What does understanding mathematics mean for teachers?: relationship as a metaphor for knowing

WHAT DOES UNDERSTANDING MATHEMATICS MEAN FOR TEACHERS?

“Each year countless legions of school children struggle through a version of mathematics denuded of passion, denuded of intellect, denuded of the intrigue that has sustained this ancient art through the millennia. Yet some few teachers, inspired by their subject, go beyond the school-math tradition of recipe and rehearsal to tinge their instruction with a sense of open-ended possibility that is the true heart of mathematics. In describing his singular journey to be in relationship with mathematics Yuichi Handa lays a pathway that ALL mathematics teachers should be invited to explore in relation to their own emergent practice of teaching.” David Kirshner, Louisiana State University “This book is the first devoted to exploring teachers’ relationship with the subject they teach, not in a professional sense but in a personal sense. The question is not ‘How much do I know about the subject?’ but rather ‘How do I interact with the subject— what kind of relationship do mathematics and I have?’.This personalized view promises to unlock new ways of thinking about knowing, and new ways of understanding and explaining the nature of classroom teaching. All teachers, from primary grades to graduate level courses, can find themselves in this eye-opening and inspiring book.” James Hiebert, University of Delaware Opening up alternative ways of thinking and talking about ways in which a person can “know” a subject (in this case, mathematics), this book leads to a reconsideration of what it may mean to be a teacher of that subject. In a number of European languages, a distinction is made in ways of knowing that in the English language is collapsed into the singular word know. In French, for example, to know in the savoir sense is to know things, facts, names, how and why things work, and so on, whereas to know in the connaître sense is to know a person, a place, or even a thing— namely, an other—in such a way that one is familiar with, or in relationship with this other. In mathematics education, the focus generally tends to be on how learners and teachers know mathematics in the savoir sense, and rarely (if explicitly) in this other connaître manner. Of course, part of the reason for this may be the absence of a clear image of what this manner of knowing mathematics would look like. Primarily through phenomenological reflection with a touch of empirical input, this book fleshes out an image for what a person’s connaître knowing of mathematics might mean, turning to mathematics teachers and teacher educators to help clarify this image. Yuichi Handa is Assistant Professor of Mathematics, California State University, Chico.

STUDIES IN CURRICULUM THEORY William F. Pinar, Series Editor Handa

What Does Understanding Mathematics Mean for Teachers? Relationship as a Metaphor for Knowing Joseph (Ed.) Cultures of Curriculum, Second Edition Sandlin/Schultz/Burdick (Eds.) Handbook of Public Pedagogy: Education and Learning Beyond Schooling Malewski (Ed.) Curriculum Studies Handbook – The Next Moment Pinar The Wordliness of a Cosmopolitan Education: Passionate Lives in Public Service Taubman Teaching By Numbers: Deconstructing the Discourse of Standards and Accountability in Education Appelbaum Children’s Books for Grown-Up Teachers: Reading and Writing Curriculum Theory Eppert/Wang (Eds.) Cross-Cultural Studies in Curriculum: Eastern Thought, Educational Insights Jardine/Friesen/Clifford Curriculum in Abundance Autio Subjectivity, Curriculum, and Society: Between and Beyond German Didaktik and Anglo-American Curriculum Studies Brantlinger (Ed.) Who Benefits from Special Education? Remediating (Fixing) Other People’s Children Pinar/Irwin (Eds.) Curriculum in a New Key: The Collected Works of Ted T. Aoki Reynolds/Webber (Eds.) Expanding Curriculum Theory: Dis/Positions and Lines of Flight Pinar What Is Curriculum Theory? McKnight Schooling, The Puritan Imperative, and the Molding of an American National Identity: Education’s “Errand Into the Wilderness” Pinar (Ed.) International Handbook of Curriculum Research Morris Curriculum and the Holocaust: Competing Sites of Memory and Representation Doll Like Letters In Running Water: A Mythopoetics of Curriculum Westbury/Hopmann/Riquarts (Eds.) Teaching as a Reflective Practice: The German Didaktic Tradition Reid Curriculum as Institution and Practice: Essays in the Deliberative Tradition Pinar (Ed.) Queer Theory in Education Huebner The Lure of the Transcendent: Collected Essays by Dwayne E. Huebner. Edited by Vikki Hillis. Collected and Introduced by William F. Pinar For additional information on titles in the Studies in Curriculum Theory series visit www.routledge.com/education

WHAT DOES UNDERSTANDING MATHEMATICS MEAN FOR TEACHERS? Relationship as a Metaphor for Knowing

Yuichi Handa California State University, Chico

First published 2011 by Routledge 711 Third Avenue, New York, NY 10017 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 Yuichi Handa 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 utilized 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 Handa,Yuichi. What does understanding mathematics mean for teachers? : relationship as a metaphor for knowing / Yuichi Handa. p. cm. — (Studies in curriculum theory) Includes bibliographical references and index. 1. Mathematics—Study and teaching—United States. I. Title. QA12.H36 2011 510.71—dc22 2010035096 ISBN 0-203-83743-6 Master e-book ISBN

ISBN: 978–0–415–88597–3 (hbk) ISBN: 978–0–203–83743–6 (ebk)

For my parents, for Bang, and for Jen

CONTENTS

Preface Acknowledgments 1

Introduction to a Phenomenon

viii xiv 1

1.9 A Tangent Prior to the Rise (and Run)

15

2

Relationship as Reciprocity: Grace

18

3

A Bringing Forth of Self: Will

57

4

Relationship as Interest

74

5

“Doing” Mathematics and its Relation to the Life Path of being a Mathematics Teacher/Educator

An Epilogue in Two Parts

104 126

Part I: Returning to the Episode of the Straitjacket

126

Part II: Relationship as a Returning: Going Past “Understanding”

131

Appendix: A Narrativized “Methodology”

135

References Index

139 147

PREFACE

When a teacher tells a student that the reason for those many hours of arithmetic is to be able to check the change at the supermarket, the teacher is simply not believed . . . The same effect is produced when children are told school math is “fun” when they are pretty sure that teachers who say so spend their leisure hours on anything except this allegedly fun-filled activity. Nor does it help to tell them that they need math to become scientists—most children don’t have such a plan. The children can see perfectly well that the teacher does not like math any more than they do and that the reason for doing it simply that it has been inscribed into the curriculum. All of this erodes children’s confidence in the adult world and the process of education. And I think it introduces a deep element of dishonesty into the educational relationship. (Seymour Papert, 1993, p. 50) I began my own quest for a doctorate in mathematics education with two ends in mind—one of economic practicality, and the other of personal meaning. It is to the latter end that this monograph pertains. As a mathematics professor at a local community college and a lecturer in mathematics at a nearby four-year university, I decided upon pursuing a doctorate with a desire for both greater economic possibilities and deeper conviction in what I was doing professionally. By “deeper conviction,” I mean to say that I carried substantial self-doubt over my career choice of being a mathematics teacher, although I could not think of a better alternative for myself. Part of this self-doubt, I believe, had to do with deep ambivalences I harbored around the entire enterprise of mathematics education itself, at least so as I could envision it at the time. That is to say, I could not entirely justify why one would deem it meaningful and/or necessary to teach on my part, or to learn on the students’ part, the discipline of mathematics.

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The standard economic argument of “You need this for your future” seemed too facile and ultimately insincere, as I knew no one else in my circle of friends outside of the math department who used any mathematics past basic arithmetic in their own lives and careers. And though I had found some enjoyment in mathematics as an undergraduate and later as a graduate student in mathematics, I could not help at times but feel as if I were inflicting upon my students something that for the most part was unenjoyable for them, and in the end useless and irrelevant for their larger life projects. But part of this sense of “uselessness” and “irrelevance” for my students, I came to believe, had to do with my own involvement—or really, lack of involvement— with mathematics. Since leaving graduate school, I found myself doing little to no mathematics. I had tried at times to study a new or old math text, as if I were a student again, but well before the end of the second or third chapter, I would lose interest and abandon the project altogether. Making a new discovery in, say, number theory—what I thought of as the easiest domain within which for me to do so—seemed to involve more of a commitment than I was willing to bring. And though I subscribed to a number of mathematical journals, I rarely read the articles or bothered with the “brainteaser” problems that most of these journals contained. And so I reasoned to myself, “If I, the teacher, am so uninvolved with mathematics, what gives me the right to impart—really, inflict—this stuff upon others?” I felt myself precisely on the teaching end of Papert’s quote that heads this chapter. I felt like a fraud, even though I was never trying to sell math as “fun” or particularly “useful.” (In fact, I shared with my students on occasion that one of the reasons why they should study mathematics is to learn how to put up with something that they do not like—that it would teach them some tolerance, essentially, for bullshit. I said this to students, neither for shock value, as a funny joke, nor to be liked, but because on some level mathematics had become for me little more than a means of living. And that, in my mind, made it bullshit. Thus, calling it for what it was (to me) felt like one way of stemming the seeming dishonesty of teaching the subject.) This is not to say that I did not know the subject well. On the contrary, I felt utterly competent in what I knew of the mathematics I was teaching; I really “knew” the content, or so I believed and still do. But that was something different from my own felt sense of disconnection from the subject. I suppose that in some naïve way, I came into doctoral studies hoping for some answers, with a certain optimism for having these misgivings resolved, whether through gathering the right sort of information, or through meeting others who might “set me straight.” And if I should fail at this, I could at least leave the field of teaching—or mathematics teaching, for I did find some meaning in the teaching end of things—with some consolation that I had at least tried. Once my studies began, I would come to find out that much of doctoral studies in mathematics education—at least as I experienced it—concerned itself

x

Preface

more with the “how” question than with the “why” question—“How could we help students to learn mathematics better?” “How effective are different teaching approaches?” “How do students learn mathematics?” and so on, and not “Why should they/we bother?” I suppose there may have been an assumption that a doctoral student in mathematics education would have already resolved the “why” question long ago, if it was ever an issue in the first place. At any rate, I was left to my own in many ways to finding ways of bringing my very personal interests to bear on my doctoral studies. At the same time, I might add that there was quite the emphasis on grappling with the question of what it meant to understand a piece of mathematics as in, “What does it mean to say that a learner, or even a teacher, truly ‘understands’ some mathematics?” Although it may not be apparent to the reader at this point, I hope to show how this question is, in fact, centrally implicated in the seemingly unrelated quest of making meaning of being a mathematics teacher. From one point of view, this book might even be considered my prolonged though indirectly routed response to that exact question. One of the first papers I wrote during this time (thankfully, unpublished) was an attempt to question and to justify the teaching of mathematics through different perspectives, each based upon a different school of philosophy (à la Idealism, Realism, Reconstructionism, Existentialism, Phenomenology, Post-Modernism, and so on). I also ended up busily tracking down published journal articles by mathematics educators that grappled with the justifiability of teaching and learning mathematics for some or all students. One paper in particular that swayed my own thinking considerably is by the mathematics educator Brent Davis entitled, “Why teach mathematics to all students?” (2001). In the article, Davis forwards what struck me at the time as an ingenious insight in noticing that behind the “why” question was in fact a “what” question. When a teacher or student asks, “Why do we teach and learn mathematics?” there is beneath it a question of, “What is it that we are teaching and learning that we call mathematics?”—that, in fact, the need to ask the “why” question is likely prompted by a dissatisfaction with the “what” that has already been presumed. Although his own answer to the “what” question—that “it is a system of interpretation”1—left me dissatisfied, I am grateful to him for helping me to look past my fixation on the “why” question in seeking meaning as a mathematics teacher.

1 To offer an elaboration, I quote Davis: “Modern curricula are, for the most part, devoid of any sense of the fact that mathematics is one of we Westerners’ principal means of interpretation—that is, of mapping categories of experience onto one another. . . I join with Lakoff and Nunez in asserting that ‘what human mathematics is’ is not a mathematical or philosophical question, but one that should be addressed through an interdisciplinary study of mind, brain, and their relation. Put differently, it is a question of the complex coupling of biology and phenomenology” (Davis, 2001, p. 23).

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Neither his insight, nor any others’ arguments for the legitimacy or illegitimacy of mathematics education,2 ultimately quelled my deep-seated unease over my position and predicament as a mathematics educator, who it seemed to me practiced little mathematics.Yet, in time I began to shift not only in my thinking, but in my general orientation toward mathematics itself.To speak of it in technical terms, it was not so much an epistemological shift, but an ontological shift in the way I stood in relation to the world, and specifically, to mathematical activity. It is a shift that I could attribute to many factors, including mathematical and nonmathematical conversations with other mathematics teachers and educators, reflecting upon my own personal enthusiasm and interests in mathematics, and even “doing” a little mathematics in ways that I could not have envisioned. One pivotal incident that contributed to this shift occurred outside of the confines of academia, out in the snowy Vermont wilderness. I had been out with a small group of friends tracking wildlife, such as deer, foxes, squirrels, and grouse—not for hunting—but simply to track and come to some deeper understanding of the surrounding terrain and the lives that populated that terrain. For myself, it had something to do with wanting some connection to the wild—to something larger than myself. One of the women in the group was telling me of an acquaintance, a bird lover and wilderness guide who had once taken her bird watching—someone whose seeming passion for birds had sparked a fascination for her in birds then. And since, she had felt attuned to the world of birds, as if having undergone a kind of initiation rite into the realm of birds through this man’s own passion, or connection. When I remarked how lucky she was to have come across such a teacher and guide, she agreed by explaining how important it was to her that her teachers be passionate about what they teach, especially if they wanted their own students to learn to love what they taught. Perhaps it was in the juxtaposition of my own desire for connection to the wilderness, and of her friend’s connection to birds? Or perhaps it was in “getting away” from the everyday routine of thinking, reading, and writing mathematics education, and of seeing things freshly? But all at once, I felt an excitement about the possibilities in considering this phenomenon in relation to teaching mathematics. I began wondering how important it was for a mathematics teacher to have that “passion,” or what I also thought of as a certain intimacy and connection with mathematics.3 By intimacy, I did not mean any comprehensive type of knowledge of the field, but more a viable and personalized relationship with what we call mathematics. Just

2 For other relevant arguments, see Chazan (1996), Huckstep (2000), Noddings (1994), Usiskin (1995), and Walkerdine (1988). 3 This is the typical argument given for why professors in colleges and universities (i.e., those who are actively “relating” to their disciplines) are in the best position to “pass on” their insights and excitement about the subject to their students.

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as the bird lover forms a particular and intimate relationship with the world of birds and the tracker with the surrounding wilderness through a kind of active engagement and co-participation, some teachers, perhaps, come to form a particular and intimate relationship with the subject that they teach, or so I began thinking. Some teachers, I know, enter the field for the love of children, and others for the love of teaching. But what happens, when one lacks the love and joy—or more generally, sense of connection, intimacy, and relationship—toward the subject? What happens when one lacks or loses that viable sense of relationship to the subject that one then is supposed to teach? Is not teaching an act of bringing another into relationship with that object (or “other”) that one is supposedly teaching? Yet, what if one has become estranged from that other (of the subject)? Can one still be considered a teacher of, or guide into, that subject? I wondered these things for a number of reasons, partly because I could at many times sense a lack of connection to the subject on my part as a mathematics teacher and wondered whether this was okay. I mean, how would this play out in my own teaching, and in the learning of my students? Was I doing my students a disservice by teaching a subject that at times I felt disconnected to? And at other times, when I did carry an excitement about a very specific subtopic, I noticed that certain pedagogical potentials availed themselves to me in ways that I could not have anticipated. Though this seemed commonsensical to me at the time, was there a way for me to explain this phenomenon in a way that would not sound fluffy and trite, but rigorous and grounded both in experience and in theory? And on the flip side, I also remember teaching a geometry course, where my own excitement and enthusiasm for the subject seemed to get in the way of my being empathetic of the students’ own lack of interest in the subject. So, it did not appear to me simply to be a case of “excitement toward subject = good teaching.” I also wondered about these things because I have been around others who appeared very connected to what they taught, and somehow this seemed to matter to me as either a student or apprentice to them. The notion of being brought into relationship with the subject, or activity, resonates for me in such instances. These introspective reflections had a powerful effect upon me. Maybe it was that I had the right image now—that of “relationship” to subject? Slowly, these reflections began to clarify for me the potential importance of having such a viable relationship to the subject of mathematics. It was an image that seemed to open up new insights for me in making meaning of being a teacher of mathematics. Yet, perhaps more importantly, it seemed to offer in a rather reflexive fashion an avenue for change in the way that I stood in relation to mathematics— to a subject that I was engaged with as a teacher. In retrospect, I recognize that my unease about my professional identity could very well have been indicative of my own strained relationship to the subject. To speak of it metaphorically—if I am not already, by speaking of it as a relationship—it was a relationship that called for a healing of sorts. What began as a quest

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xiii

to find justification for my existence as a mathematics teacher has instead turned into a pursuit of understanding, appreciating, and perhaps even invigorating a sense of relationship to mathematics. In this regard, I think of philosopher John Dewey’s comment on how we come to resolve some stubborn questions and issues: Old ideas give way slowly, for they are more than abstract logical forms and categories. They are habits, predispositions, deeply engrained attitudes of aversion and preference. Moreover, the conviction persists—though history shows it to be a hallucination—that all the questions that the human mind has asked are questions that can be answered in terms of the alternative that the questions themselves present. But in fact, intellectual progress usually occurs through sheer abandonment of questions together with both the alternatives they assume, an abandonment that results from their decreasing vitalism and a change of urgent interest. We do not solve them, we get over them. Old questions are solved by disappearing, evaporating, while new questions corresponding to the changed attitude of endeavor and preference take their place. (Dewey, 1910, p. 19) I, too, never solved my original “problem,” but somehow it has faded considerably. I attribute this fading primarily to having shifted the focus onto privileging my own relationship and connection to subject matter. What such a connection—or forming of a “relationship”—to mathematics might look like and mean is at the heart of this book. My perhaps naïve hope is that by reframing a person’s learning of mathematics less in terms of the knowledge that accrues and more in terms of the “relationship” that evolves, that some old and persistent questions will be allayed, and that fresh questions with new alternatives will arise for the benefit of educators and for the educational venture as a whole. Lastly, I would like to note that although this is not a self-improvement book in that I offer no prescriptions for behavior or activity to better establish a relationship with mathematics or any other subject, it is my hope that by presenting the shape and quality of what it means to be in relationship to a subject matter, I will have offered possibilities for the reader of such a way of being, just as others have done for me.

ACKNOWLEDGMENTS

I will do this briefly and in chronological order. I am grateful to Jim Hiebert and to Dan Chazan, who each in his own way helped to shape the various wild impulses of thought that have now become this book. Tony Whitson fed the wildness while David Blacker laughed along with it. Thank you, Max van Manen, for providing a path and for sharing the tools. I am inspired by the thinking of Deborah Schifter, David Kirshner, and Stephen Brown. Each has helped me to stay in this field. I am grateful to all the teachers, musicians, and other adventurous souls who offered and continue to offer to me a model of being in this world. Lastly, I want to thank Bill Pinar for his magnanimous support and encouragement and Naomi Silverman for her patient guidance.

1 INTRODUCTION TO A PHENOMENON

An Opening Anecdote Let be be finale of seem. (Wallace Stevens, 1923, from “The Emperor of Ice-Cream”) Not quite a straitjacket, but more a stiff suit—something emblematic of “proper decorum.” It is what I imagine myself wearing, in spirit, while conversing with Dan, a senior mathematics educator. In actuality, the atmosphere is casual, and we are both in short sleeves and shorts, sipping on coffee and tea, sitting outside of a café. And we are talking math—that is, exploring mathematical ideas in an area of elementary-level mathematics related to the division-of-fractions algorithm commonly referred to as the “invert-and-multiply” algorithm. Only, he seems to be exploring and intellectually flitting about, while I feel stifled and hampered, as if I were constrained by that metaphorical stiff and stuffy suit. It is a situation that feels atypical for me, as in most any other setting, I would likely be fluttering about myself. He is raising one mathematical conjecture after another, each moving in a direction different from the prior one. At once he is making connections to secondary mathematics, to supermarket pricing, and to various pedagogical concerns. Maybe to him, there is a certain coherence, but to me, it seems almost magical and “wild.” I notice the proverbial twinkle in his eyes. Meanwhile, I struggle just to keep up. I have come into the conversation with a pedagogical agenda: to figure out the best way to lay out a lesson plan so as to convey the mathematical justification for the algorithm to the students. So, as I listen, it is not that I simply have to understand for myself the ideas that Dan is sharing, but I am listening also as a teacher: “Do these ideas also belong in the lesson plan? If so, how might I fit them into what’s already there? Or do I need to

2

What Does Understanding Mathematics Mean for Teachers?

abandon my original approach for the lesson?” and so on. Dan, on the other hand, seems to be focused very much on the mathematics, and again, I somehow struggle to keep up. I am not as “free” to simply think mathematically as he seems to be. (Of course, he does not have to teach the lesson himself!) At the same time, I realize that I am bound not just by my pedagogical constraints. I could just as easily join him in the mathematical exploration by putting aside my teacherly concerns for now, but instead, I feel strangely hindered. It is as if something in my bearing toward the task will not give way, will not loosen. I feel like the straight man—the stooge—to his inventive excursions. I am the personification of the stiff suit, of proper decorum (but in all the worst ways!). And even though I have been trained by my education to solve mathematics problems (within reason), I realize that I have only on a few occasions made true mathematical explorations, as he appears to be doing in front of me. Further, I cannot remember ever doing so in the company of others. And suddenly, what feels to be a particular rigidity to my mathematical way of being is becoming apparent, if not to him, to myself.That is to say, I am becoming revealed to myself in the presence of a mathematical mind that appears “freer” than my own. And how shy and embarrassed I feel for this even though I try not to give any indication of what feels like a tinge of self-conscious shame. (“Why, I should not even be feeling such embarrassment and shame!” and on and on, an involuting spiral of self-consciousness wraps me in its deepening hold.) I sit here, slightly distracted from the actual content of the conversation, while taking in my predicament—a state of affairs that feels unbearably clear in focus. It occurs to me that the way in which I have been approaching the study of mathematics to a large extent has been rather orthodox, which is to say that it has lacked a certain personal flair, or a real, genuine curiosity. I have been hiding, so to speak, behind a façade of competence for most of my educational career. And this seemingly innocent interaction with Dan is revealing to me this façade and what has and has not lain behind it. In the days and weeks following, I make little to no active effort at change. Instead, I find myself returning repeatedly in thought to the image of the stiff suit, and how strange that feels to me, almost as if to relish the newfound awareness of the unwieldy façade I have been carrying all of this time. Part of the reason why I make no conscious effort at change is that I have a sense that things—that I—am already undergoing a process of change; or that dwelling upon the image of the stiff suit is what will bring about the desired transformation, as if searing the image will bring about a dramatic shift in my core psyche. It is the kind of experience that I have had in relation to other domains, such as with writing (through immersing myself in the writing of a particular author or poet whom I admire), with music (similar to writing), and even with personal growth (through meeting others who seemed to embody some kind of understanding that I lacked at the time). In each of these cases, I might use the word “inspiration” (etymologically derived from in- “in” + spirare “to breathe,” thus “to

Introduction to a Phenomenon 3

breathe in”) to capture some aspect of this experience. I think the word is particularly apt for the anecdote, for Dan’s freewheeling ways truly did breathe life into dark, decrepit parts of me that had not been ventilated in some time. And yet, in all of the scenarios I mentioned above, I can also point to painful moments of self-revelation, where I have had to behold my utter “lack” of something unnamable and its accompanying veneer of competence, as in this anecdote just told. In fact, this may be at the heart of the “breathing in” process. It may be that the airing of “seem”-ing is precisely what allows for true “be”-ing, or an authenticity of some sort, to emerge. But I also imagine it as if Dan has brought me into a different relationship with mathematics through some kind of an unconscious, though still legitimate manner of “initiation.” Or that even something has been “transmitted” from him to myself, which I experienced then as a “clear-seeing” of my utter lack of whatever it is that he transmitted my way. What he “transmitted,” though, was not any new knowledge or know-how, but a manner of being in this world, and in particular, in relation to mathematical activity.

“Relationship” to Mathematics I begin the first chapter with the episode just told for two reasons. First, I take the narrative as an entryway into the phenomenon under study. For now, I will call this phenomenon “relationship” to subject matter, which in the case of this study will focus primarily on mathematics.1 Being that the central aim of this book is to clarify and to shape a particular manner of knowing a subject matter as hinted by Dan’s way of interacting with the mathematics in contrast to my own, the anecdote offers one image (of many to be forwarded) of such a way of both knowing and not knowing the subject. It is an anecdote which I will revisit in the Epilogue. Second, the initial narrative is offered as a token indication of my own personal investment in the writing that is to follow. On a personal level, what I seek through this study is insight, not only into this phenomenon of “relationship,” but also into how “relationship” relates to what may have occurred in my own mathematical education prior to that one incident as well as what happened with Dan. From what I have since seen in working with both pre-service and in-service mathematics teachers, I believe that the kind of disconnected relationship to mathematics that I hinted at on my part in the anecdote is not entirely uncommon among mathematics majors and mathematics teachers. Thus, arriving at an

1 My choice of mathematics as the domain of inquiry is a personal one, as might be imagined by my own personal history as partially disclosed in the preface. At the same time, it is my aspiration that what I write here will be pertinent not only to those whose interest is in mathematics education but to other disciplines as well.

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What Does Understanding Mathematics Mean for Teachers?

essential understanding of what a meaningful relationship to a subject such as mathematics might be and look like, is in my mind not only a personal venture, but likely one of concern to others. I might note, for the curious, that something did change for me after the experience. The “stiff suit” eventually faded, and I found myself relating to mathematics—both in attitude and in activity—differently, and in what felt to me, with a deeper sense of connection and meaning. I found myself more involved in discussions around mathematics. I became more curious about why things were so, mathematically. But also, as a teacher, I felt a greater sense of fascination with what I was teaching. Although I do not want to give any impression that I had indeed “arrived” (at some enlightenment?), the point is that something had changed. Perhaps what I needed was to be shown that such a way of being in relationship to mathematics was possible? And perhaps being shown a possibility had its impact on me because I had sought that all along somehow, to be able to bring my own sense of curiosity to bear in my relationship to mathematics? Yet, I also began unearthing memories of mathematical explorations from my own past, most of which I had negated in my mind as an obsessive pursuit of the inconsequential—as they had led, essentially, to nothing “tangible” that I could speak of at the time. Hence, perhaps it was more accurate to say that Dan had helped me to see that a particular way of being, or of relating to mathematics, was in fact vital and viable for my own sense of selfhood as a mathematics educator, rather than being inconsequential as viewed in terms of “publishable results?” Perhaps that was the real gift, to come to see value in what I previously held to be valueless? That is, through coming to value a particular manner of relating to mathematics, I could avail myself of the possibilities of a more satisfying and meaningful engagement with the discipline.2 And maybe even, a more satisfying and meaningful relationship to the discipline is what other teachers are seeking also? Of course, one does not often seek what one has no language or mental construct for, as was the case for me. Words such as “passion” or “love” for the subject—while connected to what I have only vaguely referred to as “relationship” to subject—are oftentimes construed less as an adaptable and adoptable orientation toward subject matter, and more often as an emotive, or else inherent disposition. In turn, they become relegated in some people’s thinking (including my own) as something that one either has or does not have. At the same time, “curiosity” seems to me too narrow a term—and perhaps even too idiosyncratic a trait in that it

2 To be careful about this, I am not making the claim that valuing relationship automatically leads to a change in the way that one relates to the subject. Instead, I am claiming that valuing a particular manner of relating to mathematics opens up possibilities (that will be elaborated upon in this book) that one could then choose to avail for oneself if desired.

Introduction to a Phenomenon 5

may not be as pivotal a link for many in forging a strong sense of “relationship” to the subject. But what if there were a way to talk about a way of knowing a subject matter in such a manner that captured this sense of connection and relationship to the subject, while also being large enough to contain different modes of such connection, including curiosity? And what if this way of thinking and talking about things were also conducive to the possibilities of growing into such a way of knowing?

“Knowing” and its Constituent Aspects Here in the stillness of forest, the sun columning before me temple-ancient, that wonder is what I regret losing most; that wonder and the true names of birds. (Susan Goyette, 1998, from “The True Names of Birds”) In the English language, the verb “to know” appears a highly adaptive and widely encompassing word. It does seemingly many things, such as in the sentence, “She knows the name of that red-breasted bird, knows how to quickly solve a quadratic equation with a missing linear term, knows why the ocean looks blue, knows a personal interpretation for the Wallace Stevens’ poem ‘The Snow Man,’ as well as knowing her own children.” At once, the word “know” adapts itself from a factual knowing of information or names, to a knowing how to do something, to a knowing why something may be, to a more complex type of knowing—or understanding—likely arrived at through reflection that would even implicate the knower in the knowing,3 and lastly to a knowing that is embodied by a being-in-relationship with another. There are, of course, further categories of use for the verb (including inaccurate uses, such as when someone might say, “I know that it will rain tomorrow” when faith or belief is intended) but the point simply being that it appears to do a lot for us as communicating individuals. Although its fluidity and capacity to do as much work as it does could be considered useful and advantageous depending upon the situation—for example, in everyday conversations where one usually does not want to expend enormous

3 In contrast to the first three examples, “knowing” a personal interpretation of a poem requires that the knower bring forth his or her own life history, cultural baggage, poetic sensibilities, and so on. If one were to put someone else in his or her place, the interpretation would differ, whereas in the first three cases, the knower himself/herself is not relevant to the known fact or idea. He or she could be replaced, and we would have the same known fact or know-how. In this way, the kind of knowing as specified by knowing an interpretation of a poem is one that would implicate the knower in the knowing.

6

What Does Understanding Mathematics Mean for Teachers?

energies thinking up the most appropriate and exact word for every sentence that one utters—these same virtues become liabilities of sorts when attempting to be more explicit and precise. This may happen, for example, when one’s concern is the matter of knowing itself, as occurs in educational circles as well as in a number of other domains including philosophical inquiry. Being able to distinguish between whether a student knows that something is so, from whether he knows how to do it becomes at times an important and necessary distinction—a distinction that is otherwise obscured by the plurality of the word “know,” unless modified as with the that and how above. Of course, that is an easy case, resolved by the addition of a single modifier. Naturally, this may be more than just a semantic issue (although some might argue that all of our conceptions, and thus experiences of reality are constrained by language).Various distinctions surrounding the phenomenon and/or conceptualizations of knowledge and knowing have been made by a number of well-known educational philosophers and thinkers, including John Dewey and William James, whose works along with others’ will be addressed in subsequent chapters. Yet, one particular nuance that is almost entirely veiled in the English language (and does not offer easy resolutions such as in the previous example) but makes itself apparent when considering a number of the romantic European languages is—to use the French in this case—the distinction between savoir and connaitre.4 To know in the savoir sense is to know things, facts, names, something by heart, and even knowing how to do something, whereas to know in the connaitre sense is to know a person, a place, or an-other in such a way that one is “familiar with,” or “acquainted with” this other. It is a distinction that, as I will argue, has not been fully investigated for its educational implications.5,6

4 In Spanish, it would be saber and conocer, in German, wissen and kennen, in Italian, sapere and conoscere, and in Latin, scire and cognoscere. 5 My treatment of the two words differs rather sharply from how the editor-translators of Guy Brousseau’s writings (Balacheff, Cooper, Sutherland, & Warfield, 1997) have dealt with them. They refer to connaitre as “knowing” and savoir as “knowledge,” where the “former refers to individual intellectual cognitive constructs, more often than not unconscious; the latter refers to socially shared and recognized cognitive constructs, which must be made explicit” (p. 72). Their use builds, to some extent, upon the work of Michel Foucault (1972), who wrote in Archaeology of Knowledge, “By connaitre I mean the relation of the subject to the object and the formal rules that govern it. Savoir refers to the conditions that are necessary in a particular period for this or that type of object to be given to connaitre and for this or that enunciation to be formulated” (p. 15, footnote 2). It should be noted that I am employing the two terms, less so in the vein of defining a theory of knowing, but more for their usefulness in signifying, and even accentuating (with the contrast of savoir) a way of knowing that suggests a sense of personal connection and/or relationship. In other words, I am centrally interested in clarifying a manner of knowing that involves “relationship,” where the savoir/connaitre distinction offers a linguistic handle on the phenomenon. 6 See Appelbaum (2008) for a discussion around connaitre and savoir that is closer in tone to mine.

Introduction to a Phenomenon 7

A savoir way of knowing is straightforward. Its normal translation into English is typically “to know.” On the other hand, with connaitre, “to be familiar with” is usually given as a supplementary translation to “to know” so as to differentiate it from savoir. When one considers that connaitre is used to denote a knowing of nonobjectified others as in persons and places that one knows, then I think that it begins to hint at a kind of knowing with a different feel from the more detached savoir way of knowing. In fact, we might call it a knowing that is steeped in relationship. In that vein, it is a verb that can also be used to mean “to meet,” as in another. Even the etymological associations between “familiar” and “family” (constituted by a web of relationships)—both rooted in that which is “domestic”— suggest a link to such a characterization.7 So as to more fully explore the difference and to problematize the issues arising out of consideration of these ways of knowing, I would briefly like to offer an example from my own life. As someone who likes to look at the birds out on my balcony (where I have a bird house and a bird feeder), I consider myself to know in qualitatively different ways—that may qualify as both savoir and connaitre manners of knowing—some local birds. In the savoir manner of knowing, I am able to quickly identify most birds by sight (by shape, size, and color, and by flight

7 Although the primary phenomenon and perhaps even metaphor under study here is that of relationship to mathematics, I resort to the savoir/connaitre distinction for the way in which I am both able, somewhat loosely, to map relationships into a connaitre way of knowing, and for the way in which it sets up a contrast of that way of knowing alongside the more common sense of knowing or knowledge as in the savoir way of knowing. I might note here that there are a number of similarly paired terms—theoretical frameworks, in some cases—that others have defined and explored. Etienne Wenger’s (1998) notions of reification and participation are briefly considered later in this chapter. Phenomenologist Max van Manen (1990) writes of the gnostic and pathic dimensions of experience, which I examine in Chapter 2. Also in that chapter, I consider the commonly conceived split between cognition and affect.There is also the age-old contrast between learning and unlearning, to be touched upon in Chapter 3. A group of feminist scholars (Belenky, Clinchy, Goldberger, & Tarule, 1986) consider the differences between what they term separate and connected ways of knowing, which I briefly address in Chapter 4. A number of other constructs of relevance—that I do not address—are as follows: neo-Vygotskian theorists Lave and Wenger (1991), whose situated view of knowing offers a contrast to the traditional cognitive view of knowledge; John Dewey’s ideas of the logical and psychological (see Dewey, 1902/1956, 1933); William James’s notions of the conceptual and perceptual order (see James, 1996, pp. 47–112). In terms of expression, there is the designative vs. expressive distinction, as well as semantic and mantic qualifications. In a rather telling observation, the first term in each of the binary pairings—the savoir equivalents—as I have laid them out agree in their meanings and connotations fairly consistently. Even the latter terms share more in common among themselves than might be expected, although there is less convergence of meaning than among the first terms. Consequently, one point worth making here is that the idea of trying to explore alternate conceptualizations of knowledge and knowing against a commonly conceived image of it as in the savoir way of knowing is not new. And perhaps more importantly, the plethora of attempts also suggests the importance of such an undertaking and of seeking out a broader and fuller conceptualization of what it means to know something.

8

What Does Understanding Mathematics Mean for Teachers?

pattern when I am attentive) and sound (calls and songs). I also know some of their seasonal migration patterns, the pecking order among species when it comes to who bullies whom, their diet, and so on. But I have also come to know some birds in what might be considered a connaitre manner. They have somehow become important to me, not as an abstraction, an idea, an object, or something to simply be labeled with a species name, or even as an extension of, say, nature. Instead, one could say that I have “come into relationship with” a few of these birds. They have become an other, as opposed to an object. Something in me, my history and my being have come into relation with these distinct others. This is to say that I experience a sense of connection or intimacy, along with joy and pleasure in their company, or to the life and being that is the bird, instead of to the label or name. At the same time, I do not experience these two ways of knowing as necessarily distinct. I find that the more I am able to imbue myself in the experience of being with the various birds in my neighborhood—that is, the more I attune myself to them—the quicker I seem to be in identifying and naming them. Likewise, I have noticed that the more facts that I learn, and the more I am able to identify the birds, the more easily I am able to feel connected into this world of birds. There is a kind of interplay in how one way of knowing supports the other. It is not simply a case of privileging one over the other. Yet, I have also noticed how each way of knowing can just as well at times inhibit the development of the other. Sometimes, I am content just to watch, to enjoy their behavior without any thought of learning something in particular. While not a particularly unfortunate or onerous development, if this were all that I did, I might in fact be missing out on some of the rewards of actually studying and learning about birds. At other times, I have found myself content just to be able to name the birds, and really not paying mind to this other living being that I had already mentally “captured” with the label. This, too, would be a problem in the long run if that were all that I did, as I would then turn into some kind of a naming machine without any relation to the life surrounding me. What I hope emerges from my short description is the dynamic and complex interplay that is possible between these two ways of knowing.8 As I will return to in the Epilogue, I believe that it is this interplay—or lack thereof—that underlies my initial anecdote, particularly in terms of my own mathematical education, and perhaps, the mathematical education of many if not most in this country. But to arrive at that point, as well as to begin to understand more fully the way in which these two qualitatively different manners of knowing might interact, I believe that

8 This is to say that I do not see savoir and connaitre, necessarily, as oppositional binaries. There is not mutual exclusivity nor intrinsic conflict between the two, as will be the case with some of the other constructs I will discuss at later points. What I say about savoir and connaitre not being oppositional binaries could also be applied to Buber’s I–It and I–Thou relations as well as Wenger’s reification and participation.

Introduction to a Phenomenon 9

I must more clearly articulate—that is, in some ways establish—what a connaitre way of knowing might entail.9

A Comparison Across Disciplines As I noted earlier, to know a subject such as mathematics in the savoir sense is fairly straightforward. It corresponds with most people’s intuitions about what it means to know that subject, whereas with a connaitre way of knowing, things may not be as clear. There is not, I believe, any agreed-upon image of what that might actually look like. It is one thing to talk in suggestive and metaphorical language around such a phenomenon, especially in regards to another living being such as a bird—where “being in relationship” with that other might have some resonance despite sounding perhaps a bit on the lonesome side. It is another to be precise and specific in offering a concrete set of descriptions of the phenomenon in relation to a school subject matter, where to speak of having a “relationship” to a discipline might only be comprehensible on metaphorical terms, if that. It is all the more challenging, I believe, to conceptualize such a relationship to a subject such as mathematics, as opposed to say the arts. It has been my experience that in speaking of these ideas (where I quickly explain the savoir/connaitre distinction) with visual arts and music teachers in particular, there occurs surprisingly quickly an implicit understanding and awareness as well as sympathetic identification with the importance of knowing their art or craft in ways that correspond well with both a savoir and connaitre way of knowing as I have described them. Meanwhile, I cannot say the same of my interactions with mathematicians and mathematics teachers. In fact, a musician who did not know a piece she was playing in both ways would probably be looked upon as being somewhat deficient. If she knew it only in the savoir sense, she would be understood to have no artistry; while to be sufficient only in the connaitre sense would imply a lack of technical proficiency, or else a lack of historical, cultural, and/or musical perspective so as to be wanting in her ability to do a just interpretation to the piece. What is more, it is not just the professional musician who is expected to know his or her music as such. There appears an implicit expectation of music teachers, that they too must know more than the facts and technicalities; they must also

9 Technically, it is in seeking a deeper understanding of ways of knowing that do not preclude a connaitre type of knowing (which I take to be a restatement of a way of knowing steeped in relationship) that is the intent of the book—not just in isolating the connaitre way of knowing. This is to say that in some ways, the separation forged by a savoir and connaitre way of knowing is somewhat artificial, as there is likely not in actuality a case of a purely savoir or connaitre way of knowing.Yet, the separation does serve a useful purpose in emphasizing two aspects of knowing, where the savoir usually gets more play in both the research field and popular culture, as far as knowing and understanding school subject matter is concerned.

