Education of Primary School Mathematics (한국수학교육학회지시리즈C:초등수학교육)

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

DOI QR Code

Effects of Mathematical Justification on Problem Solving and Communication

수학적 정당화가 문제 해결과 의사소통에 미치는 영향

  • Received : 2013.11.17
  • Accepted : 2013.12.25
  • Published : 2013.12.31
Download PDF
⟨ Previous Next ⟩

Abstract

Mathematical justification is the process through which one's claim is validated to be true based on proper and trustworthy data. But it serves as a catalyst to facilitate mathematical discussions and communicative interactions among students in mathematics classrooms. This study is designed to investigate the effects of mathematical justification on students' problem-solving and communicative processes occurred in a mathematics classroom. In order to fulfill the purpose of this study, mathematical problem-solving classes were conducted. Mathematical justification processes and communicative interactions recorded in problem understanding activity, individual student inquiry, small and whole group discussions are analyzed. Based on the analysis outcomes, the students who participated in mathematical justification activities are more likely to find out various problem-solving strategies, to develop efficient communicative skills, and to use effective representations. In addition, mathematical justification can be used as an evaluation method to test a student's mathematical understanding as well as a teaching method to help develop constructive social interactions and positive classroom atmosphere among students. The results of this study would contribute to strengthening a body of research studying the importance of teaching students mathematical justification in mathematics classrooms.

수학적 정당화란 일반적으로 적절한 근거에 기초하여 자신의 주장이 참임을 보이는 과정이라고 할 수 있다. 하지만 교실 실제에서의 수학적 정당화는 사회적 상호작용을 바탕으로 수학적 의사소통을 촉진하는 역할을 한다고 할 수 있다. 이에 본 연구는 수학적 정당화 활동이 학생들의 문제해결과 의사소통 과정에 미치는 영향을 조사하고자 하였다. 이를 위해 수학적 정당화 활동이 강조되는 문제해결 중심 수업을 실시하고 문제 이해 활동, 개별 탐구 활동, 소집단 토의 활동, 전체 논의 과정에서의 수학적 정당화 활동과 의사소통 과정을 분석하였다. 연구 결과 수학적 정당화 활동은 학생들이 다양한 문제해결 방법을 찾는데 도움을 주었고 의사소통 과정을 촉진하였으며, 다양한 표현 방법을 사용하도록 자극하였다. 또한 수학적 정당화 활동은 학생들의 이해를 평가하는 방법이 될 수 있으며, 교실에서의 사회적 관계 및 역동적인 교실 문화를 형성하는데 기여하였다.

