From integers to fractions: The role of analogy in developing a coherent understanding of proportional magnitude.

Abstract

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.

Perceptions of ease and difficulty, but not growth mindset, relate to specific math attitudes

“…Specifically, the ease‐as‐possibility scale positively related to perceived ability with WNs across both studies, and perceived importance of WNs in Study 2. These findings align with empirical work indicating WN tasks are easier than fraction tasks (Fazio et al., 2014; Yu et al., 2020). Finally, the difficulty‐as‐impossibility scale negatively related to ratings of preference for percentages.…”

“…Moreover, colloquially percentages are often referred to in terms of WNs (e.g., 'I got a 100 on my test', rather than 'I got 100% of the problems correct on my test'.). Thus, individuals should be able to bootstrap WN knowledge to percentages easier than fractions (although, see Yu et al, 2020). Indeed, Siegler et al (2011) noted that a common strategy for estimating fractions on number lines is by applying percentage knowledge (e.g., ⅘ is 80% of the way across a number line from left-to-right).…”

“…Moreover, colloquially percentages are often referred to in terms of WNs (e.g., ‘I got a 100 on my test’, rather than ‘I got 100% of the problems correct on my test’.). Thus, individuals should be able to bootstrap WN knowledge to percentages easier than fractions (although, see Sidney et al., 2021 and Yu et al., 2020). Indeed, Siegler et al.…”

Leveraging Math Cognition to Combat Health Innumeracy

“…At its core, the natural number bias is a failure to engage in relational reasoning (Gentner, 1983) by considering concepts in isolation rather than in relation to one another. 5 Rational numbers and natural numbers share many common features, which can be adaptively leveraged when people attempt to apply their knowledge about natural numbers when reasoning with rational numbers (Sidney, Thalluri, et al, 2019;Yu et al, 2020Yu et al, , 2021. However, as our many examples have illustrated, overgeneralizing and misapplying the properties of natural numbers to rational numbers can lead to inaccurate estimates of rational number magnitude.…”

“…LEVERAGING MATH COGNITION numbers, fractions, ratios, decimals, and percentages early in their formal math curriculumthen according to our developmental framework (see Figure 2), they should be more likely to develop into adults who do not easily fall prey to numerical biases, thus ameliorating LEVERAGING MATH COGNITION innumeracy in adulthood. Thus, to address natural number bias at this inflection point, math curricula can be restructured such that more productive connections are made between natural number knowledge and knowledge about rational numbers (e.g., Yu et al, 2020Yu et al, , 2021.…”