10

What Does Understanding Mathematics Mean for Teachers?

have some kind of “relationship” with music as well. The presumption seems to go, “If I, the teacher, do not have some connection to music, how can I possibly pass it on to my students?” Yet, with a subject such as mathematics, to speak of having some relationship to mathematics seems fraught with ambivalence and perplexity. One could conceivably conjure up images of a professional mathematician “in relationship with” mathematics. Yet due perhaps to the logical and reasoned nature of the subject, most images of such mergings with mathematics, unlike in the arts, likely come across as eccentric, or even nutty and insane (see Hoffman, 1998; Kanigel, 1991; Wilson & Latterell, 2001), instead of being dedicated and well-trained as in the case of a professional musician. Even if such were not the image that came to mind, one must acknowledge the tension for the case of mathematics, between the personal and familiar tinge in the connaitre way of knowing and the rather impersonal and seemingly nonsubjective “face” of mathematics, at least so far as it is popularly conceptualized, and especially, as encountered by most in schools.10 When one then begins to reflect, not upon professional mathematicians where zany characters are aplenty, but upon mathematics teachers or teacher educators (such as myself), then the idea of knowing mathematics in this connaitre manner— that is, being “in relationship with” mathematics—seems an even more distant notion and image. It does not have the same ring, or resonance, as say of a music teacher being in relationship with music. Rarely does one encounter the mathematics teacher, who like their musical counterpart, continues to spend countless hours continuing to develop their craft—that is, attending to their relationship to the subject through “practice” or other disciplinary activities in connection with mathematics. Perhaps such a state of affairs can partly be attributed to the actual differences in music and mathematics themselves, or more precisely in how most persons generally interact with each.11 There is a useful theoretical framework formulated

10 Take, for example, Edward Rothstein’s (1995) characterization of mathematics against music: “ ‘Music’ seems steeped in affect; we commonly talk about music as sad or happy or angry or gentle. Music is spiritual, aesthetic, religious. Mathematicians couldn’t care less about the emotions suggested by a theorem’s proof. Has anybody ever encountered a ‘sad’ theorem, or presented an ‘angry’ proof, or inspired a courtship through abstract musings about topological spaces?” (p. 6). 11 Technically, these differences as I speak of them are not between mathematics and music at large, nor of the disciplines as necessarily practiced by the professionals in each field, but of the images as held predominantly by the rest of us—including teachers—whose contact with them occurs primarily through school, avocation, and other transmuted, and translated forms of the discipline. It is what some have referred to as “school math” (see Papert, 1993; Popkewitz, 1988;Watson, 2008) so as to differentiate it from mathematics as practiced by mathematicians. A technical term used to denote what happens to disciplines as they enter school and other didactical settings, used by epistemologists, is the term “didactical transposition” (Balacheff et al., 1997, p. 21).

Introduction to a Phenomenon 11

by learning theorist Etienne Wenger that may help with unpacking an aspect of these differences. In his book Communities of Practice (1998), Etienne Wenger discusses the interplay between participation and reification. Participation is defined as the “complex process that combines doing, talking, thinking, feeling, and belonging. It involves our whole person including our bodies, minds, emotions, and social relations” (p. 56). Meanwhile, reification is defined as the “process of giving form to our experience by producing objects that congeal this experience into thingness” (p. 58), and would include not only objects such as books, plans, and notes, but also mental objects such as concepts and even words. As an example, a book could be considered a reification of someone’s thinking, with the act of thinking being a form of participation. Even a person’s understanding of, say, a grammatical rule is a kind of reification, as long as it has solidified into something that one might call an understanding, whether correct or not. One reification of Wenger’s ideas—and of course, reifications often leave out important details as he points out—is to consider participation as process and reification as product. Thus, in terms that Wenger has introduced, one might say that music generally lends itself to participating—that is, to playing and to listening. In fact, music teachers that I have spoken with tend to portray music as a participatory act, where engaging in actual music making or else listening to music is considered the primary act in relation to music. By contrast, mathematics as most nonprofessional persons experience it and thus conceive of it seems to fit well Wenger’s notion of reification: something that one learns or knows (really in the savoir sense of it), and rarely, if ever, something that one “plays,” participates with, or simply does.12 One might wonder, though, if there is a meaningful difference between “doing” and “learning (as acquiring)” mathematics other than mere semantics? It is a question that I take up in greater depth in Chapter 5. But for now, consider the difference between the statements, “somebody is learning mathematics” and “somebody is doing mathematics.” Notice how the meaning of the word “mathematics” shifts, depending upon whether one speaks of “learning” it, or “doing” it.When I say that I am doing mathematics, the word “mathematics” comes to signify a process, activity, or act of participation, whereas when I say that I am learning mathematics, the same “mathematics” becomes a reification—that is, an object.With the latter, it becomes an actual, congealed thing to learn, different from the fluid and participatory domain of activity of the former. Furthermore, mathematics that is “discovered” or “applied”—again, typical school activities involving mathematics—also both keep it as object.

12 Incidentally, the research literature on mathematics teachers’ beliefs on mathematics generally coincides with this point, that many mathematics teachers themselves see mathematics less as a process that one engages in, or does, and more so as a static collection of knowledge—rules and/or concepts—that one learns or acquires (see Ernest, 1989; Thompson, 1985, 1992).

12

What Does Understanding Mathematics Mean for Teachers?

I offer the contrast between “doing” and “learning” mathematics at this point to show that participation in mathematics is not a readily accessible image for most as it is with music—except among professional mathematicians whose job it is to do mathematics. And perhaps this has something to do then with why “having a relationship” with mathematics may feel less customary than that of having a relationship with music. In other words, relationship in its most general sense (as well as familiarity) grows out of some kind of participating with whomever or whatever the other may be, but without readily accessible modes of participation, the resulting relationship—or the connaitre way of knowing it—will likely not ensue.

The Design of Things to Follow My comparison across the subjects of mathematics and music—and across math and music teachers—is not accidental here. Music offers what I consider to be a broad entryway into the phenomenon of a connaitre way of knowing, or “relationship with” subject matter. I mean this in two ways. On one hand, by speaking with musicians and music teachers about their relationship to music, it has helped me to calibrate my own thinking and approach in then conversing with mathematics teachers, many of whom struggled, at least initially, with conceiving of ways of knowing not savoir-like in nature. Second, the musicians and music teachers as a group tend to present fairly resonant descriptions of what entering into a relationship with subject matter looks like that I believe will help to anchor and to augment the descriptions given by the mathematics teachers around the same phenomenon. This is to say that the role of music and the music teachers as an “entryway” into the phenomenon occurs both in the methodology13 as well as in the presentation. And yet, there is one other important reason for bringing in the musicians and music teachers and their descriptions. My goal in this study is to understand and to describe a way of knowing that involves a sense of relationship, with mathematics as the setting. By looking not just at descriptions given by mathematicians and mathematics teachers, but also those offered by musicians and music teachers, I hope to be able to confirm the fit of the described knowing across these two disciplines. That is, by seeking the invariant structures of experience (with respect to the phenomenon) common to both groups, I am able to more surely arrive at an essential understanding of the phenomenon of “relationship to subject matter or discipline” itself. In this way, the music teachers might be said to provide a kind of proving ground, in addition to being an entryway for the study. What emerges then through reflection upon my sources—relevant theories and philosophies mostly from outside of mathematics, accounts of the nature of

13 The methodology I have adopted for this study is a rather loose adaptation of a hermeneutic phenomenological methodology as described in Van Manen (1990).

Introduction to a Phenomenon 13

mathematical activity and experience by mathematicians and mathematics teachers, and descriptions of the experiences of music teachers, and musicians—is a three-fold characterization of relationship. Two or “twin” aspects concern the lived experience of entering-into-relationship, with the third having to do with the resultant state of being-in-relationship with a subject, in this case, mathematics.14 Thus in Chapter 2 I will elaborate upon the first of the coupled qualities: the experience itself of being affected through the relationship. The sense here is that in the process of entering into true relationship, one undergoes experiences of a receptive or passive nature brought on by the effect of other. It is what philosopher Martin Buber (1970) refers to as “grace.”There are ways in which the study of, or engagement in, a subject matter can grace the engager—that is, act upon the self in a profound manner, whether through an aesthetic experience, a sense of pleasure or fun, an emotional catharsis, or even through an experience of coming to an understanding. There is also in this same process a bringing forth of self into the relationship, which I examine in Chapter 3. In order to be affected by subject matter, one is usually invested.There are different ways in which one “invests” the self—or brings oneself—into relationship with the subject matter. This chapter builds upon Buber’s notions of “will,” within the context of mathematics learning and knowing. Lastly, there is the state of being-in-relationship with subject matter—no longer the becoming but the being-in-relationship with the subject. It is this sense of being-in-relationship that, as I will argue, circumscribes the becoming. That is, it succeeds the joining of will and grace, while also supplying motivation toward further engagement and becoming. In the design of the chapter, though, I start with the issue of motivation (or that which precedes engagement), followed by discussion of the resultant relationship and knowing. Each of Chapters 2–4 will work to develop the shape and form of each of the three-fold characteristics while on the whole attempting to unveil the essential nature of relationship with subject matter. Yet, as hinted at in the Preface, this project of relationship is from another perspective an attempt at reconceptualizing and expanding the meaning of “understanding” (mathematics), as will become evident by the end of this book. Thus, I conclude each of these chapters (as well as the Epilogue) with a re-examination of what it means to “understand” some mathematics, as informed by the ideas within these respective chapters. In the closing chapter, I shift the focus from the phenomenon itself to how it pertains to the life path of being a teacher.Thus, Chapter 5 might be looked upon

14 I might note beforehand that I do not consider these characterizations to be discrete and mutually exclusive.That is, some qualities that I attribute to one aspect could very well easily be talked about in terms of another. Thus, there are overlapping characterizations. For the reader who is interested in a brief description of how I arrived at the three-fold characterization of “relationship” to subject matter, see Appendix.

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What Does Understanding Mathematics Mean for Teachers?

as an “applications” chapter where I use the theoretical machinery established to look more closely at the act of “doing” mathematics and how that relates to teachers. In particular, I aim to recast the idea of mathematics teachers “doing” mathematics from the standpoint of relationship. For example, it can be viewed as a way of making affordance in the relationship with mathematics for one’s curiosity and the need and/or disposition to inquire. In the Epilogue, I return to the initial anecdote between Dan and myself so as to reflect upon matters of mathematics teaching. And finally, I revisit the idea of relationship one last time, juxtaposed (ever so briefly) next to understanding, and close with an interview transcript, but viewed through the lens of relationship. Just to be sure of things, I would like to offer one last explanation on the role of some teacher comments/quotes to come in this book. As may be apparent by this point, my interests are educational in nature. Because of this, I will resort to accounts given not only by philosophers and mathematicians but also those by mathematics teachers and mathematics teacher educators, again, in the task of explicating the phenomenon at hand. Some of these teacher accounts are from the literature, and some were gathered through recorded conversations. All of the accounts are utilized for the purposes of illuminating the phenomenon of relationship with mathematics, which is to say that the emphasis is on the phenomenon, and not necessarily on any specific types of persons, such as teachers. But again, my reason for the study of this phenomenon of relationship has been from the beginning toward educational ends, and in particular, toward the quest of existential meaning-making as a teacher of mathematics (i.e., not feeling a fraud), and toward mathematics teaching. Thus, a secondary goal of this study is to begin looking at the relevance of the ideas from the perspective of being a teacher. But again to be clear about this, this is not a study about how teachers know mathematics. Instead, it is about a way of knowing mathematics that I argue has relevance to mathematics teachers and teaching. Insofar as the interplay between the primary goal of the book (i.e., the phenomenon of relationship) and the secondary one of the relevance to mathematics teachers and teaching, the general trend is as follows: Chapters 2 and 3 focus primarily on the phenomenon with sparse teacher accounts acting as simply one of a number of lenses through which to look upon the phenomenon. In Chapter 4, the teacher comments begin to predominate, especially in the first half of the chapter due to the fact that the emphasis shifts ever so slightly away from the phenomenon (of relationship that would subsume both the self and other-of-mathematics) and toward the particular, personal motivations for entering into relationship. In Chapter 5, the relevance of the ideas to being a teacher comes further to the forefront. And finally, in the Epilogue, the shift culminates as I link the ideas to mathematics teaching through my own thoughts about being a teacher.

_ 1.9 A TANGENT PRIOR TO THE RISE (AND RUN)

Before commencing with the slow theoretical climb to come, I will make one short digression—partly to motivate the ascent, and partly to frame the view from atop. On a past project, I had an opportunity to interview a handful of mathematics educators and mathematicians. I asked each what it meant to understand a mathematical concept in some “deep” manner—that is, what would it mean for them to have a “deep understanding?” Here was a theoretically important question in the field, with repercussions for both learning as well as teaching mathematics. A number of theorists have grappled with it in one form or another, from William Brownell (1945) and his ideas of “meaningful” learning, through Van Engen (1949) with his distinctions between syntactic and semantic meanings, to Richard Skemp (1976) and his conceptualizations of instrumental vs. relational understanding (see also Hiebert (1986) for his discussion on procedural and conceptual knowledge), to a handful of modern thinkers and theorists outside of mathematics education (e.g., Piaget, 1970; Vygotsky, 1978b; Lave & Wenger, 1991). My plan had been to study the meaning— from an experiential perspective (or more technically, from lived experience)—of what it meant to hold a deep mathematical understanding. While most of the interviewees offered up insightful as well as familiar responses, I found one mathematician’s responses somewhat exasperating (likely as I was just a beginning interviewer).While I could not disagree with his comments, it seemed to me that he was in some perverse way giving me nothing but runarounds to my questions, as if trying to avoid committing himself to anything tangible and concrete. For the project at hand, I made no use of the interview transcript. Yet, in returning to, and re-reading the same transcript in the context of this current project, I find that in fact he was conveying to me things that I was

16

What Does Understanding Mathematics Mean for Teachers?

incapable of hearing at the time. I could not hear what he was saying because my ear was attuned to, and maybe even fixated upon, understanding, and not on relationship. Put differently, I did not have the theoretical machinery—or I had too much of some other—to attend to the significance of his comments. I suspect that several of the teacher remarks to come in the remaining chapters of this book also cannot be fully appreciated without recourse to new theoretical lenses. Although a number of the other teacher comments in this book would similarly make the point, I would like simply to lay bare a portion of my transcript with Ponce—the mathematician—because he speaks of things that are at the heart of this inquiry, that at the same time may or may not yet be apparent. This is to say that I have an ulterior reason: there is quite a bit of theoretical work ahead—a “re-attunement” if you will—and I would like to offer an incentive for laboring through the theoretical effort, namely, that what you read from Ponce here, as well as several other teachers to follow starting in Chapter 2, will come to have a meaning and even brilliance that would not have been readily apparent prior to the toil. The plan is to reprise the same transcript again at the very end, at which point I hope that the reader will come to re-appreciate his “eloquence” on matters relating to a connaitre way of knowing that I, at one time, had such difficulty hearing. [So that you, the reader, at this point might get a sense for my impatience and irritation, recall that I was after Ponce’s description of “deep understanding.”] Yuichi: If you can think back to certain times where you felt that you understood a mathematical concept on a fairly deep and comprehensive level, please tell me about it. Ponce: I think that it’s hard for me to . . . find such an occasion. What happens always is, I find, feel that it’s actually that’s very often is that my understanding becomes better, but I don’t know whether I can say that I understood something very deeply or completely. I never had that feeling, and from all of my experience, I don’t think that it can be some final process, I mean, because you do go deeper and deeper, and you feel . . . and you usually remember those steps when you get deeper and deeper, but I’m not sure that I can really say that I understood something completely. Yuichi: So from what you said—if I’m interpreting correctly—you’re saying that you’re not content to just stop at say “I have deep understanding” because you feel that you can continue to— Ponce: Yes, in the past that’s what’s happened all the time. It seems that I understand things deeper and deeper. Yuichi: I see. Ponce: So, I don’t think I can really say that I understand something completely . . . I understand some definitions, I think, completely,

A Tangent Prior to the Rise (and Run)

17

like “What is a graph?” or “What is a set?” But even with set theory, as you know, one can ask brilliant questions about basic set theory which requires very careful language, so if you don’t use it, contradictions can be brought [about] immediately, so if you understand elementary set theory, I’m not sure . . . I understand many things about it but . . . Yuichi: Did you actually have issues with my question to some extent of trying to qualify understanding, this idea of “deep?” Ponce: Yeah, I think that it’s all relative. And when more and more time you spend in mathematics, more and more connections you see between things, and then, some applications which you might have thought you know . . . Yet of course, if you can understand this deeper . . . in addition to what I knew about this subject, I know more . . . It’s all relative, I mean. What is deep water? Yuichi: I see. So, you’re talking, and I’m rethinking my questions. Maybe I need to rephrase it somehow . . .You mentioned there not being a “final process.” But isn’t there sometimes? Couldn’t that be a “deep” understanding? Ponce: I question further and very often I cannot answer the questions I ask. With some areas, you just deliberately decide not to question anymore because there’s finite time there. You are interested in results, and you cannot dwell in one direction all the time so you have to stop it somewhere. When I stop, does that mean that . . . I understand deep and I don’t want to understand more? No, rather, in most cases it’s just that I deliberately decided to stop at this point without trying to understand things further.

2 RELATIONSHIP AS RECIPROCITY: GRACE

Relation is reciprocity. My You acts on me as I act on it. Our students teach us, our works form us. (Martin Buber, 1970, p. 67) In his book I and Thou (1970), the philosopher Martin Buber contrasts two ways of relating with other. One might also call these “attitudes” or “stances” toward the other. One of such stances, he refers to as an I–It relation, characterized by a lack of mutuality and reciprocity, wherein the I is complicit in a typical subject–object relationship.This is to say that because the other, as It in the I–It relation, does not act upon the I, it is essentially relegated to “object” status. As such, it is an instrumental way of the subject (I ) to relate to other (It). With the I–Thou relation, on the other hand, the I and the other act upon one another—through a directness and presentness brought to bear upon the interaction—resulting in a manner of co-evolvement of self and other. For Buber, it is such mutuality of effect that gives rise to “true relation.” To elucidate the distinction, I use Buber’s own illustration. He begins by describing his own I–It orientation toward a tree: I contemplate a tree. I can accept it as a picture: a rigid pillar in a flood of light, or splashes of green traversed by the gentleness of the blue silver ground. I can feel it as movement: the flowing veins around the sturdy, striving core, the sucking of the roots, the breathing of the trees, the infinite commerce with earth and air—and the growing itself in its darkness. I can assign it to a species and observe it as an instance, with an eye to its construction and its way of life.

Relationship as Reciprocity: Grace 19

I can overcome its uniqueness and form so rigorously that I recognize it only as an expression of the law—those laws according to which a constant opposition of forces is continually adjusted, or those laws according to which the elements mix and separate. I can dissolve it into a number, into a pure relation between numbers, and eternalize it. Throughout all of this the tree remains my object and has its place and its time span, its kind and condition. (pp. 57–58) Whether as an object of contemplation, both in form and movement, as an instance of a category or manifestation of some underlying law, or even as an abstraction, the tree remains for Buber an “object.” In fact, he goes so far as to call it “my object,” likely emphasizing the ways in which he himself has specified— that is, acted upon mentally without a reciprocal being-acted-upon by—the tree with its varying characterizations. Again, it is an instrumental manner of relating with the tree. Note, also, the similarity of this description to my earlier characterization of a savoir manner of knowing: a knowing of, a knowing that, a knowing how, and so on. In both, the manner of knowing or relating implies an element of distance, or absence of personal connection and what might be called intimacy. Contrast that now with how Buber describes the I–Thou relation, which could be read as being aligned more closely with a connaitre way of knowing, but also that which also does not preclude the savoir way of knowing. But it can also happen, if will and grace are joined, that as I contemplate the tree I am drawn into a relation, and the tree ceases to be an It. The power of exclusiveness has seized me. This does not require me to forego any of the modes of contemplation. There is nothing that I must not see in order to see, and there is no knowledge that I must forget. Rather is everything, picture and movement, species and instance, law and number included and inseparably fused. Whatever belongs to the tree is included: its form and its mechanics, it colors and its chemistry, its conversation with the elements and its conversation with the stars—all this in its entirety. The tree is no impression, no play of my imagination, no aspect of a mood; it confronts me bodily and has to deal with me as I must deal with it—only differently. One should not try to dilute the meaning of the relation: relation is reciprocity. (p. 58)

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What Does Understanding Mathematics Mean for Teachers?

There are two relevant elements (as one might expect in a “reciprocity”) that call for attention here. They are contained in the twin notions of “will” and “grace.” Whereas the notion of will suggests an active impulse, that of grace implies a more passive, or receptive side.1 If, as he writes, that with the joining of these two qualities one is “drawn into a relation,” one might consider these—the active bringing in of “will” or something close to it, and the receptive undergoing of “grace” or again something along such lines—as the two constituent processes involved in entering into relationship with the other. For the remainder of this chapter, I will focus on the “passive,” or receptive element, while I will attend to the “active” element in the next chapter.2 In regards, then, to this passive or receptive element, one can look to Buber’s manner of portraying this process of entering into relation: “I am drawn into a relation,” “The power of exclusiveness has seized me,” and later, “[The tree] confronts me bodily.” In each of these depictions, he chooses to express the lived experience3 of entering into the I–Thou relation in terms that places the I, or himself, as submitting to, or undergoing something that is outside of his own volition or control. Note how he is “drawn into,” “seized” and “confronted” by. That is, he allows himself to be acted upon by this specific other, or more accurately, by the relationship that consummates this specific other and himself.4 Might one say that entering-into-relationship with an-other necessarily involves such an element of being acted upon by the other? Buber’s writings suggest that, indeed, the notion of being genuinely affected or acted upon by the other in question may very well be central to an experience of coming into “true” relationship with that other. 1 By “passive,” I do not mean something necessarily negative, such as in a fear-based passivity that would be equated more with withdrawal. Perhaps “receptive” gives a better sense of what is meant, although I will use the two words interchangeably, for a primary definition of “passive” is “(adj.) acted upon rather than acting or causing action”—which gives more of the neutral tinge that I intend. In connection,Wong (2002) argues that educational research and policy have neglected this “passive” element of learning to their own detriment. 2 Naturally, one could argue that each is part of a whole and cannot be separated from the other, and that further, the iterative nature of their interaction makes such a distinction contrived, or worse, misleading.Yet, for the sake of organization and analysis, I have decided to consider the two separately, and will touch upon their interrelation later in Chapter 3. 3 For the reader who is unfamiliar with the term “lived experience,” I mean to point to the immediately lived through experience of some phenomenon (such as say “coming to know”).That is, if I ask, “What is the lived experience of ‘coming to know?’ ”, I am asking what immediate, pre-reflective, pre-conceptual aspects of the experience of coming to know assert themselves to one’s consciousness as one lives through that experience. Contrast that with simply “experience,” which tends just as much toward conceptualization as it does toward the pre-conceptual aspects of the experience. 4 Technically, I believe that it would be truer to Buber’s ontology to say that one is acted upon by the relationship with other, as opposed to saying that one is affected directly by the other, since for Buber, self and other are recognized and actualized through relationship. At the same time, the actual lived experience of this same phenomenon may go both ways, although the distinction may be subtle and rather personal. Thus, except in places when explication is crucial, I will abide by what I take to be the shorthand of “being affected by other.”

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Gnostic and Pathic So as to lay further groundwork before moving into the specifics of the mathematical discipline and the relationship of a person to that subject, I now briefly turn to another pairing of terms: gnostic and pathic. Both denote relational understandings. Whereas a gnostic way of knowing allies itself with the cognitive, the conceptual, the intellectual, the rational, and the technical—more or less, the savoir—the pathic (as in em-pathic) “is sensed or felt, rather than thought or reasoned” (Van Manen, 2002, p. 25). Yet pathic should not be interpreted as just feeling. As with Buber’s I–Thou relation, it points instead to an attitude, or stance toward an-other that, rather than being just cognitive in nature, “perceives the world in a feeling or emotive modality of being” (Van Manen & Li, 2003, p. 220). In writing of the distinction between the gnostic and the pathic aspects in the practice of nursing, Van Manen (1999) points to a personalizing quality to the pathic dimension of relating. What then makes pathic practice distinct? The difference is this: pathic thought turns itself immediately and directly to the person himself or herself. A pathic relation is always specific and unique. Even a relatively brief encounter between a patient and a health care provider can have this personal quality. A personal relation is something you can have only with a specific other. The pathic orientation meets this concrete person in the heart of his or her existence, without trying to reduce to a diagnostic picture, a certain kind of case, a preconceived category of patient, a psychological type, a set of factors on a scale, or a theoretical classification. In other words, there is something deeply personal or intersubjective to the pathic relation. That is also the reason that the pathic personal relation is easily confused with the private one. (p. 33) Note, again, the contrast posed between the dia-gnostic with its reducing, categorizing, scaling, factoring, classifying, and theorizing orientation, and the direct and “personal” pathic manner of relation. Whereas the gnostic shares the same distancing (or distance-maintaining) effect of the savoir and I–It orientations, the pathic shares the more immediate and personalizing manner of relating with and knowing the other of a connaitre way of knowing (as I have described it), as well as Buber’s I–Thou relation.5

5 I want to be careful here. Although I am aligning a number of different frameworks in this paragraph, it is not in my opinion a clean alignment. Buber is clear to state that the I–Thou relation is inclusive of the I–It relation. I have also stated that my primary intention is to clearly define a manner of knowing steeped in relationship, which I view as a knowing that would not preclude a connaitre way of knowing (which assumes on some level a savoir knowing that is also included).Yet, with the gnostic and pathic dimensions of knowing, there does tend to be a stronger sense of

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What Does Understanding Mathematics Mean for Teachers?

What I would in fact suggest underlies such a personalizing effect in the latter group is the inter-subjective (literally, “between subjects”) nature of understandings arrived through a pathic stance—that is, the subject-to-subject-ness of the pathic stance, as opposed to a subject-to-object like tone implicit in the gnostic approach. To be more explicit about this, it is that the other is privileged, not as object strictly to be acted upon, or made to understand, but instead as another who is capable of reciprocity, of bringing a shared sense of understanding, and in turn of mutually acting upon the self; and it is exactly this orientation of other-as-subject, that I would argue, gives rise to the this sense of a “personal” relationship.

Mathematics as Other: Through the Aperture of Music Music isn’t the notes on the page. That means nothing. (Stuart, choral director and piano teacher) Mathematics deals with ideas. Not pencil marks or chalk marks, not physical triangles or physical sets, but ideas (which may be represented or suggested by physical objects). (Reuben Hersh, 1998, p. 22) Such an essence falls, as it were, between the [physical and the theological], not only because it can be conceived both through the senses and without the senses. (Aristotle, speaking of the essence of mathematics in Ptolemy, cited in Dossey, 1992, p. 40) At this point, a natural concern to raise in regards to mathematics might be phrased as follows: So far, with a connaitre way of knowing, with Buber’s I–Thou relation, and with this notion of a pathic orientation, it has been understood that the other in question is, if not another sentient being, at least some entity with material existence, such as a tree (for Buber) or an actual place, such as Florence (for connaitre). But when one speaks of mathematics as this other, is that not stretching the meaning of all three of these frameworks in a way that they are not meant to be? This appears a reasonable concern, especially when one considers, in particular, the abstractness of mathematics—that is, its lack of physical and concrete essence, much less sentient existence. Still, I would assert, and will argue in this section and the next, that one can meaningfully talk about being in relationship to mathematics in a manner similar to the way that Buber talks about being in

oppositional qualities to the pairings. My purpose in forwarding this particular framework as well as a few others that too share this oppositional quality then is because despite the aforementioned misalignment, they still help to bring forward in relief, an image for what a knowing characterized by relationship could be. I am less interested in rigor here (at least so in the main body of the text) and more so in making bold, evocative strokes.

Relationship as Reciprocity: Grace 23

relationship with a tree, that I talk about knowing birds in a connaitre way, or that a nurse talks about relating to a patient with pathic understanding. That is, mathematics can meaningfully be considered an “other,” not just as mental object, but in fact, as “other” that acts upon the self who is engaged with, and in relationship with mathematics. I first begin, though, with the case of music—a helpful bridge, if you will, between the concrete others of the prior discussions and the rather “abstract” world of mathematics. When it comes to people’s experiences with music, I think it fairly common to hear someone say that a piece of music “really touches him or her,” or that a person is/was “deeply moved by the music.” Others might speak of music as “seizing” hold of them, “drawing them into” its melodies and rhythms, or even “confronting” them, as Buber described the I–Thou sense of entering into relationship with a particular tree. Naturally, there are other ways in which music acts upon the listener. In each of these cases, something occurs, where the listener (or player) lets go into the experience presented by the music, allowing for the music to reciprocate the listener’s bringing forth of commitment of attention with some effect upon the listener. That is, this thing called music—with its constituent melodies, harmonies, rhythms, feelings, moods, and ideas—exerts forth its own agency on the listener. To carry the argument further, it is not entirely unreasonable to speak of having some sense of relationship to a piece of music that one has “connected with.” There is, for many, a knowing of—or familiarity with, as in connaitre—a favorite piece of music that is more than just cognitive in nature.There is emotion, affect, and ultimately, a sense of intimacy that binds the listener to the favorite piece. One can even speak of “losing oneself ” in the music, as if, again, one is relinquishing the distancing ethos of the subject–object dyad so as to dissolve into a subject-to-subject union—the I–Thou relation—of self with music. Yet, how much like mathematics is music, especially in its otherworldliness? Even if one were to argue that music is sensual (possibly unlike mathematics) in that it is experienced through the sense of hearing, it is still at root an abstraction just as much as mathematics is. Music is not the notes on a page, as one of my musician interviewees claimed. Nor is it strictly the sounds that one hears—if one is speaking of true music—or else machines would easily replace living musicians. Instead, for most musicians and listeners of music, it is an ineffable expression of feeling, emotion, and even thought, made manifest through sound, and this places it in a world far from trees, people, and things of this world. Further, one could argue that mathematics, too, can be made manifest to the senses in the form of symbols, as well as graphs and other pictorial representations of underlying mathematical ideas. So, accessibility to the senses is not music’s mark of uniqueness against mathematics. Certainly the experience of interacting with music for most persons tends toward the affective much more so than their interactions with mathematics, and of course, affect does come into play in ways that

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What Does Understanding Mathematics Mean for Teachers?

will be explored later in this chapter. At the same time, the point for now has less to do with the weighting of affect, and more to do with the fact that both mathematics and music live in comparable spaces that bridge this physical world to another beyond the purview of the senses. And it is this commonality, I would argue, that makes consideration of mathematics as possibly legitimate an other as one could consider music to be.6

Mathematics as Platonic Other Most writers on the subject seem to agree that the typical working mathematician is a Platonist on weekdays and a formalist on Sundays. (Philip Davis & Reuben Hersch, 1981, p. 321) If the notion of considering mathematics as “other” still feels tenuous, I appeal also to the lived experiences of mathematicians in describing their own “Platonic” relationship to mathematics. But first, a cursory explanation of Platonism, or Platonic Idealism: in short, Platonism asserts that truths are perfect and eternal. It exists, has always existed, and will always continue to exist, suggesting that it exists independent of whether a person thinks of it or not; from this view, the proverbial felled tree does make a sound regardless of whether anyone is there to hear it or not. The work of man, then, for the Platonist is to discover these universal truths—or in fact, to re-discover it, as illustrated in the “Doctrine of reminiscence,” where Plato supposes that the human soul once had true knowledge before losing it by inhabiting our bodies. Mathematics, in particular, was considered a prime demonstration of the immutability of such universal truths, as concepts such as 1 + 1 = 2 appear true 6 Although I would be proceeding against the current of my argument here (which has striven to move from embodied other, through music, to mathematics), I do think it telling that even amongst musicians and composers, there is a strong Platonic sense of the “other-ness” of music, especially in composition. On one hand, it seems natural to imagine mathematics as being “hardwired” into reality—as being “just there” whether one is there to see it or not due to its seemingly necessary nature. But music, on the other hand, with its seemingly more expressive and subjective quality, comes across as less necessary—not in terms of utility, but in terms of its structure. Couldn’t a composer have just as easily written a note to be held for two counts instead of one-and-threequarters? Couldn’t a chord with four notes in it have been changed to one with just three? Aren’t these fairly arbitrary decisions dependent more on the composer’s whim or taste, and less on some underlying logical or objective necessity, leading one away from supposing an a priori nature to music? We do call it “composition,” after all, which can also mean, “to make” (close to “to invent”) in contrast to “to discover.” Yet, some musicians and composers tend to express a similarly Platonic view of music, as do the mathematicians. Richard Strauss, for example, speaks of how, “while the ideas were flowing in upon me . . . the entire musical, measure by measure—it seemed to me that I was dictated to by two wholly different Omnipotent Entities” (cited in Abell, 1964, p. 145). Similarly, Giacomo Puccini described the composition of his opera Madama Butterfly as being “dictated to me by God; I was merely instrumental in putting it on paper and communicating it to the public” (Abell, 1964, pp. 156–157).

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regardless of whether people discover them or not. Such ideas were considered to exist in a sphere of reality apart from, yet somehow accessible by, human thought. As Davis and Hersch put it, Platonism is a view that supposes that mathematics, along with other truths, are “ ‘out there somewhere,’ floating around eternally in an all-pervasive world of Platonic idea” (1981, p. 69), and waiting to be discovered.7,8,9 As such, Platonism has been criticized for its shortcomings in providing insight into how people actually learn—that is,“make contact” with and “discover” this other, eternal reality of truths—without resorting to explanations steeped in mysticism (e.g., intuition, grace, genius, and so on).10 One might see why many mathematicians (hard thinking people, in general, who would likely want to be taken seriously as scientific, rational thinkers) would not want to offer a Platonic account, when asked, of what mathematics is; and instead, end up retreating into the philosophically “safe” explanation that is formalism: that it is just a game played with formal symbols, and that any usefulness or meaningfulness of results insofar as the real world is concerned is merely coincidental. At least formalism does not appear, on face value, to undermine a rationalist stance that seems undone with the utter adoption of Platonic Idealism. Yet, to return to Davis and Hersch’s quote that heads this section, many working mathematicians do indeed act as if there were some objective or ontological reality to mathematics when they are actually engaged in the act of doing mathematics, when they do not have to explain themselves. Formalism, while sufficiently rationalist in tone for explanation and justification, fails to capture the actual experience of mathematical activity for many. Instead, in the lived experience of some mathematicians, mathematics does become other, almost as solid and present as a thing of this world. Consider the following descriptions:

7 Many prominent philosophical thinkers, including Augustine (with his brand of Religious Idealism, separating the “World of God” from the “World of Man”), Rene Descartes (with his spin of the cogito (mind) contemplating objects of thought founded in the Deity (God)), Immanuel Kant (with his notion of Euclidean geometry as exhibiting a priori understandings, as well as with his universally valid moral laws in the “categorical imperatives”), to Georg Hegel (with the search for final “Absolute Spirit” through the dialectic), all carry shades of Plato’s Idealist tendencies in locating some version of objective and eternal truth “outside” of, or prior to the human mind. 8 See Ozman & Craver (1999) for a view of the role of idealism in contemporary education. 9 According to Platonic thought, it is through the mind that one somehow reaches the desired universal truths. Thus, one criticism leveled at Platonism in regards to how its influence tends to play out in education—with reverberations for this study—regards the privileging of the intellect or cognition above the personal, social, emotion, and physical dimensions of life and learning, relegating these other aspects to a kind of subsidiary status. 10 See Davis & Hersch (1981) for a description of the role of Platonism in the history and philosophy of mathematics, written for a general audience. For a more technical treatment of the current state of philosophy of mathematics see Tymoczko (1998).

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What Does Understanding Mathematics Mean for Teachers?

I certainly had the feeling in that particular case that I was discovering it and not inventing it . . . We couldn’t have invented all that. We had discovered a structure that must have been there. At least, that’s the feeling I had; it hung together too well. (Henry Pollack, as quoted in Albers & Alexanderson, 1985, p. 243) Despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception . . . They, too, may represent an aspect of objective reality. (Kurt Godel, as quoted in Davis & Hersch, 1981, p. 319) It’s like in the case of an intimate familiarity with a person. With such a person you often know what he is going to do without having to ask . . . The (abstract) things have a life of their own, but if you understand them, you make predictions and you are pretty sure that you will eventually find whatever you foresaw . . . Like a person whom you really know and understand, (the mathematical construct) will perform a certain operation or will react in a certain way to your action. This intimacy is exactly what I had in mind: you know what is to happen without making any formal steps. Of course, as in the case of human relationships, you may sometimes be wrong! (from Sfard, 1994, p. 49) Whether considered as a structure that one discovers, as objects of set theory that one intuitively “perceives,” or a piece of mathematics that one can know “like a person whom you really know and understand,” each of these descriptions captures a sense of mathematics, not as meaningless scribbles following rules of logic, nor as just one’s mental creations, but as other that one interacts with and comes to know. Each exemplifies the Platonist orientation of the working mathematician. Perhaps, one might say that it is merely an act of the imagination—a metaphorical manner of viewing phenomenon—that allows some mathematicians to view mathematics as objectively existent other? And that ultimately, one cannot consider as legitimate, such doings on the part of the experiencer upon these imagined “others” as then conferring any meaningful “otherness” status. Yet, the point of emphasis here is not of the actual or ontological differences between a tree and mathematics, or between a city and music, or even between manners of engaging with an object of the world and of the mind, but rather, it is of the lived experience of how the other is experienced by the experiencer. Consider the fact that a person can indeed become involved with mathematics or music—in ways similarly passionate and intimate—as he or she can with others of this world. Would this not then qualify the claim that mathematics-as-other is resonant, if not in a strictly factual or ontological sense, but in regards to the lived

Relationship as Reciprocity: Grace 27

experience—that is, in the phenomenological sense—of those who would engage with mathematics? Yet, is it also not the case that the meaning of being-inrelationship with other, or of entering into an I–Thou relation, or even of engaging another pathically, are all rooted in lived experience? In other words, is this not of a matter rooted in experience as opposed to “fact”? Consider Maurice Friedman’s (1955) commentary on Buber’s thoughts in this regard. The It of I–It may equally well be a he, a she, an animal, a thing, a spirit, or even God, without a change in the primary word.Thus I–Thou and I–It cut across the lines of our ordinary distinctions to focus our attention not upon individual objects and their causal connections but upon the relations between things, the dazwischen (‘there in-between’). (p. 57) Likewise, I would argue that the experience of entering-into-relationship has less to do with the ontological status of other—i.e., what the other in the relationship is or is not—and more so to do with space “in-between”—that is, the relationship itself. To return then to the earlier-established point of being-acted-upon as a central aspect of entering-into-relationship with other, the question now becomes, “In what ways, specific to mathematics, can someone be acted-upon by this other?” It is to this question that the remainder of this chapter concerns itself.

Affect, Feelings, and Emotions Feeling is essentially the relation of my self to something. (Helmuth Plessner, as cited in Heller, 1979, p. 1) Emotion without cognition is blind, and . . . cognition without emotion is vacuous. (Israel Scheffler, 1977, p. 172) Perhaps the prototypical experience of being affected by other is exactly that of affect—that is, emotion. Consider, first, how the etymological root of the word “emotion”—from the Latin root emovere (“to set in motion”)—offers insight into the actual lived experience of what it is to experience an emotion—namely, as an experience of becoming “set in motion,” that is “being acted upon” (in this case, by the emotion itself, or by that which has stirred the emotion).11 There is also the common etymological root shared by both “affect (n.)” as in emotion and “affect (v.)” as in “being affected,” in the Latin afficere (“to act on, have influence on” or more broadly, “to do”). Thus, to be “affected” is to be “acted on,” or to be “done unto.” Similarly, an affect itself is an experience of being acted upon. 11 All etymologies are from Harper (2001).

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What Does Understanding Mathematics Mean for Teachers?

