Abstract
In this chapter, we describe and comment on how strategy flexibility or adaptivity has been defined, operationalized, and investigated from different theoretical perspectives on (elementary) mathematics learning and teaching. The resulting working definition is the selection and execution of the most appropriate solution strategy (available in one’s strategy repertoire) on a given mathematical task, and for a given individual, in a given context or situation. Then we report the scarce available empirical research indicating that strategy flexibility is an important and distinctive feature of being good at mathematics or having true mathematical expertise. In the third and final part, we argue that, because strategy flexibility has to be viewed as a disposition (involving also knowledge, beliefs, attitudes, and emotions) rather than a skill, teaching for strategy flexibility cannot be conceived as a process that one can implement until routine expertise in the use of the strategies has been obtained, but should be the goal from the start of the teaching and learning process and in an integrative way.
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Notes
- 1.
Although we are aware that some authors define the terms “adaptivity” and “flexibility” differently (Verschaffel et al., 2009), in this chapter they will be used as synonyms.
- 2.
For example, Dowker’s (1992) analysis of the role of flexibility in mathematicians’ estimations, Cortés (2003) study of flexibility in high school mathematics teachers’, engineers’ and scientists’ solutions of (systems of) algebraic equations, or Carry, Lewis, and Bernard’s (1980) analysis of the flexibility in expert solvers of algebraic equations.
References
Acevedo Nistal, A., Clarebout, G., Elen, J., Van Dooren, W., & Verschaffel, L. (2009). Conceptualising, investigating and stimulating representational flexibility in mathematical problem solving and learning: A critical review. ZDM-The International Journal on Mathematics Education, 41, 627–636.
Baroody, A. J., & Dowker, A. (Eds.). (2003). The development of arithmetic concepts and skills. Constructing adaptive expertise. Mahwah, NJ: Erlbaum.
Bisanz, J. (2003). Arithmetical development: Commentary on chapters 1 through 8 and reflections on directions. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. 435–452). Mahwah, NJ: Erlbaum.
Blöte, A. W., Van der Burg, E., & Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: Instruction effects. Journal of Educational Psychology, 93, 627–638.
Bransford, J. (2001). Thoughts on adaptive expertise. Unpublished manuscript. Retrieved from http://www.vanth.org/docs/AdaptiveExpertise.pdf
Carr, M., & Jessup, D. L. (1997). Gender differences in first-grade mathematics strategy use: Social and metacognitive influences. Journal of Educational Psychology, 89, 318–328.
Carry, L. R., Lewis, C., & Bernard, J. E. (1980). Psychology of equation solving: An information processing study (Final Technical Report). Austin, TX: The University of Texas at Austin.
Cortés, A. (2003, July). A cognitive model of experts’ algebraic solving methods. Proceedings of the International Group for the Psychology of Mathematics Education, USA, 2, 253–260.
Coulson, R. L., Feltovich, P. J., & Spiro, R. J. (1989). Foundations of a misunderstanding of the ultrastructural basis of myocardial failure. A reciprocation network of oversimplifications. The Journal of Medicine and Philosophy, 14, 109–146.
De Corte, E., Mason, L., Depaepe, F., & Verschaffel, L. (2011). Self-regulation of mathematical knowledge and skills. In B. Zimmerman & D. Schunk (Eds.), Handbook of self-regulation of learning and performance (pp. 155–172). New York: Routledge.
Dowker, A. (1992). Computational estimation strategies of professional mathematicians. Journal for Research in Mathematics Education, 23, 45–55.
Duncker, K. (1945). On problem solving. Psychological Monographs, 58(5, Whole no 270).
Ellis, S. (1997). Strategy choice in sociocultural context. Developmental Review, 17, 490–524.
Ericsson, A. K., Charness, N., Feltovich, P., & Hoffman, R. R. (2006). Cambridge handbook on expertise and expert performance. Cambridge, UK: Cambridge University Press.
Feltovich, P. J., Spiro, R. J., & Coulson, R. L. (1997). Issues of expert flexibility in contexts characterized by complexity and change. In P. J. Feltovich, K. M. Ford, & R. R. Hoffman (Eds.), Expertise in context: Human and machine (pp. 125–146). Menlo Park, CA: AAAI Press.
Geary, D. C. (2003). Arithmetical development: Commentary on chapters 9 through 15 and future directions. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 453–464). Mahwah, NJ: Erlbaum.
Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6, 105–128.
Greeno, J. G., Collins, A. M., & Resnick, L. B. (1996). Cognition and learning. In D. C. Berliner & R. C. Calfee (Eds.), Handbook of educational psychology (pp. 15–46). New York: Macmillan.
