한국수학교육학회지시리즈E:수학교육논문집 (Communications of Mathematical Education)

  • 제31권1호
  • /
  • Pages.71-84
  • /
  • 2017
  • /
  • 1226-6663(pISSN)
  • /
  • 2287-9935(eISSN)
한국수학교육학회 (Korean Society of Mathematical Education)
권호 표지
DOI QR코드

DOI QR Code

테일러급수의 이해에 대한 연구

A study on understanding of Taylor series

  • Oh, Hye-Young (Department of Mathematics Education, Incheon National University)
  • 투고 : 2016.10.19
  • 심사 : 2017.01.26
  • 발행 : 2017.02.15
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

테일러급수는 대학 전공 수학의 여러 개념을 포함하는 복잡한 구조를 가지고 있다. 이 주제는 미적분학, 해석학, 복소해석학 등의 수학뿐만 아니라 물리학, 공학 등 다른 학문에서도 유용성과 응용성을 가진 강력한 도구이다. 그러나 학생들은 이 주제의 수학적 구조를 제대로 이해하는데 어려움을 느낀다. 이에 본 연구에서는 어떻게 학생들이 테일러급수 수렴을 이해하는지를 알기 위해서 학생들의 수학적 특징을 세 유형으로 분류한다. 그 후에 테일러급수 수렴의 구조적 상(image)을 이용해서 테일러급수 수렴에 대한 이해도를 분석하고 이에 대한 결과를 제시하고자 한다.

Taylor series has a complicated structure comprising of various concepts in college major mathematics. This subject is a strong tool which has usefulness and applications not only in calculus, analysis, and complex analysis but also in physics, engineering etc., and other study. However, students have difficulties in understanding mathematical structure of Taylor series convergence correctly. In this study, after classifying students' mathematical characteristic into three categories, we use structural image of Taylor series convergence which associated with mathematical structure and operation acted on that structure. Thus, we try to analyze the understanding of Taylor series convergence and present the results of this study.

키워드

참고문헌

  1. 고효상.김두웅.김준혁.김철환.김슬기.김응상 (2013). 테일러급수를 이용한 전기 자동차 충전 대수 예측. 대한전기학회 전력기술부분회 추계학술대회 논문집. (Go, H. S., Kim, D. U., Kim, J. H., Kim, C. H., Kim, S. K., Kim, E. S.(2013). The prediction of electric vehicle charging load using Taylor series. Journal of electrical engineering and technology.)
  2. 김진환 (2014). 테일러급수 수렴에 대한 예비중등교사의 이해실태와 GeoGebra를 활용한 교수방안 탐색. 대한수학교육학회지 <학교수학>, 16(2), pp.317-334. (Kim, J. H.(2014). Exploring teaching way using GeoGebra based on pre-service secondary teachers' understanding-realities for Tayor series convergence conceptions. Journal of Korea Society Educational Studies in Mathematics , 16(2), 317-334.)
  3. 임경택.조태호.백종검.김시유 (2001). CMOS 그라운드 연결망에서 발생하는 최대 동시 스위칭 잡음의 테일러급수 모형의 분석. 대한전자공학회 하계종합학술대회 논문집, 24(1) (Im, K. T., Cho, T. H., Baek, J. H., Kim, S. Y.(2001). Taylor's series model analysis of maximum simultaneous switching noise for ground interconnection networks in CMOS systems. Journal of the institute of electronics and information engineers, 24(1).)
  4. 허민.오혜영역. Howard Eves (1995). 수학의 위대한 순간들. 서울: 경문사. (Howard Eves , Her, M., Oh, H. Y.(1995). Great Moments in Mathematics. Seoul: Kyungmoon Co.)
  5. Alcock, L.,& Simpson, A. (2004). Convergence of sequences and series: Interactions between visual reasoning and the learner's beliefs about their own role. Educational Studies in Mathematics, 57(1), 1-32. https://doi.org/10.1023/B:EDUC.0000047051.07646.92
  6. Alcock, L.,& Simpson, A. (2002a). Two components in learning to reason using definitions'. Proceedings of the 2nd International Conference on the Teaching of Mathematics, Hersonisoss, Greece, www.math.uoc.gr/-ictm2/
  7. Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-year students. International Journal of Mathematical Education in Science and Technology, 32, 487-500. https://doi.org/10.1080/00207390010022590
  8. Carlson, M.,& Bloom, I. (2005). The cyclic nature of problem solving: An emergent multidimensional problem-solving framework. Educational Studies in Mathematics, 58, 45-75. https://doi.org/10.1007/s10649-005-0808-x
  9. Kidron, I., & Zehavi, N.(2002). The role of animation in teaching the limit concept. International Journal of Computer Algebra in Mathematics Education, 9(3), 205-227.
  10. Kozma, R., & Russell, J.(1997). Multimedia and understanding: Expert and novice responses to different representations of chemical phenomena. Journal of Research in Science Teaching, 34(9), 949-968. https://doi.org/10.1002/(SICI)1098-2736(199711)34:9<949::AID-TEA7>3.0.CO;2-U
  11. Kung. D., & Speer. N.(2010). Do they really get it? Evaluating evidence of student understanding of power series. In Proceedings of the Thirteenth Conference on Research in Undergraduate Mathematics Education. Raleigh. NC; North Carolina State University.
  12. Martin. J., & Oehrtman. M.(2010). Strong metaphors for the concept of convergence of Taylor series. In Proceedings of the Thirteenth Conference on Research in Undergraduate Mathematics Education. Raleigh, NC: North Carolina State University.
  13. Martin. J.(2013). Differences between experts' and students' conceptual images of the mathematical strucrure of Taylor series convergence. Edu Stud Math.
  14. Sfard, A.(1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1-36. https://doi.org/10.1007/BF00302715
  15. Sfard, A.(1992). Operational origins of mathematical objects and the quandary of reification-The case of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of Epistemology and Pedagogy MAA Notes (Vol. 25, pp. 59-84). Washington. DC: Mathematical Association of America.
  16. Sfard, A., & Linchevski, L.(1994). The gains and the pitfalls of reification: The case of algebra. Educational Studies In Mathematics, 26(2/3), 191-228. https://doi.org/10.1007/BF01273663
  17. Stewart, J.(2008). Calculus. (6thed.).Belmont: Brooks Cole.
  18. Tall, D., & Vinner, S.(1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151-169 https://doi.org/10.1007/BF00305619
  19. Vinner, S.(1992). The function concept as a prototype for problems in mathematics learning. In G. Harel, & E. Dubinsky (Eds), The concept of function: Aspects of epistemology and pedagogy(pp. 195-213). Washington, DC: Mathematical Association of America.
  20. W.R. Wade(2010). An introduction to analysis. Pearson Education, Inc.
  21. Zandieh, M.(2000). A theoretical framework for analyzing student understanding of the concept of derivative. Research in Collegiate Mathematics Education IV. 8, 103-126.

피인용 문헌

  1. Taylor 정리의 역사적 고찰과 교수방안 vol.31, pp.1, 2017, https://doi.org/10.14477/jhm.2018.31.1.019