Keywords

References

  1. 교육과학기술부 (2011). 수학과 교육과정. 교육과학기술부 고시 제 2011-361호 [별책 8]. Ministry of Education, Science and Technology (2011). Mathematics curriculum. Notification No. 2011-361 [supplement 8] of the Ministry of Education, Science and Technology.
  2. 권성룡 (2003). 초등학생의 수학적 정당화에 관한 연구. 한국수학교육학회지 시리즈 C <초등수학교육>, 7(2), 85-99. Kwon, S. Y. (2003). A study on mathematical justification activities in elementary school. J ournal of the Korean Society of Mathematical Education Series C: Education of Primary School Mathematics, 7(2), 85-99.
  3. 김민주.권오남 (2006). 사회적 상호작용 중심의 탐구지향학습에서 나타나는 학생들의 논증과 수학적 정당화. 교육학연구, 44(1), 247-275. Kim, M. J., & Kwon, O. N. (2006). Students' argumentation and mathematical justification in inquiry-oriented learning. Korean Journal of Educational Research, 44(1), 247-275.
  4. 김성준 (2003). 패턴과 일반화를 강조한 대수 접근법 고찰. 대한수학교육학회지 <학교수학>, 5(3), 343-360. Kim, S. J. (2003). A study on approaches to algebra focusing on patterns and generalization. J ournal of Korea Society of Educational Studies in Mathematics: School Mathematics, 5(3), 343 -360.
  5. 김정하.강문봉 (2009). 초등학교 교사들의 수학적 정당화에 대한 연구. 대한수학교육학회지 수학교육학연구 , 19(3), 371-392. Kim, J. H., & Kang, M. B. (2009). A study on mathematical justification of elementary school teachers. J ournal of Educational Research in Mathematics, 19(3), 371-392.
  6. 김지영.박만구 (2011). 수학 영재 교육 대상 학생의 기하 인지 수준과 증명 정당화 특성 분석. 한국수학교육학회지 시리즈 C <초등수학교육>, 14(1), 13-26. Kim, J. Y., & Park, M. G. (2011). An analysis of justification process in the proofs by mathematically gifted elementary students. Journal of the Korean Society of Mathematical Education Series C:Education of P rimary School Mathematics, 14(1), 13-26.
  7. 서은미.류희찬 (2009). 탐구형 기하 소프트웨어 환경에서 나타나는 초등학생의 수학적 정당화. 한국교원대학교 수학교육연구소, 청람수학교육, 1(1), 39-65. Seo, E. M., & Lew, H. C. (2009). Case study of the elementary students' mathematical justification displayed in the process of solving geometry problems using dynamic geometry software. KNUE Journal of Mathematics Education, 1(1), 39-65.
  8. 서지수, 류성림 (2012). 수와 연산.도형 영역에 서 초등 3학년 학생들의 수학적 정당화 유형에 관한 연구. 한국수학교육학회지 시리즈 E<수학교육 논문집>, 26(1), 85-108. Seo, J. S., & Ryu, S. R. (2012). A study the types of mathematical justification shown in elementary school students in number and operations, and geometry. Journal of the Korean Society of Mathematical Education Series E: Communications of Mathematical Education, 26(1), 85-108.
  9. Balacheff, N. (1987). Processes of proof and situation of validation. Educational Studies in Mathematics, 18(2), 147-176. https://doi.org/10.1007/BF00314724
  10. Carpenter, T. P., Fanke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic & algebra in elementary school. Portmouth, NH: Heinmann.
  11. Cobb, P., McClain, K., & Gravemeijer, K. (2005). Learning about statistical covariation. Cognition and Instruction, 21(1). 1-78.
  12. Cobb, P., Yackel, E., & McNeal, B. (1992). Characteristics of classroom mathematics traditions: An interactional analysis. American Educational Research Journal, 29(3), 573-604. https://doi.org/10.3102/00028312029003573
  13. Conner, A. (2012). Warrants as indications of reasoning patterns in secondary mathematics classes. Paper presented at the 12th International Congress on Mathematical Education(TSG-14, 2984-2992), Seoul, Korea.
  14. Council of Chief State School Officers(CCSSO) & National Governors Association Center for Best Practices(NGA Center). (2010). Common Core State Standards for Mathematics. http://www.corestandards.org/
  15. Forman, E. A., Larreamendy-Joerns, J., Stein, M. K., & Brown, C. A. (1998). You're going to want to find which and prove it:Collective argumentation in a mathematics classroom. Learning and Instruction 8(6). 527-548. https://doi.org/10.1016/S0959-4752(98)00033-4
  16. Hanna, G. (1990). Some pedagogical aspects of proof. Interchange 21(1), 6-13. https://doi.org/10.1007/BF01809605
  17. Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1), 5-23. https://doi.org/10.1023/A:1012737223465
  18. Krummheuer, G. (1995). The ethnology of argumentation. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures(pp. 229-269). Hisdales, Nj: Erbaum.
  19. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. NY: Cambridge University Press.
  20. Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231-258. https://doi.org/10.1207/s15327833mtl0703_3
  21. Luhmann, N. (1995). Social systems. Stanford, CA: Stanford University Press.
  22. Maturana, H. R., & Varela, F. J. (1992). The tree of knowledge: The biological roots of human understanding. Boston: Shambhala.
  23. NCTM (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
  24. Polya, G. (1968). Mathematics and plausible reasoning. (2nd ed.). Princeton, NJ:Princeton University Press.
  25. Popper, K. (1963). Conjectures and refutations:The growth of scientific knowledge. London:Routledge and Kegan Paul.
  26. Rigelman, N. R. (2007). Fostering mathematical thinking and problem solving: The teacher’s role. Teaching Children Mathematics, 13(6), 308-314.
  27. Reid, D. A. (2002). Conjectures and refutations in grade 5 mathematics. Journal for Research in Mathematics Education, 33(1), 5-29. https://doi.org/10.2307/749867
  28. Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teacher. Journal of Mathematics Teacher, 15(1), 3-31.
  29. Sowder, L., & Harel, G. (1998). Types of student's justification. The Mathematics Teacher , 91(8), 670-675.
  30. Staples, M. E., Bartlo, J., & Thanheiser, E. (2012). Justification as a teaching and learning practice: Its (potential) multifacted role in middle grades mathematics classrooms. Journal of Mathematical Behavior, 31(4), 447-462. https://doi.org/10.1016/j.jmathb.2012.07.001
  31. Toulmin, S. E. (2003). The uses of argument(updated ed.). NY: Cambridge University Press. Originally published in 1958.
  32. Varela, F., Thompson, E., & Rosch, E. (1991). The embodied mind: Cognitive science and human experience. Cambridge, MA: MIT Press.
  33. Yackel, E. (2002). What we can learn from analyzing the teacher’s role in collective argumentation. Journal of Mathematical Behavior, 21(4), 423-440. https://doi.org/10.1016/S0732-3123(02)00143-8
  34. Yackel, E. (2004). Theoretical perspectives for analyzing explanation, justification and argumentation in mathematics classrooms. Journal of the Korea of Mathematical Education Series D: Research in Mathematical Education, 8(1), 1-18.
  35. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education. 27(4), 458-477. https://doi.org/10.2307/749877
  36. Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49(3), 379-402. https://doi.org/10.1023/A:1020291317178