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 …

We describe UMA (Unified Model of Arithmetic), a theory of children’s arithmetic implemented as a computational model. UMA extends a theory of fraction arithmetic (Braithwaite et al., 2017) to include arithmetic with whole numbers and decimals. We evaluated UMA in the domain of decimal arithmetic by training the model on problems from a math textbook series, then testing it on decimal arithmetic problems that were solved by 6th and 8th graders in a previous study. UMA’s test performance closely matched that of children, supporting three assumptions of the theory: (1) most errors reflect small deviations from standard procedures, (2) between-problem variations in error rates reflect the distribution of input that learners receive, and (3) individual differences in strategy use reflect underlying variation in learning parameters.

Numerical knowledge is of great and growing importance. While children are attending school, numerical knowledge is essential for learning more advanced mathematics and science, and eventually for learning computer science, psychology, sociology, economics, and a host of other subjects. After children leave school, numerical knowledge is essential not just in STEM areas but also in a wide range of other occupations. Illustratively, a survey of more than 2,000 employed people in the United States, chosen through random digit dials, indicated that 94 percent reported using math in their work, including majorities in occupations classified as upper white collar, lower white collar, upper blue collar, and lower blue collar (Handel, 2016). Moreover, numerical proficiency is related to occupational success: numerical knowledge at age seven years predicts SES at age forty-two years, even after statistically controlling for IQ, years of education, reading skill, working memory, race, and family SES (Ritchie & Bates, 2013).Reducing differences in numerical knowledge is also crucial for the goal of reducing educational and economic inequality. When children from low-income backgrounds enter kindergarten, their numerical knowledge lags a full year behind that of peers from middle class backgrounds (Jordan et al., 2006). That gap seems to have long-term consequences: Even after statistically controlling for a variety of relevant variables, the numerical knowledge of individual preschoolers’ and kindergartners’

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 …

Recent work has suggested that cross-notation understanding (e.g., 2/5 vs. 0.25) is important for math outcomes. In this study, equivalent fraction, decimal, and percent stimuli were used to examine individual differences in cross-notation and within-notation comparison in undergraduate students (N=183). Hierarchical linear regression analyses suggested that cross-notation magnitude comparison accuracy added explanatory power beyond that of within-notation magnitude comparison accuracy in predicting fraction arithmetic calculation and estimation skills, as well as ACT scores. Additionally, participants did not perceive equivalent rational numbers as equivalent when expressed in different notations (e.g., percentages were perceived as larger than equivalent fractions or decimals). Undergraduate students were also randomly assigned to one of two number line interventions: one focused on emphasizing connections among fractions, decimals, and percentages and another focused on developing fraction magnitude representations. Both interventions yielded improvements in rational number understanding, but there were some greater benefits of the cross-notation intervention.

This chapter examines the relations between children’s understanding of mathematical concepts and their ability to execute procedures that embody those concepts. Delineating how these two types of knowledge interact is fundamental to understanding how development occurs. After all, concepts and procedures are much of what children learn in the course of development, and without question, they develop in tandem rather than independently.

Rational numbers are important for advanced math outcomes but pose challenges for many children. Although a fair amount is known about how children understand fractions and decimals, little is known about children’s understanding of percentages and relations across the three rational number notations. Here, we found that many middle school students are inaccurate in comparing across fraction, decimal, and percentage notations. Moreover, cross-notation comparison skill predicted math achievement and other math outcomes beyond within-notation comparison skill. These findings suggest that helping students understand relations among fractions, decimals, and percentages is critical to high quality numerical development.

The integrated theory of numerical development provides a unified approach to understanding numerical development, including acquisition of knowledge about whole numbers, fractions, decimals, percentages, negatives, and relations among all of these types of numbers (Siegler, Thompson, & Schneider, 2011). Although, considerable progress has been made toward many aspects of this integration (Siegler, Im, Schiller, Tian, & Braithwaite, 2020), the role of percentages has received much less attention than that of the other types of numbers. This chapter is an effort to redress this imbalance by reporting data on understanding of percentages and their relations to other types of numbers. We first describe the integrated theory; then summarize what is known about development of understanding of whole numbers, fractions, and decimals; then describe recent progress in understanding the role of percentages; and …

