In mathematical pedagogy, the goals are different from those of the "ordinary" math class or of a more conceptual approach to mathematics teaching (e.g., Good, Grouws, & Ebmeier, 1983).8 In this ideal, learning what mathematics is and how one engages in it are goals coequal and interconnected with acquiring the "stuff" — concepts and procedures — of mathematics.
What is mathematics? The kind of math teaching that I am describing differs radically from "ordinary" mathematics teaching on what counts as "mathematics" — what students are supposed to learn, what matters about learning mathematics, what it means to know and to do mathematics, and where the authority for truth lies. The ordinary math class makes mathematics synonymous with computation. Students learn algorithms for computational procedures; calculational speed and accuracy are valued goals and meaning is hardly ever the focus. Problem solving generally means word problems which are little more than symbols dressed up in words and have little to do either with real life or with mathematics (Lave, 1988). By default the book has epistemic authority: Teachers explain assignments to pupils by saying, "This is what they want you to do here," and the right answers are found in the answer key. In this kind of teaching, knowing mathematics means remembering definitions, rules, and formulae, and doing mathematics is portrayed as a straightforward step-by-step process. Learning to "play by the rules" often entails a "suspension of sense-making" in school mathematics (Davis, 1983, 1986; Lave, 1988; Schoenfeld, in press b, p. 9; Stodolsky, 1985). Math is primarily a "tool" subject, one that has applications to everyday life both by ordinary people who must balance their checkbooks and by experts who in some mysterious way use mathematics to construct bridges or send rockets into space.
In more conceptual mathematics teaching, mathematics is seen as "a body of logically consistent, closely related ideas" (Good, Grouws, and Ebmeier, 1983) and knowing mathematics entails understanding those relationships. For example, when students learn about fractions, they should be led to integrate that new knowledge — e.g., calculating with fractions and ordering fractions — with their previous knowledge about natural numbers. The goal is for students to remember what they have learned, but the assumption is that they will remember more effectively if they understand what they are being taught rather than simply learning by rote. Therefore, more emphasis is placed on meaning than in the ordinary math class. Knowledge of mathematical procedures entails knowing when to use them and understanding why they work. For example, in learning to subtract with regrouping, students should know when regrouping is necessary and when it is not and should be able to explain the steps using a model such as base 10 blocks or popsicle sticks bundled in tens. Ideally, doing mathematics is portrayed as smooth and straightforward, not frustrating or uncertain — if one has this kind of meaningful understanding of the content. Epistemic authority lies less with the book than with the teacher who dispenses concepts and evaluates the correctness of students' answers.
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. Lampert (in press a) discusses how the substantive and epistemological dimensions of mathematical knowledge go hand-in-hand in this view of mathematics. She explains that, in her classroom, she tries to shift the locus of authority in the classroom — away from the teacher as a judge and the textbook as a standard for judgment and toward the teacher and students as inquirers who have the power to use mathematical tools to decide whether an answer or a procedure is reasonable.
But, she adds, students can do this only if they have meaningful control of the ideas: Students will not reason in mathematically appropriate ways about objects that have no meaning to them; in order for them to learn to reason about assertions involving such abstract symbols and operations as .000056 and a2 + b3, they need
to connect these symbols and operations to a domain in which they are competent to "make sense." For example, fifth graders could understand that if .000056 represented an imaginary amount of money, you would need about 1000 times that amount just to equal one nickel. Using money, a familiar domain, could enable eleven-year-olds to make sense of this incredibly tiny quantity and to argue about whether .000056 is more or less than .00003.
Views of teaching, learning, learners, and context. In addition to what counts as knowledge of mathematics, each of these kinds of teaching — ordinary, conceptual, and mathematical pedagogy — also implies certain embedded assumptions about the teaching and learning of mathematics: about pupils, teachers, and the context of classrooms. The ordinary math class is based on the assumption that mathematics is only learned through repeated practice and drill, that "knowing" math means remembering procedures and concepts. The teacher's role is to show pupils how to do the procedures and to give them tricks, mnemonics, and shortcuts that make it easier for the pupils to keep track of everything. Consider, for example, the old rules of thumb that rattle around in our heads (e.g., "invert and multiply" or "to multiply by 10, add a zero"). Pupils are expected to absorb and retain what they have been shown, a commonsense sensory view of learning. Conceptual mathematics teaching assumes that the teacher should do more than tell students how to do procedures and should, when possible, also emphasize the meaning of those procedures by showing how they work. The teacher plays an active role: leading, showing, directing, and structuring class time. The students play roles within this structure, answering teachers' questions, completing assignments, pursuing explorations set out by the teacher. Use of manipulatives is valued. This approach to teaching assumes that pupils will learn more if teachers are clear and direct about the content and spend more time developing student understanding and less on massive drill and practice. Still, practice remains an important component of this approach.
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.
Perhaps most significant in the classroom context is the teacher's role in guiding the direction, balance, and rhythm of classroom discourse by deciding which points the group should pursue, which questions to play down, which issues to table for the moment. The teacher in mathematical pedagogy has a critical role to play in facilitating students' learning. The teacher introduces a variety of representational systems which can be used to reason about mathematics, models mathematical thinking and activity, and asks questions that push students to examine and articulate their ideas. Finally, practice also takes on an entirely different meaning in this approach than in either of the other approaches. Here students engage in the practice of mathematics, learning what it means to do mathematics (Collins, Brown, & Newmann, 1987; Lampert, 1986, in press a; Lave, 1987; Romberg, 1983; Schoenfeld, 1985): making conjectures, attempting to prove them to other members of the community, revising and elaborating ideas.
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