the time. “Subjectivity” might be understood here in terms of how teacher and student are subject to specific discursive frames, where actions are evaluated with respect to that discursive register. Meanwhile, “culture” is a function of the perspective one has of its scope and of how one is positioned within it. Radford et al. (2008, p. 167, my emphasis) suggest a “common conceptual ground for the evolution of the students’ meanings”. They paint a sympathetic

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version of students being drawn into culture where teachers and students share construction of cultural objects. This Vvgotskian conception of learning is underpinned by a Gadamerian moderate hermeneutics.35 Here knowledge is coformed by teacher and student but primarily on the teacher’s terms rooted in past conceptions of culture. Radford, Furinghetti and Katz (2007, p. 108) make an explicit link to Gadamer: “Classroom emergent knowledge is rather something encompassed by the Gadamerian link between past and present. And it is precisely here, in the unravelling and understanding of this link, which is the topos or place of meaning, that the history of mathematics has much to offer to mathematics education”. For Gadamer we are immersed in language and so we cannot assume any distance from language to inspect how it functions.36 We understand ourselves through the categories of language, which simultaneously describe and create the world we inhabit and our relation to this. Although we can build a picture of reality, we can never access this reality directly. We always end up describing previous descriptive structurings, which results in this brand of hermeneutics having a rather conservative image in relation to other contemporary theory. This, however, is just one possible version of events, which could be conceptualised in different ways. Badiou (2009a, p. 75) questions the link between past and present arguing instead that the potentialities of the future may result from an explicit break with the past: The reactive subject filters the incorporation of becoming into the present of science according to the epistemological grids of transmission . . . That is why this subject can take the name of pedagogism: it believes it can reduce the new to the continuation of the old. A particular form of pedagogism . . . is the accretion of results laid out on the same plane in accordance with the old empiricist concept of result, so that absence of discrimination renders the present illegible.

Science, Badiou is arguing, is commonly understood merely as technology that works, as linked to “lucrative ‘applications’” (ibid). A set of knowledge with guaranteed outcomes. We know Newton works in the world we face now, so why bother with Einstein? Or more crudely, and realistically, evidenced-based, mixed methods research linked to the latest government policy document, and concerned with raising standards, is more likely to secure research funding from a given source. The Vygotskian premises, which underlie the account of Radford et al. (2008), suggest an easy entry for individuals into a “culturally embodied reflective, mediated activity”. Key figures in this model are teachers equipped to shape mathematics

35 Radford’s

positioning of teachers moves between different hermeneutic attitudes spanning all four models proposed by Gallagher: Conservative: The goal is clear for the teacher, but, generally speaking, not for the students. (Radford et al., 2007, p. 6) Moderate: a ubiquitous interpretative process where the subjective and the cultural become entangled. (p. 4) Critical and Radical collide in: a more powerful voice for the students in a situation where differing views and forces collide awaiting nonetheless new forms of divergence and resistance (Radford, et al., 2008, p. 8). 36 Brown (1996a, 2001); Gadamer (1972).

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in line with some socially approved structure, with children embracing those expectations. I argued in Chapter 6 that this Vygotskian formulation displays reproductive tendencies centred on students being brought into existing practices as the primary goal of education. Teachers and students are often alienated from cultural or pedagogical tools and that compliance with them, or accommodation even, is not the only educational choice. Radford et al. echo Vygotsky’s developmental framework in which students appropriate cultural voices. Yet Wegerif (2008, p. 355) argues that it is possible to read this alternatively as the cultural voices appropriating the students. “Vygotsky interprets differences as ‘contradictions’ that need to be overcome” (ibid, p. 347). This is not the only interpretation of an educational interaction; students or teachers may wish, consciously or otherwise, to counter the educational agenda. In Badiou’s terminology they are only subjects of truth when they act in this fashion. The model of Radford et al. (2008) is predicated on the existence of experienced teachers able to administer the classroom in the terms they outline. This assumes assent from the teachers. It also assumes their technical capacity to carry out lessons on this format, or a training course that might be able to produce that capacity in teachers more widely. The model similarly presupposes that students will be compliant participants appreciative of the teacher’s benevolence and in agreement with this external specification of learning objectives. There is an apparent assumption here that there is some notional model of good teaching that both teachers and children will recognise and support. We might object, however, that many teachers are alienated from such conceptions at least insofar as their professional or personal capacity to conceptualise in those terms will be limited, and children may react negatively to externally defined plans for them, no matter how ideal these plans are in their conception. Children, perhaps humans in general, assert their understandings of who they are through a personal exploration of the boundaries they encounter or perceive. Some empirical studies provide evidence of such alienation. My own British government-funded study37 of student teachers and new primary teachers in Britain provides interview data relating to teachers’ attempts to work within prescriptive curriculum frameworks. Aspects of this study are discussed in the next chapter. Meanwhile, another British government-funded study specifically rejected a Vygotskian perspective in its empirical focus on children not being especially attracted to the mathematics they were being offered in schools (Bibby, 2007, 2008, 2010). Further, Walls (2009) provides empirical data of children learning mathematics seen from a Foucauldian perspective on how subjectivity is produced. Here “children had come to believe that doing mathematics in the approved and uniform fashion as prescribed by the teacher, was more important than finding their own methods or developing viable variations. Alternative approaches, regarded as transgressions, were likely to result in ‘getting in trouble’” (p. 109).

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6.4 Example Four: The Cultural Theory of Objectification How might we understand mathematical truth in its encounter with humans? Does mathematics depict truth or does it merely work as analytical apparatus in some instances? I posed these two questions to myself in some work I was doing a few years ago. This was before I had seriously engaged with the work of Badiou. As a result of my more recent engagement with Badiou’s work these two questions seem to be very different. Most particularly Badiou poses a previously unfamiliar understanding of the word truth. Mathematics has been susceptible to cultural and historical turbulence in its very formation. This turbulence results in a recurrent demand to re-think terms that suggest greater permanence, like truth, or objectivity, or meaning. Some instances of mathematics education research have sought to work on this issue. Radford, Bardini, and Sabena (2007, p. 2), for example, question conceptions of learning where the learning is understood as “something mental, as something intrinsically subjective, taking place in the head”. There is a prevalent tradition in mathematics education research in which mathematical objectivity is seen as transcending cultural specificities. Radford (e.g. 2007) questions this tradition. Nevertheless, Radford’s work is not entirely uncontroversial with its recasting of objectivity as something more contextually bound. And such concerns have led Radford to be more precise in defining the term “objectivity”. Chapter 5 considered papers of a Special Issue according to how they depicted subjectivity. In a paper from that Special Issue Radford (2006a) focuses on mathematical objectivity.38 Radford, however, encounters a difficulty in reaching a notion of object that simultaneously satisfies the two audiences that he variously seeks to address, mathematicians and educators. His approach contrasts with Badiou, whose notion of an object results from an operation in which a set of elements is “counting as one”. Here I seek to outline Radford’s enquiry. I suggest that Badiou’s approach offers a solution to Radford’s difficulty. Radford outlines the classic opposition between a “time honoured tradition that meaning is the real and objective description of the intrinsic properties of objects or states of affairs” and conceptions of meaning as a subjective construct, based on “intentions that we want to convey” (p. 40). The former is well known in scientific traditions. The latter privileges the human construction. Meanwhile, Radford is not entirely convinced by newer discursive approaches and suggests: “that mathematical knowledge may still claim some sort of objectivity” (p. 41). The paper is motivated by the question: “if, in one way or another, knowledge rests on the intrinsically subjective intentions and deeds of the individual, how can the objectivity of conceptual mathematical entities be guaranteed?” (p. 39). It provides a philosophical account of mathematical objects as understood alternatively in relation to the theories of Peirce

38 Radford’s

paper appeared in a special issue of Educational Studies in Mathematics that I analysed in Chapter 5 with respect to how the terms “teachers”, “students” and “mathematics” were variously understood (see also Brown, 2008b, 2010).

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and Husserl each seen as offering a potential solution to this loss of objectivity – but with mixed results. For Peirce “reality influences our thoughts but is not created by them”. And “semiotic activity yields knowledge” (p. 42). “Peirce advocated a view according to which we inhabit a world whose objects, laws and state of affairs are intelligible and semiotically knowable, even if to know them we have to go through an unlimited process of semiosis. Truth, indeed, is the ultimate point of this process” (p. 46). The strict separation of subject and object proposed by Peirce allows the possibility of ideal objects, counter to Radford’s historical conception of mathematical entities. So viewed individual subjects would take little part in constructing the mathematical objects, reduced as they are to “transforming these cultural concepts embodied in texts, artefacts, language and beliefs into objects of consciousness” (p. 60, my emphasis). This restrictive conception of the learner is at variance with both constructivist conceptions of learning within mathematics education research and broader understandings of subjectivity throughout cultural studies, where human subjects are seen as effects of discourse. So Peirce fails on two key counts: a limited conception of the human subject; and an unsatisfactory account of mathematics’ objectivity. Husserl is presented as a potential saviour of objectivity through another route. His Idealism was centred on “the role played by intentions in our apprehension of things” (p. 47). Radford, however, is concerned that this conception of the subject is divorced from social influences: “By removing the contextual and cultural factors surrounding intentionality, Husserl’s account ends up portraying a theory of truth and meaning that is universal and beyond culture and time” (p. 51). This discussion of Peirce and Husserl provides Radford with a platform for discussing meaning in mathematics education. His core conclusion is that objectivity, whether understood by Peirce’s Realism or by Husserl’s Idealism, is an untenable enterprise. Radford (p. 39) thus opts for a notion of “contextual objectivity”, which “gives up transcendentalism” in dealing with this tension within mathematical learning. Radford, it seems, has paid a high price in which both truth and objectivity are shown the door in much the same breath: Here we abandon the idea of truth in the essentialist metaphysical tradition, according to which truth is that which remains once all that is ephemeral has been removed – an idea that goes back to Plato’s aristocratic ontology . . . We also abandon the idea of objectivity as an uncompromised access to transcendental entities (p. 60).

In negotiating the split perspective Radford ultimately chooses the side of education rather than of mathematics. Yet, the terrain can be depicted differently, so that a choice is not required in quite the same way. In a well-known anthropological example Levi-Strauss describes a village where inhabitants are divided into two sub-groups. This is discussed by Žižek (2007, p. 242). When we ask an individual to draw on a piece of paper or on sand the ground plan of his/her village (the spatial disposition of cottages), we obtain two quite different answers, according to whether he or she belongs to one of the other sub-group: a member of the

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first sub-group perceived the ground plan of the village as circular – a ring of houses more of less symmetrically disposed around the central temple, whereas a member of the second sub-group perceived his/her village as two distinct heaps of houses separated by an invisible frontier.

Žižek sees Levi-Strauss’ point as going beyond mere “cultural relativism in which the perception of social space depends on the observer’s group belonging”, or, where a helicopter could fly over to capture the “actual” disposition of the buildings. Žižek argues that there is “a fundamental antagonism that the inhabitants of the village were not able to symbolise, an imbalance in social relations that prevented the community from stabilising itself into a harmonious whole” (p. 243). Žižek supplements Levi-Strauss’ example with some others that may make the story more accessible to a contemporary audience: masculine/feminine; right/left, where understanding of the terms and the terrain that locates them is defined according to whether you are one or the other, where no objective mediation could be possible within a consistent form of language. In some respects the phenomenological experience of “objectivity” is different for each mathematician. Earlier, I wrote about subjective experiences of grappling with the term “circle”. Yet, in writing about it, I felt haunted by disapproving mathematicians from another tribe who I imagined would see “circle” as a mathematical notion, untainted by such personal constructs thereby discrediting my analysis. This might be understood as a similar antagonism according to whether one is able or not to dispense with one’s active subjective engagement to allow purely abstract entities. Mathematicians who see mathematics as an entirely abstract domain are a different breed to those attentive to its historical evolution and hence its potential immersion within the social sciences. To move from one domain to another requires a major switch in modes of thinking, from one conception of life to another. One cannot move neutrally from one form of life to another since those forms of life predicate different conceptions of who one can be. Radford’s (2006a) task was conceptualised as an attempt to articulate such a divide on a grander scale. His paper was constructed for a mathematically oriented audience protective of meaning being seen as objective. With this audience in mind he argued the case against transcendental objectivity and truth. More commonly, however, Radford’s analysis in other papers39 is addressed to an audience of teachers or teacher educators, governed primarily by educational concerns and centred on changing their own practices in line with cultural norms. Besides Radford’s own evolving perspective through time these contrasting formulations result in Radford offering two perspectives, which are not easy to reconcile. Each perspective, I suggest, risks presenting a clipped version of subjectivity, culture and hence of mathematical learning that does not account sufficiently for the diversity of educational or mathematical interest. By insisting that it becomes a choice neither side comes out of it very well. Within Badiou’s taxonomy Radford’s analysis of children’s learning in classrooms would fall squarely in a hermeneutic tradition, but with some discomfort 39 E.g.

Radford et al. (2007).

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resulting from analytical hecklers. Radford focuses on students’ expressions of their growing awareness of mathematical attributes as they pass through a succession of perspectives. Rather than supposing that objects have an a priori positivistic existence; “mathematical objects are fixed patterns of reflexive human activity incrusted in the ever-changing world of social practice mediated by artefacts.40 Mathematical objects change slowly as mathematical objects as the field of mathematics expands (although any individual human’s awareness of mathematics as a whole would be different). Mathematics objects change as pedagogical or cultural objects rather more rapidly. Mathematical generalisations achieved through processes of updating and fitting new stories to newly articulated situations would in Badiou’s theory be seen as operations through which specific unities (such as generalisations within mathematical stories) are asserted. According to Badiou, we experience truths with great frequency. Everyone including students can experience truth. Radford’s difficulty, however, may be a result of linking truth and objectivity so directly. Within Badiou’s formulation Radford’s notion of “contextual objectivity” comprises knowledge referenced to specific forms of culture. So too are the ideal objects of analytic philosophy. Badiou conceptualises a truth beyond existing cultural arrangements where neither mathematical ideas nor human subjects have settled, and can never settle. Investigational work as described above supplements the task of learning as depicted by Radford in this significant paragraph from the paper discussed above: Within this semiotic-cultural approach, an important distinction has to be made between learning and the production of new knowledge. While new cultural concepts arise from communal, reflective, mediated activities in the zone of proximal development of the culture, school learning is the process of activity and creatively transforming these cultural concepts embodied in texts, artefacts, language, and beliefs into objects of consciousness. This process, in which subject and object modify each other, is the process of meaning, the process where subjective knowing and objective knowledge merge (Radford, 2006a, p. 60).

