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ispartof: Cognitive Development vol:66 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.

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

Insight into early precursors of proportional reasoning is necessary to further our theoretical understanding of mathematical development and to guide early interventions. Although several researchers have suggested patterning as a possible precursor for proportional reasoning, there is little empirical evidence to support this assumption, particularly at a young age. To address this gap, the current study explored if patterning in 4- to 5-year-olds (n = 346) is associated with proportional reasoning one and a half years later. Two measures of patterning ability (repeating and growing patterns) and two measures of proportional reasoning (one with discrete quantities and one with a discrete and a continuous quantity) were administered, together with measures addressing general cognitive and numerical abilities. Regression analyses showed that patterning is a unique predictor of proportional reasoning ability over and above sex and general cognitive and numerical abilities. An interaction effect between pattern types and the nature of the quantities was observed: Performance on repeating patterns was uniquely related to performance on proportional reasoning with two discrete quantities, whereas performance on growing patterns was uniquely related to performance on proportional reasoning with a discrete and a continuous quantity. Theoretical implications and suggestions for future studies are discussed.

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

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...

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

When solving subtraction problems such as 83-46, children use the direct subtraction (DS) strategy (e.g., 83 - 40 = 43, 43 - 6 = 37) or the subtraction by addition (SBA) strategy (e.g., 46 + 4 = 50, 50 + 30 = 80, 80 + 3 = 83, so the answer is 4 + 30 + 3 = 37). This study is the first to use the choice/no-choice method to examine DS and SBA use in third-graders (8- and 9-year-olds) with varying mathematical achievement levels. All children (N = 66) solved a series of small difference subtractions (e.g., 72 - 64) and large difference subtractions (e.g., 94 - 8) in one choice condition (choice between DS and SBA) and two no-choice conditions (obligatory use of either DS or SBA). Results showed that, although only the DS strategy was taught to these children, in the choice condition nearly half the children already made use of the SBA strategy and that SBA was used on one in five subtractions. Whereas DS was the fastest strategy on large difference items, interestingly, children achieved a similar level of accuracy with the SBA strategy compared with DS. Finally, 1 in 5 children made problem-based adaptive strategy choices, and children were generally adaptive to their individual strategy speed. This study clearly demonstrates the nascent use of SBA in third-graders and provides evidence for the plea to introduce SBA as an alternative to DS already in the lower grades of elementary school.

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.

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

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

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.

In this chapter, we review a long-standing and comprehensive line of research on the use of the subtraction by addition (SBA) strategy when solving subtractions, in which our own research center has played an important role. We document this phenomenon in a variety of subdomains of the elementary arithmetic curriculum, ranging from single-digit mental subtraction, multidigit mental and written subtraction, to word problems in the domain of subtraction. We also review the available research that tries to link these findings about people’s use of SBA in these different curricular subdomains to their conceptual knowledge of subtraction. Taken as a whole, the reviewed research has yielded recurrent and convincing evidence for the frequent, efficient, and flexible use of SBA in the above curricular subdomains among both elementary school children and adults. Given that learners are typically taught to solve subtractions by means of direct subtraction (DS) while SBA gets little or no instructional attention, these are striking results. At the end, we reflect on some theoretical, methodological, and educational implications of these remarkable findings.

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

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

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

We examined children's number-based and digit-based strategy use profiles on multi-digit subtraction problems in relation to task performance and individual (gender, grade, mathematical achievement level) and contextual (country) characteristics. 318 third- to sixth-graders solved a multi-digit subtraction strategy use task, where half of the subtraction problems had number characteristics that stimulated the number-based compensation strategy. First, latent class analyses revealed five strategy use profiles. Most children consistently relied on one strategy (either digit-based strategies or the number-based decomposition or sequential strategies) across problems. Only a minority of children demonstrated strategy variety and flexibility – more often high achievers and children from higher grades. Second, children's strategy instruction history was related to their profile of strategy use, but this relation was less straightforward than generally assumed. Finally, consistent digit-based strategy users were more accurate than consistent decomposition strategy users and varied number-based strategy users.

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

Activities with patterns (e.g., making a necklace with alternating blue and red pearls) are common in many preschool and kindergarten settings. These activities are assumed to play an important role in children’s mathematical development. Nevertheless, most research on early mathematical development focuses on numerical and arithmetical abilities, with considerably fewer empirical studies on the role of patterning in children’s mathematical development. There is now, however, an increasing number of research studies in developmental cognitive psychology and (psychology of) mathematics education that focus on the analysis and stimulation of patterning abilities in young children. In this contribution, we discuss the recent literature on patterning abilities in children in the transition from preschool to primary school. We compare different definitions and operationalizations of patterning and elaborate on the association between mathematical patterning abilities and other domain-specific as well as domain-general cognitive abilities. Finally, interventions aimed at stimulating patterning abilities in young children are explored. Theoretical, methodological, and educational implications of the available studies as well as perspectives for further research are discussed.

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.

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.

This study investigated the relationship between kindergartners’ Spontaneous Focusing on Numerosity (SFON) and their number-related utterances during numerical picture book reading. Forty-eight 4- to 5-year-olds were individually interviewed via a SFON Imitation Task and a numerical picture book reading activity. We expected differences in the frequency of number-related utterances during picture book reading between children with a higher SFON score, providing more number-related utterances, and children with a lower SFON score. Our results showed large inter-individual differences in both kindergartners’ SFON and the frequency of their number-related utterances during picture book reading, yet SFON was not related to the frequency of number-related utterances. This unexpected result is discussed in terms of its scientific, methodological, and educational implications.

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...