Educational Studies in Mathematics

, Volume 88, Issue 2, pp 163–181 | Cite as

A fictional dialogue on infinitude of primes: introducing virtual duoethnography

Article
First Online:

Abstract

We introduce virtual duoethnography as a novel research approach in mathematics education, in which researchers produce a text of a dialogic format in the voices of fictional characters, who present and contrast different perspectives on the nature of a particular mathematical phenomenon. We use fiction as a form of research linked to narrative inquiry and exemplify our approach in a dialogue related to various proofs of infinitude of primes. We view Lakatos’ (1976) dialogue in the seminal Proofs and Refutations as an example of virtual duoethnography. We discuss the affordances of this approach as an alternative to the formal ways of presenting research in mathematics education.

Keywords

Duoethnography Virtual monologue Euclid Infinitude of primes Dialogic method Narrative inquiry Scripting tasks 
This is a preview of subscription content, log in to check access.

References

  1. Antonini, S., & Mariotti, M. A. (2006). Reasoning in an absurd world: Difficulties with proof by contradiction. 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, pp. 65–72). Prague, Czech Republic: PME.Google Scholar
  2. Bjuland, R., Cestari, M. L., & Borgersen, H. E. (2012). Professional mathematics teacher identity: Analysis of reflective narratives from discourses and activities. Journal of Mathematics Teacher Education, 15(5), 405–424.Google Scholar
  3. Buchbinder, O., & Zaslavsky, O. (2013). Inconsistencies in students’ understanding of proof and refutation of mathematical statements. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 129–136). Kiel, Germany: PME.Google Scholar
  4. Clandinin, D. J., & Connelly, F. M. (2000). Experience and story in qualitative research. San Francisco, CA: Jossey-Bass.Google Scholar
  5. Chase, S. E. (2007). Narrative inquiry: Multiple lenses, approaches, voices. In N. K. Denzin & Y. S. Lincoln (Eds.), Collecting and interpreting qualitative materials (pp. 57–94). Thousand Oaks, CA: Sage Publications.Google Scholar
  6. Chen, C.-L. (2012). Learning to teach from anticipating lessons through comics-based approximations of practice. (Doctoral dissertation, University of Michigan). Retrieved from  http://deepblue.lib.umich.edu/bitstream/handle/2027.42/91421/chialc_1.pdf;jsessionid=D816C95BE2A228037B77E101AE57068E?sequence=1. Accessed 10 Oct 2013.
  7. Clough, P. (2002). Narratives and fictions in educational research. Buckingham: Open University Press.Google Scholar
  8. De Freitas, E. (2004). Plotting intersections along the political axis: The interior voice of dissenting mathematics teachers. Educational Studies in Mathematics, 55(1–3), 259–274.Google Scholar
  9. De Freitas, E. (2008). Enacting identity through narrative: Interrupting the procedural discourse in mathematics classrooms. In J. F. Matos, P. Valero & K. Yasukawa (Eds.), Proceedings of the Fifth International Mathematics Education and Society Conference (pp. 272–282). Lisbon: Centro de Investigação em Educação, Universidade de Lisboa—Department of Education, Learning and Philosophy, Aalborg University.Google Scholar
  10. Ejersbo, L. R., & Leron, U. (2005). The didactical transposition of didactical ideas: The case of the virtual monologue. Working Group 11: Different theoretical perspectives and approaches in research in mathematics education, 1379. Retrieved October 10, 2013 from http://www.erme.tu-dortmund.de/~erme/CERME4/CERME4_WG11.pdf#page=143
  11. Euclides, I. (2002). In D. Densmore (Ed.), Euclid’s elements: All thirteen books complete in one volume, the Thomas L. Heath translation. Santa Fe, NM: Green Lion Press.Google Scholar
  12. Harel, G. (2007). Students’ proof schemes revisited. In P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 65–80). Rotterdam, The Netherlands: Sense Publishers.Google Scholar
  13. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from an exploratory study. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics III (pp. 234–283). Providence, Rhode Island: American Mathematical Society.Google Scholar
  14. Haven, K. (2007). Story proof: The science behind the startling power of story. Westport, CT: Libraries Unlimited.Google Scholar
  15. Herbst, P., Chazan, D., Chen, C.-L., Chieu, V.-M., & Weiss, M. (2011). Using comics-based representations of teaching, and technology, to bring practice to teacher education courses. ZDM—The International Journal of Mathematics Education, 43, 91–103.CrossRefGoogle Scholar
  16. Koichu, B., & Zazkis, R. (2013). Decoding a proof of Fermat’s Little Theorem via script writing. Journal of Mathematical Behavior, 32, 364–376.CrossRefGoogle Scholar
