Advertisement

ZDM

, Volume 41, Issue 1–2, pp 29–38 | Cite as

Mathematical paradoxes as pathways into beliefs and polymathy: an experimental inquiry

  • Bharath SriramanEmail author
Original article

Abstract

This paper addresses the role of mathematical paradoxes in fostering polymathy among pre-service elementary teachers. The results of a 3-year study with 120 students are reported with implications for mathematics pre-service education as well as interdisciplinary education. A hermeneutic-phenomenological approach is used to recreate the emotions, voices and struggles of students as they tried to unravel Russell’s paradox presented in its linguistic form. Based on the gathered evidence some arguments are made for the benefits and dangers in the use of paradoxes in mathematics pre-service education to foster polymathy, change beliefs, discover structures and open new avenues for interdisciplinary pedagogy.

Keywords

Beliefs Interdisciplinarity Paradoxes Pre-service teacher education Polymathy Russell’s paradox 

Supplementary material

11858_2008_110_MOESM1_ESM.doc (1.2 mb)
Supplementary Figure 1 (DOC 1278 kb)
11858_2008_110_MOESM2_ESM.doc (1.1 mb)
Supplementary Figure 2 (DOC 1125 kb)
11858_2008_110_MOESM3_ESM.doc (546 kb)
Supplementary Figure 3 (DOC 545 kb)
11858_2008_110_MOESM4_ESM.doc (32 kb)
Supplementary Table 1 (DOC 32 kb)
11858_2008_110_MOESM5_ESM.doc (34 kb)
Supplementary Table 2 (DOC 34 kb)

