Educational Studies in Mathematics

, Volume 76, Issue 2, pp 165–182 | Cite as

A mathematical experience involving defining processes: in-action definitions and zero-definitions



In this paper, a focus is made on defining processes at stake in an unfamiliar situation coming from discrete mathematics which brings surprising mathematical results. The epistemological framework of Lakatos is questioned and used for the design and the analysis of the situation. The cognitive background of Vergnaud’s approach enriches the study of freshmen’s processes at university. The mathematical analysis and the results specifically underscore the in-action definitions and the zero-definitions and highlight the need of similar mathematical experiences in education, particularly focused on defining processes and the exploration of a research problem.


Zero-definitions In-action definitions Proof-generated definitions Concept image Displacements Regular grid Discrete mathematics 


  1. Borasi, R. (1992). Learning mathematics through inquiry. New Hampshire: Heinemann.Google Scholar
  2. Brousseau, G. (1997). Theory of the didactical situations in mathematics. Dordrecht: Kluwer.Google Scholar
  3. Burton, L. (2004). Mathematicians as enquirers: Learning about learning mathematics. Berlin: Springer.Google Scholar
  4. Carlson, M. P., & Bloom, I. (2005). The cyclic nature of problem solving. Educational Studies in Mathematics, 58(1), 45–75.CrossRefGoogle Scholar
  5. De Villiers, M. (1998). To teach definitions in geometry or to teach to define? In A. Olivier & K. Newstead (Eds.), Proceedings of PME 22 (Vol. 2, pp. 248–255). Stellenbosch: RSA.Google Scholar
  6. De Villiers, M. (2000). A Fibonacci generalization: A Lakatosian example. Mathematics in College, pp. 10–29.Google Scholar
  7. Dorier, J.-L. (Ed.). (2000). On the teaching of linear algebra. Dordrecht: Kluwer.Google Scholar
  8. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In O. D. Tall (Ed.), Advanced mathematical thinking (pp. 95–123). Dordrecht: Kluwer.Google Scholar
  9. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.Google Scholar
  10. Goldin, G. A. (2004). Problem solving heuristics, affect, and discrete mathematics. Zentralblatt für Didaktik der Mathematik, 36(2), 56–60.CrossRefGoogle Scholar
  11. Grenier, D., & Payan, C. (1999). Discrete mathematics in relation to learning and teaching proof and modelling. In I. Schwank (Ed.), Proceedings of CERME 1, vol. 1 (pp. 143–155). OsnabrückGoogle Scholar
  12. Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1–2), 5–23.CrossRefGoogle Scholar
  13. Hanna, G. (2007). The ongoing value of proof. In P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 3–16). Rotterdam: Sense Publishers.Google Scholar
  14. Harel, G. (1998). Two dual assertions: The first on learning and the second on teaching (or vice versa). American Mathematical Monthly, 105(6), 497–507.CrossRefGoogle Scholar
  15. Heinze, A., Anderson, I., & Reiss, K. (2004). Discrete mathematics and proof in the high school. Zentralblatt für Didaktik der Mathematik, 36(2), 44–84 and 36(3), 82–116.Google Scholar
  16. Lakatos, I. (1961). Essays in the logic of mathematical discovery. Thesis. Cambridge.Google Scholar
  17. Lakatos, I. (1976). Proofs and refutations. Cambridge University Library.Google Scholar
  18. Larsen, S., & Zandieh, M. (2005). Conjecturing and proving as part of the process of defining. In G. M. Lloyd, M. Wilson, J. L. M. Wilkins & Behm, S. L. (Eds.), Proceedings of the 27th PME-NA.Google Scholar
  19. Larsen, S., & Zandieh, M. (2008). Proofs and refutations in the undergraduate mathematics classroom. Educational Studies in Mathematics, 67, 205–216.CrossRefGoogle Scholar
  20. Ouvrier-Buffet, C. (2003). Can the Aristotelian and Lakatosian conceptions constitute a tool for the analysis of a definition construction process? Mediterranean Journal for Research in Mathematics Education, 2, 19–36.Google Scholar
  21. Ouvrier-Buffet, C. (2006). Exploring mathematical definition construction processes. Educational Studies in Mathematics, 63(3), 259–282.CrossRefGoogle Scholar
  22. Pimm, D., Beisiegel, M., & Meglis, I. (2008). Would the real Lakatos please stand up. Interchange, 39(4), 469–481.CrossRefGoogle Scholar
  23. Ramirez Alfonsin, J. L. (2006). The Diophantine Frobenius problem. Oxford: Oxford University Press.Google Scholar
  24. Schoenfeld, A. H. (1985). Mathematical problem solving. San Diego: Academic.Google Scholar
  25. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides on the same coin. Educational Studies in Mathematics, 22, 1–36.CrossRefGoogle Scholar
  26. Tall, D. O. (Ed.). (1991). Advanced mathematical thinking. Dordrecht: Kluwer.Google Scholar
  27. Tall, D. O. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24.Google Scholar
  28. Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177.CrossRefGoogle Scholar
  29. Uhlig, F. (2002). The role of proof in comprehending and teaching elementary linear algebra. Educational Studies in Mathematics, 50(3), 335–346.CrossRefGoogle Scholar
  30. Vergnaud, G. (1996). The theory of conceptual fields. In L. Steffe, P. Nesher, P. Cobb, G. Goldin & B. Greer (Eds.), Theories of mathematical learning (pp. 219–239). Mahwah: Erlbaum.Google Scholar
  31. Vinner, S. (1991). The role of definitions in teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65–81). Dordrecht: Kluwer.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Université Paris Diderot, LDARParis Cedex 13France
  2. 2.IUFM de Créteil, UPECParis Cedex 13France

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