한국수학교육학회지시리즈E:수학교육논문집 (Communications of Mathematical Education)

  • 제34권3호
  • /
  • Pages.355-372
  • /
  • 2020
  • /
  • 1226-6663(pISSN)
  • /
  • 2287-9935(eISSN)
한국수학교육학회 (Korean Society of Mathematical Education)
권호 표지
DOI QR코드

DOI QR Code

무한급수의 이해에 대한 연구

A study on understanding of infinite series

  • Oh, Hye-Young (Department of Mathematics Education, Incheon National University)
  • 투고 : 2020.04.29
  • 심사 : 2020.06.27
  • 발행 : 2020.09.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

무한급수 개념은 학부의 전공 수학 교육과정의 중요한 주제이다. 여러 세기 동안 그것은 학습자에게 직관에 반대되는 장애를 제공했을 뿐만 아니라 해석학 연구의 중심적 역할을 해 왔다. 수학의 역사에서 무한급수 개념에 대한 이해가 미적분학 발달의 기초가 되었듯이 현재의 학생들에게 무한급수 개념에 대한 이해는 전공 수학을 학습하는 데 꼭 필요하다. 무한합의 개념을 가진 학생 대부분은 무한급수의 수렴 판정 같은 수학적 내용은 어려워하지 않으나 무한급수 개념을 부분합의 열을 이용해서 구성하는 것은 어려워한다. 이에 본 연구에서는 무한급수 개념을 구성하는 방법을 APOS 이론과 발생적 분해의 관점에서 부분합 스키마를 이용하여 분석하고자 한다. 질적 연구를 통해 급수 개념의 구성 방법을 점검해서 무한급수 지도 개선에 대한 유용한 교육적 시사점을 얻고자 한다.

The concept of infinite series is an important subject of major mathematics curriculum in college. For several centuries it has provided learners not only counter-intuitive obstacles but also central role of analysis study. As the understanding in concept on infinite series became foundation of development of calculus in history of mathematics, it is essential to present students to study higher mathematics. Most students having concept of infinite sum have no difficulty in mathematical contents such as convergence test of infinite series. But they have difficulty in organizing concept of infinite series of partial sum. Thus, in this study we try to analyze construct the concept of infinite series in terms of APOS theory and genetic decomposition. By checking to construct concept of infinite series, we try to get an useful educational implication on teaching of infinite series.

