Search

ispartof: Cognitive Development vol:66 status: published

ispartof: Journal of Numerical Cognition vol:7 issue:3 pages:479-482 status: published

The current study examined upper elementary school children’s frequent, efficient and adaptive use of direct subtraction (DS) and subtraction by addition (SBA) when mentally solving multi-digit subtractions, replicating and expanding previous research by Torbeyns et al. (2018). First, children were offered subtractions in two choice conditions in which they had to indicate whether they would use DS or SBA to solve each item. In the choice-compute condition they were allowed to make actual calculations, in the choice-decide condition they were not allowed to do so. Thereafter, two no-choice conditions were offered (mandatory use of either DS or SBA). Results showed that children made frequent, efficient and adaptive use of the untrained SBA strategy. Furthermore, children were most adaptive for task and subject characteristics in the choice-compute condition. The current study confirms previous findings on SBA, and questions current classroom practices that focus heavily on the development of DS.

The present study aimed to analyze the direction of the associations between repeating patterning, growing patterning, and numerical ability. Participants were 410 children who were annually assessed on their repeating patterning, growing patterning, and numerical ability, at ages 4, 5, and 6 years (i.e., spring 2017, 2018, and 2019). A cross-lagged panel model identified bidirectional associations between all three abilities from ages 4 to 5 years while taking into account spatial skills. From ages 5 to 6 years, both patterning abilities predicted later numerical ability, but the reverse was no longer true. Associations between performances on both pattern types also disappeared. Results highlight the unique associations between repeating patterning, growing patterning, and numerical ability, above spatial skills. ispartof: Child Development vol:92 issue:4 pages:1354-1368 ispartof: location:United States status: published

ispartof: Journal Of Educational Psychology vol:115 issue:5 pages:1-16 status: Published online

Children’s spontaneous focusing on Arabic number symbols (SFONS) has been identified as a relevant component of their early mathematical development. This study investigated whether SFONS is a sepa...

Young children show large individual differences in their tendency to focus spontaneously on numerical aspects (e.g., numerosities or Arabic number symbols) of their everyday environment. The origins of these individual differences are unclear. Given the role of the home numeracy environment (HNE) in children’s early mathematical development and the assumed link between children’s spontaneous number focusing tendencies and their numerical behavior in everyday life, it is plausible that children’s spontaneous focusing tendencies are related to their HNE. The present study aimed to test this hypothesis by longitudinally investigating children’s spontaneous focusing on numerosity (SFON) and spontaneous focusing on Arabic number symbols (SFONS) in relation to their HNE. Participants were 128 children (4- to 5-year-olds), who were followed from the second until the third year of Flemish kindergarten. In both kindergarten years, children completed a SFON and SFONS Picture task while their parents completed a home numeracy questionnaire. Correlation analyses and structural equation modeling revealed no significant associations between children’s spontaneous number focusing tendencies and their HNE, neither in second nor in third year of kindergarten. This finding suggests that children’s spontaneous number focusing tendencies are not per se related to their HNE. Various possible explanations for this unexpected finding are discussed and directions for further research on this relationship are suggested. ispartof: ZDM Mathematics Education vol:52 issue:4 pages:729-749 status: published

ispartof: Journal Of Educational Psychology vol:114 issue:1 pages:37-55 status: published

Teachers' pedagogical content knowledge (PCK) is pivotal for effective classroom instruction. Unfortunately, research on the acquisition of preschool teachers' PCK in the domain of early mathematics and how this knowledge relates to their mathematical classroom activities is scarce. The present contribution investigates (1) the acquisition of PCK in pre-service and in-service preschool teachers and (2) the association between preschool teachers' PCK and self-reported classroom activities in the domain of early mathematics. We offered a scenario-based questionnaire to 162 pre-service preschool teachers and 75 in-service preschool teachers to assess their PCK in the domain of early mathematics. Additionally, in-service preschool teachers fulfilled a questionnaire about their mathematical classroom activities. Our findings demonstrated that first-year pre-service preschool teachers scored significantly lower than second- and third-year pre-service preschool teachers. Furthermore, this latter group achieved a significantly lower PCK score than in-service teachers. Second, we observed a positive association between teachers' PCK and the classroom activities in the domain of early mathematics they reported to provide. Overall, our findings point to the interplay between PCK and effective classroom activities in preschool mathematics, and to the importance of teacher training and professional development programs, including opportunities to enhance the acquisition of this important knowledge.

