ways in which the prospective teachers’ opportunities to learn are designed may at times situate the mathematical questions within apparently pedagogical contexts (e.g., make a story problem), so that the kind of mathematical work they do in the course helps them develop mathematical proficiency in ways they can use in teaching. But the course is not about how to teach, nor about how children learn. It is explicitly and deliberately a sustained opportunity for prospective teachers to learn mathematical ideas in ways that will equip them with mathematical resources needed in teaching.

Focus on Student Thinking

The successful programs that focus on mathematics and children’s thinking are programs grounded in practice. Teachers do not learn abstract concepts about mathematics and children. In the programs, teachers look at problem-solving strategies of real students, artifacts of student work, cases of real classrooms, and the like. Furthermore, the teachers in these programs are challenged to relate what they learn to their own students and their own instructional practices. They learn about mathematics and students both in workshops and by interacting with their own students. Specific opportunity is provided for the teachers to discuss with one another how the ideas they are encountering influence their practice and how their practice influences what they are learning. Discussions in these programs are conducted in a spirit of supporting the teachers’ inquiry. The analysis of children’s thinking is not presented as a fixed body of knowledge, and the teachers engage not only in inquiry about how to apply knowledge about students’ thinking in planning and implementing instruction but also in inquiry to deepen their understanding of students’ thinking.40

The workshop described in Box 10–2 forms part of a professional development program designed to help teachers develop a deeper understanding of some critical mathematical ideas, including the equality sign. The program, modeled after Cognitively Guided Instruction (CGI), which has proven to be a highly effective approach,41 assists teachers in understanding how to help their students reason about number operations and relations in ways that enhance the learning of arithmetic and promote a smoother transition from arithmetic to algebra.42 This particular workshop was directed at illuminating students’ misconceptions about equality and considering how those misconceptions might be addressed.

Several features of this example of professional development are worth noting. The teachers focus on children’s thinking about a critical mathematical idea. Although they begin by considering how children think, the teachers

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