Making sense of fraction quotients, one cup at a time

•A case study of fraction division anchors a general discussion.•For us a model is a concrete thing, a tool to help make sense of something.•The case study: why just memorize a rule when you can make good sense of it?•The general discussion: how concepts can emerge through strategic modeling.•This paper extends prior work on models for whole number products.

On the surface, we discuss a concrete case, in which a group of learners made sense of fraction division, beginning with a specific, concrete problem that demanded fresh insight. To meet its challenge, several education undergraduates, joined later by the authors, built mutable, evolving models to support their thinking. More fundamentally, we discuss those models in some depth, as foundations for a theoretical analysis of emergent sense and meaning. The initial problem begins, in effect, as a test case to investigate, but soon, with further understanding, it emerges as a special case, to support or represent an insight that will hold in general. The kind of knowledge to be built, therefore must shift. Hence, for us, the mathematics we discuss is mathematics in the making, anchored to evolving models, arguments, and explanations.