# The nature and development of experts’ strategy flexibility for solving equations

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## 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.

## Keywords

Mathematics Instruction Prospective Teacher Conceptual Knowledge Efficient Strategy Strategy Choice## Notes

### 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.

## References

- Baroody, A. J., & Dowker, A. (Eds.). (2003).
*The development of arithmetic concepts and skills: Constructing adaptive expertise*. Mahwah, NJ: Lawrence Erlbaum.Google Scholar - 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.CrossRefGoogle Scholar - 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).Google Scholar - 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.Google Scholar - 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.Google Scholar - Dowker, A. (1992). Computational estimation strategies of professional mathematicians.
*Journal for Research in Mathematics Education,**23*(1), 45–55. doi: 10.2307/749163.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Krutetskii, V. A. (1976).
*The psychology of mathematical abilities in school children (J*. Chicago: University of Chicago Press. Teller, Trans.).Google Scholar - 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.CrossRefGoogle Scholar - Lewis, C. (1981). Skill in algebra. In J. R. Anderson (Ed.),
*Cognitive skills and their acquisition*(pp. 85–110). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - National Mathematics Advisory Panel. (2008).
*Foundations for success: The final report of the National Mathematics Advisory Panel*. Washington, DC: US Department of Education.Google Scholar - National Research Council. (2001).
*Adding it up: Helping children learn mathematics*. Washington, DC: National Academy Press.Google Scholar - 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.CrossRefGoogle Scholar - Newton, K. J., & Star, J. R. (2009). Exploring the development of flexibility in struggling algebra students. (Unpublished manuscript).Google Scholar
- 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.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - Star, J. R. (2005). Reconceptualizing procedural knowledge.
*Journal for Research in Mathematics Education,**36*(5), 404–411.CrossRefGoogle Scholar - 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.Google Scholar - 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.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar - Wertheimer, M. (1959).
*Productive thinking (Enlarged ed.)*. New York: Harper & Brothers.Google Scholar - Yakes, C., & Star, J. R. (2009). Using comparison to develop teachers’ flexibility in algebra. (Unpublished manuscript).Google Scholar