Firstly, the paper distinguishes between learning and the production of new knowledge or cultural growth. Secondly, the Vygotskian formulation see learning as the process by which cultural concepts are transformed in to objects of (individual) consciousness. These assumptions are centred on seeing the primary goal of education as being to bring students into existing practices. I am arguing, however, that learning, by individuals, or across communities, can also be conceptualised alternatively as being implicated in the production of new knowledge, where new forms are constructed against new backdrops. This echoes an instance of Radford’s later work (e.g. 2007, p. 1790), where “learning is not merely appropriating something or assimilating something; rather, it is the very process by which our human capacities are formed”.41

40 Radford

(2008a, p. 8), his emphasis. conception of subject and of human formation is a radical reconfiguration of Lacan’s notion and quite different to the Vygotskian premises underpinning Radford’s work (see Brown, 2008a).

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The student investigational work above was surfing on the fringes of schooloriented mathematical knowledge, marking out new territory against which findings could be understood and, in Badiou’s sense, incorporating new objects. This is indicative of a potential dimension of school mathematics where the student is concerned with generating new stories, of defining new generalisations, with counting new entities as one, (pursuing processes of objectification, cf. Radford, 2008a), and with mapping out the domain of mathematics in novel and unexpected ways. Learning is not understood primarily as growing alignment with more or less familiar cultural forms, or with fixed patterns of activity. The statements “the negative diagonal is all zeros and is also a line of reflection” and “No. of cycles = n − 1/log2 (nm + 1)” are both new mathematical “objects”, consequential to the students’ novel ways of structuring the activities. The set of numbers on the diagonal and the set of cycles of a particular configuration are each “counted as one” to produce these “objects”. They are verifiable with respect to particular domains of knowledge. Such assertions set a new topography such that all the “objects” in the space are seen, or created, slightly differently. Yet, as a consequence, perceptions of the particular space and of mathematical spaces more generally are unsettled. Truth may be glimpsed beyond the horizons afforded by these new perspectives.

7 Conclusion This chapter commenced with the objective of providing a theoretical approach to understanding how culture shapes and renews the mathematics that we encounter in schools. Having introduced, in the previous chapter, Lacan’s account of human subjectivity as a way of understanding how humans are implicated in changing cultural parameters, the task of this chapter was to extend this analysis to accommodate school mathematics. This chapter also sought to bring Lacan in to the present through the recent work of Badiou. Badiou’s very contemporary stance insists that philosophy today needs to take note of Lacan’s notion of the (human) subject. He also advises that we more broadly integrate the significant philosophical advances in the second half of the twentieth century. Badiou’s subject comes into being through fidelity to new ways of being that have been stimulated by encounters with truth. The cultural parameters that guide our actions, however, are so often built on existing forms of knowledge. Badiou’s conception of truth is in opposition to knowledge. Knowledge names, classifies, and orders parts of the situation according to the housing ideology, for those that it serves. Badiou’s truth relates to more equitable arrangements. Mathematics has a capacity to suggest a truth beyond the containment of its symbols. The symbolic apparatus of mathematics can be seen as just that – apparatus that mediates our access to truth, whilst never being properly representational of that truth. Mathematics has its history, has its knowledge and has its utilities. But mathematics provides analytical and critical apparatus that enable us to experiment beyond our more familiar stories.

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School mathematics, however, is often encountered as knowledge, that is, in cultural forms that have tamed their truth. Its symbols have become set in cultural images that can temper mathematical poignancy. School mathematics teaching is often in the business of enabling students to better apprehend and use socially derived mathematical apparatus. And that can drive mathematics into forms that are more easily managed in the educational contexts concerned, and accountable within the regulative apparatus that doubles to formally assess understanding of the field and student conformity with social norms. That is, in the world of teaching situations, mathematical objects are recast as pedagogical and assessment objects that result in the erstwhile mathematical definitions becoming implicated in socially governed processes. Within educational contexts, the meanings of mathematical objects are necessarily a function of the relationships within such social settings. That has always been the case. The currency in education comprises pedagogically or socially defined objects, not so much mathematical objects understood in a more platonic sense (Radford, 2008a). For example, insisting that mathematics be met through reallife situations can either provide, or block access. It can enable resonance with a practical situation that the student knows, or suffocate the underlying mathematics through restricting its power to unfold. Test items can result from enculturated mathematics that frame situated ideas more than mathematical concepts, and police mathematics in those terms. School knowledge derives from administrations trying to administer populations of teachers and children with more or less predictable results against a register of externally defined standards. On the one hand mathematical ideas are cropped to meet the needs of restrictive curriculums. On the other hand they are cladded with cultural paraphernalia that perhaps make them more identifiable. Yet the eternity and insistence of mathematics’ truth transcends its cultural appearances and the governance that those appearances can sometimes assert, such as through imposed curriculums in schools. No story is final. School mathematics is not located in a settled environment and its norms can be challenged. For a teacher viewed as an agent of culture there is a common tendency towards reproduction. The teacher’s job description can sometimes be specified in terms of their behaving themselves according to a new regulative structure. And these structures may need to be precise in their demands. Lacan’s conception of truth, as always being new, proposes for the teacher an educational alternative to being a mere purveyor of everyday discourse. Lacan’s optimistic conclusion sees perpetual renewal as the mode of life that the teacher and her students must live. The teacher and student both constantly have to revise their sense of how things are, who they are, and how they each fit in. Teaching is at least in part a trigger to stimulate enquiry beyond the clutches of present cultural configurations. It is about enabling students to encounter mathematical truth through experimentation in time. Even for a teacher charged with following a curriculum within a setting of compulsory education, mathematics itself is always in a process of evolution resisting the strictures that curriculums place upon it and, in some important respects, helping the teacher to be seditious. The teacher’s role can be recast towards enabling a future beyond her conception.

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Education, more broadly, can perform a dual role, of assimilating knowledge of culture and of providing the possibility of accessing truth. Stressing the culturally specific aspects of mathematics for social participation in that era, however, can potentially be at the expense of the experimentation and critique through which truth might be encountered. We may be insisting that truth be reached primarily through the filter of existing cultural forms. Badiou is suggesting that this may not be fast enough. As teachers we may be distracting our students from seeing options that we are too old to see. Badiou’s formulation promotes insurgency at root. It is predicated on punching holes in the knowledge that had been generated in and for previous circumstances. And it supports seeing learning as being witness and complicit to the creation of mathematics. By asserting new objects, we can assert new places for them. Learning, then, can be seen as being subject to mathematics coming into being, or reshaping to new conditions. It is about futuring our understanding of the world, with a healthy antipathy towards existing frameworks. Learning is about participation in formative processes (reshuffling) and about the apprehension of existing forms (it would be pointless to try to live without repetition). That is, learning can be aligned with participation in cultural renewal. Education, through privileging experimentation and critique, can be a trigger to a new reality, as well as being a critical or disruptive picture of existing reality. It would be about cultivating responsiveness to new circumstances. Mathematics can remain alive in the analytical apparatus we create to confront the complexity of the life we experience. And for mathematics education this is about building the capacity to mathematise in ever more creative and productive ways. In Badiou’s philosophy, new ontological possibilities are provided for the learner and teacher beyond the established states of representation in order to expand our grasp of human learning, of what it is to learn, to produce new subjectivities and new learning communities . . . the ethical imperative for pedagogy is concerned with maximising the power of learning, it is not focussed on what we are and should be, that is to say on some transcendent position towards being, but upon the potentiality and “unknown” of becoming . . . a leap into a new space, where the event of learning precipitates a new order of being and becoming that has the potential to invoke new states of experience (Atkinson, 2007).

Ultimately radical departures within school mathematics might disrupt the very relations of school life. In Chapter 2 I raised the question as to how we might think artistic objects against the backdrop of contemporary art. The chapter contemplated how we might draw an analogy between objects in art and objects in mathematics towards loosening our suppositions as to the provenance of mathematical objects. Adams (2010) argues that schools typically avoid contemporary art. He suggests that this reluctance might be down to the activities not being readily accommodated in schools, as we know them. Typically art lessons follow a traditional format where the teacher leads, guiding the students to specific cultural models of art, centred on Western notions of perspective, produced in traditional media such as paint, and so forth. Following Bourdieu, he argues that schools require very specific models of teachers and students to function effectively, where effectiveness is gauged against the reproductive capacity of schools. Expanding the frontiers of art, or re-thinking

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its objects, within school might not be possible within existing school infrastructure, modes of assessment, societal expectations of school art, conceptions of what it is to “learn” art, etc. Similarly, the domain of school mathematics is hemmed in by similar factors. As discussed the very models of mathematical apprehension are very often built into the pedagogical apparatus that are favoured. And Adams (ibid) argues that these modes of apprehension derive from restrictive social relations predicated on reproductive models of education. These models may similarly prevent mathematics adjusting to new circumstances. Atkinson (2011) argues that art education can only advance through allowing for forms of pedagogy that disrupt existing models from within. Similarly, within mathematics education, if we are to advance, we need to recognise that the familiar territory of school mathematics derives from a very selective mapping.42 If mathematics is to change from within its manifestations within school education, there must be some loosening of how work in schools is defined and held in place. We need to re-think what counts as mathematics to enable its capacity to support experimentation and critique. In the next chapter, I pursue a more explicitly political lens provided through Žižek’s engagement with Lacan. Here the notion of subject comes to the fore in discussing how teachers and students identify with the social parameters that shape their teaching and learning.

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recently discovered that my favourite walking territory the Peak District was much bigger if I did not insist on the route including a quality pub for lunch after about 3 h walking.

Chapter 8

The Political Shaping of Mathematical Learning

1 Introduction The realities of the previous generation clearly impact on the next generation. These realities can firm up in relation to familiar forms that can result in conservative modes of education centred on delivery of these forms. The making of mathematics in classrooms now can sometimes be seen as subservient to the forms created and then favoured by the previous generation, who had been making mathematics, according to their own particular image, in the circumstances they had faced. In such instances, mathematics is referenced and anchored to the last people’s models, thereby shielding the dynamic and analytical aspects of mathematical capabilities that might be invoked in the new conditions. In school education more generally, adjustments to knowledge are processed through successive curriculum models and wider educational infrastructure shaped according to the preferences of the day, ostensibly on a long march to “improvement” or to “raise standards”. Curriculums can be variously understood: as an outline of mathematical content; a guide to mathematical learning by children; a cynical ploy to make teachers and children more accountable according to a particular institutionalised account of mathematics; or, a reconfiguration of the discipline itself to meet contemporary needs, etc. The evaluation instruments used are often referenced to objects from earlier knowledge bases. The instruments of assessment carry with them the modes of governance to which we are subject. The acquisition of settled knowledge is easier to govern than a learning attitude centred in experimentation and critique. And the demands of governance can suppress new perspectives and temper intellectual responsiveness. It is not possible to understand work in teachers’ colleges or schools without some understanding of how the wider governance of education impacts on the practice of individuals. This chapter is primarily attentive to the ways in which administrative structures impact on everyday classroom practice. I have already provided a detailed book length examination of how teacher education processes and policy frameworks shape classroom mathematics, the practices of teachers and the mathematics encountered by children in primary schools.1 This supported a more 1 Brown



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political engagement with task of mathematics education research. The theoretical frame was influenced by Slavoj Žižek. Drawing on the work of both Lacan and Hegel, Žižek provides a contemporary frame for how political structures shape the actions of individuals. This provides another dimension to how the work of Lacan has been integrated into contemporary thought. The reader is directed to that book for a detailed account. In this chapter, I briefly highlight theoretical aspects of the earlier book, whilst developing those themes in relation to present concerns.2 I commence by briefly outlining how school mathematics results from policy demands. I further portray classroom mathematics as being processed and understood as cultural material. I then consider the preparation of primary teachers against a backdrop of wider school policy. Finally, I provide a case study of a teacher grappling with externally imposed regulation of her professional practice. These brief accounts of those locations think about the perspectives we take as teacher educators, teachers and researchers in conceptualising our impact as agents of change in schools and colleges.

2 The Regulation of School Mathematics 2.1 Curriculum and Assessment The earlier book documented a major nationwide mathematics curriculum initiative in terms of how individual teachers and students functioned within that framework. The two studies of the project described were specifically targeted at documenting the 5-year training experience of people becoming teachers in primary schools. The particular focus was on how mathematics came to be conceptualised as a result of these changes. Classroom mathematics is what classroom teachers say it is. That is much of the experience of school mathematics for most school students. But what teachers say depends on their interpretation of curriculum guidance and administrative procedures, and on their propensity or competence to be guided by them. The formation of what teachers say is an important dimension of understanding both school mathematics and teacher professional development in mathematics. Our study sought to document that formation and argued that the external definition of mathematics by the government has had the tendency to retain the upper hand in England, at the expense of teacher professionalism and the agency of teacher educators. It is this insistence on an externally accountable version of mathematics set within a restrictive register, which, the book argued, has countered some of the success that some universities had achieved in providing a version of mathematics that positively engaged future teachers. Teacher educators did not have the final word. The weight of the accountancy apparatus had in many instances squeezed out the possibility of more autonomous professionalism. 2 Žižek is ever present in the discussion that follows but direct quotes are minimised to reduce the complexity of discussion.

