Variations in Partitive Quotient Strategy Use by Children Who Have Been Taught the Part-Whole Fraction Sub-construct

Abstract

This paper presents findings from a study that examined the strategies that children, who had only been taught the part-whole fraction sub-construct at school, used for finding the fraction associated with solving varied partitive quotient problems. A qualitative, microgenetic research design was used involving nine year 5 (aged 9–10) children engaged in eight individual task-based interviews over a 6-week period. The data analyzed showed that across the eight tasks, six of the nine children used more than one strategy for quantifying each person’s share but, by the third task, in general, each child had settled into a regular pattern of strategy use. The analyzed variations in children’s approaches to solving the partitive quotient problems revealed instances of when and how the part-whole fraction sub-construct interfered with children’s engagement with the partitive quotient problems. Considering that, internationally, the part-whole sub-construct is still the first fraction sub-construct that many children learn in schools, the findings are significant, since they provide new, in-depth insights into emerging approaches to solving partitive quotient tasks that are influenced by children’s existing part-whole knowledge. The findings provide new evidence for intra- and inter-individual variation in strategy use and strategy selection in tasks related to the partitive quotient meaning of fractions. The paper highlights for education, the need for teaching that introduces different fraction sub-constructs to learners early in their schooling so that one meaning of fractions does not become representative of all fraction knowledge.

Learning and understanding the complexity of fractions

“…The interpretation of representation can be explained as follows: the denominator stands for the number of recipients, and the numerator is the quantity that has to be shared (Behr et al, 1993;Kieren, 1993Kieren, , 1976Marshall, 1993;Middleton et al, 2001). This process involves a minimum of two stages of interpretation: The first stage includes an interpretation of the fraction as an operation, and this means a pathway to understanding the equivalence or transfer between as the result of 1÷ 4 (Behr et al, 1983;George, 2017;Marshall, 1993). Furthermore, division is the only whole operation that includes whole numbers, where a rational number can be the outcome, and it is a way to connect and develop students' whole number understanding so they have a concept of numbers that includes rational numbers (Bright et al, 1988;Middleton et al, 2001).…”

“…A ratio can likewise be seen as a part-part relationship, where two similar units are compared. For instance, there are four boys for every five girls (George, 2017). The term ratio is not well defined, and there is no consensus about the terminology (like fractions).…”