Abstract
This study evaluated students’ representational choices while they solved linear function problems. Eighty-six secondary-school students solved problems under one choice condition, where they chose a table, a formula, or both to solve each problem, and two no-choice conditions, where one of these representations was forced upon them. Two conceptualisations of representational flexibility were used: a groupwise conceptualisation, where group-based measures of students’ fluency with the different representations were used to determine which representational choices could be considered flexible, and an individualised conceptualisation, where each individual student’s fluency with each of the representations to solve each problem type was taken into account when determining which choices could be considered flexible for that particular student. A strong correlation between groupwise flexibility and choice condition accuracy, and an even stronger correlation between individualised flexibility and choice condition accuracy were found. The implications for research and instruction are discussed.
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Notes
The choice/no-choice method was borrowed from the literature on strategy choice. This method was initially used to analyse students’ strategy choices in multiplication tasks (Siegler and Lemaire 1997), and later on in other tasks from various cognitive domains (for an overview see Luwel et al. 2009). Given the strong link that exists between strategies and representations (Borba and Confrey 1996; Tabachneck et al. 1994a), the method has recently been applied to the field of mathematical problem solving with external representations (Acevedo Nistal et al. 2010).
For purposes of readability, statistical details are only included in text for the main effects and interaction effects. For the numerous contrast analyses (e.g., pairwise comparisons between grades, problem types, etc.), we only report whether differences are statistically significant (p < .05) or not, without mentioning each time χ² scores, degrees of freedom and exact p values.
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Acknowledgments
This research was partially supported by Grant GOA 2006/01 “Developing adaptive expertise in mathematics education” from the Research Fund K.U. Leuven, Belgium, and the Grant G063709N “Representational adaptivity in mathematical thinking and learning: analysis and improvement” from the Fund for Scientific Research Flanders.
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Acevedo Nistal, A., Van Dooren, W. & Verschaffel, L. What counts as a flexible representational choice? An evaluation of students’ representational choices to solve linear function problems. Instr Sci 40, 999–1019 (2012). https://doi.org/10.1007/s11251-011-9199-9
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DOI: https://doi.org/10.1007/s11251-011-9199-9