한국수학교육학회지시리즈D:수학교육연구 (Research in Mathematical Education)

  • 제25권1호
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  • Pages.65-89
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  • 2022
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  • 1226-6191(pISSN)
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  • 2287-9943(eISSN)
한국수학교육학회 (Korean Society of Mathematical Education)
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Secondary Teachers' Views about Proof and Judgements on Mathematical Arguments

  • Kim, Hangil (STEM Education, University of Texas at Austin)
  • 투고 : 2022.01.27
  • 심사 : 2022.03.21
  • 발행 : 2022.03.31
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초록

Despite its recognition in the field of mathematics education and mathematics, students' understanding about proof and performance on proof tasks have been far from promising. Research has documented that teachers tend to accept empirical arguments as proofs. In this study, an online survey was administered to examine how Korean secondary mathematic teachers make judgements on mathematical arguments varied along representations. The results indicate that, when asked to judge how convincing to their students the given arguments would be, the teachers tended to consider how likely students understand the given arguments and this surfaces as a controversial matter with the algebraic argument being both most and least convincing for their students. The teachers' judgements on the algebraic argument were shown to have statistically significant difference with respect to convincingness to them, convincingness to their students, and validity as mathematical proof.

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과제정보

I would like to thank Drs. Eric J. Knuth, Carlos Nicholas Gomez Marchant, and Flavio S. Azevedo for their thoughtful comments on the earlier versions of the manuscript. I extend thanks to the editor in chief and anonymous reviewers for their efforts in providing constructive feedback on the manuscript and College of Education at The University of Texas at Austin for providing the funding for this study to collect the survey data.

