Geometry as Objective Science in Elementary School Classrooms: Mathematics in the Flesh (Routledge International Studies in the Philosophy of Education)

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Geometry as Objective Science in Elementary School Classrooms

Routledge International Studies in the Philosophy of Education

1. Education and Work in Great Britain, Germany and Italy Edited by Annette Jobert, Catherine Marry, Helen Rainbird and Lucie Tanguy 2. Education, Autonomy and Democratic Citizenship Philosophy in a Changing World Edited by David Bridges 3. The Philosophy of Human Learning Christopher Winch 4. Education, Knowledge and Truth Beyond the Postmodern Impasse Edited by David Carr

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Geometry as Objective Science in Elementary School Classrooms Mathematics in the Flesh

Wolff-Michael Roth

New York

London

First published 2011 by Routledge 270 Madison Avenue, New York, NY 10016 Simultaneously published in the UK by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN

Routledge is an imprint of the Taylor & Francis Group, an informa business This edition published in the Taylor & Francis e-Library, 2011. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.

© 2011 Taylor & Francis The right of Wolff-Michael Roth to be identified as author of this work has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Roth, Wolff-Michael, 1953– Geometry as objective science in elementary school classrooms : mathematics in the flesh / Wolff-Michael Roth. p. cm. — (Routledge international studies in the philosophy of education ; 27) Includes bibliographical references and index. 1. Geometry—Study and teaching (Primary) 2. Education, Primary—Philosophy. I. Title. QA461.R68 2011 372.7—dc22 2010047439 ISBN 0-203-81787-7 Master e-book ISBN

ISBN13: 978-0-415-89157-8 (hbk) ISBN13: 978-0-203-81787-2 (ebk)

Contents

List of Figures Preface Introduction: Of Hands, Flesh, and Mind

ix xiii 1

PART A Toward a Theory of Mathematics in the Flesh Introduction to Part A

15

1

What Makes a Cube a Cube? A Phenomenological Overture

19

2

From Intellectualist Metaphysics to Embodiment Epistemologies

35

3

Material Life as the Organizing Principle of Knowing

60

PART B Stories of Mathematics in the Flesh Introduction to Part B

83

4

The Flesh, Distractions, and Mathematics

87

5

Coordinating Touch and Gaze: Re/Constructing a Mystery Object

111

6

Emergence of Measurement as the Realization of Geometry

138

7

Doing Time in Mathematical Praxis

157

viii Contents

PART C Emergence of Geometry—An Objective Science Introduction to Part C

197

8

Ethno-methods of Sorting Geometrically

201

9

Reproducing Geometry as Objective Science

231

10 Rethinking Mathematical Conceptions Epilogue: From the Flesh to Society in the Mind Appendix: Transcription Conventions Notes References Index

243 260 267 269 279 287

Figures

1.1

1.2

1.3 2.1 2.2 4.1

5.1

6.1

a. Chris is oriented to the box following what we can hear to be a teacher question, “What would that box have to have to be a cube?” b. At the end of the presentation, Chris squarely gazes at his teacher, no longer “caught up” with the pizza box but into the familiar terrain of the red cube that he has manipulated so many times before.

23

a. A depiction of the Necker cube. b. A cube similar to the Necker cube, but drawn according to the laws of perspective. c. A cube similar to the Necker cube, but with “hidden” edges drawn in dotted lines.

30

In the Müller-Lyer illusion, the eye sees two lines of precisely the same length as having different length.

33

Collections of mystery objects as second-graders came to classify them (see also Chapters 8 and 9).

40

Categories (concepts) of geometrical objects according to the rules Kant established.

41

A concept map of the lesson on vertices and edges is prominently displayed on a large sheet of paper taped to the chalkboard in front of the classroom.

98

Sylvia, Jane, and Melissa (from left to right) sit at a large round table where they investigate a mystery object hidden in a shoebox to build a model of it from plasticine.

116

While Mrs. Turner is asking a question about why the mystery object in the shoebox is a cube, Ben is producing the caliper configuration with both hands, the one holding and the other measuring/gesturing.

146

x

Figures

6.2

7.1

Jane fi rst measures the height of a pizza box using the caliper (a, b) and then shows how to count the repeated application of the measure to find out how many pizza boxes it takes to make a cube by stacking (c, d).

153

Ben uses the same gestures as Mrs. Turner—in an episode involving Chris during a different lesson—to show how Post-it pads need to be stacked to produce a cube.

165

7.2

Mrs. Turner’s talk and gestures concerning a cube are coordinated with her turning toward a cube that is behind her. 166

7.3

Chris’s body and head/gaze movements come to be aligned with those of Mrs. Turner, as he re-orients to the pizza box after having just oriented away from it.

167

Chris’s movements are associated with complementary movements on the part of Mrs. Turner, as she turns allowing her to see where to Chris orients himself.

168

Mrs. Turner, in a movement complementary to that of Chris, steps away from the position that she has held, thereby allowing Chris to take it up.

169

Mrs. Turner’s pitch level picks up where Chris ends before returning into its normal range.

174

Mrs. Turner’s pitch contour repeats that previously available in Chris’s voice.

174

The sudden and non-anticipatable jump and rise of Mrs. Turner’s pitch level creates a problem for the subsequent pitch level, here “solved” by a pitch that lies halfway between the anticipatable pitch ending and the actual pitch ending.

175

Chris produces the same four-beat structure in his gesture while explaining the nature of a cube that Mrs. Turner enacts by prosodic means.

184

Cheyenne produces a four-beat structure while explaining the characteristics of a cube.

186

Cheyenne’s and Ethan’s performances of the four-beat structure are precisely coordinated.

186

Cognitive content—counting of geometric concept words—is aligned with the iconic (counting) and beat (scanting) hand gestures and with the production of the rhythmic pattern.

187

7.4

7.5

7.6 7.7 7.8

7.9

7.10 7.11 7.12

Figures 8.1

xi

The children sit in a circle. Mrs. Winter and Mrs. Turner (in black) sit with the children. Recognizable collections include the cubes (far left bottom), cones, and pyramids. Connor currently holds his object next to one of the rectangular solids.

205

Connor orients toward the collection of cubes that Kendra is pointing to.

210

Kendra gazes at her object and then looks around at the existing category systems.

247

10.2

Kendra wraps her fi ngers around the cylinder.

249

10.3

“Comparing” two cylinders.

249

10.4

The cylinder—which in Greek is kylindros, the roller— affords “rolling” actions of Kendra’s hands.

250

10.5

More comparison actions.

250

10.6

The cylinder affords many different ways of holding it.

251

10.7

A conception of conceptions.

256

8.2 10.1

Preface

This book is about early learning experiences in and with geometry. I exhibit the origins of geometry as objective science in and out of the intuitively given everyday lifeworlds of children. These lifeworlds are pre-geometric; but they are not without model objects that denote and come to anchor geometric idealities that the children will understand at later points in their lives. That is, my analyses show how geometry, an objective science, arises anew from the pre-scientific but nevertheless methodic actions of children in a structured world always already shot through with significations. More importantly, though, I present a way of understanding knowing and learning in mathematics that differs from other current approaches, although it shares aspects with other theories, most notably enactivism and embodiment approaches.1 I develop this new way of understanding with materials collected during a three-week unit of geometry in a second-grade elementary classroom. I use these materials to both articulate the contradictions and incongruences of other theories—Immanuel Kant, Jean Piaget, and more recent forms of (radical, social) constructivism, embodiment theories, and enactivism. I show how material phenomenology fused with phenomenological sociology provides answers to the problems that these other paradigms do not answer. Projects such as this book do not emerge from a vacuum, but are based on years of developing understandings. In my case, there are more than 25 years of intense study of concrete data on learning, collected under controlled and field conditions (classroom, everyday work). In the course of this analytic work, numerous theoretical frameworks that I have come to know and have worked through did not hold up to the empirical evidence, encouraging me to seek new ways of theorizing and understanding the data sources I collected. Among those theoretical paradigms that I worked extensively with are Piagetian and neo-Piagetian theories, information processing theory, artificial neural networks, radical constructivism, social constructivism, situated/distributed cognition, embodiment/enactivist theories, and cultural-historical activity theory. The theories that have stuck with me, and which I contributed to developing, are characterized by two aspects: (a) they are phenomenological, view the world from the perspective of the

xiv Preface actor (which tends not to be one of theoretical reflection on the ongoing action) and (b) they focus on the social/societal aspects of life, emphasizing that all consciousness, as the etymology of the term suggests, is knowing with and for the other. It is impossible to trace one’s understanding back to everything one has read or encountered. But across this book, the main philosophical trends that have shaped my thinking are recognized in the references—but the same reference is not likely to appear repeatedly unless there is a specific need (e.g., paraphrase) or purpose (quote). Some chapters developed from the original versions—sometimes vastly longer drafts—of three published papers. In the historically first of the three entitled “Bodily Experience and Mathematical Conceptions: From Classical Views to a Phenomenological Reconceptualization” (Roth & Thom, 2009a)2 , I developed a new theory of mathematical concepts and conceptions; I subsequently realized that this framework needed to be pushed even further. In “The Emergence of 3D Geometry from Children’s (Teacherguided) Classification Tasks” (Roth & Thom, 2009b)3, I developed the idea of how geometry as objective science can be thought of as emerging from the non-geometrical practices that children bring to school. Both papers were, taking up a comment from Hans Freudenthal concerning his own work, “a bit phenomenologically tainted.” Subsequent to writing these two papers, I became familiar with the work of the French philosopher Michel Henry and his commentaries on another French philosopher, Pierre Maine de Biran. Together with the writings of three further French philosophers—Jean-Luc Nancy, Jacques Derrida, Jean-Luis Chrétien—on the role of the body generally and the sense of touch (tact) more specifically to knowing led me to propose a radicalization of theories of mathematical knowing. In “Incarnation: Radicalizing the Embodiment of Mathematics” (Roth, 2010)4, I formulated mathematical knowing for the fi rst time in terms of the material phenomenological approach of Maine de Biran and Michel Henry. I am grateful to the three publishers for allowing me the use of materials from the articles, which I de- and recomposed into different chapters for the present purposes. In addition, I was able to develop my early ideas concerning this approach in two papers that my colleague Jennifer Thom fi rst-authored. To her I am particularly indebted for providing me with a space for developing theoretical issues to understand phenomena to which she tends to take the enactivist approach of Humberto Maturana and Francisco Varela. Throughout this book I use my own translations of original French and German texts, because the licensed translations constitute interpretations, which often have epistemologies embedded that are opposite to the one I develop here. Moreover, translations frequently do not render what is so special about the texts in their native language. For example, I use a quote that in my translation reads “[A concept therefore is] an incorporeal, even though it incarnates or effectuates itself in bodies” (Deleuze & Guattari, 1991/2005, p. 26). The official translation, however, suggests, “The

Preface xv concept is an incorporeal, even though it is incarnated or effectuated in bodies” (p. 21). In this translation, the reflexivity of the concept—which incarnates itself almost despite the person—is lost, though this passivity is precisely what we need to retain in a proper account of learning and that is highlighted in the work of Michel Henry. As I show in this book, we do not intend our intentions/intentionality; we receive them/it as a gift. This passivity, a central issue in phenomenological philosophy of the 20th century, is absent in the Anglo-Saxon literature but is central to my own approach— which follows such philosophers as Martin Heidegger, Emmanuel Levinas, Michel Henry, Jean-Luc Marion, and Jean-Luc Nancy. Many translations of philosophical books into English include a translator’s preface in which the difficulties of translation are articulated and in which the partiality of the decisions by the translators are made salient. I personally tend to carry the multiple senses of a word into English, separating/adjoining them by using a slash—e.g., Heidegger’s “die Hand reicht” becomes “the hand reaches/extends” because of the double entendre underlying the German verb—rather than choosing one or the other translation. This research was made possible by two research grants from the Social Sciences and Humanities Research Council of Canada and an institutionally administered small grant from the same Council. I am grateful to Jennifer Thom, with whom I held two of these grants; to Mijung Kim, who was involved in the data collection as a postdoctoral fellow; and to Lilian Pozzer-Ardenghi, who at the time was a doctoral student in my research team. Jennifer had taken a central role in the design of the curriculum. Mijung and Lilian helped in the videotaping. My thanks also go to the teacher and all her children for their participation in this project. Victoria, BC July 2010

But the craft of the hand is richer than we tend to think. The hand does not only grasp and catch, does not only squeeze and push. The hand reaches/extends, receives/welcomes—and not just things: the hand extends itself, and receives/welcomes itself in the hand of others. The hand holds. The hand carries. The hand draws/signs, presumably because man is a sign. . . . But the gestures of the hand run everywhere through language and precisely then with their most perfect purity when man speaks by being silent. And only in as far as man speaks does he think; not the other way around, as metaphysics still tends to think. Every motion of the hand in every one of its works carries itself through the element of thinking; every hand gesture bears itself in that element. (Heidegger, 1954, p. 51)

Introduction Of Hands, Flesh, and Mind Perception is not a sort of beginning science, and a fi rst exercise in intelligence; we have to fi nd again an exchange with the world and a presence in the world much older than intelligence. (Merleau-Ponty, 1996, p. 66, emphasis added)

“We have to fi nd again,” writes Maurice Merleau-Ponty, “a presence in the world much older than intelligence.” This quote, found after the fi rst draft of this book had already been completed, succinctly summarizes my intention. Already before fi nding this quote I had remembered an instance in my life, the story of which I have sometimes told as an example of embodiment but that now is taking on new signifi cance. I begin this introduction/book with this story, because it is exhibiting the nature of the phenomena at the heart of my present concerns: the hand, the quintessential organ of tact, as a synecdoche1 for the fl esh, which is the foundation of all senses, and, therefore, of all sense that we can make. The analytical category of the fl esh refers us to the experience of the body proper in as far as it is incarnated, a living sensuous body in which cognition is but one of the forms of consciousness and where the different forms of consciousness—affect, tact, rhythm, and other noncognitive forms—exist side-by-side. The category of the flesh (Ger. Leib, Fr. chair) is a familiar one in European phenomenology, where it has been developed as a distinction to the material body (Ger. Körper, Fr. corps) in a history of European thought that ranges from Maine de Biran to Edmund Husserl, Merleau-Ponty, Emmanuel Levinas, Didier Franck, Jacques Derrida, Jean-Luc Nancy, Jean-Luis Chrétien, and Michel Henry. Unlike the body, “flesh is not matter in the sense of corpuscles of being that are added or follow one another to form beings” (Merleau-Ponty, 1964, p. 181), it is a modality of life. Flesh, as I show in this book, is at the origin of intentions, bodily skill, and the emergence of mathematics. In particular, my concern is a rethinking of embodiment in the way it has come to us through a line of philosophical and cognitive science research on the body, the mind, and metaphors/metonymies that are said to allow complex forms of cognition to emerge from basic experiences in the world. This following instance from my life takes us right into the world of the hand that Heidegger writes about in the frontispiece, the intelligent hand that traverses language, the hand that has an intelligence of its own, an intelligence that is older and deeper than the one that comes with and from language.

2

Geometry as Objective Science in Elementary School Classrooms

IMMEMORIAL MEMORY—IN/OF THE HAND Some 20 years ago—I had graduated a few years earlier and I am already teaching at a private high school outside Toronto—I decide to call my doctoral supervisor. But as I turn to the telephone, I realize that I have forgotten her telephone number. I try remembering it, but—it may appear evident but seldom attended to in the learning science literature—precisely because I have forgotten it, I cannot aim at retrieving it in the way I would do if I wanted to retrieve a bean jar from the top of my kitchen shelf when I make a bean salad. I decide to call the telephone directory assistance in the state of Mississippi where she lives. But as I sit down and orient my fingers toward the touch pad, they dial the area code I remember but then continue and, as my fi ngers push the keys almost despite myself, a familiar melody begins to emerge from the receiver. The phone on the other side rings, and then her voice tells me not only “hello” but also that my hand indeed has found the forgotten number. In this episode, my fingers remember a telephone number that I have forgotten and cannot recall: “I” have both forgotten and not forgotten the number. Given that the telephone number is a sign of the form 1–601-###-####, it might have been possible to employ another sign to recall it, like some people make a knot in their handkerchief, an arbitrary sign constructed to recall something else just as a computer uses pointers (i.e., a deictic sign, an address) to retrieve the information that the central processor needs at the moment. But this is not the case here. In the present situation, my fi ngers remember, but they do so in an unmediated way, without a sign that points to something else, the memory, to retrieve the information. They remember in moving; the movement is their memory. The memory in the fi ngers therefore is unmediated: it is in the fi nger themselves, in the movement that they produce. It is here that we can find this memory. Because there are no signs involved, it is a special form of memory, a form of immemorial memory that has arisen from the movement itself (Maine de Biran, 2006). Or rather, the memory has arisen from the selfaffection of the flesh, which, through previous dialing, has developed the capacity to dial this number without being told so by the conscious mind. In fact any intention to dial already presupposes the capacity to move the arm, hand, and fi ngers, to reach out, and to push keys, and to dial. Thus, beneath the intentional act of my dialing, which requires a knowing consciousness of the object, there is an operative intentionality that makes the conscious dialing of the number possible. Without this capacity, without the underlying unmediated “I can dial,” there would not be an intention. That is, intention is a consequence of the capacities that my fi ngers have for movement rather than their origin and driver. In the ear, something special also has happened. The ear recognizes the melody that comes from dialing the particular number as a familiar one, even though at the moment I cannot remember (have forgotten) the number.

Introduction

3

How can the ear recognize a tone sequence as familiar even though the mind has forgotten the numbers that produce the sequence? The answer is this: memory realizes itself in the act of hearing. The recognition, therefore, somehow precedes itself. In fact, “recognition always precedes itself when we want to derive it from any content whatsoever” (Merleau-Ponty, 1945, p. 473). Recognition is not a passive event; it is an active form of consciousness. It requires that the sound striking my ear resonate with the capacity to hear; it strikes a form of memory that I have not recalled. It requires a connection with the past, a trace of the past, that is, a past in its own domain. Cognition of the melody is wound up with its recognition. Recognition requires some form of comparison between the contents of a memory and the current content of imagination: it is a fundamental mode of consciousness where the “I” is conscious of the fact that what it does is the re-production of a previous perception (Husserl, 2001). But the active form of this memory has gone lost, being evoked not in and by itself—as in the case of the fi ngers that dial—but by the original melody. Recognition and remembering are wound up into one and the same process; the difference between them has become undecidable, that is, we do not know where one begins and the other ends. “The non-intended effort that constitutes the vigil also constitutes the enduring of the I, or the identical person, and makes remembering possible” (Maine de Biran, 1859b, p. 439). The episode of my hand that remembers dialing a phone number may be discarded all too easily, as a story of rote recall of little value to mathematical knowing. But while writing this book, other episodes of my life returned to my mind, episodes, for example, in which my hands continue to write in English while I am talking to my wife who has come by my office to say goodbye for the day as she goes to work. My hands have been walking and working the keys, but this time they have not done so because of rote memory. They have been producing text all the while I am orienting to my wife and having one of the typical conversations that we have at such occasions. The hands have continued to write, productively creating text, not just repeating age-old tunes but contributing new text to whatever I am working on at the moment. How can we theorize this knowledge of the hands to write and produce text even though the mind apparently focuses on something else? In fact, the hand knows much more and tends to be ahead of myself, and when there is an error, such as an error of typing, my hand may have already finished the subsequent words before my intellectual mind realizes that that there is something wrong. And it is not so much that my hand articulates what my mind has already prefigured: My hand does in writing what my mouth does in speaking: It is ahead of the game. In each case, the articulation is faster than the intellectual mind that comes to realize what has just been said or written only after the fact. That is, the expression produced by mouth or hand is the thinking that the mind subsequently claims for itself. 2 This expression is expression of itself, not expression of something else—e.g., mind, thought—that pre-exists it.

4

Geometry as Objective Science in Elementary School Classrooms

The fi rst movements of an organ leave traces. These traces are the result of an auto-affection, the moving aspects of the flesh affecting themselves prior to any sensory capacity, and constitute memory before memory, immemorial memory. We observe such immemorial (i.e., unmediated) memory in the two cases described above, the fi ngers that remember the telephone number and the ear that recognizes the melody as a familiar one. This auto-affection is possible only in the flesh but not in the body (generally): The flesh is the locus of the originary sense-giving intuition because it is endowed with sense and affect (Henry, 2003). That is, this auto-affection precedes the sensorimotor capacities and their schemata on which constructivist and embodiment approaches ground their theories of knowing and learning: “When we begin, there already is an absolute antecedence” (Nancy, 2006, p. 59). The coming into presence, into consciousness, is at least partially enabled by incarnation: The other version of the coming calls itself incarnation. If I say verbum caro factum est (logos sarx egeneto)3, I say that in one sense that caro makes the glory of the veritable coming of the verbum. But I immediately say, in a very different sense, that verbum (logos) makes the veritable presence and the sense of caro (sarx). (p. 58) In his extended reflection on knowing and the body, the French philosopher Jean-Luc Nancy shows how the word, consciousness, and the flesh are intimately intertwined. In the tropes of tact and touch lies the unity of sense and the senses, inside and outside, material bodies and sense, subject and object, or subjectivity and objectivity.

OF BODIES AND FLESH Some readers may be tempted to attribute my writing to a constructing and constructivist mind. But, to call the text a result of my mind denies fingers “the intelligence of the integrated knowing hand, which guides as it is guided, singing from place to place, making melodies in a network of spatial contexts that are grasped and tactilely appreciated in the most intimate and still mysterious ways” (Sudnow, 1979, p. 11). The knowing hand is an improvisatory one, which “handles the keyboard by finding itself in good positions at all times to move ahead. It knows the terrain as routes to be taken for speaking a language” (p. 12). And it does so with its own rhythm, pacing, and melody, accelerating when there is a lot to say and slowing down for taking a breath. It is precisely this rhythmization, the performative of making music, writing, and talking/gesturing/doing mathematics that we need to theorize rather than engaging in stale formalisms that expel the most distinguishing features that separate the human body from all other animate and inanimate bodies. In this book, I show that—because rhythm

Introduction

5

and pacing of expressive communication, among others, constitutes a form of consciousness very different from intellectualizing consciousness—the difference between body and mind becomes undecidable. Thus, “rhythmization, just like the apprehension of a melody, includes an act radically different from a simple encounter with the rhythm-object—an essentially creative act” (Abraham, 1995, p. 73). Each creative act, “synthesizing the successive emergences, sights a phenomenon irreducible to either the mere perception of these emergences or their mechanical production: it is precisely this phenomenon that we call rhythm” (p. 73). Especially in Chapter 7, I show how rhythmicity underlies mathematical knowing, learning, and teaching. But the body I refer to is not the body theorized in the work of embodiment theorists Mark Johnson, George Lakoff, or Rafael Núñez. Theirs is an intentional body, the intention of which remains unexplained. How would their bodies intend their own intentionality? Their body is not the body that I am after. Mine is the one capable of the caress, which is not driven by “an intentionality of disclosure/unveiling, but of searching: a movement in and unto the invisible” (Levinas, 1971, p. 235). It is precisely the flesh and not the materiality of the body that is capable of the caress. Whereas two bodies may be in contact, it is only in and through the flesh that there is an excess from that which I touch and that which touches me; this excess precisely attests itself in the caress (Chrétien, 1992). It is in and through the flesh—because we are present in fl esh and blood—that we come to face a fellow human being rather than another material body. The living body inhabited by flesh is “a felt body, a body that is seen, that can be touched, render a sound when it is hit, that has an odor, this sweet odor of honey of a piece of wax, that is smooth or rough, cold or hot, dry or moist, hard or soft” (Henry, 2000, p. 157). These living bodies are very different from machines, including the living, autopoietic ones that Maturana and Varela (1980) conceive. The flesh, the incarnated body, is different from the body-thing of the physiologist, the lived but purely intentional body, the one embodied in the autopoietic living machine, or the body purely considered as expression. “[T]he problem with a living machine is basically no different from the problem of an inanimate one; both need a helmsman, a director, a master, a conductor” (Sheets-Johnstone, 2009, p. 18). Maturana and Varela’s bodies relate to the ones I am after as sheet music relates to a live performance of improvisational jazz or as dance instructions to dancing. Theirs is a body already endowed with schemata that have linguistic structure and that is the material to be transformed and proliferated by metonymic and metaphoric processes in embodiment theory.4 My living body, or, as I refer to it here, the fl esh, is always in performance; its knowing exists nowhere else but in performance, and with different forms of consciousness among which linguistic consciousness is only one (Sheets-Johnstone, 2009). This consciousness includes the rhythmization and melodies that Nicolas Abraham articulates and the jazz improvisations, writing on the typewriter keyboard that David

6

Geometry as Objective Science in Elementary School Classrooms

Sudnow has written about, and the dancing that is at the heart of the matter in Maxine Sheets-Johnstone’s work. In the same way as the hand sings with the piano, the hand thinks with the computer keyboard, the voice produces the scanting rhythms denoted by the classical meters, so my category of the flesh is used to theorize mathematical knowing, or rather, geometrical thinking. There is not thought behind expressive performance, thought which drives the body. The flesh does the thinking in and through its expression; and this expression is not an expression of something else, some intellectual phenomenon behind. Any expression is that of a living, sensing, feeling, moving body itself constituted by the flesh. The bodies in embodiment and enactivist theories do not seem to have affect, emotion, but they are characterized by formal structures, schemas, without flesh and blood. These schemas constitute the cold cognition that the various constructivist paradigms have offered to us. What I am concerned with here are real people, in fl esh and blood, with strengths, weaknesses, feelings, shortcomings, powers, and infi rmities. In a word, I am concerned with mathematical knowing of real, living, incarnate and therefore human beings. As such, human beings are subject to/of forms of experience not theorized in any other learning theory: radical passivity, a passivity that we do not choose, but which comes with life (being) itself. I could not intentionally dial my supervisor’s telephone number, the number gave itself in the movements of my hand; even the intention to call her was given to me and not intended by another intention. This radical passivity exists because bodies resist to movements—otherwise they would be moving on their own such as the uncontrolled and unwanted movements in certain illnesses (e.g., Parkinson’s)—and from the fact that we are subject to the constraints of the social and material worlds. This passivity is not theorized at all in the enactivist, embodiment, and constructivist literatures, but it underlies the phenomenon of entrainment that is central to the phenomenological sociology that I articulate in Chapters 3 and 7 to explain how the expressions of the organic body always are expressions of a socialized body. Unlike in the constructivist, embodiment, and enactivist theories, the living organic body I theorize here does not exhibit subjective knowing but always already collective knowing, knowing-with.

CHILDREN’S HANDS IN GEOMETRY, GEOMETRY IN CHILDREN’S HANDS In the previous section, I articulate forms of knowing in, of, and with the hand. I do so for three reasons. First, the hand is the most important organ in the evolution of knowing, as may be evident from the frontispiece. The hand, with its capacity to do work, touch (sense), and express (gesture) is more foundational to knowing than the eye (and associated mirrors) that has come to underlie the metaphors of knowing throughout the history

Introduction

7

of metaphysics.5 It is in and through the hand that the sense of the body becomes the body of sense in both a material phenomenological and cultural-historical activity theoretic approach.6 We can (learn to) function without sight or hearing, but without tact, the capacity of touching, we lose all sense of (and with) reality. Second, the hand serves me in a synecdochical function: It is a part of the living body that stands for all other parts of the living body that are capable of tact, contact, and contingency. I describe what students and teachers do with their hands, well knowing that it is not the hand that operates on its own. The hand is part of an integral capacity to act, an integral “I can,” the integrality of which comes from the fact that “from an enactive viewpoint, any mental act is characterized by the concurrent participation of several functionally distinct and topographically distributed regions of the brain and their sensorimotor embodiment” (Varela, 1999, p. 272). It is to this central capacity that all specific capacities are subordinated functionally; and this central capacity coordinates the specific activities as well. Thought, then, is but one phenomenon of expression among many others that can be isolated analytically from the expression of the body as a whole. In this book, the children’s hands, which build, work on, sense, and gesture with/out objects, know mathematics in the ways that Sudnow’s hands know in playing the piano, which orient toward a situation rather than playing notes. They take a stance, a position, and this stance expresses itself in the way the hands move. What I pursue in this book is a way of understanding what the children do in early geometry, which is more like getting to the melody of their mathematics rather than to the structures of the transcriptions, the musical scores and sensorimotor schemata that theoreticians put on paper. It is mathematics as performance that I am interested in describing and theorizing, the rhythms, the melodies, the hums, which always resonate when people make mathematics. Children learn to do mathematics in the way those who follow the “Suzuki method” learn to play an instrument by playing before learning to read sheet music—they learn to play by ear, in the way they learn to speak their mother tongue. In the process their bodies become mathematical, because they can feel (sense) when it (mathematics) makes sense. My hand knows the telephone number even when I remember it in the form of mathematical digits—and the recall of numbers, even my own, is tied to rhythmic and musical performance that allows me to reproduce the number that I so seldom call. The literature on knowing and learning mathematics in school classrooms—or, for that matter, science, history, and any other subjects—is too much concerned with the intellectual aspect of life. This literature forgets that in and through the children, life gives itself a mind rather than the other way around. From the Kantian gesture of focusing on the mind at the expense of everything else, knowing and learning mathematics—as it appears in many accounts of mathematics education—looks more like what we know about computers rather than what we know about how

8

Geometry as Objective Science in Elementary School Classrooms

human beings conduct their life and mobilize/enact knowing as they go. Thus, this existing mathematics education literature focuses on “interpretation” of the talk of others, on “taking things as shared,” when in fact much of our lives does not require us to interpret, or “to make meaning.” In communications with others, in written or oral form, it is irrelevant that we somehow become “aware of the meaning of the expression by doing concurrent thinking about its reference, even though, while saying the words with greater emphasis, there is the sense that what they stand for is being sought” (Sudnow, 1979, p. 52). What really matters is not that we somehow “make meaning” but that we seek “the melody of the text. In experience, this seems to reside precisely in a special manner of heightened and more extensive bodily absorption with the fi nest particulars of the unfolding sound places” (p. 52). Being means understanding; mathematically being means understanding mathematics in a practical way. Thematic interpretation is but a derivative mode, always already based in practical understanding of the world.7 But thematic interpretation develops understanding—not leading to something different, a different understanding. Rather, through interpretation, understanding comes to understand itself in an explicit way. In this book, I am precisely after this practical understanding of the world that children bring to and develop in the mathematics classroom. I am interested in understanding how this practical understanding undergirds learning and knowing, and how knowing and learning are never independent of this practical understanding. In fact, this prior, practical understanding undergirds and allows mathematics to emerge even and precisely when this understanding is inconsistent with formal mathematics. That is, other than Kantian, Piagetian, or constructivist theories, where there is a gap between knowing one’s way around the world and knowledge of the world, the approach outlined here never severs the feeling, moving, sensing person inhabiting its world in flesh and blood from the “knowledge” or “cognitions” that it is said to “have.” Even though I suggest that the essence of mathematical performance is not captured by the schemata that Piaget, enactivist, and embodiment theorists propose in varying ways as the basis of mathematical knowing, I do not mean to suggest that this essence is ineffable. My case is similar to the one in musical theory of the nuance, which cannot be captured by schema theory and Chomsky-like generative grammar (Roholt, 2010) but constitutes an aspect in which practical musical consciousness makes the difference. Thus, “hearing a B-flat(17) is not to perceive that pitch as an isolated property . . . [but] what is truly musical about a B-flat(1) is what it does in a perceived musical context” (p. 7). Rather, what the ear hears are changes that musicians characterize for each other by means of metaphors such as an F-sharp (13) or F-sharp(14) that are “bright,” “shimmering,” or “radiant” in different ways and that are further characterized by comparisons with recorded or live performances.

Introduction

9

Mathematics generally and geometry specifically emerge from the children’s hands—here serving as synecdoches of the living bodies to which they belong and which they constitute—much as music emerges from the knowing hands on the piano keyboard and much in the way my hands write seemingly on their own using a computer keyboard while my eyes are fi xed on the monitor. These hands learn to act mathematically as they move, gather, or shape objects; and they learn as the children actively search to sense the characteristics of specific geometrical forms—such as cubes, rectangular parallelepipeds (also called cuboids or rectangular prisms),8 spheres, cones, or pyramids—and the everyday objects with which they come to be associated—such as dice, pizza boxes, Post-it pads, toothpaste boxes, balls, ice creams, and reproductions of Egyptian tombs. Most importantly, what children do with their living body generally and with their hands specifically cannot be represented appropriately by means of language. Throughout this book I use drawings to show events from the position of the person operating the camera. The images should not be looked at with a linguistic consciousness, for this would reduce knowingin-action to linguistic consciousness. Rather, each image should be treated as an instruction for doing what can be seen, much like the texts and images in a cookbook are instructions for the user to enact certain steps to end up with some meal. Merely looking at the images is like merely looking at a cookbook. Without actually doing what instructions tell us, the very modes of consciousness required for doing mathematics/cooking are not mobilized unless we do it ourselves. Repeatedly, I invite readers to engage in investigations (e.g., in Chapter 1). Readers should do these investigations, because the forms of consciousness I want them to experience would not be mobilized otherwise. Watching a soccer game, golf, or ice hockey on television or reading about such games is very different than playing soccer, golf, or ice hockey: The sense is different when we play audience then when we play the game. It is precisely the sense of the game that matters, because it underpins the sense we make. The sense of the game derives from and exists in and as of the living body (flesh); and this is so not only for doing sports, dancing, playing music but also and especially for doing mathematics. The sense of mathematics emerges from a developing sense of the game for doing mathematics.

STRUCTURE OF THIS BOOK This book presents a way of understanding the emergence of mathematics generally and geometry specifically from elementary children’s engagement with cultural-historically marked objects that have their specific place in geometrical science. Whereas I am positively inclined toward enactivism— which I consider the best approach to knowing, though its implications have not yet worked out to its fullest in the mathematics education literature—I

10

Geometry as Objective Science in Elementary School Classrooms

show two important lacunae in this theory: (a) the source of intention and memory that underlie mathematical cognition and (b) the source of the inherently cultural-historical—and therefore shared—nature of knowing that any individual incarnate agent does evolve. Among those mathematics educators employing the theory, too many focus on intellectual consciousness and too few on all the other forms of consciousness that constitute integral parts of knowing. In their approach, the very distinction between body and mind is presupposed and maintained by the role of intentionality. In my approach articulated here, this integral knowing is only partially and one-sidedly represented by intellectual consciousness. This book is divided into three parts, each prefaced by an introduction of its own, and each preparing what is to come or building on what has been developed before. Thus, Part A is devoted to the articulation of theoretical issues. More specifically, using a concrete lesson fragment that features a student in the process of articulating an answer concerning his insights about geometrical objects generally and cubes particularly—for which I provide an initial reading in Chapter 1—I articulate the reigning theoretical frameworks that have been used in the philosophy of mathematics and mathematics education over the last several decades. Thus, in Chapter 2, I comment on and make reference to the positions of Immanuel Kant, Jean Piaget, Maurice Merleau-Ponty, enactivism (as articulated by Humberto Maturana and Francisco Varela), embodiment (as articulated by Rafael Núñez and his collaborators), (radical, social) constructivism (as articulated by Ernst von Glasersfeld and Paul Cobb, respectively), Hans Freudenthal, and Pierre van Hiele. In the third chapter, I develop an approach entirely grounded in material life, presenting the perspectives of material phenomenology and its emphasis on the fl esh as opposed to the body, phenomenological sociology and its emphasis of the socialization of the body, and cultural-historical activity theory, which incorporates both phenomenological and sociological thinking. In Part B, “Stories of Mathematics in the Flesh,” I present detailed analyses of learning in a second-grade mathematics classroom, where children are in the process of doing their fi rst unit on geometry. In particular, I use lesson fragments in which cubes somehow come into play—as objects of activity, comparative objects, fashioned objects, gestured objects, or objets trouvés—which allows us to see similarities and differences in knowing across the episodes. The four chapters of this Part B exhibit a development, from fragments in which children articulate issues out of their personal experiences with the world and out of their intuitions, which often do not coincide with the geometrical knowing that the teachers attempt to foster. Yet these experiences, as we can see in the unfolding sequence of the four chapters, constitute the very material from which formal geometry emerges, such as the use of normative measurement for deciding what one has to do to make a cube from a pizza box. Chapter 5, for example, presents a lesson fragment in which a student comes to realize that the object

Introduction

11

that she had modeled as a cube based on what she felt by touching the object without seeing it really is a rectangular prism with three different pairs of faces. In the fi nal chapter of Part B, I show how rhythmic phenomena in the expressive articulations of teachers and students encourage us to employ the proposed approach of mathematics in the fl esh. These rhythmic phenomena arise from a self-affection (of movement) that is central to my radicalization of embodiment, on the one hand, and is central to a phenomenological sociological approach in which entrainment ascertains the coordination of knowing across all individuals of a collectivity on the other hand. This chapter, therefore, explicitly shows where and how both embodiment/enactivist and social constructivist theories fail in contrast to the approach presented here. In Part C I anticipate the questions traditionally trained mathematics educators may have. Specifically, this third part of the book is concerned with (a) how formal geometry—here exemplified by acts of geometrical sorting—arises from the everyday methods of sorting; (b) how formal geometry as an objective science emerges from the children’s ethno- (i.e., everyday) methods of sorting; and (c) how to rethink the nature of mathematical concepts and conceptions. As the entirety of Part B, two of the three chapters employ lesson fragments that focuses on children’s classification of cubes, whereas the fi nal chapter, as a sort of extension into objects other than parallelepipeds (cuboids, cubes, rectangular prisms), uses an episode featuring the classification of cylinders. I conclude this book with an ever-so-brief epilogue, in which I take a panoptic view back over the main aspects of the argument for theorizing mathematical knowing as incarnated, as a form of mathematics in the fl esh. In the Appendix, I provide a description of the transcription conventions used throughout this book.

Part A

Toward a Theory of Mathematics in the Flesh

Introduction to Part A Spatiality may be the projection of the extension of the psychic apparatus. No other derivation probable. Instead of Kant’s a priori, conditions of our psychic apparatus. Psyche is spread out, does not know thereof. (Freud, 1999, p. 152, emphasis added)

During the fi nal days of his life, Sigmund Freud, who had spent his entire career researching psyche and the unconscious, suggests that the human psyche, heretofore thought as something immaterial, actually is spread out.1 “‘Psyche’ is body,” Nancy (2006, p. 22) notes, “and it is precisely that which escapes it.” It is precisely this escape, the process of the escape that constitutes the psyche. What is it in or of the living human body that might play this central role in who we are and what we know? Numerous phenomenological philosophers of the late 20th century discuss the flesh as the foundation and medium of tact, itself the foundation of all senses.2 Others, often coming from the arts, theater, and dance, have joined in the celebration of the living body and aliveness as the fundamental aspect of being that enables all forms of knowing, including the intellectual forms (e.g., Sheets-Johnstone, 2009). Tact relates human beings to the world, as apparent in all those concepts that are based on the same Latin root word tangēre, to touch: contact, contingency, contagion, contagious, contiguous, contiguity, contaminate, touching, tact (beat), tactile, tangent (line, distance, point, length), tangent (function), or co-tangent. Tact is the general sense. It is synonymous with sensibility itself, thereby constituting the sense of all the senses, any sense we can make, and the sense of sense itself. Psyche, the flesh, the senses, and mathematics come to be irremediably and intricately woven into a complex lacework. And it is precisely the connateness of tact with contact, contingency, contagion, and musical tact that makes our experience inherently embodied, shared, and intersubjective rather than metaphysical, monadic, subjective, and singular. As the end result of transcendence, the Kantian, constructivist mind becomes metaphysical, no longer is present in the world, but withdraws into the netherworld of its representations and constructions. Kant and all constructivist theories that ensue are concerned with the mind, which, as its etymological origin in the Proto-Indo-Germanic root men- (to think, to be busy mentally) suggests, is concerned with theorizing that which is conscious and accessible to reflection. But reflective activity makes up only part of our lives, in fact, is subordinate to a more integrative minimal unit: collectively motivated activity. That is, activity, which always realizes

16

Geometry as Objective Science in Elementary School Classrooms

collective object/motives, sublates (i.e., keeps, suspends, supersedes) and therefore integrates, the generally separate phenomena of intellect and affect. Karl Marx and Friedrich Engels, on whose theory Lev Vygotsky and Alexei Leontjew built their dialectical materialist, societal psychological theories of knowing, articulated the integration of materialist and dialectical idealist approaches into one of lived and living human praxis.3 In this approach, all higher-order cognitive functions are understood to be the result of interaction rituals in the context of concrete, sympractical work designed to fulfill collective and individual needs. In Western scholarship, however, neither phenomenology nor cultural-historical activity theory has been central to the discussions of knowing and learning in philosophy, education, and the learning sciences (cognitive science, cognitive psychology, artifi cial intelligence). Rather, if at all, the phenomenological and subject-centered Marxist theories that did affect the learning sciences are rooted in Western scholarship— including the phenomenological approaches of Edmund Husserl, Martin Heidegger, and Maurice Merleau-Ponty or the cultural-historical approach articulated by Yrjö Engeström—have influenced a broad range of research in cognitive science, artifi cial intelligence, sociology, poetics, and even mathematics. In Western cognitive science, fi rst phenomenology then the embodied cognition research showed how there could be no cognition without the human body. There is something unsatisfying and lacking, however, in the concept of the body, which undermines the very effort to ground (mathematical) knowing differently than in the private cogitations of the isolated mind that moves and processes ephemeral representations. There is a difference between the German Körper (body) and Leib (body), between the French corps (body) and chair (body) that is not thought by the concept of the body. Thus, “with respect to the body [English in the original] (in all places where it obsesses, for example, the American culture and academy), whom will we make believe without laughter that it is a trustworthy equivalent of that which we call corps, corps propre, or chair [flesh]” (Derrida, 2000, p. 79)? In this Part A, I shift the discourse from the body to the flesh to present a far more radical approach to the conceptualization of mathematical knowledge that is grounded in a material and sociological phenomenology than that provided by the embodiment and enactivist literatures on the topic. This approach is further developed in Part B and Part C of this book. In this fi rst part, though, I present the central ideas underlying this book from dialectical materialist psychology (as developed by Lev Vygotsky) to materialist phenomenology (as developed by Maine de Biran and Michel Henry) and to phenomenological sociology (Bourdieu). The close relation between cognition and the world is possible only when there is flesh—which is able to self-affect and to remember without representation. In contrast, material bodies constitute an insufficient condition for mind to emerge.

Introduction to Part A 17 In the brief Chapter 1, I provide a fragment from a lesson in a secondgrade classroom and enact a careful reading thereof. I subsequently use this lesson fragment and my reading to articulate different theories, their strengths and shortcomings with respect to the task of providing us with an understanding of the exhibited mathematical knowing and its conditions. Specifically, in Chapter 2 I articulate the different approaches to mathematics generally and to geometrical knowing specifically that have been taken in a history of theorizing mathematical thought from Kant to modern day embodiment and enactivist theories. I draw on philosophical analyses of experiences with and knowledge of cubes, wherever I could fi nd them, as a way of exhibiting where these theoretical frameworks fall short to account for geometrical knowing in real time. In Chapter 3, then, I present a way of understanding geometrical knowing, mathematics in the flesh, and how its cultural dimensions are acquired in and through participation in everyday human interactions generally and in those that occur in schools more specifically. That is, unlike the enactivist approach, in which cognitive and development is dealt with as a “strictly subject-dependent creative process” (Maturana & Varela, 1980, p. 49, my emphasis), phenomenological sociological and cultural-historical approaches recognize the essentially passive aspects of human experience that lead the flesh to produce and exhibit cultural-historical, that is, intersubjective forms of knowing rather than forms that are singular and subjective.

1

What Makes a Cube a Cube? A Phenomenological Overture Can you go out for a game of tennis, make love, repair a roof, or plant a garden short of being in the flesh? Cheshire cats might be able to accomplish such feats . . . (Sheets-Johnstone, 2009, p. 20) I stepped straightly into phenomenology, which was a bit didactically tainted. (Freudenthal, 1983, p. 210)

In the course of reading this book, readers discover that even the fi rst elementary school tasks involving children with three-dimensional shapes allow the objective nature of geometry to emerge from the events in an elementary classroom. But this objective nature of geometry emerges each and every time from the flesh, much like playing a game of tennis, making love, repairing a roof, or planting a garden. The purpose of this book is to push—in style and content a tribute to Hans Freudenthal—how we think and think about the “embodiment” of mathematical knowledge as something that only an incarnate being (i.e., being in the flesh) can accomplish. To get the point of this chapter you have to enact the following task.1 Through this exploration, I intend us to focus on the phenomenology of geometrical and spatial experience—which is very different for non-Western cultures. Find (what you have learned to be) a cube and take it in your hand. Look at the object. Do you see a cube? Of course not: you see aspects (parts) of a cube, and the nature of the aspect depends on your current perspective and the orientation of the object. But how do you know that these are aspects of a cube? To know this, you need to know that a cube expresses itself in the aspects you see given perspective and orientation. Close your eyes and feel the object. Do you feel a cube? Of course you do not. Even if the object is small so that you can enclose it entirely in one or both palms, you do not feel the twelvefoldedness of the edges, the sixfoldedness of the sides, and the eightfoldedness, the smoothness/roughness of the surfaces, the color, and so on; and especially, you do not feel all of these aspects of a cube simultaneously. If you still were to think you felt a cube, then you are mistaken, for you only hold a token of a cube rather than a type. 2 This distinction is important, as geometry is dealing only in ideal types rather than concrete tokens (Husserl, 1997a), not in the least because the exactness required by geometry cannot be achieved in nature given Heisenberg’s uncertainty principle. That is, we never perceive a cube, be it by means of

20 Geometry as Objective Science in Elementary School Classrooms one or all senses simultaneously; we can only think cubes. And when we perceive an object that we know to be a cube, we do not actually think it, that is, construct it by coordinating the input from the different senses. Rather, in each case the cube organizes itself in front of and through me. The perception and thought of a cube are independent to a certain extent, “not because perception is liable to errors, but because the concept enables us to establish the fact that the perception is wrong” (Freudenthal, 1983, p. 237). Now, following my instructions, you intended to explore the cube visually and manually. But where did this intention come from? Did you have an intention for the intention? And if you had an intention to intend, where did this intention to intend intention come from? We know that this way of thinking does not yield an answer, as it leads to infi nite regress, which, as we know from experience, is not the way in which we come to encounter and understand our own intentions. Some time in our personal histories, the cube has arrived in our consciousness, as a result of our lived experiences. But before that time, the object, the cube as cube, “does not give itself so long as its signification is not adequately fulfilled” (Marion, 2002, p. 126). This is so because the “paradigm for this impossibility is seen in even the most simple intentional object (the cube . . . whose geometric faces we never, by an essential impossibility, perceive all at once” (p. 126). How is it that our understanding of the cube as a cube has arrived even though our mind did not have the concept of a cube to see and recognize one? How, without a “plan” of what a cube is or looks like, does a constructivist mind arrive at a cube from the disparate sensual (visual, tactile) experiences that a learner may have with the object that we now know as a cube? How can the mind intend to construct a cube when it does not know what a cube is and therefore cannot intentionally aim at constructing it? This is the dilemma that Kant—the “Logodedalus,” the master artisan of the mind—has never solved. 3 But then how did we come to understand that by looking at or touching some solid square faces we were seeing/touching a cube?4 The problem for the constructivist mind is known in the cognitive sciences as the “symbol grounding problem,” that is, the problem of how mental representations are related to anything else than mental representations. 5 How does the intellectual mind know that what it has constructed relates to anything in the world? How does perception, given its theory-laden nature, ever come to see a cube when any possible experience exhibits only perspectives that never reveal the totality of things we know today about a cube (eight corners, twelve sides, six square faces intersecting at 90 degrees, etc.)? The general constructivist answer is that only fit and usefulness matter rather than real relations. But then the question is how the mind can interact with the physical world, how the abstract, metaphysical mind can manipulate the material, physical body and the senses and test its knowledge in the world? In fact, it is impossible

What Makes a Cube a Cube?

21

to understand how the constructivist mind could do anything as simple as paint in the outside world (Merleau-Ponty, 1964b). Moreover, how can the knowing subject know itself if the known cannot be anything other than an ob-ject, something thrown (Lat. iacere, to throw) in front of, before, against (Lat. ob-) the subject?6 In Kant’s constructivist project, the knowing subject and the object known are but two abstractions and a real, positive connection between the two does not exist (Maine de Biran, 1859a, b). The separation between inside and outside, the mind and body, is inherent in the constructivist approach. This is why Ernst von Glasersfeld’s (e.g., 1989) radical constructivist mind knows nothing of and about the world and remains stuck with and in its own representations that can be tested only for fit. The past decade and a half has seen a slow increase in the number of studies that argue for an embodiment approach to mathematical knowing and learning.7 Some mathematicians and mathematics educators, however, resist the idea, holding fast to the conceptualization of knowing in terms of mental representations. The purpose of this Part A of the book is to look at one mathematical event through different epistemological lenses and to show that one of the problems with embodiment is that it does not make a distinction common to phenomenological thought, the one between the body and the flesh. I suggest theorizing the flesh rather than the body as the ground of all knowing: knowledge as incarnated, enfleshed. Because it is in the flesh that mathematical knowledge is incarnate. It is in the flesh that we fi nd tact (touch), contact, contingency, contiguity, contagion, and contamination, and therefore, the ground of knowledge. It is in the flesh that we experience, know, and understand all forms of passion. The second problem of these approaches, too, will be resolved, because the living body that arises from the flesh—precisely because it is in contact with and exposed to the world, therefore to open to contagion and contamination—always is a socialized, social body, allowing us to understand why “constructions” and “embodied” or “enacted” knowing never are singular but always already cultural-historical. To anchor the discussion, I begin with a fragment extracted from video recorded in the second-grade classroom in which teachers (Mrs. Winter and Mrs. Turner) and their students have embarked on a study of the geometry of three-dimensional objects.8 I then move toward a theory of radical embodiment, or incarnation, grounded in the early work of a little-known French philosopher Maine de Biran,9 whose work has found prominence more recently in the articulation of material phenomenology 10 and who has been influential on the thought of the later Merleau-Ponty. Unlike in the embodiment literature that has recently gained some prominence in mathematics education, materialist approaches theorize knowing beginning with very primitive forms of experiences without attributing intentionality and mental character to them. These fundamentally non-intentional phenomena precede (intentional) sensorimotor actions, which are their outcomes.

22

Geometry as Objective Science in Elementary School Classrooms

“WHAT MAKES IT A RECTANGULAR PRISM INSTEAD OF A CUBE?” The Fragment 1.1 below begins after a description that Chris has provided about rectangular prisms that can have different lengths of the sides. Mrs. Turner, who is teaching the class during this part of the lesson, fi rst summarizes in her words what she has heard Chris to say, and then follows up asking about what the pizza box would have to have to make it a cube (turn 17). At this time, Chris is in the front of the classroom—as defi ned by the seating arrangements with respect to the chalkboard—where the pizza box rests next to the chalkboard; he is slightly bent over the box (Figure 1.1a). For much of the time in the lesson fragment, Chris is oriented to the pizza box and therefore with his back toward the remainder of the class. It is only toward the end of the fragment—when he points to the familiar cube in his hand—that he reorients to Mrs. Turner, gazing at her face (Figure 1.1b) while producing a more articulate utterance concerning the nature of a cube. Initially, the production may be described as inarticulate. It is punctuated by interjections and pauses. But this initial part is even more important in and to my case than the second part, which I link to the phenomenological analyses of the experience of a cube that we can fi nd in the literature. As we read the following account, we need to keep in mind that Chris speaks rather than reads from a text completed before. What he says and does emerges in real time, as an improvisation, as an expression that expresses itself rather than a thought behind. The expression, which is an expression of the body as a whole, is the thought—the verbal being only one part among other constitutive parts that all stand on their own so that they cannot be reduced to the verbal mode. Each expressive form goes with a different form of consciousness, each refl ecting reality in a different way. “Psyche is extended,” says Sigmund Freud on/near his deathbed, and thereby obliquely points us to the organism as a whole as the psychological entity we need to focus on. In a sense, therefore, Chris is as much subject to what he says as he is the subject of this saying.

Articulating an Answer and the World: An Episode After the Mrs. Turner’s utterance, there is a pause (turn 18).11 An almost simultaneous beginning of each of the two speakers follows (turns 19 & 20), with Mrs. Turner speaking quickly to complete an utterance that translates the preceding question: “what would that box have to have to be a cube” (turn 20). There is another pause before Chris begins by using some of Mrs. Turner’s words, “it would have to” and then he produces an interjection preceding a pause (turn 22). The production is punctuated by pauses of different length, “that turn,” “or this,”

What Makes a Cube a Cube?

23

Figure 1.1 a. Chris is oriented to the box following what we can hear to be a teacher question, “What would that box have to have to be a cube?” b. At the end of the presentation, Chris squarely gazes at his teacher, no longer “caught up” with the pizza box but into the familiar terrain of the red cube that he has manipulated so many times before.

“like the,” “sorta like” and then another, conversationally substantial pause of 3.04 seconds. The verbal part of Chris’ presentation continues in a stop-and-go manner punctuated by pauses, “square here an like” and then his voice fades away becoming all but inaudible “but to-nthis-squ.” Transcribed, his articulation so far has been, “um this . . . it would have to um that turn or this like these . . . sorta like . . . um square here an like.” He then turns to the teacher squarely facing her and producing a rapid utterance “it has square here and here and here and everywhere.” Fragment 1.112 17

T:

18 19

C:

20 21 22a

T: C:

so makes it a rectangular prism as opposed to a cUBe, because if it was a cube what would it have to have (0.58) um [thiss ] that box have to have to be a cube. (0.77) it would have to um ((gets a cube from chalk holder)) (0.44) that squ : are : ((a)) or this (0.21) like the: (0.62) s : : ((b))

24

Geometry as Objective Science in Elementary School Classrooms

22b

22c

((bends down to box, moves cube in direction of box, moves left hand along edge of pizza box))

22d

sorta like ((gesture [a] to [b]))

22e

(3.04) ((hand moving along edge)) um (0.23) s : : quare

22f

here

22g

an like ((rectangle)) (0.68) ((movement along edge of box)) ijst has (([a])) square (([b])) (0.45) n here= (([b])) =an=here= (([c])) =an here= (([d])) =and everywhere (([e]))

What Makes a Cube a Cube? 25

23 24

M:

(0.63) ´how could you make a cube from pizza boxes.

In the early part of the response, Chris is entirely oriented to the pizza box (Figure 1.1a). In this, a relation is established between the box (the object) and him (subject), from which is to emerge a response. His orientation exhibits the nature of the preceding utterance as a question, which he shows for everyone to see as the constitution of the second part of the turn pair that establishes the question–answer sequence. This orientation also communicates something like absorption, of the subject (Chris) with its object. Here the object is not the pizza box in itself but the question about this entity and what would make it a cube. The initial part of the “response” (turn 22) is produced together with an inarticulate production in the verbal modality. Chris moves his hand along the two edges / sides of the pizza box that are visible to one of the two cameras in the room and closer to him (turn 22c, gesture b c; turn 22d, gesture a b). There is a very long pause, and Chris then utters the word “square” (turn 22e) while he points to the two sides again (turns 22e, f). Seen and heard in response to the question “what would that box have to have to be a cube,” these communicative performances can be seen to indicate the two sides that have to be “sorta like square ((points to one rectangular side of box)) here ((points to other rectangular side of box)).” But whereas we can hear/see what he articulates and points out, we do not know what he is intellectually conscious of, what is in his mind, what representation, mental picture, or conception he might have—if such things do actually exist. In fact, all of this talk about hidden and inaccessible constructs is irrelevant in the face of the fact that he can be witnessed to be responding appropriately.

