# Cognitive Development of Proof

## Abstract

This chapter traces the long-term cognitive development of mathematical proof from the young child to the frontiers of research. It uses a framework building from perception and action, through proof by embodied actions and classifications, geometric proof and operational proof in arithmetic and algebra, to the formal set-theoretic definition and formal deduction. In each context, proof develops over the long-term from the recognition and description of observed properties and the links between them, the selection of specific properties that can be used as definitions from which other properties may be deduced, to the construction of ‘crystalline concepts’ whose properties are a consequence of the context. These include Platonic objects in geometry, symbols having relationships in arithmetic and algebra and formal axiomatic systems whose properties are determined by their definitions.

## Keywords

Knowledge Structure Euclidean Geometry Formal Proof Mathematical Thinking Mathematical Proof## Notes

## References

- Alcock, L., & Weber, K. (2004). Semantic and syntactic proof productions.
*Educational Studies in Mathematics, 56*, 209–234.CrossRefGoogle Scholar - Antonini, S. (2001). Negation in mathematics: Obstacles emerging from an exploratory study. In M. van den Heuvel-Panhuizen (Ed.),
*Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 49–56). Utrecht, The Netherlands: Utrecht University.Google Scholar - Arnold, V. I. (2000). Polymathematics: Is mathematics a single science or a set of arts? In
*Mathematics: Frontiers and perspectives*(pp. 403–416). Providence: American Mathematical Society.Google Scholar - Bass, H. (2009).
*How do you know that you know? Making believe in mathematics*. Distinguished University Professor Lecture given at the University of Michigan on March 25, 2009. Retrieved from the internet on January 30, 2011, from http://deepblue.lib.umich.edu/bitstream/2027.42/64280/1/Bass-2009.pdf. - Boas, R. P. (1981). Can we make mathematics intelligible?
*The American Mathematical Monthly, 88*(10), 727–773.CrossRefGoogle Scholar - Bruner, J. S. (1966).
*Towards a theory of instruction*. Cambridge: Harvard University Press.Google Scholar - Burton, L. (2002). Recognising commonalities and reconciling differences in mathematics education.
*Educational Studies in Mathematics, 50*(2), 157–175.CrossRefGoogle Scholar - Byers, W. (2007).
*How mathematicians think*. Princeton: Princeton University Press.Google Scholar - Byrne, R. M. J., & Johnson-Laird, P. N. (1989). Spatial reasoning.
*Journal of Memory and Language, 28*, 564–575.CrossRefGoogle Scholar - Dawson, J. (2006). Why do mathematicians re-prove theorems?
*Philosophia Mathematica, 14*(3), 269–286.CrossRefGoogle Scholar - Dehaene, S. (1997).
*The number sense: How the mind creates mathematics*. New York: Oxford University Press.Google Scholar - Dreyfus, T., & Eisenberg, T. (1986). On the aesthetics of mathematical thoughts.
*For the Learning of Mathematics, 6*(1), 2–10.Google Scholar - Dubinsky, E., & McDonald, M. A. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton (Ed.),
*The teaching and learning of mathematics at university level: An ICMI study*(New ICMI study series, Vol. 7, pp. 273–280). Dordrecht: Kluwer.Google Scholar - Duffin, J. M., & Simpson, A. P. (1993). Natural, conflicting and alien.
*The Journal of Mathematical Behavior, 12*(4), 313–328.Google Scholar - Duffin, J. M., & Simpson, A. P. (1995). A theory, a story, its analysis, and some implications.
*The Journal of Mathematical Behavior, 14*, 237–250.CrossRefGoogle Scholar - Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics.
*Educational Studies in Mathematics, 61*, 103–131.CrossRefGoogle Scholar - Epp, S. (1998). A unified framework for proof and disproof.
*Mathematics Teacher, 91*(8), 708–713.Google Scholar - Fischbein, E. (1993). The theory of figural concepts.
*Educational Studies in Mathematics, 24*, 139–162.CrossRefGoogle Scholar - Flener, F. (2001, April 6).
*A geometry course that changed their lives: The Guinea pigs after 60 years*. Paper presented at the Annual Conference of The National Council of Teachers of Mathematics, , Orlando. Retrieved from the internet on January 26, 2010, from http://www.maa.org/editorial/knot/NatureOfProof.html - Flener, F. (2006).
*The Guinea pigs after 60 years*. Philadelphia: Xlibris Corporation.Google Scholar - Freudenthal, H. (1973).
*Mathematics as an educational task*. Dordrecht: Reidel Publishing Company.Google Scholar - Goldin, G. (1998). Representational systems, learning, and problem solving in mathematics.
