The Mathematical Education (한국수학교육학회지시리즈A:수학교육)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

An investigation in learnability of counter-examples in secondary school mathematics textbooks

고등학교 수학 교과서에서의 반례에 대한 학습가능성 탐색

  • Received : 2013.08.05
  • Accepted : 2014.02.15
  • Published : 2014.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

In recent years, there has been increasing interest in the pedagogical importance of counter-examples that contradict statements about mathematics education research and the curriculum revision process for high school mathematics courses. Using a literature research method, this study analyzed views about counter-examples according to a method of approach to statements and the classification of counter-examples and their criteria. The study also described the learnability of the content of counter-examples presented in Korean secondary school mathematics textbooks. The results showed that generating many counter-examples enables learners to understand mathematical concepts exactly, construct links between mathematical contents, and have flexible thoughts about mathematical objects. Considering the learnability of counter-examples, the contents of counter-examples in school mathematics textbooks are needed for mathematics teachers and students to generate numerous counter-examples and verify the justification of generating counter-examples in various manners.

Keywords

References

  1. 교육과학기술부 (2011). 수학과 교육과정: 교육과학기술부 고시 제 2011-361호 [별책 8]. 교육과학기술부. (Ministry of Education, Science and Technology (2011). Mathematics Curriculum.: MEST announcement 2011-361 [Separate version 8]. Seoul: MEST.)
  2. 김수미, 정은숙 (2005). 범례 제시를 통한 도형 개념 지도 방안. 수학교육학연구, 15, 401-417. (Kim, S. M. & Jung, E. S. (2005). Building Geometrical Concepts by Using both Examples and Nonexamples. The Journal of Education Research in Mathematics, 15, 401-417.)
  3. 김수환 외 (2009). 고등학교 수학 I. 서울: 교학사. (Kim, S.H. et al. (2009). High school mathematics I. Seoul: Kyohaksa.)
  4. 김수환 외 (2009). 고등학교 수학 II. 서울: 교학사. (Kim, S.H. et al. (2009). High school mathematics II. Seoul: Kyohaksa.)
  5. 김해경 외 (2009). 고등학교 수학 I. 서울: 더 텍스트. (Kim, H.K. et al. (2009). High school mathematics I. Seoul: The text.)
  6. 김해경 외 (2009). 고등학교 수학 II. 서울: 더 텍스트. (Kim, H.K. et al. (2009). High school mathematics II. Seoul: The text.)
  7. 양승갑 외 (2009). 고등학교 수학 I. 서울: 금성출판사. (Yang, S.K. et al. (2009). High school mathematics I. Seoul: Kumsung.)
  8. 양승갑 외 (2009). 고등학교 수학 II. 서울: 금성출판사. (Yang, S.K. et al. (2009). High school mathematics II. Seoul: Kumsung.)
  9. 우무하 외 (2009). 고등학교 수학 I. 서울: 박영사. (Woo, M.H. et al. (2009). High school mathematics I. Seoul: Pakyoungsa.)
  10. 우무하 외 (2009). 고등학교 수학 II. 서울: 박영사. (Woo, M.H. et al. (2009). High school mathematics II. Seoul: Pakyoungsa.)
  11. 우정호 외 (2009). 고등학교 수학 I. 서울: 두산동아. (Woo, J.H. et al. (2009). High school mathematics I. Seoul: Doosandonga.)
  12. 우정호 외 (2009). 고등학교 수학 II. 서울: 두산동아. (Woo, M.H. et al. (2009). High school mathematics II. Seoul: Pakyoungsa.)
  13. 류희찬 외 (2009). 고등학교 수학 I. 서울: 미래엔컬처그룹. (Lew, H.C. et al. (2009). High school mathematics I. Seoul: Mirae N Culture.)
  14. 류희찬 외 (2009). 고등학교 수학 II. 서울: 미래엔컬처그룹. (Lew, H.C. et al. (2009). High school mathematics II. Seoul: Mirae N Culture.)
  15. 윤재한 외 (2009). 고등학교 수학 I. 서울: 더 텍스트. (Yoon, J.H. et al. (2009). High school mathematics I. Seoul: The text.)
  16. 윤재한 외 (2009). 고등학교 수학 II. 서울: 더 텍스트. (Yoon, J.H. et al. (2009). High school mathematics II. Seoul: The text.)
  17. 이강섭 외 (2009). 고등학교 수학 I. 서울: 지학사. (Lee, G.S. et al. (2009). High school mathematics I. Seoul: Jihaksa)
  18. 이강섭 외 (2009). 고등학교 수학 II. 서울: 지학사. (Lee, G.S. et al. (2009). High school mathematics I. Seoul: Jihaksa)
  19. 이동원 외 (2009). 고등학교 수학 I. 서울: 법문사. (Lee, D.S. et al. (2009). High school mathematics I. Seoul: Bobmunsa.)
