Skip to main content

An Ode to Imre Lakatos: Quasi-Thought Experiments to Bridge the Ideal and Actual Mathematics Classrooms


This paper explores the wide range of mathematics content and processes that arise in the secondary classroom via the use of unusual counting problems. A universal pedagogical goal of mathematics teachers is to convey a sense of unity among seemingly diverse topics within mathematics. Such a goal can be accomplished if we could conduct classroom discourse that conveys the Lakatosian (thought-experimental) view of mathematics as that of continual conjecture-proof-refutation which involves rich mathematizing experiences. I present a pathway towards this pedagogical goal by presenting student insights into an unusual counting problem and by using these outcomes to construct ideal mathematical possibilities (content and process) for discourse. In particular, I re-construct the quasi-empirical approaches of six!4-year old students’ attempts to solve this unusual counting problem and present the possibilities for mathematizing during classroom discourse in the imaginative spirit of Imre Lakatos. The pedagogical implications for the teaching and learning of mathematics in the secondary classroom and in mathematics teacher education are discussed.

This is a preview of subscription content, access via your institution.


  1. Australian Education Council. (1990). A national statement on mathematics for Australian schools. Melbourne, VC: Australian Educational Council.

  2. Bohman, T. (1996). A sum packing problem of Erdos and the Conway-Guy sequence. Proceedings of the American Mathematical Society, 124(12),3627–3636.

  3. Brodkey, J.J. (1996). Starting a Euclid club. Mathematics Teacher, 89(5), 386–388.

    Google Scholar 

  4. Dienes, Z.P. (1960). Building up mathematics. London: Hutchinson Education.

    Google Scholar 

  5. Dienes, Z.P. (1961). An experimental study of mathematics learning. New York: Hutchinson & Co Ltd.

    Google Scholar 

  6. Doerr, H. & Lesh, R. (2003). A modeling perspective on teacher development. In R. Lesh & H. Doerr (Eds.), Beyond Constructivism (pp.125–125). New Jersey: Lawrence Erlbaum Associates.

  7. English, L.D. (1998). Children’s problem posing within formal and informal contexts. Journal for Research in Mathematics Education, 29(1), 83–206.

    Google Scholar 

  8. English, L.D. (1999). Assessing for structural understanding in children’s combinatorial problem solving. Focus on Learning Problems in Mathematics, 21(4),63–82.

    Google Scholar 

  9. Fawcett, H.P. (1938). The nature of proof. Thirteenth yearbook of the NCTM. New York: Bureau of Publications, Teachers College, Columbia University.

    Google Scholar 

  10. Fomin, D., Genkin, S., & Itenberg, I. (1996). Mathematical Circles (Russian Experience). American Mathematical Society.

  11. Gardner, M. (1997). The last recreations. New York: Springer-Verlag. Goldbach, C. (1742) Letter to L. Euler, June 7,1742. Retrieved January 11, 2004 from

  12. Guy, R.K. (1982). Sets of integers whose subsets have distinct sums, Theory and practice of combinatorics, Annals of discrete math, 12, 141–154. North-Holland, Amsterdam.

  13. Hogendijk, J.P. (1996). Een workshop over Iraanse mozaïken. Nieuwe Wiskrant, 16(2), 38–42.

  14. Hung, D. (1998) Meanings, contexts and mathematical thinking: the meaning-context model. Journal of Mathematical Behavior, 16(3),311–344.

    Google Scholar 

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

    Google Scholar 

  16. Lesh, R. & Doerr, H. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning and problem solving. In R. Lesh & H. Doerr (Eds.), Beyond constructivism (pp. 3–34). New Jersey: Lawrence Erlbaum Associates.

  17. Lewis, B. (2003). Taking perspective. The Mathematical Gazette, 87(510),418–431.

    Google Scholar 

  18. Maher, C.A. & Kiczek, R.D. (2000). Long term building of mathematical ideas related to proof making. Contributions to Paolo Boero, G. Harel, C. Maher, M. Miyasaki (organisers), Proof and Proving in Mathematics Education. ICME9 -TSG 12. Tokyo/Makuhari, Japan.

  19. Maher, C.A. & Martino A.M. (1996a) The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–244.

    Article  Google Scholar 

  20. Maher C.A.& Martino A.M. (1996b) Young children invent methods of proof: The “Gang of Four.” In P.Nesher, L.P. Steffe, P. Cobb, B. Greer & J. Goldin (Eds.), Theories of mathematical learning, (pp. 431–447). Mahwah, NJ: Lawrence Erlbaum Associates.

  21. Maher, C.A., & Martino, A.M. (1997) Conditions for conceptual change: From pattern recognition to theory posing. In H. Mansfield (Ed.), Young children and mathematics: concepts and their representations. Durham, NH: Australian Association of Mathematics Teachers.

  22. Maher, C.A. & Speiser, B. (1997). How far can you go with block towers? Stephanie’s intellectual development. Journal of Mathematical Behavior 16(2),125–132.

    Google Scholar 

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

  24. Rotman, B. (1977). Jean Piaget: Psychologist of the real. Cornell University Press.

  25. Sriraman, B. (2003a). Mathematical giftedness, problem solving, and the ability to formulate generalizations. The Journal of Secondary Gifted Education, 14(3), 151–165.

    Google Scholar 

  26. Sriraman, B. (2003b). 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 

  27. Sriraman, B. (2004a). Discovering a mathematical principle: The case of Matt. Mathematics in School, 33(2),25–31.

    Google Scholar 

  28. Sriraman, B. (2004b). Reflective abstraction, uniframes and formulation of generalizations. Journal of Mathematical Behavior, 23(2), 205–222.

    Article  Google Scholar 

  29. Sriraman, B. (2004c). Discovering Steiner triple systems through problem solving. The Mathematics Teacher, 97(5),320–326.

    Google Scholar 

  30. Sriraman, B. (2004d). Re-creating the renaissance. In M. Anaya, C. Michelsen (Eds.), Proceedings of the topic study group 21 -Relations between mathematics and others subjects of art and science: The 10 th International Congress of Mathematics Education, Copenhagen, Denmark (pp. 14–19).

  31. Sriraman, B. & Adrian, H. (2004a). The pedagogical value and the interdisciplinary nature of inductive processes in forming generalizations. Interchange: A Quarterly Review of Education, 35(4), 407–422.

    Google Scholar 

  32. Sriraman, B. & English, L. (2004a). Combinatorial mathematics: Research into practice. Connecting research into teaching. The Mathematics Teacher, 98(3),182–191.

    Google Scholar 

  33. Sriraman, B. & Strzelecki, P. (2004a). Playing with powers. The International Journal for Technology in Mathematics Education, 11(1), 29–39.

    Google Scholar 

  34. Van Maanen, J. (1992). Teaching geometry to 11 year old “medieval lawyers.” The Mathematical Gazette, 76(475), 37–45.

    Google Scholar 

  35. Wheeler, D. (2001). Mathematisation as a pedagogical tool. For the Learning of Mathematics, 21(2), 50–53.

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Bharath Sriraman.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sriraman, B. An Ode to Imre Lakatos: Quasi-Thought Experiments to Bridge the Ideal and Actual Mathematics Classrooms. Interchange 37, 151–178 (2006).

Download citation


  • Combinatorics
  • conjecture
  • counting
  • generalization
  • Lakatos
  • mathematization
  • mathematical structures
  • pedagogy
  • problem solving
  • refutation
  • teacher education