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

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## Abstract

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.

## Keywords

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

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## References

- Australian Education Council. (1990).
*A national statement on mathematics for Australian schools*. Melbourne, VC: Australian Educational Council.Google Scholar - Bohman, T. (1996). A sum packing problem of Erdos and the Conway-Guy sequence.
*Proceedings of the American Mathematical Society*,*124*(12),3627–3636.Google Scholar - Brodkey, J.J. (1996). Starting a Euclid club.
*Mathematics Teacher*, 89(5), 386–388.Google Scholar - Dienes, Z.P. (1960).
*Building up mathematics*. London: Hutchinson Education.Google Scholar - Dienes, Z.P. (1961).
*An experimental study of mathematics learning*. New York: Hutchinson & Co Ltd.Google Scholar - 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.Google Scholar - 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 - 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 - Fawcett, H.P. (1938).
*The nature of proof. Thirteenth yearbook of the**NCTM*. New York: Bureau of Publications, Teachers College, Columbia University.Google Scholar - Fomin, D., Genkin, S., & Itenberg, I. (1996).
*Mathematical Circles**(Russian Experience)*. American Mathematical Society.Google Scholar - 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 http://www.mathstat.dal.ca/~joerg/pic/g-letter.jpgGoogle Scholar - 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.Google Scholar - Hogendijk, J.P. (1996). Een workshop over Iraanse mozaïken.
*Nieuwe**Wiskrant, 16*(2), 38–42.Google Scholar - Hung, D. (1998) Meanings, contexts and mathematical thinking: the meaning-context model.
*Journal of Mathematical Behavior, 16*(3),311–344.Google Scholar - Lakatos, I. (1976).
*Proofs and refutations*. Cambridge, UK: Cambridge University Press.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - 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 - National Council of Teachers of Mathematics. (2000).
*Principles and**standards for school mathematics:*Reston, VA: NCTM.Google Scholar - Rotman, B. (1977).
*Jean Piaget: Psychologist of the real*. Cornell University Press.Google Scholar - 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 - 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 - Sriraman, B. (2004a). Discovering a mathematical principle: The case of Matt.
*Mathematics in School, 33*(2),25–31.Google Scholar - Sriraman, B. (2004b). Reflective abstraction, uniframes and formulation of generalizations.
*Journal of Mathematical Behavior*, 23(2), 205–222.CrossRefGoogle Scholar - Sriraman, B. (2004c). Discovering Steiner triple systems through problem solving.
*The Mathematics Teacher, 97*(5),320–326.Google Scholar - 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).Google Scholar - 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 - Sriraman, B. & English, L. (2004a). Combinatorial mathematics: Research into practice. Connecting research into teaching.
*The**Mathematics Teacher, 98*(3),182–191.Google Scholar - Sriraman, B. & Strzelecki, P. (2004a). Playing with powers.
*The**International Journal for Technology in Mathematics Education*, 11(1), 29–39.Google Scholar - Van Maanen, J. (1992). Teaching geometry to 11 year old “medieval lawyers.”
*The Mathematical Gazette*, 76(475), 37–45.Google Scholar - Wheeler, D. (2001). Mathematisation as a pedagogical tool.
*For the Learning of Mathematics, 21(2)*, 50–53.Google Scholar