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An investigation of algorithm justification in elementary school mathematics Weinstein, Marian S.

Abstract

It was the purpose of this study to determine by experimental procedures whether there are any significant differences either in computational skill with an algorithm or in ability to extend that algorithm among elementary school pupils taught a mathematical algorithm by different methods of justification. The four types of justification methods were: pattern, algebraic, pattern followed by algebraic, and algebraic followed by pattern. A pattern justification is one based on an analog to two-dimensional phyiscal actions, whereas an algebraic justification is one based on the algebraic principles for rational numbers, as well as the rules of logic. Differences in performance among treatment groups were examined for four algorithms varying in both mathematical operation accomplished by the algorithm and in complexity of the algorithm. The latter is determined by the number of steps and processes required for its execution. The two simple algorithms were multiplication of a fraction and a mixed number and comparison of fractions using the cross-product rule. The two complex algorithms were conversion of a fraction to a decimal and calculation of the square root of a fraction. Three classes were given a strictly pattern justification, three a strictly algebraic justification, one a pattern followed by algebraic justification, and one an algebraic followed by pattern justification for one simple and one complex algorithm. The same assignment was made for the other simple and complex algorithm pair. In total, 16 grade five classes participated in the experiment. The results of the multiple analysis of covariance indicated no significant differences in the case of all four algorithms tested among the comparisons between students taught by a strictly pattern approach and students taught by a strictly algebraic approach. However, there is evidence to indicate that students taught by an algebraic approach, as a group, tend to do better on extension tests than their pattern taught counterparts and that students taught by a pattern method, as a group, tend to do better on simple algorithm computation tests than their algebraically taught counterparts. No trend is evident in terms of performance of groups on the complex algorithm computation tests. Furthermore, the existence of a significant between-groups-within-treatments effect indicates the strong possibility of a teacher by treatment interaction which might be further investigated. Although some significant differences were found among the algebraic followed by pattern and pattern followed by algebraic comparisons in favor of the algebraic followed by pattern groups, these results must be considered in light of the possibility that a group difference is what is being indicated confounded with a treatment difference. There did appear a trend, although nonsignificant, indicating that the algebraic followed by pattern taught students performed better, in general, on the extension tests. Finally, the data indicated the plausibility of a model for research into algorithm learning in elementary mathematics which incorporates two dimensions--type of justification provided for the algorithm and complexity of the algorithm-- as useful determinants of student performance on computation and extension tests based on that algorithm.

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