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Mathematical Proof and Genre Theory

  • David Matthew BowersEmail author
  • Valentin A. B. Küchle
Article

To write an intro

In seventeen syllables

Is very diffi—

Although the words above follow a traditional haiku’s 5-7-5 pattern, they may be more aptly described as a joke—indeed a silly one. Yet in essence, it describes a key message of this paper: a genre is more accurately defined and analyzed by the interaction between its form and the intent it pursues than by its form alone. Using this modern understanding of what it means to be a genre, we explore the genre of mathematical proof and borrow from genre theory to pose questions about proof that have not yet been formulated from this conceptual foundation.

Mathematics that is generative and relevant is, at its heart, creative. Yet the primary means for developing, communicating, and sanctioning new knowledge—proof—often fails to support and reflect the importance of creativity in mathematics. It is with deference to creativity that we seek to apply a conceptual apparatus that originated in the humanities for the study of literature,...

Notes

References

  1. Adamchik, V., & Wagon, S. (1997). A simple formula for π. American Mathematical Monthly 104(9), 852–855. Retrieved from http://www.jstor.org/stable/2975292.
  2. Bazerman, C. (1994). Systems of genres and the enactment of social intentions. In Freedman, A. & Medway, P. (eds.), Genre and the New Rhetoric, pp. 1–22. London: Taylor & Francis.Google Scholar
  3. Bowers, D. M. (2019). Rhetorical experiment in empathy: Mathematics and mathematical practice amongst the Nacirema. In J. Subramanian (ed.), Proceedings of the 10 th International Mathematics Education and Society Conference (pp. 294–302). Hyderabad, India.Google Scholar
  4. Brown, J. R. (2008). Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures (second edition). New York: Routledge.zbMATHGoogle Scholar
  5. Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher 18(1), 32–42.CrossRefGoogle Scholar
  6. Coe, R. M. (1994). “An arousing and fulfilment of desires”: The rhetoric of genre in the process era—and beyond. In Freedman, A. & Medway, P. (eds.), Genre and the New Rhetoric (pp. 181–190). London: Taylor & Francis.Google Scholar
  7. Czocher, J. & Weber, K. (in press). Proof as a cluster category. To appear in Journal for Research in Mathematics Education. Google Scholar
  8. Dawkins, P. C., & Weber, K. (2017). Values and norms of proof for mathematicians and students. Educational Studies in Mathematics 95(2), 123–142. Retrieved from  https://doi.org/10.1007/s10649-016-9740-5.
  9. Delgado, R. & Stefancic, J. (2001). Critical Race Theory: An Introduction. New York: University Press.Google Scholar
  10. Doyle, T., Kutler, L., Miller, R., & Schueller, A. (2014). Proofs without words and beyond: PWW and mathematical proof. Retrieved from https://www.maa.org/press/periodicals/convergence/proofs-without-words-and-beyond-pwws-and-mathematical-proof.
  11. Freedman, A. & Medway, P. (1994). Locating genre studies: Antecedents and prospects. In Freedman, A. & Medway, P. (eds.), Genre and the New Rhetoric (pp. 1–22). London: Taylor & Francis.Google Scholar
  12. Frentz, T. S., & Farrell, T. B. (1976). Language-action: A paradigm for communication. Quarterly Journal of Speech 62, 333–349.CrossRefGoogle Scholar
  13. Greeno, J. G. (1998). The situativity of knowing, learning, and research. American Psychologist 53(1) 5–26.CrossRefGoogle Scholar
  14. Greer, B. & Mukhopadhyay, S. (2012). The hegemony of mathematics. In Skovmose & Greer (eds.), Opening the cage: Critique and Politics of Mathematics Education (pp. 229–248). The Netherlands: Sense.Google Scholar
  15. Holdener, J. A. (2002). A theorem of Touchard on the form of odd perfect numbers. American Mathematical Monthly 109(7), 661–663. Retrieved from  https://doi.org/10.1080/00029890.2002.11919899.
  16. Ivanič, R. (1998). Issues of identity in academic writing. In Writing and Identity: The Discoursal Construction of Identity in Academic Writing (pp. 75–106). Amsterdam: John Benjamins.CrossRefGoogle Scholar
  17. Kline, M. (1977). Why the Professor Can’t Teach. New York: St. Martin’s Press.Google Scholar
  18. Lakoff, G. (1987). Women, Fire, and Dangerous Things: What Categories Reveal About the Mind. Chicago: University of Chicago Press.CrossRefGoogle Scholar
  19. Lakoff, G. and Núñez, R. E. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.zbMATHGoogle Scholar
  20. Littlewood, J. (1953). A Mathematician’s Miscellany. London: Methuen.zbMATHGoogle Scholar
  21. Miller, C. R. (1984). Genre as social action. Quarterly Journal of Speech 70, 151–167.CrossRefGoogle Scholar
  22. Miller, C. R. (1994). Rhetorical community: The cultural basis of genre. In Freedman, A. & Medway, P. (eds.), Genre and the New Rhetoric (pp. 67–78). London: Taylor & Francis.Google Scholar
  23. Paré, A. & Smart, G. (1994). Observing genres in action: Towards a research methodology. In Freedman, A. & Medway, P. (eds.), Genre and the New Rhetoric (pp. 146–154). London: Taylor & Francis.Google Scholar
  24. Pearce, W. B., & Conklin, F. (1979). A model of hierarchical meanings in coherent conversation and a study of indirect responses. Communication Monographs 46, 76–87.CrossRefGoogle Scholar
  25. Pressman, E. R., Young, J., Thomas, J., Montgomery, M. (producers), & Brown, M. (director). (2015). The Man who Knew Infinity [motion picture]. United Kingdom: Warner Bros.Google Scholar
  26. Reid, D. A. (2002). Conjectures and refutations in grade 5 mathematics. Journal for Research in Mathematics Education 33(1), 5–39.CrossRefGoogle Scholar
  27. Richardson, L. (2000). Writing: A method of inquiry. In Denzin, N. K. & Lincoln, Y. S. (eds.), Handbook of Qualitative Research (pp. 923–948). Thousand Oaks: Sage Publications.Google Scholar
  28. Schifter, D. (2009). Representation-based proof in the elementary grades. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (eds.), Teaching and Learning Proof Across the Grades: A K–16 Perspective (pp. 71–86). New York: Routledge.Google Scholar
  29. Schooler, J. W., Ohlsson, S., & Brooks, K. (1993). Thoughts beyond words: When language overshadows insight. Journal of Experimental Psychology 122(2), 166–183.CrossRefGoogle Scholar
  30. Selden, A., & Selden, J. (2014). The genre of proof. In M. N. Fried & T. Dreyfus (eds.), Mathematics & Mathematics Education: Searching for Common Ground (pp. 248–251). Dordrecht: Springer. Retrieved from  https://doi.org/10.1007/978-94-007-7473-5.
  31. Simonton, M. J. (2012). Introduction to Cultural Anthropology. Dubuque: Kendall Hunt Publishing Company.Google Scholar
  32. Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society 30(2), 161–177.MathSciNetzbMATHCrossRefGoogle Scholar
  33. Weaver, J. A., & Snaza, N. (2017). Against methodocentrism in educational research. Educational Philosophy and Theory, 49(11), 1055–1065.CrossRefGoogle Scholar
  34. Weber, K. (2014). Proof as a cluster concept. In C. Nicol, S. Oesterle, P. Liljedahl, & D. Allan (eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (vol. 5, pp. 353–360). Vancouver: PME.Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Michigan State UniversityEast LansingUSA

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