Factors contributing to children’s understanding of fractions: quantitative reasoning and arithmetic
Keywords: Arithmetic
Fractions
Non-verbal reasoning
Quantitative Reasoning

The aim of this study was to investigate the origin of children’s understanding of fractions by testing a theoretical model in which quantitative reasoning and arithmetic knowledge are proposed as independent and significant factors for learning fractions. The model also includes non-verbal ability as a moderator of how much children learn when they participate in fractions lessons.
Quantitative reasoning is the ability to use relations between quantities to come to new conclusions; for example, knowing that a cake shared fairly among three children and two cakes shared fairly among six children produces equivalent amounts of cake. Arithmetic knowledge is the ability to understand number relations in the operations of addition, subtraction, multiplication and division.
This study used a combination of longitudinal and intervention methods involving three steps: (1) the development of measures and (2) an analysis of their predictive strength in the context of a longitudinal study; and (3) an intervention study.
Participants in the longitudinal study were 9-year-olds (N=124; Year 4) who had received limited formal instruction on fractions. They completed two tests, the Quantitative Reasoning and the Arithmetic Knowledge test, which were subjected to a confirmatory factor analysis to investigate the statistical independence of the measures. They also completed a non-verbal ability test, which was used as a control in the longitudinal predictions. Children were assessed a second time six months later using the same measures to investigate whether the factor structure of the measures was invariant over time. At the final assessment point, 12 months later, at the end of Year 5 and after formal instruction about fractions in school, participants answered a sample of items from a standardised Fractions test (CSMS Fractions) (Hart, Brown, Kerslake, Kuchemann, and Ruddock, 1985).
The confirmatory factor analyses showed a better fit of a two-factor model than of a one-factor model, supporting the hypothesis that the two theoretically different abilities, quantitative reasoning and arithmetic, can be differentiated statistically. Hierarchical regressions (N=105) investigated the independence and relative strength of quantitative reasoning and arithmetic in the prediction of performance in the CSMS Fractions items after school instruction. The regressions showed that each measure contributed independently to the prediction of performance in the CSMS Fractions items, after controlling for non-verbal ability; quantitative reasoning was the stronger predictor of the two.
The intervention study aimed to assess the impact of two fraction intervention programmes on the CSMS Fractions test: one intervention promoted quantitative reasoning and the other promoted arithmetic skills. Both groups were compared to an unseen control group, who participated in the assessments but did not receive any intervention from the researcher. A total of 103 Year 4 children in four schools participated. All children were pre-tested before random allocation to one of three groups. Each of the intervention groups participated in 8 weekly half-hour sessions with a researcher in small groups outside the classroom. Analyses of covariance, controlling for pre-test performance and non-verbal ability, showed a significant effect of the quantitative reasoning intervention: Cohen’s d effect size=0.44 at the immediate post-test and d=0.45 at the delayed post-test. The arithmetic intervention had a significant impact only at the delayed post-test (d=0.28). Pre-test performance did not moderate children’s learning in the intervention, whereas non-verbal ability acted as a significant moderator at the delayed post-test.
Results confirm that quantitative reasoning and arithmetic are different mathematical abilities. Both make independent contributions to learning fractions; quantitative reasoning is a stronger predictor. This result is consistent with training results, as quantitative reasoning training yields the best performance in fractions learning. This learning remained stable after several weeks and non-verbal reasoning moderates their learning in the long term.