Abstract:
The aim of the study was to investigate 5th and 6th grade students’ understanding of proportional and non-proportional situations. The study also looks at reasons of erroneous solutions in proportional and non-proportional problems depending on the type of numbers used in the problems as integer ratio or non-integer ratio. Data collection instrument was a 12-item test that included four types of word problems, additive, constant, proportional comparison (PC), and proportional missing-value (PM), each with an integer and non-integer ratio. One hundred and twenty 5th and 101 6th grade students in a private school participated in the study. 5th and 6th grade students solved proportional and non-proportional situational problems with different success rates. In detail, constant problems were solved with the lowest success rate, while proportional missing-value problems with the highest success rate in both grades. When the use of erroneous strategies was calculated in percentages, the tendency to overuse proportional strategy in non-proportional situations was observed in both grade levels. The study also examined number effect on students’ success rate in proportional and non-proportional situations. The analysis showed that, fifth grade students’ success rates in integer and non-integer numbered problems were significantly different in only additive problems. However, in the 6th grade the success rates differed significantly in additive and PC problems. Additionally, 5th grade students’ choice of the methods significantly differed depending on the number change in only additive problems, while 6th grade students’ choice of the strategies significantly differed in constant and PC problems. In constant problems when problems included integer ratios 5th and 6th grade students tended to use proportional methods, and when problems included non-integer ratios they tended to prefer additive methods. Use of “other” method in the 5th grade also increased significantly by the number change in the problems. In additive problems, when numbers changed from integer ratio to non-integer ratio, there was a significant difference in the overuse of proportional methods. However, the expected difference in the overuse of additive strategies in proportional problems when numbers form non-integer ratios was not observed. Qualitative findings revealed two important points. One of these points was students’ understanding about “building-up” strategy. One another point was students’ decision of the solution strategy without a full understanding of the problem. Frequently, students had a tendency to rely their solution strategy on the relation between numbers in the problems. Besides, students’ difficulty in explaining their way of thinking, and students’ beliefs about mathematical problems supported their insufficient understanding of the problem.
