Getting at the mathematics: Sara’s journal

In this paper, we discuss issues in planning and conducting research into mathematics learning. We emphasize two central themes: (a) the learners ’ mathematics (especially the issues and ideas, in given problem situations, that learners choose to think about and to present, and (b) the kinds of knowledge that learners may be building (including their ideas about what mathematics is, and how people do, learn, use, communicate and understand it). While the first theme is at least partly mathematical, the second interweaves cognition and epistemology. Anchored in student data from an extended classroom teaching experiment in the mathematics of change, we focus on key choices needed to build narratives to help researchers capture, in detail, how students understand their own engagement with their mathematics. According to Brousseau (1997), for example, learning takes place (or perhaps fails to take place) in the course of student-teacher interactions. These are seen as governed by (frequently tacit) understandings (the didactic contract) about what is to be learned, how learning can be demonstrated, and about the respective responsibilities of learner and teacher. To understand an act of learning, in this view, one needs to understand not onl