This study presents a patterning intervention for 5-year-olds, who were in their third year of preschool in Flanders (Belgium), with two key elements: A focus on the structure of patterns, and the inclusion of growing patterns (e.g., ABAABAAAB) in addition to repeating patterns (e.g., ABABAB). We evaluated the impact of this intervention on patterning and numerical ability in a quasi-experimental pretest-intervention-posttest design with a business-as-usual control group. The intervention consisted of a 20 week program with children in the patterning condition receiving weekly 30 min of patterning activities. After the intervention, the patterning group (n = 73) outperformed the control group (n = 74) on both repeating and growing patterns. We observed no transfer effect to children's numerical ability.

Background Early patterning competence has recently been identified as an important precursor of mathematical development. Whereas the focus of this research has been on children’s ability regarding repeating patterns, children might also differ in their spontaneous attention to patterns. Aims The present study aimed to explore 4‐ to 5‐year olds’ Spontaneous Focusing On Patterns (SFOP) and its association with their patterning and mathematical ability. Sample Participants were 378 children (Mage = 4 years 10 months; 191 boys) from 17 preschools. Methods Spontaneous Focusing On Patterns was measured with a construction task in which children had to build a tower with 15 blocks of three different colours. The constructions of the children were grouped into three categories (i.e., pattern, random, and sorting). We additionally administered tasks assessing their patterning ability, mathematical ability, spatial ability, and visuospatial working memory. Results When building a tower, 37% of the preschoolers spontaneously created a pattern, 49% made a random construction, and 14% sorted the blocks per colour. Preschoolers who spontaneously created a pattern had better patterning and mathematical ability than children in the random group. Group differences in patterning ability and spatial skills accounted for the difference in mathematical ability. Conclusions The current data suggest that children’s spontaneous attention to patterns is an important component of their mathematical ability. Children’s spontaneous pattern constructions may provide opportunities to discuss and practice patterns in preschool settings or at home, but more research is required to further analyse the role of SFOP in early mathematical development. ispartof: British Journal Of Educational Psychology vol:90 issue:3 pages:1-17 ispartof: location:England status: Published online

Many studies have investigated the association between children’s spontaneous attention to number and their early mathematical abilities. This work has presented number in a non-symbolic format by exclusively using numerosities as their stimuli. Therefore, little is known about children’s spontaneous attention for Arabic number symbols. We aimed to address this gap by exploring, for the first time, whether we can observe individual differences in young children’s tendency to spontaneously focus their attention on Arabic number symbols (SFONS) using a newly developed Picture task. We investigated SFONS and its association with spontaneous focusing on numerosity (SFON), numerical abilities, and teacher ratings of mathematical competence in the three years of Flemish kindergarten. One hundred and eleven kindergartners (2 years 4 months – 6 years 2 months of age) completed a battery of tasks to measure their SFONS and SFON, as well as a range of numerical abilities (i.e., Arabic numeral identification, verbal counting, and counting objects). Kindergarten teachers were asked to rate their children’s mathematical competence on a 4-point Likert scale. Findings revealed that children in the three kindergarten years under study largely differed in their tendency to spontaneously attend to Arabic number symbols in the pictures they had to describe. We also observed differences in the levels of SFONS and SFON between the three kindergarten years. Although SFONS and SFON were not strongly associated in the three kindergarten years, significant associations were found between children’s SFONS and their numerical abilities in kindergarten year 1 and 2, and teacher ratings of their mathematical competence in kindergarten year 3. Together, these results suggest that SFONS is a relevant component of children’s early mathematical development. ispartof: Early Childhood Research Quarterly vol:48 pages:111-121 status: published