To re-approach things from an experiential perspective, emotions are not something that one does, or acts to create.12 Instead, the core experience of emotion is that of being done unto. One does not make feelings, but simply feels them. In short, one undergoes feeling. Naturally, a person can conjure a thought that then triggers an emotion, but even then, the core experience of the emotion itself, I would suggest, is that of an undergoing. In other words, with the thought the emotion is triggered, releasing it to affect the one experiencing the emotion. To speak of most persons’ relationship to music (say as a listener) for example, I would assert that it is largely the emotional aspect of the experience of listening to music that creates a sense of connection and intimacy with a piece of music. The music touches us. It makes us cry, or brings joy to our hearts. Or else it casts a mood over (some of the more melancholic of) us. Each of these emotional experiences “sets in motion” in the listener some process that then either binds or repels the listener in relation to the music. In the favorable case, it fosters a sense of relationship. (In the unfavorable case, it sets up a kind of antagonism that is probably more common a thing for most with mathematics.) As the philosopher Agnes Heller writes,“To feel means to be involved in something” (1979, p. 1). And thus for many, “feeling the music” becomes tantamount to becoming involved in it, or, in some kind of relationship with it. Of course, this is not a book strictly on emotion nor one purely on relationship, but one on trying to understand a way of knowing with overtones of relationship and familiarity to the object of knowing. My concern here is with the phenomenon of relationship to mathematics—and in particular, to a manner of knowing mathematics with something more than just the “cognitive.” What I have argued for so far is that there is a being-acted-upon element of experience involved in such a way of knowing. Further, as I have claimed, feeling or emotion is one prototypical example of such an experience. So, the challenge here is to try to connect—or dissolve the disconnection between, depending upon the perspective—emotion to knowing, instead of considering emotion in a vein entirely separate from cognition.13 Phenomenologically, one could argue that the separation between what we call cognition and emotion, to begin with, is entirely conceptual, and falsely conceptual at that—that is, it contradicts the actual lived experience of either. Feelings do not arise without conceptualization (or pre-conceptualizations), and thinking does not occur without some feeling (such as the desire to make sense of something). As Ludwig Wittgenstein expressed it:“Emotions are expressed 12 Technically speaking, I am deliberately leaving out the role of intention—or what one brings forth—in the experience of emotion or of feeling, which I intend to address in Chapter 3. The treatment of emotion in this chapter is exactly on the “ ‘undergoing’ ” aspect of the experience of emotion. 13 It turns out that the entire notion of separating knowing and emotion is fiercely debated among psychologists. See Ratner (1989) for a critique of the Naturalistic argument that does assume the separation between the two.

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in thoughts . . . A thought rouses emotions in me” (Zettel, as cited in Heller, 1979, p. 24). Further, in Buddhist epistemology—a phenomenologically based philosophy in itself—there is not the sharp distinction made between cognition and emotion found in Western psychology. Instead, both are subsumed under the label of “mental states.” It is, as a Buddhist friend puts it, “pointless to make a distinction between the two in Buddhism with its aim of Nirvana because they’re both forms of mental defilements, or delusion.” In fact, even from a neurological point of view, the brain’s circuitry does not cleanly distinguish between cognition and emotion, as the two are intertwined in biology also. There is no “emotion” part of the brain that is entirely free from cognitive processing (see, for example, Davidson & Irwin, 1999). Yet, even without resorting to a blurring of our commonly held distinction between the cognitive and emotive, there is another way then of linking the two. It involves looking at very specific emotions that commonly surface in relation to intensely “cognitive” acts, such as during scientific or mathematical activity.

The Cognitive Emotions In such a vein, philosopher of education Israel Scheffler (1977) has coined a term—“the cognitive emotions”—to denote the intertwining of emotions in the act of knowing, or more specifically, to refer to those emotions that service cognition and at the same time are contingent upon the cognitive act. Two examples that he elaborates upon are the joy of verification and the significance of surprise. They are both considered by Scheffler to be cognitive emotions—to be distinguished from a noncognitive emotion—because neither would result without a cognitive supposition of some kind. Just as one cannot be disappointed without expectations, the cognitive emotions cannot occur without some epistemological expectation in place. The two examples he cites happen in fact to correspond to when such suppositions turn out to be confirmed by subsequent experience, and when there is a mismatch between expectation and experience. A wonderful quote, in its unabashed excitement and exuberance, embodying the joy of discovery comes from Johannes Kepler (1619) upon his discovery of the elliptical orbit of the planet— The wisdom of the Lord is infinite; so also are His glory and His power.Ye heavens, sing His praises! Sun, moon, and planets glorify Him in your ineffable language! Celestial harmonies, all ye who comprehend His marvelous works, praise Him. And thou, my soul, praise thy Creator! It is by Him and in Him that all exists. That which we know best if comprised in Him, as well as in our vain science. To Him be praise, honor and glory throughout eternity. (p. 1085)

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Such delight and rapture (likely undergirded by a “healthy fear of God”) that is expressed by Kepler upon his discovery—the verification of some instinct he no doubt held for a mathematical order in the heavens. Although a great deal more subdued than Kepler, the mathematician G.H. Hardy (1940/1969) expresses the significance of surprise also. In regards to the proofs for both the irrationality of – √2 and for the set of prime numbers being infinite, he writes, that “there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form . . . and this is true too of the proofs of many much more difficult theorems” (p. 113). One might not consider this an expression of emotion, per se (although one might argue that for some mathematicians, that’s about as good as you’ll get); and yet, it is a description of an effect of a mathematical proof upon the one reading, studying, or creating the proof. It is the effect of surprise—the significance of surprise, to use Scheffler’s terms—as experienced by the mathematician engaged with the “good proof.” Put differently, despite Hardy’s rather impersonal manner of articulating the element of “unexpectedness,” the surprise does not itself lie in the proof, anymore than the proof contains tears or laughter, but rather, it is that the proof exerts the effect of surprise upon the one engaging with the proof. The mathematics educator Stephen Brown relates his own sense of wonder and amazement at the connection between the inclination of a tangent to a curve and the area under the same curve, as forged by the Fundamental Theorem of Calculus. The feeling of amazement comes, not simply from appreciating the result, but because it stands in stark contrast to prior-held suppositions. He explains: Why am I amazed by the connection even after I have engaged in a problem solving act that states precisely and even proves the nature of the connection? Explanations and solutions help, but no matter how many there are, I am still haunted by the earlier intuition that the ideas are independent of each other. Proofs and problem solving activities are one thing. Expectations and intuitions are another. (Brown, 2001, p. 107) Naturally, one could speak of other dimensions of affect that are prevalent in response to mathematics, both in the positive as well as the negative vein. On one hand, there is the pure, unadulterated excitement and joy of mathematical discovery, a sense of flight and freedom, or what a number of mathematicians studied by Leone Burton (2004) in her book Mathematicians as Enquirers: Learning about learning mathematics termed “euphoria.” Consider the following descriptions of such affective responses to mathematics: one from the mathematician Georg Cantor (in a Hardy-esque, impersonally declarative tone), one from the author Edward Rothstein, who was at one time a student of mathematics, another from a mathematician from Burton’s study, and another by a high school mathematics teacher.

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The essence of mathematics lies in its freedom. (trans. from Cantor, 1883) Actually the sensation of doing mathematics is, at times, an almost giddy one, of extravagant freedom, and frightening possibility. (Rothstein, 1995, p. 38) When I think I have solved a problem I go completely euphoric. Immediately, I have to communicate this excitement. Sometimes, I am wrong and have to start again. (Burton, 2004, p. 86) I could remember preparing for lessons thinking, “Wow, this is incredible! I understand why this proof applies to all natural numbers, but it doesn’t apply to irrational numbers.” (a high school maths teacher) Whereas the first three could be classified as examples of what Abraham Maslow (1964/1970) has called the “peak experience” or what Mihaly Csikszentmihalyi (1990) has referred to as “flow” or the “flow state,” the last has a slightly different feel. Although similar to the first three in its sense of excitement, I would instead put this in the class of experiences described by Gestalt psychologist Wolfgang Kohler as an “aha” experience—a topic to which I intend to return later in this chapter. And lest one think that such excitement is purely an affective response without cognitive implications, consider the evaluative role that emotions can also play as exemplified in the following: J. Robert Oppenheimer once remarked that he had a definite way of judging the importance of a new idea in theoretical physics: he judged its importance by the amount of excitement he felt when the idea first occurred to him. (Davis, 1992, p. 229) Again, to briefly summarize up to this point, I have so far argued, based upon Buber’s notion of “true relationship,” that one important feature of coming into a way of knowing steeped in relationship would involve a capacity for “being acted upon” by other, and in turn, I have asserted that affect, or the experience of emotion in response to other is one representative actualization of such a “being acted upon” by such an other. Meanwhile, conventional thought would keep emotion separate from “knowing.” This is not to say that the interaction between the two is not generally considered (although only vaguely, I would suggest). But rather, the fact that one thinks of the two as interacting already shows an a priori theoretical bias of individuating one from the other. To then advance an image of a way of knowing more in line with “connaitre” then would mean a blurring of such distinction. Scheffler’s notion of the cognitive emotions begins to do just that, by unraveling the conventional fixation on dichotomizing cognition

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and emotion. It portrays a manner of “knowing” that is not relegated entirely to the cognitive. And that brings us current.The task at this point is two-fold: it is to continue to lend actual and specific shape to the ways in which one is indeed “acted upon” by mathematics by those who would interact with the subject, while also keeping an eye on how these instances then inform the manner of knowing itself. Rather than thinking that emotional experiences are sometimes a part of mathematical knowing and sometimes not, I will in fact argue that mathematical knowing that involves emotional experiences are of a qualitatively different character than those that are not. In other words, the quality of knowing is itself different because one has “been acted upon” by the other that is mathematics through emotional (or aesthetic) experiences. It is a knowing, that as will be seen, is different from a savoir knowing.

Frustration On what is perhaps the other end of the emotional spectrum from the affective states discussed thus far, a commonly expressed feeling among mathematicians and nonmathematicians alike in relation to mathematical activity is that of frustration. I have written previously (Handa, 2003) of the phenomenon of frustration, and of its seemingly inextricable role in mathematical engagement. One intriguing connection that can be drawn between frustration and the previously noted cognitive emotions—the joy of verification and the feeling of surprise—is the pivotal role of expectations and suppositions. The etymological derivation of frustration (from the Latin frustrare, meaning “to deceive or to disappoint”) points to an underlying feature of frustration, as being borne out of mis-perceived suppositions, or pre-conceptions. Whether it is in being “deceived” or “disappointed,” I either have entered a situation with a false idea, or worse (insofar as my emotional plight is concerned), I have brought to bear my own hopes in those ideas so as to experience disappointment from the then thwarted hopes.This is to say that the experience of frustration can be borne out of an ill fit between cognitive or epistemological expectations and the surprising brunt of experience, or of learning, which is to say that it, too, might be considered a cognitive emotion in the context of learning a subject such as mathematics. Of course, mathematics—and especially school mathematics—is rather stark in this regard. Unlike a school subject such as say music (and to a lesser degree, many of the humanities), where one could rather casually fall back upon a relativist argument (e.g., “beauty is in the eye of the beholder”) to gloss over what in fact may be poor taste or even a poor “argument,” mathematics more often than not presents its counterpoint to false suppositions in the barest of guises.The “truths” (results) of mathematics cannot as easily be denied, unless by sheer denial or ignorance.14 14 Naturally, this may be one of the draws of mathematics for some—that, one cannot be entirely deluded in one’s abilities for long. I touch upon this issue in more depth in Chapter 4. See Chazan & Ball (1999) for a contrary view.

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So it is not surprising that frustration appears a fairly common part of the mathematical experience for mathematicians and nonmathematicians alike.To the question of whether frustration can be subverted in his work, Pogo, a professional mathematician I interviewed in a prior study, related to me— You just sort of live with it. Most of it can’t [. . .] I think that it’s part of the work, to be able to work your way through it. It may be only one aspect, but I think that’s important. Interestingly, Pogo expresses a kind of frustration as deception as he explains that at times he just sits there “being extremely frustrated [. . .] thinking I had it, congratulating myself, going for long walks, and coming back and realizing I didn’t have it.” In the same vein, with intimations of the self-deception of frustrare (in unanticipated “dead ends”), Stephen Brown (1990) writes: One has to be prepared to be disappointed, frustrated, angry, as one comes to appreciate not only that schemes and plans lead to dead ends, but that a supposed problem being investigated may make little sense. (p. 20) In recapitulating the experiences of mathematics teachers who are engaged in the “doing” of mathematics, Deborah Schifter (Schifter & Fosnot, 1993) also comments: Doing mathematics can be pleasurable, even exhilarating, but it does not always feel good. Being challenged, encountering novelty, confronting one’s misconceptions—in short, building new and stronger understandings— typically involves bewilderment and frustration. (p. 11) Here is a description of frustration as a product of confronting one’s own prior misconceptions, as arising out of a mismatch between one’s suppositions and subsequent experience. And this sounds right, that both a sense of wonder and frustration can arise out of this same set of circumstances. In particularly receptive moods or modes of being, the mismatch might be transmuted into a sense of wonder and delight—such as in Stephen Brown’s reaction to the Fundamental Theorem of Calculus. On the other hand, when less receptive and perhaps more driven—whether by external pressures or more internalized ones—the same misalignment between preconceptions and actuality might be pressed into a sense of frustration. Considering then the various external and internalized pressures confronting mathematicians, teachers, and students alike, one could see why frustration—just as much, if not more, than a sense of wonder— may be such a commonly experienced aspect of mathematical activity for most. I might note here that I do not judge such pressures as intrinsically good or

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bad, desirable or not. For example, a striving toward excellence, as captured in the notion of the Greek arête, which one may deem a desirable trait for learning is one such type of internalized pressure. This then could lead to an experience of frustration.15 This, in turn, is to say that frustration is neither an intrinsically good or bad phenomenon, but rather, one that can surely be tempered and balanced by an attitude and stance of receptivity. Clearly, too much of frustration is not helpful. As I wrote in a previous study— When it comes to the phenomenon of frustration as deception/disappointment, if one begins with an acknowledgement and acceptance that struggle, and possibly even frustration, may ultimately be an intrinsic part of the experience of mathematical activity, then it becomes clearer to see that what sets apart those who are successful from those who are not may not necessarily be that the former persevere through their frustrations, for even the most accomplished will pull away from problems. What does seem to separate them is that some return to the problem. On the other hand, those who do not return to the scene of their struggle are likely those who do not find success with, or meaning in mathematical activity.Thus (and obviously), too much frustration could thwart the impulse to return to a problem, leading to the end of engagement for some. (p. 27) At the same time, one can also speak of becoming receptive, in a larger sense, toward the experience of frustration itself (as necessarily intertwined with mathematical activity), as both Brown and Schifter’s comments suggest. One expression of such receptivity is conveyed by an elementary teacher (Lia), in embracing confusion as an inevitable element of the learning process: If you aren’t confused, it means you already know the thing to be learned. If you are confused it means that learning is about to take place, or that the opportunity exists . . . When I myself am a student, if the confusion isn’t present I push myself to go deeper or broader in order to find it. This is

15 The neuroscientist Richard Davidson (in Goleman, 2003) explains how different parts of the frontal lobes of the brain—predominantly the left (for positive) vs. right (for negative)—become activated depending upon whether subjects are shown pictures designed to activate positive or negative emotions.When it comes to the emotion of anger, he notes that while internal (unexpressed) anger states, outwardly directed anger states such as rage, as well as frustration that is associated with sadness (or crying, such as in babies) show the right-sided activation of the frontal lobes associated with negative emotions, the kind of frustration as might be experienced in what psychologists call “approach behavior”—as typified by attempting to solve a very difficult math problem—shows the opposite and “positive” left frontal activation. As he explains, “In adults, the same kind of anger has been studied in people trying to solve a very difficult math problem. Though the tough math problem is very frustrating, there is an active attempt to solve the problem and meet the goal. Again, when this happens, we find left frontal activation.This is a type of anger that, in the West, we might call constructive: anger associated with an attempt to remove an obstacle” (p. 199).

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pretty easy to do most of the time, because I still feel very insecure as a student of mathematics. Similarly, another elementary teacher speaks of “mathematical confusion” as even being “tantalizing” when the confusion—and likely, the accompanying frustration—leads to new understandings: I got a tantalizing introduction to how it feels to be thrown into mathematical confusion over something as apparently simple as dividing one fraction by another. I discovered the meaning behind the operations I had memorized and forgotten many years ago. I experienced what it feels like when math makes sense. I had created my own understanding. (in Schifter, 1996c, p. 67) These last two accounts suggest a link—perhaps inextricable in nature—between frustration and the process of coming to know in mathematical terms. What I hope to have conveyed in speaking of frustration and its role in mathematical activity is first its consideration as yet another cognitive emotion, and second as one of perhaps many links between affect and knowing. By consideration of frustration as an emotional response to mathematical activity, it becomes, then, a form of “being affected” by the subject.

The Secondary Task: The Nature of Knowing But this begs the question—to now turn to the secondary task as mentioned earlier—of “How do these experiences of emotion play into the qualitative nature of the knowing?” In other words, if one is acted upon—say in the sense of undergoing some affective charge in relation to the mathematical activity—how then would the knowing be informed by such undergoing? One intriguing notion is posited by Jerome Bruner (1990), who, in citing the work of psychologist Frederic Bartlett, writes: . . . [Bartlett] insists in Remembering that what is most characteristic of “memory schemata” as he conceives them is that they are under the control of an affective “attitude” . . . Indeed, Bartlett goes further than that. In the actual effort to remember something, he notes, what most often comes first to mind is an affect or a charged “attitude”—that “it” was something unpleasant, something that led to embarrassment, something that was exciting. The affect is rather like a general thumbprint of the schema to be reconstructed. “The recall is then a construction made largely on the basis of this attitude, and its general effect is that of a justification of the attitude.” Remembering serves, on this view, to justify an affect, an attitude. The act of recall is “loaded,” then, fulfilling a “rhetorical” function

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in the process of reconstructing the past. It is a reconstruction designed to justify. (p. 58) Bruner suggests an interesting role reversal here. Instead of considering affect as a kind of appendage or side-kick that would bolster cognition, Bruner presents a re-conceptualization of cognition—that is, the “cognitive” act of remembering—as the servant to, or even the offshoot of an affective “attitude.” In other words, the issue is not one of supplementing learning with emotions that then might lead to greater recall and thus, learning, but that what one calls “learning” may indeed be a function—that is, a “justification”—of an experience charged with affect. According to this perspective, responses to mathematics such as joy, surprise, euphoria, and even frustration, then, become not just positive or negative “side effects” to mathematical activity, but are, in fact, the foundational grounding to actually knowing mathematics. In cognitive science research (see for example Bjork, 1994), one widely accepted result is that learning tends to be more robust after some struggle with the relevant material. That is, one knows the material better if there is that struggle involving some affective charge. (Naturally, the charge could be positive as well.) Here, there is an assumption of cognition/learning as being defined on its own, as if it could stand without affect (but standing better with it). Meanwhile, what Bruner suggests, while not contradicting such results, would lead to a dramatically, though subtly, different interpretation: that with struggle and the associated affective charge, a kind of milieu, or basis, for what we call “learning” or “knowing” is created. And further, without the affective state, there is, in fact, less of a reason for the justification that is the remembrance or recall even to be. Affect then becomes the grounds, and not just the prop, for memory in learning. Needless to say, without real reason, the learning would necessarily be fragile, as the research suggests. This points to a dramatically contrary conception of knowing from the standard one (that would leave out the affective element among others to be discussed in subsequent sections and chapters) toward one that would include these personal dimensions of knowing as integral, in fact, and not simply supplemental or supportive to the act of knowing. And because they are indeed personal dimensions, aspects such as affect (and others such as aesthetics) admit for the knowing to be of a more relationship-like nature. The aim of the book as a whole, again, is to loosen conventional notions of knowing as rooted entirely in the savoir kind of knowing (or “knowledge”), and at the same time, secure an image of something different, inclusive of a connaitre way of knowing. In this regard, Bruner’s comments begin to hint at how indeed the experience of emotion—as a subcategory of “being affected” by other—can be conceptualized as complicit in the act of knowing, and this possibility, in turn, begins the broadening of what it might mean to “know.”

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The Aesthetic Experience Such mathematical creations . . . have an eerie beauty. They display an intriguing balance between the inventive and the concrete. Often, there is in their creation a crucial moment on which the overall argument turns . . . we are presented with an argument that seems thoroughly conventional. Yet the familiar suddenly becomes strange. Like a walker who is so intent on moving his feet that he forgets to look around, the mathematician is transported and transformed. The apprehension of the beautiful has something to do with this experience. . . . (Edward Rothstein, 1995, p. 145) One rather obvious class of grace-like responses to mathematics-as-other that I have yet to address explicitly is the aesthetic response. By an aesthetic response, I mean the experience or apperception of beauty. In his Critique of Judgment (1790/1987), Immanuel Kant raises the notion of aesthetic pleasure as being inherently “disinterested” in the object of beauty.16 For Kant and later for Schopenhauer, the true “pleasure of beauty” is one of momentary freedom from strivings and desire (in particular, for the existence of the object that one would call “beautiful”).17 It is a manner of appreciating without grasping for the object of beauty that was anticipated in the writings of author Karl Philipp Moritz: But what brings us pleasure without actually being useful we call beautiful . . . I have to take pleasure in a beautiful object purely for its own sake; to this end the lack of an exterior purpose must be replaced by an inner purpose; the object must be something perfect in itself. (as cited in Hammermeister, 2002, p. 29) Basing his comments on the aesthetic theory of James Joyce—who likely was influenced by Schopenhauer himself—the mythologist Joseph Campbell expresses a similar sentiment: 16 Of Kant’s four “moments” of aesthetic judgment, this first one of “ ‘disinterestedness’ ” is perhaps the one most questioned and critiqued by other philosophers since Kant’s time. As Burnham (2006) explains, “To pick three examples, Kant’s argument is rejected by those (Nietzsche, Freud) for whom all art must always be understood as related to will; by those for whom all art (as a cultural production) must be political in some sense (Marxism); by those for whom all art is a question of affective response (expressionists).” 17 Unlike purely subjective pleasures (“ ‘pleasure of the agreeable’ ”) where one might prefer pie to cake—and thus, would take some interest in there being pies—or else pleasure derived from concepts in the good (“ ‘pleasure of the good’ ”) where one might be interested in the good manifesting somehow in the world, aesthetic pleasure was considered by Kant to be the only kind of pleasure that stayed “ ‘disinterested’ ” in the existence of its object. In the way that the arising of the cognitive emotions is dependent upon cognitive suppositions, one might say that the experience of aesthetic pleasures—that is, the pleasure component—is dependent upon some aesthetic perception. In other words, we take pleasure in something because we judge it beautiful, rather than judging it beautiful because we find it pleasurable.

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Joyce makes a distinction between what he calls “proper art” and “improper art.” By “proper art,” he means that which really belongs to art. “Improper art,” by contrast, is art that’s in the service of something that is not art: for instance, art in the service of advertising. Further, referring to the attitude of the observer, Joyce says that proper art is static, and thereby induces esthetic arrest, whereas improper art is kinetic, filled with movement: meaning, it moves you to desire or to fear and loathing. (from Osbon, 1991, p. 246) The phenomenological aesthetician Roman Ingarden (1961) describes the mechanism for this “aesthetic arrest” and “disinterest” as involving an unfettering of the aesthetic “present” moment from “the things and affairs of the surrounding real world” (p. 310): As I have already remarked, the phase of our actual “present” is normally “rimmed” with a reminiscence of bygone experience of not long ago and somehow connected, more or less loosely, with our “present” as well as with a perspective of our “nearest” future, which, however, we usually do not clearly realize . . . [Under an aesthetic experience] our new “present” . . . loses all distinct connection with our direct past and future within the course of our “daily life.” It forms a secluded whole, which we include into the course of our life only ex post, when the aesthetic process is over. Because of this extinguishing and/or removal of ties to that which surrounds the aesthetic moment, and a consequent “narrowing of the field of consciousness with reference to this world,” the urge for utility and exterior purpose fades somehow and loses its usual importance. I point to such a “disinterested” vision of the aesthetic response for the way in which it resonates with the noninstrumental manner of relating to other in Buber’s I–Thou relationship.18 Joyce’s “improper art,” as well as Kant’s nonaesthetic pleasures, both imply the object being considered as existing for something other than for its inherent nature, whereas with “proper art,” the pleasure of beauty, and the aesthetic moment, the object of beauty is freed up simply to seize and to confront the one, and to induce “aesthetic arrest”—that is, to affect the one, and in turn, to become useless in a manner of speaking (for any instrumental purposes). The aesthetic experience then, in this sense, is when the need for justification and rationale—Moritz’s “external purposes”—fall away, leaving one to behold other or object for nothing other than the experience of the beholding itself.This 18 There is also an intriguing poetic relationship between aesthetic pleasure and Buber’s characterization of the receptive aspect of true relationship in “grace,” when one considers that the “Graces” of Greek mythology—the three sister goddesses, Aglaia, Euphrysyne, and Thalia—were the dispensers of charm and beauty.

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is likely what G.H. Hardy (1940/1969) had in mind when he made his oft-quoted and often misconstrued comment about the “uselessness” of mathematics: It is undeniable that a good deal of elementary mathematics—and I use the word “elementary” in the sense in which professional mathematicians use it, in which it includes, for example, a fair working knowledge of the differential and integral calculus—has considerable practical utility. These parts of mathematics are, on the whole, rather dull; they are the parts which have the least aesthetic value. The “real” mathematics of the “real” mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly “useless” (and this is as true of “applied” as of “pure” mathematics. It is not possible to justify the life of any genuine professional mathematician on the ground of the “utility” of his work). (pp. 119–120) The “uselessness” here sounds that of a defiant exaltation of the aesthetic element of mathematical activity—likely in response to those who would value mathematics for more utilitarian purposes. Also, relevant to my own personal narrative from the preface, this sense of “external purposes” falling away helps me to interpret my earlier obsession with justifying and rationalizing mathematical activity as indicative of my own lack of aesthetic—and “disinterested”—relationship with mathematics, as I have already implied. This is, at least for me, a personal corroboration of the importance of noninstrumental modes of relating with mathematics, especially for one such as myself (and others) who would seek to sustain some professional affiliation with the study of mathematics.19 Kant’s notions of the aesthetic experience (toward other or object) as involving a “freeing up” from longings and strivings also fits well the focus of this chapter on the more receptive domains of experience. One could say that the aesthetic response is yet another manner of a “being acted upon” by other, this time in being shaken from one’s strivings. In describing what he calls the “preliminary emotion of the aesthetic experience,” Ingarden (1961), characterizes the aesthetic process as exactly a “passive” experience: As I have said, this quality “strikes” us, it imposes itself on us. Here I want to point out that the perception of this quality is passive and, in this phase of experience, a fleeting one, and, moreover, of such a kind that, if the whole process were in this phase suddenly interrupted, we ourselves probably 19 This might be considered a rather obvious and silly (because it is so obvious) claim to make, but consider that the common rationale given for the teaching and study of mathematics is one of economic empowerment, or else usefulness toward the study of other subjects such as the sciences—both, instrumental orientations toward the discipline and subject.

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would not know how to answer the question what kind of quality it had been: we receive the impression of it, we experience it rather than perceive it . . . In its very beginning it is a state of an excitement with the quality, which has imposed itself on us in the object perceived. While not realizing, in the first moment, distinctly what kind of quality it is, we feel only that it has allured us to itself, impelled us to give attention to it, to possess it in a direct, intuitive contact. (p. 309) Note in this description—as it was with Buber’s account of his I–Thou encounter with the tree—the passive stance of the one experiencing the aesthetic: “this quality ‘strikes’ us, it imposes itself on us,” “an excitement . . . has imposed itself on us,” “it has allured us to itself, impelled us . . .” What I take as implicit in Ingarden’s depiction of this the preliminary emotions, is that in an aesthetic experience, one has to allow for the other to act upon oneself for it to become started—that the capacity for receptivity, or surrender to experience is a necessary condition for the aesthetic process. It is a sentiment echoed by John Dewey in Art as Experience (1934), when he writes, “There is . . . an element of undergoing, of suffering in its large sense,20 in every experience. Otherwise there would be no taking in of what preceded” (p. 41). And later, he adds, “The esthetic or undergoing phase of experience is receptive. It involves surrender” (p. 53). Here, Dewey is quick to point out that such receptivity is different from mere passivity in the vein of withdrawal due to fear or other preoccupations. Such “mere” passivity, according to him, would only cause one to be overwhelmed by other, or by the experience entirely, and in turn would not allow one to take in—that is, receive—the other. Sure enough, these notions of “taking in,” “receiving,” “undergoing,” “suffering,” “receptivity,” and “surrendering” all point to the self-same sense of allowing for oneself to be acted upon by other. But to this, Dewey also adds that in all esthetic perception “there is an element of passion” (p. 49), and here, we arrive at a crucial junction in this exploration. It is to the nature of passion.

Passion, as an Undergoing If we remind ourselves at this point of the foreshadowing of “passion” in the Latin patior for suffering and enduring, we come to understand passion as a particular aspect of a journey when the determination of will or the foresight of reason no longer can make their contribution. It would refer us to that aspect of the journey when the channel through which we must pass becomes too turbulent and too swift for deliberate and premeditated action and when the good helmsman, after having done all he can, ties himself to 20 Suffering, in its large sense, is “to undergo or experience; to allow or permit” (Flexner, 1980).

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the wheel and entrusts himself to the waves to carry him. At the height of passion the water steers the ship. (Bernd Jager, 1978, pp. 350–351) My own inquiry began with a great interest in what it might mean for a mathematics teacher to be passionate about mathematics. What I imagined as “having a relationship to a subject” was my own way of describing a sense of passion for the subject. I chose to steer clear from using the term itself (until now) because I thought that it carried unwanted connotations, particularly in regards to emotive personality traits, or else to mental states denoting intense desire. It seemed too easy, if I was not careful, to conflate my own image of passion (as I will speak it) with something more outward in expression, or else with a kind of crazed desire. So, I settled upon approaching the phenomenon from a slightly removed vantage point of “relationship” as well as the savoir/connaitre distinction. But again, it is to passion-toward-subject that I began my own inquiry, which is to say that it lies at the heart of this inquiry. At this point, I believe that I have laid out enough of the conceptual pieces to draw up an image of passion that is possibly more useful and relevant for the educational venture than it being just an emotive expression of excitement or of strong craving. Recall that the pathic modality of being/knowing involved a taking in, or receiving of other that was direct and immediate; and that it involved a manner of surrendering toward an intersubjective understanding—similar to the way in which one might surrender toward the life of a conversation (and not necessarily the other) as opposed to controlling its trajectory.21 Consider then that the words passion and pathic both have close etymological associations, rooted in meanings such as “to suffer, endure” (from the Latin pati) and “what befalls one” (from the Greek pathos). If one considers the broader meaning of suffering as did Dewey (as “to undergo or experience; to allow or permit”), then the phenomenology of passion as a capacity for and allowance of surrendering toward the dynamic of “what befalls one” emerges. In this view, to be passionate toward other is to take a deeply pathic stance toward other. Accordingly, I would argue that the heart of passion has the least to do with doing or effecting anything upon anyone or anything else (in contrast to the usual “rah rah” vision of passion), and has mostly, instead, to do with working up the capacity to receive other—that is, to allow for other to affect oneself deeply. It is, as Dewey writes, to “summon energy and pitch it at a responsive key in order to take in” (1934, p. 53). The deeper into one’s being this taking in or allowance, the

21 Consider how in normal, everyday conversations, one can both assert and relinquish control of the trajectory of the conversation by how one holds onto and loosens one’s grip on one’s own agenda. To be an adept and gracious conversationalist—and most pertinent to the current discussion, to experience learning from the conversation—is to know this play between assertion and relinquishment, between will and grace.

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deeper indeed is the passion. Put differently, passion, then, is the readiness to surrender oneself toward experience—toward the relationship between self and other that is the I–Thou relationship.22 What I have so far articulated in relation to Dewey’s ideas on the aesthetic experience and the phenomenon of undergoing, for the most part, is in following science educator David Wong’s (2002) interpretation of Dewey’s writings. Where he takes things from here, I find intriguing. In particular, he points out that the capacity to undergo, or to relinquish control is a form of risk-taking, and as such demands courage. Of course—although he does not say it—it is a bit of a banality to say that learning takes courage.23 What intrigues me, though, is this notion that passion requires great courage—that is, the allowing of oneself to be taken by other (to feel, to undergo) itself requires great courage. I am reminded of a scene from the movie Pushing Tin (1999), where a staid and frustrated character is challenged by another of a more “passionate” disposition to stand at the end of an airplane runway, to be tossed about by a descending plane’s backdraft. At first terrifying, the experience of being pitched through the air with utter loss of control becomes, in the end, a freeing experience for the character.The experience also becomes a metaphor for the kind of letting go and undergoing allowed that is required of living a passionate life. In educational circles, such privileging of courage in letting go (of control) feels starkly different from the prizing of more seemingly “active” dispositions such as tenacity and persistence—i.e., “overcoming” as opposed to “undergoing”—that are commonly bandied about as being of prime importance in the educative process for the learner to possess. Yet, I think this is an important insight—again, not so much that learning takes courage, but that feeling and passion require courage because they both involve risks that are inherent in relinquishment of control. They involve the possibility of becoming unseated, moved, and even transformed by other—that is, “being affected” by this other.Yet, real learning requires feeling. That was Bruner’s point. Consequently, it is because learning and cognition require feeling, and because feeling (including passion) requires letting go or letting be, and because letting go requires risk-taking, and because risk-taking requires courage—because of this

22 William Pinar (2010) uses the concept of “self-shattering” to describe the consequences of this same surrender to experience. He writes, “For me, such intensification of experience implies selfshattering insofar as the boundaries of the self dissolve into the aesthetic experience that extricates us from submersion in the banal, the provincial, and presses us into the world.” 23 What is mostly meant by courage is something referred to as “intellectual courage” which I find dissatisfying as a descriptor for the reason that it again buttresses cognition (“intellect”) with emotion (“courage”), and as I will immediately address, it fails to explain adequately the causal links between the affective nature of courage and the more cognitively tinged nature of “learning.”

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chain of causality that runs through feeling and emotions, and only because it does, I would argue—does learning itself require courage.This is to say that courage is implicated in learning precisely because affect is implicated in learning. But it is not a courage to speak up in a classroom when confused, or to openly air one’s thoughts for criticism from others, although these are important too. It is, instead, the courage to become invested in the subject to the point where joy, frustration, and any from the spectrum of emotions can arise in the interaction. It is the courage to be “acted upon” by the relationship with the subject.

Passion and Aesthetics: Being Seized by a Problem For God’s sake, I beseech you, give it up. Fear it no less than sensual passions because it, too, may take all your time, and deprive you of your health, peace of mind, and happiness in life. (Wolfgang Bolyai [in referring to Euclid’s famed parallel postulate], cited in Davis & Hersch, 1981, pp. 220–221) If one considers passion as the capacity for suffering in its “larger sense,” or as an undergoing, then mathematician Wolfgang Bolyai’s admonition (above) for his mathematician son, Janos (who failed to heed his father’s advice in the end), does not at all seem overdramatized a reaction to the effect that mathematics can have on some. Bolyai’s words convey an element of passion that one might call a “being seized,” “taken,” or “captured” by a problem, or even “obsession.” It is similar to the portrayal of the great Carl Friedrich Gauss, and of the nature of his mathematical involvement: Part of the riddle of Gauss is answered by his involuntary preoccupation with mathematical ideas . . . As a young man, Gauss would be “seized” by mathematics. Conversing with friends he would suddenly go silent, overwhelmed by thoughts beyond his control, and stand staring rigidly oblivious of his surroundings. Later he controlled his thoughts—or they lost control over him—and he consciously directed all his energies to the solution of a difficulty till he mastered it. (from Bell, 1965, p. 254) Likewise, Bernhard Reimann’s mathematical absorption would take a hold of him to the point of illness— I became so absorbed in my investigation of the unity of all physical laws that when the subject of the trial lecture was given to me I could not tear myself away from my research. Then, partly as a result of brooding on it, partly from staying indoors too much in this vile weather, I fell ill. (from James, 2002, p. 185)

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Richard Wertime (1979) describes the nature of such “intellectual passions” (and possibly, obsessions) when he speaks of the nature of problems. Note in his portrayal, the emphasis on both the personal dimension of what makes a problem a problem, and of the aspect of “being seized by”—that is, surrendering toward— the problem. There is an essentially binding or promissory dimension to the act of facing a problem. A problem is not an entity, which has its existence independent of a person (regardless of the illusion fostered by textbooks full of problems); it is an intensely personal and passionate affair, one which is deceptively hard to break off at will. Moreover, problems are hard work. They require attention and courage, and they involve a significant act of self-surrender which can seriously jeopardize the individual’s sense of himself. (pp. 192–193) Again, I point the reader to the way in which Wertime qualifies the nature of a “passionate affair”: “deceptively hard to break off at will,” “require[s] . . . courage,” “involve[s] a significant act of self-surrender,” “jeopardize[s] the individual’s sense of [perhaps, control of] himself.” Yet, Wertime’s claim that “a problem is not an entity, which has its existence independent of a person,” is a crucial turn in thought—one with great relevance to this inquiry. In this view, what makes a problem problematic for a person rests not inherently in the problem—contrary to the intent of curriculum designers who, for example, think that real-world contexts somehow turn an otherwise meaningless problem into an “authentic” one.24 Instead, it is, as Wertime writes, “an intensely personal and passionate” matter. In a similar vein, Stephen Brown has also pointed out in his writings that problems, indeed, do not exist independently of a person’s own personal interests and agenda—an issue to which I return later in Chapter 5.What makes it “passionate” is, in fact, dependent exactly upon the subjective experience of “being seized” by the problem. Being a subjective experience, the locus of activity—at least so in

24 Related to this, Whitson, Julien, & Matusov (2002) refer to a manner of student involvement whereby even purportedly “authentic” problems are reduced by those working at them to “schoolish” (as in “foolish”) assignments, and not authentically engaging, or personally meaningful problems. To quote, “We have concluded that the effectiveness of the problem in our classes was not determined solely by the quality of the problem itself, in terms of its design, substantive relevance, or other characteristics of the problem. Success was determined, rather, by whether or not the students ‘owned’ the problem as a real problem for them, as the problem with which they really were engaged. In some sections, this happened. In other sections of our courses, although students were indeed working on the problem as a class assignment, the problem they were actually engaged with was the ‘schoolish’ problem of how to complete this course requirement with an acceptable grade for this assignment” (p. 2).

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regards to what makes one’s engagement with a problem a passionate one—is in the space of interaction between the person and the problem, and not exclusively in the problem. Thus, what may be a problem to one person—that is, a genuinely engaging and “gripping” one—may be just a meaningless exercise to another. Yet, for a number of mathematicians—and possibly most—it may very well be that what “grips” their intellectual passion is exactly the aesthetic dimensions: the aesthetic undergoing. As mathematician Roger Penrose (1974) writes, “Basically, the motivations turn out often to be ultimately aesthetic ones” (p. 266). Other mathematicians, including Jacques Hadamard (1945) and John von Neumann (1956), have also voiced similar views of what holds a mathematician to mathematics. So, here is a vision of passion as “being seized” by a problem, and of the aesthetic experience as entwined with such a passion.25 Of course, in the preceding sections, I characterized both passion and the aesthetic response as a capacity for deep receiving and allowance for a taking in. Hence, the link between the two is not entirely surprising. In fact, I believe it confirms commonly held intuitions about the interrelation between the two. The popular image of the passionate artiste, or passionate lover (or both together in one), attest to an image of passion and the responsiveness to beauty as being somehow coupled together. To quickly summarize this section to this point, I have argued that aesthetic experiences constitute yet another class of grace-like “undergoing” experiences, similar to the affective responses of the preceding section. In addition, I have suggested passion and obsession as variants of, or relations to, the aesthetic response. Within these latter discussions, I touched upon issues of courage (as implicated in learning) and of seizure by problems as depending, not solely upon the problem, but upon the space between the problem and the problem-solver.What I have yet to do is to speak specifically about aesthetic responses to mathematics, as an example of a grace-like response to mathematics.

Aesthetic Responses to Mathematics To the question of what it might mean for a person to be passionate toward mathematics, the answer might very well be tied up with degrees of receptivity toward the subject, especially as manifested in aesthetic appreciation or affective sensitivity to the subject. Having already elaborated upon the affective element, and briefly touched upon the passionate and obsessive in specific relation to mathematics as well, I continue with a focus on the aesthetic aspect.

25 What happens as a consequence of such “seizing” is addressed in Chapter 4. The focus, again, for this chapter is in delving into phenomenon of the undergoing itself, and its constituent aspects— not yet, its aftermath.

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Naturally, there is great overlap between the affective and aesthetic facets of mathematical activity. Mathematician Henri Poincare speaks of the aesthetic as a feeling, and as an emotional sensibility: It may be surprising to see emotional sensibility invoked a propos of mathematical demonstrations which, it would seem, can interest only the intellect. This would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance.This is a true aesthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility. (Poincare, 1956, p. 2047) Or, consider mathematician Roger Penrose’s description of mathematical elegance, which should call to mind Scheffler’s cognitive emotions—namely, the significance of surprise: Elegance is more or less synonymous with simplicity . . . I think perhaps one should say it has to do with unexpected simplicity, where one imagines that things are going to be complicated but suddenly they turn out to be much simpler than expected . . . And there is the element of connecting together different ideas that one had not expected to be related. (1974, p. 268) Although one could fill an entire book with the possible varieties of aesthetic responses to mathematics—and there are many, as in the appreciation for symmetry, order, transcendence, insight, economy, fit, subtlety, transparency, connections, and so on—I will turn instead to those aspects of aesthetics that are most relevant to this inquiry: to how the aesthetic intertwines with issues of knowing.