Senses and Sense-Making Intellectualist approaches reduce expressions of knowing to mental representations and therefore to an external relation of the subject to the

26

Geometry as Objective Science in Elementary School Classrooms

object.13 But the hand movement that we see in the transcribed fragment is not subject to the same mode, the verbal; we have to think it differently than as a performance that the student consciously controls. As something that we see, it addresses visual-perceptive consciousness rather than intellectual consciousness, two very different aspects of knowing that cannot be reduced one to the other. We therefore should not hastily assume that underlying this movement there is anything similar to an adult thought, a conception, which has a hard time expressing itself in words. We cannot even assume that Chris has heard the question in the way we hear (read) it. We also know that the pizza box and later the cube appear in the same presence, but we do not know whether this presence is present to Chris, whether and how these objects are present in and to his intellectual consciousness, a presence that requires re-presentation. But how should we then understand the hand movement? How come he does move in the way he does? How and why does the hand move along the edges, pointing, while uttering words that do not constitute a coherent and “logical” argument? The response to this question takes us to an instant that even precedes the spatial bodies that are the phenomenological primitives on which geometrical understanding is built. There is knowing (-how) and memory in the body, which comes from an “I can,” a power to act in this way without conscious refl ection. Chris’s hands are able to move and he can intend moving this way because the living body remembers that it can, even prior to any conception and conscious intention.14 The hand and fi ngers move along edges, across the box, and along another edge. And to proceed in this manner, this body needs to know prior to all intention that such movements are within its range of possibilities—much in the way my hand and fi ngers moved and in this movement remembered my supervisor’s telephone number. Chris gazes, but we do not know whether he is intellectually conscious of this seeing and which aspects constitute a part of the guiding of movement in the unity of perception movement. The episode points us, in fact, to a time before words, a time “before” time, before intentionality, when in the fi rst hand-arm movements, Chris, long before he is conscious of being Chris, realizes the possibility to move the hands voluntarily to reach out and touch. Before a hand can reach out to touch, it has to “know” (has to have the know-how) that it can reach; and this knowing cannot come but from the movement itself, for it precedes even the recognition of anything such as a material body. Before I can intend to reach for something and touch it, I have to (immanently) know I can reach at all and that in touching I will be affected. From Aristotle to the present day, scholars have recognized the primacy of touch in human experience: “no living being can survive, in the world, for an instant without touching, which is to say, without being touched” (Derrida, 2000, p. 161). Whereas it is possible to live without seeing, hearing, tasting, and smelling, “we will not survive one instant without being with contact, in contact” (p. 161). The sensations from the eyes and other

What Makes a Cube a Cube?

27

external organs come to be coordinated by tact, the only sense spread across the living, organic body. And with the movements and coordination of movements of eyes and hands, the world begins to emerge from tact, that is, contact. Chris’s present experience is based on the coordination of hands with eyes, so that seeing the pizza box and moving the hand along one edge/face, then along another edge/face, is but a realization of the coordination of hands and eyes and the concrete realization of the ability of moving them. Certainly of equal if not greater importance to the radical approach to embodiment outlined in Chapter 3 is the second part of the explanation, in which the rectangular prism of the pizza box comes to be set against the cube (“but to-n-this . . .”). But Chris does not just contrast the rectangular prism (pizza box) with the cube or state some properties that one has but the other does not. Rather, uttering nothing more than “it has square here,” and, as he rotates the cubical object touching different surfaces, says “here” each time, he ends in the totalizing statement “and everywhere.”

The Visible and the Invisible In the literature on the phenomenology of perception, philosophers have noted that knowing a cube means knowing what it will look like when the relative orientation between subject and object changes. Here, even though Chris does not produce a formal defi nition of the cube, he does in fact enact what is the requisite experience for knowing what a cube is. He rotates the cube, exposing its different, initially hidden faces so that he comes to look at them “squarely,” allowing him to see the different squares. Moreover, he says “it just has square,” and then his left hand turns the cube and his right hand points while uttering “here . . . and here . . . and here and everywhere” (turn 22g). A constructivist mind would produce a plan and then move the limbs accordingly. But we do not have any evidence that there is a constructed plan driving what Chris does. It would be plain out wrong if we were to say that he has a “conception” of the cube and assume that this conception exists in verbal-intellectual form, as a “word meaning.” Even less must we assume that he has some formal conception or knowledge a priori from which he could derive the properties of a cube.15 At this stage, what he makes available to us in and through his body as a whole is the experiential fact that the object that goes with the sound transcribed in the conventions of the International Phonetics Alphabet as “kju:b” can be turned to exhibit squares “everywhere.” He exhibits orientation to the cube, and, thereby, he exhibits both position and dis-position toward this intentional object. Moreover, we do not know about an explicit intention to rotate the object, but, seeing the object rotate in his hands, we know of his hands’ ability to make the cube rotate and exhibit its different faces. And Chris makes available to us a description of what he sees: squares everywhere.

28 Geometry as Objective Science in Elementary School Classrooms We often hear mathematics teachers and mathematics educators speak about students who know something but who cannot tell it. In the research on gestures, there are indications that students may have one or two underlying conceptions that express themselves in their hand gestures and that may (or may not) be accompanied by words that express the same or less advanced conceptual content.16 Gesticulation is unconsciously produced, so that this aspect of body movement cannot be theorized as falling under intellectualizing consciousness and verbal forms of cognition. The point of the description in the previous paragraph is precisely that: Embodied cognition and enacted mind cannot mean that the body, too, is subjected to the reign of intellectualism. A phenomenologically and ethnographically adequate description must take into account what is actually there, not what the theoretical presuppositions that underpin a constructivist view of the mind want us to see there. Even talk about the body is insufficient, because the cube and the pizza box also are material bodies, and we need to distinguish them, incapable of cognition, from Chris, expressing knowing heterogeneously across his organic, lived, living, and alive body. Rather than thinking about the material body as the ground of cognition, therefore, we might better think about and theorize the flesh—capable of passion and caress—which is the source of the body in cognition (see Chapter 3). This problem exists particularly if researchers tell us “meanings” behind the gestures, some core idea, which is the very point we need to question.

TOWARD INCARNATION Rather than thinking of gestures in terms of knowledge or conceptions, which confuses explicit and implicit knowledge forms, knowing what and knowing how, I shift the theoretical discourse toward that which is immanent. Gesticulations are not present in linguistic consciousness, and it is the nonconscious production that I focus on here in this episode; the production of gesticulation goes alongside with an inarticulation in the verbal expression. But to say that he has an “understanding” taken in the common sense of “to comprehend by knowing the sense of words used” would be an overstatement, precisely because of the absence of words. Something else is immanent in his movements. In the present situation, what is immanent in Chris’ hand and finger movement precisely is the memory of the movements he is capable of producing, and neither memory nor the production requires verbal thought and intellectual consciousness. There is an “I can” immanent to the power of the arm/hand/finger to move in this way; and this “I can,” prior to any “I,” is, in material phenomenology, the result of an auto-affection by means of which the muscles become empowered and know how to move. Auto-affection is possible only in the flesh, not in a medical-material body per se. As a result, conscious thought “is born with the manifestation that it renders visible without knowing or wanting it, and perhaps without

What Makes a Cube a Cube?

29

even being able to do so” (Marion, 2002, p. 265, emphasis added). That is, originally—in phylogenesis and ontogenesis—conscious thought is given rather than intended. Even in adulthood, illumination, insight, and novel ways of thinking are given to us rather than intended. Conscious thought may in fact follow and emerge from the expressions of the moving hands, arms, and body parts. The “powerlessness to stage the phenomenon . . . can be understood as our abandoning the decisive role in appearing to the phenomenon itself” (p. 132). But once produced for a fi rst time, every repeated movement produced with an intention essentially encloses or presupposes the reminiscence of a power, or the cause (of immediate effects) that is inseparable from it (Maine de Biran, 1952). Once my hand was capable of the movement of dialing the telephone number, it was also capable of remembering it without requiring another sign than itself. My hand’s capacity to move stands for itself; this allows the reproduction of the movement without intellectualizing consciousness. In the same way, as soon as the hand and arms know to move parallel to a line or face that the eyes concurrently see, there is a capacity to gesture in this manner without requiring the intentions of an intellectualizing consciousness. This knowing exists in and through the flesh, which remembers in doing what it can do. For could we do mathematics—or “go out for a game of tennis, make love, repair a roof, or plant a garden short of being in the flesh?”—in any other way?

THE CUBE: PHENOMENOLOGICAL PRAXIS We may learn more about what makes a cube a cube—i.e., how we see a cube even when there clearly is none—by means of phenomenological praxis. Such praxis is designed to learn about a phenomenon by creating within the same subject multiple experiences of some situation or with some object and then disclosing the conditions that make us have one or another experience. Here, this inquiry concerns the conditions and work of visual perception that allows us to see something as a cube.17 Take the drawing in Figure 1.2a. Readers familiar with the psychology of perception will recognize it as my rendering of a Necker cube.18 This drawing allows us to investigate perception and how we come to see what we see. Upon fi rst sight, you may see a cube, if you see a cube at all, from slightly above and from the front right. But, if you see a cube, you might actually see one from below and front left. These two perceptions are the two spatial configurations that are seen in psychological experiments, where they are categorized as “cognitive illusions.” Rather than wondering about illusions, let us engage in the praxis of perception to fi nd out what is at the origin of the experience of seeing the cube in one or the other way (i.e., from below or from above). We may do so by, for example, by exploring how to switch quickly back and forth from the cube seen slightly from above to the other one seen from below.

30

Geometry as Objective Science in Elementary School Classrooms

Figure 1.2 a. A depiction of the Necker cube. b. A cube similar to the Necker cube, but drawn according to the laws of perspective. c. A cube similar to the Necker cube, but with “hidden” edges drawn in dotted lines.

To begin with, allow the first cube to appear, for example, the one that you see from below and extending into the back toward the left. Then intend seeing the other one until you see it. Move back to see the fi rst; return to the second. You might also do this: look at the first cube, the one seen from the bottom and extending into the back toward the left. Close and open you eyes—but intend to see the other cube upon opening the eyes again. Practice until you can switch between the two in the rapid fl icker of the eyelids. Once you achieve this, observe what is happing with your eyes during the flicker. That is, how can you generate this or that experience voluntarily? You may notice that if you place your eyes to the lower left corner that appears within the set of lines and then move toward a non-present vanishing point to the left (“along the surface”)—this may be along the edge leading from the “front” vertex toward the back left—then the cube becomes instantly apparent. Similarly, focusing on the equivalent vertex further up and to the right and then moving along the edge “backward” to a nonexisting vanishing point allows you to see a cube from the top. That is, unbeknownst to your intellectual consciousness, the movement of the eye from one of the two vertices toward a non-existing vanishing point in the back to the left or right of the diagram creates one or the other experience. This, therefore, is a statement about how the experience of seeing a cube is produced even if we do not attend to it. If the eyes do not make these movements, then the cubes do not appear and the lines remain on a fl at surface. You can push this experience further—but this is difficult and requires considerable practice. The question we attempt to answer is this: How do we see one and the same cube over an extended time? Or, equivalently, is the eye movement from the vertex to the corresponding vanishing point necessary for us to see a cube? To reach an answer, fixate, for example, the lower vertex. Or, equivalently, attempt to have both cubes appear at the same time. You may not be able to achieve this feat on your first few attempts—psychologists generally use equipment that allows an image to be fixed on the retina. But as soon as you achieve this feat—that is, the eye fixed so that the image falls on the

What Makes a Cube a Cube?

31

same spots on the retina—you will notice that the figure dissolves completely and you won’t see but a dark grey or black field. You no longer see lines. That is, as soon as the eye no longer moves, you cannot see the lines and even less a cube.19 To see a cube, the eye needs to move back and forth between the cube and some other place that constitutes the ground against which the cube appears as the figure. In one sense, the cube is a cube because the eye finds it again upon moving away, and to generate the cube, my eye has to move from the vertex to its corresponding vanishing point. The upshot of this investigation is this: We do not just see or recognize a cube because its mirror image is produced on the retina. Rather, our eyes have to do work, and associated with this work there are changes on the retina. Based on the changing images, and based on prior experience, we have learned to see cubes. We can see cubes because our eyes know what they have to do to make a cube appear. It is in the non-perceived movement of the eye that the distension between the cubical figure and the ground occurs and that the former comes to detach itself from the latter. But we should not think of the image as something standing before the ground, as if projected against a screen; rather, in the image the ground is rising to us (Nancy, 2003). It is not merely, as enactivist theorists would say, that the organism is bringing forth a world—the world gives itself to the organism, which learns how to make any figure reappear. Experiences with cubes as cubes and drawings of cubes in a social context allow children to see the drawings as instances of three-dimensional cubes. But newly born children do not see perspective at all so that the world appears flat to them. This is not surprising given that the retina is two-dimensional (though curved). But how would we be able to see in three dimensions if in fact what registers on the retina has only two dimensions? This is where the sense of touch, tact (from Lat. tangēre, to touch), comes in, which is the one sense that develops in the contact (from Lat. contingēre, composite of con- + tangēre, to touch each other) with the world. But contact also means contingency, which has precisely the same etymological origin. And it has the same etymological origin as contiguity—i.e., (physical or non-physical) proximity—with a continuous material body, and even the proximity of impressions and ideas in temporal and spatial terms. It is out of the first movements of our hands/arms as newly born babies that a sense of space evolves together with a sense of proximity and distance. Vision comes to be coordinated with touch such that the eye learns to see spatiality: from its immanent movements, saccades, and the changes in the two-dimensional images on the retina. Babies learn to perceive perspective after beginning to move about (or to move parts of the bodies, including the eyes) and experience the changes in the visual field coordinated with their own movements. The perception of a drawing as a particular type of cube is determined by the context—i.e., by that into which it is woven (Lat. con-, with, together + texēre, to weave)—and, therefore, with which it is contiguous. Take, for instance, a cube that has been sketched using the theory of perspective drawing from painting, where parallel lines of an object actually meet when extended (Figure 1.2b). In this figure, there is a single vanishing point: take a ruler and follow

32

Geometry as Objective Science in Elementary School Classrooms

the lines of the top square backward, they will intersect in one point. If you do the same exercise as before with this cube (Figure 1.2a), but for the corner in the lower center left, then the cubical nature disappears: you see a part of a pyramid narrowing toward the front and right. The cube no longer is a cube when the vanishing point is changed. This experience, however, is cultural-historically contingent. It was the German artist Albrecht Dürer who introduced perspective into painting. He used a device consisting of a small obelisk and a screen placed between it and the object to be painted (there exists a famous self-portrait showing him at work in this manner). Aiming at the object via the tip of the obelisk gave him a line of sight, and where this line intersected with the screen, he would place the points corresponding to the object.20 We can further change the context of the cube by drawing dotted those lines that would be hidden if the cube were solid (Figure 1.2c). In this case, the eye is almost forced into a viewpoint from above because there are two contextual cues that the eye uses to let us see one rather than the other possible cube: the dotted lines and the perspective drawing. That the eye is “forced” to do something even when we know better is a familiar experience with illusions, such as the one that goes under the name of Müller-Lyer illusion (Figure 1.3). We do see the two straight lines as having different lengths even though we know that they are precisely of the same length. We do not have to be psychologists doing experiments to figure out why the two lines look different. A simple phenomenological study similar to the one with the cube helps us out. When I had the idea of doing this investigation, I used a graphics program and began by reproducing the diagram.21 Then I added others and, in pair-wise configurations, looked at the line by itself paired with first one of the two arrow configurations, then with the other. I removed one arrow in each, then the other. I removed the upper half of each arrow in one then in the other of the two parts of Figure 1.3. You will notice that your eye moves from the line to the arrow, being confined in the lower case and opening up in the upper case. If not before, then the insight in what the eyes do will come for you in and with the last step. You are likely seeing that the upper figure, once you remove the upper half of the two arrows, looks like a truncated rail line moving into the distance with the horizontal line as a railroad tie. The lower part of the figure looks like a railroad tie from which the (“parallel”) rails move backward. That is, in the upper part of the figure, the “tie” is in the back, whereas in the lower part the “tie” is in the front. Now here is the clincher. The eye has learned, through experiences with parallel lines running into the distance—railroad tracks, roads, long city streets—that things in the back look smaller than they do when we stand close to them. If there are lines of equal size, one further in the back than the other, then it will be seen larger than the one closer to us. If you pay close attention to what happens as you look at the upper or lower part of Figure 1.3, then you notice—as said above—that the eye moves between the line and the background, part of which are the arrows. That is, each line is seen with respect to the arrows, which, in their orientation, make the eye move as it remembers from parallel

What Makes a Cube a Cube?

33

Figure 1.3 In the Müller-Lyer illusion, the eye sees two lines of precisely the same length as having different length.

lines and how they appear. This remembering goes unbeknownst to intellectual consciousness but is the kind of immemorial memory that has arisen with/from an original auto-affection. These phenomenological investigations show that there is a lot of immanent work underlying the perception of a cube and that of an organization of lines in two-dimensional drawings. For us to see a three-dimensional object when there is only a two-dimensional drawing and only a two-dimensional “recording” device (i.e., the retina), our eyes have to do work. The perception arises from this work. But this work does not require a mental plan. The eye does it on its own. There is a plan of some kind, but not of the kind that is separate from the doing itself. The memory of the movement required for seeing a cube lies in the immanent movement itself. That is, there is a perceptual consciousness that does not require the same kind of mediation as our intellectual consciousness, which requires and is enabled by signs (representation) that point to something other than themselves. To see an object as a cube, the eye has to do the corresponding work. Neuroscientific work concerning the role of mirror neurons suggests that these types of neurons not only are active when we do something but also when we see someone else do the same thing or when we imagine doing this thing. 22 That is, when we imagine a cube during a geometry exercise even without actually seeing one or a drawing thereof, these same mirror neurons that are active when we see, hold, and touch one, would also be active when we just imagine one. But these mirror neurons evolve with original experiences. They might just be the natural scientists’ explanation for the auto-affection of the flesh that the phenomenological analysis described in Chapter 3 has worked out. 23 Up to this point, I treat the perception of the cube as cube as unproblematical. Most of my readers will likely have done the same. However, we have to ask, “What would we have seen if we had never seen a cube before, that is, if we had been members of a culture where the mathematical idea (concept) of a cube does not yet exist?” A historical analysis of geometry in

34

Geometry as Objective Science in Elementary School Classrooms

the Greco-Roman culture shows that the figures of geometry, the cubes, triangles, prisms, or pyramids are ideal, limit figures that have resulted from continual refi nement of objects that the proto-geometers had found in their real lived-in world (Husserl, 1997a). The real objects humans encountered before the origin of geometrical science were not constant, they fluctuated, making self-identity and self-sameness doubtful; these objects of immediate intuition and originarily intuitive thought, however, are forever lost. But with improvements in the available technologies, especially the technology of measurement—see the comments regarding the origin of the caliper in Chapter 6—objects were subject to continuous refi nement, smoothing, straightening, flattening, and so on. As a result, objects were pushed further and further, subject to what the technological processes allowed. From this increasing perfection have arisen the ideas of ideal objects—for example, the idea of a cube, an object with six identical faces, 12 edges of identical length, and all faces intersecting at identical, right angles. The cube thereby has become a cultural object, which, as all cultural achievements and acquisitions, is “objectively recognizable/knowable and available, without requiring that their sense formation be repeated and explicitly renewed” (p. 25). These ideal objects are apperceived in the sensible embodiments culture makes available: speech, writing, objects. What we perceive, therefore, when we do the experiment with the Necker cube, is not some “raw” phenomenon, but a cultural object with a cultural history that has been lost. We no longer have to constitute the object in the manner that the Greeks had to evolve the idea of the cube as a limit object but rather, the limit object itself is available to us in such figures as the Necker cube or, in the context of the geometry of the second-grade class at the heart of this book, in the quasi-ideal cubes, spheres, cones, cylinders, or pyramids that the teachers have brought to the classroom.

2

From Intellectualist Metaphysics to Embodiment Epistemologies But I soon had to exchange the phenomenological thread for a methodological one . . . what follows now is simply mathematics or, as far as it might be valued as phenomenology, it is one with its object at a very high level, the phenomenology of a quite advanced mathematics. (Freudenthal, 1983, p. 210)

Unlike Freudenthal in the introductory quote of this chapter, I am not concerned simply with mathematics (the quote is from a chapter on geometry) but with the epistemology of mathematics (geometry). The following should be understood in the spirit of the quote but with the extension that I am concerned with the epistemology of mathematics, or rather, with that of geometry. I begin the overview of epistemological approaches by sketching the position that presents mind in a metaphysical manner, consisting of representations that are grounded prior to all experience or that in some unspecified way have been abstracted from experience. In fact, the intellectualist vision “is inseparable from the belief in the dualism of mind and body, spirit and matter” and “originates from an almost anatomical and therefore typically scholastic viewpoint on the body from outside” (Bourdieu, 1997, p. 160). Embodiment approaches have arisen from and are grounded in phenomenological studies of experience. These approaches are of interest because modern day neuroscientific studies that have appeared around the turn to the 21st century have essentially confi rmed the analyses of perception and social cognition that phenomenological philosophers have advanced in the fi rst half of the 20th century.1 I wrote this chapter for inclusion in this book, because reviewers of the various articles that led me to its writing complained that I was giving short shrift to Kant, Piaget, embodiment theory, or enactivism—necessitated, I felt, by the limited amount of space that journals have/make available for developing a truly founded argument. Here, I use the fragment in Chapter 1 as a touchstone to articulate where the various theories no longer can explain what is happening. I begin this chapter with the discussion of the contributions Immanuel Kant made to the understanding of how we know mathematics generally and geometry particularly because of the influence this work has had on epistemology. From Kant I move to the work of Jean Piaget, the perhaps most influential psychologist on cognitive development, who was strongly influenced by Kant’s constructivism. Both Kant and Piaget, using as they do the metaphor of construction for understanding

36

Geometry as Objective Science in Elementary School Classrooms

learning and cognitive development, influenced more recent epistemologies, including both radical and social constructivism.

IMMANUEL KANT: THE LOGODAEDALUS According to classical philosophy and in much of metaphysically oriented cognitive research to the present day, Chris does not display knowing in the lesson fragment. In Kant’s intellectualist constructivist approach, the statements of geometry are synthetically a priori and are recognized with “apodictic,” that is, clear and incontrovertible certainty, and therefore constitute absolute truths. Seeing a cube requires, in the intellectualist approach, “the thought of the cube as a solid made of six equal faces and twelve equal lines that intersect at right angles—and depth is nothing but the coexistence of the faces and equal lines” (Merleau-Ponty, 1945, pp. 305–306). It is the mind that does not know the body or flesh. This is so since space itself has to exist a priori, because otherwise “you could not construct any synthetic proposition whatsoever regarding external objects” (Kant, 1956, p. 91). For Kant, all propositions of geometry are cognized synthetically a priori, because there is nothing that we could say with apodictic certainty about a cube, such as the one Chris is in the process of describing. “How,” Kant asks, “could you affi rm that that which lies necessarily in your subjective conditions to construct a triangle must also necessarily belong to the triangle in itself?” (p. 91). Readers familiar with the writings of radical constructivists will immediately recognize how the subject in this approach is caught in its own constructions that at best can be tested for viability. The separation between external bodies, the material world, and the mind has been made prior to any critique of reason. But precisely because the mind is separate from the material body and the material world, it cannot test its own constructions in a world. The body, which is the “tool” for constructing empirical knowledge about the world, plays no role in establishing knowledge characterized by certainty. It is not surprising, therefore, that “‘Kant the philosopher’ has nothing to say about the flesh, about the philosopher’s flesh, about his ‘union of the soul and the body’” (Derrida, 2000, p. 51). We can say a lot about objects of experience, such as triangles or cubes, but nothing at all about the thing in itself (Ger. Ding an sich) that constitutes the very possibility for the experience. As to the forming of objects, Kant says nothing about it in his critique of pure reason. 2 Although touch is the clumsiest of all senses, it is the most important one, as it is the one to which the others have to be referred to produce empirical knowledge of a physical body (Kant, 1964). Kant does not discuss, as far as I know, how we know a cube; but he does say that we would not be able to form an experiential concept of a physical body if we were not to have the sense of touch. We do not know what he might have said

From Intellectualist Metaphysics to Embodiment Epistemologies

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concerning Chris’s explanation of what distinguishes the rectangular prism from a cube. Kant’s paradigmatic geometric object is the triangle. For Kant, the category (idea) of a triangle exists before (a priori) the apperception of a triangle, even when it is the fi rst time that an individual subject perceives an entity such as a cube. For Kant, Chris cannot perceive the cube or rectangular prism prior to knowing one. If he recognizes a cube as a cube, he knows or at least has some knowledge of it. But we might ask Kant this: A few years earlier, how did Chris know that he was looking at a cube when he gazed at a dice for a fi rst time? How did Chris achieve a synthesis of the manifold experiences that he has had with a dice so that he comes to know it as a cube? For Kant, the object does not just exist: it has to exist as a figural synthesis of the manifold. This figurative synthesis of the manifold is transcendental and precedes experience. Figurative synthesis is the transcendental synthesis of imagination, where imagination is the ability to represent an object in intuition without its presence. 3 Kant discards any possibility that the concept of an object, such as a cube, could arise from experience: “Our representations have to be given prior to the analysis thereof; it is impossible for conceptions to emerge analytically from content” (Kant, 1956, p. 116). Alone the synthesis of diversity is the origin of cognition. Kant’s response to my question is clear: Chris cannot have constituted the conception of a cube from experience. Kant studies the human mind in the abstract, separated from the subject in which this mind embodied and that gives rise to this mind in and through its participation with others in the world. For Kant, the (visual) perception of an object, as appearance, requires the synthetic unity of the manifold in a given intuition, because it alone allows the manifold sameness in the concept of a magnitude. All appearances are necessarily seen as aggregates, but this apprehension is a successive synthesis: “I cannot imagine a line, as short as it may be, without drawing it in thought, i.e., to draw it from one point successively, and in this way, making it available to intuition” (Kant, 1956, p. 205). This successive synthesis of productive intuition is the foundation of mathematics of extension, that is, geometry.4 The concept of a triangle is connected to the possibility of such a thing; and this requires space to be a formal a priori condition of external experience. The productive synthesis that allows us to construct Kant’s triangle or Chris’s cube in intuition is the same that is required by the apprehension of its appearance when making an experience-based concept. Whereas Kant suggests that tact—which for him only exists at the tip of the fi ngers—establishes a relation between sensibility and the world, he does not provide us with suggestions about how (intellectual) intuition and the world are related. He does not show how anything that appears in necessarily subjective intuition can be related to anything that is external. The Logodaedalus does in fact write about the body, such as when he treats the five senses (Kant, 1964). He considers sensibility, which consists of two parts: sense and the power of imagination (Ger. Einbildungskraft).

38 Geometry as Objective Science in Elementary School Classrooms The fi rst is the faculty of intuition in the presence of the object; the second exists in intuition even in the absence of the object. These senses are divided in an outer part, where the body is affected by physical objects—such as the effect that the pizza box and cube have on Chris—and an inner part, which is affected by the mind. He uses the term sensation to denote the consciousness of a representation through sense. The sense of touch is the most important one, as, for Kant, only human beings are endowed with it in their fi ngertips—the equivalent antennae of insects being capable of sensing presence only, but not the presence of form. That is, Kant would say that Chris is capable of recognizing a cube because of his fi ngertips, whereas an animal would only be able to note the presence of something material but not the presence of a shape such as the cube or the rectangular shape of the pizza box. Touch is the most important because it is “the only one of immediate external perception” (p. 447), and even though it is the coarsest one. Without this sense we would be unable to form any concept of shape, which means that the other two primary senses, sight and hearing, are modeled on the sense of touch. Cognition in and of experience is grounded in the sense of touch. There is only one place in his complete works where the body is actually connected to the mind in a way of interest to embodiment and enactivist literatures, that is, where the difference between the two becomes undecidable (i.e., syncopic). At the very end of his life, Kant (1964) analyzes such issues as wit and the laughter that goes with it, recognition of the sublime and the shudder that accompanies it, or the physical aspects of horror children experience when told nurses’ tales. In discussing wit, Kant suggests that the sublime causes disequilibrium in the innards, which, as the tension releases and the innards vibrate, bring about laughter. The way out of the aporia that led to the separate mind and body actually lies here, that is, in phenomena where the difference between the living organic body and mind are undecidable. 5

IMMANUEL KANT AND THE CONSTRUCTION OF MATHEMATICAL CONCEPTIONS Conceptions are an important way of thinking about mathematical knowing. In Chapter 10, I propose a reformulation of this category. I therefore briefly sketch in this section how the Logodaedalus thought about it. Kant’s (1956) account of mathematics is based upon the activity of constructing mathematical objects in pure intuition (time and space). In yielding objects for mathematics, our intuition contributes in an essential way to the formulation of mathematical truths. To construct a concept means: to represent the corresponding ideas a priori. The construction of a concept requires a non-empirical intuition, which consequently, as intuition, is a singular object, but nevertheless a construction of the concept; the universality for

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all sorts of ideas that belong to the same concept are expressible in some representation. Thus, the subject is said to “construct” a triangle in representing the concept corresponding to the object either through imagination in pure intuition or, following it, also in empirical intuition (e.g., in a paper-and-pencil drawing). But in each case, the representation is completely a priori, without having borrowed the pattern from any bodily/ embodied experience whatsoever. The individual figure is empirical and serves to express the concept in its generality. This is so because in empirical intuition, one always considers the action of constructing the concept— which is independent of many determinations, for example, the size, the sides and angles—and thereby abstracts from the differences that do not affect the concept. In Kant’s approach, the uppermost laws of nature lie within ourselves, that is, within our reason. We cannot therefore seek the general laws in nature, mediated by experience, but conversely, have to seek the general, orderly nature of nature in the possibility of the senses and those in the mind-based conditions of the possibility of experiences. Kant realized that however abstract concepts may be, and therefore, however much they may have been abstracted from experience, they are associated with image-like ideas,6 the purpose of which is to make them suitable for practical use even in the case that they have not been derived from such experience. For how could we otherwise associate sense and reference to conceptions if these were not grounded in some intuition—which always has to be somehow exemplified in the possibility of an experience? Kant distinguishes between transcendental and empirical concepts: the former are related to things (Ger. Dinge) in general including those created by mind itself (“noumena”), whereas the latter can be related to phenomena (“phaenomena”), that is, objects of possible experiences. To exemplify his distinction in the context of geometry, for example, he uses the example of the cone. Thus, the “conic figure can be visualized without any empirical help, merely based on the concept, but the color of the cone requires that it had been given in one or another experience” (Kant, 1956, p. 614). This, therefore, is a requirement that Kant has for Chris to have a conception of a cube. He somehow makes it up in his mind, based on intuition, and Chris’s experience in the world has nothing to do with the conception. Kant suggests that a triangle can be constructed on the basis of an a priori, nonempirical intuition that it takes at least three intersecting lines to enclose/ produce a planar figure. But in contrast to philosophers, mathematicians draw on representations, which allow them to view the concrete but only in a representation that they have constructed a priori. Again, Kant uses the concept of triangle as an example, pointing out that facing the question about the sum of the interior angles, the philosopher will ponder the term and not achieve new knowledge, whereas the mathematician will construct a triangle and exhibit for every person caring to witness the classical proof that the sum is 180° (in the case that the full circle comes to 360°). An

40 Geometry as Objective Science in Elementary School Classrooms empirical proof or even the necessity for the angle sum to be 180° would, so Kant, never be possible (as it always leaves open the possibility that one fi nds another triangle in which this is not the case). For Kant (1958), concepts are organized hierarchically into genera and species, which indicate relative orders. For example, in the context of the lesson described in Chapters 8–10, the second-grade students were to learn concepts of three-dimensional geometry by classifying different objects, including rectangular solids, solids with curvature, cones, pyramids, and so on (Figure 2.1). These solids can be grouped. Thus, because in both cube and pizza box the faces stand at right angles, Chris is confronted with a particular class of object, and he currently distinguishes different genera. Kant’s theory implies that the highest concept is the one that cannot be a species (e.g., “solid”), and the lowest concept as the one that cannot be genus (e.g., “cube”). The extent of a concept is larger if it encompasses more things and therefore allows thinking more with it. The highest concept (“conceptum summum”) is that from which nothing can be abstracted further without making it disappear altogether. A lower concept is not contained in a higher one; for it does contain within itself more than the higher one. But the former is contained within the latter, because the higher contains the epistemic ground of the lower. Moreover, Kant emphasizes, a concept is not more encompassing than another because it contains more below itself—for this we cannot know—but insofar as it contains not only the other concept but additional concepts as well. What is valid to or what contradicts higher-order concepts is also valid for lower concepts that are contained within the higher concept; and (b) conversely, what is valid for or contradicts lower-order concepts also is valid for or contradicts higherorder concepts. All of this then leads us to hierarchies of more or less inclusive concepts (Figure 2.2).

Figure 2.1 Collections of mystery objects as second-graders came to classify them (see also Chapters 8 and 9).

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Figure 2.2 Categories (concepts) of geometrical objects according to the rules Kant established.

EPISTEMOLOGY AND PSYCHOLOGY: JEAN PIAGET AND MODERN CONSTRUCTIVISM With respect to the psychology of geometry, Jean Piaget realized a program of research that took on many of the Kantian presuppositions, including those of space. Piaget tended to read the transcriptions of interviews with children in terms of adult rationality, adult understandings of space.7 Accordingly, he would have listened to Chris wondering about which aspects of the talk related to our adult rationality (of a geometer), and he might have theorized what he thought to be the underlying mental structure and processes that led to the observed gestures and utterances. For Piaget, space can be considered entirely independently from its content, consistent with a scientific tradition that dates back to Galileo and Descartes; and ideas about certain geometrical properties, such as parallelism, precede their perception. He supposes that the “science of this independent space is the science of pure geometry” (Piaget, 1970, p. 60), which, in his take, is the version articulated in the Erlangen program of the mathematician Felix Klein.8 Thus, for example, Piaget claims that the ontogenetic development of geometric knowledge goes from topological intuitions and operations to Euclidean metric intuitions. Consistent with Kant, he proposes that the child’s fi rst intuitions are topological, concerned with dividing and ordering space rather than measuring it. Piaget suggests that geometric figures, in contradistinction to time, can be perceived as a whole—as if Chris could see the cube as a whole rather than just a subset of its faces. Temporal duration, Piaget (1970) notes, cannot be apprehended at once. He therefore suggests that knowledge of space “is more direct and simpler from the psychological point of view than time” (p. 61).9 But here he also distinguishes himself from Kant,

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who, as I show above, posits the synthesis of geometric objects even as simple as a line. But Piaget’s schema theory is a direct descendant of Kant’s (1956) German Schema, a “mere product of imagination or intuitive force” (p. 189), although Piaget has developed theories in which cognitive structures about space emerge as the result of the child’s interaction with the world. But, in these theories, as in constructivism more generally, the world is pregiven, independent of knowledge and experience, and, for Piaget, exists with objective structures. Thus, in these theories, the child is “an enactive agent, but an enactive agent who evolves inexorably into an objectivist theorist” (Varela, Thompson, & Rosch, 1991, p. 176). Piaget’s theory of the development of a conception of space therefore is not universal, as studies among various aboriginal peoples show that their spatio-temporal conceptions differ radically from those commonly found in Western cultures. For Piaget, knowing is all about the construction of représentations—in English rendered as “conception”—but which are Vorstellungen in Kant’s sense, intuitions, or mental objects.10 Thus, for example, Piaget suggests that the recognition of a square when a lozenge shape is actually perceived requires a reconstruction of the figure in the mind so that it is head on; and the figures (images) in mind are necessary for the theory of transformations—part of the Erlangen program—to be applicable.11 Accordingly, this feat requires the projective coordination of two perspective-dependent perceptions and it requires the recognition that the object has equal sides and equal angles (Piaget & Inhelder, 1967). But this way of thinking is a direct consequence of the Kantian theory; and there are alternatives to thinking in this way, necessitated by the fact that our experience is not consistent with the mental rotation. Thus, we know that we do not have to rotate mentally an object to fi nd out that it is a cube rather than some other rectangular parallelepiped: We see the cube however it is oriented at the time and “all perceptions are implicated in my present ‘I can’” (Merleau-Ponty, 1964a, p. 290). Chris does not have to mentally rotate the thing that is in his line of sight while he reorients from the pizza box to gaze at Mrs. Turner, to whom he articulates the differences between the pizza box and a cube. This momentary appearance of the object allows him later to go for the cube (see the analysis thereof in Chapter 7) rather than for some other object on the chalk tray and to use it as part of his explanation (Fragment 1.1). That is, Piaget’s description is not consistent with our experience of perceiving geometrical objects. But if we want to understand why Chris does what he does, which is at the heart of his learning and development, we need a phenomenological account of his experience, of what is salient to him at this moment and therefore explains why he does what he does. Merleau-Ponty holds against Piaget not to (be able to) provide such an account, because everything Piaget sees is in view of adult rationality, in view of a mathematician’s Euclidian-cultural perception. This makes him focus on what children do not do or know rather than what they actually perceive and do.

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Piaget’s representational take on the perception of space and spatial object constitute a bone of contention with the phenomenological approach in mathematics education and philosophy. Thus, a phenomenological approach to mathematics and mathematical cognition is critical of the constructivist position. We do not have to worry about concepts but rather use everyday language while doing things with particular objects. In this way, “one can explore the geometrical domain without forming concepts so widely that fi nally over-ripe concepts drop in one’s lap” (Freudenthal, 1983, p. 226). As a consequence, geometry is “forced upon us”: The world of boxes, including the pizza box, and the world of cubes and rectangular prisms—Freudenthal writes about parallelepipeds—that Chris is to talk about are very different worlds. Thus, “a box is a box, but not in the way a cube is a cube, or a sphere a sphere. . . . Cubes are similar to each other, as are spheres. Boxes are not” (p. 229). Freudenthal critiques Piaget to be concerned with formalisms rather than with the experience of the world and how it changes: sometimes from the richer (i.e., rigid bodies, boxes) to the impoverished classifications (i.e., parallelepipeds), at other times from the impoverished classifications to the richer ones (e.g., car VW, Duck, Peugeot car). Freudenthal also suggests that Piaget is not so much concerned with understanding the child’s conception as with the traces (early signs) of an adult rationality that can be identified in the child. He proposes instead activities where children build motor capacities while paving with congruent tiles, “fitting activities,” which prepare them for systematic geometry before having any words (Freudenthal, 1971). Most importantly to the present argument, however, once systematic geometry is reached, the motor sensations “will not become redundant. They are raw material for geometrical thinking” (p. 423). That is, whereas Piaget’s subject abstracts from concrete operations to reach formal operation, Freudenthal’s subject continues to draw on the motor sensations in geometrical thinking. Or, as I would say, doing geometry intelligibly requires motor sensations in addition to verbal (geometric) consciousness. Piaget’s approach is intellectualist and subjectivist. The problem of all intellectualist—i.e., constructivist—approaches is that they are based on the idea that action is preceded by a premeditated and explicit plan, Kant’s Vor-stellung (advance positing), a vision typical of scholasticism. But if these actions are premeditated, how can the Piagetian subject learn something new given that it already knows its actions conceptually? Precisely because the intellect is the source of actions, it implements an existing program. Piaget’s child subjects, then, would not need to go out and manipulate objects, because anything they can do with objects is already prefigured. We can frame the problematic in terms of intentionality: To act and to learn intentionally means that the object (of action, learning) has to be known so that it can be intended. But for this reason, the subject does not have to act or learn, because the object already would be

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known. They could, in a very Kantian manner, just cogitate and figure out everything from their sensorimotor programs. Piaget never dealt with this learning paradox in his theory. Only a theoretical program in which human agents never are completely subjects of their own practices can get us out of this bind. Only a non-intentional theory that contains passivity and givenness as an integral moment can get us out of the constructivist bind in which a subject never can get beyond the confi nes of its own constructions (Henry, 2003).

JEAN PIAGET AND RADICAL CONSTRUCTIVISM Piaget and the (radical) constructivists he influenced do not take Kant onboard in a wholesale manner. They do in fact critique Kant’s approach in the Critique of Pure Reason. In the constructivist approach, the subject is constructing itself; and the emerging subject’s (cognitive) structure is at the heart of the human apprehension of the world. What we see and hear Chris to exhibit then is the result of a construction that has gone on before; he exhibits a form of knowledge that can be traced back to structures in his mind. Piaget reproduces Kantianism by emphasizing that human beings are actively involved in the world and have the potential for unlimited selfdevelopment. Radical constructivists go a step further, emphasizing (a) that cognizing beings do not have access to the world and its phenomena; (b) that cognizing beings are closed with respect to information; and (c) that all that a cognizing being can do is testing its constructions for fit (von Glasersfeld, 1984). From the perspective of radical constructivism, knowledge, no matter how it is defined, “is in the heads of persons, and that the thinking subject has no alternative but to construct what he or she knows on the basis of his or her own experience” (von Glasersfeld, 1995, p. 1, emphasis added). Piaget and radical constructivists attribute everything that Chris exhibits during his interaction with Mrs. Turner to mental structures; and they would likely fi nd something in the mind that would explain how my hand remembered the telephone number of my doctoral supervisor. What saves Piaget in respect to radical constructivist is his adherence to a reality out there so that our conceptions describe reality, whereas von Glasersfeld’s subjects are caught in their subjectivity. Even those conceptions that Kant thought to exist a priori, such as space and time, are, according to radical constructivists (in this following Piaget), the result of constructions as the individual engages with and extends its lifeworld. In radical constructivism, the subject has to wonder whether its thinking somehow corresponds with others, that is, whether it can take its own constructions as shared with others.12 For Piaget, abstraction is a key element in the construction of conceptions and even more so in the construction of mathematical operations, as the mind engages in lifting structural properties from concrete

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experiences—e.g., moving about and differently arranging pebbles and counting each configuration—to get to the properties in themselves, such as commutativity (in whichever order a collection of objects is counted, the total number remains constant). Piaget refers to this process as refl ective abstraction, which denotes the fact that the mind reflects on its own operations with concrete objects and then extracts from these reflections general principles. With respect to the development of geometrical notions, Piaget suggests that children—in contrast to the history of geometry, which moved from Euclidean metric geometry to projective geometry and to topology—begin by developing topological intuitions fi rst. Among the early intuitions are those of ordering and dividing space. The latter is precisely the beginning for Kant, who, supposing space as an a priori condition of experience, has the subject derive geometrical objects without any phenomenal experience through divisions of this space. The construction of an empirical (sensual) concept requires perceiving that which is unique and unchanging about the geometrical figures that Chris and his classmates manipulate, feel out, build, and sort (Figure 2.1) so that it can be left behind as the non-unique (that which they share) in the comparison to arrive at abstract classificatory schemata (e.g., Figure 2.2). Piaget showed that children of about four years of age, asked to copy a circle, a square, and a triangle, will draw shapes that all look about the same, that is, do not preserve the geometric properties of the different shapes. Asked to draw a cross, they do draw something that is or at least resembles a cross. That is, the children exhibit a topological intuition of closed curves that have been copied as closed curves versus open shapes that have been copied as open shapes. The children of this age also correctly copy relations, for example, small circle within a larger circle, small circle outside of the larger circle, and small circle lying across the circumference of the larger circle. The difference between Piaget and Kant, therefore, is the presupposition of the latter that space is given a priori as the condition of experience as such, whereas the former shows that space itself is a result of constructions based on experiences. Piaget and those following him make a difference between concrete concepts (constructed during the sensorimotor, pre-operational, and operational stages) and abstract knowledge (formal operational stage). For Kant, however, every concept is an abstract concept. We are held not to give privilege over the abstract or concrete use of concepts, for we recognize very little in many things with very abstract concepts and with concrete concepts recognize a lot in rather few things. Kant suggests that all mathematical and experience-based concepts have to be constructed synthetically, so that—in contrast to some recent proposals—empirical concepts cannot be defi ned, because these are inherently open and unfi nished. Only those concepts can be completely defi ned synthetically that are arbitrary, such as truly transcendental mathematical concepts.

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The approaches taken by Kant and Piaget after him lead to considerable theoretical problems. One of the most important was the disjunction between conceptions and their realization in experience—a problem cognitive scientists refer to as the “symbol grounding problem” and which mathematics educators articulate in the distinction between knowing a concept and applying a concept. This problem disappears only when conception and incarnate experiences are theorized such that the order in the senses—and therefore the sense of concepts—arises from the senses themselves (Waldenfels, 1999). A related problem is that of the process of abstraction: To abstract a common property from some experiences, a property that characterizes the conception, the conception already needs to exist to be able to classify according to its characteristic property. That is, for Chris to construct the cube based on experiences with cubes, he needs to know the cube beforehand because he could not extract properties from the object as being those of a cube. Piaget is forced to have the minds of his children develop toward an adult rationality, which is that of an objective, mathematical world. Accordingly, the child develops conceptions grounded in invariant (logical) structures typical of adult rationality; such a Piagetian approach cannot show how conceptions arise from the wild, undomesticated, and “untamed” child-like experiences in a nevertheless structured and inherently intelligible world shot through with significance. Piaget’s vision of Chris is a teleological one, where he has to reach the mathematically correct conception unless he simply is incapable of attaining formal reason.

CRITIQUES OF PIAGET AND VAN HIELE ON THE DEVELOPMENT OF GEOMETRICAL THINKING Mathematics educators understand the development of geometrical and spatial thinking among children in terms of two core theories.13 In Piaget’s, as articulated in the two previous sections, geometrical concepts emerge from primitive topological notions and relationships such as proximity, separation, order, and enclosure (Piaget, Inhelder, & Szeminska, 1948a, 1948b). Concerning the grouping of shapes and the emergence of simple geometrical concepts, these authors’ research shows that given simple shapes four- to five-year-old children “can put them into little collections on the basis of shape” (Piaget, 1970, p. 27) but that they place them in figural collections, which, when changed, lead children to “think that the classification has been changed if the design has changed” (p. 27). When they are “a little older,” children will “be able to make little piles of the similar shapes” (p. 27). Piaget’s work by itself and through its descendants (e.g., von Glasersfeld, 1991) has influenced much of present-day mathematics education research on learning and development. However, Piaget’s work has not

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gone unchallenged even by those who recognize and acknowledge the enactive perspective he takes. There are numerous studies providing evidence for very different developmental trends on some of the basic geometrical tasks Piaget used. One of the difficulties with Piaget’s work is that the spontaneous, open, non-permanent, irreversible, incommensurable, polythetic, and polyvalent modes of the child’s perception, action, and comprehension are interrogated only in terms of their nature as rudiments of object permanence—as measurable physical time, mathematical space, and selfreflexive subjectivity (Meyer-Drawe, 1986). In this sense, Piaget’s model is one in which the developing individual is assessed in terms of what she or he cannot do relative to the rational adult scientist (e.g., “he is not able to understand the relationship of class inclusion,” Piaget, 1970, p. 27). It is a deficiency model of the human mind in its development (“he is not able”), one that also emphasizes external causes for the child’s development toward the competencies specified by the curriculum (“class inclusion”). The drawback of the Piagetian approach is that only those material (concrete) practices and actions are made thematic that can be transformed operatively, that is, through internalized logical sequences for which the child is not yet ready (“his classifying ability is still preoperational,” p. 27). A second drawback of Piaget’s approach is that it does not evaluate children’s actions in their own right, as intelligible and accounted-for choices, but always with respect to adult physicists. Thus, in this approach the child inhabits a world that is reduced to monothetic (and therefore unambiguous) perspectives and the child encounters the things/objects of/in the world in terms of their permanence (invariant structures). Throughout this book I conceive each child as competent being in its own right, making sense of its lifeworld in drawing on all the resources perceptually salient to him or her in the situation. We can only understand what a child experiences—or a person at any other age for that matter—when we relate what it does and how it does it to its own world; it is only with respect to its world that the child can ask/hear questions and it responds again within and with respect to its world. Thus, in Chapter 4, I present several lesson fragments where the disjunction between the adult world and the child world remains unnoticed on the part of the participants. I think about the child as a rational being, as rational as the adult, with the difference that there are different (perceptual) resources available in the child’s lifeworld and to its reasoning. I do not think of Piaget’s children in deficit terms, but that they “put [simple shapes] into little collections on the basis of shape” because this is the rational thing to do within their world. Foreshadowing the events that I describe in Chapters 8 and 9, a child who perceives color differences may classify a blue and a red object differently, though both of them may be classified as cubes from a geometrical perspective. The child who classifies by color therefore is rational and grounded rather than deficient: In providing reasons for its classification, the child exhibits accountability and accounting procedures. It is precisely because they are rational and grounded that

48 Geometry as Objective Science in Elementary School Classrooms the children—as the ancient Greek adults—come to understand geometry based on their non-geometrical experiences. Largely as an alternative to the Piagetian stage theory, which primarily refers to cognitive development, Pierre van Hiele (1986) developed his own model of learning geometry.14 Whereas Piaget’s theory does not attend to how children evolve language—the shared medium and tool of children and their instructors—the van Hiele model suggests that at each level, learners develop a language specific to the tasks they are confronted with. In the Piagetian model, the human mind necessarily (teleologically) develops to specific endpoints given by the classical logic on which mature scientific thought is built as a reflection of the laws of the material world. The van Hiele model, instead, places emphasis on learning processes that are specific to particular historical periods. In the van Hiele model, the learner is theorized to move through five levels of understanding and geometrical thought: (a) gestalt-like visualization through increasingly sophisticated levels of recognition (Level 0); (b) description and analysis (Level 1); (c) informal deduction and abstraction (Level 2); (d) formal deduction (Level 3); and (e) proof/rigor (Level 4). These levels are sequential and, as such, the learner must have achieved all lower levels before being able to benefit from instruction at a particular level. Although the van Hiele model has not been formulated in terms of age, some mathematics educators suggest that the children in K–3 will be approximately at the level of visualization (Level 0) and children in fourth and fifth grade begin to conduct analyses (Level 1). One study shows that in a group of nine eighth-graders, one student had achieved complete competence of Level 1 (mode was at intermediate competence), whereas the mode of the distribution for achieving Level 2 was at “low” (Gutiérrez, Jaime, & Fortuny, 1991). Relevant to the present work (especially Chapters 8 through 10), sorting, identifying, and describing shapes are part of Level 0 tasks, whereas classifying objects using properties of shapes constitutes a Level 1 task. Children in second grade therefore should not be able to benefit from instruction in classifying—tasks that are at the center of Chapters 8 through 10—because the children are theorized to be at Level 0. Moreover, at Level 1, children may insist that a square is not a rectangle, whereas they make this integration at Level 2, which is the level of abstraction. In the sample lesson fragment, Chris distinguishes the rectangular prism from the cube and provides a description of what the former has to have to make it consistent with the concept of a cube. As I show throughout this book, Chris and his peers not only describe the rectangular prism as a flattened cube but they also come to describe how to fi nd out how many pizza boxes one has to stack to arrive at a cube (see Chapter 6). It is not surprising to me—based on the empirical results of the present studies—that the van Hiele model has not been without criticism. For example, whereas in its original statements the different levels were held to be distinct, subsequent investigations failed to detect developmental

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discontinuities. Thus, the same students may operate at multiple levels simultaneously depending on task particulars or they may fl ip-flop back and forth between different levels. In other instances, the students seem to deploy levels preferentially depending on task. More importantly, when the van Hiele scheme is applied to verbal protocols from students who had ceased to study geometry, the individuals seemingly regressed. Other researchers found that although students appear to be demonstrating progress with respect to identifying familiar shapes, they may encounter difficulties with unfamiliar shapes. In light of the fact that it is difficult to determine exactly the van Hiele levels of a performance, it is understandable that some mathematics educators suggest that the theory lacks clarity with respect to the shaping of the phenomenological exploration at the fi rst and founding levels of geometrical thinking. For example, questions are raised about the nature of the actions students should perform to move from one level to the next.