Young children’s early repeating patterning abilities are important foundations for their later mathematical development. Prior studies on young children’s repeating patterning abilities have been conducted exclusively in developed countries differing in economic, societal and educational characteristics from developing countries. In this study, we analyzed the performance and the errors of 4- to 5-year-olds on three repeating patterning tasks, and we investigated the association between children’s repeating patterning abilities and their early mathematical abilities in a developing country, Ecuador. A total of 60 preschoolers and 56 kindergartners completed a repeating patterning test and an early mathematics test at the end of the school year. Results first indicated rather low repeating patterning performances of both preschoolers and kindergartens, with large inter-individual differences in these performances. Kindergartners had significantly better patterning abilities than preschoolers. Second, error analyses indicated that children made different types of errors when solving the patterning tasks. The majority of errors were pattern-related errors, with preschoolers making more errors that were not pattern related than kindergarteners. Third, children’s patterning abilities were positively associated with early mathematical abilities in both grades. These results complement current theoretical models of young children’s mathematical development and inform educational policy and practices in this domain for developing countries. ispartof: European Journal Of Psychology Of Education vol:36 issue:4 pages:1-20 status: Published online

Teachers’ pedagogical content knowledge (PCK) importantly contributes to instructional quality and student outcomes. We aimed to complement the limited insights into preservice preschool teachers’ PCK, and its association with opportunities to learn (OTL), during teacher training. We offered 162 first-to-final-year preservice preschool teachers from two different teacher training institutes in Flanders (Belgium) a recently developed scenario-based instrument addressing students’ understanding of preschoolers’ (mis)conceptions and appropriate instructional strategies in the domain of early mathematics. Our findings revealed quantitative differences between first-year students’ PCK on the one hand and second-year and third-year students’ PCK on the other hand. We did not observe any quantitative differences in PCK between second-year and third-year students. Additional analyses on students’ errors pointed to qualitative differences between first-, second- and third-year students’ PCK. We interpret these findings in view of students’ OTL during teacher training, and discuss their theoretical and methodological implications for future work in the domain of teacher competence. ispartof: ZDM: The International Journal on Mathematics Education vol:52 issue:2 pages:1-12 status: Published online

We investigated the use of the subtraction by addition strategy, an important mental calculation strategy in children with different levels of mathematics achievement. In doing so we relied on Siegler’s cognitive psychological model of strategy change (Lemaire & Siegler, 1995), which defines strategy competencies in terms of four parameters (strategy repertoire, distribution, efficiency and selection), and the choice/no-choice method (Siegler & Lemaire, 1997), which is essentially characterized by offering items in two types of conditions (choice and no-choice). Participants were 63 11-12-year-olds with varied mathematics achievement levels. They solved multi-digit subtraction problems in the number domain up to 1,000 in one choice condition (choice between direct subtraction or subtraction by addition on each item) and two no-choice conditions (obligatory use of either direct subtraction or subtraction by addition on all items). We distinguished between two types of subtraction problems: problems with a small versus large difference between minuend and subtrahend. Although mathematics instruction only focused on applying direct subtraction, most children reported using subtraction by addition in the choice condition. Subtraction by addition was applied frequently and efficiently, particularly on small-difference problems. Children flexibly fitted their strategy choices to both numerical item characteristics and individual strategy speed characteristics. There were no differences in strategy use between the different mathematical achievement groups. These findings add to our theoretical understanding of children’s strategy acquisition and challenge current mathematics instruction practices that focus on direct subtraction for children of all levels of mathematics achievement. ispartof: Journal of Numerical Cognition vol:4 issue:1 pages:215-234 status: published

The number line estimation task is widely used to investigate mathematical learning and development. The present meta-analysis statistically synthesized the extensive evidence on the correlation between number line estimation and broader mathematical competence. Averaged over 263 effect sizes with 10576 participants with sample mean ages from 4 to 14 years, this correlation was r = .443. The correlation increased with age, mainly because it was higher for fractions than for whole numbers. The correlation remained stable across a wide range of task variants and mathematical competence measures (i.e., counting, arithmetic, school achievement). These findings demonstrate that the task is a robust tool for diagnosing and predicting broader mathematical competence and should be further investigated in developmental and experimental training studies. ispartof: Child Development vol:89 issue:5 pages:1467-1484 ispartof: location:United States status: published

El objetivo del presente estudio fue examinar la influencia del nivel socioeconómico de los niños de jardín infantil en el desarrollo de sus habilidades numéricas tempranas y su enfoque espontáneo en el número (SFON), después de controlar por variables cognitivas de dominio general, como inteligencia y memoria de trabajo, y por edad. Al iniciar el jardín infantil, 355 niños y niñas resolvieron una batería de pruebas para evaluar sus habilidades numéricas tempranas, SFON, inteligencia y memoria de trabajo. Los resultados indican que existen diferencias individuales en las habilidades numéricas tempranas y el SFON de los niños ecuatorianos al iniciar el jardín infantil. Además, el nivel socioeconómico se correlacionó positivamente con las habilidades numéricas tempranas, pero no con el SFON. Al final se discuten las implicaciones teóricas, metodológicas y prácticas del estudio.

