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Investigating the substance and acceptability of empirical arguments: The case of maximum-minimum theorem and intermediate value theorem in Korean textbooks

  • Received : 2024.01.31
  • Accepted : 2024.03.06
  • Published : 2024.03.31

Abstract

Mathematical argument has been given much attention in the research literature as a mediating construct between reasoning and proof. However, there have been relatively less efforts made in the research that examined the nature of empirical arguments represented in textbooks and how students perceive them as proofs. Cases of point include Intermediate Value Theorem [IVT] and Maximum-Minimum theorem [MMT] in grade 11 in Korea. In this study, using Toulmin's framework (1958), the author analyzed the substance of the empirical arguments provided for both MMT and IVT to draw comparisons between the nature of datum, claims, and warrants among empirical arguments offered in textbooks. Also, an online survey was administered to learn about how students view as proofs the empirical arguments provided for MMT and IVT. Results indicate that nearly half of students tended to accept the empirical arguments as proofs. Implications are discussed to suggest alternative approaches for teaching MMT and IVT.

Keywords

References

  1. Aberdein, A. (2019). Evidence, proofs, and derivations. ZDM Mathematics Education, 51(5), 825-834. https://doi.org/10.1007/s11858-019-01049-5 
  2. Alcock, L., & Inglis, M. (2008). Doctoral students' use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69(2), 111-129. 
  3. Bae, J., Yeo, T., Cho, B., Kim, M., Chun, H., Cho, S., & Byun, D. (2018). Mathematics II. Keumsung Publication. 
  4. Balacheff, N. (1988). Aspects of proof in pupils' practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216-235). Hodder & Stoughton. 
  5. 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 
  6. Bergwall, A., & Hemmi, K. (2017). The state of proof in Finnish and Swedish mathematics textbooks-Capturing differences in approaches to upper-secondary integral calculus. Mathematical Thinking and Learning, 19(1), 1-18. https://doi.org/10.1080/10986065.2017.1258615 
  7. 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 
  8. Bieda, K., Conner, A., Kosko, K. W., & Staples, M. (Eds.). (2022). Conceptions and consequences of mathematical argumentation, justification, and proof. Springer.
  9. Cai, J., & Cirillo, M. (2014). What do we know about reasoning and proving? Opportunities and missing opportunities from curriculum analyses. International Journal of Educational Research, 64, 132-140. https://doi.org/10.1016/j.ijer.2013.10.007 
  10. Cai, J., Ni, Y., & Lester, F. K. (2011). Curricular effect on the teaching and learning of mathematics: Findings from two longitudinal studies in China and the United States. International Journal of Educational Research, 50(2), 63-64. https://doi.org/10.1016/j.ijer.2011.06.001 
  11. 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 
  12. Coe, R., & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students. British Educational Research Journal, 20(1), 41-53. https://doi.org/10.1080/0141192940200105 
  13. Conner, A. (2008, July 6-13). Argumentation in a geometry class: Aligned with the teacher's conception of proof [Topic Study Group (Vol. 12)]. 11th International Congress on Mathematics Education, Monterrey, Mexico. 
  14. Conner, A., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). Identifying kinds of reasoning in collective argumentation. Mathematical Thinking and Learning, 16(3), 181-200. https://doi.org/10.1080/10986065.2014.921131 
  15. Davis, J. D., Smith, D. O., Roy, A. R., & Bilgic, Y. K. (2014). Reasoning-and-proving in algebra: The case of two reform-oriented U.S. textbooks. International Journal of Educational Research, 64, 92-106. https://doi.org/10.1016/j.ijer.2013.06.012 
  16. Epstein, D., & Levy, S. (1995). Experimentation and proof in mathematics. Notices of the AMS, 42(6), 670-674. 