The Generative Aesthetic I have a very keen sense of the beautiful and when I found my equation I knew I was right. (Dirac, as quoted in Penrose, 1974, p. 267 ) In considering how the aesthetic might be implicated in issues of knowing, I now turn to a class of aesthetic responses to mathematics that mathematics educator Nathalie Sinclair (2004) calls the “generative aesthetic.” These are moments of aesthetic experience in the midst of mathematical inquiry that guide “the actions and choices that mathematicians [or nonmathematicians] make as they try to make sense of objects and relations” (p. 270). The case I gave earlier of Robert Oppenheimer’s excitement (assuming that it is an aesthetic response of excitement) as indicating the potential importance of an idea stands as an illustration of this generative aesthetic.

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One of the examples given by Sinclair is of a mathematician who stops at the factorization 137×73—noticing its pattern of symmetry—in the process of proving the claim that in the infinite sequence 10001, 100010001, 10001000100001 . . ., there are no primes.26 The aesthetic appeal of the symmetric factorization leads the mathematician to believe correctly that there is something there. His aesthetic response to the mathematics becomes partially generative of a solution. It is a response not entirely rooted in logic, but in an aesthetic appreciation. Sinclair also points to mathematician Douglass Hofstadter’s perception of harmony upon noticing a set of parallel lines in solving a difficult mathematical problem, and of how he would have failed to notice a crucial relationship involved had the resulting lines not been parallel. In this example, too, a kind of aesthetically motivated heuristic—of following where beauty and harmony would lead— is implied in the “cognitive” act of solving a mathematics problem. The mathematician Henri Poincare lends insight into the generative aesthetic in his discussion of mathematical creation: Among the great numbers of combinations blindly formed by the subliminal self, almost all are without interest and without utility; but just for that reason they are also without effect upon the esthetic sensibility. Consciousness will never know them; only certain ones are harmonious, and, consequently, at once useful and beautiful.They will be capable of touching this special sensibility of the geometer of which I have just spoken, and which, once aroused, will call our attention to them, and will bring them into consciousness. (Poincare, 1956, p. 2048) This “special sensibility” that Poincare speaks of, as selecting out the “harmonious” and “beautiful” in working through the muck of endless other combinations and possibilities is the generative aesthetic, similar to the mathematicians’ in Sinclair’s examples. As it appears, it is an aesthetic that is complicit—as was affect, previously—in the act of coming to know. Accordingly, besides the aesthetic experience being another well-documented concretization of being-affected by other in the relationship between self and mathematics, the generative role that it can take in mathematical activity attests to its consideration as an aspect of knowing.27 Of course, this connection between the aesthetic and the act of knowing is just barely hinted at by my discussion of it so far. In the closing and upcoming section of this chapter, I will turn to a more explicit elaboration of the link between knowing and the family of “undergoing”

26 Originally in Silver & Metzger (1989). 27 Note how the generative aesthetic overlaps with a kind of mathematical intuition that is specific to an aesthetic sensibility.

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facets of experiences, including the aesthetic and affective. In this articulation, I will approach knowing through the phenomenon of understanding as an undergoing experience—that is, through the lived-experience of coming to an understanding.

Understanding as an “Undergoing” Experience I have found you an argument; I am not obliged to find you an understanding. (Samuel Johnson, in Boswell’s Life of Johnson, cited in Noddings & Shore, 1984, p. 68) Having proposed affective and aesthetic responses as two major categories of “undergoing” or grace-like experiences in regards to mathematics, I will offer one other type of experience as being in that class. That is the experience of coming to an understanding of some mathematics. Now here, I am invoking what may turn out to be a crucial and rather subtle shift in regards to what constitutes an understanding. What I propose is to reconsider understanding from its regular meaning of having or acquired some “deep and connected” knowledge (i.e., “big picture”), or else its more radical conceptualizations as a knowing how-to-act in particular settings—both signifying states of having something or having attained to something—to one of an undergoing experience or event. It is to view understanding from its lived experience perspective, and to see the moment(s) of understanding, then, as constituting a being-affectedby-other type phenomenon. Re-consider Penrose’s comment quoted earlier: Elegance is more or less synonymous with simplicity . . . I think perhaps one should say it has to do with unexpected simplicity, where one imagines that things are going to be complicated but suddenly they turn out to be much simpler than expected . . . And there is the element of connecting together different ideas that one had not expected to be related. Although he speaks of elegance and unexpected simplicity and connections— all of which could be subsumed under an aesthetic response—his comment of “connecting together different ideas that one had not expected to be related,” is also a moment of understanding. It is an instance of seeing a group of ideas in relation to one another. Among the various aesthetic responses to mathematics, perhaps most notably those of drawing connections between seemingly disparate ideas, have strong reverberations of arriving at an understanding of the mathematics. In describing an aesthetic bent which one could regard as being sympathetic with a mathematical aesthetic, author Robert Pirsig (1974) describes its purpose as “bring[ing] order out of chaos and mak[ing] the unknown known” (p. 61). I believe that this would resonate with many who have engaged with

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mathematics as describing a kind of aesthetic experience that is relatively common in mathematics, that at the same time merges with the phenomenon of coming to know. In responding to a prompt on mathematical beauty, Patricia, a former middle school mathematics teacher, describes her experience of it in terms that suggest its merging of both affect and understanding of mathematics— It was more of an emotional sense than a sense of looking at physical beauty. It was a sense of—it’s hard to describe—getting something, seeing the pieces fall into place, seeing the puzzle fit together. We see in Patricia’s description, an attempt at capturing the experience of beauty, that then shifts through a pointing toward affect (“It was more an emotional sense than a sense of looking at physical beauty”), then toward a depiction of understanding (“It was a sense of . . . getting something”). In her one statement, she delineates the entire arc of ideas discussed in this chapter. Further, “seeing the pieces fall into place” (a passive, allowing for description) juxtaposed next to “getting something” (an active doing type of depiction) recalls Buber’s joining of grace and will as constituting the process of coming into relationship, or in this case, understanding. Again, if I have not been clear, I suggested earlier in the chapter that what one might call an “aesthetic response” is a variety of an undergoing type experience, of “being affected” by the other that is mathematics.Yet, I have now pointed to a way of viewing the lived experience of understanding, or coming to know, as similar, if not synonymous, with at least some varieties of the aesthetic experience. Thus, I am suggesting that understanding be regarded, similar to aesthetics, as a manner of being affected, or touched by mathematics.

Understanding as “Aha” Students, they’ve asked me, “When will I know that I got it?” I say, “You’ll know it in here” (pointing to his heart). (Ed, college math teacher) Of course, we speak of those vital instances of understanding as an undergoing experience when we speak of them as “aha” experiences. These are the moments when an understanding comes over us—a coming together of sorts, where we stand receptive, ready to surrender to the experience. It is understanding as a feeling of closure, as expressed by Frank Smith (2002): This agreeable feeling of closure is part of what I regard as a uniquely human characteristic—a sense of good fit, or sense of appropriateness. This sense isn’t much talked about, but it is crucial to the way we make sense of anything. It tells us when we can stop puzzling, when we have found a solution.

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An example is finding the right piece for a hole in a jigsaw puzzle, when we say, “Aha,” and can move on. (p. 21) But maybe, all real understandings contain to some degree such “aha” moments. Or, one could push this further and, in fact, reconsider understanding as that manner of experience which has been informed by the undergoing experience of an “aha” variety. Rather than conceive of understanding as a having-of-knowledge or knowing-of-action, “understanding” would refer instead to a subjectively felt and defined feeling or sense of things—a “qualitative meaning,” to use Dewey’s words— that tells us that we know. In this view, then, understanding is when we can say, “aha.” Naturally, understanding that is subjectively defined as such is one that is personally tinged. It involves the person with the mathematics. As I will articulate and elaborate upon in the remainder of this chapter, it is a view of understanding that carries intimations of the personal and intimate nature of the connaitre way of knowing. It is understanding as an undergoing experience. To give clearer shape to the discussion, I now turn to a passage from an elementary teacher (Lisa) who describes the importance of understanding to her: I began by thinking that I couldn’t solve it because I couldn’t get my mind around it. I knew it was a division problem but couldn’t explain why and therefore couldn’t tell what to divide into what. Then I went to simple algebra, because I knew I could solve the problem that way, from memory. Whenever I go to algebra it is a cop-out. It means that I can’t derive what the problem is about at its core or what to do about it just based on “logic” and the information given . . . What interests me about this is how dissatisfied I feel . . . when I can’t derive an answer “logically.” I don’t care how valuable math instinct (unconscious) is, it doesn’t help me feel that I understand, in any kind of profound way, what’s going on in a math problem. Knowing what to do isn’t any good. I need to know why. ( from Schifter & Fosnot, 1993, p. 127) What Lisa calls “knowing why” is intertwined with a feeling of understanding for her. Even though she is able to figure the problem using algebra, the feeling of dissatisfaction lets her know that she has not understood the mathematics. She has not yet experienced that feeling of surrender to an understanding. To be clear about this, I am not just suggesting that with understanding usually come certain feelings, but rather, I am reiterating the idea that “understanding” be interpreted in terms of whether it involves an “undergoing” element of experience—that is, a feeling of understanding—and if not, that it be considered as mere memorization (i.e., “knowing by heart”) or something else that is different from what Ed (in the comment that headed this subsection) pointed to as a knowing in or with the heart. In other words, I am speaking of a knowing that is accompanied by a feeling of having understood.

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What is of interest in Lisa’s account—and with consideration of understanding as involving an undergoing—is of how “understanding” then becomes defined in relation to the person involved in the understanding. Conceivably, there will be those who will, in fact, experience a feeling of satisfaction with an algebraic solution—likely, persons whose grasp of algebra is itself grounded in a sense of understanding and is more than just algorithmic.Yet, since it does not for her, she has not experienced a feeling of understanding. This is to say that understanding as such is not rooted entirely in the “what” that is known, but that instead, understanding is just as much rooted in the experience of the knower; it involves the interaction between the “what” and the knower’s experience. The mathematics that lends itself toward an experience of understanding for one person may not for another. Such a characterization of understanding, then, implicates the knower as it depends also upon that knower. In fact, one could say that it brings the knower into relationship with the known, or that it connects the two.The feeling of connection to mathematics that usually accompanies an experience of understanding could in fact be construed as attesting to such a state of affairs.28

One Teacher’s Expression of Understanding as an “Undergoing”-type Experience: a Brief Case Study of Eve29 Eve, a former high school mathematics teacher, expresses a similar sense of understanding as a “buying into,” when she says, “I don’t buy into the idea that the evens and integers have the same cardinality.” The language, itself, bespeaks a sense of personal investment that distinguishes itself from the removed stance toward other or object of a strictly savoir type of knowing.

28 One pragmatic consequence of the reconceptualization of understanding to that of understandingas-(an undergoing)-experience has to do with the pivotal role of “understanding” in the educational enterprise. In short, understanding is the holy grail of education; it is what educators aim for. Thus, when understanding is conceived as something that one can have or attain, then that becomes the goal. Engagement, pleasure, aesthetics, effort, all must serve the attaining of some knowledge state called understanding. But if one adopts a view of understanding-as-experience, then the goal shifts to one of helping students to have such undergoing experiences of understanding. The accrual of knowledge is pursued not as an end in itself, but as a means of increasing the likelihood of (the experience of) understanding. The broader goal of education is then not a uniform knowledge state of across-the-board expertise on the part of a student body, but a personal and whole relationship—defined by these experiences of understanding—to subject matter on the part of each student within that student body. It sets a different bar for students and for educators. 29 By focusing a little more closely upon one teacher’s comments, I hope to begin showing the power of the ideas discussed so far, insofar as hearing what a learner or teacher may be saying in regards to knowing mathematics. The intent, though, is not to set a precedent for the remainder of the book. Instead, this one case should be looked upon as a hint at potential future research based upon the ideas, in addition, to further elaboration of the ideas being developed here.

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I don’t buy it really. I’ve seen the proof, but there’s something in my experience that it fundamentally troubles me, for example, that the evens is the same cardinality as the integers. That troubles me, and I can make an argument for why they’re the same, but that doesn’t deal with my issues, my conflict. I would call attention, first, to her language, in how it depicts her lack of an undergoing. As long as the given argument does not deal with her issues and her conflicts, there is not the experience of understanding. Hence, there is no “buying into” the idea for her; she is otherwise unwilling to surrender to the facts as given (that the evens and the integers share the same cardinality). Note, further, the distinction she draws between being able to provide a mathematical argument—which in conventional circumstances could very well be taken in itself as an “understanding”—and with her unease in accepting that then as a personally satisfying understanding, enough so as to “buy it.” When I ask her to further elaborate upon her meaning of “buying into,” she explains: I think mathematics, for me to “buy it,”—I don’t mean looking at a proof— but I mean to actually “buy” something. I don’t just see an argument, but I think that there’s an important intuitive aspect to buying something. It has to feel right. Here, the intuitive (related to the generative aesthetic as suggested earlier) and the affective converge with understanding as “buying into” the idea. Again, it hints at a view of understanding as far more than knowing some piece of information, but as involving a feeling of having gotten it—a coming together of sorts. To further elaborate, Eve continues with her explanation with another mathematical example: For example, what I can’t buy with the Cantor set30—it’s not the measure thing [of it being measure zero]—but the fact that you’re not just left with endpoints . . . In the Cantor set, you remove your middle third, for example,

30 The Cantor set is taken as follows: Begin with the interval [0, 1]—that is, all real numbers between 0 and 1 including the endpoints. Now, imagine slicing that interval into thirds, where you remove the open interval (1/3, 2/3)—that is, the middle third (but leaving the endpoints at 1/3 and 2/3). You are now left with the thirds on the left and on the right (including the endpoints). Now, do the same thing with each of these thirds. Slice them into thirds (or ninths, in relation to the original interval), and remove the middle (always retaining the endpoints), and repeat this process ad infinitum—the Cantor set being those points that are never removed in this process. The comment concerning the “uncountable” nature of the Cantor set refers to the counterintuitive fact that the Cantor set is not an infinite set of (end)points that one can simply “count off ” as one could with the natural numbers (i.e., 0,1,2,3, . . .). Instead, the Cantor set has just as many points (i.e., same cardinality) as the original interval [0, 1] while in fact containing no intervals.

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and you have your two [new] endpoints [referring to the points at the inner edge of the segments still remaining], and these endpoints never leave. They’re in the Cantor set forever. And you keep taking out the middle third, and once you have an endpoint, it’s always in the [Cantor] set, but the set of endpoints is countably infinite. So, there’s an uncountable number of other points that are in there that aren’t endpoints [owing to the fact that the Cantor set itself is uncountable] . . . So, there are an uncountable number of non-endpoints, so if they’re non-endpoints, then they must be between [the endpoints], so I’m having trouble reconciling [this]. So, if it’s not an endpoint, it must be between endpoints, and there has to be sort of this continuum between those endpoints. At this point, I enter the conversation as a fellow inquirer and ask— Me:

Why does it have to be a continuum? I’ve not studied Cantor sets to that extent, but to me it doesn’t seem to necessarily have to be an interval. Oh, but for it to be uncountable [maybe it has to be in the form of an interval] . . . But wait, is that the only way to represent uncountability—through an interval? Eve: I don’t think so, but maybe it’s just that there’s another idea that I need to know before I can [get it]. Me: So, it’s the irreconcilability of ideas that’s at the root of “not buying?” Eve: Yes, yes. And so I understand, and you can show me mathematically that there are—and I can convince myself that the measure is zero. I can do that. It feels funny, but I can do it. But I just—my gut tells me that there can only be endpoints left.You know, so there’s some idea that’s not . . . it doesn’t, or it’s just too contrary to my [mathematical] experience, or there’s some idea that I need to have that I don’t have yet . . . So I need something that reconciles this conflict. And experience is not doing it for me so far, so maybe I haven’t struggled with an idea long enough. Yeah, so I don’t buy it because—and I talked [earlier] about the difference between a proof that proves and a proof that gives understanding. I follow the proof that proves. I follow it, but it doesn’t give me understanding. So, I’m at a point where I can say, “I see your proof, but you know, it hasn’t helped me. It doesn’t give me understanding. And so, until it achieves the understanding part, reconciles these things for me, I’ll follow it. I’ll say it’s logical, but I’m not going to buy it.” Eve forwards the ideas of proofs-that-prove and proofs-that-explain (see Hanna, 1990). In response, I posit an image of a proof-that-proves being like a path that moves from point A to point B without ever taking into account some conflicting

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idea held by Eve over at point C, whereas the proof-that-explains does go through that other point C, so as to reconcile the potential conflict for her. But of course, for someone else who may not have the conflicting idea over at point C, the same proof-that-proves (that moves simply from point A to B without recourse to C) could very well explain or shed understanding for that person. Further, if this other individual held another so-called conflicting idea at a point D that the proof-thatexplains (to Eve) failed to go through, then that same proof would no longer be a proof-that-explains for this other individual, implying a very personalized sense of coming to an understanding of an idea. Although the theoretical construct of proofs-that-prove and proofs-thatexplain have generally attributed explanatory power to the proofs themselves, my conversation with Eve sheds insight on the fact that what may be a proof-thatexplains to one person could very well be a proof-that-proves to another, and vice versa—that the explanatory power of a proof is in fact contingent upon the ideas brought forth by the person engaging with the proof, which the proof may or may not reconcile. It leads one to ask, about so-called proofs-that-explain, “For whom does it explain?”—not simply whether it does or not. At least, explanation is not a function solely of the proof as sometimes glossed over in theoretical considerations. Similar to Lisa’s earlier account, we have arrived at a view of understanding that is personal and rather intimate, for an understanding as such is effected exactly when a person’s own set of intuitive understandings, in the form of skepticisms and conflicting ideas, have been quelled and reconciled by the mathematics or explanation that has been offered. It is this event that we call understanding. It also recalls Wertime’s characterization of a “problem (being a genuine problem)” as being dependent not just upon the objective qualities of the problem itself, but upon the interaction between the problem and the very particular and personal qualities that the problem-solver brings to the interaction. In fact, one might conjecture a connection between problems that seize and the personalized, undergoing understandings as articulated.31

Subjective Universality A curious question to ask is how the subjective and experiential elements have so easily been laid by the wayside in conventional conceptualizations. In other words, how is it that the “pull” of a problem or the explanatory power of a proof is so easily assumed to reside in the problem or proof itself, or that understanding becomes a matter of gaining a hold of objectively defined bits—whether connected or not—of information? In short, what lies behind the tendency to take the person out of the knowing, and instead project these subjective elements onto the objective aspects?

31 I take up further exploration of this conjecture in the next two chapters.

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In a comment that brings together understanding with the aesthetic, Edward Rothstein hints at a dynamic that may underlie this tendency away from experience, away from subjectivity: In coming to understand something or make sense of it, we must be attentive to “proportion” and analogy in their most general sense.We must create metaphors as we work, exercising, in Aristotle’s phrase, the “intuitive perception of similarity in dissimilar things.” This is, in essence, the activity of mathematics.That activity also has something to do with our experience of beauty itself. It touches on the mechanism by which we come to know beauty and “feel” the aesthetic. What we “feel” at such moments is the analogy of part and whole, object and other object, relation and relation. This is one reason that in moments of aesthetic transport we assert the universality of the beautiful: we are feeling something not inchoate but precise and seemingly beyond contradiction. (1995, p. 168) This propensity toward asserting “universality”—in this case, of the mathematically beautiful as if beauty were a property of the object and not of our personal perceptions (e.g., “beauty is in the eye of the beholder”)—may relate to, and perhaps even motivate the projection of the subjective feeling of understanding onto objective knowledge, the explanatory power of proofs onto the proofs, and the problematic nature of problems onto the problems themselves. It shares some semblance to what Immanuel Kant, in speaking of the aesthetic experience, described as subjective universality: This universality is distinguished first from the mere subjectivity of judgments such as “I like honey” (because that is not at all universal, nor do we expect it to be); and second from the strict objectivity of judgments such as “honey contains sugar and is sweet” because the aesthetic judgment must, somehow, be universal “apart from a concept.” Aesthetic judgments behave universally, that is, involve an expectation or claim on the agreement of others—just “as if ” beauty were a real property of the object judged. If I judge a certain landscape to be beautiful then, although I may be perfectly aware that all kinds of other factors might enter in to make particular people in fact disagree with me, nevertheless I at least implicitly demand universality in the name of taste.The way that my aesthetic judgments “behave” is key evidence here: that is, I tend to see disagreement as involving error somewhere, rather than agreement as involving mere coincidence. (Burnham, 2006) It may very well be that this phenomenon of subjective universality underlies not just judgments of beauty, but most experiences of an undergoing variety. It at least

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offers a plausible explanatory mechanism for how the subjective facets of the various phenomenon as discussed become overshadowed by seeming universality—leading to the projection (of beauty, explanatory power, clarity, and even meaning) onto the object at hand. For example, it appears to account for G.H. Hardy’s earlier quote (“In a good proof there is a very high degree of unexpectedness . . .”), as if the unexpectedness were in the proof, and not in Hardy’s subjective experience. In short, the notion of subjective universality appears to address the class of undergoing experiences—including understanding—and helps to locate them where they would rightfully belong: in that in-between space between sheer objectivity and subjectivity. Understanding, for example, admits elements of the objective (as it is not purely a matter of taste), while very much involving the subjective, at least in the way I have portrayed it. (Yet, is not the image of this in-between space correspondent to that of a connaitre way of knowing—of a way of knowing that is personally shaded between objective knowledge and subjective experience?) It appears then that subjective universality captures something essential about the lived experience of understanding, and also of why understanding so easily becomes conceptualized apart from the person. It acknowledges the individual and experiential elements of understanding, while also offering an explanatory mechanism for how the seeming universality of what we understand (“It’s so clear and obvious!”) may underlie then the tendency to diminish, in our minds, the subjective aspect of the phenomenon. In explicating the relation of the personal to the universal, it supports a view of understanding, then, as an event—as that moment of experience when something happens to us, when mathematics stirs a recognition, moves us into a felt-sense of knowing and of having arrived at a place of connection to itself.

3 A BRINGING FORTH OF SELF: WILL

Perception of relationship between what is done and what is undergone constitutes the work of intelligence. ( John Dewey, 1934, p. 45) “Nisi credideritis, non intelligitis.”—unless ye believe, ye shall not understand. (St. Augustine) I began the last chapter with Buber’s characterization of an I–Thou relationship. From that discussion emerged the twin notions of “will” and “grace” as constituent aspects of the process of entering into relationship, with the latter being the focus of the chapter. In this chapter, I now take up the more active (on the part of the self) element of “will.” Another way in which to think of the two would be as what happens to one (grace) and what one does (will).1 Whereas “grace” implies the self being affected by other—or self being affected by relationship with other—“will” suggests an image of what the self brings forth into the interaction with other.2 I resort to this sense of a “bringing forth of self ” into the relationship as the guiding image or metaphor of this chapter. Naturally, what one brings forth and how one is affected are entwined in a complex and iterative dynamic. It is what William James called “flights and 1 Who one is will be addressed in the next chapter. 2 This is not to say that “will” is to be construed necessarily as self doing unto other, but more simply, as self doing, or self bringing forth. For example, a person can bring forth an attitude of receptivity so as to receive other. In this case, there is no doing unto other, but simply the doing of the embodying of receptivity. While not a doing unto other, it is still distinct from the undergoing, or being affected-by-other of the previous chapter, for it is what one brings forth into the interaction.

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perchings.” To quote John Dewey (1934), who equates the flights with the doing and the perchings with the undergoing— The flights and perchings are intimately connected with one another; they are not so many unrelated lightings succeeded by a number of equally unrelated hoppings. Each resting place in experience is an undergoing in which is absorbed and taken home the consequences of prior doing, and, unless the doing is that of utter caprice or sheer routine, each doing carries in itself meaning that has been extracted and conserved. ( p. 56) That is, the quality of each undergoing is shaped in some capacity by the manner in which one has brought forth one’s “will.” In turn, that undergoing will inform and influence the way in which the “will” is subsequently brought forth. We have had glimpses in the last chapter of specific ways in which each can affect the other. For example, what holds some mathematicians to the discipline— or, what motivates their continued doing of it—are the aesthetic experiences and moments of understanding. In the other direction, we saw that what a person brings to the table with a mathematical problem, in fact, informs the interaction between person and problem, and dependent upon that, there may be a resultant “seizure” on the part of the person by the problem or not. Similarly, an experience of understanding (the “aha”) involves both what is brought forth by the learner as well as the mathematical explanation that is meant to stir in the learner such an understanding. To focus on the direction of doing to undergoing, a receptive stance may, in fact, be deemed necessary for the undergoing experience itself.3 In this regard, then, it is clear—and common sense, really—that an undergoing just does not happen by sheer happenstance. It comes about, at least partly, from what is brought forth by the person who experiences the undergoing. One might consider such a bringing forth as a kind of precondition—or what I would call a readying or preparation4 of self—for the experience of undergoing.

3 In the broader context of learning in general, the manner of receptivity that one might speak of is the capacity to “learn from experience—capable, that is, of acknowledging the inadequacies of our initial beliefs, and recognizing the need for their improvement” (Scheffler, 1977, p. 182). Certainly, such receptivity is at the heart of the educative process. Consider the etymological roots of education—from the Latin educare meaning, “to lead out” or “to lead.” Thus, to be educated is to be “led,” implying a receptive state as Scheffler describes. Who or what leads the one being educated is a different question altogether, which I only touch upon in the Epilogue, although in the context of this discussion, I am interested in keeping the focus on other as mathematics (or some other discipline) itself. 4 Note that I am using the word “preparation” in a way different from the idea of planning and anticipating for, thinking ahead, or rehearsing. Instead, I mean for it to mean something closer to “precondition.” Only, I prefer the word “preparation” because the word implies an active element, whereas precondition feels too static for the act of readying oneself for the undergoing.

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Buber called it “will.” Others have referred to it as commitment, exertion, discipline, effort, intention, persistence, incubation (of incubation and illumination) and so forth. In relation to problem-solving, mathematics educator Caleb Gattegno (1981) called it “affectivity”: This requires that we recognize that problem solving is not essentially an intellectual activity . . . the finding of the solution is a much more complex travail in which affectivity is the force that keeps us working on the problem and gives us the stamina to continue working on it. “Staying with the problem” is the most important feature of problem solving, and yet one that is almost never discussed. (p. 43) Gattegno’s point is that exertion of will—or a kind of emotional work in “staying with the problem”—buttresses the intellectual activity that is otherwise more manifest, and as such may be at the crux of problem-solving. I might also liken it to a kind of will-to-struggle, or even a willingness to experience frustration. It is this emotional work that precedes—again, acts as the readying of self for—the eventual undergoing experience. Ian Winchester (1990) speaks of it as a “slow and careful preparation”: The flash of understanding is invariably the fruit of a slow and careful preparation, the development, through fits and starts, of a skilled, practical competence. (p. 70) Winchester’s comment links the preparation with the “flash of understanding,” or what one brings forth with the consequent experience of understanding. Phenomenologist Ingarden (1961), in a different vein, points to the fact that the passive aspect of the aesthetic experience is embedded in an otherwise active life: However, it should not be supposed . . . that an aesthetic experience is a purely passive and noncreative “contemplating” of a quality . . . as opposed to an “active” practical life. On the contrary, it is a phase of a very active, intensive, and creative life of an individual. (p. 313) And in a broader sense of it, phenomenologist Mikio Fujita (2002) simply calls it “waiting”: However, only in the world of becoming is it also possible that “what is waited for” which has so far been nebulous, may become distinct because of a certain way or certain possible ways of “how we wait.” For instance, if a poem comes to a poet, the poem depends on the way the poet has waited for the poem. It is not that a certain poem comes to the poet regardless of

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what he or she is doing—thinking, working, resting or daydreaming—in short, how he or she is living. In addition, although an inspiration might come to us at a seemingly unexpected moment, it does not come to anyone at any time. “If the thunderbolt hits you, you must have been prepared for that,” said Antoine de Saint-Exupery. (p. 133) Whatever the name one gives it, something approximating Buber’s notion of will is being addressed in each of the comments above. Still, I think it useful to return to Buber’s original passage, so as to begin distinguishing “will” in the specific sense as used by Buber: But it can also happen, if will and grace are joined, that as I contemplate the tree I am drawn into a relation, and the tree ceases to be an It. Buber speaks of will, not in just any sense, but as can be joined with grace. So as to bring out this image of will in relief, I offer a stark contrast or two from my own past: As a high school junior, I prepared for the SATs by memorizing a list of 3,000 vocabulary words by sheer will. That is, I “hardwired” the words and definitions into my brain by the use of a primitive computer program I wrote, which acted as a virtual flashcard shuffler—a truly barebones and brute-force operation, if there ever was one! There were also the times in college when I attacked mathematics problems with a kind of determination as might be captured in the intent to conquer and overcome the problem. I worked alone, and there was something (again) brutal and punishing in the way I went about the task—it could even call to mind the idea of an “abusive” (or “love-hate”) relationship, a topic to which I will return in the epilogue. “Will” as exemplified through such war metaphors—or a “brute force” brand of it that is mostly about overcoming—has a different feel from a view of will as Buber forwards, of a will that can be joined with grace. I think it of relevance that two contrary approaches to mathematical problem-solving are spoken of as being “brute force” and “elegant,” as in “I solved the problem by brute force,” or “Your solution is rather elegant.” Might one speak similarly of will, in terms of one that is brute force, and another that would give rise to elegance (which, of course, is of an aesthetically tinged experience of undergoing)? In fact, I would offer that the version of will as illustrated in my personal examples is all about overcoming and doing unto the problem or situation, and as such preserves a subject–object relation between self and other, whereas Buber’s vision of will leaves room for grace (or undergoing), in turn allowing for a subject-subject relation to emerge. Further, while I may have accrued knowledge in the savoir sense of it in my doings, the kind of “will” as I was bringing forth may have indeed impeded the development of a connaitre type of knowing exactly because it could not be joined with grace.That is, it precluded me from being “drawn into a (true) relation.”

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“Staying with the Problem” Out of time we cut “days” and “nights,” “summers” and “winters.” We say what each part of the sensible continuum is, and all these abstract whats are concepts. The intellectual life of man consists almost wholly in his substitution of a conceptual order for the perceptual order in which his experience originally comes. (William James, 1996, pp. 50–51) While I may have contrasted Buber’s view of will with what it is not, I have yet to inquire into its own textures and meanings. Thus, a question that I would like to explore in this and the next two subsections is, “What exactly is this will—this bringing forth of self—that is different from an overcoming view of it, that instead would allow for it to entwine and join with grace, and that would support, rather than impede the development of a relationship between self and other?” Further, “What are its constituent aspects?” In this regard, Gattegno’s image of “staying with the problem” offers what I believe is a resonant image of will that is strong yet adaptive—that is, resolute yet open to grace. It implies both the act of bringing oneself forward to the problem, as well as sticking with it. In the latter sense, it bespeaks a sense of staying with and supporting the problem-solving process, and of holding a tension between the two poles of (1) forcing the issue (i.e., the overcoming will) and (2) just giving up and withdrawing from the problem (i.e., no will). In a mathematical context, it would be distinguishable from plunging ahead with a set strategy or algorithm without thought or pause for fit (with its forced quality), and on the other hand from sitting back, hoping for an answer to magically appear, or waiting for the teacher’s solution (with its meek, passive quality). This manner of withholding forced resolutions while attending to the problem at hand without withdrawal calls to mind Keats’ notion of “negative capability,” as in the capacity to be “in uncertainties, mysteries, doubts, without any irritable reaching after fact and reason” (quoted in Dewey, 1934, p. 33).While Keats considered it a prime trait for poetic genius, John Dewey (1933) regarded it as the basis for reflective thinking: Reflective thinking . . . involves willingness to endure a condition of mental unrest and disturbance. Reflective thinking, in short, means judgment suspended during further inquiry; and suspense is likely to be somewhat painful. (p. 13) The image of one willing “to endure a condition of mental unrest and disturbance” and contend with the “somewhat painful” suspension of judgment, while also ready to undergo the trouble of “further inquiry” offers a lived-in description

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of what it might mean to “stay with a problem.” It explicates the quality of embracing, and of holding a space for uncertainty while also moving toward resolution. It is an image of exerting effort and will, but also, of subjecting oneself to the process of reflective thought, as if to afford a kind of agency on the part of the problem (or problem-solving process). I might liken it to the metaphor of listening to a problem, in the way that one listens to another person with what I would call its “active receptivity.” Listening is an arduous (active) process as most of us know, but it is also receptive. Similarly, to “listen” to a problem (or to the problem-solving process) would be akin to allowing for aspects of the problematic situation—with its inherent solutions—to emerge to one’s perception. In fact, such a phenomenon of emergence (of solutions) may be an undeniable consequence of a stance willing to endure suspense and uncertainty—a stance, again, of active receptivity. The notion of negative capability is one to which Dewey later returns in Art and Experience in examining exactly this, the receptive stance. Dewey points to receptivity as involving both a “reconstructive doing” as well as a “taking in.” The “doing” would correspond with this discussion on will, with the “taking in” corresponding to the undergoing aspect of experience as discussed in the last chapter. The doing involves what he calls perception, in contrast to recognition: Recognition is perception arrested before it has a chance to develop freely. In recognition there is a beginning of an act of perception. But this beginning is not allowed to serve the development of a full perception of the thing recognized. It is arrested at the point where it will serve some other purpose, as we recognize a man on the street in order to greet or to avoid him, not so as to see him for the sake of seeing what is there. (1934, p. 52) Note how Dewey links recognition (or the lack of negative capability) to the issue of instrumentality. Dewey seems to be suggesting that the impetus to instrumentalize the other is complicit in—or may even be another descriptor for—the act of recognition, or of premature reification. In this regard, it recalls the spirit of instrumentality laden in Joyce’s “improper art” or Kant’s nonaesthetic pleasures from the last chapter. Each fails to afford a taking in of other as is. Dewey’s recognition, as such, also aligns itself with the I–It relation, of seeing other as object or It. Dewey continues: Bare recognition is satisfied when a proper tag or label is attached, “proper” signifying one that serves a purpose outside the act of recognition . . . It involves no stir of the organism, no inner commotion. (1934, p. 55) The notion of recognition hints at a potential conflict inherent between the desire for knowledge (as name, tag, or even mental object) and the sustaining of interaction—

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or perception—so as to foster a sense of relationship with the object/other of interest. It is the latent tension extant between the savoir and connaitre ways of knowing, or between what education philosopher Heesoon Bai (2001) describes as “looking as identification and seeing as in-dwelling,” whose difference hinges upon what she calls, “intense, total, and sustained attention” (p. 13). It is a distinction that recalls my original illustration of a strictly savoir manner of knowing, where I was able to name various birds but without a sense of connection—or a “stirring of inner commotion”—in response to the life around me. The urge to name and to capture mentally the birds by prematurely arresting perception toward them is exactly what recognition entails. In contrast, perception is the act of “reconstructive doing” through which— Consciousness becomes fresh and alive . . . An act of perception proceeds by waves that extend serially throughout the entire organism. There is, therefore, no such thing in perception as seeing or hearing plus emotion. The perceived object or scene is emotionally pervaded throughout. (p. 53) The emotionality of perception recalls the various characterizations of undergoing experiences described in the last chapter. In particular, Dewey’s point of emotion being more than just an addition to seeing or hearing, is similar to Bruner’s suggestion that affect, too, is more than just an appendage to knowing. Dewey continues: Perception is an act of the going-out of energy in order to receive, not a withholding of energy. To steep ourselves in a subject-matter we have first to plunge into it . . . We must summon energy and pitch it at a responsive key in order to take it in. (p. 53) Here, Dewey is clear in explicating that perception is in fact a doing—a “going-out of energy in order to receive.” It is a plunging in for the purpose of taking in, an exertion of will for the receiving of grace. Here, then, is a contrary image of will to that of an overcoming variety. It is will as still a “doing,” but one where the doing is directed outwardly but also toward the self—in restraining the tendency toward recognition, toward instrumentalizing other, toward premature reification, and toward withdrawal. Simultaneously, one is sustaining perception, interaction, and attention—that is, nurturing a state of connectedness of self with other. Will, in this way, may be regarded as an intensified withholding and waiting, but also an enduring and a staying of the course wrought out by the interrelation between self and other (as problem or circumstance).

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Desirable Difficulties Faces the Blizzard, faces the storm Was the heart of the circle Nature formed A covenant born. (Jack Gladstone, 1997, from “Faces the Blizzard”) In his liner notes to the song “Faces the Blizzard,” singer-songwriter Jack Gladstone explains the inspiration for the song: My song to honor the North American Plains Bison. Upon being caught in the ferocity of a winter storm, Buffalo Chief would lead his people into the wind. By walking into the storm, they would get through the storm before those herds who wandered aimlessly or fled the storm’s advance. A behavioral trait ultimately selected for, and an extremely powerful metaphor for humankind as well. (1997) The behavior of the North American Plains Bison presents an apt metaphor for the current discussion also. “Walking into the storm” in the context of mathematics may be akin to an earlier description offered by the elementary teacher Lia: If you aren’t confused, it means you already know the thing to be learned. If you are confused it means that learning is about to take place, or that the opportunity exists . . . When I myself am a student, if the confusion isn’t present I push myself to go deeper or broader in order to find it. Or one could “face the storm.” In the words of a mathematician from Burton’s study: “When I am stuck, I hope I have learnt to welcome stuckness almost as the natural state of a mathematician” (p. 59). Tellingly, 55 of the 70 mathematicians interviewed in Burton’s study referred to the state of being stuck. Similarly, mathematics educators John Mason, Leone Burton, and Kaye Stacey write in their book Thinking Mathematically (1982),“Everyone gets stuck. It cannot be avoided, and it should not be hidden. It is an honourable and positive state, from which much can be learned” (p. 49). The message here, of course, is that difficulties can be welcomed.Whether one actively seeks difficulties, or welcomes them once there, both attest to a stance that would view difficulties as necessary, if not in fact desirable.5 This, in itself, marks what for most persons would be an act of will—again, directed toward self, or in 5 I think it important to consider both ways of phrasing the phenomenon (“seeking” and “welcoming” the difficulties), as there may be a correspondence, respectively, to a “masculine” and “feminine” (not male and female) modality of being in the world.

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this case, toward one’s attitude toward the difficulties. What I would suggest, though, is that this, too, is a brand of will that would meet with grace or subsequent undergoing, specifically in the experience of understanding and learning. If one reconsiders the “aha” experience (that moment of grace), that moment of release inherent in such an experience occurs exactly because it is preceded by a state of tension or mental difficulty. If there were not tension, there would be no release. One might even go so far as to suggest that the greater the buildup, the greater the emotional sense of liberation upon resolution. In cognitive psychology, such tension preceding learning is referred to as “cognitive conflict.” It is the “honourable and positive state,” the desirable difficulty insofar as learning is concerned. Cognitive psychologists have demonstrated in a variety of ways that long-term performance, including the ability to transfer learning to novel but related environments, is enhanced by introducing difficulties for learners through various means, such as through varying the conditions of practice, providing contextual interference, and reducing feedback to the learner (see Bjork, 1994). What is of interest is that such difficulties for the most part, in fact, impede performance and learning in the short term. As such, the desirability of such difficulties is not likely to be apparent to the learner himself, who more than likely will foreground instead the errors and confusion, and further, rely on short-term gains as the measure of learning. On the other hand, if such difficulties are removed, long-term learning suffers (most often unknowingly for the learner) while short-term performance improves, (mis)leading the learner to perceive learning as taking place:6 Rapid progress in the form of improved performance is reassuring to the learner, even though little learning may be taking place, whereas struggling and making errors are distressing, even though substantial learning may be taking place. (Bjork, 1994, p. 194) Thinking in terms of short-term and long-term gains in learning appears to give another perspective on will, and suggests another way of characterizing will as would support an undergoing. Indeed, it appears that sacrificing short-term gains for long-term advantage may be at the heart of what it would mean to look upon difficulties as desirable. Privileging long-term gains affords an outlook of embracing the difficulties of frustration, confusion, and the like. But not by accident or mere coincidence, it also evokes Fujita’s “waiting,” Dewey’s “perception” that would hold off on hasty “recognition,” and Winchester’s “slow and careful preparation.” In other words, it appears another way of talking about this shifty entity—this will as can be joined with grace.