The home numeracy questionnaire (HNQ) developed by LeFevre et al. (2009) is a self-report instrument to assess the home numeracy of young children. In this questionnaire, young children’s parents have to indicate how often their child participates both in direct (e.g., counting down: 10, 9, 8, …) and indirect numeracy activities (e.g., playing board games with die or spinner). Based on a principal component analysis, LeFevre and colleagues found a four factor structure of the HNQ. Using a sample of 176 Ecuadorian parents, we investigated the validity and reliability of the HNQ in a developing country. A confirmatory factor analysis indicated that, with minor changes to the model, the same four factor model could be found as by LeFevre et al. (2009). This study not only confirmed the validity and reliability of the HNQ, but also showed its usefulness for investigating young children’s home numeracy beyond developed countries

Mathematics-related affect comprises an individual’s attitudes, beliefs, emotions and motivations towards mathematics. These affective constructs have been widely studied in mathematics education and cognitive psychology and have been shown to be related to cognitive outcomes such as performance on a range of mathematical tasks. However, it is not yet clear how these constructs develop, or how they relate to cognitive factors in young children who are in the early stages of learning mathematics. As such, the current special issue focuses on mathematics-related affect in primary school children aged 4 to 10 years. It brings together five recent empirical studies and two discussant articles looking at the development of attitudes, beliefs, emotions and motivations towards mathematics, and the relations between affective and cognitive factors in these young age groups. In this introductory paper, we provide some brief historical context, followed by a rationale for the special issue, and an overview of its structure and scope. ispartof: Educational Studies in Mathematics vol:100 issue:3 pages:201-209 status: published

In this study, we aimed to address two gaps in research on early mathematical patterning, namely the lack of attention (1) to growing patterns and (2) to the association between different aspects of patterning and numerical ability. Participants were 400 four-year olds from a wide range of socio-economic backgrounds. Children’s patterning and numerical ability were assessed by means of individual tasks. The patterning tasks assessed their performance on three patterning activities (i.e., extending, translating, and identifying the pattern’s structure) for two types of patterns (i.e., repeating and growing). The numerical measure included a set of eight well-known numerical tasks. We additionally controlled for individual differences in spatial ability and visuospatial working memory. Results indicated an effect of both activity and patterning type on children’s patterning performance, as well as an interaction between both. Furthermore, children’s performance on four out of six patterning tasks uniquely contributed to their numerical ability above age, spatial ability, and visuospatial working memory. These findings support the importance of specific pattern types and patterning activities in the early stage of children’s mathematical development and give directions for further educational practices.

Despite the general agreement regarding the importance of stimulating young children’s early mathematical skills, mathematical learning opportunities in preschool are limited. Educational technology (ET) may be an effective tool to address this problem. Taking into account the weaknesses of previous reviews, we conducted a systematic and comprehensive review of the research literature on the effectiveness of ET in early mathematics education, critically analyzing the findings of studies adhering to a media comparison approach (comparing an ET condition to a non-ET condition or another ET condition) versus studies following a value-added approach (comparing at least two ET conditions which only differ with respect to one feature in the ET or in its implementation). Reviewing 54 studies, we systematically analyzed (1) the effectiveness of ET, (2) the features of the ET and ET implementation associated with ET effectiveness, and (3) child characteristics associated with ET effectiveness. Our analyses indicated that media comparison studies comparing an ET to a non-ET condition provide evidence for the effectiveness of ET for enhancing young children’s mathematical competencies. Second, value-added studies pointed to ET implementation features associated with ET effectiveness, namely teacher support during ET use and an individual grouping structure. Finally, both media comparison and value-added studies revealed that ET effectiveness is associated with children’s performance in the ET and with their prior knowledge. To further advance the field, value-added studies on features of the ET and its implementation are required, as well as studies focusing on important child characteristics that are associated with ET effectiveness. ispartof: International Journal of Child-Computer Interaction vol:27 pages:1-26 status: published

