한국수학교육학회지시리즈A:수학교육 (The Mathematical Education)

  • 제59권1호
  • /
  • Pages.81-99
  • /
  • 2020
  • /
  • 1225-1380(pISSN)
  • /
  • 2287-9633(eISSN)
한국수학교육학회 (Korean Society of Mathematical Education)
권호 표지
DOI QR코드

DOI QR Code

수학 문제 만들기 유형에 따른 가추 유형과 가추에 동원된 사고 전략 분석

Analysis of abduction and thinking strategies by type of mathematical problem posing

  • 투고 : 2020.02.03
  • 심사 : 2020.02.24
  • 발행 : 2020.02.29
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

본 연구는 학생이 만든 수학 문제에 따른 가추 유형과 가추에 동원된 사고 전략에 대해 알아보았다. 중학교 2학년 4명의 학생이 네 개의 과제에 대해 문제 만들기 활동을 하여, 동치문제, 동형문제, 유사문제를 만들었다. 동치문제의 경우 조작적 가추가 주로 발현되었다. 동형문제와 유사문제는 조작적 가추, 이론적 가추, 창의적 가추가 모두 발현되다. 가추에 동원된 사고 전략으로, 조작적 가추는 대상으로 인식하기, 패턴 찾기, 숫자나 그림으로 변환하기, 유추하기, 반대로 생각하기, 결합하기, 제거하기의 사고전략이 나타났다. 이론적 가추에서는 수학적, 경험적, 확산적 지식을 활용하는 사고전략이 나타났다. 창의적 가추에서는 상승화 전략, 형식화 전략, 창조적 전략이 나타났다. 결합하기, 제거하기, 수학적, 경험적, 타교과 지식의 활용, 문제와 직접적 관련이 없는 규칙을 접목시켜 만드는 창조적 전략은 본 연구에서 새롭게 도출된 사고 전략이다.

This study examined the types of abduction and the thinking strategies by the mathematics problems posed by students. Four students who were 2nd graders in middle school participated in problem posing on four tasks that were given, and the problems that they posed were classified into equivalence problem, isomorphic problem, and similar problem. The type of abduction appeared were different depending on the type of problems that students posed. In case of equivalence problem, the given condition of the problems was recognized as object for posing problems and it was the manipulative abduction. In isomorphic problem and similar problem, manipulative abduction, theoretical abduction, and creative abduction were all manifested, and creative abduction was manifested more in similar problem than in isomorphic problem. Thinking strategies employed at abduction were examined in order to find out what rules were presumed by students across problem posing activity. Seven types of thinking strategies were identified as having been used on rule inference by manipulative selective abduction. Three types of knowledge were used on rule inference by theoretical selective abduction. Three types of thinking strategies were used on rule inference by creative abduction.

