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Recent advances in the research on strategy selection and execution in mathematics problem solving.

Saturday, October 01st,   2011 [14:20 - 16:00]

SY_18. Recent advances in the research on strategy selection and execution in mathematics problem solving

Imbo, I. 1 & Luwel, K. 2, 3

1 Department of Experimental Psychology, Ghent University, Belgium
2 Centre of Educational Research and Development, Hogeschool-Universiteit Brussel, Belgium
3 Centre for Instructional Psychology and Technology, Katholieke Universiteit Leuven, Belgium

A growing body of research has shown that people use multiple strategies to solve a wide range of cognitive tasks, such as arithmetic, reading, decision making, currency conversion, et cetera. The present symposium brings together five contributions that focus on the selection and execution of cognitive strategies in mathematics problem solving. All contributions report on recent advances regarding strategic behavior in different mathematical tasks ranging from simple and complex addition and subtraction over computational estimation to determining numerosities. Two contributions focus on alternative methodologies for identifying individuals' strategies in addition to the classical verbal trial-by-trial verbal reports. The first talk presents the 'operand recognition paradigm' that allows identifying arithmetic strategies based on the degradation of memory traces. The second talk provides neurophysiological evidence indicating that different strategies activate different brain areas. This study revealed a strong association between the verbal strategy reports and the brain areas that were activated. A third contribution tackles a completely new issue in the study of strategy choices, namely within-item strategy switching. Within-item strategy switching refers to the extent to which individuals switch strategies after they have judged that a particular strategy might not be the most appropriate one for solving a specific problem. Two other contributions investigate a number of variables that might shed a better light on the way individuals select and execute cognitive strategies. One talk focuses on the role of culture, response language, and working memory, whereas the other one looks at intelligence-related differences in strategy selection and execution. Finally, the last talk also spends attention to a possible improvement in children's strategy selection and execution by contrasting two types of feedback, namely outcome feedback vs. strategy feedback.



SY_18.1 - The operand recognition paradigm as a method to investigate individuals arithmetic strategies

Thevenot, C.

Department of Psychology, University of Geneva, Geneva, Switzerland

The rationale of the operand recognition paradigm (ORP) is to infer the arithmetic strategies used by individuals from the time they take to recognize the operands of a problem after they have solved it (Thevenot, Castel, Fanget, & Fayol, 2010). This rationale is motivated by the fact that algorithmic procedures degrade the memory traces of the operands, whereas they remain intact after a solution process by retrieval of the result from long-term memory (Thevenot, Barrouillet, & Fayol, 2001). First, we will describe the main results obtained with the ORP on mental addition and subtraction. We will show that, whereas individuals with low and high abilities in arithmetic use the same strategies for small and large problems (i.e., retrieval and calculation procedures respectively), they differ in the way they solve medium problems (e.g., 8 + 5 or 13 - 8). While higher-skilled individuals can retrieve the result of such problems from long-term-memory, lower-skilled individuals have to resort to reconstructive strategies. Second, we will present an alternative interpretation of our effects, which has been formulated by Metcalfe and Campbell (2010; in press). The authors suggest that the differences we obtain in recognition times between small and large problems could be due to difficulty-related switch costs rather than strategy use. We will explain why this alternative interpretation is a very bad candidate in order to account for our results.

SY_18.2 - Neurophysiological validation of verbal strategy reports in mental arithmetic

Grabner, R. H. 1 & De Smet, B. 2

1 Institute for Behavioral Sciences, Swiss Federal Institute of Technology, Zurich, Switzerland
2 Department of Educational Sciences, Katholieke Universiteit Leuven, Belgium

There is a long tradition of behavioral research showing that arithmetic problems are solved with various strategies including the retrieval of the correct solution from memory (fact retrieval) and the application of procedural strategies (e.g., transformation or counting). The applied strategies can be inferred from the size of the presented problems (e.g., small problems are frequently solved by fact retrieval) or can be directly assessed by means of trial-by-trial verbal strategy self-reports. However, the validity of these verbal strategy self-reports has been repeatedly questioned, and most investigations on the neural correlates of these strategies rely on the problem size rather than the strategy report approach. In the presented electroencephalography (EEG) study, we investigated the validity of both approaches by evaluating their association with the brain responses during mental arithmetic. To this end, 19 adults had to solve small and large addition and subtraction problems and to indicate the applied strategy (fact retrieval vs. procedure use) after each problem by means of verbal strategy reports. Analysis of event-related (de-)synchronization (ERD/ERS), a measure of task-related changes in oscillatory EEG activity, revealed a general convergence of verbal strategy reports and the problem size approach. Fact retrieval was accompanied by higher left-hemispheric theta ERS, while procedural strategies were reflected in higher widespread ERD in the lower alpha band and bilateral parietooccipital ERD in the upper alpha band. A direct comparison of the EEG data from both approaches showed a stronger association of the ERS/ERD data with the verbal strategy self-reports than with problem size. These findings suggest a higher sensitivity of verbal strategy self-reports to arithmetic strategies and thus provide the first neurophysiological evidence for the validity of verbal strategy self-reports.

