Journal of the Korean School Mathematics Society (한국학교수학회논문집)

The Korean School Mathematics Society (한국학교수학회)
권호 표지
DOI QR코드

DOI QR Code

Students' Problem Solving Based on their Construction of Image about Problem Contexts

문제맥락에 대한 이미지가 문제해결에 미치는 영향

  • Received : 2020.02.21
  • Accepted : 2020.03.22
  • Published : 2020.03.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this study, we presented two geometric tasks to three 11th grade students to identify the characteristics of the images that the students had at the beginning of problem-solving in the problem situations and investigated how their images changed during problem-solving and effected their problem-solving behaviors. In the first task, student A had a static image (type 1) at the beginning of his problem-solving process, but later developed into a dynamic image of type 3 and recognized the invariant relationship between the quantities in the problem situation. Student B and student C were observed as type 3 students throughout their problem-solving process. No differences were found in student B's and student C's images of the problem context in the first task, but apparent differences appeared in the second task. In the second task, both student B and student C demonstrated a dynamic image of the problem context. However, student B did not recognize the invariant relationship between the related quantities. In contrast, student C constructed a robust quantitative structure, which seemed to support him to perceive the invariant relationship. The results of this study also show that the success of solving the task 1 was determined by whether the students had reached the level of theoretical generalization with a dynamic image of the related quantities in the problem situation. In the case of task 2, the level of covariational reasoning with the two varying quantities in the problem situation was brought forth differences between the two students.

본 연구에서는 고등학교 2학년 학생 3명을 대상으로 기하 영역의 두 가지 과제를 제시하여 학생들이 문제 상황에서 문제해결 초기에 갖는 이미지의 특성을 파악하고 각 학생들의 이미지가 문제를 해결하는 동안 어떻게 변화하며 영향을 미치는지 밝히고자 하였다. 첫 번째 과제에서 학생 A는 문제해결 과정 초기에 정적인 이미지(유형1)를 가지고 있었지만, 후에 동적이면서도 문제 상황에서의 양들 사이의 불변의 관계를 인식한 유형3으로 발전하였고 학생 B와 학생 C는 문제해결 과정 전반에 걸쳐 유형3으로 관찰되었다. 첫 번째 과제에서 학생 B와 학생 C의 문제맥락에 대한 이미지에 차이점이 발견되지 않았지만 두 번째 과제에서는 분명한 차이를 드러내었다. 두 번째 과제에서 학생 B와 학생 C 모두 문제맥락에 대한 동적인 이미지를 가지고 있었지만 학생 B의 경우 양들 사이의 불변의 관계를 인식하지 못하였고 학생 C는 불변의 관계를 인식하는 잘 발달된 양적 구조를 가지고 있었다. 이에 따라 각 과제의 문제해결 성공 여부가 좌우되었는데, [과제1]에서는 문제 상황에서의 양들에 대한 동적인 이미지를 갖고 이론적 일반화 수준에 도달했는지의 여부에 의해서, [과제2]의 경우에는 문제 상황에서의 두 양에 대한 공변 추론 수준에 따라 학생들 간의 차이가 발생하였다.

