The Korea Society Of Educational Studies In Mathematics

Current Issue

Journal of Educational Research in Mathematics - Vol. 30

[ Special ]
Journal of Educational Research in Mathematics - Vol. 30, No. SP1, pp.115-134
Abbreviation: JERM
ISSN: 2288-7733 (Print) 2288-8357 (Online)
Print publication date 31 Aug 2020
DOI: https://doi.org/10.29275/jerm.2020.08.sp.1.115

Teachers’ Views about the Role of Examples in Proving-related Activities
Eric Knuth*, ; Hangil Kim** ; Orit Zaslavsky*** ; Rebecca Vinsonhaler** ; Damon Gaddis** ; Luis Fernandez**
*Professor, University of Texas at Austin, USA
**Graduate Student, University of Texas at Austin, USA
***Professor, New York University, USA

Correspondence to : Email: eric.knuth@austin.utexas.edu, hangil_kim@utexas.edu, orit.zaslavsky@nyu.edu, rkv@austin.utexas.edu, damongaddis@utexas.edu, lmfg1992@utexas.edu

Please cite this article as: Knuth, E., Kim, H., Zaslavsky, O., Vinsonhaler, R., Gaddis, D., & Fernandez, L. Teachers’ views about the role of examples in proving-related activities.


Abstract

Examples play a critical role in mathematical practice, in general, and in proving-related activities (e.g., developing conjectures, exploring conjectures, justifying conjectures), in particular. Yet, despite the critical role examples play in proving-related activity, we contend that students typically receive very little, if any, explicit instruction on how to become more deliberate and strategic in their use of examples. The goal of the study reported here was to explore teachers’ beliefs about the role examples play in proving-related activities, and the instructional practices they implement to foster the development of students’ abilities to strategically think about and productively use examples. Fifty-four middle school mathematics teachers responded to a series of on-line survey questions that focused on the role and use of examples during proving-related classroom activities. We found that many teachers have limited views of what it means to use examples strategically during proving-related activities, and that they tended not to provide explicit instruction designed to help students learn to strategically think about and productively use examples during their engagement in proving-related activities. The findings suggest the need for both professional development and curricular resources to support teacher efforts to help their students learn to strategically think about and productively use examples during proving-related activities.


