Teaching Mathematics

Toward a Sound Alternative

Brent Davis

• Publisher: Routledge
• Year: 2013
• eBook ISBN:
  9781136520914
• Topic: Education
• Subtopic: Education General

This book presents an approach to the teaching of mathematics that departs radically from conventional prescription-oriented and management-based methods.  It brings together recent developments in such diverse fields as continental and pragmatist philosophy, enactivist thought, critical discourses, cognitive theory, evolution, ecology, and mathematics, and challenges the assumptions that permeate much of mathematics teaching. The discussion focuses on the language used to frame the role of the teacher and is developed around the commonsense distinctions drawn between thought and action, subject and object, individual and collective, fact and fiction, teacher and student, and classroom tasks and real life. The discussion also addresses the question of how mathematics teaching can be reformed to better suit current academic and social climates. Making use of the theoretical framework of enactivism, the book explores the subject through an account of a middle school teacher's appreciation and understanding of her role. Teaching mathematics, as both the report of this teacher's experience and the discussion make clear, demands an embracing of ambiguity, uncertainty, complexity, and moral responsibility. Courses for Adoption
Education: Mathematics for Elementary Teachers, Methods for Teaching Elementary Schools, Methods for Teaching Secondary Schools, Curriculum Studies, Critical Pedagogy Special Features *Elucidates the importance and relationship between theory and practice. Employs reflective teaching techniques to focus students on their own learning, knowledge, and understanding of mathematics.Details a collaborative venture that traces the development of new thinking and insights about math teaching and learning. *A fine blending of theory with practice.

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Chapter 1

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Close Your Eyes and Listen

Conceptual Underpinnings

Just as nature finds its way to the core of my personal life and becomes inextricably linked with it, so behavior patterns settle into that nature, being deposited in the form of a cultural world. Not only have I a physical world, not only do I live in the midst of earth, air and water, I have around me roads, plantations, villages, streets, churches, implements, a bell, a spoon, a pipe. Each of these objects is molded to the human action which it serves. Each one spreads round it an atmosphere of humanity.

—Maurice Merleau-Ponty¹

I close my eyes, and the planet is auditory only: tree branches twist into tubas and saxes, are caught by large hands that press down valves, and everywhere on this ranch I hear feral music—ghostly tunes made not by animals gone wild but by grasses, sagebrush, and fence wire singing.

—Gretel Ehrlich²

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Section A
Enactivism

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[O]rganism and environment enfold into each other and unfold from one another in the fundamental circularity that is life itself.

—Francisco Varela, Evan Thompson, and Eleanor Rosch³

I listen, because it reminds me again and again that the whole world runs by rhythms I have not yet learned to recognize.

—Thomas Merton⁴

Much of the recent activity in the field of mathematics education has consisted in efforts to negotiate a series of impassable dichotomies—dichotomies that seem to be the direct and inevitable consequences of a collision between traditional objectivist perspectives and more recent subjectivist proposals. In this section, I examine the origins of the mode of bipolar thinking that has given rise to these tensions, seeking not to resolve them but to understand them more deeply. It is by endeavoring to develop such understandings, I suggest, that we open up possibilities for not merely healing the “gaps,” but for side-stepping the mode of thinking (and acting) out of which they arise.

In beginning the discussion in this manner, I do not wish to suggest that the dichotomies that we construct and through which we make sense of the world are unhelpful, but that they should not be regarded as absolute. Thought and language, it can easily be argued, are founded on our capacity to separate “this” from “that”— and so, to deny the possibility of such distinctions is to render human experience completely incomprehensible. It is thus that, rather than suggesting that mind/body, self/other, fact/faction, knower/known and other binaries are false—a move which is, in itself, dichotomizing—I invoke the pragmatist measure of truth and argue that such constructions are valid insofar as they are useful.

The point I attempt to make in this section, then, is that, in terms of mathematics teaching, many dichotomies have outlived their utility, having locked us in a modern mind-set which posits us as essentially autonomous entities: Not only are we isolated from one another, but we are set apart from the universe. The foundation of this sort of dualistic thought is the topic I turn to presently. That discussion serves as a precursor to a brief introduction of an alternative orientation to issues of identity and cognition which, for the moment, I will describe as embracing complex and unpredictable evolutionary dynamics rather than imposing orderly and calculable mechanical processes.