Keywords

References

  1. 교육부(2015). 수학과 교육과정. 교육과학기술부 고시 제2015-74호 [별책 8].
  2. 김남희, 나귀수, 박경미, 이경화, 정영옥, 홍진곤(2011). 수학교육과정과 교재연구. 서울: 경문사.
  3. 우정호(1999). 학교수학의 교육적 기초. 서울: 서울대학교출판부.
  4. 우정호, 정영옥, 박경미, 이경화, 김남희, 나귀수, 임재훈(2006). 수학교육학 연구방법론. 서울: 경문사.
  5. 권영인, 서보억(2007). 고등학교 도형의 방정식 단원에서 논증기하의 활용에 대한 연구. 한국수학교육학회 시리즈 E. 수학교육논문집, 21(3), 451-466.
  6. 김근배, 최옥환, 박달원(2018). 유추와 분석적 방법을 활용한 타원 초점 작도. 한국학교수학회논문집, 21(4), 401-418.
  7. 김성준(2002). 대수적 사고와 대수 기호에 관한 고찰. 수학교육학회, 12(2), 229-245.
  8. 김희, 김선희(2010). 기하 증명에서 기호의 역할과 기호 중재에 의한 직관의 형성. 수학교육학연구, 20(4), 511-528.
  9. 나귀수(1997). 기하 개념의 이해와 적용에 관한 소고. 수학교육학연구, 7(2), 349-358.
  10. 나귀수(2009). 분석법을 중심으로 한 기하 증명 지도에 대한 연구. 수학교육학연구, 19(2), 185-206.
  11. 도정철, 손홍찬(2015). GSP를 사용한 기하수업에서 수준별 학생의 논증기하와 해석기하의 연결에 관한 연구. 한국학교수학회논문집, 18(4), 411-429.
  12. 마민영, 신재홍(2016). 대수 문장제의 해결에서 드러나는 중등 영재 학생간의 공변 추론 수준 비교 및 분석. 학교수학, 18(1), 43-59.
  13. 박종희, 신재홍, 이수진, 마민영(2017). 그래프 유형에 따른 두 공변 추론 수준 이론의 적용 및 비교. 수학교육학연구, 27(1), 23-49.
  14. 반은섭, 신재홍, 류희찬(2016). 오마르 카얌(Omar Khayyam)이 제시한 삼차방정식의 기하학적 해법의 교육적 활용. 학교수학, 18(3), 589-608.
  15. 반은섭, 류희찬(2017). 동적 기하 환경을 활용한 문제 해결 과정에서 변수 이해 및 일반화 수준 향상에 관한 사례연구. 수학교육학연구, 27(1), 89-112.
  16. 손홍찬(2011). GSP를 활용한 역동적 기하 환경에서 기하적 성질의 추측. 학교수학, 13(1), 107-125.
  17. 양은경, 신재홍(2014). 개방형 기하 문제에서 학생의 드래깅 활동을 통해 나타난 수학적 추론 분석. 수학교육학연구, 24(1), 1-27.
  18. 양은경, 신재홍(2015). 역동적 기하 환경에서 중등 영재학생들의 합동변환 활동에 대한 발생적 분해. 수학교육학연구, 25(4), 499-524.
  19. 장혜원(2013). Byrne의 'Euclid 원론'에 기초한 증명 지도에 대한 연구. 수학교육학연구, 23(2), 173-192.
  20. 정영우, 김부윤(2015). 기하 증명에서의 대표성에 관한 연구. 수학교육학연구, 25(2), 225-240.
  21. Boyer, C. B. (1946). Proportion, equation, function: Three steps in the development of a concept. Scripta Mathematica, 12, 5-13.
  22. Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352-378. https://doi.org/10.2307/4149958
  23. Carlson, M. P., Smith, N., & Persson, J. (2003). Developing and connecting calculus students' notions of rate of change and accumulation: The fundamental theorem of calculus. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the Joint Meeting of PME and PMENA (Vol. 2, pp. 165-172). Honolulu, HI: CRDG, College of Education, University of Hawai'i.
  24. Confrey, J. & Smith, E. (1994). Exponential functions, rate of change, and the multiplicative unit. Educational Studies in Mathematics, 26, 135-164. https://doi.org/10.1007/BF01273661
  25. Davydov, V. V. (1990). Types of generalisation in instruction: Logical and psychological problems in the structuring of school curricula (Soviet studies in mathematics education, Vol. 2; J. Kilpatrick, Ed., J. Teller, Trans.).Reston, VA: National Council of Teachers of Mathematics. (Original work published 1972)
  26. Dindyal, J. (2004). Algebraic thinking in geometry at high school level: Students' use of variables and unknowns. In I. Putt, R. Faragher & M. McLean (Eds.), Mathematics education for the third millennium: Towards 2010 (Proceedings of the 27th annual conference of the Mathematics Education Research Group of Australasia, Townsville) (pp. 183-190). Sydney: MERGA, Inc.
  27. Dindyal, J. (2007). The need for an inclusive framework for students' thinking in school geometry. The Montana Mathematics Enthusiast, 4(1), 73-83.
  28. Duval, R. (2002). Representation, vision, and visualization: Cognitive functions in mathematical Thinking. Basic Issues for Learning. In F. Hitt (Ed.), Representations and mathematics visualization (pp. 311-336). Mexico: PME-NA-Cinvestav-IPN.
  29. Ellis, A. B. (2011). Algebra in the middle school: Developing functional relationship through quantitative reasoning. In J. Cai, & E. Knuth (Eds.), Early algebraization (pp. 215-238): Springer-Verlag Berlin Heidelberg.
  30. Herbert, K., & Brown, R. (1999). Patterns as tools for algebraic thinking. In B. Moses (Ed.), Algebraic thinking: Grades K - 12 (pp. 123-128). Reston, VA: National Council of Teachers of Mathematics.
