Abstract Understanding fractions is critical to mathematical development, yet many children struggle with fractions even after years of instruction. Fraction arithmetic is particularly challenging. The present study employed a computational model of fraction arithmetic learning, FARRA ( F raction A rithmetic R eflects R ules and A ssociations; Braithwaite, Pyke, and Siegler, 2017), to investigate individual differences in children’s fraction arithmetic. FARRA predicted four qualitatively distinct patterns of performance, as well as differences in math achievement among the four patterns. These predictions were confirmed in analyses of two datasets using two methods to classify children’s performance—a theory-based method and a data-driven method, Latent Profile Analysis. The findings highlight three dimensions of individual differences that may affect learning in fraction arithmetic, and perhaps other domains as well: effective learning after committing errors, behavioral consistency versus variability, and presence or absence of initial bias. Methodological and educational implications of the findings are discussed.

Last. Luke Rinne(UD: University of Delaware)H-Index: 11

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The efficacy of a research-based fraction sense intervention for sixth graders with or at risk for mathematics difficulties (N = 52) was examined. The intervention aimed to build understanding of fraction magnitudes on the number line. Key concepts were taught with a narrow range of denominators to develop deep understanding. The intervention was centered on a visual number line in the meaningful context of a color run race. Students were randomly assigned to the fraction sense intervention (n =...

#1Ilyse Resnick(PSU: Pennsylvania State University)H-Index: 11

#2Luke Rinne(UD: University of Delaware)H-Index: 11

Last. Nancy C. Jordan(UD: University of Delaware)H-Index: 42

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Reasoning about numerical magnitudes is a key aspect of mathematics learning. Most research examining the relation of magnitude understanding to general mathematics achievement has focused on whole number and fraction magnitudes. The present longitudinal study (N = 435) used a 3-step latent class analysis to examine reasoning about magnitudes on a decimal comparison task in 4th grade, before systematic decimals instruction. Three classes of response patterns were identified, indicating empirical...

Last. Robert S. Siegler(CMU: Carnegie Mellon University)H-Index: 100

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Many children fail to master fraction arithmetic even after years of instruction. A recent theory of fraction arithmetic (Braithwaite, Pyke, & Siegler, 2017) hypothesized that this poor learning of fraction arithmetic procedures reflects poor conceptual understanding of them. To test this hypothesis, we performed three experiments examining fourth to eighth graders' estimates of fraction sums. We found that roughly half of estimates of sums were smaller than the same child's estimate of one of t...

: Fraction arithmetic is among the most important and difficult topics children encounter in elementary and middle school mathematics. Braithwaite, Pyke, and Siegler (2017) hypothesized that difficulties learning fraction arithmetic often reflect reliance on associative knowledge-rather than understanding of mathematical concepts and procedures-to guide choices of solution strategies. They further proposed that this associative knowledge reflects distributional characteristics of the fraction ar...

Many students’ knowledge of fractions is adversely affected by whole number bias, the tendency to focus on the separate whole number components (numerator and denominator) of a fraction rather than on the fraction's magnitude (ratio of numerator to denominator). Although whole number bias appears early in the fraction learning process and under speeded conditions persists into adulthood, even among mathematicians, little is known about its development. Performance with equivalent fractions indic...

Last. Kelly Trezise(University of Melbourne)H-Index: 5

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This article gives an introduction to latent class, latent profile, and latent transition models for researchers interested in investigating individual differences in learning and development. The models allow analyzing how the observed heterogeneity in a group (e.g., individual differences in conceptual knowledge) can be traced back to underlying homogeneous subgroups (e.g., learners differing systematically in their developmental phases). The estimated parameters include a characteristic respo...

Last. Robert S. Siegler(CMU: Carnegie Mellon University)H-Index: 100

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: Many children fail to master fraction arithmetic even after years of instruction, a failure that hinders their learning of more advanced mathematics as well as their occupational success. To test hypotheses about why children have so many difficulties in this area, we created a computational model of fraction arithmetic learning and presented it with the problems from a widely used textbook series. The simulation generated many phenomena of children's fraction arithmetic performance through a ...

Last. Hans Gruber(University of Regensburg)H-Index: 29

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Contemporary theories of expertise development highlight the crucial role of deliberate practice in the development of high level performance. Deliberate practice is practice that intentionally aims at improving one’s skills and competencies. It is not a mechanical or repetitive process of making performance more fluid. Instead, it involves a great deal of thinking, problem solving, and reflection for analyzing, conceptualizing, and cultivating developing performance. This includes directing and...

Last. Nancy C. Jordan(UD: University of Delaware)H-Index: 42

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: The present study investigated the development of fraction comparison strategies through a longitudinal analysis of children's responses to a fraction comparison task in 4th through 6th grades (N = 394). Participants were asked to choose the larger value for 24 fraction pairs blocked by fraction type. Latent class analysis of performance over item blocks showed that most children initially exhibited a "whole number bias," indicating that larger numbers in numerators and denominators produce la...

Although error avoidance during learning appears to be the rule in American classrooms, laboratory studies suggest that it may be a counterproductive strategy, at least for neurologically typical students. Experimental investigations indicate that errorful learning followed by corrective feedback is beneficial to learning. Interestingly, the beneficial effects are particularly salient when individuals strongly believe that their error is correct: Errors committed with high confidence are correct...

#2Jake McMullen(UTU: University of Turku)H-Index: 13

Last. Michelle Hurst(U of C: University of Chicago)H-Index: 5

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Understanding fractions and decimals requires not only understanding each notation separately, or within-notation knowledge, but also understanding relations between notations, or cross-notation knowledge. Multiple notations pose a challenge for learners but could also present an opportunity, in that cross-notation knowledge could help learners to achieve a better understanding of rational numbers than could easily be achieved from within-notation knowledge alone. This hypothesis was tested by r...

Last. Lara Nugent(MU: University of Missouri)H-Index: 18

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The study tested the hypotheses that boys will have an advantage learning the fractions number line and this advantage will be mediated by spatial abilities. Fractions number line and, as a contrast, fractions arithmetic performance were assessed for 342 adolescents, as was their intelligence, working memory, and various spatial abilities. Boys showed smaller placement errors on the fractions number line (d = -0.22) and correctly solved more fractions arithmetic problems (d = 0.23) than girls. W...

This study is devoted to identifying a solution algorithm for standard fractions as one of the tasks that allow investigating the level of human adaptability to the cognitive load. The influential factor for a successful solution was the number of stages, and for an unsuccessful one - their duration. It was revealed that the solution success/failure correlated with the spectral power values and ratio in the theta- and alpha-diapasons of the EEG. The successful solution is accompanied by the main...

Understanding fraction magnitudes is especially important in daily life, but fraction reasoning is quite difficult. To accurately reason about fraction magnitudes, adults need to monitor what they know and what they do not know. However, little is known about which cues adults use to monitor fraction performance. Across two studies, we examined adults’ trial-by-trial fraction estimates, confidence judgments, and ratings of fraction familiarity. Adults were more confident when their estimates wer...