CLASSICAL PHENOMENOLOGY Phenomenology constituted a sharp contrast to the efforts of Kant, who reduced all knowing to the mind. The fundamental affi rmation Husserl (e.g., 1997b) made concerning the Kantian approach to perception is this: things are given in the fl esh (Ger. Leib, as distinct from Körper, body).15 This means that the perception is given to my flesh and by my flesh. To be given in the flesh simultaneously means a mode and a destination of the giving. My actual perception of the cube from a particular angle is a function of all the other faces that are co-intended, perspectives that are possible and therefore can be anticipated in the movements of the cube in the hand. Because the perception thereby is a function of the current point of view, “my perception would take another trajectory if I was to touch the cube or to modify my position with respect to it. All actuality implies its potentialities” (Franck, 1981, p. 47). These potentialities depend on the movements that the organic body can take. But where the intentionality that underlies these free movements of the body comes from is not yet resolved. Although Husserl emphasizes the role of the flesh as opposed to the body, some of the dominant directions in phenomenology, including the one taken by the early Merleau-Ponty, have not taken up this distinction. In the phenomenological analysis of experiences with a cube, the presentations of the aspects of the cube and the kinesthetic experiences are not following parallel trajectories. Rather, they are playing together, interacting, such that the aspects have no sense of being (other then, perhaps, because they are continually demanded by the kinesthetic-sensible situation as a whole, by way of each modification of the general kinesthesis). From my perspective, “I never see as equal the six faces of the cube, even if it is made of glass, and yet the word ‘cube’ makes sense, the cube

50 Geometry as Objective Science in Elementary School Classrooms itself, the real cube beyond sensible appearances, has its six equal faces” (Merleau-Ponty, 1945, p. 235). Chris sees and talks about a cube even though it presents itself only under certain aspects, which change from the instant when it was resting on the chalk tray to the instant while he is holding it in and rotating it with his hands. This experience, in phenomenological terms, can be described in these terms: “As I move around it, I see the frontal face, which was a square, transform itself, then disappear, while the other sides appear and become each square each in its turn” (p. 235). But this experience of the different aspects of the cube is but “an occasion to think the total cube with its six equal and simultaneous faces, the intelligible structure that makes it reasonable” (p. 235). I can think the non-visible faces precisely because I exist in flesh in blood, for it is because of the flesh that the “hidden face of the cube shines some place as much as the one I have in sight and co-exists with it” (MerleauPonty, 1964a, p. 182). “And it is for my flesh, my body of vision, that the cube becomes possible” (p. 253). That is, in contrast to Piaget’s way of thinking, Merleau-Ponty does not require Chris to make mental rotations to recognize the cube as a cube when he perceives it on the rest of the chalkboard. The cube, given to him in his flesh, is implied when Chris sees but some aspect of it. The cube obtains its depth as I investigate the object with my gaze, not, as Kant suggests, as an intellectual synthesis, “since depth does not posit the multiplicity of perspective appearances foregrounded by analysis and only sees that multiplicity against the background of the stable thing” (Merleau-Ponty, 1945, p. 306). For Merleau-Ponty, the cube genetically arises from a state in which it is not only invisible but also unthinkable. Prior to knowing cubes, Chris cannot see a cube, that is, constitute it from his perception. Prior to them being a unity with different parts, those aspects of our experiences from which the parts emerge as parts are not even parts of something. The unity of the phenomenon does not come from a synthesis, recognition, or synthesis of recognition: The unity of the phenomenon is grounded in unity of the body. For it would be difficult otherwise to “bring together the notion of the number six, the notion of ‘side,’ and that of equality, and link them together in a formula that is the defi nition of a cube” (p. 236). The definition of a “cube” in such a manner would pose problems. Symbolic thought is blind, a point recognized in the concept of the symbol grounding problem in the cognitive sciences. Neither Kant nor the modern-day constructivists provided a solution to the problem how anything that appears in consciousness and intuition is connected to things and phenomena that appear in our perception. Thus, “one emerges from blind, symbolic thought only by perceiving the singular spatial being that bears all these predicates together” (p. 236). One of the claims Piaget made is that we perceive objects such as a cube in terms of its projections, which we then mentally rotate such as to construct the object to which these projections belong. For Chris, when his gaze strikes the object on the chalk tray behind Mrs. Winter, this would

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have meant rotating whatever falls onto his eyes until he gets a cube. But how would Chris know that the projection is that of a cube rather than of some other solid? Did he really mentally rotate the object and then decide that it was a cube? How could he have known the hidden faces (views) if he did not already know that the object was a cube? From a phenomenological perspective, we do not experience projections of the cube, for example, the projections of a side viewed from a certain angle. What we do perceive is a side. “I do not construct the idea of the geometrical projection that explains the perspectives, but the cube already exists in front of me and unveils itself to me through them” (p. 237, my emphasis). Thus, MerleauPonty contradicts Piaget, who requires a mental transformation to occur in which the perceived shape is mentally transformed—e.g., by means of a mental rotation—into a cube. But Chris does perceive a cube when his gaze falls upon it; and, while holding and rotating it, he actually exhibits for others to see that there are square faces everywhere. We know nothing of any mental transformations, if they exist at all. In fact, as the experiences of James Watson and Francis Crick during the discovery of the DNA molecule structure show, rotating physical shapes on a table may be preferable and leads to results that mental rotation could not achieve.16 The cube is an experiential fact, not something constituted (constructed) in and by the mind and then connected to my bodily experiences so that it can be tested for viability. The cube and its world are given to me, not merely as entities external to my living body, but together with and in my lived and living body, which thereby comes to be constitutive of the world and the cube. If there is anything like a cube with twelve edges, eight vertices, and six sides, “and if I can rejoin the object it is not because I constitute it from the inside: it is that I dive into the thickness of the world with my perceptual experience” (p. 236). But, as pointed out in Chapter 1 and as elaborated in Chapters 5 and 6, Husserl (1997a) shows that the geometrical cube does not exist in the world, as it is only a limit-idea. “The cube with six equal faces is the limit-idea by means of which I express the carnal presence of the cube, which is there under my eyes, under my hands, in its perceptual evidence” (Merleau-Ponty, 1945, p. 236). It is not that there is cognition of the cube and some world outside, somehow connected by an operation of mind. Rather, my organic body (i.e., the flesh) and the material cube are the conditions of each other. What then is the unique act that allows me to see the possibility of all the different appearances? Intellectualists respond that the thought of the cube as a solid has six equal faces and 12 edges that intersect at right angles— and depth is nothing other than the co-existence of the faces and the equal edges. But, suggests Merleau-Ponty, we are given nothing but a defi nition of depth that really is but a consequence. “The six faces and the twelve equal edges do not make the total sense of depth and, to the contrary, this defi nition makes no sense without depth. The six faces and the twelve edges cannot co-exist simultaneously for me unless there is depth” (p. 306).

52 Geometry as Objective Science in Elementary School Classrooms Merleau-Ponty was critical of Piaget, as other phenomenological philosophers and educators after him. He holds against the latter the apparent self-sufficiency of the adult world and rationality. He points out the differences between the child’s and the adult’s world that appear in Piaget’s theory, suggesting that the early forms of thought are not to be overcome—“eradicated” some modern day constructivists say—because they are merely unsophisticated (Fr. barbares, “barbaric”) forms of thought. Rather, these early forms of thought remain “an indispensible acquisition underlying that of maturity if there is to be for the adult one intersubjective world” (p. 408, emphasis added). For constructivist mathematics educators to this day—including those of the social brand—this world is one that is at best taken-as-shared and never truly intersubjective. Chris’s and our understanding of the cube is merely taken-as-shared rather than shared, a conception that differs within the two approaches concerning the collective dimension of all knowledge articulated in Chapter 3, that is, the conceptions articulated within phenomenological sociology and cultural-historical activity theory.

ENACTIVIST AND EMBODIMENT THEORIES

Philosophical Underpinnings Opposing the Kantian and constructivist view of a cognizing agent representing a world outside of it in the form of representations constructed by the agent, enactivist and embodiment literature proposes the alternative of a subject that enacts a world (Maturana & Varela, 1980). In his preface to one of the two papers in Maturana and Varela’s book, the British cyberneticist Stafford Beer comments on the limitations of the Kantian system and expresses the hope that we are getting somewhere in theorizing cognition if his remarks annihilate the Kantian game of categories and the disciplines founded upon them. But, as I suggest, Maturana and Varela, like Piaget, inappropriately ground their epistemology in biology, for the emergence of historical culture during anthropogenesis has changed the rules of and for cognitive development (Holzkamp, 1983).17 Very different laws underlie knowing and learning and their development than those that are valid in the biological domain. Moreover, choosing biology as their ground, Maturana and Varela already submit to the dualism—res extensa and res cogitans—that is characteristic of the sciences, the presupposition underlying their specialization, and the foundation for naturalistic psychology. Scientific thinking and its dualism is precisely the origin of “the incomprehensibility of the problems of reason” (Husserl, 1997a, p. 67). It is this dualism that creates a world of scientific bodies and a world of the mind, that “excludes the flesh” (p. 87); and this exclusion eliminates our sensually intuited, sensible world from all considerations of knowing.18

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Maturana and Varela consider cognition as a feature of a self-organizing, that is, autopoietic system or machine.19 The organism—i.e., the learner— and the environment are structurally coupled to, interact with, and adapt to each other. The environment is considered to be a trigger that sets into motion events and changes that are determined by the structure of the organism: “changes that result from the interaction between the living being and its environment are brought about by the disturbing agent but determined by the structure of the disturbed system” (Maturana & Varela, 1992, p. 96). Because organism and environment are co-implicated, the organism constitutes a source of perturbations for changes that are determined by the structures of the environment. Enactivist mathematics educators take up this approach in considering the learning environment as providing but triggers that start changes, which are determined by processes, and states proper to the learners their ways of interpreting the environment and their ways of making sense. Thus, according to this perspective, what we see while watching Chris is the result of a process of self-organization, that is, an autopoietic process in the course of which Chris has been learning and developing to the point where he articulates the characteristics of a cube. However, the structure of this knowing is determined entirely by characteristics attributable to him, and the environment served merely as a trigger to get the process of change going. The essence of the enactivist theory, therefore, is fundamentally constructivist, though there are some differences compared to the Piagetian position. Central to the enactivist approach is the presupposition that what is relevant in cognition, knowing, and learning is not the “representation of a pregiven world by a pregiven mind but is rather the enactment of a world and a mind on the basis of a history of the variety of actions that a being in the world performs” (Varela et al., 1991, p. 9). An increasing number of mathematics educators share this conviction; and, as the present work shows, rightly so to some extent. However, it is perhaps the emphasis on the mind that makes these educators continue to separate the mind and the body; and certainly there is a distinction between the learner and his environment. In one approach to embodiment, embodied kinesthetic image schemas— e.g., container, path-whole, or source-path-goal—that have originated in bodily experience are said to be the materials for subsequent metaphorization (Johnson, 1987). The notion of schema, however, is problematic from the perspective of a material phenomenology. Johnson explicitly uses the term schema in Kant’s sense (see his Chapter 2). But for Kant, schema are in the mind, not in the body; and even if they were in the body, they could be related to the mind only because of a construction, because the philosopher Kant has nothing to say about the relation of the body and thought. Based on her phenomenology of movement, Sheets-Johnstone (2009) has very similar concerns with Johnson’s notion of embodied image schemata. The problem is that the schema is a non-sensible image or structure, which

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means it is already an abstraction from the sensible, itself an abstraction of something even more foundational that allows sensing—and therefore sense-making—to emerge. Thus, how the structural elements of bodily experiences are transformed to become integral aspects of discourse has not been articulated. The schemas are assumed to consist of structural elements that “have a basic logic, and can be metaphorically projected to give structure to a wide variety of cognitive domains” (Varela et al., 1991, p. 177). In a strong sense, only language (Gr. logos) is related to logic (Gr. logos).20 The basic “logic,” therefore, imposes linguistic structure on the schemata, but this linguistic structure on an evolutionary scale is to be explained. The logocentrism underlying this line of research is evident in relevant publications that contain nothing but words (e.g., Lakoff & Johnson, 1999) despite the fact that other scholars recognize the existence of forms of consciousness that are very different from words and that require forms of engagement radically different from reading to be understood. The forms of consciousness underlying improvisational dance cannot be reduced to words, constituting a form of thinking in movement (SheetsJohnstone, 2009). Moreover, rhythmic consciousness and perceptive consciousness, for example, are very different from linguistic consciousness and cannot be translated into language. The form of consciousness typical of a soccer midfielder, whose pass reaches the winger or striker even under the most adverse conditions—rain, snow, wind—is very different from the consciousness of the applied mathematician or physicist calculating the trajectory of a soccer ball. The sense of the game characterizing soccer is radically different and irreducible to the sense of the game of the applied mathematician. The embodied image schemas are conceptualized and exemplified in some abstract concepts—e.g., container schema that underlies such concepts as IN and OUT. These schemata mediate between language and spatial perception (Lakoff & Núñez, 2000). Image schemata are defi ned as perceptual primitives that organize experiences in which spatial relations come into play. But, as Sheets-Johnstone (2009) points out, the problem with the embodied image schemata is that “bodily experience enters only after linguistic fact” (p. 224), whereas we require forms of understanding that are unmediated by language, such as the self-affection of movement, archetypal relations that ground the “embodied image schemata” such as containment (in/out). Underlying Chris’s presentation in Fragment 1.1, therefore, we might recognize image schemata that express what the student does—as he says very little. But his actions, to be possible at all, have to be based on “archetypal corporeal-kinetic forms and relations” (p. 225) on which the embodied schemata might be based and on the basis of which subsequent metaphorization takes place. Metaphorization occurs by means of conceptual metaphors, which constitute mappings that preserve the (inferential) structure of some well-understood source domain to a target domain. For example, the container schema underlies Boolean logic and

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its realization in Venn diagrams. Perhaps to assuage the charges from the constructionist camp, Lakoff and Núñez then add that meaning and cognition are neither subjective and isolated nor externally determined. The authors describe “conventionalized meaning” as shared by means of image schemata and conceptual projections and conceptual practices. “Meaning” therefore is socially constructed; but it is not arbitrary because constrained by biological processes “that take place in the ongoing interaction between mutually constituted sense-makers and the medium in which they exist” (Lakoff & Núñez, 2000, p. 53). Embodied image schemata still have the intentional potential, that is, they are enacted when the agent has the intent to move or act. But where does this intention come from?

Embodiment/Enactivism in Mathematical Cognition and Mathematics Education There are different camps for grounding enactivist research in mathematics education. Some researchers follow the biological approach of Maturana and Varela, others ground their work in phenomenology, mainly Merleau-Ponty, whereas others again presuppose the schematic encoding of primary bodily experiences. For constructivist and other mathematicians/mathematics educators, it is difficult to conceive of knowing as incarnate. The simplistic view of embodiment, thought as something corporeal, has been both anticipated and critiqued as Aristotelian and scholastic: “it is absurd to submit to pure understanding the mixture of understanding and body” (Merleau-Ponty, 1964b, p. 55). Because such linkages are made in the literature, I do understand that some mathematicians and mathematics educators have trouble accepting the “embodiment” hypothesis. As long as the body and its experiences are but the ground for abstractions toward a metaphysical content, there is no reason to consider them in theories of the mind: body and embodied experiences are but crutches to the mind, left behind when abstraction has occurred. 21 This will only change once it is recognized that only beings in fl esh and blood think as part of conducting life and that without the flesh nothing happens at all. There are embodiment/enactivist studies that investigate, for example, the relation between doing a walk along a meter tap and the temporal modulation of this walk. Other studies of the same ilk make claims about experiences of motion and conceptual understanding of motion or between iconic gestures and mathematical conceptions. Whereas this linkage and transfer between physically walking and talking about a walk may have actually occurred and therefore be phenomenally real, it does not yet provide us with an explanation of how something as diverse as walking and mathematically talking/modeling the walk should be linked in any manner. There is no theoretical mechanism that would explain why a person’s kinesthetic involvement necessarily brings about a deep, conceptual

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understanding of distance, time, and speed; and such experiences do not yet show that the physical experiences are a necessary prerequisite of mathematical understanding. What is the nature of an understanding that allows such diverse things as kinesthesis and conception to be commensurable and mutually constitutive? An important paper that theorizes the embodied nature of mathematics provides a long list of what embodied cognition does not mean (Núñez, Edwards, & Matos, 1999). Embodied cognition is not simply about the conscious experience of one’s body, or conscious experience of its influence on actions, or the manipulation of real or ideal (virtual) things, or about the connection and situatedness in subject matter teaching. Rather, the authors articulate an understanding of embodied cognition as an epistemology about the nature of human ideas and their organization in lived reality. The authors then provide a paradigmatic example of how human beings learn to balance, and the idea of balance comes to be metaphorized into language. They suggest that the “meaning of balance is intimately related to our experience of bodily systemic processes and states of being in the world” (p. 51) as it is to image-schematic structures that organize those experiences and make them significant for us. We are not told, however, what “meaning” is, especially not the “meaning of balance,” as concept or phenomenon—and this is a clincher here, as meaning tends to be related theoretically to words (see Chapter 8 and the events that follow Connor’s question “What do you mean?”). 22 In various places, Núñez, alone and in collaborations, uses the notion of function and the diffi culties in teaching it as a paradigmatic case (e.g., Núñez, 2008). Thus, there is the idea of natural continuity, on one hand, in which a function is understood much in the way of motion along a continuous trajectory similar to a person traveling along a road to get from point A to point B. This idea is contrasted, on the other hand, with the “Cauchy-Weierstrass” defi nition of a function that builds on the experiences of gaplessness (a function as an ensemble of points between which there are no gaps) and the preservation of closeness (delta-epsilon condition). What we are not provided with is an analysis that there can be in experience natural continuity despite gaps—every night, we loose consciousness to sleep and yet experience our lives as continuities. Crucial to the demonstration of the embodied nature of mathematics are the gesticulations, here by a mathematician during lectures in university classes (Núñez, 2009). However, gesticulations alone are insuffi cient, because in many theories they are the products of underlying conceptions, and even Núñez treats them as expressions of conceptual metaphors. That is, the embodied cognition literature appears to overemphasize, again, intellectual consciousness, cognition, over the experience that arises from a life given in and to the fl esh. What a radical embodiment approach requires is a theory in which the body (or, as I argue here, the flesh) is a necessary condition of mathematical thinking

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especially when it is said to be about abstract things. The mere presence of gestures does not imply necessity.

Rapprochement and Distancing In many ways, I am sympathetic to the enactivist and embodiment literatures. But over the years, I became aware of two shortcomings of this approach with respect to the present work. First, it is unclear how the linkage between body and abstract mathematics is produced because the “mechanisms linking abstract mathematics to that experiential grounding are conceptual metaphor and conceptual blending” (Lakoff & Núñez, 2000, p. 102), processes that need to be explained by a radically embodiment approach because they are already cognitive. Second, there is very little that this embodiment approach has to say about the role of society and culture and the shared nature of mathematical knowledge that would allow it to be reproduced over and over again, and thereby allow its objective nature to emerge. There is very little in Where Mathematics Comes From that would point to the essential role of the collective (culture) as the origin of any form of cognition. Concerning the first issue, the position articulated in the next chapter differs from Varela et al. (1991), who use a long excerpt from the early Merleau-Ponty to articulate and ground their position on the embodiment of mind, especially concerning the intertwining of intention and the object world perceived. But Merleau-Ponty revised his position considerably after reading Maine de Biran, though remaining critical of the latter—in fact, already in his phenomenology of perception, Merleau-Ponty (1945) notes that “the objective body is not the truth of the phenomenal body, that is, the truth of the body such as we live it; it is nothing but the latter’s impoverished image” (p. 493). At the end of his life, Merleau-Ponty began but an attempt in overcoming the opposition of the body that is feeling—the living body—and the body that is felt—the organic, material body. Both embodiment theory and enactivism focus on the agent, whose intentional, perception-guided actions in the world bring about changes in the agent. The approach has nothing to say about how intention develops, on the one hand, and how we can come to perceive and know aspects that cannot be predicted based on what we currently know. That is, the approach has nothing to say about emergent phenomena of consciousness. It has nothing to say, either, about the fundamental experience of discovering the unknown, which is a discovery precisely because it cannot be predicted on the basis of the organism’s current state. As phenomenological analyses show, we have such experiences, for example, when we confront novel artwork, experiences that can be understood only through the phenomenon of givenness.23 To understand passion, insights, and learning, we require a non-intentional phenomenology (Henry, 2003). New perceptual phenomena are as much due to structures outside the agent as they are due to structures within

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the agent—the distinction between outside and inside in explanations of cognition are not helpful and constitute an expression of the problem that scholars want to address (Mikhailov, 2001). We are not only the agential subjects of but also subjected to the new, which teaches us to see and hear in novel ways. In this way, John Cage’s piano piece entitled 4’33’’ teaches us to hear silence, which tends to disappear into the background when we listen to regular music; and Paul Klee teaches us to see an imaginary main line when there are only two secondary lines apparent. The category of passivity, however, allows us to understand how something like a cube suddenly becomes a visual and perceptual experience even though the learner could not have anticipated it precisely because s/he did not know it (as per Merleau-Ponty’s analysis mentioned above). It is a phenomenon that creates and occurs at “the borderline situation in which the alien is identical with one’s own and one’s own exists as an experienced reality of Other” (p. 26). The description Maturana and Varela (1992) provide—a description taken up widely among “enactivist” mathematics educators, whereby the structural changes within the organism are only triggered by the environment but determined by structures within the organism itself—is problematic. Thus, for the authors, the “nervous system is a strictly deterministic system whose structure specifies the possible modes of conduct” (Maturana & Varela, 1980, p. 46), which disallows radically new forms of behavior to emerge. The description is problematic because there is no place for culture as a determining moment in and of individual cognition. The entire description of knowing is centered on the individual organism (in mathematics education, the learner) and his/her intention, but there is little to suggest why individual learners all come to reproduce culture rather than constituting constructivist monads informationally closed to their environment. The description would be accurate only if the organism and the environment could be described independently, that a separation of variables describing one and the other can be brought about. 24 But this would not be the case if the variables cannot be separated. For many dynamical systems, especially those that describe cognition in motion, such a separation cannot be conducted. Thus, changes in cognition cannot be reduced to structures inherent in the organism alone. One part of the total system—e.g., the environment—can “enslave” other parts, here, the learner. As I outline in Chapter 3, such “enslavement” is an integral aspect of the way in which the individual lived/living body comes to be a socialized body in phenomenological sociology. The determinations of changes in the organism by outside influences require a different approach to biology and cognition, an approach that is consistent with the catastrophe theoretic approach and its manner of describing the emergence and evolution of structure. 25 Thus, essential aspects in the evolution of biological organism, culture, and cognition cannot be modeled based on characteristics of the organism, the changes in which are merely triggered but not determined by environmental

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conditions. We cannot overcome the Cartesian distinction between body and mind, inside and outside, by suddenly making these opposite interact or even transact: to overcome traditional conceptions of mind, the very distinction between inside and outside has to be the result of the inner workings of the system itself. For me, the problematic nature of enactivism and embodiment theory is apparent time and again in the way these approaches come to inform the field of teaching and research. Thus, I found it quite amusing to hear an “enactivist” colleague, grounded in the work of Maturana and Varela, ask a student, who had just completed building the model of an invisible object based on her sense of touching, “Did you build [your model] in your mind?” If enactivism means a continuation of the dominance of the mind, then it is not something I am interested in. That is, in terms of my hand that remembered the telephone number, the enactivist mathematics educator would have asked me whether I remembered the number in my mind rather than in and with my hands. The mind that is required to do well on written tests, the linguistic mind and the associated intellectualist consciousness, is, in my understanding, only a subset of the forms of consciousness we produce. “Mind” understood as intellectual consciousness only refers in an oblique and partial, one-sided manner to the total reflection of material reality. There are other forms of consciousness equally and more important, and it is precisely those other parts, such as affective consciousness and rhythmic consciousness that allow us to coordinate with others to form society, which, in French, is a corps social, a social body. In the approach presented next, this social body shapes my individual body, socializing my body. We do need a different way of theorizing the relation of the body and mind, for “the exclusive consideration of the extended body by contemporary thought and especially by scientific thought developed after Galileo bars our access to the genuine body” (Henry, 2004, p. 344). Moreover, the exclusive consideration of the extended body—Descartes’ res extensa— “keeps us from conceiving it as having speech, as having the speech of the life that speaks in every living body” (p. 344). This language of the body includes the spontaneous gestures of everyday life, dance, mime, sports, music, and so on. Even a hand given as part of a greeting, a hug, and open arms as part of a welcome constitute forms of communication as much, and perhaps even more, as the words we use in those contexts. The human body is not just another object, and not just another organic or living body— plants and animals have them, too, and use them, according to zoosemiotics and phytosemiotics, to exchange information—but we have to think the body precisely through the notion of the “pathetic flesh of our living corporeality” (pp. 344–345).

3

Material Life as the Organizing Principle of Knowing What memory or, better, remembering is about, for those sequences of movements that enterprises like songs and poems and talk are, with no categories or summarizers that store particular sequences of places and paces in their specificity—what this species of remembering is remains a deep puzzle. It will not be fathomed until we focus on pace-placed movements and their bodily organization. (Sudnow, 1979, p. 31)

In a meditation on the placement of the hands—in both playing piano and typing a text on the type-writer—Sudnow asks himself how the hand knows it should use this rather than that finger for an opening note/word if there is no explicit instruction by the mind beforehand. How do the hands know—in playing piano, writing on the keyboard, making iconic and deictic gestures over and about geometric objects—without being told so, without following a plan that they have received beforehand? How did my own hand remember my supervisor’s telephone number and how do my hands write when I have a conversation at the same time? My hunch—informed by such works as that of Michel Henry and Pierre Maine de Biran—is that this memory is the one that arises from the auto-affection of the flesh, which thereby remembers without requiring the sign-mediated memory of the mind. This immanent capacity underlies everything I do, not only with my hands, but also in the recognition of the movements of the hands of others. This, as recent cognitive science research has shown, exhibits itself in the phenomenon that movements are recognized only when we are already able to enact them or hear them in the case that the actions are always accompanied by an action-specific sound.1 Moreover, this is consistent with the fact that we can see and even hear in the actions of others the intentions underlying their actions, and such intentions can only arise if the auto-affection has already occurred such that a movement knows to reproduce itself.

WHY MATERIALIST APPROACHES? The approaches to cognition from Kant to Piaget and constructivism focus on mind independent of the body; these are metaphysical approaches, which lead us to the belief that mind works independently of the world. They also make us think of mind as working independently of the body

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generally and independent of the emotions specifically, which are, in important ways, shaped by collective emotional contexts. A materialist approach inherently shifts our attention to the fact that life has begun without consciousness, without the mind. At the same time, life has given itself mind and consciousness. This means that we should not end up holding one or the other position, a material physical or metaphysical one, but that these two aspects of human life mutually constitute each another. In this chapter, I articulate material phenomenological approaches— from philosophy and sociology—that allow us to understand the practical knowing of the body that is the source of any abstract knowing. In fact, rather than theorizing thought and talk/speech/language, I suggest theorizing thinking/expressing, but to do so from a performative perspective that is grounded in the explicitly human experiences of the world. This thinking is not that of an enactivist system (living machine), not that of a metaphorized sensorimotor schema, and not that of the abstracted (formal) operation— thought “segregated from the fullness of life, from the personal needs and interests, the inclinations and impulses of the thinker” (Vygotsky, 1986, p. 10). It is the thinking of a real person, in fl esh and blood, who contributes to bringing to life mathematics lessons in real time and who exhibits in activity the very methods for making the activity rational and accountable, despite (and perhaps because of) the “mumbles, stumbles, malapropisms, metaphors, tics, seizures, psychotic symptoms, egregious stupidity, strokes of genius, and the like” (Rorty, 1989, p. 14) that characterize everyday talk, including that which we can hear in the offices of mathematicians. Concerning the role of affect, Maturana and Varela’s (1980) “living machine” does indeed have something like it in the form of an “emotional tone.” Yet, this emotional tone “does not specify a particular conduct, but determines the nature of the interaction” (p. 46, my emphasis). This is inconsistent with the neuropsychology and sociology of emotion, as in the cultural-historical approach, where it is known that affect does indeed contribute to specifying the possible modes of conduct, which Maturana and Varela strictly attribute to the nervous system. This is the perspective of an “old approach” that “precludes any fruitful study of the reverse process, the influence of thought on affect and volition” (Vygotsky, 1986, p. 10). And, as I articulate especially in Chapter 7, it precludes understanding how affective phenomena at the collective level are mediated by and mediate interaction rituals such that the behavior of the individual comes to be entrained to the collective behavior. At some time in the evolution of the human species, thought as we know it today did not exist; nor was there anything like a self-understanding of the human agent as a sub-ject confronting the ob-ject, that which is thrown before the subject and stands over and against it. How has thought arisen in the absence of thought? How can a constructivist organism aim at getting for itself a human mind when it does not know a human mind or Piaget’s adult rationality? How does such an organism

62 Geometry as Objective Science in Elementary School Classrooms give itself intentionality, when any such giving requires—following Husserl’s analyses—requires intentionality? To understand the emergence of gesticulation and the precursors of mathematical understanding, it is useful to draw on materialist approaches. Rather than grounding themselves and their constructs in reified common sense discourses, materialist approaches are concerned with the evolutionary and cultural-historical conditions that lead to the emergence of experiences and discourses about such phenomena as space, cubes, geometry, mind, and so on. The point is that mathematical structures are not abstracted away from experiences but that the experiences are immanently present in the same way that the mirror neurons of motion are active when we see someone move. Seeing someone move gives us the feel of the movement, and this is the ground of understanding—it is empathic and sympathic. The impact this has on how we think about conceptions are articulated in Chapter 10.

MOVING FROM THE BODY TO THE FLESH Although Merleau-Ponty (1945) has attempted to break away from the metaphysical dualism between the mind and the world (body), the essence of his phenomenology of perception retains the opposition between consciousness and its objects. Recognizing that perception is more than the result of an intentionality going outside to find a world, the philosopher realizes in his unfi nished work from the end of his life that the difference between the active and the passive body is undecidable: “my activity is identically passivity” (Merleau-Ponty, 1964a, p. 181).2 Not only do other people see us, but also so do, as many painters have been saying, the objects themselves, in which are expressed our own cultural-historical ways of seeing. This is so because “the very existence of mind is possible only at the borderline [self, other] where there is a continual coming and going of one into the other” (Mikhailov, 2001, p. 20, original emphasis). Chris not only sees the pizza box and the cube, he knows himself to be seen, which he makes available to all when he turns his body and face to gaze at the teacher. This interlacing of the world—which, in and through my entire being is both experienced and experiencing itself, and my consciousness—is possible only in the flesh. The interlacing occurs, precisely, as we listen: “To be listening is to be at the same time outside and inside, to be open from without and from within, hence from one to the other and from one in the other” (Nancy, 2007, p. 14). Thus, relevant to consciousness is not my material body, but the flesh that has the dual capacity to affect (act) and to be affected (suffer). For, “if there is a flesh, that is to say, if the hidden face of the cube shines somewhere as well as the one I see, and exist with it, and if I see the cube, I am also part of the visible, I am visible from elsewhere” (Merleau-Ponty, 1964a, p. 182). My flesh and the flesh of the world are but

Material Life as the Organizing Principle of Knowing

63

two sides of the same phenomenon: “if it and I, together, are taken as the same ‘element’—do we have to say of the seeing or the seen?—this coherence, this visibility in principle, is carried across all instantaneous discordance” (p. 182). It is precisely in the flesh that seeing and being seen, touching and being touched cross over and constitute the unity of the material world and the consciousness in which it appears. My left hand that explores the cube can be felt by the right hand, so that the former is both touching and being touched in the same way the cube is touched by it. It sees itself seeing, and feels itself feeling. It is because of the flesh that seeing and touching are interrelated, because both are activities of the same “I can”; it is because of the flesh that being seen and touched occur to the same “I suffer.” The two maps, the one of the senses by means of which the world impinges upon me, and the one of my projects, by means of which I come to act upon the world, are internally complete parts of the same Being (Merleau-Ponty, 1964b). All flesh is capable of being—and recognizing itself as—a body, but not all bodies are flesh (e.g., the cube). Although he conceives of the flesh as the lynchpin of consciousness, Merleau-Ponty did not have the time to further develop the role of the flesh to a knowing consciousness. 3 He did not resolve, for example, how intentional seeing and touching can come about in the fi rst place. This intention is not “natural” but the result of an object orientation, which deaf-blind children, for example, have to be explicitly taught—an unfamiliar box placed in their hand does not stimulate exploration but would be just dropped (see below). This unfi nished work would be taken up, in different ways, by dialectical materialist approaches to psychology and material phenomenology. The duplicity of feeling, which derives from this crossing over, at the borderline, makes it possible that the (mental) image and the world it resembles are but two sides of the same coin, the inside of the outside and the outside of the inside. It is because of this reason that we can recognize a cube when we see it in the world, rather than being a Kantian, constructivist mind forever wondering how our subjective representations relate to the world. This duplicity can also be seen in the movements of Chris: Just as many painters have felt that the things look at them rather than them looking at the things, so it is the rectangular prism (pizza box) and the cube that shape the gestures Chris exhibits. The three-dimensional objects fi nd their expression in the hand movements that before any conceptualization have explored such objects and enabled the capacity to reproduce the shapes in gestural form. With Merleau-Ponty we might say that Chris’s gestures constitute the continual birth of the rectangular prism (pizza box) and the cube. One day in the future, we may imagine him having found his articulation and, without gesture, talking about these geometrical figures in ways that constructivist mathematics educators might say that he has “constructed” “meaning.” Vision is neither a mirror nor a thought that in a constructivist way tests itself in the world: “vision is a conditioned thought,

64 Geometry as Objective Science in Elementary School Classrooms it is born ‘at the occasion’ from what arrives in the body, which ‘excites’ the former to think” (Merleau-Ponty, 1964b, p. 51, my emphasis). That is, the outside is not just stimulating the body (system) to change but it arrives in the body, shaping it, making it a socialized body that can reproduce and reproduces society (corps social) precisely for this reason.

MATERIAL PHENOMENOLOGY

Flesh and the Auto-Affection of Life The flesh has been variously defi ned, but always as an identity of two processes (e.g., seeing/seen, touching/touched, feeling/felt, hearing/heard) and most recently “as the identity . . . of the affected with the affecting” (Marion, 2002, p. 231). This flesh, because it feels itself, thereby has the potential to auto-affect itself prior to any intention. From this auto-affection of the flesh arises the sense of “I can,” an arm or hand moves because it can; the eye searches and sees because it can move and submit itself to visual impression. This is what Merleau-Ponty (1964a) intuited near the end of his life when he said that my activity is essentially passivity. In his notes, he marks that he has to come back to the analysis of the cube. This analysis would show that incarnation, the flesh, is the necessary condition for a world: “My vision and my body themselves emerge from the same Being, which is, among others, cube” (p. 252, my translation, original emphases). I do not merely perceive a cube, out there, but, in the perception, I perceive myself, my perception. It is the object that I later have come to name a cube that has taught my eyes and hands to follow its outlines so that they recognize it when seeing or feeling it again. “It is for my flesh, my body of vision, that there can be the cube itself” (p. 253). But, “the cube as such, the one with six equal faces, exists only for a non-situated gaze” (p. 252). The flesh is the precondition, because without it, there would be no experience of bodies, including the material body of the flesh, which, in touching, recognizes itself to be of the same order as the other bodies surrounding it. It is the flesh, with its capacity of tact (i.e., sense of touch), contact (i.e., touched and being touched), contingency, that is, to be the ground of all senses, sense-making efforts, and, therefore, knowledge. Flesh, because of its capacity of tact and contact, also is subject to contamination and contagion, and, thereby, capable to be affected by other flesh. Affect, therefore, is characteristic of the flesh, “because even representative sensation . . . is affection” (p. 288). The flesh, seat of the senses, also is the source of the movement undergirding sense impressions, so that movement is immanent to the activation of the senses, especially that of touch, the privilege of which is shared with all other senses. It is the centripetal force that holds us together in one rather than projecting us into the separate realms of bodies and (metaphysical)

Material Life as the Organizing Principle of Knowing 65 minds. We learn to see (in) three dimensions, even when presented with flat photos, drawings, pictures, and paintings, precisely because of the association of vision with the experience of movement and touch. In the material phenomenological approach, touch is the primary sensemaking organ (Maine de Biran, 2006). Although tact is characteristic of the body as a whole, the hand has become its paradigmatic synecdoche. It is from the movement of the fi ngers, which adjust themselves over the cube, envelope it at multiple points, run successively over its faces, slide with ease over its edges and follow their direction, that Chris’s hands recognize the cube. There is a unique resistance with different impressions and the surface abstracts itself from the cube, “the contour from the surface, the line from the contour” (p. 61). When his hands move again over the cube, and when his fi nger points to the different faces that expose themselves as squares, which he names squares, they do so with the memory of previous experiences of holding the cube. The diminishing effort involved in repeating experiences “not only is the primary condition, but also the complete and abridged form of consciousness” (Ravaisson, 1838, p. 19). If Chris is able to recognize a cube visually, this experience itself is grounded in the coordination of the perceptual sense with that of touch. It is precisely because the hand combines motility with sensibility, precisely because it is flesh, that “it opens a quarry to intelligence and furnishes it with its most solid materials” (Maine de Biran, 2006, p. 61). Near the beginning of my analysis in Chapter 1, there is a description of Chris’s “response” to the teacher’s question, and I suggest that it is essentially built on (gestural) movement and touch. In touch, there are actually two important aspects. First, there is a form of resistance that the flesh experiences to itself that has to be overcome by the effort that puts the hand in motion. Second, there is the sense of touch that affects the hand giving it the sense impressions of and from the surface, the roughness of the surfaces, the sensation of hot and cold, the way in which the surfaces curves. Traditionally, the sense impressions have been thought mainly as constituting a primordial. But in the senses, there is an intention. This intention floats in traditional theories, because these do not indicate how such intention (to collect “information” by means of the senses) comes to be possible in the fi rst place. It is precisely the self-affection of the flesh, the memory of the movement in the movement itself that provides the possibility for the intention.4 It is because of this auto-affection that my hand/fi ngers were able to remember my supervisor’s telephone number even without the help of another sign. This memory is not just rote, passive, but it is active, enabling my hand to act. Because of the movement that experiences itself, the hand “knows” to enact the movement required to sense by means of touch. It is the resisting continuum that renders manifest reality. This living body now is referred to (by phenomenological philosophers) as the fl esh. It is a living/ organic body with the powers to discover itself as a material body among other material bodies. It is a body before (intentional) sense impressions,

66 Geometry as Objective Science in Elementary School Classrooms and, therefore, a lived/living body that exists before the world. That is, the flesh precedes the body that appears in the embodiment and enactivist literature, characterized as it is by bodily sensorimotor schemas and intention. If there are such schemas and intentions, then they are produced by the flesh, which, because of its immemorial memory, cannot be structured like a traditional sign, Kant’s “non-sensible image.” Sensorimotor schemas, as images, refer to something else, something which they image (Lat. root imit-āri, to copy), a capacity to move; the motion of the flesh refers only to itself. Sound production, which eventually leads to language, is but one of the expressions of life that arises from the auto-affection of the flesh. Thus, like the original gesture, the original “cry belongs to the immanence of life as one of its modalities in the same way as the suffering which the cry bears within itself is one of life’s modalities” (Henry, 2003, p. 341).

Auto-Affection and Periodicity Every movement that I have executed during a fi rst time with an object, I can recognize later when I take the same solid in my hand; this recognition has no other condition than the movement itself, which now becomes a form of “sign,” which is not really a sign because it only stands for itself. This is precisely the same as in rhythmic consciousness. Rhythmic noise is not noise on which rhythm has been superimposed—it is impossible to perceive rhythm and noise separately. Because rhythmic consciousness is the result of an auto-affection in a noisy environment, it plays a central part in the sociological phenomenon of entrainment, which, in Chapter 7, is a central part of my explanation of knowing and learning in Chris’s mathematics class. This is where my approach differs from those embodiment and enactivist theorists who work with schemas. In the theory that Maine de Biran initially articulated and that Henry subsequently worked out, there is no schema that would mediate the enactment of a movement, drive the body to move itself. It is the movement itself that serves as the sign, the power of self-movement that the effort inscribes during a fi rst occurrence. Thus, “all movements executed by the hand, all positions that it has taken by running along the solid, may be repeated at will in the absence of the solid” (Maine de Biran, 1859b, p. 147). This is precisely what we encounter in the chapters of Part B, where hand movements are observed that, as gestures, repeat previous hand movements but now in the absence of the object. That is, we might say—a bit tongue in cheek and in opposition to the common-sense expression relating willing mind and weak flesh—that it is the flesh that is strong and the origin of the much more feeble mind. Rhythmicity and other periodic phenomena are interesting within the present perspective, because they involve forms of consciousness that require the active involvement on the part of the “perceiver,” who re/produces the phenomenon also coming from and existing elsewhere. Because the flesh,

Material Life as the Organizing Principle of Knowing 67 contiguous with other aspects of the world, is open to contamination and contagion, it tends to resonate in synchrony with other periodic phenomena. It is capable of re/producing these period phenomena and, therefore, to be part of interaction rituals that underlie the formation of society. I return to these aspects below in the section on phenomenological sociology.

Auto-Affection and Memory The next time any originary movement is executed, the renewed effort will be less, and the motor capacities that have enacted the movement cannot but recognize the difference as its own will, intention because “the memories of an act encloses within it the sensation of the power required to repeat it” (Maine de Biran, 1859b, p. 474). The same movement, in repeating itself, by orienting itself to the same solid, its forms and its diverse qualities, will permit me to recognize this solid without that this recognition would have any other condition than the movement itself. My fi ngers dialed my supervisor’s telephone number, therefore, without any other condition than the movement itself; and my ear recognized the melody, because a similar autoaffection of the ear, which does not require any other condition than the pitch movement itself. Chris recognizes the faces of the cube, whatever its direction while resting on the chalk tray or while turning in his hands, because the movements of his eyes produce a recognition that underlies his seeing a cube rather than something else. This form of memory has to be distinguished from a second kind of memory, the one that we are more familiar with, a memory mediated by some sign, a form of representation, whether it is an internal one (word, image) or an external one (knot in handkerchief, note on paper). This second form of memory depends on the fi rst, which is the trace of the way in which the memory was formed in the fi rst place: “To say the truth, these memories themselves are nothing but the free awakening of the thought of possibilities that we say are sleeping within these” (Henry, 2000, p. 208). It is important here to observe that the memory in the movement is “of an entirely different order, foreign to the thought, to say the truth, to any representation, to any memory, which are the immemorial power of my pathic flesh” (p. 208).5 Even without mediation, my eyes know how to move so that I see a cube even when there is only a two-dimensional drawing of it in the form of a Necker cube (Figure 1.2a). The memory of the cube then lies at the interior of the movements that experience, feel themselves, and move by themselves that we have to seek/fi nd an experience that only belongs to it, the movement. It is precisely then that we no longer have the distinction between mind and a material body—mind is the body itself, memory in the movement rather than in some schema or representation that is used to bring the movement about. Mind does not act on the body or instruct it to do what it has to do. The possibilities in and of flesh constitute the most ancient form of memory.

68

Geometry as Objective Science in Elementary School Classrooms By laying the tangible object aside, we may conceive how the traces it has left can live again in the internal sense, by the exercise of the same activity that has concurred during their formation in the fi rst place. And fi rst, the module or instrument that serves to determine these forms always is present: all the movements executed by the hand, all the position that it has taken in passing over the solid, can be repeated voluntarily in the absence of the solid. These movements are the signs of the diverse elementary perceptions relative to the initial qualities inseparable from resistance. They can serve to recall the ideas corresponding to it, and this recall, executed by means of the available signs, constitutes the memory properly said. There is therefore a true memory of tangible forms. (Maine de Biran, 1859b, p. 147, emphasis added)

This form of thinking is important, as we require nothing but the movement itself in the absence of any representational and representative intentionality, even in the absence of the five senses.

Emergence of Habitus, Intentionality, Conceptual Knowledge Most importantly, as the effort in executing the movement diminishes, and the consciousness required executing it, that is, as the movement becomes a habit, this movement does not leave the sphere of intelligence (Ravaisson, 1838). “It does not become the mechanical effect of an external impulse,” such as a mental plan, “but the effect of a penchant that follows the will” (p. 28). It does not, in modern day parlance, become recorded in some rote memory. This penchant is a tendency, which inherently implies intelligence, toward the end that the will gave to itself. In more recent phenomenological analyses the ensemble of such tendencies is denoted by the term habitus thereby constituting a system of structured structuring dispositions. It is a form of immediate intelligence, a sense of the game, where nothing separates subject and object. It is precisely this aspect that is not theorized in current embodiment and enactivist research: The fact that the original movement is not lost but continually present in and through the habitus it formed. In this way, “the influence of the habitus also extends to the highest and purest regions of heart and mind” (p. 40). Phenomenological analyses of spatial objects, from Maine de Biran to the present-day (e.g., Chapter 1), show that geometrical objects are not apprehended all at once. In this, material phenomenology is consistent with the enactivist/embodiment approach. Where they diverge is in the fundamental starting point, which for a material phenomenology lies in an originary passivity that precedes all intentionality underlying sense-making efforts of the incarnate agent. Prior to the intention or will to explore the world, prior to any sense-making effort, there are experiences that allow intentionality, and therefore agency, to emerge. Intentionality is the relation between a subject and an object, and such a relation could not exist

Material Life as the Organizing Principle of Knowing 69 without the self-affection that enables the transitive subject | object relation in activity. Prior to any schemata that could be metaphorized, there has to be an immanent self-affection arising from auto-movement that precedes a correlation of the movement with sensation. This self-affection precedes any sensing, sensation, and sense-making effort. Schemas are the result of a dialectical movement involving part and whole. Even the simplest figure, to be figure, requires the movement of the eye. Maine de Biran (1859b) proposes a different approach. Using a solid cube as an example, he asks how an idea of the cube can precede the experience of the cube as a totality. How would it possible to have a conception a priori of all the different forms of experience that are possible with a cube? How would it be possible to have an idea of all the relationships that can be abstracted from the cube by means of analysis once we know what a cube is? To someone who has not yet learned to see, the cube would appear as a square or as whatever figure appears depending on the relative positions of observer and object. Maine de Biran then considers a blind person who learns to touch. For such a person, the idea of a cube as a whole could not exist prior to the distinct perception of its parts. There is a true synthesis at work, as the composite whole does not precede the composition of its parts: “these two modes of generation of which we speak therefore are reduced to the same” (p. 301). When the idea of the composite has formed, it is possible to return to the simple and to consider it separately; but this simple has been the starting point for the composite whole, here, the cube. The two modes of constitution, from part to whole and whole to part “therefore are reduced to the same [mode]” (p. 301). There is a memory of the simple in the series of coordinated movements, and this memory is in the flesh, which auto-affects itself. Maine de Biran suggests that this not only is how sensory synthesis occurs but also that it is a true emblem of what occurs in all forms of reasoning properly speaking. From a constructivist position, thought is the seat of knowledge and intention. The question is how anything without extension, thought or consciousness, can have an effect on something as material as the body. Maine de Biran points out that it is not the body that matters, because it is the result of processes even more fundamental than the body that is said to be the seat of embodied knowledge. The original corporeity is not my body that I can touch and feel as my hand moves along it. The original corporeity is situated in the hand that does the touching and feeling, itself enabled by its originary movements. The question now is how this hand that is the instrument (means) that allows us to know has itself come to be known? This question is important, as the hand, as a material body, has itself come to be known to me as a body.6 How is a mobile organ, the hand, able to move itself without being known? How can a hand move itself intentionally in a certain direction without an intention? It is the immanence of all its powers of my flesh in the flesh that the latter becomes the place of an immemorial memory. This is the memory of a body that remembers each time when it takes a cube, turns

70 Geometry as Objective Science in Elementary School Classrooms it in the hand, remembering not only how it feels but how it looks as a result of the turning. The resultant movement cannot be thought in terms of the “displacement of an objective organ, it is not given to any ‘remembrance’ properly speaking, to any representation, to any thought: it is the self-movement of a prehensive power revealed to itself in the pathic self-donation of my originary corporeity” (Henry, 2000, p. 207).

Life, Auto-Affection, and Community Thinking knowing in terms of auto-affection may lead readers to think that the material phenomenological approach takes us into the same kind of solipsism that constructivism has taken us. However, the material phenomenological approach does not just think how the individual knows and learns but is essentially concerned—at least since Merleau-Ponty—with the way in which culture mediates what we do and, therefore, what we learn. The problem only arises if we take inside and outside as given, when in fact, the differentiation between inside and outside is itself a result of processes, for the outside is “‘outside’ for us precisely by virtue of the fact that we experience it, and it acquires meaning as such ‘within’” (Mikhailov, 2001, p. 20). In contrast to embodiment and enactivist theories, which focus on the exteriorized body, material phenomenology theorizes life and how it accomplishes itself in and through the lived/living bodies; life is thought as the principle of everything; and this life is precisely what constitutes the common basis of the community and its individual members. Thus, “what the community and its members have in common is not something, this or that, this little piece of land or that profession, but the way in which the things are given to them” (Henry, 1990, p. 161). What is common to all members of the human community is this auto-affection of life, which is the basis and essence of the subjectivity that we experience. Auto-affection and selfhood are irreducible. All members of a community therefore have this in common: “the coming to itself of life in which everyone comes to itself as the Self that it is” (p. 177). Auto-affection radically individualizes so that it is precisely this subjectivity of life that is common to all members of the human community—across all their inherent differences. Thus, the transcendental condition of the possibility of any conceivable Self and any conceivable me, that is, its immanent generation in the absolute Life, reveals itself to be identically the condition of the transcendental possibility of the relation if each Self and all the others, the condition of the experience of others. (Henry, 2003, p. 204, original emphasis) That is, what any community has in common is its content: transcendental life. Most importantly, because affect is that which characterizes life generally, affectivity is also that which ties together any community, constitutes the very possibility of community: “Suffering, joy, desire or love, and even

Material Life as the Organizing Principle of Knowing 71 resentment or hate carry within a unifying power infi nitely larger than that attributed to reason” (p. 206). Thus, it is precisely affectivity that underlies the human community. In Chapter 7 I show how periodic phenomena, which give rise to resonance, exist all over the mathematics classroom of which Chris and Mrs. Turner are integral members. Periodic phenomena, as research shows, are essentially affective phenomena, they underlie the ways in which affect—which underlies interaction rituals—comes to coordinate and align the members of a community. A community, in essence, is affective. Here, affect is transmitted by means of contagion and contamination as we come entrained—i.e., affected from without and within—to the mood that characterizes the collective of which each “I” is a constituent part.7 An interesting case for thinking the community from the individual— which leads any theory of the community into a cul de sac—relates to intentionality. This same concept is implicit in all theories of learning that I discuss in Chapter 2, including those that bear most affi nities with my own, embodiment and enactivist theories. In the sensorimotor act, intention is already presupposed, as the organism reaches out to sense the world and thereby makes and comes to its senses. Intention presupposes the subject intending to reach out toward the object, that is, it presupposes a double exteriority: the world outside the subject and the presence of a future state of the material body (i.e., a re-presentation). But this intentionality cannot begin until after the dawn of the day; intentionality presupposes the “I can” underlying the transitive relation between the subject and the object it reaches out for. Intentionality therefore is always too late when it comes to the ipseity of the subject. But auto-affection occurs without and prior to any representation of the world, prior to any ipseity of the subject, and, therefore, prior to any intentionality. Life is given to us. Being-there is essentially Being-with, which means that community is not the result of an effort of individuals to bind together in and as a community but that the community is an essential condition for selfhood and intentionality to emerge. The origin of humans “is nothing other than the affective, sensegiving relation of our animal forebears, in the first instance, toward one another” (Mikhailov, 2001, p. 26, original emphasis). The ego and its other, the alter ego, have a shared essence and common birth. We all experience the same pathos, which is a pathos-with, an essential passivity whereby life, intentionality, and thought are given to rather than intended by us.

THE LIVING BODY AND THE SOCIAL SPACE

Phenomenological Sociology, Sociological Phenomenology Up to this point, it may have appeared as if I were moving myself into the same cul de sac into which so many other scholars interested in epistemological questions have walked—how the individual comes to know and

72 Geometry as Objective Science in Elementary School Classrooms how this knowing relates to culture and society. There is in fact a close relation between the social and material space that comprehend an individual and the comprehension of this individual of the social and material space, who in fact contributes to constituting this space (Bourdieu, 1997). The comprehension of the social and material space initially and above all is practical (i.e., not constructed), emerging from engagement with the (material, social) beings that populate this space. “In this way, they are one and the other in life and they know it before being and knowing themselves” (Henry, 2003, p. 205). All human beings, above and before all are alive, exist in flesh and blood, and this form of existence is common to all. It leads to a material inclusion; and it is because of this “material inclusion—often unnoticed and suppressed—and what follows from it, that is, the incorporation of social structures in the form of dispositional structures . . . that I acquire a practical knowledge and mastery of the encompassing space” (Bourdieu, 1997, p. 157). It is therefore unlikely that Chris would come to rediscover geometry entirely on his own. Chris does not live in ancient pre-Euclidean Greece and prior to the emergence of cubes, spheres, cones, cylinders, or pyramids; rather, he lives in a world where the idealities of geometry are embodied in the objects and talk that surround him and are integrated in the existing signifying relations. He therefore comes to know the cube as cube as a result of the multiple relations in which interaction rituals make specific structures come to light and become salient. Salience is not something naturally given, but is a result of interactions, of which the distinction between the individual and the world is another one of the results. That is, material space becomes salient within social space, which is both a condition and a result of interactions in which each human being participates. The lived/living body is the principle of collectivization: It “is subject to a process of socialization of which individuation is itself the product, the singularity of the ‘me’ being fashioned in and by social relations“ (p. 161). The lived/living body is subject to such a process, because it has “the (biological) property of being open to the world, thus exposed to the world, and, thereby, susceptible to be conditioned by the world” (p. 161). As a result, it comes to be “shaped by the material and cultural conditions of existence in which it is placed from the beginning” (p. 161). That this is not recognized is a problem that arises when the lived/living body is treated merely as an exteriorized material thing among other exteriorized material things, leading to the perception that the human (social) sciences can be transformed into natural sciences. But there is a tight integration between the two forms of spaces, as “social space tends to be translated into physical space” (p. 161), and, conversely, physical space is translated into social space (e.g., physical space in church, courts of law, school, and so on are the result of and resources in the exercise of power). In a MerleauPontian gesture, Bourdieu notes that the “world is comprehensible, immediately endowed with sense, because the body, which, because of its senses and its brain, has the capacity to be present at the outside of itself” (p. 163).

Material Life as the Organizing Principle of Knowing 73 It is exteriorized, outside of itself precisely with the re-presentations, which constitute exteriority as such. This exteriority is the origin of the inside/ outside separation, the origin of the world and “me,” a situation where being and the world exist and stand in a being-in-the-world relation that Heidegger used as his starting point. From the contact to the world, which constitutes exposure to contamination and contagion, the body comes to be impressed and durably modified. From the exposure to regularities—any social regularity, such as speech, itself being associated with a material regularity (sound)—that are engaged by means of corporeal knowledge arises a practical comprehension that is very different from “the intentional act of conscious decoding that is normally designated by the idea of comprehension” (p. 163). To understand the practical comprehension of the world, one has to abandon mentalism and intellectualism—without abandoning the possibility that there can be nonconceptual forms of organization in which language does not intervene (e.g., ability to form grammatically correct sentences without knowledge of formal grammar). To comprehend practical comprehension, we need to pursue alternatives to “mechanical materialism” and “idealist constructivism.” These alternatives have to allow for generative agency, but this agency is that of a “socialized body, which invests in its practice socially constructed organizational principles that have been acquired in the course of a socially situated and dated experience” (p. 164). “The practical sense is that which allows to act as required . . . without posing or executing a ‘you must,’ a rule of conduct” (p. 166). It constitutes a practical sense inhabited by the world it inhabits. Habitus, a system of structured structuring dispositions, allows us to recognize a situation as making sense based on a practical, corporeal sense of anticipation of the tendencies of the field. In contrast to the systemic approaches in sociology, which tend to decompose moments of society into structural, interacting entities, Bourdieu uses the concept of the field. It corresponds to habitus, with which it stands in a constitutive relation: The field shapes habitus but habitus recognizes and produces the field. “In a field, agents and institutions constantly struggle, according to the regularities and the rules constitutive of this space of play” (Bourdieu & Wacquant, 1992, p. 102). Habitus, the system of dispositions, emerges because of the exposition that characterizes our situation in the world generally and the fields in which participate more specifically. The practical knowledge arises not from an external relation to the world but exists in the mutually constitutive relation between field and habitus. That is, the mathematical dispositions that the students in the class I researched, such as Chris, develop therefore are an outcome of the interaction rituals that constitute the field of his classroom and school. These social relations are the origin of any higher-order cognitive functions Chris, as any other child, will develop (Vygotsky, 1978). In Chapter 7, I show how periodic phenomena such as pitch and rhythm constitute the phenomena on which the ritualistic dimension of social interactions is built.

74 Geometry as Objective Science in Elementary School Classrooms Pizza box and cube are not just material things out there; they are integral aspects of the patterns in a human society. At fi rst, there is a practical comprehension, and it is this practical comprehension that we can see in the episode presented in Chapter 1. Chris moves his left hand along one of the sides of the pizza box, then along its other side, neither one of which is square. He subsequently takes the cube and points to its different faces, while saying that all of them are square. Even if he does not yet verbally articulate a response in some abstract way, for example, by saying “to get a cube, you need to stack a few pizza boxes,” we see—in the series of actions that point to the non-square sides of one object followed by the comment—that the cube consists of square faces. The practical sense of the habitus is the principle of practical comprehension. It guarantees an immediate relation of engagement, by means of the senses, which gives sense to the world. In everyday parlance, many expressions refer to this alignment between what a person does and the world, including “to have a sense of the game” or “to be in tune with the situation/others.” In the same way that intentionality emerges from the “I can” of the motor system of the flesh, sense and collective intentionality emerge from the way in which habitus can anticipate the changes in the field of which it is an integral part. There is common sense because we have senses in common, because we are flesh before we are bodies.

