Joke Torbeyns

Conceptualizing, Investigating, and Enhancing Adaptive Expertise in Elementary Mathematics Education.

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.RésuméIl y a quelques années, Hatano faisait le partage entre l’expertise routinière et adaptative, et plaidoyait avec force en faveur du développement et de la réalisation des programmes d’instruction qui visent spécialement ce dernier type d’expertise. Dans cette contribution nous réfléchissons sur un aspect de l’adaptativité, à savoir l’utilisation adaptative des stratégies de solution dans l’arithmétique de l’école primaire. Dans la première partie de cet article nous donnons quelques réflexions conceptuelles et méthodologiques sur la question d’adaptativité. Plus spécifiquement, nous analysons de façon critique les définitions et les opérationnalisations de l’adaptativité stratégique qui tiennent compte non seulement des caractéristiques de la tâche et de l’individu, mais nous plaidons aussi pour un concept et une approche méthodologique qui impliquent également le contexte socioculturel. La deuxième partie comporte quelques considérations éducatives concernant les questions pourquoi, quand, pour qui, et comment obtient-on l’expertise adaptive dans l’éducation élémentaire de mathématiques.

Strategy development in children with mathematical disabilities: insights from the choice/no-choice method and the chronological-age/ ability-level-match design.

This study investigated the strategy characteristics and development of children with mathematical disabilities (MD) in the domain of simple addition and subtraction, in terms of Lemaire and Sieglers model of strategic change, using the choice/no-choice method and the combined chronological-age (CA)/ability-level (AL)-match design. Four groups of children, matched on either CA or mathematical AL, solved a series of 36 problems with the bridge over 10 in four conditions. In the choice condition, children could choose between retrieval, decomposition to 10, and counting on to solve each problem. In the retrieval, decomposition, and counting conditions, they had to answer the same 36 problems using retrieval, decomposition to 10, and counting on, respectively. The results revealed clear differences in the frequency, efficiency, and adaptiveness with which the CA-matched children applied the available strategies. In contrast, we observed no differences in strategy frequency, efficiency, and adaptiveness between the AL-matched children. These results support the hypothesis that the strategy development of children with MD is marked by a delay rather than a specific deficit. Moreover, this study further documents the value of the methodology used to study childrens strategy use and development in the domain of simple arithmetic.This study investigated the strategy characteristics and development of children with mathematical disabilities (MD) in the domain of simple addition and subtraction, in terms of Lemaire and Sieglers model of strategic change, using the choice/no-choice method and the combined chronological-age (CA)/ability-level (AL)—match design. Four groups of children, matched on either CA or mathematical AL, solved a series of 36 problems with the bridge over 10 in four conditions. In the choice condition, children could choose between retrieval, decomposition to 10, and counting on to solve each problem. In the retrieval, decomposition, and counting conditions, they had to answer the same 36 problems using retrieval, decomposition to 10, and counting on, respectively. The results revealed clear differences in the frequency, efficiency, and adaptiveness with which the CA-matched children applied the available strategies. In contrast, we observed no differences in strategy frequency, efficiency, and adaptiveness between the AL-matched children. These results support the hypothesis that the strategy development of children with MD is marked by a delay rather than a specific deficit. Moreover, this study further documents the value of the methodology used to study childrens strategy use and development in the domain of simple arithmetic.

Simple Addition Strategies in a First-Grade Class With Multiple Strategy Instruction

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 childrens strategies with the 4 parameters of Lemaire and Sieglers (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 results in light of international reform mathematics efforts to promote childrens adaptive expertise.

The Development of Early Numeracy in Europe

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.

Solving Subtraction Problems by Means of Indirect Addition

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.

Strategic competence: Applying Siegler's theoretical and methodological framework to the domain of simple addition