SY_18.3 - Within-item strategy switching in arithmetic in children, young, and older adults

Ardiale, E. 1, 2 & Lemaire, P. 1, 2

1 Laboratoire de Psychologie Cognitive, Universites d'Aix-Marseille, Marseille, France
2 Centre National de la Recherche Scientifique, France

We tested whether participants switched strategies while they are solving problems, and age-related changes in such within-item strategy switching. Children, young, and older adults performed a computational estimation task (e.g., provide estimates of 58 x 72, with either a rounding-down strategy, 58 x 72 = 50 x 70 = 3500, or a rounding-up strategy, 58 x 72 = 60 x 80= 4800). Participants had the possibility to switch strategies if they judge the current strategy not the most appropriate while solving each problem. Children (third, fifth, and seventh-graders), young and older adults were tested. The main findings showed that (a) all participants switched strategy within-item and chose the best strategy more often than chance, (b) when participants switched strategies they gave more accurate responses, (c) all participants tended to switch strategy most often when problem and strategy characteristics guided such strategy switching. Moreover, age-related differences were found regarding (a) mean proportions of strategy switches, (b) skills at switching strategy when most appropriate, and (c) relative costs/benefits associated with strategies. Implications of these findings concern not only formal models of strategy selection (e.g., they speak to interruption mechanism during strategy selection and strategy execution uniquely assumed by a computational model such as SCADS*). They also concern age-related differences regarding cognitive flexibility underlying strategic behaviours.

SY_18.4 - How do culture, computational skill and response language influence strategic math behavior?

Imbo, I. 1 & LeFevre, J. 2

1 Department of Experimental Psychology, Ghent University, Belgium
2 Centre for Applied Cognitive Research, Carleton University, Canada

In a recent study, Imbo & LeFevre (2009) tested the effects of working-memory load on complex addition solving (e.g., 58 + 76) in three different cultures (Belgians, Canadians, and Chinese). The Chinese participants were faster than the Belgians, who were faster and more accurate than the Canadians. The Chinese also required fewer working-memory resources than did the Belgians and Canadians. However, the Chinese chose less adaptively from the available strategies than did the Belgians and Canadians. In this talk, we present a follow-up study, in which we tested the effects of culture, response language, and computational skill on strategy efficiency and strategy adaptivity. Three groups of participants were tested on a computational estimation task (e.g., 42 x 57 = ?) in no-load and load conditions: 40 Belgian-educated adults who answered in their first language (Dutch), 40 Chinese-educated adults who answered in their first language (Chinese), and 40 Chinese-educated adults who answered in their second language (English). The Belgians, who had weaker computational skills than the Chinese, were less efficient in solving the estimation problems than the Chinese answering in their first language. The Chinese who answered in English were slower than the Chinese who answered in Chinese, and this difference was larger for difficult strategies and under working memory load. Strategy adaptivity was lower in the Chinese groups than in the Belgian group. That is, the Chinese were less likely to choose the strategy that produced the best estimate, and especially so when their working memory was loaded. The efficiency results are interpreted in terms of the encoding complex model, whereas the explanation for the adaptivity results is based on cultural differences in educational history.

SY_18.5 - Effects of intelligence and feedback on children's strategy adaptivity

Luwel, K. 1, 2 , Foustana , A. 3 , Papadatos, Y. 3 & Verschaffel, L. 2

1 Centre of Educational Research and Development, Hogeschool-Universiteit Brussel, Belgium
2 Centre for Instructional Psychology and Technology, Katholieke Universiteit Leuven, Belgium
3 Department of Special Education and Psychology, University of Athens, Greece

A test-intervention-test design was used to investigate the effect of intelligence and feedback on the four parameters of children's strategic competence (i.e., repertoire, frequency, efficiency and adaptivity). Children of three intelligence groups (i.e., low, medium and high intelligent children) had to determine different numerosities of coloured blocks that were presented in a square grid. They could solve this task by either using an addition strategy by which the coloured blocks in the grid were added, or an insightful subtraction strategy by which the number of empty squares was subtracted from the total number of squares in the grid. In Test Session 1 (TS1), we assessed children's initial strategic competence. In the intervention session (IS), children were provided with feedback on each trial. Half of the children from each group received outcome feedback (OFB), which informed them about the accuracy of their answer, whereas the other half received strategy feedback (SFB), informing them about the appropriateness of the selected strategy. In Test Session 2 (TS2), we tested for possible effects of the provision of feedback on each of the four parameters of strategic competence. The three intelligence groups exhibited large differences on all strategy parameters in TS1. In TS2, there was a strong decrease in these group differences for all parameters due to the provision of feedback. Furthermore, it was found that SFB was somewhat more beneficial than OFB, especially for the parameters involving strategy selection. We can conclude that high intelligent children show a greater strategy flexibility than less intelligent children but that the provision of feedback can lead to a dramatic improvement in the (flexible) use of strategies in less intelligent children.

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