Research in Mathematical Education (한국수학교육학회지시리즈D:수학교육연구)

Korean Society of Mathematical Education (한국수학교육학회)
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What Distinguishes Mathematical Experience from Other Kinds of Experience?

  • Received : 2015.09.30
  • Accepted : 2016.03.23
  • Published : 2016.03.31
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Abstract

Investigating students' lived mathematical experiences presents dual challenges for the researcher. On the one hand, we must respect that students' experiences are not directly accessible to us and are likely very different from our own experiences. On the other hand, we might not want to rely upon the students' own characterizations of what constitutes mathematics because these characterizations could be limited to the formal products students learn in school. I suggest a characterization of mathematics as objectified action, which would lead the researcher to focus on students' operations-mental actions organized as objects within structures so that they can be acted upon. Teachers' and researchers' models of these operations and structures can be used as a launching point for understanding students' experiences of mathematics. Teaching experiments and clinical interviews provide a means for the teacher-researcher to infer students' available operations and structures on the basis of their physical activity (including verbalizations) and to begin harmonizing with their mathematical experience.

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References

  1. Burton, D. M. (2007). The history of mathematics: An introduction (6th ed.). New York, NY: McGraw Hill. ME 1988g.04424 for the 1st edition (1998)
  2. Confrey, J. & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. J. Res. Math. Educ. 26(1), 66-86. ME 1995f.03828 https://doi.org/10.2307/749228
  3. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In: D. Tall (Ed.), Advanced Mathematical Thinking (pp. 95-123). Dordrecht, Netherlands: Kluwer Academic Publishers. ME 1993b.02231
  4. Dubinsky, E.; Dautermann, J.; Leron, U. & Zazkis, R. (1994). On learning fundamental concepts of group theory. Educ. Stud. Math. 27(3), 267-305. ME 1995d.02441 https://doi.org/10.1007/BF01273732
  5. Hackenberg, A. J. (2005). Construction of algebraic reasoning and mathematical caring relations. Unpublished doctoral dissertation, University of Georgia.
  6. Hackenberg, A. J. (2010). Mathematical caring relations in action. J. Res. Math. Educ. 41(3), 236-273. ME 2011f.00174
  7. Leron, U.; Hazzan, O. & Zazkis, R. (1995). Learning group isomorphism: A crossroads of many concepts. Edu. Stud. Math. 29(2), 153-174. ME 1998b.01189 https://doi.org/10.1007/BF01274211
  8. Norretranders, T. (1998). The user illusion: Cutting consciousness down to size (J. Sydenham, Trans.). New York: Viking Penguin. [Original work published 1991.]
  9. Norton, A. (2015). What Distinguishes Mathematical Experience from Other Kinds of Experience? In: O. N. Kwon, Y. H. Choe, H. K. Ko, and S. Han (Eds.), The International Perspective on Curriculum and Evaluation of Mathematics: the Proceedings of the KSME 2015 International Conference on Mathematics Education held at Seoul National University, Seoul 08826, Korea; November 6-8, 2015, Vol. 3 (pp. 266-273). Seoul: Korean Society of Mathematical Education.
  10. Norton, A. & Boyce, S. (2013). A cognitive core for common state standards. J. Math. Behav. 32(2), 266-279. ME 2013c.00398 https://doi.org/10.1016/j.jmathb.2013.01.001
  11. Piaget, J. (1970). Structuralism (C. Maschler, Trans.). New York: Basic Books.
  12. Piaget, J. & Inhelder, B. (1967). The child's conception of space (Trans. F. J. Langdon & J. L. Lunzer). New York: Norton (Original work published in 1948).
  13. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on process and objects on different sides of the same coin. Educ. Stud. Math. 22(1), 1-36. ME 1992a.03697
  14. Sfard, A. & Linchevski, L. (1994). The gains and the pitfalls of reification-the case of algebra. Educ. Stud. Math. 26(2-3), 191-228. ME 1995d.02425 https://doi.org/10.1007/BF01273663
  15. Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences 4(3), 259-309. https://doi.org/10.1016/1041-6080(92)90005-Y
  16. Steffe, L. P. & Olive, J. (Eds.) (2010). Children's fractional knowledge. New York: Springer. ME2010c.00118
  17. Steffe, L. P. & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In: A. E. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267-306). Mahwah, NJ: Erlbaum. ME 2000e.03223
  18. Tall, D.; Thomas, M.; Davis, G.; Gray, E. & Simpson, A. (1999). What is the object of an encapsulation of a process? J. Math. Behav. 18(2), 223-241. ME 2002c.01850 https://doi.org/10.1016/S0732-3123(99)00029-2
  19. Ulrich, C. L. (2012). Additive relationships and signed quantities (Doctoral dissertation). Retrieved from: http://www.libs.uga.edu/etd/
  20. von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. London: Routledge Falmer. ME 1995b.00710
  21. Wilkins, J. L. M. & Norton, A. (2011). The splitting loope. J. Res. Math. Educ. 42(4), 386-416. ME 2011e.00238 https://doi.org/10.5951/jresematheduc.42.4.0386