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Spatial-temporal-enactive structuring in combinatorial enumeration

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Abstract

Lockwood has argued that taking a set-oriented perspective is critical for successful combinatorial enumeration. To date, however, the research literature has not yet captured the cognitive processes involved in taking such a perspective. In this theoretical paper, we elaborate the constructs of spatial structuring and spatial-numerical linked structuring from within our constructivist theoretical perspective to provide a cognitive account of how students conceptualize and generate combinatorial composites. Our elaborations emerged from analyzing and reflecting on data from two constructivist teaching experiments examining undergraduate students’ combinatorial concepts, mental actions, and strategies. We define multiple constructs in our theoretical approach, including spatial-temporal-enactive (or S*-) structuring, two specific forms of S*-structuring (intra-composite and inter-composite structuring), numerical structuring, and S*-numerical linked structuring. We then illustrate how our framework can be used to analyze students’ physical/mental actions and operations when engaging in combinatorial inquiry. Critically, our findings suggest that successful and meaningful combinatorial enumeration depends on constructing intra-composite S*-structuring for single combinatorial composites, then abstracting, reflecting on, generalizing, and linking this intra-composite structuring to an inter-composite structuring to generate the set of combinatorial composites. We conclude with a discussion of how our paper contributes to existing research on combinatorial reasoning.

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  1. Department of Teaching and Learning, The Ohio State University, Columbus, OH, USA

    Joseph Antonides & Michael T. Battista

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  1. Joseph Antonides
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Correspondence to Joseph Antonides.

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Antonides, J., Battista, M.T. Spatial-temporal-enactive structuring in combinatorial enumeration. ZDM Mathematics Education 54, 795–807 (2022). https://doi.org/10.1007/s11858-022-01403-0

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