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The role of generic examples in teachers’ proving activities

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Abstract

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.

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Notes

  1. 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.

  2. 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.

  3. 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.

  4. 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.

References

  • 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.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Bills, L., & Rowland, T. (1999). Examples, generalisation and proof. Research in Mathematics Education, 1(1), 103–116.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative theory. New Brunswick, Canada: Aldine Transaction.

    Google Scholar 

  • Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.

    Article  Google Scholar 

  • Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5–23.

    Article  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • 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.

    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 

  • Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389–399.

    Article  Google Scholar 

  • Knuth, E. (2002a). Proof as a tool for learning mathematics. Mathematics Teacher, 95(7), 486–491.

    Article  Google Scholar 

  • Knuth, E. (2002b). Teachers conceptions of proof in the context of secondary school mathematics. Journal for Research in Mathematics Education, 5(1), 61–88.

    Google Scholar 

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

    Google Scholar 

  • Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15, 227–289.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

    Google Scholar 

  • National Governors Association. (2010). Common core state standards for mathematics. Washington, DC: Council of Chief State School Officers.

    Google Scholar 

  • Reid, D., & Knipping, C. (2010). Proof in mathematics education: Research, learning, and teaching. Rotterdam: Sense Publishers.

    Book  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • Rowland, T. (2001). Generic proofs: Setting a good example. Mathematics Teaching, 177, 40–43.

    Google Scholar 

  • 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.

    Google Scholar 

  • Schoenfeld, A. H. (1994). What do we know about mathematics curricula? The Journal of Mathematical Behavior, 13(1), 55–80.

    Article  Google Scholar 

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

    Google Scholar 

  • Stylianides, G. J. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9–16.

    Google Scholar 

  • Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks. Mathematical Thinking and Learning, 11(4), 258–288.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Yopp, D. A., & Ely, R. (2016). When does an argument use a generic example? Educational Studies in Mathematics, 91(1), 37–53.

    Article  Google Scholar 

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Acknowledgments

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.

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Authors and Affiliations

  1. Department of Mathematics and Science Education, Adiyaman University, Altinsehir Mah, Merkez, 02000, Adiyaman, Turkey

    Muhammed Fatih Dogan

  2. College of Information Studies, University of Maryland, Patuxent 2109G, 4161 Fieldhouse Drive, College Park, MD, 20742, USA

    Caro Williams-Pierce

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  1. Muhammed Fatih Dogan
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  2. Caro Williams-Pierce
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Correspondence to Muhammed Fatih Dogan.

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Dogan, M.F., Williams-Pierce, C. The role of generic examples in teachers’ proving activities. Educ Stud Math 106, 133–150 (2021). https://doi.org/10.1007/s10649-020-10002-3

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