Representations as site of the tension between abstract and concrete in mathematical practice: University students at work with a spirograph

In this paper, we propose to study representations as site of encounter with the lively tension between abstract and concrete in mathematical practice. In so doing, we question visions of practice as a sequence of moves towards abstraction that see mathematical concepts as fixed and disembodied, detached from the physical, and learning in terms of developmental trajectories. Beyond a tendency to reduce abstract and concrete to a dualism conceiving of more of one as less of the other, we are more interested in what learners do and how this mobilises practice as a flow of activity across abstract and concrete. We offer the case of some university students working with a spirograph to experiment with this approach to mathematical practice.