  17. Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  18. Leavy, P. (2013). Fiction as research practice: Short stories, novellas, and novels. Walnut Creek, CA: Left Coast Press.Google Scholar
  19. Leikin, R., & Winicki-Landman, G. (2001). Defining as a vehicle for professional development of secondary school mathematics teachers. Mathematics Teacher Education and Development (MTED), 3, 62–73.Google Scholar
  20. Leron, U. (1985). A direct approach to indirect proofs. Educational Studies in Mathematics, 16(3), 321–325.CrossRefGoogle Scholar
  21. Leron, U., & Hazzan, O. (1997). The world according to Johnny: A coping perspective in mathematics education. Educational Studies in Mathematics, 32(3), 265–292.Google Scholar
  22. Lloyd, G. (2006). Preservice teachers’ stories of mathematics classrooms: Explorations of practice through fictional accounts. Educational Studies in Mathematics, 63(1), 57–87.Google Scholar
  23. Long, C. T. (1987). Elementary introduction to number theory. Engelwood Cliffs, NJ: Prentice-Hall.Google Scholar
  24. Mason, J. (1998). Researching from the inside in mathematics education (pp. 357–377). Dordrecht, The Netherlands: Springer.Google Scholar
  25. Mason, J. (2002). Researching your own practice: the discipline of noticing. London, UK: Routledge Falmer Press.Google Scholar
  26. Mason, J. (2010). Mathematics education: Theory, practice and memories over 50 years. For the Learning of Mathematics, 30(3), 3–9.Google Scholar
  27. Mason, J., & Watson, A. (2009). The Menousa. For the Learning of Mathematics, 29(2), 33–38.Google Scholar
  28. Mus, S. (2012). The case for fiction as qualitative research: Towards a non-referential ground for meaning. Ethics and Education, 7(2), 137–148.Google Scholar
  29. Nardi, E. (2008). Amongst mathematicians: Teaching and learning mathematics at the university level. Dordrecht, The Netherlands: Springer.Google Scholar
  30. Netz, R. (1999). The shaping of deduction in Greek mathematics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  31. Norris, J., & Sawyer, R. D. (2012). Toward a dialogic methodology. In J. Norris, R. D. Sawyer, & D. Lund (Eds.), Duoethnography: dialogic methods for social, health, and educational research (Vol. 7, pp. 9–39). Walnut Creek, CA: Left Coast Press.Google Scholar
  32. Norris, J., Sawyer, R. D., & Lund, D. (Eds.). (2012). Duoethnography: dialogic methods for social, health, and educational research (Vol. 7). Walnut Creek, CA: Left Coast Press.Google Scholar
  33. Pimm, D., Beisiegel, M., & Meglis, I. (2008). Would the real Lakatos please stand up. Interchange, 39(4), 469–481.CrossRefGoogle Scholar
  34. Rapke, T. K. (2014). Duoethnography: a new research methodology for mathematics education. Canadian Journal for Science, Mathematics and Technology Education, 14(2), 172-186.Google Scholar
  35. Ribenboim, P. (1996). The new book of prime number records. New York: Springer.CrossRefGoogle Scholar
  36. Spindler, J. (2008). Fictional writing, educational research and professional learning. International Journal of Research & Method in Education, 31(1), 19–30.CrossRefGoogle Scholar
  37. Van Dormolen, J., & Zaslavsky, O. (2003). The many facets of a definition: The case of periodicity. The Journal of Mathematical Behavior, 22(1), 91–106.Google Scholar
  38. Zaslavsky, O., & Ron, G. (1998). Student’s understanding of the role of counter-examples. In A. Oliver & K. Newstead (Eds.), Proceedings of 22nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 225–232). Stellenbosch, South Africa: PME.Google Scholar
  39. Zazkis, D. (2014). Proof-scripts as a lens for exploring students’ understanding of odd/even functions. Journal of Mathematical Behavior, 35, 31–43.CrossRefGoogle Scholar
  40. Zazkis, R., Liljedahl, P., & Sinclair, N. (2009). Lesson plays: Planning teaching vs. teaching planning. For the Learning of Mathematics, 29(1), 40–47.Google Scholar
  41. Zazkis, R., Sinclair, N., & Liljedahl, P. (2009). Lesson play—A vehicle for multiple shifts of attention in teaching. In S. Lerman & B. Davis (Eds.), Mathematical action & structures of noticing: Studies inspired by John Mason (pp. 165–178). Rotterdam, The Netherlands: Sense Publishers.Google Scholar
  42. Zazkis, R., Sinclair, N., & Liljedahl, P. (2013). Lesson play in mathematics education: A tool for research and professional development. Dordrecht, The Netherlands: Springer.Google Scholar
  43. Zazkis, R., & Zazkis, D. (2014). Script writing in the mathematics classroom: Imaginary conversations on the structure of numbers. Research in Mathematics Education, 16(1), 54–70.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Faculty of EducationSimon Fraser UniversityVancouverCanada
  2. 2.Department of Education in Science and TechnologyTechnion—Israel Institute of TechnologyHaifaIsrael

Personalised recommendations