*The Journal of Mathematical Behavior, 17*(2), 137–165.CrossRefGoogle Scholar - Gray, E. M., & 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 - Harel, G., & Sowder, L. (1998). Student’s proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.),
*Research in collegiate mathematics education III*(pp. 234–283). Providence: American Mathematical Society.Google Scholar - Harel, G., & Sowder, L. (2005). Advanced mathematical thinking at any age: Its nature and its development.
*Mathematical Thinking and Learning, 7*, 27–50.CrossRefGoogle Scholar - Healy, L., & Hoyles, C. (1998).
*Justifying and proving in school mathematics*. Summary of the results from a survey of the proof conceptions of students in the UK (Research Report, pp. 601–613). London: Mathematical Sciences, Institute of Education, University of London.Google Scholar - Henderson, D. W., & Taimina, D. (2005).
*Experiencing geometry: Euclidean and Non-Euclidean with history*(3rd ed.). Upper Saddle River: Prentice Hall.Google Scholar - Hershkowitz, R. & Vinner, S. (1983). The role of critical and non-critical attributes in the concept image of geometrical concepts. In R. Hershkowitz (Ed.),
*Proceedings of the 7th International Conference of the International Group for the Psychology of Mathematics Education*(pp. 223–228). Weizmann Institute of Science: Rehovot.Google Scholar - Hilbert, D. (1900).
*The problems of mathematics. The Second International Congress of Mathematics*. Retrieved from the internet on January 31, 2010, from http://aleph0.clarku.edu/∼djoyce/hilbert/problems.html - Hilbert, D. (1928/1967). The foundations of mathematics. In J. Van Heijenoort (Ed.),
*From Frege to Gödel*(p. 475). Cambridge: Harvard University Press.Google Scholar - Hoffmann, D. (1998).
*Visual Intelligence: How we change what we see*. New York: W.W Norton. 1998.Google Scholar - Housman, D., & Porter, M. (2003). Proof schemes and learning strategies of above-average mathematics students.
*Educational Studies in Mathematics, 53*(2), 139–158.CrossRefGoogle Scholar - Johnson, M. (1987).
*The body in the mind: The bodily basis of meaning, imagination, and reason*. Chicago: Chicago University Press.Google Scholar - Joyce, D. E. (1998).
*Euclid’s element*s. Retrieved from the internet on January 30, 2011, from http://aleph0.clarku.edu/∼djoyce/java/elements/elements.html - Klein, F. (1872).
*Vergleichende Betrachtungen über neuere geometrische Forschungen*. Erlangen: Verlag von Andreas Deichert. Available in English translation as a pdf at http://math.ucr.edu/home/baez/erlangen/erlangen_tex.pdf(Retrieved from the internet on January 30, 2011). - Koichu, B. (2009). What can pre-service teachers learn from interviewing high school students on proof and proving? (Vol. 2, 9–15).*Google Scholar
- Koichu, B., & Berman, A. (2005). When do gifted high school students use geometry to solve geometry problems?
*Journal of Secondary Gifted Education, 16*(4), 168–179.Google Scholar - Kondratieva, M. (2009). Geometrical sophisms and understanding of mathematical proofs (Vol. 2, pp. 3–8).*Google Scholar
- Lakoff, G. (1987).
*Women, fire, and dangerous things: What categories reveal about the mind*. Chicago: Chicago University Press.Google Scholar - Lakoff, G., & Johnson, M. (1999).
*Philosophy in the flesh*. New York: Basic Books.Google Scholar - Lakoff, G., & Núñez, R. (2000).
*Where mathematics comes from: How the embodied mind brings mathematics into being*. New York: Basic Books.Google Scholar - Lawler, R. W. (1980). The progressive construction of mind.
*Cognitive Science, 5*, 1–34.CrossRefGoogle Scholar - Lénárt, I. (2003).
*Non-Euclidean adventures on the Lénárt sphere*. Emeryville: Key Curriculum Press.Google Scholar - Leron, U. (1985). A direct approach to indirect proofs.
*Educational Studies in Mathematics, 16*(3), 321–325.CrossRefGoogle Scholar - Lin, F. L., Cheng, Y. H. et al. (2003). The competence of geometric argument in Taiwan adolescents. In
*Proceedings of the International Conference on Science and Mathematics Learning*(pp. 16–18). Taipei: National Taiwan Normal University.Google Scholar - MacLane, S. (1994). Responses to theoretical mathematics.
*Bulletin (new series) of the American Mathematical Society, 30*(2), 190–191.Google Scholar - Maher, C. A., & Martino, A. M. (1996). The development of the idea of mathematical proof: A 5-year case study.
*Journal for Research in Mathematics Education, 27*(2), 194–214.CrossRefGoogle Scholar - Mason, J., Burton, L., & Stacey, K. (1982).
*Thinking mathematically*. London: Addison Wesley.Google Scholar - Meltzoff, A. N., Kuhl, P. K., Movellan, J., & Sejnowski, T. J. (2009). Foundations for a new science of learning.
*Science, 325*, 284–288.CrossRefGoogle Scholar - National Council of Teachers of Mathematics. (1989).