참고문헌

  1. Alcock, L., & Inglis, M. (2008). Doctoral students' use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69(2), 111-129. https://doi.org/10.1007/s10649-008-9149-x
  2. Almeida, D. (2000). A survey of mathematics undergraduates' interaction with proof: some implications for mathematics education. International Journal of Mathematical Education in Science and Technology, 31(6), 869-890. https://doi.org/10.1080/00207390050203360
  3. Australian Curriculum, Assessment and Reporting Authority. (2015). Australian Curriculum: Mathematics. Retrieved from https://australiancurriculum.edu.au/f10-curriculum/mathematics/
  4. Balacheff, N. (1988). Aspects of proof in pupils' practice of school mathematics. In D. En Pimm (Ed.), Mathematics, teachers and children (pp. 216-235). London, England: Hodder & Stoughton.
  5. Balacheff, N. (2002). A researcher epistemology: A deadlock for educational research on proof. In Proceedings of the 2002 international conference on mathematics: Understanding proving and proving to understand (pp. 23-44). Department of Mathematics: National Taiwan Normal University.
  6. Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The teaching of proof. In L. I. Tatsien (Ed.), Proceedings of the international congress of mathematicians (Vol. III, pp. 907-920). Beijing, China: Higher Education Press.
  7. Basturk, S. (2010). First-year secondary school mathematics students' conceptions of mathematical proofs and proving. Educational Studies, 36(3), 283-298. https://doi.org/10.1080/03055690903424964
  8. Bell, A. (1976). A study of pupils' proof-explanations in mathematical situations, Educational Studies in Mathematics, 7, 23-40. https://doi.org/10.1007/BF00144356
  9. Bieda, K. N. (2010). Enacting proof-related tasks in middle school mathematics: challenges and opportunities. Journal for Research in Mathematics Education, 41(4), 351-382. https://doi.org/10.5951/jresematheduc.41.4.0351
  10. Bieda, K. N., Ji, X., Drwencke, J., & Picard, A. (2014). Reasoning-and-proving opportunities in elementary mathematics textbooks. International Journal of Educational Research, 64, 71-80. https://doi.org/10.1016/j.ijer.2013.06.005
  11. Blanton, M. L., & Stylianou, D. A. (2014). Understanding the role of transactive reasoning in classroom discourse as students learn to construct proofs. The Journal of Mathematical Behavior, 34, 76-98. https://doi.org/10.1016/j.jmathb.2014.02.001
  12. Chazan, D. (1993). High school geometry students' justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24(4), 359-387. https://doi.org/10.1007/BF01273371
  13. Coe, R. & Ruthven, K. (1994). Proof practices and construct of advanced mathematics students. British Educational Research Journal, 20, 41-53. https://doi.org/10.1080/0141192940200105
  14. Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington DC: Council of Chief State School Officers.
  15. Creswell, J. W., & Poth, C. N. (2016). Qualitative inquiry and research design: Choosing among five approaches. Thousand Oaks, CA: Sage publications.
  16. Department for Education. (2014). Mathematics programmes of study: Key stage 4 (National curriculum in England). Retrieved from https://www.gov.uk/government/uploads
  17. de Villiers, M. D. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17-24.
  18. de Villiers, M. (1999). Rethinking proof with the Geometer's Sketchpad. Emeryville, CA: Key Curriculum Press.
  19. Dickerson, D. S., & Doerr, H. M. (2014). High school mathematics teachers' perspectives on the purposes of mathematical proof in school mathematics. Mathematics Education Research Journal, 26(4), 711-733. https://doi.org/10.1007/s13394-013-0091-6
  20. Ellis, A. B., Lockwood, E., Dogan, M., Williams, C., & Knuth, E. J. (2013). Choosing And Using Examples: How Example Activity Can Support Proof Insight. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th conference of the international 2-265 group for the psychology of mathematics education (Vol. 2, pp. 265-272). Kiel, Germany: PME.
  21. Ellis, A. B., Ozgur, Z., Vinsonhaler, R., Dogan, M. F., Carolan, T., Lockwood, E., Lynch, A., Sabouri, P., Knuth, E., & Zaslavsky, O. (2019). Student thinking with examples: The criteria-affordances-purposes-strategies framework. The Journal of Mathematical Behavior, 53, 263-283. https://doi.org/10.1016/j.jmathb.2017.06.003
  22. Epstein, D., & Levy, S. (1995). Experimentation and proof in mathematics. Notices of the AMS, 42(6), 670-674.
  23. Hanna, G., & Jahnke, H. N. (1999). Using arguments from physics to promote understanding of mathematical proofs, In O. Zaslavsky (ed.), Proceedings of the twenty-third conference of the international group for the Psychology of Mathematics Education (Vol. 3, pp. 73-80). Haifa, Israel: PME.
  24. Hanna, G. (2000). Proof, explanation and exploration: an overview. Educational Studies in Mathematics, 44(1/2), 5-23. https://doi.org/10.1023/A:1012737223465
  25. Harel, G., & Sowder, L. (1998). Students' Proof Schemes: Results from Exploratory Studies. Research in Collegiate Mathematics Education III, 7, 234-282.
  26. Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396-428. https://doi.org/10.2307/749651
  27. Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24, 389-399. https://doi.org/10.1007/BF01273372
  28. Hoyles, C. (1997). The Curricular Shaping of Students' Approaches to Proof. For the Learning of Mathematics, 17(1), 7-16.
  29. Kim, H. (2020). Embracing proving into everyday lesson by problem posing. In Sacristan, A. I., Cortes-Zavala, J. C. & Ruiz-Arias, P. M. (Eds.). Mathematics education across cultures: Proceedings of the 42nd meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Mexico.
  30. Kim, H. (2021). Problem posing in the instruction of proof: Bridging everyday lesson and proof. Research in Mathematical Education, 24(3), 255-278. https://doi.org/10.7468/JKSMED.2021.24.3.255
  31. Knuth, E. J. (2002a). Teachers' Conceptions of Proof in the Context of Secondary School Mathematics. Journal of Mathematics Teacher Education, 5, 61-88. https://doi.org/10.1023/A:1013838713648
  32. Knuth, E. J. (2002b). Secondary school mathematics teachers' conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405. https://doi.org/10.2307/4149959
  33. Knuth, E. J., Choppin, J., & Bieda, K. (2009). Middle school students' production of mathematical justifications. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 153-170). New York, NY: Routledge.
  34. Knuth, E., Kim, H., Zaslavsky, O., Vinsonhaler, R., Gaddis, D., & Fernandez, L. (2020). Teachers' views about the role of examples in proving-related activities. Journal of Educational Research in Mathematics, 30(3), 115-134. https://doi.org/10.29275/jerm.2020.08.sp.1.115