Recent evidence indicates that young children's spontaneous focusing on numerosity (SFON) uniquely contributes to their early numerical abilities. This study complements previous findings by validating the relation between young children's SFON and their early numerical abilities in a developing country, namely Ecuador. We analysed 355 Ecuadorian 5- to 6-year-olds' SFON in relation to their early numerical abilities at the start of kindergarten, controlling for children's socio-demographic (socio-economic status, age) and general cognitive (working memory, intelligence) characteristics. Our results evidence the unique contribution of Ecuadorian kindergartners' SFON to their early numerical abilities, controlling for children's working memory, intelligence, socio-economic status, and age. Our findings support the validity of previous findings on the unique contribution of SFON to young children's early numerical abilities in developed countries for developing countries. Additionally, they raise timely questions for further theoretical and methodological studies on young children's numerical development worldwide, in developing and developed countries. Statement of contribution What is already known? SFON uniquely contributes to early numerical abilities. However, this is only documented in Finnish samples. Previously only limited control for domain-general cognitive and socio-demographic characteristics. What does the study add? SFON uniquely contributes to early numerical abilities in Ecuadorian 5- to 6-year-olds. This unique contribution remains after controlling for WM, IQ, SES, and age. Evidence for the universal nature of the association between SFON and early numerical ability.

Numerical competencies acquired in preschool are foundational and predictive for children's later mathematical development. It remains to be determined whether there are gender differences in these early numerical competencies which could explain the often-reported gender differences in later mathematics and STEM-related abilities. Using a Bayesian approach, we quantified the evidence in favor of the alternative hypothesis of gender differences versus the null hypothesis of gender equality. Participants were 402 4- to 5-year-old children attending preschool in Flanders (Belgium). Children were selected via stratified cluster sampling to represent the full range of socioeconomic backgrounds. All children completed eight numerical tasks (verbal counting, object counting, numeral recognition, symbolic comparison, nonsymbolic comparison, nonverbal calculation, number order, dot enumeration). Results supported the gender equality hypothesis, and this evidence was substantial for seven of the eight numerical tasks. Preschoolers' early numerical competencies are characterized by gender equality. They probably do not explain later-reported gender differences.

© 2017 The British Psychological Society. We aimed to investigate upper elementary children's strategy use in the domain of multidigit division in two instructional settings: the Netherlands and Flanders (Belgium). A cross-sectional sample of 119 Dutch and 122 Flemish fourth to sixth graders solved a varied set of multidigit division problems. With latent class analysis, three distinct strategy profiles were identified: children consistently using number-based strategies, children combining the use of column-based and number-based strategies, and children combining the use of digit-based and number-based strategies. The relation between children's strategy profiles and their instructional setting (country) and grade were generally in line with instructional differences, but large individual differences remained. Furthermore, Dutch children more frequently made adaptive strategy choices and realistic solutions than their Flemish peers. These results complement and refine previous findings on children's strategy use in relation to mathematics instruction. Statement of contribution: What is already known? Mathematics education reform emphasizes variety, adaptivity, and insight in arithmetic strategies. Countries have different instructional trajectories for multidigit division. Mixed results on the impact of instruction on children's strategy use in multidigit division. What does this study add? Latent class analysis identified three meaningful strategy profiles in children from grades 4-6. These strategy profiles substantially differed between children. Dutch and Flemish children's strategy use is related to their instructional trajectory. ispartof: British Journal of Developmental Psychology vol:36 issue:2 pages:169-187 ispartof: location:England status: published

Mathematical skills are crucial to success in modern western societies. We are continuously confronted with mathematical information and tasks in both our professional and private daily lives. Duri...