  17. Fan, L. (2013). Textbook research as scientific research: Towards a common ground on issues and methods of research on mathematics textbooks. ZDM Mathematics Education, 45(5), 765-777. https://doi.org/10.1007/s11858-013-0530-6 
  18. Fawcett, H. (1938). The nature of proof. Columbia University Teachers College Bureau of Publications.
  19. Fujita, T., & Jones, K. (2014). Reasoning-and-proving in geometry in school mathematics textbooks in Japan. International Journal of Educational Research, 64, 81-91. https://doi.org/10.1016/j.ijer.2013.09.014 
  20. Grouws, D. A., Smith, M. S., & Sztajn, P. (2004). The preparation and teaching practices of U.S. mathematics teachers: Grades 4 and 8. In P. Kloosterman & F. Lester (Eds.), The 1990 through 2000 mathematics assessments of the National Assessment of Educational Progress: Results and interpretations (pp. 221-269). NCTM.
  21. Harel, G., & Sowder, L. (1998). Students' proof schemes: Results from exploratory studies. In E. Dubinsky, A. Schoenfeld and J. Kaput (Eds.), Research in collegiate mathematics education III (Vol. 7) (pp. 234-282). American Mathematical Society. 
  22. Hong, S., Lee, J., Shin, T., Lee, C., Lee, B., Shin, Y., Jeon, H., Kim, H., Kwon, B., Choi, W., & Kang, I. (2018). Mathematics II. Jihaksa Publication. 
  23. Hwang, S., Kang, B., Yoon, K., Lee, K., Kim, S., Lee, M., Kim, W., Park, M., & Park, S. (2018). Mathematics II. MiraeN Publication. 
  24. Jones, K. (2010). The role of the teacher in teaching proof and proving in geometry. Proceedings of the British Society for Research into Learning Mathematics, 30(2), 62-67. 
  25. Kim, H. (2021). Problem posing in the instruction proof: Bridging everyday lesson and proof. Research in Mathematical Education, 24(3), 255-278. https://doi.org/10.7468/jksmed.2021.24.3.255 
  26. Kim, H. (2022a). Secondary teachers' views about proof and judgements on mathematical arguments. Research in Mathematical Education, 25(1), 65-89. https://doi.org/10.7468/jksmed.2022.25.1.65 
  27. Kim, H. (2022b). Teacher noticing for supporting student proving: Gradual articulation. Journal of Educational Research in Mathematics, 32(1), 47-62. http://doi.org/10.29275/jerm.2022.32.1.47 
  28. Kim, W., Cho, M., Bang, K., Yoon, J., Shin, J., Lim, S., Kim, D., Kang, S., Kim, K., Park, H., Shim, J., Oh, H., Lee, D., Lee, S., & Jung, J. (2018). Mathematics II. Visang Publication. 
  29. Knipping, C., & Reid, D. A. (2019). Argumentation analysis for early career researchers. In G. Kaiser & N. Presmeg (Eds.), Compendium for early career researchers in mathematics education, ICME-13 monographs (pp. 3-31). Springer. https://doi.org/10.1007/978-3-030-15636-7_1 
  30. Knuth, E. J. (2002). 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 
  31. Knuth, E. J., Choppin, J., & Bieda, K. (2009). Middle school students' production of mathematical justifications. In D. Stylianou, M. Blanton & E. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 153-170). Routledge. 
  32. 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. 
  33. Knuth, E., Zaslavsky, O., & Kim, H. (2022). Argumentation, justification, and proof in middle grades: A rose by any other name. In K. Bieda, A. Conner, K. Kosko, & M. Staples (Eds.), Conceptions and consequences of mathematical argumentation, justification, and proof (pp. 129-136). Springer International Publishing. 
  34. Koestler, C., Felton, M. D., Bieda, K., & Otten, S. (2013). Connecting the NCTM process standards and the CCSSM practices. NCTM. 
  35. Koh, S., Lee, J., Lee, S., Cha, S., Kim, Y., Oh, T., & Cho, S. (2018). Mathematics II. Sinsago Publication. 
  36. Korea Textbook Research Foundation (n.d.). Downloading the list of the approved titles. Retrieved September 11, 2023 from https://www.kotry.kr/ 
  37. Kwon, O., Shin, J., Jeon, I., Kim, M., Kim, C., Kim, T., Park, J., Park, J., Park, J., Park, C., Park, H., Oh, K., Cho, K., Cho, S., & Hwang, S. (2018). Mathematics II. Kyohaksa Publication. 
  38. 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). Erlbaum. 