  20. 이동원 외 (2009). 고등학교 수학 II. 서울: 법문사. Lee, D.S. et al. (2009). High school mathematics II. Seoul: Bobmunsa.
  21. 이만근 외 (2009). 고등학교 수학 I. 서울: 고려출판. (Lee, M.G. et al. (2009). High school mathematics I. Seoul: Koryobook.)
  22. 이만근 외 (2009). 고등학교 수학 II. 서울: 고려출판. (Lee, M.G. et al. (2009). High school mathematics II. Seoul: Koryobook.)
  23. 이정곤, 류희찬 (2011). 예비교사들을 대상으로 한 증명활동과 반례생성 수행결과 분석 : 수열의 극한을 중심으로. 수학교육학연구, 21, 379-398. (Lee, J.G. & Lew, H.C. (2005). Preservice Teachers' Writing Performance Producing Proofs and Counterexamples about Limit of Sequence. The Journal of Education Research in Mathematics, 21, 379-398.)
  24. 이종희, 이지현 (2009). 상위권 고등학생들의 수학적 정당화와 반증의 유형에 대한 사례연구. 교과교육학연구, 13, 633-652.
  25. Lee, C.H. & Lee, J.H. (2009). A Case Study RegardingHigh-ranking Highschool Students' Mathematical Justificationand Disproof. Subject Pedagogical Research, 15(3), 757-776.
  26. 이준열 외 (2009). 고등학교 수학 I. 서울: 천재교육. (Lee, J.Y. et al. (2009). High school mathematics I. Seoul: Chunjae Education.)
  27. 이준열 외 (2009). 고등학교 수학 II. 서울: 천재교육. (Lee, J.Y. et al. (2009). High school mathematics II. Seoul: Chunjae Education.)
  28. 정상권 외 (2009). 고등학교 수학 I. 서울: 금성출판사. (Jung, S.K. et al. (2009). High school mathematics I. Seoul: Kumsung.)
  29. 정상권 외 (2009). 고등학교 수학 II. 서울: 금성출판사. (Jung, S.K. et al. (2009). High school mathematics II. Seoul: Kumsung.)
  30. 최용준 외 (2009). 고등학교 수학 I. 서울: 천재교육. (Choi, Y.J. et al. (2009). High school mathematics I. Seoul: Chunjae Education.)
  31. 최용준 외 (2009). 고등학교 수학 II. 서울: 천재교육. (Choi, Y.J. et al. (2009). High school mathematics II. Seoul: Chunjae Education.)
  32. 황석근 외 (2009). 고등학교 수학 I. 서울: 교학사. (Hwang, S.G. et al. (2009). High school mathematics I. Seoul: Kyohaksa.)
  33. 황석근 외 (2009). 고등학교 수학 II. 서울: 교학사. (Hwang, S.G. et al. (2009). High school mathematics II. Seoul: Kyohaksa.)
  34. 황선욱 외 (2009). 고등학교 수학 I. 서울: 좋은책신사고. (Hwang, S.W. et al. (2009). High school mathematics I. Seoul: Sinsago.)
  35. 황선욱 외 (2009). 고등학교 수학 II. 서울: 좋은책신사고. (Hwang, S.W. et al. (2009). High school mathematics II. Seoul: Sinsago.)
  36. Alcock, L. & Weber, K. (2005). Using warranted implications to read and understand proofs. For the Learning of Mathematics, 25(1), 34-38.
  37. Balacheff, N. (1991). Treatment of refutations: Aspects of the complexity of a constructivist approach to mathematics learning. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 89-110). Dordrecht: Kluwer Academic Publishers.
  38. Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The Teaching of Proof. Paper presented at the International Congress of Mathematicians, Beijing, China. Volume III, 1-3, 907-920.
  39. Carpenter, T., & Franke, M. (2001). Developing algebraic reasoning in the elementary school: Generalization and proof. In H. Chick, K. Stacey, J. Vincent, & J. Vincent. (Eds.), Proceedings of the Twelfth International Commission on Mathematical Instruction (Vol. 1, pp. 155-162). Melbourne: University of Melbourne.
  40. Council of Chief State School Officers and National Governors Association. (2011). Common core state standards initiative [CCSSI] : Preparing America's students for college and career [Data file]. Retrieved from http://www.corestandards.org.
  41. Epp, S. (1998). A unified framework for proof and disproof. Mathematics Teacher, 91(8), 708-713.
  42. Hanna, G. (1991). Matheamtical proof. In D. Tall (Ed.), Advanced mathematical thinking (pp. 54-61). The Netherlands: Kluwer Academic Publishers.
  43. Hanna, G. (1995). Challenges to the importance of proof. For the Learning of Mathematics, 15(3), 42-49.
  44. Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24, 389-399. https://doi.org/10.1007/BF01273372
  45. Knuth, E. (2002). Teachers' conceptions of proof in the context of secondary school mathematics. Journal of Mathematics Teacher Education, 5(1), 61-88. https://doi.org/10.1023/A:1013838713648
