Advertisement

Educational Studies in Mathematics

, Volume 93, Issue 2, pp 137–153 | Cite as

Categorizing and promoting reversibility of mathematical concepts

  • Martin A. Simon
  • Melike Kara
  • Nicora Placa
  • Hakan Sandir
Article

Abstract

Reversibility of concepts, a key aspect of mathematical development, is often problematic for learners. In this theoretical paper, we present a typology we have developed for categorizing the different reverse concepts that can be related to a particular initial concept and explicate the relationship among these different reverse concepts. We discuss uses of the typology and how reversibility can be understood as the result of reflective abstraction. Finally, we describe two strategies for promoting reversibility that have distinct uses in terms of the types of reverse concepts they engender. We share empirical results which led to our postulation of these strategies and discuss their theoretical basis.

Keywords

Reversibility Mathematical concepts Reflective abstraction Task design 

Notes

Acknowledgments

This work was supported by the National Science Foundation under Grant No. DRL-1020154. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References

  1. Adi, H. (1978). Intellectual development and reversibility of thought in equation solving. Journal for Research in Mathematics Education, 9, 204–213.CrossRefGoogle Scholar
  2. Biddlecomb, B., & Olive, J. (2000). JavaBars [Computer software]. Retrieved from http://math.coe.uga.edu/olive/welcome.html
  3. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.Google Scholar
  4. Carpenter, T. P., Fennema, E., Peterson, P. L., & Carey, D. A. (1988). Teachers’ pedagogical content knowledge of students’ problem solving in elementary arithmetic. Journal for Research in Mathematics Education, 19, 385–401.CrossRefGoogle Scholar
  5. Carpenter, T. P., & Moser, J. M. (1983). The acquisition of addition and subtraction concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 7–44). New York: Academic.Google Scholar
  6. Dreyfus, T., & Eisenberg, T. (1996). On different facets of mathematical thinking. In R. J. Sternberg & T. Ben-Zeev (Eds.), The nature of mathematical thinking (pp. 253–284). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  7. Greer, B. (2011). Inversion in mathematical thinking and learning. Educational Studies in Mathematics, 79, 429–438.CrossRefGoogle Scholar
  8. Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28, 383–432.CrossRefGoogle Scholar
  9. Hackenberg, A. J., & Tillema, E. S. (2009). Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. The Journal of Mathematical Behavior, 28, 1–18.CrossRefGoogle Scholar
  10. Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence (A. Parsons & S. Milgram, Trans., 5th ed.). New York: Basic Books.Google Scholar
  11. Krutetskii, V. A. (1976). The psychology of mathematical abilities in school children (J. Teller, Trans., J. Kilpatrick & I. Wirszup, Eds.). Chicago: University of Chicago Press.Google Scholar
  12. Lamon, S. J. (2007). Rational numbers and proportional reasoning: Towards a theoretical framework for research. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 629–667). Charlotte, NC: Information Age Publishing.Google Scholar
  13. Montangero, J., & Maurice-Naville, D. (1997). Piaget or the advance of knowledge. New York: Psychology Press.Google Scholar
  14. Norton, A., & Wilkins, J. L. M. (2009). A quantitative analysis of children’s splitting operations and fraction schemes. The Journal of Mathematical Behavior, 28, 150–161.CrossRefGoogle Scholar
  15. Norton, A., & Wilkins, J. L. M. (2012). The splitting group. Journal for Research in Mathematics Education, 43, 557–583.CrossRefGoogle Scholar
  16. Olive, J., & Steffe, L. P. (2002). The construction of an iterative fractional scheme: The case of Joe. The Journal of Mathematical Behavior, 20, 413–437.CrossRefGoogle Scholar
  17. Piaget, J. (1950). La réversibilité de la pensée et les opérations logiques. Bulletin de la Société Française de Philosophie, 44, 137–164.Google Scholar
  18. Piaget, J. (2001). Studies in reflecting abstraction (R. L. Campbell, Trans.). Sussex: Psychology Press.Google Scholar
  19. Piaget, J., Apostel, L., & Mandelbrot, B. B. (1957). Logique et équilibre. Presses Universitaire de France.Google Scholar
  20. Ramful, A., & Olive, J. (2008). Reversibility of thought: An instance in multiplicative tasks. The Journal of Mathematical Behavior, 27, 138–151.CrossRefGoogle Scholar
  21. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114–145.CrossRefGoogle Scholar
  22. Simon, M. A. (2013). Issues in theorizing mathematics learning and teaching: A contrast between Learning Through Activity and DNR research programs. The Journal of Mathematical Behavior, 32, 281–294.CrossRefGoogle Scholar
  23. Simon, M. A. (2015). Learning Through Activity: Analyzing and promoting mathematics conceptual learning. In K. Beswick, T. Muir, & J. Wells (Eds.), Proceedings of the 39th conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 51–65). Hobart, Australia: PME.Google Scholar
  24. Simon, M. A., Placa, N., & Avitzur, A. (2016). Two stages of mathematical concept learning: Further empirical and theoretical development. Journal for Research in Mathematics Education, 47, 63–93.CrossRefGoogle Scholar
  25. Simon, M. A., Saldanha, L., McClintock, E., Karagoz Akar, G., Watanabe, T., & Ozgur Zembat, I. (2010). A developing approach to studying students’ learning through their mathematical activity. Cognition and Instruction, 28, 70–112.CrossRefGoogle Scholar
  26. Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3–39). Albany: State University of New York Press.Google Scholar
  27. Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge. The Journal of Mathematical Behavior, 102, 1–41.Google Scholar
  28. Steffe, L. P., & Olive, J. (Eds.). (2010). Children’s fractional knowledge. New York: Springer.Google Scholar
  29. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. Kelly (Eds.), Research design in mathematics and science education (pp. 267–307). Hillsdale, NJ: Erlbaum.Google Scholar
  30. Tunc-Pekkan, Z. (2015). An analysis of elementary school children’s fractional knowledge depicted with circle, rectangle, and number line representations. Educational Studies in Mathematics, 89, 419–441.CrossRefGoogle Scholar
  31. Tzur, R. (2004). Teachers’ and students’ joint production of a reversible fraction concept. The Journal of Mathematical Behavior, 23, 93–114.CrossRefGoogle Scholar
  32. Von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Washington, DC: Falmer.CrossRefGoogle Scholar
  33. Weber, E., & Thompson, P. W. (2014). Students’ images of two-variable functions and their graphs. Educational Studies in Mathematics, 87, 67–85.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Martin A. Simon
    • 1
  • Melike Kara
    • 1
  • Nicora Placa
    • 1
  • Hakan Sandir
    • 2
  1. 1.New York UniversityNew YorkUSA
  2. 2.Gazi ÜniversitesiMerkezTurkey

Personalised recommendations