References

  1. Annells, M. (2006). Triangulation of qualitative approaches: Hermeneutical phenomenology and grounded theory. Journal of Advanced Nursing, 56(1), 55–61. doi: 10.1111/j.1365-2648.2006.03979.x.CrossRefGoogle Scholar
  2. Australian Education Council (1990). A national statement on mathematics for Australian schools. Melbourne: Australian Educational Council.Google Scholar
  3. Ball, D. L. (1990). The mathematical understandings that pre-service teachers bring to teacher education. The Elementary School Journal, 90, 449–466. doi: 10.1086/461626.CrossRefGoogle Scholar
  4. Cobb, P. (1988). The tension between theories of learning and theories of instruction in mathematics education. Educational Psychologist, 23, 87–104. doi: 10.1207/s15326985ep2302_2.CrossRefGoogle Scholar
  5. Ernest, P. (1989). The impact of beliefs on the teaching of mathematics. In Keitel, C., Damerow, P., Bishop, A., & Gerdes P., (Eds.), Mathematics, education and society (pp. 99–101). Paris: UNESCO Science and Technology Education Document Series No 35. Google Scholar
  6. Ernest, P. (1991). The philosophy of mathematics education. London: Falmer Press.Google Scholar
  7. Fenstermacher, G. D. (1978). A philosophical consideration of recent research on teacher effectiveness. In L. S. Shulman (Ed.), Review of research in education (pp. 157–185). Ithasca (IL): Peacock.Google Scholar
  8. Goldin, G. A. (2000). Affective pathways and representations in mathematical problem solving. Mathematical Thinking and Learning, 17(2), 209–219. doi: 10.1207/S15327833MTL0203_3.CrossRefGoogle Scholar
  9. Goldin, G. A. (2002). Affect, meta-affect, and mathematical belief structures. In G. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: a hidden variable in mathematics education? (pp. 59–72). Dordrecht: Kluwer.Google Scholar
  10. Grigutsch, S. (1996). Mathematische Weltbilder” bei Schülern: Struktur, Entwicklung, Einflussfaktoren. Dissertation. Duisburg: Gerhard-Mercator-Universität Duisburg, Fachbereich Mathematik.Google Scholar
  11. Leathman, K. (2006). Viewing mathematics teachers’ beliefs as sensible systems. Journal of Mathematics Teacher Education, 9(1), 91–102.CrossRefGoogle Scholar
  12. Leder, G. C., Pehkonen, E., & Törner, G., (Eds.), (2002). Beliefs: A hidden variable in mathematics education? (Vol. 31). Dodrecht: KluwerGoogle Scholar
  13. Merleau-Ponty, M. (1962). Phenomenology of perception (C. Smith, Trans.). London: Routledge & Kegan Paul.Google Scholar
  14. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston: Author.Google Scholar
  15. Padula, J. (2005). Mathematical fiction—it’s place in secondary school mathematics learning. The Australian Mathematics Teacher, 61(4), 6–13.Google Scholar
  16. Pajares, M. F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62(3), 307–332.Google Scholar
  17. Root-Bernstein, R. S. (1989). Discovering. Cambridge: Harvard University Press.Google Scholar
  18. Root-Bernstein, R. S. (1996). The sciences and arts share a common creative aesthetic. In A. I. Tauber (Ed.), The elusive synthesis: aesthetics and science (pp. 49–82). Netherlands: Kluwer.Google Scholar
  19. Root-Bernstein, R. S. (2000). Art advances science. Nature, 407, 134. doi: 10.1038/35025133.CrossRefGoogle Scholar
  20. Root-Bernstein, R. S. (2001). Music, science, and creativity. Leonardo, 34, 63–68. doi: 10.1162/002409401300052532.CrossRefGoogle Scholar
  21. Root-Bernstein, R. S. (2003). The art of innovation: Polymaths and the universality of the creative process. In L. Shavinina (Ed.), International handbook of innovation (pp. 267–278). Amsterdam: Elsevier.CrossRefGoogle Scholar
  22. Sriraman, B. (2003a). Can mathematical discovery fill the existential void? The use of conjecture, proof and refutation in a high school classroom. Mathematics in School, 32(2), 2–6.Google Scholar
  23. Sriraman, B. (2003b). Mathematics and literature: Synonyms, antonyms or the perfect amalgam. The Australian Mathematics Teacher, 59(4), 26–31.Google Scholar
  24. Sriraman, B. (2004). Mathematics and Literature (the sequel): Imagination as a pathway to advanced mathematical ideas and philosophy. The Australian Mathematics Teacher, 60(1), 17–23.Google Scholar
  25. Sriraman, B. (2005) Re-creating the Renaissance. In Anaya, M., & Michelsen, C. (Eds.), Relations between mathematics and others subjects of art and science. Proceedings of the 10th International Congress of Mathematics Education, Copenhagen, Denmark, pp. 14–19.Google Scholar
  26. Sriraman, B., & Dahl, B. (2008). On bringing interdisciplinary ideas to gifted education. In Shavinina, L. V. (Ed.), The international handbook of giftedness. Heidelberg: Springer Science (in press).Google Scholar
  27. Thompson, A. G. (1984). The relationship of teachers' conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15(2), 105–127.CrossRefGoogle Scholar
  28. Thompson, A. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In Grouws, D. A. (Ed.), Handbook of research on mathematics teaching and learning (pp. 127–146). Englewood-cliffs: Prentice Hall International.Google Scholar
  29. Törner, G. (2002). Mathematical beliefs. In Leder, G. C., E. Pehkonen, E., & Törner, G. (Eds.), Beliefs: a hidden variable in Mathematics education? (pp. 73–94). Dordrecht: Kluwer.Google Scholar
  30. Törner, G., & Sriraman, B. (2007). A contemporary analysis of the six “Theories of Mathematics Education” Theses of Hans-Georg Steiner. Invited paper for special issue of ZDM- The International Journal on Mathematics Education in memoriam Hans- Georg. Steiner 39(1):155–163.Google Scholar
  31. Romme, M. A. J., & Escher, A. D. M. A. C. (1993). The new approach: A Dutch experiment. In M. A. J. Romme & A. D. M. A. C. Escher (Eds.), Accepting voices (pp. 11–27). London: MIND publications.Google Scholar
  32. Sainsbury, R. M. (1995). Paradoxes. Cambridge: Cambridge University Press.Google Scholar
  33. Wedege, T., Skott, J. (2006). Changing Views and Practices: A study of the KappAbel mathematics competition. Research Report: Norwegian Center for Mathematics Education & Norwegian University of Science and Technology (pp. 274). Trondheim.Google Scholar

Copyright information

© FIZ Karlsruhe 2008

Authors and Affiliations

  1. 1.Department of MathematicsThe University of MontanaMissoulaUSA

Personalised recommendations