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Most pervasive of all in shaping school mathematics was the National Numeracy Strategy. This comprised a detailed framework governing mathematics teaching in all schools, introduced as a central plank in the government’s apparatus to raise standards following a mediocre performance in international comparative tests. The Strategy became a hegemonic success, convincing teachers to follow its demands, despite its apparent downgrading of the teachers’ own voices. The Strategy prescribed in previously unimaginable detail the conduct of mathematics lessons in schools. Its key features were as follows: an emphasis on calculation, especially mental calculation; a three-part template for daily mathematics lessons, starting with 10–15 min of oral/mental arithmetic practice, then direct interactive teaching of whole classes and groups, and finally 10 min of plenary review; detailed planning using a suggested week-by-week framework of detailed objectives, specified for each year group; a systematic and standardised national training programme, run by newly appointed local consultants using films3 and slides to demonstrate “best practice”. The level of detail was quite overwhelming leaving the individual teacher with little space for manoeuvre. Mathematical ideas outside of this frame no longer seemed to feature prominently in teachers’ accounts of their own teaching. Professionalism had come to be defined more in terms of being able to do the job of implementing the curriculum. The curriculum was tightly shaped around student performance on a very specific style of exam question. Personal aspirations seemed to have been re-routed through the official language that had become the filter through which the teacher understood his or her own more personal aspirations of being an educator. This resulted in a very specific conception and form of mathematics, closely mirroring the style of TIMSS test items, being centre stage in English schools. The British government policy in some sense met part of its target in moving England’s league table position for 8th grade students in mathematics from 18th to 7th between 2003 and 2007, the period immediately after our study’s empirical findings were collected.4 Our study, however, reported a strange result. It seemed that the largest mathematics initiative in England in recent years had not resulted in trainees and new teachers being able to discuss mathematics outside of the administrative frame. The trainees and new teachers followed the scheme in a largely compliant fashion. The teaching of mathematics, the book argued, had become commodified into a boxed up version, with predictable forms of theme presentation and associated questions and test items. Despite much effort by government agencies to raise the profile of mathematics, the trainees and new teachers were only able

3 This

use of films is to be described in the third subsection. a 2010 policy document the new coalition government chose, bizarrely, to ignore the 2007 TIMSS result that revealed a significant improvement on the 2003 TIMSS result, in favour of citing the 2006 PISA result that showed a decline in relation to the earlier PISA result. It seemed to be in their interest to paint a drop in standards to justify their new policy ambitions (Department of Education, 2010, pp. 46–47). This switch in relative fortunes in relation to TIMSS and PISA might be attributed to government policies shifting attention away from the problem-solving dimension more prominent in the later measurement. 4 In

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to discuss mathematics as a set of classroom procedures. When asked what skills they needed to be a mathematics teacher in the primary school, the responses from trainees and new teachers seemed not to emphasise skill in mathematics itself. They seemed to speak mainly about empathy with pupils and classroom organisation in administering the scheme. The concern for administering the scheme effectively had crowded out richer mathematical discussion in teacher education courses. Success in the league table had seemingly been achieved through some curious policy decisions: by downgrading teachers’ professional role, ignoring researchers,5 and redefining the content of school mathematics to the limited ambitions of TIMSS. Another immediate practical concern for teacher education was also raised by this observation. There were some explicitly mathematical challenges associated with the course such as a new test to check the trainees’ mathematical competence and associated preparation. These aspects, however, rather than raising mathematical capabilities in a clear way, activated renewed anxieties among the trainees without apparent benefit in terms of added classroom functionality. There was also some indignation resulting from the implied inadequacy of their school mathematics qualifications. The agitation linked to this test provoked an interesting question. Should then, given weaknesses in the trainees’ mathematical profiles, mathematics learning for trainees become a bigger part of an already crowded initial training programme, when the administrative attitude among new teachers appeared to support an adequate level of functionality in the classroom? Counter-intuitively, more mathematics for the trainees during the training process in this wider scenario seemed as likely to cause difficulties, as it was to solve them. A conjecture that we explored centred on trainees not developing a particularly sophisticated discourse of mathematics prior to entering university. Talking about mathematics was not a significant feature of their world as a school student, especially so in a curriculum aiming at success in closed format exams. As they learned to talk about it in the university environment, mathematics was not seen independently of the discourses of teaching it. These discourses were firmly centred in classroom administration. The trainees did not become more sophisticated in discussing mathematics itself. Mathematical content knowledge, insofar as it was encountered, as in the new mathematics skills test for teachers, was seen as something completely different. This version of mathematics jarred with the language of pedagogically oriented mathematics they learned in university, a language that had also enabled them to shield their anxiety in relation to the subject. This prompts a wider question to do with how school mathematics sits in relation to mathematics more generally. Where is its location? That is, school mathematics manifests itself as so many pedagogical devices, curriculum material, exam questions, textbooks, worksheets and so forth. A casual glance at TIMSS test items reveals a very specific version of mathematics, centred on basic skills, short closed

5 In a survey commissioned by the British Educational Research Association little evidence was found to support the government’s assertion that its policies were informed by research, if only because there was not adequate research to support such an assertion (Brown & McNamara, 2001).

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questions, in bland “real life” situations. It is a version of mathematics fit for a very limited purpose. In this sense mathematics is “a totality made up of subsets”6 (e.g., test-maths, university mathematics, workplace mathematics, creative mathematics, research enquiry). These subsets may not readily talk to each other across the partitions that define them. The TIMSS format is echoed in school tests and a host of materials widely available to parents designed to prepare children for such tests. That is how children frequently encounter mathematics in schools. That is how mathematics is marketed. Yet, given earlier discussion in this book, there is a more profound issue about the ontological status of mathematics in general, in its interface with humans. Seen outside of the school context how might we imagine mathematics existing otherwise? In its interface with humans it can only exist tangibly as cultural material, often within models to support practice in a multitude of domains. Let us work outwards from this specific case of mathematics in a teacher education context towards making some more general statements about the status of mathematics as a school subject. The effectiveness of any teacher education initiative targeted at the enhancement of classroom practice is dependent on how the style of training is matched to the specific professional and administrative needs. School mathematics has many faces. Conceptions of mathematics are various, situated, and governed by multiple possible political preferences. Learning outcomes in the subject are dependent on the learning theories and assessment instruments being applied. Mathematics can be many things and there cannot be a best way of measuring it since it is not a singular entity. The activities of primary school mathematics relate closely to the organisational and philosophical underpinning of primary education more generally. Training in mathematical content as such may well not be sufficient or even useful in this context if it is not understood in terms of pedagogic need and the administrative structure that supports it. The mathematics test for teachers, designed to lift standards, triggered anxiety among trainees, undermining the successful image work that had been achieved by universities in re-casting mathematics as a subject that could be enjoyable. Two conceptions of mathematics were in conflict here. Similarly, the content orientation of the new policy put teachers back at the front of the class. Here they re-adopted the didactic styles of teaching in schools that had triggered so many of the anxieties that had blighted the school experience of so many of the trainees we had interviewed.7 Nevertheless, the curriculum guidance did appear to be effective in allowing teachers to get on with the job, in a curriculum where numeracy had regained some prominence after a period of downgrading in English schools.8 6 Gabriel

and Žižek (2009, p. 15).

7 Some recent research points to a negative correlation between test scores achieved through

transmission teaching and mathematical enjoyment (Pampaka, Williams, Hutcheson, Wake, Black, Davis et al. (in press). 8 On 27 June 2009 BBC news found a brief gap in the coverage of Michael Jackson’s death to interview two former ministers about the barely announced decision to abandon the National Numeracy Strategy in English schools after nearly a decade of it defining mathematics in exacting

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Briefly, the research that we had carried out showed that the link with a more explicitly mathematical discourse was a rather marginal concern as teachers had proceeded through their training. School mathematics had been bureaucratised and held in place by administrative regulation, which produced compliance and resistance, in different forms. Training was referenced to that bureaucracy. Pedagogical strategies as defined in the curriculum framework had become normalised as though they were the substance of mathematics itself. And in some senses they indeed had become that, since mathematics had become to be known through material caricatures, such as particular results through standard methods. Through the fixing of habitual conceptions, school mathematics had been mapped into a specific set of procedures, through which mathematics was loved or hated. These procedures anchored mathematics. It was these caricatures of mathematics that had tangible effects on classroom practices and the prescriptions set in respect of them. How might we understand this state of affairs through the filter of Žižek’s theoretical approach against the themes of this book?

2.2 Mathematics as a Commodity Mathematics underpins many social practices. These practices can be interpreted in many ways. The on-going historical formation of mathematics amidst social practices is marked by a fluidity of meanings. Meanings relate to local circumstances and the meanings variously assigned to mathematical forms adjust as circumstances change. Symbolic or social objects are generated, through which mathematics is understood. Mathematics itself, however, is more than a social practice. There is something more significant that needs to be accounted for. It is a system of thought that transcends mere utility. Yet, the common assertion of mathematics being “useful” knowledge (in Badiou’s sense) influences and shapes the way in which mathematics is pursued, administered, funded and developed as an intellectual discipline, and consequently how we come to know it. Mathematics is a vast field, and different aspects can be wheeled in to support alternative educational priorities (the useful aspects for industrialists, the measurable aspects for evaluators, new discoveries for frontier mathematicians, basic skills to support everyday activities, fascinating ideas for lively minds, etc.). Mathematics has an existence beyond specific manifestations, but in so many educational contexts mathematics has become a product of its cultural manifestations and known through them, exists

detail. History was being rapidly re-written to explain successive shifts in policy. Kenneth Baker claimed that he had known from the outset that it would fail as classroom decisions need to be left with teachers. Meanwhile the former minister who had introduced the Strategy (Estelle Morris) agreed that it was time to move on, to focus instead on having an “evidence base” to support classroom decisions made variously by teachers and schools. Yet the swings and roundabouts of party politics construct and follow evidence in fickle ways that weakens any attempt to link policy to research, even if it were possible to pin down research in a consensual fashion. The decision however as to how school mathematics is defined remains open. The pedagogical layers remain in place obscuring a more abstract mathematics that may be concealed.

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through them, and grows through them. More generally, the statements that seek to locate mathematical phenomena as something tangible in this way so often become the statements that police its boundaries.9 And this policing can mask the parts of mathematics that exceed its administrative housing. In short, we need to be attentive to how mathematics functions within these wider constructions of knowledge. Meanwhile, mathematics at a micro level in the school context operates, at least partially, through the guise of pedagogic devices. We need to understand how the devices work, and what they import, rather than supposing that they act neutrally, or that mathematical concepts can be perceived directly. Mathematical content in schools is a function of its cultural housing. This housing can obscure, or make, any supposed content. It can give mathematics its sense. Mathematics is being defined as a set of objects, concepts, skills, procedures and competencies, which partition mathematics in to so many discrete components, results ultimately in mathematical knowledge presenting itself as if it were so many commodities exchanged in the educational marketplace. This tangible presentation is qualitatively different to notions of mathematics understood in a more abstract manner where “The sense looks after itself”.10 There is a tension in this search for mathematics, being or not being, being there or not being there,11 perhaps merely as symbols, maybe in a more tangible cultural embodiment. The term “commodity”, however, requires some unfolding towards getting at this tension.12 Mathematics is desired by society. This desire to have mathematics gets expressed as a demand for something more tangible within many educational locations, such as a set of particular skills, acquisition of particular symbolic forms, or a curriculum of a certain form. This activates some questions: What is concealed in the forms of mathematics crafted as tangible commodities in this way? Or rather, what is embedded in the materials and practices through which people encounter that which is called “mathematics” in the classroom? What are the dominant commodity forms in mathematics education? How do they govern our practices, and influence our analyses of those practices?13

9 Gattegno

(1963); Freudenthal (1978). (1989). 11 In Badiou’s terms, this is the contrast between pure being and “existing”, which he equates to being there, in a world. 12 The notion of commodity in the work of Marx, and dream in the work of Freud, Žižek argues, are both symptoms of something else. With both commodity and dream we should “avoid the properly fetishistic fascination of the ‘content’ supposedly hidden behind the form: the ‘secret’ to be unveiled through analysis is not the content hidden by the form (the form of commodities, the form of dreams) but, on the contrary, the ‘secret’ of this form itself. The theoretical intelligence of the form of the dreams does not consist in penetrating from the manifest content to its ‘hidden kernel’, to the latent dream thoughts; it consists in the answer to the question; why have the latent dream thoughts assumed such a form, why were they transposed in to the form of a dream?” (Žižek, 1989, p. 11) Similarly, in relation to commodities, Žižek argues that capitalist economic analysis suppresses alternative understandings of the value of labour. 13 Williams (in press) discusses theoretical conceptions of “use” and “exchange” values of mathematics education within our “audit” culture (Strathern, 2000) from a Cultural Historical Activity Theory perspective. 10 Tahta

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The commodified forms of mathematics have become the cultural currency used to measure and classify mathematical thinking.14 The need for accountability in mathematical learning results in specific transformations of the mathematical teaching and learning around these commodified forms. These caricatures can be produced, for example, to assess mathematical activity for the purposes of making comparisons within particular skills-based frameworks. Or, they may derive from attempts to organise our understanding: The geometrical explorations described earlier entailed continuous understanding being processed as discrete explanations, to enable the activity to be accounted for. Accordingly, (school) mathematics itself is being defined around specific reifications of intellectual processes. These reifications are culturally specific. Such organisation of mathematics perhaps points to fantasies of what mathematics is “really” like.15 Nevertheless, school curricula can persistently emphasise skills for audit purposes rather than deeper appreciation. The curricula provide particular ways of structuring reality. Within such curricula mathematics may be supposed to be anchored in a set of key mathematical objects (number bonds, formula for an area of a rectangle, bar graphs, etc.). This defines value in specific ways and only takes account of certain measurable features. In turn, those measurable features come to be seen as mathematics itself, or at least condition this mathematics within the particular environment. Open-ended problem solving or the investigations described in the last chapter, for example, may not fit that model of education, with its associated assessment procedures, its school infrastructure, its given supply of teachers, etc. Particular forms of mathematics, and particular ways of being mathematical, are excluded. The influence of a more traditional conception of mathematics remains strongly present in the sense that this mathematics provides the system against which the correctness of school mathematics is judged. Yet, the point in the analysis here is not to target the underlying mathematics as the ultimate quest but rather to question why mathematical activities in the classroom have assumed the social forms that they have. That is, why have they become commodities with a given form? The commodification of mathematics in the construction of curricula can extend in to how the training of teachers is understood. Pertaining to our studies mentioned above: How, for example, do societal desires to incorporate mathematics in life get translated into teacher training programmes of the forms they currently take? What desires are being expressed in teacher-training guidance provided for universities and their students? Mathematics, as it is manifested in primary classroom activities, is a symptom of many things. That is, it is not only a symptom, for example, of “proper mathematics”, a higher or more abstract mathematics of the sort a university academic mathematician would suppose and the symbolic products of such activity. 14 Žižek often tells the story of a cop finding a drunk looking for his keys under streetlight, when in

fact he had lost his keys on the dark side of the street. When the cop asks why he is looking in the wrong place, the drunk defensively retorts: “Because this is where the light is” (cited by Taylor, 2010, p. 13). 15 In Lacanian terms these fantasies are centred in the mathematics that we suppose we used to have, a false nostalgia that taints what we have now.

