Seeing an Exercise as a Single Mathematical Object: Using Variation to Structure Sense-Making

In this theoretical paper we take an exercise to be a collection of procedural questions or tasks. It can be useful to treat such an exercise as a single object, with individual questions seen as elements in a mathematically and pedagogically structured set. We use the notions of 'dimensions of possible variation' and 'range of permissible change', derived from Ference Marton, to discuss affordances and constraints of some sample exercises. This gives insight into the potential pedagogical role of exercises, and shows how exercise analysis and design might contribute to hypotheses about learning trajectories. We argue that learners' response to an exercise has something in common with modeling that we might call 'micro-modeling', but we resort to a more inclusive description of mathematical thinking to describe learners' possible responses to a well-planned exercise. Finally we indicate how dimensions of possible variation inform the design and use of an exercise.