The Mathematical Education (한국수학교육학회지시리즈A:수학교육)

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

DOI QR Code

A study on understanding the deduction system in the proof

증명에서 연역 체계 이해에 관한 연구

  • Received : 2012.10.31
  • Accepted : 2013.01.25
  • Published : 2013.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

To help students understand the deduction system in the proof, we analyzed the textbook on mathematics at first. As results, we could find that the textbook' system of deduction is similar with the Euclid' system of deduction. The starting point of deduction is different with each other. But the flow of deduction match with each other. Next, we searched for the example of circular argument and analyzed. As results, we classified the circular argument into two groups. The first is an internal circular argument which is a circular argument occurred in a theorem. The second is an external circular argument which is a circular argument occurred between many theorems. We could know that the flow of deduction system is consistent in internal-external dimension. Lastly, we proposed the desirable teaching direction to help students understand the deduction system in the proof.

Keywords

References

  1. 강문봉 (1992). 분석법에 대한 고찰. 대한수학교육학회 논문집 2(2), 81-93.(Kang, M.J. (1992). An educational study on analysis. Journal of the Korea Society of Educational Studies in Mathematics 2(2), 81-93.)
  2. 강문봉 (1993). Lakatos의 수리철학의 교육적 연구. 박사학위논문, 서울대학교.(Kang, M.J. (1993). An educational study on the Lakato's philosophy of Mathematics. Doctoral dissertation, SNU.)
  3. 강미광 (2010). 유클리드 기하학에서 삼각형의 합동조건의 도입 비교, 수학교육 49(1), 53-65.(Kang, M.K .(2010). A study on the comparison of triangle congruence in euclidean geometry. The Mathematical Education 49(1), 53-65.)
  4. 김흥기 (2001). 중학교 수학에서 증명을 위한 공리 취급에 관한 연구, 수학교육 40(2), 291-315.(Kim, H.K. (2001). A note on treatment of axioms for proof in middle school mathematics. The Mathematical Education 40(2), 291-315.)
  5. 나귀수 (1998). 증명의 본질과 지도 실제의 분석-중학교 기하단원을 중심으로-. 박사학위논문, 서울대학교.(Na, G.S. (1998). An analysis of the nature of proof and practice of proof edcuation: focused on the middle school geometry. Doctoral dissertation, SNU.)
  6. 나귀수 (2009). 분석법을 중심으로 한 기하 증명 지도에 대한 연구, 수학교육학연구 19(2), 185-206.(Na, G.S. (2009). Teaching geometry proof with focus on the analysis. The Journal of Education Research in Mathematics 19(2), 185-206.)
  7. 류성림 (1998a). 수학교육에서 '증명의 의의'에 관한 연구, 수학교육 37(1), 73-85.(Ryu, S.R. (1998a). A study on the meaning of proof in mathematics education. The Mathematical Education 37(1), 73-85.)
  8. 류성림 (1998b). 피아제의 균형화 모델에 의한 증명의 지도 방법 탐색. 박사학위논문, 한국교원대학교.(Ryu, S.R. (1998b). A study on the teaching method of proofs based on piagetian equilibration model. Doctoral dissertation, KNUE.)
  9. 류성림 (1998c). 전형식적 증명의 의미와 교수학적 의미에 관한 연구, 대한수학교육학회지 논문집 8(1), 313-326.(Ryu, S.R. (1998c). A study on the meaning of preformal proof and its didactical significance. Journal of the Korea Society of Educational Studies in Mathematics 8(1), 313-326.)
