4: Developing Mathematical Ideas: A Program for Teacher Learning¹

The principles that, in our view, underlie effective teaching practices are such abstract directives as: know the mathematics, center your lesson on student thinking, and try to discern a logic in students’ mistakes. For most teachers, learning to translate these principles into competent management of the contingent, moment-to-moment character of classroom interaction is a process fraught with both intellectual and emotional challenges. Developing Mathematical Ideas (DMI) is a curriculum for K-8 teacher learning designed in response to this challenge. It uses case discussion to model how close attention to student thinking and a deep grasp of the mathematics at issue can lead teachers to a more coherent, hence more confident, approach to their instruction. DMI consists of seven modules, each focused on a particular mathematical theme studied over eight 3-hour seminar sessions. It integrates case discussion with opportunities for participants to explore relevant mathematics, investigate the mathematical thinking of their students, analyze lessons from innovative curricula, and consider related research from the education literature. This chapter excerpts the discussion of a single case from the second session of a seminar, Reasoning Algebraically About Operations, to demonstrate how teachers can come to recognize that questions about pedagogical strategy should be based on and follow from consideration of the mathematics content and student thinking.