There is a growing awareness that many children are not developing fast and accurate retrieval-based strategies for solving single-digit addition problems. In this study we individually assessed 166 third and fourth grade children to identify a group of children (called accurate-min-counters) who frequently solved simple single-digit addition problems using a min-counting strategy and were accurate using it. We investigated if these children were adaptive when it came to using retrieval for simple addition and if they were disadvantaged when it came to demonstrating mental computational flexibility with multi-digit addition. We found accurate-min-counters represented over 30% of participants. These children were often incorrect when they were required to use retrieval for simple addition and were less flexible than most peers with mental computation strategies. The findings indicate that educators should be …

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 …

Supplementary materials to: Tian, J., Leib, E. R., Griger, C., Oppenzato, C. O., & Siegler, R. S. (2022). Biased problem distributions in assignments parallel those in textbooks: Evidence from fraction and decimal arithmetic. Journal of Numerical Cognition, 8(1), 73–88. https://doi.org/10.5964/jnc.6365

The number one plays a special role in mathematics because it is the identity element in multiplication and division. The present findings, however, indicate that many middle school students do not demonstrate mathematical flexibility representing one as a fraction. Despite possessing explicit knowledge of fraction forms of one (e.g., 95% of students indicated that 36/36 = 1), most students did not recognize and apply knowledge of fraction forms of one to estimate numerical magnitudes, solve arithmetic problems, and evaluate arithmetic operations. Specifically, students were less accurate in locating fraction forms of one on number lines than integer forms of the same number; they also were slower and less accurate on fraction arithmetic problems that included one as a fraction (e.g., 6/6 + 1/3) than one as an integer (e.g., 1 + 1/3); and they were less accurate evaluating statements involving fraction forms of one than the integer one (e.g., lower accuracy on true or false

In the target article, Xing and colleagues (2021) claimed that 6-to 8-year-olds who spontaneously referenced the midpoint of 0–100 number lines made more accurate magnitude estimates and scored higher on a standardized math achievement test than other children. Unlike previous studies, however, the authors found no relation between accuracy on the number line estimation task and a dot discrimination task used to assess the Approximate Number System (ANS). These findings, the authors claim, constitute evidence against the idea that children’s numerical magnitude understanding entails representational change. We disagree.In the literature on the development of numerical magnitude understanding, the “gold standard” assessment is the number-line estimation task (Schneider et al., 2018, Siegler and Opfer, 2003, Siegler et al., 2009). Unlike numerical comparisons (“Which is larger--N1 or N2?”) or …

Supplementary materials to: Schiller, L. K., Fan, A., & Siegler, R. S. (2022). The power of one: The importance of flexible understanding of an identity element. Journal of Numerical Cognition, 8(3), 430–442. https://doi.org/10.5964/jnc.7593

In diesem Kapitel untersuchen wir den Verlauf der pränatalen Entwicklung – einer Zeit erstaunlich schnellen und dramatischen Wandels. Wir werden Störeinflüsse und Umweltgefahren, die den sich entwickelnden Fötus schädigen können, betrachten. Danach behandeln wir in Kürze den Prozess der Geburt und besondere Verhaltensaspekte des Neugeborenen. Schließlich diskutieren wir Probleme, die mit geringem Geburtsgewicht und Frühgeburt einhergehen.

Im Zentrum des vorliegenden Kapitels steht die Entwicklung fundamentaler Konzepte, die sich in den allermeisten Situationen als nützlich erweisen. Konzepte sind geistige Vorstellungen über die Struktur von Gegenständen, Ereignissen, Eigenschaften oder Beziehungen auf der Basis ihrer Ähnlichkeit. Diese Konzepte lassen sich zwei Gruppen zuordnen. Eine Gruppe wird verwendet, um Dinge zu klassifizieren, die in der Welt vorkommen: Menschen, Lebewesen insgesamt und unbelebte Objekte. Die zweite Gruppe umfasst die Dimensionen, mit deren Hilfe wir unsere Erfahrungen repräsentieren: Raum (wo etwas auftrat), Zeit (wann es auftrat), Kausalität (warum es passierte) und Zahl (wie oft es passierte).

