Connecting Symbolic Fractions to Continuous Proportion Using a Fraction Card Game

• In Event: From Proportions to Symbolic Rational Numbers

Abstract

Fractions are a notoriously difficult math topic. However, a growing body of work suggests that infants and young children have relatively sophisticated intuitions about non-symbolic representations of proportional quantities well before fraction instruction (e.g. Boyer et al., 2008; Hurst & Cordes, 2018; McCrink & Wynn, 2007). Based on these findings, researchers have proposed that children’s non-symbolic understanding of fractions can be leveraged to support their symbolic fraction learning (e.g., Boyer & Levine, 2015; Matthews & Hubbard, 2016). Despite an increasing discussion about the importance of non-symbolic representations of fractions, it is not clear what kind of representations should be used and how they should be introduced to most effectively connect non-symbolic and symbolic fraction information.

In the current study we address this question using a fraction card game to teach 1st and 2nd grade children (N = 195) about fractions. Children were told they would play a fraction card game with the experimenter and that on each turn, whoever had the card with the bigger fraction would get to keep both cards. After the child decided which of the two fractions they thought was bigger, they checked their answer by constructing rectangular area models to depict the fractions using a condition-specific approach (Figure 1A). In the Actively Divided condition, we dynamically divided area models into equal-sized units, in the Pre-Divided condition we used area models that looked like the end-state of the Actively Divided condition, and in the Non-Divided condition we used continuous representations of the fraction magnitude that was not divided into unit-sized parts. At pre- and post-test, children were assessed on mapping symbolic and visual representations of fractions (Figure 1B), comparing fraction magnitudes, and judging equivalent fractions via symbols (Figure 1C) and pie charts (Figure 1D).

Overall, we find that actively dividing area models in a way that highlights the denominator unit best supports young children’s fraction learning (Table 1). In particular, children in the Actively Divided condition significantly improved from pre- to post-test on Symbolic-to-Non-Symbolic Mapping (p = 0.003, partial eta squared = 0.13), Symbolic Fraction Comparison (p < 0.001, partial eta squared = 0.18), Pie Chart Comparison (p = 0.01, partial eta squared = 0.11), and Pie Chart Equality (p = 0.02, partial eta squared = 0.09). The other two conditions, on the other hand, showed less consistent improvement and in some cases significantly lower improvement than the Actively Divided condition. Analyses of the game-play training point to the possibility that the greater benefit of the Actively Divided condition may stem from its ability to help children confront their misconceptions when reasoning about discrete numerical information in the context of fractions (e.g., Ni & Zhou, 2005).

Taken together, these findings provide evidence that area models can be effective for introducing children to fraction concepts. Specifically, drawing children’s attention to the meaning of the numerator and denominator through the iterative unit partitioning can help children connect their intuitions about non-symbolic fractions with fraction symbols, potentially mitigating children’s common fraction misconceptions and errors.

Authors

• Jacob Butts, University of Chicago
  Presenting Author

• Michelle A Hurst, University of Chicago
  Non-Presenting Author

• Susan Cohen Levine, University of Chicago
  Non-Presenting Author