What Does Understanding Mathematics Mean for Teachers?

Relationship as a Metaphor for Knowing

Yuichi Handa

• Publisher: Routledge
• Year: 2013
• Print ISBN: 9781138017764
• eBook ISBN: 9781136870521
• Edition: 1
• Topic: Education
• Subtopic: Education General

This book opens up alternative ways of thinking and talking about ways in which a person can "know" a subject (in this case, mathematics), leading 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. 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.

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 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 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 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 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 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, f...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Contents
• Preface
• Acknowledgments
• 1. Introduction to a Phenomenon
• 1.9 A Tangent Prior to the Rise (and Run)
• 2. Relationship as Reciprocity: Grace
• 3. A Bringing Forth of Self: Will
• 4. Relationship as Interest
• 5. “Doing” Mathematics and its Relation to the Life Path of Being a Mathematics Teacher/Educator
• An Epilogue in Two Parts
• Appendix: A Narrativized "Methodology"
• References
• Index