Analogy Facilitates Fraction Learning

• In Event: Analogy and Cognitive Development

Abstract

In cognitive development, learning more than the input provides is a central challenge. This challenge is especially evident in learning the meaning of numbers. Integers – and the quantities they denote – are potentially infinite, as are the fractional values between every integer. Yet children’s experiences of numbers are necessarily finite. How can children bootstrap their limited experience to learn the magnitudes of all those numbers they never encounter?

Analogy is one mechanism to bootstrap representational change far beyond the training space. According to structure-mapping theory (Gentner, 1983), alignment between base and target problems facilitates comparison and abstraction of underlying structures. Indeed, previous work indicated that alignment between small, familiar integers and large, unfamiliar integers on number lines fosters a more precise representation of large numbers (Thompson & Opfer, 2010).

Our study hypothesized that analogy between integers and fractions on equivalent number-lines (e.g., 3 on a 0-8 number-line and 3/8 on a 0-1 number-line) also improves children’s understanding of fractional magnitudes. To test this, forty-seven 8-to-14 year-olds (M = 10.8 y, SD = 1.6) were randomly assigned to one of two groups: ‘Alignment’ and ‘No Alignment’. Children in the Alignment group were given integer and fraction number-lines simultaneously and vertically aligned, whereas children in the No Alignment group received the same problems sequentially, thereby blocking the ability to compare the problems (Figure 1). Both groups of children received the same assessments-- Number-to-Position (NP) and Position-to-Number (PN) tasks--across pretest, training, and posttest. In neither group were children instructed to compare the problems nor did any receive feedback on their answers.

We modeled children’s error rates on each training and posttest task with separate linear mixed-effects regression to test for condition effects, controlling for age and pretest. As expected, alignment of fractions and integers decreased errors with fraction estimates (NP: B = .63, SE = .12, p < .001; PN: B = 1.12, SE = .11, p < .001). More importantly, the training effect persisted on a posttest, where integers were not presented (NP: B = .26, SE = .11, p < .05; PN: B = .33, SE = .14, p < .05), suggesting that children were able to make the spontaneous analogy between integers and fractions. Interestingly, comparison between integers and fractions on number-lines also facilitated a more precise representation of integers during training (NP: B = .56, SE = .09, p < .001; PN: B = .10, SE = .11, p = .337) and posttest (NP: B = .31, SE = .11, p < .01; PN: B = .17, SE = .14, p = .235), indicating that the development of integer and fraction understanding is not independent, but rather interacts with each other.

Our results suggest that spatial alignment between integers and fractions facilitates spontaneous analogy, which leads to a better understanding of fractional magnitudes. These findings have an important educational implication. Rather than a holistic understanding of fractional magnitude being hurt by integer knowledge, pre-existing integer knowledge can help fraction learning when instruction draws on the power of good analogies.

Authors

• Shuyuan Yu, Ohio State University
  Presenting Author

• Marta Mielicki, Kent State University
  Non-Presenting Author

• Charles Fitzsimmons, Kent State University
  Non-Presenting Author

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

• John E Opfer, The Ohio State University
  Non-Presenting Author