Understanding how environments influence learning requires attending not only to what is present but also to what is absent. In the context of mathematics learning, this means attending not only to problems that children encounter frequently in textbooks but also to ones that appear rarely. We present research in this article showing that students perform surprisingly poorly on seemingly simple fraction and decimal arithmetic problems that are seldom seen in textbooks. Next, we describe imbalanced distributions in textbooks of mixed notation arithmetic and comparison problems, and we hypothesize similar relations between the frequency of those types of problems and student accuracy on those tasks. Finally, we review findings about relations between textbook input and student performance in whole number arithmetic and mathematical equality, and we propose a hypothesis regarding when imbalanced …

Bevor wir uns den Forschungsarbeiten zur Entwicklung der Intelligenz zuwenden, geht es zunächst darum zu klären, was Intelligenz ist. In den meisten Bereichen der kognitiven Entwicklung wie Wahrnehmung, Sprache und Begriffsverstehen werden altersbezogene Veränderungen geprüft. Aber die Intelligenzforschung interessiert sich auch für individuelle Unterschiede zwischen Kindern gleichen Alters. Fragen zur Intelligenzentwicklung werden aus gutem Grund kontrovers diskutiert, da sie sehr grundlegende Aspekte betreffen: die Rolle von Vererbung und Umwelt, den Einfluss ethnischer Unterschiede, die Effekte von Reichtum und Armut und die Möglichkeit zu Fortschritten. Daneben werden neuere Intelligenztheorien vorgestellt, die einen größeren Bereich menschlicher Fähigkeiten umfassen. Zu den wichtigsten Intelligenzleistungen von Kindern gehört der Erwerb schulischer Fähigkeiten wie Lesen …

Wesentlich für das Verständnis der Entwicklung ist die Kenntnis der biologischen Grundlagen, die Verhalten beeinflussen. Der Schwerpunkt dieses Kapitels liegt auf den wichtigsten biologischen Faktoren, die vom Augenblick der Befruchtung bis in die Pubertät hinein im Spiel sind: die Vererbung und der Einfluss der Gene, die Entwicklung und frühe Funktion en des Gehirns sowie wichtige Aspekte der körperlichen Entwicklung und Reifung. Jede unserer Körperzellen trägt das genetische Material, das wir bei der Befruchtung geerbt haben und das unser Verhalten lebenslang beeinflusst. Alle unsere Verhaltensweisen werden jeweils vom Gehirn gesteuert. Alles, was wir irgendwann im Leben tun, wird über den Körper vermittelt, der sich in der frühen Kindheit und in der Adoleszenz sehr schnell und drastisch, in anderen Lebensphasen langsamer und subtiler verändert.

This study investigated relations between the distribution of practice problems in textbooks and students’ learning of decimal arithmetic. In Study 1, we analyzed the distributions of decimal arithmetic practice problems that appeared in 3 leading math textbook series in the United States. Similar imbalances in the relative frequencies of decimal arithmetic problems were present across the 3 series: Addition and subtraction more often involved 2 decimals than a whole number and a decimal, but the opposite was true for multiplication and division. We expected children’s learning of decimal arithmetic to reflect these distributional biases. In Studies 2, 3, and 4, we tested the prediction that children would have more difficulty solving types of problems that appeared less frequently in textbooks, regardless of the intrinsic complexity of solving the problems. We analyzed students’ performance on decimal arithmetic from an …

In diesem Kapitel betrachten wir die besondere Beschaffenheit der Interaktionen mit Gleichaltrigen und ihre Auswirkungen auf die soziale Entwicklung von Kindern. Zuerst besprechen wir theoretische Perspektiven zur besonderen Rolle der Interaktionen mit Gleichaltrigen. Nachfolgend betrachten wir Freundschaften, die engste Form solcher Beziehungen. Danach betrachten wir die Interaktionen von Kindern innerhalb der größeren Gruppe von Peers. Diese Beziehungen werden nicht zusammen mit Freundschaften behandelt, weil sie für die Entwicklung von Kindern eine etwas andere Rolle zu spielen scheinen, besonders was das Ausmaß an Vertrautheit untereinander betrifft. Zuletzt geht es um den Einfluss der Eltern in Bezug auf die Peer-Beziehungen von Kindern.

Im vorliegenden Kapitel werden fünf besonders einflussreiche Theorien der kognitiven Entwicklung untersucht: die Theorie von Jean Piaget, der Informationsverarbeitungsansatz, die domänenspezifischen Ansätze und Kernwissenstheorien, die soziokulturelle Perspektive sowie die Perspektive dynamischer Systeme. Wir erläutern hierzu die ihnen zugrunde liegenden theoretischen Annahmen über das Wesen von Kindern, ihre zentralen Fragestellungen und geben praktische Beispiele für ihre pädagogische Anwendbarkeit.

