Different Presentations of a Mathematical Concept Can Support Learning in Complementary Ways

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Previous research has found that different presentations of the same concept can result in different patterns of transfer to isomorphic instances of that concept. Much of this work has framed these effects in terms of advantages and disadvantages of concreteness or abstractness. We note that mathematics is a richly structured field, with deeply interconnected concepts and many distinct aspects of understanding of each concept, and we discuss difficulties with the idea that differences among presentations can be ordered on a concrete-abstract dimension. To move beyond this, we explore how different presentations of a concept can affect learning of subsequent concepts and assess several distinct aspects of understanding. Using the domain of elementary group theory, we teach adult participants a group operation using a visuospatial or an arithmetic presentation. We then teach them concepts that build upon this operation. We demonstrate that our presentations differentially support learning complementary aspects of the system presented. We argue that these differences arise from the fact that each presentation supports learning by connecting to different systems of reasoning learners are already familiar with, and that it is these connections to extant knowledge systems, rather than differences in concreteness versus abstractness, that determine whether a presentation will be helpful. Furthermore, we show that presenting both presentations and encouraging participants to recognize the relationship between them improves performance without requiring additional time, at least for some participants.

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