  46. Ko, Y. (2010). Proofs and Counterexamples: undergraduate students' strategies for validating arguments, evaluating statements, and constructing productions. Doctoral dissertation, University of Wisconsin-Madison, Wisconsin.
  47. Ko, Y., & Knuth, E. J. (2013) Validating proofs and counterexamples across content domains: Practices of importance for mathematic majors. Journal for Mathematical Behavior, 32, 20-35. https://doi.org/10.1016/j.jmathb.2012.09.003
  48. Komatsu, K. (2010). Counter-examples for refinement of conjectures and proofs in primary school mathematics. Journal of Mathematical Behavior, 29, 1-10. https://doi.org/10.1016/j.jmathb.2010.01.003
  49. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press.
  50. Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29-63. https://doi.org/10.3102/00028312027001029
  51. Lin, F. L. (2005). Modeling students' learning on mathematical proof and refutation. In H. Chick, K. Stacey, J. Vincent, & J. Vincent. (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 3-18). Melbourne: University of Melbourne.
  52. Lithner, J. (2004). Mathematical reasoning in calculus textbooks exercises. The Journal of Mathematical Behavior, 23, 405-427. https://doi.org/10.1016/j.jmathb.2004.09.003
  53. Mason, J., & Klymchuk, S. (2009). Using counter-examples in calculus. Imperial College Press: London.
  54. Mesa, V. (2010). Strategies for controlling the work in mathematics textbooks for introductory calculus. Research in Collegiate Mathematics Education, 16, 235-265.
  55. Mesa, V., & Griffiths, B. (2012). Textbook mediation of teaching: an example from tertiary mathematics instructors. Educational Studies in Mathematics, 79 (1), 85-107. https://doi.org/10.1007/s10649-011-9339-9
  56. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
  57. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
  58. Peled, I., & Zaslavsky, O. (1997). Counter-examples that (only) prove and counter-examples that (also)explain. FOCUS on Learning Problems in Mathematics, 19(3), 49-61.
  59. Polya, G. (1973). How to solve it. Princeton University Press. Princeton, NJ.
  60. Schmidt, W. H., McKnight, C. C., Valverde, G., Houang, R. T., & Wiley, D. E. (1996). Many visions, many aims, 1: A cross-national investigation of curricular intentions in school mathematics. Dordrecht: Kluwer.
  61. Schoenfeld, A. H. (2009). Series editor's forward: The souk of mathematics. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and Learning proof across the grades: A K-16 perspective (pp.xii-xvi). New York, NY: Routledge.
  62. Selden, A. & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4-36. https://doi.org/10.2307/30034698
  63. Thurston, W. P. (1995). On proof and progress in mathematics. For the Learning of Mathematics, 15(1), 29-37.
  64. Travers, K. J., & Westbury, I. (1989). The IEA Study of mathematics I: Analusis of mathematics curricula. Oxford: Pergamon Press.
  65. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: learners generating examples. Lawrence Erlbaum Associates
  66. Weber, K. (2010). Mathematics majors' perceptions of conviction, validity, and proof. Mathematical Thinking and Learning, 12, 306-336. https://doi.org/10.1080/10986065.2010.495468
  67. Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 227-236). Reston, VA: National Council of Teachers of Mathematics.
  68. Zaslavsky, O., & Ron, G. (1998). Students' understanding of the role of counter-examples. In A. Olivier, &K. Newstead (Eds.), Proceedings of the 22nd conference of the international group for the psychology of mathematics education, (Vol. 4, pp. 225-232). Stellenbosch, South Africa: University of Stellenbosch.
  69. Zazkis, R., & Chernoff, J. E. (2008). What makes a counterexample exemplary? Educational Studies in Mathematics, 68(3), 195-208. https://doi.org/10.1007/s10649-007-9110-4