Using Teacher Portfolios to Enrich the Methods Course Experiences of Prospective Mathematics Teachers

School Science and Mathematics, Dec 2004 by Hartmann, Christopher

[The students] need to be more organized and I don't know if they could do it on their own. Some of our students are lower level and have a lot of organizational problems. All of them really benefit from having [the problem] laid out how it was asked (Portfolio interview 1).

Appendixes C and D demonstrate one way that Margaret added structure to her students' mathematical explorations. Appendix C contains the original tasks from the CMP curriculum. As shown in Appendix D Margaret revised these materials by adding structures to help her students record their results. First, she added a table to organize the students' initial exploration of the Match/No Match game. At the bottom of her worksheet she also provided four spaces for the students' to record their response to Question B. Through these revisions Margaret provided scaffolding for her students' investigations by reducing the need for them to construct their ownrepresentations. By doing so she also reduced the mathematical demands of the tasks. For example, in Question B she provided a space for each possible response, reducing the students' need to identify the number of permutations in the sample space.

When I first encountered Margaret' s task revisions during my midterm review of her portfolio, I reflected on the fact that research identifies a tendency for teachers to reduce the cognitive complexity of mathematical tasks in ways that inhibit learning. Stein, Grover, and Henningsen ( 1996) reported that this tendency is both relatively common in mathematics classrooms and one of the obstacles to using NCTM Standards-based curricula (National Council of Teachers of Mathematics, 1989), such as CMP, as a catalyst to improve mathematics instruction. My awareness of this research helped me use Margaret's portfolio to assess her needs as a learner. Furthermore, by focusing my attention on her revisions, the "Things I Modified" section of Margaret's portfolio helped me reflect on both her instructional practices and her progress toward meeting the goals of the course. Since I perceived (based on evidence in their portfolios) that several of the prospective teachers were revising tasks based on similar reasoning, my awareness of the evidence in Margaret's portfolio enabled me to help all of the prospective teachers inquire into the ways that teacher beliefs can impact student learning in mathematics classrooms.

Using Portfolio Artifacts to Inform Practice

My midterm review of the student teachers' portfolios helped me identify particular problems of mathematics teaching they might investigate in their classrooms and discuss during the biweekly discussion of cases of student thinking. For example, Margaret's portfolio indicated that she was revising curriculum materials in order to help her students to successfully complete mathematical tasks (e.g., appendixes C and D). In response I asked the prospective teachers to collect evidence from their classrooms to address the following question: "What distinguishes a successful student of mathematics from a struggling student of mathematics in your classroom?" I presented this prompt to the class to guide their preparation for their next discussion of cases of student thinking. My goal was to focus the student teachers' attention on the implicit beliefs about teaching and learning mathematics that guide their instructional practices.