Practical Comprehension, Habitus The intellectualist perspective is inseparable from the dualism of body and soul, mind and matter. Its origin comes from the quasi-anatomical, scholastic view of the body in its exteriority. The problem is that the body comes to be thought as driven by an intellectual understanding—and this problem is not eliminated at least in the educational applications of the enactivist approach—rather than by forms of knowing that are proper to it. When we theorize the body as the locus of analysis, then the problem is that it becomes a body among other bodies, isolated from its environment, driven by a mind that is in control over the body. Enactivism does not preclude this reading of the mind as being in control over the body while enacting mathematical concepts. There are two issues central to my approach, thematic in my retheorizing knowing in terms of the flesh. First, the material phenomenological approach gives (non-linguistic) reason to the flesh, which knowingly enacts recognizable social practices without requiring formal knowledge of the social practices. Second, because the flesh is outside of itself, open and exposed to the world, it has the capacity to be fashioned by the material and societal conditions of the world. It is precisely in this capacity to be fashioned all the while fashioning the world that there is a double form of learning that embodiment, enactivist, and (radical, social) constructivist approaches do not currently make thematic. It is not that the individual subject enacts its own world: Because the flesh is in relation with

Material Life as the Organizing Principle of Knowing 75 the world, in the various societal spaces that constitute structured fields, it enacts a social rather than a singular world.8 The world is immediately comprehensible—by a practical rather than intellectual understanding of how the world works—because it “has the capacity to be outside of itself with its senses and mind” (Bourdieu, 1997, p. 163). It is not just that the sense “acquires” somehow “information” about the outside world; but rather, acting in and enacting the world leaves lasting traces in the flesh—immemorial memory—that generates the movements. That is, as we have seen from the phenomenology of Maine de Biran, simply acting auto-affects the movement-producing organism, which gives an immemorial memory. Movements, however, are not just produced by the organism but are subject to the current conditions on both sides of any borderline we may draw between inside and outside.9 The resulting traces, the immemorial memory, therefore is marked both by the reigning conditions as much as by the history of the subject (“I can”). Because the flesh is outside of itself, it has the capacity to be durably fashioned, not just at random, but by the (societal, material) structures of the field in which it participates. In such participation, the subject acquires a system of disposition, a habitus, which allows it to practically anticipate the regularities of the field. Thus, students and teachers in my mathematics classroom learn to anticipate what will happen even though they may have no intellectual knowledge of their practical understanding. Acting in societal settings, fields, is possible because of a “bodily knowing (connaisance par corps) that assures it a practical comprehension of the world completely different from the intentional act of conscious deciphering that is normally associated with the idea of comprehension” (p. 163). We might therefore understand the relationship of habitus and field in a dialectical manner: exposition (to the world) leads to disposition and disposition leads to (specific forms of) exposition. To comprehend practical comprehension, we have to place ourselves outside intellectualism and its alternative of thing and consciousness. We have to get “outside of mechanical materialism and idealist constructivism; that is to say, more precisely, we have to get rid of mentalism and intellectualism that conceived the practical relation to the world as a ‘perception’ and this perception as a ‘mental synthesis’” (Bourdieu, 1997, p. 163). This requires us to build a materialist theory that is capable of recognizing the active aspect of practical comprehension. The body that is fashioned and learns, but it does not do so on its own, with the subjectivist penchant that enactivist and embodiment theories tend to take; the experiences are that of a flesh, which from the very beginning is socialized and socializes itself. When the children in the second-grade mathematics classroom make sense, then they are not individualist constructors of subjective knowledge; they are fashioned by the world they inhabit and become fashioned in acting. This fashioning occurs in interaction rituals and their characteristic periodicities—which, as I show in Chapter 7, include rhythm, pitch, and complementary occupation of space.

76 Geometry as Objective Science in Elementary School Classrooms The concept of habitus was created to overcome the reductionisms in other approaches. This habitus is a cultural-historical set of dispositions that shape how/what we perceive and enact. These dispositions, therefore, are structuring, allowing us to produce patterned actions and recognize in certain ways; and they are structured because they have been shaped in and by previous participation in interaction rituals. Habitus allows us to act without reflecting, because it spontaneously produces actions, based on the immemorial knowing that exists in the flesh. Thus, the flesh can act without express intention and calculation, appropriate to the issues at hand. Because of this, the agent never is completely (conscious) subject of his/her actions. The practical sense therefore does not require reflection to produce actions and exhibit practical understanding that only appear to follow rules, when in fact they exhibit properties of the field and the associated habitus. The practical sense is therefore capable of “quasi-bodily anticipation of the tendencies immanent to the field and the conduct of all the isomorphic habitus, like in a well-trained team or in an orchestra, they are in communication with” (Bourdieu, 1997, pp. 166–167, emphasis added). The societal order inscribes itself in the flesh, more or less dramatically, together with an affectivity that is ever present while participating in the social world. For Bourdieu, “[t]he most serious social injunctions are not addressed to intellect but to the body, treated as a memory pad” (p. 169). The living body, because it is in contact with the world and therefore exposed, is disciplined physically, leading to a discipline of the mind. The living body is fashioned somatically and psychically in the same act so that corporeal discipline comes to be lived as mental discipline. “This psychosomatic action is often exerted through emotion and suffering, psychological or even physical” and can be found “as much in everyday pedagogic action (‘sit up straight,’ ‘hold your knife in your right hand’) as in rites of institution” (Bourdieu, 1997, p. 169). As I show in Chapter 7, such injunctions, too, are integral aspects of the mathematics classroom in which Chris and his peers come to act geometrically.

CULTURAL-HISTORICAL ACTIVITY THEORY AND THE SENSUALITY OF LIFE At this point, before closing out this chapter, I cannot but briefly articulate the affi nities between the approach articulated in this book and the one that has been worked out in Marxist psychology. In fact, throughout his work, Henry (e.g., 2004) points to the affi nities between material phenomenology, which is concerned with understanding life, and the theories of Marx, who thinks of “life as having productive powers all the while it is pathetic, one can say that it experiences itself continually and never exits from this condition” (p. 296). Material activity and material relations with other people constitute, for both Marx and Henry, the real language of

Material Life as the Organizing Principle of Knowing 77 life; and without this language of life there would be no other language. As material phenomenology, Marxist psychology places primacy on knowing as arising from the sensuous aspects of life generally and of labor specifically. The material aspects of life come to be reflected/refracted in consciousness (Leontjew, 1982). Every action, every operation executed by the individual who participates in collectively motivated activity changes the person. This is so because every operation and every action of the body requires energy, which derives from the material resources of the body, and therefore expends the body. This expenditure changes the body; and the changes leave traces—in the material body, consciousness, emotion, and so on. In Das Kapital (Marx & Engels, 1973), the authors explicitly articulate the distinction between body and flesh in their discussion of the distinction between commodities and commodity owners. The latter, in addition to having bodies, also have the five (or more) senses and a soul. In labor, the flesh comes to be fashioned; and only flesh is capable of labor. Bodies are exchangeable, are dead, and may be commodities; flesh cannot be exchanged (Marx & Engels, 1983). Human beings become exchangeable precisely when they are reduced to bodies that can be traded, like slaves. The working subject, the individual, not only is organic flesh but discovers itself as inorganic body: “It is not only the organic flesh, but also the inorganic nature as subject” (p. 396). This living body is continuously reproduced and developed and its relation to the inorganic body is of double, that is, subjective and objective nature. From this perspective, we do not just see Chris exhibit and express knowing, but, more importantly, he is transformed simultaneously. Expression of knowing is transformation, and, therefore, learning. At fi rst, such a statement may surprise. But, upon reflection, it is not so surprising at all. We all know how attempting to express something to a colleague or a student, we actually come to better understand what we attempt to express or teach. How can this be the case? Precisely because expression (reproduction) and transformation, from the dialectical materialist position that Marx and Engels developed, stand in a mutually constitutive (dialectical) relation. This is the very position Bakhtine [Volochinov] (1977) takes on language, which is transformed every time a word is used. That is, word use means transformation of language, and therefore, transformation of language users—“it is the expression that organizes the mental activity” (p. 123). A living language is a language that changes because it is living in use; and a language that nobody speaks is dead and remains the same. Dialectical materialist approaches theorize knowledge as the result of labor specifically and practical engagement with the world more generally. First there is praxis; then there is awareness; and fi nally there are theory and theorizing thought: “The production of ideas, conceptions, of consciousness initially is immediately interlaced with material activity and the material intercourse of man, language of real life” (Marx & Engels, 1969, p. 26). Consciousness is reflection of material life and world: “consciousness

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cannot be something else than conscious being, the being of humans is their real life process” (p. 26). Russian psychology (e.g., Vygotsky, 1989) has taken on these ideas, fundamentally and explicitly following Marx and Engels in grounding the emergence of language and consciousness in the need for communication: “Language comes into existence, as does consciousness, from the need, the genuine necessity of the interrelation with other human beings” (Marx & Engels, 1969, p. 30). The form of consciousness at stake, here, is linguistically mediated consciousness rather than the other forms that I also bring into my argument, especially in Chapter 7, such as rhythmic consciousness. Because of his lack of words, we cannot ascribe to Chris in Fragment 1.1 a linguistic consciousness and linguistic forms of knowing. In contrast, perceptual and rhythmic consciousness may be identified in his performance. In dialectical materialist psychology, ideas (thoughts) do not come readymade but grow as a person engages in speaking (Vygotsky, 1986). Speaking and thinking are theorized as two processes that do not determine each other but that develop together and in relation to each other, perhaps more like two branches of a kiwi vine that intertwine and mutually stabilize each other as they extend further and further away from the trunk. This approach was extended to include gesticulations, which not only stand in relation to the verbal productions but also to thought more generally (McNeill, 2002). McNeill explicitly grounds his work in both Vygotsky and Merleau-Ponty, because both suggest that thought and expression constitute an irreducible unit of which gesture and language are moments, that is, external aspects that one-sidedly capture the unit as a whole. Importantly, there is a higherorder unit that sublates verbal and gestural productions and thought—a unit that Vygotsky has called “word meaning.” In my research laboratory, we have extended this approach further to communicative production more generally, including, in addition to words and gesticulations, prosody, body position, body orientation, perceptual markers in the setting, and so on. This Vygotskian idea therefore allows us a new appreciation of Fragment 1.1. Together with the inarticulate utterances, Chris also produces pointing and iconic gestures. But because gesticulations are produced without linguistic consciousness—linguistic consciousness deriving from words and utterances—what we see is thought in the making. This thought is not just verbal; verbal thought is but a “vanishing” moment of an integral form of knowing, of which gestures, prosody, body position and movement, and other forms of expressions are but one-sided manifestations. We may think of the thought expressed in Fragment 1.1 in terms of a seed—which, in dialectical materialist theory, represents the abstract that concretizes itself (the possibilities of its genome) in different ways depending on the context. In this analogy, the seeds constitute general possibilities. They have the same kind of relation to the fi nally articulated thought that a minute seed has to a majestic oak tree. The seed constitutes the general, because it would develop different characteristics in a different context because of

Material Life as the Organizing Principle of Knowing 79 gene–environment interactions. Vygotsky (1986) suggests that thought and expression constitute environments for each other, each mediating the development of the other. Words and gesture stand in a similar relation, each being the expression of a different form of consciousness: linguisticintellectual and perceptive. Because of this, we cannot tell what thought is by using words alone, because any elaboration would be falsifying what is happening at the moment. We do see, for example, pointing gestures and movements along the edges of the pizza box. In turn 22e, then, the utterance of the sound-word that we hear as “square,” and a pointing roughly to one of the narrow sides of the pizza box already followed along in a previous gesture. Then Chris utters “an like,” associated with a gesture to the other narrow side of the box (turn 22f). To understand the earliest point that the thought may have articulated itself in very undeveloped form, McNeill (2002) suggests reading the event backward from the ultimate idea articulated. One can then see certain markers characteristic of the fi nal production that have appeared before. These repetitions are referred to as catchments. Consistent with philosophical theories on repetition worked out throughout the 19th and 20th century (Nietzsche, Bakhtin, Derrida, Deleuze), these repetitions are not understood as identities, but as different, because their function is different. Catchments are markers of an idea under continual transformation from abstract to concrete and that can be identified by going backwards until the point where they appeared to have originated (at least heuristically); this beginning instant is denoted by the term growth point. It is where the idea begins in the most general but also most undeveloped manner. If we proceed in this manner, then the idea of the squares on all sides is available in Chris’s production, though unarticulated, in the initial gestures where he moves his hand alongside the pizza box. However, these gestures are only potentialities that arise from a form of understanding totally prior to all representation. They go “deep” to where there are no image schema, no representations, but only the memory of a living/lived body itself. We may ask, what might it be that allows the gesture along the edges of the pizza box (turn 22c, from b to c, and 22d, from a to b)? Of course, some readers might say, this is because of hand-eye coordination that allows the subject of action to move the hand along a line visible to him. But there is more to it. One part lies in the capacity to see the edge as an edge, which itself is the result of experiences children have very early in their lives. This perception is enabled by—and itself enables—the movement of the hand, which is a movement in what comes to be space. Perception of the line is dependent on the ability to move along the line, initially with the hand and later on merely by means of the movements of the eye (muscles). Perception and self-movement are but two sides of the same coin and mutually presuppose each other. But even this cannot be the end of the analysis, for the hand or eye cannot have an intention to move unless there is a form of memory that has no form of representation other than the self-affection

80 Geometry as Objective Science in Elementary School Classrooms in and of the movement itself. The work of Meshcheryakov (1979), who trained deaf-blind children so that they could function as part of society, allows us to understand the emergence and growth of many of the characteristics that make us human. Deaf-blind children do not automatically or innately explore their environment as other children. They do not have a propensity to find out and explore the world. They exist, as Meshcheryakov reports, in a vegetative state: In the early literature on deaf-blind children they are often referred to as (mobile) vegetables. Meshcheryakov shows in his work with deaf-blind children that orientational investigatory behavior and intention are not innate but emerge as actions directed toward knowledge of some object that has already been part of previous practical activity. The association between feeling and acting has to be developed in the deaf-blind children in an active manner. For example, a teaspoon is used to feed a child. The first spoonful of food is poured into the mouth while the child is entirely passive. But the child has to grab hold of the food on the second spoon with the top teeth and lip so that some food will stay in the mouth when the spoon is retrieved. In this way, the child is able to learn the association between moving parts of her mouth, the feeling of food, and the subsequent “reward” derived from swallowing and digestion. To learn to eat with the spoon requires not only feeding with the spoon but also additional help whereby an adult guides the child in exploring the spoon that has been feeding it. Meshcheryakov also writes about a girl who has to be taught to stand up when already over fouryears-old. The training begins by allowing her to associate the feeling of pressure on her soles—a nurse pushing—and slowly being lifted. It is with the feeling of constant pressure and support on the soles that the child first begins to unbend her legs. The child learns to associate the pressure on her soles with her own muscular movements. In this experience, the Russian psychologists has achieved precisely what Merleau-Ponty (1964b) derives from his phenomenological consideration: If the eyes cannot see the body and if the hands are not touching the body as body—i.e., if there were no self-reflexive, living body—then there is no experience of a human body and there is no humanity. According to Meshcheryakov, the children he has worked with have become human only when this self-reflexivity was enabled and the children felt their actions (e.g., moving lip, the seizing, feeling of food) and the results thereof (e.g., the food in their mouth). In sum, therefore, cultural-historical activity theory constitutes an approach that is consistent with the material phenomenological analysis of knowing in the flesh, which is a modality of life. However, it has not been articulated as a theory of the flesh and how consciousness of the body as the seat of the subject is itself the outcome of the self-affection of the flesh— though, as stated, the idea of labor transforming the flesh is immanent in the materialist dialectical approach.

Part B

Stories of Mathematics in the Flesh

Introduction to Part B The body is a natural higher mathematician. All that mind does is give expression to the body’s capacities. . . . Dividing in half, in the manner of the symmetrical body’s ways, is part of the calculus, topology, trigonometry, and algebra my body does. Mathematics is perhaps the purest form of human thought, not because its pictures struggle toward a perfect concept of nature, but because its pictures have their origin in the ways of the body in its most elemental capacities. A descriptive foundation for mathematics demands . . . studies of the embodied calculus of such accomplishments as reaching for a doorknob and getting there on time. (Sudnow, 1979, pp. 78–79)

In this quote that opens Part B of this book, Sudnow articulates a relationship between (the English) body and mathematics based on his own experiences as a jazz musician and music teacher. It is an articulation that in a striking way reminds us of Marx’s and Henry’s language of life, which constitutes the necessary condition for any other form of language. In this way, Sudnow expresses an intuition that philosophers have articulated since Greek antiquity about the relationship between mathematics and the soul (psyche, mind), one that the German philosopher Gottfried Leibniz captured in a quote provided in Chapter 7 between the soul’s unconscious counting through music.1 Sudnow’s articulation is ambivalent, though, because we may read it from, and consistent with, an enactivist perspective on mathematical knowing, where doing mathematics is likened to the competencies underlying the reaching for a doorknob or getting to some place on time. The quote can also be read from, and consistent with, a constructivist perspective, whereby the bodily knowing is but a steppingstone to a metaphysical mind that somehow transcends the confines of the material body. Thus, mind is said to give expression to the body’s capacities, which allows mind to develop and then function independently of the capacities from which it derives. Moreover, the body is described as the origin of mathematical pictures, which can be read from a Piagetian position in which those material origins are left behind as soon as the mind has reached the formal operational level. Embodiment theorists also may read this quote from, and consistent with, their perspectives: the body originates the pictures (images) that are transformed by metaphors and metonymies into linguistic forms of the verbal mind. In and with this part of the book, however, I provide materials that allow us to read this quote in a different way, one that is consistent with

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material and sociological phenomenology, that is, with the idea of mathematics in the flesh. In this second part of the book, I exhibit aspects of knowing and learning that are not theorized in other approaches. First, the episodes mobilized across the four chapters—and culminating with the analyses featured in Chapter 7—are inconsistent with constructivist approaches because they all exhibit the irreducibility of living/lived body and the metaphysical mind. There is no mind that directs the body. There is no mind the contents of which come to be constructed, or contents that are delivered to others by means of speech. My approach has affi nities with embodiment and enactivist theories. However, as we read through this Part B and to Chapter 7, it becomes evident that mathematics remains in the flesh and is not transcended (i.e., “constructed” to constitute abstract mind) to become independent of the body, which serves merely as a means and origin to arrive at a metaphysical mind. More akin to the enactivist approach, with which my approach of mathematics in the flesh shares the most similarities, I emphasize that mathematics only exists in and through performances in which the difference between the living/lived body—i.e., the flesh—and the metaphysical mind is undecidable. That is, we cannot talk about the mathematical mind without talking about the mathematical body, and we cannot talk about the mathematical body without also talking about the mathematical mind. Any expression is material, any communicative expression is produced by the material body, including the body of sound that we hear as a word, such as the sounds that the conventions of the International Phonetics Association would make us transcribe as “kju:b” (cube), “skwɛǝ” (square), or “rektæ'ŋgjulǝ 'priz(ǝ)m” (rectangular prism). To explain why the body produces sounds that are related to the metaphysical, the two need to be intimately connected, exist in an inner rather than external relation that opposes the body and the mind. This inner relation exists when the difference between body and mind is undecidable, that is, in other words, when we cannot say while analyzing any particular situation of mathematical knowing whether it is due the body or due to the mind. It is at this point, then, that mind and body are irreducible modalities, externalities, and expressions of life. Though it is like enactivism in many ways, my approach differs from it in two significant dimensions. First, in enactivism, the intention—which underlies movement for the purpose of changing, sensing (getting to know), or signifying (symbolizing) the world—is assumed and its origin remains unquestioned. Only a material phenomenological approach theorizes this origin, which comes from an auto-affection of fi rst, cumbersome movements. This auto-affection, as the root word affect suggests, has an important passive aspect that is not theorized in/by enactivism. Only the flesh, capable of action and passion, has the capacity to self-affect; and this selfaffection is the source of the living body as a body.2 This same self-affection underlies the rhythmic phenomena that I exhibit in Chapter 7 and that structure anything that is produced in and through the living body,

Introduction to Part B

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including its expressions. That is, rhythmic consciousness, which lies at the borderline of action and passion, is a phenomenon of the flesh rather than that of the enactivist body, which is already structured by sensorimotor schemas, and, therefore, according to a logos. Rhythmic consciousness is important because it underlies sociality, the socialization of bodies, because the flesh, enabling tact, underlies contact, contamination, and contagion. Second, bodily expression is thought not an afterthought (e.g., SheetsJonstone, 2009). It is thought not an expression of thought, that is, a thought represented in different bodily forms (expressions). But this thought cannot be reduced to verbal knowing and linguistic consciousness, which only constitute external moments of the phenomenon as a whole. The second difference with enactivist literature comes from my emphasis on the living/lived body, which is not the result of individual constructions, movements, or sensorimotor actions. Rather, it is a social/socialized body, one that comes to be shaped in and through participation in collective events, that is, in and through participation in society, a social body (corps social). Chapter 4 gives several examples of episodes where the children’s actions initially do not conform to what the adults expect. More importantly, adults often do not realize that what is apparent to the children is very different from what is apparent to themselves. Yet it is precisely out of such experiences that children come to reason geometrically. Thus, for example, I present in Chapter 5 an extensive analysis of a lesson fragment that leads to the emergence of coordination between the sense of touch of one hand, and the sense of touch of the other hand that is working in concert with vision. More specifically, the lesson fragment captures the activity in which children model a mystery object hidden in a box, where they can only touch but not see it. Based on their sense of touch, children are asked to produce a model from modeling clay. In Chapter 6, I provide an analysis of how measurement, an important component of geometry emerges from hand movements that initially served to manipulate and shape some object, then take on epistemic function (i.e., to fi nd out about the object), and fi nally exhibit symbolic function. Once such hand gestures have obtained symbolic function, they can be combined with other symbolic functions, such as counting, which allows measurement, the repeated application of a standard, to emerge. I show how the social constraints, which require students to be accountable through the accounts of what they do, lead them to counting as a way of doing geometry in a more unambiguous manner. Finally, I exhibit in Chapter 7 how rhythmic phenomena entrain specific performances of others so that the collective event constrains the movements of each constitutive member. That is, any expression, subject as it is to social constraints of communication, is a social rather than an individual expression. Bodily expressions are understood because the living/lived body producing them is a socialized body. That is, even if new movements, gestures, rhythms were to emerge, they would be subject to the phenomenon

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of entrainment and thereby come to be subject to and shaping collective rhythms (knowing). We see such beginnings in Chapter 4, where we can see some of the bodily expressions that are unexpected and inconsistent within the categories of mathematical expressions—e.g., a pointing fi nger gesture through the face of a cube when asked to show a straight edge. I exhibit the “forces” to which such expressions are subjected, which lead to changes in these expressions. But it is only in Chapter 7 that I articulate the theoretical concept of entrainment, in the appropriate context of rhythmic phenomena, which allows us to understand mathematics in the flesh to be a social phenomenon, characteristic of living/lived individual and social bodies.

4

The Flesh, Distractions, and Mathematics

In this chapter I focus on lesson fragments during which children come to articulate forms of knowing that are inconsistent with the mathematical ideas that the two teachers present in the classroom intend to foster. I do this not to denigrate what children know but rather to articulate this knowing as the very ground and material for all mathematical knowing that subsequently emerges. In their everyday of knowing, we actually fi nd the very methods that allow any higher form of knowing to emerge in and through interactions with other members of society (their peers, teachers). In this, therefore, I radically depart from the approach expressed in Piaget’s work and in conceptual change research, both of which take deficit perspectives. For example, throughout Piaget’s work we can fi nd characterizations of children’s where explanations such as “lack of exploration,” “a general defi ciency,” “still passive,” “cannot analyze,” and “is incapable of abstraction” abound. In the constructivist conceptual change literature, we can fi nd expressions such as “eradicating” children’s ideas and using pedagogical strategies so that they “abandon” what they know. Piaget and others regard children through the lens of adult scientifi c rationality that is the ultimate goal of a person’s development. A very different approach to understanding children comes to us from a book on the talk’s body, which really is the body’s talk: “The child understands the talking around him in the way he understands it. The child’s understanding is not imperfect” (Sudnow, 1979, p. 39). The author continues to suggest that if the child’s understanding were imperfect, then so would be that of amateur attendees in jazz clubs, and, by extension, those of most concertgoers, spectators of sports, and every other event where some audience follows the play of professionals. Giving reason to the child also means giving (a very different form of) reason to its organic, living/lived body, or rather, the flesh from which its body springs forth. If we stand back and look at the events in Chris’s classroom, we may learn how much Piaget perhaps misunderstood children because, as many commentators have pointed out over the years, he gazed at their

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expressions through the lens of mature science and mathematics. Thus, when we look closely at many lesson fragments in my database, we get the impression that children actually do not hear the adults in the way they intend to be heard—which requires an adult rationality. At the same time, the adults appear not to listen to what children have and intend to say but hear them through their own intentions and adult forms of rationality. But the teachers and students in this classroom use one and the same language, one and the same English, which they make available to each other. There is a form of monolingualism at work, one that is of the other, whose ear determines what is heard.1 Yet somehow they pull off these lessons by mobilizing, to paraphrase Garfi nkel (1967), their everyday activity of doing school mathematics as their method for making this activity accountable to each other in observable (visibly, hearably) rational and reportable (for all practical purposes) as the organization of school mathematics activities. It is this mutually observable and accountable nature of the joint activity that allows the ultimate convergence of the different ways of hearing and speaking. We know that those listening to talk tend to hear through and with respect to their own rationality even if, as in the case of the computer program ELIZA, the current “speaker” simply reflects back aspects of the previous speaker’s utterance. Garfi nkel shows that listeners make sense of answers to their questions even if these answers have been produced at random without any underlying rationality (i.e., “rhyme or reason”). 2 The child develops because it experiences the world and things “as the jointly sensed and jointly thought attention of adults to himself, and then as the attention of his peers, of books, or of television to him” (Mikhailov, 2001, p. 27).

ABOUT SALIENCE IN CHILDREN’S AND ADULT’S OBSERVATIONS In this episode, I show how teachers and students have conversations where hearings diverge and yet conversations come about. As a result, there may be contradictions—as different observers of the fragment involving Chris have suggested to me—which dissolve when we understand the participants to be talking about different things yet hear the other as speaking about the same thing that is salient to them. That is, very different objects are salient to Mrs. Turner and Chris, and what each hears is consistent with what the hearer rather than the speaker attends to. Yet what matters for understanding the unfolding event is just this: how participants hear others, how they re/act, and the enchainment of their contributions in their common turn-taking ritual. Each tends to hear what the other says in terms of their own horizon, the objects and events that appear within it. In the course of presenting this case, I articulate further to the perspective on mathematics in the flesh.

The Flesh, Distractions, and Mathematics 89 This fragment immediately precedes the one featured in Chapter 1—and that I discuss throughout Part A of this book. We may gloss the fragment in this way. In response to a question about additional things to be said about faces (of geometric objects), Chris not only asks for a turn at talk but also for the permission to walk to the front of the class. After arriving there, he talks, in reference to a pizza box, about rectangular prisms that have different sides, some longer and other shorter. He describes the pizza box as a flat cube. Rather than hearing him talk about the sides, Mrs. Turner responds to his description of a flat cube. But Chris returns to the description of the different sides, only for Mrs. Turner to return to the pizza box as rectangular prism and it not being a cube. That is, in this instance there appear to exist different hearings, which exhibit the different concerns of the participants (Mrs. Turner, Chris). However, the observation of the child arises directly from the perceptual appearances of the sides of the box, which come to be marked in and through the performative of the iconicdeictic gestures. When Chris arrives at the chalkboard, he picks up a pizza box resting on a storage shelf, and utters, “I think this one’s more like a rectangular prism” (turn 05). He then explains, “because this one’s longer” while moving his left hand along one of the sides of the box. He then shifts the box into his left hand and, while moving his right hand along the pizza box face orthogonal to the fi rst (turn 05), continues his explanation: “like longer than this one, this one’s” (turn 05). Mrs. Turner produces an interjection of acceptance acknowledgment (“uh hm”), which gives Chris the space to continue. He does so after a brief pause. Fragment 4.1 00 01 02 03

T: T: C: T:

what ELSe about faces can we describe. chrIS? i=m gonna tell you this. ((he gets up and walks to front)) ^OHkay:. ((lifts arm and shoulder, as if saying, “what can you do about this”)) 04 (5.29) ((Chris walks to the front)) 05 C: um. (0.87) ((arrives at board)) i thINk this ones ((picks up pizza box)) more like a rectangular prism, because () this ones ((hand moves along one edge, Fig.)) like lon: ger than this one; this ones: ? ((rH moves down, up, down right face, Fig.))

90 Geometry as Objective Science in Elementary School Classrooms 06 T: uh hm 07 (0.44) 08 C: ((places hand on top of pizza box)) ((looks up, facing Mrs. Turner, then puts box down)) 09 (3.18) 10 T: so the rectang()ular pr=prism might be a fl at cUBE? 11 (0.22) 12 C: um h[m ] 13 T:

[it] could be like a cube thats bin (0.22) ((hand movements showing the closing of the distance between two palms, Fig.)) flatten[ed] ((hands retract))

14 C:

[uh] hm bu:t () ((lH moves down face of box; then holds box with lH)) s:::ohm::e are like long=an some are shorter than the other one[ss] ((rH movement along right edge, then rH movement along edge pointing to Mrs. T, then again rH movement along rivght face)) ((looks up to her)) 15 T: [uh] hm 16 (4.91) ((Chris places box, no longer looks at her, busy placing the box even after she has started again)) 17 T: so makes it a rectangular prism as opposed to a cUBe,

Chris continues, at a pace slower than normal, drawing out each phoneme produced, “it’s like a fl at cube” (turn 08). Simultaneously, he places his hand on the square cover of the pizza box. He then reaches out to return the pizza box to its place. There is an extended pause (3.18 seconds) before Mrs. Turner takes the next turn at talk, uttering, with some stuttering and repeats, “So the rectangular prism might be a fl at cube?” (turn 10). Chris, who has picked up the pizza box again, interjects an “um hm.” This allows Mrs. Turner to continue, which she does before Chris has completed. She suggests, “It could be like a cube that has been fl attened,” while producing a two-handed gesture of two palms moving against each other, as if compressing an object between the two hands, which comes to be fl attened in the process (turn 05). Chris, overlapping Mrs. Turner in the way she has done with him, sets up a contrast: “hm but.” At this time, his left hand moves along the left narrow face of

The Flesh, Distractions, and Mathematics 91 the box, which it grabs when it reaches the bottom end. Then, following some initial hesitation, he provides another description of entities (“some”) that are shorter than other ones. While doing so, the right hand fi rst moves down the right narrow face of the pizza box, then moves toward Mrs. Turner to produce a gesture along it from left to right. The hand comes around the edge, and just as he utters the contrast again (“than”), the right hand repeats the fi rst gesture along the right face (turn 14). Mrs. Turner produces an interjection, “uh hm,” and then she stops. A long pause develops, which Mrs. Turner brings to an end in concluding what she has heard Chris to say: “so, that makes it a rectangular prism as opposed to a cube” (turn 17). Very explicitly, therefore, Mrs. Turner tells Chris, other members of this community, and the analysts watching the tape that she hears Chris speak about the pizza box, which is a rectangular prism rather than a cube. Chris points out, however, therein contrasting her summary, that he is speaking about the fact that some faces are longer than others, and this feature makes the pizza box a rectangular prism. This particular prism therefore is more like a fl at cube. But, he says, thereby setting up the contrast with what Mrs. Turner has said, that the sides he is pointing to and gesturing along are of different length. One is longer than the other. His comment is about the faces, and the difference between associated faces, has as its consequence a different overall shape. He contrasts his description to the one she provides, and thereby articulates a disagreement, which is enacted in the verbal and manual gestures. In this fragment, Chris tells Mrs. Turner and his class—and, vicariously, the researchers overhearing the talk—something about faces. He does so a fi rst time, articulating that some faces are longer than others. While talking, his hands move along two of the narrow faces of the pizza box that are orthogonal to each other. The movements are precisely coordinated with the verbal indices “this one,” “this one,” and “this one [i]s.” There is one right-hand movement not associated with a verbal index but precedes it, thereby falling together with the comparison (turn 05). The right-hand movement up to the starting point, however, falls together with a verbal index, and there is another repetition as if wanting to produce the correct coordination between verbal and hand gesture during the second downward movement along the face. How is such coordination possible, and, how is the recognition of a discoordination possible so that it can be subsequently corrected? Such coordination would be a tremendous task for a (conscious) mind operating like a computer, because it would have to do all the calculations for the two movements and then bring about the precise timing between the two. For traditional psychology, “this unity was constructed, it referred to intelligence and memory” (Merleau-Ponty, 1996, p. 63). But the task would be a lesser challenge if the whole lived/living body itself were to be understood as an expression, not the expression of something behind

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it—e.g., some cognition happening independently of the body movement. That is, the task would be a lesser challenge if we were to consider the movement itself as the thinking and remembering (Henry, 2000; SheetsJohnstone, 2009). The lived/living body thereby comes to be an expression that expresses itself as “the subjective reality of an inner voice, born of its externalization for the other, and thus also for oneself as for the Other within oneself” (Mikhailov, 2001, p. 17, original emphasis). This body is a unit precisely when it is thought as the fl esh endowed with the capacity of tact throughout its lived/living body thought as the flesh. It is therefore not surprising when the phenomenological analysis reveals that “my perception is not the sum of visual, tactile, auditory data, I perceive in an undivided manner with my whole being, apprehend but one structure of the thing” (Merleau-Ponty, 1996, p. 63). This unity is a “unity of style, which, though it is immediately perceivable, has nothing of a strict and unsurprising coherence of the concerted products of a plan” (Bourdieu, 1980, p. 174, original emphasis). Conversely, but for the same reason, we easily detect the absence of such unity, for example, when the voices we hear do not fit to the bodies we see in dubbed movies, news reports, or the broadcast from the Canadian Parliament when the translator comes in as soon as the current speaker switches into the other national language. 3 Merleau-Ponty discusses precisely this phenomenon familiar to those of us who speak both languages, the one in which the actors speak and the one that has been dubbed in. The hand of an expert typist knows in and from the feel that the fi nger has hit the wrong letter without looking at the typed page, just as we (immanently) know from the feel of our body that we are tripping. There is more involved in Chris’s expression while the hands move along the pizza box. Because the hands move along the box, it has to be present to/for him. Chris sees the box and gestures along it. But it is not the result of an “interpretation” of the box—this again would take too much calculation. The coordination of seeing and pointing occurs at a different level. And at that level I do not have to think like a computer to produce the verbal and hand gestures. I am not the spectator of these movements: They come forth from my living/lived body. Underneath the speaking intention, with an intentional object that is yet to be developed in clarity, there is an unintended effort, an operative intention that allows the speaking and gesturing movements to begin even in the absence of a precise (or precisely formulated) object (content of talk, expression). The coordination of speech and gesture is possible because the practical competencies precede refl ective cognition—calculations and deductions— because of dispositions “inscribed as much in the postures and folds of the body . . . as in the automatisms of language and thought” (Bourdieu, 1980, pp. 176–177). The unity of tact throughout our living bodies, characteristic of the flesh, holds the different parts together. When the coordination is not achieved, a correction is produced much faster than

The Flesh, Distractions, and Mathematics 93 any analysis and mental correction could take place, “on the spot, in the twinkling of an eye, and in the heat of the moment, that is, in conditions that exclude distance, withdrawal, perspective, delay, detachment” (p. 137). The non-linguistic consciousness for and of such discoordination is possible precisely because of the self-affection of the flesh that does not need to distance itself from itself to know—just as my hand did not need my mind to remember the digits of a telephone number to call up my supervisor. The knowing in and of the lived/living body is different from the intellectual consciousness so that it is possible for a difference to emerge between our doing and our telling what we do. In this fragment, Chris and Mrs. Turner use the term “fl at cube” as a way of characterizing and categorizing a particular object in their perceptual field. It is an everyday way of talking that makes immediate sense even though it is not correct mathematically. That it makes sense can be taken from the fact that the “fl at cube” as a way of talking/thinking about some parallelepipeds was not a one-time event in this classroom. It also emerged in other lesson fragments, such as when, in Chapter 5, Melissa brings it together with a cube in her hand as a model for a fl at cube hidden from sight. It also emerged from an interaction between a group comprised of Oisin, Ben, and Ethan and Mrs. Turner (associated with a fragment in Chapter 7). This term might have become an unintended resource for future sense-making efforts, because the fl at cube is a particularization of the cube. In geometry, however, the cube is a particularization of the rectangular prism featured in this classroom specifically and the parallelepiped more generally.

DO EDGES AND VERTICES DISTINGUISH A CUBE? In the preceding section and lesson fragment, we see how the faces of a cube are features that visual-perceptually protrude in Chris’s lifeworld, as he makes them available in his responses to and for the benefit of his audience (including us). Perceptually (visually), there are aspects that stand out and that children point out as particular—even though for the adult rationality, these may be features that occur in all or most of the objects that exist in this classroom. These perceptions are not the result of “interpretations” or interpretive constructions, not in the way these concepts are used and theorized in the literature on interpretations (hermeneutics).4 Rather, these perceptions are ways in which our lifeworlds are given to each of us: In these perceptions things stand out as in a clearing. For children, what can be seen in this clearing often differs from the adult things and clearings. Each person acts towards the things that appear in their clearing. It should therefore not surprise us that the children may act in ways that are at odds with adult ways of seeing; yet their ways are intelligible as soon as we see them in terms

94 Geometry as Objective Science in Elementary School Classrooms of the things that they perceive and bring forth in their actions. If we carefully look, we can actually fi nd out about what and how they see. Thus, even though a teacher may intend to elicit comments on how a cube distinguishes itself, as seen within her adult rationality, a child may articulate aspects salient to her that are standing out and surprising; and these aspects may perhaps be annoying to the teacher, because they are inconsistent with standard mathematics to be taught. In this section, we encounter such an instant in the second-grade mathematics classroom under investigation. The following fragment is part of a lesson in which Mrs. Turner asks students about the models that they have built of mystery objects (see also Chapters 5 and 6). These mystery objects were placed in a shoebox where they could not be seen but only be felt; the students were asked to build the models based on what they felt. To complete the task, students within a group had to agree on a model for the mystery object. The fragment is part of an episode that begins with the designation of Ben’s group (Ben, Bavneet, Joel) as the current presenters of the model they have arrived at, and ends when Mrs. Turner designates another group. Ben has been presenting the group’s case. As part of the exchange with the group, Mrs. Turner asks, while holding up the demonstration cube normally kept on the chalk tray, whether there “is something else that tells you it is a cube?” (turn 32). After a pause, Ben produces an interjection; then there is another long pause, before Ben utters “yea,” only to reverse himself following another pause, “no, ah, no” turn 38). 5 Mrs. Turner queries; Ben answers with “no”; and the teacher responds, producing a “no” with rising intonation. Here, Bavneet announces, “I know,” and, when there is a pause developing, the teacher invites her, “Yes?” (turn 46). “Yes,” Bavneet says, completing the response to Mrs. Turner’s question/invitation about anything else that tells them it is a cube. Invited, Bavneet articulates some features that stand out in her lifeworld. Fragment 4.2 (Bavneet; Ben.mov 28 29 30 31 32

33 34 35 36 37

T: b: J: T:

B: B:

are all the same? ((:b)) do you agree with that? ya. ((Mrs. T gazes at Joel)) yap. ((Mrs. T’s fi nger points to Joel)) (0.27) okay (0.19) and (1.03) anYthing else that tells you it is a cUBE? ((holds up the cube from the chalkboard)) (1.05) um (2.15) yea (0.58)

The Flesh, Distractions, and Mathematics 95 38 39 40

41 42 43 44 45 46 47 48

49 50 51 52

53

B: T:

B: T: b: T: b:

T: b:

T:

54

55

b:

56 57

(0.47) ((has begun to walk toward chalkboard, turns to Ben)) hum? ah no ((shakes head)) (0.43) no? ((picks up the cube)) i know (0.43) yes? (0.41) yes because you can feel the edges ((touches 2 vertices of cube in Mrs. Turner’s hand, Fig.)) (1.46) what ´about them. (1.10) they have vertices ((touches 3 vertices, Fig.)) =they have vertiCE:S, yes? ((Baveneet nods)) (1.08)

and (0.65) they have ((gets up, takes cube in lH, “feeling”/ “caressing” around one face, Fig.)) (2.07) (0.86)

T:

((:B)) hOW did you know; im looking at your model here. ((points to her model))

96 Geometry as Objective Science in Elementary School Classrooms “You can feel the edges” (turn 48), she says while touching two of the vertices (see drawing). With falling intonation, Mrs. Turner utters, both a question and an invitation, “what about them?” After a pause, Bavneet continues, “They have vertices” and points to three vertices (turn 52). Mrs. Turner exactly repeats the words Bavneet has used and then, with rising intonation utters, “Yes?” After another considerable pause, Bavneet begins offering a response, “and they have,” and gets up to take the cube from Mrs. Turner. Her right hand index fi nger repeatedly moves back and forth over one of the faces, almost as if she were caressing it (turn 55); she exhibits a questioning facial expression. At the end of a long silence, Bavneet responds with very low speech volume—as if she were apologizing for the contradiction to what she has stated during turn 44—“I don’t know” (turn 55). This ends the interaction abruptly, as Mrs. Turner stops this interaction by shifting to her right and by directly addressing Ben, uttering what comes to be another question. Here, Bavneet tells us that the edges stand out as something special, appropriate to the question she has heard Mrs. Turner ask. But, as apparent in the latter’s query (turn 50), Bavneet’s utterance is not a response to what Mrs. Turner has intended asking; it might be a response if Bavneet had responded to the second question offered: “what about [the edges]?” Bavneet gives it a try, “they have vertices.” In fact, she gesturally points to three of the four vertices associated with the top square, thereby also making salient two or three of the four edges. But apparently this, too, is not enough, as Mrs. Turner merely repeats the constative statement and then, with rising intonation, offers an invitation to go on, “Yes?” (turn 53). Bavneet makes another attempt, but then gives up, “I don’t know”; and Mrs. Turner, in turning to Ben, gives up pursuing any further understanding what might have led Bavneet to announce that she knows something else about the cube. The edges do stand out in visual perception, and so do the vertices. Bavneet articulates this salience in moving her arm forward until her index finger touches one or the other vertex for a total of five times. But this salience is not the one sought after, which would be the one telling everyone that the mystery object is a cube. Vertices and edges are salient; and this fact is not the one that Mrs. Turner appears to question. For, in asking “what about them?,” she actively acknowledges the presence and salience of the edges; and she does the same for the vertices, both acknowledging and asserting their presence. But despite this salience, she is pursuing one that Bavneet realizes is hidden from her: “I don’t know.” Mrs. Turner turns to Ben, and, as seen in Chapter 6, allows the bringing forth of measurement in geometry. From the normative perspectives of a person concerned with enacting curriculum, this fragment may constitute an instant of missed opportunities for working out salience and how it pertains to the lesson at hand. There was an opportunity—at least as seen from the sidelines where the analyst stands—for teasing out why edges and vertices alone do not distinguish cubes from other parallelepipeds. It might have been possible to tie

The Flesh, Distractions, and Mathematics

97

Bavneet’s articulations to the events in the group that had its turn before Ben’s, where during the modeling exercise there was a long debate about whether the mystery object is a rectangular prism or a cube (see Chapter 5). At this point, however, we do not know what is salient to Mrs. Turner while she talks to Bavneet. In the heat of the moment, she may have had a general feeling (sense) that Bavneet’s contribution is an interruption of the exchange that she has engaged in with Ben. What is salient in children’s perception is important, as it is precisely this perception that they use as a resource in making sense. What is salient is not the result of a “construction,” but rather something that is given to us—recall from Chapter 1 how our eyes do make three-dimensional cubes appear even though we are not aware of their activity. In one of my science education research studies, I had found out that the students in a 12th-grade physics course were divided—without knowing so—about whether there was movement or no movement in their teacher’s science demonstration (Roth, 2006). But those students who saw movement used the observation to make sense of the teacher’s subsequent explanations of the theory in the same way that those students who did not see movement used what they perceived to make sense of the teacher’s explanation. The result was conceptual confusion, because the same teacher explanation was heard (by some) as explaining movement and (by others) as explaining the absence of movement. Knowing what is salient for another person requires some careful attention to their expressions, and, perhaps, explicit engagement with them. As we see in the above fragment involving Chris, Mrs. Turner apparently is not aware what is salient to Chris and what he is talking about; and, in the present instance, that which is salient to Bavneet does not fit into the flow of the lesson and remains without conclusion. Salience also is the central feature in the next episode.

OF FACES AND EDGES (HOW THE EDGES OF A CUBE APPEAR) In this fragment from a lesson on edges, we notice how edges appear differently to a student than the teachers intended. Unlike what we see in the preceding sections, we observe in the present section the pedagogy that the teachers mobilize in interaction to make the features appear differently. A drawing of a pyramid with labels on the different parts—edges, vertices, and faces—prominently features at the chalkboard (Figure 4.1), just behind where Mrs. Winter has positioned herself. She interacts with one of the students, Thomas, who articulates for us how certain shapes stand out and how, to the fi nger, an edge appears triangular. They appear triangular to the fi nger, which, in moving along, can feel the edge. But Mrs. Turner and Mrs. Winter want him to see the edge as straight edge, as opposed to the one of the circular face of the cylinder, which is “curved [round].” Both teachers apparently work hard on getting something else appear for Thomas.

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Figure 4.1 A concept map of the lesson on vertices and edges is prominently displayed on a large sheet of paper taped to the chalkboard in front of the classroom.

After Mrs. Turner has solicited a vote on the nature of edges and articulated the results—“some people say all the edges are all the same” and— “edges are different”—she asks for a showing of hands of those who “had their hands up for some edges to be different from other edges” (turn 01). Thomas is one of the students who raises his hand, and, after a pause, Mrs. Turner calls on him. There is a long pause before Thomas responds to the “who” question by saying “everyone.” There is another pause, and then Mrs. Turner asks Thomas, “How would you describe this edge” while running her finger along the edge of the cube that is facing the class generally and Thomas particularly. There is yet another pause, and then, in a slow delivery, Thomas utters “well” (turn 08). Mrs. Turner reaches forward to hand the cube to Thomas, who is sitting in the first row of desks, asking him to “describe it.” Fragment 4.3a (Thomas) 01

T:

02 03 04 05

T: t:

okay (0.63) wHO had their hands up for some edges to be different from other edges. (1.81) ((Mrs. T looks around the classroom, points to Thomas)) thomas. (2.62) everyone.

The Flesh, Distractions, and Mathematics 06 07

T:

08 09

t:

10

T:

11 12 13

T:

14

t:

15 16

t:

17 18 19 20

T: W:

21

t:

22

W:

99

(2.36) thomas; hOW would you describe this edge. ((Runs her fi nger along an edge of the cube)) (1.31)

i ask you to run your finger along it; ((Mrs. T reaches the cube toward Thomas, who leans forward,)) (0.79) ((Thomas gets cube from Mrs. T)) how would you describe it. (0.58) ((Thomas intently looks at the cube, Fig.))

well (0.85) because um; (2.30) (0.39) like (2.30) ((gets up from his seat with cube in hand)) like this one. ((points to the word vertices in the diagram on chalkboard with a pyramid on which the different parts of an object Fig.)) (0.52) yES. ((Fig.) (0.79) vertices? ((she points to the top vertex in the diagram)) yea. (0.36) like this on [mine ] [but your] instead of your if you start on one verticee; put that finger on that verticee. (0.82) your finger (0.26) point (0.62) one verticee (0.48) and rUN:it along that edge to the other verticee (0.77) okay? (1.24) whats that feel like; what kind of edge is that. ((Mrs. W runs his finger along the edge))

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Thomas intently looks at the cube, which he holds in both of his hands (turn 13). His interjections alternate with pauses, which can be heard as hesitation, and, after a particularly long one, he describes, in what we can hear to be a confident delivery, that he observes “it” as “like a sort of vertex” (turn 14). He then gets up and walks toward the chalkboard to point to the word “vertices” on the parts of a geometrical object exemplified in the pyramid (turns 16, 18). From pointing to the diagram, the hand moves down to one of the vertices of the cube (turn 16) and then points back up to the “vertices” in the diagram (turn 18). Mrs. Turner responds affirmatively, and Mrs. Winter utters with rising intonation, “Vertices?,” to which Thomas provides an affirmative second turn (turn 21). He adds, “like this on mine.” However, this does not appear to be correct, as Mrs. Winter’s talk overlaps with his, beginning her turn with an adversative conjunction, “but your” (turn 22). The sense of adversity is strengthened when Mrs. Winter, in a second attempt, again marks opposition with the adverbial phrase “instead of your,” only to start a third time with an opposition-marking conditional, “if you start.” In this fragment, we observe how, after some initial hesitations, Thomas confidently articulates an observation about vertices. There is a form of generalization, whereby he categorizes the vertices of his cube in terms of the vertices on the drawing, which are vertices in general (though also the particular vertices of the figure in the drawing). Thomas talks about the vertices that stand out on his cube and then gets up and points to the term “vertices” (in the diagram), from which there are arrows to the different vertices of a pyramid. The generalization that we observe first is from the specific three-dimensional object in his hand, an imperfect cube, to a two-dimensional diagram. Second there is a generalization from a material object to an ideal object. Third, there is a generalization of a feature from the category of cubes to one that is also found in other categories. That is, in his pointing, there is a triple generalization. But this generalization meets an apparent opposition. Mrs. Winter orients toward Thomas and takes some time to grab the cube from him while she begins to talk. She asks him to put a finger on the vertex where she is placing her finger—i.e., she is placing her finger as a non-verbal instruction for where his finger is to go (turn 22, a). Thomas’s index finger hits the vertex, but as he raises his gaze to look toward the class, his hand slips off. Mrs. Winter follows with her hand and, when she has succeeded grabbing hold of the index finger (turn 22, b) utters “your finger.” She moves the finger to the vertex while uttering “point,” but the hand moves away again. Mrs. Winter pulls it back to the vertex (turn 22, c) and utters “one verticee.” She then pulls the finger along the edge to another vertex, while describing the process in words, “run it along the edge to the other verticee.” There is a pause, which she follows with an “okay?” marked by a rising intonation. During the speaking pause, she runs the finger again along the edge. There is another pause, and, while uttering with rising intonation toward the end, “What’s that feel like?,” she runs the finger along the edge for a third time. Mrs. Winter runs the finger yet another time along the edge while uttering, again with rising intonation, “What kind of edge is that?” She retracts her

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hand; and, in the pause that follows (turn 22), Thomas runs his finger twice more along the edge. An expression of consternation follows, and then, while gazing toward the floor, he scratches his head—as if he were asking, “What do you want from me?” (turn 23). Fragment 4.3b (Thomas) 22

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23 24

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25 26

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27 28 29

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[but your] instead of your if you start on one verticee; put that finger on that verticee. ((Fig. a)) (0.82) your fi nger ((grabs fi nger, Fig. b)) (0.26) point (0.62) ((holds fi nger to vertex, Fig. c)) one verticee (0.48) and rUN:it along that edge to the other verticee (0.77) okay? (1.24) whats that feel like; what kind of edge is that. ((Mrs. W runs his fi nger along the edge))

(4.22) ((scratches his head, Fig.)) ((Mrs. W gets the cylinder)) kay, () where (0.57) all right thIS time; (0.68) put your finger OUT ((Thomas puts finger out, Mrs. W takes it and places it on the edge of the cylinder)) (1.12) what does thAT one feel like. ((moves his fi nger along the edge)) (0.88) it feels like um; (4.26) ((Mrs. W moves his fi nger repeatedly around the circumference)) ((Thomas has questioning look, Fig.)) does it feel the sAME or does it feel different; (0.32) feel different. (0.93) what is different about those two edges. (0.77)

102 36

Geometry as Objective Science in Elementary School Classrooms t:

37 38

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39

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40 41 42 43

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44 45

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46 47

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48 49

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because um this one is round and this one is ap (0.48) isa square (1.63) ((Thomas looks up to Mrs. W, as if looking for confi rmation)) if you jUSt did ONE of them ((moves fi nger up and down an edge of the cube)) if you jUSt did ONE of them ((moves fi nger up and down an edge of the cube)) (0.79)((Thomas nods)) what would it (0.64) f::eel like. (1.19) feel like um this one (0.65) ((moves finger up and down the edge, Fig.)) is um () triangle, (1.83) ((Mrs. W gets a pyramid)) that would be the triangle. (0.82) triangle and (0.57) and this one feels like um; (1.14) a triangle ((moves around the triangular face of the pyramid)) (0.95) right; not using a sh:ape word. (0.22) so we=re not using a tRIangle CIRcle or squARE. ((rhythmically counting, beat gesture))

Mrs. Winter turns around to get a cylinder from the tray of the chalkboard, makes a false start (“where”), then suggests “all right this time,” and fi nally asks Thomas to put his finger out gesturing (“put out the fi nger”). As before, she captures his index fi nger and, just as she says “what does that one feel like,” moves the fi nger around the circular edge of the object (turn 26). Thomas begins a response by repeating part of Mrs. Winter’s phrase, “it feels like.” He stops; and a long pause develops while Mrs. Winter moves his fi nger twice around the circumference. Gazing emptily in front of him, Thomas has raised his hand to his chin facing the cylinder in Mrs. Winter’s hand (turn 29). He appears lost. At this point, Mrs. Turner enters the conversation, beginning a questionresponse sequence, “Does it feel the same or does it feel different?,” to which the second part comes forth almost immediately, “feel different” (turn 32). There is a pause before Mrs. Turner offers another question, “what is different about those two edges?,” to which there is a defi nite second part clearly informed by what is in front of his eyes: “this one is round and this one is a square.” During the fi rst part, Thomas follows the circumference of the cylinder with his fi ngertip, then moves it to the cube and tracks the edge around the entire top square. During the ensuing

The Flesh, Distractions, and Mathematics 103 pause, Thomas’s gaze moves upward toward Mrs. Winter’s face, as if he were looking for confi rmation. But Mrs. Winter begins with a conditional, “if you just did one of them” and moves her finger down and up one of the cube’s edges. There is a pause, and another utterance with question structure, “What would it feel like?” accompanied by her finger running down and up the edge of the cube that faces Thomas. He completes the question-response pair by describing, “feel like this one is triangle.” As the utterance unfolds, he runs his fi nger five times down and up the edge closest to him on the cube that Mrs. Winter is tending toward him (turn 43). But, while turning toward the chalkboard tray, Mrs. Winter clamps the cylinder under her left arm, picks up a tetrahedron and makes a constative statement, “that would be the triangle” (turn 45). Thomas runs his fi nger up and down the cube’s edge while uttering again, “triangle,” then turns and moves his fi nger around one of the triangular faces of the tetrahedron twice spaced by a pause, while uttering, “and this one feels like a” and “triangle.” After a pause, Mrs. Winter acknowledges, “right,” and continues, “not using a shape word.” She elaborates, rhythmically beating down with her right hand while unfolding the fi ngers as if counting from one to three while articulating the shape words, “we are not using a triangle, circle, or square” (turn 49). (I return to this instance in Chapter 7.) In this second part of the fragment, Thomas provides a second, third, and fourth response to the original question about what he feels when running his fi nger along the edge. In the first of these three attempts, he runs the index fi nger around the circumferences of the circular and square faces and tells everyone listening—after being encouraged to state the differences in the tact—that the fi rst feels round and the second square. His hand in fact describes a curve in the fi rst case, while changing direction three times in the case of the square. His words are accurate descriptions of the movements that his hand accomplishes at the same time. In the second attempt, Thomas runs his fi nger repeatedly along the edge and then describes what he feels: The edge does feel (tri-) angular. But again, Mrs. Winter does not evaluate his response positively. They begin a third attempt, in which Thomas uses the same verbal description for what he feels moving up and down the edge of the cube and when following the triangular circumference of the tetrahedron. In this situation, the verbal description is accurate and consistent with his previous articulations. The edge of the cube feels triangular, and the circumference also feels like a triangle. But again, the response apparently is not appropriate, as Mrs. Winter tells him that they are not using shape words. After Mrs. Winter states that they are not using shape words and exemplifies the kinds of words not to be used (this performance is further analyzed in Chapter 7), a pause unfolds during which she runs her index fi nger three times down and up the edge of the cube. She offers up a question, “Can someone help us out?” but there is no response. She offers up another invitation to a question-response pair, “he said this one is”

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(turn 53) leaving an answer slot in a grammatically unfi nished sentence. But again, there is a pause. Mrs. Winter produces yet another invitation, initially providing a response for her second question, “What we said, round” and then uttering, while pointing to the edge of the cube, “What would that one be?” (turn 55). She calls on Alicia, who proffers a response, “straight,” which Mrs. Winter, after a pause, follows up with an invitation for indications of agreement (“Can we agree?”). The response is silence, so Mrs. Winter continues, “this one is straight, this edge is round” while moving her fi nger fi rst along the edge of a cube and then along the edge of the cylinder. Fragment 4.3c (Thomas) 49

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50 51 52 53

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54 55

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56 57 58 59 60

61 62 63 64 65 66

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67 68

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70

right; not using a shape word. (0.22) so we are not using a triangle circle or square. ((rhythmically counting, beat gesture)) (1.93) ((Mrs. W moves fi nger 3 times up and down the edge of the cube)) can someone help us out? (0.70) he said thIS one is ((moves fi nger around the circumference of cylinder)) (1.74) ((moves to look at his face)) however what we said; rOUNd (1.12) what would this one bE:? ((feels edge of cube)) (0.35) ^alICia. straight. (1.03) can we agree? (0.44) this one is straight ((moves fi nger along edge of cube)) (0.83) thIS () edge ((moves around cylinder)) () is round. (0.44) would you agree with that thomas? (0.38) .h uh yea. (0.44) so which one is the straight (0.80) edge. (0.37) this one ((moves fi nger down the face of the cube)) is it down hERE? showm put your fi nger down straight edge. (1.51) ((Thomas moves fi nger down the middle of the cube face, Fig.))

