*Br J Educ Psychol ; 92(2): e12471, 2022 Jun.*

##### RESUMO

BACKGROUND: Adaptive expertise is a highly valued outcome of mathematics curricula. One aspect of adaptive expertise with rational numbers is adaptive rational number knowledge, which refers to the ability to integrate knowledge of numerical characteristics and relations in solving novel tasks. Even among students with strong conceptual and procedural knowledge of rational numbers, there are substantial individual differences in adaptive rational number knowledge. AIMS: We aimed to examine how a wide range of domain-general and mathematically specific skills and knowledge predicted different aspects of rational number knowledge, including procedural, conceptual, and adaptive rational number knowledge. SAMPLE: 173 6th and 7th grade students from a school in the southeastern US (51% female) participated in the study. METHODS: At three time points across 1.5 years, we measured students' domain-general and domain-specific skills and knowledge. We used multiple hierarchal regression analysis to examine how these predictors related to rational number knowledge at the third time point. RESULT: Prior knowledge of rational numbers, general mathematical calculation knowledge, and spontaneous focusing on multiplicative relations (SFOR) tendency uniquely predicted adaptive rational number knowledge, after taking into account domain-general and mathematically specific skills and knowledge. Although conceptual knowledge of rational numbers and general mathematical achievement also predicted later conceptual and procedural knowledge of rational numbers, SFOR tendency did not. CONCLUSION: Results suggest expanding investigations of mathematical development to also explore different features of adaptive expertise as well as spontaneous mathematical focusing tendencies.

##### Assuntos

Logro , Conhecimento , Feminino , Humanos , Masculino , Matemática , Instituições Acadêmicas , Estudantes*J Exp Psychol Learn Mem Cogn ; 2021 Sep 30.*

##### RESUMO

To explain children's difficulties learning fraction arithmetic, Braithwaite et al. (2017) proposed FARRA, a theory of fraction arithmetic implemented as a computational model. The present study tested predictions of the theory in a new domain, decimal arithmetic, and investigated children's use of conceptual knowledge in that domain. Sixth and eighth grade children (N = 92) solved decimal arithmetic problems while thinking aloud and afterward explained solutions to decimal arithmetic problems. Consistent with the hypothesis that FARRA's theoretical assumptions would generalize to decimal arithmetic, results supported 3 predictions derived from the model: (a) accuracies on different types of problems paralleled the frequencies with which the problem types appeared in textbooks; (b) most errors involved overgeneralization of strategies that would be correct for problems with different operations or types of number; and (c) individual children displayed patterns of strategy use predicted by FARRA. We also hypothesized that during routine calculation, overt reliance on conceptual knowledge is most likely among children who lack confidence in their procedural knowledge. Consistent with this hypothesis, (d) many children displayed conceptual knowledge when explaining solutions but not while solving problems; (e) during problem-solving, children who more often overtly used conceptual knowledge also displayed doubt more often; and (f) problem solving accuracy was positively associated with displaying conceptual knowledge while explaining, but not with displaying conceptual knowledge while solving problems. We discuss implications of the results for rational number instruction and for the creation of a unified theory of rational number arithmetic. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

*PLoS One ; 16(8): e0255283, 2021.*

##### RESUMO

Chinese children routinely outperform American peers in standardized tests of mathematics knowledge. To examine mediators of this effect, 95 Chinese and US 5-year-olds completed a test of overall symbolic arithmetic, an IQ subtest, and three tests each of symbolic and non-symbolic numerical magnitude knowledge (magnitude comparison, approximate addition, and number-line estimation). Overall Chinese children performed better in symbolic arithmetic than US children, and all measures of IQ and number knowledge predicted overall symbolic arithmetic. Chinese children were more accurate than US peers in symbolic numerical magnitude comparison, symbolic approximate addition, and both symbolic and non-symbolic number-line estimation; Chinese and U.S. children did not differ in IQ and non-symbolic magnitude comparison and approximate addition. A substantial amount of the nationality difference in overall symbolic arithmetic was mediated by performance on the symbolic and number-line tests.

##### Assuntos

Cognição , Matemática , Criança , Pré-Escolar , China , Humanos , Masculino*New Dir Child Adolesc Dev ; 2020(173): 83-100, 2020 Sep.*

##### RESUMO

Although almost everyone agrees that the environment shapes children's learning, surprisingly few studies assess in detail the specific environments that shape children's learning of specific content. The present article briefly reviews examples of how such environmental assessments have improved understanding of child development in diverse areas, and examines in depth the contributions of analyses of one type of environment to one type of learning: how biased distributions of problems in mathematics textbooks influence children's learning of fraction arithmetic. We find extensive parallels between types of problems that are rarely presented in US textbooks and problems where children in the US encounter greater difficulty than might be expected from the apparent difficulty of the procedures involved. We also consider how some children master fraction arithmetic despite also learning the textbook distributions. Finally, we present findings from a recent intervention that indicates how children's fraction learning can be improved.

