The role of perception in mathematical reasoning

Robert L. Goldstone (reporting work conducted with David Landy)

Algebraic reasoning is typically considered to be a paradigmatic case of widespread, amodal calculation involving rule-based, symbolic transformations. However, we report evidence that it is strongly influenced by perceptual grouping. Participants judged the validity of a set of equations that tested their ability to apply order of operations rules (e.g. multiplication precedes addition). Accuracy was greatest when non-mathematical grouping pressures were consistent, rather than inconsistent, with the mathematical grouping. Manipulating perceptual grouping through physical spacing resulted in a six-fold change in errors, and other reliable grouping effects were found based upon alphabetic proximity, connectedness, implied closure, and functional form. Effects of order of operation on spatial grouping were also found in participants¹ arithmetic productions. Learned order of precedence in mathematics influences our automatic deployment of attention and early eye movements. Our cumulative evidence suggests that one way we act in a cognitively sophisticated manner is by ³rigging up² our perceptual system to produce desirable results that are in accord with formal systems. We conclude that symbolic reasoning may generally be more visual and external than usually supposed. Accordingly, we propose a computational model of algebraic reasoning that focuses on perceptual processes of grouping, matching, marking, and attentional shifting.