The Flesh, Distractions, and Mathematics 105 71

W:

72 73

t: W:

74 75 76 77 78 79 80 81

T: J: W:

now when yOU=re doing this we are going right through the face. (0.68) here is our strAIGht edge ((she grabs his fi nger and moves it along the edge of the cube, Fig.)) now go through the rOUNd edge ((Thomas puts finger on the circular edge of cylinder, moves it around))

good for you. ((looks toward Cheyenne)) (1.03) cheyenne () you need to put that shirt properly ((Thomas begins to move back)) (0.70) jane nOW (0.68) um (1.19) ((bends down to ear of Thomas)) ((Thomas moves to his seat))

Mrs. Winter orients herself toward Thomas and asks him whether he can agree with the statement, which he affirms. But Mrs. Winter follows up asking him to identify the straight edge. Thomas’s hand moves forward, touches the face of the cube, and the index finger moves downward in a straight line while Thomas says, “this one.” But with a rising intonation and a grammatically formed question emphasizing a location “here,” Mrs. Winter utters, “Is it down here?” and then reiterates the request for showing a straight edge. It is not just a reiteration, but being reiterated, it also is an evaluation that the intervening response has been inappropriate. Thomas again moves his finger down the center of the face of the cube (turn 70). Mrs. Winter describes what he is doing as “going straight through the face.” She grabs his finger and places it on the edge of the cube; and while moving his finger down the edge, she articulates “here is our straight edge” and then invites him “to go through the round edge” (turn 71). With a very low volume—which we can hear as a timid attempt—Thomas says “this one” while pointing to the edge of the cylinder and then moving it around the edge. The response is immediate, embodying a positive evaluation, “Good for you.” There are some brief comments concerning appropriate classroom behavior (see comments in Chapter 7), and the beginning of a teacher-student exchange between Mrs. Turner and Jane. Bending down to bring her mouth near his ear, Mrs. Winter softly invites Thomas to sit down, which constitutes of the work of bringing this episode to a close and marking this closure.

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In the course of this fragment, Thomas has exhibited a lot of knowing in action and his responses are in many ways appropriate, articulating and expressing precise responses to the questions that are posed to him. But these responses are not the ones the teachers apparently want to hear, as evident in the repeated attempts at eliciting a different answer, attempts that come to an end when the apparently desired response has been given. The positive evaluations and the turning to a different topic are two markers that suggest that what has happened just before is appropriate, whereas the recognizable repetitions, rearticulated beginnings, and oppositive expressions all are markers that what has preceded does not constitute the desired response or course of action. For example, Thomas does in fact move his fi nger through a straight line, thereby enacting a movement that has the verbal expression “is straight” as an appropriate predicate, but in this lesson on edges, the teachers apparently want him to apply the predicate “is straight” to the edge. That is, the unfolding lesson emergently exhibits the intention to articulate some aspects at the expense of all those others that perceptually stand out. Moreover, there is also an intention to distinguish between the responses that might be appropriate here—and in other parts of the curriculum—from those that are appropriate for this lesson, which is one about edges. One of the achievements of this fragment, therefore, is to allow this intention to emerge for the students from the apparently chaotic display of geometrical knowledge, which, though appropriate generally, is not appropriate in this special context. In this fragment, therefore, teacher intentions come to be marked all over the classroom to be perceived publicly. This is so even when the teachers do not state their intention—children’s recognition of round (curved) and straight edges. Like anger, shame, hate, love, and so on, intentions are not hidden psychological facts but are “types of behavior or styles of conduct visible on the outside. These are over the face or in the gestures and not hidden behind them” (Merleau-Ponty, 1996, p. 67). Intentions can be heard and seen in the (discursive, practical) actions of the teachers and in the style of their conduct, where they are available to the children. But intentions to act emerge from the self-affection of the flesh; and only someone capable of the action can recognize intentions, that is, someone who has experienced the same self-affection of the flesh.6 In fact, without apparent intentions, children would not orient toward what we consider to be aspects of the world. This was quite clear in the education of deaf-blind children in the Soviet Union (see Chapter 3), who did not exhibit an orientational reflex when an object was placed in their hands. They did not automatically orient to the objects, “What is this?” or explore the objects through tactile investigation. Rather, these children learned to explore from the hands of their caretakers and educators, who held the children’s hand in their own. Children learned the intention to explore with their hands from the explorations of the hands of their educators.

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There are interesting linguistic dimensions to be coordinated with the perceptual experiences that emerge from this lesson. Thus, for example, the adjective round can be used with different senses, an important one being circumferential, pertaining to a circle. But in this way, Thomas has used the adjective when he describes what he feels to be a “circle.” Another sense is curvilinear, curved, and forming part of a circle; and it is in this sense that the adjective is to be used at this instant in this classroom. A similar differentiation might have been employed to get to the object of the lesson if the line had been described as a line, a straight line, or as linear. There is also a coordination of distinction to be made between what it feels to be touching an edge as distinguished to the feeling of the movement. The differences are not made explicit in this lesson fragment, and it is left to Thomas and the other students to figure out for themselves the difference.

TEACHING: IN FLESH (AND BLOOD) In this chapter, we see three students articulate aspects salient in their own worlds but that are not the ones that need to be standing out in the world of the mathematics curriculum that the two teachers have planned. But it is in and through the interactions that new aspects come to be disclosed to the children in a process essentially grounded on what they currently know. How do such changes in the lifeworld occur? How do the students come to disclose new aspects that add to, expand, and transform their own lifeworlds? The world as I come to perceive it in the course of my experiences, my lifeworld, consists of and is fi lled with things. Things touch other things, and the places where they touch are articulations, joints and distinctions simultaneously. Because the world and its things are externalities, they may appear differently depending on the context—like light may appear as a wave or particle, or value may appear as use-value or exchangevalue. Articulations apparent in my perception can be articulated, that is, named and distinguished in talk. Verbal articulations, therefore, mark and denote (material) articulations in and of our lifeworld. In the course of interacting with others in a world fi lled with things that others use and refer to, children’s lifeworlds change to become increasingly those inhabited by their older fellow beings. Therefore, it is inside his lifeworld, that is, inside his own fi eld of subjectivity, that Thomas develops who he is. “Outside that field, relegated to the external space of perception (and hence other) he has nothing, nor can there by anything that is his own” (Mikhailov, 2001, p. 27). He would be the isolated subject of constructivism, “if he were himself not one of the charged ‘items’ in a fi eld common to all, stuffed with the affective purposes of people’s dealings with one another and continuously recharged by them” (p. 27). In the school classroom, his peers, the books, the physical organization,

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and every other salient aspect is “stuffed with affective purposes” of real people acting in fl esh and blood. Throughout the three fragments with see Mrs. Turner and Mrs. Winter moving, scanting, and engaging with their entire beings in teaching this lesson. When Mrs. Winter reaches out to grab Thomas’s hand and fi nger, it is contact with a living human hand, not just contact between bodies. When Thomas moves up and down the edge of the cube, there is a particular feel that offers itself to him: It feels like a triangle. He would not have needed to reach out and touch and move repeatedly up and down the edge if he had already known what it feels like. In reaching out, ready to be touched in touching—contact always implies the touching with, touching and being touched—he opens himself up to the possible experiences that extend what he can anticipate. In contact, he is affected; and so it is precisely affect that we need to understand in teaching and learning. He has taken a risk in allowing himself to be touched by the edge of the cube. It is the edge that gives itself to him, as he gives himself to it, to allow a new feel to emerge that he can become conscious of. But when Mrs. Winter takes his hand, it is the hand of another human being moving his fi nger along the edge. It is a human hand, already shaped by culture, which now moves his fi nger along a culturally charged “item” in a cultural field common to them. The relevant features of this item have been identified culturally, but, as made salient in the act of teaching, they are not yet apparent in the perception of the student. Human hands exhibit intentions, as do human-produced “noises” (i.e., talk), and children are still in the process of coming to know the intentions that are immanent in actions of others. It is in the contact of another human hand that is contagious, his own hand movement becomes “contaminated,” and in being affected, his affect comes to be socialized. No longer is it just the cube that gives itself, it is the cube and feel given to him in the course of another person’s actions. The movement is not just any arbitrary one, but one inhabited by a reason and intention, even though these may yet have to be disclosed and become apparent to Thomas. He is not just moving his fi nger along the edge, but another human being (living body) moves it with him. His movement comes to be shaped by his teacher’s movements—it is apt to take up not only her intention but the collective knowing and intention that it exhibits and marks. In her hands, his hand is allowed to develop the intentionality for exploring the straight line of an edge rather than any other straight line. Intention and intentionality are not given: they require the self-affection of the flesh, which knows how to move prior to all linguistic forms of knowing of such movement. But these movements do not just come about. In the present instance, Mrs. Winter makes them happen with her own guiding hands that exhibit a particular intentionality to Thomas specifically but to anyone caring to observe generally. The voice that comes with the guiding hand is the voice of another human being. It is not just a mechanical sound: It is a voice addressing Thomas,

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with a particular rhythm, intonation, and various temporal features (see Chapter 7). It continuously modulates itself together with the sound envelope that we can hear as words. At times the intention of the teacher’s talk may not be apparent, and in such situations, we can observe the questioning or consternate expression in and through Thomas’s body. From a culturalhistorical activity theoretic perspective, this voice, as all the other things in this classroom, “are experienced as his own affective purposeful response to the plenitude and wholeness of beings as it addresses him, and as it is cognitively perceived by him and by all the ‘remote organs of his life’” (Mikhailov, 2001, p. 27, original emphasis). It is not just that he is/feels cognitively addressed, but the address is inherently affectively charged by the “affective thought of all those who address him with their own co-thinking and co-feeling, and to whom he addresses his own unique being” (p. 27). It is because of the senses that the flesh is outside of itself, where it is apt to be fashioned by the regularities that it encounters. There are regular features when the teacher speaks, and in the way she counts out the shapes not to be used while moving the hand/arm in beat fashion (see Chapter 7). But these regularities in themselves do not invite investigation of the geometrical object held in front of Thomas’s eyes. There is an investigative intention that children have to acquire. And this they do not do on their own. As the work of Alexander Meshcheryakov with deaf-blind children showed, the investigative intention is learned in the course of purposeful activity, which, in the present case, is the collective learning activity itself. It is a form of orientation that Thomas, Bavneet, and Chris learn in the enactment of the lesson. It is not just a situation between individual students and the teacher. For example, present in the classroom are not just Thomas and Mrs. Winter. They talk, but they do so not just for themselves. Their talk is directed toward the other, therefore involving Mrs. Turner and Thomas’s classmates as audience. In fact, in the course of this lesson fragment Thomas orients himself toward his classmates repeatedly. His gaze turned toward the class seven times during the 2 minute 37 second fragment; and it does so as if he were looking for his peers’ evaluation of what he has done and said. He also uttered “everyone” as a next turn to Mrs. Turner’s question about who has raised the hand regarding the existence of differences between “some edges” and “other edges.” In saying “everyone,” he articulates his perception of others and their responses to the question. What he says, shows, and does is not just for himself, but also for the others generally and Mrs. Winter particularly. And what he says is evaluated, something that he would find out in the course of this fragment if he were not already familiar with this kind of turn of events. In the lesson fragment, we observe how Thomas’s movements come to be shaped precisely because he has a body, which is open to the world to be fashioned. Because of this openness in and to society (corps social), this living body (corps) has the capacity and opportunity to be socialized. The

110 Geometry as Objective Science in Elementary School Classrooms expressions of others become his expressions; the intentions exhibited by the teachers become his intentions. This fashioning occurs as Thomas’s movements follow or are complementary to those of Mrs. Winter. There are like movements, such as when Mrs. Turner moves her fi ngers to the same vertices that Bavneet has touched immediately before or when Mrs. Winter moves her fi nger up and down the straight edge of a cube before and after Thomas has had repeated turns. There are also complementary movements, especially when the teacher’s hands and those of students are oriented to the same object, such as when Mrs. Winter tends the cube, cylinder, and tetrahedron toward Thomas, who works on and gestures in its direction; and such as when Mrs. Turner holds the cube in the direction of Bavneet, who expresses and exhibits the salience of the vertices. These constitute moments of a community of co-movers, and the two teachers precisely function to select among all the movements those that defi ne the good ones (both verbal and gestural) from those that are less appropriate. In our classroom, as in music making, “there is a social world, an organization of ways of doing such movements, and an organization of ways of regarding them” (Sudnow, 1979, p. 4). It therefore is not as we sometimes fi nd in the embodiment, enactivist, and constructivist literatures that the individual somehow develops new insights by metaphorical and metonymical extensions from their bodily schema. First, it is in and with their flesh that they come to inhabit a world that gives itself and that they come to enact. Here, it is precisely the flesh that comes to be fashioned. The children’s bodies become mathematical bodies not only learning movements but also acquiring intentions that underlie these movements designed to displace, feel, point to, or gesturally represent objects. In this way, the children acquire a general style of bodily movement, which comes to be of a complexity that is not easily described by formalisms such as the metaphorization of basic sensorimotor experiences in diagrams that embodiment theorists sometimes use. Rather, as has been said about reading, we “learn to deal with the essential shapes of these figures, acquiring gestural modes of looking—of timing, of reaching for places in contexts of places at eye, or reaching to traverse the text with just the right sort of coverage” (Sudnow, 1979, p. 17). It is precisely these “ways of ‘reading’ that transcend the idiosyncratic requirements of some particular physics arrangement” (p. 17).

5

Coordinating Touch and Gaze Re/Constructing a Mystery Object Mappings have still to be viewed phenomenologically more broadly, but this much can be said, that if mappings present themselves in any geometric context whatsoever, they are fi rst of all mappings of restricted parts of space, which can be indicated or filled by bodies. (Freudenthal, 1983, p. 231)

Many philosophers consider contact and tact as the condition for human forms of knowing to emerge. Whereas we can do without the other senses—beautifully exemplified in the life story of Helen Keller or of Meschcheryakov’s students who became university professors despite their deaf-blind nature—we could not ever do without touch. The other senses come to be coordinated with and enabled when we are with contact, and in contact. Yet, perhaps because we experience the world as one given to all of our senses simultaneously, we tend to be unaware that the senses are not inherently coordinated and that what we “see”—i.e., perceive and understand—is not necessarily what we know when we can merely touch. There is a mapping that we learn when we are in contact with, as Freudenthal writes in the introductory quote, “restricted parts of space, which can be indicated or filled by bodies.” Although I understand the nature of phenomenological investigations differently from Freudenthal, I too use this term to denote my investigations into the problem of mathematics and its relation to the flesh. We tend to assume that gaze and touch work in unison inherently. But this, as this chapter shows, is not the case. One lesson in particular brought out that this is not the case inherently, allowing us to reflect upon the relation between the different sense modalities. For this lesson, Mrs. Turner had placed “mystery” objects (cube, rectangular prism, cone, etc.) into shoeboxes. She had cut a round hole in one side of each box just large enough to allow pushing a hand inside. On the inside of the hole, she had taped a plastic bag so that the children could not “cheat” by looking inside; but reaching through the hole into the plastic bag allowed them to feel the mystery object. Whereas we might assume that it is an easy task to reach and feel the object inside the box to know what it is, it turns out not to be the case. In many student groups on my videotapes, students differ in what they feel when reaching into the same shoebox. For example, Nathan builds a pyramid. He is sure that he felt the bumps of the edges of a pyramid. But in the end, it turns out that there is a cone in the shoebox. Cheyenne, working in the same group, has a little flat

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tree-shaped model rather than a pyramid; and Geena has bits and pieces of plasticine rather than some recognizable geometrical object. Thus, students in the same group may differ in assessing what the mystery object is that they feel. In this chapter, I follow one group of students—Jane, Melissa, and Sylvia—in the process of building (a) model(s) for their mystery object. Their individual models differ initially and what they feel when reaching into the box and touching the mystery object comes to change over time. The purpose of this chapter, therefore, is to push a deeper understanding of the relationship between the sense of touch, the sense of vision, and gestures, on the one hand, and the language students have available for describing their experiences, on the other hand. Here, as in other chapters, I take the transcript to be a protocol of the activity through the lens of the children, who exhibit what is relevant to them for each other and to the occasional adult interacting with them. This is not to mean, however, that their inquiry is unmediated by the adult world, that is, by the current cultural-historical conditions and forms of relations. The objects in the classroom, the discourses of teachers and research assistants, and the language the children have available all mediate what the children can learn and how they learn it. The transcript I use, therefore, constitutes a protocol of the children’s effort to make it through the task. By proceeding in this manner, we eschew the kinds of troubles that characterize the work of Piaget and his coworkers, problems articulated, among others, by Hans Freudenthal.

ON MAPPING In the chapters of Part A, I show and make an argument for the grounding of mathematics in the flesh. I articulate how this position differs from that of the constructivists, to whom the role of the body is an epiphenomenon, a ground for arriving at an abstraction, which, for Piaget, was prefigured by a form of logic inherent in nature and our bodies. As a consequence, he “misses the extraordinary conceptual reality; infants think in bodily terms” (SheetsJohnstone, 2009, p. 368). Mind, to Piaget, is something metaphysical, and the process of abstraction allows us to move from embodied operations to the formal operations in the metaphysical realm; mind comes with the power to think in words or clear images. For embodiment theorists, mappings also do occur. Thus, fundamental experiences somehow are metaphorized— from Gr. meta-, between, and phérein, to carry—literally carried between two domains to take on linguistic/mathematical form in the second. Núñez (2009), for example, shows how during a mathematical proof on the chalkboard a mathematician uses gestures to express the source-path-goal schema that is consistent with the ε-δ definition of continuity. What we observe here is a correlation between a mathematical concept and a linguistically structured explanation; what we do not observe is a necessary condition of the body in the development of this concept. The schema, inherently, already is a

Coordinating Touch and Gaze 113 structure of mind and does not take us back to the original bodily experience, the fact that human beings, as other animate beings, think in movement. The existence of such a condition has to be shown, because everyday experiences often contradict mathematical structures and knowledge so that the question how mathematics can emerge from contradictory experiences.1 That is, in the embodiment literature we do not find studies that investigate how formal mathematics emerges even though mathematical structures contradict everyday experiences. In science education, for example, there is a whole body of literature concerning “misconceptions,” “alternate frameworks,” “naïve ideas,” and other categories concerned with elucidating fundamental everyday understandings and phenomenological primitives. The solution this literature has offered to teachers is this: Design strategies that “eradicate” or make children “abandon” their “naïve ideas.” But the real question is how everyday, mundane experience constitute the very ground of mature mathematical knowledge even though it might contradict the formal mathematics. I return to this question in Chapters 8 and 9. Even if eradication and abandonment were the phenomena at issue, these processes still would have to be mobilized by the person’s mind articulated to be a deficient one. In this chapter, I present a case from a task where groups of students build models of mystery objects inside shoeboxes that cannot be seen but only felt through an opening. That is, the children are engaged in producing/evolving models for something that they cannot see but only touch. This task is of the same nature as the one Piaget and his co-workers used to study the recognition of shapes by means of haptic perception. A child is presented with an object and touches or feels it but is not allowed to see it. In Piaget’s work, the children are asked to name, draw, or point out from a collection of objects the one it feels. He considers the task to require the translation of “tactile-kinaesthetic impressions from an invisible object into a spatial image of a visual kind” (Piaget & Inhelder, 1967, p. 18). There are other mappings possible, however, which do not require the “construction” of visual-spatial images. Thus, in the present instance, one child (Melissa) will come to recognize her first model as inappropriate while comparing original and model by means of simultaneously touching both (one with left, the other with right hand). Here, I focus on the investigation of one group—Jane, Melissa, and Sylvia. Melissa has three models available as a reference among which to select the one best corresponding to the one she can feel, including the one that she has formed herself. The three-dimensional objects are such that they require what in Piaget’s work constitutes a second form of the task, where the child, because of the complexity of the object, no longer can make an identification by simply touching it but is forced to engage in tactile exploration. It is after what he terms “third stage” around the age of 6.5 to seven years that the child supposedly achieves the synthesis of complex forms. Models are of special importance in geometry, because they function as cultural objects that allow us to apperceive the idealities of geometry

114 Geometry as Objective Science in Elementary School Classrooms without having to go through the original processes underlying the formation of their sense.2 Thus, for example, the cube as an ideal object with six precisely square faces of the same size either parallel to one another or at a 90° angle is a historical accomplishment. We do not have to return to these formation processes of the establishment of the idealities, much as we do not have to know about electricity, semi-conductors, and liquid crystals to use a computer with an LCD monitor. It is in this way that sensual models “are embodiments of sedimented significations in the methodical praxis of mathematicians” (Husserl, 1997a, p. 26, emphasis added). Interestingly, therefore, the present task requires children to make a model of a model, that is, their plasticine models are models of objects that are already models of the ideas of cubes, cylinders, spheres, pyramids, cones, and so on. In a reflection on the experience of cubes—which shares a lot with the phenomenological inquiry that I provide in Chapters 1 and 3—Freudenthal (1983) suggests that the “mental image of a cube seems to differ considerably from the visual one prescribed by the theory of perspective” (p. 242). Actual cubes always present themselves under a perspective, but this is not so for the ideal cubes in our imagination. The image of a cube “involves as much as one needs to recognize, to make, to produce, and to reproduce cubes. It includes six faces, though one cannot see more than three at a time and may be unsure about the actual number, four, or six, or eight” (p. 242). Based on this and similar reflections, Freudenthal articulates a strong critique of Piaget’s position on children’s development of geometrical understandings generally and space particularly. He suggests that Piaget “dogmatically interpreted [the] Erlanger program” (p. 231), which he characterizes as a bodice for the adult mathematician and as an oversized suit for the child. The aspects of this Erlanger program “function as blinkers” (p. 232) in Piaget’s laboratory investigation. According to Piaget, ontogenetic development proceeds along a trajectory characterized by the adjectives topological, projective, affine, similar, and congruent, or “from the poor to the rich structures, from large groups of automorphisms to small ones” (p. 232). A model is another material object that reproduces certain aspects of the thing it comes to stand for. It is a reproduction, which, for geometrical objects, is an important feature. They are to build a model from plasticine. The intended task structure is one of a mapping, where the students touch/ feel a three-dimensional object and build a model that is of the same kind but in which size, color, or position in space (i.e., on the table) do not matter. Such mappings belong into affi ne transformations of the type f: [x1,x 2 ,x3] → [α1x1, α2x 2 , α3x3], with the special condition in the current task that α1 = α2 = α3 (or something close to it) so that, for example, a cube remains a cube (rather than becoming a rectangular prism), a sphere remains a sphere (rather than becoming

Coordinating Touch and Gaze 115 an ellipsoid), and so on.3 Here I follow the discussion of the phenomenology of boxes and their transformations that Freudenthal presented in opposition to Piaget’s work. As the analysis featured below shows, each of the three girls in the group has a different model, two of which are “rectangular prisms,” that is, parallelepipeds in which all sides are rectangles (Sylvia, Jane), and one is a cube (Melissa). There is some debate with Melissa to assist her in touching/feeling that the mystery object is more like the rectangular prisms than like the cube she has built. But it is not until Mrs. Turner requires them to arrive at one and the same model—there is only one object in the box to be modeled—and the fact that other groups have already completed the task that the three girls push toward a common model. In the extended episode at the heart of this chapter, therefore, there are repeated occasions where a girl touches the mystery object again, sometimes holding her own or another model in the left hand while touching the mystery object in the shoebox with the other. Melissa comes to change what she senses following a particular instruction that leads her to directly compare her model and the mystery object. The entire episode—marked as such by the beginning of the task engagement to the point where Mrs. Turner moves into another activity, discussing the results of the children’s modeling tasks—is cut up into eight fragments. Three of these fragments precede the one in which Mrs. Turner joins the three girls and asks them to come up with one single model rather than the two (three) they have arrived at up to that point; there are two further fragments preceding the instant when the teacher announces to the entire class that students should read until the few groups are done who still work on their task.

“IT’S A CUBE . . . I’M MAKING MINE A CUBE” Jane, Melissa, and Sylvia are working together on this task. They have received their shoebox containing a mystery object. Just after they arrive at the big round table where they are going to work during this task and where they occupy just one quarter of its circumference (Figure 5.1), Sylvia takes her turn at reaching into the box to touch/feel the object it contains. When she is done, Jane pulls the box closer to her only to push it over to Melissa, who takes a turn at reaching into the box, where she touches/ feels around for about 11 seconds. After having worked her plasticine for a while—in the apparent attempt to make it more pliable—Jane also reaches into the box. Sylvia takes another turn, reaching deep into the box, and then announces that she feels it (turn 001).4 Melissa responds, “Feel it, eh?” and then continues to categorize it, “I have felt it’s a cube” (turn 002). Jane grimaces, as if questioning this description of the object as a cube, and Sylvia announces that “it is not a cube” (turn 004), a description Jane accepts by ascertaining the same fact, “I didn’t feel a cube” (turn 006), followed by a reconfi rmation on the part of Sylvia. A pause unfolds.

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Figure 5.1 Sylvia, Jane, and Melissa (from left to right) sit at a large round table where they investigate a mystery object hidden in a shoebox to build a model of it from plasticine.

Melissa says that she has felt a cube, and then announces a reason: She has checked the sides, “like that,” while showing how she had checked. Holding her cubical model between thumb and index fi nger of the right and left hands, a “caliper configuration,” she gesturally exhibits what she has done to check the shape (turn 009). She shows for each of the three dimensions the same distance between thumb and index fi nger. In fact, she exhibits at this instant symmetry with respect to 90° rotations around two of the three axes through the center of opposite sides. In response, Jane puts her left hand back into the box and Melissa instructs her to “feel around,” continuing after a pause, “to feel it.” Jane continues reaching into the box, and then announces, “if I feel the top it seems like its square, but if I feel the side, it seems like rectangle” (turn 015). Melissa disagrees with her, “I don’t feel a rectangle” (turn 016). The two descriptions contradict: Melissa firmly suggests that she has felt a cube, and shows how she has ascertained the shape, and Jane is equally convinced that the top is a square but “the side” is not. Fragment 5.1a 001 002 003 004 005 006 007

S: M: S: J: S:

i feel it feel it eh? i have felt its a cu:be () (1) ((J grimaces, questioningly?)) no its not a cube ((shakes head while rH in box)) () i didnt feel a cube. me either.

Coordinating Touch and Gaze 008 009

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(3) i did. (1) i checked the sides like that. ((Fig., caliper grip on each of 3 sides)) ((puts lH into box)) you should feel around. (2) to feel it (3) ((puts lH into box)) if i feel the top it seems like its square but if i feel the side it seems like rectangle i dont feel any rectangle. (2)

While Jane and Sylvia work on their models, Melissa accompanies her shaping of the cube with utterances, explaining that she is making a cube, exhibiting the cube-shaped plasticine object in her hand. Fragment 5.1b 018 019 020 021 022 023 024 025 026 027 028 029 030 031 032

M: J: M: M: M: M: M: M:

i=m making mine like this. ((shapes it into cube)) i think its oblong. (2) i am making mine a cube (6) if its the if its the same. ((pounds on plasticine)) (2) to make a (??) (11) ((lifts ‘cube’, lets it fall)) h: .h: its alive (4) you see (2) basically the same basically basically (1)

Sylvia then announces what she thinks “it” to be, holding up her model of the mystery object. After a while, Lilian (the research assistant fi lming the group) utters, “What’s this?” to which Jane responds by suggesting that “it is a little bit flat”; and Sylvia shows her model again. But Melissa insists, uttering, while holding up her model, “big cube” (turn 038), and later continuing, “I think it is a cube” (turn 041); in both instances, she emphasizes the category name, “cube.” Lilian points out that “there is only one” and asks, “which one you think is in the box” (turn 039). She encourages the three girls to “touch it a little more” (turn 042).

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Fragment 5.1c 033 034 035 036 037 038

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039

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040 041 042 043 044

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045 046 047 048

049 050 051 052 053 054 055 056

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they=re all like (3) now you feel it? ((holds hands fl at, rubs, Fig.)) feel it feel the side of it how fl at it is youll see (1) see how long it is. (2) its shaped like this oh:: (9) oh you cant feel it like that

M:

(14) ((after feeling for a long while, M turns her gaze to the “cube” ((pulls rH out)) i thi i still think it is a cube.

S:

J: J: M:

057 058

its this ((holds up her model)). (3) whats this its a little bit flat i think its like this ((holds her model)) i think its like this ((holds, stares at her “cube”)) (0.85) big=cUBe but there is only one; which one you think is inside the box. actually i think i think it is a cU:Be may be you guys need to touch it a little more (3) ((pushes box to M)) feel feel it ((M reaches inside the box)) (1)

Jane pushes the shoebox toward Melissa asking her to feel it (turn 044) and, as Melissa reaches into the box, asks whether she can feel it. Sylvia makes a hand gesture rubbing the palms of her hands together, while saying, “feel it, feel the side of it how fl at it is” (turn 048). Here, the verb “feel” can be heard in two ways, as an instruction on how to touch and as an instruction of what to feel in touching. She thereby invites Melissa to touch and to feel in a particular way, that is, consistent with her own model that lies in front and slightly to the right of her (turn 048). Moreover, in Sylvia’s gesture, we observe the parallelism that underlies the defi nition of parallelepipeds—facing faces are parallel. The gesture therefore captures—much better than the word “rectangular

Coordinating Touch and Gaze 119 prism”—the nature of the mystery object as it comes to be represented in and by Sylvia’s hands. Melissa apparently continues to touch/feel about. Jane watches her intently uttering “see how long it is,” and she continues, while pointing to her own model, “it’s shaped like this” (turn 052). There are longer pauses, while Melissa continues to feel and while the two others are shaping their plasticine, until Melissa announces, “I still think it is a cube” (turn 058). That is, at this point, after having had her right hand in the box for 47 seconds apparently touching/feeling the object inside, Melissa announces again that she thinks it is a cube.

“A TYPE OF RECTANGLE . . . I HAVE IT AS A CUBE . . . IT’S THE SAME EVERYWHERE” In response to Melissa’s announcement that she still thinks the mystery object is a cube, Sylvia proposes to check and reaches into the box. Operating the camera, Lilian asks Melissa why she thinks it is a cube (turn 060). Using the same iconic gesture as before, the caliper grip, Melissa says “it’s the same” and “it’s the same everywhere” while holding the cube between the thumb and index. As she talks, Melissa rotates the cube and exhibits the caliper configuration once for each of the three dimensions of the cube (turn 061). It is a cube, because the dimensions shown by the constant caliper configuration are the same everywhere. But her two peers disagree, announcing that they do not consider the object to be a cube (turns 067) or reaching into the box (turn 063). Fragment 5.2a 058 059

M: S:

((pulls rH out)) i thi i still think it is a cube. ((S picks the box, turns it, reaches in)) let me check

060 061

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062 063

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064 065

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why do you think it is a cube. cause like the same ((turns cube and has caliper grip with thumb/index Fig.)) (1) its the same () everywhere (2) where is it ((reaches into the box)) () i cant feel it now () ough (2) i say a cube ((gazing at cube, then at S, who still feels)) (3)

066 067 068

J: M:

i dont think its a cube. i think a cube. ((S still with lH in box))

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069 070

M:

(4) i need me to measure it ((tosses cube in hands, gazes at it))

Melissa suggests making a measurement (turn 070), but Sylvia invites her to touch the object again and to touch it in the way she concurrently shows with her hand gesture (turn 071), a caliper configuration. But in Sylvia’s caliper, the thumb comes to stand against the other four fi ngers, forming a plane. In this configuration, the gesture bears an iconic relation to her own model, where a large “top” area contrasts a rather low height of the rectangular parallelepiped (“rectangular prism”). Throughout the remainder of this Fragment 5.2b, Melissa insists on her description of the mystery object as a cube: “I have it as a cube. It’s a cube. I think it’s a cube” (085). But Melissa and Jane resolutely maintain that the mystery object is not a cube. Whereas some readers may get the sense from the transcription that Melissa is opposed merely for playful purposes, the intonations and the seriousness in which the three engage each other as available from watching the videotape contradicts such a hearing. Fragment 5.2b 071

S:

072

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073

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074 075 076

M: S:

077 078 079 080 081 082

M: S: J: M: L: S:

083

L:

why dont you come here touch () touch it (1) feel the flat parts ((rubs palms of hand as in turn 048, M reaching, gazing at M)) and you touch it like this ((makes the caliper, Fig.)) () youll see.

((reaches rH into the box, feels about)) (6) yea (1) its a cube ((reaching into box) he? i dont think its a cube i dont i DO why you dont think it its a cube. because it has flat faces () so it has rectangular parts ((picks up model, holds, turns)) like thi:s yea

Coordinating Touch and Gaze 121 084

S:

085

M:

like this ((holds model in lH, runs rH palm)) and it has squares here ((strikes with palms)) () a[nd ] [i th]ink a type of rectangle (1) a type of rectangle ((gazes at S)). () i have it as a cube. ((turning to J)) its a cube. () i think its a cube. ((squarely addresses J, who gazes back; J has hand in box, feeling))

In both the iconic caliper configuration and in the description of requiring a measurement, Melissa articulates yet unused resources: use of measurement. Measurement takes us in fact back to the beginning of geometry: It has been the phenomenological origin of geometry as a science. Yet Piaget—in the attempt to follow the Erlangen program of Felix Klein concerning topology as the organizing framework for the human conception of space—insisted on measurement as following spatial relations. But, to do justice to Piaget, the rotational symmetry Melissa exhibits to show that the three orthogonal edges are of the same length does fit the Erlangen program of characterizing geometrical knowledge in terms of group theory. However, in the caliper configuration that Sylvia produces, spatial relations is prefigured, contextualized by the configuration of the four fingers in the form of a plane surface, indicating the measurement to be taken as the height over the plane. How precisely measurement emerges from such gestures is the focus of Chapter 6. “I STILL THINK IT’S (NOT) A CUBE” Following Melissa’s last announcement that the mystery object is a cube, that she thinks it is a cube, Lilian asks her about the defining characteristics of a cube: “What does it have to have to be a cube?” Melissa delineates, “It has to have the same sides and faces” (turn 089) and, following a query about whether this is what she felt, says that she “felt all around it the same . . . as my cube” (turn 092). That is, she again expresses sameness under rotation or properties of the octahedral symmetry group. In response, Jane pushes the shoebox in front of her, an invitation that Melissa accepts by putting her hand back into it. Jane invites her to feel the side that is facing at her (turn 095) and Sylvia comments that “it can’t be a cube” (turn 096), a statement Jane confirms by affirmation. Melissa states again her opposing description. Fragment 5.3 086

M:

087 088

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[i th]ink a type of rectangle (1) a type of rectangle ((gazes at S)). () i have it as a cube. ((turning to J)) its a cube. () i think its a cube. ((squarely addresses J, who gazes back; J has hand in box, feeling)) (1) what does it have to have to be a cube.

122 Geometry as Objective Science in Elementary School Classrooms 089

M:

090 091 092

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093 094 095 096 097

M: J: S:

098

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099 100 101 102 103

M: S: S:

104 105

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106 107 108

L:

((gazing at cube, turns in hands, caliper grip)) has to have the same (2) sides. (1) (2) and did you (2) [feel ] [i felt] all () around it the same ((points to different faces)) (1) as my cube. (4) ((J pushes box to her)) lets see: ((rH into box)). ((lH into box)) feel feel this side thats pointing at you

((J works on plasticine, S & M look at the object in her hand)) ((gaze at model in S’s hand)) (3) ((rH in box)) think i think its a cube.

(5) i think its this shape ((holds up a thin book, lH, rH covering lower part of it)) you cut it down a bit its like this ((places lH as if with knife at some part on the book, hidden by box)) (1) do you think its like this. (4)

After a longer pause, Sylvia holds up a thin booklet, and, covering its lower part with the four fingers of each hand, announces that the object looked like it. Jane agrees and adds that it, the mystery object, would look like the booklet if the bottom half were cut off from it (turn 103). Sylvia gestures such a cutting by holding the palm of her hand vertically (its normal vector parallel to the table), as if she had a knife in her hand with which she is cutting the booklet in about half. But Melissa still announces that she thinks it is a cube. In this fragment, we fi nd a continuation of the debate, Melissa insisting on a cube-shaped nature of the mystery object, and the other two holding that it has a different shape. Sylvia produces another model, and Jane affi rms that the mystery object has the shape indicated in the iconic gesture. In these gestures, they exhibit different symmetries than the one that Melissa articulates. The two elaborate this new model, which has approximately the same shape as Sylvia’s plasticine model (somewhat flatter). They have produced a transformation of the type f :[x1,x 2,x 3 ] → [α12 x1, α 22 x2, α 32 x3 ],

where the coefficients are all the same, but different from the one that characterizes the transformation into the plasticine model. It will turn out—in the

Coordinating Touch and Gaze 123 discussion that follows this episode and that I do not represent here (similar to the one where Bavneet talks about the cube as featured in Chapter 4)—that Sylvia’s model is identical in size when compared to the mystery object. That is, in her transformation α1 = α2 = α3 = 1. Here, Sylvia and Jane, together, have produced yet another transformation of the mystery object, this time into a model of a different material (plasticine) and of different size, but maintaining the same relation between the three dimensions. That is, although the alphas are the same within the first and second models, that is,

α11 = α 12 = α 13 and α12 = α 22 = α 32 they are different between the two models. But their relations continue to be the same so that α 11 α 12 α 13 = = α 12 α 22 α 32

“SO YOU STILL THINK IT’S A (SQUARE) CUBE?” Melissa now appears to pursue a different tack. She announces to be making pretty edges while pinching the edges of her cube and crimping them. She laughs. But Jane suggests putting her hand into the box and gives instruction to “try to fi nd the long face” so that she “can feel the square face”; and Jane continues that this would allow her to “feel each side.” Melissa exhales strongly, making a face, as if in exasperation. She appears to feel about with her right hand in the shoebox while Jane gazes intently at her face, as if in expectation of something to happen.5 Fragment 5.4a 109

M:

110 111

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112 113 114 115 116

M: J:

im making pretty edges ((pinches edges)) () hī hī hī: ()((edges with pinches)) i am making it like this ((J gazes at her object, then moves to get closer look)) () hī hī hī () .hhī (3) fi rst like try to fi nd it there try to feel the long face () so you can feel the square face. and then you can feel each side on it. ()

(9) ((M feels about the box))

() ((M feels, J gazes at her face, as if in expectation of something to happen))

124 Geometry as Objective Science in Elementary School Classrooms 117 118 119 120 121 122

J:

J: M:

feel it at its sides again. ((M gazes at her cube)) (8) ((M begins to fiddle with model)) so you still think its a cube? ((nods)) uh hm () i sort of like i still think its a cube. (9) ((J reaches into box, lH, shrugs shoulders))

Jane asks Melissa if she still thinks the object to be a cube, as if testing whether her “teaching” (scaffolding) has brought about a desired effect: for Melissa to recognize the mystery object as something different than the cube as that she is currently modeling it. Melissa responds that she “still think[s] it is a cube.” Jane reaches into the box, which moves a bit as she feels about, then shrugs her shoulders making a face as if in disagreement with Melissa. Fragment 5.4b 122 123 124 125 126 127 128 129 130

S:

S: M: J:

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131 132 133 134 135 136 137 138 139

M:

S: M: S:

T:

(9) ((J reaches into box, lH, shrugs shoulders)) i think its this ((brings model to mouth, J gazes at her)) (10) so its (2) flat ((pads it)) () so what else we supposed to do now well when you are fi nished you have to do a () read a book ((points to M’s object)) ((M rH in box)) okay (mea? un?) (2) she did ((pointing to another group)) (25) ((orientation to another group; S reaches into box)) i still think it is a square cube ((gazes at her cube, S still feels)) (7)

() what yes: () this ((rH holds up her model, lH feels, she gazes into space before her, J looking at her)) (3) ((approaches group)) have you all agreed what your object is?

ARTICULATING THE OBJECT/MOTIVE Up to now, the three girls have contrasted the different models that they have built. When Mrs. Turner arrives at their table, she articulates the differences between the models and suggests that there is only one object in the box. She points out that as a group they have to fi gure out which

Coordinating Touch and Gaze

125

one of the three models most accurately corresponds to the object in the box. The fragment begins with Mrs. Turner’s question whether they agree on the nature of the mystery object and then invites the children to articulate what they “think” is in the shoebox. Melissa describes her model as a cube, Jane categorizes hers as a “rectangular prism,” and Sylvia says that she has forgotten the name (turn 150). After a pause, Mrs. Turner says that it “will be a rectangular prism as well” (turn 152), and thereby categorizes the two models the same. She does so again when she lines up all three objects, placing Sylvia’s and Jane’s models next to each other and setting Melissa’s a bit apart. She says, “this is a cube and these are rectangular prisms.” That is, although the two models have different proportions, suggesting different factors αi in the transformation—in one model at least, the three alphas are not the same—they are also members of the same category, “rectangular prisms.” Fragment 5.5 139

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140 141 142 143 144

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145 146

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147 148 149 150 151 152

J: T: S: T:

((approaches group)) have you all agreed what your object is? yea we we ((turns about toward T)) we think its this okay () okay take a piece of () ((pushes box aside)) lets take a look you all think it[s [cube] okay melissa thinks its a cube ((points in direction)) what do you think it is jane a rectangular prism ((:S)) and what do you think it is ((points)) (3)

(3) this one will be a rectangular prism as well ((turns the object)) okay ((takes object from J)) bt lets take a look ((lines up all 3 objects)) (3) they all look like different types of something this is a cube ((points)) and these are rectangular prisms right? ((places S’s and J’s objects together, Fig.))

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162

M:

.hh .hh ((reaches for box)) but there is only one object in here but we are trying to () i think we are () it all has () the same ((makes caliper grip)) ((has taken box)) i cant fi nd it just a minute (2) you dont really () oh there it is () okay (3) i=m gonna take this i think it is () let me just touch it to make sure () okay i touched it now (2) you as a group need to figure out what it feels more like () does it feel more like ((holds up S’s)) more like a rectangular prism like this ((picks up J’s)) like this or ((picks up M’s)) more like a cube ((holds it up)) i think it is a cube ((reaches for her object)) i (??) you have to come together and decide what its gonna be because its only one object in here ((pounds on box, gazes at S)) right? yea and as a group you have to say () WE think it is () because okay i=m really thinking its a cube ((gazes at her object))

Melissa laughs (turn 153), and—in response to the teacher’s statement that there is only one object in the shoebox—reiterates: “it all has the same.” At the same time, she repeats the caliper confi guration that she already has articulated before (turns 009, 061). That is, she focuses on and exhibits for others the rotational symmetry she feels, which is exhibited in her model. In the subsequent turn, Mrs. Turner then suggests to be touching the mystery object herself and invites the group members to come to a consensus, whether it “feels more like a rectangular prism or more like a cube” while holding up Sylvia’s and Melissa’s models, respectively. Melissa again announces that the mystery object is a cube. Sylvia says something (inaudible on the videotape); and Mrs. Turner, while hitting the shoebox, suggests that they have to “decide what it’s gonna be, because it’s only one object in here.” She continues by emphasizing that they have to come to a group decision (“as a group,” and “WE think”). In this case, Mrs. Turner marks the two rectangular prisms as different “more like this,” holding up Sylvia’s model, “or more like this,” holding up Jane’s model. Then she marks the two as different from Melissa’s model, which she denotes by the term “cube.” Mrs. Turner has already articulated previously that Sylvia’s model also is a rectangular prism, the same name that Jane has used for denoting her own model. That is, Mrs. Turner articulates that there are differences between the models, which thereby are different models of the same object.6 But because there is only one object in the shoebox, only one kind of model is allowed.

Coordinating Touch and Gaze 127 The teacher works with students so that the object/motive is changed from the one they have pursued, each making a model, to each making a model that is consistent with all the other models in this group. That is, much as the intentions underlying movement and sensing, which are the result of the auto-affection of the flesh, the intention embodied in the activity is not inherently given but is something that the children have to realize in and through their actions. Whether they actually do so is an empirical matter. More importantly, this collective intention actually gives sense to the goal-directed, intended actions (Leontjew, 1982). But it is only through such intended actions that the motives of activity come to be realized. In not distinguishing between the two types of rectangular parallelepipeds (“rectangular prisms”), Mrs. Turner articulates the motive of the activity to be a recognition as a parallelepiped rather than a particular transformation, one in which the proportion of the different dimensions is maintained. In her practice, it is the geometric category—rectangular prism versus cube—that is of importance rather than the particulars of the transformation. In fact, there is an interesting contradiction at the heart of this event. As can be seen in the image accompanying turn 152, there is a difference between Sylvia’s model and Jane’s model. In the end, and unbeknownst to all involved at the moment, Sylvia’s model is nearly identical with the mystery object in all dimensions. All three factors α of the transformation will be not only equal but also identical to 1. But in Jane’s model, the factors clearly are not identical, as her model differs from Sylvia’s not only in size but also in the relation of the different sides. That is, the multiplication factors α i were different from each other α 1J ≠ α 2J ; α1J ≠ α 3J ; α 2J ≠ α 3J .

This, too, is the case for Melissa. That is, α 1M ≠ α 2M ; α 1M ≠ α 3M ; α 2M ≠ α 3M .

But, in the discursive practices that Mrs. Turner exhibits, the factors are sufficiently similar in Jane’s case so that the resulting transformation still yields a “rectangular prism,” whereas in Melissa’s case, the factors for the three dimensions are clearly different leading to a cube. Now a cube also is a rectangular prism, though one of a special kind and with different symmetries, here exhibited in the gestures and rotations that Melissa performs. But in this classroom, there is a distinction between the two types of objects that is encouraged throughout the lessons (see, e.g., Chapters 8 and 9); and this distinction is one of the conditions for the students’ bodies to develop socialized flesh rather than individualsubjective mental constructions.

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“THE SAME EDGES, AND THAT FEELS SQUARE . . . I STILL THINK IT’S THE SAME AS MINE” Fragment 5.6 begins as the earlier fragments have ended. Melissa insists on the mystery object being a cube, as in her model, whereas Jane and Sylvia suggest otherwise. Jane then reaches back, even pushing both hands into the hole of the shoebox, saying that she “can feel it,” and describes what she can feel: “flat top squares and sides flat” (turn 173). She pushes the box over to Melissa, who again reaches into the shoebox with her right hand while holding on to her plasticine model in the left, turning it between her fi ngers. She appears to be asking whether it matters if the model is exactly the same (“right as thing as”), then suggests that “this is actually a flat cube,” and concludes that she “think[s] it’s the same” (turn 175). Jane responds after a while: “They are not the same” (turn 177). Here, Melissa offers a possibility for transforming her earlier description. On one hand, she still uses the word “cube” to classify the mystery object, but, on the other hand, she describes it as a “flat cube,” which is one of the descriptions that has been used previously in public discussions, including on the previous day when Chris was talking about the pizza box as an instance of a flat cube (see Chapters 1–3, 4). Fragment 5.6 162 163 164 165 166 167 168 169 170 171 172 173

M:

J: M: S: M: J:

174 175

M:

176 177 178 179

J: M: L:

180

M:

S: J: M:

i=m really thinking its a cube ((gazes at her object)) (1) i dont think like janes (??) (1) ((:S)) do you think we need it flattened i=m mak think it is this though (14) ((J reaches into box, feels)) ((???)) ((leans to her, bringing ear closer)) what? do you think its that one? ((points to J’s)) i think its this ((gazes at her own)) i can feel it ((gazes at M)) () with fl at top squares and sides fl at ((both hands in hole)) (3) ((J pushes box to M)) is it matter; has to be a right as thing as ((holds up cube)) because this is actually a flat cube ((gazes at cube, lH; feels in box with rH)) yea:: () .h ha ha () i think its the same (2) they are actually not the same sEE it has to be the same ((M gazes at L)) so what you do with your plasticine has to be like what you have in the box i feel i feel like the same and it is still flat

Coordinating Touch and Gaze 129 181 182 183 184

M:

185 186

L: M:

187 188 189 190 191 192 193 194

L: M: L: M: L: M:

J:

S:

(1) no it is not ((mutual gaze)) (3) and it has ((holds cube, gazes at her cube)) and it has the same the same what? the same edges. () ((turns her cube)) and () that feels () square (1) ((feels about)) and it has ((J approaches her object, M gazes at it, looks at her own)) (1) and this is ((gazes)) it has same it is a it is the same ((pushes back box)) all the faces are the same? uh hm did you check all around? ((nods vigorously twice)) and they feel all the same? i still think its the same s=mine (4) ((S reaches lH into box)) it isnt the same

At this point, Lilian enters the discussion saying that “it” “has to be the same,” where it is not evident that the three iterations of “the same” refer to the same entity. Lilian, having been present during Mrs. Turner’s instructions, repeats the conditions of this task: Students in a group have to have “the same” model—the same despite their differences. Melissa may have suggested that the two models are the same, because a flat cube still has something in common with a cube (a “flat cube” still is a cube modified by an adjective), but Jane figures them to be different. As the conversation unfolds, it becomes clear that Melissa is talking about the three sides of the object as being the same, as in her cube, which she is holding up in response to Lilian’s question (turn 185). Lilian asks whether all the sides are the same, and Melissa not only confirms this but also states that “it” “is the same” as expressed in the model that she has built. She then finds her assessment contradicted by Sylvia, who insists that “it isn’t the same” (turn 194). “WE DON’T CARE ABOUT THE SIZE IF IT’S THE SAME” In this fragment, both Sylvia and Jane reach into the box and comment on the fact that it, Melissa’s model, is not the same as the one they are presently touching—as Melissa understands what they are saying. Jane takes Melissa’s model from the latter’s hands, and then, while holding the model in her left hand, reaches into the box with her right hand (turn 200). Melissa suggests a conjunctive reason (“’cause”) concerning the size of her object, but Jane rejects the reason, as “we don’t care about the size” and adds, “it’s the shape” (turn 203). She places the open palm of her left hand on the top of the cube, then rotates the cube and feels the side now on the top. During this time, the other hand is in the box; Melissa gazes at her intently. Jane then announces, “It does

130 Geometry as Objective Science in Elementary School Classrooms not feel the same,” thereby describing in words for the others what she has done with her hands: compared the two objects. That is, she has touched/felt the object, rotated it, and has touched/felt it again. When she says it is not the same, she in fact articulates that the mystery object is not the same following a 90° rotation. Jane articulates evidence of the different rotational symmetries that the two objects belong to. In fact, we can see in Jane an articulation of an orientation with her entire body, one hand intending to feel the mystery object, the other hand feeling the model visibly in front of them. The attitude is one of investigation as part of which two objects—one of which is proposed as a model of the other—come to be compared. But Melissa insists, “It’s still the same as mine, it’s a cube” (turn 211). Her statement again meets the opposition of both Jane (turn 213) and Sylvia (turn 214), yet Melissa continues to insist, the vehemence of her response expressed in increased speech intensity and pitch (turn 215). At this point, all three continue to press on their models, working the plasticine, as if attempting to increasingly approximate the surface to that of an ideal object. Sylvie pulls the box again, reaches into it and apparently feels the object. Nearly a minute goes by in this manner, when all of a sudden Mrs. Turner announces that those who are finished should read while the others complete their task. Fragment 5.7 194 195 196 197 198 199 200

S: M: S: J:

201 202

M:

203

J:

204 205 206

J: M:

207

J:

it isnt the same (4) fEEL it ((reaching her object toward S)) (7)

(1) see can i feel it ((grabs object, reaches into box rH, holds object in lH)) (1) cause its small ((gazes intently)) that one is smaller ((pointing to hers)) but we dont care about the size it is the shape (3) ((gazes in air, lH flat on cube, rotates it 90 degrees, feels new top, Fig.)) this one (4) ((pointing to top square)) it does not feel the same that it could be ((takes cube back, presses on top)) ((still with lH in box)) no not the bumps

Coordinating Touch and Gaze 208 209 210 211 212 213 214 215

M: M:

J: S: M:

131

(3) ((J pushes box back toward M)) cause its the same exactly as mine (1) its still the same as mine ((Jane gazes at cube)) (1) its a cube () actually it sort of () cause a cube is (?) ((:M)) yEA cubes arent ((J gazes at her)) (??)

