My premise is that understanding mathematics involves a mélange of knowledge, beliefs, and feelings about the subject. Central is propositional and procedural knowledge of mathematics — that is, understandings of particular topics (e.g., fractions and trigonometry), procedures (e.g., long division and factoring quadratic equations), and concepts (e.g., quadrilaterals and infinity), and the relationships among these topics, procedures, and concepts (Davis, 1986; Hiebert & Lefevre, 1986; Skemp, 1978). This substantive knowledge of mathematics is what is most easily recognized by others as "subject matter knowledge." Another critical dimension, however, is knowledge about mathematics. This includes understandings about the nature of knowledge in the discipline — where it comes from, how it changes, and how truth is established. Knowledge about mathematics also includes what it means to "know" and "do" mathematics, the relative centrality of different ideas, as well as what is arbitrary or conventional versus what is necessary or logical, and a sense of the philosophical debates within the discipline.
A subject which, in mainstream American culture, elicits, alternately, awe, anxiety, and intense dislike, mathematics is a domain of feelings as well as knowledge (Bassarear, 1986; Brandau, 1985; Buerk, 1982; McLeod, 1986). Understanding mathematics is colored by one's emotional responses to the subject and one's inclinations and sense of self in relation to it. The three preceding paragraphs sweep broadly over a vast conceptual issue: what "understanding mathematics" means and includes. The discussion has barely sketched the domains and boundaries. How would one comprehensively map the essential substantive knowledge of mathematics? And what is entailed in "understanding" any of it? The first question — about a map — is one I did not attempt to address in this work. I chose instead to sample purposefully specific mathematical topics and to explore how the teacher candidates understood the particular underlying concepts and procedures. The second question — about what understanding mathematics for teaching means and includes — was, however, a significant question for this study. Below I discuss three issues of critical concern in exploring prospective teachers' knowledge of and about mathematics:
(1) the fact that prospective teachers' "knowledge" is not necessarily true,
(2) the difference between tacit ways of knowing and explicit conceptual knowledge in doing mathematics, and
(3) connectedness of knowledge as a crucial dimension of mathematical understanding.
Describing prospective teachers' understandings of mathematics as "knowledge" is problematic, for they of course "know" things that are wrong. Some of the prospective teachers whom I interviewed, for instance, thought that 7 ÷ 0 = 0, that squares were not rectangles, and that doing mathematics meant adding, subtracting, multiplying, and dividing. Although false from a disciplinary point of view, and therefore not "knowledge" (Scheffler, 1965), these ideas were what they "knew." Using the term "understanding" (instead of "knowledge") might be one acceptable way of dealing with the fact that people's "knowledge" is not always true. However, with the goal of preparing teachers to teach mathematics, teacher educators have a dual focus: We are interested in understanding not only in what the ideas mean to them as knowers, but also in appraising their understanding in light of standards of disciplinary knowledge. To avoid the problem of referring to incorrect notions as "knowledge," I considered talking about their understandings, right or wrong, as "ideas" (a term I use in referring to the teacher candidates' knowledge about teaching, learning, and learners). This term applied better to issues about the nature of mathematics than it did to specific mathematical topics, however. It seemed odd to talk about specific substance — that 1 3/4 ÷ 1/2 = 3 1/2 , for instance — as an "idea." Knowledge is also not an all or nothing matter (Nickerson, 1985). What does one say about the knowledge of a person who says that "you can't divide by zero"? This is true, but of interest here is also how she understands it — as an arbitrary "fact" or as a logical consequent of other mathematical ideas and principles. She may, by way of explanation, say that "it's just one of those things you have to remember," "zero can't do anything to a number," "it's undefined." Or she may prove her assertion by comparing division by 0 to division by 2 or by using the inverse relationship of multiplication and division. In each case, she would reveal significantly different things about her understanding of division by zero as well as her understanding of mathematics more broadly. These differences matter in teaching mathematics from the perspective of mathematical pedagogy in which the goals include helping students understand mathematics as a discipline .
What matters are the qualitative dimensions of prospective teachers' knowledge — what they know and how they think about it. Still, the truth value of their ideas is equally critical, for teachers are responsible for helping their pupils access disciplinary knowledge. Consequently, "knowledge," "understanding," "belief," and "idea" are all used in my discussions, albeit cautiously and with qualification.
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