The study of mathematics is apt to commence in disappointment. The important applications of the science, the theoretical interest of its ideas, and the logical rigour of its methods, all generate the expectation of a speedy introduction to processes of interest. . . Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it. . . .The reason for this failure of the science to live up to its great reputation is that its fundamental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances. Accordingly, the unfortunate learner finds himself struggling to acquire a knowledge of a mass of details which are not illuminated by any general conception. (Whitehead, 1911/1948, pp. 1-2)
Alfred North Whitehead opens his book, An Introduction to Mathematics (1911/1948) by lamenting the trivial encounters that most people have with mathematics. In this thin but dense volume, he sets out to correct the "pedantry" of traditional mathematics teaching by helping his readers to develop an appreciation for the power of mathematics as a system of human thought. In a similar spirit, Michael Guillen (1983), a contemporary research mathematician and physicist, argues that those who are alienated from mathematics are "deprived of being able to consider for [themselves] the manifold and unique ways in which mathematics bears on our far-flung human concerns, including questions about God" . Passionately, Guillen writes about his desire to ignite readers with his excitement about mathematics, to persuade them that mathematics is a form of human imagination and vision accessible to all.
Despite the good intentions and the passions of mathematics lovers like Whitehead and Guillen, the school mathematics experience of most Americans is and has been uninspiring at best and mentally and emotionally crushing at worst. "Mathematical presentations, whether in books or in the classroom, are often perceived as authoritarian" (Davis & Hersh, 1981, p. 282). Ironically, the most logical of the human disciplines of knowledge is transformed through a mis-representative pedagogy into a body of precepts and facts to be remembered "because the teacher said so." Despite its power, rich traditions, and beauty, mathematics is too often unknown, misunderstood, and rendered inaccessible. The consequences of traditional mathematics teaching have been documented; they include lack of meaningful understanding, susceptibility to "mathematical puffery and nonsense" (Schoenfeld, in press b), low and uneven participation, personal dread.
As an experienced mathematics teacher and late blooming mathematics amateur, I, too, worry about our culture's mathematical disenfranchisement and disenchantment. Because this dissertation grows out of my concern for improving the character and quality of mathematics teaching, I begin this first chapter by exploring mathematical pedagogy — a vision of mathematics teaching and learning quite different from what students encounter in most classrooms.
Mathematical pedagogy, so named because disciplinary mathematics is at its core, affords students the opportunity to develop both understanding and power in mathematics. It is an approach to teaching math that emphasizes making sense and having control in situations that involve quantitative and spatial reasoning. In order to give the reader a vision of mathematical pedagogy, this chapter begins with a story from a third grade class. Examining what is going on in this class takes us on a short trek into the philosophy of mathematics and the inherent relationship between epistemology and pedagogy. To highlight the distinctive features of a mathematical pedagogy, I then compare this view of mathematics teaching with two other dominant views of mathematics teaching.
Rooted in mathematics itself, the goal of a mathematical pedagogy is to help students develop mathematical power and to become active participants in mathematics as a system of human thought. To do this, pupils must learn to make sense of and use concepts and procedures that others have invented — the body of accumulated knowledge in the discipline — but they also must have experience "doing" mathematics, developing and pursuing mathematical hunches themselves, inventing mathematics, and learning to make mathematical arguments for their ideas (see Romberg, 1983). Good mathematics teaching, according to this
view, should eventually result in meaningful understandings of concepts and procedures, as well as in understandings about mathematics: what it means to "do" mathematics and how one establishes the validity of answers, for instance. I embrace this vision of mathematical pedagogy in reaction to the mindless mathematics activity that prevails in so many schools and results in so little meaningful student learning (Erlwanger, 1975; Wheeler, 1980).
Mathematical pedagogy, however, is founded on yet another view of mathematics. On one hand, just as in conceptual mathematics teaching, "meaningful understanding" is emphasized. Students are helped to acquire knowledge of concepts and procedures, the relationships among them, and why they work. The goals are different, however. For example, learning computational skills is valued as much for what students can learn about numbers, numeration, and operations with numbers than as an end in itself. On the other hand, mathematical pedagogy also explicitly emphasizes not only the substance of mathematics but also its nature and epistemology (see Davis, 1967). Just as central as understanding mathematical concepts and procedures is understanding what it means to do mathematics, being able to validate one's own answers, having opportunities to engage in mathematical argument, and seeing value in mathematics beyond its utility in familiar everyday settings.
Mathematical pedagogy assumes that students must be actively involved in constructing their own understandings, in discovering and inventing mathematics. The basis for this emerges directly from a largely constructivist epistemology of the discipline. Mathematical pedagogy also takes a group orientation to classroom learning: The model is not of a teacher facilitating the learning of individual students. Instead, this approach uses the classroom as a mathematical community; learning involves collaboration among individuals. Although this emphasis on the group overlaps the current vogue for cooperative learning (e.g., Slavin, 1978), the warrant for this orientation in mathematical pedagogy derives from a view of mathematics
as a disciplinary community.
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