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It is, perhaps, as significantly referenced to social practices, or to how we understand ourselves as being human. For example, primary mathematics may be crafted to be consistent with the wider primary curriculum and models of classroom organisation, valuing other things such as quality interactions between pupils, activities that keep children quiet, pupil autonomy and appropriate classroom behaviour. But, similarly, in being influenced by the form of curricula preferences, comparative test items, etc., primary mathematics is crafted according to a particular conception of education (of teachers, of learners, and of mathematics), derived from a particular conception of society.16 Mathematical activity does not solely locate mathematical ideas. Mathematical activity is also a manifestation of social activity more generally.17 This entails linking the mathematical activities (seen as activities governed by certain procedures, rules, performance criteria, etc) with other social discourses, including those specifically related to mathematics. The meanings assigned to mathematical discourses are functions of their relationships with the other discourses with which they are entwined.18 This softens any assumption that the activities in school are anchored in specific or unique mathematical concepts. Rather, we need to attend to the reification of such supposed concepts as they unfold in alternative ways of life. This would move us away from any universal conceptions of what mathematics should be about, but instead alerts us to the historical and social processes that generated classroom mathematics in the forms it now takes. How might we unravel these alternative conceptions? Firstly, we have the mathematical content or concept. Secondly, we have its manifestation in a culturally located pedagogical form, perhaps as identified in a curriculum, presented by a teacher, or produced by a student. Thirdly, there are the more unconscious desires that bring us to these phenomena in the first place, yet these desires resist precise symbolisation, and can be disguised by more tangible phenomena.19 Desire might be associated with the intrinsic excitement of mathematics to those who are absorbed in the mathematics as mathematics (the joy of solving a problem, finding a mathematical solution to a non-mathematical problem, detecting mathematical coherence in theories of mathematics, etc). This desire may, however, also be linked to the administrative control issues for those in government, the relationship with children for young teachers, or the supposition that mathematics gives us control over our lives, etc. The original desires for including mathematical ideas in our everyday lives, it seems, are being reconfigured, or re-routed, to meet specific caricatures of mathematics now demanded. Desire could, for example, result 16 Adams

(2010) discusses this issue in the context of art education. develops an idea first explored elsewhere (Brown, 1987, pp. 171–174). 18 Presmeg (e.g. 2002, 2003) discusses this issue further. 19 For Freud the dream has a “structure (that) is always triple. This comprises the manifest dreamtext, the latent dream content or thought and the unconscious desire articulated in a dream” (Žižek, 1989, p. 13). We could regard the mathematical activity, or teaching device, as analogous to the manifest content (the form). The latent content is the supposed mathematical content. Yet the desire at work is constituted through a host of contributory factors. 17 This

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from attempts to reconcile alternative demands, such as the teacher wanting children to do mathematics in a particular way, or meeting curriculum targets, or following other externally imposed rules. In our studies, and more widely, an administrative language of how teaching should be conducted, replaced a more direct focus on the mathematical content of the children’s work. This administration could be seen as an alternative discursive apparatus that released the teacher from needing to be more explicitly mathematical in describing the children’s learning. This alternative, however, seemed to result in a perverse pleasure achieved in following the rules to “get it right”. Nevertheless, this apparatus can shape classroom activity in mathematics, and hence shape the way in which children understand mathematics. Mathematics will always exceed its specific manifestations. This surplus over such manifestations locates a disappointment that we cannot be precise. There is more to it than has been pinpointed in the constructions of knowledge that govern our lives. The name “mathematics” that has been used liberally and variously throughout this book locates something that is more than the sum of descriptions of the activities in which it arises.20 Yet, it has proved quite impossible to adopt a consistent perspective on what mathematics “is”. In Žižek’s terms, the guarantor of the identity of any particular object is “the retroactive effect of naming itself. It is the name itself, the signifier, which supports the identity of the object”.21 Mathematics has the quality of “a pure signifier that designates, and at the same time constitutes, the identity of a given object beyond the variable cluster of its descriptive properties”.22 “Mathematics” in schools is only being accessed indirectly through descriptions of the activities taking place around it. The sum of those aspects is not the whole, but can affect the whole. The name “mathematics” alone remains in the intersection of all the descriptive features that we may assign to mathematics. The name holds the place. All other descriptive features of mathematics fall short of locating mathematics itself. It is this surplus, or gap, that results in motivation, or desire; the life (or truth) of mathematics as a system of thought that has an existence beyond specific cultural instances.

3 Regulation Through Teacher Education The earlier book noted how so many new entrants to the profession had experienced difficulties with the subject in their own schooling. These difficulties with mathematics hampered their emergent sense of becoming a teacher in school, especially when it came to preparing to teach mathematics. Our research was shaped around how the student and new teachers themselves constructed their rationales against the backdrop of a complex situation. Given that they needed to embrace an amalgam of

20 “Mathematics”

serves as a “‘pure’ meaningless ‘signifier without the signified’” (Žižek, 1989, p. 97). 21 Žižek (1989, p. 95), original emphasis. In Badiou’s terms this is equivalent to counting as one. 22 Žižek (1989, p. 98). Žižek cites Kripke and his notion of a “rigid designator”.

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complex stories, the project asked how they managed to do this. How did they process the multiple demands that they encountered in to a mode of professional life to which they could subscribe? The project revealed that the external definition of teacher was prescriptive in this context. Teacher subscription to professional life was sometimes replaced by what seemed to be mere compliance. Yet this compliance sometimes conferred its own pleasures. Teachers quite enjoyed following rules. Many found the strict direction a positive relief given their anxieties with mathematics. They could just deliver mathematics according to the given description. Their very professional identity was understood in terms of their compliance with the curriculum frameworks and the associated materials that framed mathematics in very specific ways. They identified with those definitions of what it was to be a teacher and understood themselves and their practice in those terms. For mathematics teaching in the locations described, teacher development was not seen primarily in terms of person-centred mathematical development. The teachers’ own engagement with mathematics was successively squeezed out of the training model as ever more training was carried out in schools. Teaching was understood more in terms meeting externally defined competencies. The working week that teachers’ experienced across the profession was filled with tasks resulting directly from regulative demands.23 The style of lesson planning was prescribed in such detail in the policy documents that teachers generally had much less scope for their own input. Their professionalism had seemingly been eroded. Yet in the book we conjectured that external demands were less precise than their initiators claimed. The control technology was not reliable in terms of building mathematical achievement, no matter how strenuously it was applied. But similarly we argued that the personal voice of the trainees became polluted almost beyond recognition with the felt need to meet others in some shared agenda. Demands for trainees entering the profession included, for example: meeting school requirements; meeting university requirements; being popular with children; pleasing parents; building an enjoyable conception of mathematics; performing adequately on a mathematical skills test; achieving personal aspirations; following wider curriculum framework adequately; getting through school inspections, minimising teacher and pupils’ anxieties relating to mathematics; and, not least, teaching up to 10 other curriculum subjects. The trainees’ accounts of “who they were” were built out of attempts to speak to these multiple agendas, with more or less commitment. That is, their accounts of “who they were” as teachers were built out of their answers to the questions on how they were meeting these wider demands which in many senses they wanted to satisfy. And this desire to please might be understood as a perceived alignment with collectively defined ambitions. For example, a teacher may need to believe that she is making things better for her pupils for her to be able to function in her professional role. It is this personal need in determining professional identity that can predominate over any actual externally imposed performative criteria, or any actual alignment with a collectively defined

23 Large

numbers of teachers leave the profession fairly early in their careers.

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ideological programme. The teacher may not need, however, to reach a final resolution of such dilemmas and may continue to work with many such notions variously activated according to demands made in different professional contexts. Or she may subscribe to the customary ways that surround her and see that participation as supportive of the common good.24 Mathematics was a very small part of this wider picture. How might we read such states of affairs theoretically? Lacan’s human subject is split and tells different stories to different people, hoping that they do not meet up to compare notes. In Lacanian terms, the ego is an inauthentic agency working to conceal an apparent lack of unity. Lacan depicts a human subject who is not fully aware of his or her personal boundaries. The human subject is seen as incomplete, split, divided and never quite getting to a final resolution of these conflicting dimensions. As an individual I am forever trying to complete the picture I have of myself in relation to the world around me and in relation to the others who also inhabit it. Žižek deploys this idea with respect to how ideologies function. He suggests that our compliance with ideologies is in what we do rather than in what we say we do. There is an irreconcilable gap between what we think we do and what we are doing. This gap shields our awareness from the effect of our own actions. Žižek combines Althusser with Lacan in portraying the individual as not having an essential identity. Rather the individual more or less identifies with a range of discourses that are not necessarily compatible. These identifications identify the individual. I define who I am through my mode of participation with the multiple conversations taking place around me. In our study the new teacher was asserting a teacher identity centred on identification with the administrative structure that defined mathematics. In Althusserian terms the trainees and teachers are interpellated by ideology. They speak the ideology with which they currently identify; in this case it was centred in the National Numeracy Strategy that defined mathematics and associated teacher practices. The Strategy provided a safety net for teachers nervous about the subject. Yet the teachers are immersed in ideology without being fully aware of this. This entails a tricky meeting of personal aspirations and an expectation that he or she will be told how to teach. Agency on the part of the trainee mingles with dependency shaped by the form of the external demands encountered. The account of power offered by Althusser is partial, however, since interpellation fails to capture

24 An

ideology closer to home was noted in an example offered earlier. It is often supposed by mathematics education researchers that exam results, enjoyment of mathematics and functionality with mathematics are commensurate goals. But what sustains this supposition? Is it supportable across large populations of teachers and students? What would be needed to support it? Many instances of small-scale research support the commensurability in the researchers’ own practices. Such researchers may be relative experts in their field and one may anticipate good practice. Yet large-scale studies across diverse bodies of teachers supporting this commensurability are harder to locate. Indeed some recent instances of large-scale studies suggest that the three ambitions can pull in different directions. It may be necessary to choose between them in the short term. Nevertheless, collective ambitions such as the desire to “raise standards” can result in improvements being judged on a singular scale oblivious to the conflicts that this can suppress.

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the flow of emotion or conscience activated in trainees in their efforts to comply with the demands they face.25 In Lacan’s model the human subject is always trying to please someone. How do I imagine that others see me? How does that compare with how I see myself? There is a gap between how one is and how one might be. For this reason Žižek departs from Althusser’s account.26 Such a perspective, in Žižek’s view, results in people being held in place by the way in which they are described in multiple and various ways within the symbolic network. Teachers are externally defined and are recognised only to the degree to which they comply with these role determinations. There is little space for them to have their own say within this. No scope to reflect on themselves as subjects. Thus Žižek (1989) argues that such an account brings us too close to the world. We become immersed in its structures27 but without knowing it, since we cannot see outside of the structures. In this scenario, the individual is regulated according to established forms of social practice.28 Žižek follows Lacan’s theory that conceptualises divided subjects as being both included in and excluded from discourse. The discourses in which we are immersed do not provide a full vocabulary to describe who we are. The discourses contain impossibilities and specifically do not allow us to complete descriptions of our identities or our desires. This produces a lack, a lack that the subject seeks to bridge. This is experienced in Lacan’s construct as nostalgia for a lost self that never existed. That is, the subject is a retroactive construction. Self-image, he claims, is delusory and acts as a misleading filter through which the world is understood. There is no “true” version, however, the possibility of investigating the delusions remains open. Žižek seeks to reintroduce some human agency into the picture through distancing the perspective assumed.29 This perspective is shaped around a “fantasised” structuring of reality. I explore this more in the final section through a detailed example. In Žižek’s scenario trainees would still find themselves immersed in socially acceptable ways of describing their own practice. And teacher identity would still be predicated on a fragmented self that is having difficulty in or refusing to reconcile all the conflicting discourses acting through that self. Yet Žižek identifies this 25 Butler

(1997, p. 107). characterises such post-structuralism, also manifested in the work of Derrida and Foucault, as more generally as referring to a state in which “we all live in a world of signs referring to other signs” where “there is no final reality” (1989, p. 27). 27 “Interpellated”, in Althusser’s terminology, or “subjectified” in Foucault’s. 28 As Atkinson (2007) puts it: “Here the subject is almost pre-determined by the power of discourse and social practices in that they mark out a place for the subject”. This squeezes out any space for subjective intervention. 29 Žižek (1989, p. 175) contrasts Lacan’s notion of a divided subject with Foucault’s late work, which was concerned with articulating the different modes by which individuals assume their subject positions. In Foucault’s analysis, the subject creatively surfs from one subject position within a discourse to another to produce different effects, to craft a technology of self (Foucault, 1997). Whereas Lacan focuses on a subject who exceeds discourses, “the failure of its representation is its positive condition”. That is, the subject thrives though not being pinned down in a clear definitive statement. This would be a prison. 26 He