6 This phenomenon appears related to the earlier cited notion of “schoolish engagement,” where students subvert the long-term goal of learning by prioritizing the short-term goals of school activities (e.g., passing the test, getting the grades).

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Repetition, or Repetition? One last notion I would like to explore in conjunction with Buber’s will is that of repetition. Although not mentioned by any of the mathematics teachers I personally spoke with, it is a theme that was repeated by most all of the musicians I interviewed, who spoke of it in varying regards to the process of coming into relationship with a piece of music. One wonders the reason for its omission among the mathematics teachers, though one could venture a guess of it being some combination of the unpopularity of the idea in mathematics education— especially in its closeness to the dreaded rote learning—and of the difference in how mathematics is learned (while a piece of music is usually mastered). It is to the former concern to which I now turn, from which I will also touch upon the latter distinction in this chapter and the next. I would suggest that repetition and rote have become synonymous, or merged, in the minds of some educators—that is, they have become conflated. While some would call for a “return to basics” where repetition and practice would govern the pedagogical practices of mathematics teachers, others would do away entirely with such “mind-numbing” visions of mathematics teaching. What appears, though, to underlie any such dialogue around the role of practice and repetition in mathematics learning is the conflation of repetition with rote.While one camp would accept rote as a necessary evil for the sake of mastery through repetition and practice, the other would throw out the virtues associated with repetition so as to avert mindless rote learning. Yet, surely repetition need not be allied with routine, dullness, and unimaginative learning—in other words, with rote learning. Or maybe that is not entirely clear? So as to help extricate repetition from rote, I will presently argue that repetition can involve the faculty of will as envisioned by Buber in leading to a sense of relationship—that is, it can work synonymously with will-as-would-be-joinedwith-grace in leading to understanding. In order to pursue this argument, I begin with two of the more resonant descriptions as offered by the musicians. Stuart, a pianist and choral director, explains the role of repetition as follows: The more I play a song, or sing a song, or conduct a song, the more I become attached to it, or it becomes more a part of me. So then, it’s as if I develop a relationship with the music—so to speak. Here is an image of a repeated act that brings not dullness of mind, but deepened “attachment”—a sense of relationship.7 In a similar vein, Mark, a professional pianist as well as piano teacher, has his students listen repeatedly to a CD of the music that the student would eventually play:

7 One oftentimes finds a similar relationship toward repetition in religious as well as academic ritual.

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They would listen to [a piece of music] till they could play it in their head . . . The idea here is they listen to the music so much it becomes a part of them. So understanding where they must go with the piece, that’s done. They know that. Even if they hadn’t memorized it, they know the feel of the piece, and they know where it needs to go, so now, the effort is on yes, accurately playing it, and developing all of the things that contribute to technique and the expressiveness. But over and above all of that, because they’ve listened to it, they’ve made it a part of themselves. Again, Mark’s students will have engaged repeatedly with a recording to deepen their understanding and feel for a piece of music. It is a reiterated act that feels nothing like the meaninglessness associated with rote. In both Stuart’s and Mark’s accounts, what is made explicit is that the repetitive act and a sense of closeness to the music can go hand in hand. The vision of repetition in their accounts entails a bringing of oneself over and over to the task at hand, so as to foster a sense of “oneness” or intimacy with the music. Yet, I think this is how it can happen in relation to ideas and concepts also. No doubt most of us have experienced how repeated exposure to an idea—or in particular, to a difficult idea—can gradually reveal, or “open up” new aspects and distinctions surrounding that idea not noticed before by us. It is a process whereby one’s understanding becomes progressively fine-grained, drawing one further into what could be called an intimacy with the idea. It is a phenomenon rooted in repetition that contrasts dramatically with the otherwise rote version of repetition. This begs the question: how is it that such divergent meanings can arise from the same repetitive act? Further, is the consequence of repetition rooted more so in something other than the act (of repetition) itself? Agnes Heller (1979) explores in her own writings on the phenomenology of feelings and involvement, the ways in which the meaning of the repetitive act can shift depending upon whether it has the character of means or is an end per se for the individual. First, she points to the low personal involvement level associated with repetitive actions and thinking of a purely instrumental nature, which she terms “repetition as a means”: I get dressed, that is I tie my shoes, I put on my clothes, I have done it a thousand times already.While I perform this act I do not think about it, I think of other things: things into which I am involved (my assignments, a date, etc.). The degree of my involvement is independent of the type of action. (pp. 9–10) Later, she offers the contrast: The discus-thrower practices every part of the movement, which later

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becomes spontaneous and repetitive; but these practiced movements, even if a thousand times repeated, become inseparable from the act of throwing the discus, from the entire process. If the throwing the discus is a pleasure in itself for him or if he intends to win in discus-throwing events (direct or indirect positive involvement) . . . the intensity of involvement, as opposed to involvement in repetitive acts of a purely instrumental nature, can increase in direct relation with repetition. (p. 10) As Heller would suggest, those actions (including thinking) that are performed primarily as a means to some other end—such as doing math drills not to learn but for one’s grades—generally lead to low personal involvement and investment, or else to a form of involvement typified by boredom. They become, in essence, rote. Yet, if the task being repeated is an end per se for the individual, this same task—even repetitious math problems, for example, for the person intent on mastering the topic at hand—can generate positive personal involvement that would overshadow any rote facet of the repetitive act. In turn, the kind of “oneness” or intimacy as described by the musicians can begin to emerge. Note how the intent toward mastery—mentioned earlier by the musicians— appears key to what shifts the quality of the repetitive act. For in bringing forth such intent or posturing, a task as simple as say factoring of polynomials can begin to unveil its complexities. Whereas for the student who approaches each new polynomial as a “means,” the task may remain qualitatively static (and different only quantitatively—i.e., the numbers change), for the learner who would seek mastery, each new problem offers the potential for an unveiling of deepened understanding. New number relationships are revealed. Connections to prior mathematical topics become apparent. A whole new world begins to disclose itself to such a learner, as the learner undergoes the grace of revelation and understanding. This is to say that what changes between the two scenarios is not the repetitive nature of the tasks themselves—i.e., there is no intrinsic meaning, or lack thereof in the repetitive act—but instead, the meaning is attributable to some synergy between the posturing as brought forward by the actor and the task itself. Insofar as deciding the quality of the experience and the meaningfulness of the task (as much as the task will afford), it is the posturing as brought forth by the actor that is under the actor’s control. Consequently, such posturing appears also another way of speaking of Buber’s will. In regards to repetition and rote, one way then to differentiate between the two is to say that repetition affords the possibility of giving way to grace, while rote does not, as rote is will that would not meet with grace, or understanding. It is of relevance to the larger discussion that instrumentalizing the task at hand appears complicit in decreasing involvement, and vice versa for the intrinsic mode of relating. Whether it is Buber’s I–Thou relationship, Kant’s aesthetic pleasure, Dewey’s perception (over recognition), or the nature of repetition as an end, the

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recurrent theme of intrinsic-valuing-of-other vs. instrumentalizing-of-other asserts itself yet again. One approach toward a “repetitive” pedagogy in mathematics can be found in Hong Kong mathematics teaching (see Hawkins, 1994; Ko & Marton, 2004). Asian mathematics educators describe their lessons as “repetition with variation,” which uses systematic variation so as to make “it possible for students to discern what is tacitly understood, by means of contrasting non-instances” (Ko & Marton, 2004, p. 84). Through exploring the interplay between variation and invariance, the invariant mathematical properties are surfaced in a form of instruction reliant on repetition. In terms of the current discussion, one could construe such a pedagogical approach as externalizing and explicating that which would otherwise be implicit for the learner who views the repetitive act as an end. In other words, it is a form of instruction that attempts to give to a wider range of students the same experience of emergence of phenomenon and ideas as experienced by the student who brings himself/herself over and over to the same task, noticing deeper and finer details with each passing; but these details would be revealed instead through the teacher’s careful sequencing of problems, each with slight variations from the prior problems. From a different but related vantage point, cognitive psychologists Carl Bereiter and Marlene Scardamalia (1989) refer to what is brought forward by the actor in learning contexts as intention. They point to two studies (Owen & Sweller, 1985; Resnick & Neches, 1984) where students are given problems that embody mathematical concepts (e.g., devising operations with blocks that correspond to symbolic operations in arithmetic, or showing symbolic operations corresponding to operations with the blocks) with the expectation that the students will acquire a deeper understanding of the underlying concepts through engagement in solving the problems. Both studies conclude that the assumed learning is not as certain as expected, and in fact, occurs only among those children who were “trying to learn.” Bereiter and Scardamalia render the nature of such intentional learning: As we interpret it, the children who were “trying to learn” were not simply investing extra effort in trying to solve the problems they are presented. Instead, they were dividing their effort between solving those problems and solving other, unassigned problems, which were problems having to do with the state of their own understanding of the phenomena. These latter were learning problems. (pp. 365–366) What is brought forth by the children “trying to learn” is not a mindless, overcoming will, but something more intelligent—the intention toward understanding and mastery, or what Ellen Langer (1989) calls mindfulness. Similarly, with the musicians I interviewed, it is likely the same intention toward mastery that is

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brought forth so as to foster the sense of deepening familiarity and relationship with the music as realized. It is of educational interest that whether the learning is through the form of repetition or problem-solving—usually the two pedagogical methods held in contrast to one another—each can lead both to desirable and undesirable states for the learner.Whether those states are similar between the approaches (and even across others) is open to question, although at a glance, it appears that repetition can indeed lead to something approximating a sense of familiarity or relationship. In this regard, it appears that there may be a powerful and meaningful consequence of bringing oneself over and over to a task or to an idea not necessarily wrought out by insufficiently repeated involvement. On the other hand, without the will or intention as brought forth by the learner, we all know that repetition can deteriorate into rote, and also that problem-solving can yield seeming, or even genuine engagement, but without real learning. Note that I am not sounding the conventionally claimed virtues of repetition for the sake of automaticity, but for that something else that I have so far referred to as a sense of relationship or familiarity (or of a connaitre way of knowing)—something otherwise overlooked in most educational discourse.8

Understanding as Verstehen: “Standing before” so as to Unlearn Mathematics In this chapter, I have explored the meaning and nuances of a will as would be joined with grace in the image of “staying with a problem,” in negative capability and Dewey’s notion of perception (vs. recognition), in the seeking and welcoming of difficulties, in intentional learning, and finally, in a manner of mindful repetition (vs. a rote one). As I have suggested at the beginning of this chapter, each affords an experience of undergoing—the topic of the previous chapter—whether as in an affective or aesthetic response, or as an experience of understanding or “aha.” The point being that, as Buber suggests, it is when “will” and “grace” are joined that one arrives at true relation—not one without the other. 8 All of this is to suggest that rather than wondering which and what type of instruction is more “effective” over some other, it would perhaps behoove educators to reexamine the sought ends— that is, what one would mean by “effective” or even of desirable consequence—and in turn to consider the affordances and constraints of the different instructional approaches available to educators in pursuit of the reconsidered goals. For example, one could ask, “When a learner is able to bring forth proper effort/will/intention, what differing consequences follow depending upon the instructional approach?” Further, “What are the ways in which the learner’s will or intentionto-learn is best mobilized within each instructional approach?” Or in a more general sense, “What are the ways in which a learner’s will is best brought forward?” (This could be construed as the classic, “How best to motivate” question.) While I have raised a loose conjecture or two around the first question, I will leave the second question for future inquiries (for it is much too large a question for this book). As for the third and last question, the next chapter, on some level, can be read as a reframing and subsequent response to it, while also being an attempt to give fuller shape to the image of a knowing steeped in relationship.

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In closing the last chapter, I suggested an image of understanding as connaitre, or as an undergoing event animated, informed, and structured by personal involvement. Yet, just as “grace” without “will” offers only partial description of the phenomenon of entering into relationship, understanding as an undergoing, I would suggest, presents also only a “half-picture.” Thus, in the spirit of pondering the other “half ”—that is, the complement to understanding as undergoing—I consider the notion of understanding as the modern German verstehen (deriving from Middle High German, verstan, meaning, “to stand before, or in front of ”).9 Note the closeness of its etymological imagery to Gattegno’s “staying with a problem,” as well to singer-songwriter Jack Gladstone’s “(Bison) facing the storm.” Further, verstehen’s “standing before” describes well what underlies mindful repetitive action. In such a repetitive act, a person brings his or her own being, again and again, so as to “stand before” some idea in order to experience the unveiling of meaning and understanding. All suggest an exertion of will and effort coupled with an affordance of agency toward the other that one is facing, standing before, staying with, so as to make allowance for emergence. It is will as would join with grace. At the same time, there is another shade of meaning implied by understanding as “standing before.” It is that to “stand before” is also to put oneself on the line. Standing before does not just imply a persevering through trials, and overcoming them. It also intimates a putting of oneself “out there”—not slinking away in a comfortable hideout by sitting near—but standing oneself in a place that is vulnerable to the influence of the other. It is the risk-taking and courageousness required of learning and of passion, which I spoke of in Chapter 2. In mathematical terms, it could take the form of sharing one’s preliminary thinking with a colleague, or of unearthing one’s own assumptions to oneself. Part of what can happen when one learns something—not just in an additive sense of tacking more knowledge onto what one already knows (the savoir knowing), but in a transformative sense that would involve a qualitative shift in one’s knowledge state (understanding as connaitre)—is that one loosens one’s hold of old ideas and concepts for sounder ideas and concepts. This is so especially in the case when one may have held such ideas closely and dearly up until that time. In a manner of speaking, one unlearns such concepts (or preconceptions). But is that not the nature of learning itself—that it is a matter of stepping out of one view into another? In order to make that move, there is, phenomenologically, that moment when one lets go of the old, when one unlearns what is no

9 In phenomenology, verstehen is a complementary “tool” to introspection. Whereas through introspection, the phenomenologist looks into his or her own subjective processes, it is through verstehen that the phenomenologist moves closer to the subjective experiences of others. Stated more broadly, verstehen is the process of subjective interpretation (see Truzzi, 1974). I will be exploring instead the connotations as arising from its etymological roots.

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longer necessary and is in fact an impediment to learning. Part of “standing before” a problem, then, involves not just seeking out solutions, but being subject to a surfacing of one’s assumptions and thoughts, and a consequent seeing of their relevance, irrelevance, or even hindrance. This is to say that what is in fact sometimes missing in the theorizing of understanding and learning is the unlearning— the questioning and loosening—of brought-in assumptions that are a core part of the process of learning. For example, many mathematical “misconceptions” as held by students are oftentimes not simply incorrect conceptions that are readily corrected, whether by a teacher or by the unwitting student. Instead, they are deeply grooved misunderstandings that, in fact, can stand in the way of actual understanding unless “unlearned,” for example, by cognitive dissonance (Flavell, 1963; see also Balacheff, 1990; Movshovitz-Hadar & Hadass, 1990). But cognitive dissonance occurs, excepting cases of sheer happenstance, only when one’s false understandings have been surfaced, whether to oneself or to another facilitating one’s learning. Certainly, unlearning (as opposed to learning as it’s conventionally imagined) is not a trivial task for most learners to undertake unless they are willing to put their ideas forward, whether in discussion, on paper, or even in their own minds to themselves—that is, unless they are willing to “stand before” experience. Otherwise, preconceptions and assumptions will hardly be questioned so as to lead to their “unlearning.” There will not be the cognitive dissonance that could lead to the eventual learning. I would like to explore one example to give further shape to what I mean here. In speaking of heuristics for problem-solving, the famed mathematician George Polya (1957) mentions the importance of understanding the problem. Although useful for the enculturated and savvy problem-solver, I think that it misses something for the moderately trained neophyte (i.e., “A little knowledge is a dangerous thing”): for him, he does not always know what pertains to the problem and what does not. He does not oftentimes know what he knows about the problem because he also knows a lot of extraneous mathematics. He enters a problematic situation, clutching onto his favorite memorized formulas, schemes, and strategies that have worked in the past. Thus, part of what he knows may have nothing to do with the problem at hand, but he will not be able to distinguish whether it does or not.This is the position of many unseasoned problem-solvers who “throw the kitchen sink” at some problems—problems that in the expert’s view would require just a moment’s thought, as the expert problem-solver will readily assess what is given by the problem.The nonexpert, on the other hand, is cluttered with too much knowledge that is irrelevant to the business at hand so as to be deaf to the problem, in a manner of speaking. He cannot so easily be clear about what he knows about the problem. Consequently, what can be important for such a problem-solver is to relax— that is, to “unclutch,” or to loosen his hold on the ideas and conceptions that he holds onto so tightly so as to give room for apperception of what is given by the problem itself. And of course, such an “unclutching” is what I have been calling

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“unlearning.” It is not that one is actually “unlearning” something so as to forget it, but that one is putting aside what is in the way of perceiving what the problem offers and does not offer. But the question arises: how does such unclutching, loosening, or “unlearning” transpire? I think that the key for the learner is in looking carefully not just at the problem (the other), but also at the self. He must acknowledge and consider all of the knowing and knowledge brought forth into the situation by himself. What Polya’s formulation skips over, I believe, is the examination of self that must occur much prior to even knowing what is given by the problem. It is not so simple as just listing what one knows about the problem, if one cannot attend to the problem—that is, if one is burdened by all that one brings forward in trying to grapple with a problem.Thus, one must also list what one is bringing forward into the interaction with the problem.What formulas, what tricks, what results, and what predilections does one bring? What assumptions (that are not even mentioned in the problem)? These must be brought forward, sometimes because they are useful, but more often than not for the unseasoned problem-solver, because they are in the way; they are in need of being let go of, and temporarily “unlearned.” And it is in bringing them forward (whether in discourse and conversation, on paper, or explicitly in one’s mind) that they can be examined and recognized as relevant or else irreconcilable with the reality of the problem at hand. Thus, to “stand before” a problem, would bespeak this sense of putting oneself—with one’s own thinking, presumptions and ideas—before the problem, so as for the self to withstand the scrutiny of experience. In this way, one can either utilize, or “unlearn,” the ideas and learnings that one brings forth so as to make, in turn, affordances for the undergoing of an understanding.10 In this manner, understanding-as-verstehen gives rise to understanding-as-connaitre, “standing before” merges with undergoing, and “will” joins with “grace,” thereby extending the scope of depiction and meaning of what it means to “understand” something, in turn, becoming a way of telling the larger story of “understanding,” and of relationship to mathematics.

10 One tactic to facilitate such “unlearning” is to unfamiliarize the familiar. Mathematics teacher educators, for example, try to help pre-service teachers to get past certain preconceptions that would stand in the way of deeper learning when they introduce base systems other than base 10. The point of such instruction is not to have teachers who can add numbers in base 7, but instead, it is to sufficiently unfamiliarize the situation for the pre-service teachers so that they cannot rely on formulaic thinking in grappling with the underlying structure of the numbering system itself. It is such formulaic thinking that teacher educators would want the pre-service teachers to unlearn—at least temporarily so that it would not get in the way of deeper learning. Such teacher educators would want the pre-service teachers to look past what they think they know (e.g., “It’s called the tens place because that’s what it’s always been”) to the actual concepts underlying the numbering system (e.g., “It’s called the tens place because the numeral in that place stands for the number of tens that have been counted”).

4 RELATIONSHIP AS INTEREST

The conventional focus of mathematics educators on epistemological issues (i.e., dealing with questions of knowledge, programs of study, learning, and instruction) eclipses the fact that education is an ontological issue (involving us in questions of existence, being, and identity) . . . what one knows and what one does cannot be dissociated from who one is. As Maturana and Varela concisely phrase it: Knowing is doing is being. (Brent Davis, 1995, pp. 6–7) In the previous two chapters, I elaborated upon two constituent aspects of the process of entering-into-relationship with mathematics. In brief, the idea was that through the merging of “grace” and “will,” one entered, and was brought into relationship with, the other of mathematics. The focus in those chapters was on what occurs—that is, the actual lived experience—in and around the moments of “connection,” where such moments begin to define a sense of relationship to subject matter. As the focus so far has been on the process of connection—or what I have earlier called the “becoming” of coming into relationship—two related questions arise: first, what motivates such engagements and becomings, and second, how to speak of the actual manner of relationship that results? More or less, what precedes and succeeds the joining of grace and will? In regards to the former, surely it is not the case, simply, that all who approach mathematics attempt to engage it in equal measure. Accordingly one might ask, what are some distinguishing characteristics, either in the self, or in the interaction between self and mathematics, that would lead one to “stay with” mathematics and to become receptive to the “undergoing” elements of mathematical activity, or not to “stay” and instead to abandon?

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And as for the question of what follows the joining of will and grace, it is a matter of characterizing the resulting relationship (or knowing) between mathematics teacher/educator and mathematics. In other words, how to delineate the nature of “true relation” that emerges? What comes to be, as I will suggest subsequently, is no longer the becoming but the being-in-relationship, no longer a process of relationship, but a state of relationship characterized by an intertwining of self with other. So as to facilitate my elaboration of both concerns, I will resort to an image of relationship-as-interest to anchor the chapter. Although to say that one “is interested” in a subject such as mathematics is usually a weaker statement than to say that one “has a passion for” or “has a connection to” that subject, looking more closely at its etymological root of inter (“between”) + esse (“to be,” thus “to be between” from the Latin) reveals a statement of being usually veiled in colloquial discourse. Thus, to “be interested in” mathematics is to imply that one is literally “in,” or “in between” the subject—a statement, not of occurrence or happening, but of actuality and state of being. But also, this image of interest as “being in-between” is echoed in the vernacular, as we say that we are into something, as in “I am really into Non-Euclidean Geometry” when we are very much involved or passionate about it. Conversely, to say that “math is interesting to me” (or else that “one has mathematics as an interest”) is to connote that mathematics is “in” or “in between” oneself (or that one has mathematics as an “in between”). Note, then, how with inter-esse, the self and the other/object-of-interest are entwined, one in the other. In such a way, they are implicated in an intertwining of self and other, where each is “in” the other. It is this state of being intertwined that, as I will argue, provides both an explanatory mechanism for the motivation to engage with mathematics (à la will and grace) as well as a description of what results through such engagement.

Intertwining with Other and its Relation to Motivation: That which Precedes the Joining of Will and Grace I would call to mind the fact that “interest” usually denotes an inherent curiosity and motivation toward the object of interest. That is, to state that one is “interested” in a subject such as mathematics is usually taken to mean that one is intrinsically drawn toward engagement with the subject. Under such considerations, one might ask, in what manner is an “intertwining between self and other” (from inter-esse) related to the notion of intrinsic motivation? In this section, I offer one possibility. In Chapter 3, I focused on the will-to-struggle or to grapple with the matter at hand as one aspect of what one “brings forth” into relationship. Yet, what one brings forward in any kind of interaction, in a larger sense, is the self with more than just the will and the mind, but also its values, beliefs, dispositions, tendencies,

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culture, interests, feelings, needs, ambitions, and so on—more or less, the whole person (or however much of that whole person one chooses to bring forward). One way, then, to rethink the question of intrinsic motivation is to invoke an image of mathematics, or the relationship with mathematics, as intertwining self and mathematics—as in how mathematics elicits (mathematics into self )1 and receives (self into mathematics) various aspects of self, including the will. That is, I will present an argument that part of what contributes to, or supports a bringing forth of will and an opening up to grace in a learner—i.e., intrinsically motivates a learner—may be in how these various aspects of self are afforded interaction in relationship with mathematics. Part of that argument will entail an examination of how different images of mathematics-as-presented lead to the self (or aspects of the self) being elicited and received by mathematics differently. Before moving directly into the discussion at hand with other-as-mathematics, I will again begin with the case of music so as to give an accessible, and perhaps even universal, image of how one may become “entwined” in relationship with a disembodied other.Take the phenomenon of what Edward Rothstein (1995) calls “identification” with music: There is, of course, a venerable tradition in which music is thought of as little more than pleasurable sensation; at the end of the twentieth century that is generally all most people require of it. This pleasure is often rooted in the listener’s sense of “identification” with the music: feeling that Bruce Springsteen, say, or Hector Berlioz understands and precisely expresses the listener’s inner, unarticulated (but deeply felt) sentiments. For many people, music is most successful when it mirrors the listener’s state of mind, or when it manages convincingly to create one. (pp. 82–83) When I “identify” with a piece of music, it is not just that I feel my feelings mirrored, but that I feel my feelings both stirred and received by the music. The music touches me (music in me), and in turn, I am able to “pour forth” my feelings into the music (me into music—e.g., consider blues or country music for the forlorn or heartbroken). If I take such an account as a common and prevalent one, then as Rothstein writes, the pleasure of listening to music consists (for most of us) in being able to experience an entwining of one’s feelings with music. As a contrast, imagine if one’s feelings were somehow untouched and barred from the interaction. How much more difficult it would be to feel a sense of connection, or of relationship with music, at least for many of us. And perhaps part of that reason may be because so many of us identify ourselves with our 1 Here I am resorting to an image of mathematics as coming into the self in speaking of it eliciting or stimulating aspects of self, and the complementary image of self as entering mathematics in speaking of the self being received by mathematics.

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feelings. “If music touches and receives our feelings, it touches and receives us,” or so the experience of listening to music feels. But also, it may be because to feel— or to find an avenue for expression of our feelings—is a need; and it is this need that is attended to in relationship with music. Yet, it is conceivable to speak of this relationship with music as becoming intertwined with other aspects of ourselves, such as, say, a disposition for rhythm. While a sense of musical rhythm would serve little to no use in mathematical endeavors, it would be of great benefit in playing or appreciating music. The relationship with music would sustain and privilege this aspect of ourselves, while mathematics would spit it out, and bar it entry namely for lack of use or relevance. Depending upon one’s identification with this trait, the subsequent reaction to music and to mathematics may differ dramatically. Further, one could imagine that different music would receive different aspects of a person. For example, some find J.S. Bach’s music cold to emotion but welcoming of a higher sensibility, while Chopin’s music may call forth a yearning for the beautiful, and so forth. Rather than saying that music-at-large intertwines with certain specific aspects of a person, it would perhaps be more accurate to say that of the many forms of music, different manifestations of it entwine with different aspects of self. So, a way of rethinking a person’s intrinsic motivation toward an undertaking would be in how deeply that person identifies with those aspects of self that are entwined in such activity.2 To offer a rather simplistic example, suppose that I have a strong need or identification with a disposition for “certainty” (or logical necessity). Suppose further that mathematics provides a milieu within which such core aspects of my self are stimulated and received—in other words, honored and even fostered. Undoubtedly, I will be inclined to exert will in fostering a further connection with the other of mathematics. But this, I would suggest, is exactly what is meant by being “intrinsically” motivated toward mathematical activity. On the other hand, I could be presented a version of mathematics that de-emphasizes such cleanness of thought that could then lead, in turn, to my becoming demotivated, or no longer intrinsically motivated. Of course, another student with little to no identification to such traits could experience different, or even the exact opposite trends. This is to say that being “intrinsically” motivated relies upon what one brings forth, but importantly, also upon the other—or the particular facet or “face” of other—that one is presented with.3

2 I will refer to such strongly identified aspects of self as “needs” in places, perhaps conflating what psychologists call true human needs with what is felt by certain individuals as something strongly identified with or desired. I would also call attention to the common etymological roots of “identity” and “identification” (with needs). 3 Here, I would point out that this perspective, more or less, stands productive dispositions (Kilpatrick, Swafford, & Findell, 2001) on its (conceptual) head by acknowledging the so-called productivity of a disposition, not on any qualities inherent to the disposition itself, but to how it interacts with the specific version of other (of mathematics) that is presented.

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Consider, for example, the following account of a student’s relationship to mathematics: Math does make me think of a stainless steel wall—hard, cold, smooth, offering no handhold, all it does is glint back at me. Edge up to it, put your nose against it, it doesn’t give anything back, you can’t put a dent in it, it doesn’t take your shape, it doesn’t have any smell, all it does is make your nose cold. I like the shine of it—it does look smart, intelligent in an icy way. But I resent its cold impenetrability, its supercilious glare. ( from Buerk, 1982, p. 19) The student offers a vibrant yet stark portrayal of mathematics that is “cold” and emotionless, that in turn would rebuff her feelings. Certainly, it is not the vision of mathematics held by all.Yet, it captures a perspective of mathematics that would likely not have arisen had her particular expectations and feelings—that is, what she brought forth—been elicited and/or received in the interplay between self and other-as-mathematics.4 Naturally, one would not expect such a student to be intrinsically motivated toward mathematical learning and activity. And wherein lies the cause? I would conjecture that it is in the mismatch between who she is (in the needs and dispositions that she carries strong identification with) and the version of mathematics presented to her, for just as music has many faces, mathematics, too, as I will soon illustrate, appears adaptable to the different needs and dispositions brought forth by the learner. The point being that “intrinsic” motivation should not be considered in the self so much as hinging on the synergy between self and other, and in particular, in how that self is afforded an intertwining in that synergy with other. To summarize the chapter to this point, I have raised the questions of what motivates and what succeeds the joining of will and grace.Thus far, I have focused on the former query, by proposing the intertwining of self and mathematics (as suggested by inter-esse) as one explanatory mechanism for such motivation. Further, it is an explanation that decenters motivation strictly from the individual, and displaces it onto the interplay between that individual and other—that is, onto the burgeoning (or dwindling) relationship between self and other. Where the discussion will now lead is to specific examples of these intertwinings, so as to give fuller understanding of how they may both motivate, and in turn, lead toward deeper connection.

4 In a similar vein, mathematics educator Peter Appelbaum (2008) uses the psychological terms egosyntonic and egodystonic to describe, respectively, when mathematics (or other) is in harmony and alignment with the needs and goals of the self and when it is in conflict with one’s concept of self.

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A Few Examples of “Intertwinings” Before proceeding with a few illustrations of how self and other-as-mathematics can intertwine, I would like to forewarn the reading of what may be a noticeable shift in the role of the teachers’ voices. Up until this point, the focus has primarily been on the phenomenon of entering-into-relationship, with the teacher comments acting as one of just a number of sources for supporting the development of ideas. In the next few sections that are concerned with the shape of the intertwinings, the teacher comments will emerge to the foreground, not only because they offer the richest portrayals of intertwinings that I have on hand, but also because in speaking of motivations, dispositions, needs, and being, I have approached ever so closer to the person (i.e., the self), a step removed from the phenomenon of relationship. In other words, with the joining of will and grace, self and mathematics are subsumed in relationship. But prior to that, there is self, and there is mathematics. And because my ultimate interests are of the relevance of these ideas to teaching and teachers (as expressed in the Introduction), I have chosen to amplify here the teachers’ voices. On another note, the examples that follow should not be construed as an exhaustive or representative cataloguing of “intertwinings” with mathematics. Instead, I offer the examples to give clearer shape to what I mean by an “intertwining” between what I have called “aspects of self ” and mathematics.They are the examples that emerged during my conversations with the mathematics teachers and teacher educators.

Challenge and Arête One quality given affordance in mathematical activity that was mentioned by some of the teachers was of the need or attraction to personal challenge. For one of the elementary teachers, Michelle, who would become the math specialist in her school, mathematics eventually became the grounds for “overcoming [her prior] math phobia.” Although the language arts appealed to her also, it did not provide for her the experience of “overcoming,” whereas in mathematics, each time she did “overcome,” she could feel herself becoming stronger. For Jan, a current mathematics teacher educator and one-time high school teacher, it was not just mathematics at large, but challenging mathematics that had meaning for her. When I ask her to explain what it is about the challenge that draws her, she responds— I guess one thing is that most of school I just feel like I breezed by. I did what people said. It wasn’t that challenging, right? I’m not going to say that it was boring, per se, but in some ways, it kind of was. I remember my 6th grade math class, and just like the whole thing I felt was just really boring. It was like doing the same things over and over again. So, when we

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did something new that really pulled something different out of me, then I felt like [a part of me] wasn’t being used before. Even though it was harder, I liked the value of it. Maybe I’m only saying it in retrospect . . . no, at the time, I really remember enjoying it . . . It just wasn’t as interesting when I didn’t have to think any way deeply about it. I didn’t think I was getting as much out of it. As Jan speaks of the “enjoyment” in being challenged, she points to something that may be true about challenges in general, that part of facing a difficulty is that new aspects of oneself can become surfaced. Or, to borrow from her phrasing, they “pull something different” out of ourselves. These heretofore undiscovered parts of ourselves that emerge out of a challenge—in this case, elicited by mathematical engagement—signify a manner of transformation reminiscent of Buber’s grace. It is what the other teacher implies also when she speaks of “overcoming” or becoming “stronger.” Both teachers are speaking of an undergoing that is evoked by will—a will in the form of seeking and accepting a challenge. Both teacher accounts, to speak of it from the perspective of this chapter, portray mathematics as eliciting and/or receiving their strivings for excellence— or what the Ancient Greeks called arête. It is mathematics, and not the other disciplines, that stirs and receives such strivings for these teachers. Naturally, mathematics is not the only subject that calls forth a sense of challenge (or raises frustration), but for these teachers, it is where such needs were given expression. The quality of arête, in fact, relates to what a few other teachers mentioned as their simple love of games and puzzles as their primary draw to mathematics. Ethno-athematics educator Ubiratan D’Ambrosio explains that the Greek conception of mathematics was motivated by the same ethos of arête that gave rise also to the Olympic Games, as a celebration of human excellence5— Let me elaborate on the importance given by the Greeks to the famous problems [e.g., trisection of the angle, quadrature of the circle, and duplication of the cube] and to their solution strictly with ruler and compass: the only explanation for this is that they were interested in something else. The same with the so-called philosophical impasses of infinity and the irrational . . .The problems that were philosophical impasses to the Greeks are indicators that Greek mathematics was very close to the idea of a game . . . By playing these games they were approaching the gods, both intellectual games . . . and the physical games. The big objectives of athletics were not to win the medals as a reward for fitness but to get closer to the gods.The athlete, as an athlete, and

5 In fact, their notion of competition is rooted in such a striving toward excellence, as implied in the etymological roots of the word (from the Latin, com-petere: “to strive together (toward excellence)”), as opposed to a “surpassing others” or “beating the other down at all costs” mentality.

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the philosopher are following the same practice, though with different drives. It’s the same thing that doing mathematics was not aimed at solving daily, everyday, problems . . . but to be closer to the gods. (Ascher & D’Ambrosio, 1994, p. 38) Thus, in naming a love of puzzles and games, the teachers intimate the Ancient Greek conceptualization of mathematics—a conceptualization that could be said to receive their need for intellectual challenge and play. For them, mathematics is far from the impenetrable “stainless steel wall” of the earlier account, and instead, calls forth for participation and involvement.

Social Connection Another quality that would spur participation and involvement, cited by many of the teachers, was the importance of communication and interaction with others.6 For

6 Although I could not generalize from such a small sample, nevertheless, the strong female bias (seven of ten female teachers, and only one of eight male teachers) in raising the importance of social interaction stands in line with the work of feminist scholars Belenky, Clinchy, Goldberger, and Tarule (1986). In Women’s Ways of Knowing, they propose a theory of how women’s epistemological development moves through a series of five stages, from passivity to empowerment and integration. In the fourth of the five stages—called “Procedural Knowledge”—the authors propose that a person begins to develop procedures for “acquiring, validating, and evaluating knowledge claims” (Goldberger, 1996, p. 5) in one of two ways. They call these “separate” and “connected” knowing, of which it is the latter that they speculate as likely being more commonly practiced among women. Whereas separate knowers come to know through impersonal, disinterested, “objective” lenses, such as through a discipline, connected knowers come to know through the lens of another person—that is, through empathetic interaction with another. Although the authors point out that both types of knowers stand to benefit from group interaction, it is the connected knowers who would, in fact, likely require the “intermediaries” of other persons in interacting with an “impersonal” subject such as school mathematics (unlike say literature with the personalities and voices of the authors). Also, Belenky and her colleagues refer to the connaitre (“personal”, “intimate,” and “involving equality”) and savoir (“separation” and “mastery over it”) distinction, as well as the differences in understanding and knowledge, as being allied with the connected and separate ways of knowing, respectively. Although there are strong parallels and overlapping ideas between what they have written and what I am writing about here, the primary differences rest in (1) the fact that they speak of both ways of knowing as “procedures,” whereas my treatment focuses instead on the phenomenon of relationship to other, and would just as easily admit the experience (the undergoing) as well as the lasting outcome in personhood (Chapter 4) as part of the phenomenon just as much as the procedure or “work” aspect (see Clinchy, 1996, pp. 209–211); and (2) their epistemology is founded on Peter Elbow’s notions of the “doubting game” and the “believing game” as underlying separate and connected knowing, respectively (see Elbow, 1973). To play the “doubting game” is to enter a situation, more or less, as a skeptic, whereas to play the “believing game” is to enter an encounter ready to empathize, or to see the situation from the other’s perspective—that is, ready to believe. On the other hand, when I speak of having a sense of relationship, or connection to a subject such as mathematics, I make no distinction whether one’s orientation to other is as skeptic or as empathizer, but in whether there is the struggle, the

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some, their relationship with mathematics was shaped from as far back as they could remember through interaction with family and classmates, while for others, eventually being able to “talk math” led, in varying degrees, to a transformation of their relationship to the subject. For many of this latter group, mathematics until some point was a subject that did not afford social and/or personal interaction, thus disconnecting math with who they were as social and empathetic beings. One way of grasping the importance held by some for social interaction is to think of it as a need for empathetically coming to know others, as in who they are, how they think, and so forth. Consider the humanities for example, say music or literature. Studying and talking about a prelude by Chopin or a poem by Poe involves interaction with both the artist (as person) as well as the person that one is in conversation with. In listening to a piece of music or reading a poem, one comes to know not just the piece but the artist behind the work. Likewise, in listening to a fellow classmate’s thoughts and feelings on a piece of music or a poem, one cannot help but hear something of the other’s own worldview. There is plenty of humanity—of self-expression and coming to know others— involved. In contrast, mathematics as usually presented constrains self-expression, affording conversations that focus less with knowing how others think and feel, and more so with following and executing the rules of mathematics. If there is a social element, it tends away—and not toward—mathematics so that one can usually distinguish between discourse on mathematics and socially bonding discourse. In contrast, consider Patricia’s portrayal of an important social and mathematical interaction with a colleague, and note where she places the real intrigue and captivation for her: I took discrete math a few semesters ago, and I really enjoyed what I was doing . . . And every now and then, I’d run into these walls where I couldn’t see a thing. It might not even be a hard thing. It could be an easy thing. I would go to Jim . . . But I wouldn’t listen necessarily to how he was helping me to do the math. It was almost like I was instead looking at how he was thinking in a more like, “What’s he doing? What’s going on in his head?” . . .

“emotional work,” or the investment of will, and its subsequent undergoing, “aha,” or moment of grace. A knowing that does not touch the personhood of the knower in such a way, I would relegate to a savoir way of knowing, or to a fragile sense of relationship. In contrast, a connaitre way of knowing, or a sense of relationship with other, as I have been describing it, can arise out of both the procedure of separate and connected knowing. For example, when I work to make fine distinctions in thought or argument using disinterested reason, I do not experience distance from the ideas, but in fact and in an ironic vein, I experience a closeness and intimacy with the ideas. I seem to extricate my personal feelings from the act of cerebral engagement, yet in the experience of coming to an understanding, I am returned to my feelings in the form of joy, confidence, and satisfaction.

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I remember one time, I was going through my notes and trying to tell him how I could do this and that, and all of a sudden, he’s going, “Well, why here.Why are you looking at this? You can think this through.What are you thinking about?” It was almost like by working on the problem together, I was almost getting insight into his brain, into his thinking process, into his ability. It was like gaining insight into someone’s thought process. To watch it, it was almost like . . . it was almost a different road, a different path—I didn’t even know people even thought like this . . . It was almost like seeing into a different world . . . just like all of a sudden, it was like this tunnel had been opened, and like there’s a hole there, like if I drilled a hole right here, and there was a secret garden over there. It was like going through this opening, and seeing this whole other way, this whole other parallel universe . . . I could like suddenly see through this opening into Jim’s thought process . . . It felt like an opening, like another path to explore . . . I’m seeing into his brain, almost. Patricia’s interaction with Jim appears to defy categorization as being either strictly mathematical or just social. The mathematical conversation does not entirely force out an intimately personal element, nor are they simply chitchatting away about their personal lives. But rather, the mathematically driven interaction is allowing for Patricia, in fact, to better know Jim’s person through increased familiarity with his mathematical thinking process. Her comments point to a growing familiarity with Jim’s “thought process” and his “brain,” but also to a “different path,” a “different world,” even a “secret garden.” If one flips the Heideggerian notion that every mode of being in the world is a way of knowing the world, then we can say that for Patricia to come to know the way Jim knows and thinks about mathematics is to come to know his mode of being in the world—that is, to enter his world—a world, whose discovery for her, is of obvious delight and satisfaction. Contrast the richness of socializing experience of Patricia’s description with the earlier description of mathematics as a “stainless steel wall.” Somehow, mathematics and mathematical activity as experienced by Patricia allows for her to bring forward a need and/or disposition for empathetic, personal interaction (or one could say that that aspect is elicited through the mathematical activity). This is to say that her social being is afforded an intertwining in her interaction with mathematics (or that version of it), whereas it is likely not for the student who sees math as “hard,” “cold,” and “impenetrable.”