  10. 류희찬, 류성림, 한혜정, 강순모, 제수연, 김명수, 천태선, 김민정 (2009a). 중학교 수학 1. 서울: 미래엔 컬처그룹.(Ryu, H.C., Ryu, S.R., Han, H.J., Kang, S.M., Je, S.Y., Kim, M.S., Cheon, T.S., & Kim, M.J. (2009a). Middle school mathematics 1. Seoul: Mirae N Culture.)
  11. 류희찬, 류성림, 한혜정, 강순모, 제수연, 김명수, 천태선, 김민정 (2009b). 중학교 수학 2. 서울: 미래엔 컬처그룹.(Ryu, H.C., Ryu, S.R., Han, H.J., Kang, S.M., Je, S.Y., Kim, M.S., Cheon, T.S., & Kim, M.J. (2009b). Middle school mathematics 2. Seoul: Mirae N Culture.)
  12. 박영훈, 여태경, 김선화, 심성아, 이태림, 김수미 (2009a). 중학교 수학 1. 서울: 천재문화.(Park, Y.H., Ye, T.G., Kim, S.H., Sim, S.A., Lee, T.R., & Kim, S.M. (2009a). Middle school mathematics 1. Seoul: Chunjaemunhwa.)
  13. 박영훈, 여태경, 김선화, 심성아, 이태림, 김수미 (2009b). 중학교 수학 2. 서울: 천재문화.(Park, Y.H., Ye, T.G., Kim, S.H., Sim, S.A., Lee, T.R., & Kim, S.M. (2009b). Middle school mathematics 2. Seoul: Chunjaemunhwa.)
  14. 박우석 (2008). 제르멜로와 공리적 방법, 논리연구 11(2),1-56.(Park, W.S. (2008). Zermelo and the axiomatic method. The logical research 11(2), 1-56.)
  15. 박은조, 방정숙 (2005). 수학 교사들의 증명에 대한 인식, 한국학교수학회논문집 8(1), 101-116.(Park, E.J., & Pang, J.S. (2005). A survey on mathematics teachers' cognition of proof. Journal of the Korea School Mathematics Society 8(1), 101-116.)
  16. 신항균, 이광연, 윤혜영, 이지현 (2009a). 중학교 수학 1. 서울: 지학사.(Sin, H,G., Lee, K.Y., Yoon, H.Y., & Lee, J.H. (2009a). Middle school mathematics 1. Seoul: Jihaksa.)
  17. 신항균, 이광연, 윤혜영, 이지현 (2009b). 중학교 수학 2. 서울: 지학사.(Sin, H,G., Lee, K.Y., Yoon, H.Y., & Lee, J.H. (2009b). Middle school mathematics 2. Seoul: Jihaksa.
  18. 우정호 (1998). 학교 수학의 교육적 기초. 서울: 서울대학교출판부.(Woo, J.H. (1998) The educational base of school mathematics. Seoul: SNU Publishing.)
  19. 우정호, 박교식, 박경미, 이경화, 김남희, 임재훈, 박인, 이영란, 고현주, 김은경 (2009a). 중학교 수학 1. 서울: 두산동아.(Woo, J.H., Park, G.S., Park, G.M., Lee, G.H., Kim, N.H., Lim, J.H., Park, I., Lee, Y.R., Ko, H.J., & Kim, E.G. (2009a). Middle school mathematics 1. Seoul: Dusandonga.)
  20. 우정호, 박교식, 박경미, 이경화, 김남희, 임재훈, 박인, 이영란, 고현주, 김은경 (2009b). 중학교 수학 2. 서울: 두산동아.(Woo, J.H., Park, G.S., Park, G.M., Lee, G.H., Kim, N.H., Lim, J.H., Park, I., Lee, Y.R., Ko, H.J., & Kim, E.G. (2009b). Middle school mathematics 2. Seoul: Dusandonga.)
  21. 유윤재 (2004). 공리의 문화적 의미, 한국수학사학회지 17(1), 119-125.(Yoo, Y.J. (2004). The cultural meaning of axioms. Journal for History of Mathematics 17(1), 119-125.)
  22. 이준열, 최부림, 김동재, 송영준, 윤상호, 황선미 (2009a). 중학교 수학 1. 서울: 천재교육.(Lee, J.Y., Choi, B.R., Kim, D.J., Song, Y.J., Yoon, S.H., & Hwang, S.M. (2009a). Middle school mathematics 1. Seoul: Chunjae education.)
  23. 이준열, 최부림, 김동재, 송영준, 윤상호, 황선미 (2009b). 중학교 수학 2. 서울: 천재교육.(Lee, J.Y., Choi, B.R., Kim, D.J., Song, Y.J., Yoon, S.H., & Hwang, S.M. (2009b). Middle school mathematics 2. Seoul: Chunjae education.)
  24. 이지현 (2011). 중학교 기하에서의 공리와 증명의 취급에 대한 분석, 수학교육학연구 21(2), 135-148.(Lee, J.H. (2011). An analysis on the treatment of axiom and proof in middle school mathematics. The Journal of Education Research in Mathematics 21(2), 135-148.)