This study explored the development of Ecuadorian Kindergartners’ spontaneous focusing on numerosity (SFON) during the Kindergarten year, as well as the contribution of early numerical abilities to this development. 100 Kindergartners coming from 10 classrooms received two SFON tasks, one at the beginning and one at the end of the school year, and an early numerical abilities achievement test at the beginning of the school year. Results first demonstrated limited SFON development during the Kindergarten year, with inter-individual differences in and intra-individual stability of children’s SFON tendency. Second, both children’s SFON tendency and their early numerical abilities at the start of the Kindergarten year were predictively related to their SFON tendency at the end of the year. Our results do not only add to our theoretical understanding of SFON in young children, but also inform educational policy and practices in the domain of early mathematics education in Ecuador, as they provide building blocks for optimizing the educational goals and curricula for Kindergarten mathematics. ispartof: European Journal of Psychology of Education vol:32 issue:3 pages:449-462 status: published

BACKGROUND: Young children's spontaneous focusing on numerosity (SFON) as measured by experimental tasks is related to their mathematics achievement. This association is hypothetically explained by children's self-initiated practice in number recognition during everyday activities. As such, experimentally measured SFON should be associated with SFON exhibited during everyday activities and play. However, prior studies investigating this assumed association provided inconsistent findings. AIMS: We aimed to address this issue by investigating the association between kindergartners' SFON as measured by two different experimental tasks and the frequency of their number-related utterances during a typical picture book reading activity. SAMPLE: Participants were 65 4- to 6-year-olds in kindergarten (before the start of formal education). METHODS: Kindergartners individually participated in two sessions. First, they completed an action-based SFON Imitation task and a verbal SFON Picture task, with a short visuo-motor task in between. Next, children were invited to spontaneously comment on the pictures of a picture book during a typical picture book reading activity. RESULTS: Results revealed a positive association between children's SFON as measured by the Picture task and the frequency of their number-related utterances during typical picture book reading, but no such association for the Imitation task. CONCLUSIONS: Our findings indicate that children with higher SFON as measured by a verbal experimental task also tend to focus more frequently on number during verbal everyday activities, such as picture book reading. In view of the divergent associations between our SFON measures under study with everyday number activities, the current data suggest that SFON may not be a unitary construct and/or might be task-dependent. ispartof: British Journal of Educational Psychology vol:88 issue:4 pages:550-565 ispartof: location:England status: published

In the last decades, strategy variability and flexibility have become major aims in mathematics education. For children with mathematical learning disabilities (MLD) it is unclear whether the same goals can and should be set. Some researchers and policy makers advise to teach MLD children only one solution strategy, others advocate stimulating the flexible use of various strategies, as for typically developing children. To contribute to this debate, we compared the use of the subtraction by addition strategy to mentally solve two-digit subtractions in children with and without MLD. We used non-verbal research methods to infer strategy use patterns, and found that both groups of children switch between the traditionally taught direct subtraction strategy and subtraction by addition, based on the relative size of the subtrahend. These findings challenge typical special education classroom practices, which only focus on the routine mastery of the direct subtraction strategy. publisher: Elsevier articletitle: Subtraction by addition in children with mathematical learning disabilities journaltitle: Learning and Instruction articlelink: http://dx.doi.org/10.1016/j.learninstruc.2013.11.001 content_type: article copyright: Copyright © 2013 Elsevier Ltd. All rights reserved. ispartof: Learning and Instruction vol:30 pages:1-8 status: published

Background. In the last decades, children’s understanding of mathematical principles has become an important research topic. Different from the commutativity and inversion principles, only few studies have focused on children’s understanding of the addition/subtraction complement principle (if a − b = c, then c + b = a), mainly relying on verbal techniques. Aim. This contribution aims at deepening our understanding of children’s knowledge of the addition/subtraction complement principle, combining verbal and non-verbal techniques. Sample. Participants were 67 third- and fourth-graders (9-10-year-olds). Methods. Children solved two tasks in which verbal reports as well as accuracy and speed data were collected. These two tasks differed only in the order of the problems and the instructions. In the looking back task children were told that sometimes the preceding problem might help to answer the next problem. In the baseline task no helpful preceding items were offered. The looking back task included 10 trigger-target problem pairs on the complement relation. Results. Children verbally reported looking back on about 40% of all target problems in the looking back task; the target problems were also solved faster and more accurately than in the baseline task. These results suggest that children used their understanding of the complement principle. The verbal and non-verbal data were highly correlated. Discussion. This study complements previous work on children’s understanding of mathematical principles by highlighting inter-individual differences in 9-10-year-olds’ understanding of the complement principle and indicating the potential of combining verbal and non-verbal techniques to investigate (the acquisition of) this understanding. ispartof: British Journal of Educational Psychology vol:86 issue:3 pages:382-396 ispartof: location:England status: published

