Lieven Verschaffel

The predictive value of numerical magnitude comparison for individual differences in mathematics achievement.

Although it has been proposed that the ability to compare numerical magnitudes is related to mathematics achievement, it is not clear whether this ability predicts individual differences in later mathematics achievement. The current study addressed this question in typically developing children by means of a longitudinal design that examined the relationship between a number comparison task assessed at the start of formal schooling (mean age=6 years 4 months) and a general mathematics achievement test administered 1 year later. Our findings provide longitudinal evidence that the size of the individuals distance effect, calculated on the basis of reaction times, was predictively related to mathematics achievement. Regression analyses showed that this association was independent of age, intellectual ability, and speed of number identification.

Working memory and individual differences in mathematics achievement: A longitudinal study from first grade to second grade

This longitudinal study examined the relationship between working memory and individual differences in mathematics. Working memory measures, comprising the phonological loop, the visuospatial sketchpad, and the central executive, were administered at the start of first grade. Mathematics achievement was assessed 4 months later (at the middle of first grade) and 1 year later (at the start of second grade). Working memory was significantly related to mathematics achievement in both grades, showing that working memory clearly predicts later mathematics achievement. The central executive was a unique predictor of both first- and second-grade mathematics achievement. There were age-related differences with regard to the contribution of the slave systems to mathematics performance; the visuospatial sketchpad was a unique predictor of first-grade, but not second-grade, mathematics achievement, whereas the phonological loop emerged as a unique predictor of second-grade, but not first-grade, mathematics achievement.

The Effect of Semantic Structure on First Graders' Strategies for Solving Addition and Subtraction Word Problems.

In a longitudinal investigation, data were collected on the problem representations and solution strategies of 30 first graders who were given a series of simple addition and subtraction word problems (Verschaffel, 1984). The children were interviewed three times during the school year, and data obtained on their solution strategies and on the influence of problem structure on the strategies. The results complement those of recent related research, especially the work of Carpenter and Moser (1982, 1984). More precisely, the influence of problem structure on childrens solution strategies appears even more extensive and decisive than that described by previous researchers.

Symbolizing, modeling and tool use in mathematics education

Introduction and overview K. Gravemeijer, et al. Preamble: from models to modelling K. Gravemeijer. Section I: Emergent Modeling. Introduction to Section I: Informal representations and their improvements B.van Oers. The mathematization of young childerns language B.van Oers. Symbolizing space into being R. Lehrer, C. Pritchard. Mathematical representations as systems of notations-in-use L. Meira. Students criteria for representational adequacy A. diSessa. Transitions in emergent modeling N. Presmeg. Section II: The Role of Models, Symbols and Tools in Instructional Design. Introduction to Section II: the role of models, symbols and tools in instructional design K. Gravemeijer. Emergent models as an instructional design heuristic K. Gravemeijer, M. Stephan. Modeling, symbolizing, and tool use in statistical data analysis P. Cobb. Didactic objects and didactic models in radical constructivism P.W. Thompson. Taking into account different views: three brief comments on papers by Gravemeijer and Stephan, Cobb and Thompson C. Selter. Section III: Models, Situated Practices, and Generalization. Introduction to Section II: models, situated practices, and generalization L. Verschaffel. On guessing the essential thing R. Nemirovsky. Everyday knowledge and mathematical modeling of school word problems L. Verschaffel, et al. On the development of human representational competence from an evolutionary point of view: from episodic to virtual culture J. Kaput, D. Shaffer. Modeling reasoning D. Carraher, A. Schliemann. Index.

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.

Not Everything Is Proportional: Effects of Age and Problem Type on Propensities for Overgeneralization

Previous research (e.g., De Bock, 2002) has shown that-due to the extensive attention paid to proportional reasoning in elementary and secondary mathematics education-many students tend to overrely on proportional methods in diverse domains of mathematics (e.g., geometry, probability). We investigated the development of misapplication of proportional reasoning with the age and the educational experience of students. A paper-and-pencil test consisting of several types of proportional and nonproportional arithmetic problems with a missing-value structure was given to 1,062 students from Grades 2 to 8. As expected, students tended to apply proportional methods in cases in which they were clearly not applicable. Although some errors of overapplication were made in the 2nd grade, their number increased considerably up to Grade 5 in parallel with the growing proportional reasoning capacity of the students. From Grade 6 on, students started to distinguish more often between situations when proportionality was applicable and when it was not, but even in 8th grade, a considerable number of proportional errors were made. The likelihood of error varied with the type of nonproportional mathematical model underlying the word problems.

Teaching Realistic Mathematical Modeling in the Elementary School: A Teaching Experiment with Fifth Graders.

