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 engage with mathematics in extended problem explorations.