Pre-Service Teachers’ Criteria for Evaluating Mathematical Arguments That Include Generic Examples

Abstract

This study investigated how pre-service teachers evaluate mathematical arguments including generic examples. By using written responses of 71 PSTs, the results revealed six criteria used by PSTs, which were being explanatory, being general, correctness, mode of representation, mode of argumentation, and structure of the argument. The criteria suggest what PSTs considered and might value while evaluating arguments. Also, PSTs found deductive arguments more convincing than generic examples arguments. While evaluating arguments with generic examples nature, PSTs considered generic example with visual representation more valid and convincing than with numeric representation. PSTs seemed to be relatively adept at evaluating arguments; however, many had difficulty with identifying the structure of the generic examples. Overall, this study suggests a more coherent approach for integrating generic examples in teacher education programs and directions for further research.

Keywords

• Reasoning and Proof,Proof evaluation,Generic examples

References

1. Alibert, D. & Thomas, M. (1991). Research on mathematical proof. In D. Tall (Ed.) Advanced Mathematical Thinking (pp. 215-230). Kluwer: The Netherlands.
2. Balacheff, N. (1988). Aspects of proof in pupils' practice of school mathematics. In D. Pimm (ed.), Mathematics, teachers and children (pp. 216-235). London: Hodder & Stoughton.
3. Ball, D., Hoyles, C., Jahnke, H., & Movshovitz-Hadar, N. (2002). The teaching of proof. Paper presented at the International Congress of Mathematicians, Beijing, China. In L. Tatsien (Ed.), Proceedings of the international congress of mathematicians (Vol. 3, pp. 907–920). Beijing: Higher Education Press.
4. Ball, D.L., Thames, M.H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.
5. Bell, A. (1976). A study of pupils’ proof – explanations in mathematical situations. Educational Studies in Mathematics, 7, 23-40.
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.
7. Bills, L. (1996). The Use of examples in the teaching and learning of mathematics. In Puig L. & Gutierrez A. (Eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 81–88). Valencia, Spain: PME.
8. Bleiler, S. K., Thompson, D. R., & Krajčevski, M. (2014). Providing written feedback on students’ mathematical arguments: Proof validations of prospective secondary mathematics teachers. Journal of Mathematics Teacher Education, 17(2), 105-127.

9. Department for Education. (2014). Mathematics programmes of study: key stages 1 and 2: National curriculum in England. Retrieved from https://www.gov.uk/government/publications/national-curriculum-in-englandmathematics-programmes-of-study
10. Dogan, M. F. (2015). The nature of middle school in-service teachers' engagements in proving-related activities. [Unpublished doctoral dissertation]. University of Wisconsin-Madison, USA.
11. Dogan, M. F. (2019). The nature of middle school in-service teachers’ engagements in proving-related activities. Cukurova University Faculty of Education Journal, 48(1), 100-130.
12. Dogan, M. F., & Williams-Pierce, C. (2019). Supporting teacher proving practices with three phases of proof. Teacher Education Advancement Network Journal, 11(3), 48-59.
13. Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: strategies for qualitative theory. New Brunswick: Aldine Transaction.
14. Hanna, G. (2018). Reflections on proof as explanation. In Advances in Mathematics Education Research on Proof and Proving (pp. 3-18). Cham, Springer.
15. Harel, G. & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Issues in mathematics education: Vol. 7. Research in collegiate mathematics education III (pp. 234-283). Providence, RI: American Mathematical Society.
16. Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.
17. Isler, I. (2015). An Investigation of Elementary Teachers’ Proving Eyes and Ears. Unpublished Doctoral dissertation. The University of Wisconsin-Madison, USA.
18. Knuth, E. (2002). Teachers conceptions of proof in the context of secondary school mathematics. Journal for Research in Mathematics Education, 5(1), 61-88.
19. Ko, Y. Y., & Hagen, C. J. (2013). Conviction and validity: Middle school mathematics teachers’ proof evaluations. In M. Martinez, & A. Castro Superfine (Eds.). Proceedings of the Thirty-Fifth Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 801-804). Chicago, IL.
20. Ko, Y. Y., & Knuth, E. J. (2013). Validating proofs and counterexamples across content domains: Practices of importance for mathematics majors. The Journal of Mathematical Behavior, 32(1), 20-35.
21. Lannin, J. K., Ellis, A. B., & Elliott, R. (2011). Developing essential understanding of mathematical reasoning for teaching mathematics in prekindergarten-grade 8. Reston, VA: National Council of Teachers of Mathematics.
22. Leron, U., & Zaslavsky, O. (2013). Generic proving: Reflections on scope and method. For the Learning of Mathematics, 33(3), 24–30.
23. Lovin, L. A., Cavey, L. O., Whitenack, J. W. (2004) Evidence and justification: Prospective K–8 teachers’ proof-making and proof-evaluating. In D. E. McDougall and J. A. Ross (Eds.). the Proceedings of the Twenty-Sixth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 1201–1208). Toronto, Canada.
24. Martin, W.G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal of Research in Mathematics Education, 20(1), 41-51.
25. Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15, 227-289.
26. Morris, A. K. (2007). Factors affecting pre-service teachers' evaluations of the validity of students' mathematical arguments in classroom contexts. Cognition and Instruction, 25(4), 479-522.
27. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
28. National Governors Association (2010). Common Core State Standards for Mathematics. Washington, DC: Council of Chief State School Officers.
29. Powers, R. A., Craviotto, C., & Grassl, R. M. (2010). Impact of proof validation on proof writing in abstract algebra. International Journal of Mathematical Education in Science and Technology, 41(4), 501-514.
30. Reid, D., & Knipping, C. (2010). Proof in mathematics education: Research, learning, and teaching. Rotterdam: Sense Publishers.
31. Reid, D., & Vallejo Vargas, E. (2018). When is a generic argument a proof? In A. J. Stylianides, & G. Harel (Eds.). Advances in mathematics education research on proof and proving (pp. 239–251). Cham: Springer International Publishing.
32. Rø, K., & Arnesen, K. K. (2020). The opaque nature of generic examples: The structure of student teachers’ arguments in multiplicative reasoning. The Journal of Mathematical Behavior, 100755.
33. Rowland, T. (2001). Generic proofs: Setting a good example. Mathematics Teaching, 177, 40-43.
34. Rowland, T. (2002). Generic proofs in number theory. In S. R. Campbell & R. Zazkis (Eds.), Learning and teaching number theory: Research in cognition and instruction (pp. 157–183). Westport, CT: Ablex Publishing.
35. 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.
36. Stylianides, A.J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38, 289-321.
37. Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9-16.
38. Stylianides, G. J., & Stylianides, A. J. (2008). Proof in school mathematics: Insights from psychological research into students' ability for deductive reasoning. Mathematical Thinking and Learning, 10(2), 103-133.
39. Stylianides, G. J., Stylianides, A. J., & Weber, K. (2017). 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). Reston, VA: National Council of Teachers of Mathematics.
40. Suominen, A. L., Conner, A., & Park, H. (2018). Prospective mathematics teachers’ expectations for middle grades students’ arguments. School Science and Mathematics, 118(6), 218-231.
41. Yopp, D. A., & Ely, R. (2016). When does an argument use a generic example? Educational Studies in Mathematics, 91(1), 37-53.
42. Weber, K. (2010). Mathematics majors’ perceptions of conviction, validity, and proof. Mathematical Thinking and Learning, 12(4), 306-336.