Individual Differences in Nonsymbolic Ratio Processing Longitudinally Predict Algebraic Reasoning in Children

• In Event: New Developments in the Science of Numerical Thinking

Abstract

Algebra is considered a gatekeeper for advanced mathematics (Gaertner et al., 2014). Studies of children and adults suggest that performance in algebra is positively associated with fraction knowledge (DeWolf et al., 2015; Siegler et al., 2012). More recent work has suggested that the ability to perceive and compare perceptual ratios—the ratio processing system (RPS)—may be indirectly associated with algebra performance (Matthews et al., 2016). Lewis and colleagues (2016) argued that the RPS may serve as a foundation for symbolic fraction processing, and the ability to process symbolic fraction magnitudes may support better fraction knowledge, which, in turn predicts algebra performance. We tested these predictions by investigating the longitudinal association between nonsymbolic and symbolic ratio processing, fraction knowledge, and algebra in two groups of children (2nd and 5th graders).

We followed 96 younger participants from 2nd to 4th grade and 66 older participants from 5th to 7th grade. In Year 1 (Y1), participants completed a ratio comparison task. They were presented with two ratios and instructed to identify the larger. There were three different notations: two line ratios (LL), two symbolic fractions (FF), or one line ratio and one symbolic fraction (MX). In Year 2 (Y2), participants completed a grade appropriate Fraction Knowledge Assessment (FKA). In Year 3 (Y3), they completed a grade appropriate Algebraic Reasoning Task (ART). We conducted hierarchical regression models including LL, FF, and MX as ART predictors in a first step, and entering FKA in a second step.

Results are summarized in Table 1. In younger participants, the model with all the three notations as predictors explained 20% of the variance in algebra (p<.001). Examining the contribution of each notation separately, nonsymbolic ratio processing alone positively correlated with algebra (p<.05), but not mixed (p=.59) or symbolic (p=.15) comparisons. When fractions performance was included in the model, the explained variance increased to 45% (ΔR = .25, p<.001). Participants with higher fractions scores also had higher scores in algebra (p<.001). However, the association between nonsymbolic fractions and algebra was no longer significant (p=.10).

In older participants, the model with the three notations of the ratio comparison task as predictors explained 15% of the variance in algebra (p<.01). Higher scores in symbolic fractions processing alone positively correlated with algebra (p<.05), but not nonsymbolic (p=.83) or mixed (p=.31) comparisons. When fractions performance was included in the model, the explained variance increased to 54% (ΔR=.39, p<.001). Participants with higher fractions scores also had higher algebra scores (p<.001). However, the association between symbolic fractions and algebra was no longer significant (p=.57).

Consistent with our predictions, nonsymbolic ratio processing was a stronger predictor of algebraic reasoning than symbolic fraction processing for our younger participants, but the opposite pattern was observed for our older participants. Our findings support the RPS model (Lewis et al., 2016), and suggest a developmental shift from nonsymbolic ratio processing to symbolic fraction processing. These results may have theoretical and practical implications, as they help identify key cognitive mechanisms that helps support the acquisition of algebraic reasoning.

Authors

• Isabella Starling Alves, University of Wisconsin - Madison
  Presenting Author

• Percival G Matthews, University of Wisconsin - Madison
  Non-Presenting Author

• Edward M. Hubbard, University of Wisconsin - Madison
  Non-Presenting Author