Abstract
Research has identified two core difficulties many students have with fractions: First, they often struggle with processing fraction magnitudes, and second, they rely on natural number concepts in fraction problems (“Natural Number Bias”, NNB). Yet, the relation between these two difficulties is not well understood. Moreover, while most studies of the NNB relied on analyses of the whole samples, there is empirical evidence that the occurrence of the NNB differs between student subgroups. In the present study, we investigate individual students’ profiles of the occurrence of the natural number bias and their ability to process fraction magnitude, using a dynamic assessment that utilizes continuous diagrams on touchscreen devices. We analyze data of 234 low-achieving 6th-grade students from Germany who completed a symbolic fraction comparison task, and a fraction magnitude estimation task with continuous circle and tape diagrams. A cluster analysis on the comparison task revealed three distinct clusters: a Typical Bias cluster (better performance on symbolic fraction comparison items congruent to natural number-based reasoning), a Reverse Bias cluster (better performance on items incongruent to natural number-based reasoning), and a No Bias cluster (similar performance on congruent and incongruent items). Only students in the No Bias cluster but not students in the other clusters demonstrated a distance effect in symbolic fraction comparison, suggesting fraction magnitude processing. Linear mixed models on the percent absolute error in the magnitude estimation task revealed significantly lower percent absolute error for students in the No Bias cluster compared to students in the other two clusters. Moreover, students in the No Bias cluster were significantly slower to solve both fraction comparison and fraction magnitude estimation tasks than students in the other clusters. These results suggest that the occurrence of the natural number bias and the ability to process fraction magnitude are closely related. The continuous representations used in our digital assessment tools appeared to be suitable for assessing both the natural number bias and fraction magnitude processing.
Highlights
Plenty of research has shown that many students struggle with learning of rational numbers, of fractions (e.g., Behr et al, 1983; Siegler et al, 2011; Lortie-Forgues et al, 2015)
We were interested in individual profiles of Natural Number Bias” (NNB), and in the interplay between an NNB and fraction magnitude processing
Based on current literature and these findings, we suggest a tentative model of competence in fraction magnitude processing that could be empirically evaluated in further research: (1) On the lowest level, students show a persistent NNB with no fraction magnitude processing
Summary
Plenty of research has shown that many students struggle with learning of rational numbers, of fractions (e.g., Behr et al, 1983; Siegler et al, 2011; Lortie-Forgues et al, 2015). Two major difficulties seem to be that students (1) are not sufficiently able to understand and process fraction magnitudes, and (2) rely on natural number principles when reasoning about rational numbers, causing Natural Number Bias (see Ni and Zhou, 2005 and see section “The Natural Number Bias as a Source of Individual Errors in Solving Fraction Problems”). While both difficulties have been discussed in the literature, there is still little evidence about the relation between the two. The present study assesses individual students’ profiles (i.e., student subgroups) of natural number bias and investigates how these profiles are related to students’ ability of processing fraction magnitude
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