*J Exp Child Psychol ; 213: 105210, 2022 01.*

##### RESUMO

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 reanalyzing three published datasets involving fourth- to eighth-grade children from the United States and Finland. All datasets included measures of rational number arithmetic, within-notation magnitude knowledge (e.g., accuracy in comparing fractions vs. fractions and decimals vs. decimals), and cross-notation magnitude knowledge (e.g., accuracy in comparing fractions vs. decimals). Consistent with the hypothesis, cross-notation magnitude knowledge predicted fraction and decimal arithmetic when controlling for within-notation magnitude knowledge. Furthermore, relations between within-notation magnitude knowledge and arithmetic were not notation specific; fraction magnitude knowledge did not predict fraction arithmetic more than decimal arithmetic, and decimal magnitude knowledge did not predict decimal arithmetic more than fraction arithmetic. Implications of the findings for assessing rational number knowledge and learning and teaching about rational numbers are discussed.

##### Assuntos

Idioma , Aprendizagem , Criança , Finlândia , Humanos , Matemática , Estados Unidos*Cogn Sci ; 45(10): e13048, 2021 10.*

##### RESUMO

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect "online" effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem-solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving.

##### Assuntos

Metacognição , Humanos , Conhecimento , Matemática , Resolução de Problemas , 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).

*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(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*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*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*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(3): e0152115, 2016.*

##### RESUMO

Study sequence can have a profound influence on learning. In this study we investigated how students decide to sequence their study in a naturalistic context and whether their choices result in improved learning. In the study reported here, 2061 undergraduate students enrolled in an Introductory Psychology course completed an online homework tutorial on measures of central tendency, a topic relevant to an exam that counted towards their grades. One group of students was enabled to choose their own study sequence during the tutorial (Self-Regulated group), while the other group of students studied the same materials in sequences chosen by other students (Yoked group). Students who chose their sequence of study showed a clear tendency to block their study by concept, and this tendency was positively associated with subsequent exam performance. In the Yoked group, study sequence had no effect on exam performance. These results suggest that despite findings that blocked study is maladaptive when assigned by an experimenter, it may actually be adaptive when chosen by the learner in a naturalistic context.

##### Assuntos

Currículo/estatística & dados numéricos , Aprendizagem/fisiologia , Autocontrole/psicologia , Estudantes/psicologia , Adolescente , Comportamento de Escolha/fisiologia , Avaliação Educacional/estatística & dados numéricos , Feminino , Humanos , Masculino*Cognition ; 149: 40-55, 2016 Apr.*

##### RESUMO

The idea that cognitive development involves a shift towards abstraction has a long history in psychology. One incarnation of this idea holds that development in the domain of mathematics involves a shift from non-formal mechanisms to formal rules and axioms. Contrary to this view, the present study provides evidence that reliance on non-formal mechanisms may actually increase with age. Participants - Dutch primary school children - evaluated three-term arithmetic expressions in which violation of formally correct order of evaluation led to errors, termed foil errors. Participants solved the problems as part of their regular mathematics practice through an online study platform, and data were collected from over 50,000 children representing approximately 10% of all primary schools in the Netherlands, suggesting that the results have high external validity. Foil errors were more common for problems in which formally lower-priority sub-expressions were spaced close together, and also for problems in which such sub-expressions were relatively easy to calculate. We interpret these effects as resulting from reliance on two non-formal mechanisms, perceptual grouping and opportunistic selection, to determine order of evaluation. Critically, these effects reliably increased with participants' grade level, suggesting that these mechanisms are not phased out but actually become more important over development, even when they cause systematic violations of formal rules. This conclusion presents a challenge for the shift towards abstraction view as a description of cognitive development in arithmetic. Implications of this result for educational practice are discussed.

##### Assuntos

Desenvolvimento Infantil , Cognição , Formação de Conceito , Conceitos Matemáticos , Criança , Feminino , Humanos , Masculino , Percepção*Front Psychol ; 4: 980, 2013.*

##### RESUMO

Graphs and tables differentially support performance on specific tasks. For tasks requiring reading off single data points, tables are as good as or better than graphs, while for tasks involving relationships among data points, graphs often yield better performance. However, the degree to which graphs and tables support flexibility across a range of tasks is not well-understood. In two experiments, participants detected main and interaction effects in line graphs and tables of bivariate data. Graphs led to more efficient performance, but also lower flexibility, as indicated by a larger discrepancy in performance across tasks. In particular, detection of main effects of variables represented in the graph legend was facilitated relative to detection of main effects of variables represented in the x-axis. Graphs may be a preferable representational format when the desired task or analytical perspective is known in advance, but may also induce greater interpretive bias than tables, necessitating greater care in their use and design.