School Mathematics (대한수학교육학회지:학교수학)

The Korea Society of Educational Studies in Mathematics (대한수학교육학회)
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Analysis of Transforming Mathematical Representation Shown in the Class of Composite Function Using the CAS

CAS 공학을 사용한 합성함수 수업에서 나타난 수학적 표상 전환 과정에 대한 분석

  • Received : 2015.01.28
  • Accepted : 2015.03.04
  • Published : 2015.03.31
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Abstract

This study examined mathematics class using the CAS(Computer Algebra Systems, CAS) targeted for high school first grade students. We examined what kind of transforming of representations got up according to mathematics subject contents at this classroom. This study analyzed 15 math lessons during one month and the focus of analysis was on the classroom teacher. In particular, for transformations among representations this study mainly investigated from theoretical frameworks such as transparent and opaque representation of Lesh, Behr & Post(1987), descriptive and depictive representation of Kosslyn(1994). According to the results of this study, CAS technology affected the transforming of representations in high school math class and this transforming of representations improved the students' thinking and understanding of mathematical concepts and provided the opportunity to create the representation of individual student. Such results of this study suggest the importance of CAS technology's role in transforming of representations. and they offer the chance to reconsider the fact that CAS technology could be used to improve students' ability of transforming representations at the mathematics class.

본 연구는 일반계 고등학교 1학년 학생들을 대상으로 컴퓨터 대수시스템(Computer Algebra Systems, CAS), 즉 CAS 공학을 사용한 수학수업을 교사의 표상 전환 중심으로 살펴보았다. 이 수업에서 수학교과 내용별로 어떤 표상 전환이 일어났는지를 한 달 진행된 CAS 공학을 사용한 15차시 수업 중 합성함수의 개념 도입부분에 해당하는 두 차시의 수학 수업을 중심으로 분석하였다. 특히 Lesh, Behr, & Post(1987)의 투명 표상과 불투명 표상 사이의 전환과 Kosslyn(1994)의 설명 표상과 묘사 표상 사이의 전환을 중심으로 살펴보았다. 본 연구의 결과 CAS 공학은 일반계 고등학교 수학 수업에서 표상 전환을 도와주었으며 교사 개인의 표상을 만들어내는 기회를 제공하였다. 그러나 표상 전환의 기회가 모두 교수-학습의 목적에 맞게 사용되지는 않았다. 이러한 결과는 수학 수업에서 CAS 공학에 의한 표상 전환의 새로운 역할의 중요성과 교사 역할의 중요성을 재고하는 기회를 제공할 것으로 기대된다.