  31. Hoffer, A. (1981). Geometry in more than proof. Mathematics Teacher, 74, 11-18.
  32. Kaput, J. (1995). Long term algebra reform: Democratizing access to big ideas. In C. Lacampagne, W. Blair, & J. Kaput (Eds.), The Algebra Initiative Colloquium (pp. 33-52). Washington, DC: U.S. Department of Education.
  33. Kieran, C. (1996). The changing face of school algebra. In 8th International Congress on Mathematical Education, Selected Lectures (pp. 271-286). S.A.E.M. THALES.
  34. Mason, J. H. (2002). Generalisation and algebra: Exploiting children's powers. In L. Haggerty (Ed.), Aspects of teaching secondary mathematics: Perspectives on practice (pp.105-120). London: RoutledgeFalmer.
  35. Merriam, S. B. (1998). Qualitative research and case study applications in education. San Francisco, CA: Jossey-Bass.
  36. Mitchelmore, M. (1993). Abstraction, generalization and conceptual change in mathematics. Hiroshima Journal of Mathematics Education, 2, 45-57.
  37. Mitchelmore, M. C., & White P. (1995). Abstraction in mathematics: Conflict, resolution and application. Mathematics Education Research Journal, 7, 50-68. https://doi.org/10.1007/BF03217275
  38. Mitchelmore, M. C., & White P. (1999). Learning mathematics: A new look at generalisation and abstraction. Referred paper at the combined conference of the Australian and New Zealand Associations for Research in Education, Australia.
  39. Moore, K. C., & Carlson, M. P. (2012). Students’ images of problem contexts when solving applied problems. The journal of Mathematical Behavior, 31(1), 48-59. https://doi.org/10.1016/j.jmathb.2011.09.001
  40. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
  41. Saldanha, L. A., & Thompson, P. W. (1998). Re-thinking co-variation from a quantitative perspective: Simultaneous continuous variatin. In S. B. Berensen &, K. R. Dawkins, M. Blanton, W. N. Coulombe, J. Kolb, K. Norwood, & L. Stiff(Eds.), Proceedings of the 20th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 298-303). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
  42. Schoenfeld, A. H. & Arcavi, A. (1988). On the meaning of variable. Mathematics Teacher, 81, 420-427.
  43. Smith, J., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95-132). New York: Erlbaum.
  44. Steffe, L., & Izsak, A. (2002). Pre-service middle-school teachers' construction of linear equation concepts through quantitative reasoning. In D. Mewborn, P. Sztajn, D. White, H. Wiegel, R. Bryant, & K. Noony (Eds.), Proceedings of the Twenty-Fourth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 1163-1172). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
  45. Thompson, P. W. (1989). A cognitive model of quantity-based algebraic reasoning. Paper presented at the annual meeting of the American Educational Research Association.
  46. Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181-234). Albany, NY: SUNY Press.
  47. Thompson, P. W. (2016). Researching mathematical meanings for teaching. In English, L., & Kirshner, D. (Eds.), Handbook of international research in mathematics education (pp. 435-461). London: Taylor and Francis.
  48. Thompson, P. W., Hatfield, N., Joshua, S., Yoon, H., & Byerley, C. (2017). Covariational reasoning among U.S. and South Korean secondary mathematics teachers. The Journal of Mathematical behavior, 48, 95-111. https://doi.org/10.1016/j.jmathb.2017.08.001
  49. Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421-456). Reston, VA: National Council of Teachers of Mathematics.
  50. Usiskin, Z. (1988). Conceptions of school algebra and uses of variable. In A. F. Coxford & A. P. Shulte (Eds.), The ideas of algebra, K-12 (1988 Yearbook of the National Council of Teachers of Mathematics, pp. 8-19). Reston, VA: NCTM.