Communications of Mathematical Education (한국수학교육학회지시리즈E:수학교육논문집)

Korean Society of Mathematical Education (한국수학교육학회)
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A case study on inquiry activities of synthetic division through analogies

유추를 통한 조립제법 탐구활동 사례 연구

  • Jung, Milin (Graduate school of Dept. of Math. Education, Korea University) ;
  • Whang, Woo Hyung (Dept. of Math. Education, Korea University)
  • Received : 2014.01.10
  • Accepted : 2014.02.14
  • Published : 2014.02.15
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Abstract

The purpose of the study was to investigate the aspects of analogy of high school student's thinking process revealed in the inquiry activity with synthetic division. The case study method of qualitative research was conducted with two high school 10th grade students. Structure-mapping model(SMM) of Gentner and similarity frames which were proposed by other researchers were utilized to analyze the data. Two students used analogy as a tool and they could discover synthetic division of more than 2 degrees, but they revealed different levels of mathematics discovery depending on the different degree of analogical thinking. Surface similarity in the process of inquiry activity played a vital role in analogical thinking. We asked students to explore and discover analogy based on structure similarity. Analogy based on the systematic approach made it possible to predict upper domain. Analogy based on the procedure similarity induced internalization. We could conclude that analogy has instrumental, heuristic and reflective characteristics.

본 연구의 목적은 조립제법 소재의 탐구활동에서 나타난 고등학생들의 사고 과정을 분석하여 유추의 양상을 조사하는 것이다. 인문계 고등학교 1학년 학생 2명을 대상으로 질적 사례연구로 수행되었다. 자료의 분석을 위하여 연구자가 제안한 유사성 분류 틀과 Gentner의 Structure-mapping Model(구조사상 모형, 줄여서 SMM)을 이용하였다. 두 학생 모두 유추를 도구로 사용하여 2차 이상의 조립제법을 발견하였으나, 유추적 사고의 능력에 따라 수학적 발견에 차이를 보였다. 탐구활동 과정에서 표면 유사성은 유추에서 중요한 역할을 수행하였다. 구조 유사성에 근거한 유추는 학생들도 수학자처럼 탐구하고 발견할 수 있도록 하였으며, 체계성의 원리에 의한 유추는 다른 영역에 대한 예측과 설명을 가능하게 하였고, 절차 유사성에 의한 유추는 내면화를 이끌어 냈다. 또한 유추의 성격이 도구적, 발견적이고 또한 반성적이라는 결론을 얻었다.

