A study of children's understanding of fraction size and its relationship to proportional reasoning

One purpose of this study was to investigate children's understanding of fraction size; specifically, their ability to order fractions and to relate the size of a fraction to whole numbers. Other purposes were to examine the relationships between children's understanding of fraction size and their knowledge of the multiplicative relationship between numbers as measured by a proportional-reasoning task and to explore the commonality among children's strategies for tasks involving multiplicative structures.Three different written tasks, two fraction-size tasks and one proportion-reasoning (PR) task, were administered to sixty fourth- and fifth-grade children from three schools in east-central Illinois. The two fraction-size tasks assessed children's ability in relating the size of fractions to whole numbers (FW) and comparing the size of fraction pairs (FF). The proportion-reasoning tasks were based on Noelting's orange-juice tasks and required knowledge of the multiplicative relationship between numbers. Twelve students from each grade were individually interviewed to explore their strategies.The results revealed the following: (1) Children's performance on the fraction-and-whole-number-relation (FW) tasks revealed several misconceptions, including the dominance of whole-number concepts, detachment of conceptual understanding from mathematical procedures, using fraction models without knowledge of fraction size, and deficiency of estimation skill. Children's performance on the fraction-fraction comparison tasks (FF) tasks revealed an additive type of thinking. Results also showed that children performed better on the FF tasks than on the FW tasks. (2) There was a significant correlation between children's performance on the two fraction-size tasks and their knowledge of the multiplicative relationships between numbers measured by the proportional-reasoning task. (3) Children employed parallel strategies across different types of tasks involving fractions or ratios. Children used parallel strategies on the two fraction-size tasks. Moreover, these strategies were reliably associated with levels of performance on the proportional-reasoning tasks. Put differently, strategies that required multiplicative thinking were related to success on the fraction-size and proportional-reasoning tasks.The findings implied that instruction of fractions needs to help children recognize the differences between the concept of whole numbers and the concept of fractions, develop an understanding of the size of a fraction in relation to other numbers including whole numbers, and construct an understanding of the multiplicative relationship. Efforts need to be taken to help children build the connections between the concrete representations of fractions and the ideas of fractions as numbers.U of I OnlyETDs are only available to UIUC Users without author permissio