East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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On the design of a teaching unit for the exploration of number patterns in Pascal graphs and triangles applying theoretical generalization.

이론적 일반화를 적용한 파스칼 그래프와 삼각형에 내재된 수의 패턴 탐구를 위한 교수단원의 설계

  • Kim, Jin Hwan (Department of Mathematics Education Yeungnam University)
  • Received : 2024.01.26
  • Accepted : 2024.02.28
  • Published : 2024.02.29
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Abstract

In this study, we design a teaching unit that constructs Pascal graphs and extended Pascal triangles to explore number patterns inherent in them. This teaching unit is designed to consider the diachronic process of teaching-learning by combining Dörfler's theoretical generalization model with Wittmann's design science ideas, which are applied to the didactical practice of mathematization. In the teaching unit, considering the teaching-learning level of prospective teachers who studied discrete mathematics, we generalize the well-known Pascal triangle and its number patterns to extended Pascal triangles which have directed graphs(called Pascal graphs) as geometric models. In this process, the use of symbols and the introduction of variables are exhibited as important means of generalization. It provides practical experiences of mathematization to prospective teachers by going through various steps of the generalization process targeting symbols. This study reflects Wittmann's intention in that well-understood mathematics and the context of the first type of empirical research as structure-genetic didactical analysis are considered in the design of the learning environment.