Curiosity Chris: That old cat killer, curiosity? Yeah? Something so deeply embedded in our psyches that it screams to us from ancient myths of Pandora, Eve, Lot’s wife.

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Joel: Eve lost paradise. Lot’s wife was turned into a pillar of salt. Chris: Hey, knowledge doesn’t come cheap my friend. Good or bad, curiosity is woven into our DNA like tonsils, or like the opposable thumb. It’s the fire under the ass of the human experience. (From the television show, Northern Exposure) The most vital and significant factor in supplying the primary material whence suggestion may issue is, without doubt, curiosity. The wisest of Greeks used to say that wonder is the mother of all science. ( John Dewey, 1933, pp. 30–31) For a number of teachers, curiosity and its related traits (e.g., a desire for understanding and for inquiry and exploration) stood as key attributes, that when able to be interwoven into their interactions with mathematics helped forge deepened interest in the subject. For the sake of this discussion, I will draw a distinction between two brands of curiosity that in actuality may overlap and intertwine in important ways. The purpose for the split, though, is to tease out a particular nuance that can easily become obscured, especially, in mathematics education. The first manner of curiosity is what I would refer to as the desire, and sometimes even the need, to understand. It is that tendency to ask “why?” It is a kind of curiosity that Thomas Green (1971) has called, “the capacity to wonder how and why [which] has its roots in a kind of ignorance. It stems from our recognition that there is some feature of the world we do not understand but, given time, can comprehend. The capacity to wonder how or why stems from the fact that our knowledge is incomplete” (p. 198, as cited in Brown, 2001). But also, because such curiosity is based upon incomplete knowledge, it is also usually satisfied, and thus satiated by the obtaining of the missing knowledge. Whereas this type of curiosity may be motivated by not knowing, there is another—a “curiosity for curiosity’s sake,” if you will—that is rooted in the desire and need for risk and for exploration. In fact, “curiosity” may be a kind of misnomer. Instead, it might be called a disposition and need for adventure, exploration, creativity, inquiry, and risk.7 Unlike the first kind of curiosity, it would likely not be dependent upon whether any missing knowledge was had or not. Whereas the former is about understanding a given situation, the latter is more about exploring, or creating new (and ungiven) terrain. The former has as its end,

7 Brown (2001) refers to the work of Green (1971) in contrasting “wondering how and why” with “wondering at,” the latter, which Brown describes as “essentially an admission of amazement” (p. 105). Although what I am calling a “disposition and need for adventure, exploration, creativity, inquiry and risk” leads to similar ends and arguments as what Brown does with “wondering at,” I have chosen my own long and awkward phrasing for what I consider its explicitness of description.

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a grasp of what is; the latter concerns itself with what might or could be. In regards to mathematics, the former would be akin to understanding a topic such as calculus “inside and out,” answering all the “whys,” whereas the latter would be closer to doing actual research and inquiry in the subject.8 Or, the former could be associated with solving problems, whereas the latter is more about looking for the problems themselves. The point here is that both fall well within the conventional meaning and image carried of what it means to be “curious,” while at the same time, there is clearly some divergence between the two. Because of the heavy emphasis in mathematics education on “understanding” a piece of mathematics, I believe that most educational discourse leans toward the former brand of curiosity that would have “understanding” as an end, in the process overshadowing the latter kind that would involve exploration, inquiry, creativity, and risk, not just for the sake of understanding, but for the sake of—well—curiosity itself. Naturally, the two can work in complementary fashion, and most likely do in many instances. For now, I would like to illustrate how each aspect of curiosity can “intertwine” with a teacher’s relationship with mathematics. The latter brand of curiosity will be implicated in the discussion on “doing” mathematics, the topic of the next chapter.

Curiosity 1 Early in my interview with Nadine, an elementary teacher, she mentions her prior “math anxiety,” and of how she “had built a wall” with mathematics, recalling an earlier description of mathematics as a “stainless steel wall.” When I ask where the “wall” may have come from, she responds: A lot of it, teachers just want students to accept this, do it this way, and be done with it . . . I have vivid memories of asking questions and just getting, “Don’t worry about that. Just do it this way.” I wanted to understand . . . how that happened, and why that worked. And all that sort of thing . . . but I never considered that as a failing on the part of the teacher. I considered it a failing on the part of me. “Could this be another experience where nobody else is asking these questions—everybody else must get it. It must be me!” That’s probably as an adult and as a teacher now, I have grown to understand this . . . I’ve taken a lot of personality profiles in my adult life that showed that I am a question asker, an analyzer, and so part of that was matching my personality with these teachers’ personalities that just needed to get through the subject: “Don’t worry about that.” And that’s not the way I work.

8 Note how most teacher preparation programs concern themselves with the former, and less so with the latter.

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Yet, when she is given an opportunity to understand—in a problem-solving context—her reaction to mathematics shifts— Well, it was very exciting, almost—even if it was just straight math, it was fun. It was like winning at any game, like this giant puzzle. Okay, I just put the last piece in. I won! So it was very exciting to really understand what I was doing, and then for whatever reason there is about myself, or about people in general, when I understood what I was doing, the memorization of how to do it just came from that. To offer an interpretation of Nadine’s narrative, an aspect of Nadine is the person who “analyzes and questions,” and so tends to ask “why” questions. In her mathematical training (prior to teaching), she was not given the opportunity to bring that part of her selfhood into interaction with mathematics. This is to say that the vision of mathematics as presented to her did not allow for involvement with this strand of her being—this curiosity. Thus, an integral part of who she may be as a person was neither elicited nor received by mathematics. Her description of building “a wall” then aptly captures the lived experience of her way of relating to mathematics. Her former description also hearkens to an image of will without grace. Yet, when she is given the opportunity to ask and to understand the “whys,” she finds the experience exciting—reminiscent of will that would join with grace—and also liberating, of the curiosity in her which would now be given play in mathematical activity. Further, she finds that images that she had of herself as having math anxiety are replaced instead with a self-image of being a competent problem-solver. In fact, it was in “problem-solving” that the part of her that questions, analyzes, and so on was given voice (within a mathematical context). For those, like Nadine, who seek to understand and to ask “why,” then, being presented a vision of mathematics that would honor such tendencies could very well be experienced as both stimulating, as well as receiving of such aspects of the person, thereby leading to a state of intrinsic motivation, as I have argued.9

Curiosity 2 Patricia, current teacher educator and ex-middle school teacher, describes a time when as a graduate student in mathematics education, she explored composite functions—

9 Some mathematics educators, including Hiebert et al., (1997) who write, “Understanding is also important because it is one of the most intellectually satisfying experiences” (p. 2), would likely argue that the need or desire for understanding is a universally cherished trait or tendency.

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I can remember we were examining families of functions, and we were doing transformations of families of functions, and we started playing around with what would happen if we multiplied these together, what would happen if we did this to this, and part of it was this collaborative experience . . . the two of us together just saying, “What if we did it this way? What if we did it that way?” And just that moment when we both just got it. I think we were looking at a composite function—we were looking at graph forms: “What happens in a composite function?” That idea that one function is more powerful in a composite function, and it’s going to be the one that drives the resulting graph. Note how she and her partner are experimenting, asking questions, playing with different mathematical scenarios. Note also the “aha”—the experience of understanding—that sweeps over her and her partner. Without prompting from me, what had primarily been a descriptive account turns reflective: You know, when I had done composite functions in calculus, it was just sticking this in here or there. And I used to love that symbolic manipulation, and I didn’t see any point in it, but then all of a sudden . . . realizing, “I don’t just memorize this?” And it’s not just step 1 to step 2 to step 3, but it is . . . it is an exploration. I guess that’s interesting that that word exploration was always an appealing word to me. I always considered myself a person that loved adventure, loved to be outside. To me exploring math, being outside, taking hikes, walking up and down . . . It was suddenly that idea of seeing the mind as exploring . . . It’s funny because . . . it just hit me that exploring is a big part of it. I thought it was more about an affinity for beauty . . .There are lots of problems I don’t get finished, but I enjoy doing. The journey is as good as getting there. And I can just let a problem sit and come back to it. Patricia portrays her love of “adventure” and “exploring” as being a common thread of interest among different activities, including mathematics. Being able to engage mathematically where this love of exploration is given play in that engagement offers meaning and satisfaction to her—it is a motivating factor. The part of her that has an affinity for exploration and inquiry is afforded an intertwining with mathematics. In such a way, she is into the mathematics (of composite functions), and in some ways, the mathematics is in and a part of her (to the point of her being able to recall the excitement and importance of an experience from some time ago). Thus, as with the other individual portrayals of inter-esse, the “intertwining” of Patricia’s curiosity and math suggests not only a motivation for further engagement, but also an image of a viable, resultant relationship between Patricia and mathematics—a point to which I will return to later in this chapter when I focus my discussion on what results from the joining of will and grace.

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Certainty and Cleanness of Thought I was good at integrating and differentiating things in a mechanical sense. Somehow I liked it. I kept fooling around with it . . . There was something about abstraction. I liked the cleanness, the security of ideas. (Paul Halmos, in Albers & Alexanderson, 1985, p. 123) The family of traits associated with logical necessity (certainty, cleanness, and also exactness and precision), as mentioned earlier in my illustrative example, was referred to by a number of teachers as part of the initial draw to mathematics for them: It was like getting your dishes done. It was clean; it was neat. They were put away. They were done. You could do it over and over again. When I was younger, I sort of liked the cleanness of it, the uncomplicated simplicity of math . . . Mathematics was a “clean” science, where chemistry was a very messy science. I, too, might supply my own experiences in regards to this trait, as much of my enjoyment of mathematics as a K-12 student came because it was so clean, and in particular, because it felt so divorced from the messiness of opinion and interpretation. As an immigrant who struggled initially with the English language, I felt empowered by this subject that seemed not to depend upon rhetorical and artistic capabilities (which I felt I sorely lacked), but simply upon “clean” execution of intricate procedures and rules, and sometimes cleverness. A good deal of mathematics for me in my K-14 education was fairly mechanical in nature—in a way that I liked. Like “getting your dishes done.” One could sit with only pencil and paper and work on some problems—chug away like a machine while singing along to a song on the radio.The utter simplicity and cleanness of “accomplishing” so much without too much mental strain felt satisfying to me in its own way. In comparison, writing an essay—whether in my native language or in English— required so much thinking and deliberation. So much high anxiety! Yet, I would eventually experience a shift in the nature of my relationship to mathematics, as already alluded to in the preface and the introductory anecdote that opened the first chapter. It was not so much that I stopped valuing clarity and even the satisfaction of “clean” execution—nor have I lost the enjoyment in completing fairly menial tasks—but I inevitably sought more from my relationship with mathematics. I sought a place within which I could bring to bear my own sense of curiosity, sociability, and appreciation for beauty, among other aspects of my developing self. As I changed, I needed for mathematics to change with me.

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Similarly, a few of the above-quoted teachers would make explicit mention of a transition, though different in their details, in their manner of relating to mathematics as well. Because these narratives of change—in the way our interests interact with mathematics—surface the seeming malleability of mathematics as experienced and perceived by a learner, and ultimately because this has to do with issues of motivation, I will briefly explore Lewis’s story so as to shed light upon this phenomenon.10

A Case Study of Lewis, and of the Different “Faces” of Mathematics Lewis taught mathematics at high school level for a few years before becoming a mathematics teacher at the community college level. He began his interview by relating his early satisfactions with mathematics: Lewis: Math was sort of satisfying because it made sense in a satisfying way, and I appreciated that, where I could take other courses, like history; and you could have different points of view, and you could both be correct. It was just a matter of how you chose to slice the bread, you know, but in math, particularly at that level [of ] x + x = 2x. There wasn’t too much. That was satisfying. But then again, maybe it was satisfying because my dysfunctional growing up years, I was looking to do something that was that way. Me: You mean like, it had a certainty? Lewis: Yeah, some certainties. Some reassurance that tomorrow the answer to this problem would be the same if it was correct. Tomorrow, the answer would be the same as the answer today. It’s not going to evolve and change. Later in the conversation, Lewis mentions his experiences at Stanford. As a student of both George Polya and Morris Kline, and through exposure to a problemsolving approach to mathematics, his satisfaction with mathematics as a place where he could expect the “same” right answer day after day begins to give way to an appreciation for the multiplicities inherent in mathematical problem-solving. In a similar vein, teaching students from other cultures, and thus being confronted with different approaches to problem-solving, leads him to broaden his scope on the nature of mathematics. Contrary to his initial description of mathematics as a place where “it’s not going to evolve and change,” later in the interview, Lewis relates that for him, “math is a human endeavor that is evolving even as we sit in a class and take a

10 This “shift,” I would conjecture, occurs not just in relation to this particular trait of certainty, but to other traits as well.

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class.” When I point out the contrast between his early and later views of mathematics, he responds— You’re right . . . So, in some cases, a problem can have different interpretations and have different answers, so that’s another aspect of math that I would bring out today in my teaching, but earlier, my response was in where I was at that time. And at that time, the reassurance and the stability was probably something I was struggling [with], psychologically.11 I would call attention to how Lewis looked to mathematics, initially, as a discipline that would offer the satisfaction of a sure answer that could not be contradicted by personal opinion, or change with the passing of time. Mathematics presented for him a place where such a “psychological” need was received. That is, his early experiences with mathematics afford an entry point for Lewis of relating with mathematics—albeit a “dysfunctional” one perhaps—but, nevertheless, one that could mature as his own psychological needs developed. Yet, creating new inroads into the world of mathematics as one matures and progresses through various adult “developmental stages”12 so as to gain a new vision of mathematics—or a renewed commitment to it—is a deep and arduous challenge (see Buerk, 1982, for a similar argument, specifically, for women learners of mathematics). As it stands, as with Lewis, the process is most likely a matter of personal and professional determination and luck—for the most part, a hit-andmiss (or)deal. The point, pertinent to the larger discussion, being that motivation framed in

11 With its eyes on pedagogical implications, the conventional interpretation of Lewis’s narrative would look favorably upon the shift in his conceptualization of mathematics. In W.G. Perry’s (1981) terms, he has moved from a “dualistic” (right or wrong) to a “multiplistic” (different interpretations) conception of mathematics. Certainly, for a teacher to present a “dualistic” version of mathematics to his or her students would bar entry for those not “in line” with the “correct way” of thinking, whereas a multiplistic view would likely open up the mathematical dialogue for a greater number of students. Thus, thinking in terms of the pedagogical ramifications, one can easily cast his early relationship to mathematics as less than desirable. I should further note that a mathematics teacher’s move from a “dualistic” to “multiplistic” view of mathematics is not so straightforward as that of say someone involved in the arts, where one can, in fact, go from a right/wrong mentality toward one characterized by multiplicity, say in the form of an interpretive turn. Although there are certain topics and results in mathematics such as Gödel’s incompleteness theorem, Cantor’s ideas around infinity, Euclid’s parallel postulate, the continuum hypothesis, or in a larger vein, axiomatics itself, where something akin to the arts “version” of multiplicity occurs, most of mathematics as experienced by most mathematics teachers and educators still lead to starkly right and wrong answers. Thus, the form of “multiplicity” as commonly encountered in mathematics, as alluded to by Lewis and a few other teachers, is not one of relativism, or a multiplicity of correct answers, but of a multiplicity of solution methods or paths toward the one, or set of (still) correct answer(s). 12 Or it could be the case for some that as one’s view of mathematics changes, one’s development itself is brought along. Here is a variation of the old saw of the Piagetian (see Piaget, 1970) “development pulling the learning” versus the old Vygotskian (see Vygotsky, 1978a) “learning pulling the development.”

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terms of an intertwining is not a static conception. As the person changes, the interplay between self and mathematics must also co-evolve somehow for there to be continued “motivation,” which is to say that mathematics (as perceived) must also change. But as can be seen in Lewis’s narrative, there is a certain adaptability to mathematics itself, in how it appears to support both the need for certainty and the urge for multiplicity, in how it can shift in its perceived and experienced embodiment, from one guise to another. It is not just in Lewis’s account, but as I listened to each teacher’s narrative, I was struck again and again by the divergences in how one could be engaged by mathematics, and by how large mathematics could be, in accommodating and absorbing—as if adapting its interface with, or shifting its appearance to—the many divergent inclinations and human needs as expressed by the teachers. And of course, in some cases, I noted how it did not.

Self as Reflected in Life Projects and Trajectories One is reminded of Maturana’s notion that a person in action is a node in many simultaneous and interpenetrating “conversations” (of which “mathematics” is only one). (Thomas Kieren, 2000, p. 229) One way to conceptualize an adolescent’s lack of curricular engagement (an alternative to student “laziness” or “lack of ability”) is to suggest there is a mismatch between the students’ sense of their trajectory and the curricular content of the course. (Daniel Chazan, 2000, p. 50) Until now, I have been describing an “intertwining” of self and mathematics, in terms of the dispositions and needs of self in relation to mathematics.Yet, another way to speak of the “self ”—or to use the common parlance in educational research, “identity”—is to consider the life project or trajectory of an individual as reflecting, if not embodying, some aspect of this “self.” To give an example, consider the teenager who has always been “good” at mathematics, and who envisions a future for herself in a profession that requires a certain degree of mathematical competency, such as, say, engineering. The story that she tells herself, and possibly others (and others may tell about her to herself and to others) is that “she is good at math” or that “she will become an engineer,” and so on. This is an aspect of who she is. She may also have other strands of a life trajectory, involving for example her personal life. The point, though, is that if one were to approach her and ask, “Who are you?” her response would likely include some narrative involving her mathematical proficiency, inclinations, and ambitions. That is, she is (at least partly) the trajectory upon which she is traveling.

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Yet, because such “life histories and trajectories” can be told—that is, externalized in narrative—these stories that are told about the self, either by oneself or others, also might be looked upon as being reflective of what I have been calling “self.”13 Or stated more rigorously, such stories can be thought of as—depending upon the philosophical persuasion—either who the person is, or reflective of who the person is. In either case, we have a way of talking about the “self ” as more than an abstraction, and instead as a narrative around an individual’s life project. To return then to the central concern thus far—of the relation of “intertwining” and motivation—I would now re-approach this same intertwining by viewing the self through the narratives of life projects and trajectories. I begin with an illustrative telling of a classroom episode by Michael, a college professor of philosophy and education: In high school I did fairly well in math but was not very attracted to it. I associated math with school algebra and geometry. But in grade 12 we had this youngish teacher who seemed to embody mathematics in a manner that was truly fascinating. I recall this one day in particular. The math teacher is writing a differential equation of some sort on the blackboard in order to show how and why a particular formula needs to be employed. And we are about to work through it together as a class. We have become used to his approach that appears quite improvisational, almost whimsical. The teacher tells us again that he is less interested in the calculation than in the concept on which the equation is based. As he finishes writing the equation with chalk on the board I already feel a pleasant anticipation. I love his casual style. While writing the teacher tells some story. I am mesmerized by his calligraphic writing style and the movements of his hand. The letters and numbers almost appear playfully on the board— not at all like a dreary math lesson. Then the teacher steps back and looks with a mischievous smile over his shoulder. He returns to the board, smiles again and erases and changes one letter. “Just testing,” he jokes, “glad you caught that one—we could have been here forever.” We all laugh with him. The teacher explains the several parts of the underlying formula and that we will need to test it. He steps 13 By my phrasing, I recognize that I am veering somewhat close to the philosophically unpopular, and perhaps unsound, “essentialist” position that would assume some underlying entity called the “self ” as reflected in narrative. My intent, in not attempting to be explicitly nonessentialist, is primarily for communication purposes, as I believe the essentialist view is in fact closer to what Bruner calls “folk knowledge.” And so, by loosening the discourse in regards to the philosophical rigor around the constitution of the “self ”—in considering narrative as reflective, as opposed to somehow defining such a “self ” —I am hoping to lead more smoothly for the nontechnical reader to the larger issue of whether such narratives (as reflective of “self ”) then intertwine with mathematical knowing. For those readers who are interested in a discussion of identityas-narrative, and of a project studying mathematics learners through that particular lens, see Sfard & Prusak (2005).

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back again, cocks his head sideways a bit, and adopts a waiting posture. We all laugh at his playful demeanor. Math has become some secret text that we are all helping to decode.The teacher goads us on to risk ourselves with making suggestions as to how to reduce the equation. Soon the board is full of letters, numbers and lines and crossed out portions. Individual students call out suggestions or corrections. And the teacher jokes, thanks us, keeps asking further questions. We are slowly working through the equation but as we get to the end something appears wrong. This is not the correct answer. We all cheer mockingly. But the teacher grins his enigmatic smile and shrugs his shoulder:“Well working is for the stupid. The answer does not really matter.” Not only is my admiration for this teacher unshakable, I thoroughly enjoy this moment. I love this math. It is for thinking people and I want to be a thinking person. Then, suddenly, one of the bright math students puts up his hand: “Sir, I think that the formula you are trying to test is wrong to begin with. Should there not be an x square in the denominator?” Now the teacher seems somewhat uncertain and visibly embarrassed. “Well, let’s get back to it tomorrow,” he says. As I walk out of the class, math seems slightly less glamorous. Similar to the earlier portrayals where mathematics either engaged or failed to engage the various needs and dispositions (as aspects of the self), Michael’s story is illuminating in light of how his life project of becoming a “thinking person” is at first given affordance, and then later moderated by the happenings in the classroom. One might point to the lack of confluence between Michael’s life trajectory and what generally occurs for him in mathematics classrooms (this episode notwithstanding) as striking at the root of his un-interested attitude toward mathematics (“In high school I did fairly well in math but was not very attracted to it”). That is, his ultimate lack of motivation toward the subject is synonymous with a lack of intertwining between his life trajectory and (his study of) mathematics.14 On the other end, recall Lewis, the college math professor I interviewed from the last section. His love of the “human” in the humanities becomes apparent when he describes his attraction to the study of history, and also of traveling and acquainting himself with the cultures and histories of the places he visits. His interest in mathematics, in turn, takes on a shape that brings in his own interest in the human, captured partly by his privileging of the beauty of human reasoning in mathematics, as a teacher. Further, coming to understand—so that he might bring

14 A number of mathematics education researchers have considered life trajectories in relation either to motivation or to the nature of students’ engagement with mathematical ideas, including Chazan (2000), Peressini et al., (2002), and Sfard & Prusak, (2005).

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alive to his students—the historical and human aspects of mathematical thinking and achievement becomes a key impetus for him as both a learner and teacher of mathematics. One might say that by intertwining his own interest in the human and the historical in with his understanding of mathematics, Lewis is able to foster a truer interest in the subject of mathematics for himself than if he had not.

A Case Study of April: On the Life Project of Becoming a More “Effective” Teacher Although I would be broaching upon a theme from the next chapter (in its emphasis on the life path of being a teacher), I want to offer one other illustration for how a life trajectory and mathematical trajectory can intertwine. The purpose will be to give fuller shape to the image of intertwining trajectories while also reiterating its link to the motivation to engage with mathematics. In the latter part of this illustration, I will also be transitioning into the second query of this chapter, that of beginning to characterize the resultant knowing from the joining of will and grace. I would take the case of April, an elementary teacher. The trajectory, or life project, in question is that of being a teacher. April began as an untrained teacher with an interest in reading. Meanwhile, her “mathematical” trajectory—separate from her concerns as a teacher—could be characterized for the most part as being relatively uninvolved. She describes her own mathematical learnings as having been mostly “easy” and “uninteresting,” and at times, even “boring.” By the time she becomes a teacher, it is no longer “a puzzle” to her, at least so in the context of her teaching. Her interest in the subject, though, is ultimately “awakened,” not through advanced coursework in mathematics or through other conventional avenues of interacting with the subject, but through her concern and curiosity about her students’ learning— And then I think I sort of woke up to math. I thought, “Wait a minute! How are these kids understanding math?” . . . “How do children come to understand—really understand—math?” Once she brings her focus upon whether the children are “really understanding” the math, April begins to recognize and acknowledge her own shortcomings as a teacher— April:

It was so clear to me that the kids weren’t learning anything much, when you taught the old-fashioned traditional way. And so I thought, “I have to do something about this because they’re not learning.” And it’s obvious to me that I could teach this stuff over and over and over, and they still don’t get it.

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When you say the “old traditional way,” what are some of the characteristics that you’re specifically thinking of? You know, if you were teaching subtraction, the teacher would stand at the board and say, “Here is a subtraction problem, and here is how you do it. Put the big number on top, put the little number on the bottom. Start over here with the ones. This takeaway this, can’t do it. Cross this out.” And I was really good at doing that—I mean, I’m really good at delivering that kind of instruction, and the kids still couldn’t remember it—I mean some of them could, but a lot of them couldn’t. I always felt like it was about a third of them that couldn’t remember it, and didn’t know what they were doing. And it made me feel like, “obviously, I’m a failure here. This isn’t working!”

But to teach differently, and more effectively, would require further and deeper learning of mathematics itself—a task to which she applies herself.15 As such, it is the desire to teach in a way that “works”—this “teacherly” trajectory—that becomes the motivation for April to further her understanding of mathematics. Unlike Michael, who does not find in mathematics a subject for the “thinking person,” April finds in mathematics a subject through which she can find the thinking persons, but within her students. Her eventual interest in mathematics, and its meaningfulness to her, is interwoven with her “teacherly” trajectory of someone who is “intrigued” with her students’ thinking. Put differently, her motivation toward mathematical engagement dovetails with her life project of becoming a more successful teacher. Of course, this is not to say that her relationship to mathematics is chiefly instrumental. Although it may be motivated by the desire to becoming a better teacher, coming to understand a piece of mathematics through effort and struggle carries the excitement and the experience of understanding—that is, the joining of will with grace. She still carries a genuine enthusiasm about mathematics. When I ask what it is like for her to deeply understand a piece of mathematics, she responds—

15 Mathematics educator, Deborah Schifter, points to the need for such mathematical development, as a teacher attempts a shift in pedagogy that would be centered on student thinking: “An algorithm-based instruction whose goal for students is accurate recall of computational procedures does not produce in its practitioners the need to continue learning. By contrast, the practitioner centering mathematics instruction on student thinking must continue to develop as mathematical thinker—making new mathematical connections as he or she engages students in exploration and discussion; as manager of classroom process—inventing new ways to leverage students’ powers of reasoning as he or she is confronted with their surprising notions; and as monitor of student learning—discovering new aspects of children’s mathematical thought” (Schifter, 1996b, p. 4).

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Yeah. It’s amazing. It feels fabulous to . . . find out that [various mathematical ideas] make sense, and that I can actually figure it out for myself . . . It’s an incredible feeling of power. It’s just so amazing that . . . I thought that if when you were trying to find the area of something you multiplied the length times the width, and you got the area. And so I thought that you multiplied a line times a line, and magically it turned into space. I had no idea what was going on at all. And when I realized that, I thought, “I can’t believe I thought that!” I thought, “You did line times line, and it changed into space.” I didn’t really understand what was going on at all. It’s really incredible. Stories such as April’s show how an intertwining of a life trajectory or life project with the mathematical learning can lead to deepened interest in the subject. Yet, such an interest, I would argue, also captures the shape of what results through a process of entering into relationship. It describes the consequence of the joining of will and grace—not just the motivation for engagement, but also the result of such commitment. Consider, for example, how April characterizes the nature of “understanding” a piece of mathematics— [It’s when] I really understand the math so well, that whatever any kid says, like today, when he said the thing, I can sort that out in my mind and know how to work with it. I think that’s how I know. In fact, I know that I understand prime numbers and multiples and stuff. I understand that that way, much better than I understand polygons. I mean, a lot of the lessons in the book about polygons, I kind of understand why they want me to do it that way, but the whole time, I’m thinking,“I’m not sure what the kids are going to get out of this.” And so I don’t know until I see them working. Even like the thing that they were doing today with those power polygons, I’m thinking, “What exactly are they supposed to be learning from this?” I can’t see that they like doing it. I can see that it’s useful in some ways, but . . . I need to understand what the big picture is. So I kind of need to know what’s the big idea behind polygons that the kids are supposed to be learning that I’m supposed to understand too. And I’m not sure I understand it yet. For her, “understanding” a piece of mathematics is inextricably linked to her actions as a teacher. However, it is not simply that her understanding informs her actions. Instead, what it means to “understand” is defined for her in terms of how that understanding can be leveraged—or appropriated (to use a popular term)—in the classroom. More specifically, it means being able to sort out her students’ questions and comments around the topic in question. It is an “understanding” that is contextualized and situated in her actions as a teacher. It is different from “understanding” so as to pass tests, or simply to know something “inside and out” in some

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decontextualized sense. It is, instead, a knowing that involves who she is and what she does as a teacher in the classroom. In this way, the interweaving of her “teacherly” trajectory—or an aspect of who she is—and of her “mathematical” trajectory can be seen to capture more than just her motivation for mathematical engagement, but also her resultant knowing and relationship to mathematics. To close the chapter, I will now turn to closer examination of this claim, of intertwining trajectories—or inter-esse tinged knowings—as depicting the consequence of the joining of will and grace. In the process, I will also attempt to give clearer and fuller shape to such resultant knowings.

Inter-esse as the Joining of Will and Grace To draw the connection from will and grace to inter-esse, I turn to reconsideration of the colloquial treatment of the word “interest.” When I say that I am “interested” in mathematics, this speaks both about my potential motivation toward engagement, but also about what may be the result of certain interactions with mathematics, for if I had not had some favorable interactions with mathematics, I would likely not state an interest in the subject. Yet, such “favorable interactions” that would give rise to an interest in mathematics is just what I would characterize as the joining of will and grace. In other words, what is experienced as “favorable” is likely some form of grace mixed in with struggle, or an allowance for a bringing forth of will leading to an undergoing. Grace could be something as simple as the feeling of enjoyment and excitement, the surprise of contradiction to the “aha” of understanding, or the sublime experience of beauty, each arising out of some effort and/or struggle; and in terms of will, I have depicted in this chapter the accounts of some teachers whose interest in mathematics is elicited by their sense of challenge and strivings for excellence—a calling forth of their will that would eventually give way to the satisfaction of a challenge overcome. In these ways and more, “interest” does in fact suggest a consequence of will and grace. I would argue, further, that interest as inter-esse—that is, the intertwining between self and mathematics—gives a more concrete image of the resulting interest. Take, for example, the story of Patricia around her love of adventure and exploration (i.e., “Curiosity 2”). She is able to intertwine such aspects of her selfhood in with her relationship with mathematics, but these intertwinings no doubt come as a result of mathematical experiences that called upon exertion of will (effort) and likewise contain “undergoing” moments of understanding as in the one episode she describes. What comes of these mathematical engagements that would privilege her sense of adventure and curiosity is the interweaving of who she is with her vision of what mathematical activity can be; it is exactly this interweaving that describes “interest,” as I have so far suggested. Similar comments might be made of the stories and comments made by the other teachers in this chapter. The process of arriving at a relationship—imagined

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as an intertwining—with mathematics that would elicit or receive important needs and dispositions, or else align with a life project, would surely be marked by moments of connection spurred by struggle. In fact, the most vibrantly described intertwinings appear to have been those that were forged and arrived at—as opposed to say just fallen into by happenstance—indicating the effect of will and of grace. In this way, one could say that inter-esse epitomizes some essential element of what both precedes and succeeds the bringing forth of will and surrendering to grace. It is the state-of-being that circumscribes such moments of becoming. Said differently, the experience of connection and of coming into “true relation” (à la Buber)—i.e., the joining of will and grace—appears couched in a larger milieu of “interest,” of intertwinings between self and mathematics, that shapes and is shaped, in turn, by the quality of that connection.

Understanding as Inter-esse A squirrel is trying to figure out whether it is safe to cross a stream. To him, it is a raging current, and he will drown there. But the stream is only up to the fetlocks of a horse. This is not a question of definitions; it’s a matter of how to act effectively in the world. We must determine whether it’s appropriate for our platoon to cross the water. Here we cannot rely on information from the horse or squirrel: we must insert ourselves into the process of measurement, gauge ourselves in comparison to its elements.Thus what we are in any situation—for example, “able to cross” or “not able to cross”—is only partially a function of our relative size. What matters is how we fit the situation—whether we are big in that water.The answer we get will be very different from the answer someone else gets. (Sun-Tzu, 2003, pp. 83–84) The quote above points to the role of the self (or I ) in relationship with the world, and how it informs the world that is seen. Rather than assuming an objective and equally accessible world for all, it underscores the ways in which the world becomes unprecedented—and in turn the self as well—through differences in the way one stands in relation to the world, or to “other.” The point that I intend to establish in the next few pages is that knowing mathematics—at least so in an interested manner—also requires consideration of the knower in the knowing. It, too, is not simply about accruing an objectively definable lot of knowledge, but about recognizing the “fit” (as in degree of intertwining) between knower and known. I would proceed with the discussion, first in broad strokes, by recounting both Lewis and April’s stories, of their intertwining trajectories with mathematics.Their stories depict a relationship to mathematics that is particular to their own life projects, where through that particularity, their knowings could very well be said to have become personalized.This is to say that they know, or understand, mathematics in a way that is rather unique to their own life interests, and accordingly, each of

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them, indeed, has somehow become implicated in the knowing itself. Their slant on mathematics is no longer generic and objectively definable, and instead, involves who they are as more than just a nonspecific learner of mathematics. In allowing for the subject to matter to them as more than a distinct strand of interest unrelated to their other life trajectories, their own persons (in the aspect of their trajectories) have become entwined with their knowing of mathematics. To probe more closely and with finer detail into the issue at hand, I would now turn to two excerpts from my conversation with Richard, a former middle school mathematics teacher and current mathematics teacher educator. In the first, he explains an instance of coming to an understanding of triple integrals: Richard: I remember a physics course that was . . . an elective . . . and we got to choose, what we as a group of senior physics majors wanted to study, and we wanted to study cosmology. So we did a course in cosmology, and had to do papers. My topic had to do with the interstellar medium, and had to read a bunch of papers about that. And I remember discovering how important mathematics was to that. We were taking triple integrals to sort of try to get to some understanding of what the density of matter was of the interstellar medium. And I remember being fascinated by this application of mathematics. I remember thinking, “Wow. So this is triple integrals.” It made sense to me in the context of the problem I was writing about. So there was another interesting moment, as I think back. I became sort of re-interested, if you will, in mathematics per se. Me: As an application? Richard: As an application, and the power of mathematics in a context that was interesting. I was pretty interested in astronomy. In fact, I had entered college thinking that being a physics major was the smart way to becoming an astrophysics major in graduate school, and ultimately an astronomer. The straight interpretation (and my initial one in the transcript) would be to associate Richard’s excitement with finding a “real-life” application for the mathematics.16 But as Richard clarifies, it is in making “sense” of triple integrals in a context that was of interest to him and in alignment with his life project—specifically, of becoming an astronomer—that is significant to him. Note, also, how he qualifies his consequent response to the understanding as becoming “re-interested” in (or to editorialize, “re-intertwined” with) the subject. 16 Here is the kind of interpretation that works hand in hand with the tendency of some teachers and many textbooks to include “real-life” applications in the instruction for generating excitement and enthusiasm.

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Contrast Richard’s understanding with (1) a hypothetical learner who may have shared a similar learning experience around triple integrals but without the slightest life interest in cosmology, and (2) another hypothetical learner who may have the same life interests, but had come to an understanding of triple integrals only in relation to nonastronomical ideas. For the first learner, the experience as recounted by Richard would most likely not have the same degree of significance and meaningfulness. Learning about triple integrals in the abstract (though in connection with other mathematical ideas) may be just as relevant—and maybe even less bothersome—to such a person as learning about it in regards to the interstellar medium.With the second learner, s/he may arrive at some grasp of the concept of triple integrals, sufficient for most all purposes, but s/he would likely not carry the same degree of personal involvement and investment as had Richard. To use a set of terms I used in earlier chapters, s/he would know it in a savoir fashion, but not necessarily so in a connaitre-like manner. Something in his/her knowing would be less intimate, more distant than it was for Richard. Similar to a few of the earlier accounts, what Richard’s learning episode points to is the personal connection to an understanding that can occur when it interweaves a life trajectory in that knowing. In a second excerpt, Richard describes an understanding that in yet a different manner “personalizes” the knowing. There was a problem where I took a class in Analysis . . . [and the] expectation was that people would do it pretty symbolically. I actually found a geometric representation of it. Not just geometric, but actually that connected with some arithmetical structures that we had been working with—you know, in a simpler situation. So I presented it, not cause I was trying to be different. It was just my way of seeing this problem, and he actually showed that to the class. He said, “Here’s a rather unusual proof that’s completely different from the standard way of thinking about this.” He didn’t say “elegant,” but “very interesting in that it is so very different from the way that the professional way thinks about this problem, and [it] opens up another window on the mathematics of the problem.” It wasn’t as elegant or shorter, but it connected to other pieces of knowledge in ways that—I had used something that was near and dear to me. Something that I had used in my interactions with middle school . . . I had tried to really understand [the shorter symbolic proof ] every step of the way, but I don’t know that I really understood the internal . . .—it was never quite mine. I had memorized the steps, and tried to understand each one as deeply as I could. But boy, that was not the route I [would] have taken to get there. It’s a wonderful proof, but the ownership is a different kind of ownership. For Richard, it is not just that the “geometric” solution is his, and that the shorter method is not. But rather, his solution method uses “something that was near and dear” to him, something from his interactions as a middle school teacher. It

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came from his life (though mathematically informed) experiences. His solution may not have been as “elegant” as the “professional” one, but it was borne out of his struggles and his own mathematical history, leading to a different kind of “ownership”—as he tries to explain—than his grasp of the “professional’s” solution. Perhaps, in this case, it is not just ownership, but also relationship. In this, I am reminded of a comment made to me by a number of musicians, of how one’s life experiences can be brought into one’s musical expression and appreciation; and how for the true artist, it must be brought into play. Camille, a pianist, explained to me that until she had gone through certain life experiences— in particular, difficult ones—she could not understand Beethoven’s music, and thus, could not play his work as she eventually came to see how it should be played. Stuart, a choral director and pianist himself, expresses a similar idea as follows: It’s not just that we experience the sound, but what it’s doing is it’s resonating in our hearts with our own personal experiences. And so if we’re singing a song, or if we’re playing it, or even if we’re just physically hearing, it connects with a personal experience that we’ve had. And usually the painful ones. Or the most joyful ones: having a baby, or a divorce, or a death in the family, or somebody that we love that is sick, or you lost your job . . . The ability to connect with the inner struggles, the inner pains, the inner triumphs—the good things too . . .The more I play a song, or sing a song, or conduct a song, the more I become attached to it, or it becomes more a part of me. So then, it’s as if I develop a relationship with the music—so to speak. The sense of “relationship” for Stuart develops from the interweaving of life experiences and emotions with that of his interactions with music. Similarly, Richard’s interactions with mathematics that would afford a bringing in of his “outside” mathematical experiences might be illustrative of a knowing steeped in relationship. In this regard, I am reminded also of a passage by mathematics educator Stephen Brown (2002): Using the array of solutions to Gauss’ arithmetic progression problem, I had [my students] select what they thought were the most elegant ones. One student chose what appeared to me to be a very messy algebraic solution. When asked to describe what made it elegant, she confided that after spending a long amount of time coming up with a rather complicated way of combining odd and even terms for any progression, she wanted to find a way of honoring the process rather than streamlining it in favor of some final step. She wanted to find a way that would encapsulate how she had gotten there rather than the actual “there” was. In some sense then, she was interested in capturing the personal genealogy of the problem—how it unfolded in her own analysis. (pp. 4–5)

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In both Brown’s student and Richard’s accounts, the real interest for each is in knowing the ideas in a way where the self ’s struggle—or “personal genealogy” as Brown calls it—is interwoven in the actual knowing.17 The word that comes to mind in both narratives is “intimacy.” To be intimate is to be acquainted with the details in only the way that one can personally know, almost always arising out of experience. And this should not be confused with the notion of experiencing things kinesthetically or “concretely.” But rather, I mean that one’s interaction and learning be embedded in what might be called a “heightened sense of experience”—what I have been referring to as the joining of will and grace. It is what John Dewey called an experience—as to be distinguished from simply experience. Educational philosopher David Hawkins (1973), explains— Human beings are always engaged in something which William James called flights and perchings. Sometimes, though less often, they are not only momentarily engaged, but are engrossed, concentratedly. John Dewey celebrated the continuity and culmination of this sort of engagement with his famous distinction between experience and an experience, that which orders and unifies, which is consummatory, which brings enhancement in its wake. Dewey elsewhere saw this kind of phase as a precondition both of artistic expression and scientific achievement, coming only, as he said, after long absorption in a subject matter, which is fresh. (p. 10) In the case of our two narratives, what results is not necessarily artistic expression or scientific achievement—although the resulting excitement appears as palpable as if they were—but instead, what comes of their engagement is a knowing where their own persons are entwined in the story of their mathematical discoveries. One could describe it as the self being ensconced in the narrative that is the knowing. Yet, it would be just as true to say that the ideas and concepts that are known as such are in turn ensconced in personal story, where it is not just the finalized knowledge itself that Richard and Brown’s student are able to offer but also their personal interaction with the development of the idea. To recall the notion that the self

17 Here is a curious passage of relevance from Jacques Hadamard (1945) in The Mathematician’s Mind: “The historians of the amazing life of Evariste Galois have revealed to us that, according to the testimony of one of his schoolfellows, even from his high school time, he hated reading treatises on algebra, because he failed to find in them the characteristic traits of inventors . . . [Similarly, Jules Drach] always wishes to refer to the very form in which discoveries have appeared to their authors. On the contrary, most mathematicians . . . prefer, when studying any previous work, to think it out and rediscover it by themselves” (p. 11). What it seems that both Galois and Drach are after is the same sense of this “personal genealogy” of the mathematics, and not simply the cut-and-dried “results.” It would point to one way of somehow making more “personal” the ideas. But also, those mathematicians who would prefer to rediscover it by themselves may in fact be motivated by making “personal” their knowing in their own manner.

