Abstract

Much of the power of mathematics comes from its generality and ability to unify prime face dissimilar domains. The same combinatorics formula applies to sealing wax, cabbages, and kings with no customization needed, or even permitted. By one account, analytic thought in math and science requires developing deep construals of phenomena that run counter to untutored perceptions. This approach draws an opposition between superficial perception and principled understanding. In this talk, I advocate the converse strategy of grounding mathematical reasoning in perception and action. I will describe empirical evidence for perceptual changes that accompany learning in mathematics. In arithmetic and algebraic reasoning, we find that proficiency involves executing spatially explicit transformations to notational elements. People learn to attend mathematical operations in the order in which they should be executed, and the extent to which students employ their perceptual attention in this manner is positively correlated with their mathematical experience. People produce mathematical notations that they are good at reading. Perceptual and attentional processes are tailored to fit mathematical requirements. Thus, for reasoning in mathematics (and science, but that’s another talk), relatively sophisticated performance can be achieved not only by ignoring perceptual features in favor of deep conceptual features, but also by adapting perceptual processing so as to conform with and support formally sanctioned responses. These “Rigged Up Perceptual Systems” (RUPS) offer a promising strategy for achieving educational reform.