From Integers to Fractions: The Role of Analogy in Developing a Coherent Understanding of Proportional Magnitude

• Published in: Developmental psychology Vol. 58; no. 10; pp. 1912 - 1930
• Main Authors: Yu, Shuyuan, Kim, Dan, Fitzsimmons, Charles J, Mielicki, Marta K, Thompson, Clarissa A, Opfer, John E
• Format: Journal Article
• Language: English
• Published: United States American Psychological Association 01.10.2022
• Subjects: Computation; Elementary School Students; Fractions; Logical Thinking; Mathematical Logic; Mathematics Skills; Number Concepts
• ISSN: 0012-1649; 1939-0599
• DOI: 10.1037/dev0001398

Children display an early sensitivity to implicit proportions (e.g., 1 of 5 apples vs. 3 of 4 apples), but have considerable difficulty in learning the explicit, symbolic proportions denoted by fractions (e.g., "1/5" vs. "3/4"). Theoretically, reducing the gap between representations of implicit versus explicit proportions would improve understanding of fractions, but little is known about how the representations develop and interact with one another. To address this, we asked 177 third, fourth, and fifth graders (M = 9.85 years, 87 girls, 69% White, 19% low income) to estimate the position of proportionally equivalent integers and fractions on number lines (e.g., 3 on a 0-8 number line vs. 3/8 on a 0-1 number line, Study 1). With increasing age, children's estimates of implicit and explicit proportions became more coherent, such that a child's estimates of fractions on a 0-1 number-line was a linear function of the same child's estimates of equivalent integers. To further investigate whether preexisting integer knowledge can facilitate fraction learning through analogy, we assigned 100 third to fifth graders (M = 10.04 years, 55 girls, 76% White) to an Alignment condition, where children estimated fractions and integers on aligned number lines, or to a No Alignment condition (Study 2). Results showed that aligning integers and fractions on number lines facilitated a better understanding of fractional magnitudes, and increased children's fraction estimation accuracy to the level of college students'. Together, findings suggest that analogies can play an important role in building a coherent understanding of proportions.