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gap between the cover story and the conflict as a potential site for resistance where a more autonomous individual identity could be asserted. The cover story is a function of self-image resulting from the hegemonic grip. This however is a delusory enterprise that can be substituted. But for this to be achieved it is necessary to find ways of enabling trainees to “see outside the frame”.30 Recent government policy moves such as the National Numeracy Strategy in Britain have attempted to provide a comprehensive picture such that the dominant discourses are all shaped by the policy. This makes it difficult to see outside of this particular frame. The teacher’s role is so tightly defined. Alternative ways of understanding education seem peculiar against this backdrop. Yet these must be given space in the teacher education programme if the programme is to be more than mere training for a given regime (Chapter 4 described an initiative that enabled teachers in secondary schools to see beyond the National Numeracy Strategy). Or space will be made by teachers’ discontent with their situation. Trainees and new teachers aligned themselves more or less successfully in a multitude of discourses. There were apparent drives that motivated their attempted participation in these discourses. There was seemingly a perverse pleasure to be achieved through performing correctly within a given regulatory frame. There were pleasures associated with believing that one’s actions were for the common good. One task related to trainee teachers was taming of the beast of mathematics. The trainees in our study actively reduced their emotional range in their dealings with it. They confined it to certain linguistic modes that kept it in check as a threat to their psyches. It was bolted down as being a very definite affair. The regulation implicit in the official curriculum guidance appeared to be positively enjoyed in what might be seen as its constraint of the teachers’ own professional integrity. The guidance shaped the teacher’s sense of self as a mathematics teacher. The next section pinpoints more directly how mathematics was framed in these arrangements. Maybe Britain is a special case at this point in time. The state of affairs does not seem to bode well for longer-term teacher initiated professional development. Where and how will new curriculum development ideas come from if teachers have been prepared largely for compliance with the current model? In Badiouian terms they are being equipped with knowledge. They are not being enabled to engage in the experimentation and critique that might enable the emergence of new ways of thinking commensurate with changing circumstances. The administration will need to decide later on a new set of rules since the professional judgement of teachers has not been incorporated in to the model. Curriculum packages have a limited shelf-life and training must prepare for later professional adjustment, to avoid a proliferation of civil servants of a time-specific governmental truth.31 Policy initiatives might alternatively promote improved practice that transcends the conceptualisations embedded within specific government initiatives. That shift 30 Žižek

offers the example of capitalism as the wider frame that embraces the limits of the individual’s understanding of the world. Capitalism’s grip, Žižek contends, is such that it appears that there is no alternative scenario. 31 This paraphrases Foucault with his specific conception of truths that are more akin to Badiou’s notion of knowledge.

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might keep alive the debates that negotiate the boundaries of mathematical activity in the classroom and how those boundaries might reshape in response to even broader evolving social demands such as economic and intellectual necessity. It would be unfortunate if the prevailing conception of teacher development reached further towards the preference of providing new rules, with the teacher understanding their own professional development in terms of following those rules more effectively.

4 Regulation Through School Policy Such British government intervention in education over the last decade or more has been widely interpreted as an assault on the professional integrity of teachers, where teachers themselves are required to police their own practice against externally defined agenda.32 Part of the armoury of accountability procedures still holding on for now entails schools carrying out self-assessments, governed by the mantra: “the school that knows and understands itself is well on its way to solving any problems it has. The school that is ignorant of its weaknesses, or will not, or cannot face up to them is not well managed”.33 Detailed procedures were provided, instructing how such assessments were to be carried out in each school. Brenda was a head teacher in a primary school who carried out practitioner research for a professionally oriented doctorate in education with me. By recording her own actions she was able to reflect on how these actions impacted on the definition of her professional role and on the terrain in which this was enacted. Previously as deputy in the same school she had been responsible for self-assessment and this provided the theme for her research. In assessing lessons presented by her staff she followed the procedures. She had also attended 3-day training programme for such assessors. Yet Brenda found herself asking whether such procedures could be carried out to the letter. As part of her research she sought to experiment with the template provided to experience and report on the affective dimension of such compliance. Below there are some extracts from Brenda’s research diary, produced during her attendance at the training sessions. These extracts point to the difficulties she experienced in occupying the official discourse, as it were, whilst reconciling this with more personal reflections made during the practitioner research process. This very distinction between discursive styles, however, proved somewhat tricky as the language of her more personal accounts increasingly made use of the official vocabulary. It became less clear whether her personal use of language embraced or resisted her deepening immersion in the official vocabulary. Throughout the three days participants were expected to watch a number of videoed extracts of lessons and produce “Lesson Evaluation Forms”. Initially the descriptions and marks on a scale 1 to 7 varied considerably. We worked in groups but were asked not to confer initially. I was mildly reprimanded for talking during one video, studious attention was necessary. We 32 Bottery

and Wright (1996). (2000, p. 150).

33 OfSTED

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were expected to complete our forms and then confer. Some wrote furiously others were more reticent. However, it was in the scoring that the course leader was interested. Each table was asked for a group conclusion, which involved considerable discussion on most tables. The initial video split the group I was working with and some members had to mark the level down, including me. All members had to move some of their marks in one direction or another. Critical argument ensued, “How can you say that?” and “How can that tell you that?” One trainee wrote that, “There are some weaknesses in teaching that mean that some pupils do not understand”. Comments from the group included, “How do you know?” and “Do you not find that some pupils don’t understand in every lesson?” Indeed, one comment was if they all understood perhaps the lesson would not be stretching enough. However after much discussion a group decision was reached. All the results were collected, next the trainer gave the “right grades”. Members of our group were pleased that we were close and there were a few smug comments of “I told you.” This approach continued over five videos of lessons. Interestingly much of the verbal evaluation was unfavourable. A teacher’s voice intonation was heavily criticised and one lesson was “torn to bits”. My neighbour stated, “I would never allow my lessons to be recorded as even the good lessons are found at fault.” It seemed that negative remarks were easier to give than praise or that the lessons were particularly poor. The latter was not reinforced by the marks, which were generally around average. The official grades were lower to start than the majority given by the groups and no lesson was pronounced as excellent in any area.

Such comments point to an apparent demand for compliance in line with tightly specified procedures that were seen as producing the correct results. Brenda felt that her own professionalism was on the line with her conceptions of what constituted good education not fitting well with the new official doctrine, centred on explicit achievement objectives: Objectives are overt, but have little influence on the quality of teaching. A teacher’s objective might be to move the Level Three children up to Level Four in Maths. But having the objective does not move the children, whereas the interaction is the essence of practice. Yet my role dictates that I set objectives that are meant to improve practice.

She felt at a complete loss when her then head teacher declared: “Surely you know a good lesson from a bad one?” As Brenda admitted: “This was difficult to answer because simplistically I felt I did”. Yet this new doctrine seemed to be as much about teachers learning to accept control of their practice as it was about producing better lessons. And the account seemed to point at some of the teachers deriving pleasure from working “correctly” within the exercises. Yet Brenda remained ambivalent: I appeared to be uncomfortable with the ideology proposed by the government. I wanted to strive to be a better teacher. I felt I had it within my power to work out the best routes to take. Now I was to be told which route to take and how to travel it. How would I accommodate the mismatch between my perceived reality and this proposed ideology? Would it entail a change in my ideological stance? Or could I work with a mismatch?

Yet Brenda’s attempt at working through these conflicts led to some surprising results. One of my (new) roles was Assessment and Evaluation Coordinator in which I was regarded as very efficient. I had computer files full of graphs recording and predicting results in tests. In interviews with the LEA inspector responsible for target-setting satisfaction with the analysis of results was always forthcoming. Official reports of visits always left me with a feeling of satisfaction. I felt that I was doing a good job. . . Indeed the document contained

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a number of health warnings, such as “it is important to recognise that data analyses provide few, if any, answers” and “despite all the fuss about targets and the use of data, it is important to remember that children do not grow taller by being measured”. My cynicism was probably softened by these remarks. I actually enjoy playing with numbers and data, so after voicing the negative aspects to the course leader I completed the necessary work satisfactorily. In fact my diary entries say little about this section of the training. The effect of performance tables on teacher and children was important to me and I had voiced my opinions, which were met with nods from other managers. . . .(but) here I (had) accepted an imposed idea that I felt did not really address educational improvement. Yet I gained pleasure through the praise I received about my competence.

These thoughts point to an increasingly fatalistic acceptance of the new strictures where small pleasures are derived from getting the game right or finding space for “playing with numbers”. Brenda is not so much embracing or resisting the strictures as becoming part of them at practical level whilst still attempting to distance herself from them at an intellectual level. Her very identity became a function of governmental apparatus where individual and social could no longer be readily distinguished. Žižek pinpoints how the individual presents herself as a manifestation of the social order: The field of social practices and socially held beliefs is not simply on a different level from individual experience, but something to which the individual has to relate, which the individual himself has to experience as an order which is minimally “reified”, externalised . . . the gap between the individual and the “impersonal” social dimension is to be inscribed back within the individual himself: this “objective” order of the social substance exists only insofar as individuals treat it as such, relate to it as such (Žižek, 2006a, p. 6, his emphasis).34

Brenda had seemingly become part of the social structure from which she was seeking to distance herself: “My first performance management objective was to gain accreditation in school self-evaluation. This was to enable me to evaluate teachers’ performance for performance management and for the school’s general improvement”. This extract is quite striking in its resplendent use of the official discourse. Words such as “performance management”, “accreditation”, “school selfevaluation”, and “improvement” are sprinkled liberally. Yet it is not clear how much irony is intended. In being obliged to use such terms so often in her professional life, where her own performance is evaluated through a particular style of usage, is it possible for Brenda to remain distanced from the words that she uses? Žižek (1989, pp. 27–33) had earlier argued that contemporary ideological structures only function successfully if subjects act with a certain degree of cynical distance. This cynical distance acts as intellectual insulation against the ideology individuals act out. Ideological functioning, he remarks, is premised on a growing gap between our discursive production and associated activity, there is no necessary relationship between reality and its symbolization.35 Ideologies incorporate their own cynicism

34 This 35 cf.

echoes the Hegelian analysis offered in Chapter 6. Laclau and Mouffe (2001).

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in advance, since such openness undermines attempts to distance oneself and be critical from a supposed outside. (I)n different historical circumstances it would undoubtedly have subversive effects; today however, in the era of cynicism, ideology can afford the secret of its functioning (its constitutive idiocy, which traditional, pre-cynical ideology had to keep secret) without in the least affecting its efficiency (Žižek, 1996, p. 200, his emphasis).

Meanwhile, full on, face-value compliance, he has argued, leads to a collapse of the structure. He cited the example of Jaroslav Hašek’s The Good Soldier Švejk, in which a soldier creates havoc by following rules to the letter.36 Cynical distance, he had further argued, is a way “to blind ourselves to the structuring power of ideological fantasy: even if we do not take things seriously, even if we keep an ironical distance, we are still doing them”.37 In Žižek’s notion of fantasy: “It is not so much in what we believe as in our external practices that fantasy is to be found”.38 In situations of ideological immersion we have no choice other than to assume a cynical or ironical distance that sustains at a practical level, the ideology perhaps being resisted at an intellectual level. Yet, Žižek later teases out a more precise distinction between the two positions of cynic or ironist and considers the potential for movement between these. He remarks that these two positions each activate their own potential deadlock: The cynic reduces ideological chimeras to raw reality, he is in search of the real ground of elevated ideological fictions; whereas the ironist entertains a suspicion that perhaps reality itself is not real but always already structured as a fiction, dominated, regulated by an unconscious fantasy. Each of these two attitudes involves its own trap; the cynic’s, a naive belief in ultimate reality outside the cobweb of symbolic fictions; the ironist’s, the opposite: the reduction of reality itself to a fiction. (Žižek, 1996, pp. 207–208)

Brenda had attempted to play it straight but with seditious intent in a situation that seemingly presented itself as a totalitarian coup of the discursive territory. That is, she sought to be a Good Soldier Švejk figure, following the rules to the letter to see if such tactics could rumble the core by revealing in a direct way their ridiculous functioning if pursued with mechanical compliance. Yet the very structures promoted a cynical distance in a number of ways. For example, by offering health warnings about their own shortcomings, or by occupying positions of “common sense” that so obviously concealed strategic moves, or by distracting her with pleasurable tasks where she could achieve localised success, Brenda could not fail to move to and fro between consent and denial, but in so doing she enacted at a practical level the very structures that she sought to discredit at an ideological level. She would have needed either monumental cynical powers, or fantastic powers of compliance, to not assume a cynical or ironical position. Brenda’s desire to hold on to her own professional aspirations was translated into a social language shaped around this new 36 Žižek

(1997, p. 22). This character might be seen as a prototype for Forest Gump also discussed by Žižek (1996, pp. 200–201). 37 Žižek (1989, p. 33), his emphasis. 38 Butler (2005, p. 5).

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order, which in a sense fixed the parameters and thereby supported the success of the new social order. And the pursuit within these parameters became the new source of pleasure. The official guidance kept tricking the teacher in to believing that she was already using the required language. The teacher was attempting to distance herself from complete immersion in the official discourses but found herself sucked in through a need to use a language and associated practices that were officially sanctioned in her communication with colleagues. The new documentation had also masked itself in a language of common sense. The virtue of new practices was presented as selfevident. The teacher became aware of how their use of language transformed the parameters that had given rise to this use and as a result the teachers became caught in an uncertain space between what Žižek defines as cynical and ironical positions. Nevertheless, the individual can achieve pleasure (jouissance) through experimenting with this discursive material and with one’s relation to it. And it is the pursuit of this pleasure that shapes the portrayals of self that gear into the outer world. Brenda, by attempting to distance herself intellectually from the official discourses found satisfaction in this attempted distancing which was always impossible at a practical level. But Brenda’s ultimate conclusion about carrying out her own research was pessimistic: why research good practice when the government is telling me what it is? To conclude: Brenda told me a story reminiscent of the dilemma faced by the New Zealand principal and the fictional Baltimore teacher described in Chapter 3. Working as she was in a school serving one of the most deprived districts in Greater Manchester she was obliged to attend to school performance statistics demanded of all schools. Consequently, she had invested heavily in preparing students for government tests and ensuring that they all attended. On the morning of the mathematics test she was horrified to see that there were three absentees. She promptly drove to their houses to round them up, only to discover that one of the 10-year olds concerned was totally drunk, having been at a party all night to celebrate his father’s release from prison. It was at this point Brenda, exasperated, asked herself “What am I doing?” It seemed she was crafting reality to produce statistics. And this was indeed a losing battle since the annual statistics typically portrayed the best teachers as being concentrated in Richmond upon Thames, a wealthy suburb of London. Yet for Lacan the restrictiveness of discourses gives and takes. It is what stimulates desire to reach beyond current ways of making sense, to look for something new. Brenda decided in favour of early retirement, taking her traditional British canal boat through the canals of Europe for a few years.


