Coordinating Visual and Analytic Strategies: A Study of Students' Understanding of the Group D 4

Abstract

This study contributes to the ongoing discussion of visualization and analysis in mathematical thinking. On the basis of data gathered from clinical interviews with 32 students in their first abstract algebra course, we consider the tasks of listing the elements of the dihedral group D4 and finding the product of two such elements. These problems can be solved either using a “visual” approach of transforming a square or an “analytic” approach of multiplying permutations. Rather than clearly preferring either a visual or analytic strategy, most students in our study used some combination of these approaches. Our results suggest that the conventional analyzer/visualizer dichotomy may not be an appropriate classification scheme for describing learning processes or for designing instruction. We propose an alternative model, the Visualizer/Analyzer or VA model, that assumes visualization and analysis to be mutually dependent in mathematical problem solving, rather than unrelated opposites. Our model provides one description of how this mutual dependence might function. We end by considering how pedagogical approaches might be designed in consonance with this model to help students coordinate visual and analytic thinking.

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