The Role of a Learning Community in Changing Preservice Teachers' Knowledge and Beliefs about Mathematics Education. Research Report 91-1.

Commonly, preservice elementary teachers bring to their professional studies deeply rooted ideas about teaching and learning mathematics. These ideas are embedded in the content knowledge, the pedagogical experiences, and the epistemological orientation of prospective teachers. They view mathematics as a fixed body of knowledge that is best learned by memorizing facts and rules and procedures for applying them to textbook exercises. They view the role of the teacher as carrying out goals determined by text material, providing demonstrations and examples of tasks to be completed, and checking assignments for completeness and accuracy. They expect their teacher preparation program to provide the techniques to make teaching efficient and effective. This conception of mathematics education contrasts with the nature and the creation of knowledge in the discipline [Davis and Hersh, 1981], and it denies children's natural capacity for and interest in understanding mathematical ideas [Resnick, 1983; Riley, Greeno & Heller, 1983; Romberg & Carpenter, 1986]. Further, it conceives of teaching as a matter of technical competence rather than reflection and decision making based on what children are coming to know. The literature on the impact of professional study on teachers' beliefs points to the difficulty in overcoming ingrained notions developed during previous school experiences [Ball, 1988; Feiman-Nemser, 1983; Tabachnick, Popkewitz & Zeichner, 1979-80; Zeichner, Tabachnick, & Densmore, 1987]. If we are to cause prospective teachers to rethink these beliefs, we must create situations where these beliefs are faced and reconsidered. This demands powerful interventions that challenge and yet are safe situations in which students can take mathematical, emotional, and intellectual risks. Creating a community of learners with shared responsibility for learning holds the promise of providing such an environment [National Council of Teachers of Mathematics, 1989a, 1989b; Schwab, 1976]. What is the building of a community of learners likely to contribute to learning to teach mathematics? We have ample evidence that learning in isolation from interaction with others is likely to result in students' constructing mathematical worlds that have little fit with the accepted "truths" of the discipline [e.g., Erlwanger, 1973]. One might extrapolate that learning to teach in isolation from the experience of personal interactions with others' exploring the discipline itself would lead to equally impoverished views of what it means to teach mathematics. Thus, the creation of a community in which one's private world is exposed has the potential to challenge the learner's currently held views and lead to the construction of more acceptable and powerful views. It is through the give and take that one-to-one community begins. Opening up of oneself to community can, as Schwab [1975] puts it, happen "in one and only waythrough speech, by talk" [p. 32]. He argues that classrooms must be "rich in occasions for this symbolic exchange" for it is the " eans by which children convey and receive recognitions of their personality [,..]convey[...]the promise of mutual support in difficulty" and "find sources of help and occasions for the giving of help" [p. 2]. We believe that these ideas are equally valid with preservice teachers. The act of receiving help, of being nurtured, is important. Of equal importance, especially to preservice teachers, is giving help. However, herein lies one of the traps-distinguishing between help given by telling, which results in dependent learners, and help given by questioning and collaborating, which results in empowered learners. Do teachers insist on telling learners how to solve a problem or do they give them ownership of ideas by contributing prompts, questions, counterexamples to wrong dir ctions, or strategies for thinking about a problem situatio ? The former moves students and teacher quickly through material, but only a few truly encounter the potential of the ideas embedded in a problem and its solution. The latter approach has the potential to open ideas up to more of the community and shape the understanding of the on who gives help. The emphasis on meaning making is key to changing the curr nt conceptions of preservice teachers about mathemat cs. I students are to build mathematical schema they can use in a flexible way to approach new problem situations, then they must develop the disposition to seek ways to make sense of new ideas. Establishing a classroom where arguments are made to support conjectures, and where the criterion for what makes sense is determined by students and teacher working together, is likely to engender in students a very different view of mathematics from the typical rule-and-procedure orientation. Alibert [1988] provides a picture of classroom discourse that is supportive of what we believe is important for preservice teachers to experience in the making of mathemat-