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could be spoken of as the narrative around a life project, then it is truly the intertwining, or confluence of two narratives—that of the self’s trajectory and that of the mathematical understanding—that would characterize an interested (as in inter-esse) knowing, or what I have termed “understanding as inter-esse.” Certainly, it is not mere coincidence that the etymological root of understand (“to stand in the midst of ” from the Old English understandan) evokes a mental picture similar to that of inter-esse (“to be between”). To recall Chapters 2 and 3, I examined the notion of understanding as both an “undergoing” experience and a “standing before” (of verstehen). What transpires from this process of understanding is a transformation of self-in-relation-to-other that would culminate in interest (or inter-esse): an understanding that would intertwine the personhood of the knower with the object being known. Thought of in this way, one could say that to truly understand is to become interested, and that to be entwined as such, is to be in relationship.

5 “DOING” MATHEMATICS AND ITS RELATION TO THE LIFE PATH OF BEING A MATHEMATICS TEACHER/EDUCATOR

This study so far has concerned itself with exploring a way of knowing that I have described as being steeped in relationship, or as a connaitre way of knowing. And although I have used accounts of mathematics teachers and mathematics teacher educators to bolster the image of these expanded images of knowing, in some sense, much of what I have discussed has not yet addressed the particularities associated with being a teacher of mathematics. That is to say, other than parts of Chapter 2 (i.e., Eve’s transcript) and Chapter 4 (e.g., Lewis’s and April’s stories), I have delved into this particular way of knowing largely at a level nonspecific to teachers. Meanwhile, the details of how that knowing interacts, specifically, with the life path of being a teacher have been relegated primarily to the background. In this closing chapter, I would like to reverse this trend, and foreground now the teacherly specificities as would pertain to having a “relationship” to mathematics. Put differently, this chapter might be looked upon as a kind of “applications” chapter, where I use some of the ideas developed so far to recast particular issues surrounding the personhood of being a mathematics teacher. I will do so, first, through re-exploring the topic of “doing mathematics”—a topic I broached in Chapter 1—and move from there toward characterizing a pedagogically relevant “doing” of mathematics as a form of mathematical engagement with the potential for bringing about a viable “relationship” between teacher and discipline. Of course, “doing” mathematics is already a prized activity, both among mathematicians and mathematics educators. In fact, it is what mathematicians do. Yet, in regards to mathematics teachers, I would argue that it tends to be valued more toward instrumental ends—specifically of knowledge accrual and teaching effectiveness—and less for intrinsic reasons, as a rewarding and meaningful endeavor in itself. This is to say that “doing” mathematics is something cherished more so for

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its effect on teaching (with teacher as savoir knower of the subject), and less so for its effect on the teacher, or teacher’s relationship to mathematics (which, naturally, could also impact teaching, but simultaneously bring the teacher’s personhood into consideration). My intent here is to offset what I perceive as an imbalance in emphasis by bringing the latter to the fore. I will attempt to do so, first, by outlining what I take to be the commonly accepted rationale for why teachers might value doing mathematics (an argument, which incidentally, I do not disagree with). From there, I will lay out an alternative justification for its practice—a substantiation steeped in the theoretical machinery established in the earlier chapters. Within this second rationale, I will address what I mean by doing mathematics, as say contrasted with studying or learning mathematics. Lastly, I will close the chapter by offering concrete images of what doing mathematics for teachers, as opposed to professional mathematicians, could be, and how these engagements hold relevance to the life path of being a teacher.

Why do Mathematics?—One Rationale I feel that I’m first and foremost a teacher, and then I’m a teacher of mathematics. (Beatrice, high school mathematics teacher) Feelings are not incidental inconveniences on the way to a better teaching practice; curiosity, pride, and affection and concern for their students are among the reasons teachers keep at it. (Deborah Schifter, 1993, p. 195) I begin with a clarification, or a redefinition: I take it to be that a mathematics teacher/educator is a person whose identity enfolds within itself (at least) two aspects: one, an interest in the subject of mathematics itself, and two, an interest in teaching, by which I mean a concern with pedagogy as well as with student learning, engagement, and growth. Thus, for a “mathematics teacher” there is involvement in mathematics, as well as having a kind of “love” for teaching, learning, students, and other “teacherly” aspects.1 The sentiment, as voiced by Beatrice (above), is one that implies such a conception, but specifically, with an emphasis on the latter aspect.2 Consider also the following two statements by educational researchers Joan Talbert and Milbrey McLaughlin (2002) in their study of US high school teachers:

1 One might argue that this separation is rather artificial, as it may be. My purpose in separating these two is for purposes of discourse and analysis. Incidentally, I do not claim opposition or mutualexclusivity between these aspects. 2 Incidentally, it was a sentiment voiced by a number of the elementary teachers I interviewed.

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Our research supports [the] claim that teachers’ ultimate career rewards lie in their success in engaging students in their classes. (p. 327) Incentives and rewards that matter most for teachers are their success with students, their engagement in course content, and the quality of their colleagues. (p. 328) In particular, I would point to the understandable importance, and meaningfulness, to most teachers of student engagement and success. Clearly, motivating and helping students to learn the subject is a prime motivator for many teachers in continuing their own professional development, both as pedagogues as well as learners of mathematics. We saw in the last chapter, in the case study of April, of how her goal of becoming a more effective teacher—specifically, her desire to better understand her students’ thinking—led to her further development as a mathematical learner and thinker, and in turn, of how “understanding” a piece of mathematics for her came to have a meaning particular to her teaching actions and to who she is as a teacher in the classroom. The point being that the “teacherly” aspect or trajectory of a mathematics teacher/educator should be accounted for when considering what kind of mathematics they know and how they know it. By “what kind of mathematics they know,” I am referring to the products of their knowledge, or what I have been calling savoir knowledge. In this regard, there have been important theoretical and practical advances in the field of general and mathematics-specific educational research. Educator Lee Shulman (1986) proposed a category of content knowledge specific to teaching in pedagogical content knowledge, which includes— for the most regularly taught topics in one’s subject area, the most useful representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations—in a word, the ways of representing and formulating the subject that makes it comprehensible to others . . . Pedagogical content knowledge also includes an understanding of what makes the learning of specific topics easy or difficult: the conceptions and preconceptions that students of different ages and backgrounds bring with them to the learning of those most frequently taught topics and lessons. (p. 9) This is subject matter knowledge that has been “transformed” (Shulman, 1986) or somehow bent toward pedagogical ends. It is the amalgamation of content and pedagogical knowledge. For the case of mathematics, it is what mathematics educator Deborah Ball and mathematician Hyman Bass (2000) have called “pedagogically useful mathematical understanding.” In actuality, theirs is an extension of pedagogical content knowledge in that it subsumes, in addition to the kind of

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knowledge as specified in Shulman’s description, also the way in which that knowledge must be held by teachers in order for it to be pedagogically useful. It is not simply the case that teachers must have thorough pedagogical content knowledge, as they argue, but it must also be understood in ways so as to be usable in teaching. One example they give is of the need for teachers’ knowledge to be easily “decompressed” or “unpacked,” where their own tacit understanding can instead be made explicit at a level of granularity appropriate for the learners. As they describe this “decompressed” knowing— Its logical steps are small enough to make sense for a particular learner or a whole class, based on what they currently know or do not know . . . This kind of knowledge is quite clearly mathematical, yet formulated around the need to make ideas accessible to others. (p. 99) Holding mathematical knowledge in this way, Ball and Bass argue, affords teachers the capacity not only to produce understandable explanations for students, but also to see and to hear from the students’ perspective—their errors as well as unconventional insights—in ways that overly compressed and tacit ways of knowing would not. Certainly, this way of holding knowledge appears well suited for a teacher. Mathematics educator Liping Ma (1999) also touches upon something like a mathematics-specific pedagogical content knowledge when she describes “knowledge packages,” as “a group of pieces of knowledge” bound together for the purpose of “promot[ing] a solid learning of a certain topic” (p. 19). Differing from simply a logical or structural bounding together of mathematical ideas—that say a mathematician might know—knowledge packages contain instead “key ideas” that would anchor the learning of surrounding ideas, the sequence of development through related concepts, an interweaving of procedural and conceptual topics, and “concept knots” where important and related concepts converge. As Ball and Bass (2000) write of mathematics-specific pedagogical content knowledge such as Ma’s notion of knowledge packages— This kind of understanding is not something a mathematician would have, but neither would it be part of a high school social studies teacher’s knowledge. It is special to the teaching of . . . mathematics. (p. 87) In this way, pedagogical content knowledge—as well as its mathematical “offspring,” pedagogically useful mathematical understanding and knowledge packages— appear to be forms of knowledge and of knowing that on some level intertwine the “teacherly” interests and motivations of mathematics teachers with the mathematical ones. But I would call attention to the fact that it is still savoir knowledge.

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It is to the question of how teachers might develop such mathematical understandings that “doing mathematics” enters the picture. Both Ball and Bass, as well as Ma, propose various forms of engagement with mathematical content—some of which would fall into the category of “doing mathematics”—as avenues for developing these pedagogically useful mathematical understandings.3 I want to be quick to say that none of these authors claims that the merit in “doing mathematics” lies solely in its helpfulness in developing mathematically specific pedagogical content knowledge.That is, they do not exclude other possible reasons for why it may be beneficial for teachers. At the same time, I would point to the fact that this particular attitude toward doing mathematics tends to emphasize its usefulness around teacher knowledge development, which as the thinking goes, would affect teaching effectiveness. Although I am not suggesting that there is anything wrong with such an interpretation, there is an alternative or complementary reason for why a teacher may want to do mathematics. And this has to do with the teacher—not so much as holder of (savoir) knowledge—but as representative of, guide into, and even lover of, subject matter. In short, it has to do with the teacher’s own relationship to mathematics.

“Doing” Mathematics: A Complementary Rationale Interwoven Perhaps boredom sets in as answers are given to questions that were never asked. (D. Bob Gowin, as cited in Brown & Walter, 1990, p. 3) In the preceding chapters, I have characterized a teacher’s relationship to mathematics as being borne out of a joining of will and grace. The will would be the effort, persistence, “stayings,” strivings and struggles brought forth toward engagement by the teacher, whereas the grace entails the attendant feelings of excitement, surprise, satisfaction, of aesthetic appreciation, and of revelation in understanding a piece of mathematics. Further, I have elaborated upon both the motivation for entering into, as well as the resulting relationship arising out of such experiences and engagements, as captured by an intertwining of the teacher’s personhood—in the needs, dispositions, and life projects and trajectories of the teachers—with that of their manner of relating with and knowing mathematics. In light of this described image of “relationship,” the point that I will be attempting to develop in the remainder of this chapter is that an alternative rationale for doing mathematics is exactly in fostering a more viable relationship to the subject for the teacher. To use a set of terms introduced earlier, the merit of doing mathematics is not just for gaining savoir knowing—though it be 3 It is worth noting that neither proposes taking more coursework in advanced topics, or reading advanced mathematics texts—the kind of rationale one might consider when looking at most teacher preparation programs.

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pedagogically inclined—but also for developing a connaitre knowing of the subject, that is a sense of intimacy and connection to the subject. But before getting too far ahead of myself, I had better define more clearly what I mean by “doing” mathematics. In a nutshell, I take it that a person is doing mathematics when he or she is producing mathematics.4 I would contrast this with that of being a consumer of mathematics. To produce is to create or to discover, deriving from the Latin producere (meaning “bring forth” or “draw out”). Specifically, one draws out what is not there or apparent, to begin with. In contrast, to consume is to take in that which is given. The essential divergence in meanings between producing and consuming, I would point to, has to do with the state—given or not—of the “object” in question. Generally speaking, if it requires a drawing forth, one can say that it is produced. On the other hand, if it is already given, it is already produced, and hence the interaction would more strongly be characterized by a consuming (or else ignoring, as oftentimes happens with mathematics). This notion of producing mathematics is echoed further in the idea of authoring mathematics. Deriving from the French autor (“father”), which in turn is obtained from the Latin auctorem, literally meaning “one who causes to grow,” to author then carries meanings similar to the “drawing forth” of producing. That is, when one is “author” of something, one is in the role of helping to grow and to draw forth that something. When that something is mathematics, then one becomes an “author” of some piece of mathematics, carrying reverberations of what it means to hold a certain mathematical author-ity around that topic. Consider further that one typically “fathers” another with whom one is in relationship. Consequently, this connection between “doing” mathematics and “fathering” mathematics (through the link of “producing” mathematics) hints at an underlying potential for fostering a sense of relationship in the doing of mathematics. I should note that I am not making a distinction whether the mathematics that is produced, or authored, is new to the larger mathematics community or not. For example, the Indian-born mathematician Srinivasa Ramanujan (see Kanigel, 1991) produced a large volume of mathematics that had already been established continents away without his knowledge. I think that most mathematicians would still consider his mathematical engagement as a case of doing mathematics. But similarly, a mathematics teacher who happens to “produce” a result well-known to other teachers, I am suggesting, is still involved in a form of doing mathematics as long as the arrival to that result involves a drawing forth of ideas not given to the inquiring teacher. In this way, doing mathematics, as I am speaking of it, concerns the nature and quality of interaction—that is, the process of producing—and not the product of that interaction. This “quality of interaction” may be marked for the mathematical 4 If I were being technically minded, I would qualify it as the “conscious act of producing mathematics,” so as to preclude having to say that calculators and computers also “do” mathematics.The reader may be interested in one mathematician’s take on the same question (see Devlin, 2005).

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do-er by an accompanying sense of excitement and satisfaction associated with “growing” and “drawing forth” something new—whether to oneself or to others— recalling the affective aspect of grace, discussed in Chapter 2. Lastly, I would not automatically equate any particular activities—such as raising conjectures, solving problems, or doing basic-drill problems—as being or not being a form of doing mathematics, for it is not to the form, but to the actual act of whether there is a “drawing forth” of mathematics that the point of emphasis lies. Naturally, different activities do in fact afford varying degrees of a “drawing forth” (dependent, also, upon the learner, naturally). And ultimately, these “variances” do point to differing depths of engagement and experience—that is, a “doing” of mathematics. This, in turn, could lead to greater or lesser undergoings, whether in affective, aesthetic, obsessive, or “aha”-like responses to mathematics. For example, activities such as reading and studying mathematics texts, listening to lectures, doing “practice” drill problems—the typical activities found in K-16 classrooms—appears largely to make consumers out of those who would engage in such activities. If a student were to take in all that the texts, lectures, and problems were offering, s/he would primarily be taking in that which is given.This is not to say that there is no room for the doing of mathematics within such activities. A bright or reflective learner might “draw out” his or her own connections between given pieces of information—connections not pointed to by the text or teacher.That act in itself, I would qualify as a production—as contrasted with consumption—of knowledge. It would be a momentary break from consuming to producing mathematical knowledge for such a learner. But clearly, most K-16 texts, lectures, and practice problems offer relatively few and short-lived opportunities as such—while also being heavily reliant on the initiative brought forth by the learner—and thus, are limited in their capacity to evoking the excitement of “doing” in the learner. As a second category of activities, a student might be engaged in “problemsolving” activities.This could range from playing mathematical “games,” to solving math puzzles, to grappling with “open-ended” problems that yield numerous solution methods. Because the solutions to these are generated by the problem-solver and are not given, they are thus “drawn out” and new to the problem-solver. In this regard, activities such as these appear to afford greater producing of mathematics for the learner than the previous class of activities considered. At the same time, one could argue that because the solutions to such problems are implicit in the problems themselves, the “produced” solutions are in fact indirectly given. In other words, even the so-called “open ended” problems have determined outcomes; there are just a myriad of them. When the problem is given, the accompanying solutions, to some extent, could be said to be implicitly given also. Thus, the actual “drawing out” may be considered partial, or incomplete, making the doing-ness of it, somewhat unsatisfactory—unsatisfactory, not just in terms of its limited sweep, but also for the problem-solver insofar as his or her affective response is concerned, specifically in regards to feelings of ownership. The solution may be his or hers, but the problem may never have been.

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If, in fact, solving a given problem leads to what I have called a partial “doing” of mathematics, then to actually “draw forth” the problem itself (as well as the solution) might be looked upon as a way of completing or furthering the doing-ness involved. One would be producing not only the solution, but also the problem.The latter would correspond to what is more commonly referred to as “finding a question” or “posing a conjecture.” Thus, mathematical activity that includes this process of finding a problem or question, in addition to solving it, appears to afford the greatest degree of mathematical “doing,” at least of the types of activities considered so far. In fact, for many, this is what doing mathematics means: finding a mathematical question and answering it. Consider the following two statements: A student isn’t really doing mathematics unless she is asking herself questions and solving problems. Everybody agrees about that. (Brousseau, as cited in Balacheff et al., 1997, p. 79) Posing questions, making and proving conjectures, exploring puzzles, solving problems, debating ideas, contemplating the beauty of mathematical structures—these constitute the “doing of mathematics”. (Schifter & Fosnot, 1993, p. 12) Neither Guy Brousseau nor Deborah Schifter—both mathematics educators— leaves out the primacy of asking and posing questions in mathematical “doing.” (And they are speaking in reference, not to professional mathematics, but to school activity!) For both of them—and perhaps for many mathematicians as well— doing mathematics requires both “asking questions and solving problems.” It necessarily involves the finding and posing of questions and conjectures as part of the larger arc of mathematical production or “doing.” Although I would not want to say that an activity such as the reading of a text affords no opportunities for the “production” and “doing” of mathematics, the traversal of the entire arc from explicit problem-posing to problem-solving as a “drawing forth” appears more complete and involved, and thus, potentially more authentically engaged. Again, it is what most mathematicians do when they say that they are doing mathematics. And because it involves the “drawing forth” of even the problem by the do-er, there is greater affordance for feelings of “ownership” of not just the solution but the entire process of engagement with the topic.

Ownership to Inter-esse Yet, this feeling of ownership, I would instead call inter-esse: the intertwining of self with the object of interest. Here is how I mean it. A problem or question that is given, whether by another (authority) or by a text, rarely has the same continuity to experience that a question arising out of one’s own stream of experiences has.The act of finding a question involves engaging one’s (mental) environment. Contrary to common expression, questions do not just come “out of nowhere” (excepting the case when they are

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given by another). They come arising out of our experiences and our engagements with ideas and our surroundings. This is to say that they come out of our “lives”: out of the confluence of our different life trajectories. Contrast this with the typical experience of being given a problem, whether from a teacher, a facilitator, or from a text. Even though both a good teacher/ facilitator and a good text will strive for coherence and continuity, so that the questions asked appear to be linked to prior discussions and questions, most teacher, facilitators, and texts can fathom at best only a small fraction of the entirety of our experiences.Thus, the linkage of questions given to that of our life streams, in comparison to questions generated out of us (and our lives) will more than likely tend toward the tenuous. Put differently, questions that are given to us tend not to intertwine with our various life trajectories as richly and with the same depth, leaving us less “inter-ested.” We may call this “ownership” of the question, but I think that description misses the mark somehow, as it is not an issue of “having” or “possessing.” What happens with questions that we “draw forth” from out of experience is that they are borne out of our lives, and in turn we “father” (or author) them. Yet, we do not possess or own them. Instead, they have a life of their own (a bit like children whom we might “help to grow”), while also being woven into the life stream of our lives. In this way, we are “intertwined” with these questions and possibly with where they will lead. This is to say that we are in relationship.5

Asking Questions and Posing Problems6 as a “Staying With” Mathematics Yet, there is another way in which a “doing” of mathematics that is inclusive of finding problems leads not just to a savoir knowing, but also to a connaitre knowing, to relationship. Consider first what the culmination of savoir is. It is in the accumulation of knowledge and of know-how. Now, consider what the height of connaitre, or relationship, might be. With what I have discussed so far, I would propose that it is in the prolongation and continuance of a state of participation, of process, of connection, of staying with other. A sense of relationship peaks in a persistence of connection between self and other. Now, consider the contrast between finding a problem and solving a problem along those terms. When one solves a problem, one is essentially done with the

5 Further, “ownership” supposes too much agency on our part, as well as an I–It orientation toward the question. As discussed in Chapter 2, the lived experience of grappling with a truly meaningful question involves a sense also of being owned by the question. These questions that arise out of our life streams, I would argue, have greater potential for “seizing” us in ways that given problems can only do so on a hit-and-miss basis. 6 There is quite a substantial literature in mathematics education pertinent to “problem posing” (see for example Crespo, 2003; English, 1997a, 1997b; Silver & Cai, 1996).

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process. Without recourse to seeking out more problems, the interaction with mathematics comes to an end. On the other hand, with finding problems, there are two ways in which participation is maintained. One is that problems call for solutions. The natural follow-through to finding a problem is to solve it. In this way, one stays mathematically involved. But there is also the fact that with finding problems and questions, there is never a definitive end. As Stephen Brown and Marian Walter (1990) write— Given a situation in which one is asked to generate problems or ask questions—in which it is even permissible to modify the original thing—there is no right question to ask at all. Instead, there are an infinite number of questions and/or modifications and, as we implied earlier, even they cannot easily be ranked in an a priori way. (p. 5) This sense in which the act of finding problems breaks away from what Brown and Walter later call the “right way syndrome” points to an important dynamic involved in problem finding. In contrast to solving problems with “rightness” of solutions, finding problems does not involve an a priori “rightness” or “wrongness.” Thus, as one begins to look at mathematics as a place of raising questions and conjectures, interactive possibilities begin to open up.This is to say that mathematics becomes a domain in which one’s curiosity becomes welcomed and received, as one is freer to explore and to “be creative.” That is, doing mathematics as such affords and even privileges one’s curiosity to be at play in one’s mathematical engagement. But also, as the possibilities for interaction become manifold, problem-finding affords continuation of mathematical participation. This is to say that one is never done with asking questions, as long as one privileges this type of activity in mathematical interaction. In this way, doing mathematics, inclusive of problem-seeking, offers a vision of mathematical activity that allows one greater opportunities for staying in participation with mathematics. It is a staying reminiscent of Gattegno’s “staying with the problem”—from Chapter 3, an aspect of Buber’s “will”—but in this case, it is a staying with the mathematics that would give rise to the problem as well as to the solution. In short, it hints at possibilities for the culmination of a connaitre way of knowing, of relationship. Another way to put it—and perhaps more accurate to lived experience—is to say that this kind of “doing” of mathematics is an avenue for continually returning to mathematics. Consider the following comment made by a teacher: I have learned that I need to work hard not to shut down when I get the answer to something, but to do the hard work of asking myself more questions after I think I’ve found the answer . . . I am rediscovering the discipline to make myself wonder. (Schifter, 1996a, p. 496).

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Here, the teacher cites the “hard work of asking [herself] more questions” even after finding an answer. Normally, if one has found an answer, the task is over. Mathematical activity may cease. But for this teacher, the asking of questions becomes a way of returning to the mathematics even after resolution. Similarly— and as I have just argued in the prior section—“doing” mathematics over its entire arc of possibilities from asking questions to resolving the questions provides a manner of engagement allowing for continual returning. The reader may further recall that in Chapter 1, I pointed to the shift from saying,“He studies/learns/reads mathematics” to “He does mathematics.” Mathematics goes from object to process, from thing-to-possess to participatory engagement. What is captured in the language is what I have argued, that in doing mathematics, there is availed for the do-er continual opportunities for the returning toward mathematics—that is, a process-focused engagement.To reprise a comment I made in Chapter 1 of this book: Relationship in its most general sense (as well as familiarity) grows out of some kind of participating with whomever or whatever the other may be, but without readily accessible modes of participation, the resulting relationship— or the connaitre way of knowing it—will likely not ensue. In this way, the real value for teachers in “doing” mathematics may be in supporting a recurrent returning to mathematics—that is, in fostering a sense of connection and relationship to the subject that they are to teach.

The Predicament for Teachers: Feasibility and Relevance Perhaps to the practicing mathematician there is little new here, just common sense (with a lot of cluttered metaphorical prettiness). Yet, for mathematics teachers without mathematics research experience, there arises a challenge in terms of practice. It is one thing to speak of the importance of doing mathematics. It is another to make that act both feasible and relevant to the practicing teacher. Here are two different concerns—feasibility and relevance—that I see as a potential challenge for teachers. In regards to feasibility, I would ask, where have mathematics teachers ever had encouragement to pose their own mathematical questions? Does not most K-16 education consist of solving problems posed by others (e.g., the teacher, the text, the test, and so on)? In this way, many teachers’ experiences of doing mathematics, insofar as their preparatory experience is concerned, have \been incomplete at best; many have rarely if ever experienced the full arc of doing mathematics, beginning from the posing of a problem all the way to its solution. Where this leaves many mathematics teachers, I believe, is without a viable image of doing mathematics that appears workable and within reach. Instead, it feels a world away. The conventional image of it involves focusing on a narrowly defined topic and applying oneself for years and years to advance far enough to

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arrive at the frontiers of new knowledge, just to start the actual “doing” or producing of mathematics. Yet, in this vision of it, there is too great a cost for too little a connection to the life of being a teacher. Essentially, there is no relevance, for what would be the point of coming up with a new theorem in some obscure field of mathematics for most teachers? Further, the idea of continuing to study out of advanced texts (as I attempted on many occasions) loses its pertinence to the life trajectory of being a teacher. For example, reading a text on functional analysis would find little connection and application to the daily life of teaching pre-algebra, algebra, pre-calculus, and even calculus.7 In fact, mathematics may be one of the few school subjects where many, if not most, of its teachers are not at least part-time practitioners of the discipline. Most mathematics teachers who graduate with a bachelor’s degree in the subject have likely never had significant experiences of “doing” mathematics.To a certain extent, it would be true to say that the majority of those who end up doing mathematics do so only after a long period of study in the field—a kind of apprenticeship, if you will—and begin such “doing” only at the point when they engage in mathematical research as doctoral candidates in mathematics. And of course, other than university mathematicians, most teachers do not obtain doctorates in the field. Contrast this state of affairs with many of the other school subjects. One could readily conjure an image of a language arts teacher who continued to enjoy reading, or else writing on the side, as ways of engaging in the practice of the discipline, and staying connected to the subject as more than just a pedagogue. I have known art teachers who maintain a studio art practice. All of the music teachers I interviewed for this project played either professionally or as avocation, continuing to stay invested as musicians. Many computer science teachers have opportunities for staying active as programmers in their own right. In a somewhat lesser sense, laboratory components of most science classes allow for the teachers to continue “doing” some science.8 So again, without a viable image of what doing mathematics can look like, many mathematics teachers are left without a potentially important recourse for fostering a sense of relationship to the subject. What I believe is called for, consequently, is a vision of “doing” mathematics with relevance to a teacher’s life—that is, a vision for “doing” mathematics that is pedagogically relevant. Such “doing” could be considered a special subcase, or subactivity, of the kind of mathematical doings that one would expect of research mathematicians. So as to emphasize the pedagogical bent, I will refer to it as doing pedagogically relevant mathematics— 7 Some might argue that teachers should know far more than what they are teaching, but I am not speaking of what teachers should know, but rather, what type of activity has tangible bearing—and a sense of relevance—upon their daily teaching lives. 8 Even the idea of a science teacher reading science so as to stay abreast of the latest advances in the field does not seem as unlikely as a math teacher keeping up with the advances in his or her own field, although this concerns issues of staying in touch with one’s field, and less so with “doing” the disciplinary practices.

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something, incidentally, that I would distinguish from Shulman’s pedagogical content knowledge or any of its discipline-specific derivations, in its stress on the “doing” and experiential element. At the same time, such doing could very well lead back to the gaining of these pedagogically motivated forms of content knowledge as suggested in the first rationale for doing mathematics (à la, Ball & Bass and Ma). So as to lend shape to what doing pedagogically relevant mathematics could look like, I will briefly examine three cases of such “doing” on the part of teachers, and in turn, specify some of the underpinnings that may characterize this specialized form of mathematical “doing.” Naturally, I will not claim that these cases exhaust the list of possibilities, or that the characterizations I draw are necessarily comprehensive. Instead, my intent is to begin to give a sense for what doing pedagogically relevant mathematics involves, and in particular, point to its connections to the life path of being a teacher.

Doing Pedagogically Relevant Mathematics: Where the Problems Come From and Where They Remain Every teacher should do research and should have had training in doing research. I am not saying that everyone who teaches trigonometry should spend half his time proving abstruse theorems about categorical teratology and joining the publish-or-perish race. What I am saying is that everyone who teaches, even if what he teaches is high-school algebra, would be a better teacher . . . if he did research in and around high-school algebra. (Paul Halmos, 1975, p. 469) I begin by quoting one of countless “reader reflections” (over the years)9 from one of the NCTM practitioner journals (Berry, 2003)— While we were discussing the topic of the convergence or divergence of alternating series in my calculus class, the standard conversation regarding the sum of (1) + 1 – 1 + 1 – 1 + . . .

9 I want to note that the choice of this example is somewhat arbitrary. It was one of a number of “reader reflections” and articles in the one issue from which I could have chosen—the choice being decided partly by its relative brevity, and partly by its content focus on secondary-level mathematics (so as to contrast against the second example). And still, I could have looked through other issues, which I did not. Instead, I picked it out from the very first practitioner journal I chose rather arbitrarily. The point being that contrary to what I suggested earlier—of there not being a viable vision of doing mathematics for teachers—it does appear that some teachers are in fact doing mathematics, and have been doing so for some time.

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ensued. After exploring the different results that we can obtain from rearranging expression (1), such as (1 – 1) + (1 – 1) + . . . = 0 and 1 + (– 1 + 1) + (– 1 + 1) + . . . = 1, we arrived at the usual conclusion that the series diverged. A student then noticed the connection between expression (1) and the series for ∞ 1 (2) ––––– = 1 – x + x2 – x3 + . . . = ∑ (–1)n xn 1+x n=0

if x is allowed to equal 1. Of course, expression (2) is valid only for |x| < 1. Together, however, the class and I used the following argument to “prove” that expression (1) does in fact converge: 1 1 + 1 – 1 + 1 – 1 + . . . = lim (1 – x + x2 – x3+ . . . ) = lim ––––– = –– x →1x →1- 1 + x 2 where x → 1- means approaching 1 “from below,” in other words, approaching 1 from such values as 0.9, 0.99, 0.999, . . . This method of approaching 1 does not violate the |x| < 1 convergence criterion for expression (2). The error in reasoning here is well-hidden (to me, as well as to my students) and can generate some wonderful discussions on the depth and subtleties of mathematics reasoning and areas where many a pitfall occurs. (pp. 611, 639) Now, consider the following abridged description also, originally from a mathematics teacher/educator’s journal, which later became part of an unpublished manuscript: The Problem: Arrange the digits 5, 6, 7, 8, and 9 into a three-digit number and a two-digit number so that the product is as large as possible . . . My interest in this problem arose while I was tutoring a 7th grader.We were flipping through the book Math Mind Stretchers by Ann Fisher and came across this and similar problems. My student began his work on the problem by making educated guesses and finding the products. Just as I would have, he placed the largest digit (9) in the hundreds place. As he worked though different possibilities, I quickly realized that I didn’t know the answer or an elegant way of determining what was in fact the largest product. My approach to the problem would have been very similar to my student’s. I began thinking that there must be a more elegant method of solution. Additionally, I was surprised by the answer in the back of the book; 875 × 96 results in the largest product! As is often the case, the answer generated more questions: Why is it better to have the largest digit in the tens place of the second number instead of the hundreds place? Does this

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pattern hold for any five digits? What if the digits are not consecutive? What if some of the digits repeat? What if the goal is to find the smallest product? My initial approach to the problem was guess and check. I would select five digits and try different arrangements until I found what I believed was the largest product.The most difficult part of this approach was not knowing if I had in fact obtained the largest product. Several times I had what I thought was a winner only to, a few days later, find a better arrangement. Without trying all 120 possible arrangements of five digits, could I guarantee that I had in fact obtained the largest possible product? As I experimented with various sets of numbers, a pattern emerged. For five digits, it seemed that the following arrangement resulted in the largest product: 2nd Largest

3rd Largest

5th Largest

Largest

4th Largest

With conjecture in hand, I set out to prove this. Along the way I had several insights into the multiplication algorithm and multiplication itself. Early in my work I realized why my initial guesses did not result in the largest product; by examining the multiplication algorithm I noticed how different arrangements affect the product. Switching the position of two digits results in gains in some places and losses in others. By maximizing the gains and minimizing the losses, the product is increased. Most surprisingly, I found that the above arrangement does not always result in the largest product. Another arrangement results in a larger product when the two largest digits are equal! Before getting into the final proof, I will outline my journey . . . [I will cut it off here, for this example is given only to give a flavor for the kind of inquiry that is possible at the elementary level of mathematics. And it is only one of a multitude! For the interested reader, she does go off into both an informal proof that analyzes the multiplication algorithm, as well as a formal proof resorting to the use of algebra and variables.] I begin with these two examples to make two preliminary claims. First and foremost, we have here teachers (or with the help of students in one) posing mathematical questions—one at the secondary and another at the elementary level of mathematics curriculum. In the first example, the question is implicit in that the teacher and his students have basically answered the question of,“How can we use equation (2) to restate the result of equation (1)?” And of course, the question(s)

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are made explicit in the latter account. The point being that these are not pedagogical questions—such as, how best to teach an idea or how to explain it—but truly mathematical ones. And although the completion of the arc of mathematical doing in arriving at a solution is not made explicit in either description (the latter simply because I cut it off), we can assume the case for both by what is implied. Secondly—besides a certain degree of difficulty—there is a feature to these explorations that set it apart from the predominance of mathematicians’ doing of mathematics (or say a doctoral dissertation in mathematics). It is that the questions and/or the problematizing of a mathematical situation arise out of the teacher’s teaching. It is not that each teacher went off to an advisor or to a textbook to find an intriguing problem, or that they labored in a specific area of mathematics for years to find these problems, which they have reported on, but rather, a problematic situation arose within a teaching (or tutoring) situation, from where the teacher allowed for himself/herself to be drawn into a mathematical inquiry. In the first example, the teacher appears to carry the inquiry through with the class, whereas in the second, the teacher explores further in her own time, separate from her student.The carry-through may be different, but in both, the question— not the literal problem, but the questioning stance toward the problem—arises within a pedagogical context. That is to say, these explorations are to some extent pedagogically motivated. Naturally, “pedagogical motivation” extends past the scope of these two examples. For example, even if a teacher had been planning a lesson (i.e., not with any students present) and had come upon a mathematical concept that required further investigation which the teacher then proceeded to explore, such a motivation, I would argue, as being pedagogically contextualized. This is to say that one need not be in the presence of actual students for a mathematical doing to be motivated by pedagogical concerns. Yet, as long as the problems are motivated within pedagogical contexts, one could characterize such mathematical inquiries as enfolded within the larger context of a teacher’s life; they are borne out of that life trajectory. Or to recall the idea of interest (or inter-esse) as an intertwining, we would have mathematical doings with the potential for entwining a teacher’s teacherly trajectories with the mathematical ones. As I argued in Chapter 4 about such intertwinings, these pedagogically motivated mathematical doings should in turn hold greater “interest” for the teacher.

Doing Pedagogically Relevant Mathematics: Sense Making and Meaning Making I would now proceed with further elaboration of how a doing of mathematics might intertwine with a teacherly trajectory through another example. Consider the following mathematical statement:

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13 + 23 + 33 + . . . + n3 = (1 + 2 + 3 + . . . + n)2

Here is a result that is readily proven by mathematical induction with a bit of symbol pushing (the reader may consider giving it a quick try). In fact, one would expect to find such a problem in the secondary or college curriculum as a practice problem for proof by induction. The question, though, that Richard, the former middle school mathematics teacher and current mathematics teacher educator from the last chapter, asked himself was whether he could “conceptualize” the result. In this case for him, it meant, could he come up with a visual representation of the statement? Before proceeding with a discussion of how Richard’s doing of mathematics is pedagogically relevant, I should offer up what Richard did.Yet, just as Richard left the solution up to me so that I could “have fun with it,” in the same spirit, I will draw up only the beginnings of such a “picture proof ” for the reader as well. The idea, first, is that (1 + 2 + 3 + . . . + n)2 lends itself toward a very natural diagram (say for n = 4).

FIGURE 5.1 Visual representation for (1 + 2 + 3 + . . . + n)2 [ i.e., (1 + 2 + 3 + . . . + n) (1 + 2 + 3 + . . . + n)] for n = 4.

Similarly, 13 + 23 + 33 + . . . + n3 can readily be depicted visually (again with n = 4):

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FIGURE 5.2

Visual representation for 13 + 23 + 33 + . . . + n3, for n = 4.

As one tries to understand the reason for the equality of volume between the two figures, it soon becomes clear that one need show how each of the shaded regions in the upper diagram forms a cube (for example, the largest, outermost shaded region would form a 4 × 4 × 4 cube). I hope the interested reader, at this point, might take a few minutes to work out the rest of the proof, but also pay attention to other patterns or results that one observes.10 Now, returning to Richard’s mathematical “doings”—the key step, of course, is that he asked himself a mathematical question. Notice that for him, it was not so much about discovering a new result. In fact, it was a result that he already knew— For instance, the sum of cubes, the sum of the first n cubes. That’s an easy formula to derive . . . I know the answer to it, or I can look up the answer to it, but it’s the deriving of the answer, and can you make sense of the

10 So a few silly little patterns that emerged during my mucking around with the drawings were (1) The sum of the first n consecutive odd integers is a perfect square (or n2 = 1 + 3 + 5 + . . . + (2n – 1), equivalently, 1 + 3 + 5 + . . . + n = (n + 1)2/4); (2) A quick corollary: the difference of squares (e.g., a2 – b2) can be written as the sum of consecutive odd integers (and its converse, the sum of any consecutive odd integers can be written as the difference of squares); (3) The sum of the first n consecutive even integers is equal to the product of consecutive integers (or 2 + 4 + 6 + . . . + n = a (a + 1), where a = n/2); (4) The difference of cubes can be written as a sum of consecutive odd integers (but not its converse); and (5) the simplest and most relevant to Richard’s problem: 1 + 2 + 3 + . . . + (n – 1) + n + (n – 1) + . . . 3 + 2 + 1 = n2. Although these are not particularly deep or difficult results, their verbal and symbolic expressions nevertheless contain little of the seeming transparency of ideas that the drawings convey. In fact, the diagrams almost trivialize these results. In this regard, I recall a doctoral mathematics course taught by the mathematician Donald Saari, who reconceptualized mathematical voting theory in terms of geometric diagrams (see Saari, 1995). While we were proving Arrow’s “Impossibility” Theorem and others that entail rather difficult mathematics (and some that cannot be proven otherwise), these same results seemed to give way with as much effort as drawing a few lines here and there, as if a smart 10th grader could do it and understand it, as it required at most standard ideas from calculus, linear algebra, and basic high-school geometry. I remember thinking, “This is pretty easy stuff. I can’t believe Kenneth Arrow got a Nobel Prize for this!” Of course, later, another “high-caliber” mathematician/scientist, Duncan Luce, confided to me, “Yes, using Saari’s methods makes the mathematics much simpler, but of course, that’s Saari’s genius.”

