Recent evidence suggests that we have an intuitive number sense and that visuospatial processes may ground simple mathematical reasoning, but higher level mathematical cognition is often assumed to depend only on the manipulation of symbolic expressions, governed by a set of rules and logical axioms. To assess rule vs. visuospatial thinking in a higher level mathematical domain, we asked undergraduates to solve trigonometry problems and to report their use of rules, mnemonics, and visuospatial representations including the unit circle, right triangle, and sine and cosine waves. Use of the unit circle was reported most commonly, and was associated with better performance, even after controlling for the extent and recency of trigonometry experience. While unit circle users took more time, their performance was robust to problems that rule users tended to fail. Our findings suggest that even higher level mathematical cognition is more than just the manipulation of symbolic expressions.
Mickey, K.W. (2018). Understanding trigonometric relationships by grounding rules in a coherent conceptual structure (Doctoral dissertation). Retrieved from https://purl.stanford.edu/fn607qb0406 [pdf | web]
Mickey, K.W. & McClelland, J.L. (2017). The unit circle as a grounded conceptual structure in precalculus trigonometry. In D.C. Geary, D.B. Berch, R. Ochsendorf, & K.M. Koepke (Eds.), Acquisition of complex arithmetic skills and higher-order mathematics concepts (pp. 247-269). London, UK: Academic Press. [pdf | web]
Alibali, M., Kalish, C., Rogers, T. T., Massey, C., Kellman, P., Sloutsky, V., McClelland, J.L., & Mickey, K.W. (2015). Connecting learning, memory, and representation in math education. In Noelle, D. C., Dale, R., Warlaumont, A. S., Yoshimi, J., Matlock, T., Jennings, C. D., & Maglio, P. P. (Eds.), Proceedings of the 37th Annual Conference of the Cognitive Science Society (pp. 19-20). Austin, TX: Cognitive Science Society. [pdf | web]
The typical structure of equations influences how we learn the meaning of the equal sign. Previous studies have shown that as students gain experience with addition problems, they actually perform worse on certain problems, before eventually improving. We seek to explain this trajectory with gradual implicit learning, without explicit representation of strategies or principles. Our parallel distributed processing model is nevertheless able to simulate several phenomena observed in how children learn mathematical equivalence: not only how successful performance develops, but also what strategies are used and how equations are encoded.
Mickey, K.W. & McClelland, J.L. (2014). A neural network model of learning mathematical equivalence. In P. Bello, M. Guarini, M. McShane, & B. Scassellati (Eds.), Proceedings of the 36th Annual Conference of the Cognitive Science Society (pp. 1012-1017). Austin, TX: Cognitive Science Society. [pdf | web]
The environment can constrain the way we think and act within it. Such an influence has been largely ignored within the domain of spatial language, which has largely focused on objects and their identities, independently of the environments in which they occur. To investigate whether the environment also has an influence, we instructed participants to place a located object either near or far from a reference object within survey perspectives of manipulated 3D environments. When a geographical feature in that environment was present and had meaningful semantic content, it systematically altered the distance, direction and orientation of the placements, with these alterations well beyond the range expected based on a geometric definition of the spatial term. This environmental influence is consistent with a situated view of cognition.
Mickey, K.W., Carlson, L.A., & Freundschuh, S.M. (2010). Location, location, location: Environmental constraints on interpreting spatial terms [Abstract]. In S. Ohlsson & R. Catrambone (Eds.), Proceedings of the 32nd Annual Conference of the Cognitive Science Society (p. 579). Austin, TX: Cognitive Science Society. [web]