키워드

참고문헌

  1. Brown, S. I., & Walter, M. I. (2005). The art of problem posing(3rd. Ed. e-book). Lawrence Erlbaum Associates Publisher.
  2. Conner, A., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). Identifying kinds of reasoning in collective argumentation. Routledge, Mathematical Thinking and Learning, 16(3), 181-200. https://doi.org/10.1080/10986065.2014.921131
  3. Creswell, J. W. (2009). Research design: Qualitative, quantitative, and mixed methods approaches (3rd ed.). Thousand Oaks, CA: Sage Publications.
  4. Eco, U. (1983). Horns, hooves, insteps: Some hypotheses on three types of abduction. In U. Eco & T. Sebeol(eds.), The sign of three : Dupin, Holmes and Peirce (pp. 198-220). Bloomington, IN : Indiana University Press.
  5. Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: the importance of qualification. Educational Studies in Mathematics, 66(1), 3-21. https://doi.org/10.1007/s10649-006-9059-8
  6. Jung, S. M. (1993). Three concepts of discovery. The Korean Society for Cognitive Science, 4(1), 25-49.
  7. Kim, P. S. (2005). Analysis of thinking process and steps in problem posing of the mathematically gifted children. The Journal of Elementary Education, 18(2), 303-334.
  8. Kim, S. H. & Lee, C. H. (2002). Abduction as a mathematical resoning. The Journal of Educational Research in Mathematics, 12(2), 275-290.
  9. Kim, S. H. (2004). Semiotic consideration on the appropriation of the mathematical knowledge. Doctoral dissertation. Ehwa Womans University, Korea.
  10. Kim, W. K., Jo, M. S., Bang, G. S., Bae, S. K., Ji, E. J., Im, S. H., Kim, D. H., Kang, S. J., & Kim, Y. H. (2015). Mathematic 1, Seoul: visang.
  11. Krulik, S. & Rudnick, J. A. (1992). Reasoning and problem solving: A handbook for elementary school teachers. the United States: NCTM.
  12. Kwon, Y. J., Jeong, J. S., Kang, M. J. & Kim, Y. S. (2003). Research article : A grounded theory on the process of generating hypothesis-knowledge about scientific episodes. Journal of the Korean Association for Research in Science Education, 23(5), 458-469.
  13. Lee, H. (1989). Syllogism and dialectic. Kyungnam Journal of Philosophy, 5, 3-30.
  14. Lee, M. H. (2020). Analysis of Abduction Types and Thinking Strategies on Mathematics Problem Posing. Doctoral dissertation, Kangwon National University.
  15. Lee, Y. H. & Kahng, M. J. (2013). An analysis of problems of mathematics textbooks in regards of the types of abductions to be used to solve, The Journal of Educational Research in Mathematics, 23(3), 335-351.
  16. Magnani, L. (2001). Abduction, reason, and science: Process of discovery and explanation. New York: Kluwer Academic/Plenum Publishers.
  17. Ministry of Education (2015). Mathematical curriculum. Seoul: Author.
  18. Merriam, S. B. (1988). Qualitative research and case study applications in education. San Francisco: Jossy-Bass.
  19. Na, G. S. (2017). Examining the problem making by mathematically gifted students. School Mathematics, 19(1), 77-93.
  20. National Council of Teachers of Mathematics (NCTM, 1989). Curriculum and evaluation standards for school mathematics. Reston, Va.
  21. Oh, P. S. & Kim, C. J. (2005). A theoretical study on abduction as an inquiry method in earth science. Journal of the Korean Association for Science Education, 25(5), 610-623.
  22. Oh, P. S. (2006). Rule-inferring strategies for abductive reasoning in the process of solving an earth-environmental problem. Journal of the Korean Association for Science Education, 26(4), 546-558.
  23. Paik, S. Y. (2016). Teaching & learning of mathematical problem-solving. Seoul: Kyungmoon.
  24. Pease, A., & Aberdein, A. (2011). Five theories of reasoning: Interconnections and applications to mathematics. Logic and Logical Philosophy, 20(1-2), 7-57.
  25. Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23-41. https://doi.org/10.1007/s10649-006-9057-x
  26. Pedemonte, B., & Reid, D. (2011). The role of abduction in proving processes. Educational Studies in Mathematics, 76(3), 281-303. https://doi.org/10.1007/s10649-010-9275-0
  27. Peirce, C. S. (1958). Collected papers of Charles Sanders Peirce Vols I-VI. C. Hartshorne & P. Weiss (Eds.). Cambridge, MA: Harvard University Press.
  28. Peirce, C. S. (1980). Collect papers of Charles Sanders Peirce Vols VII-VIII. In B. Arthur (Ed.). Cambridge, MA: Harvard University Press.
  29. Polya, G. (1957), How to solve it(2nd ed.). NY.: Doubleday & Company, Inc.
  30. Psillos, S. (2000). Abduction: Between conceptual richness and computational complexity. Retrieved Jan. 20, 2018, from http://users.uoa.gr/-psillos/PapersI/85-Abduction%20(Kakas%20&%20Flach)%20chapter.pdf
  31. Reed, S. K. (1989). Constraints on the abstraction of solutions. Journal of Educational Psychology, 81, 532-540. https://doi.org/10.1037/0022-0663.81.4.532
  32. Reys, R. E., Lindquist, M. M., Lambdin, D. V., Smith, N. L., Rogers, A., Falle, J., Frid, S. & Bennett, S.(2012). Helping children learn mathematics. (1st Australian Ed.). Australia. Milton, Qld: John Wiley & Sons.
  33. Rivera, F. D., & Rossi Becker, J. (2007). Abduction in pattern generalization. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31st Conference of the international group for the Psychology of Mathematics Education(vol.4, pp. 97-104), Seoul, Korea.
  34. Silver, E. A., Mamona-Dows, J., Leung, S. S., & Kenney, P. A. (1996). Posing mathematical problems. Journal for Research in Mathematical Education, 27(3), 293-309. https://doi.org/10.2307/749366
  35. Stickles, P. R. (2006). An analysis of secondary and middle school teacher's mathematical problem posing. Doctoral dissertation, Indiana University. the United States.
  36. Toulmin, S. E. (1958). The uses of argument. Cambridge: Cambridge University Press.
  37. Weber, K., & Alcock, L. (2005). Proof validation in real analysis: Inferring and checking warrants. The Journal of Mathematical Behavior, 24(2), 125-134. https://doi.org/10.1016/j.jmathb.2005.03.003