Skip to main content
SpringerLink
Log in

The Long-Term Cognitive Development of Reasoning and Proof

  • Chapter
  • First Online:
Explanation and Proof in Mathematics

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Aberdein, A. (2005). The uses of argument in mathematics. Argumentation, 19, 287–301.

    Article  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • 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 

  • Alcock, L., & Weber, K. (2005). Proof validation in real analysis: Inferring and checking warrants. Journal of Mathematical Behavior, 24, 125–134.

    Article  Google Scholar 

  • 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 

  • Asghari, A. (2005). Equivalence. PhD thesis, University of Warwick.

    Google Scholar 

  • Chin, E.-T. (2002). Building and using concepts of equivalence class and partition. Ph D, University of Warwick.

    Google Scholar 

  • Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 95–123). Dordrecht: Kluwer.

    Google Scholar 

  • Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Hilbert, D. (2000). Mathematical problems. Bulletin of the American Mathematical Society, 37(4), 407–436 (Original work published 1900).

    Google Scholar 

  • Hoyles, C., & Küchemann, D. (2002). Students’ understandings of logical implication. Educational Studies in Mathematics, 51(3), 193–223.

    Article  Google Scholar 

  • 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 

  • Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66(1), 3–21.

    Article  Google Scholar 

  • 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 

  • 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 

  • Lakoff, G. (1987). Women, fire and dangerous things. Chicago: Chicago University Press.

    Google Scholar 

  • Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41.

    Article  Google Scholar 

  • 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 

  • Piaget, J., & Inhelder, B. (1958). Growth of logical thinking. London: Routledge & Kegan Paul.

    Google Scholar 

  • 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 

  • 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 

  • Raman, M. (2002). Proof and justification in collegiate calculus. Unpublished doctoral dissertation, University of California, Berkeley.

    Google Scholar 

  • Rasmussen, C., Stephan, M., & Allen, K. (2004). Classroom mathematical practices and gesturing. Journal of Mathematical Behavior, 23, 301–323.

    Article  Google Scholar 

  • Saussure (compiled by Charles Bally and Albert Sechehaye). (1916). Course in General Linguistics (Cours de linguistique générale). Paris: Payot et Cie.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior, 21, 459–490.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Tall, D. O. (ed). (1991). Advanced mathematical thinking. Dordrecht, Holland: Kluwer.

    Google Scholar 

  • 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 

  • Tall, D. O. (2004a). The three worlds of mathematics. For the Learning of Mathematics, 23(3), 29–33.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • Toulmin, S. E. (1958). The uses of argument. Cambridge: Cambridge University Press.

    Google Scholar 

  • Toulmin, S. E., Rieke, R., & Janik, A. (1979). An introduction to reasoning (2nd ed.). New York: Macmillan.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Weber, K., & Alcock, L. (2005). Using warranted implications to understand and validate proofs. For the Learning of Mathematics, 25(1), 34–38.

    Google Scholar 

  • 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 

  • Zeeman, E. C. (1960). Unknotting spheres. Annals of Mathematics, 72(2), 350–361.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Education, University of Warwick, Coventry, CV4 7AL, UK

    David Tall

Authors
  1. David Tall
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juan Pablo Mejia-Ramos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Tall .

Editor information

Editors and Affiliations

  1. Ontario Institute for Studies in, University of Toronto, Bloor Street W. 252, Toronto, M5S 1V6, Canada

    Gila Hanna

  2. FB Mathematik, Universität Duisburg-Essen, Essen, 45117, Germany

    Hans Niels Jahnke

  3. Inst. Philosophie, Universität Bochum, Universitatsstr. 150, Bochum, 44801, Germany

    Helmut Pulte

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Tall, D., Mejia-Ramos, J.P. (2010). The Long-Term Cognitive Development of Reasoning and Proof. In: Hanna, G., Jahnke, H., Pulte, H. (eds) Explanation and Proof in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0576-5_10

Download citation

Publish with us

Policies and ethics

Search

Navigation