School Mathematics (대한수학교육학회지:학교수학)

The Korea Society of Educational Studies in Mathematics (대한수학교육학회)
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Analysis on the Argumentation in Exploring the Pick's Formula Using the Geoboard of Graphing Calculator in Math-Gifted 5 Grade Class

초등영재학급을 대상으로 그래핑 계산기의 지오보드를 활용한 Pick 공식의 탐구 과정에서 나타난 논증활동의 분석

  • Received : 2016.02.10
  • Accepted : 2016.03.14
  • Published : 2016.03.31
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Abstract

This study was to find characteristics of argumentation derived from a discourse in a math-gifted 5 grade class, which was held for finding a Pick's formula using Geoboard function of TI-73 calculator. For the analysis, a video record of the class, transcript of its voice record, and activity paper were used as data and Toulmin's argument schemes were applied as analysis standard. As a result of the study, we found that the graphing calculator helped the students to create an experimental environment that graphing a grid-polygon and figuring out its area. Furthermore, it also provided a basic demonstration through 'data->claim' composition and reasoning activities which consisted of guarantee, warrant, backing, qualifier and refutal for justifying. The basic argumentation during the process of deriving the Pick's theorem by the numbers of boundary points and inner points was developed into a 'collective argumentation' while a teacher took a role of a conductor of the argumentation and an authorizer on the knowledge at the same time.

이 연구는 5학년 영재반 수업에서 TI-73 그래핑 계산기의 지오보드를 사용하여 Pick의 공식을 찾아가는 과정에서 나타난 수업담화로부터 논증과 논증활동의 특성을 알아보고자 하였다. 분석을 위한 자료는 수업 비디오, 음성녹음록, 활동지가 있으며 Toulmin의 논증 도식을 분석의 준거로 사용하였다. 연구 결과 그래핑 계산기의 지오보드는 주어진 조건의 다양한 격자다각형에 대한 넓이를 계산해줌으로써 실험과 관찰의 환경을 조성하고 '자료${\rightarrow}$주장'의 구성과 이의 정당화를 위한 보증, 지지, 한정어, 반박의 논증활동을 유발시키는 도구적 역할을 하였다. 경계점의 수와 내점의 수로 Pick의 공식을 유도할 때 '집단적 논증'의 방식이 나타났으며 교사는 논증활동을 지휘하는 역할, 지식을 판단하는 권위자의 역할을 하였다.