Keywords

References

  1. 고상숙․김규상 (2003). 분수 학습에서 정신모델 구성을 위한 유추의 역할, 한국수학교육학회지 시리즈 E <수학교육 논문집>, 15(1), 105-111.
  2. 김연수 (1999). 기저유추원의 두 처리방식이 아동의 유추적 문제해결에 미치는 연령별 효과. 서울대학교 석사학위 논문.
  3. 김용태․신봉숙․최대욱․이순희 (2005). 유추를 통한 분수 연산에 관한 연구, 한국수학교육학회지 시리즈 E <수학교육 논문집>, 19(4), 715-731.
  4. 김진호 (2005). 수학자가 수학을 탐구하듯이 학습자도 수학을 탐구할 수 있는 방안 모색, 한국수학교육학회지 시리즈 A <수학교육>, 44(1), 87-101.
  5. 박미미․이동환․이경화․고은성 (2012). 유추에 의한 문제제기 활동을 통해 본 통계적 개념 이해, 대한수학교육학회지 수학교육학연구, 22(1), 101-115.
  6. 박현정․이종희 (2006). 중학생들이 수학 문장제 해결 과정에서 구성하는 유사성 분석, 대한수학교육학회지 수학교육학연구, 16(2), 115-138.
  7. 서보억 (2009). 유추를 활용한 진법의 확장에 대한 수학탐구활동, 국제수학영재교육세미나프로시딩, 14, 39-42.
  8. 신은아 (2004). 귀납적 추론과 유추의 교과서 사례분석. 경북대학교 석사학위논문.
  9. 양기열․이의진 (2011). 수학영재학생들의 유추를 통한 이차곡면의 탐구활동 분석, 영재교육연구, 21(2), 269-286. https://doi.org/10.9722/JGTE.2011.21.2.269
  10. 이경화 (2009). 영재아들의 세 유형의 유추 과제 해결, 대한수학교육학회지 수학교육학연구, 19(1), 45-61.
  11. 이승우 (2001). 학교 수학에서의 유추와 은유. 서울대학교 석사학위 논문.
  12. 이종희․김선희 (2002). 인수분해 문제 해결과 유추, 대한수학교육학회지 <학교수학>, 4(4), 281-599.
  13. 조성남․이현주․주영주․김나영 (2011). 질적연구방법과 실제. 서울: 그린.
  14. 조용환 (1999). 질적 연구: 방법과 사례. 서울: 교육과학사.
  15. 최남광․유희찬 (2009). 영재교육에서 유추를 통한 데카르트 정리의 도입가능성 고찰, 대한수학교육학회지 수학교육학연구, 19(4), 479-491.
  16. 한인기․이상근 (2000). "유추"를 활용한 기하학습 자료 개발, 한국수학교육학회지 시리즈 F <수학교육 학술지지>, 5, 165-174.
  17. Alexander, P. A., White, C. S., & Daugherty, M. (1997). Children's use of analogical reasoning in early mathematics learning. In Lyn D. English(Ed.), Mathematical reasoning: Analogies, metaphors, and images(pp. 117-147). Mahwah, NJ: Lawrence Erlbaum Associates.
  18. Bassok, M. (1990). Transfer of domain-specific problem-solving procedures. Journal of Experimental Psychology: Learning, Memory, and Cognition, 16(3), 522-533. https://doi.org/10.1037/0278-7393.16.3.522
  19. Bassok, M., Wu, L. & Olseth, K. L. (1995). Judging a book by its cover: Interpretative effects of content on problem-solving transfer. Memory & Cognition, 23(3), 354-367. https://doi.org/10.3758/BF03197236
  20. Bourbaki, N. (1950). The architecture of mathematics. The American Mathematical Monthly, 57(4), 221-232. https://doi.org/10.2307/2305937
  21. Carpenter, P. A., Just M. A., & Shell, P. (1990). When one intelligence test measures: A theoretical account of the processing in the raven progressive matrices test. Psychological review, 97, 404-431. https://doi.org/10.1037/0033-295X.97.3.404
  22. Chen, Z. (1995). Analogical transfer: From schematic pictures to problem solving. Memory & Cognition, 23(2), 255-269. https://doi.org/10.3758/BF03197226
  23. Chen, Z. (2002). Analogical problem solving: A hierarchical analysis of procedural similarity, Journal of Experimental Psychology: Learning, Memory, and Cognition, 28, 81-98. https://doi.org/10.1037/0278-7393.28.1.81
  24. Chen, Z., Yanowitz, K. L., & Daehler, M. W. (1995). Constraints on accessing abstract source information:Instantiation of principles facilitates children's analogical transfer. Journal of Educational Psychology, 87(3), 445-454. https://doi.org/10.1037/0022-0663.87.3.445
  25. Creswell, J. W. (2005). 질적 연구방법론: 다섯 가지 전통. (조흥식, 정선욱, 김진숙, 권지성 역). 서울: 학지사. (원저 Qualitative inquiry and research design: Choosing among five traditions은 1998년 CA: Sage Publication에서 출판).
  26. Denney, N. W. (1972). A developmental study of free classification in children. Child Development, 43, 221-232. https://doi.org/10.2307/1127885
