Research in Mathematical Education (한국수학교육학회지시리즈D:수학교육연구)

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

DOI QR Code

High school students' evaluation of mathematical arguments as proof: Exploring relationships between understanding, convincingness, and evaluation

  • Hangil Kim ( Samchun Middle School)
  • Received : 2024.01.23
  • Accepted : 2024.04.02
  • Published : 2024.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

Researchers continue to emphasize the centrality of proof in the context of school mathematics and the importance of proof to student learning of mathematics is well articulated in nationwide curricula. However, researchers reported that students' performance in proving tasks is not promising and students are not likely to see the need to prove a proposition even if they learned mathematical proof previously. Research attributes this issue to students' tendencies to accept an empirical argument as proof for a mathematical proposition, thus not being able to recognize the limitation of an empirical argument as proof for a mathematical proposition. In Korea, there is little research that investigated high school students' views about the need for proof in mathematics and their understanding of the limitation of an empirical argument as proof for a mathematical generalization. Sixty-two 11th graders were invited to participate in an online survey and the responses were recorded in writing and on either a four- or five-point Likert scale. The students were asked to express their agreement with the need of proof in school mathematics and to evaluate a set of mathematical arguments as to whether the given arguments were proofs. Results indicate that a slight majority of students were able to identify a proof amongst the given arguments with the vast majority of students acknowledging the need for proof in mathematics.