In this fragment, Jane evolves a direct method of comparing the model with the original. Up to this point, the three students tended to have either worked on their object or reached into the shoebox to feel the mystery object. In doing both actions simultaneously, the body becomes a living (literal) link between the original and the model, the source and target domain. The mapping f :[x1,x 2,x 3 ] → [α12 x1,α 22 x 2, α 32 x 3 ]

is symbolized in the living body that makes a connection. But this living body is one only because of the tact that is spread throughout, making the feeling in the right hand and the feeling of the left hand the feeling of one and the same person. This unity of the tact is characteristic of the flesh. By feeling each side of both objects, Jane can sense the different multiplication factors that exist between what she has in her right hand, the mystery object, and what she feels in her left hand and has visually available. Whereas Piaget is worried about the size in children’s understandings of affi ne transformations (parallelograms, as implemented in the Nuremberg scissors), Jane here announces that size does not matter. It is the shape. Piaget considers it to be a shortcoming of the child who draws consecutive stages of the scissors with different lengths of the sides. But then he might marvel at these children concerned with preservation of the symmetry group, which an object of a particular shape has independent of its size. In this sense, the models Jane and Sylvia built are the same, for they both belong to the dihedral D2h symmetry group, whereas Melissa’s model exhibits the octahedral symmetry Oh.7 THE CUBE DISAPPEARS FROM THE HAND The teacher has suggested that several groups already finished their task. Only a few groups are still working at their models. Sylvia announces that they are not done but that they “still have to figure out” (turn 220). As before, Melissa repeats her position: “I think it is a cube” (turn 221), but hears that Jane is not convinced and responds, “I think it feels like one

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though” (turn 224). Sylvia then proposes to her a way of testing; she articulates what Jane has done before, that is, directly comparing the original with the model. Fragment 5.8 220 221 222 223 224 225

S: M: J: M: S:

226

M:

227 228

S:

229 230

J:

231

M:

232 233

M:

234 235 236 237

M: S:

guess we are not done yet we still have to figure out i think its a cube (1) ((looks at her model)) what () i will i think it feels like one though. okay. look what you can do. you can take your object and () and feel like with that one ((hits box l) and take it as you feel like its really the same. ((shakes each hand separately)) ((reaches for J’s object)) can i feel with this to see its the same? ((reaches into box, rH, holds J’s object, lH)) (2) no i (want?) you to take here in your hand ((holds her object)) (3) feel this part ((touches square face)) and go in here ((touches the box)). ((places object on narrow edge, turns it over 90 degrees at a time, touches with index fi nger; J watches intently)) (11) oh oh ((lifts gaze)) ((turns over J’s object)) (5) ((M makes bunny mouth, returns object to J, grabs her own)) i=m making it else ((begins to reshape her object)) (3) so you can just make it like this ((shows her own)) ((M makes her object to resemble that of J and S.))

Sylvia presents the instruction what to do not only by verbal means: “You can take your object and feel like with that one . . . and feel like it is really the same” (turn 225). But, while doing so, she holds her own model in the right hand and hits the shoebox with her left hand before making a “feeling” movement with it (Figure 5.1). In response, Melissa asks Jane for her model to feel with it and to see whether it is the same (turn 226). She holds Jane’s model in her left hand, while reaching with her right hand into

Coordinating Touch and Gaze 133 the shoebox. Sylvia instructs Melissa how to hold and manipulate Jane’s object using her own model as an example. Melissa touches different parts of the model with her left hand, intently gazing at it. Jane gives further instruction where to feel, directly pointing to the surface and touching it with her flat open palm (turn 230). Melissa turns the model over from resting with its large face on the table, then turns it so that one of its narrow edges comes to rest on the table. At this point she begins to rotate the object by 90°, placing her index finger on the narrow face (turn 230). She rotates the model another 90°, again placing her finger on the narrow face. She repeats these movements five more times so that she has nearly turned it over twice in its entirety. What she does is in fact exploring similarity under a particular kind of operation, 90° rotation. If the four narrow sides were the same, the mystery object/model would be a “flattened cube,” that is, a square prism with a dihedral D4h symmetry; if adjacent pairs are not the same, the resulting object/model would be cuboid with a dihedral D2h symmetry. Melissa raises her head, puckers her lips, returns Jane’s model, begins to grin, picks up her own model, squeezes it, then begins to work it into a different shape. She formulates8 the intent of her actions: “I’m making it else” (turn 235); and Sylvia suggests a little while later, “So you can make it like this” (turn 235). At this instant, the cube has emerged for her in her touch. She announces the surprise, first in the interjection, then in her face, and then by means of the action with which she begins to change her model. About one and one half minutes later, Melissa’s model has approximately the same shape as the models of her peers. In this final part of the episode, the relationship between Melissa and the natural object in the shoebox changes. Before that moment, she feels it and names and describes (verbally and gesturally) what she has felt as a cube and explains—using the “caliper” gesture—that she feels all sides to be the same. She states that a condition of the cube is that the edges and sides are the same. But what she senses changes when she directly “compares” what she has in her right hand, the mystery object, with the object she has in her left hand, Jane’s model. It is this object—which she can feel and see—that mediates her relation to the one she only feels, hidden from sight in the shoebox. That is, in this situation the ultimate shape of the mystery object that she holds in her hand does not emerge as the result of an immediate perception but it is one that is mediated by another artifact that another student had made in the hereand-now of the lesson. But the object they modeled is itself not a geometrically ideal one but a fabricated object that has been denoted by the category name “rectangular prism,” initially by Jane and confirmed by the teacher.

ON INTENTIONS AND REPRESENTATIONS By taking Jane’s model and reaching into the shoebox to engage in a comparison, Melissa exhibits willingness to open up and learn. She does exhibit surprise and changes her own model immediately after this expression. At

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a minimum, she has had to open up to an experience that she initially held to be other: feeling the mystery object as something else than she has done before. She opens up to be impressed, and the comparison constellation allows the nature of what she senses to change. In reaching into the shoebox, Melissa exhibits the capacity to move her arm, hand, and fi ngers to engage in touching. Her living/lived body already knows how to move, which is the precondition for intending to reach out and learn by sensing. But learning by sensing also means that the flesh is ready to be impressed in and by the contact with a specific section of the material world around her; and she is impressed by what she feels, affected by the way in which the object responds to her pressure. And the resulting affect is evidenced in the articulation of surprise. This shaping has occurred, involving culture in different places. On one hand, there is the mystery object, itself an objet trouvé that the teacher has placed within the box and reachable only through a plastic bag. Yet these objects in the shoeboxes are not quite objets trouvés, for in this classroom, they stand for the idealizations of geometry: these are the cubes, spheres, prisms, pyramids, and so on that scientific geometry is about. On the other hand, culture, in the form of a categorized fabricated object, Jane’s model, allows Melissa to learn to feel. It is as if Jane’s hand were guiding here, though it is “only” the rectangular prism that Jane has produced. Melissa’s feeling with the right hand is mediated by the left hand, itself touching an object that is the result of Jane’s touch. This object is not just some lifeless product, it is a product in which Jane’s sense has come to be crystallized and found objective form. This allows Melissa to learn coordinating the unseen object with one that is visible, and in the process, her sense of touch comes to be shaped. Something else happens in the episode, though not emphasized as such in my description. Throughout the entire time, the three girls work the surfaces of their plasticine models in apparent attempts to straighten out the bumps, to make them smooth, to shape and polish their surfaces so that the “human-made” models increasingly come to approximate the ideal geometrical world. This strikingly resembles the retrospective account of how the primal geometers polished artifacts to align their surfaces with the emerging ideas of ideal geometrical objects (Husserl, 1997a). In fact, the increasing capacity of working and polishing surfaces and the increasing idealization of their objects emerged hand-in-hand. My videotapes, therefore, exhibit the efforts of artisan-engineers already attuned to a science, geometry, which they are only in the process of learning, a science that deals in idealities rather than in objects with bumps and unshapely edges. Idealizations are the end product of continuous refi nement processes that work on entities of our mundane, everyday experiences. Tact, the sense of touch, is characteristic of the flesh rather than of bodies, as all flesh also has a body but not all bodies are flesh. Moreover, bodies do not have intentions, which are the results of auto-affection experienced by the flesh in movement. Without this intention underlying the reach of the hand to

Coordinating Touch and Gaze 135 contact the mystery object, the sensorimotor intentions that are so prominent in enactivist accounts would not be able to emerge and exist. As described in Chapter 3, such intentions do not exist in deaf-blind children, where they have to be actively taught. But such sensorimotor intentions do not merely arise in the auto-affection of the flesh; it is in understanding the collective motive, which gives intention to the activity as a whole, which gives sense to and calls for the intentioned sensorimotor exploration of the mystery object in the box. What the three girls learn is, therefore, the intentional motive in this form of activity that their participation in this lesson realizes. The episode presented in this chapter allows us to reflect on the position Piaget takes with respect to the recognition of shapes. Perception, to him, is the knowledge of objects that results from direct contact. The concepts of representation or imagination pertain to instances where some object is imagined or when it is present in parallel to perception. Representation itself “is a system of meanings or significations embodying a distinction between that which signifies and that which is signified” (Piaget & Inhelder, 1968, p. 17). The image is held to be an imitation that is internalized, resulting from motor activity, “even though its final form is that of a figural pattern traced on the sensory data” (p. 17).9 The mental image, because of this dual nature, tends to move back and forth between the two characteristics. We do not know what is in the mind of Melissa, whether she has any particular visual image in her mind that she looks at and compares to the mystery object. But she has built a representation, a cube, and she denotes the cube as cube, specifying its characteristics both in words and in measurement terms. That is, we see her compare a model (image) in the left hand with the target object in the right hand rather than forming an image in her mind. Even when she explicates why she thinks the mystery object to be a cube, she uses her own cube together with gestures as a means of articulating her “idea.” It is as material as the mystery object itself. Melissa does not merely indicate visual criteria for a cube, which might be articulated in the words “same edges” and “same faces,” but she also provides gestural representations of what is to be done to ascertain that an object is a cube. She produces the caliper configuration with her hand and moves it to the different edges corresponding to the x, y, and z axes in a mathematician’s spatial representation of her model. Because the edges are the same following 90° rotations of the model around two of these axes—it does not change under the corresponding transformation—the object has to be a cube. She suggests to have done it to the mystery object as well. Her articulations, which are consistent with the model directly available to vision and touch, turn out to be inconsistent with the mystery object once she has had the opportunity to directly compare it with another model of different shape. Piaget appears to be right, if only with respect to the importance of experiences that can be described in terms of symmetry groups. As the episode unfolds, we can observe the different types of representations that the three students evolve. At the very end, it becomes obvious that the models the three had built when Mrs. Turner fi rst approaches

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their table constituted a Euclidean congruent representation (Sylvia), an (approximate) similarity relation (Jane), and an affi ne relation (Melissa). Congruence allows change in position and orientation, whereas other characteristics must not change, preserving angles and lengths of geometrical objects. That is, Sylvia’s model embodies Euclidean congruence, which preserves the object under translations, rotations, and reflections. The other two models embody affine congruence, since the lengths of the original are not preserved in the models. The idea of congruence is itself a culturalhistorical achievement that arises with the limit figures of geometry, that is, with ideality, for in empirical praxis there is not absolute precision. Only ideal figures are exactly self-same: Only ideal cubes have exactly the same extensions, 12 edges and six faces. In the real lived in world, such objects of experience do not exist. The discovery that brought geometry into being was that these “elementary shapes, singled out in advance as universally available . . . because of the method that produces them, are intersubjectively and univocally determined” (Husserl, 1997a, p. 26). This method includes the provision of accounts. The objectivity of geometry, its intersubjective and unequivocal nature arises from the very methods already available in the children’s everyday manner of holding each other accountable. Even experiences are reproducible, even if mediation is required, such as when both Jane and Melissa use a second, visibly available object to compare it to the mystery object that they cannot see. In fact, in this comparison, the existing models serve precisely the same function as they do in scientific research: to be tested for their viability during experimentation. Iconic representation does not require identity of the relative dimension, yet the cube would be excluded as an iconic representation of the parallelepiped in the shoebox. Melissa’s model certainly is a symbolic representation, even including some iconic ones as well, for example, rectilinearity of faces, orthogonality of adjacent sides, and parallelism of opposing sides. This kind of reproduction falls under a class denoted by the adjective “compository” (Freudenthal, 1983, p. 244), because it is combination of iconic parts and accounts for flexibilities. It is as important to geometry for reproducing objects as other forms of reproduction. These different forms of representation do “not at all bear witness to defective mental objects” (p. 244). This episode develops over what might be considered a long period of time considering a relatively simple task. There is a temporality at work, which is produced in and through the actions of the children. In fact, it is their action that temporalizes the children and what they do. This temporality is of a different order than the reversible time of the physical sciences, and it is of a different pace than the one enacted by a knowing consciousness in which time does not really exist. It is the temporality of a search, where the outcome is not known, and, therefore, where there is no ultimate measure that would indicate to the children whether they are “on the right track.” This temporality of real sensuous labor of mathematics not only is absent in the theories of Kant, Piaget, and the (radical, social)

Coordinating Touch and Gaze 137 constructivists but also from that of embodiment and enactivist theorists. But it is a central aspect of my own way of thinking mathematical knowledge in terms of knowing in the flesh, which is an important reason for coming back to the question of temporality—as in rhythm, pace, frequencies—in Chapter 7.

6

Emergence of Measurement as the Realization of Geometry

Geometry, as the etymology of the term denotes, derives from the Greek words ge, earth, and metria, measuring. That is, a central aspect of proto-geometry has been the act of measuring—though subsequently, once the Greeks were doing geometry scientifically, measurement does not appear to have played a major role. Historically, however, geometry as mathematical science arose from the pre-scientific, intuitively given world, from the “first very primitively and then artistically exercised method of determination by surveying and measuring in general” (Husserl, 1997a, p. 26, original emphasis). The children featured in this book, however, have not been born into the same pre-scientific world. Theirs is a world in which idealities, embodied in and mediated by concrete cultural artifacts, is part of the everyday experience of growing up. They are not living in a pre-geometric world where kúbos (cube) was a die for playing with, kúlindros (cylinder) a roller, sphaîra (sphere) a ball, pyramis (pyramid) a royal tomb of Egypt, and kírkos (circle) a round or ring. At that time, all of these words pertained to real objects in the real world of the Greek. But, even though the children today do not rediscover geometry in the way that the ancient Greek did, they still do discovery work. It is measurement that allows the objects of the intuitively, non-objectively given world, to become intersubjectively available in the same way to every member of the collective. It is the art of measurement that—to paraphrase Husserl—practically discovers the possibility to choose certain empirical basis forms as measures and to use them to practically determine in an unequivocal manner their relation to other bodily forms. In this chapter, I describe how measurement emerges in this mathematics classroom at different places and times exhibiting the different functions of hand/arm movements just articulated. The emergence of measurement is tied to the practical, experienced need to be accountable for any claim, statement, or argument made. In this way, the children in the second-grade classroom reproduce geometry as an objective science precisely because measurement and accountability make objects intersubjectively available, that is, make them inter-objective. MEASUREMENT AND LIVED PRACTICAL WORK In the geometry lessons that are presented throughout this book, the issues at hand are not just related to shape or aspects of shape (vertices, edges,

Emergence of Measurement as the Realization of Geometry 139 faces) and their features (straight, curved). Rather, the metric aspects of geometry become characteristic features of the interaction between participants. They become issues, as we see in Chapter 5, precisely when teachers or peers ask students to render accountable this or that position they are taking. For example, in the preceding chapter, we observe repeated instances where others (Sylvia, Jane, Mrs. Turner, and Lilian) challenge Melissa to make her position on the nature of the mystery object—which she has articulated to be a cube in her model and in her talk—visible, rational, and reportable for the purposes at hand. Much of the three students’ activity consists in producing ways of accounting for the process of transformation between what they feel when they touch the mystery object and the different models they build thereof. Because they are asked to produce only one model—i.e., that all three girls featured in the preceding chapter have to have the same model—the activity structure encourages them to produce rational accounts for the transformations that they are produce. Measure can be understood as the conceptual tool by means of which two entities can be compared in numerical terms (Crump, 1990). Once such a tool exists, it can be applied to make numerical comparisons by assigning this unit “to every member of the class to which the measure is applicable” (p. 73). For example, the measure of length can be applied to anything that has at least one spatial dimension, a line or, rather, an idealized line. In the previous chapter, both Melissa and Sylvia use what I call the “caliper confi guration” to measure or to symbolize having measured the linear dimensions of the objects at/in the hand. The hand— being our principal organ of tact and contact, allowing us to manipulate and sense the world—may well be the original measurement instrument on which all forms of calipers have been modeled. Thus, the Kpelle in Liberia—at least during the 1960s when the research was performed— still use the handspan and multiples thereof as a means for measuring short distances (Gay & Cole, 1967). Whereas the handspan may be a fi xed unit, the distance between thumb and index fi nger may be modified to constitute the unit of comparison. In fact, any other fi nger could be used for that purpose: Whenever required in garden or other handiwork, I use the distances between right-hand thumb and the other fi ngers of an open, slightly spread hand as good approximations of four-, six-, eight-, and nine and one half-inch units (with an accuracy of 10% or better). The movable, “natural” caliper also emerges as a geometrical practice within this classroom generally in response to challenges concerning the size (length) of some object. We see such instances in the preceding chapter, when Melissa uses the “caliper configuration” to make a case for the mystery object to be a cube, as embodied in her model. Hand/arm movements (gestures) may have different functions. Some movements serve to operate upon the world. They do work and therefore are referred to in the French cognitive science literature as gestes ergotiques, translated as “ergotic gestures” in English translations of these papers. The

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adjective ergotic, deriving from the Greek word ergon (work), is used to refer to the transformative aspect that the hand movement has on the world. Other hand/arm movements have epistemic function because their function is to find out about the world—to feel surface characteristics, temperature, or to measure. My own research has shown that in the course of science investigations, hand movements change their character so that iconic hand movements (gesticulations) emerge from earlier ergotic and epistemic hand movements (Roth, 2003). During the task described in Chapter 5, the children in this class find out (learn) about a mystery object merely by means of contact and tact. Finally, there are hand/arm movements with symbolic functions, such as those movements used in pointing and descriptive (iconic) gestures. For example, Sylvia deploys an iconic gesture to indicate that the mystery object in their shoebox is low in vertical dimension—distance between thumb and palm—relative to the two other dimensions symbolized by the hand. The advantage of such a description—a continuity of hand/arm movements that have ergotic, epistemic, and symbolic function—derives from the fact that it allows for a continuity in the transformation of the hand movements: from doing work, and thereby exploring the world, or exploring the world alone, to using gestures to refer to aspects of the world, including, reflexively, to the hand movements that previously have changed the world or explored it. In fact, it is during the work that new movements are generated as variations on earlier ones, and as necessitated by the aspect of the world at (in) hand. In an auto-affection of the flesh, the new movements become a form of memory required for intention and subsequent symbolic use of the movement as a whole or parts thereof.1 As athletes know, fi nding the right movement may take a considerable amount of time and training, but once found, it will be remembered rather easily and for a long time: One does not easily forget how to ride a bicycle or how to row a skiff after having learned it. In the changeover from ergotic to epistemic and symbolic movements, the differences between consecutive states are undecidable and syncopic, in other words, it is impossible to say where one state ends and the next one begins. This is so because, from the perspective of the recipient, the hand may be doing one, the other, or both. That is, because the two types of movements cannot be distinguished, they are syncopal in nature. Thus, rather than theorizing a change in the type of hand/ arm movement, the same hand/arm movement is understood as taking on different functions. We thereby obtain a continuous trajectory of signs that begins with the auto-affection of the flesh, which then takes on sign function as the movements fi rst are used for epistemic purposes and fi nally for symbolic purposes. Moreover, the relations between the movements do not require embodied image schematic representation as it does in embodiment and enactivist theories. Here the movement represents itself, if it represents anything at all. The flesh auto-affects itself as it enacts the movements for a first time, giving it the capacity to intentionally effect the movements for symbolic purposes in subsequent situations.

Emergence of Measurement as the Realization of Geometry 141 THE CALIPER CONFIGURATION: BETWEEN HOLDING AND COMPARING In this first section, we return to Jane, Melissa, and Sylvie engaged in the task of building a model for the contents of the shoebox. Already at the very beginning of the episode, the different results that Jane’s, Melissa’s, and Sylvia’s explorations have yielded are made evident. Melissa announces that what she has felt is a cube, but her two teammates express doubt (Jane’s grimace) and oppositional views: “It is not a cube” (Sylvia), “I didn’t feel a cube” (Jane), and “me neither” (Sylvia). Responding to the opposition, Melissa provides an account of how she has arrived at the conclusion that the mystery object is a cube. She says, “I checked the sides like that” while taking her cubical model between thumb and index finger (turn 009). She rotates the cube—as can be gauged from the position of the grey-shaded face in drawing accompanying turn 009—and applies the caliper configuration to another dimension of her cube; she rotates it yet another time and exhibits the caliper configuration along the third dimension. In this configuration, the positions of the fingers resemble the caliper—a term whose origins are uncertain, but which some hypothesize to be the French word calibre, size of a bullet. The measurement device, however, has been in use for nearly 3,000 years and has been developed independently in southern Europe (Greece, Italy) and in China. Though coming in various shapes, as compass or Vernier caliper, these precisely resemble the configuration that Melissa’s fingers exhibit for us as the way in which she compares (“measured”) the three different dimensions of the mystery object as shown in her account of what she has done with the hand hidden from view. Fragment 6.1 (from Fragment 5.1a) 001 002

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i feel it feel it eh? i have felt its a cu:be ()

(1) ((J grimaces, questioningly?)) no its not a cube ((shakes head while rH in box)) () i didnt feel a cube. me either. (3) i did. (1) i checked the sides like that. ((caliper grip on each of 3 sides, Fig.)) ((puts lH into box)) you should feel around.

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What we can observe here, in fact, is a movement from a simple perceptual discrimination that Jane and Sylvia offer—in saying, “I didn’t feel a cube”—to a substitution by a comparing operation. Melissa moves the hand, establishes a unit distance in opening the caliper to match the length of the fi rst dimension, and then takes this unit to the sides of the two faces orthogonal to the fi rst and orthogonal to each other. Here we can also see a transformation from something that originally may have served as a way of holding on to the object, a grip, and allowing it to emerge into a standard (sign) applied to other dimensions as well. Or, in other words, the same measure is consistent with the length of the edge after a 90° rotation of the object (cube). That is, Melissa also exhibits symmetry properties of the object under specific types of transformations. In this situation the caliper configuration—clearly that of the left hand—simultaneously serves to hold the cube, that is, it does work. The difference between the movement doing work and doing measure (epistemic movement) is undecidable and therefore syncopal in nature. Moreover, this sign can subsequently emerge, as we see below, into a gesture even in the absence of the object itself (here an edge with its end points). It is precisely when the same unit is applied repeatedly to an entity and associated with counts that true measurement will emerge. Initially, all three girls have made models, and for a short while, they simply state that the mystery object resembles their own. But then, after the research assistant present (Lilian) reminds them that there is only one mystery object inside the box and that therefore there can be only one model, fi rst ways of accounting for what they feel inside the box are produced. For example, Sylvia asks Melissa, who has her hand inside the box, “you feel it? Feel it, feel the side of it how flat it is, you’ll see” (Chapter 5, turn 048). In this way, Sylvia instructs Melissa how to go about noting what she has noted before, which has led her to build a flat model. It is precisely when you “feel the side of it,” that you can have the experience of the flatness. Of course, this description is as far from the feel as is the description “knead the dough until smooth” is from feeling the dough in the way required in a successful realization of a bread recipe. The recipe is but a rational account of what the baker has done with her hands. In the same way, Sylvia has made her actions of modeling—or, rather, the “feel of it”—rational and reportable for the purposes at hand. She thereby has exhibited the accountable nature of her actions. Repeatedly, Melissa provides rational accounts for what she has done, all of which are based on a comparison using a standard unit: “I checked the sides like that” + caliper configuration of three orthogonal edges (turn 009); “Cause like the same ((caliper grip)) it’s the same everywhere” (turn 061); “but we are trying I think we are—it all has the same” (makes caliper grip) (turn 152); and “It has the same edges, it is the same” (turns 184, 186). Melissa explicitly articulates measurement as the principle on which to base the account (turn 070). Sylvia, too, uses the caliper in an iconic way

Emergence of Measurement as the Realization of Geometry 143 to show how a shorter distance between the parallel caliper legs separates two long and wide faces. In this instance, it is not the more scientific (mathematical) measurement that wins out but the “feel” of the object in the hand, the nuance. But the simple articulation of the feel does not get the three girls out of the deadlock. Jane and Sylvia variously articulate the feel of the mystery object: “didn’t feel like a cube,” “oblong” (turn 019), “top square” (turn 015, 084, 111), “side rectangle” (turn 015, 082), “can’t be a cube” (turn 096), “flat” (turn 036, 048, 082, 125, 166), “not the same” (turn 194, 205), a double-handed gesture with a flat object separating the palms (turn 048), and “long” (turn 050). Rather, just as in the case of rock musicians communicating the feel of a particular musical phrase by playing it or by reminding each other of some other band’s playing the phrase, the turning point comes when Melissa engages in a direct comparison of the feel. The direct comparison later enables her to simultaneously feel the mystery object in the right hand and the model in the left hand, and, therefore, she could feel the difference as well.

DOING THE CALIPER: FROM EPISTEMIC TO SYMBOLIC MOVEMENT As long as students use the caliper grip to hold the object they measure and talk about, the difference between the ergotic and symbolic aspects remains undecidable, syncopic. Thus, we may not be able to decide—based on a videotape, photograph, or live event—whether a hand movement/position is to hold the object, to fi nd out something about it, or to symbolize holding or fi nding out. A certain form of abstraction does in fact occur when the hand produces a caliper shape in the absence of the object measured. Here, a differentiation has occurred, then, between the ergotic function of the grip or the epistemic function of the touch, on one hand, and the caliper configuration as a symbolic denotation of the measurement process, on the other hand. This, therefore, becomes the reference of some unspecified thing. It is measurement as such. In the following fragment from the same lesson as the preceding one, Ben’s group is asked to state the nature of the mystery object in the shoebox. In the course of the interaction with Mrs. Turner, the students generally—but Ben particularly—are asked to articulate evidence for the contention that the mystery object looks like the model they constructed: a cube. Initially the issue comes to be articulated as one in which the students are asked to distinguish their model from a rectangular prism. Part of the episode not reproduced here (turns 33–56, see Fragment 4.2) involves the teacher and Bavneet, before Mrs. Turner turns back to Ben and asks him to articulate why his model contains all squares rather than rectangles or oblongs. It is at that point that Ben articulates measurement. But let us look at the entire lesson fragment (excluding the exchange with Bavneet).

144 Geometry as Objective Science in Elementary School Classrooms Following Mrs. Turner’s question offer, “what makes you think it is a cube?,” Ben unfolds a response, punctuated by pauses and interjections (turns 05–09). While contrasting the “one side” with the “other,” his right hand produces two gestures that suggest sides orthogonal to each other, the first in a vertical direction to the desktop, the second parallel to it (turn 08). Uttering the final part of his statement with rising intonation—using the same gestures as Ben but in the reverse direction, first the one parallel to the desktop then the one orthogonal (turn 13)—Mrs. Turner questions its contents, and Ben affirms. Mrs. Turner makes three long steps, stretches out to catch hold of a rectangular prism on the chalk rest. She repeats, like a stutterer, “like a,” while returning to her previous position next to Ben’s table. Here she completes the first part of a question-response pair, “like a rectangular prism where one side is the same as the other?” (turn 13). As the utterance unfolds, Mrs. Turner first holds the parallelepiped out in front of her while she names it “rectangular prism,” and then touches fi rst one of the square sides with her right palm, then the opposing side with her left palm. Fragment 6.2a 01

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bens group. (0.20) ben ´what does your group think thIS is: in here. a cube a cube what makes you think its a cUBE? (0.92) um; (1.15) ((T shakes head, opens palm “inviting” explanation)) .hh that; (0.20) um; (1.43) it um, (0.21) one sides the same as the otHER. ((Fig.)) (0.95) ONE side is the same as the otHER? yea. (0.53) okay ((walks to right, picks up parallelepiped)) (0.57) like a like a like a (0.52) rectangular prISM where one side is the same as the other? ((touches facing “square” sides, Fig.)) yea. (0.86) (:B) no; =so why is it not a rectangular prISM then. (0.25)

Emergence of Measurement as the Realization of Geometry 145 19 20 21 22

23 24 25 26 27 28 29 30 31 32

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they are all the same(0.71) theyr whats all the [same.] [i: th]ink its the same. ((turns toward object at chalkboard, points, Mrs. T follows his fi nger with her gaze)) (1.31) ((Mrs. T walks to objects, picks the cube)) um because um the sides are all the same; (0.89) ((Mrs. T returns to her position)) okay (0.43) so all the faces, yea are all the same? ((:b)) do you agree with that? ya. ((Mrs. T gazes at Joel)) yap. ((Mrs. T’s fi nger points to Joel)) (0.27) okay (0.19) and (1.03) anYthing else that tells you it is a cUBE? ((holds up the cube from the chalkboard))

Ben affi rms, but, after a pause, Joel, who has turned to Ben, negates. Mrs. Turner continues, “so why is it not a rectangular prism then?,” which Ben completes as a question-response pair, “they are all the same” (turn 19). Apparently beginning another utterance that possibly confi rms the preceding one, Mrs. Turner then produces what can be heard—because of its grammatical structure—as a question but which intonationally— because of its falling pitch toward the end—is a statement, “What’s all the same?” Overlapping her, Ben says, “I think it’s the same,” while pointing toward the chalkboard where, on the tray, there lies a cube. After a longer pause that unfolds as Mrs. Turner walks to the cube on the tray and returns with it holding it up for everyone to see. Ben continues, “because the sides are all the same” (turn 24). Mrs. Turner seeks confi rmation by restating Ben’s phrase but by modifying the “sides” into “faces,” “so all the faces are all the same” (turns 26, 28). Ben is responding in a brief pause that she leaves in the middle of the utterance. She then turns to Bavneet: “Do you agree with that?” Both Bavneet and Joel affi rm. At this point, Mrs. Turner produces a query that leads into an exchange with Bavneet (Chapter 4), who points out the vertices on the cube. Mrs. Turner then turns again in the direction of Ben and (thereby) addresses him. Mrs. Turner begins this third part of the episode (Fragment 6.2b) by describing Ben’s model, “this looks like square, square, square,” and she then continues, “how did you know to do square, square, square, square,” and repeats herself, “how did you know square, square, square, square?” (turn 57). But, after an unfolding pause, it is Joel who begins a second turn, “um square, square”; Mrs. Turner continues, however, as if she had not fi nished her previous turn, “not rectangle, rectangle,

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Figure 6.1 While Mrs. Turner is asking a question about why the mystery object in the shoebox is a cube, Ben is producing the caliper configuration with both hands, the one holding and the other measuring/gesturing.

rectangle” and then compares this to the results of another group, “like they, right Jonathan?” Across the three turns at talk, from the third cluster of repetition of “squares,” over Joel’s attempt, and to the beginning of the question about rectangles, Ben has brought his hands up from his lap and, making the caliper configurations, holds the cube fi rst up high, then brings it to eye level (Figure 6.1). Jonathan affi rms. Mrs. Turner orients back in the direction of Ben, “How do you know that?” Punctuated by pauses and interjections, Ben begins a second turn and then, as his hand moves up from his lap above the desktop forming a caliper configuration, he states, “I measured it” and then lifts his head and directs his gaze at Mrs. Turner (turn 67). Fragment 6.2b 57

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((:B)) hOW did you know; im looking at your model here. ((points to Joel’s model)) (0.50) and (0.32) this looks like square square square square how did you know t do square square square square; hOW [did you know square square square square; [((Ben brings up hands with cube, gestures, Fig. 6.1)) (1.18) ((Mrs. T shrugs shoulder, looks at Joel)) um square square =not rect]angle rectangle rectangle like (0.30) thEY ((points to back of classroom, Ben & Joel turn around)) have rectangle rectangle rectangle or oblong oblong oblong; right jonathan? (0.24) yea. so () how did you know to go square square square square; (0.33) all along the sides and not o:blONG o:blONG o:blONG; ((moves head in same rhythm)) (1.36)

Emergence of Measurement as the Realization of Geometry 147 65 66 67

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hOW did you know that. (0.71) um (0.32) well we (0.49) um (0.47) i measured ((caliper configuration, Fig. a)) it um ((looks up, Fig. b) um [ the ] [how di]d you measure; um because i took the (?cular) and i (0.30) made thumbs (0.45) how long it was ((caliper configuration, Fig. a)) and then i (0.43) turned it to the other side and ((caliper configuration, Fig. b)) it was the same. (0.62) so youve felt with your fINger? (0.28) and you ran along and you and you felt so you were able to measure (0.32) okay. (0.27) () yOUR () group () built THIS ((picks up Joel’s “stick” model)) (0.60) and this is interesting you built a (0.83)

(0.45) a cUbe and this is how they built it. ((holds it up))

Mrs. Turner begins a second turn while Ben is still speaking, “how did you measure?” At this point, Ben produces a visible and audibly rational account that will be satisfactory for the present purposes—as per the positive evaluation that Mrs. Turner produces in repeating what Ben has said in constative form, adding an interjection of assent, “okay.” Ben provides an account in which he articulates having felt the length of the mystery object, while his hand takes on a caliper configuration placing the end of his thumb on the table: “made thumbs” (turn 69). While uttering, “then I turned it to the other side” he brings up and twists his arm, rotates the hand in caliper configuration as if he had moved it from one to another side of the cube intersecting with the fi rst at 90 degrees (turn 69). He states the result of the comparison: “and it was the same” (turn 69). Mrs. Turner then turns and begins what is going to be a recognizably different issue, and, in so doing, also affi rms that the previous topic has been completed (turn 73).

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Here, (the idea of) measurement emerges from a situation in which the children first provide a qualitative description of sides that feel the same. Mrs. Turner summarizes this initial account in terms of repeated sequences of the shape word “square, square, square, square.” She then requests an account: “How did you know that?” (turn 65). In a way similar to Melissa in the previous section, and similar to his own prior operations, Ben produces a caliper configuration together with the verb “measure” to account for what his hands have done while touching/feeling the mystery object for the purpose of building a model thereof. However, unlike the other two instances, he employs gestures independently of the object in his hands. That is, whereas Melissa in Chapter 5 and he previously formed the caliper configuration with the model in their hands, the hand movement here is entirely symbolic. In the preceding situations, the caliper configuration also has had symbolic (sign) function while the hand held the object. That is, the caliper configuration may have had its origin in the fact of holding the object—it has ergotic function, i.e., does work—and subsequently emerges to have purely symbolic function. At the transition points, there are instances where the difference between the two functions is undecidable, belonging to both functions simultaneously. In addition, rather than just using a caliper configuration, Ben “made thumbs,” which he uses to establish “how long it was.” He claims to have “made thumbs” while holding the distal phalange of the thumb against the desktop. Here, the caliper is replaced by a part of the thumb, which serves as a fixed unit of measure that no longer is derived from a preceding gesture but introduces a new element. The distance between the configurable legs (fingers) of the caliper now comes to be turned into a fixed length element that serves as the unit for comparing (at least) two different dimensions of the mystery object. Having a fixed measure may be both a fi rst in the emergence of comparison, such as when we take a stick hold it to an object, place a finger to produce a reference unit length, and hold this length to the comparison object. This in fact already is a generalization from holding two entities to be compared in length directly to one another. (Connor employs such direct comparison in Chapter 8, in an episode from the very fi rst lesson in a curricular unit that Mrs. Winter and Mrs. Turner denote by the term “geometry.”) Or it may cognitively follow the caliper, which itself might emerge from the use of the hand as an inherently modifiable “instrument” with which to compare two entities. The use of body parts as units of measurement is not surprising given the fact that such practices reach far back in human history: the foot, the (hand)span, the handbreadth, the sailor’s fathom, the mile (from Lat., mille passuum, a thousand paces), the German Elle (corresponds to the cubit, from Lat. cubitus, elbow to end of the middle fi nger), and the French pouce (thumb, one inch) all are measurement units that derive their names from body parts or bodily actions. Excavations in the Indus Valley reveal that measurement instruments have been in use since the fi fth millennium BCE, and measuring instruments have been in use during the

Emergence of Measurement as the Realization of Geometry 149 construction of many major cities during the golden era between 1750 and 2300 BCE. On the scale of human and cultural evolution, these are rather brief time scales, suggesting that measurement is quite a recent and complex cognitive feat. In the present instance, too, measurement re-emerges from the everyday experiences of the children when they are held accountable for their claims and actions.

FROM COMPARING TO COUNTING: QUANTITATIVE MEASUREMENT In the previous sections, we see the caliper configuration being used to hold an object, to feel an object, and to symbolize a measurement made between two different dimensions. The rudiments of measurement therefore have emerged. But in all instances there is the same, though adjustable standard measure that serves to compare two different items. However, true measurement comes into being if a particular measure comes to be applied repeatedly to the same dimension, that is, if a standard length comes to be associated with counting. The length of an object, thereby, becomes a multiplicity, because “the representation of a totality of given objects,” here the length unit, “is a unity in which the representations of single objects are contained as partial representation” (Husserl, 2003, p. 21). That is, true measurement is related to the application of number, which is but another term to denote multiplicity. In this section, I describe the emergence of this advanced practice, the coincidence of measure and counting. In the following lesson fragments, we can see that the suggestion of measurement emerges following a particular investigative structure that itself emerges from the exchanges between the adults in the room. The measurement arises from qualitative indications and slowly turns into an incarnate suggestion of the number of boxes needed in a stack to turn Chris’s pizza box into a cube. We have already been to this episode before. It follows the instances in which Chris has fi rst shown how there are different sides to a pizza box (Chapter 4), which then turns into an explanation of what he has to do to turn the pizza box into a cube (Chapters 1–3). Following Chris’s turn in response to Mrs. Turner’s question “what would that box have to have to make it a cube?”, I (R = Roth) utter, “how could you make a cube from pizza boxes?” (turn 24). There is a pause, an interjection, and then Chris says with low speech volume, “I don’t know.” There is another pause, and then Mrs. Winter produces an utterance grammatically structured like a question, “What would we need to do to make this pizza box into a cube?” (turn 28); but Mrs. Turner repeats what she says, before Chris completes it as a question-response pair, “take a whole bunch and stack them on top” (turn 30). He initially places his hand on the pizza box currently held by Mrs. Turner, and then moves it upward until it reaches a height approximately corresponding to the sides of the square box (turn 30).

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Fragment 6.3a 22

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it would have to um ((gets a cube from chalk holder)) (0.44) that tu:rn: or this (0.21) like the:(0.62)s:: ((bends down to box)) kinda like (3.04) kinda ((hand moving along edge)) (0.23) s::quare here an here like ((rectangle)) (0.68) ((movement along edge of box)) it just has squares (0.45) here and here and everywhere (0.63) ´how could you make a cube from pizza boxes. (1.46) u::m ((looks at pizza box)) (1.14) (1.83) whose got an idea how could you (0.43) what would we need to dO: to make this pizza box into a [cube ] [in or]der to make this into a cube ((holds up pizza box)) what would we need to do to it take a whole bunch and stack them on top ((shows with hand gesture, Fig.)) (0.58) in order to do what, (0.89) in order to make it square (0.49) okay (0.66) you had a word the other day on the board; didnt you? (1.66) you stA:ck th[EM:] [yes] you stack them. ((shows with her hand above pizza box, Fig.)) (0.44)

Rather than accepting/evaluating, Mrs. Turner utters, “In order to do what?,” which Chris reifi es as a question-response pair, “In order to make it square” (turn 34), which is followed by the teacher’s evaluative turn, “Okay.” I (in the process of videotaping) offer a question, which, when there is no response, I answer myself, “you stack them,” thereby repeating a term Chris has already used and a term that Mrs. Turner had written on the chalkboard during the previous lesson. She affi rms, “Yes, you stack them” (turn 41).

Emergence of Measurement as the Realization of Geometry 151 Here, first Chris then Mrs. Turner and then Chris again (turn 44) show how stacking pizza boxes would lead to a cube. That is, the thickness of the cube is immanent in these members’ descriptions and stacking several of them leads to a new object that has about the same dimensions in all directions. Qualitatively, therefore, the repeated measurement is immanent in the description of what one has to do, but is not yet articulated as a manner of giving a precise description how many pizza boxes are actually needed to do this. Yet the extension of the cube literally is performed and felt as Chris moves his arm and hand upward. In her next utterance, Mrs. Winter provides the seed for such thought to emerge. Mrs. Winter then articulates wondering about something (“how many pizza boxes”), but Chris is talking simultaneously to Mrs. Turner, explaining what one had to do, using the same hand gesture from the box upward as before (turns 41, 44). A student calls out “ten” while Chris is still talking, and then Mrs. Turner repeats and fi nishes the issue about the number needed to make a cube. A student calls out “one,” and after a pause, Chris suggests grinning from ear to ear, “A whole bunch of pizzas” (turn 50). Mrs. Turner solicits more input, apparently without making an evaluating comment, though Daphne revises her fi rst number “four” when Mrs. Turner utters with an upward intonation “do you think four?” This is thereby heard as querying the truth of the number, with a subsequent student revision. Kendra offers a number “two thousand,” which the two teachers take with surprise interjections. But Mark makes a suggestion, which Mrs. Winter affi rms as having “it right down” (turn 68). Fragment 6.3b 43 44

45 46 47 48 49 50 51 52 53 54 55 56 57

W:

well what i was [wondering how many pizza boxes]. C: (0.43) kendra? (0.84) two thOUSand. (0.29) = =OH: my goodness. two thousand that=ll be way longer. i think we () i () i marks got it right down the end an really long rectangular pris[m].

In this situation, the number of boxes required to make a cube from the pizza box goes unquestioned and without justification required. Some responses are qualitative, such as when Chris raises his hand from the box to some distance visually about the same as the side of the square. He also suggests “a whole bunch of pizzas,” which is associated with a smile and could have been heard/taken as a joke: You could eat a “whole bunch” of pizzas and then make a cube from the empty boxes. The suggestion no longer would have been mathematical, but would have placed the response in the gustatory domain. The numerical suggestions include 1, 4, 5, and 2,000. Mrs. Turner then calls on Jane, thereby opening what would turn out to be the final part not only of the fragment but also of the entire episode (variously discussed in Chapters 1–3, 4, and here). In her turn, Jane proposes a way of establishing how many pizza boxes there are needed suggesting a count; and Mrs. Turner will subsequently denote what Jane has done as “measuring.” At first, Jane gets up to move close to the box, gesturing along the edge facing her (Figure 6.2a). She then grabs the box between her thumb and index finger (Figure 6.2b) in the “caliper configuration and then moves the hand across the top (square) of the box (Figures 6.2c, d). She repeats the gesture, initially stating that “you go like that” and then articulates, “and count how many” (turn 70). That is, she articulates determining the number of boxes needed by taking its thickness and counting how many times this “thickness” it would take to make the same as the side of the square. Fragment 6.3c 69 70

T: J:

[j]ANe? yea um (0.39) i think if you wanted () TO make it exactly like a cube, (0.43) you have to know how ((Fig. 6.2a)) (0.45) like ((Fig. 6.2b)) (1.07) how long this is ((shows thickness of box in Jennifer’s hands)) and then ((Fig. 6.2c)) () you take that ((Fig. 6.2b, takes ‘measure’ of thickness)) and that and go like that ((makes walking movement

Emergence of Measurement as the Realization of Geometry 153

71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87

with fi ngers across the box, Fig. 6.2c, 6.2d)) and [count] how many.[ R: [UM ] [did everyone ´get that; S4: (0.48) T: jane said that we would have to measure? (0.69) how much (0.21) thIS is (1.09) and then measure, (0.61) J: ((gets up, points across the box)) you go like this and then count it. W: and count it and then what. (0.56) J: and then you can (0.36) and then you know how much to put (0.20) on W: how many to stack on top to make a cUBe vERy interesting. (0.20) ^cheyenne. (0.40) C: um i think it will take ten boxes. I think. T: ten boxes. but, W: you know what, we will work it out. T: YES we=ll do it okay. W: we=ll work that out how many pieces to make this a cube. T: thank you very much ((gestures Chris to get back to his place))

Figure 6.2 Jane first measures the height of a pizza box using the caliper (a, b) and then shows how to count the repeated application of the measure to find out how many pizza boxes it takes to make a cube by stacking (c, d).

154 Geometry as Objective Science in Elementary School Classrooms My turn following Jane (turn 71), which addresses itself to the class as a whole (“everyone”), including its teachers, indicates that there is something remarkable that has gone on, but something which others may not have remarked: “Did everyone get that?” Mrs. Turner rearticulates, thereby affirming, what Jane has said. But she does not exactly repeat. Mrs. Turner introduces the term “measure,” which replaces the term “count” in Jane’s account. There Mrs. Turner stops, giving rise to a silence, which Jane breaks articulating her method again, “you go like this,” gesturing repeated applications of the caliper across the box, and completes: “and count it.” Mrs. Turner asks for the subsequent action, which Jane responds to by saying that then “you know how much to put on to make a cube” (turn 79). Mrs. Turner states an evaluation, which begins by reiterating/affirming the response, “how many to stack on top to make a cube” and then follows it with a more explicit evaluation, “very interesting” (turn 80). There is one more turn, in which Cheyenne offers another number, before the two teachers collaboratively bring the fragment to a close by announcing a deferral of the actual measurement to another instance in the course (“We will work it out.”).

FROM ACCOUNTABILITY TO ACCOUNTS AND COUNTING In this chapter, we see how measurement, the origin and heart of geometry, arises from the events in a second-grade classroom when the children are asked to produce accounts of what they have done or to provide accounts of what they “think.” These accounts allow them to become accountable and to make their mathematics accountable. It is precisely in and through these accounts that thinking becomes something that is not relegated to the mind, but becomes a public event visibly and hearably rational and reportable for all practical purposes of this mathematics classroom. It is not that they have to take these accounts as shared as various social constructivist mathematics educators tend to describe it: In and as of account, what is being said in this classroom is a cultural possibility and therefore inherently shared. The children participate in this collective event, their individual performances driven by the collectivity as much as by their own intentions, which inherently reflect the collective dispositions of the present field. Thus, holding the children accountable leads to thinking as a public, and therefore collective and collectively modifiable event. Participation in the event entrains and fashions the lived/living body so that any higher order function (thinking) that is the result from this participation—a form of interaction—is itself shaped by the process, that is, is itself societal through and through. That is, measurement is not just something evolving in individual bodies. It is enabled by the auto-affection of the flesh, which permits epistemic movements that are made available to others as intelligible, inherently shared and shareable expressions. Epistemic movements and counting together allow the emergence of measurement.

Emergence of Measurement as the Realization of Geometry 155 The events in this classroom also suggest a possible organization of sequences that might facilitate the emergence of measurement as a practice to be used in situations other than the particular context in which we encounter it here. Thus, we observe here a development, which begins when children handle objects and build models (with their hands). From their hand movements holding the objects they emerge hand movements that have epistemic function, such as when Melissa or Ben use them to measure the length of different sides of their mystery objects and of the models thereof. At the same time, as hand movement, it is inherently shared with others who can see in the performance of movement with intention. 2 Later on, the same hand movements may take on symbolic function, such as when Ben uses the caliper configuration to refer to the measurement and comparison has enacted before. The initial epistemic hand movement now is a symbolic hand movement with interactive function, as it demonstrates to the teacher and all the peers what has been done to ascertain the nature of the mystery object. Finally, the hand movements can be connected together into chains of movements, where each link is associated with a count of one. The total length thereby comes to be a multiplicity (grammatically expressed in plural expressions such as “20 centimeters” or “eight unit lengths”) as much as it is a unity (grammatically expressed in the singular of “the length”). In this association of repeated hand movements and a counting procedure emerges measurement. Any movement, because of the capacity of the flesh to auto-affect itself, becomes a possibility for learning and symbolization. Even machines change when they do work, and in a sense one can consider the abrasions that occur a form of memory—e.g., the memory that batteries are said to develop when they are not entirely depleted before recharging. But only the flesh is capable of transforming ergotic and epistemic movements into symbolic ones for its own purposes. Measurement tends to emerge in a culture when there is a need for it. Measurement therefore must have some utility (Davis & Hersh, 1981). In the present situation, the utility arises from the fact that it is a way of comparing and being rationally accountable. In this instance, the need emerges from the question, “How many . . .?” setting up (provoking) an answer in the form of a count or number. The method Jane articulates for producing the answer does precisely that: It shows how taking a reference length and counting how many times it fits into the side of a square. As soon as such action is used to mark off a stick, a measuring instrument such as a meter will have emerged. Measurement instruments evolved precisely to be “templates for shaping and polishing surfaces or reckoning material alignments and lengths” (Lynch, 1991, p. 80). That is, as we see in the present case, the everyday world of the children is the beginning of mathematization and is subsequently substituted by a mathematical world. This development corresponds to the historical process that led Galileo to the “surreptitious substitution of the mathematically substructed world of idealities for the only real, actually perceptually given, ever experienced and experienceable

156 Geometry as Objective Science in Elementary School Classrooms world—our everyday lifeworld” (Husserl, 1997a, p. 52). We see already in Chapter 5 how the three girls work the surfaces of their models—i.e., “polish” them—so that their artifacts come to be aligned with the ideal surfaces of the geometrical object in the shoebox. Here, they further idealize their experience, thereby contributing to the transformation of their real-lived in world comes to map onto the ideal world of the geometer. That is, mathematization and idealization occur simultaneously.

7

Doing Time in Mathematical Praxis Praxis unfolds in time and it has all the correlative characteristics, such as irreversibility, which synchronization destroys; its temporal structure, that is, its rhythm, its tempo, and above all its orientation, is constitutive of its sense. (Bourdieu, 1980, p. 137) Rhythmizing consciousness does not apprehend its object in the same way as unembellished perception does. (Abraham, 1995, p. 21)

The intellectualist approach according to which practical understanding of the world is governed by mind confuses presence in the world (Being) with the presence of the present of world, which appears in the form of things (beings), re-presentations. Thus, with the emergence of re/presentations that can be used to make something present over and over again, time has been expelled from common accounts of cognition. Thus, intellectual consciousness represents practice, synchronizes its moments, and thereby destroys the sense characteristic of praxis as it unfolds in real time.1 Practical understanding does not require formal knowledge: For example, children’s language is grammatical even prior to their encounter with formal grammar. The approach I take here, the one of mathematics in the flesh, is precisely designed to address this recurrent problem to knowing in mathematics education research, whatever the brand. Practical comprehension is comprehension in and through the flesh, enacted without the conscious mind as master and prior to any embodied image schemas. It is knowing in the flesh acquired through the flesh, which comes to be marked in participating in an inherently structured, societal and material world, from its beginning that even precedes the actual birth of a child.2 It is precisely because temporal and rhythmic features derive from the auto-affection of the flesh that the difference between the living/lived body and mind becomes indistinguishable: There is no sense possible without the auto-affection of the flesh. It makes no longer sense to operate with the Cartesian distinction maintained in the embodiment/enactivist literature, as both material body and metaphysical mind are but modalities of the flesh. The notion of praxis, which is based on knowing as performance, goes together with the notion of the living/lived body, the flesh. As soon as we move from the diachronic nature of praxis to its synchronic description in terms of knowledge, we loose all developmental aspects and necessities in the same way that synchronic linguistics loses the phenomenon of linguistic change.

158 Geometry as Objective Science in Elementary School Classrooms Time and temporality are central features of human praxis, presenting themselves in the guise of pacing, rhythms (body, voice), and vocal frequencies. These are important, not only because they structure exchanges and therefore practical consciousness (e.g., memory that is facilitated and recalled more easily with rhythmic features) but also because they serve to communicate and share emotions. Thus, “patterns of body alignment, eye gaze, speech hesitations or flow, loudness as well as overt expressions” (Collins, 2004, p. 110) communicate emotions, pride, shame, and so forth. These are precisely the features that create the alignment and mutual focus that we can observe in this second-grade classroom when following the events on a micro-scale. That is, “emotion never is a psychic and internal fact, but a variation in our relation with others and with the world that can be read from our bodily attitude” (Merleau-Ponty, 1996, p. 67). Rhythm and other periodic features structure the places we inhabit and fields that we enter: These features are written all over the fields and therefore are available to every member of the setting. In this chapter, I present a series of analyses that exhibit the interlacing of speakers and listeners. This interlacing can be observed in the complementarity of body movements, the production and reproduction of periodic features (rhythm, intonation) across interaction participants and across expressive modalities within and across the living/lived bodies of members to a setting. This interlacing underlies entrainment, the feature by means of which we come aligned with others not only on the micro-scale but also at the scales of individual development.

