No abstract

No 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.

Educational material often includes engaging perceptual information. However, this perceptual information is often extraneous and may compete with the deeper to-be-learned structure, consequently hindering either the learning of relevant structure or its transfer to new situations. This hypothesis was tested in 4 experiments in which 6-to 8-year-old children learned to read simple bar graphs. In some conditions, the bars were monochromatic (i.e., No Extraneous Information), whereas in other conditions, the bars consisted of columns of discrete countable objects (i.e., Extraneous Information). Results demonstrated that the presence of extraneous information substantially attenuated learning; participants tended to count the objects and failed to acquire the appropriate strategy. The interference effects decreased with age. These findings present evidence of how extraneous information affects learning of new mathematical knowledge. Broader implications of these findings for understanding the development of the ability to filter task-irrelevant information and for educational practice are also discussed.

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.