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Chapter 9

Concluding Remarks

How then do we relate to mathematics? Is it part of us? Or are we part of it? Do we shape mathematics? Or does it shape us? Or are mathematics and humans completely separate entities, occupying completely different domains, that have no association? I propose that it is not possible to settle on a final answer to any of these questions. Perspectives on these questions cannot be fixed. Engagement with the world can take many diverse forms, as can our mathematical or scientific reflections on that world. In his 2005 novel Measuring the World Thomas Kehlmann provides a fictionalised account of the lives of Alexander von Humboldt and Carl Friedrich Gauss, who are depicted as vanguard figures in the quest to build a scientific attitude to the world. It provides an insightful story of how our physical world came to be conceptualised as a mathematised entity, and how those individuals experienced newness, in themselves, and in the world. Humboldt’s life, in truth and fiction, was all about experience in the pursuit of science. He initiated the modern scientific point of view by charging around the world carrying theodolites, sextants, barometers, thermometers, clocks, compasses and every other measuring instrument then available; circumnavigating the world, climbing mountains, being tied to ships’ masts to measure waves, experimenting on his own body, making maps, monitoring astronomical movement.1 Gauss, meanwhile, was depicted as seeing much of Humboldt’s extreme science as a sheer waste of time. Gauss was content to sit at home and look out of the window,2 perhaps with a

1 During a fictional stay by Gauss at Humboldt’s house, Humboldt did “a couple of experiments on the fluctuation of the earth’s magnetic field, before, dictating a memorandum about the costs and possible uses of breeding seals in Warnemünde, writing four letters to two Academies, talking with Daguerre about the apparently insoluble problem of fixing images chemically on copper plates, drinking two cups of coffee, resting for 10 min and then proof reading three chapters of the account of his (world) journey, with footnotes about the flora of Cordillera. He had discussed the order of the upcoming evening reception at the Choral Hall with the secretary of the Society of Natural Scientists, written a short memorandum on the pumping of groundwater for the new Mexican prime minister, and replied to letters of enquiry from two biographers (pp. 183–184)”, all before Gauss got up and grumpily demanded breakfast. 2 Bibby (2010) has reported that the mathematician Marcus de Sautoy also found gazing out of a window to be highly productive. “But [she continues] gazing out of the classroom window is


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telescope and a clear sky, and speculate. Yet, this speculation detected major mathematical or physical phenomena such as the curvature of space, the magnetism of the earth and the like without needing to move from his front room. Could one possibly choose whether mathematics as a body of symbolic apparatus is to be brought into existence through bodily experience in a physical world, or through mere contemplation?3 The empirical is a specific mode of symbolising experience resulting from a history of synthesising sensual engagement. Physical experience and symbolic experience need each other in documenting our spatial world, and we all need both. The world is a function of how we make sense of it, but so are we ourselves a function of how we make sense of ourselves. We are subject to our own stories. On confronting new frontiers in the physical world such as the unimaginably large and the unimaginably small, we are reliant on modes of signification that transcend human scale. Yet, we still require mediation that calls both on our experience of the physical world and our capacity to symbolise. The universal set of mathematical knowledge is always in the process of being expanded and rearranged, which has an impact on the status and meaning of knowledge, and their elements, previously included. Mathematical statements may be correct, but the domains that house mathematics are constantly on the march. These moves can change how humans relate to the scientific knowledge, or how logic relates to conceptions of the world. Our own very constitution as human subjects is in part a function of how we are constructed with respect to scientific categories. The space we occupy has been created in response to a world conceptualised by our ancestors. And the demands they have left for us are expressed in their terms. This is where Lacan’s model of subjectivity reaches out to an expansive world that cannot be pinned down for a moment.4 Lacan depicts the human subject as being a function of the symbolic world that precedes him. Yet that symbolic world is constantly being repaired or modified by the humans that it defines. New circumstances result in old stories no longer working quite so well, in depicting the scientific world, and in depicting the human subject. We can be slow to realise that the stories have lost their poignancy. Lacan, however, remains optimistic that humans will always emerge from these creaky structures to live again. In Lacan’s thinking, any given symbolic statement produced as knowledge has a limited shelflife, not because the statement becomes wrong as such but because its situation or

often seen as a threat to the teacher’s authority, and anyone caught doing the gazing is likely to be reprimanded. Time is being wasted; you are not thinking!” 3 Extensive work in mathematics education on the theme of gesturing brings these two domains together (e.g. Radford, 2003; Roth & Thom, 2008). 4 “The subject is dependent on the articulated chain represented by science’s acquired knowledge. The subject has to take his place there, situate himself as best he can in the implications of that chain. He constantly has to revise all the little intuitive representations he has come up with, and which becomes part of the world, and even the so-called intuitive categories. He’s always having to make some improvements to the apparatus, just to find somewhere to live. It’s a wonder he hasn’t been kicked out of the system by now. And that is in fact the goal of the system. In other words, the system fails. That is why the subject lasts.” (Lacan, 2008/1967, p. 81)

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scope of possible situations has changed. The statement’s relation to truth changes through usage and in becoming familiar. Our depictions of the world change that world materially since our construction of the world is understood through the evolving descriptive categories we apply to our actions. And those constructions are built in our own image, a reflection as they are of our comprehension of the world in our own terms, and as experienced through our bodies. Crucially, we reorient ourselves through our relational encounters with others and how we imagine them interpreting and acting in response to our deeds in that shared world. Except, for Badiou, truth itself is never captured substantially. Knowledge can grant or deny access. Knowledge may enable us to approach truth, or it may also obscure its presence. In the first instance, I earlier introduced the example of Galileo needing to be recontextualised against contemporary knowledge.5 Whilst black holes would have troubled Galileo, somehow Galileo did provide us with a connection to eternal truth that has not been diminished in some respects, despite much processing as culture, and a lot of human experience in the intervening 342 years. Similarly, take the teaching of Shakespeare in English schools. Despite the massive cultural industry that is Shakespeare, his work continues to provide access to some enduring truths about humanity and art more generally despite the passage of nearly half a millennium. To suggest that his plays depict a way of life that no longer applies would be to miss the point. In the second instance, new knowledge often only works with respect to specific cultural domains and locks us into particular modes of making sense of the world. According to Lacan, we produce images of ourselves as having notional control that fix knowledge and block renewal.6 We are content to live with our current version of events. In the novel, and in real life, Humboldt’s swashbuckling adventures of physical endurance were reported as a list of measurements, idealistic descriptions of the physical world, devoid of the human angle, dismissive of the people he encountered, as though they were not part of the picture. It is one thing to live, experience and create mathematics. It can be quite a different matter to report on it and preserve it as history or as scientific knowledge serving people. Mathematics can support an intellectual attitude that we adopt in building analytical apparatus for the complex challenges that we face in life. In some of the activities described above the point was for the students to experience the creation of an object as a result of following through their own conjectures. By converting a complex problem into something manageable the intellectual demand was activated. It was not about learning about specific mathematical objects or standard results as such. It was about learning to create mathematical objects more generally by organising information in an intelligent way. The challenge was to map the territory of 5 Similarly Lacan (2008/1967, p. 74) suggests, after Gödel’s Incompleteness Theorem that proved the inherent limits of any axiomatic method, that “even arithmetic turns out to be a basket; I’m not saying it is double bottomed, but there are lots and lots of holes in the bottom.” 6 In Lacan’s terminology, the Imaginary is the persistent refuge from the failure of the Symbolic to fully explain things in a more universal fashion.

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mathematics as if for the first time, rather than supposing the well-known attractions need to be seen first. The activities entailed watching mathematical objects coming into being, and getting to understand how that happened, so that it could be made to happen again in different situations. It was about watching objects appear in a newly understood world. The world was being reconfigured as a new world to accommodate newly thought, albeit sometimes familiar, objects. Mathematics can also be a body of knowledge that we draw on in our intellectual contemplations. This, however, can lead to teachers and students working their way through the syllabus trying to cover everything instilling a dependency on the teacher who knows what the student needs to know. The teacher is quite capable of helping them at any point, to see it in the teacher’s way. And students set their target in the teacher’s terms. “Teachers feel pressure to ‘cover the material’, students feel pressured to learn quickly, and disciplinary structure supposes the possibility of discrete knowledge, a knowledge that can be learned in time” (Britzman, 1998, p. 54). This imitation of understanding is no match for the students having reached a conclusion through their own exploration. Mathematics is constructed, preserved and signified through apparatus that is necessarily cultural and hence temporal. New and existing mathematical phenomena derive their meanings from how they now relate to an ever-expanding mathematical knowledge base. Or, more mundanely, school knowledge derives from administrations trying to administer populations of teachers and children with more or less predictable results against a register of externally defined standards. Meanwhile, for individuals, mathematical constructions are held in place by incomplete accounts of school learning. And teachers in schools working with children will, like all of us, have specific and restricted historical and mathematical conceptions in some areas of their knowledge. On the one hand, mathematical ideas are cropped to meet the needs of restrictive curriculums. On the other hand, they are cladded with cultural paraphernalia through which we get to know them. Mathematics has sometimes been depicted as a field comprising ideal objects held in place by the technologies that have been developed to provide access. This technology is culture- and time-dependent implying a twofold task for students – learning the present cultures of mathematics for immediate social participation, and also access to truth through experimentation and critique. School mathematics teaching is often in the business of enabling students to better apprehend and use socially derived mathematical apparatus. And that can drive mathematics into forms more easily managed in the educational contexts concerned, and accountable within the regulative apparatus that doubles to formally assess understanding of the field and student conformity with social norms. That is, in the world of teaching situations, mathematical objects are recast as pedagogical and assessment objects that result in the erstwhile mathematical definitions becoming implicated in socially governed processes. Within educational contexts the meanings of mathematical objects are necessarily a function of the relationships within such social settings. That has always been the case. The currency in education comprises pedagogically or socially defined objects, not so much mathematical objects understood in a more platonic

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sense.7 In a specific case being considered, geometry has been converted into particular linguistic forms for accountancy purposes or formal recognition, such as tests or exams. This can compromise aspects of geometrical learning in the way Gattegno highlighted, such as where continuous experience of certain geometric forms is prematurely seen in terms of discrete categorisation, which may obscure or close off potential apprehensions of spatial phenomena. Teachers and students also find themselves understood in terms of discrete categories with respect to their engagement with mathematical phenomena. Their actions are partitioned according to a discrete mark up of the mathematical terrain. Teachers are not teachers in themselves but teachers subject to particular cultural specifications. They need to be employed in a job with certain social expectations, working practices and responsibilities that restrict how others read their actions and indeed how they assess their own practice. Specifically, they work to curriculums that mark out the field of mathematics in particular ways. Student engagement with mathematics is assessed according to how recognisable it is against this frame. The “that’s me” is forced into alignment with the “that’s it” within an externally defined register that defines “learners”, “teachers”, “mathematics” and the relations between them. This book has not so much been concerned with whether particular teaching strategies were achieving good pedagogical results in a particular context. Such a call would depend on the specific educational regime and the way in which it frames mathematical objects. Rather, the issue has been to do with how mathematical objects were located in these activities, and how those locations suggest interfaces with humans and their respective tasks or roles more generally. Mathematics education research itself meanwhile seeks to inform the social interactive processes that locate but also transform the objects concerned. This task can be understood from a range of perspectives that can mark out various operational levers, not just changes to teacher practice. And as researchers we need to be aware of how our work is governed and formatted by a range of agencies, from employers allowing limited space between other duties, to funding agencies being specific about the perspectives they want, to research assessment exercises or journals defining what is of value to the research community. But more generally we need to be attentive to the assumptions built in to our work that restrict our scope of interest.

7 Radford (2008c, p. 456) argues that Plato’s work on asserting a theory of knowable forms was motivated by “the restoration of the old aristocratic values. . .(against an ethics inspired by a) . . .’sense of relatively and temporal flux’ (quoting Levi).”