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patterns. I think the sum of the first n cubes, when you look at that, the answer to that is amazing. It has to do with the square of the sum of the first n integers . . . Well, I knew the result, and I knew that the result was pretty amazing, so I said, “How can I conceptualize the result?” Accordingly, his “doing” is focused less on proving the actual statement of equivalence, and more so on “conceptualizing” and “making sense” of the result.Whereas proving equivalences aligns with what one might expect from the “doings” of a mathematician, making transparent the conceptual underpinnings of a mathematical fact, I would argue, is one that naturally intertwines, in particular, with the life trajectory of being a teacher. I will let Richard narrate his reasons: I don’t know how I came to do it . . . I think it was just wanting to understand things more deeply. [For example], here’s something I found out that I didn’t understand well. A model for division of fractions. I was teaching . . . And I wanted to come up with a poster to illustrate the addition of fractions, the subtraction of fractions, the multiplication of fractions—and I had ways to do that. But then the division of fractions, I didn’t have a good model. I realized that I had multiple degrees, and no good models for the division of fractions. What could that picture look like? So I’m making this poster to use in the class the next morning. “Gosh, what would be a good picture?” So, that was one of the sparks, of “Okay, you understand it algorithmically, but what sense does that make?” So making sense of the familiar, of things that were known algorithmically, making sense of things conceptually became sort of a passion. Again, for students, because you want the students to know, because you wanted the student to have access, but you also personally wanted the satisfaction of understanding it more deeply, or in different ways, or with a different representation. I want to point out a number of issues touched upon by Richard. First is that he voices a dual motivation for his mathematical inquiries. At once, he wants “the satisfaction [for himself ] of understanding it more deeply, or in different ways” but he also wants for students “to know” and “to have access” to the mathematics. His inquiry serves both his mathematical curiosity and his teacherly concerns of reaching his students. It offers an intertwining of his mathematical and teacherly trajectories. In regards to the former, it is not just about accruing pedagogical content knowledge, but also about experiencing the satisfaction—the undergoing, the grace—of understanding. Simultaneously, it is also about doing mathematics in a way that carries relevance for him as a teacher of students. The relevance occurs through “making sense of the familiar” and “making sense of things conceptually” by which he is potentially able to offer students “access” to the ideas. Now, here is a telling phrase—“making sense”—that suggests more than meets the eye. To “make sense” of mathematics conventionally stands as a synonym for

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“conceptualizing” or “understanding” some piece of mathematics. But within Richard’s mathematical doings, we see a more literal and phenomenologically resonant meaning, which is that by “making sense,” Richard is making sensorily available (the mathematical meanings), in this case visually. What is originally an abstract, symbolic statement in 13 + 23 + 33 + . . . + n3 = (1 + 2 + 3 + . . . + n)2, through Richard’s explorations, has been lent visual meaning. He has more or less mapped an abstract statement into a visually salient form; he has made “sense” of it.11 In light of such “mappings,” consider the following commentary on Plato’s (good old) “allegory of the cave” by Edward Rothstein (1995)— Once completing the movement upward, Socrates notes, dialectic also involves a return, downward through the line. The descending philosopher no longer becomes absorbed by the images of the lower worlds; instead he sees them as distant representations and interpretations of primary Forms. Looking again at the lower worlds is like being presented with a completed theory and then finding its applications . . . The descent in Plato’s cave journey is, we might argue, the mapping of abstraction into the concrete. Music is given meanings, mathematics applications. (p. 238) Although, perhaps, to some an antiquated view of a teacher—as one who “brings” the truth down from on high—Rothstein’s description of the descending philosopher appears to reflect the journey taken by Richard.12 The higher symbolic forms have been brought into the sensuous domain, the abstract into the visually representable. It is a movement of mind that gives Richard a greater sense for what is happening mathematically, but also provides his students with potentially greater accessibility to the ideas as well. Richard, himself, may have facility, and more importantly, trust with the abstract notations, but his students may not. By bringing the abstraction down into a visually available form, his students may now have greater say in the involved reasoning. Naturally, they also may not. The point, though, is that Richard’s inquiry points toward a pedagogical potential. In 11 It is likely to the visual senses in particular that mathematics most naturally lends itself. There are mathematics educators who strive to give embodiment (e.g., kinesthetic senses) to mathematical ideas (see, for example, Nemirovsky et al., 2004). As for hearing, I am not aware of how complex mathematical ideas can be made more transparent through conversion to sound, although in a different vein, most musicians could be said to be making some basic mathematical relations available to the auditory senses, including some who deliberately made such efforts, such as the classical composers Erik Satie, Claude Debussy, and Bela Bartok in their particular use of Fibonacci numbers and the golden ratio within their compositions (see Adams, 1996; Howat, 1983; Lendvai, 1971). Meanwhile, I am not aware of mathematical doings that would map abstract mathematical concepts onto taste or smell. 12 It is of some interest that Rothstein chooses to call this mapping of the abstract into the concrete as finding an application for mathematics, as it corresponds to mathematician Hyman Bass’s characterization of mathematics-specific pedagogical content knowledge as a form of “applied” mathematics.

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this way, “making sense” of mathematics suggests a form of mathematical doing that is entwined with pedagogical interests. I would also suggest that “sense making” as such falls under a larger class of activity that one could call “meaning making.” That is, one could make transparent the meanings of mathematical concepts to more advanced students without making them available to the senses (turning “sense making” into a special case of such “meaning making”). For example, one could try to explain an algebra concept without recourse to algebra, using only arithmetic but not necessarily drawings. If the students already had solid understandings of arithmetic operations, then meaning could be provided for them without resorting to diagrams.To quote mathematics educator, Daniel Chazan (2006), who himself engaged in a bit of mathematical doing in deriving a calculus result using only algebra (as his students had not yet learned calculus)13— As a teacher, if one wants to take students’ ideas seriously, one must be able to construct mathematical arguments with the concepts and understandings that students have at their disposal, even if those concepts and tools are not the most powerful mathematical tools for making relevant arguments. Only in this way can teachers convince their students by mathematical, rather than authoritarian means . . . The teacher asks: “what kind of argument could I generate if I assumed I did not have access to advanced methods, but stuck to the methods available to my students at this point in their academic careers?” (p. 6) Doing pedagogically relevant mathematics in this sense is a bit like reconsidering mathematical results that one may have already taken for granted, and reworking them from the students’ point of view. It is mathematical engagement by a teacher at the level appropriate to the subsequent pedagogy (and not reliant on remote mathematical machinery). It is to put oneself in the student’s place, in terms of the tools and knowledge available for mathematical inquiry. But this is part of the work of being a teacher: to take the part of the student. Such work, then, calls upon the teacher to not just “do” any old mathematics, but to “do” mathematics while being a teacher.14

13 See Chazan (2006) for a full description of the mathematical exploration, or Chazan (2000, pp. 18–21) for a statement of the problem itself. 14 I recognize here the overlap between what I am calling the pedagogically relevant doing of mathematics and what Ball & Bass (2000) have referred to as the “unpacking” of mathematics done by some teachers. I would suggest first that unpacking be considered a subset of this manner of doing mathematics. The first two examples of such doing of mathematics I have given in this chapter do not involve unpacking but simply exploring and doing mathematics, but still, with relevance to a teacher’s life. But more importantly, I would argue that there are two important and related distinctions in the point of emphasis between the two. First concerns the duration of involvement. Most images of unpacking involve the teacher doing so on-the-spot in the classroom, or else, with a few

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In this way, whether the mathematical activity is borne out of a teaching episode (as with the first two examples) or is aimed at pedagogical potentialities (this last example), or both, what we have is an image of doing mathematics that interweaves the teacherly strand of a teacher’s identity in with the doing—a doing that suggests a pathway toward a deepening relationship with mathematics for teachers.

extra minutes or hours outside of the classroom. Doing mathematics can take weeks, even months. Granted, these are just images of each activity.Yet, what is pointed at by this distinction is that with unpacking, the emphasis is on the knowledge that is unpacked. A teacher generally unpacks so that that teacher can hold the mathematics in its unpacked state for pedagogical purposes. With doing, on the other hand, the emphasis is on the act itself—thus, the prolonging. A teacher who is doing mathematics is doing so because there is intrinsic pleasure, satisfaction, or undergoing in the experience itself. The motivations may stem from pedagogical concerns, but the action itself has as much nonpedagogical (and instead ontological) repercussions as it does pedagogical for the doer as a mathematizing individual.

AN EPILOGUE IN TWO PARTS

Part I: Returning to the Episode of the Straitjacket (from pp. 10–13) A teacher who is not “carried away” by something is no teacher at all, for it is in being carried away that one is moved to carry others away, too. (David Blacker, 1997, p. 100) The goal is the same, no matter what the age of the student; someone must be there to listen, respond, and add a dab of glue to the important words that burst forth. The key is curiosity, and it is curiosity, not answers, that we model. (Vivian Gussin Paley, 1986, p. 127) Recall the opening episode between Dan and me.While he appeared to be flitting from one mathematical conjecture to another, I felt fettered in intellectual restraints. In actuality, he was “doing” mathematics, for he was “drawing out” different leads from the invert-and-multiply algorithm while intuiting connections to other areas of mathematics. Meanwhile, I was after a lesson plan, and not after the production of any greater mathematical knowledge or insight. In fact, our interaction would not have had the impact on me that it did had I too held a viable vision of mathematical doing—one that would have allowed me to join him in exploration. Instead, mathematics had by that time become for me a domain where my own sense of curiosity and creativity had little room to flourish. There was no intertwining between these aspects of myself and mathematics. I had, more or less, become “overtrained” in the poor sense of the word, to where I had gained greater (savoir) knowledge while remaining entrenched in a state of (connaitre) deadness to

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the subject. That is to say, somewhere in my educational training, I had simply decided (or felt forced into a decision) that I would find creative outlets elsewhere—such as in music, poetry, philosophy, writing, and pedagogy even. A deep and meaningful sense of inquiry and of personal enjoyment would be had in these other realms of interest. It was not such a big deal to me, as again, I felt competent in what I knew, or so I had convinced myself. Meanwhile, I questioned the value of mathematics teaching and learning as related in the preface, blind to any connection between my feelings of inauthenticity and fraud as a teacher, and this unspoken feeling of estrangement from the subject. Only, it was in the presence of another’s intellectual liveliness thriving within a mathematical context, that I was confronted by my own distance to the subject. Maybe more importantly, I was being shown that one could indeed bring forth one’s sense of curiosity to bear—or to play—in mathematics. One need not publish in a mathematics journal to feel unapologetic about being mathematically animate and alive. I was at once enthralled (for the possibilities) and unnerved (for what it showed to me of my current state of being), for Dan’s example was presenting possibilities that I had long ago discarded—the possibility that one could be intellectually curious while thinking mathematics. It is likely no coincidence, in this regard, that “curiosity” and “cure” share the same etymological root in the Latin cura, meaning “care, concern.” Perhaps there is a clue given in how to cure that which is not well, or broken—that is, through reintroducing the elements of care and of curiosity. Many of the teachers I spoke with cited curiosity (in one of its myriad guises) as that crucial trait, that when reintroduced into their relationship with mathematics, renewed and reawakened their interest in mathematics. My own relationship to mathematics, which had clearly barred both care and curiosity, was one in need of greater care, but also a cure, or healing. Descriptors such as “love-hate” relationship, or even “abusive” relationship felt resonant to how I learned and dealt with mathematics. I could perform and look great on the outside, in just about any classroom setting whether as a doctoral student or as the teacher, but when I was alone, there was no love, no curiosity, no real care or desire to return to mathematics. Clearly, some kind of healing in my relationship to mathematics was called for, although unbeknownst to me until this time. I would call attention to the fact—if it is not a bit of a surprise already—that I am speaking of healing within a context of learning. Under a savoir conception of learning (which is how it mostly is in the field) the concern by and large tends toward “deeper” and “more robust” learning; but when one includes also a connaitre way of knowing, or relationship, one can speak meaningfully of notions such as “abusive” relationships and of healing. It is not simply a matter of deepening or solidifying one’s knowledge in relationship. Instead, one heals what requires mending, or else nurtures what is already present.This is to say that with relationship, a new set of metaphors and images assert themselves, in turn, raising new

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queries insofar as teacher learning and teacher preparation are concerned (as well as with student learning). Although it is outside of the scope of this investigation to explore in any empirical manner such ramifications, I would leave it to future work to extend what I will now consider only briefly and conjecturally. One such ramification has to do with the role of the mentor in such healing. That is, how might a mentor or teacher help to effect such healing in the mentored? If I take the anecdote between Dan and me as a case in point, I think then that it has something to do with issues of authenticity in the teacher/guide; and for the student, the necessity of trust in the guide, as well as having a model for how also to trust the mathematics. Recall my own self-doubts and feelings of inauthenticity as a teacher of mathematics—as one who did not practice what I “preached.” Consider also the etymological roots of the word “authentic,” deriving from the Greek autos (“self ”) + hentes (“doer, being”). To be authentic in this sense is to do or to be oneself. In my case, I could talk and teach all the mathematics I needed, but I did not do or embody (the “being” of ) mathematics. I was not mathematically authentic. In contrast, Dan was, in fact, embodying and doing mathematics. That is where the trust came in for me—I could see and sense something authentic, next to which, I could feel the sting of my own lack thereof. But besides being openly curious about the mathematics, he also appeared to me to be having fun and enjoying himself. And I think this is crucial, not just for the apparent reason that I, like many other humans, seek fun and enjoyment wherever I can.1 Instead, recall my comments from Chapter 2, where I suggested that both feeling and passion require courage because they require a relinquishment of control. To quote, “They involve the possibility of becoming unseated, moved, and even transformed by other—that is, ‘being affected’ by this other.” Similarly, curiosity requires risk, as it too involves being unseated by what one finds. In a manner of speaking then, someone who is able to feel excitement and to be curious about mathematics could be said to be capable of giving himself or herself over to the subject.To some extent, then, these are ways of embodying and exhibiting trust in the subject. As the one being “mentored” (although implicitly in the situation), had I simply been told “to enjoy and to have fun with mathematics,” or else to “be more curious and to take more risks”—instead of laying witness to a surrendering toward mathematics—there would clearly not have been the same meaning for me. The real significance of the episode had to do with being present to a letting go into mathematical exploration and a trusting of the mathematics by another. I could then see and acknowledge its possibility. Also important for me, it was not by some mathematical celebrity, but by a fellow mathematics educator. That way, I could not rationalize to myself, “Well, he’s already got all the knowledge in the world to 1 Consider that Bruner (1979) suggested that one of two primary criteria in determining whether something is worth teaching was “Whether the knowledge gives a sense of delight . . .” (p. 109).

An Epilogue in Two Parts 129

have the kind of confidence required for that kind of freeness of mind.” Instead, I knew I had the know-how, just not the inclinations he displayed. What Dan presented for me was a model of mathematical behavior and being that (I hate to admit) I could more or less imitate. Of course, some of us do not think much of imitation (including myself with my healthy-sized ego), but consider what phenomenologist Max van Manen (2002) writes of it: Indeed, this imitational process (mimesis) is the meaning of learning. In early English, to “learn” meant to teach or to let learn, as well as to learn. It would then be correct to say that someone could “learn” someone to learn something. In the Dutch language,“to learn” (leren) is still used interchangeably for teaching and learning. “Teacher” is leraar; “student” is leerling. Etymologically, to learn means to follow the traces, tracks or footprints of one who has gone before. In this sense, the teacher or parent who is able to “let learn” therefore must be an even better learner than the child who is being “let learn”.2 (p. 28) Dan’s “teaching” to me, thus, was in him learning, or letting learn himself. And what I gathered through his example was of how to let learn for myself—that is, I gained a vision of how to let go into mathematical exploration. I could trust him as a teacher because I saw him trusting the mathematics enough to “undergo” the excitement of inquiry. This is different from trusting that algorithms and procedures will work, or that an answer will eventually come. I already held that kind of trust. Instead, this was about trusting that mathematics would receive my own creative, exploratory, and curiosity-laden inclinations. That is what I needed to take in and to witness in action. I see in Dan’s actions and embodiment (which are not considered explicitly pedagogical in the conventional sense) a complementary vision for teaching to the usual one of knowing well and conveying clearly the subject. It is the capacity to be carried away, or taken by the subject. Some may call it passion, or excitement. I would call it the taking up a stance of receptivity and curiosity that would lead to the undergoing experience of grace—or simply, the capacity to let learn. Regardless of its designation, it pertains to teaching precisely because a teacher 2 In a similar vein in regards to the notion of “letting learn,” philosopher Martin Heidegger (1954/1968) writes, “And why is teaching more difficult than learning? Not because the teacher must have a larger store of information, and have it always ready. Teaching is more difficult than learning because what teaching calls for is this: to let learn. The real teacher, in fact, lets nothing else be learned than—learning. This conduct, therefore, often produces the impression that we properly learn nothing from him, if by ‘learning’ we now suddenly understand merely the procurement of useful information.The teacher is ahead of his apprentices in this alone, that he has still far more to learn than they—he has to learn to let them learn. The teacher must be capable of being more teachable than the apprentices” (p. 15).

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who does not trust mathematics enough to let go to it, and to be truly excited and taken by it, cannot offer excitement about the subject to his or her own students. Such a teacher cannot do so because he or she cannot model, and thus present an embodied image for how to risk letting go into mathematical inquiry—to let learn—and become overtaken by the excitement and enthrallment of such experience. This is to say that without a relationship to mathematics that allows for play, creativity, curiosity, and personal meaning to be at play, a teacher will not hold a vision of mathematics that is compelling enough for his or her students to do the same. The real significance, then, of a teacher’s sense of authenticity may be due exactly to this idea that the meaning of learning is in mimesis (imitation). A teacher helps to bring about healing in a student’s relationship to the subject by presenting a model (as “self doers” of mathematics) of a healed relationship himself or herself.3 Without that, what else is the teacher but a peddler of inert knowledge? If one accepts such a premise, then teachers and teacher educators are indeed called upon to nurture, or else outright heal their own relationships to mathematics; and further, to have some sense of authenticity as mathematical do-ers in their own right. Teachers cannot content themselves with simply knowing the subject well, but instead, must aspire to give themselves over to their own subject, an idea that educational philosopher David Blacker describes as reliving and re-invoking the myth of the original teacher and mentor: Mentor, who in Homer’s Odyssey is able in his own way to give himself over to his discipline (of wisdom)— Wisdom [as Athene, goddess of wisdom] speaks through Mentor, not from him, and so is heard; Mentor’s proper role is to vanish into wisdom for the sake of wisdom’s pupil, as the pupil searches for his past and for his future. (1997, p. 19) Similarly, a mathematics teacher must also have the capacity to “lose” himself or herself into mathematics—into the I–Thou relationship that would subsume both self and mathematics. Mathematical authenticity emerges through “vanishing” into the mathematics. Only then might the call of mathematics (and not just the teacher’s urgings) be heard by the student. This then may be at the heart of what it means for a teacher/mentor to bring a student into relationship with the other of mathematics. He or she finds a way to “get out of the way” so that the student might meet the subject directly; and the teacher steps aside, precisely, by entering into relationship with—that is, by joining in with, not removing oneself from— the mathematics at hand.

3 From such a view, the teacher (or teacher educator) is less a disseminator of information, or facilitator toward construction of new knowledge, but a healer of unhealed relationships with mathematics. Under such a paradigm, the central dictum must be the Biblical, “Healer heal thyself.”

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The phenomenon of authenticity (or inauthenticity) then is not simply a matter of personal valuation and esteem, but also one with pedagogical implications; it is a potential indicator for the viability of the relationship between educator and discipline—a relationship that one either models for others as healed, or tucks away under empty rationalizations for the study of the field (as I once did). It is an idea that is pitched by Richard, one of the teachers I interviewed. I will leave it to him to have the last say here, for he speaks eloquently of authenticity, of doing mathematics, and of being a learner of mathematics as a teacher and teacher educator— So, I think we, hopefully first and foremost, are students of teaching and learning in the domain of mathematics. And celebrate the fact that we’ve got a manageable domain. And don’t pretend that we’re mathematicians. But do learn more about mathematics whenever, and learn to do mathematics. And that’s an essential piece here. If we don’t learn to do mathematics in some authentic way . . . not as a mathematician would do it necessarily, but maybe making a conjecture, testing a conjecture, understanding some little piece of something more deeply, hopefully something that’s important in the school culture, school mathematics, that the people (students) we work with have to understand. But I think that if we don’t practice doing mathematics, whatever our horizon is, then you’ll get in trouble because it’s inauthentic. “I’m propounding a discipline I don’t practice.” You have to walk the walk in whatever way that you can.

Part II: Relationship as a Returning: Going Past “Understanding” In mathematics you don’t understand things.You just get used to them. (Attributed to the mathematician John von Neumann) Of the various ideas in this book, the theme of “returning to” is a recurrent one. In Chapter 2, I pointed out that what holds many mathematicians to the discipline is exactly the experiences of grace in beauty and understanding, that these undergoings are what draw them back to mathematics. Even in regards to the phenomenon of frustration, there is an element of returning involved. Recall an earlier comment I made concerning frustration in mathematical activity— If one begins with an acknowledgement and acceptance that struggle, and possibly even frustration, may ultimately be an intrinsic part of the experience of mathematical activity, then it becomes clearer to see that what sets apart those who are successful from those who are not may not necessarily be that the former persevere through their frustrations, for even the most accomplished will pull away from problems. What does seem to separate

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them is that some return to the problem. On the other hand, those who do not return to the scene of their struggle are likely those who do not find success with, or meaning in mathematical activity. Then in Chapter 3, I elaborated upon repetition-as-an-end-per-se as being an aspect of “will.” But of course such repetition is a form of “returning”: No doubt most of us have experienced how repeated exposure to an idea—or in particular, to a difficult idea—can gradually reveal, or “open up” new aspects and distinctions surrounding that idea not noticed before by us. It is a process whereby one’s understanding becomes progressively fine-grained, drawing one further into what could be called an intimacy with the idea. From Chapter 4, the image of “intertwinings” (as in inter-esse) between self and mathematics hints at trajectories that weave in and out—that is, move away from and then return back toward one another.These intertwinings then depict continual returnings between self and mathematics. And of course, in Chapter 5 I characterized the doing of mathematics as a mode of mathematical engagement that afforded ongoing opportunities for returning to mathematics. Yet, is it not the case that such “returning to” lies at the heart of relationship, for what is a true relationship but that place where one returns to again and again? Is that not how relationships are formed, through continual returnings? Indeed its etymological derivation (from the Latin relationem, meaning “a bringing back, or returning”) hints at this innate (and even secret) meaning. If one takes seriously such linguistic origins, then, it is not unreasonable to take things a step further, and in fact, to speak of relationship as being in the returning itself—not the place that one returns to, but the act of returning. Not the thing, but the action; not the reification, but the participatory act itself. I am in relationship to that which I return to when I am returning. It is with such an understanding of relationship that I would like now to reprise the interview transcript with the “frustratingly evasive” mathematician I presented earlier, just before the theoretical climb of Chapter 2. The first time round with Ponce’s comments, I interpreted his responses as being “perversely” noncommittal.Yet, what seemed to be perverse then, now sounds to me profound. What I recognize is that the notion itself of “deep” understanding (that I was after then) arises from a view of knowing steeped in savoir. For Ponce, and others like him, who would stay in relationship and would continue in their returning toward mathematics, it is a matter, not of (achieving) depth, but of deepening. Listen, again, to what he says. Listen carefully. What he speaks of, unflinchingly and repeatedly, is of his recurrent returning to mathematics, of a persistence of connection and engagement; in this way, he speaks of relationship. (He also speaks of my own feelings about this current inquiry.)

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Yuichi: If you can think back to certain times where you felt that you understood a mathematical concept on a fairly deep and comprehensive level, please tell me about it. Ponce: I think that it’s hard for me to . . . find such an occasion. What happens always is, I find, feel that it’s actually that’s very often is that my understanding becomes better, but I don’t know whether I can say that I understood something very deeply or completely. I never had that feeling, and from all of my experience, I don’t think that it can be some final process, I mean, because you do go deeper and deeper, and you feel . . . and you usually remember those steps when you get deeper and deeper, but I’m not sure that I can really say that I understood something completely. Yuichi: So from what you said—if I’m interpreting correctly— you’re saying that you’re not content to just stop at say “I have deep understanding” because you feel that you can continue to— Ponce: Yes, in the past that’s what’s happened all the time. It seems that I understand things deeper and deeper. Yuichi: I see. Ponce: So, I don’t think I can really say that I understand something completely . . . I understand some definitions, I think, completely, like “What is a graph?” or “What is a set?” But even with set theory, as you know, one can ask brilliant questions about basic set theory which requires very careful language, so if you don’t use it, contradictions can be brought [about] immediately, so if you understand elementary set theory, I’m not sure . . . I understand many things about it but . . . Yuichi: Did you actually have issues with my question to some extent of trying to qualify understanding, this idea of “deep?” Ponce: Yeah, I think that it’s all relative. And when more and more time you spend in mathematics, more and more connections you see between things, and then, some applications which you might have thought you know . . . Yet of course, if you can understand this deeper . . . in addition to what I knew about this subject, I know more . . . It’s all relative, I mean. What is deep water? Yuichi: I see. So, you’re talking, and I’m rethinking my questions. Maybe I need to rephrase it somehow . . .You mentioned there not being a “final process.” But isn’t there sometimes? Couldn’t that be a “deep” understanding? Ponce: I question further and very often I cannot answer the questions I ask. With some areas, you just deliberately decide not to question anymore because there’s finite time there. You are interested in

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results, and you cannot dwell in one direction all the time so you have to stop it somewhere. When I stop, does that mean that . . . I understand deep and I don’t want to understand more? No, rather, in most cases it’s just that I deliberately decided to stop at this point without trying to understand things further.

APPENDIX: A NARRATIVIZED “METHODOLOGY”

The point of the creative act of expression is to arrive at a target that is not there: once the target is there, the expression has been accomplished. (Albert Hofstadter, 1965, p. 35) Of course, not all stories are successful. There are good stories and mediocre stories and downright bad stories. How are they to be judged? If they do not aim at a static or “literal” reality, how can we discern whether one telling of events is any better or more worthy than another? The answer is this: a story must be judged according to whether it makes sense. (David Abram, 1996, p. 265) The purpose of this section is to offer a brief narrative of how I arrived at the three core “themes” (Chapters 2, 3, and 4) that constitute the heart of this book. One might consider it a condensed and narrativized “methodology” section, as if in a technical report, although I will not refer to any explicit brand of methodology in the narrative.1 I might remind the reader that the aim of the entire study from the outset was to shape or even to define, through both theoretical and empirical means, a knowing steeped in relationship, and in particular, to mathematics. Implicit in shaping and defining some thing or idea is the understanding that one does not have a clear idea of it to begin with. As such, Hofstadter’s quote above offers what I feel is an apt image for the difficulties inherent in such an undertaking.

1 Instead, I refer the interested reader to Brown’s (1989) “cognitive aesthetics” and to Van Manen’s (1990) hermeneutic phenomenology, although I, myself, have been lax in my adherence to either methodology.

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This same tension is expressed in a question asked of me at an early stage of the inquiry, “If you don’t quite know exactly what it is, how do you know what to look for?” The short answer to the question is to say that it has been through an iterative process of triangulation across different “points of reference”—self-reflective, theoretical, philosophical, empirical, and even anecdotal—that I have arrived at the ideas in this book. The longer answer is to draw a parallel to how teachers come to arriving at the “big pictures” of a subject such as algebra. By “big picture,” I mean not only the overarching organizational structure of a subject—that is, the larger framework as a static entity—but also knowing in a dynamic and heuristic sense how and where things fit within that larger framework. One might liken it to knowing how a closet is organized, where the socks go, where the pants go, and so on.Then, when someone throws you a turtleneck sweater, you know where to hang it, or to fold it and tuck it away. For some teachers, gaining a big picture of a subject is in coming to know how to “sort out” within one’s mind, the various student comments that are made around the topic at hand. Similarly, my task, partly, in this work was coming up with such a big picture of relationship. It was to find some organizational scheme for the phenomenon of relationship to subject matter—based upon the “points of reference” mentioned earlier—that would allow me to tuck away different aspects of the experience/ phenomenon fairly systematically, while also not becoming overwhelmed with having too such places of storage. In this sense, what I sought were constituent aspects of the phenomenon. What follows is an elaboration of this process. I began with four “points of reference” in regards to some inkling of an idea of “relationship” to subject matter, and specifically to mathematics. First, there were my own experiences around the phenomenon, and reflections upon those experiences, the most relevant of which I have already elaborated upon in both the Preface and Chapter 1. Second, I studied written accounts of mathematicians’ relationship to mathematics, both autobiographical as well as second- and thirdperson descriptions. Third, I looked to the theoretical literature: mostly the philosophical and sociological, much of which were enfolded within the three middle chapters. Lastly, I turned to artists’ and musicians’ accounts of their own relationship to their discipline and craft, so as to fine-tune my own sense of what “relationship” to a discipline might mean. I did so mostly through informal conversation, which later became more formalized in the form of clinical interviews with six music teachers, all but one of whom were also active as professional musicians. My intent at this point was to seek out a pattern of meanings emergent from these four sources. It was important, in particular, that I seek such emergent meanings. Rather than my encroaching upon the information with a priori definitions (I may have had a priori assumptions, but nothing so clear as a definition), the attitude was one of seeking out common structures of meaning that evolved out of the so-called “data.”

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Thus, using those four sources, I tentatively settled upon a series of definitions of what I thought could constitute the core features of a relationship to mathematics. These “core features” were little more than my best guesses of what relationship to mathematics might be. They included such themes as: • • • • •

taking delight in math passion toward math privileging process over product finding expression of being through mathematical activity having a capacity for “staying with” a problem.

At one point, I had created a list of up to 10 such qualities that I thought would best define a person’s interest and connection to the subject. Having now made explicit to myself my expectations of what I might find, I began conducting clinical interviews—conversational explorations, really—with mathematics teachers to see how well these features would arise in the narrative offered by the teachers. I could have chosen mathematicians, engineers, scientists, or even students of mathematics, but I was ultimately interested in how these ideas might relate to mathematics teachers and to teaching, and so, my choice. For selection, I sought out eight math teachers, all but one of whom met the following two criteria: (1) that they taught mathematics for more than 10 years, and (2) that they expressed excitement about being a mathematics teacher. I took these two criteria as the best proxy for what I considered an interest in, or connection to mathematics, again, as a teacher; I was also interested in the sense of relationship on the part of teachers as regards familiarity (à la, the standard translation of “connaitre”), especially, developed over a course of time. I proceeded to ask the teachers for a narrative of their own relationship, not just to teaching, but also specifically to mathematics, from as far as back as they could remember to the present moment. The narratives turned out quite contrary to my expectations, as many of the expected “features” never quite emerged. Yet, the teachers also spoke of alternative ways in which they were making meaning of their relationship to mathematics as teachers that I had not anticipated (an example being that most spoke of the excitement in coming to an understanding, which I had originally not heard as an instance of a grace-like response to mathematics). My task at that point was to seek out the commonality among the descriptions. It was in trying to capture some essential commonalities and cohesion among the narratives that the three themes that constitute the heart of this book emerged.This is to say that the three themes satisfied the following criteria: the smallest and fewest possible such characterizations of connection and relationship to mathematics that would be large enough to both contain and order the ideas within the narratives while also lending insight into the phenomenon at hand (a kind of “basis,” if you will—à la, linear algebra). In this way, I sought a nonredundant list of themes (i.e., the

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“independence”) without also having to sidestep, or disregard any portion of what was found empirically (i.e., the “spanning” property)—not just among the eight mathematics teachers’ descriptions, but also among the six music teachers’ narratives, as well as my own, along with what was in the literature already—and in that endeavor, I found that the three themes offered the “big picture” structure that I sought. But also, the three themes registered for me a resonance, or felt sense, of trueness to the phenomenon at hand. Contrary to my sense of unease with the seemingly arbitrary and disconnected qualities of the bulleted list of themes, these three characterizations offered for me a felt sense of a convergence of ideas. I might note that I continued with clinical interviews of mathematics teachers— I recruited 10 more for the task, five of whom currently work as mathematics educators (i.e., teachers of teachers)—and continued with gathering narratives of their relationship to mathematics. These later interviews were conducted not for the sake of establishing greater validity, but more for understanding with greater clarity each of the three themes. Thus, this second wave of mathematics teachers and mathematics teacher educators were selected according to how well I believed that they could potentially inform any one or more of the themes. Again, if I have not been explicit enough in stating so, my gathering of the teachers’ accounts should not be looked upon as verification of the validity of the ideas of this book. Neither should their accounts be considered representative of a larger population of teachers. Instead, they should be considered as offering shape and form, as well as impetus, to my own reflections upon what it might mean for a person to be in relationship to the subject. This study, then, would be best understood as an attempt to give shape to what it would mean to be in relationship to mathematics. At the same time, it also suggests a way of framing—that is, thinking about—the cluster of phenomenon that surrounds such a way of knowing. As the quote by David Abram that heads this appendix section implies, I will have told a successful story if it helps to make sense, in this case, of a phenomenon of relationship to mathematics.

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INDEX

Abram, D. 138 active receptivity 62 aesthetic experience 37–45 aesthetic pleasure 37, 68 aesthetic responses 45–8, 49 aesthetic theory 37 affect 27 “allegory of the cave” 123 Art as Experience 40, 62 authoring mathematics 109 Bai, H. 63 Ball, D. 106–7 Bartlett, F. 35 Bass, H. 106–7 Bereiter, C. 69 Blacker, D. 130 Bolyai, W. 43 Brousseau, G. 111 Brown, S. 30, 33, 44, 101, 113 Brownell, W. 15 Bruner, J. 35–6 Buber, M. 13, 18–19 Buddhist epistemology 29 Burton, L. 30, 64 Campbell, J. 37 Cantor, G. 30 Chazan, D. 124 cognition 36 cognitive conflict 65

cognitive dissonance 72 cognitive emotions 29–32 Communities of Practice 11 confusion 34–5 connaitre 6, 7, 9, 16, 22, 36, 63, 71, 109, 112, 127 Critique of Judgment 37 Csikszentmihalyi, M. 31 D’Ambrosio, U. 80 decompressed knowing 107 Dewey, J. 40, 58, 61, 102 “Doctrine of reminiscence” 24 “doing” mathematics 11–12; asking questions and posing problems 112–14; doing pedagogically relevant mathematics 115–25; feasibility and relevance 114–16; interwoven complementary rationale 108–11; ownership to inter-esse 111–12; rationale 105–8; relation to life path of being a mathematics teacher/educator 104–25 doing pedagogically relevant mathematics 115–25; sense making and meaning making 119–25; visual representation for (1 + 2 + 3 + . . . + n)2 120; visual representation for 13 + 23 + 33 + . . . + n3 121; where the problems come from and where they remain 116–19

148

Index

emotion 27 “euphoria” 30 feeling 27–9, 42–3 formalism 25 Friedman M. 27 frustration 32–5 Fundamental Theorem of Calculus 30, 33 Gattegno, C. 59 Gauss, C.F. 43 generative aesthetic 46–8 Gladstone, J. 64, 71 gnostic 21–2 Green, T. 84 Hardy, G.H. 30, 39 Hawkins, D. 102 “heightened sense of experience” 102 Heller, A. 28, 67–8 Hofstadter, D. 47 I and Thou 18 I–It relation 18–19, 62 “improper art” 38, 62 Ingarden, R. 38 “intellectual passions” 44 intentional learning 69 inter-esse 75, 87, 119; ownership to 111–12; tinged knowings 97–8; understanding as 98–103 intertwinning 79–91, 132; certainty and cleanliness of thought 88–9; challenge and arête 79–81; curiosity 83–7; with other and its relation to motivation 75–8; social connection 81–3; teacher and different faces of mathematics 89–91 intertwinning trajectories 94 intimacy 19 intrinsic motivation 77–8 I–Thou relation 18, 22, 27, 57, 68 James, W. 57–8 Joyce, J. 37, 38 justification 36 Kant, I. 37, 55 Kepler, J. 29–30

knowing 36; and its constituent aspects 5–9; qualitative nature 35–6 knowing mathematics 28 “knowledge packages” 107 Kohler, W. 31 Langer, E. 69 learning 36 learning mathematics 11–12 Ma, L. 107 MacLaughlin, M. 105–6 Maslow, A. 31 Mason. J. 64 mathematical confusion 35 mathematical elegance 46 Mathematicians as Enquirers: Learning about learning mathematics 30 mathematics: aesthetic experience of 37–45; aesthetic responses to 45–8; affect, feelings, and emotions 27–36; bringing forth of self 57–73; certainty and cleanliness of thought 88–9; challenge and arête 79–81; comparison across disciplines 9–12; core features of relationship to 137; curiosity 83–7; design and methodology 12–17; desirable difficulties 64–5; different faces of 89–91; “doing” in relation to the life path of being a mathematics teacher/educator 104–25; examples of intertwinings 79–91; generative aesthetic 46–8; gnostic and pathic 21–2; inter-esse tinged knowings 97–8; intertwining with other and its relation to motivation 75–8; knowing and its constituent aspects 5–9; learning 11; narrativized methodology 135–8; nature of knowing 35–6; as other through the aperture of music 22–4; passion and aesthetics 43–5; passion as an undergoing 40–3; as platonic other 24–7; relationship as a returning 131–4; relationship as interest 74–103; relationship as reciprocity 18–56; relationship phenomenon 1–17; relationship to 3–5; repetition 66–70; returning to 113–14; self as reflected in life projects and trajectories 91–8; social

Index

connection 81–3; “standing before” so as to unlearn 70–3; “staying with” 112–14; “staying with the problem” 61–3; subjective universality 54–6; “undergoing”-type experience case study 51–4; understanding as ‘Aha’ experience 49–51; understanding as an undergoing experience 48–56; understanding as inter-esse 98–103; understanding as verstehen 70–3; usefulness of 39 mathematics teacher/educator: feasibility and relevance as predicament for 114–16; redefinition of 105; relation of “doing” mathematics to 104–25 Mentor 130 Mikio, F. 59 mimesis 129 mindfulness 69 Moritz, K.P. 37 music: and mathematics 22–4, 76–7 “negative capability” 61, 62, 70 nonaesthetic pleasure 62 Oppenheimer, J. R. 31, 46 partial “doing” of mathematics 111 participation 11 pathic 21–2 pedagogical content knowledge 106 “pedagogical motivation” 119 “pedagogically useful mathematical understanding” 106–7 Penrose, R. 45, 46 perception 62, 63, 65, 68, 70 phenomenon of authenticity 131 Pirsig, R. 48 Platonic Idealism see Platonism Platonism 24–5 Poincare, H. 46, 47 Polya, G. 72–3 producing mathematics 109 “proper art” 38 Pushing Tin 42 Ramanujan, S. 109 receptivity 62

149

reciprocity 18–56 recognition 62, 65, 68, 70 “reconstructive doing” 63 reflective thinking 61 reification 11 Reimann, B. 43 Remembering 35–6 repetition 66–70 returning to mathematics 113–14 “right way syndrome” 113 Rothstein, E. 30, 37, 55, 76, 123 savoir 6, 7, 9, 21, 36, 63, 71, 106, 112, 127 Scardamalia, M. 69 Scheffler, I. 29 Schifter, D. 33, 111 Shulman, L. 106 Sinclair, N. 46–7 Skemp, R. 15 social connection 81–3 “special sensibility” 47 Stacey, K. 64 “staying with the problem” 59, 61–3, 70, 71 Subject–object relationship 18 subject-to-subject 22 subjective universality 54–6 Talbert, J. 105–6 true relationship 31 understanding: “aha” experiences 49; inter-esse 98–103; undergoing experience 48–56; verstehen 70–3 unlearning 72–3 van Engen, H. 15 Van Manen, M. 21, 129 verstehen 70–3 “waiting” 59–60, 65 Walter, M. 113 Wenger, E. 11 Wertime, R. 44 Winchester, I. 59 Wittgeinsten, L. 28–9