This paper examines a particular problem in teacher education: helping prospective elementary teachers learn to teach mathematics. As a mathematics teacher educator, my goal is to help my students learn to do something different from--and better than--what they experienced as pupils in mathematics classes. My problem is also framed by the kinds of experiences with mathematics and with the teaching and learning of mathematics that my students have had before I ever meet them and the ways in which those experiences influence the trajectories on which they move in becoming teachers. Although teacher educators sometimes speak of preservice teacher education as the first stage in learning to teach, nothing could be further from the truth. In fact, before they take their first professional course, future mathematics teachers have already clocked over 2,000 hours in a specialized "apprenticeship of observation" (Lortie, 1975, p. 61) which has instilled not only traditional images of teaching and learning but has also shaped their understandings of mathematics. My students are also almost all women, as are most prospective elementary teachers, a fact that is significant, given what we know about the widespread alienation of girls from mathematics during their precollege education. In considering the problem, I focus on the role of a methods course in helping prospective elementary teachers learn to teach mathematics. A methods course is a particular curricular occasion, one that is different from other kinds of teacher education courses in some significant ways. It is about acquiring new ways of thinking about teaching and learning. But it is also about developing pedagogical ways of doing, acting, and being as a teacher. And it is about a particular subject matter--one that brings its own set of issues, different from those in writing or social studies, for instance. What do my students bring to my course and what should I do in trying in influence the direction they move after leaving it, as they develop into teachers of mathematics?
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