Learning fractions is a critical step in children’s mathematical development. However, many children struggle with learning fractions, especially fraction arithmetic. In this article, we propose a general framework for integrating understanding of individual fractions and fraction arithmetic, and we use the framework to generate interventions intended to improve understanding of both individual fractions and fraction addition. The framework, Putting Fractions Together (PFT), emphasizes that both individual fractions and sums of fractions are composed of unit fractions and can be represented by concatenating them (putting them together). To illustrate, both “3/9” and “2/9+ 1/9” can be represented by concatenating three 1/9s; similarly, 2/9+ 1/8 can be represented by concatenating two 1/9s and one 1/8. Interventions based on the PFT framework were tested in 2 experiments with fourth, fifth, and sixth grade children. The …

Wir geben in diesem Kapitel einen Überblick über einige der einflussreichsten allgemeinen Theorien der sozialen Entwicklung. Diese Theorien sind auf die Erklärung vieler wichtiger Entwicklungsaspekte gerichtet, beispielsweise Emotionen, Motivation, Persönlichkeit, Bindung, Selbst, Beziehungen zu Gleichaltrigen, Moral und Geschlecht. In diesem Kapitel beschreiben wir vier grundlegende Theorien, und zwar psychoanalytische Theorien, Lerntheorien, sozialkognitive Theorien und ökologische Theorien. Wir erläutern die jeweiligen Kernannahmen und wichtigsten Befunde.

Wir verwenden Symbole, um unsere Gedanken, Gefühle und Wissensbestände zu repräsentieren und um diese anderen Menschen mitzuteilen. Die Fähigkeit zum Symbolgebrauch befreit uns von der Gegenwart und versetzt uns in die Lage, von früheren Generationen zu lernen und über die Zukunft nachzudenken. In diesem Kapitel konzentrieren wir uns zuerst und vorrangig auf den Erwerb des wichtigsten Symbolsystems, der Sprache. Hierzu gehört sowohl eine Beschreibung der Sprachentwicklung als auch eine Diskussion der Theorien, die versuchen, diesen Prozess zu erklären. Wir werden danach den Umgang von Kindern mit nichtsprachlichen Symbolen, z. B. mit Bildern und Modellen, betrachten.

Emotionen sind ein grundlegender Teil der menschlichen Erfahrung, aber wie Kinder lernen, ihre Gefühle und ihr Verhalten zu regulieren, kann lebenslange Konsequenzen haben. In diesem Kapitel untersuchen wir die Entwicklung von Emotionen sowie die Entwicklung der Fähigkeit von Kindern, ihre Gefühle und das mit ihnen zusammenhängende Verhalten zu regulieren. Darüber hinaus betrachten wir Zusammenhänge zwischen dem Temperament der Kinder und ihrem Verhalten sowie Verbindungen zwischen emotionalem Stress und psychischer Gesundheit.

In diesem Kapitel werden wir uns zunächst damit befassen, wie sich zwischen Kindern und wichtigen Bezugspersonen eine Bindung entwickelt. Anschließend werden wir untersuchen, auf welche Weise die Entwicklung von Bindungen die emotionale Basis für die kurz- und langfristige Entwicklung des Kindes legt. Wir werden sehen, dass der Bindungsprozess eine biologische Grundlage zu haben scheint, sich aber in Abhängigkeit vom familiären und kulturellen Kontext auf unterschiedliche Weise entfaltet. Im Anschluss daran werden wir ein verwandtes Thema untersuchen – die Entwicklung des kindlichen Bewusstseins des Selbst, also ihr Selbstverständnis, ihre Selbstidentität und ihr Selbstwertgefühl. Auch wenn viele Faktoren diese Entwicklungsbereiche beeinflussen, bildet die Qualität der frühen Bindungen doch die Grundlage dafür, wie sich Kinder in Bezug auf sich selbst fühlen, einschließlich ihres …

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.

In diesem Kapitel behandeln wir die Entwicklung in drei eng miteinander verwandten Bereichen: der Wahrnehmung, dem Handeln und dem Lernen. Unsere Darstellung konzentriert sich vorrangig auf die frühe Kindheit, da sich während der ersten beiden Lebensjahre eines Kindes in allen drei Bereichen außerordentlich schnelle Entwicklungen vollziehen. Ein weiterer Grund hat damit zu tun, dass die Entwicklung in den genannten Bereichen während dieser Lebensphase besonders eng miteinander verwoben ist: Beginnt ein Kind beispielsweise zu krabbeln und dann zu laufen, wird zunehmend mehr von der Welt für es zugänglich so dass es immer mehr entdecken und lernen kann.

