Robert Siegler

Background Although the generation of errors has been thought, traditionally, to impair learning, recent studies indicate that, under particular feedback conditions, the commission of errors may have a beneficial effect. Aims This study investigates the teaching strategies that facilitate learning from errors. Materials and Methods This 2‐year study, involving two cohorts of ~88 students each, contrasted a learning‐from‐errors (LFE) with an explicit instruction (EI) teaching strategy in a multi‐session implementation directed at improving student performance on the high‐stakes New York State Algebra 1 Regents examination. In the LFE condition, instead of receiving instruction on 4 sessions, students took mini‐tests. Their errors were isolated to become the focus of 4 teacher‐guided feedback sessions. In the EI condition, teachers explicitly taught the mathematical material for all 8 sessions. Results Teacher time‐on …

Over the last decades, theoretical perspectives in the interdisciplinary field of the affective sciences have proliferated rather than converged due to differing assumptions about what human affective phenomena are and how they work. These metaphysical and mechanistic assumptions, shaped by academic context and values, have dictated affective constructs and operationalizations. However, an assumption about the purpose of affective phenomena can guide us to a common set of metaphysical and mechanistic assumptions. In this capstone paper, we home in on a nested teleological principle for human affective phenomena in order to synthesize metaphysical and mechanistic assumptions. Under this framework, human affective phenomena can collectively be considered algorithms that either adjust based on the human comfort zone (affective concerns) or monitor those adaptive processes (affective features …

Growing evidence points to the predictive power of cross-notation rational number understanding (eg, 2/5 vs. 0.25) relative to within-notation understanding (eg, 2/5 vs. 1/4) in predicting math outcomes. Though correlational in nature, these studies suggest that number sense training emphasizing integrating across notations may have more positive outcomes than a within-notation focus. However, this idea has not been empirically tested. Thus, across two studies with undergraduate students (N= 183 and N= 181), we investigated the effects of a number line training program using a cross-notation approach (one that focused on connections among fractions, decimals, and percentages) and a within-notation approach (one that focused on fraction magnitude representation only). Both number line approaches produced positive effects, but those of the cross-notation approach were larger for fraction magnitude estimation and cross-notation comparison accuracy. Together, these results suggest the importance of an integrated approach to teaching rational number notations, an approach that appears to be uncommon in current curricula.

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 …

We examined the development of numerical magnitude representations of fractions and decimals from fourth to 12th grade. In Experiment 1, we assessed the rational number magnitude knowledge of 200 Chinese fourth, fifth, sixth, eighth, and 12th graders (92 girls and 108 boys) by presenting fraction and decimal magnitude comparison tasks as well as fraction and decimal 0–1 and 0–5 number line estimation tasks. Magnitude representations of decimals became accurate earlier, improved more rapidly, and reached a higher asymptotic accuracy than magnitude representations of fractions. Analyses of individual differences revealed positive relations between the accuracy of decimal and fraction magnitude representations at all ages. In Experiment 2, we presented an additional set of 24 fourth graders (14 girls and 10 boys) with the same tasks but with the decimals that were being compared varying in the number …

Growing evidence highlights the predictive power of cross-notation magnitude comparison (eg, 2/5 vs. 0.25) for math outcomes, but the underlying mechanisms remain unknown. Across two studies, we investigated undergraduates’ cross-notation and within-notation comparison skills given equivalent fractions, decimals, and percentages (Study 1, N= 220 and Study 2, N= 183). We found participants did not perceive equivalent rational numbers equivalently. Cluster analyses revealed that approximately one-quarter of undergraduates exhibited a bias to select percentages as larger in cross-notation comparisons. Compared with the other cluster of undergraduates who showed little-to-no bias, the percentages-are-larger bias cluster performed worse on fraction number line estimation and fraction arithmetic (exact and approximate), as well as reporting lower SAT/ACT scores. Hierarchical linear regression analyses demonstrated that cross-notation comparison accuracy accounted for variance in SAT/ACT beyond within-notation accuracy. Mediation analyses revealed a potential mechanism: stronger cross-notation knowledge equips individuals to evaluate the reasonableness of solutions. Together, these results suggest the importance of an integrated understanding of rational number notations, which may not be fully assessed by within-notation measures alone.

Imbalances in problem distributions in math textbooks have been hypothesized to influence students’ performance. This hypothesis, however, rests on the assumption that textbook problems are representative of the problems that students encounter in classroom assignments. This assumption might not be true, because teachers do not present all problems in textbooks and because teachers present problems from sources other than textbooks. To test whether distributions of problems that students encounter parallel distributions of textbook problems, we analyzed fraction and decimal arithmetic problems assigned by 14 teachers over an entire school year. Five of the six documented biases in textbook problem distributions were also present in the classroom assignments. Moreover, the same biases were present in 16 of the 18 combinations of bias and grade level (4th, 5th, and 6th grade) that were examined in assignments and textbooks. Theoretical and educational implications of these findings are discussed.

To advance understanding of the gap in fraction learning between students in high-achieving East-Asian countries and the United States, we examined both intended curricula (i.e. standards) and implemented curricula (i.e. textbooks) in East Asia and the United States. Many similarities were present in both standards and textbooks. However, U.S. students began studying fractions earlier and studied them over more grades, and East-Asian instruction was more concen trated and included more mathematically challenging problems. Additionally, U.S. standards and textbooks tended to contextualize problems and emphasize the part–whole and measurement models of fractions, whereas East-Asian curricula tended to teach fraction concepts within the context of multiplicative reasoning and to teach fraction operations as an extension of whole-number operations. Educational implications of the findings about input …