Recent research has convincingly documented elementary school childrens tendency to neglect real-world knowledge and realistic considerations during mathematical modeling of word problems in school arithmetic. The present article describes the design and the results of an exploratory teaching experiment carried out to test the hypothesis that it is feasible to develop in pupils a disposition toward (more) realistic mathematical modeling. This goal is achieved by immersing them in a classroom culture in which word problems are conceived as exercises in mathematical modeling, with a focus on the assumptions and the appropriateness of the model underlying any proposed solution. The learning and transfer effects of an experimental class of 10and 11-year-old pupils--compared to the results in two control classes-provide support for the hypothesis that it is possible to develop in elementary school pupils a disposition toward (more) realistic mathematical modeling.

Number and Arithmetic

During the last decade several major shifts have occurred in the conceptualisation of mathematics as a domain, of mathematical competence as a goal for instruction, and of the way in which this competence should be acquired through schooling. This chapter begins with a summary of the general characteristics and principles underlying the ongoing world-wide reform of mathematics education. Afterwards it documents and illustrates how these general characteristics and principles permeate a major domain of the mathematics curricula for the elementary school, called ‘Number and Arithmetic’. Five related topics within this domain are discussed, namely: number concepts and number sense, the meaning of arithmetic operations, mastery of basic arithmetic facts, mental and written computation, and word problems as applications of the numerical and arithmetical knowledge and skills.

The CLIA-model: A framework for designing powerful learning environments for thinking and problem solving

A major challenge for education and educational research is to build on our present understanding of learning for designing environments for education that are conducive to fostering in students self-regulatory and cooperative learning skills, transferable knowledge, and a disposition toward competent thinking and problem solving. Taking into account inquiry-based knowledge on learning and recent instructional research, this article presents the CLIA-model (Competence, Learning, Intervention, Assessment) as a framework for the design of learning environments aimed to be powerful in eliciting in students learning processes that facilitate the acquisition of productive knowledge and competent learning and thinking skills. Next, two intervention studies are described that embody major components of this framework, one focussing on mathematical problem solving in primary school, and a second one relating to self-regulatory skills in university freshmen. Both studies were carried out in parallel with the development of the framework, and were instrumental in identifying and specifying the different components of the model. They yielded both promising initial support for the model by showing that CLIA-based learning environments are indeed powerful in facilitating in students the acquisition of high-literacy learning results, especially the acquisition and transfer of self-regulation skills for learning and problem solving.RésuméUn défì important pour l’éducation et la recherche pédagogique est de développer en partant de notre compréhension actuelle de l’apprentissage des environnements éducationnels susceptibles de promouvoir chez les étudiants des habiletés d’apprentissage autorégulatrices et collaboratives, des connaissances transférables, et une disposition orientée vers le raisonnement et la résolution de problèmes compétents. Tenant compte des résultats de la recherche sur l’apprentissage et l’enseignement, cet article présente le modèle “CLIA (Competence, Learning, Intervention, Assessment)” comme cadre de référence pour concevoir des environnements d’apprentissage visés à stimuler chez les étudiants des processus d’apprentissage qui facilitent l’acquisition de connaissances productives et des habiletés compétentes d’apprentissage et de raisonnement. Ensuite, deux recherches d’intervention sont présentées qui représentent les composantes majeures du modèle CLIA: une expérience porte sur la résolution de problèmes mathématiques dans l’enseignement primaire, et la seconde a pour objet les habiletés autorégulatrices chez des étudiants en première année de l’université. Ces recherches ont été réalisées en parallèle avec le développement du modèle, et ont été instrumentales pour l’identification et la spécification des différentes composantes du modèle. Les deux investigations ont apporté du support initial pour le modéle en montrant que des environnements d’apprentissage basés sur le cadre de référence CLIA sont en effet stimulants pour faciliter chez les étudiants l’acquisition de résultats d’apprentissage d’ordre supérieur, spécialement l’acquisition et le transfert d’habiletés d’autorégulation de l’apprentissage et de la résolution de problèmes.

The Illusion of Linearity: Expanding the Evidence towards Probabilistic Reasoning.

Previous research has shown that – due to the extensive attention spent to proportional reasoning in mathematics education – many students have a strong tendency to apply linear or proportional models anywhere, even in situations where they are not applicable. This phenomenon is sometimes referred to as the ‘illusion of linearity’. For example, in geometry it is known that many students believe that if the sides of a figure are doubled, the area is doubled too. In this article, the empirical evidence for this phenomenon is expanded to the domain of probabilistic reasoning. First, we elaborate on the notion of chance and provide some reasons for expecting the over generalization of linear models in the domain of probability too. Afterwards, a number of well-known and less-known probabilistic misconceptions are described and analysed, showing that they have one remarkable characteristic in common: they can be interpreted in terms oft he improper application of linear relations. Finally, we report on an empirical investigation aimed at identifying the ability of 10th and12th grade students to compare the probabilities of two binomial chance situations. It appears that before instruction in probability, students have a good capability of comparing two events qualitatively, but at the same time they incorrectly quantify this qualitative insight as if the variables in the problem were linked by a linear relationship. Remarkably, these errors persist after instruction in probability. The potential of this study for improving the teaching and learning of probability, as well as suggestions for further research, are discussed.