  39. Lakatos, I. (1976). Proofs and refutations. Cambridge University Press. 
  40. Lee, J., Choi, B., Kim, D., Lee, J., Jeon, C., Jang, H., Song, Y., Song, J., Kim, S., & Kim, M. (2018). Mathematics II. Chunjae Publication. 
  41. Lew, H., Sunwoo, H., Shin, B., Cho, J., Lee, B., Kim, Y., Lim, M., Han, M., Nam, S., Kim, M., & Jung, S. (2018). Mathematics II. Chunjae Publication. 
  42. McCory, R., & Stylianides, A. J. (2014). Reasoning-and-proving in mathematics textbooks for prospective elementary teachers. International Journal of Educational Research, 64, 119-131. https://doi.org/10.1016/j.ijer.2013.09.003 
  43. Ministry of Education [MoE]. (2015). 2015 revised mathematics curriculum. MoE. 
  44. MoE. (2022). 2022 revised mathematics curriculum. MoE. 
  45. Miyakawa, T. (2017). Comparative analysis on the nature of proof to be taught in geometry: The cases of French and Japanese lower secondary schools. Educational Studies in Mathematics, 94(1), 37-54. https://doi.org/10.1007/s10649-016-9711-x 
  46. National Council of Teachers of Mathematics [NCTM] (2000). Principles and standards for school mathematics. NCTM. 
  47. Otten, S., Males, L. M., & Gilbertson, N. J. (2014). The introduction of proof in secondary geometry textbooks. International Journal of Educational Research, 64, 107-118. https://doi.org/10.1016/j.ijer.2013.08.006 
  48. Park, K., Lee, J., Kim, J., Nam, C., Kim, N., Lim, J., Yoo, E., Kwon, S., Kim, S., Kim, J., Kim, K., Yoon, H., Koh, H., Yoon, H., Kim, Y., Kim, H., Lee, K., Cho, Y., Lee, J., & Yang, J. (2018). Mathematics II. Donga Publication. 
  49. Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of 'welltaught' mathematics courses. Educational Psychologist, 23(2), 145-166. https://doi.org/10.1207/s15326985ep2302_5 
  50. 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 
  51. Schoenfeld, A. H. (2009). The soul of mathematics. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. xii-xvi). Routledge. 
  52. Stein, M. K., Remillard, J., & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319-370). NCTM. 
  53. Selden, J., & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29(2), 123-151. https://doi.org/10.1007/BF01274210 
  54. Stylianides, G. J. (2008). Investigating the guidance offered to teachers in curriculum materials: The case of proof in mathematics. International Journal of Science and Mathematics Education, 6, 191-215. https://doi.org/10.1007/s10763-007-9074-y 
  55. Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks. Mathematical Thinking and Learning, 11(4), 258-288. https://doi.org/10.1080/10986060903253954 
  56. Stylianides, G. J. (2014). Textbook analyses on reasoning-and-proving: Significance and methodological challenges. International Journal of Educational Research, 64, 63-70. https://doi.org/10.1016/j.ijer.2014.01.002 
  57. Stylianides, G. J., & Stylianides, A. J. (2014). The role of instructional engineering in reducing the uncertainties of ambitious teaching. Cognition and Instruction, 32(4), 374-415. https://doi.org/10.1080/07370008.2014.948682 
  58. Stylianou, D. A., & Blanton, M. L. (2011). Connecting research to teaching: Developing students' capacity for constructing proofs through discourse. The Mathematics Teacher, 105(2), 140-145. https://doi.org/10.5951/mathteacher.105.2.0140 
  59. 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 
  60. Toulmin, S. (1958). The uses of argument. Cambridge University Press. 
  61. Umland, K., & Sriraman, B. (2014). Argumentation in Mathematics. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 46-48). Springer Dordrecht. https://doi.org/10.1007/978-94-007-4978-8_11 
  62. Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R. T. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks. Springer Science & Business Media. 
  63. Weber, K. (2014). Proof as a cluster concept. In P. Liljedahl, C. Nicol, S., Oesterle, & D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (pp. 353- 360). PME. 
  64. 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/10.5951/jresematheduc.27.4.0458