  27. English, L. D. (1997). Analogies, metaphors, and images: Vehicles for mathematical reasoning. In Lyn D. English(Ed.), Mathematical reasoning: Analogies, metaphors, and images(pp. 3-18). Mahwah, NJ: Lawrence Erlbaum Associates.
  28. Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht: D. Reidel Publishing Company.
  29. Gholson, B., Smither, D., Buhrman, A., Duncan, M. K., & Pierce, K. A. (1997). Children's development of analogical problem-solving skill. In Lyn D. English(Ed.), Mathematical reasoning: Analogies, metaphors, and images(pp. 149-190). Mahwah, NJ: Lawrence Erlbaum Associates.
  30. Gentner, D. (1982). Are scientific analogies metaphors? In David S. Miall(Ed.), Metaphor: Problems and perspectives(pp. 106-132). Atlantic Highlands, NJ: Humanities Press.
  31. Gentner, D. (1983). Structure-mapping: A theoretical framework for analogy. Cognitive science, 7, 155-170. https://doi.org/10.1207/s15516709cog0702_3
  32. Gentner, D. (1989). The mechanism of analogical learning. In Stella Vosniadou & Andrew Ortony (Eds.), Similarity and analogical reasoning. New York: Cambridge University Press. 199-241.
  33. Gentner, D., & Markman, A. B. (1997). Structure mapping in analogy and similarity. American psychologist, 52, 45-56. https://doi.org/10.1037/0003-066X.52.1.45
  34. Grick, M. L., & Holyoak, K. J. (1980). Analogical problem solving. Cognitive Psychology, 12, 306-355. https://doi.org/10.1016/0010-0285(80)90013-4
  35. Grick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15, 1-38. https://doi.org/10.1016/0010-0285(83)90002-6
  36. Holyoak, K. J. (2005). Analogy. In Holyoak, K. J. & Morrison, R. G.(Eds.), The cambridge handbook of thinking and reasoning(pp. 117-141). New York: Cambridge University Press.
  37. Holyoak, K. J., & Koh, K. (1987). Surface and structural similarity in analogical transfer. Memory & Cognition, 15(4), 332-340. https://doi.org/10.3758/BF03197035
  38. Holyoak, K. J., & Thagard, P. (1989). Analogical mapping by constraint satisfaction. Cognitive Science, 13, 295-355. https://doi.org/10.1207/s15516709cog1303_1
  39. Holyoak, K. J., & Thagard, P. (1997). Mental leaps: Analogy in creative thought. Cambridge, MA: MIT Press.
  40. Hummel, J. E., & Holyoak, K. J. (1992). Indirect analogical mapping. In Proceedings of the Fourteenth annual conference of the cognitive science society(pp. 516-521). Hillside, NJ: Lawrence Erlbaum Associates.
  41. Hummel, J. E., & Holyoak, K. J. (1997). Distributed representations of structure: A theory of analogical access and mapping. Psychological Review, 104, 427-466. https://doi.org/10.1037/0033-295X.104.3.427
  42. Hummel, J. E., & Holyoak, K. J. (2003). A symbolic-connectionist theory of relational inference and generalizayion. Psychological Review, 110, 220-264. https://doi.org/10.1037/0033-295X.110.2.220
  43. Hunt, E. B. (1974). Quote the raven? Nevermore! In Lee W. Greg(Ed.), Knowledge and cognition(pp. 129-157). Hillside, NJ: Lawrence Erlbaum Associates.
  44. Inhelder, B., & Piaget, J. (1964). The early growth of logic in the child. New York: Norton.
  45. Keane, M. T., & Costello, F. (2001). Setting limits on analogy: Why conceptual combinations is not structural alignment. In Dedre Gentner, Keith J. Holyoak, & Boicho N. Kokinov(Eds), The analogical mind : perspectives from cognitive science(pp. 287-312). Cambridge, MA: MIT Press.