##### Assuntos

Desenvolvimento Infantil , Aprendizagem , Conceitos Matemáticos , Matemática/educação , Criança , Desenvolvimento Infantil/fisiologia , Humanos , Aprendizagem/fisiologia*Cogn Psychol ; 123: 101333, 2020 12.*

##### RESUMO

Three rational number notations -- fractions, decimals, and percentages -- have existed in their modern forms for over 300 years, suggesting that each notation serves a distinct function. However, it is unclear what these functions are and how people choose which notation to use in a given situation. In the present article, we propose quantification process theory to account for people's preferences among fractions, decimals, and percentages. According to this theory, the preferred notation for representing a ratio corresponding to a given situation depends on the processes used to quantify the ratio or its components. Quantification process theory predicts that if exact enumeration is used to generate a ratio, fractions will be preferred to decimals and percentages; in contrast, if estimation is used to generate the ratio, decimals and percentages will be preferred to fractions. Moreover, percentages will be preferred over decimals for representing ratios when approximation to the nearest percent is sufficiently precise, due to the lesser processing demands of using percentages. Experiments 1, 2, and 3 yielded empirical evidence regarding preferences that were consistent with quantification process theory. Experiment 4 indicated that the accuracy with which participants identified the numerical values of ratios when they used different notations generally paralleled their preferences. Educational implications of the findings are discussed.

##### Assuntos

Compreensão , Formação de Conceito , Matemática , Resolução de Problemas , China , Humanos , Modelos Teóricos , Estudantes/psicologia , Estados Unidos*Cogn Psychol ; 112: 81-98, 2019 08.*

##### RESUMO

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 (Fraction Arithmetic Reflects Rules and Associations; 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.

##### Assuntos

Individualidade , Aprendizagem , Conceitos Matemáticos , Criança , Escolaridade , Humanos , Modelos Psicológicos , Psicologia da Criança*J Exp Psychol Learn Mem Cogn ; 44(11): 1765-1777, 2018 Nov.*

##### RESUMO

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 arithmetic problems children encounter. To test these hypotheses, we examined textbooks and middle school children in the United States (Experiments 1 and 2) and China (Experiment 3). We asked the children to predict which arithmetic operation would accompany a specified pair of operands, to generate operands to accompany a specified arithmetic operation, and to match operands and operations. In both countries, children's responses indicated that they associated operand pairs having equal denominators with addition and subtraction, and operand pairs having a whole number and a fraction with multiplication and division. The children's associations paralleled the textbook input in both countries, which was consistent with the hypothesis that children learned the associations from the practice problems. Differences in the effects of such associative knowledge on U.S. and Chinese children's fraction arithmetic performance are discussed, as are implications of these differences for educational practice. (PsycINFO Database Record (c) 2018 APA, all rights reserved).

##### Assuntos

Comportamento de Escolha/fisiologia , Aprendizagem/fisiologia , Conceitos Matemáticos , Resolução de Problemas/fisiologia , Adolescente , Fatores Etários , Análise de Variância , Associação , Criança , Compreensão , Feminino , Humanos , Masculino , Matemática*Dev Sci ; 21(4): e12601, 2018 Jul.*

##### RESUMO

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 the two addends in the problem. Moreover, children's estimates of fraction sums were no more accurate than if they had estimated each sum as the average of the smallest and largest possible response. This weak performance could not be attributed to poor mastery of arithmetic procedures, poor knowledge of individual fraction magnitudes, or general inability to estimate sums. These results suggest that a major source of difficulty in this domain is that many children's learning of fraction arithmetic procedures develops unconstrained by conceptual understanding of the procedures. Implications for education are discussed.

##### Assuntos

Compreensão , Matemática , Criança , Feminino , Humanos , Conhecimento , Aprendizagem , Masculino*Dev Sci ; 21(2)2018 03.*

##### RESUMO

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 indicated that between fourth and eighth grade, whole number bias decreased, and reliance on fraction magnitudes increased. These trends were present on both fraction magnitude comparison and number line estimation. However, analyses of individual children's performance indicated that a substantial minority of fourth graders did not show whole number bias and that a substantial minority of eighth graders did show it. Implications of the findings for development of understanding of fraction equivalence and for theories of numerical development are discussed.

