Classification of Efficient Calculation Problems and the Effect of Instruction Using an Abstract Strategy

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Calculation problems such as 1+2+3+97+98+99 can be solved rapidly and easily by using certain techniques; we call these problems "efficient calculation problems." However, many students do not always solve them efficiently. To address this problem among students, this study developed a new teaching method. The first study sought to examine underlying subcategories of efficient calculation problems. To accomplish this, multidimensional scaling and cluster analysis to a similarity matrix obtained from expert judgments were conducted. As a result, we classified 20 efficient calculation problems into 8 categories. The second study examined the effect of an intervention on 59 eighth grade students and 52 fifth grade students. The students were instructed to use an abstract strategy that stated, "Think carefully about the whole expression," and were then taught the solution to the problem. The results indicated that the eighth grade students solved similar problems efficiently after the intervention, while the fifth grade students did not. The results also suggested that the effect of the intervention was greater among students with sufficient basic calculation skills.



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