Potential Gaps during the Transition from the Embodied through Symbolic to Formal Worlds of Reflective Symmetry

Abstract

Even though reflective symmetry is heavily embedded in the everyday, learners continue to experience challenges when they mathematize concepts from this informal/everyday context. In this article we argue that symmetry exists in nature, it can also be symbolized algebraically and it can be abstracted into the world of axioms and theorems. We problematize this multiple nature of symmetry which on one hand is supportive and on the other acts as a contributory factor to learners’ gaps in knowledge. Tall’s three worlds of mathematics helped us to show the transition of symmetry from the embodied through symbolic to the formal world and the inherent gaps attributed to the shifts in thinking thereof. We then used this same framework to analyse learners’ responses to a reflective symmetry task. The results show that many learner responses could be explained explicitly by the lack of flexibility in the applicability of experiences in the embodied world of reflective symmetry. Learners’ responses were deeply rooted in the embodied world, which indeed remains helpful in some situations but tended to confuse them in others, hence inhibiting further application. The article recommends that teachers need to understand these subtle changes so that they can address the challenges explicitly.