##### Assuntos

Viés , Aprendizagem , Matemática , Adulto , Atenção , Criança , Compreensão , Feminino , Humanos , Conhecimento , Estudantes*Psychol Rev ; 124(5): 603-625, 2017 Oct.*

##### RESUMO

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 small number of common learning mechanisms operating on a biased input set. The biases were not unique to this textbook series-they were present in 2 other textbook series as well-nor were the phenomena unique to a particular sample of children-they were present in another sample as well. Among other phenomena, the model predicted the high difficulty of fraction division, variable strategy use by individual children and on individual problems, relative frequencies of different types of strategy errors on different types of problems, and variable effects of denominator equality on the four arithmetic operations. The model also generated nonintuitive predictions regarding the relative difficulties of several types of problems and the potential effectiveness of a novel instructional approach. Perhaps the most general lesson of the findings is that the statistical distribution of problems that learners encounter can influence mathematics learning in powerful and nonintuitive ways. (PsycINFO Database Record

##### Assuntos

Aprendizagem , Matemática , Ensino , Simulação por Computador , Humanos , Modelos Teóricos*J Learn Disabil ; 50(6): 614-620, 2017.*

##### RESUMO

Learning fractions is difficult for children in general and especially difficult for children with mathematics difficulties (MD). Recent research on developmental and individual differences in fraction knowledge of children with MD and typically achieving (TA) children has demonstrated that U.S. children with MD start middle school behind their TA peers in fraction understanding and fall further behind during middle school. In contrast, Chinese children, who like the MD children in the United States score in the bottom one third of the distribution in their country, possess reasonably good fraction understanding. We interpret these findings within the framework of the integrated theory of numerical development. By emphasizing the importance of fraction magnitude knowledge for numerical understanding in general, the theory proved useful for understanding differences in fraction knowledge between MD and TA children and for understanding how knowledge can be improved. Several interventions demonstrated the possibility of improving fraction magnitude knowledge and producing benefits that generalize to fraction arithmetic learning among children with MD. The reasonably good fraction understanding of Chinese children with MD and several successful interventions with U.S. students provide hope for the improvement of fraction knowledge among American children with MD.

##### Assuntos

Desenvolvimento Infantil/fisiologia , Aprendizagem/fisiologia , Conceitos Matemáticos , Matemática/educação , Criança , China , Discalculia/fisiopatologia , Feminino , Humanos , Masculino , Estados Unidos*Annu Rev Psychol ; 68: 187-213, 2017 Jan 03.*

##### RESUMO

In this review, we attempt to integrate two crucial aspects of numerical development: learning the magnitudes of individual numbers and learning arithmetic. Numerical magnitude development involves gaining increasingly precise knowledge of increasing ranges and types of numbers: from nonsymbolic to small symbolic numbers, from smaller to larger whole numbers, and from whole to rational numbers. One reason why this development is important is that precision of numerical magnitude knowledge is correlated with, predictive of, and causally related to both whole and rational number arithmetic. Rational number arithmetic, however, also poses challenges beyond understanding the magnitudes of the individual numbers. Some of these challenges are inherent; they are present for all learners. Other challenges are culturally contingent; they vary from country to country and classroom to classroom. Generating theories and data that help children surmount the challenges of rational number arithmetic is a promising and important goal for future numerical development research.

##### Assuntos

Logro , Desenvolvimento Infantil/fisiologia , Cognição/fisiologia , Aprendizagem/fisiologia , Matemática/educação , Pré-Escolar , Humanos*PLoS One ; 11(10): e0165243, 2016.*

##### RESUMO

We examined whether playing a computerized fraction game, based on the integrated theory of numerical development and on the Common Core State Standards' suggestions for teaching fractions, would improve children's fraction magnitude understanding. Fourth and fifth-graders were given brief instruction about unit fractions and played Catch the Monster with Fractions, a game in which they estimated fraction locations on a number line and received feedback on the accuracy of their estimates. The intervention lasted less than 15 minutes. In our initial study, children showed large gains from pretest to posttest in their fraction number line estimates, magnitude comparisons, and recall accuracy. In a more rigorous second study, the experimental group showed similarly large improvements, whereas a control group showed no improvement from practicing fraction number line estimates without feedback. The results provide evidence for the effectiveness of interventions emphasizing fraction magnitudes and indicate how psychological theories and research can be used to evaluate specific recommendations of the Common Core State Standards.

##### Assuntos

Conhecimento , Jogos de Vídeo , Criança , Feminino , Humanos*Dev Sci ; 19(3): 341-61, 2016 May.*

##### RESUMO

The integrated theory of numerical development posits that a central theme of numerical development from infancy to adulthood is progressive broadening of the types and ranges of numbers whose magnitudes are accurately represented. The process includes four overlapping trends: (1) representing increasingly precisely the magnitudes of non-symbolic numbers, (2) connecting small symbolic numbers to their non-symbolic referents, (3) extending understanding from smaller to larger whole numbers, and (4) accurately representing the magnitudes of rational numbers. The present review identifies substantial commonalities, as well as differences, in these four aspects of numerical development. With both whole and rational numbers, numerical magnitude knowledge is concurrently correlated with, longitudinally predictive of, and causally related to multiple aspects of mathematical understanding, including arithmetic and overall math achievement. Moreover, interventions focused on increasing numerical magnitude knowledge often generalize to other aspects of mathematics. The cognitive processes of association and analogy seem to play especially large roles in this development. Thus, acquisition of numerical magnitude knowledge can be seen as the common core of numerical development.

