Number Lines, but not Area Diagrams, Support Children’s Fraction Division Problem Solving

• In Event: 2-082 - External Representations in Mathematical Thinking and Learning

Integrative Statement

In practice, teachers use a variety of external representations to facilitate fraction learning, such as learning about fraction division. Although developing fluency with multiple representations may indeed be advantageous (Rau & Matthews, 2017), commonly-used area diagrams (e.g., circles) have different problem solving affordances than number line diagrams. Following the integrated theory of whole number and fractions development (Siegler et al., 2011), we propose that number lines are more effective at supporting children’s deep understanding of fraction operations than area models. The integrated theory suggests that number lines help children represent the magnitude of all rational numbers. Number lines may also better support children’s fraction magnitude understanding (e.g., Hamdan & Gunderson, 2017) and facilitate transfer across whole number and fraction concepts. Thus, we investigated whether asking 123 fifth and sixth grade students to solve fraction division problems using a number line or area diagram resulted in more accurate problem solving than without any diagram present, and whether reasoning with a number line was more likely to lead to accurate problem solving and mathematically-sound conceptual models of division than reasoning with area diagrams, both circular and rectangular.  

Each child was randomly assigned to a between-subjects diagram condition (see Figure 1) as they solved 18 fraction division problems, one per page without feedback (see Table 1): (a) circular area (n=33), (b) rectangular area (n=29), (c) number line (n=31), and (d) no diagram provided (n=30). The problems were introduced by a researcher demonstrating “how you can show your work” using a whole number division example (6÷2) by using the same diagram as their randomly assigned experimental condition. Children rated the difficulty of each problem immediately after solving it. Children’s written work was coded for presence of the correct answer, whether the magnitude of each operand was represented accurately, and whether the relationship between operands accurately reflected quotative or partitive division.

As expected, number lines afforded accurate problem-solving. Children who solved fraction division problems in the number line condition had higher rates of accuracy than children in the circular condition, t(115)=-2.23, p=.03, ΔR2=.04, rectangular condition ,  t(115)=-2.48, p=.01, ΔR2=.04, and the no visual model condition, t(115)=-2.37, p=.02, ΔR2=.05, controlling for problem order, grade, and gender. Furthermore, the odds of consistently generating a division model among children in the number line condition were nearly 10 times greater, OR=9.83, 𝛘2(1)=16.01, p<.01, than among children in the circular condition, and over five times greater, OR=5.30, 𝛘2(1)=8.60, p<.01, than those in the rectangular condition. Finally, despite our finding that the circular condition did not support children’s accuracy, children in the circular condition rated problems as being significantly less difficult than children in the other conditions, t(115)=-2.69, p<.01, ΔR2=.06 (see Table 1).

Overall, our findings suggest that number lines uniquely support children’s reasoning and problem solving with fraction division problems. This finding provides support for the integrated theory and demonstrates how external representations can shape children’s thinking. Looking across accuracy and difficulty, number lines may be “worth” their perceived difficulty, whereas circles may be unhelpful despite seeming more intuitive.

Authors

• Pooja Gupta Sidney, University of Kentucky
  Presenting Author

• Clarissa A Thompson, Kent State University
  Non-Presenting Author

• Ferdinand Rivera, San Jose State University
  Non-Presenting Author