Greer, B. (1997). Modelling reality in mathematics classrooms: The case of word problems. Learning and Instruction, 7, 293–307.
Hacker, D. J. (1998). Definitions and empirical foundations. In D. J. Hacker, J. Dunlosky, & A. Graesser (Eds.), Metacognition in educational theory and practice (pp. 1–24). Mahwah, NJ: Erlbaum.
Hatano, G. (1982). Cognitive consequences of practice in culture specific procedural skills. The Quarterly Newsletter of the Laboratory of Comparative Human Cognition, 4, 15–18.
Hatano, G. (2003). Foreword. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. xi–xiii). Mahwah, NJ: Erlbaum.
Hatano, G., & Inagaki, K. (1992). Desituating cognition through the construction of conceptual knowledge. In P. Light & G. Butterworth (Eds.), Context and cognition (pp. 115–133). Hemel Hempstead, UK: Harvester Wheatsheaf.
Hatano, G., & Oura, Y. (2003). Reconceptualizing school learning using insight from expertise research. Educational Researcher, 32, 26–29.
Hecht, S. A. (2002). Counting on working memory in simple arithmetic when counting is used for problem solving. Memory and Cognition, 30, 447–455.
Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up. Helping children learn mathematics. Washington, DC: National Academy Press.
Koninklijke Nederlandse Akademie van Wetenschappen. (2009). Rekenonderwijs op de basisschool: Analyse en sleutels tot verbetering [Elementary mathematics education: Analysis and keys for improvement]. Amsterdam: KNAW.
Lemaire, P., & Reder, L. (1999). What effects strategy selection in arithmetic. The example of parity and five effects on product verification. Memory and Cognition, 27, 364–382.
Luwel, K., Foustana, A., Papadatos, Y., & Verschaffel, L. (2011). The role of intelligence and feedback in children’s strategy competence. Journal of Experimental Child Psychology, 108, 61–76.
Luwel, K., Lemaire, P., & Verschaffel, L. (2005). Children’s strategies in numerosity judgment. Cognitive Development, 20, 448–471.
Luwel, K., Torbeyns, J., Schillemans, V., Onghena, P., & Verschaffel, L. (2009). Strengths and weaknesses of the choice/no-choice method in research on strategy use. European Psychologist, 14, 351–362.
Luwel, K., Torbeyns, J., & Verschaffel, L. (2003). The relation between metastrategic knowledge, strategy use and task performance: Findings and reflections from a numerosity judgement task. European Journal of Psychology of Education, 18, 425–447.
Luwel, K., Verschaffel, L., Onghena, P., & De Corte, E. (2003). Flexibility in strategy use: Adaptation of numerosity judgement strategies to task characteristics. European Journal of Cognitive Psychology, 15, 247–266.
Milo, B., & Ruijssenaars, A. J. J. M. (2002). Strategiegebruik van leerlingen in het speciaal basisonderwijs: begeleiden of sturen? [Strategy instruction in special education: Guided or direct instruction?]. Pedagogische Studiën, 79, 117–129.
Muis, K. R. (2007). The role of epistemic beliefs in self-regulated learning. Educational Psychologist, 42, 173–190.
Muis, K. R. (2008). Epistemic profiles and self-regulated learning: Examining relations in the context of mathematics problem solving. Contemporary Educational Psychology, 33, 177–208.
National Council of Teachers of Mathematics. (1989). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Retrieved from http://standards.nctm.org/document/index.htm
National Mathematics Advisory Panel. (2008). Foundations for Success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.
Ontwikkelingsdoelen en eindtermen. Informatiemap voor de onderwijspraktijk. Gewoon basisonderwijs. (1998). [Standards. Documentation for practitioners. Elementary education.]. Brussel, Belgium: Ministerie van de Vlaamse Gemeenschap, Departement Onderwijs, Afdeling Informatie en Documentatie.
Payne, J. W., Bettman, J. R., & Johnson, J. R. (1993). The adaptive decision maker. Cambridge, UK: Cambridge University Press.
Perkins, D. (1995). Outsmarting IQ: The emerging science of learnable intelligence. New York: The Free Press.
Rittle-Johnson, B., Star, J. R., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology, 101(4), 836–852.
Schillemans, V., Luwel, K., Onghena, P., & Verschaffel, L. (in press). The influence of presentation order on individuals’ strategy choice. Studia Psychologica.
Selter, C. (1998). Building on children’s mathematics. A teaching experiment in grade three. Educational Studies in Mathematics, 36, 1–27.
