Negative Instances in Mathematical Concept Acquisition: Transfer Effects Between the Concepts of Commutativity and Associativity.

Negative instances have been considered by mathematicians to be essential to the understanding of advanced mathematical concepts (Gelbaum & Olmsted, 1964; Steen & Seebach, 1970). Dienes (1964) argues for the use of negative instances in the teaching of mathematics to elementary and secondary school children. Bereiter and Engleman (1966) and Markle and Tiemann (1970) state explicitly that all instructional sequences designed for concept learning should include negative instances. Yet, in a review of over 250 experimental studies in concept attainment, Clark (1971) found 26 studies that reported a debilitating effect for negative instances on concept attainment and only 11 that did not report a debilitating effect for negative instances. Markle and Tiemann (1972) state that, despite such laboratory evidence, positive and negative instances "are of equal importance in the teaching of real concepts." In a review of research, Bourne and Dominowski (1972) support the contention that for simple, conjunctive concepts a series of positive instances is to be favored over any mixture of positive and negative instances, but that for more difficult concepts, such as disjunction, the advantage of positive instances is either absent or negative instances are favored. Both Clark (1971) and Bourne (1967) agree that the research in concept attainment is restricted in nature and that the concepts studied in schools are quite different. There is some evidence from recent educational research that negative instances are valuable components of classroom instruction (Tennyson, Woolley, & Merrill, 1972; Shumway, 1971).