In recent work in cognitive science, it has been proposed that cognition is a self-organizing, dynamical system. However, capturing the real-time dynamics of cognition has been a formidable challenge. Furthermore, it has been unclear whether dynamics could effectively address the emergence of abstract concepts (e.g., language, mathematics). Here, we provide evidence that a quintessentially cognitive phenomenon-the spontaneous discovery of a mathematical relation-emerges through self-organization. Participants solved a series of gear-system problems while we tracked their eye movements. They initially solved the problems by manually simulating the forces of the gears but then spontaneously discovered a mathematical solution. We show that the discovery of the mathematical relation was predicted by changes in entropy and changes in power-law behavior, two hallmarks of phase transitions. Thus, the present study demonstrates the emergence of higher order cognitive phenomena through the nonlinear dynamics of self-organization.

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Recent work in embodied cognition has proposed that representations and actions are inextricably linked. The current study examines a developmental account of this relationship. Specifically, we propose that children's actions are foundational for novel representations. Thirty-two preschoolers, aged 3.4 to 5.7 years, were asked to solve a set of simple gear-system problems. Participants' motions and verbalizations were coded to establish the strategies they used. The preschoolers initially solved the problems by simulating the turning and pushing of the gears. Subsequently, most participants discovered a new representation of the problems: the turning direction of the gears alternates. Results show that the number of actions that embodied alternation information, during their simulation of the system, predicted the later emergence of the higher-order representation (i.e. that the gears alternate turning direction). Thus, it appears that the preschoolers discovered a new representation based on their own actions. These results are consistent with the developmental embodiment hypothesis: actions are central to the emergence of new representations.

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In this Brief Report, we share the main findings from our line of research into embodied cognition and proof activities. First, attending to students' gestures during proving activities can reveal aspects of mathematical thinking not apparent in their speech, and analyzing gestures after proof production can contribute significantly to our understanding of students' proving practices, particularly when attending to dynamic gestures depicting relationships that are difficult to communicate verbally. Second, directing students to produce physical actions before asking them to construct a mathematical proof has the potential to influence their subsequent reasoning in useful ways, as long as the directed actions have a relationship with the proof content that is clearly meaningful to the students. We discuss implications for assessment practices and teacher education, and we suggest directions for future research into embodied mathematical proof practices.

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