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Genericity, Conviction, and Conventions: Examples that Prove and Examples that Don’t Prove

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Advances in Mathematics Education Research on Proof and Proving

Part of the book series: ICME-13 Monographs ((ICME13Mo))

Abstract

The four chapters in this section address issues related to the use of examples in proof and proving . Several questions arise from reading these chapters. I structure this chapter around some of these questions: First questions related to the nature of (mathematical) examples and their sources, then questions related to generic proving, including the subjective nature of generic proof , different levels of genericity, and how students may view generic arguments . I conclude with some observations regarding rigor and evidence .

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References

  • Alcock, L., & Weber, K. (2010). Referential and syntactic approaches to proving: Case studies from a transition-to-proof course. In F. Hitt, D. Holton, & P. Thompson (Eds.), Research in collegiate mathematics education VII (pp. 93–114). Washington: AMS.

    Chapter  Google Scholar 

  • Aricha-Metzer, I., & Zaslavsky, O. (2017). The nature of productive and non-productive example-use in proving. Journal of Mathematical Behavior. http://dx.doi.org/10.1016/j.jmathb.2017.09.002.

  • 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 and Stoughton.

    Google Scholar 

  • Dreyfus, T., Nardi, E., & Leikin, R. (2012). Forms of proof and proving in the classroom. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 191–214). New York: Springer.

    Google Scholar 

  • Ellis, A. B., Lockwood, E., Williams, C., 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 International Group for the Psychology of Mathematics Education (pp. 135–142). Kalamazoo, MI: Western Michigan University.

    Google Scholar 

  • Goldenberg, P., & Mason, J. (2008). Shedding light on and with example spaces. Educational Studies in Mathematics, 69(2), 183–194.

    Article  Google Scholar 

  • Harel, G. (2013). Intellectual need. In K. Leatham (Ed.), Vital directions for mathematics education research (pp. 119–151). New York: Springer.

    Chapter  Google Scholar 

  • Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III (pp. 234–283). Washington, DC: Mathematical Association of America.

    Chapter  Google Scholar 

  • Harel, G., & Tall, D. (1991). The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematics, 11(1), 38–42.

    Google Scholar 

  • Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.

    Article  Google Scholar 

  • Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education (pp. 70–95). NY: Cambridge University Press.

    Chapter  Google Scholar 

  • Hershkowitz, R., & Vinner, S. (1983). The role of critical and non-critical attributes in the concept image of geometrical concepts. In R. Hershkowitz (Ed.), Proceedings of the 7th Conference of the International Group for the Psychology of Mathematics Education (pp. 223–228). Weizmann Institute, Israel: PME.

    Google Scholar 

  • Iannone, P., Inglis, M., Mejía-Ramos, J. P., Simpson, A., & Weber, K. (2011). Does generating examples aid proof production? Educational Studies in Mathematics, 77, 1–14.

    Article  Google Scholar 

  • Knuth, E., 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). New York, NY: Routledge.

    Google Scholar 

  • Knuth, E., Kalish, C., Ellis, A., Williams, C., & Felton, M. (2012). Adolescent reasoning in mathematical and non-mathematical domains: Exploring the paradox. In V. Reyna, S. Chapman, M. Dougherty, & J. Confrey (Eds.), The adolescent brain: Learning, reasoning, and decision making (pp. 183–209). Washington, DC: American Psychological Association.

    Chapter  Google Scholar 

  • Knuth, E., Zaslavsky, O., & Ellis, A. (2017). The role and use of examples in learning to prove. Journal of Mathematical Behavior. http://dx.doi.org/10.1016/j.jmathb.2017.06.002.

  • Leron, U., & Zaslavsky, O. (2013). Generic proving: Reflections on scope and method. For the Learning of Mathematics, 33(3), 24–30.

    Google Scholar 

  • Love, E., & Mason, J. (1992). Teaching mathematics: Action and awareness. Milton Keynes: Open University.

    Google Scholar 

  • Mac Lane, S. (1986). Mathematics form and function. New York, NY: Springer.

    Book  Google Scholar 

  • Mason, J. (2017). Relationships between proof and examples. Journal of Mathematical Behavior. http://dx.doi.org/10.1016/j.jmathb.2017.07.005.

  • Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277–290.

    Article  Google Scholar 

  • Mills, M. (2014). A framework for example usage in proof presentations. Journal of Mathematical Behavior, 33, 106–118.

    Google Scholar 

  • Movshovitz-Hadar, N. (1988). Stimulating presentations of theorems followed by responsive proofs. For the Learning of Mathematics, 8(2), 12–19, 30.

