Affordances of fractions and decimals for arithmetic.

Liu, Qiushan; Braithwaite, David W.

doi:10.1037/xlm0001161

Cited by 2 publications

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“…Of course, some arithmetic problems are conceptually more difficult than others. For example, division of fractions appears harder to conceptualize than other arithmetic operations with fractions (Ma, 1999), and multiplication of decimals is more difficult to understand than addition of decimals (Liu & Braithwaite, 2022). However, the present results suggest that such conceptual considerations are not necessary to explain many between-problem variations in error rates.…”

Section: Discussionmentioning

confidence: 99%

A unified model of arithmetic with whole numbers, fractions, and decimals.

Braithwaite,

Siegler

2024

Psychological Review

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This article describes UMA (Unified Model of Arithmetic), a theory of children’s arithmetic implemented as a computational model. UMA builds on FARRA (Fraction Arithmetic Reflects Rules and Associations; Braithwaite et al., 2017), a model of children’s fraction arithmetic. Whereas FARRA—like all previous models of arithmetic—focused on arithmetic with only one type of number, UMA simulates arithmetic with whole numbers, fractions, and decimals. The model was trained on arithmetic problems from the first to sixth grade volumes of a math textbook series; its performance on tests administered at the end of each grade was compared to the performance of children in prior empirical research. In whole number arithmetic (Study 1), fraction arithmetic (Study 2), and decimal arithmetic (Study 3), UMA displayed types of errors, effects of problem features on error rates, and individual differences in strategy use that resembled those documented in the previous studies of children. Further, UMA generated correlations between individual differences in basic and advanced arithmetic skills similar to those observed in longitudinal studies of arithmetic development (Study 4). The results support UMA’s main theoretical assumptions regarding arithmetic development: (a) most errors reflect small deviations from standard procedures via two mechanisms, overgeneralization and omission; (b) between-problem variations in error rates reflect effects of intrinsic difficulty and differential amounts of practice; and (c) individual differences in strategy use reflect underlying variation in parameters governing learning and decision making.

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Section: Discussionmentioning

confidence: 99%

A unified model of arithmetic with whole numbers, fractions, and decimals.

Braithwaite,

Siegler

2024

Psychological Review

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“…Interestingly, over the last decade, a growing body of research focused on the learning of fractions-i.e., one format of rational numbers (Braithwaite et al, 2017(Braithwaite et al, , 2018(Braithwaite et al, , 2022Hansen et al, 2017;Jordan et al, 2013;Q. Liu & Braithwaite, 2022;Schneider & Siegler, 2010;Siegler et al, 2011;Tian et al, 2020;Tian & Siegler, 2017).…”

Section: Introductionmentioning

confidence: 99%

Basic mathematical skills and fraction understanding predict percentage understanding: Evidence from an intelligent tutoring system

Spitzer¹,

Garcia²,

Moeller³

2022

Preprint

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In several countries, students learn basic mathematical skills first, followed by fractions before learning percentages. Previous longitudinal studies observed that basic mathematical skills (e.g., arithmetic) are significant predictors of understanding fractions, which is critical for later academic success. However, it is still unclear whether this predictive power generalizes to other types of rational numbers such as percentages. Here, we analyzed a large longitudinal dataset (2.798 students; 472.626 problem sets) from an online learning environment to predict performance in fraction understanding as well as processing percentages based on basic mathematical skills (i.e., arithmetic, measurement units, and geometry). As students learned fractions before percentages, we additionally assessed whether understanding fractions predicted processing percentages. We observed that performance in arithmetic, measurement units, and geometry significantly predicted performance on percentages and fraction understanding. We also found transfer effects from fractions to processing percentages. This indicates that processing percentages and understanding fractions similarly build on basic mathematical skills. Moreover, our results point towards the idea that knowledge attained on one format of rational numbers (fractions) may transfer to another format of rational numbers (percentages).

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Affordances of fractions and decimals for arithmetic.

Cited by 2 publications

References 50 publications

A unified model of arithmetic with whole numbers, fractions, and decimals.

A unified model of arithmetic with whole numbers, fractions, and decimals.

Basic mathematical skills and fraction understanding predict percentage understanding: Evidence from an intelligent tutoring system

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