The Long-Term Cognitive Development of Reasoning and Proof

  • David Tall
  • Juan Pablo Mejia-Ramos


This paper uses the framework of “three worlds of mathematics” (Tall 2004a, b) to chart the development of mathematical thinking from the thought processes of early childhood to the formal structures of set-theoretic definition and formal proof. It sees the development of mathematical thinking building on experiences that the individual has met before, as the child coordinates perceptions and actions to construct thinkable concepts in two different ways. One focuses on objects, exploring their properties, describing them, using carefully worded descriptions as definitions, inferring that certain properties imply others and on to coherent frameworks such as Euclidean geometry through a developing mental world of conceptual embodiment. The other focuses on actions (such as counting), first as procedures and then compressed into thinkable concepts (such as number) using symbols such as 3 + 2, ¾, 3a + 2b, f (x), dy/dx; these operate dually as computable processes and thinkable concepts, termed procepts, in a developing mental world of proceptual symbolism. These may lead later to a third mental world of axiomatic formalism based on set-theoretic definition and formal proof. In addition to charting the development of proof concepts through these three worlds, we use the theory of Toulmin to analyse the processes of reasoning by which proofs are constructed.


Euclidean Geometry Formal Proof Mathematical Thinking Mathematical Proof Structure Theorem 
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  1. Aberdein, A. (2005). The uses of argument in mathematics. Argumentation, 19, 287–301.CrossRefGoogle Scholar
  2. Aberdein, A. (2006a). The informal logic of mathematical proof. In R. Hersh (Ed.), Eighteen unconventional essays on the nature of mathematics (pp. 56–70). New York: Springer.CrossRefGoogle Scholar
  3. Aberdein, A. (2006b). Managing informal mathematical knowledge: Techniques from informal logic. In J. M. Borwein, W. M. Farmer (Eds.) Lecture notes in computer science: vol 4108, Mathematical knowledge management (pp. 208–221). New York: Springer.Google Scholar
  4. Alcock, L., & Weber, K. (2005). Proof validation in real analysis: Inferring and checking warrants. Journal of Mathematical Behavior, 24, 125–134.CrossRefGoogle Scholar
  5. Alcolea Banegas, J. (1998). L’Argumentació en matemàtiques. In E. C. Moya (Ed.), XII Congrés Valencià de Filosofia (pp. 135–147). Valencia: Diputació de Valencià.Google Scholar
  6. Asghari, A. (2005). Equivalence. PhD thesis, University of Warwick.Google Scholar
  7. Chin, E.-T. (2002). Building and using concepts of equivalence class and partition. Ph D, University of Warwick.Google Scholar
  8. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 95–123). Dordrecht: Kluwer.Google Scholar
  9. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.CrossRefGoogle Scholar
  10. Evens, H., & Houssart, J. (2004). Categorizing pupils’ written answers to a mathematics test ­question: ‘I know but I can’t explain’. Educational Research, 46, 269–282.CrossRefGoogle Scholar
  11. Forman, E., Larreameny-Joerns, J., Stein, M. K., & Brown, C. A. (1998). “You’re going to want to find out which and prove it”: Collective argumentation in a mathematics classroom. Learning and Instruction, 8(6), 527–548.CrossRefGoogle Scholar
  12. Gray, E., & Tall, D. O. (1994). Duality, ambiguity and flexibility: a proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–141.Google Scholar
  13. Hilbert, D. (2000). Mathematical problems. Bulletin of the American Mathematical Society, 37(4), 407–436 (Original work published 1900).Google Scholar
  14. Hoyles, C., & Küchemann, D. (2002). Students’ understandings of logical implication. Educational Studies in Mathematics, 51(3), 193–223.CrossRefGoogle Scholar
  15. Inglis, M., & Mejia-Ramos, J. P. (2008). Theoretical and methodological implications of a broader perspective on mathematical argumentation. Mediterranean Journal for Research in Mathematics Education, 7(2), 107–119.Google Scholar
  16. Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66(1), 3–21.CrossRefGoogle Scholar
  17. Knipping, C. (2003). Argumentation structures in classroom proving situations. In M. A. Mariotti (Ed.), Proceedings of the 3rd Congress of the European Society for Research in Mathematics Education. Bellaria, Italy: ERME.Google Scholar
