Harvard Educational Review
  1. Teaching Mathematics

    Toward a Sound Alternative

    By Brent Davis

    New York: Garland, 1996. 324 pp. $21.95 (paper).

    Occurring somewhere between the surety of the known and the uncertainty of the unknown, the act of listening is similar to the project of education. It is, after all, when we are not certain that we are compelled to listen. Our listening is always and already in the transformative space of learning. (p. xxiv)

    Brent Davis moves mathematics into the realm of the auditory, where language and speech are the focus. This is not to say he attempts to fracture mathematics learning into auditory and visual components, but, rather, to recenter the learning of mathematics away from its primary focus on the visual to what has been a secondary means of learning, the auditory. When I first began to grasp his contentions, I remembered so many lectures on mathematics; weren't these auditory learning situations? The fact is that in mathematics lectures, the focus is not on what is said by the teacher, but on what is written on the chalkboard. For example, a teacher might say, "Given any linear equation represented in standard form, the slope of the line described by this equation is given as the coefficient of the variable `x,' which is represented by the variable `m'." However, they would likely also write: y = mx + b.

    Clearly, mathematics instruction focuses on the visual, but then the argument could be that mathematics is inherently visual. Why then focus on "word problems," mathematics problems based on language? Although learning and using mathematics requires both the visual and the auditory, the auditory, Davis claims, has been ignored.

    Davis's book is not a simplistic one that merely gives various examples of possible aurally based lessons. He is meticulous in how he develops his framework of mathematics education. He begins from four root terms, listening, mathematics, teaching, and learning, and through five chapters develops his vision in a complete and comprehensive manner.

    In chapter one, he begins by developing the theoretical framework upon which he bases his teaching. It is also apparent from the succeeding chapters that his "theory" is influenced by his own practice in mathematics education as an assistant professor of Curriculum Studies at the University of British Columbia. Chapter one is composed of three parts — Enactivism, Hermeneutics, and Listening. In the Listening section, Davis expands the definition of listening: "It is by listening — by attending to the person's action and situation, and not just to his or her voice — that one comes to know the other" (p. 36). Through this expanded definition, Davis reiterates his earlier contention of the separateness, yet inseparability, of the auditory and the visual. But by "attending to a person's actions and situation," Davis does not mean to look, but to listen, to hear what a person is doing, to what the person is also hearing. Davis ends this chapter by providing a framework of three distinct yet not exclusive "`modes' of listening" (p. 52): Evaluative, Interpretive, and Hermeneutic. He describes and gives examples of each.

    Having established his listening framework, Davis moves to the subject of mathematics in chapter two. Here he develops his interpretation of mathematics and its relationship to curriculum. He names mathematics as ecological mathematics, which, he is careful to state, is not to be conflated with environmental organizations such as Greenpeace. Davis begins by giving a broad definition of ecology:

    Ecology is about interrelationships and interconnections. It involves an attunement to codependencies, mutual effects, and codetermination — in essence, to the fundamental intertwining of all things. When we speak of ecology, then we speak of everything that shapes our being — their effects on us and ours on them. Folding back to earlier discussions of enactivism, hermeneutics, and listening, such notions are found in an awareness of this sort of deep ecological interweaving. (p. 58)

    Then, he connects mathematics to human ecology and human existence. He directly addresses what so many mathematics students from grade school have asked over the years, "What do I need this for?" by unfracturing mathematics, by making clear the connections between mathematics and life. But he is also clear that mathematics is limited in both its purpose and meaning to humanity. Mathematics is only a part of life within the framework he develops, and he concentrates on in this part. In the next two sections of chapter two, Davis moves through his ideas about mathematics curriculum and its relationship to teacher education and teaching, providing examples of his modes of listening and tying together his ideas about teaching mathematics with a listening orientation.

    In chapter three, Davis provides a discussion of his understanding of teaching and how listening, mathematics, and teaching coalesce and mingle. In three sections — Culture Making, Artistry, and Pedagogy — he gives the reader a clear understanding of his formulation of mathematics education through listening, once again returning to his modes of listening structure.

    In Chapter Four, Davis explores his ideas about learning in three sections: Knowing, Understanding and Meaning, and Play. In the fifth and final chapter, Davis reaffirms the connections he had been making among listening, mathematics, teaching, and learning. In the chapter's three sections — The Nature of Teaching, Assessment, and Mathematics Teaching as Listening — he returns to his modes of listening structure, applying it in a different context.

    In writing this review, I must have used "visual" metaphors at least five times and went back to ask why? I removed these visual metaphors because I wanted to concentrate on the aural. Learning and expressing through sight are often predominant in our thinking, and I believe that paying more attention to sound, to listening, to hearing can open our ears to a whole new pedagogical approach to mathematics and, indeed, to other subjects. I believe that Brent Davis, through this book, has done that.

     R.H.T.
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    Book Notes