Of all the parts of this book, this section is perhaps the most densely written. As such, I must ask for the indulgence and patience of those for whom some of the ideas are new. Once again, I have structured the writing so that each of these notions will be further developed in a context where they are applied to an issue of mathematics teaching. My main purpose here, in addition to situating some of the ideas philosophically, is to announce some of the central principles that will be used in the subsequent discussions.

Our Modern Heritage

The predominant epistemological perspective of the “modern” era was first announced by mathematician and philosopher René Descartes in the seventeenth century. Descartes, a contemporary of Galileo, Bacon, and Locke, and a predecessor of Newton, articulated two breaks from earlier perspectives on knowledge and modes of inquiry—perspectives which he rejected as inconsistent and unreliable mixtures of fact and fancy.

The first point of departure was on the issue of method, whereby Descartes denounced tradition, hearsay, mysticism, and religion as he called for the pre-eminence of the “natural light” of (mathematical) reason. Voicing a disdain for all other intellectual authorities, Descartes argued that all previous speculation should be rejected until indubitable principles, against which all other knowledge claims could be measured, were derived. In calling for this shift to a particular and narrow conception of reason, Descartes introduced many concepts and arguments which are foundational to modern science and analytic philosophy.

In this regard, perhaps his most noted contribution is his cogito— “I think”—which also marks his second break with tradition. Briefly, in his quest for a certain foundation for his epistemological system, Descartes arrived at the self-evident and self-verifying truth of the statement, “I think,” and this axiom became the solid ground on which he sought to verify or refute all other knowledge.

It is important to note that Descartes' project was built upon a distrust of the evidence of the senses—a suspicion that was inherited from the ancient Greeks.⁵ Because one's knowledge of the world was always and inevitably filtered through untrustworthy sensory organs, one could never know—in any absolute way—the “truth” of the (external) universe. At best, one could build better and better mental representations of the physical world, and the process of assembling those representations demanded a persistent attitude of questioning—an attitude that Descartes introduced as foundational to scientific inquiry. This “method of doubt” was offered as a screen to sift out truths from those knowledge claims that could not be validated. Rational thought was thus offered as a way of knowing that was both superior to and independent of a reliance on the senses. Descartes' model of reason—and the one that was to become the model of rationality in the modern world—was found in geometry, a discipline that offered a process of verification that Descartes regarded as the only route to unimpeachable fact.⁶ For him, geometric reasoning offered a means of deducing the nature of the entire universe from foundational principles, with each deductive step bound to preceding steps in an irrefutable sequence of logical moves. It was thus that, according to Palmer, truth for Descartes was “more than merely the conformity between knower and known, it [was] the subject's rational certainty of this conformity.”⁷ Rational reflection (“rationalism”) rather than empirical observation (“empiricism”) was the key to knowledge.

In establishing the cogito as the foundation and geometric reason as the means of construction—that is, in specifying both the axioms and the logic of this manner of determining truth—Descartes advanced a mode of dualistic thought that permeates modern perspectives of the universe.⁸ Positioning the radical subject (i.e., the modern ideal of “self” or identity as solitary, coherent, and independent of context) as the reference point for all that is known, for example, compelled him to propose the existence of at least one object—an Other—that was independent of himself and relative to whom he could situate himself as part of an objective world. Stated in different terms, in distinguishing the figure of the “self,” Descartes also distinguished the background of the “not-self” which was collected under such names as “other” and “world.” Thus arose the fundamental subject-object dichotomy. Paradoxically, this dichotomy also provided the impetus for the empiricist tradition which, contra Descartes' rationalist proposal, relies on observation and experiment as the basis of knowledge production.

Another split initiated by Descartes' thought was the dichotomizing of mind and body. In arguing that thinking is the basis for all truth, and hence of existence, Descartes was suggesting that a person is essentially a thinking thing—one that is capable of conceiving of itself as existing without a body. Put differently, it is not essential in Descartes' formulation that we have a corporeal existence. Of course, this mind/body separation finds its roots in earlier philosophical and religious thought. However, in giving it a rational “scientific” basis, Descartes set the stage for a series of tensions that now, collectively, serve as a pervasive and resilient backdrop to much, and perhaps most, of Western academic thought.