Siegler, R. S. (1996). Emerging minds: The process of change in children’s thinking. Oxford, UK: Oxford University Press.
Siegler, R. S. (1998). Children’s thinking. Upper Saddle River, NJ: Prentice Hall.
Siegler, R. S. (2000). The rebirth of children’s learning. Child Development, 71, 26–35.
Siegler, R. S., & Araya, R. (2005). A computational model of conscious and unconscious strategy discovery. In R. V. Kail (Ed.), Advances in child development and behavior (Vol. 33, pp. 1–42). Oxford, UK: Elsevier.
Siegler, R. S., & Lemaire, P. (1997). Older and younger adults’ strategy choices in multiplication: Testing predictions of ASCM using the choice/no-choice method. Journal of Experimental Psychology. General, 126, 71–92.
Spiro, R. J. (1980). Constructive processes in prose comprehension and recall. In R. J. Spiro, B. C. Bruce, & W. F. Brewer (Eds.), Theoretical issues in reading comprehension (pp. 245–278). Hillsdale, NJ: Erlbaum.
Star, J. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404–411.
Star, J. R., & Newton, K. (2009). The nature and development of experts’ strategy flexibility for solving equations. ZDM-The International Journal on Mathematics Education, 41, 557–567.
Star, J. R., Rittle-Johnson, B., Lynch, K., & Perova, N. (2009). The role of prior knowledge in the development of strategy flexibility: The case of computational estimation. ZDM-The International Journal on Mathematics Education, 41, 569–579.
Star, J. R., & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31, 280–300.
Straker, A. (1999). The national numeracy project: 1996–99. In I. Thompson (Ed.), Issues in teaching numeracy in primary schools (pp. 39–48). Buckingham, UK: Open University Press.
Sweller, J., van Merrienboer, J. J. G., & Paas, F. G. W. C. (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10, 251–296.
Thompson, I. (1999). Getting your head around mental calculation. In I. Thompson (Ed.), Issues in teaching numeracy in primary schools (pp. 145–156). Buckingham, UK: Open University Press.
Threlfall, J. (2002). Flexible mental calculation. Educational Studies in Mathematics, 50, 29–47.
Torbeyns, J., Verschaffel, L., & Ghesquière, P. (2005). Simple addition strategies in a first-grade class with multiple strategy instruction. Cognition and Instruction, 23, 1–21.
Treffers, A., De Moor, E., & Feijs, E. (1990). Proeve van een national programma voor het reken/wiskundeonderwijs op de basisschool. Deel 1. Overzicht einddoelen [Towards a national curriculum for mathematics education in the elementary school. part 1. Overview of the goals]. Tilburg, the Netherlands: Zwijsen.
Van der Heijden, M. K. (1993). Consistentie van aanpakgedrag [Consistency in solution behavior]. Lisse, the Netherlands: Swets & Zeitlinger.
Van Dooren, W., Verschaffel, L., & Onghena, P. (2002). The impact of preservice teachers’ content knowledge on their evaluation of students’ strategies for solving arithmetic and algebra word problems. Journal for Research in Mathematics Education, 33, 319–351.
Verschaffel, L., De Corte, E., Lamote, C., & Dhert, N. (1998). Development of a flexible strategy for the estimation of numerosity. European Journal of Psychology of Education, 13, 347–270.
Verschaffel, L., Greer, B., & De Corte, E. (2007). Whole number concepts and operations. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 557–628). Greenwich, CT: Information Age Publishing.
Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24, 335–359.
Wittmann, E. Ch., & Müller, G. N. (1990–1992). Handbuch Produktiver Rechnenübungen: Vols 1 & 2 [Handbook of productive arithmetic exercises: Vols. 1 & 2]. Düsseldorf und Stuttgart, Germany: Klett Verlag.
Wittmann, E. Ch., & Müller, G. N. (2004). Das Zahlenbuch [The book of numbers. Mathematics]. Düsseldorf und Stuttgart, Germany: Klett Verlag.
Acknowledgments
This study was funded by Grant GOA 2006/01 “Developing adaptive expertise in mathematics education” from the Research Fund Katholieke Universiteit Leuven, Belgium.
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Verschaffel, L., Luwel, K., Torbeyns, J., Van Dooren, W. (2011). Analyzing and Developing Strategy Flexibility in Mathematics Education. In: Elen, J., Stahl, E., Bromme, R., Clarebout, G. (eds) Links Between Beliefs and Cognitive Flexibility. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1793-0_10
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