Transfer of Mathematical Knowledge: The Portability of Generic Instantiations

Abstract

Abstract— Mathematical concepts are often difficult to acquire. This difficulty is evidenced by failure of knowledge to transfer to novel analogous situations. One approach to this challenge is to present the learner with a concrete instantiation of the to‐be‐learned concept. Concrete instantiations communicate more information than their abstract, generic counterparts and, in doing so, they may facilitate initial learning. However, this article argues that extraneous information in concrete instantiations may distract the learner from the relevant mathematical structure and, as a result, hinder transfer. At the same time, generic instantiations, such as traditional mathematical notation, can be learned by both children and adults and can, in turn, allow for transfer, suggesting that generic instantiations result in a portable knowledge representation.

Benefits of “concreteness fading” for children's mathematics understanding

“…Thus, children who solved a concrete problem as their first problem (concreteness fading and concrete) were less likely to make an error than were children who solved an abstract problem first (concreteness introduction and abstract), c 2 (1, N ¼ 63) ¼ 22.91, p < .001. This is consistent with previous research that suggests initial learning and performance may benefit more from concrete materials relative to abstract materials (Kaminski, Sloutsky, & Heckler, 2009).…”

“…In support of this point, children who solved a concrete problem first were less likely to make an error than children who solved an abstract problem first. However, if given only concrete instruction, children's knowledge may become too tied to the concrete context and may not transfer to dissimilar situations (e.g., Kaminski et al, 2009). Indeed, children in the concrete condition did not transfer their situated knowledge of math equivalence to the symbolic context.…”

Using analogy to improve abstract conditional reasoning in adolescents: not as easy as it looks

“…The fact that this condition produced a significant decrement in the ability of the better reasoners to make logical inferences specifically with abstract content clearly illustrates the difficulty of using analogical transfer in the case of complex abstract reasoning processes. The results of these two studies are consistent with recent empirical work, suggesting that experience with familiar problems does not necessarily help in understanding abstract principles (e.g., Kaminski et al 2009).…”

Future Directions for Theory and Research with Instructional Manipulatives:Commentary on the Special Issue Papers

“…Using notions from embedded (Martin and Schwartz 2005) and embodied (Barsalou 1999(Barsalou , 2008Glenberg and Robertson 2000) cognitive viewpoints, the authors propose that under certain circumstances perceptually rich and interactive manipulatives may improve transfer of learning. This perspective may help clarify the contradictory findings in the literature that perceptually rich features hinder transfer while in others these features facilitate transfer (Kaminski et al 2009;McNeil et al 2009;Sloutsky et al 2005). According to Pouw, van Gog, and Paas, learning with manipulatives is a complex interplay between structuring and restructuring the environment (i.e., embedded cognition) and a cognitive system consisting of manipulations of perceptual symbols (i.e., embodied cognition).…”

“…On one end of the continuum are grounded representations that share the visual appearances of objects they represent (e.g., pizzas to represent fractions) while on the other end are idealized representations with superficial characteristics stripped away (e.g., plastic circles to represent fractions). Related research with manipulatives is inconsistent with studies suggesting that grounded manipulatives may inhibit learning relative to idealized representations (Kaminski et al 2009;McNeil and Jarvin 2007).…”