키워드

참고문헌

  1. Ko, J. H. (2012). Analysis of error type occurred in limit of sequence and infinite series, Master's thesis, The graduate school of education at Aju University.
  2. Kwak, B. J. (2006). A study on special convergence test of infinite series, Master's thesis, The graduate school of education at Inje University.
  3. Kim, G. Y. (2007). Infinite series of real number, Master's thesis, The graduate school of education at Dongkuk University.
  4. Kim, S. D. (2010). Infinite series, Master's thesis, The graduate school of education at Dongkuk University.
  5. Kim, C. J. (2010). Analysis of misconception occurred in infinite series learning, Master's thesis, The graduate school of education at Hankuk University of Foreign studies.
  6. Kim, H. J. (2002). A study on convergence test of infinite series, Master's thesis, The graduate school of education at Hanyang University.
  7. Park, H., Kim, S. & Kim, W. (2007). On the applications of the genetic decomposition of mathematical concepts-In the case of ${\mathbf{Z}}_n$ in abstract algebra-, J. Korea Soc. Math. Ed. Ser. A: The Mathematical Education, 44(4), 547-563.
  8. Shin, J. H. (2017). Understanding and practice of qualitative research method, Emotinbooks
  9. Lee, S. Y. (2007). Infinite series of function, Master's thesis, The graduate school of education at at Dongkuk University.
  10. Lee, J. S. (2016). Pre-teacher's cognition on application of mathematics history and implicit mathematics history application types of middle school mathematics textbooks, Master's thesis, The graduate school of education at Ewha Womans University.
  11. Cho, Y. H. (2008). A study on understanding of concept of infinite series: For high school seniors, Master's thesis, The graduate school of education at Konkuk University.
  12. Ji, Y. C. (2006). Concept image and error on limit of infinite sequence and series, Master's thesis, The graduate school of education at Korea University.
  13. Choi, J. Y. (2012). Analysis and suggestion of infinite series chapter from middle school 1 st grade mathematics textbooks from the 2007 revised curriculum: Centered on infinite geometric series, Master's thesis, The graduate school of education at Kyunghee University.
  14. David M. Burton (2013). The History of Mathematics : An Introduction, Kyungmoon Co.
  15. Howard Eves/ Her, M., Oh, H. Y. (1995). Great Moments in Mathematics, Kyungmoon Co.
  16. Keith Devlin. (1996). Mathematics: the science of patterns, Kyungmoon Co.
  17. Arnon, I., Cottrill, J., Dubinsky, E., Oktac, A., Fuentes, S. R., Trigueros, M. & Weller, K. (2014). APOS Theory, Springer.
  18. Asiala, M., Brown, A., DeVries, D.J., Dubinsky, E., Mathews, D. & Thomas, K. (1996). A framework for research and development in undergraduate mathematics education, In J. Kaput, E. Dubinsky. & A. H. Schocnfeld(Eds.), Research in collegiate mathematics education II (pp.1-32), Providence, RI: American Mathematical Society.
  19. Bagni, G. T. (1996-7). Storia della Mathematica, I-II-III, Bologna: Pitagora.
  20. Bagni, G. T. (2000). Difficulties with series in history and in the classroom, In J. Fauvel & J. van Maanen (Eds.), History in mathematics education: The ICMI study (pp. 82-86). Dordrecht, the Netherlands: Kluwer.
  21. Bagni, G. T. (2005). Infinite series from history to mathematics education, International Journal for Mathematics Teaching and Learning [on-line journal], posted June 30, 2005, University of Plymouth, U.K. http://cimt.plymouth.ac.uk/journal/default.htm.
  22. Bagni, G.T. (2007). Didactics and history of numerical series, 100 years after Ernesto Cesaro's death (1906): Guido Grandi, Gottfried Wilhelm Leibnitz and Jacobo Riccati. Retrieved from http://www.syllogismos.it/history/GrandiJointMeeting.pdf.
  23. Baker, B., Cooley, L. & Trigueros, M. (2000). The schema triad-a calculus example, Journal for Research in Mathematics Education, 31, 557-578. https://doi.org/10.2307/749887
  24. Dubinsky, E. (1991). Reflective Abstraction in Advanced Mathematics Thinking, In Tall, D. (Eds.), Advanced Mathematics Thinking, Kluwer Academic Publishers, 류희찬.조완영.김인수 역(2003), 고등수학적사고, 경문사, 127-165.
  25. Dubinsky, E. & MacDonald, M. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research, In D. Holton et.(Eds.), The teaching and Learning of Mathematics at University Level: An ICMI Study, Kluwer Academic Publishers, 273-280.
  26. Fishbein, E., Tirosh, D., & Melamed, U. (1981). Is it possible to measure the intuitive acception of a mathmetical statement? Education Studies in Mathematics, 12, 491-512. https://doi.org/10.1007/BF00308145
  27. Leibniz, G. W. (1715). Acta Eruditorum Supplementum 5, 'Epist. G.G.L. ad V. claris. Ch. Wolfium'.
  28. Mamona, J. C. (1990). Sequences and series-sequences and functions: students' confusions, International Journal of Mathematical Education in Science and Technology, 21, 333-337.
  29. Martinez-Planell, R., Gonzalez, A. C., DiCristina, G. & Acevedo, V. (2012). Students' conception of infinite series, Educational Studies in Mathematics, 81(2), 235-249. https://doi.org/10.1007/s10649-012-9401-2
  30. McDonald, M. A., Mathews, D. & Strobel, K. (2000). Understanding sequences: A tale of two objects, In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education IV (pp. 77-102). Providence, RI: American Mathematical Society.
  31. Stewart, J. (2008). Calculus (6th ed.), Belmont: Brooks Cole.