  46. Ma, L. (2002). 초등학교 수학 이렇게 가르쳐라. (신현용, 승영조 역). 서울: 승산, (원저 Knowing and teaching elementary mathematics: Teacher's understanding of fundamental mathematics in China and the United States는 Mahwah, NJ: Lawrence Erlbaum Associates에서 출판).
  47. Markman, A. B. & Gentner, D. (1993). Structural alignment during similarity comparison. Cognitive Psychology, 23, 431-467.
  48. Markman, A. B. & Gentner, D. (1997). The effects of alignment on memory. Psychological Science, 8, 363-367. https://doi.org/10.1111/j.1467-9280.1997.tb00426.x
  49. Medin, D., & Ortony, A. (1989). Psychological essentialism. In Stella Vosniadou & Andrew Ortony(Eds.), Similarity and analogical reasoning(pp.179-196). New York: Cambridge University Press.
  50. Merriam, S. B. (2001). Qualitative research and case study applications in education(2nd ed.). San Francisco:Jossey-Bass Publishers.
  51. Miles M. B., & Huberman, A. M. (1994). Qualitative data analysis: A sourcebook of new methods(2nd ed.). Sage Publications, CA: Thousand Oaks.
  52. Novic, L. R., & Holyoak, K. J. (1991). Mathematical problem solving by analogy. Journal of experimental psychology: Learning, Memory, and Cognition, 17, 398-415. https://doi.org/10.1037/0278-7393.17.3.398
  53. Pierce, K. A. & Gholson, B. (1994). Surface Similarity and Relational Similarity in the Development of Analogical Problem Solving: Isomorphic and Nonisomorphic Transfer. Developmental Psychology, 30(5), 724-737. https://doi.org/10.1037/0012-1649.30.5.724
  54. Polya, G. (1954). Mathematics and Plausible Reasoning, Vol. I: Induction and analogy in mathematics. London: Oxford University Press.
  55. Reitman, W. R. (1965). Cognition and thought: An information-processing approach. New York: Wiley.
  56. Richland, L. E., Holyoak, K. J., & Stigler, J. W. (2004). Analogy use in eighth-grade mathematics classrooms. Cognition and instruction, 22, 37-60. https://doi.org/10.1207/s1532690Xci2201_2
  57. Ross, B. H. & Kilbane, M. C. (1997). Effects of Principle Explanation and Superficial Similarity on Analogical Mapping in Problem Solving. Learning, Memory, and Cognition, 23, 427-440. https://doi.org/10.1037/0278-7393.23.2.427
  58. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, Fla.: Academic Press..
  59. Smith, L. B. (1989). A model of perceptual classification in children and adults. Psychological Review, 96, 125-144. https://doi.org/10.1037/0033-295X.96.1.125
  60. Spearman, C. (1923). The nature of intelligence and the principles of cognition. London: MacMillian.
  61. Spradly, J. P. (1988). 참여관찰방법. (이희봉 역). 서울: 대한교과서주식회사, (원저 Participant observation은 1980년 New York: Holt, Rinehart and Winston에서 출판).
  62. Sternberg, R. J. (1977a). Component processes in analogical reasoning. Psychological Review, 84, 353-378. https://doi.org/10.1037/0033-295X.84.4.353
  63. Sternberg, R. J. (1977b). Intelligence, information processing, and analogical reasoning: The componential analysis of human abilities. Hillside, NJ: Lawrence Erlbaum Associates.
  64. VanLehn, K. A. (1983). Felicity conditions for human skill acquisition : validating an AI-based theory. Doctoral dissertation, Massachusetts Institute of Technology.
  65. Vygosky, L. S. (1962). Thought and language. Cambridge, MA: MIT Press.
  66. Wharton, C. M., Holyoak, K. J., Downing, P. E., Lange, T. E., Wickens, T. D., & Melz, E. R. (1994). Below the surface: Analogical similarity and retrieval competition in reminding. Cognitive Psychology, 26(1), 64-101. https://doi.org/10.1006/cogp.1994.1003