In this study we investigated the variability, frequency, efficiency, and adaptiveness of young children’s strategy use in the domain of simple addition by means of the choice/no-choice method. Seventy-seven beginning second-graders, divided in 3 groups according to general mathematical ability, solved a series of 25 simple additions in 3 different conditions. In the first condition, children could choose whatever strategy they wanted to solve each problem. In the second and third condition, the same children had to solve all problems with one particular strategy, respectively adding up to 10 and retrieval. The results demonstrate that second-graders as a whole choose adaptively between retrieval, decomposition, and counting strategies when solving simple additions, and that they use these strategies neither equally frequently nor equally efficiently. Furthermore, our results indicate that children with different mathematical ability use generally the same strategies to solve these problems, but differ in the frequency, accuracy and adaptiveness with which they apply these strategies. Finally, this study documents the value of the choice/no-choice method to assess the adaptiveness of young children’s strategy use in the domain of early arithmetic.RésuméDans cet article, nous examinons différentes caractéristiques des stratégies cognitives utilisées par de jeunes enfants afin de résoudre des problèmes d’addition initiale. Les stratégies sont caractérisées et étudiées d’apprès le cadre conceptuel et méthodologique proposé par Siegler (Lemaire & Siegler, 1995; Siegler & Lemaire, 1997). Il s’agit notamment de la variabilité, la fréquence et l’efficacité des stratégies utilisées ainsi que de la faculté d’adaptation des enfants dans la sélection de stratégie. Soixante-dix-sept élèves en deuxième année de l’école primaire ont calculé une série de 25 problèmes d’addition initiale dans 3 conditions différentes. Dans la première condition, les élèves pouvaient utiliser une stratégie à leur choix. Dans la deuxième et la troisième condition, les mêmes enfants étaient obligés à résoudre toutes les additions soit par la stratégie de décomposition jusqu’à 10 soit par la stratégie de mémorisation. Les résultats montrent que des élèves de deuxième année primaire se servent de plusieurs stratégies (la stratégie de mémorisation, décomposer, compter) selon le type de problèmes d’addition initiale. De plus, ces stratégies different selon leur fréquence et leur efficacité. Quant à la compétence mathématique des enfants, nous pouvons conclure que celle-ci n’influence pas le répertoire des stratégies utilisées, mais qu ’elle a toutefois un effet substantiel sur la fréquence, l’efficacité et la faculté d’adaptation dans leur choix stratégique. Finalement, notre étude illustre la valeur du cadre méthodologique de Siegler qui permet d’étudier la faculté d’adaptation de jeunes enfants dans leurs choix stratégiques en mathématiques.

Strengths and Weaknesses of the Choice/No-Choice Method in Research on Strategy Use

About 10 years ago, Siegler and Lemaire (1997) introduced the choice/no-choice method as a means of obtaining unbiased estimates of performance characteristics of cognitive strategies. They also illustrated the possibilities of this method and discussed its potential extensions for cognitive (developmental) research. The present article provides a critical discussion of the strengths and weaknesses of the choice/no-choice method in research on strategy choice and strategy change, starting from an overview of the studies that have been done with this new method. We end with a general evaluation and some issues for further research.

Development of Early Numeracy in 5- to 7-Year-Old Children: A Comparison between Flanders and The Netherlands.

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 interpreted in terms of differences in the goals, the content, and the organization of preprimary and primary school mathematics education in both countries.

The relation between metastrategic knowledge, strategy use and task performance: Findings and reflections from a numerosity judgement task

In the research literature several positions concerning the role played by metacognition in adaptive strategy choice can be distinguished. While many authors adhere so-called metacognitive models of strategy choice and strategy change, others have questioned the extent to which metacognitive factors are associated with strategy choice and task performance and have proposed alternative theoretical frameworks wherein strategy choices are described in terms of associative models. In the present article we report data coming from a larger research project on the development of children’s numerosity judgement strategies and skills. The experimental task involved judging numerosities of colored blocks presented in a rectangular grid. Participants were 59 second grade and 50 sixth grade children, whose strategic performance data — obtained by means of a systematic analysis of their response-time patterns — were compared with interview data collected at the end of the experiment. The major result of this comparison is that not only the children from the oldest age group, but also the children from the youngest age group showed clear evidence of metacognitive awareness and understanding of different aspects of their strategic performance.RésuméDans la littérature, il existe plusieurs positionnements sur le rôle joué par la métacognition dans la sélection adaptative des stratégies. Si nombreux auteurs souscrivent aux soi-disants modèles métacognitifs de sélection des stratégies et de changement de stratégie, d’autres scientifiques ont mis en question la mesure dans laquelle des facteurs métacognitifs sont associés à la sélection des stratégies et la performance de la tâche et proposent d’autres cadres de référence théoriques alternatifs qui déterminent la sélection des stratégies comme des modèles associatifs. La présente article discute les données d’une étude plus étendue sur le développement chez enfants des stratégies et les aptitudes pour déterminer des quantités. La tâche de l’expérience consiste en déterminer un nombre de blocs colorés représentés dans une grille carrée. Les participants sont 59 élèves de la deuxième année (de l’école primaire) et 50 élèves de la sixième année, dont les résultats de la performance stratégique — déterminés par l’analyse systématique des séquences dans leurs temps de réaction — ont été comparés aux résultats des interrogations faites à la fin de l’expérience. Le résultat le plus important qui sort de cette comparaison est que aussi bien les enfants du groupe d’âge plus âgé, que ceux du groupe d’âge le plus jeune ont montré une nette conscience métacognitive et compréhension de différentes aspects de leur performance stratégique.

The Acquisition of Preschool Mathematical Abilities: Theoretical, Methodological and Educational Considerations

This is an Accepted Manuscript of an article published by Taylor & Francis Group in Mathematical Thinking and Learning on 7/05/2015, available online: http://www.tandfonline.com/10.1080/10986065.2015.1016810.