Abstract
Largely absent from the emerging literature on flexibility is a consideration of experts’ flexibility. Do experts exhibit strategy flexibility, as one might assume? If so, how do experts perceive that this capacity developed in themselves? Do experts feel that flexibility is an important instructional outcome in school mathematics? In this paper, we describe results from several interviews with experts to explore strategy flexibility for solving equations. We conducted interviews with eight content experts, where we asked a number of questions about flexibility and also engaged the experts in problem solving. Our analysis indicates that the experts that were interviewed did exhibit strategy flexibility in the domain of linear equation solving, but they did not consistently select the most efficient method for solving a given equation. However, regardless of whether these experts used the best method on a given problem, they nevertheless showed an awareness of and an appreciation of efficient and elegant problem solutions. The experts that we spoke to were capable of making subtle judgments about the most appropriate strategy for a given problem, based on factors including mental and rapid testing of strategies, the problem solver’s goals (e.g., efficiency, error-free execution, elegance) and familiarity with a given problem type. Implications for future research on flexibility and on mathematics instruction are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baroody, A. J., & Dowker, A. (Eds.). (2003). The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah, NJ: Lawrence 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. doi:10.1037/0022-0663.93.3.627.
Carry, L. R., Lewis, C., & Bernard, J. E. (1979). Psychology of equation solving: An information processing study. Austin, TX: The University of Texas (Final Technical Report).
Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5, 121–152. doi:10.1207/s15516709cog0502_2.
Cortés, A. (2003). A cognitive model of experts' algebraic solving methods. In Proceedings of the international group for the psychology of mathematics education (Vol. 2, pp. 253–260). USA.
Dowker, A. (1992). Computational estimation strategies of professional mathematicians. Journal for Research in Mathematics Education, 23(1), 45–55. doi:10.2307/749163.
Ericsson, K. A., & Charness, N. (1994). Expert performance: Its structure and acquisition. The American Psychologist, 49(8), 725–747. doi:10.1037/0003-066X.49.8.725.
Ericsson, K. A., Krampe, R., & Tesch-Romer, C. (1993). The role of deliberate practice in the acquisition of expert performance. Psychological Review, 100, 363–406. doi:10.1037/0033-295X.100.3.363.
Krutetskii, V. A. (1976). The psychology of mathematical abilities in school children (J. Chicago: University of Chicago Press. Teller, Trans.).
Larkin, J., McDermott, J., Simon, D. P., & Simon, H. A. (1980). Expert and novice performance in solving physics problems. Science, 208(4450), 1335–1342. doi:10.1126/science.208.4450.1335.
Lewis, C. (1981). Skill in algebra. In J. R. Anderson (Ed.), Cognitive skills and their acquisition (pp. 85–110). Hillsdale, NJ: Lawrence Erlbaum Associates.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: US Department of Education.
National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
Newton, K. J. (2008). An extensive analysis of elementary preservice teachers’ knowledge of fractions. American Educational Research Journal, 45(4), 1080–1110. doi:10.3102/0002831208320851.
Newton, K. J., & Star, J. R. (2009). Exploring the development of flexibility in struggling algebra students. (Unpublished manuscript).
Nistal, A. A., Van Dooren, W., Clarebout, G., Elen, J., & Verschaffel, L. (2009). Conceptualising, investigating, and stimulating representational flexibility in mathematical problem solving and learning: A critical review. ZDM-International Journal on Mathematics Education. doi:10.1007/s11858-009-0189-1.
Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574. doi:10.1037/0022-0663.99.3.561.
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(1), 71–92. doi:10.1037/0096-3445.126.1.71.
Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411.
Star, J. R., & Madnani, J. (2004). Which way is best? Students’ conceptions of optimal strategies for solving equations. In D. McDougall & J. Ross (Eds.), Proceedings of the 26th annual meeting of the North American chapter of the international group for the psychology of mathematics education (pp. 483–489). Toronto: University of Toronto.
Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18, 565–579. doi:10.1016/j.learninstruc.2007.09.018.
Star, J. R., & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31, 280–300. doi:10.1016/j.cedpsych.2005.08.001.
Van Dooren, W., Verschaffel, L., & Onghena, P. (2002). The impact of pre-service teachers’ content knowledge on their appreciation of pupils’ strategies for solving arithmetic and algebra word problems. Journal for Research in Mathematics Education, 33(5), 319–351. doi:10.2307/4149957.
Van Dooren, W., Verschaffel, L., & Onghena, P. (2003). Pre-service teachers’ preferred strategies for solving arithmetic and algebra word problems. Journal of Mathematics Teacher Education, 6(1), 27–52. doi:10.1023/A:1022109006658.
Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2007). Developing adaptive expertise: A feasible and valuable goal for (elementary) mathematics education? Ciencias Psicologicas, 2007(1), 27–35.
Wertheimer, M. (1959). Productive thinking (Enlarged ed.). New York: Harper & Brothers.
Yakes, C., & Star, J. R. (2009). Using comparison to develop teachers’ flexibility in algebra. (Unpublished manuscript).
Acknowledgments
Thanks to Martina Kenyon, Jennifer Rabb, and Nira Gautam for their help in coding and analyzing the data reported here. This project was supported by a grant from Temple University to the second author.
Author information
Authors and Affiliations
Corresponding author
Additional information
Paper submitted to the special issue of ZDM: International Journal of Mathematics Education.
Appendix: Algebra assessment
Appendix: Algebra assessment
Rights and permissions
About this article
Cite this article
Star, J.R., Newton, K.J. The nature and development of experts’ strategy flexibility for solving equations. ZDM Mathematics Education 41, 557–567 (2009). https://doi.org/10.1007/s11858-009-0185-5
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11858-009-0185-5