To elaborate, in constructing the world on the foundation of the cogito, Descartes articulated more than the separations of mind from body, self from other, and representation from reality—all of which might be described as manifestations of a mind/body dualism. In addition to the essential distinction between mental and physical objects (with the consequent priority being assigned to the former), Descartes also contributed to the foundation of a host of other dichotomies, including knower versus known, organism versus environment, human versus nature. Further, the Cartesian orientation contributed to a view of the Self as a unified coherent subject: an autonomous entity that is isolated from others, independently constituted, essentially static, and able to maintain its integrity through diverse experience.

Other consequences of this perspective, which I shall heretofore refer to as “modernist,” included an empirical emphasis on the trustworthiness of methods used to develop knowledge (i.e., more accurate representations of reality). As such, method came to be seen in increasingly mechanical and technological terms; the universe, correspondingly, was reduced to a similarly technical form. Today, machine metaphors frame and reveal Western perspectives on the universe, the earth, nature, our bodies, and—ultimately, with the development of the computer—our minds. With this technical mind-set, the aim of inquiry has grown beyond die desire to better our understandings. The primary goal is now to control the objects of our inquiry. As Palmer elaborates, with the widespread acceptance of Descartes' conclusion that “the world has meaning only with respect to man,” our relationship to the world is no longer cast in terms of open responsiveness, but in “restless efforts to master it.”⁹

And, perhaps most significantly, with thought being afforded priority over being in Descartes' cogito, epistemic issues began to overshadow ontological concerns—a reversal that has had profound implications for our modern conceptions of both knowledge and education. I will return to a further exploration of the consequences of modernist philosophies in later chapters that deal more specifically with these topics.

Before moving on, though, an important point of clarification should be made. In this analysis of Descartes' work and the manner in which it figures into current thinking, my intention is not to denigrate his conclusions. Rather, I believe that he was a luminary, a critical figure in our cultural heritage. His contributions to our philosophical, mathematical, and scientific thought—not to mention our commonsense—have greatly enabled the technologies that permit our current standards of living. However, as I develop in this text, the truths of Descartes, while well-suited to the circumstances of the seventeenth century, do not fit well with the situations in which we find ourselves today. It is thus time to interrogate his legacy, asking ourselves about what we have been taking for granted and how our assumptions have shaped our perceptions and actions. Prompted by a host of crises, ranging from the personal to the planetary, we are coming to the slow realization that we must behave differently— and the capacity to alter our action, I believe, rests on a willingness to explore different ways of thinking.

Foundations of an Alternative

It is interesting to note the prevalence in research reports of the claim that particular methodologies or perspectives are “anti-dualist.” Reacting to a philosophical backlash against Cartesian (modernist or analytic) bipolar thinking, researchers and theorists are quick to point out that they have not succumbed to a “this-or-that” way of thinking. Nevertheless, within mathematics education at least, there seems to be an irresistible tendency to grant priority to one or the other of the “real” known (material or abstract) or the ideal knower—tendencies which, like the favoring of either empirical or rational modes of inquiry, find their origins in the same system of beliefs.

And so, in spite of their apparent diametric opposition, these modernist perspectives can quite easily be shown to be on the same rational loop—a loop which begins with the epistemic primacy of “I think.” The consequences of dualistic ways of thinking, along with extensive critique of such thought, are offered by a group of theorists who tend to be gathered under the title of “postmodernism” (although not always by their own choosing). Unfortunately, while postmodern discourses have offered valuable critiques of Descartes' legacy, it seems that one of the precepts of postmodernism—that is, that the quest for new groundings is doomed to failure—has been profoundly misinterpreted as suggesting that we can say very little about anything. Not surprisingly, this conclusion has prompted numerous and zealous attempts to destroy the foundations of existing structures—thus demonstrating the temporal and contextual nature of all knowledge—while offering in their place the unsteady (and unsatisfactory) ground of fallibilist, relativist, and in...

Table of contents

• Cover
• Half Title
• Critical Education Practice
• Full Title
• Copyright
• Dedication
• Contents
• Series Editor's Preface
• Foreword
• Acknowledgements
• Front Word
• Setting the Tone-Introduction
• Chapter 1 Close Your Eyes and Listen-Conceptual Underpinnings
• Chapter 2 An Ear to the Ground-The Subject Matter
• Chapter 3 Stood on One's Ear-The Educational Endeavor
• Chapter 4 All Ears-Cognition
• Chapter 5 Playing It by Ear-Teaching
• Back Word
• Notes
• Bibliography
• Index