Koen Luwel

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.

Does switching between strategies within the same task involve a cost

In two experiments, participants had to switch regularly between two cognitive strategies of a different complexity in the context of a numerosity judgement task. Expt 1 comprised bivalent stimuli (i.e. allowing the application of the two strategies), whereas Expt 2 involved univalent stimuli (i.e. allowing the application of only one strategy). Both experiments revealed that switching between strategies entailed a cognitive cost that was reflected in longer response times on switch compared to non-switch trials but not in reduced accuracy. The size of this switch cost did not differ as a function of strategy complexity but tended to diminish as a strategy became more appropriate for solving a particular problem. We discuss the extent to which current theories of task switching can account for these findings.

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.

Using segmented linear regression models with unknown change points to analyze strategy shifts in cognitive tasks

Some years ago, Beem (1993, 1995) described a program for fitting two regression lines with an unknown change point (Segcurve). He suggested that such models are useful for the analysis of a variety of phenomena and gave an example of an application to the study of strategy shifts in a mental rotation task. This technique has also proven to be very fruitful for investigating strategy use and strategy shifts in other cognitive tasks. Recently, Beem (1999) developed SegcurvN, which fitsn regression lines with (n - 1) unknown change points. In the present article we present this new technique and demonstrate the usefulness of a three-phase segmented linear regression model for the identification of strategies and strategy shifts in cognitive tasks by applying it to data from a numerosity judgment experiment. The advantages and shortcomings of this technique are evaluated.

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 role of intelligence and feedback in children's strategy competence.

A test-intervention-test study was conducted investigating the role of intelligence on four parameters of strategy competence in the context of a numerosity judgment task. Moreover, the effectiveness of two feedback types on these four parameters was tested. In the two test sessions, the choice/no-choice method was used to assess the strategy repertoire, frequency, efficiency, and adaptivity of a group of low-, average-, and high-intelligence children. During the intervention, half of the participants from each intelligence group were given outcome feedback (OFB), whereas the other half received strategy feedback (SFB). The pretest data showed large differences among the three intelligence groups on all four strategy parameters. These differences had disappeared at the posttest due to a particularly strong improvement on all strategy parameters in the low-intelligence group. Furthermore, it was found that SFB was more beneficial than OFB for all parameters involving strategy selection. These results indicate that intelligence plays an important role in childrens strategy use and suggest that strategy feedback can be a powerful instructional tool, especially for low-intelligence children.

Comparing apples and pears in studies on magnitude estimations

The present article is concerned with studies on magnitude estimations that strived to uncover the underlying mental representation(s) of magnitudes. We point out a number of methodological differences and shortcomings that make it difficult drawing general conclusions. To solve this problem, we propose a taxonomy by which those studies could be classified, taking into account central methodological aspects of magnitude estimation tasks. Finally, we suggest perspectives for future research on magnitude estimations, which might abandon the hunt for the mathematical model that explains estimations best and turn, instead, to investigate the underlying principles of estimations (e.g., strategies) and ways of their improvement.

Strategic aspects of children’s numerosity judgement

A previous study (Luwel, Verschaffel, Onghena, & De Corte, 2000) revealed that 2nd and 6th graders use at least 3 different kinds of strategies for determining different numerosities of blocks presented in square grids: (a) an addition strategy by means of which (groups of) blocks are counted (and added), (b) a subtraction strategy in which the number of empty squares is subtracted from the total number of blocks in the grid (i.e. the anchor), and (c) an estimation strategy, whereby the number of blocks is determined in a quick but imprecise way. Although 6th as well as 2nd graders used the clever subtraction strategy, the majority of 2nd graders had serious trouble with the correct determination of the anchor, resulting in huge numerosity judgement errors. Since this finding seriously complicated the testing of a number of hypotheses in that study, we replicated the above study but presented the children information about the grid size. This manipulation led to a substantial increase in the proportion of appropriate subtraction strategy users and made it possible to investigate the effect of several subject and task variables on the frequency, accuracy, and adaptiveness with which the different strategies were applied. Results are discussed in terms of the conceptual framework of Lemaire and Siegler (1995) regarding strategic change.RésuméUne étude antérieure (Luwel, Verschaffel, Onghena, & De Corte, 2000) a démontré que des élèves en deuxième et en sixième année (de l’école primaire) se servent d’au moins 3 différents types de stratégies pour déterminer différentes quantités de blocs représentés dans une grille carrée: (a) une stratégie d’ addition par laquelle des (groupes de) blocs sont comptés (et additionnées), (b) une stratégie de soustraction par laquelle on soustrait le nombre de carrés vides de la totalité des blocs représentés dans la grille (i.e., l’ancre) et (c) une stratégie d’estimation dans laquelle on détermine assez rapidement le nombre de blocs, mais de façon imprécise. Quoique tant les élèves de la deuxième année que ceux de la sixième utilisent la stratégie de soustraction de manière intelligente, la majorité des élèves de la deuxième année éprouvent toutefois de sérieuses difficultés dans la détermination correcte de l’ancre, ce qui aboutit à un taux très élevé d’erreurs dans la determination des quantités.Comme ces résultats ont considérablement compliqué le contrôle d’un certain nombre d’hypothèses dans ladite étude, nous avons repris cette expérience tout en fournissant aux élèves des informations supplémentaires sur la taille de la grille.Cette manipulation a abouti à une hausse substantielle de la proportion des élèves qui optaient pour la stratégie de soustraction, qui était la plus appropriée. Elle a également permis l’examen de l’effet de plusieurs variables (liées au sujet et à la tâche) sur la fréquence, la précision et la faculté d’adaptation par laquelle les différentes stratégies ont été mises en œuvre. Les résultats sont discutés suivant le cadre conceptuel de Lemaire et Siegler (1995) concernant le changement de stratégie.