RHYTHM, PACE (TEMPORALITY), AND SENSE In Chapter 4, I point out and insist upon the consideration of the pauses in the exchanges between Mrs. Turner and the students. For example, her exchange with Thomas is marked by pauses and interjections that apparently do not add to the content of talk but serve interactional functions for dealing with pauses. These pauses are important not only because they give structure to and shape the temporality of the event but also because they are resources for participants and onlookers to make attributions about the knowledge of a person. For example, if there is a long pause after a teacher question to a particular student, this might be heard as evidence of lack of knowledge or as insecurity on the part of the student to proffer a response. Research conducted in the 1980s on pauses following teacher questions showed that in general, teachers tend to wait less than a second before moving on, asking another question, or pointing at a specific student to respond. That is, the teachers tend to increase the pace by breaking silences. Teachers are driving the rhythm and pacing of a lesson independently of the question whether students are actually following. Rhythm and pace are important in the constitution of society because they lead to the entrainment of individuals into collective movement and

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emotional alignment. Central to the production of rhythmic and temporal coordination is the living/lived body. Flesh is with contact, that is, in contact with the world. Contact is tact with: tact, the sense of touch. Related to the fi rst signification, tact also is the behaviorist term for an utterance evoked in contact with others and the world. Surprisingly, perhaps, tact not only refers to the sense and to a verbal response, but also to temporality: tact as stroke in beating time. We may create a new sense for the term contact, as the tact common to two or more people. Temporality, (gestural) beats, rhythm, sense, and verbal contact are some of the threads underlying this chapter. Rhythm can be understood as the production of form under the constraints of time and temporality. As form, rhythm is the product of self-reference, auto-affection. This allows a polythetic expression—i.e., produced across the different modalities of the body—to be constituted monothetically, as one unit (idea), in the auto-perception of the voice and movements of the speech organs in the organic body of the speaker. It simultaneously makes possible the monothetic constitution of the expression in the perception of another’s voice and auto-affection of the auditory organs in the organic body (flesh) of the listener. The constitution of semantic forms, sense and signification, thereby becomes indistinguishable from the auto-affection of the flesh. Affectivity then is the impossibility to distinguish the constitution of semantic forms from the sensation of one’s own flesh (Gumbrecht, 1988). In this way, rhythmic features come to constitute a “consensual domain of the fi rst order” on which language and the production of sense rests as a “consensual domain of the second order” (pp. 725–726). It is only when we can exhibit the irreducibility of these two orders that “embodiment” come to be plausible alternatives to constructivism. Because “embodiment” suggests the very division that we want to overcome, I suggest that incarnation in fact is the plausible alternative to all forms of constructivism. The upshot of the rhythmic basis of expression is that it does not make sense to listen to semantic content only. To listen only for the contents of words is like listening to psychologists or physiologists talk about hearing: They, being Cartesians par excellence, reduce this experience to recordings in the cilia and decoding mechanisms of the brain. They have already enacted the reduction that leads to the distinction of body and mind, as they concern themselves not with life but with the externalized modality of the material body. This leads us to a disembodied form of hearing. Living/ lived hearing, however, is something different, because “the spoken word— the one I proffer or hear—is pregnant with signification that can be read in the texture of the linguistic gesture itself” (Merleau-Ponty, 1960, p. 144). As a result, “a hesitation, a change of voice, the choice of a certain syntax is sufficient signification” (p. 144). Therefore, reducing a gesture to language or to some linguistically articulated schema—in the way embodiment theorists tend to do—leaves us with very little; and it leads us to accounts of knowing that are just as disembodied as the constructivist explanation. 3

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But if we were merely listening to the word-contents of the sounds that we make in communication, then teachers would hardly be able to make sense of students’ utterances; and, students, too, would hardly be able to learn. Even the simple sequence “this is a cube” could not be understood independent of other bodily expressions, which allow us to hear a question (“This is a cube?”), a constative statement of a fact (“This is a cube.”), an order (“This is a cube!”), and so on. These other bodily expressions also allow us to ascertain whether the speaker is certain, speaks with confidence, is intimidated, or tentatively proffers an answer. The temporality and rhythm of speech is integral to the expression and recognition of knowing. Hearing and speaking in real time means being attuned to and being in tune with the world across the entire spectrum of means that we produce and use to communicate, including movements, positions, prosody (speech intensity, pitch, speech rate, and many other aspects of speech), facial expressions, and so on. Moreover, it is not that we listen to the “meaning” of a word but rather, as Merleau-Ponty (1960) shows in his analyses, communicative intentions are written all over the situation. The way in which we orient corporally to the signification as speakers and audiences is implicit, “and does not suppose any thematization, no ‘representation’ of my body or the milieu” (p. 145). For example, Mrs. Turner and Mrs. Winter are listening for the pauses in the child’s speech, nervousness, expressions, thereby coming to hear what the child says not by decoding some sounds to hear words to be interpreted to find their “meaning.” In listening, they actually use their living/lived body to “take hold of a course of movements, which makes listening a course of activity itself” (Sudnow, 1979, p. 83). In listening, we do not have time to “interpret” what another person says, but we understand in hearing; we do not hear a sound and then “interpret” it as coming from a Harley motorcycle, but we hear a Harley approaching us or moving away. Rhythmic features structure the ways in which we communicate and, therefore, structure the ways in which we interact with others: They make possible interactions as the rituals on which society is based because rituals enable such material processes as entrainment, resonance, and synchrony. Thus, “successful conversational ritual is rhythmic: one speaker comes in at the end of the other’s turn with split-second timing, coming in right on the beat as if keeping up a line of music” (Collins, 2004, p. 69). Moreover, resonance does not only underlie the coordination between people, but more importantly, it underlies the constitution of sense: “sound and sense mix together and resonate in each other, or through each other” (Nancy, 2007, p. 7). A fundamental resonance therefore also is fundamental to (the constitution of) sense, which requires us to seek sense in sound, as well as look for sound and resonance in sense. Rhythmic phenomena, therefore, are not merely the characteristics of individuals but precisely of collectives: collective, ritualistic features mark interactions. This poses the question how rhythmic alignment is achieved in the face of the apparent demands that interpretation would make on cognition. In

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response to this question we may note that rhythmic phenomena are available throughout our bodies, in the beat of the heart, the circadian rhythms of sleeping and waking, diurnal hormonal changes, breathing, contractions of and along the entire digestive tract from esophagus to the colon, the vibrations of our vocal cords, and so on. Because rhythm is a carnal phenomenon, arising from the auto-affection of the flesh—the heart knows to beat in the way our hand knows and remembers to grasp after having achieved it for a first time—it is possible to produce rhythmic phenomena even though everyday situations are improvised, are not machine-like implementation of routines. It is precisely this spontaneity of communication and of the constitution of everyday life that is inaccessible to traditional psychology and metaphysical approaches to cognition. It is the experience of the rhythmic movement to itself that lies at the bottom of our experience of the constitution of sense. Rhythmic phenomena in humans and human relations cannot be thought as perceptive categories alone. Any given rhythm has to be thought as arising from the tension that exists between materially given, objective phenomena in the setting and the rhythmization of the perceived on the part of the listener: Such phenomena, as cognition, sit on the borderline between inside and outside, in fact, constituting the borderline itself. Rhythm has a performative dimension: It is actively produced as much as received and reproduced. Because of its performative nature, rhythm also implies disturbance, breaks, pauses, difference, and discontinuities—phenomena involved in syncopation. It is precisely for this reason that rhythm can be thought as the interplay of discontinuity and continuity, regularity and irregularity (Plessner, 1981); without the disturbance of rhythm, we would not be able to perceive rhythm—in the same way that we would not perceive figure without the continual change of the focal between figure and ground (see Chapter 1). Central to the present articulation of mathematics in the flesh is the intermodality of rhythm, which therefore is consistent with the phenomenological observation that my perception is not an intellectual integration, a sum of perceptions, but a holistic experience held together by tact. My living/lived body, the flesh, is rhythmically organized—it is therefore not surprising that many mundane expressions refer to rhythmic phenomena: “to be in/out of tune,” “to be in/out of sync,” “to have the groove,” “to have the beat,” or “I resonate with you.” Most if not all educators will be able to accept that communication involves more than words; and they will be able to accept that teachers use “information” from these other modes to make sense of classroom events. Similarly, educators will be able to accept that children, too, draw on these other modes in their attempts at making sense of what is happening. Especially those educators who commit to enactivist and embodiment approaches to cognition will emphasize all those other modes as being aspects of communication and cognition. However, this is not the point that I make throughout this book. In other approaches, the body

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continues to be thought separately from the mind in the sense that it provides the resources for metaphoric and metonymic extensions. These are bodies enabled by “embodied (mental) image schemas,” that is, bodies that already are viewed/thought separately from mind. That is, it is supposed that cognition is embodied in the sense that it is only through the body that mind can have developed. This, however, still allows the mind to be a separate entity, as the symbolic world now functions as if on its own. (Maturana and Varela’s machine metaphor does not help overcoming the divide between schematic plans and improvised situated actions.) The world of the body has been transcended in the metaphoric and metonymic extension and the bodily “schemas” now are no longer necessary constituents of thought or rather the sense we make in thinking. My point is different. We come to a truly performative account only if the body is a necessary condition of sense, that is, if the difference between the body of sense and the sense of the body is undecidable. In this case it makes no longer sense to speak of the embodiment of mind, because the very distinction takes us away from understanding the phenomenon in the same way that we do not understand light if we pit against each other the two forms in which it externalizes and expresses itself, as wave or as particle (corpuscle). The living/lived body is not a machine, and any patterned (schematic) account of concrete action is valid only after the fact, because human actions fundamentally are improvisations, including the acts of speaking. It is not a machine because its self-knowledge does not come from a feedback loop but from the auto-affection that leads to what Sheets-Johnston (2009) describes as the identity of thinking and movement. Moreover, it is not just a material body, as I argue throughout this book, but the flesh capable of auto-affection that constitutes a form of immemorial memory, which enables intention and symbolic memory. It is precisely because of this immemorial memory that we do not need schemas to do what we do and to do it, recognizably, over and over again. We perform the movements each time, not because of a schema that is somehow encoded in our mind, but because the movement can be repeated over and over again without requiring a representation—in the form of a mental or sensorimotor schema (plan). Just recall how my hands have recalled the telephone number, not because of a mental representation that it enacted, not because they were acting according to a schema, but because of an immemorial memory that even I was not aware of. Hand gestures employed in speaking are but vestiges of the history of thinking that also appear rather than being constitutive of the sense. Typically, we may fi nd approaches to integrating speech and gesture, where both are attributed to some cognitive model that is described by linguistic means. Thus, if those other means (modes) of communication come to be described by verbal means, then they have been reduced to language. Those other modes are subsidiary to language and linguistic means, which thereby come to be the essence of thought. Thinking is properly incarnate when all these other means are but aspects of a

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greater unit. This greater unit manifests itself in the various modes only one-sidedly. Therefore neither the whole nor the parts other than language are reducible to language. Moreover, we have to theorize thinking (moving) as a creative and spontaneous process the essence of which is not the enactment of a program (plan), the schema. Spontaneity etymologically derives from the Lat. sponte, of one’s own accord, and therefore denotes actions that spring forth entirely from their own (natural) impulse (rather than from a plan, schema). To capture the spontaneity of communicating and thinking, we need to theorize thinking and speaking in terms of performances, improvisations, which subsequently may be assessed in terms of the degree to which it corresponds to some score. It is not the schemas that cause the performance, like the musical score does not cause the musical performance, or like the recipe does not cause the particular events in a kitchen.

INTERLACING OF BODIES In Chapter 3, I articulate how in and through participation in society (corps social) we come to have socialized bodies (corps socialisé). What we do and signify is immediately endowed with sense because a socialized body produces it. This socialized body is the result of participation in social fields. By participating in social fields, we are shaped in a double way. First, our existing habitus allows/makes us see the social and material structures in particular ways, thereby shaping in which way we participate; and, second, our actions leave traces, change our bodies. We can see the continual adaptation of persons acting in this mathematics classroom to other persons and artifacts, themselves associated with cultural histories and a mutual shaping with cultural practices.4 Throughout this book, we encounter students in exchanges with other students in movement interacting over and with objects. During such exchanges, they listen, and in listening, they have to open up to be affected (to hear), which makes them exist outside and inside at the same time—and it is precisely in the capacity to be outside that their living bodies come to be fashioned. This mathematics classroom is a field—a field within the field of the school—where children are part of interaction rituals involving material structures and objects; and such interaction rituals entrain and therefore shape the children’s living/lived bodies. In fact, the physical discipline that our living/lived bodies are subjected to make them disciplined bodies, living/lived bodies that exhibit, in their practices, academic disciplines (Roth & Bowen, 2001). The children we observe in the preceding chapters of this book develop disciplined (living, organic) bodies capable of exhibiting the discipline required by the mathematical discipline they study. Accordingly, living/lived mathematical bodies are made in the process of doing mathematics in schools. The children sit on their chairs; and when they

164 Geometry as Objective Science in Elementary School Classrooms do not do so properly, they are reminded that sitting has to occur in a particular manner to be appropriate for this classroom. Thus, when Cheyenne has taken her T-shirt and moved it over the backrest—so that the latter directly comes into contact with her skin—Mrs. Winter reminds her, “Cheyenne, you need to put that shirt properly” (Fragment 4.3c, turn 75). Similarly, when Brandon’s feet “dangle” into the circle where there are geometrical objects grouped, Mrs. Winter reminds him, “Brandon, you gotta get your feet up” (Fragment 8.5, turn 33). The children sit, oriented toward the chalkboard, filled with writing or, sometimes, with posters that themselves are inscribed. Throughout this book, children interact with familiar objects, such as pizza and toothpaste boxes, Post-it blocks, tin cans, and cardboard rolls, which they learn to associate with the geometrical objects that the teachers introduce as “cubes,” “rectangular prisms,” “spheres,” “cylinders,” and the likes. In fact, what the teachers materially produce are sounds, and these sounds come to be associated with material objects that give rise to particular visual and tactile experiences. That is, sound, sight, and touch are the senses relevant to mathematics lessons; and what they produce and reproduce contributes to the constitution of mathematical sense.5 More important than the interlacing of the flesh with the material bodies of cubes, rectangular prisms, pyramids, spheres, and the like are those where the flesh, social bodies, comes to be intertwined with other social bodies. Thus, we see students engage in exchanges with other students when Jane talks to Melissa about how to model the mystery object inside the shoebox in front of them, their actions intertwine (Chapter 5). Jane pushes the shoebox to Melissa, who then reaches inside; or Jane and Melissa, taking turns, orient each other to the models they have built, sometimes working with different and sometimes with the same object. Interlacing occurs, for example, while Melissa is reaching into the shoebox while Sylvia iconically gestures the caliper configuration, thereby “directing” Melissa what she has to orient to with her hand in the shoebox. And interlacing of their bodies occurs when Jane touches the top of her model while Melissa explores it with her left hand and while exploring the mystery object in the shoebox at the same time. Such interlacing, which is an expression of coordination of living/lived bodies in movement, constitutes thinking in movement. We also see direct contact between a teacher and a student, when Mrs. Winter takes Thomas’s fi nger and moves it along the edges of the cube so that he can learn to identify the discourse of a “straight edge” with the visual and tactile impressions of a material straight edge (Chapter 4). At other times, a teacher holds an object such that a student can access it sensorily in particular ways. Thus, Mrs. Winter holds out a cube so that Thomas can see it in a particular way: Here, it is the face that he orients to and moves his fi nger across in response to the teacher’s instruction, “so which one is the straight edge?” (Fragment 4.3c, turn 66–68) or

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when she asks him “what kind of edge is that” while holding out the cube (Fragment 4.3b, turn 22–23). But the teachers’ gestures, too, leave their traces, for these are central aspects in/of the communication. In Fragment 4.1, Mrs. Turner moves her hands in a particular manner, palms facing each other and moving closer and farther away together with her talk about a rectangular prism being something like a fl attened cube (turn 13). We subsequently see students also produce similar hand gestures, such as when Ben shows what to do so that a Post-it pad would become a cube (Figure 7.1). In one instant, we observe Ben producing a hand gesture that iconically exhibits one side and the top face of a cube, and only fractions of a second later, Mrs. Winter also produces these two gestures (Fragment 6.2a, turn 08). And, to state a fi nal example, Chris and Mrs. Turner produce precisely the same hand gesture above the pizza box to show what would have to happen to turn it into a cube (Fragment 6.3, turns 41, 44). But we should not take this coordination as evidence for the fact that the audience interprets a speaker, because perceptual consciousness is very different from linguistic consciousness (Vygotsky, 1986) and therefore does not use the means and processes underlying interpretive processes that require linguistic consciousness (to produce explanation). Intellectual interpretation is unnecessary, because, according to Sheets-Johnstone (2009), such movement is thinking and no further intellectualization is required. We can understand such instances of interlacing as forms of resonance, where particular temporal patterns come to be reproduced. This reproduction sometimes is directly coordinated in exchanges between two subjects, who take complementary roles and places; sometimes this reproduction occurs later in time, as if it were the echo of a previous movement. Living/lived bodies come to be moving about as part of communication between participants that exhibit a complex interlacing of space and orientations. The living/lived body becomes an expression in this configuration. In the following fragment, Chris has turned his back to

Figure 7.1 Ben uses the same gestures as Mrs. Turner—in an episode involving Chris during a different lesson—to show how Post-it pads need to be stacked to produce a cube.

166 Geometry as Objective Science in Elementary School Classrooms

Figure 7.2 Mrs. Turner’s talk and gestures concerning a cube are coordinated with her turning toward a cube that is behind her.

the class and the teacher while placing the pizza box underneath a toothpaste box, the place from where he had taken it. Mrs. Turner makes a (summarizing) statement concerning rectangular prisms and cubes, “so that makes it a rectangular prism as opposed to a cube” (turn 17). She continues producing a clause: “so if it was a cube.” In the second part of this utterance, her upper body rotates sideways and to the right, her head turns even further so that her gaze comes to be directed to a cube currently placed on the chalk rest (Figure 7.2). That is, in this movement, her gaze moves from the pizza box to the cube and then reorients toward Chris and the pizza box. As she utters the fi nal part of this turn 17, she points in the direction of the pizza box. But Chris is still oriented to it so that he cannot see the deictic gesture (Figure 7.3a). Fragment 7.1 (from 4.1) 16

17

T:

18 19 20

C: T:

21 22

C:

(4.91) ((Chris places box, no longer looks at her, busy placing the box even after she has started again)) so makes it a rectangular prism as opposed to a cUBe, because if it was a cube what would it have to have (0.58) um [thiss ] that box have to have to be a cube. (0.77) it would have to um ((gets a cube from chalk holder)) (0.44) that squ:are: ((a)) or this (0.21) like the:(0.62)s:: ((b))

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Figure 7.3 Chris’s body and head/gaze movements come to be aligned with those of Mrs. Turner, as he re-orients to the pizza box after having just oriented away from it.

Chris then brings his upper body up, gazing toward the cube, which is, from his perspective, to the left and behind Mrs. Turner (Figure 7.3b). He comes up a bit further and then moves his head to the left, his gaze direction now intersecting with Mrs. Turner’s deictic gesture direction precisely at the pizza box (Figure 7.3c). Here, then, Mrs. Turner’s hand moves in a manner that can be recognized as a pointing gesture. The movement comes to be interlaced with the movement of Chris’s head and eyes. The two movements are oriented toward what may be called the focal artifact or object of joint attention. When we observe the instant in real time, it is so fast that we come to realize that any hope of modeling this fragment by an interpretive (constructive) mind, which takes in and processes information to subsequently produce an action, would fail. The constructivist mind would not have enough time to call up the embodied image schema that it then tells the body to enact. Such a model of human interaction and joint production of attention is computationally so expensive that it would take orders of magnitude longer to produce the event. “The position of another as other-than-myself is impossible if it is consciousness that has to do it” (Merleau-Ponty, 1960, p. 152). Linguistic consciousness is not only too slow but also would make contradictory demands in that it involves “constituting him as constituting, and constituting as constituting in respect to the very act through which I constitute him” (p. 152). Incarnation, Merleau-Ponty suggests, cannot be classified as simple psychological phenomenon precisely because it requires the perception of others. Chris’s gaze is directed for a while to the pizza box. Then, just as he begins a turn at talk, his head turns, gaze directed toward the cube (as

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previously) his right leg moves forward as he stretches to reach for the red cube on the chalk tray. As he comes close to Mrs. Turner, she moves her left leg backward, bringing her body around and a step away from the chalkboard, thereby creating space for Chris to get to the cube (Figure 7.4). Both are now oriented toward the cube, a cultural-historically charged object in the discipline of geometry. Their two bodies, as in a dance, have both oriented them to a geometrical object in a coordinated movement that makes space (teacher) and uses the now available space to access the object. In fact, when we look at their body positions, placements, and orientations across the sequence as a whole, we notice that as a result, Mrs. Turner ends up about one meter away from Chris, where a complex of leg movements fi rst takes her a step away from the chalkboard and then, as her right leg moves backward followed by a body rotation and a backward movement of the left leg puts her in a new position (Figure 7.5). In this new position, her body is oriented as previously in the direction of the class, with her head turned toward Chris. But the ensemble of geometrical objects on the chalk tray has now become perceptually accessible and reachable rather than being hidden behind Mrs. Turner’s body. Conversely, the poster featuring the parts of geometrical objects exemplified by a pyramid now, from the perspective of the class, now comes to be hidden behind her. In subsequent parts of this same lesson, when she makes reference to these parts, she moves again to make the visual access possible for all students. In these instances, Mrs. Turner’s and Chris’s living/lived bodies move together and in response to each other in the same way a pair of dancers move in a coordinated dance without having to intellectualize their next step or movement.

Figure 7.4 Chris’s movements are associated with complementary movements on the part of Mrs. Turner, as she turns allowing her to see where to Chris orients himself.

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Figure 7.5 Mrs. Turner, in a movement complementary to that of Chris, steps away from the position that she has held, thereby allowing Chris to take it up.

In this section, I exhibit the ways in which the movements of participants in mathematics lessons come to be interlaced, interact, produce interactions in their actions, and actions in interactions. A hand gesture on the part of one person entrains the head and body movement of another, a forward step and gaze direction toward a particular mathematical object comes to be entangled with the sideward and backward steps of another person, oriented not only to the geometrical object but to the space another person requires for accessing the object. But all these movements are so fast that they would be prohibitive in a model that is built on interpretation, a process that involves explanation and therefore is mediate rather than immediate; it would be too slow if movement were thought as enactment of embodied image schemas in the mind that order the body how to move. Such coordination of living/lived bodies in space requires, according to Merleau-Ponty, a form of consciousness very different from the linguistic one that is so important in constructivist theories. Merleau-Ponty’s findings have been confi rmed more than once in more recent neuroscientific and cognitive scientific studies. Thus, one study in cognitive science concerning the actions in the game of Tetris shows that if the player were to interpret the shape and position of the object currently visible on the screen prior to moving the falling object (rotating it, moving it sideways), the actions would be far slower than actually observed (Kirsh & Maglio, 1994). The authors offer a model in which actions have pragmatic and epistemic dimensions and are not the result of prior computations (interpretations) but rather serve to both act in and change the world and, in the process, develop a practical understanding of it. The Tetris player’s “epistemic” movement is thinking. A similar attunement to others, the setting, and to mathematical objects is observable in the events presented here, as there is a continuous

170 Geometry as Objective Science in Elementary School Classrooms reorientation between boxes understood as concrete examples of rectangular prisms and objects created to stand in for the mathematical idea of a cube. The capacity to be attuned to the changing objects of attention, both materially and ideally, means knowing to communicate mathematically, to understand mathematics in very practical ways. The different bodies that appear in my description are not equivalent. The cube on the chalk tray or the pizza and toothpaste boxes return to their places after having been used. Their part in the events leaves no traces in them. These bodies are unaffected—apart from some wear observable only over periods of time much longer than the curriculum. On the other hand, the living/lived bodies of humans are similar, their movements being coordinated, and precisely because it takes energy and because use wears them, the events come to be inscribed in them. As my long-term research in fish hatcheries showed, even the most “boring,” routine jobs—e.g., feeding fish, washing fish ponds—have observable long-term effects in the living/lived bodies of the workers, which come to exhibit increasing mastery of the work at hand: the way they move the scoop while throwing feed or what they can actually perceive in fish movement. That is, even though the effects are not immediately observable and noticeable, they bring about changes in the flesh of the workers, that is, they produce adaptation to the activity and therefore learning. Schema theory only describes the structure of the movement, not the qualitative differences in the movements that distinguish the different levels of expertise. These living/lived bodies, therefore, have to be theorized differently, they are not the same kinds of bodies; they are not bodies that come about in a modification by means of the adjectives “living” and “lived.” This is why I propose the concept of the flesh, endowed with senses, which is the source of the human body as recognized by the person inhabiting the body. The singularity of the “I” is itself the product of all the interactions, a result of the flesh that recognizes its organic body as a body among bodies. We learn because of the flesh, as our living bodies come to act in a social world, already structured in multifarious ways, including lifeless objects that are associated with particular cultural practices that themselves change in the history of the culture. Always acting in a world already full of significations, the flesh is in a social world all the while the social world is in the flesh. “The structures of the world are present in the structures . . . that agents mobilize in understanding. When the same history pervades both habitus and habitat, dispositions and position . . . history communicates in a sense to itself, reflects upon itself” (Bourdieu, 1997, p. 180). I noticed this tight interlacing of positions and dispositions (for teaching) while observing pairs of teachers work together over longer periods of time, about two to three months.6 The teachers came to resemble one another in what they were doing, from intonation, to use of specific words, to moving about and positioning in the classroom, and interactional styles. And nothing of this alignment had been present in the linguistic consciousness of the

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participating teachers until my research mapped it out for different modalities (e.g., prosody, positioning, movement, orientations, etc.). In the fragments featured here, we already have another dimension that is central to human interactions: time and temporality generally and the pacing of social events specifically. Thus, Chris’s and Mrs. Turner’s movements are not only coordinated in space but also, and more importantly, they are coordinated in time. It is as if a hidden choreography were at work, but a choreography that gives substantial amount of improvisational freedom to the performers and nobody knows future actions even seconds hence. Even the agents themselves do not know what they will be doing precisely. Mrs. Turner steps to the side, even though she did not know this only seconds before. But in stepping aside, she opens the access to the cube at precisely the instant that Chris is reaching for it. Both have already oriented to the cube, though neither has been in a position to note this orientation of the other, which was (partially) hidden because of the way in which they were currently oriented. (Chris gaze is directed toward the cube presently not visible to Mrs. Turner, and she orients to the cube while Chris is preoccupied by the attempt to get the pizza box back into its place.) Such coordination, rather than being the result of interpretations, can be better understood as resonance phenomena, thinking in movement. Resonance, a phenomenon of coordination of movement, leads us to the ritualistic aspects of interactions, which we thereby come to understand as interaction rituals. In the next section, I turn to a number of different features of interaction rituals that are all concerned with time.

INTERLACING OF PACING, PITCH, PITCH CONTOURS We are not tuned to sounds from which we extract words, but rather we hear words. However, we do not just hear words, but we hear words spoken tentatively, almost like questions; we hear questions even though the grammatical form of the utterance is a statement, and, conversely, we hear questions even though intonationally an utterances is produced like a statement; and we hear/see whether a student appears to know but cannot express herself, or whether the student does not appear to know and therefore the expression is not fluent. As teachers, we do all of this with a spontaneity that no constructivist theory can explain—even schema theory would require prior interpretation and the identification of a relevant sensorimotor schema that would then tell the body how to act. It is precisely from this spontaneity that we can learn something about knowing that we could not know otherwise. In the previous section we see how it is precisely because acting and sensing go together that people can coordinate their mutual orientations to objects of joint attention, body positions and orientations, gestures, and head movements. In this section we return to the same instant of the

172 Geometry as Objective Science in Elementary School Classrooms lesson but add a few more lines of transcript. One aspect I explicitly point out in Chapter 1 is the coordination of the hand movement and speech. Thus, in turn 14 (Fragment 7.2), the hand movements are coordinated with the verbal expressions (i.e., with movements of different components of the vocal tract, lungs, and so on); and where there might be a danger of misunderstanding, we observe the repetition of a gestural expression (i.e., movement) precisely coordinated with the speech expression (i.e., movement). Such coordination, further associated with the coordination of the living body in space and the coordination with other (living) bodies, would be a feat so complex that the computational mind processing representations would be lost in the face of the task. But this coordination is much more easily understandable if we think in terms of resonance phenomena, coordinated movements, where periodic features come to align with each other as long as the characteristic frequencies of the individual participants are not too different (in which case “synchrony” is not achieved). There is further coordination, however, which makes the interlacing of the bodies both tighter and more complex. This is why phenomenological approaches consider the entire living/lived body pervaded by tact as one expression, not merely as the expression of something thought beforehand and then made available to others. It is therefore that “my spoken words surprise me and teach me my thoughts” (Merleau-Ponty, 1960, p. 144). All of the different expressions I produce have an “immanent sense, which do not come from an ‘I think,’ but from an ‘I can’” (p. 144). This “I can,” before all thought and linguistic mediation, is a characteristic feature of the flesh. An expression therefore never is the expression of something else that went on before, thought, but the expression constitutes the very phenomenon of thinking. “For the speaking subject, expressing is to become conscious” (p. 146). This precisely is the central idea why cognitive scientists and artificial intelligence researchers interested in the role of space in cognition speak of epistemic actions: Actions not only change some aspect of the lifeworld but also constitute a form of making sense.7 Fragment 7.2 13

T:

14

C:

[it] could be like a cube thats bin (0.22) ((hand movements showing the closing of the distance between two palms)) flatten[ed] ((hands retract)) [uh] hm bu:t () ((lH moves down face of box; then holds box with lH)) s:::ohm::e are like long=an

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some are shorter than the other one[ss]

15 16

T:

17

T:

18 19 20 21 22

C: T: C:

((rH movement along right edge, then rH movement along edge pointing to Mrs. T, then again rH movement along right face)) ((looks up to her)) [uh] hm (4.91) ((Chris places box, no longer looks at her, busy placing the box even after she has started again)) so makes it a rectangular prism as opposed to a cUBe, because if it was a cube whAT would it have to have (0.58) um [thiss ] that box have to have to be a cube. (0.77) it would have to um ((gets a cube from chalk holder)) (0.44) that squ:are: ((a)) or this (0.21) like the:(0.62)s:: ((b))

22

Here, I orient readers to another phenomenon of central importance to the way in which social action and interaction becomes possible, and in which trouble comes to be known: features of speech, or, as it is known to specialists, prosody. Prosody includes speech intensity, speech rate, and, importantly, pitch (as well as other aspects of sound production, such as the fi rst major frequency other than pitch, generally referred to as F1). Previous research—my own and that of others—has shown that in confl ict situations, the pitch levels of two adjacent speakers tend to be very different, moving toward higher and higher pitch levels as the confl ict aggravates.8 On the other hand, in classrooms that are evaluated to exhibit a great degree of solidarity among students and with the teacher, the pitch levels of adjacent speakers tend to align. New teachers, after working together with the regular and more experienced teachers, tend

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to use the same prosodic features. New teachers generally and teachers new to a class specifically tend to become attuned to the students so that their pitch levels begin where the previous student speaker has ends; and students, in a similar manner, tend to fall into the pitch and reproduce pitch contours of the preceding teacher utterance. This is also observable in the present fragment, noticeable in the three turn changes (turns 13–14, Figure 7.8, turns 14–15, Figure 7.7, and turns 19–20, Figure 7.6). A typical example of the joining of the pitch levels between two speakers can be observed in the change over between turns 19 and 20. Mrs. Turner, who follows Chris, not only overlaps his speech but her pitch also aligns with his, leading to continuity in the melodic line of the two speakers (Figure 7.6). Another kind of “pick up” exists when the second turn employs the same pitch contour, leading not to a direct continuation of the main frequency but to the melodic line that is produced toward

Figure 7.6 Mrs. Turner’s pitch level picks up where Chris ends before returning into its normal range.

Figure 7.7 Mrs. Turner’s pitch contour repeats that previously available in Chris’s voice.

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the end of the previous utterance such as between turns 14 and 15 (Figure 7.7). The effect is that we hear the same melody produced in and by the two sound performances, which, depending on other aspects of the context, may be heard as confi rmation or as mocking of the previous speaker/utterance. Sometimes, however, the coordination becomes difficult, such as when, for example, the previous utterance continuously falls as in a statement only to suddenly rise at the end, heard as an emphasized word. That is, the intonation and the content of the utterance signal the unfolding of a constative, which allows an anticipated end point for the pitch at the end of the utterance. If the pitch, however, rises suddenly, not as the intonational flag marking a question but as emphasis, the next speaker is—unconsciously—in a quandary, having anticipated a lower ending of the pitch, which suddenly ends high. Perceptive consciousness, being of different nature than verbal consciousness, allows a rapid solution to the problem. One of the ways in which the problem may be solved is to begin halfway between the two, the anticipated endpoint of the pitch and the actual endpoint. Such a situation occurs in the change over between turns 13 and 14 (Figure 7.8). Chris comes in with his pitch about halfway between the projected endpoint of the utterance that precedes his own and the actual endpoint, by means of which Mrs. Turner emphasizes the word “flattened.” Emphases may be produced by different means, such as rapid momentary increases in pitch levels, by means of noticeable increases or decreases in speech intensity, or by means of changes in speech rate. But all of these features also are involved in the production of another structure of classroom life, rhythm, which my own research in innercity science classrooms is a way of aligning and coordinating members

Figure 7.8 The sudden and non-anticipatable jump and rise of Mrs. Turner’s pitch level creates a problem for the subsequent pitch level, here “solved” by a pitch that lies halfway between the anticipatable pitch ending and the actual pitch ending.

176 Geometry as Objective Science in Elementary School Classrooms of the same community and to both express and produce a sense of solidarity. In the next section, I turn to precisely this feature, which produces phenomena in which any conceived difference between some mind and a body becomes undecidable. In other words, there is an alignment of different forms of consciousness that cannot be reduced to each other.

RHYTHM AND PACE Musica est arithmetica nesciendi se numerare animi (Music is the arithmetic of the soul that is unaware of its own counting). —Gottfried Leibniz (quoted in Sambursky, 1973, p. 172) Rhythm is the incarnation of a formal cognition in an intuitive content. (Abraham, 1995, p. 24) Once we train our ear to hear rhythmically, we can begin to recognize the intricate complexity of communication in real time. I use but two instances from the fragment exhibited in the previous sections to illustrate how pacing and rhythm are produced together with mathematical content. The difference between the production of punctuated movements and the conceptual production (i.e., thinking) is undecidable. (The vocal production that lies at the origin of the words we hear is itself a completely corporeal phenomenon.) Rhythm is not just added to the speech: It is the rhythm of the speech and therefore cannot be perceived separately or detached from speech. Rhythm is not an epiphenomenon but it is the phenomenon without which speech would not exist. In communication there exists a “constitutive tension between the phenomenon of ‘rhythm’ and the dimension of ‘sense’” (Gumbrecht, 1988, p. 715, original emphasis). The rhythm is immanent to the production of communicative expression. But I can become aware of this rhythm, such as when I use meter notation (i.e.,–, ‿) to indicate on which syllables there are major emphases and on which there are minor emphases. In this sense, rhythm can be abstracted and represented with beat gestures, tapping, or by means of appropriate and widely used signs. In this case, the rhythm has been separated from its substrate and has become a transcendental object in its own right. The point is that in the production of speech, which is impossible to reproduce here, the rhythm is an immanent feature and integral part of the production of mathematical communication. In a strong sense, therefore, to understand how a student or teacher understands, readers have to reproduce the communication using the fragments in the same way that a musician would use a score to play a piece of music. Readers have to reproduce for themselves the phenomenon by enacting the transcriptions, including the stress patterns indicated; and thereby, in their own performances, readers experience these immanent features.

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In turn 14, Chris produces a next turn to the constative statement that the pizza box, denoted as an instance of a rectangular prism: “could be like a cube that’s been flattened” (turn 13, Fragment 7.2). I already exhibit above the coordination between the expression of geometrical content— the sides that are of different length therefore making this pizza box something different from a cube that has been flattened—and the hand gestures. In this turn 14, we also observe the production of rhythm. Fragment 7.3 14

C:

[uh] hm bu:t () ((lH moves down face of box; then holds box with lH)) s:::ohm::e are like long=an

some are shorter than the other one[ss ]

This fragment may be transcribed using meter notation, which exhibits rhythm, including the syncopation that occurs. We can see that there are two phrases, and across the phrases the da-da-dam (‿ ‿ –) rhythm typical for constative statements, where the “da” (“‿”) and the “dam” (“–”) correspond to weak and strong emphasis, respectively. | 1.11 s | 1.44 s | uh hm but () s:::ohm::e are like long=an

‿ ‿

–

‿

‿

–

‿

–

| 1.10 | 1.41 s | some are shorter than the other one[ss ]

–

‿

–

‿

‿

–

This is a three-quarter rhythm with the strong emphasis on the third beat. In questions, the strong emphasis tends to occur fi rst with the two

178 Geometry as Objective Science in Elementary School Classrooms weak emphases following. In the two phrases, Chris also produces gestures, one along each of the two orthogonal sides of the rectangular pizza box. The end of each phrase falls together with the main gesture. In the second phrase, there is a gesture that precedes the principal one, which looks the same, but it comes “off beat.” Perhaps not surprisingly, it is repeated so that it comes to fall together with the right place; but this surprise itself is surprising, as the whole performance has ended faster than even the beginning of an intellectual interpretive process would allow. Both main phrases are of almost the same length—differing by only three one-hundredths of a second. A close look at the second phrase reveals that there are more syllables, but, as the corresponding timing measure exhibits, this greater number of syllables is produced in the same amount of standard time. In a musical notation, this corresponds to two lined eighth notes (♫).9 To really appreciate the performance, readers ought not just silently read transcript and its rhythm for its cognitive content, but actually perform it, read it aloud while stressing as the meter indicates. Only then does the rhythmic consciousness, which is a performative, actually cognize what is happening: “To be able to speak about rhythm, one has to actually produce it, coproduce it in perception, in the action of the body, duplicate it, experience the process of the rhythmic in ones own living body/flesh” (Brüstle, Ghattas, Risi, & Schouten, 2005, p. 27). We also note the two syncopations preceding each main part of the respective phrases, which constitutes these parts as off the regular da-da-dam beat (i.e., ‿ ‿ –). However, overall, a new rhythm emerges, in which the syncopated and un-syncopated parts are integrated to form a new rhythm that is repeated in each of the two lines (Abraham, 1995). Together with this rhythmic moment of the performance, Chris produces a semantic-conceptual moment. Here, it is a correction of a constative statement that Mrs. Turner has made as her summary of what he has said before. Following the marker of contrast “but,” Chris describes and gesturally highlights the differences that may be observed in the sides. The two parts of the comparative statement, each pertaining to one side of a box, have the same overall rhythm. The rhythm is structured as the composition of a regular bar and one with syncopation (missing of a beat of the 3/4 rhythm, or a 2/4 beat). Aligned with the content and rhythm are the hand/arm gestures. In this performance, therefore, the difference between purely mathematical content and purely bodily performance is undecidable; speech with mathematical content and rhythmic movement of speech are but two sides of the same coin. Mathematical communication cannot be reduced to conceptual content; it is irremediably tied to the living/lived body, its rhythms and melodies. This a social body that is inherently in an interaction ritual, for which the communication is produced; but as expression, it is thought, personal and social thought simultaneously. This thought, however, generally is not cogitated beforehand but is the result of an improvised movement expression, which, as it unfolds in time, leads to the movement (i.e., development) of thought.

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Teachers, too, produce highly rhythmic mathematical communication, and, in fact, it is one aspect of the structured classroom setting that is structured by and structures the actions of the children. These structures, in turn, entrain and thereby change the living/lived bodies lastingly. This can also be seen in the repeatedly analyzed fragment, from where I extract Mrs. Turner’s turn 17. Fragment 7.4 17

T:

so makes it a rectangular prism as opposed to a cUBe, because if it was a cube whAT would it have to have

Across the complex utterance—which consists of a statement, a comparison, a clause, and a question—we observe the 3/4 tact with the major stress at the end (da-da-dum) or at the beginning (dum-da-da). We see, for example, in the last part (i.e., the question), that the “what” is stressed, as this is typical for grammatically marked questions, followed by the two unstressed beats, which is the classical meter of dactyl: dum-da-da. This pattern is repeated once, falling together with the grammatical end of the sentence. The segments are of approximately the same length, though the syncopation that occurs in the fi rst verse lengthens it; this syncopation occurs as the main emphasis is missing, and we observe the beginning of the da-da-dum pattern—the classical meter of anapaest—that is constant for the remainder of the performance. (1.69) | (1.51) | so that makes it a a rectangular prism | as opposed to a cUBe, | ‿ – ‿ ‿‿‿– ‿‿ ‿‿ – | – | ‿ ‿ – | (1.45) | (1.24) bcause if it was a cu::be | what would it have to have| ‿ ‿ –‿ ‿ – ‿ ‿ – ‿ ‿ | | –

In the second line, we also observe a syncope, which occurs between the conditional clause that constitutes the fi rst half and the question that constitutes the second half of the line. Throughout this book, we can observe questions that are implemented by an emphasized interrogative that begins the phrase or utterance, with the pitch falling toward the end as it is generally observed in constative utterances. From a rhythmic perspective, the interrogative “what” would fall on a weak emphasis, but there is a strong emphasis because of the question to come. This unexpected strong beat changes the rhythm and we therefore get the syncopation where two strong emphases follow each other, enabled by a particularly elongated production of the word “cube,” which by itself takes 0.53 seconds. In both lines of the transcript, the faster phrase is heard as a clause, and this hearing

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is brought about by the faster than normal pace (allegro). As before, the conceptual aspects are inseparable from those that are purely related to the production of the utterance. Mathematical content, clauses that explicate, comment upon, or modify the main clause are recognizable in and by the different prosodic means that implement them. These features precisely exhibit the competence, which would not be detected if a speaker were to confuse or interchange main with subsidiary clauses, produce discoordination between gestural and verbal production. In fact, when gestures and speech are misaligned, researchers recognize both a lower level of cognitive development and the readiness to proceed to a higher level.10 That is, coordinated expression means that the living/lived body as a whole produces canonical or non-canonical mathematics, recognizable as such precisely in the alignment of the performative aspects of mathematical communication rather than by some form of magical appearance of inaccessible mental constructions and mental representations. Mrs. Turner, too, produces gestures while speaking even though Chris cannot see them—not surprising, perhaps, given that even congenitally blind individuals gesture when communicating with other congenitally blind individuals. She opposes the palms of her hand and produces the same gesture three times, the fi rst time with about half the amplitude as the second and third time. The transcript with the rhythms also indicates the beat part of this gesture (the closure), the reverse occurring in the non-underlined parts separating the two closing gestures. We observe that the main gestures—the ones with the large gap closing—fall together with the main emphases, occurring together with the sound heard as the word “cube.” In a way that bears striking resemblance with the performance of Chris, there is an offbeat gesture, and it is repeated in the same phrase together with the main emphasis. How is such coordination possible? Phenomenological analyses suggest that we take position in and orient to the world, and the living/lived body as a whole becomes expression. It constitutes an “I can” that is spread throughout the flesh rather than limited to this or that organ the movement of which needs to be coordinated with another organ. This is so because “I engage myself with my body among the things, they coexist with me as incarnated subject, and this life among the things has nothing in common with the construction of scientific objects” (MerleauPonty, 1945, p. 216, emphasis added). It is precisely habitus—immemorial memory—that allows the world to emerge in and from our actions by the manner “it orients to the world, brings to an attention that, like the one of the long jumper who concentrates, is an active and constructive bodily tension toward the imminent future” (Bourdieu, 1997, p. 173). In the previous fragment, we see how rhythmic features characterize the speech, as they are produced with and structure the same material that produces sounds, which we hear as words. These rhythmic features cannot be thought apart from the material that makes the sound. But, we may also observe explicit repetition at the level of language, generally structured a

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second time by the rhythmic features in the body of the sound. The same explicit features can also be found in other modes of communication, such as when the students’ (and teachers’) gestures take up and are shaped by the same recurrences in structure. Thus, for example, in Fragment 7.5 (which repeats Fragment 6.2b), Mrs. Turner, in producing the fi rst part of a question-answer sequence, utters the same word four times (turns 57, 60, 63). In turn 57, she produces three series in each of which the word “square” appears four times, “square, square, square, square.” During the fi rst sequence, her index fi nger twice moves along the four top edges of Joel’s cube model, each square circumference gesture coinciding with the performance of a “square, square” articulation. As she utters the second sequence, her index fi nger repeats the circumference gesture, including three 90° turns, making the gesture recognizably consistent with the “square” description that she utters. During the third series, the video shows Ben holding up his model, exhibiting the caliper configuration repeatedly while rotating the cube. Fragment 7.5 (Fragment 6.2b) 57

T:

58 59 60

J: T:

61 62 63

J: T:

64

((:B)) hOW did you know; im looking at your model here. ((points to Joel’s model)) (0.50) and (0.32) this looks like square square square square how did you know t do square square square square; hOW [did you know square square square square; [((Ben brings up hands with cube, gestures, Fig. 6.1)) (1.18) ((Mrs. T shrugs shoulder, looks at Joel)) um square square =not rect]angle rectangle rectangle like (0.30) thEY ((points to back of classroom, Ben & Joel turn around)) have rectangle rectangle rectangle or oblong oblong oblong; right jonathan? (0.24) yea. so () how did you know to go square square square square square; (0.33) all along the sides and not o:blONG o:blONG o:blONG; ((moves head in same rhythm)) (1.36)

There is a longer pause. Joel then utters “square” twice, immediately succeeded by Mrs. Turner, who now produces twice a sequence consisting of three “rectangle” utterances and one sequence of three “oblong” articulations, which describe the model of Jonathan’s group. She then turns again toward Ben and repeats two sequences, one consisting of five articulations of “square” and a second sequence, which, in contrast, consists of three articulations of “oblong” (turn 63). The fi rst sequence is accompanied by

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a beat gesture where the left-hand index finger produces four beats, while apparent to all pointing to each of the four vertices of a square. Beginning with the fi fth and last “square” and into the subsequent pause, the head produces four beats. The performance that produces the three iterations of “oblong” each includes a precisely coordinated beat gesture of the head, which, with each beat, also moves further to the left as if it were inscribing the beats on an invisible sheet of music (though in the reverse direction of the normal notation). That is, the repetitions of geometry words are precisely coordinated with each repetition of a beat gesture (fi nger, head), that is, movements of the living/lived body. When we now do an analysis in terms of the strong and weak emphases, the highly rhythmic aspects of Mrs. Turner’s performance become clearly evident. As an example, I represent turn 57 using the poetic meter notation. hOW did you know; im looking at your model here. (0.50)

‿

–

‿

‿ ‿

–

‿ ‿

–

‿

–

‿

and (0.32) this looks like square square square square

‿

.

‿

‿

‿

–

‿

‿

–

how did you know t do square square square square;

–

‿ ‿

–

‿

‿

‿

‿

–

hOW [did you know square square square square;

–

‿ ‿

–

‿

‿

–

‿

We notice in the fi rst line of the following representation how the question is structured by the familiar dum-da-da rhythm. There is a break as the next part comes to be a clause (“I’m looking at your model here”). The clause is hearably produced by a substantial change in speech intensity (volume), which decreases to 1/4 of its original (from 75.7 dB as the mean intensity over the fi rst part to 69.7 dB as mean intensity of the clause part of the utterance). The clause is produced, not surprisingly given its constative nature, in the typical da-da-dum rhythm. It turns out that in this situation the rhythm is maintained across the break as the previous part ends on the strong emphasis in the word “know.” Here, then the change in speech intensity affords the perception of the ending of the clause. In the second line, the constative is produced embodying the da-da-dum pattern, the third line, which is marked by the grammatical interrogative “how,” follows the dum-da-da pattern in the same way as line four. In each case, there is a weak emphasis on the fi rst two articulations of “square,” followed by a strong emphasis on the third iteration of the word, and ends on an utterance of the word with weak emphasis.

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We note a change to three iterations in the sequences constituted by “rectangle” and “oblong.” But we observe a simultaneous shift in the rhythm, which now takes on the dum-da-da form (REC-tan-gle), certainly entrained by the normal stresses of the word in the English language (i.e., ‘rɛktaŋgl). The musical rhythm changes to the dum-da entrained by the normal stresses of “oblong” (i.e., trochee in OB-long). In the real time production of speech, there is an interaction of the musical meter with the metrical feet typical of poetry. But in poetry, especially in its written form, there are no pauses, no constraints that correspond to those associated with movements in space and the time required to complete a movement of the living body that interacts with the speaking movement. We therefore observe pauses in the speaking mode while other events occur, such as making a few steps to pick up an object. Pauses are places for syncopation to occur, where one rhythm changes into another one and where the transition cannot be anticipated and the instance of the syncopation, the turning point or bifurcation, belongs to both rhythms. On the other hand, rhythmic and other prosodic features, such as pitch and pitch contours, may be precisely timed with hand/arm or other bodily gestures (nods, pointing with chin) and other movements. Sociologists have identifi ed entrainment as an important aspect of coordination among members of a collective, with strong affective components (Collins, 2004). The term entrainment derives from the physical sciences, where it has been noticed that two pendulum clocks mounted on the same wall not too far from each other tend to align the frequencies of their pendulum swings. The two pendulums, mediated by the common mount, constitute a coupled oscillator system with the same frequencies for both; but the two clocks are in anti-phase, that is, when one pendulum is at its right-most position, the other is at its left-most position. Entrainment makes sense as soon as we accept the materiality of communication; and materiality comes with the possibility of consensual domains of different order. The primary domain is that on which the coupling (synchronization) between members to the interaction occurs, that is, the periodic phenomena (pitch, F1, rhythm), and at the second consensual level do we get the coordination at the level of sense. Resonance, as Nancy (2007) suggests, is the foundation of sense. In my own research referred to above conducted among teachers who teach together for a few months, I observed the entrainment of periodic features at different levels. For example, two teachers move in anti-phase with respect to positions they take up in the front of the classroom; their pitch levels adjust to one another, and their pitch contours take on the same shape. In this second-grade mathematics classroom, we fi nd in children’s gestures the same kind of rhythms that are characteristic features of the teachers’ talk. For example, we observe children who do not point

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to just three but four of the faces of a cube. In Chapter 1, we encounter Chris, who produces an answer to the question about other features of faces of geometrical shapes not yet mentioned in the whole-class conversation. Fragment 7.6 reproduces the turns in which Chris shows and articulates that on the cube there are squares “everywhere,” whereas the corresponding indexical gestures are reproduced in Figure 7.9. (Chris, Cheyenne) Fragment 7.6 (Fragment 1.1) 22g

23

an like ((rectangle)) (0.68) ((movement along edge of box)) ijst has (([a])) square (([b])) (0.45) n here= (([b])) =an=here= (([c])) =an there=(([d])) =and everywhere (([e])) (0.63)

In turn 22g, we observe a fi rst constative statement “it just has square” with a double indexical gesture consisting of a weakly and a strongly emphasized movement (beat) of the curved right hand to the face. There is then a repetition of the indexical gesture to the same face, falling together with the emphatic part of the utterance “here,” followed by the rotation of the cube and consecutive indexical gestures to other faces simultaneous with the productions of “here,” “there,” and “eve”[rywhere]. There are therefore four indexical gestures (underlined parts of words) falling together with four strong emphases, as shown in the following transcription of the rhythm.

Figure 7.9 Chris produces the same four-beat structure in his gesture while explaining the nature of a cube that Mrs. Turner enacts by prosodic means.

Doing Time in Mathematical Praxis ijst has

‿

‿

185

square (0.45) n here=an=here=an=there=and

–

‿

–

‿

–

‿

–

‿

everywhere

‿

–

–

In this representation, we note how the rhythmic features in the vocal production parallel those in the gestural movement, the strong emphases in the former falling together with the indexical gesture (underline), whereas the weak emphases fall together with the retraction of the right hand and rotational movement of the cube to exhibit the next face. The one indexical gesture produced together with a weak emphasis itself exhibits considerably smaller amplitude of the movement toward the cube face than the other iterations. The same phenomenon can be observed in Fragment 7.7, excerpted from a discussion concerning the group of objects that Ben, Cheyenne, and Ethan collected to go together with the “geometrical standard,” a cube, that they had received as the fi rst instance of their collection representing the category. Fragment 7.7 68 69 70 71 72 73 74 75

T: C: E: C:

76

T:

77 78

C:

[but its] nOT a cube because why (0.98) because its not like ((picks up red cube)) (1.14) ((holding up yellow cube)) like having sides (0.72) like thIS: ((strikes each of x, y, z faces)) (2.66) ((Cheyenne places palm on side of red cube)) doesnt have all those different sides. okay. thank you. (0.35)

Producing the response-part to the answer slot Mrs. Turner’s utterance (i.e., “but it’s [the Post-it pad] not a cube because why”) leaves for any one of the three students to fi ll, Cheyenne begins, “because it’s not straight,” while picking up a cube from the collection. There is a pause, which Ethan, holding another cube, brings to a close uttering “like having sides” (turn 72). There is another pause, which Cheyenne brings to an end announcing, “like this,” and then indexically denotes four faces by striking a face and rotating the cube to expose the next one to be struck (Figure 7.10). The indexical gestures are punctuated and highlighted by long retractions of the hand, producing both rhythm and particular strong emphases for each movement. That is, in this performance,

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we observe the same four repetitions of the weak-strong pattern as in Chris’s presentation of the cube—interestingly contrasting what one might expect for the three orthogonal faces of the generalized rectangular prism (those parallel to the x-y, y-z, and x-z planes in the Cartesian coordinate system). Perhaps even more surprising, as Cheyenne and Ethan both respond verbally to Mrs. Turner, they both strike faces of their respective cubes with

Figure 7.10 Cheyenne produces a four-beat structure while explaining the characteristics of a cube.

Figure 7.11 Cheyenne’s and Ethan’s performances of the four-beat structure are precisely coordinated.

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the palms of their right hands. Without gazing in the direction of the other, each directing the gaze at the cube in his/her own hand, they strike a face of the cube in perfect synchrony (Figure 7.11). That is, their respective striking of the faces enacts the same rhythm until this as occurred for the fourth time. At this point, Ethan raises his head, turns it slightly to his right, and directs his gaze at the cube that Cheyenne has in her hands. Not only do they each produce the fourfold pattern, but also they do so in synchrony (in phase). Strong rhythmic features can be observed throughout the lessons associated with situations other than indexical gestures. For example, when telling Thomas that he could not use shape words, Mrs. Winter (Chapter 4) rhythmically lists the shape words while counting with the hand, which is rhythmically beating while counting from one to three beginning with the thumb to which index and middle fi nger are added with each beat (Figure 7.12). Fragment 7.8 (Fragment 4.3) 49

W:

right; not using a sh:ape word. (0.22) so we=re not using a tRIangle CIRcle or squARE. ((rhythmically counting, beat gesture))

Figure 7.12 Cognitive content—counting of geometric concept words—is aligned with the iconic (counting) and beat (scanting) hand gestures and with the production of the rhythmic pattern.

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The transcription that includes the indication of strong and weak emphases exhibits the rhythmicity and the coordination of geometrical concept words, emphases, and counting beat gestures. We note in the second line of the transcription the coincidence of strong emphases and counting beat gestures. We also note the similarity in the structure of a strongly emphasized “not” followed by another stressed syllable (USing, GONna), which constitutes an instant of syncopation, as the anticipated rhythm changes, now beginning a familiar dum-da-da. In the fi rst line, we observe another syncopation as two consecutive syllables receive strong emphasis (shAPe, word), a sequence that we can hear as fi lled with tension. The time per syllable ratio is 0.27 seconds/syllable, which is approximately the length of the pause following the utterance. right; NOT USing a sHApe wOrd (0.22)

‿

–

–

‿ ‿

– – | |

‿

1 2 3 so we=re NOT GONna use=a tRIAngle a CIRcle or squARE.

‿ ‿

‿ ‿ ‿ – ‿ ‿ – – ‿ – – | |

In the second line, the dum-da-da pattern is maintained to the end of the utterance. In fact, the parallel in the performances of the fi rst and the second line projects the coming of a list, which its own realization both at the conceptual and at the rhythmic levels. That is, the rhythm allows anticipating the list to come, sustaining its unfolding, and hearing its ending (Selting, 2004). However, the pace is different, as the speech rate increases to 0.18 seconds/syllable. We hear this as a much faster paced speech. Grammatically, the phrase is merely an elaboration of what has been said before, at a slower pace, which gives the constative statement concerning what is not to be done greater gravity. It has been noted that practical understanding—as its correlatives of mood and attunement—temporalizes itself in anticipating the future based on past experience (Heidegger, 1977b); that is, in other words, practical understanding here expresses itself in appropriate changes of the rate of movement. We observe this at operation here, where the practical understanding underlying the performative and semantic aspects of the expression temporalizes itself, that is, not only produces itself in time but also produces temporality itself. Here we see an integral performance, where incarnate features of talk with non-cognitive content and features with cognitive content are irremediably intertwined. It no longer makes sense to speak of these as two aspects, but they may at best be one-sided expressions of the same phenomenon produced in and by the flesh. One performance produces the sound material, part of which tends to be considered in traditional research as if it were a metaphysical matter, that is,

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precisely a non-material matter. But the non-cognitive mo(ve)ment of the performance not only is inseparable from but also irreducible to the cognitive mo(ve)ment.

