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The effect of teaching heuristics on the ability of grade ten students to solve novel mathematical problems Pereira-Mendoza, Lionel

Abstract

The study investigated whether it was possible to teach students to apply at least one of the heuristics of examination of cases or analogy to novel mathematical problems, and whether learning to-apply heuristics could be better accomplished by having students focus solely on the heuristics, rather than also involve them with the simultaneous learning of mathematical content. Two factors were considered in the design of the study. The first factor considered employing treatments designed to meet different objectives: one instructional treatment was designed to teach heuristics only (H); another combined the instruction in heuristics with the teaching of content (HC); and a third treatment utilized content only (C), without reference to heuristics. The latter treatment was used as an experimental control. The second factor concerned employing distinct instructional vehicles (materials) for the treatments, namely algebraic (A), geometric (G) and mathematically neutral (N) materials. A sample of grade ten boys was selected for the study. Nine self-instructional booklets were prepared, corresponding to each of the treatment x vehicle combinations. The subjects were randomly assigned to one of the nine groups. At the end of the ten day instructional period (one class period per day) two transfer tests, one algebraic and the other geometric, were administered to all subjects. Data from 189 subjects (21 per group) were analysed by ANOVA. On the algebraic test the H treatment was significantly better than both the HC and C treatments, and. there was no significant difference between the HC and C treatments. The conclusion was that only the H treatment can be claimed to be effective in teaching students to apply at least one heuristic to a novel algebraic problem. On the geometric test both the H and HC treatments were significantly better than the C treatment, and there was no significant difference between the H and HC treatments. The conclusion was that both the H and HC treatments were effective in teaching students to apply at least one heuristic to a novel geometric problem. An analysis of the pattern of heuristic application on the two tests revealed that while both heuristics were employed on the algebraic test there was little evidence of analogy being used on the geometric test. The investigator suggested that the effectiveness of the HC treatment on the geometric test may have been due to the fact that the HC treatment was more effective in teaching examination of cases than teaching analogy, and the geometric test proved more amenable to the application of examination of cases. There were no significant differences involving the vehicles. Thus it seemed reasonable to conclude that the instructional vehicle employed was not an important factor in whether students learned to apply heuristics. Finally, an exploratory non-parametric (x²) analysis for the H treatment indicated that subjects who were in the top third on either mathematical or reading achievement (or both) scored substantially higher than the bottom third.

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