Skip to main content
SpringerLink
Log in

A fictional dialogue on infinitude of primes: introducing virtual duoethnography

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. In this claim, we attempted to borrow the language from Euclid’s Definitions in Book 7. However, as a reviewer pointed out, according to Netz (1999), the language is a modern “filling out” of the language of Euclid.

  2. Note reference to [VII.31] on the margins of the proof in Fig. 1.

  3. See, for example, http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html.

  4. Delta is a student-character from Proofs and Refutations by Lakatos.

  5. Starting the list of primes with first primes is mentioned explicitly in several modern interpretations, such as Ribenboim (1996), see Fig. 3, or implicitly, for example, in Long (1987).

  6. Fermat was a lawyer and amateur mathematician.

  7. Mason and Watson (2009) presented their personal mathematical investigation in a fictional dialogue between Menousa, Xanthippe (Meno’s and Socrates’ wives, respectively), and a slave girl as “an extension and exploration of what occurs in the Meno mathematically” (p. 32), sticking closely to the style of the original Socratic dialogue.

References

  • 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.

  • 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.

  • 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.

  • Clandinin, D. J., & Connelly, F. M. (2000). Experience and story in qualitative research. San Francisco, CA: Jossey-Bass.

    Google Scholar 

  • 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.

  • 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.

  • Clough, P. (2002). Narratives and fictions in educational research. Buckingham: Open University Press.

    Google Scholar 

  • 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.

  • 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.

  • 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

  • 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.

  • 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.

  • 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.

  • Haven, K. (2007). Story proof: The science behind the startling power of story. Westport, CT: Libraries Unlimited.

  • 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.

    Article  Google Scholar 

  • Koichu, B., & Zazkis, R. (2013). Decoding a proof of Fermat’s Little Theorem via script writing. Journal of Mathematical Behavior, 32, 364–376.

    Article  Google Scholar 

  • Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Leavy, P. (2013). Fiction as research practice: Short stories, novellas, and novels. Walnut Creek, CA: Left Coast Press.

  • 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 

  • Leron, U. (1985). A direct approach to indirect proofs. Educational Studies in Mathematics, 16(3), 321–325.

    Article  Google Scholar 

  • Leron, U., & Hazzan, O. (1997). The world according to Johnny: A coping perspective in mathematics education. Educational Studies in Mathematics, 32(3), 265–292.

  • Lloyd, G. (2006). Preservice teachers’ stories of mathematics classrooms: Explorations of practice through fictional accounts. Educational Studies in Mathematics, 63(1), 57–87.

  • Long, C. T. (1987). Elementary introduction to number theory. Engelwood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  • Mason, J. (1998). Researching from the inside in mathematics education (pp. 357–377). Dordrecht, The Netherlands: Springer.

    Google Scholar 

  • Mason, J. (2002). Researching your own practice: the discipline of noticing. London, UK: Routledge Falmer Press.

    Google Scholar 

  • Mason, J. (2010). Mathematics education: Theory, practice and memories over 50 years. For the Learning of Mathematics, 30(3), 3–9.

  • Mason, J., & Watson, A. (2009). The Menousa. For the Learning of Mathematics, 29(2), 33–38.

    Google Scholar 

  • Mus, S. (2012). The case for fiction as qualitative research: Towards a non-referential ground for meaning. Ethics and Education, 7(2), 137–148.

  • Nardi, E. (2008). Amongst mathematicians: Teaching and learning mathematics at the university level. Dordrecht, The Netherlands: Springer.

  • Netz, R. (1999). The shaping of deduction in Greek mathematics. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • 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 

  • 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 

  • Pimm, D., Beisiegel, M., & Meglis, I. (2008). Would the real Lakatos please stand up. Interchange, 39(4), 469–481.

    Article  Google Scholar 

  • Rapke, T. K. (2014). Duoethnography: a new research methodology for mathematics education. Canadian Journal for Science, Mathematics and Technology Education, 14(2), 172-186.

  • Ribenboim, P. (1996). The new book of prime number records. New York: Springer.

    Book  Google Scholar 

  • Spindler, J. (2008). Fictional writing, educational research and professional learning. International Journal of Research & Method in Education, 31(1), 19–30.

    Article  Google Scholar 

  • 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.

  • 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.

  • Zazkis, D. (2014). Proof-scripts as a lens for exploring students’ understanding of odd/even functions. Journal of Mathematical Behavior, 35, 31–43.

    Article  Google Scholar 

  • Zazkis, R., Liljedahl, P., & Sinclair, N. (2009). Lesson plays: Planning teaching vs. teaching planning. For the Learning of Mathematics, 29(1), 40–47.

  • 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.

  • Zazkis, R., Sinclair, N., & Liljedahl, P. (2013). Lesson play in mathematics education: A tool for research and professional development. Dordrecht, The Netherlands: Springer.

  • 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.

Download references

Author information

Authors and Affiliations

  1. Faculty of Education, Simon Fraser University, Vancouver, Canada

    Rina Zazkis

  2. Department of Education in Science and Technology, Technion—Israel Institute of Technology, Haifa, Israel

    Boris Koichu

Authors
  1. Rina Zazkis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boris Koichu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rina Zazkis.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zazkis, R., Koichu, B. A fictional dialogue on infinitude of primes: introducing virtual duoethnography. Educ Stud Math 88, 163–181 (2015). https://doi.org/10.1007/s10649-014-9580-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-014-9580-0

Keywords

Search

Navigation