OF RHYTHMS AND MELODIES: MATHEMATICS IN THE FLESH AND IN COLLECTIVE BODIES [M]elody—is capable of exploiting the fascination created by the rhythm and of playing an imaginary duration on the tightened strings of consciousness. (Abraham, 1995, p. 24) In this chapter, we observe phenomena that are not explicable in most if not all currently available theories about knowing and learning. We observe events in which the difference between body and mind, the non-cognitive and the cognitive, becomes undecidable. It therefore makes no longer sense to maintain the Cartesian division inherent in the concept of the embodiment of mind; it makes no sense to maintain the idea of embodied image schemas as the source of the expressive movements in mathematical communication. In fact, the introductory quote to this section suggests that (verbal) consciousness exploits the rhythm features, a fact that is evident in the previous section (e.g., the difference between main and subsidiary clause). Non-cognitive rhythmic, prosodic, gestural, and motive phenomena are constitutive of the cognitive phenomena. The same sound material that psychologists and (mathematics) educators traditionally have reduced to “words” and their contents also are the material for non-cognitive rhythmic phenomena that require different forms of consciousness, including rhythmic consciousness and perceptive consciousness. The mathematical content—supposedly contained in or denoted by the words—actually piggybacks on the other phenomena: It cannot be thought independently from these others. It is therefore better to think cognition—the one we are familiar with in traditional psychology and (mathematics) education—as subordinate to a more comprehensive unit, for example, activity in the activity theoretic sense. There are two important ideas, therefore, that arise from the types of analyses that I exemplify here. First, the rhythmic, prosodic, gestural, and motive phenomena from which cognitive phenomena cannot be separated are consistent with the enactivist claims that knowing is performed and cannot be separated from moving about and participating in the world. However, I already point out in Chapter 3 that enactivism falls short because it does not explain the emergence of movement intention itself, which is done in the radical phenomenological approach. The immanent capacity to move—i.e., an immanent “I can move”—is the source of intentional movement. Because there is no place for theorizing the irreducible nature of (organic) body | (metaphysical) mind, traditional constructivist

190 Geometry as Objective Science in Elementary School Classrooms theories—from Kant to Piaget and (radical, social) constructivism—fall short in providing useful explanations for knowing and learning where the contents of words are subordinated to more comprehensive phenomena. Embodiment theories, too, appear to allow cognitive phenomena to transcend the body, as metaphorization and metonymization transcend the bodily schemata that are the origin of thought. This, therefore, requires a very different approach to thinking mathematical concepts and conceptions. In Chapter 10, I propose such a new way, where the mathematical sound-words stand in a synecdochical relation to the conceptual performances that they name. The forms of knowing and cognizing—those that we tend to consider independent of everything else that happens in a human body while communicating mathematical ideas—are something like the melody that exploits the fascination created by the rhythms in and of the body, not the rhythms that our linguistic consciousness points out but those that we produce and recognize with rhythmic, perceptive, and other forms of consciousness. Second, the phenomenon of entrainment shows that the evolution of mathematics in the flesh is not the result of the actions of the subject; it is not the result of an individual construction; it is not the result of an agent who changes as a function of the coupling with the world; and it is not the result of some subject’s metaphorization and metonymization of individual bodily schemata. Rather, each individual student functions in concert with others, and, importantly, with the teacher. We see how the various frequency-related phenomena—rhythms in gestures, voice, body orientations, and body positions, pitch, pitch contours—shape the collective activity; and this shaping is indissociable from affect. In being shaped, affected by others, we are literally affectively attuned to others and the field that we jointly constitute. That is, there is a coordination of the individual participation with the collective body and coordination occurs at a level not accessible or ruled by linguistic consciousness. Any bodily schema, if it actually were to emerge and exist, therefore would have an irreducible collective dimension. That is, the individuals are not free in developing just any form of knowing—by means of the traces that their participation in joint activity leaves in the individual auto-affecting flesh—but their knowing is constrained by, and takes on the aspects of collective knowing. Precisely because the body is exposed to the world—outside of itself with its senses—it can be lastingly impressed by the world. This impression occurs by means of resonance phenomena—i.e., phase synchronization—that are possible because of the periodicities that underlie the production and perceptive reproduction of communicative phenomena in both the primary and secondary consensual domains. This may become clearer with the following examples, one from the animal world, the other from a study I conducted in science classrooms. In southern Asia, observations of fi reflies conducted during the 1960s have shown that as night falls, swarms form as the fl ies gather in trees.

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Initially, they fl icker at random. But as the night goes on, the individual fl ickering comes to be coordinated with the flickering of others so that in the end, the flickering of the entire swarm is coordinated until all of them fl icker in unison. In this situation, we observe coordination across the collective at a scale much larger than the two clocks on the common wall. But such coordination does not occur between the swarms of different species of fi reflies. A similar phenomenon of coordinated rhythms has been observed in science classrooms (Roth & Tobin, 2010). The rhythms projected by a student talking from the back of the classrooms could be observed in the rocking of a leg, the beating of a pencil on the desk top, and in other body movements of students who could not see the speaker. As the latter changes the rhythm of her speech, the rhythm of the other students and in other modalities changed as well. Thus, as the rhythm accelerated while a confl ict between the student and her teacher evolved, the embodied bodily rhythms of other students accelerated as well. Conversely, as the rhythm in the student’s verbal articulation slowed down, the rhythmic performances of the other students slowed down as well. Interestingly, therefore, the synchronization of rhythms occurred across individuals and across modalities. Of course, such coordination especially across the changes could not occur if the other students had to extract the rhythmic phenomenon from the student’s performance, interpret it, and then implement their own rhythm. In the end, therefore, I observed a sort of implicit collusion of all those who have been formed and have formed the same field, that is, have been exposed to the same or similar societal and material conditions. “This collusion is the foundation of a practical intercomprehension, the paradigm of which could be the understanding that establishes itself among the partners on the same team, but also, despite the antagonism, between the players that are involved in the same match” (Bourdieu, 1997, p. 173). The practical comprehension therefore constitutes an esprit de corps, a visceral adhesion of the individual socialized body to a society, that is, a social corps (corps social). Rhythm and pitch play important roles in interaction rituals, appealing to and having sense (value) in the affective tonality that they call forth. As the term highlights, interaction constitutes a ritual; and rituals are build on cyclic phenomena, including rhythm (beat) and pitch (speech frequencies). As periodic phenomena, they enable entrainment, that is, the coordination of the frequencies among different “subsystems” that are but two sides of one whole irreducible “system.” All participants contribute to the emergence of the social phenomenon so that it would be as inappropriate to ask how any one individual contributes as it is to ask what the sound is of one hand clapping. The importance of rituals, rhythms, beats, and sound frequencies comes from the fact that they necessitate the individual human being to produce these in a manner that does not require cognitive cross-modal coordination and it provides the means for coordination within a human collective.

192 Geometry as Objective Science in Elementary School Classrooms The individual is a whole, only parts of which are cognitive. This cognitive part is but a part of the whole, subject to constraints that also determine the temporalities of the other parts (gestures, poetic rhythm). It has been noted that human beings are not cultural dopes that implement practices with machine-like routine (Garfi nkel, 1967). Rather, human beings creatively produce actions that only after the fact can be described to have been consistent or inconsistent with some rule—in the same way that we can determine only after the fact whether a piano player has played a tune correctly or incorrectly and whether she has played it in the consensually correct manner. Similarly, to draw on music as a metaphor, lessons are not enactments of scores but resemble improvisational jazz jams or improvisational dance sessions, where any individual contribution cannot be predicted with any precision. What teachers and students do, therefore, cannot be described as the enactment of a program, the transformation of a score into music. In everyday conversations, there is spontaneity at work that cannot be captured by intellectualist approaches and formal theories of knowing. We need to take this spontaneity as a model to understand the worlds of difference between the planned curriculum and the living/lived (“enacted”) curriculum. But the fact that something comes off the lessons should surprise no more than that there is enjoyable music produced in a jazz improvisation session. For understanding the intersubjectivity of the teaching-learning event we need an incarnate “description of the course of gestures instead of a cognitive attack on the program of minds” (Sudnow, 1979, p. 83). This also has consequences for the way in which we think about knowing, which always is a knowing-with rather than a knowing-of one or the other. The participants’ understanding, a phenomenon of the second-order consensual domain, is entirely based on their synchronization of mood, attunement, and practical understanding that occurs as a phenomenon of the fi rst-order consensual domain. When Mrs. Winter and Thomas communicate, each of them does not just exteriorize what they previously have thought for themselves. Even if they had done so, all consciousness—as its etymology con-, with, and sciēre, to know suggests—is consciousness for the other as it is consciousness for myself. Thomas speaks for the benefit of Mrs. Winter, using a language that he has received from the generalized other including from Mrs. Winter. For the teacher to know what Thomas knows, the latter’s knowing has to be public, available to every other participant, communicated in words, gestures, prosody, and every other available means of inherently carnal expression. What Thomas can carnally express has to be intelligible, and it therefore has to be already a possibility of expression in and for the collective as well. That is, what Thomas expresses in and through his performance are the expressions of a socialized body, lastingly impressed and shaped by participating with others in producing the world. He is part of a community, which constitutes “an affective aquifer and everybody drinks the same water from this source and

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well that he is himself” (Henry, 1990, p. 178). But we often do not know that we drink the same water and that we are not different from others who do precisely the same. Similarly, when Mrs. Winter speaks, she addresses Thomas: She talks for his benefit. What she says has to be intelligible to him so that what we fi nd expressed when she speaks, gestures, moves her body, intonates, and so on is not just her knowing but the knowing of Thomas as well. If he were not able to comprehend, then Mrs. Winter’s effort at communicating would not make sense at all. “When I speak or when I comprehend, I experience the presence of others within me or of myself in others” (Merleau-Ponty, 1960, p. 157). As a result “when I comprehend, I no longer know who speaks and who listens” (pp. 157–158). When Mrs. Winter communicates, she has to have the hope that her expression is intelligible. That is, what is expressed in and through her body also is a form of knowing-with. Much of this intercomprehension depends precisely on the ritualistic aspects of the relation in which they mutually fi nd themselves. The analyses in this chapter allow us to strengthen the argument for the material phenomenological approach to mathematical knowing and learning that I propose in this book. Rhythmic and perceptual phenomena require forms of consciousness that differ from linguistic consciousness, and because rhythms and perceptions are irreducible to the linguistic consciousness, all forms are subject to a more overarching unit.11 Moreover, rhythmic phenomena are of interest, as they are the result of auto-affection and the pervasive nature of tact, which are, as I present in Chapter 3, at the heart of the material phenomenological approach, making it distinct from that presented by the embodiment or enactivist literatures. Rhythmic phenomena particularly require bodies, as we see from the examples of fi reflies and clocks, but rhythmic phenomena that change the rhythms require the flesh, which is sensitive to changes and adapt to the changes that it produces itself. Thus, for example, “beats are not seconds or any other ‘standard’ unit of time. Instead, these are self-generated units that are used, in turn, as a kind of temporal ruler to measure the durationally varied events that are actually generating them—a nice example of self-reference” (Bamberger & diSessa, 2003, p. 128). Vocal beats, the rhythm in and of speaking, is internal to the speech sound itself, constituting a relation among speech sound events; yet it is not a conceptual phenomenon, but, instead, an entirely carnal and incarnated phenomenon. But bodies generally do not auto-affect: It is precisely the flesh that has this capacity. The conception of the flesh has the features that are required for an understanding of rhythm and perceptual consciousness as a lived—produced and perceived—phenomena irreducibly tied together by tact that is the ground of all senses that we use to make sense. Tact, shared by all senses, holds it all together because “it supposes an interior milieu, the flesh” (Chrétien, 1992, p. 147). In this chapter, I articulate different aspects of movement performances. I do so to be able to show how complex the expression of mathematical

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knowing is in and through communication. This communication is a complete, whole phenomenon, a unit that is highly aggregated, merged, and fused. There are no different performances in multiple modes that are simply put together such as if different players are put together to form a band to add another instrument to the melody. Rather, being produced in and through the same living body, the same intention, the mathematical expression cannot be thought independently of all the other aspects of this one expression. This same, then, goes for the perception of the expression as a whole. We do not just hear words or concepts, but we experience (hear, feel) a child proffer an answer tentatively, or with anxiety, or in a distraught manner. That is, as in musical hearing, “we do not piece together a hearing, putting it together out of the separate features we can name—a paste-up collage of, for instance, pitch, duration, accent, timbre, register” (Bamberger, 1996, p. 39). Rather, in perception, as apparent from the description of hearing an answer as a tentative answer, what we hear is a complex phenomenon in which cognitive and affective dimensions are irremediably intertwined. The second-order consensual domain is tied to and a function of the fi rst-order consensual domain. We hear and see one performance rather than separate properties that we have to construct and interpret before recombining them. The one performance, the performance being one in and through the one living/lived body that brings it about is the minimum unit that allows us to conceptualize what we actually hear and see. But this one living/lived body is inseparable from all the other living bodies so that there is “only one sphere of intelligibility where everything is intelligible to others and to oneself over the ground of this primordial intelligibility that is that of pathos” (Henry, 1990, p. 179). But to cognitively understand how this performance achieves what it achieves, we have to take it apart. The danger is, however, that we might think the performance consists of elements somehow and by some organ coordinated in time and content. There is a different way of thinking about the relation of the whole and its parts: They are mutually constitutive. Thus, the mathematical expression is a whole, which takes this form precisely because of the presence of all its parts. But each part is a part of the whole, not an independent element. It is a part that takes its form from the form of the whole—the part is a plural singular, constituted as such by the nature of the whole as a singular plural. Thus, we cannot and must not understand the part of an expression as an independent element (e.g., gesture) that can be reduced to, and understood in terms of, another element (e.g., language). We cannot express gesture in language, which, as gesture, only denotes the whole of practical understanding and intention in a synecdochical way.

Part C

Emergence of Geometry— An Objective Science

Introduction to Part C

Geometry is one of the oldest formalized mathematical domains. Having evolved in what came to be the earliest mathematical communities from mundane, pre-geometric experiences in a world of immediately comprehensible three-dimensional objects, it has been “handed down” and developed through the continual reproduction of its shared structures in incarnate geometrical activity. It is therefore not surprising that the Principles and Standards for School Mathematics (NCTM, 2000) identify geometry as one of the content standards for grades K–12. As well, in the more recent Curriculum Focal Points for Mathematics in PreKindergarten through Grade 8 (NCTM, 2007), emphasis is placed on K–4 students’ abilities to: identify geometrical ideas in their world; describe, model, draw, compare, and classify shapes according to their properties; investigate and analyze the composition and decomposition of two-dimensional and three-dimensional figures; and relate geometric ideas to measurement ideas. The question we may ask is how geometry as objective science comes to be reproduced in and through the actions of new generations? The psychologists’ answers to this question has changed over the years, from conditioning to information transfer to personal knowledge construction and to the personal construction of knowledge previously achieved while working in a collective. However, as I show in Chapter 2, these attempts in explaining knowing and learning have shortcomings, especially with respect to the role of cultural-historical acquisitions of humankind and their reproduction in and through the actions of new generations. Especially in Chapter 3 and again in Chapter 7, I describe—in the terms of phenomenological sociology—the shaping living/lived bodies undergo in interaction rituals. Now, the verb “shaping” sounds very agential and purposeful, as if teachers such as Mrs. Winter and Mrs. Turner intentionally acted to affect the bodies of the children, disciplining them. Both act in the very mundane ways in which teachers tend to act. The children in the second-grade classroom, too, act in the way children at this age and this grade level tend to act. They participate in “age-appropriate” tasks. What I show in Chapters 8 and 9 is, though, how this everyday way of acting entails a shaping of what students learn to do, and, therefore, how the

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very practices of a second-grade mathematics classroom shapes the living/ lived bodies of the children, “disciplining” them in a manner so that they may subsequently exhibit the discipline of mathematics. This is not the result of a continuous punitive action, but the very ways of indicating, for example, that color and shape are not allowed as task-relevant properties shapes the ways in which children view the objects at hand. Their habitus, the structured structuring dispositions that make them see and act in certain ways, change so that they learn to see similarities and differences underlying adult forms of geometry. What the children come to do, their practices, emerges from and is shaped by the interaction rituals in which they participate. That is, in a very real and mundane way, the interaction rituals constitute the foundation of the knowing that the students learn to exhibit: That is, to paraphrase Vygotsky (1978), the higher-order cognitive functions observed as a consequence of this geometry curriculum are the results of the interaction rituals that the children participated in. Vygotsky’s work—his concept of the zone of proximal development in particular—has often been used to suggest that there is some construction of knowledge in social situations, and these social constructions are subsequently constructed on the part of the individual. But in my approach, there is no individual construction of social constructions necessary. Rather, the very forms of participation are transformed in and through participation. That is, the children’s ways of experiencing truly underlie their learning, as their current participation constitutes precisely their knowing, which is transformed in its very articulation. Our living bodies are transformed and shaped in and through lived, sensuous and sense-making labor, which exhibits what we know: It is not after participation that children are changed by somehow transferring to the inside what they have experienced outside. It is precisely in the interaction ritual that they change, that is, learn. It is precisely in exchanges with others, where they listen, that they are “at the same time outside and inside” (Nancy, 2007, p. 14). The very expression of knowing in and through the living/lived body as a whole is changing knowing, an idea that—writing about the relation between thought and speech—Marxist social psychology has taken up entirely into its theories. In this third part of the book, I focus on geometrical classifications of objects as a context for investigating the role of lived experience and the living body (flesh) in the reproduction of geometry as objective science. The creation of categorization systems within mathematics has received interest among mathematics educators specifically and learning scientists more generally. I specifically use the term emergence to characterize the appearance of geometry at the cultural-historical, sociogenetic, and ontogenetic levels, because the constitutive linkage between geometrical classes (classifications) and geometrical actions occurs in a context that is or has been nongeometrical. Because the term emergence denotes an unforeseen occurrence or a state of things that arises unexpectedly, it is appropriate for situations of the type I describe where one cannot predict whether individual students

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and whole classes will achieve what the curriculum sets forth as a goal. In the lessons described and analyzed in Chapters 8 and 9, children appropriate geometrical classifications by learning to observationally distinguish between their peers’ and their own proper and improper geometrical classificatory acts. As in other work that focuses on mathematical learning in social situations, the collaborations required in making these situations the organized events that they are lead to emergent phenomena including, in Chapters 8 and 9, geometrical classifications of three-dimensional objects specifically and geometry lessons generally. Geometry, classification, and the classification of geometrical objects are integral aspects of recent curriculum documents in mathematics education. Such curriculum documents, however, leave open how the work of classifying objects according to geometrical properties can be accomplished given that the knowledge of these properties is the planned outcome of the curriculum or lesson. The fundamental question of the following chapters therefore is this: How can a lesson in which children are asked to participate in a task of classifying regular three-dimensional objects be a geometry lesson given that the participating second-grade children do not yet classify according to geometrical properties (predicates)? In my analyses, I focus on the incarnate collective work that leads to the emergence of the geometrical nature of this lesson. Thus, I articulate both the collective and the individual work by means of which the lesson outcomes—the complete classification of a set of “mystery” objects according to geometrical (shape) rather than other properties (color, size, “pointy-ness”)—are achieved. In the process, I show how second-grade children reproduce geometrical work, while operating in a division of labor with their teachers, to produce a particular set of geometrical practices (sorting three-dimensional objects according to their geometrical properties) for the fi rst time. In Chapters 8 and 9, I address three questions of how the classification of objects according to geometrical properties comes about and in what the work consists of that leads them there: “How does the ‘proper’ grouping of a collection of objects emerge from the collective task involvement of what recognizably is a second-grade classroom?” “How does this task involvement recognizably become a mathematics lesson?” and “What role do bodily experiences play in the process?” In Chapter 10, then, I propose a reconceptualization of mathematical concepts, which we simultaneously ground in philosophical considerations and empirical materials. I draw on an episode from the same lesson in which children (learn to) sort objects according to geometrical properties, here focusing on an episode involving a cylinder. In this reconceptualization, conceptions do not exist as disembodied, decontextualized, and transcendental ideas but only in concrete realizations of experiences and relations to other experiences.

8

Ethno-methods of Sorting Geometrically

How do children come to sort objects according to geometrical properties? How do children come to sort geometrically given that they tend to sort, as many studies show, according to properties such as size or color? These questions are especially important given that in the “intuitively given surrounding world . . . we experience not geometrical-ideal bodies but precisely those bodies that we actually experience, with that content that is the actual content of experience” (Husserl, 1997a, p. 23). That is, the question in this chapter will be how children come to classify objects (bodies) according to geometrical properties when they do not experience, at fi rst, geometrical-ideal bodies? And yet, if they do classify in and through classification experiences, what children bring to the classroom are the very methods, grounds, and materials upon which truly geometrical classification is built, however inadequate educators might deem these “preconceptions” to be. Educators often pay lip service to the ideas, conceptions, discourses, and so on that children bring with them to the classroom. Constructivist educators interested in the topic of conceptual change often write about “replacing” and “eradicating” misconceptions as the goal of instruction and about the difficulties to “eradicate” misconceptions.1 Piaget’s work has been read from this perspective as an attempt to show how children’s “barbaric thoughts” (Merleau-Ponty, 1945, p. 408) come to be replaced through the assimilation of and accommodation to new information, which leads to a “domestication” of children’s of untamed thoughts. Thus, much of educational thought, including the writings of Piaget, is about what children cannot do rather than what they actually do, what the world looks like to them. For example, pertaining to classification, he writes that the child “is not able to understand the relationship of class inclusion” (Piaget, 1970, p. 27). He describes children’s thinking as “primitive,” pointing out that “there is a very primitive ordering structure in children’s thinking, just as primitive as the classification structure” (p. 28). A phenomenological approach to learning recognizes that children’s forms of thought, because these reflect their experiences, are the necessary conditions for adult rationality. That is, this approach recognizes

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the everyday lifeworld as the foundation of sense, attempting to understand the experienced and experienceable that is immediately given to perception and the originarily intuitive thinking grounded in such experiences. One social science, though, does acknowledge the role of mundane thought to the constitution of everyday social situation: ethnomethodology. In this and the following chapter, I show how everyday methods underlie and are constitutive of formal geometry, that is, how the lifeworld is the actual foundation of geometrical science. I specify the nature of the lived work by means of which children produce mathematics and mathematics lessons. Ethnomethodology is homologous with its own subject matter, concerned as it is—in the present instance—with the articulation of the methods that people (ethno-) use to bring about everyday society, including everyday elementary mathematics lessons. These are considered ongoing achievements of members of society conceived as practical actors who (a) themselves are practical analysts of the world and others and (b) use whatever resources are available to constitute the sense of the ongoing everyday practical activity in and of which they are a constitutive part. This implies that the intentional and recognizable production of a lesson as a geometry lesson presupposes geometrical knowledge. In this sense, geometry as a subject and subject matter can only be, from the perspectives of children who do not yet know geometry, an emergent property. But more so than those who work within an ethnomethodological approach, I am also interested in understanding mundane classroom life in the cultural-historical nexus of the subject matter to be taught.

CATEGORIZATION, CLASSIFICATION Much of the existing (learning science) research on classification provides taxonomies that—even when taking an agent-oriented perspective—are constructions of the researcher rather than grounded in what actors in situations make available to each other. In contrast, I am interested here in the sensuous (living/lived) work that leads to the classifi cation of a set of “mystery” objects in an elementary classroom. I am interested in the production of predicates that constitute the objects in a collection as members of the same class, which, as the participating teachers’ telos, are to be characterized by geometrical properties. In this, I take an approach that differs from classical and traditional studies of children’s mathematical (geometrical) practice. I am concerned with providing a description of the “authoritative and accountable” practices in classrooms that—reflexively with the actions observable therein—are recognized by competent members (mathematicians, mathematics educators, and teachers) as mathematics lessons. I am particularly interested in the role of the organic body and other aspects of mundane everyday life that

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mediate how individuals and collectives elaborate their mathematical knowledgeability. Studies from an ethnomethodological perspective on coding and categorization in the natural sciences and in the social sciences show that to accomplish coding/categorization, coders require knowledge of the very organized ways in which the situation operates and from which their tobe-coded object (fish specimen, rock piles, hospital records) derives. That is, coders/classifiers do not code the object in, of, and for itself. Rather they code it in terms of aspects that belong to the surroundings/context from which an object is taken as a whole. This knowledge is consulted explicitly, that is, accountably, whenever problems in the classification become apparent. Relative to a geometry lesson featuring a classification task, therefore, children can be expected to learn (about) the intentions for classifying in addition to learning how to classify. An important realization of past research on classification was that the participants used “ad hoc” considerations when evaluating their questions about how to classify, which denotes the use of “‘et cetera,’ ‘unless,’ ‘let it pass,’ and ‘factum valet’ (i.e., an action that is otherwise prohibited by a rule is counted correct once it is done)” (Garfi nkel, 1967, pp. 20–21). Such ad hoc considerations also are at work when scientists evaluate coding and classification instructions such as the ones found in field guides for the identification of plants and animals in studies of classification among scientists. To understand the process by means of which geometry emerges from the incarnate, sensuous experience in the course of a lesson that Mrs. Winter, the lead teacher in this lesson, explicitly marks as a mathematics lesson (“we start a brand new unit in geo, math today”), I provide a step-by-step analysis of one episode that in many ways characterizes structural properties of the 22 sorting episodes that made for the largest part (55 min) of this 80-minute lesson. Here, as throughout this book, I denote by “episode” a clearly, participant-delimited event that begins with the teacher’s nomination of a student as having a turn (e.g., “Connor” or “who’s next?”) and by a clear transition to another student (e.g., “next one . . . Ben”). When necessary, Mrs. Winter reminds students about who currently has a turn, for example, admonishing a student who makes a contribution without having a turn (e.g., “It’s Sylvia’s turn” or “Kendra, he has to do his own thinking”). To provide a suitable context for understanding the episode analyzed, I fi rst provide an analysis of the logical structure of the task and then a description of how Mrs. Winter presents the steps and procedures she anticipates to follow.

THE LOGICAL STRUCTURE OF THE TASK In this classification task, each child gets a turn until all mystery objects have been placed (Figure 2.1). Mrs. Winter highlights at the beginning of

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the lesson, and in introducing the new topic of their mathematics work, that geometry is about shapes and the “math words about shapes” that go with them. Although she asks for and accepts from the children labels for the emerging groups (“squares,” “cube,” “tube,” etc.), and although she accepts children’s classification criteria generally, Mrs. Winter emphasizes that “color” and “size” are the two words that they cannot use to “do geometry.” Shape is a most complex phenomenon that must be dealt with in the qualitative representation of space. In Euclidean geometry, a large square and the same square with a slight nick in one side are different. In fact, geometry is not about the objects children fi nd in their everyday world; even the engineered geometrical objects that the teachers have brought to the classroom are but models that tend toward but never achieve the ideal objects that geometry is about. Shape is important in commonsense reasoning because very often the shape of an object is functional. From a logical perspective, the structure of the task requires the children to sort the objects {x, y, z . . .} such that they end up with collections within each of which the objects can be described by one or more predicates that differ from the predicates in another group. As a whole, the collection of mystery objects forms a membership categorization device to the extent that it can be used to classify further mystery objects. This device has the structure [cube/sphere/cone/. . . /pyramid] (Schegloff, 2007). The children’s task, mediated by the teacher’s input, consists precisely in developing a device (and the terms that come with it) based on their perceptual categorizations of material objects such that a set of predicates allows a mapping of the former to the latter. For a fi rst group, classification can be expressed in terms of the predicative statement ∀x(x . . . predicate[s]). Because such predication constitutes a form of generalization, it has been denoted as a stage in the conceptual development from (raw) stimulus to science. In the present situation, the task requires children to group objects according to similarities and differences: All objects within a group share one or more common predicates, which differ across the different groups (collections). But similarity is not easy to pin down. Similarity based on properties is problematic because, “[w]hen it comes to enumerating properties, we don’t know where to begin. The notion of a property, for all its seeming familiarity, is as dim a notion as that of similarity” (Quine, 1987, p. 159). Moreover, the things we encounter in the surrounding everyday world are not stable but “fluctuate, in general and in their properties” (Husserl, 1997a, p. 24). This has the consequence that identity with itself, self-sameness at any instant and across time, and sameness with other objects “are merely approximate” (p. 24). Watching the video of the lesson, we may note how children wrestle with the question whether two objects are the same or different: They both resemble a pyramid, but one has a slightly rounded top rather than a sharply pointed vertex. That is, whereas

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adults might have classified both as pyramids, the differences are salient to the children, who therefore wrestle with the very problematic that Husserl articulates. A phenomenological study of classification, therefore, has to understand how the world looks through children’s eyes and it has to view the process of classification as problematic through these very eyes.

SETTING UP THE SORTING TASK During the entire sorting task constituted by the 22 sorting episodes (student turns), the children sit in a large circle on the floor, Mrs. Winter and Mrs. Turner (dressed in black) are seated on chair s near the basket with the black plastic bag from which the mystery objects are to be drawn (Figure 8.1). The videotape of the lesson shows the children forming a large circle; all objects and categorizations therefore are perceptually accessible to all participants and everybody else can hear the current speaker and see his or her gesticulations. In her opening statements at the beginning of the task, Mrs. Winter asks the students whether they remember a conversation from the previous week about the new subject matter that they are to begin with the present lesson. Off camera, a student proffers “geometry,” which Mrs. Winter accepts by

Figure 8.1 The children sit in a circle. Mrs. Winter and Mrs. Turner (in black) sit with the children. Recognizable collections include the cubes (far left bottom), cones, and pyramids. Connor currently holds his object next to one of the rectangular solids.

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uttering the word again. This then leads to an articulation of the task, sorting, and the special conditions for this sorting task: it cannot be done by color or by size. Fragment 8.1 01 02

M: W:

03 04 05 06 07 08 09

J: W: O: W:

. . . and shape, and (serial?) and color. yea those are math words. those are the math words about the shape. good for you. anyone else want to tell us what they know about geometry? (3.59) it was a long weekend (0.29) what do you think jonathan? [what do] [size ] (0.96) it has something to do with size. anything else. oshin? color. so can we tell a shape by its color? and in fact we are going to talk about that so we may just as well talk about it right now. when we do any kind of sorting activity today, we are not going to do those by color and we are not going to do them by size. do you want me to say that again. when we ask you to sort things today, we are ´not using color as a category, and we are ´not using ´si:ze.

Quite innocuously, Mrs. Winter asks “can we tell a shape by its color?” She articulates the purpose of this lesson, to “tell shapes,” which in fact is the constant reminder to students after they have placed their object “Stop. Explain your thinking.” Fragment 8.2 10

W:

here is what you need to think about. i=m going to put a shape on this piece of paper. and you can look at this shape. you might already know something about that shape. and then i am going to ask people to come up and without–its a mystery bag—so without seeing it ((bends over to demonstrate pulling an object)) you have to decide does that shape ((points to pink paper on floor)) match or belong in the same group with the one i put out or any of the other ones out or does it need its own new colored piece of paper to sit on. so the fi rst shape that i am going to put out ((bends over to pull shape from the black plastic bag; pulls a cube and places it on the pink sheet)) there is the fi rst shape.

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S:

207

((Places another sheet of paper, brown)). okay, the next person who comes out has to decide does their shape fit in with that shape ((points to her cube)) or does it deserve its own color or its own category. its own place to be. ((She calls on the fi rst student.)) what were the two things we said we are not going to sort by. color and size

Mrs. Winter articulates the rules of the game. 2 These rules explicitly specify the predicates that cannot be used, and, therefore, the kind of grounding discourse students can or cannot use for explaining their reasons. In this, the rules specify the “language game” that is in play. In this opening episode, Mrs. Winter thereby introduces the children to the task, which includes the articulation of rules (predicates) that cannot be used in sorting the mystery objects children are going to pull from the black plastic bag. In the course of the lesson, these rules do not come to determine what students do. Rather, Mrs. Winter often encourages students to redo and rearticulate something until a point where the rules come to be descriptively adequate for what students individually and collectively have done. The point of the task is not merely to put an object with a given collection or to use it to begin a new collection. Rather, the task is to evolve category-bound predicates that allow verbal distinctions to be made for the different groups (collections, categories) of objects that are associated with different names that they come up with for the emergent groups and that Mrs. Turner records on strips of paper that she then places next to the objects. Some differences legitimately place items into different groups; other differences cannot be made part of legitimate predicates for drawing distinctions. Some differences are allowed in this “game,” evidenced by the apparently (perceptibly) different objects in the different collections that form classes; and that these objects form classes can be seen from the fact that they (legitimately) are placed on the same colored sheet.

Connor’s Turn3 During this episode, it is Connor’s turn to place his object with one of the existing collections or on an empty sheet that the teacher has placed on the floor just as Connor begins. He places his object on the empty sheet. Here I present a description and analysis of the lived work that makes these different classifications emerge from the sequential classroom talk. I show how the classification and the sequential aspects of talk are constitutive. The unit of analysis is the entire “turn” (episode) and the classification that is achieved in the end emerges from the sensuous labor completed over the course of the episode.

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Here, I use the term, “emergence” to denote the contingent nature of the present achievements, which could not be anticipated because there are other ways of classifying these same objects: Cubes are special cases of rectangular solids, which in turn are special cases of prisms (in the collection there are also triangular and hexagonal prisms); presence or absence of acute angles; number of faces (e.g., sphere = one; cone = two; cylinder = three; tetrahedron = four; . . .); and all are part of the same topological category. My analysis exhibits the children’s work of classifying, which is distributed across people and objects. It includes the walking about from collection to collection, the picking up, holding, feeling, turning, comparing, manipulating, placing, observing, etcetera of the object; and it includes attending to signals from others, such as pointing and emblematic “stop” gestures. It includes the ongoing a priori (instruction, orders, rules) and a posteriori descriptions of courses of actions that the teacher and others articulate. It includes alternative articulations, placements, and comparisons others provide. It is precisely this lived work that shapes the children’s bodies, disciplines them so that they exhibit the discipline of mathematics (geometry).

GETTING STARTED: A FIRST CLASSIFICATION At the beginning of this episode, Mrs. Winter reiterates for the class—as both teachers do throughout this lesson—the various rules for and of their “game,” sorting mystery objects (turn 01). In the present situation, it is the rule about having to articulate the thinking underlying a classification of the mystery object, which fi rst has to come from the student pulling it from the plastic garbage bag before others can articulate their own thinking. Furthermore, another rule consists of picking the fi rst object a student touches rather than feeling around to select an object. There is a pause during which Connor’s upper arm can be seen to be moving—everything from slightly above the elbow to the hand has disappeared in the bag. Members can see this as feeling around—as Mrs. Winter’s admonition to “take the fi rst one [he] feel[s]” indicates (turn 03). That is, Mrs. Winter requests Connor to act such that after the fact her admonition is an adequate description of what he has done. Adequateness always is determined after the fact because the living/lived body acts on its own based on its immanent knowing; in and through doing the relevant work repeatedly, the level of adequacy increases. Here, although a rule is rearticulated in turn 01, Connor acts in a way that Mrs. Winter (and others?) sees as not taking the fi rst one he felt (“take the fi rst one you feel”). It takes another little while before Connor pulls a mystery object, which he briefly and demonstratively holds up for others to see and which his classmates acknowledge with extended (2.18 s) “ohs” and “ahs” (turn 05).

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Fragment 8.3 01

02 03 04 05 06

07 08 09 10

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´BEfore we close this one out. remember that its connors turn first to tell us his thinking. (0.49) you have to wait until we ask for more ideas. (1.03) just take take the fi rst one you feel. (2.97) ((Connor appears to feel around in the black bag)) W: take the fi rst one you feel. (1.68) Ss: ((multiple ohs and ah’s, 2.18 s)) W: now look at the grou:ps: ((pointing around the circle of objects already there)) does it ´belong to another ´group (0.67) O:r. (0.29) ´can you start a new group with that. ((Connor facing the objects, looks at his object, looks around the circle, (0.67) K: um::: ((Kendra moves forward, points to the group of cubes [Fig. 8.2])) (0.62) W: ^kendra. (0.34) he has to do his own [thinking.] ((Connor moves to a new sheet, places his object, and walks away toward his seat)) C? [brandon.]

Just before Mrs. Winter begins to utter what we can hear as an instruction, Connor already has shifted his gaze across several collections; he directly gazes in the direction of the teacher’s hand after she has started to talk, moving her hand in a circular fashion, her index fi nger pointing in the general direction of the various collections on the floor. There already are 12 objects out on the floor in different collections distributed over six sheets, five of which are associated with labels (“ball,” “cone,” “tube,” “rectangular,” and “squares, cube”). There also is one empty sheet in the case that a student decides his or her mystery object does not fit with existing collections. Mrs. Winter now (and again) articulates for Connor what she expects him to do, “now look at the groups” (she is pointing around in a circle to the existing sheets with objects), and she continues, “does it belong to another group or can you start a new group with that?” Connor looks at his object, then—visibly to all present as he orients his head and upper body—gazes at the collections on the floor. At that time, Kendra, who is situated about two meters behind Connor is moving up, looks at his object, emits an extended “um” sound, attracts his attention (he orients towards her, Figure 8.2), and she demonstratively points to a collection containing two objects associated with the labels “squares” and “cubes” (turn 08). Some students clearly gaze in her direction; Melissa, sitting opposite to Connor in the circle waves her pointed finger toward the opposite direction. Connor briefly turns his gaze into the direction that

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Kendra points before reorienting to his right and toward beginning his own group of objects on the yellow sheet lying in the direction where Melissa has pointed. At the same time, there is a brief pause during which Kendra retreats and then the teacher, turning toward her, utters Kendra’s name with a slightly raised voice and says, with an intonation that members can hear as a teacherly admonition, “he has to do his own thinking” (turn 10). In fact, the utterance repeats, in a modified way, an instruction or rule already stated at the beginning of the episode: It is his turn fi rst to articulate his thinking, and the fact that Kendra is now pointing does not allow him to do his own thinking (fi rst). Mrs. Winter’s utterance “he has to do his own thinking,” in fact, constitutes normative work in its double orientation toward Kendra and Connor. It highlights the sensuous labor currently required of Connor, “to do his thinking,” and it asks and is understood as such by Kendra— articulated in her retreat—to keep quiet. The reprimand is recognized as such not merely because of the register in which it is delivered but also because of the way in which this turn is inserted within a (pedagogical) turn previously assigned to another student (Connor). It is recognized and exhibited as such in the retreat and silence Kendra displays. It is but one of the ways in which students’ bodies come to be affected and therefore socialized bodies (see Chapter 7). Kendra retreats so that, together, Mrs. Winter and her student establish the order properly described as “Connor’s turn.” Mrs. Winter reiterates the conditions of the task, which

Figure 8.2 Connor orients toward the collection of cubes that Kendra is pointing to.

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requires an explication of the reasons for an object’s placement, and the individual whose turn it is has to do so without help. That is, by repeating a previous instruction, Mrs. Winter marks Kendra’s previous pointing as a violation of the rules (instructions); and Kendra’s retreat is consistent with this reading. In other words, Mrs. Winter’s comment might have created a dilemma for Connor, if he wants to place the object in the category Kendra pointed to, it might mean that he has not done “his own thinking.” Both children, by responding with appropriate actions (Kendra retreating, Connor placing his object), therefore exhibit their understanding of these conditions. In this situation, therefore, the emerging participation framework between Connor and Kendra—one that children appear to choose naturally for accomplishing mathematical tasks collectively—is inhibited in favor of another one that is familiar in school situations: a student and the teacher as principal agents and the other students (as well as Mrs. Turner) as onlookers and witnesses. Although the classifications achieved in the two possible forms of division of labor may be the same, the trajectory of the lessons likely would be different, though in an unpredictable way. In this sense, therefore, this moment also constitutes a possible branching point in the lesson, which therefore will unfold as a function of the situated choice—here by the teacher—to enable the second form of division of labor rather than the first. The utterance “can you start a new group with that?” (turn 06) can be heard as an invitation to start a new group. In turn 10 Connor does this despite the possible tension that might be perceived between his action and Kendra’s pointing to the “square, cube” group. She does not just point but repeatedly jettisons her hand forward, in an apparent attempt—she does not just stick out the index fi nger—to attract Connor’s attention to the pointing. Kendra’s actions also constitute a form of sensuous, living/lived labor that we need to take into account to understand the episode, and in fact, it constitutes a form of instruction that might be glossed thus: “Put your object to the group toward which I am pointing,” which, as it will turn out and unbeknownst to the participants at this instant, is the one that the teacher will be satisfi ed with at the end of this episode. But at this instant, Connor does act consistent with the condition that he has to do his own, unaided classifi cation and thinking.

FEEDBACK AND FURTHER INSTRUCTIONS While Mrs. Winter is turned toward and talking to Kendra, Connor gets up, walks toward the empty sheet, picks up his mystery object and produces, as he is placing the object, a new collection. He then moves toward his place in the circle of students, thereby indicating his turn to

212 Geometry as Objective Science in Elementary School Classrooms be completed, when Mrs. Winter interpellates him, “Now, before you go, you have to explain your thinking” (turn 13), holding her hand out in a way that knowledgeable individuals recognize as stop (turn 13). In this, Mrs. Winter makes it known that from her point of view more needs to be done before Connor can sit down. She continues by articulating, while pointing to the mystery object, that he has to explain why the mystery object is placed in its own group (turn 13). She does not merely ask Connor to articulate his thinking but more precisely, she asks him to explain why the object gets its own group, emphasizing the word “own” by pronouncing it with signifi cantly greater speech intensity (volume) than the other surrounding words. Without hesitating, Connor, who has walked back to his object and has pointed to it, suggests, “’cause this one is sort of bigger than the other ones.” He picks up the object and begins to walk when Mrs. Winter interpellates him again, “Connor, just a minute, stop for a sec.” She then asks him to “remember” (the rules): “we are not counting size . . . and we are not counting color” (turn 16). Fragment 8.4 12 13

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14

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15 16

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17 18 19 20

C: W:

(1.08) NOW. befORE YOU GO ((Holds right hand out in “stop” configuration [Fig. ])) you have to explain your thinking.´why does it get its ´OWN group now. cause this ↑ `one is ↑ sort of (0.32) bigger than the other ones? ((He has walked back and now stands over his object, looking down on it.)) that s[ize. ] [connor] ((Connor picks up his object)) just a minute stop for a sec. ((Hand held out in “stop” position)) (0.53) remember WE:R not telli counting ´si::ze (0.62) ((pulls up left-hand index fi nger with right index)) ↓ cou:nting ^co:lor. ((pulls middle fi nger, as in counting 1 then 2)) ((in “giving a lecture tone”, pulling on his fi nger for 1 and 2)) ((Connor is looking at her until she is done, with his object in his hand.)) (0.50) (0.21) . (0.99) SSo ´how is it this? (0.24) different now.

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(0.83) or the same as other shapes. (0.23)

In turn 13, Mrs. Winter requests Connor to explain his thinking before he can go, that is, return to his place. She brings forward her hand with palm open in the way drivers are signaled to come to a stop (turn 13)—one of the many ways in which the children’s living/lived bodies (flesh) come to be affected because of the structures of the field. Connor, who has already stopped—sign that the injunction has worked, that his body has come to be fashioned a bit—comes to face his teacher at this very instant. This appears to be a form of gesture found in classrooms more generally, whereby a teacher, like a crossing guard delaying oncoming traffic, directs the attention of the class/group as a whole to a particular issue (Koschmann, Glenn, & Conlee, 1997). In uttering, Connor completes what we can hear as a request-response pair; he thereby may learn to articulate the grounds for acting in this rather than another way. He may learn to be accountable for his actions, even if the account is not delivered each and every time. Here specifically Connor is put in the situation, as Mrs. Winter articulates not wanting to have the result of the classification or any answer from Kendra (“he has to do his own thinking”). Mrs. Winter particularly asks Connor to articulate the grounds for his decision to place the object on its own. Rather than reworking his explanation, which Mrs. Winter has evaluated negatively (“we are not counting size”), Connor enacts what is called a preference for self-repair. That is, he repairs the choice rather than the explanation. Connor hesitates (turns 17–19). In asking how his shape is similar to (turn 22) or different from (turn 20) existing shapes, Mrs. Winter provides a specification of the task and therefore, a specification for the lesson to become a geometry lesson rather than some other lesson—for example, an art lesson where perspective and apparent size are of importance, or where students are to learn about color and the illusion of depth. Connor provides a predicate but Mrs. Winter stops him before articulating that he has to remember not to count size or color. There are pauses and some stuttered utterances on Connor’s part, which the teacher follows by asking how “it” and “this” is different “now.” It is precisely here that the teacher asks the student to make a distinction between, on the one hand, his earlier explanation grounded in his familiar experience with the world and his everyday competencies and, on the other hand, what is to become the specifically mathematical explanation fostered in and by means of this lesson. The “now” follows the articulation that size and color are not counted—i.e., the predicates in terms of these two categories though possible are disallowed—so that others, consistent with mathematics, can subsequently emerge, after this singular point, and with them the nature of the practices that make this observably a mathematics lesson.

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Again, there is a pause (turn 21), providing the teacher with an opportunity to articulate the task in a new, alternate way: specify the similarities between this and the other (collections of) objects. In turns 20–22, Mrs. Winter in effect asks Connor to classify an object that he already has classified. Since the fi rst time, however, she has noted color and size as illegitimate predicates—“improper” (knowledge) categories—the latter having been used by Connor to put his object into a group of its own. By asking him to classify again rather than to further clarify and elaborate his rationale—as she does in those instances where the object ultimately remains in the collection where students have placed them (i.e., the “correct” one)—she is inviting Connor to do (try) the sorting again. She does not invite him to classify or rethink, but in fact asks him how “it” is “different now.” Saying “now” distinguishes this moment from an earlier one, the one prior to having made the selection of the group on its own. Since then, she has rearticulated the rules of the sorting game: Neither color nor size is to be used. Just prior to her turn, Connor articulates size as the predicate that distinguishes his object from all the others, thereby deserving its own group; that is, Mrs. Winter has articulated a rule that is inconsistent with the one Connor used. “Now” means that Connor is asked to classify his object again given that his previously stated predicate is inconsistent with the rules of the game, which therefore requires another predicate for making a decision. In fact, it is not the repeated statements that cause Connor to make the right classification, a way in which some researchers might want to characterize this instant. He is asked repeatedly to try (a different) classification until such a point that the rule is descriptively adequate of what he has done before and until Mrs. Winter appears to be satisfied. At this point he would be allowed to return to his seat. Moreover, at this point his actions could have been described as consistent with the rules. If the utterance of a rule were able to cause student actions, then it would be surprising why Mrs. Winter has had to reiterate repeatedly these corrections that cumulatively shape the living/lived bodies of the children in the course of the lessons—whenever a student uses color and size, or whenever someone else provides hints for classification before the student whose turn it is has the opportunity to explain his/her thinking. In the course of the lesson, therefore, the restatements of instructions and rules become resources for understanding that the classification has to be redone; and when these restatements do not occur, the foregoing classificatory action can be considered appropriate and consistent with the rules of and instructions for the sorting game.

A SECOND ATTEMPT The episode continues with a pause before Connor utters two interjections, followed by another pause. During this time, he walks away from the colored

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sheet having picked up his object, and in a long curving trajectory moves around the other collections, gazing at them in passing and toward the sheet labeled “rectangular.” He holds his object directly against one of the two larger objects (turn 24), which we (members of the culture) can see as a comparison: He brings the object with his right hand to the others and passes his index finger of the left hand over from the object on the floor to his own. We then hear him make the comparison that we have seen him enact physically: “tiny bit, I say it is a tiny bit different” while shaking his head twice sideways as if in negation and looking the teacher squarely in the face (turn 24). As his fingers run over the surfaces, he is in a position to sense the flatness and smoothness each of the two rectangular (square) and painted surfaces presents, that is, he can sense the bodies in the way and with the content that they are really experienced rather than the geometrical-ideal bodies he is to learn about. The lived difference between the objects is therefore made available to the others in the room, most importantly for Mrs. Winter, to whom he is orienting himself, as shown in his gaze direction. At least it appears to be that way. For, after a brief pause, Mrs. Winter questions him again, “but how?” (turn 26). There is another brief pause, and, with a subdued voice, Connor utters that he “can’t really say how.” Fragment 8.5 24

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25 26

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km ↑ this ((walks in a curved path toward the collection of “rectangulars,” Fig)) one::s:sorta ((holds his object against two others)) (0.91) ny bit, (0.83) i say it is a tiny bit . (0.60) but ^how::. ((head moves down in “teacherly” manner)) (0.72) ((shrugs shoulders))

. (1.33)

[I WA]NT you look at that block ((points to block)) and i want you to take it to each group ((points around circle)) and i want you to see:: (0.56) whether it looks the same as any of the other groups ((Connor holds his object directly next to one of the rectangular prisms)) or if it is different from all. (0.32) brandon you gotta get your ´feet up.

216 Geometry as Objective Science in Elementary School Classrooms 34 35

C:

(5.68) ((Connor holds his object next to the cubes, quickly next to the cone, then the pyramid)) i ↑ thinki::t probably: go (0.98) THIS one. ((places it with the two cubes))

Here, Connor has provided proof, visible and audible to everyone: the reason for his previous classification. Connor says that his object is “a tiny bit different” while holding his object next to the collection of rectangular solids. And this tiny difference may—but does not have tomake all the difference, for in the sphere of the merely typical, the likeness of one thing with other things always is approximate. Mrs. Winter requests Connor to state how they differ and Connor responds that he “can’t really say” and then tentatively—see the rising pitch toward the end of the utterance marking questions—offers a possible answer (“It’ll be bigger?” [turn 30]). She begins her turn before Connor has ended, requesting the student to look at his object and to walk around and compare it with all the collections so that he may determine “whether it looks the same as any of the other groups or if it is different from all” (turn 31). We can hear the “I can’t really say” as an indication of his experienced inability to provide what the teacher is asking for and, simultaneously, as a request to be instructed in how to arrive at the expected response. Mrs. Winter does precisely that in providing instructions for how to go about comparing the mystery object to all others and to decide whether “it is different from all.” It is only in this latter situation that Connor’s original choice of beginning a new collection is legitimate. In fact, Mrs. Winter is asking Connor to do what he already has done—and observably so, as he moves his fi nger repeatedly across the ridge between the two objects and thereby enacts the comparison. He has scanned all of the collections and has decided that his object should be placed on a separate sheet of paper, being in a category of its own. He has passed by all the other collections and has gone to the one with the largest objects. He directly holds his object to the two largest ones in the collection, passing his fi nger from one to the other thereby showing the relation in juxtaposing his mystery objects with two others, making visible his test of the size comparison, and verbally articulating that his actual, lived experience of different heights. He can feel it: The difference is real. Asking him to do this again does not make sense unless the outcome of the fi rst time has yielded an inappropriate result. Thus, by stating the condition after Connor has started a new collection, Mrs. Winter implicitly calls his action and decision into question, which differs from her previous evaluation that only reminded Connor to justify his solution. Without taking up the intervening student’s suggestion (not completely recoverable from either of the two soundtracks), the teacher begins another instruction. Pointing around to the six collections, Mrs. Winter says “I

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want you to look at that block and I want you to take it to each group and I want you to see whether it looks the same as any of the other groups or if it is different from all” (turn 31). In this, she rearticulates instructions already provided in turn 06, using pointing gestures in the same way marking each collection on the floor with a beat (rhythmic) gesture while slightly pausing. At the same time as Mrs. Winter talks, Connor has compared again his object to the two larger rectangular ones. In fact, she asks Connor to do what he has done already twice, first perceptually, then while walking to the collection of rectangular objects. In giving the instructions again, we can see that which has occurred between the two situations as inconsistent with the description that the instruction provides prospectively for what is anticipated to happen. It is a way of bringing about another sequence of actions that then can be compared again against its prospectively provided description (i.e., the “instruction”). And engaging in this sequence again leaves a trace in Connor’s body, shaping what and how Connor will act at some future instant. This and the previous section can be viewed as the organization of classroom talk and environments that has been termed I-R-E-C sequence, which involve a situations where a student reply does not lead to a positive (accepting) assessment by the teacher (Macbeth, 2004). Thus, following an initiation of the turn (I), a student replies (R) with the statement of trouble all the while having the opportunity to initiate or correct in the same turn. In the next turn (E), the teacher then evaluates and initiates a correction. Finally, the student or teacher corrects (C) in the fourth turn of the sequence. Although it is possible that the I-R-E-C sequence can sometimes be found in its simplest form, it “can on any actual occasion show expansion to an ‘nth’ turn” (p. 710). Mrs. Winter’s utterance in turn 16 thereby sets up the sequence, in providing Connor with the opportunity to locate trouble in the previous turn 14. This is (partially) completed once Connor has placed his object in a collection where it will be once he sits down and the next student gets a turn. Mrs. Winter continues to evaluate (turns 20, 22), and Connor self-corrects in turn 24. The continuation of the evaluation, and therefore the assessment that this is a sequence, further receives support from the fact of the repeated utterance of “now,” which both mark points in time and generate a parallel structure. In each case, the “now” marks relevant prior actions. This line of analysis, therefore, suggests that repair might be one of the most important aspects of producing intended learning outcomes. In this instant, Connor does not merely participate in the I-R-E-C sequence. Rather, his active participation produces and reproduces this interactional form, shaping and reshaping the ways in which he participates now and in the future. But if the patterns of his actions change (i.e., his practices), then his living body has changed. It is through expansions of the basic pattern that a student comes to respond (act) in the “proper” way, which therefore is a clear expression of the shaping of his living/

218 Geometry as Objective Science in Elementary School Classrooms lived body in interaction ritual. The student’s living/lived body comes to be socialized, but the social body (society) itself is changed because in the intuitively given world, self-sameness and self-identity are never given, always only approximate.

CLASSIFICATION: REQUESTING AND PRODUCING AN ACCOUNT Connor turns around, holds his object right next to each of the two objects in the “square, cube” collection, then to the taller of the two objects in the “tube” collection, then gets up and, in passing, holds his object to the object on the “cone”-labeled sheet. He stops short of the “pyramid” and begins to talk, “I think it probably go this one” (turn 35) while placing his object right next to the larger cubic object (of about the same size). Here, rather than making defi nitive statement, he uses the adverb “probably,” which expresses a high degree of likelihood but not certainty. In fact, it is a device that allows members subsequently to change a statement more easily than when they have committed themselves fi rmly to one or another choice. Although this is the place where the teachers ultimately want the object to be placed Mrs. Winter requests a reason, “and can you tell us why you think that?” Fragment 8.6 35 36 37 38 39 40 41 42 43 44 45

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i ↑ thinki::t probably: go (0.98) THIS one. ((places it with the two cubes)) (0.46) and can you ´tell us why . (1.29) cauz:: these are more squares. (0.40)