• Journal of the Korean School Mathematics Society
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• v.11 no.4
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• pp.609-627
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• 2008

The purpose of this research was to investigate the relationship between the students' strategy types and the recognition for proportional situations. The students' strategy types which were based on the results of ratio and proportion tests were divided into an additive type, a multiplicative type, and a formal type. This research analyzed the students' activities of categorization when were given the proportional problems and nonproportional problems to the students. And it also explored how to develop students' recognizing for the discrimination between the proportional situations and nonproportional situations. The results was the following. First, the students didn't discriminate the proportional situations and the nonproportional situations in the initial state but they came to discriminate little by little. Secondly, the students didn't discriminate the direct proportions and the inverse proportions until the last stage. Third, the multiplicative type was outperformed more than the formal type in solving the ratio and proportion problems but the formal type was outperformed more than the multiplicative type in discriminating between proportional situations and nonproportional situations. These results are interpreted as showing that solving ratio and proportion tasks and recognizing proportional situations are different aspects of proportional reasoning and it is necessary to understand multiplicative strategy with formal strategy in recognizing proportional situations.

• School Mathematics
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• v.10 no.4
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• pp.603-623
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• 2008

The purpose of this research was to gain a better understanding of the characteristics of students' mathematical representations using additive strategy in ratio and proportion tasks. The additive strategy is the erroneous one used most often among the strategies reported in solving ratio and proportion tasks. It is a problem solving strategy that preserves the difference from one ratio to another. Students' additive strategies were categorized into four parts: subtracting without considering units of quantities, comparing the numbers that represent the whole subtracted from the part and same part, adding the difference, and subtracting the difference. In order to change from additive strategy to multiplicative strategy, the researcher asked to find out the unit quantity and found the characteristics of students' mathematical notations in the following: Firstly, the students made the number which they wanted by multiplying and adding same numbers. Secondly, they represented the mid-points between natural numbers. Thirdly, they related $a{\div}b$ to decimal number, not $\frac{a}{b}$. Fourthly, they were inclined to divide the larger number with the smaller number without understanding the context of the problem. These results are interpreted as showing that lower level of performance in the dividing operation with the notations of fraction hinders the transformation from additive strategy to multiplicative strategy.