Numerous studies have examined how the brain processes numbers. However, although much attention has been paid to whole numbers, little is known about fractions, despite their pervasive use and the large number of people who have difficulty in learning them. We hypothesized that effective processing of fractions relies on conceptual knowledge. To test this hypothesis, we recorded functional magnetic resonance imaging signals from 68 participants (34 females and 34 males) when they performed a magnitude comparison task (whole number vs. fraction) with different levels of difficulty (short-distance number pairs vs. long-distance number pairs), and examined whether brain regions that handle conceptual knowledge are more involved in processing fractions than in processing whole numbers. Spatial patterns for brain activity related to processing fractions and whole numbers differed greatly in the left …

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 …

With How Children Develop, students get an up-to-date, topically-organized introduction to child development, presented by researchers and teachers who themselves are guiding the field into new directions. The authors emphasize fundamental principles, enduring themes, and important recent studies, avoiding excessive detail and making typically difficult topics easier to grasp. This multi-media pack contains the print textbook and LaunchPad access for an additional£ 5 per student.LaunchPad is an interactive online resource that helps students achieve better results. LaunchPad combines an interactive e-book with high-quality multimedia content and ready-made assessment options, including LearningCurve, our adaptive quizzing resource, to engage your students and develop their understanding.

To test the hypothesis that a higher tendency to spontaneously focus on multiplicative relations (SFOR) leads to improvements in rational number knowledge via more exact estimation of fractional quantities, we presented sixth graders (n = 112) with fraction number line estimations and a novel task in which numerical information embedded in narratives could be estimated as fractions. Consistent with our main hypothesis, we found that SFOR tendency predicted both forms of fraction estimation. However, the relation between SFOR and fraction magnitude comparisons was mediated by fraction estimation, both on the number line and on whole number relations embedded in narrative vignettes. Thus, a higher tendency to recognize multiplicative relations in non-explicitly mathematical situations may contribute to increases in the precision with which students encode fractional relations in everyday contexts, both as …

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 …

Adaptive expertise is a valued, but under-examined, feature of students' mathematical development (e.g. Hatano & Oura, 2012). The present study investigates the nature of adaptive expertise with rational number arithmetic. We therefore examined 394 7th and 8th graders’ rational number knowledge using both variable-centered and person-centered approaches. Performance on a measure of adaptive expertise with rational number arithmetic, the arithmetic sentence production task, appeared to be distinct from more routine features of performance. Even among the top 45% of students, all of whom had strong routine procedural and conceptual knowledge, students varied greatly in their performance the arithmetic sentence production task. Strong performance on this measure also predicted later algebra knowledge. The findings suggest that it is possible to distinguish adaptive expertise from routine expertise with …

Although developmental science has always been evolving, these times of fast‐paced and profound social and scientific changes easily lead to disorienting fragmentation rather than coherent scientific advances. What directions should developmental science pursue to meaningfully address real‐world problems that impact human development throughout the lifespan? What conceptual or policy shifts are needed to steer the field in these directions? The present manifesto is proposed by a group of scholars from various disciplines and perspectives within developmental science to spark conversations and action plans in response to these questions. After highlighting four critical content domains that merit concentrated and often urgent research efforts, two issues regarding “how” we do developmental science and “what for” are outlined. This manifesto concludes with five proposals, calling for integrative, inclusive …

Children's failure to reason often leads to their mathematical performance being shaped by spurious associations from problem input and overgeneralization of inapplicable procedures rather than by whether answers and procedures make sense. In particular, imbalanced distributions of problems, particularly in textbooks, lead children to create spurious associations between arithmetic operations and the numbers they combine; when conceptual knowledge is absent, these spurious associations contribute to the implausible answers, flawed strategies, and violations of principles characteristic of children's mathematics in many areas. To illustrate mechanisms that create flawed strategies in some areas but not others, we contrast computer simulations of fraction and whole number arithmetic. Most of their mechanisms are similar, but the model of fraction arithmetic lacks conceptual knowledge that precludes …