1 Different presentations of a mathematical concept can support learning in complementary ways Andrew

Previous research has found that different presentations of the same concept can result in different patterns of transfer to isomorphic instances of the same 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 vs 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. Educational Impact And Implications Statement: The details of how a concept is taught can have far-reaching effects on students’ learning. Using abstract algebra with adult subjects, we show that two presentations of a concept that connect to different types of students’ prior knowledge can have advantages and disadvantages for later learning that builds on the target concept. We show that one possible solution to the dilemma of choosing which presentation to use is giving students both concepts and explaining how they are related to each other. In summary, when designing pedagogical materials, we should consider not only how they affect learning of the present concept, but also how they support learning of future concepts, and use multiple complementary presentations rather than searching for a single ideal one. DIFFERENT PRESENTATIONS OF A MATHEMATICAL CONCEPT 3 Different presentations of a mathematical concept can support learning in complementary ways