Jennifer A. Kaminski

The Advantage of Abstract Examples in Learning Math

Undergraduate students may benefit more from learning mathematics through a single abstract, symbolic representation than from learning multiple concrete examples.

The advantage of simple symbols for learning and transfer

A goal of successful learning is the transfer of learned knowledge to novel situations. However, spontaneous transfer is notoriously difficult to achieve. In this research, we argue that learning and transfer can be facilitated when knowledge is expressed in an abstract, generic form. In Experiments 1 and 2, undergraduate students learned two isomorphic domains, which were based on the same algebraic group, with one domain expressed in a more abstract, generic form and the other expressed in a more concrete form. In both experiments, transfer from more abstract to more concrete was greater than the reverse. In Experiment 3, undergraduate students learned the same algebraic group under varying degrees of concreteness. Our results demonstrate that the use of perceptually rich, concrete symbols may hinder learning. This research indicates that concreteness may have substantial learning and transfer costs, whereas abstractness may have benefits.

The Cost of Concreteness: The Effect of Nonessential Information on Analogical Transfer

Most theories of analogical transfer focus on similarities between the learning and transfer domains, where transfer is more likely between domains that share common surface features, similar elements, or common interpretations of structure. We suggest that characteristics of the learning instantiation alone can give rise to different levels of transfer. We propose that concreteness of the learning instantiation can hinder analogical transfer of well-defined structured concepts, such as mathematical concepts. We operationalize the term concreteness as the amount of information communicated through a specific instantiation of a concept. The 5 reported experiments with undergraduate students tested the hypothesis by presenting participants with the concept of a commutative mathematical group of order 3. The experiments varied the level of concreteness of the training instantiation and measured transfer of learning to a new instantiation. The results support the hypothesis, demonstrating better transfer from more generic instantiations (i.e., ones that communicate minimal extraneous information) than from more concrete instantiations. Specifically, concreteness was found to create an obstacle to successful structural alignment across domains, whereas generic instantiations led to spontaneous structural alignment. These findings have important implications for the theory of learning and transfer and practical implications for the design of educational material. Although some concreteness may activate prior knowledge and perhaps offer a leg up in the learning process, this benefit may come at the cost of transfer.