Keywords

References

  1. 교육인적자원부(2007). 수학과 교육과정[별책8]. 교육인적자원부.
  2. 교육과학기술부(2011). 2009 개정 수학과 교육과정[별책8]. 교육과학기술부.
  3. Arcavi, A. (1994). Symbol sense: Informal sensemaking in formal mathematics. For the Learning of Mathematics, 14(3), 24-35.
  4. Diezmann, C. & English, L. (2001). Promoting the use of diagrams as tools for thinking. In A. Cuoco (Ed.), The roles of representation in school mathematics. 2001 Yearbook of the National Council of Teachers of Mathematics (pp. 77-89). Reston, VA: National Council of Teachers of Mathematics.
  5. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95-123). Boston: Kluwer Academic Publishers.
  6. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103-131. https://doi.org/10.1007/s10649-006-0400-z
  7. Even, R. (1998). Factors involved in linking representations of functions. Journal of Mathematical Behavior, 17, 105-122. https://doi.org/10.1016/S0732-3123(99)80063-7
  8. Friedlander, A. & Tabach, M. (2001). Promoting multiple representations in algebra. In A. Cuoco (Ed.), The roles of representation in school mathematics 2001 Yearbook of the National Council of Teachers of Mathematics (pp. 173-184). Reston, VA: National Council of Teachers of Mathematics.
  9. Goldenberg, E. P. (2003). Algebra and computer algebra. In T. F. James, A. Cuoco, C. Kieran, L. McMullin, & R. M. Zbiek (Eds.), Computer Algebra systems in secondary school mathematics education (pp. 9-30). Reston, VA: National Council of Teachers of Mathematics.
  10. Goldin, G. A. (2002). Representation in mathematical learning and problem solving. In L. English (Ed.), Handbook of international research in mathematics education (pp. 197-218). Mahwah, NJ: Lawrence Erlbaum Associates.
  11. Heid, M. K. (2003). Theories for thinking about the use of CAS in teaching and learning mathematics. In T. F. James, A. Cuoco, C. Kieran, L. McMullin, & R. M. Zbiek (Eds.), Computer Algebra systems in secondary school mathematics education (pp. 33-52). Reston, VA: National Council of Teachers of Mathematics.
  12. Heid, M. K., Thomas, M. O. J., & Zbiek, R. M. (2013). How might computer algebra systems change the role of algebra in the school curriculum. In M. A. Clements, A. J. Bishop, C. Keitel, J. Kilpatrick & K. S. Leung (Eds.), Third international handbook of mathematics education (pp. 597-641). New York: Springer.
  13. Janvier, C. (1987). Representation and understanding: The notion of funtion as an example. In C. Janvier (Ed.), Problem of representation in the teaching and learning mathematics (pp. 67-72). Hillsdale, NJ: Lawrence Erlbaum Associates.
  14. Kaput, J. (1987). Representation systems and mathematics. In C. Janvier (Ed.), Problem of representation in the teaching and learning mathematics (pp. 19-26). Hillsdale, NJ: Lawrence Erlbaum Associates.
  15. Kaput, J. (1991). Notations and representations as mediators of constructive processes. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 53-74). Dordrecht, The Netherlands: Kluwer Academic Publishers.
  16. Kosslyn, S. M. (1994). Image and brain. Cambridge, MA: MIT Press.
  17. Lamon, S. J. (2001). Presenting and representing: From fractions to rational numbers. In A. Cuoco (Ed.), The roles of representation in school mathematics. 2001 Yearbook of the National Council of Teachers of Mathematics (pp. 41-52). Reston, VA: National Council of Teachers of Mathematics.
  18. Lesh, R., Behr, M., & Post, M. (1987). Rational number relations and proportions. In C. Janvier (Ed.), Problems of representation in the teaching and learning mathematics (pp. 41-58). Hillsdale, NJ: Lawrence Erlbaum Associates.
  19. Mandler, J. (1983). Representation. In P. Mussen (Ed.), Handbook of child psychology, Vol. 3, Cognitive development (pp. 420-494). New York: John Wiley & Sons.
  20. McCallum, W. G. (2003). Thinking out of the box. In T. F. James, A. Cuoco, C. Kieran, L. McMullin, & R. M. Zbiek (Eds.), Computer Algebra systems in secondary school mathematics education (pp. 73-86). Reston, VA: National Council of Teachers of Mathematics.
  21. Meira, L. (1998). Making sense of instructional devices: The emergence of transparency in mathematical activity. Journal for Research in Mathematics Education, 29, 121-142. https://doi.org/10.2307/749895
  22. Mesquita, A. L. (1998). On conceptual obstacles linked with external representation in geometry. Journal of Mathematical Behavior, 17(2), 183-195. https://doi.org/10.1016/S0364-0213(99)80058-5
  23. Muis M. A., Stephen J. H., & James J. K. (2008). From static to dynamic mathematics: historical and representational perspectives. Educational Studies in Mathematics, 68, 99-111. https://doi.org/10.1007/s10649-008-9116-6
  24. NCTM. (2000). Principles and standards for school mathematics. Reston, VA: The Author.
  25. Nuria G. & Guida de A. (2009). Social representation as mediators of practice in mathematics classrooms with immigrant students. Educational Studies in Mathematics, 70, 61-76.
  26. Olson, D. & Campbell, R. (1993). Constructing representations. In C. Pratt & A. F. Garton (Eds.), Systems of representation in children (pp. 11-26). New York: John Wiley & Sons.
  27. Paivio, A. (1986). Mental representations: A dual coding approach. New York: Oxford University Press.
  28. Piaget, J. (1962). Play, dreams, and imitation in children. New York: Norton.
  29. Pierce, R., Stacey, K., & Wander, R. (2010). Examining the didactic contract when handheld technology is permitted in mathematics classroom. ZDM-International Journal of Mathematics Education, 42, 683-695. https://doi.org/10.1007/s11858-010-0271-8
  30. Sfard, A. (1991). On the dual nature of mathematical conceptions. Educational Studies in Mathematics, 22, 1-36.
  31. Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification-the case of function. In E. Dubinsky & G. Harel (Eds.), The concept of funtion: Aspects of epistemology and pedagogy (pp. 59-84). Washington DC: MAA.
  32. Skemp, R. (1986). The psychology of learning mathematics. New York: Penguin Books.
  33. Verschaffel, L., Corte, E. D., Jong, T. D., & Elen, J. (2010). Use of representations in reasoning and problem solving. London and New York: Routledge.
  34. Werner, H. & Kaplan, B. (1964). Symbol formation. New York: John Wiley & Sons.
  35. Zazkis, R. & Gadowsky, K. (2001). Attending to transparent features of opaque representations of natural numbers. In A. Cuoco (Ed.), The role of representation in school mathematics. 2001 Yearbook of the National Council of Teachers of Mathematics (pp. 146-165). Reston, VA: National Council of Teachers of Mathematics.
  36. Zbiek, R. M. & Heid, M. K. (2001). Dynamic aspects of funtion representations. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the 12th ICMI on the future of the teaching and learning of algebra (pp. 682-689). Melbourne, Australia: The University of Melbourne.