This study analyzed children’s use of mental computation strategies and the standard algorithm on multi-digit subtractions. Fifty-eight Flemish fourth-graders of varying mathematical achievement level were individually offered subtractions that either stimulated the use of mental computation strategies or the standard algorithm in 1 choice and 2 no-choice conditions. In the choice condition, children could apply their preferential strategy on each item; in the no-choice conditions, they had to solve all items with mental computation and the standard algorithm, respectively. Results revealed that children of all achievement levels applied the standard algorithm remarkably frequently and efficiently, even on subtractions that were intended to evoke mental computation. Moreover, children did not fit their strategy choices to the numerical characteristics of the items, but high and above-average achieving children based their strategy choices on their individual mastery of the different strategies. We discuss the theoretical, methodological, and instructional implications of these results. ispartof: European Journal of Psychology of Education vol:31 issue:2 pages:99-116 status: published

Subtractions of the type M − S = ? can be solved by various strategies, including subtraction by addition. In this study, we investigated children’s use of subtraction by addition by means of reaction time analyses. We presented 106 third to sixth graders with 32 large non-tie single-digit problems in both subtraction (12 − 9 = .) and addition format (9 + . = 12). We examined the fit of three regression models, which represented the consistent use of direct subtraction, of subtraction by addition and of flexibly switching between both strategies based on the relative size of the subtrahend. Findings revealed that children did not switch flexibly between the two strategies, as adults do, but that they rely on direct subtraction for problems presented in subtraction format and on subtraction by addition for problems in addition format. We end with the major theoretical, methodological and educational implications of these results.

The present study investigates adults' use of addition to solve two-digit subtractions. Inspired by research on single-digit arithmetic, we first examined regression models in which different problem characteristics predicted participants' reaction times. Second, we compared performance on two-digit subtractions presented in 2 presentation formats, i.e., the standard subtraction format (81-37=.) and an addition format (37+.=81). Both methods lead to the conclusion that the participants switched between direct subtraction and subtraction by addition depending on the relative size of the subtrahend: If the subtrahend was smaller than the difference, direct subtraction was mainly used; if the subtrahend was larger than the difference, subtraction by addition was the dominant strategy. However, this performance pattern was only observed when the distance between the subtrahend and the difference was large; when the subtrahend and the difference were close to each other, there was no subtrahend-dependent selection of direct subtraction vs. subtraction by addition. These data indicate that theoretical models of people's strategy choices in subtraction should include the relative size of the subtrahend as an important factor in the strategy selection process.

This study examined 25 university students' use of addition to solve large single-digit subtractions by contrasting performance in the standard subtraction format (12-9=.) and in the addition format (9+.=12). In particular, we investigated the effect of the relative size of the subtrahend on performance in both formats. We found a significant interaction between format, the magnitude of the subtrahend (S) compared to the difference (D) (S>D vs. S

Some years ago, Hatano differentiated between routine and adaptive expertise and made a strong plea for the development and implementation of learning environments that aim at the latter type of expertise and not just the former. In this contribution we reflect on one aspect of adaptivity, namely the adaptive use of solution strategies in elementary school arithmetic. In the first part of this article we give some conceptual and methodological reflections on the adaptivity issue. More specifically, we critically review definitions and operationalisations of strategy adaptivity that only take into account task and subject characteristics and we argue for a concept and an approach that also involve the sociocultural context. The second part comprises some educational considerations with respect to the questions why, when, for whom, and how to strive for adaptive expertise in elementary mathematics education.