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Index

A Adams, J., 169–170, 179 Advisory Committee on Mathematics Education, 64 Aesthetic, 54–55, 59, 137 Alexander, R., 59, 191 Algebra, 7, 18, 156 Allen, W., 120, 122 Allison, A., 66 Althusser, L., 69–70, 72, 84, 100, 109, 123, 135–136, 182–183 Analytical apparatus, 2, 4, 6, 17, 33, 96, 133, 137–138, 144, 150, 163, 169, 193 Appel, S., 110 Apprehension, 3, 6, 17–18, 43, 47–48, 50, 53, 63, 90, 103, 112–113, 115, 118, 135, 140, 146–148, 160, 164, 169–170, 195 Argyris, M., 66 Art, 53–56, 59, 61, 116, 120, 140, 141, 169–170, 179, 193 Art education, 170, 179 Art Review, 53 Askew, M., 66 Aspirations, 3, 6–7, 58–59, 64, 69–73, 81, 96, 104, 142, 173, 181–182, 188 Assessment, 21, 33, 58, 64–65, 69, 89–91, 97, 103–104, 108, 122, 124, 128, 142, 151, 168, 170, 171–176, 178, 185–186, 194–195 Atkinson, D., 16, 80, 84, 110, 140, 143, 160, 169–170, 183 Australia, 66 Aztecs, 26 B Bachelard, G., 135 Badiou, A., 16, 44, 48–49, 87, 106, 108, 113, 131–150, 152, 155–158, 160–163, 165–167, 169, 176–177, 180, 184, 193

Baker, K., 176 Baldino, R., 110 Banwell, C., 151 Banyankore, 10–11 Barceló, C., 44 Bardini, C., 96, 163 Baron, S., 64 Barthes, R., 45, 87 Bartolini Bussi, M., 17, 94 Barwell, R., 60 Bazzini, L., 94 Being, 2–6, 10–11, 13–18, 21, 25, 28, 30–31, 33, 36–39, 44–46, 48–50, 53–55, 58, 60–61, 64–65, 69–72, 74–76, 78–81, 84–90, 92–94, 96–101, 104–109, 112–117, 119–122, 124, 126, 128–129, 133, 136–140, 142–152, 154–162, 165–169, 172–173, 175–184, 187–189, 191–192, 194–195 Bell, A., 75 Bellos, A., 87, 156 Berger, P., 90, 145 Bernstein, 94 Bhabha, H., 123 Bibby, T., 66, 110, 117, 162, 191 Biesta, G., 110 Big Other, 114 Binary divide, 112, 123 Biology/biological, 2, 49, 84, 90, 103, 105, 107–108, 112–114, 144 Bishop, A., 13 Black, L., 60, 69, 175 Blanton, M., 71, 116 Bohl, J.V., 68, 71, 121 Bohr, N., 140 Boni, M., 17 Bootle, R., 58 Bottery, M., 185 Bourdieu, P., 94, 118, 169

209

210 Bourriaud, N., 54–55 Bowers, J. S., 94 Boyask, R., 64 Bradford, K., 9–16 Brain, 2, 18, 61, 109, 112, 114 Breen, C., 110 The Brief History of Time, 56 British Educational Research Association, 174 British secondary schools, 18 Britzman, D., 110, 139, 194 Brown, T., 16, 22, 38, 45–46, 55, 63–64, 66–68, 72, 80, 84, 86–88, 92, 94, 100, 110–111, 115, 120–122, 128, 130, 144, 151–152, 161–163, 166, 171, 174, 179 Bruss, N., 116 Bryans, M., 66, 121 Buese, D., 120 Bullock, K., 66 Burns, C., 66 Burton, L., 94 Butler, J., 72, 87, 89, 113, 183, 188 C Cabral, T., 110 Calculus, 27–33, 45–46 Cantorian set theory, 144 Capitalist, 58, 177 Carter, G., 71, 116 Chauvet, 141 Cho, D., 110, 117, 191 Choral Hall, 191 Circle, 10–19, 21–22, 25–27, 39, 42–43, 47–48, 50, 57, 121, 124, 146–148, 165 Clarke, K., 68 Classroom, 5–6, 9, 18, 25, 43, 55–56, 58, 63–66, 75, 77–79, 83, 90–92, 96–99, 104, 110, 113–114, 116, 121, 136, 150, 158, 161–162, 165, 171–172, 174–180, 185, 191 Classroom practice, 5, 113, 171, 175–176 Clegg, J., 33 Climate, 60, 151 Cobb, P., 94 Cognitive, 2, 65, 84, 87, 90, 92, 94, 105–109, 113, 124, 126, 149, 158 Cohen, D., 68, 70 Colapietro, V., 90 Colebrook, C., 87–88 Cole, M., 71, 93, 116 Commodified, 7, 130, 173, 178 Commodity, 176–180 Common ground, 49, 158, 160 Comparative test, 6, 69, 93, 158, 173, 179

Index Concept, 11, 14, 50, 97, 107, 111–112, 118, 145, 147–149, 156, 161, 179 Confrey, J., 93, 110 Consensus/consensual, 7, 49, 65, 67, 69–71, 82, 88, 100, 134, 142–143, 147, 176 Construction, 4, 16–17, 20, 28, 38, 44–45, 47, 50, 60, 74, 84, 86, 90–91, 103–104, 114, 116, 127, 129, 143, 145–148, 159–161, 163, 177–178, 180, 183, 193–194 Constructivism, 6, 68, 93, 124, 143 Contemporary art, 4, 53–55, 169 Context, 2, 5, 9, 33, 46, 49, 63, 66, 73–76, 78–82, 85, 88, 92, 96, 101, 106, 110–113, 124, 129, 131, 133–134, 146, 151, 157–159, 175, 177, 179, 181, 195 Continuous, 17, 22, 25–26, 28, 33, 38–39, 48, 50, 96, 148–149, 178, 195 Control, 7, 45, 47, 51, 58, 60–61, 86, 90, 94, 106, 114, 118, 130, 142, 151, 179, 181, 186, 193 Control technology, 60, 90, 94, 106, 130, 181 Cooper, B., 94 Cortes, V., 87 Creation of the animals: “Where are the dinosaurs?”, 44 Credit Crunch, 58, 60 Critchley, S., 145 Cultural Historical Activity Theory (CHAT), 177 Culture, 2, 4, 10, 13, 17, 44, 54, 71, 83, 91, 100, 104, 121–122, 126, 129–130, 142, 157–158, 160–161, 164–169, 177, 193–194 Curriculum, 4–6, 9, 11, 51, 63–82, 84, 86, 91–92, 96–97, 100, 118, 121, 124, 126, 128, 142, 144, 151, 155, 160, 162, 168, 171–177, 179–181, 184 Cycle/cyclical, 15–16, 39, 48 D Damascus, 143 D’Amore, B., 110, 124 Davis, P., 60, 69, 175 Day, C., 64, 115 De Freitas, E., 55, 94 De Lange, J., 73 Deleuze, G., 135 Demands, 3, 6, 9, 57, 60, 72, 84, 88, 92, 100, 104, 109, 114–115, 118–119, 131, 142, 168, 171–173, 180–183, 185, 192 Department for Education and Skills, 64

Index Derrida, J., 17, 46, 50, 79, 86, 110, 112, 134–135, 183 De Sautoy, M., 191 Desire(s), 108–110, 114–117, 119, 123, 125–126, 177–183, 188–189 Development, 2, 4–5, 9–11, 25, 45, 50, 64–67, 69, 71, 73, 77, 86, 91–92, 106, 108, 115–118, 131, 133, 140, 149, 158, 162, 166, 172, 181, 184–185 Devine, N., 68, 128 Differentiation, 28, 31–32, 46 Discourses, 7, 14–15, 46, 50, 72, 79, 82–83, 85, 89, 93, 97, 100, 111, 114, 116–117, 124, 128–129, 141–143, 159, 164, 168, 174, 176, 179, 182–185, 187, 189 Discrete, 7, 17–18, 22, 25, 28, 38–39, 48, 50, 123, 149, 177–178, 194–195 Discursive, 45, 51, 71–72, 74–76, 79, 81, 83–84, 86–89, 91, 93, 101, 106, 110–115, 117, 119, 124, 128–129, 131, 134, 136, 138, 140, 160, 163, 180, 185, 187–189 Discursive formation, 71 Dorment, R., 54 The Doubling modulo, 138, 151 Dowling, P., 94 Dream-text, 179 Dreyfus, T., 55 Dunne, M., 94 Duval, R., 90, 95, 99–100, 126, 129 E Eade, F., 74, 130 Eagleton, T., 87 Economic, 54, 58, 64, 71, 74, 142, 177, 185 Economic and Social Research Council, 64, 74 Educational Studies in Mathematics, 5, 83, 93, 110, 163 Ego, 106–109, 114, 119–120, 124, 182 Egyptian, 26 Einstein, A., 49, 140, 142–143, 161 Eisenberg, T., 55 Ellipse, 19–21, 39, 42–43, 146 Emancipation, 86 Emergency medicine, 86 Emergency Medicine Journal, 86 Emerson, C., 117 Emotion, 124, 126, 128, 183 England, 2, 39, 55, 58–59, 64, 68–69, 72, 144, 151, 172–173 England, J., 16, 80, 84, 110–111 English National Numeracy Strategy, 73

211 Environment, 2, 4, 6, 47, 53, 55, 58–59, 63, 85, 90, 96, 115, 117, 127, 133, 147, 149, 152, 159–160, 168, 174, 178 Ernest, P., 90–91, 93–95, 99 Euclid, 17, 57, 126, 137, 141–142 Exam (ination), 4, 33, 55–56, 64, 69, 114, 128, 139–140, 146, 159, 171, 173–174, 182 F Fairclough, N., 91, 113 Fantasies, 60, 108, 110, 113, 115, 119–120, 123–125, 178, 183, 188 Farrow, M., 120 Felman, S., 110 Feltham, O., 135, 146 Fennema, E., 68 Ferguson, N., 58 Ferrara, F., 147 For the love of God, 53 Foucault, M., 71, 84–86, 110–111, 134–136, 138, 183–184 Freud, A., 107, 110–111, 119, 177, 179 Freudenthal, H., 73, 75, 177 Freudenthal Institute, 74–75 Freud, S., 106–109 Fullan, M., 66 Function, 2, 4, 6–7, 16, 31, 44, 49, 51, 53–54, 56, 63, 73, 78–79, 82, 84, 86, 89, 105, 108, 112–114, 116–117, 124, 126, 130, 137, 139–140, 145–146, 148–149, 159–160, 168–169, 177, 181–182, 184, 187, 192, 194 Furinghetti, F., 161 G Gabriel, M., 45, 137, 175 Gadamer, H.-G., 134, 161 Gade, S., 117 Galileo, 44, 143, 193 Gallagher, S., 86, 138, 161 Galton, M., 66 Gap, 30, 70, 76, 80, 117, 119, 125, 143, 175, 180, 182–184, 187 Gattegno, C., 17–18, 50, 149–150, 155–157, 177, 195 Gauss, C. F., 191 Gee, J.P., 113 Geist, P.K., 68 Geometry(ic/al), 3–4, 7, 10, 13, 16–18, 25, 28, 33, 38, 43, 45–46, 50–51, 56, 121–122, 126, 137, 140, 142, 146–149, 178, 195 Giddens, A., 106 Gödel’s Incompleteness Theorem, 193

212 Goldenberg, P., 68 The Good Soldier Švejk, 188 Goos, M., 71, 116 Government, 3, 6, 55–56, 59, 66, 69, 72–74, 76, 82, 84, 86, 100, 115, 120, 128, 131, 161–162, 172–174, 179, 184–187, 189 Government Inspections, 128 Government Inspections for training colleges, 128 Grammar, 2, 139, 145 Graph, 12, 14, 26, 28, 31–32, 35, 37, 59, 65, 86, 89, 107, 127, 157–162, 166–167, 178, 186, 191 Gravemeijer, K., 74 Graven, M., 94 Green, M., 48 Grosz, E., 108 Groves, J., 120 Gu, Q., 64, 115 Guzman, J., 158 H Habermas, J., 86, 91 Hallward, P., 49, 135–137 Hammersley, M., 79 Handal, B., 66 Hanley, U., 63, 74, 120 Hardy, T., 110 Hargreaves, D., 65 Harrington, A., 74, 138 Harvey, D., 1, 87 Hašek, J., 188 Hawking, S., 48, 56–59, 129 Hegel, G., 111–113, 129, 172, 187 Hegemonic, 173, 184 Heggs, D., 87 Heidegger, M., 134 Heinz, K., 68, 107 Heisenberg, W., 140 Henriques, J., 110 Herbel-Eisenmann, B., 87 Hermeneutics conservative, 96 critical, 86 moderate, 86, 161 radical, 86 Herrington, A., 66 Heywood, D., 38–43 Hickey, D., 66 Hirst, D., 53–55 Historical, 4, 6, 13, 17–18, 43–44, 47, 50, 63, 85–86, 103, 110, 116, 134, 139, 141,

Index 144, 147, 158–159, 163–165, 176–177, 179, 188, 194 Historicity, 17, 46 History, 4, 44, 47, 49, 54, 56, 59, 61, 65, 70, 79, 98, 136, 140–141, 155–156, 161, 167, 176, 192–193 Hodge, R., 91, 113 Hoffmann, M., 97 Hollway, W., 110 Holzman, L., 130 Homer, S., 111 Hughes, R., 53 Human, 1–4, 6, 14, 16–18, 27, 41, 43–44, 47–48, 51, 53, 56–57, 60–61, 69, 71, 73, 82, 87–88, 91, 97–98, 103–113, 115–117, 119–120, 122–125, 129, 133–135, 138, 140, 142, 146–147, 149, 151, 156–157, 160, 162–164, 166–167, 169, 175, 179, 182–183, 191–193, 195 Husserl, E., 13–14, 17, 43, 45–46, 122, 147, 164 Hutcheson, G., 60, 69, 175 I Id, 107 Ideal, 6, 16–17, 21, 25, 43, 46–48, 50, 59, 70, 126, 148, 162, 164, 166, 194 Identification, 17, 54, 67, 69–70, 72, 79, 82, 98, 105, 107–108, 117, 121–123, 125, 127, 182 Identity, 6, 65, 80, 82, 98, 180–184, 187 Ideological state apparatus, 70 Ideology (ical/ly), 7, 59, 69–73, 80, 82, 84, 86, 103, 107, 121, 123, 125, 130, 135–136, 143, 160, 167, 182, 186–188 Imaginary (Lacanian), 48, 120, 122–126, 130, 147, 193 Improvement, 65–67, 69–71, 73, 80–82, 99, 123, 171, 173, 182, 187, 192 Initial Teacher Training, 55 Inspections, 128, 181 Integration, 28, 31–32, 37–38, 46, 96 Intensity, 48 Interpellated, 84, 182–183 Interpellation, 123, 182 Interpretant, 92, 111, 127 Investigation(s), 68, 92, 136–138, 142, 150–153, 166–167, 178 J Jackson, M., 175 Jackson, S., 21 Jagodzinski, J., 110

