This paper explores how in-service teachers enrolled in a graduate proof course interpret, understand, and use generic examples as part of their proving and justification activities. Generic examples, which are capable of proving and justifying with strong explanatory power, are particularly important for teachers considering teaching proof in their classrooms. The teachers in our study used generic examples to produce three types of proof: example-based arguments enhanced with generic language; incomplete generic examples; and complete generic examples. We found that teachers conflate generic examples and visual representations, prefer visual generic examples for teaching, and consider a generic example with symbolic representation to be more convincing than a generic example without. We conclude with implications for secondary school teaching, as well as suggestions for future professional development efforts.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
This research was conducted in the USA, where K-12 refers to formal education beginning with early childhood (Kindergarten) and concludes with 12th grade (when students are generally around 18 years of age). 12th grade is the last grade before college.
Although there is an ongoing debate as to whether a generic example can be used to fully prove a theorem (e.g., Balacheff, 1988; Harel & Sowder, 1998; Leron & Zaslavsky, 2013; Rowland, 2002), our focus is on how a classroom community uses, perceives, and learns from generic examples in their proving activities. Consequently, this debate, important though it is, is beyond the scope of this paper.
In the analysis, we did not separate GEs produced before and after the instructor introduced the terminology, because we found no difference before and after.
One important note is that individual teachers would present their group’s work to the class as a whole, and consequently it will sometimes appear in the data that one teacher is presenting her own work, but in actuality, she is presenting her entire group’s work.
Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). London, UK: Hodder & Stoughton.
Ball, D., Hoyles, C., Jahnke, H., & Movshovitz-Hadar, N. (2002). The teaching of proof. In L. I. Tatsien (Ed.), Proceedings of the international congress of mathematicians (vol. III, pp. 907–920). Beijing, China: Higher Education Press.
Bieda, K. (2010). Enacting proof-related tasks in middle school mathematics: Challenges and opportunities. Journal for Research in Mathematics Education, 41(4), 351–382.
Bills, L., & Rowland, T. (1999). Examples, generalisation and proof. Research in Mathematics Education, 1(1), 103–116.
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. Accessed 04 May 2018.
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.
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.
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.
Fiallo, J., & Gutiérrez, A. (2017). Analysis of the cognitive unity or rupture between conjecture and proof when learning to prove on a grade 10 trigonometry course. Educational Studies in Mathematics, 96(2), 145–167.
Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative theory. New Brunswick, Canada: Aldine Transaction.
Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.
Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5–23.
Hanna, G. (2018). Reflections on proof as explanation. In A. J. Stylianides & G. Harel (Eds.), Advances in mathematics education research on proof and proving: An international perspective (pp. 3–18). Cham, Switzerland: Springer Nature.
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, Research in collegiate mathematics education III (vol. 7, pp. 234–283). Providence, RI: American Mathematical Society.
Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.
Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389–399.
Knuth, E. (2002a). Proof as a tool for learning mathematics. Mathematics Teacher, 95(7), 486–491.
Knuth, E. (2002b). Teachers conceptions of proof in the context of secondary school mathematics. Journal for Research in Mathematics Education, 5(1), 61–88.
Leron, U., & Zaslavsky, O. (2013). Generic proving: Reflections on scope and method. For the Learning of Mathematics, 33(3), 24–30.
Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15, 227–289.
Ministry of Instruction, University and Research (MIUR). (2012). Indicazioni nazionali per il curricolo della scuola dell’infanzia e del primo ciclo di istruzione. Rome, Italy: MIUR.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Governors Association. (2010). Common core state standards for mathematics. Washington, DC: Council of Chief State School Officers.
Reid, D., & Knipping, C. (2010). Proof in mathematics education: Research, learning, and teaching. Rotterdam: Sense Publishers.
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, Switzerland: Springer International Publishing.
Rowland, T. (2001). Generic proofs: Setting a good example. Mathematics Teaching, 177, 40–43.
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.
Schoenfeld, A. H. (1994). What do we know about mathematics curricula? The Journal of Mathematical Behavior, 13(1), 55–80.
Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38, 289–321.
Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9–16.
Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks. Mathematical Thinking and Learning, 11(4), 258–288.
Stylianides, G. J., & Silver, E. (2009). Reasoning-and-proving in school mathematics: The case of pattern identification. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K−16 perspective (pp. 235–249). New York, NY: Routledge.
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.
Tall, D. O., & Mejia-Ramos, J. P. (2009). The long-term cognitive development of different types of reasoning and proof. In G. Hanna, H. N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives (pp. 137–149). New York, NY: Springer.
Yin, R. K. (2006). Case study methods. In J. Green, G. Camilli, & P. Elmore (Eds.), The handbook of complementary methods in education research (pp. 111–122). Mahwah, NJ: Lawrence Erlbaum Associates.
Yopp, D. A., & Ely, R. (2016). When does an argument use a generic example? Educational Studies in Mathematics, 91(1), 37–53.
Thanks to Eric Knuth, Amy B. Ellis, Percival G. Matthews, Ana Stephens, and Rosemary Russ, the dissertation committee, that guided this data collection with the first author. Thanks to Jordan T. Thevenow-Harrison for their repeated reviews of this manuscript throughout the process, to Elise Lockwood for their helpful comments on an earlier version of this manuscript, and to the editor and the anonymous reviewers for their immensely helpful feedback. We are deeply grateful to the instructor who shared her classroom and her expertise and the teachers who shared their learning.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Dogan, M.F., Williams-Pierce, C. The role of generic examples in teachers’ proving activities. Educ Stud Math (2020). https://doi.org/10.1007/s10649-020-10002-3
- Generic examples
- Proof and justification
- Teacher education
- Visual and algebraic representations