This study aimed at analysing traditionally taught children’s acquisition and use of shortcut strategies in the number domain 20–100. One-hundred-ninety-five second, third, and fourth graders of different mathematical achievement levels participated in the study. They were administered two tasks, both consisting of a series of two-digit additions and subtractions that maximally elicit the use of the compensation $$\left( {45 + 29 = \_;45 + 30 - 1 = 75 - 1 = 74} \right)$$ and indirect addition strategy ( $$71 - 68 = \_;\,68 + 2 = 70,\,70 + 1 = 71$$ , so the answer is 2 + 1 or 3). In the first task, children were instructed to solve all items as accurately and as fast as possible with their preferred strategy. The second task was to generate at least two different strategies for each item. Results demonstrated that children of all grades and all achievement levels hardly applied the compensation and indirect addition strategy in the first task. Children’s strategy reports in the second task revealed that younger and lower achieving children did not apply these strategies because they did not (yet) discover these strategies. By contrast, older and higher achieving children appeared to have acquired these strategies by themselves. Results are interpreted in relation to cognitive psychological and socio-cultural perspectives on children’s mathematics learning.

Hace algunos años, Hatano diferenció entre el nivel de experto de aprendizaje de rutina y el adaptativo e impulsó el desarrollo y la implementación de ambientes de aprendizaje que tengan como objetivo el tipo adaptativo de experiencia por sobre el de rutina. En esta contribución, nos enfocamos en un aspecto de la adaptabilidad, la llamada adaptativa del uso de estrategias de solución en aritmética de educación primaria. En la primera parte de este artículo, brindamos algunos enfoques conceptuales y metodológicos en los asuntos de adaptabilidad. Más específicamente, hacemos una revisión crítica de las definiciones y operacionalizaciones de estrategias de adaptabilidad que solo toman en cuenta tareas y características del sujeto. Argumentamos la necesidad de un concepto y un enfoque que también incluya el contexto sociocultural. La segunda parte, comprende algunas consideraciones educativas con respecto a las preguntas de porque, cuando, para quien o cómo se procura la búsqueda de nivel de experto adaptativa en la educación de las matemáticas para la escuela primaria.

This study aimed at (a) constructing a reliable and valid test to assess Ecuadorian 4–5-year olds’ number and arithmetic skills; (b) providing empirical data on Ecuadorian 4–5-year olds’ number and arithmetic skills; and (c) confronting these children’s actual performances with the performances expected by national experts in this domain. We administered the Test of Early Number and Arithmetic (TENA), developed on the basis of the Ecuadorian mathematics standards to 86 Preschoolers and 127 Kindergartners and asked 10 experts to evaluate TENA’s validity and predict children’s performances on it. Results supported the overall reliability and validity of the TENA. Furthermore, we observed differences in number and arithmetic competencies between and within Preschoolers and Kindergartners, but not between boys and girls. Finally, experts overestimated children’s performances on the test. The scientific and practical implications of these results are discussed.

The aim of the study was to analyze the development of children's adaptive expertise in computing sums and differences up to 100.We defined the adaptive nature of children's strategy choices on the basis of problem (addition, subtraction), achievement, and strategy performance (accuracy, speed). Sixty-nine 2nd graders of high, above-average, or below-average mathematical achievement level solved 4 types of additions and subtractions before and after regular classroom instruction in the topic. In both assessments, children answered the problems in a choice condition, in which they could choose between the jump strategy (45 + 28 = __; 45 + 20 = 65, 65 + 8 = 73) and the split strategy (45 + 28 = __; 40 + 20 = 60, 5 + 8 = 13, 60 + 13 = 73) on each problem, and in two no-choice conditions, where they had to solve all problems with either the jump or the split strategy. Our results revealed differences in children's strategy choices on both assessments. The choices of high achievers indicated that they fit thei...

In this study, we investigated the fluency with which first graders of different mathematical achievement levels applied multiple, school-taught strategies for finding arithmetic sums over 10. We characterized children's strategies with the 4 parameters of Lemaire and Siegler's (1995) model of strategy change (strategy repertoire, distribution, efficiency, and selection) using the choice/no-choice method (Siegler & Lemaire, 1997). Eighty-three first graders who had been taught 2 reasoning strategies (decomposition-to-10 and tie) solved a series of near ties over 10 (such as 8 + 7 = .) in 4 different conditions. In the choice condition, children could choose between the decomposition-to-10 and the tie strategy on each near tie. In the 3 no-choice conditions, they had to solve all sums with decomposition-to-10, tie, and retrieval strategies, respectively. High-achieving first graders applied these strategies more efficiently but not more adaptively than did their lower achieving peers. We interpret these re...