##### Assuntos

Desenvolvimento Infantil/fisiologia , Cognição/fisiologia , Compreensão/fisiologia , Conhecimento , Matemática , Criança , Pré-Escolar , Humanos , Lactente , Modelos Teóricos*Dev Psychol ; 52(5): 746-57, 2016 05.*

##### RESUMO

Development of fraction number line estimation was assessed longitudinally over 5 time points between 4th and 6th grades. Although students showed positive linear growth overall, latent class growth analyses revealed 3 distinct growth trajectory classes: Students who were highly accurate from the start and became even more accurate (n = 154); students who started inaccurate but showed steep growth (n = 121); and students who started inaccurate and showed minimal growth (n = 197). Younger and minimal growth students typically estimated both proper and improper fractions as being less than 1, failing to base estimates on the relation between the numerator and denominator. Class membership was highly predictive of performance on a statewide-standardized mathematics test as well as on a general fraction knowledge measure at the end of 6th grade, even after controlling for mathematic-specific abilities, domain-general cognitive abilities, and demographic variables. Multiplication fluency, classroom attention, and whole number line estimation acuity at the start of the study predicted class membership. The findings reveal that fraction magnitude understanding is central to mathematical development. (PsycINFO Database Record

##### Assuntos

Atenção/fisiologia , Desenvolvimento Infantil/fisiologia , Compreensão/fisiologia , Matemática , Resolução de Problemas/fisiologia , Estudantes/psicologia , Criança , Feminino , Humanos , Estudos Longitudinais , Masculino , Testes Neuropsicológicos*J Exp Psychol Learn Mem Cogn ; 42(1): 1-16, 2016 Jan.*

##### RESUMO

We examined, on a trial-by-trial basis, fraction magnitude comparison strategies of adults with more and less mathematical knowledge. College students with high mathematical proficiency used a large variety of strategies that were well tailored to the characteristics of the problems and that were guaranteed to yield correct performance if executed correctly. Students with less mathematical proficiency sometimes used strategies similar to those of the mathematically proficient students, but often used flawed strategies that yielded inaccurate performance. As predicted by overlapping waves theory, increases in accuracy and speed were related to differences in strategy use, strategy choice, and strategy execution. When asked to choose the best strategy from among 3 possibilities-the strategy the student originally used, a correct alternative, and an incorrect alternative-students with lower fraction knowledge rarely switched from an original incorrect strategy to a correct alternative. This finding suggests that use of poor fraction magnitude comparison strategies stems in large part from lack of conceptual understanding of the requirements of effective strategies, rather than difficulty recalling or generating such strategies. (PsycINFO Database Record

##### Assuntos

Comportamento de Escolha , Conceitos Matemáticos , Formação de Conceito , Escolaridade , Feminino , Humanos , Masculino , Resolução de Problemas , Competência Profissional , Testes Psicológicos , Estudantes , Universidades , Adulto Jovem*J Exp Child Psychol ; 129: 68-83, 2015 Jan.*

##### RESUMO

We compared knowledge of fraction concepts and procedures among sixth and eighth graders in China and the United States. As anticipated, Chinese middle school children had higher knowledge of fraction concepts and procedures than U.S. children in the same grades, and the difference in procedural knowledge was much larger than the difference in conceptual knowledge. Of particular interest, national differences in knowledge of fraction concepts were fully mediated by differences in knowledge of fraction procedures, and differences between the knowledge of Chinese and U.S. children were most pronounced among the lowest achieving children within each country. Based on these and previous results, a theoretical model of the mutually facilitative interaction between conceptual and procedural knowledge of fractions is proposed and discussed.

##### Assuntos

Formação de Conceito , Matemática , Adolescente , Aptidão , Criança , Desenvolvimento Infantil , China , Compreensão , Escolaridade , Feminino , Humanos , Masculino , Estados Unidos*J Exp Child Psychol ; 123: 53-72, 2014 Jul.*

##### RESUMO

We examined relations between symbolic and non-symbolic numerical magnitude representations, between whole number and fraction representations, and between these representations and overall mathematics achievement in fifth graders. Fraction and whole number symbolic and non-symbolic numerical magnitude understandings were measured using both magnitude comparison and number line estimation tasks. After controlling for non-mathematical cognitive proficiency, both symbolic and non-symbolic numerical magnitude understandings were uniquely related to mathematics achievement, but the relation was much stronger for symbolic numbers. A meta-analysis of 19 published studies indicated that relations between non-symbolic numerical magnitude knowledge and mathematics achievement are present but tend to be weak, especially beyond 6 years of age.