The effects of concreteness on learning, transfer, and representation of mathematical concepts

The present research investigated the effects of concrete and generic instantiations on learning and transfer of a mathematical concept. The current work distinguished two types of concreteness. Irrelevant concreteness is extraneous information that is unrelated to the to-be-learned concept and has been shown in previous research to hinder both learning and transfer. Relevant concreteness involves symbols or storyline that help communicate the to-be-learned conceptual structure. In a series of experiments, undergraduate college students learned instantiations of a mathematical group that were generic, relevantly concrete, or irrelevantly concrete. Relevant concreteness was found to promote quick learning. However, the benefit of relevant concreteness did not extend to transfer. Relevant concreteness hindered recognition and alignment of structure between the learned domain and a novel isomorph which resulted in transfer failure. With slightly protracted training, the generic instantiation was learned equally well as the relevantly concrete one. Most importantly, the generic instantiation resulted in successful structural alignment and transfer. The benefit of the generic instantiation was also demonstrated for children. In a separate experiment, after learning a relevantly concrete instantiation, participants were unable to recognize learned structure when presented with novel elements. However, participants who learned the generic or irrelevantly concrete instantiation were able to recognize structure in the context of novel elements, suggesting that relevant concreteness obfuscates the analogy between the domains. When participants were given the correspondence of elements across learning and transfer domains, relevantly concrete instantiations did result in successful transfer. Furthermore, learning a generic instantiation was shown to have an advantage for transfer over learning multiple instantiations. Finally, relational diagrams that convey global structure resulted in transfer to isomorphs as well as modification and transfer to non-isomorphic systems of the same structure category. This research argues that relevant concreteness may give a leg-up in the learning process. However, this benefit comes at the cost of transfer. Learning a generic instantiation can provide a direct route to forming an abstract internal representation that can facilitate transfer.