Group actions at the fingertip: A 4E approach to mathematical cognition in Vanuatu sand drawing

Using recent ethnomathematical methods that combine mathematics and anthropology, this article explores mathematical cognition (MC) through the 4E framework (embodied, embedded, extended, and enactive) beyond classical approaches to numeration and Euclidean geometry. The research focuses on ”sand drawing” practices within the oral tradition culture of Sia Raga on Pentecost Island (Vanuatu) — an ephemeral art consisting of creating symmetrical figures on the ground using one finger, without lifting it and returning to the starting point. To address the cognitive processes at work in memorizing and creating these artifacts, the study analyzes various forms of ”external representations”. First, those of ethnomathematicians who, to analyze this practice, construct algebraic models based on group theory and group actions. Second, those mobilized by the practitioners themselves, including traces on the ground, drawing rules, and the narratives that accompany them. These representations are then examined through the lens of practitioners’ dynamic interactions with their environment, particularly with the landscape of affordances that structures it. As illustrated by the polysemous concept of cycle, our findings highlight congruencies between the different external representations and suggest that our model captures cognitive structures that reaffirm the embodied, embedded, extended, and enactive dimensions of mathematical activity. By developing a group-theoretical model of sand drawing practice, this work suggests that group actions may offer new avenues for research on mathematical cognition. The tacit nature of this type of knowledge allows for exploring educational perspectives in mathematics, particularly the potential of bodily movements, diagrammatic representations, and cultural artifacts. • 4E framework applied to sand drawing reveals mathematical cognition beyond Euclidean geometry. • Algebraic modeling captures cognitive structures in Vanuatu oral tradition practice. • Ethnomathematical methods offer new avenues for exploring mathematical ontogenesis. • Cultural artifacts enhance education through diagrammatic representations and embodied action.