Academic journal article Memory & Cognition

The Dynamics of Insight: Mathematical Discovery as a Phase Transition

Academic journal article Memory & Cognition

The Dynamics of Insight: Mathematical Discovery as a Phase Transition

Article excerpt

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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Among the most compelling issues in psychology is how the cognitive system can spontaneously leap from one structure to another. For example, as children are learning to solve addition problems (e.g., 4 1 2 5 ?), they initially raise the appropriate number of fingers to represent the two addends (e.g., four on one hand, two on the other) and count them all. However, after using this strategy for a while, children spontaneously discover a new strategy: They raise the appropriate number of fingers for the smaller addend (e.g., two), and then begin counting from the larger addend (e.g., four, five, six; Siegler & Araya, 2005). Such discoveries or insights are particularly interesting phenomena, in part because the change in structure appears to be driven by the activity of the system itself. There is no external agent guiding the individual toward a new organization, nor is there an internal plan that contains the new structure in miniature. Explaining the emergence of new structures is a serious challenge for cognitive theory. Chronicle, MacGregor, and Ormerod (2004), for example, noted that information-processing approaches had not made substantial progress in explaining new structures (i.e., insights) during problem solving. The central question is, how can a functioning system can suddenly self-organize into a new configuration in the absence of any external supervision or internal blueprint.

In previous work, we found evidence for just such a spontaneous change in cognitive structure during a simple problem-solving task (Dixon & Kelley, 2006). Participants were asked to solve gear-system problems by predicting the turning direction of the final gear, given the turning direction of the first gear (see Figure 1). After solving the problems with lower level strategies, many participants spontaneously discovered a mathematical relation-parity- that afforded a higher order solution to the problems (Dixon & Bangert, 2004; Dixon & Kelley, 2006, 2007). (The parity of the number of gears in the system [i.e., odd, even] determines whether the final gear turns in the same direction as the driving gear.) The participants rarely made errors prior to discovering parity. Furthermore, since the gear displays were static, the relation could not be extracted from the movement of the gears. The new relation appears to arise from the participants' own activity. Gentner and Namy (1999) investigated a similar phenomenon in which representational change occurs from the child's own actions. They showed that when children repeatedly compare objects during classification, they begin to detect their common dimensions, a process sometimes called progressive alignment. Gentner has proposed that repeated alignment is a central process in cognitive development (e.g., Gentner, Loewenstein, & Hung, 2007). In the present study, we address the emergence of new cognitive structure from one's own activity as an instance of self-organization. …

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