Keywords: Proving, Teacher beliefs, Example use

References
1. Alcock, L., & Inglis, M. (2008). Doctoral students’ use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69, 111–129.
2. Antonini, S. (2006). Graduate students’ processes in generating examples of mathematical objects. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, 57-64.
3. Aricha-Metzer, I., & Zaslavsky, O. (2019). The nature of students’ productive and non-productive use of examples for proving. Journal of Mathematical Behavior, 53, 304-322.
4. Bieda, K. (2010). Enacting proof-related tasks in middle school mathematics: Challenges and opportunities. Journal for Research in Mathematics Education, 41, 351–382.
5. Bieda, K., Ji, X., Drwencke, J., & Picard, A. (2014). Reasoning-and-proving opportunities in elementary mathematics textbooks. International Journal of Educational Research, 64, 71–80.
6. 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.
7. Cirillo, M. (2011). “I’m like the Sherpa guide”: On learning to teach proof in school mathematics. In B. Ubuz (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 241–248). Ankara, Turkey: Middle East Technical University.
8. Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Council of Chief State School Officers.
9. Department for Education (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.].
10. Ellis, A. E., Lockwood, E., Williams, C. C. W., Dogan, M. F., & Knuth, E. (2012). Middle school students’ example use in conjecture exploration and justification. In L. R. Van Zoest, J. J. Lo, & J. L. Kratky (Eds.), Proceedings of the 34th Annual Meeting of the North American Chapter of the Psychology of Mathematics Education (pp. 135-142). Kalamazoo, MI: Western Michigan University.
11. 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.
12. Epstein, D., & Levy, S. (1995), Experimentation and proof in mathematics. Notice of the AMS, 42(6), 670–674.
13. 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). Charlotte, NC: Information Age Publishing.
14. Healy, L. & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.
15. 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.
16. Knuth, E. (2002a). Proof as a tool for learning mathematics. Mathematics Teacher, 95(7), 486–490.
17. Knuth, E. (2002b). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405.
18. 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). New York, NY: Routledge.
19. Knuth, E., Choppin, J., & Bieda, K. (2009b). Proof in middle school: Moving beyond examples. Mathematics Teaching in the Middle School, 15(4), 206–211.
20. Knuth, E., Kalish, C., Ellis, A., Williams, C., & Felton, M. (2011). Adolescent reasoning in mathematical and non-mathematical domains: Exploring the paradox. To appear in V. Reyna, S. Chapman, M. Dougherty, & J. Confrey (Eds.), The adolescent brain: Learning, reasoning, and decision making. Washington, DC: American Psychological Association.
21. Knuth, E., Zaslavsky, O., & Ellis, A. (2019). The role of examples in learning to prove. Journal of Mathematical Behavior, 53, 256-262.
22. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge University Press.
23. Leron, U., & Zaslavsky, O. (2013). Generic proving: Reflections of scope and method. For the Learning of Mathematics, 33(3), 24-30.
24. Lockwood, E., Ellis, A., & Knuth, E. (February, 2013). Mathematicians’ example-related activity when proving conjectures. In S. Brown, G. Karakok, K. Hah Roh, & M. Oehrtman (Eds.), Proceedings of the Sixteenth Annual Conference on Research in Undergraduate Mathematics Education, 16-30.
25. Lockwood, E., Ellis, A. B., & Lynch, A. G. (2016). Mathematicians’ example-related activity when exploring and proving conjectures. International Journal of Research in Undergraduate Mathematics Education, 2(2), 165-196.
26. Lynch, A., & Lockwood, E. (2019). A comparison between mathematicians’ and students’ use of examples for conjecturing and proving. Journal of Mathematical Behavior, 53, 323-338.
27. Korean Ministry of Education. (2015). Revised Korean mathematics curriculum. Seoul, Korea: Ministry of Education.
28. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
29. Ozgur, Z., Ellis, A., Vinsonhaler, R., Dogan, M., & Knuth, E. (2019). From examples to proof: Purposes, strategies, and affordances of example use. Journal of Mathematical Behavior, 53, 284-303.
30. Philipp, R. (2007). Mathematics teachers’ beliefs and affect. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 257-315). Charlotte, NC: Information Age Publishing.
31. Polya, G. (1954). Induction and analogy in mathematics. Princeton, NJ: Princeton University Press.
32. Reid, D., & Knipping, C. (2010). Proof in mathematics education: Research, learning, and teaching. Rotterdam, The Netherlands: Sense Publishers.
33. 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.
34. Sowder, L., & Harel, G. (1998). Types of students’ justifications. Mathematics Teacher, 91, 670–675.
35. Stylianides, A. J., Bieda, K. N., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In A. Gutiérrez, G. C. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education(pp. 315-351). Rotterdam, The Netherlands: Sense Publishers.
36. Stylianides, G. (2009) Reasoning-and-proving in school mathematics textbooks. Mathematical Thinking and Learning, 11(4), 258-288.
37. Stylianides, G. J., & Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40, 314–352.
38. 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.
39. 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. Reston, VA: National Council of Teachers of Mathematics.
40. Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for research in Mathematics Education, 39(4), 431-459.
41. Weber, K., & Mejia-Ramos, J. (2011). Why and how mathematicians read proofs: An exploratory study. Educational Studies in Mathematics, 76, 329-344.
42. Zaslavsky, O. (2018). Genericity, Conviction, and Conventions: Examples that prove and examples that don’t prove. In A. J. Stylianides and G. Harel (Eds.), Advances in mathematics education research on proof and proving, ICME-13 Monographs (pp. 283-298). Cham, Switzerland: Springer International Publishing AG.
43. 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). New York, NY: Springer.