Estimation of 'real' numerosities in elementary school children

Groups of mathematically strong and weak second-, fourth- and sixth-graders were individually confronted with numerosities smaller and larger than 100 embedded in one-, two- or three-dimensional realistic contexts. While one third of these contexts were totally unstructured (e.g., an irregular piece of land jumbled up with 72 cars), another third had a clear structure (e.g., a 16x4 rectangular parking lot completely filled with 64 cars), and a last third had a “semi-structure” (e.g., the same 16x4 parking lot but with a number of cars missing). Besides analyzing the effects of different task and subject variables on pupils’ accuracy and response-time data, the study involved and analysis of their estimation strategies, with an emphasis on multiplicative strategies that profited by some of the tasks’ geometrical (semi-)structure. It was found that many children actually made use of such strategies, that using these strategies did however not always led to accurate estimations, and that their frequency and efficiency increased with age.RésuméDes groupes d’élèves forts et d’élèves faibles en mathématiques issus de classes de deuxième, quatrième et sixième année ont été individuellement confrontés à des quantités numériques supérieures et inférieures à 100, présentées dans des contextes réalistes uni-, bi- ou tridimensionnels. Si un tiers de ces contextes était tout à fait non structuré (p.e. un territoire irrégulier couvert de 72 voitures), un tiers des contextes était nettement structurés (p.e. un parking rectangulaire de 16x4 rempli entièrement par 64 voitures) et finalement un tiers des contextes semi-structurés (p.e. le même parking 16x4 où certaines voitures font défaut). L’étude fait non seulement l’analyse des effets des différentes tâches et des variables des sujets sur l’efficacité et le temps de réponse, elle présente également une analyse de leurs stratégies d’estimation, avec un accent sur les stratégies multiplicatives qui bénéficient de la structure géométrique (semi)-structurée. L’étude a montré que beaucoup d’enfants se servaient de telles stratégies et que l’utilisation de ces stratégies n’aboutit pourtant pas toujours à des estimations exactes et que la fréquence et l’efficacité évoluaient en fonction de l’âge.

Benchmark-based strategies in whole number line estimation

In this study, we used verbal protocols to identify whether adults spontaneously apply quartile-based strategies or whether they need additional external support to use these strategies when solving a 0-1,000 number line estimation (NLE) task. Participants were assigned to one of three conditions based on the number of external benchmarks provided on the number line. In the bounded condition only the origin and endpoint were indicated, the mid-point condition included an additional external benchmark at 50%, and in the quartile condition three additional external benchmarks at 25%, 50%, and 75% were specified. Firstly, participants in the bounded condition reported to spontaneously apply quartile-based strategies to calibrate their estimates. Moreover, participants frequently relied on the external benchmarks for creating internal benchmarks at the mid-point, quartiles, and even octiles of the number line. Secondly, overall estimation accuracy improved as the number of external benchmarks increased, and target numbers close to external benchmarks were estimated more accurately and with less variability. Thirdly, the use of a larger variety in benchmark-based strategies was positively related to NLE accuracy. In summary, this study provides evidence that the NLE task induces more sophisticated strategy use in participants than initially anticipated.