    Google Scholar 

  • Pauletti, K., & Zaslavsky, O. (in preparation). Mathematicians’ views on the acceptance of generic arguments of students.

    Google Scholar 

  • Pedemonte, B., & Buchbinder, O. (2011). Examining the role of examples in proving processes through a cognitive lens: The case of triangular numbers. ZDM: The International Journal on Mathematics Education, 43(2), 257–267.

    Google Scholar 

  • Peled, I., & Zaslavsky, O. (1997). Counter-examples that (only) prove and counter-examples that (also) explain. FOCUS on Learning Problems in Mathematics, 19(3), 49–61.

    Google Scholar 

  • Reiss, K., Heinze, A., Renkl, A., & Groβ, C. (2008). Reasoning and proof in geometry: Effects of a learning environment based on heuristic worked-out examples. ZDM The International Journal on Mathematics Education, 40(3), 455–467.

    Google Scholar 

  • Rissland Michener, E. L. (1978). Understanding understanding mathematics. Cognitive Science, 2(4), 361–383.

    Article  Google Scholar 

  • Rissland Michener, E. L. (1991). Example-based reasoning. In J. F. Voss, D. N. Parkins, & J. W. Segal (Eds.), Informal reasoning in education (pp. 187–208). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Rowland, T. (1998). Conviction, explanation, and generic examples. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 65–72). Stellenbosch, South Africa: University of Stellenbosch.

    Google Scholar 

  • Rowland, T. (2001). Generic proofs in number theory. In S. Campbell & R. Zazkis (Eds.), Learning and teaching number theory: Research in cognition and instruction (pp. 157–184). Westport, CT: Ablex Publishing.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Skemp, R. R. (1987). The psychology of learning mathematics: Expanded American edition. Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.

    Google Scholar 

  • 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.

    Google Scholar 

  • Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14, 293–305.

    Article  Google Scholar 

  • Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356–366.

    Article  Google Scholar 

  • Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah: Erlbaum.

    Google Scholar 

  • Weber, K. (2010). Proofs that develop insight. For the Learning of Mathematics, 30(1), 32–36.

    Google Scholar 

  • Weber, K., & Mejía-Ramos, J. P. (2015). On relative and absolute conviction in mathematics. For the Learning of Mathematics, 35(2), 15–21.

    Google Scholar 

  • Yopp, D. A., & Ely, R. (2016). When does an argument use a generic example? Educational Studies in Mathematics, 91, 37–53.

    Article  Google Scholar 

  • Zaslavsky, O. (2005). Seizing the opportunity to create uncertainty in learning mathematics. Educational Studies in Mathematics, 60, 297–321.

    Article  Google Scholar 

  • Zaslavsky, O. (2010). The explanatory power of examples in mathematics: Challenges for teaching. In M. K. Stein & L. Kucan (Eds.), Instructional explanations in the disciplines (pp. 107–128). New York: Springer.

    Chapter  Google Scholar 

  • Zaslavsky, O. (2014). Thinking with and through examples. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36, (Vol. 1, pp. 21–34). Vancouver, Canada: PME.

    Google Scholar 

  • Zaslavsky, O. (2017). There is more to examples than meets the eye: Thinking with and through examples in different settings. Journal of Mathematical Behavior. http://dx.doi.org/10.1016/j.jmathb.2017.10.001.

  • Zaslavsky, O., Knuth, E., & Ellis, A. (Eds.) (forthcoming). The roles of examples in proving and learning to prove: Special Issue. Journal of Mathematical Behavior.

    Google Scholar 

  • Zaslavsky, O., Nickerson, S., Stylianides, A., Kidron, I., & Winicki-Landman, 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: Springer.

    Google Scholar 

  • Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. Journal for Research in Mathematics Education (JRME), 36(4), 317–346.

    Google Scholar 

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Authors and Affiliations

  1. New York University, New York, USA

    Orit Zaslavsky

  2. Technion Haifa, Haifa, Israel

    Orit Zaslavsky

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  1. Orit Zaslavsky
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Correspondence to Orit Zaslavsky .

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Editors and Affiliations

  1. Faculty of Education, University of Cambridge, Cambridge, United Kingdom

    Andreas J. Stylianides

  2. Department of Mathematics, University of California San Diego, La Jolla, California, USA

    Guershon Harel

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Zaslavsky, O. (2018). Genericity, Conviction, and Conventions: Examples that Prove and Examples that Don’t Prove. In: Stylianides, A., Harel, G. (eds) Advances in Mathematics Education Research on Proof and Proving. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-70996-3_20

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  • DOI: https://doi.org/10.1007/978-3-319-70996-3_20

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  • Online ISBN: 978-3-319-70996-3

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