Paper presented at the annual meeting of the Jean Piaget Society, Amsterdam, May 31 – June 2.

As design researchers of mathematical education, we are interested in how to promote students’ embodied dynamic development toward culturally accepted mathematical practices, in our case proportionality. In line with enactivism, we assume that knowing is doing and aim to implement environmental constraints that enable learners to self-organize relevant mathematical behavior flexibly. But what are learners’ mathematical behaviors, how do these self-organize, and can this be altered effectively by task constraints? In order to answer these questions for learning proportionality we developed an embodied (tablet) task with different bodily and environmental constraints. Learners were asked in Piagetian task-based clinical interviews to “crack the code”: keep an object green by moving your fingers along x- and γ- axes. The object became green when the relative position of the fingers was = γ=1/2 x. They were instructed to either: (1) move the fingers along parallel axes (↕↕) or orthogonal axes (↕↔); and (2) keep green two bars, a full-screen or a scaling rectangle. For a subset of learners, we repeatedly observed, marked, and named the real-time unfolding of their mathematical behaviors — actions (moves), perceptions (gaze), and descriptions (think-out-loud) — and focused on changes within and across learners and tasks. The results showed a variety of doings not yet described in the literature. The occurrence of behaviors changed nonlinearly: certain (combinations of) actions, perceptions and descriptions became more or less prominent, largely in a self-organizing (deterministic) way, characterized by different time-scales and degrees of stability. Some doings occurred uniquely in/with bodily and environmental constraints and could be interpreted as different facets of the mathematical practice of proportionality. We will discuss the implications of our findings for future designs of environmental constraints to further promote mathematical dynamics.