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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.

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.

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

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

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

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.

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...

Rational numbers are amongst the most difficult topics in the elementary and secondary school curriculum and teaching them requires an appropriate knowledge base of teachers to properly deal with students' difficulties. We investigated prospective teachers' content knowledge (CK) and pedagogical content knowledge (PCK) on rational numbers, the relationship between CK and PCK, and differences in CK and PCK among prospective elementary teachers (trained as general classroom teachers) and lower secondary teachers (trained as subject-specific classroom teachers). The results revealed gaps in prospective teachers' CK and PCK, a positive correlation between CK and PCK, and a better CK but not PCK for secondary compared to elementary school teachers.

Young children’s spontaneous focusing on numerosity (SFON) predicts their later mathematical competencies. In this study we investigated the development of SFON in Ecuadorian kindergartners as well as the contribution of early numerical abilities and the quality of mathematics education to this development. The participants were 100 kindergartners drawn from 10 classrooms. Children 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. The quality of mathematics education was assessed twice via the COEMET instrument. Results demonstrated limited SFON development during the kindergarten year, with large individual differences in and highly consistent SFON performances. Additionally, children’s SFON development during the kindergarten year was predicted by their SFON tendency and early numerical abilities at the start of the year. The quality of mathematics education did not contribute to children’s SFON development. The scientific and practical implications of these results are discussed.

Children’s Spontaneous Focusing On Numerosity (SFON) predicts later mathematics performance. This association is assumed to rely on children’s self-initiated practice in number recognition during everyday activities, which would enhance their further mathematical development. Consequently, SFON in experimental tasks should be associated with SFON during everyday activities. The present contribution aims to enhance our understanding of this association by critically discussing the major results of two recently conducted studies on the association between SFON in experimental tasks and SFON during picture book reading. Study 1 revealed no association between children’s SFON in an Imitation task and their number-related utterances during numerical picture book reading. Study 2, in which we contrasted two different SFON tasks and their association to picture book reading, revealed a positive association between children’s SFON in the Picture task (but not in the Imitation task) and their number-related utterances during picture book reading. Theoretical, methodological, and educational implications are discussed.

© 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

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

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

This study analysed children's use of mental computation strategies and standard written algorithms in the domain of multi-digit addition and subtraction, using the choice/no-choice method. Twenty-one Flemish fourth-graders (M Age =9y10m) solved problem-items that either stimulated the use of mental computation strategies or a standard written algorithm, in one choice and two no-choice conditions. In the choice condition, children could use their preferred strategy on each item. In the two no-choice conditions, they had to solve all items with, respectively, a mental computation strategy (no-choice mental condition) and the standard written algorithm (no-choice written condition). The results revealed that children tended to use the written algorithm on multi-digit sums, even on sums that evoke a mental computation strategy. Moreover, they also used the standard written algorithms with high efficiency and flexibility on sums that elicit a mental computation strategy. We discuss the theoretical, instructio...

This study examined the development of indirect addition as an alternative for solving multidigit subtractions, by means of two learning environments in traditionally schooled third-graders in Flanders (Belgium). Thirty-five third-graders, who did not demonstrate mastery of indirect addition, participated in an Explicit (n ¼ 20) or Implicit (n ¼15) learning environment that aimed to encourage the development of indirect addition. Over a period of six weeks, children participated in nine individual sessions: four practice sessions, three test sessions, one transfer session, and one retention session. The results revealed that throughout the study indirect addition was rarely used, even by the children in the explicit learning environment. However, when indirect addition was used, it was executed very efficiently. 2009 Elsevier Ltd. All rights reserved.

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.

Subtraction problems of the type a – b = ? can be flexibly solved by various strategies, including the indirect addition strategy ("how much do I have to add to b to get at a?"). Little research has been done on the use of the indirect addition strategy with multi-digit numbers. The present literature review entails a summary of three recent and closely related studies conducted by the authors on this issue. The results of our first study revealed that young adults efficiently and flexibly applied indirect addition on 3-digit subtractions. The results of our second and third study showed that elementary school children seldom used indirect addition on 2-digit subtractions, despite its computational efficiency. This held true even in children who received school-based instruction in the strategy. We end with a discussion of some theoretical, methodological, and educational implications of the studies being reviewed.

The goal of this special issue is to report on recent theoretical, empirical, methodological, and instructional developments regarding the four fundamentally important arithmetic principles related...

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.

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

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...

This article describes a limited longitudinal European study of young children’s early numeracy development within three testing cycles, onaverage, at the mid-point and towards the end of their fifth to sixth year and again at the mid-point of their sixth to seventh year. Assessment was carried out using the Utrecht Early Numeracy Test (ENT) (Van Luit, Van de Rijt and Pennings, 1994). The multilevel modelling method of analysis used for the study provided an extension of multiple regression to incorporate the hierarchical structure of the data collected, with boys and girls of different social-economic status, nested within different institutions within different countries. The results showed that the ENT was a useful tool for international comparison. The finding that differences between countrieswere negligible was surprising bearing in mind that the English pupils werein formal schooling throughout the testing cycle, the Belgian, German, Greek and Dutch children from the mid-point, and the Slovene children, not at all.

This article describes a research project about the development of early numeracy in 5- to 7-year-old Flemish children compared to their Dutch peers. A group of Flemish children was tested 3 times by means of a slightly adapted Flemish version of the Utrecht Early Numeracy Test, and their results were compared to the development of those of a comparable group of Dutch children. The first and the second assessment took place, respectively, at the middle and at the end of the last year of preprimary education, and the third assessment at the middle of the first year of primary education. The results indicate that the Early Numeracy Test is a moderately reliable and construct valid instrument to assess the general early numeracy. At the time of the first assessment Flemish children scored worse on the test than their Dutch peers. However, they showed a faster progression between the first and the third assessment, resulting in a catch up with their Dutch peers at the third assessment. The results are interpr...

no abstract, Number sense is crucial in daily life and therefore an important aspect of education. The development of number sense recently attracted the attention of an increasing number of researchers in the domains of cognitive and developmental (neuro)psychology, (psychology of) mathematics education, special education, and educational neuroscience. In this article, we integrate and discuss the major results of these numerous studies on children’s acquisition of number sense. We start with the definition of the concept of number sense. Afterwards, we summarize the major findings on children’s acquisition of number sense from infancy to the early school years. Next, we provide an overview of recent intervention studies aiming at stimulating the development of number sense in young children. We end with a summary of the major findings and future issues for studies in this domain