  18. Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: interaction in classroom cultures (pp. 229–269). Hillsdale: Erlbaum.Google Scholar
  19. Lakoff, G. (1987). Women, fire and dangerous things. Chicago: Chicago University Press.Google Scholar
  20. Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41.CrossRefGoogle Scholar
  21. Peirce, C. S. (1932). Collected Papers of Charles Sanders Peirce, Vol. 2: Elements of Logic. In C. Hartshorne, & P. Weiss (Eds.) Cambridge, MA: Harvard University Press.Google Scholar
  22. Piaget, J., & Inhelder, B. (1958). Growth of logical thinking. London: Routledge & Kegan Paul.Google Scholar
  23. Pinto, M., & Tall, D. O. (1999). Student constructions of formal theory: giving and extracting meaning. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 65–73). Haifa, Israel: PME.Google Scholar
  24. Pinto, M., & Tall, D. O. (2002). Building formal mathematics on visual imagery: a case study and a theory. For the Learning of Mathematics, 22(1), 2–10.Google Scholar
  25. Raman, M. (2002). Proof and justification in collegiate calculus. Unpublished doctoral dissertation, University of California, Berkeley.Google Scholar
  26. Rasmussen, C., Stephan, M., & Allen, K. (2004). Classroom mathematical practices and gesturing. Journal of Mathematical Behavior, 23, 301–323.CrossRefGoogle Scholar
  27. Saussure (compiled by Charles Bally and Albert Sechehaye). (1916). Course in General Linguistics (Cours de linguistique générale). Paris: Payot et Cie.Google Scholar
  28. Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.CrossRefGoogle Scholar
  29. Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior, 21, 459–490.CrossRefGoogle Scholar
  30. Tall, D. O., Gray, E., Bin Ali, M., Crowley, L., DeMarois, P., McGowen, M., et al. (2001). Symbols and the bifurcation between procedural and conceptual thinking. Canadian Journal of Science, Mathematics and Technology Education, 1, 81–104.CrossRefGoogle Scholar
  31. Tall, D. O. (ed). (1991). Advanced mathematical thinking. Dordrecht, Holland: Kluwer.Google Scholar
  32. Tall, D. O. (2002). Differing modes of proof and belief in mathematics. In International Conference on mathematics: understanding proving and proving to understand (pp. 91–107). Taipei, Taiwan: National Taiwan Normal University.Google Scholar
  33. Tall, D. O. (2004a). The three worlds of mathematics. For the Learning of Mathematics, 23(3), 29–33.Google Scholar
  34. Tall, D. O. (2004b). Thinking through three worlds of mathematics. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 281–288). Bergen, Norway: PME.Google Scholar
  35. Tall, D. O. (2005). The transition from embodied thought experiment and symbolic manipulation to formal proof. In M. Bulmer, H. MacGillivray & C. Varsavsky (Eds.), Proceedings of Kingfisher Delta’05: Fifth Southern Hemisphere Symposium on Undergraduate Mathematics and Statistics Teaching and Learning (pp. 23–35). Australia: Fraser Island.Google Scholar
  36. Tall, D. O. (2006). A life-time’s journey from definition and deduction to ambiguity and insight. Retirement as process and concept: a festschrift for Eddie Gray and David Tall (pp. 275–288). Prague. ISBN 80-7290-255-5.Google Scholar
  37. Toulmin, S. E. (1958). The uses of argument. Cambridge: Cambridge University Press.Google Scholar
  38. Toulmin, S. E., Rieke, R., & Janik, A. (1979). An introduction to reasoning (2nd ed.). New York: Macmillan.Google Scholar
  39. Weber, K. (2004). Traditional instruction in advanced mathematics courses: a case study of one professor’s lectures and proofs in an introductory real analysis course. Journal of Mathematical Behavior, 23, 115–133.CrossRefGoogle Scholar
  40. Weber, K., & Alcock, L. (2005). Using warranted implications to understand and validate proofs. For the Learning of Mathematics, 25(1), 34–38.Google Scholar
  41. Yackel, E. (2001). Explanation, justification and argumentation in mathematics classrooms. In M. van den Heuval-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, vol 1 (pp. 9–23). Utrecht, Holland: PME.Google Scholar
  42. Zeeman, E. C. (1960). Unknotting spheres. Annals of Mathematics, 72(2), 350–361.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institute of Education, University of WarwickCoventryUK

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