*Curriculum and evaluation standards for school mathematics*. Reston: NCTM.Google Scholar - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston: NCTM.Google Scholar - Neto, T., Breda, A., Costa, N., & Godino, J. D. (2009). Resorting to Non–Euclidean plane geometries to develop deductive reasoning: An onto–semiotic approach (Vol. 2, pp. 106–111).*Google Scholar
- Núñez, R., Edwards, L. D., & Matos, J. F. (1999). Embodied cognition as grounding for situatedness and context in mathematics education.
*Educational Studies in Mathematics, 39*, 45–65.CrossRefGoogle Scholar - Paivio, A. (1991). Dual coding theory: Retrospect and current status.
*Canadian Journal of Psychology, 45*, 255–287.CrossRefGoogle Scholar - Papert, S. (1996). An exploration in the space of mathematics educations.
*International Journal of Computers for Mathematical Learning, 1*(1), 95–123.Google Scholar - Parker, J. (2005).
*R. L. Moore: Mathematician and teacher*. Washington, DC: Mathematical Association of America.Google Scholar - Pinto, M. M. F. (1998).
*Students’ understanding of real analysi*s. PhD thesis, University of Warwick, Coventry.Google Scholar - Pinto, M. M. F., & 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: Technion - Israel Institute of Technology.Google Scholar - Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how we can represent it?
*Educational Studies in Mathematics, 26*(2–3), 165–190.CrossRefGoogle Scholar - Poincaré, H. (1913/1982).
*The foundations of science*(G. B. Halsted, Trans.). The Science Press (Reprinted: Washington, DC: University Press of America).Google Scholar - Rav, Y. (1999). Why do we prove theorems?
*Philosophia Mathematica, 7*(1), 5–41.CrossRefGoogle Scholar - Reid, D., & Dobbin, J. (1998). Why is proof by contradiction difficult? In A. Olivier & K. Newstead (Eds.),
*Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 41–48). Stellenbosch, South Africa: University of Stellenbbosch.Google Scholar - Reiss, K. (2005)
*Reasoning and proof in geometry: Effects of a learning environment based on heuristic worked-out examples*. In 11th Biennial Conference of EARLI, University of Cyprus, Nicosia.Google Scholar - Rodd, M. M. (2000). On mathematical warrants.
*Mathematical Thinking and Learning, 2*(3), 221–244.CrossRefGoogle Scholar - Rota, G. C. (1997).
*Indiscrete thoughts*. Boston: Birkhauser.Google Scholar - Senk, S. L. (1985). How well do students write geometry proofs?
*Mathematics Teacher, 78*(6), 448–456.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.CrossRefGoogle Scholar - Shapiro, S. (1991).
*Foundations without foundationalism: A case for second-order logic*. Oxford: Clarendon.Google Scholar - Tall, D. O. (1979). Cognitive aspects of proof, with special reference to the irrationality of √2. In
*Proceedings of the 3rd Conference of the International Group for the Psychology of Mathematics Education*(pp. 203–205).Warwick, UK: University of Warwick.Google Scholar - Tall, D. O. (2004). Thinking through three worlds of mathematics. In M. J. Hoines and A. Fuglestad (Eds.),
*Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education*(Vol.4, pp. 281–288). Bergen, Norway, Bergen University College.Google Scholar - Tall, D. O. (2008). The transition to formal thinking in mathematics.
*Mathematics Education Research Journal, 20*(2), 5–24.CrossRefGoogle Scholar - Tall, D. O. (2012, under review).
*How humans learn to think mathematically*.Google Scholar - Tall, D. O., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity.
*Educational Studies in Mathematics, 12*(2), 151–169.CrossRefGoogle Scholar - Van Hiele, P. M. (1986).
*Structure and insight*. New York: Academic.Google Scholar - Van Hiele-Geldof, D. (1984). The didactics of geometry in the lowest class of secondary school. In D. Fuys, D. Geddes, & R. Tischler (Eds.),
*English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele*. Brooklyn: Brooklyn College (Original work published 1957).Google Scholar - Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge.
*Educational Studies in Mathematics, 48*(1), 101–119.CrossRefGoogle 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.
*The Journal of Mathematical Behavior, 23*, 115–133.CrossRefGoogle Scholar - Yevdokimov, O. (2008). Making generalisations in geometry: Students’ views on the process. A case study. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano & A. Sepulova (Eds.),
*Proceedings of the 32nd Conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 441–448). Morelia, Mexico: Cinvestav-UMSNH.Google Scholar - Yevdokimov, O. (in preparation).
*Mathematical concepts in early childhood*.Google Scholar