Keywords

References

  1. 강남화, 이은경 (2013). 논변, 논의 그리고 논증 개념의 명료화를 위한 문헌조사연구. 한국과학교육학회지, 33(6), 1119-1138. https://doi.org/10.14697/JKASE.2013.33.6.1119
  2. 강영란 (2015). 계산기를 활용한 초등 수학 영재의 교실 활동에 관한 활동이론적 분석, 미출판 박사학위논문, 영남대학교 대학원, 경산.
  3. 강현영, 송은영, 조진우, 이경화 (2011). 수학.과학수업 교실문화 분석연구의 신뢰도 검증방법에 대한 고찰: 구성주의 수업관찰 프로토콜을 중심으로. 학습자 중심 교과 교육연구, 15(6), 643-667.
  4. 교육부 (2015). 2015 개정수학과 교육과정. 교육부.
  5. 김민주, 권오남 (2006). 사회적 상호작용 중심의 탐구지향학습에서 나타나는 학생들의 논증과 수학적 정당. 한국교육학회 교육학연구, 44(1), 247-275.
  6. 맹승호, 박영신, 김찬종 (2013). 논증 담화분석 연구의 방법론적 고찰: 논증활동의 협력적 구성과 인식적 실행의 분석을 중심으로. 한국과학교육학회지, 33(4), 840-862. https://doi.org/10.14697/JKASE.2013.33.4.840
  7. 박교식 (2007). 정사각형 칠교판의 일곱 조각으로 만들 수 있는 볼록 다각형의 탐색. 수학교육학 연구, 17(3), 221-232.
  8. 이윤경 (2016). 고등학교 확률통계 담화분석: Mehan의 이론, Toulmin의 논증패턴, Peirce의 가추법을 중심으로, 미출판 박사학위논문, 영남대학교 대학원, 경산.
  9. Berland, L. K., & Hammer, D. (2012). Framing for scientific argumentation. Journal of Research in Science Teaching, 49(1), 68-94. https://doi.org/10.1002/tea.20446
  10. Chinn, C., & Anderson, R. (1998). The structure of discussions intended to promote reasoning. The Teachers College Record, 100(2), 315-368.
  11. Conner, A. (2008). Argumentation in a geometry class: Aligned with the teacher's conception of proof. Retrived from http://math.coe.uga.edu/OLIVE/EMAT8990FYDS07/Conner%20Arg%20Geometry.pdf
  12. Boero, P. (1999). Argumentation and mathematical proof: A complex, productive, unavoidable relationship in mathematics and mathematics education. Retrived from http://www.lettredelapreuve.org/OldPreuve/Newsletter/990708Theme/990708ThemeUK.html.
  13. Douek, N. (1999). Some remarks about argumentation and mathematical proof and their educational implications. Retrived from http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/papers/g1-douek.pdf.
  14. Erduran, S., Simon, S., & Osborne, J. (2004). Tapping into argumentation: Developments in the application of Toulmin's argument pattern for studying science discourse. Science Education, 88(6), 915-933. https://doi.org/10.1002/sce.20012
  15. Evens, H., & Houssart, J. (2004). Categorizing pupils'written answers to a mathematics test question: I know but I can't explain. Educational Research, 46(3), 269-282. https://doi.org/10.1080/0013188042000277331
  16. Hanna, G. (1995). Challenges to the importance of proof. For the Learning of Mathematics, 15(3), 42-49.
  17. Hoyles, C., & Kuchemann, D. (2002). Students' understanding of logical implication. Educational Studies in Mathematics, 51(3), 193-223. https://doi.org/10.1023/A:1023629608614
  18. Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66(1), 3-21. https://doi.org/10.1007/s10649-006-9059-8
  19. Kieran, C., Forman, E., & Sfard, A. (2001). Bridging the individual and the social: Discursive approaches to research in mathematics education. Educational Studies in Mathematics 46(1), 42-49.
  20. Kim, H., & Song, J. (2006). The features of peer argumentation in middle school students' scientific inquiry. Research in Science Education, 36(3), 211-233. https://doi.org/10.1007/s11165-005-9005-2
  21. Knipping, C. (2003). Argumentation structures in classroom proving situations. Retrived from http://lettredelapreuve.org/OldPreuve/CERME3Papers/Knipping-paper.pdf.
  22. Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb, & H. Bauersfeld (eds.), The Emergence of Mathematical Meaning: Interaction in Classroom Cultures (pp. 229-269). Hillsdale: Lawrence Erlbaum Associates.
  23. Kutzler, B. (2003). CAS as pedagogical tools for teaching and learning mathematics. In J. T. Fey, A. Cuoco, C. Kieran, L. McMullin, & R. M. Zbiek (Eds.), Computer algebra systems in secondary school mathematics education (pp. 53-71). Reston, VA: NCTM.
  24. Lampert, M., & Cobb, P. (2003). Communication and language. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to NCTMs principles and standards (pp. 237-249). Reston, VA: NCTM.
  25. Magalhas, M., & Martinho, M. H. (2012). The role of graphical calculator in developing mathematial argumentation. Retrived from http://www.icme12.org/upload/UpFile2/TSG/1308.pdf.
  26. Maloney, J., & Simon, S. (2006). Mapping children's discussions of evidence in science to assess collaboration and argumentation. International Journal of Science Education, 28(15), 1817-1841. https://doi.org/10.1080/09500690600855419
  27. Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed?. Educational Studies in Mathematics. 66(1), 23-41. https://doi.org/10.1007/s10649-006-9057-x
  28. Scandrett, H. (2008). Using Geoboards in Primary Mathematics: Going... Going... Gone?. Australian Primary Mathematics Classroom, 13, 29-32.
  29. Toulmin, S. E. (2003). The uses of argument (Updated edition). NY: Cambridge University Press.
  30. Toulmin, S. E. (2006). 논변의 사용 (고현범, 임건태 역). 서울: 고려대학교출판부.
  31. Trouche, L. (2005). Calculators in Mathematics Education: A rapid evolution of tools with differential effects. In D. Guin, K. Ruthven, & L. Trouche (Eds.), The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument (pp. 9-39). New York: Springer.
  32. Weber, K., & Alcock, L. (2005). Using warranted implications to understand and validate proofs. For the Learning of Mathematics, 25(1), 34-38.
  33. Weber, K., Maher, C. A., Powell, A. B., & Lee, H. S. (2008). Learning opportunities from group discussions: Warrants become the objects of debate. Educational Studies in Mathematics, 68(3), 247-261. https://doi.org/10.1007/s10649-008-9114-8
  34. Yackel, E. (2001), Explanation, justification and argumentation in mathematics classrooms. Retrived from http://files.eric.ed.gov/fulltext/ED466631.pdf.