Wim Van Dooren

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.

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.

Dual Processes in the Psychology of Mathematics Education and Cognitive Psychology

Research in the psychology of mathematics education has been confronted with the fact that people blatantly fail to solve tasks they are supposed to be able to solve correctly given their available domain-specific knowledge and skills. Also researchers in cognitive psychology have encountered such phenomena. In this paper, theories that have been developed in both fields to account for these findings are discussed. After giving a summary of the state of the art in both fields, we argue that bringing together these largely separately developed (sets of) theories creates opportunities for both domains and we suggest a way in which this can be done.

Are secondary school students still hampered by the natural number bias? A reaction time study on fraction comparison tasks

Rational numbers and particularly fractions are difficult for students. It is often claimed that the ‘natural number bias’ underlies erroneous reasoning about rational numbers. This cross-sectional study investigated the natural number bias in first and fifth year secondary school students. Relying on dual process theory assumptions that differentiate between intuitive and analytic processes, we measured accuracies and reaction times on fraction comparison tasks. Half of the items were congruent (i.e., natural number knowledge leads to correct answers), the other half were incongruent (i.e., natural number knowledge leads to incorrect answers). Against expectations, students hardly made errors on incongruent items. Longer reaction times on correctly solved incongruent than on correctly solved congruent items indicated that students were indeed hampered by their prior knowledge about natural numbers, but could suppress their intuitive answers.

From Addition to Multiplication ... and Back: The Development of Students' Additive and Multiplicative Reasoning Skills.

This study builds on two lines of research that have so far developed largely separately: the use of additive methods to solve proportional word problems and the use of proportional methods to solve additive word problems. We investigated the development with age of both kinds of erroneous solution methods. We gave a test containing missing-value problems to 325 third, fourth, fifth, and sixth graders. Half of the problems had an additive structure and half had a proportional structure. Moreover, in half of the problems the internal and external ratios between the given numbers were integer, while in the other cases numbers were chosen so that these ratios were noninteger. The results indicate a development from applying additive methods “anywhere” in the early years of primary school to applying proportional methods “anywhere” in the later years. Between these two stages many students went through an intermediate stage where they simultaneously applied additive methods to proportional problems and proportional methods to additive problems, switching between them based on the numbers given in the problem.

The Impact of Illustrations and Warnings on Solving Mathematical Word Problems Realistically

The present research investigated whether an illustration and/or a warning could help students to (a) build a situational model of the problem situation and (b) solve problematic word problems (P-items) that require realistic thinking more realistically. In 2 similar studies conducted in Turkey and Belgium, the authors presented 10- to 11-year-old children with several P-items. These problems were accompanied with an illustration that depicted the problem situation and/or a warning that alerted that some items may be nonstandard. Contrary to the authors’ expectation, findings from both studies showed that neither the illustration nor the warning, or even the combination of both manipulations, had a positive impact on the number of realistic reactions.

Just Answering … or Thinking? Contrasting Pupils' Solutions and Classifications of Missing-Value Word Problems

Upper primary school children often routinely apply proportional methods to missing-value problems, even when it is inappropriate. We tested whether this tendency could be weakened if children were not required to produce computational answers to such problems. A total of 75 sixth graders were asked to classify 9 word problems of three types (3 for which proportionality is an appropriate model, 3 implying an invariant additive relationship, and 3 for which the result is constant) and to solve a parallel version of this set of problems. Half of the children first found the solution and then did the classification task (SC-condition), while for the others the order was the opposite (CS-condition). On the word problem test, children often overused proportional methods, but those in the CS-condition performed better than those in the SC-condition, suggesting a positive impact of the classification task. On the classification task, most pupils took into account the underlying mathematical models, but they did not always distinguish proportional from non-proportional problems. Students in the SC-condition performed worse than those in the CS-condition, suggesting that solving the word problems first negatively affects later classifications.

Using graphical notations to assess children’s experiencing of simple and complex musical fragments

The aim of this study was to analyze children’s graphical notations as external representations of their experiencing when listening to simple sonic stimuli and complex musical fragments. More specifically, we assessed the impact of four factors on children’s notations: age, musical background, complexity of the fragment, and most salient sonic/musical parameter. One hundred and sixteen children — 8—9-year-olds and 11—12-year-olds with and without extra music education — were exposed to six fragments that differed from one another in terms of complexity and the most salient sonic parameter. Their notations were categorized by means of a classification scheme that differentiated between (a) global notations, which represent the fragments in a holistic way, and (b) differentiated notations, which try to capture one or more sonic/ musical parameters in their temporal unfolding. As expected, we found a significant impact of age and music education, with older children and children with extracurricular music education generating more differentiated notations. Furthermore, complex sounding fragments elicited much fewer differentiated notations than simple ones. We also found significant interaction effects between subject and task variables. Finally, we found a correlation between the sophistication level of children’s representations of the simple and complex fragments.

On the misinterpretation of histograms and box plots

Recent studies have shown that the interpretation of graphs is not always easy for students. In order to reason properly about distributions of data, however, one needs to be able to interpret graphical representations of these distributions correctly. In this study, we used Tversky’s principles for the design of graphs to explain how 125 first-year university students interpreted histograms and box plots. We systematically varied the representation that accompanied the tasks between students to identify how the design principles affected students’ reasoning. Many students displayed misinterpretations of histograms and box plots, despite the fact that they had the required knowledge and time to interpret the representations correctly. We argue that the combination of dual process theories and Tversky’s design principles provides a promising theoretical framework, which leads to various possibilities for future research.