Advertisement

ZDM

, Volume 41, Issue 5, pp 663–679 | Cite as

Flexible use of symbolic tools for problem solving, generalization, and explanation

  • Lisa B. Warner
  • Roberta Y. Schorr
  • Gary E. Davis
Original Article

Abstract

We provide evidence that student representations can serve different purposes in the context of classroom problem solving. A strategy used expressly to solve a problem might be represented in one way, and in another way when the problem is generalized or extended, and yet in another way when the solution strategy is explained to peers or a teacher. We discuss the apparent long-term memory implications this has regarding the preferences that students have for their original versus later developed representations, and how these preferences relate to the use of representational flexibility in classroom settings.

Keywords

Flexibility Representation Flexible thinking Problem solving 

Notes

Acknowledgments

The authors would like to thank the editor and three anonymous reviewers for a number of invaluable suggestions.

References

  1. Beishuizen, M., van Putten, C. M., & van Mulken, F. (1997). Mental arithmetic and strategy use with indirect number problems up to one hundred. Learning and Instruction, 7(1), 87–106. doi: 10.1016/S0959-4752(96)00012-6.CrossRefGoogle Scholar
  2. Carey, D. A. (1991). Number sentences: Linking addition and subtraction word problems and symbols. Journal for Research in Mathematics Education, 22(4), 266–280. doi: 10.2307/749272.CrossRefGoogle Scholar
  3. Cohen, N. J. (1984). Preserved learning capacity in amnesia: Evidence for multiple memory systems. In N. Butters & L. R. Squire (Eds.), The neuropsychology of memory (pp. 83–103). New York: Guildford Press.Google Scholar
  4. Eichenbaum, H. (2002). The cognitive neuroscience of memory. Oxford: Oxford University Press.CrossRefGoogle Scholar
  5. Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–141.Google Scholar
  6. Hatano, G. (2003). Foreword. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. xi–xiii). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  7. Heirdsfield, A. M., & Cooper, T. J. (2002). Flexibility and inflexibility in accurate mental addition and subtraction: Two case studies. The Journal of Mathematical Behavior, 21, 57–74. doi: 10.1016/S0732-3123(02)00103-7.CrossRefGoogle Scholar
  8. Karmiloff-Smith, A. (1994). Precis of beyond modularity: A developmental perspective on cognitive science. The Behavioral and Brain Sciences, 17(4), 693–745.CrossRefGoogle Scholar
  9. Karmiloff-Smith, A. (1995). Beyond modularity: A developmental perspective on cognitive science. Cambridge, MA: The MIT Press.Google Scholar
  10. Klein, T., & Beishuizen, M. (1994). Assessment of flexibility in mental arithmetic. In J. E. H. van Luit (Ed.), Research on learning and instruction of mathematics in kindergarten and primary school. Doetinchem: Graviant Publishing Company.Google Scholar
  11. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  12. Ormrod, J. E. (2004). Human learning (4th ed. ed.). Upper Saddle River, NJ: Pearson.Google Scholar
  13. Perkins, D. N., & Salomon, G. (1992). Transfer of learning. International encyclopedia of education (2nd ed.). Oxford: Pergamon Press.Google Scholar
  14. Schorr, R. Y., Warner, L. B., Geahart, D., & Samuels, M. (2007). Teacher development in a large urban district: The impact on students. In R. Lesh, J. Kaput, & E. Hamilton (Eds.), Real-world models and modeling as a foundation for future mathematics education (pp. 431–447). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  15. Shore, B. M., Pelletier, S., & Kaizer, C. (1990). Metacognition, giftedness, and mathematical thinking. Budapest: European Council for High Ability.Google Scholar
  16. Squire, L. R., & Kandel, E. R. (2008). Memory: From mind to molecules. Greenwood Village, CO: Roberts & Co.Google Scholar
  17. Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18, 565–579.CrossRefGoogle Scholar
  18. Stigler, J. W., & Hiebert, J. (1999). The teaching gap—Best ideas from the world’s teachers for improving education in the classroom. New York: The Free Press.Google Scholar
  19. Thorndike, E. L., & Woodworth, R. S. (1901). The influence of improvement in one mental function upon the efficiency of other functions. Psychological Review, 8, 247–261. doi: 10.1037/h0074898.CrossRefGoogle Scholar
  20. Threlfall, J. (2002). Flexible mental calculation. Educational Studies in Mathematics, 50, 29–47. doi: 10.1023/A:1020572803437.CrossRefGoogle Scholar
  21. Tulving, E. (1985). Elements of episodic memory (Oxford Psychology Series). New York: Oxford University Press.Google Scholar
  22. Tulving, E., & Craik, F. I. M. (2000). The Oxford handbook of memory. New York: Oxford University Press.Google Scholar
  23. Vakali, M. (1984) Children’s thinking in arithmetic word problem solving. Journal of Experimental Education, 53(2), 106–113.Google Scholar
  24. 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(3), 335–359.CrossRefGoogle Scholar
  25. Warner, L. B., Davis, G. E., Alcock, L. J., & Coppolo, J. (2002). Flexible mathematical thinking and multiple representations in middle school mathematics. Mediterranean Journal for Research in Mathematics Education, 1(2), 37–61.Google Scholar

Copyright information

© FIZ Karlsruhe 2009

Authors and Affiliations

  • Lisa B. Warner
    • 1
  • Roberta Y. Schorr
    • 1
  • Gary E. Davis
    • 2
  1. 1.Rutgers UniversityNewarkUSA
  2. 2.University of Massachusetts DartmouthDartmouthUSA

Personalised recommendations