  35. Knuth, E., Zaslavsky, O., & Ellis, A. (2019). The role and use of examples in learning to prove. The Journal of Mathematical Behavior, 53, 256-262. https://doi.org/10.1016/j.jmathb.2017.06.002
  36. Knuth, E., Zaslavsky, O., & Kim, H. (2022). Argumentation, justification, and proof in middle grades: A rose by any other name. In Bieda, K., Conner, A., Kosko, K., & Staples, M. (Eds.). Conceptions and Consequences of Mathematical Argumentation, Justification, and Proof (pp. 129-136). Springer.
  37. Kwon, O., Kim, A., & Cho, H. (2012). An analysis on the curriculum and teaching methods of Korean mathematics education departments. Mathematics Education, 51(3), 281-300. https://doi.org/10.7468/mathedu.2012.51.3.281
  38. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge, UK: Cambridge University Press.
  39. Leron, U., & Zaslasvsky, O. (2013). Generic Proving: Reflections on scope and method. For the Learning of Mathematics, 33(3), 24-30.
  40. Lockwood, E., Ellis, A., & Knuth, E. (2013). Mathematicians' example-related activity when proving conjectures. In Brown, S., Karakok, G., Roh, K., & Oehrtman, M. (Eds.). Proceedings for the Sixteenth Annual Conference on Research on Undergraduate Mathematics Education (Vol.1, pp. 16-28). Denver, CO: Northern Colorado University.
  41. Lockwood, E., Ellis, A., Knuth, E., Dogan, M. F., & Williams, C. (2013). Strategically chosen examples leading to proof insight: A case study of a mathematician's proving process. In Martinez, M. & Castro Superfne, A. (Eds.). Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 245-252). Chicago, IL: University of Illinois at Chicago.
  42. Lynch, A. G., & Lockwood, E. (2019). A comparison between mathematicians' and students' use of examples for conjecturing and proving. The Journal of Mathematical Behavior, 53, 323-338. https://doi.org/10.1016/j.jmathb.2017.07.004
  43. Mason, J. & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277-289. https://doi.org/10.1007/BF00312078
  44. Ministry of Education. (2011). Revised Korean mathematics curriculum. Seoul, Korea: Ministry of Education.
  45. Ministry of Education. (2015). Revised Korean mathematics curriculum. Seoul, Korea: Ministry of Education
  46. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
  47. Porteous, K. (1990). What do children really believe? Educational Studies in Mathematics, 21, 589-598. https://doi.org/10.1007/BF00315946
  48. Polya, G. (1954). Mathematics and plausible reasoning: Induction and analogy in mathematics (Vol. I). Princeton, NJ: Princeton University Press.
  49. Reid, D. A., & Knipping, C. (2010). Proof in mathematics education: Research, learning, and teaching. The Netherlands: Sense Publishers.
  50. Rowland, T. (2002). Generic Proofs in Number Theory. Learning and Teaching Number Theory: Research in Cognition and Instruction, 157-183.
  51. Sandefur, J., Mason, J., Stylianides, G. J., & Watson, A. (2013). Generating and using examples in the proving process. Educational Studies in Mathematics, 83(3), 323-340. https://doi.org/10.1007/s10649-012-9459-x
  52. Schoenfeld, A. (1986). On having and using geometric knowledge. In J. Hiebert (Ed.) Conceptual and procedural knowledge: The case of mathematics. (pp. 225-263). Hillsdale, NJ: Lawrence Erlbaum Associates.
  53. Schoenfeld, A. (1988). When good teaching leads to bad results: The disasters of "well-taught" mathematics courses. Educational Psychologist, 23(2), 145-166. https://doi.org/10.1207/s15326985ep2302_5
  54. Schoenfeld, A. H. (1994). What do we know about mathematics curricula? The Journal of Mathematical Behavior, 13(1), 55-80. https://doi.org/10.1016/0732-3123(94)90035-3
  55. Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4-36. https://doi.org/10.2307/30034698
  56. Senk, S. (1985). How well do students write geometry proofs? The Mathematics Teacher, 78(6), 448-456. https://doi.org/10.5951/MT.78.6.0448
  57. Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. The Journal of Mathematical Behavior, 15(1), 3-31. https://doi.org/10.1016/S0732-3123(96)90036-X
  58. Sowder, L., & Harel, G. (1998). Types of students' justifications. Mathematics Teacher, 91, 670-675. https://doi.org/10.5951/MT.91.8.0670
  59. Sowder, L., & Harel, G. (2003). Case studies of mathematics majors' proof understanding, production, and appreciation. Canadian Journal of Math, Science & Technology Education, 3(2), 251-267. https://doi.org/10.1080/14926150309556563
  60. Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289-321.
  61. Stylianides, A. J., & Stylianides, G. J. (2009). Proof constructions and evaluations. Educational Studies in Mathematics, 72(2), 237-253. https://doi.org/10.1007/s10649-009-9191-3
  62. Stylianides, G. J. (2007). Investigating the Guidance Offered to Teachers in Curriculum Materials: The Case of Proof in Mathematics. International Journal of Science and Mathematics Education, 6(1), 191-215. https://doi.org/10.1007/s10763-007-9074-y
  63. Stylianides, G. J., & Stylianides, A. J. (2009). Facilitating the Transition from Empirical Arguments to Proof. Journal for Research in Mathematics Education, 40(3), 314-352. https://doi.org/10.5951/jresematheduc.40.3.0314
  64. Stylianides, G. J., Stylianides, A. J., & Weber, K. (2016). Research on the teaching and learning of proof: Taking stock and moving forward. In Cai, J. (Ed.). Compendium for Research in Mathematics Education (pp. 237-266). Reston, VA: NCTM.
  65. Tall, D. (1999). The cognitive development of proof: Is mathematical proof for all or some? Developments in School Mathematics Education around the World, 4, 117-136.
  66. Thompson, D. R., Senk, S. L., & Johnson, G. J. (2012). Opportunities to learn reasoning and proof in high school mathematics textbooks. Journal for Research in Mathematics Education, 43(3), 253-295. https://doi.org/10.5951/jresematheduc.43.3.0253
  67. Walther, G. (2006). Illuminating examples - an aspect of simplification in the teaching of mathematics. International Journal of Mathematical Education in Science and Technology, 17(3), 263-273. https://doi.org/10.1080/0020739860170302
  68. Zack, V. (1999). Everyday and mathematical language in children's argumentation about proof, Educational Review, 51(2), 129-146. https://doi.org/10.1080/00131919997579
  69. Zaslavsky, O. (2018). Genericity, conviction, and conventions: examples that prove and examples that don't prove. In A. Stylianides, & G. Harel (Eds.). Advances in mathematics education research on proof and proving (pp. 283-298). Cham, Switzerland: Springer International Publishing AG.