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References

  1. 김남희 (1997). 일반화의 의미와 구성에 대한 이해. 수학교육학연구, 7(1), 445-458.
  2. 김진환, 박교식 (2006). 예비중등교사의 수학화 경험을 위한 교수단원의 설계: 수 분할 모델의 탐구. 한국학교수학회논문집, 9(1), 57-76.
  3. 김진환, 박교식 (2008). 예비중등교사의 수학화 학습을 위한 교수단원의 설계: 분할모델과 일반화된 피보나치수열 사이의 관계 탐구. 수학교육학연구, 18(3), 373-389.
  4. 김진환, 박교식 (2009). 일반화된 피보나치수열의 탐구를 위한 예비중등교사용 교수단원의 설계. 학교수학, 11(2), 243-260.
  5. 김진환, 박교식, 이광호 (2006). 일정한 차를 갖는 수 분할 모델의 탐구를 위한 예비중등교사용 수학화 교수단원의 설계. 학교수학, 8(2), 161-176.
  6. 박교식 (2003). 수학화 교수.학습을 위한 소재 개발 연구: 격자 직사각형의한 대각선이 지나는 단위 정사각형의 수와 그 일반화. 수학교육학연구, 13(1), 57-75.
  7. 박교식 (2006). 수학화 교수.학습을 위한 교수단원 디자인 연구: 브레트슈나이더 공식의 재발명. 학교수학, 8(3), 327-339.
  8. 박교식외 (2016). 고등학교 수학. 동아출판.
  9. 박교식외 (2019). 고등학교 확률과 통계. 동아출판.
  10. 반은섭, 류희찬 (2017). 동적 기하 환경을 활용한 문제해결 과정에서 변수이해 및 일반화 수준 향상에 관한 사례연구. 수학교육학연구, 27(1), 89-112.
  11. 박종안, 서승현, 이재진, 이준열 (2018). 이산수학. 경문사.
  12. 조진석 (2021). 수학화 경험을 위한 교사교육 교수단원 설계: 수학에서의 방향(orientation) 개념 탐구. 한국초등수학교육학회지, 25(1), 43-59.
  13. 최근배 (2017). 초등영재 학생의 수학화 학습을 위한 교수단원 설계: 삼.사각형의 등주문제 탐구. 수학교육논문집, 31(2), 223-239.
  14. Akinwunmi, K., Hoveler, K. and Schnell, S. (2014). On the Importance of Subject Matter in Mathematics Educatin: A Conversation with Erich Christian Wittmann. Eurasia Journal of Mathematics, Science & Technology Education, 10(4), 357-363.
  15. Arnon, I., Cottrill, J. Dubinsky, E., Oktac, A., Roa, S., Trigueros, M., & Weller, K. (2014). APOS Theory: A Framework for Research and Curriculum Development in Mathematics Education, Springer, NY, Heidelberg, Dondrecht, London.
  16. Ball, D. et al. (2008). Content knowledge for teaching: What makes it special?. Journal of Teacher Education, 59 (5), 389-407.
  17. Cooney, T., Davis, E. and Henderson, K. (1975). Dynamics of Secondary School Mathematics. Houghton Mifflin.
  18. Dorfler, W. (1991). Forms and Means of Generalization in Mathematics. In Bishop, A.J. (ed.), Mathematical Knowledge: Its Growth Through Teaching, Kluwer Academic Publishers, 63-85.
  19. Freudenthal, H. (1973). Mathematics as an Educational Task. D. Reidel Publishing Company.
  20. Freudenthal, H. (1983). Didactical Phenomenology of Mathematical Structures. Kluwer Academic Publishers.
  21. Freudenthal, H. (2008). 프로이덴탈의 수학교육론, 우정호외 공역, 서울: 경문사(영어 원작은 1991 출판)
  22. Fujita, T. and Yamamoto, S. (2011). The development of children's understanding of mathematical patterns through mathematical activities. Research in Mathematics Education 13(3), 249-267.
  23. Fung, C. I. (2016). Developing mathematics teaching and mathematics teachers. In M. Nuhrenborger et al. (Eds.), Design science and its importance in the German mathematics educational discussion. Springer Open.
  24. Iwasaki, H. and Yamaguchi, T. (1997). The Cognitive and Symbolic Analysis of the Generalization Process. Proceedings of the 21st Conference of the PME, vol.3, 105-112.
  25. Iwasaki, H. and Yamaguchi, T. (2003), The Incorporation of Generalization in Teaching Units. International Journal of Curriculum Development and Practice, 5(1), 25-35, Japan Curriculum Research and Development Association.
  26. Iwasaki, H. and Yamaguchi, T. (2005), Design, Practice and Evaluation of Teaching Units: Research Based on the Separation Model of Generalization. Journal of Science Education in Japan, 29(2), 133-145.
  27. Klein, F. (1968). Elementary mathematics from an advanced standpoint: arithmetic.algebra.analysis. EE. R. Hendric, C.A. Noble (trans.) New York: Dover Publications. (원작은 1924년 출판).
  28. LeBlanc, J. F. (1975). The mathematics methods program and elementary teacher preparation program in mathematics. In H. Bauersfeld et al. (Eds), Proceedings of the Conference on Tendencies and Problems of Teacher Education in Mathematics Vol. 6, pp. 393-423. Bielefeld: Schriftenreihe des IDM.
  29. Nuhrenborger, M. et al., (2016). Design Science and Its Importance in the German Mathematics Educational Discussion. ICME-13 Topical Surveys.
  30. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15 (2), 4-14.
  31. Simon, H. A. (1970). The sciences of the artificial. Boston: MIT-Press.
  32. Treffers, A. (1987). Three dimensions: a model of goal and theory description in mathematics education. Dordrecht: D. Reidel Publishing Company.
  33. Wittmann, E. (1984). Teaching units as the integrating core of mathematics education. Educational Studies in Mathematics 15(1), 25-36.
  34. Wittmann, E. (1995). Mathematics education as a design science. Educational Studies in Mathematics 29(4), 355-374.
  35. Wittmann, E. (1999). Designing teaching: The Pythagorean theorem. In T. J. Cooney, et al., Mathematics, Pedagogy, and Secondary Teacher Education, 97-165. Portsmouth, NH: Heinemann.
  36. Wittmann, E. (2001). Developing Mathematics Education in a Systemic Process. Educational Studies in Mathematics 48(1), 1-20.
  37. Wittmann, E. (2019). Understanding and organizing Mathematics Education as a Design Science-Origins and New Developments. HIROSHIMA JOURNAL OF MATHEMATICS EDUCATION 12(13), 13-32.
  38. Wittmann, E. (2021). Teaching Units as the Integrating Core of Mathematics Education, Connecting Mathematics and Mathematics Education, Springer.