Inducing Mathematical Concepts from Specific Examples: The Role of Schema-Level Variation

Inducing Mathematical Concepts from Specific Examples: The Role of Schema-Level Variation David W. Braithwaite (dwbraith@indiana.edu) Robert L. Goldstone (rgoldsto@indiana.edu) Indiana University, 1101 E. 10 th Street Bloomington, IN 47405 USA familiar contexts (Nunes, Schliemann, & Carraher, 1993). In such contexts, learners can apply intuitions from everyday life to help in understanding the mathematical ideas involved. Abstract presentation of mathematical ideas therefore risks sacrificing learning for the sake of transfer. It may, then, be desirable for learners to encounter mathematical ideas in a way that leverages their intuitive understanding of specific examples, while also drawing attention to the abstract structure present in those examples. Research on analogy suggests that this goal might be achieved through presentation of multiple specific examples followed by comparison (Gentner, Loewenstein, & Thompson, 2003; Gick & Holyoak, 1983). Comparing examples encourages learners to align their corresponding elements, and thereby to notice their common relational structure. Awareness of this structure, in turn, can facilitate understanding of new cases with the same structure. Thus, learning mathematical ideas by studying and then comparing multiple examples may enable learners to gain intuitive accessibility without losing generality. The question then arises as to how the examples which will instantiate a mathematical concept during learning are to be chosen. Central to this question is the issue of how much, and in what ways, the examples should differ from each other. If, as the above research suggests, learners induce concepts that incorporate commonalities among the examples, it seems desirable that the examples should share the mathematical structure in question, but should not share other extraneous details. Extraneous commonalities might be misunderstood as part of the concept to be learned, limiting learners’ ability to generalize (Medin & Ross, 1989), and so defeating the purpose of using multiple examples in the first place. These observations suggest that extraneous aspects should be systematically varied across examples, while holding mathematical structure constant. The present study investigates the effects on mathematical concept learning of a particular type of variation among examples: variation at the level of “semantic schemas.” This term here refers to structures more general than specific examples but less general than mathematical structure. Consider the three combinatorics problems shown in Figure 1. Problems (a) and (b) share a schema, termed “Objects Selected in Sequence” (OSS), in which a sequence of selections is made from a fixed set of options. Problem (c), by contrast, belongs to a different schema, termed “People Choosing Options” (PCO), in which several people each choose once from a fixed set of options. Abstract Previous research suggests that comparing multiple specific examples of a general concept can promote knowledge transfer. The present study investigated whether this approach could be made more effective by systematic variation in the semantic content of the specific examples. Participants received instruction in a mathematical concept in the context of several examples, which instantiated either a single semantic schema (non-varied condition) or two different schemas (varied condition). Schema-level variation during instruction led to better knowledge transfer, as predicted. However, this advantage was limited to participants with relatively high performance before instruction. Variation also improved participants’ ability to describe the target concept in abstract terms. Surprisingly, however, this ability was not associated with successful knowledge transfer. Keywords: mathematics; analogy; comparison; schemas; instruction; transfer Introduction Part of the power of mathematics lies in its generality. The same mathematical formulae may be used to understand the growth of slime molds or the accumulation of interest from investments, the probabilities of hands in poker or outcomes of scientific experiments, and the oscillations of mechanical or electromagnetic systems. In order to fully realize this power, however, learners must be able to recognize and apply mathematical concepts in contexts different from those in which they were learned – that is, to transfer their mathematical knowledge from learned to novel contexts. Learners’ difficulties in achieving such transfer are well- documented (Novick & Holyoak, 1991; Ross, 1987). One reason may be that, when a general idea is learned in the context of specific examples, learners’ concepts become tied to the details of the examples, inhibiting their ability to recall the concept or apply it correctly when faced with cases that do not share similar details (Ross, 1987). This difficulty may be especially strong when the examples are presented in a perceptually detailed format (Kaminski, Sloutsky, & Heckler, 2008), and is likely to be more serious for domain novices than experts (Novick & Holyoak, 1991). One way to address this difficulty is to present mathematical ideas in abstract form, without specific examples. Such an approach has indeed been shown to promote transfer in some cases (Kaminski et al., 2008). However, in other cases, learners have experienced serious difficulties with abstractly-presented mathematics, despite being competent with the same mathematics encountered in