Developmental Progression of Fraction Magnitude Representations

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Abstract

Introduction: The estimation of numerical magnitudes is central to the understanding of fractions and whole numbers (Siegler, Thompson, & Schneider, 2011). Previous research has documented that whole number magnitude representations transition from a logarithmic (in which the psychological distance between small numbers is exaggerated) to linear representation (Siegler & Booth, 2004; Siegler & Opfer, 2003). In contrast, the development of fraction magnitude involves linear representations that become increasingly accurate between sixth and eighth grade and does not shift from logarithmic to linear representations (Siegler et al., 2011). By sixth grade, children have had several years of fraction instruction. Very little is known about the nature of fraction magnitude representation in late elementary school, when the bulk of fraction instruction takes place (Council of Chief State School Officers & National Governors Association Center for Best Practices, 2010). It is crucial to learn more about this time period in order to understand how fraction magnitude representation develops as students begin to receive fraction instruction. We examined whether a logarithmic to linear transition would be present during the early acquisition of fractions knowledge. Specifically, we examined fraction magnitude representations in fourth grade prior to fraction instruction, immediately after fraction instruction, and one year later, at the end of fifth grade.
Method: Participants were 366 students who were assessed longitudinally at three time points: Winter Grade 4, Spring Grade 4 and Spring Grade 5. For the number line estimation task, children were presented with fraction estimation problems (0-1 number line: 9 problems, 0-2 number line: 19 problems).
Results: Percent absolute error (PAE) was used to index the accuracy of number line estimation. A 3(Grade: Winter 4th, Spring 4th, Spring 5th) X 2 (Number line: 0-1 or 0-2) repeated measures ANOVA revealed a main effect for grade, F(2, 364) = 172.61, p < .001, ηp2 = .49. Post-hoc analyses using Bonferroni correction revealed that 5th grade estimates (PAEs = 16%) were more accurate than Spring (PAEs =19%) and Winter (PAEs=25%) 4th grade estimates (all ps < .001). Estimates were more accurate on the 0-2 than 0-1 number line, PAEs = 19% and 21%, F(1, 365) = 37.90, p < .001, ηp2 = .09.

Author

• Vinaya Rajan, University of Delaware