Keywords

References

  1. Alcock, L., & Inglis, M. (2008). Doctoral students' use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69(2), 111-129.
  2. Australian Curriculum, Assessment and Reporting Authority [ACARA]. (2022). Critical and creative thinking (version 8.4). https://www.australiancurriculum.edu.au/f-10-curriculum/general-capabilities/critical-and-creative-thinking/
  3. Balacheff, N. (1988). Aspects of proof in pupils' practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216-238). Hodder & Stoughton.
  4. 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
  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. Bieda, K. (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
  7. Bills, L., & Rowland, T. (1999). Examples, generalisation and proof. Research in Mathematics Education, 1(1), 103-116. https://doi.org/10.1080/14794809909461549
  8. Buchbinder O., & Zaslavsky O. (2018). Strengths and inconsistencies in students' understanding of the roles of examples in proving. Journal of Mathematical Behavior, 53, 129-147. https://doi.org/10.1016/j.jmathb.2018.06.010
  9. 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
  10. 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
  11. Council of Chief State School Officers [CCSSO] (2010). Common core state standards for mathematics. Author.
  12. Creswell, J. W., & Poth, C. N. (2016). Qualitative inquiry and research design: Choosing among five approaches. Sage publications.
  13. Department for Education [DoE] (2014). Mathematics programmes of study: Key stage 4 (National curriculum in England). [Retrieved March 17, 2020, from https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/331882/KS4_maths_PoS_FINAL_170714.pdf.].
  14. Ellis, A., Ozgur, Z., Vinsonhaler, R., Carolan, T., Dogan, M., Lockwood, E., Lynch, A., Sabouri, P., Knuth, E., & Zaslavsky, O. (2019). Student thinking with examples: The criteria-affordances-purposes strategies framework. Journal of Mathematical Behavior, 53, 263-283. https://doi.org/10.1016/j.jmathb.2017.06.003
  15. Epstein, D., & Levy, S. (1995), Experimentation and proof in mathematics. Notice of the AMS, 42(6), 670-674.
  16. Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(2), 9-24.
  17. Hanna, G. (2020). Mathematical proof, argumentation, and reasoning. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp.561-566). Springer. https://doi.org/10.1007/978-3-030-15789-0_102
  18. Harel, G., & Sowder, L. (1998). Students' proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III (Vol. 7, pp. 234-283). American Mathematical Society.
  19. Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 805-842). Information Age Publishing.
  20. Healy, L. & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396-428.
  21. Herbst, P., & Brach, C. (2006). Proving and doing proofs in high school geometry classes: What is it that is going on for students? Cognition and Instruction, 24(1), 73-122. https://doi.org/10.1207/s1532690xci2401_2
  22. Hong, Y., & Son, H. (2021). A study on the recognition and characteristics of mathematical justification for gifted students in middle school mathematics. Journal of the Korean School Mathematics, 24(3), 261-282. https://doi.org/10.30807/ksms.2021.24.3.002
  23. Iannone, P., Inglis, M., Mejia-Ramos, J. P., Simpson, A., & Weber, K. (2011). Does generating examples aid proof production? Educational Studies in Mathematics, 77, 1-14. https://doi.org/10.1007/s10649-011-9299-0
  24. Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43(4), 358-390. https://doi.org/10.5951/jresematheduc.43.4.0358
  25. Kang, S., & Shin, B. (2022). An analysis of middle school students' deductive reasoning in proof lessons: Focused on commognition perspective. Journal of Curriculum and Evaluation, 25(1), 143-172. https://doi.org/10.29221/jce.2022.25.1.143
  26. Kim, C. (2012). A study on the completeness of 'the understanding' in the generalization process and justification-Centered on the arithmetical, geometric and harmonic average-. The Mathematical Education, 51(4), 377-393. https://doi.org/10.7468/mathedu.2012.51.4.377
  27. 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
  28. Kim, H. (2022). 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
  29. Kim, H., Kim, S., & Kwon, J. (2014). 6th grade students' awareness of why they need mathematical justification and their levels of mathematical justification. The Mathematical Education, 53(4), 525-539. https://doi.org/10.7468/mathedu.2014.53.4.525
  30. Knuth, E. (2002a). Proof as a tool for learning mathematics. Mathematics Teacher, 95(7), 486-490. https://doi.org/10.5951/MT.95.7.0486
  31. Knuth, E. (2002b). Secondary school mathematics teachers' conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405. https://doi.org/10.2307/4149959
  32. Knuth, E., Choppin, J., & Bieda, K. (2009a). 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.
  33. Knuth, E., Choppin, J., & Bieda, K. (2009b). Proof in middle school: Moving beyond examples. Mathematics Teaching in the Middle School, 15(4), 206-211.
  34. Mingus, T., & Grassl, R. (1999). Preservice teacher beliefs about proofs. School Science and Mathematics, 99(8), 438-444. https://doi.org/10.1111/j.1949-8594.1999.tb17506.x
  35. Ministry of Education [MoE] (2015). 2015 Revised Korean mathematics curriculum. Author.
  36. MoE (2022). 2022 Revised Korean mathematics curriculum. Author.
  37. Na, G. (2014). Mathematics teachers' conceptions of proof and proof-instruction. Communications of Mathematical Education, 28(4), 513-528. https://doi.org/10.7468/jksmee.2014.28.4.513
  38. National Council of Teachers of Mathematics [NCTM] (2000). Principles and standards for school mathematics. Author.
  39. NCTM. (2009). Guiding principles for mathematics curriculum and assessment. Author.
  40. Reid, D., & Knipping, C. (2010). Proof in mathematics education: Research, learning, and teaching. Sense Publishers.
  41. Roh, E., & Kang, J. (2015). A study on the understanding of limitations of experiential viewpoints for 9th grade students. The Mathematical Education, 54(1), 13-30. https://doi.org/10.7468/mathedu.2015.54.1.13
  42. 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.
  43. 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
  44. Song, S., Huh,J., & Lim, J. (2006). Analysis on the types of mathematically gifted students' justification on the tasks of figure division. Journal of Educational Research in Mathematics, 16(1), 79-94.
  45. Sowder, L., & Harel, G. (1998). Types of students' justifications. Mathematics Teacher, 91(8), 670-675. https://doi.org/10.5951/MT.91.8.0670
  46. Stylianides, A. J. (2011). Towards a comprehensive knowledge package for teaching proof: A focus on the misconception that empirical arguments are proofs. Pythagoras, 32(1), 1-10. https://doi.org/10.4102/pythagoras.v32i1.14
  47. Stylianides, A. J., Bieda, K. N., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In A. Gutierrez, G., C. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 315-351). Sense Publishers. https://doi.org/10.1007/978-94-6300-561-6_9
  48. Stylianides, G. (2009) Reasoning-and-proving in school mathematics textbooks. Mathematical Thinking and Learning, 11(4), 258-288. https://doi.org/10.1080/10986060903253954
  49. 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
  50. Stylianides, G. J., Stylianides, A. J., & Shilling-Traina, L. N. (2013). Prospective teachers' challenges in teaching reasoning-and-proving. International Journal of Science and Mathematics Education, 11, 1463-1490. https://doi.org/10.1007/s10763-013-9409-9
  51. Stylianides, G. J., Stylianides, A. J., & Weber, K. (2016). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 237-266). NCTM.
  52. Weber, K. (2010). Mathematics majors' perceptions of conviction, validity, and proof. Mathematical Thinking and Learning, 12(4), 306-336. https://doi.org/10.1080/10986065.2010.495468
  53. Zaslavsky, O., Nickerson, S., Stylianides, A., Kidron, I., & Winicki, G. (2012). The need for proof and proving: Mathematical and pedagogical perspectives. In G. Hanna, & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 215-229). Springer. https://doi.org/10.1007/978-94-007-2129-6_9