"Concrete" Examples a Fraction Too Abstract

Abstract

After examining the supporting online Material for the Education Forum by J. A. Kaminski et al ., “The advantage of abstract examples in learning math” (25 April, p. [454][1]), I doubt that abstract examples are better. Rather, the study shows that the effectiveness of teaching examples depends less on whether they are “generic” or “concrete” (a somewhat confused distinction) and more on the examples' affordances that lead students to (or distract from) the learning goal. In the study, students must learn the abstract mathematical concept of mod 3. Teaching such a concept requires appropriate modeling. The study's “generic” examples (circles, diamonds, and flag shapes) and the assessment examples both serve this purpose well. Each set of examples uses three distinct symbols, which help students master the specific, limited concept. A set of shapes does not lead to confusion because it does not introduce numerosity, which is irrelevant to mod 3. By contrast, the three “concrete” examples seem geared to teaching the broader, generalized concept that includes mod 4, mod 5, and so on. These examples (such as measuring cups 1/3, 2/3, or 3/3 full) easily generalize to higher integer orders by increasing the number of symbol states (e.g., by dividing the cups into fourths, fifths, and so on, instead of thirds), but invite confusion when introducing the study's limited concept. This interferes with the lesson. Arguably, the intuitive appeal to numerosity in these “concrete” examples (invoking specific fractions) makes them more, rather than less, abstract than the “generic” example (using only shapes and no numbers). The symbols used in the assessment, although showing hints of numerical associations, do not obviously invoke numerosity. Moreover, the assessment is almost identical to the “generic” example. It is no wonder that students who learned using the “generic” example performed better on the assessment than those who learned using the “concrete” example(s). The numerical—even abstract—nature of the “concrete” examples distracted students instead of helping them. The study clearly demonstrates the possibility of constructing confusing “concrete” examples of a particular concept, but we cannot infer that “generic” examples are always superior to “concrete” examples. Whether a confusing example is “concrete” or “generic” is beside the point. [1]: /lookup/doi/10.1126/science.1154659