Motivation 

To explore how physical representations of symbolic relationships affect accuracy in symbolic reasoning.

Symbolic Reasoning

Symbolic reasoning is a higher order thinking skill that allows us to reason about abstract structures and relationships without relying on a concrete, physical context (e.g., evaluating whether a + b * c + d = d + c * b + a). Because symbolic representations are still physically written on paper, screens, etc., Landy and Goldstone argue that they are subject to biases based on their physical representations, despite differences in physical representations being completely irrelevant to symbolic reasoning. For the equation above, most people would more quickly evaluate whether a + b*c + d  =  d + c*b + a, even though the space between symbols does not affect their relationship. Another example is data visualization. Despite representing the same data, different graph designs can affect how viewers interpret data.

Due to biases from physical representations, notation formats are important for representing symbolic structures. In prior work, Landy and Goldstone found that people who write equations on paper use spatial proximity to visually indicate grouping. They had less space around multiplication signs than addition signs, and they put the most space around equal signs. To explore this phenomenon, the authors conducted a series of four experiments to examine the effect of non-mathematical grouping on accuracy in order-of-operations-based symbolic reasoning.

General Method

Landy and Goldstone asked students to evaluate equivalencies in order of operations problems, like the one given above. They gave participants 240 equations with four letters on each side and following either +*+ or *+* structures between letters. Sensitivity of equations was manipulated in all experiments. Sensitive equations were those that were incorrect but would be correct, or vice versa, if participants did addition before multiplication. Insensitive equations were those that would be correct or incorrect regardless of the order of multiplication and addition. The experimental design was within-subjects, so all participants received all permutations. Participants were asked to answer the questions as quickly as possible without sacrificing accuracy.

Experiment 1

Additional manipulation: Experiment 1 used three physical layouts. The consistent equations visually grouped letters that were multiplied closer than letters that were added (a*b + c*d); the neutral equations had the same spacing throughout (a * b + c * d); and the inconsistent equations visually groups letters that were added closer than letters that were multiplied (a * b+c * d).

Results: In general, accuracy was higher for valid equivalencies (86%) than invalid ones (82%). An interaction of grouping and sensitivity revealed that grouping substantially affected accuracy on only sensitive equations. Therefore, grouping affected performance on only questions in which the order of operations mattered for determining equivalency, suggesting that visual grouping affects implementation of order of operations, despite having no formal symbolic meaning.

Experiment 2

Additional manipulation: Experiment 2 used a non-mathematical visual cue to group part of the equation together (i.e., imagine an oval around a + b but designed to not be confused with parentheses). All equations used the neutral spacing from Experiment 1 and added visual cues that were either consistent or inconsistent with the order of operations, or the equation had no visual cue and was neutral.

Results: The same as in Experiment 1, grouping substantially affected accuracy on only sensitive equations, suggesting that visual grouping that are clearly not mathematical can affect implementation of mathematical problem solving.

Experiment 3 and 4

The third and fourth experiment followed the same type of structure. A variable was manipulated to be consistent, neutral, or inconsistent between the visual representation and the symbolized relationship. Participants continued to perform best on consistent representations that were sensitive to order errors, suggesting that physical representations of symbolic concepts affects our processing of non-physical concepts.

Why this is important

This research suggests that symbolic reasoning is not immune to influence from physical representations of symbols. With a bit of a stretch, this means that the visual, spatial, and gestural ways that we represent abstract concepts can affect how to process and reason about those concepts, even if they should be completely unaffected. This reminds me of David Weintrop’s work on block-based and text-based programming in which he argues that novices find block-based programming easier to understand. Perhaps this is partially because block-based programming provides groupings that are consistent between physical and symbolic forms while text-based programming constrains the grouping to neutral. I don’t know how block-based languages were originally designed, but maybe we could learn about the optimal visual design of programming languages by examining how programmers write out their code on a piece of paper. It might even be an argument to teaching programming sometimes with exaggerated, consistent physical representations by using a whiteboard.

Landy, D., & Goldstone, R. L. (2007). How abstract is symbolic thought?. Journal of Experimental Psychology: Learning, Memory, and Cognition, 33(4), 720-733.

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