Index Johnson, J., 66 Johnston-Wilder, S., 146 Jones, L., 16, 111 K Katz, V., 161 Kehlmann, T., 191 Kieran, T., 68 Kilroy, D., 86 Kington, A., 64, 115 Kinsel, M., 68 Klein, M., 110 Kline, M., 13 Knowledge, 16, 33, 35, 44, 46–47, 50, 56, 78, 86, 91–92, 97, 103–104, 122–123, 127, 132–147, 150–152, 155, 157–158, 160–161, 163–164, 166–169, 171, 174, 176–177, 180, 184, 192–194 Kress, G., 91, 113 Kuhn, T.S., 48 Kynigos, C., 66 L Lacanian, 104, 109, 111, 119–126, 131, 141–144, 149, 178, 182 Lacan, J., 5, 15, 88, 91, 97, 104–133, 135, 141–144, 149–150, 157, 166–168, 170, 172, 182–183, 189, 192–193 Laclau, E., 69, 99–100, 113, 187 Lakatos, I., 135 Lakoff, G., 147 Language, 1, 10–11, 13–14, 16, 43–51, 72, 83, 86, 89, 91, 105, 109, 111, 115, 117, 121–123, 125, 131, 134–135, 138, 141–143, 145, 147, 155, 158, 161, 164–166, 173–174, 180, 185, 188–189 Lappas, D., 147 Latent dream, 177, 179 Lather, P., 7 Lave, J., 93 Leader, D., 120 Lee, Y., 93 Lemke, J., 90 Lerman, S., 93 Leslie, E., 68, 128 Levi-Strauss, C., 111, 164–165 Liberati, S., 44 Linguistic(s/ical), 1, 4, 10, 13–15, 28, 44, 47, 50, 89, 97, 105, 111–112, 116–117, 122, 127, 134–135, 142–143, 145, 184, 195 Literary theory, 87 Lloyd, G.M., 68

213 Logics of Worlds, 145 LOGO, 47, 157 Lopez-Real, F., 66 Lovlie, L., 88 Luckmann, T., 90, 145 Lyotard, J.-F., 134–135 M Madness, 85 Malson, H., 88 Manchester Metropolitan University, 74 Mancunian, 32–38 Manifest, 160, 174, 177, 179 Manifest dream content, 179 Mason, J., 16, 50, 143, 146 Mathematical investigations, 150–155 Mathematics, 1–18, 31, 43, 45–46, 49–51, 53–61, 63–75, 79, 81–101, 105–189, 191–195 Mathematics in Context (MiC), 73–82 Math wars, 68, 143 McNamara, O., 45–46, 55, 63, 64, 66, 67–68, 72, 84, 92, 110, 120, 144, 151, 162, 171, 174 Meaghan, M., 138 Meaning, 140–141 Measuring the World, 191 Medical, 86, 127–128 Medicine, 86 Milbank, J., 131 Miller, J.-A., 111 Millett, A., 66 Mind, 112, 115–116, 127, 138, 147, 149, 151, 157, 165 Miranda, I., 158 Mirror stage (stade du miroir), 117 Morgan, C., 90–91, 94–95, 99–101 Morris, E., 176 Mouffe, C., 82 Mouffe, M., 87, 100, 187 Moutsios-Rentzos, A., 17 Myhill, D., 66 Myth(ologise), 45 N Nancy, J.-L., 87 Narcissistic, 108 Narrative(s), 5, 16, 44–47, 89, 108, 113 National Centre for Excellence in the Teaching of Mathematics, 64 National Council of Teachers of Mathematics (NCTM), 68, 97 National Curriculum, 64, 75, 128

214 National Curriculum for Initial Training, 128 National Numeracy Framework, 73 National Numeracy Strategy, 73–76, 128, 173, 175, 182, 184 Neighbourhood Renewal Strategy, 128 Nelson, B., 68 Nemirovsky, R., 147 Newman, F., 130 New teacher, 6, 59, 71, 72, 120, 173–174, 180, 182, 184 Newton, I., 17, 28, 48, 140, 142, 161 New Zealand, 56, 58–59, 66, 128, 189 Ng, S., 66 Nolan, K., 44, 120 Nordtug, B., 88 Numeracy, 55, 73–76, 128, 173, 175, 182, 184 Numeracy Skills Test, 55, 128 Numeracy Skills Tests for teachers, 128 Nunes, T., 160 Núñez, R., 147 O Object(ive/ification), 2, 16, 47, 54, 60–61, 74–76, 80–81, 90, 96, 100, 108, 121, 126, 128, 134, 150–151, 155, 157–158, 162–163, 165–167, 173, 186–187 Office for Standards in Education, 64 Official, 67, 71–72, 120, 173, 184–187, 189 Ogden, 155–157 Ollerton, M., 152 Ongstad, S., 90–91, 95–96, 99–100 Ontology, 145, 164 Otte, M., 90, 94–95, 99–100, 123, 126 P Pacific, 68, 128 Paiti, M., 68, 128 Pampaka, M., 175 Papert, S., 157 Paris, 143 Parker, I., 38 Passeron, J.-C., 118 Patton, P., 138 Pedagogy, 56, 76, 110, 139–140, 157, 169–170 Peirce, C. P., 5, 83, 88, 90–92, 98, 110–112, 127–129, 163–164 Peirce, C. S., 83 Peirce’s approach to the self, 90 Pellegrino, J., 66 Perceptions, 2, 4, 18, 43, 54, 63, 66, 74, 85, 106, 140, 167 Performance, 2, 7, 18, 55–59, 64, 103, 114–115, 120, 126, 173, 179, 187, 189

Index Performative, 57, 181–182 Peters, M., 110 Phenomenology(ies), 15, 45, 165 Philosophical(ly), 2, 6, 49, 94, 96, 101, 131, 133–134, 138, 142–143, 145, 163, 175 Philosophy (ers), 6, 67, 75, 82, 93–94, 98, 106, 111–113, 129, 131, 134–135, 140, 142–144, 160, 166–167, 169 Physics, 4, 17, 49, 56–57, 140, 143 Piaget, J., 5, 6, 88, 90–93, 106, 109–111, 113–117, 119, 122, 129–130, 133, 135, 147–149, 157 Picasso, 141 PISA, 69, 173 Pitt, A., 110 Planet(ary), 38–43, 48 Policy, 6, 64–66, 72, 79, 82, 84, 89, 92, 94–96, 98–100, 128, 131, 144, 161, 171–176, 181, 184–189 Political, 2, 6, 64–65, 104, 113, 132, 142, 170, 171–189 Political science, 87 Post-modernism, 90 Post-structuralism, 86, 88, 94, 111, 183 Practitioner, 9, 16, 50, 185 Presmeg, N., 83, 88, 90–91, 95–97, 99, 110, 179 Problem solving, 68–69, 71, 89, 143, 152, 173, 178 Professional development, 4, 172, 184–185 Professionalism, 6, 72, 172–173, 181, 186 Pryor, J., 66 Psychoanalytic (al/is), 5, 106–108, 110, 112–113, 122, 141–142 Psychology, 5, 15, 60, 85, 88–93, 103, 105–110, 113–114, 116, 119, 124, 130, 160, 197 Psychology of Mathematics Education, 107 Pythagoras, 11 Pythagorean Theorem, 17 Q Qualifications and Curriculum Development Agency, 64 Qualitative unity, 137 R Radford, L., 14, 17, 63, 88, 90, 94–96, 99, 110, 116, 118, 147, 150, 158, 160–168, 192, 195 Rancière, J., 100 Rao, N., 66

Index Realistic Mathematics Education (RME), 73–75, 80, 97 Reality, 1, 8, 14, 47, 49, 58, 68, 73, 79, 105, 107–108, 117, 119, 125, 134, 140, 143, 157, 161, 164, 169, 178, 183, 186–189 Real (Lacanian), 122–125 Real life, 59–60, 78, 97, 114, 120, 155, 157, 159, 175, 193 Rees, G., 64 Reflect, 84, 116, 130, 183, 185 Reflective, 9–13, 16, 25, 50, 61, 122, 147, 161, 166 Reform, 2, 4, 65, 67–71, 73, 82, 104, 121, 124 Remillard, J., 66–68, 71, 86, 121 Renewal, 2, 6–7, 48, 50–51, 103–132, 133–170, 193 Represent(ation), 1, 12, 39, 75, 100, 107, 111, 127, 129, 139–140, 142, 147, 169, 183 Research, 2–3, 5–7, 9, 14–16, 50, 57, 59, 63–69, 71–74, 79–80, 83–101, 105–133, 146, 150–151, 158–159, 161, 163–164, 172, 174–176, 180, 182, 185, 189, 195 Rhodes, V., 66 Ricoeur, P., 44–45, 86 Rigid designator, 180 Robertson, J., 110 Romberg, T., 66 Roth, W.-M., 93, 147, 192 Runyankore (language), 10, 13–14 S Sabena, C., 96, 163 Saenz-Ludlow, A., 83, 90–91, 96, 99, 110 Sahin, C., 66 Sammons, P., 64, 115 Saunders, K., 151 Saunders, L., 64 Saussure, F. de, 5, 97, 111 Schön, D., 73 School mathematics, 2, 50–51, 56, 58–60, 63–64, 69, 73, 91, 94, 103, 142, 155, 157, 167–170, 172–180, 194 Schubring, G., 17 Self, 1, 14, 17, 46, 48, 72, 89–90, 95, 97, 105–106, 108–109, 112–113, 116–117, 119–125, 129, 144, 149, 183–185, 187, 189 Semiotic(s/ical), 5, 83, 90–92, 94–95, 97–100, 109–111, 127–130, 164, 166 Senger, E., 66 Series, C., 56, 59 Shakespeare, W., 193

215 Shape, 16–27 Sharp, J., 56 Sign, 92, 96, 111, 127–128, 159 Signified, 14, 45, 47, 59, 83, 112, 180, 194 Signifier, 14–15, 45, 59, 112, 117, 124, 126–127, 180 Sila’ila’i, E., 68, 128 Simon, M., 68 Simpson’s rule, 29, 38, 46 Sinclair, N., 17, 54–55, 137, 149 Skemp, R., 91 Skott, J., 66, 121 Smees, R., 64, 115 Smolin, L., 44 Social practices, 87, 134–135, 147, 150, 166, 176, 179, 183, 187 Social sciences, 7, 99, 103, 113, 165 Socio-cultural, 10, 14 Solomon, Y., 38–43 Sonego, S., 44 Space, 16–27 Spatial, 9–51 Spyrou, P., 17, 147 Sri Lanka, 64, 69 Stables, A., 66 Standardised Attainment Tests, 128 Standardised training programme for teachers administering the Strategy, 128 State apparatus, 70 St. Paul, 143 Steffe, L., 68 Steinbring, H., 90, 94–95, 99–100 Stephens, J., 128 Stobart, G., 64, 115 Stories, 2, 15, 31, 38, 44–47, 54, 58, 60, 70–71, 74, 84, 86, 105, 112–113, 115, 119–120, 135, 141–143, 147, 165–168, 178, 181–182, 184, 189, 191–192 Strathern, M., 177 Stronach, I., 60, 103 Subject, 3–4, 6, 13, 46, 57, 60, 64, 67, 72, 84, 87, 89, 97, 104–134, 144, 149–150, 157, 160–161, 164, 166–167, 169–171, 174–175, 180, 182–183, 192, 195 Subjectivation, 54, 85 Subject(ivity), 9–15, 83–99 Super-ego, 108 Susskind, L., 44 Swan, M., 66 Symbolic (Lacanian), 105–106, 110, 112–116, 120, 122–125 Symbolic violence, 87

216 Symptom, 7, 85, 108, 120, 122, 126–127, 177–178 Sztajn, P., 66, 71 T Tahta, D., 110, 118, 151, 157, 177 Taylor, C., 64 Taylor, P., 178 Teacher Development Agency, 64 Teacher education, 2, 6, 91, 94, 98–100, 111, 171, 174–175, 180–185 Test(s/ing), 3, 6, 33, 50, 55–56, 58–60, 64, 69–70, 93, 108, 128, 139, 158, 168, 173–175, 179, 181, 186, 189, 195 Thom, J., 147, 191–192 Time, 14, 16–17, 39, 43–45, 47–48, 140, 149, 164–165, 168, 184, 192, 194 TIMSS, 69, 173–175 Todd, S., 110 Torrance, H., 66, 74 Trace technology, 16 Tradition, 45, 107, 134, 144, 151, 163–165 Trainee, 2, 4, 6, 50, 55, 71–72, 112, 152, 182, 184, 186 Trends in Mathematics and Science Study, 51 Triantafyllou, D., 17 Truth, 6, 70, 79, 82, 96, 100, 108–109, 119, 126, 133–170, 180, 184, 191, 193–194 Tsatsaroni, A., 93 Tzur, R., 68 U Uganda, 2–3, 9, 47, 64, 121, 126 Umaki, S., 68, 128 Unconscious, 109, 112, 124, 179, 188 Unconscious desire, 179 United Kingdom, 9, 66, 69, 71 United States, 67–68, 124, 143 Unity, 90, 115, 120, 123, 145–146, 182 University, 48, 60, 74–75, 124, 139, 174–175, 178, 181 University of Wisconsin, 74–75 Urwin, C., 110 V Valero, P., 99 Valli, L., 120 van Hiele model, 147

Index van Zoest, L.R., 68, 71, 121 Venn, C., 110 Visser, M., 44 Voluntary Services Overseas, 9 von Glasersfeld, E., 93 von Humboldt, A., 191 Vygotsky, L., 5, 71, 88, 90–91, 93, 96, 109–119, 122, 130, 149–150, 157–162, 166 W Wagner, D., 87 Wake, G., 60, 69, 93, 175 Walkerdine, V., 60, 69, 91, 94, 97, 110, 114 Walls, F., 94, 110, 162 Walshaw, M., 94, 110 Watson, J. D., 49 Wegerif, R., 162 Wenger, E., 94 Werckmeister harmonies, 39 Westbrook, S., 71, 116 Whitty, G., 64 Whole Class Interactive Teaching (WCIT), 75, 77–78, 81–82, 138, 151 Wiley, N., 111 Williams, J., 49, 60, 68–69, 93, 128, 175, 177 Williams, R., 49 Wilson, D., 110, 130 Wilson, S.M., 68 The Wire, 59 Wittgenstein, L., 49, 128, 134 Woit, P., 48 World (Badiou), 145 Wright, D., 185 Wright, N., 185 X Xu, G., 93 Z Zevenbergen, R., 99 Žižek, S., 14, 45, 57, 59, 69–70, 81, 87, 89, 100, 106, 112–113, 115, 120–122, 125, 127, 129, 131–132, 134, 137, 164–165, 170, 172, 175–180, 182–184, 187–189 Zone of Proximal Development (ZPD), 117–118, 158, 166

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