  25. 조정수, 이정자 (2006). 증명보조카드를 활용한 중학생의 증명지도에 관한 연구, 한국학교수학회논문집 9(4), 521-538.(Jo, J.S. & Lee, J.J. (2006). A study on teaching mathematical proofs of the middle school students using the 'proof assisted cards', Journal of the Korea School Mathematics Society 9(4), 521-538.)
  26. 최보영 (2005). '식의 계산' 단원에서 수학 학습부진아의 오류 분석과 교정에 관한 연구: 고등학교 1학년을 대상으로. 석사학위논문, 이화여자대학교.(Choi, B.Y. (2005). A study of error analysis and correction toward students who are underachievers in calculation of numerical formula: on the basis of the first year of the high school student. Master's thesis, EWU.)
  27. 최지선 (2003). 중등학교 수학 학습에서 나타나는 오개념에 대한 고찰. 석사학위논문, 서울대학교.(Choi, J.S. (2003). A study on the misconceptions in the learning of the secondary school mathematics. Master's thesis, SNU.)
  28. 國宗進(1992). 圖形의 證明指導. 東京: 明治圖書株式會社.
  29. Almeida, D. (1996). Justifying and proving in the mathematics classroom. Philosophy of Mathematics Education Newsletter 9. http://www-didactique. imag.fr/preuve/Resumes/Almeida/POME9Almeida.html
  30. Bell, A. W. (1976). A study of pupils proofexplanations in mathematical situations, Educational Studies in Mathematics, 7, 23-40. https://doi.org/10.1007/BF00144356
  31. Bell, A. W. (1979). The learning of process aspects of mathematics, Educational Studies in Mathematics, 10, 361-387. https://doi.org/10.1007/BF00314662
  32. Borasi, R. (1996). Reconceiving Mathematics Instruction: A Focus on errors, New York: Ablex.
  33. Dodes, I. A. (1966). Mathematics: Its structure, logic and method. In E. G. Begle(Ed.), The role of axiomatics and problem solving in mathematics. Boson, MA: Ginn and Company.
  34. Fawcett, H. P. (2006). 증명의 본질(장경윤, 류현아, 한 세호 역). 서울: 경문사. (원저는 1966년 출판).
  35. Fishbein, E., & Kedem, I. (1982). Proof and certitude in the development of mathematical thinking. In. Vermandel(Ed). Proceedings of the Sixth International Conference for the Psychology of Mathematical Education. Antwerp: PME.
  36. Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1), 5-23. https://doi.org/10.1023/A:1012737223465
  37. Harel, G., & Sowder, L.(1998). Types of students' justifications. Mathematics Teacher, 91(8), 670-675.
  38. Heath, T. (1981). A History of Greek Mathematics, vol. II. New York: Dover.
  39. Lehman, H. (1980). An Examination of Imre Lakatos's Philosophy of Mathematics. The Philosophical Forum, 12, 33-48.
  40. Martin, W. G., & Harel, G. (1989). Proof Frames of Preservice Elementary Teachers, Journal for Research in Mathematics Education, 20, 1.
  41. Schoenfeld, A. H. (1994). Reflections on doing and teaching mathematics. In A. H. Schoenfeld(Ed.), Mathematical thinking and problem solving (pp.53-70). Hillsdale, NJ: Erlbaum.
  42. Sierpinska, A. & Lerman, S. (1996). Epistemologies of Mathematics and of Mathematics Education. In Alan J. Bishop(Eds.), International Handbook of Mathematics Education. Dordrecht: Kluwer Academic, 827-876.
  43. Williams, E. (1990). An Investigation of Senior High School Students' Understanding of the Nature of Mathematical Proof, Journal for Research in Mathematics Education, 11, 3.