Cognitive development of proof. (English) Zbl 1247.97020
Hanna, Gila (ed.) et al., Proof and proving in mathematics education. The 19th ICMI study. Berlin: Springer (ISBN 978-94-007-2128-9/hbk; 978-94-007-2129-6/ebook). New ICMI Study Series 15, 13-49 (2012).
Summary: This article 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.
For the entire collection see [Zbl 1234.00015].
For the entire collection see [Zbl 1234.00015].
MSC:
| 97E50 | Reasoning and proving in the mathematics classroom |
| 97C30 | Cognitive processes, learning theories (aspects of mathematics education) |
| 97D30 | Objectives and goals of mathematics teaching |
Keywords:
mathematical proof; cognitive development; perception and action; geometric proof; operational proof; formal set-theoretic definition; formal deductionReferences:
| [1] | Alcock, L.; Weber, K., Semantic and syntactic proof productions, Educational Studies in Mathematics, 56, 209-234 (2004) |
| [2] | 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. |
| [3] | Arnold, VI, Polymathematics: Is mathematics a single science or a set of arts?, Mathematics: Frontiers and perspectives, 403-416 (2000), Providence: American Mathematical Society, Providence · Zbl 0980.00004 |
| [4] | 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. |
| [5] | Boas, RP, Can we make mathematics intelligible?, The American Mathematical Monthly, 88, 10, 727-773 (1981) |
| [6] | Bruner, JS, Towards a theory of instruction (1966), Cambridge: Harvard University Press, Cambridge |
| [7] | Burton, L., Recognising commonalities and reconciling differences in mathematics education, Educational Studies in Mathematics, 50, 2, 157-175 (2002) |
| [8] | Byers, W., How mathematicians think (2007), Princeton: Princeton University Press, Princeton · Zbl 1123.00003 |
| [9] | Byrne, RMJ; Johnson-Laird, PN, Spatial reasoning, Journal of Memory and Language, 28, 564-575 (1989) |
| [10] | Dawson, J., Why do mathematicians re-prove theorems?, Philosophia Mathematica, 14, 3, 269-286 (2006) · Zbl 1123.00004 |
| [11] | Dehaene, S., The number sense: How the mind creates mathematics (1997), New York: Oxford University Press, New York · Zbl 1041.00504 |
| [12] | Dreyfus, T.; Eisenberg, T., On the aesthetics of mathematical thoughts, For the Learning of Mathematics, 6, 1, 2-10 (1986) |
| [13] | Dubinsky, E.; McDonald, MA; Holton, D., APOS: A constructivist theory of learning in undergraduate mathematics education research, The teaching and learning of mathematics at university level: An ICMI study, 273-280 (2001), Dordrecht: Kluwer, Dordrecht · Zbl 1057.00005 |
| [14] | Duffin, JM; Simpson, AP, Natural, conflicting and alien, The Journal of Mathematical Behavior, 12, 4, 313-328 (1993) |
| [15] | Duffin, JM; Simpson, AP, A theory, a story, its analysis, and some implications, The Journal of Mathematical Behavior, 14, 237-250 (1995) |
| [16] | Duval, R., A cognitive analysis of problems of comprehension in a learning of mathematics, Educational Studies in Mathematics, 61, 103-131 (2006) |
| [17] | Epp, S., A unified framework for proof and disproof, Mathematics Teacher, 91, 8, 708-713 (1998) |
| [18] | Fischbein, E., The theory of figural concepts, Educational Studies in Mathematics, 24, 139-162 (1993) |
| [19] | 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 |
| [20] | Flener, F., The Guinea pigs after 60 years (2006), Philadelphia: Xlibris Corporation, Philadelphia |
| [21] | Freudenthal, H., Mathematics as an educational task (1973), Dordrecht: Reidel Publishing Company, Dordrecht · Zbl 0246.00013 |
| [22] | Goldin, G., Representational systems, learning, and problem solving in mathematics, The Journal of Mathematical Behavior, 17, 2, 137-165 (1998) |
| [23] | Gray, EM; Tall, DO, Duality, ambiguity and flexibility: A proceptual view of simple arithmetic, Journal for Research in Mathematics Education, 26, 2, 115-141 (1994) |
| [24] | Harel, G.; Sowder, L.; Schoenfeld, A.; Kaput, J.; Dubinsky, E., Student’s proof schemes: Results from exploratory studies, Research in collegiate mathematics education III, 234-283 (1998), Providence: American Mathematical Society, Providence · Zbl 0991.00501 |
| [25] | Harel, G.; Sowder, L., Advanced mathematical thinking at any age: Its nature and its development, Mathematical Thinking and Learning, 7, 27-50 (2005) |
| [26] | 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. |
| [27] | Henderson, DW; Taimina, D., Experiencing geometry: Euclidean and Non-Euclidean with history (2005), Upper Saddle River: Prentice Hall, Upper Saddle River |
| [28] | 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. |
| [29] | 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 · JFM 31.0418.01 |
| [30] | Hilbert, D. (1928/1967). The foundations of mathematics. In J. Van Heijenoort (Ed.), From Frege to Gödel (p. 475). Cambridge: Harvard University Press. |
| [31] | Hoffmann, D., Visual Intelligence: How we change what we see (1998), New York: W.W Norton, New York |
| [32] | Housman, D.; Porter, M., Proof schemes and learning strategies of above-average mathematics students, Educational Studies in Mathematics, 53, 2, 139-158 (2003) |
| [33] | Johnson, M., The body in the mind: The bodily basis of meaning, imagination, and reason (1987), Chicago: Chicago University Press, Chicago |
| [34] | Joyce, D. E. (1998). Euclid’s elements. Retrieved from the internet on January 30, 2011, from http://aleph0.clarku.edu/∼djoyce/java/elements/elements.html |
| [35] | 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). |
| [36] | Koichu, B. (2009). What can pre-service teachers learn from interviewing high school students on proof and proving? (Vol. 2, 9-15).* |
| [37] | Koichu, B.; Berman, A., When do gifted high school students use geometry to solve geometry problems?, Journal of Secondary Gifted Education, 16, 4, 168-179 (2005) |
| [38] | Kondratieva, M. (2009). Geometrical sophisms and understanding of mathematical proofs (Vol. 2, pp. 3-8).* |
| [39] | Lakoff, G., Women, fire, and dangerous things: What categories reveal about the mind (1987), Chicago: Chicago University Press, Chicago |
| [40] | Lakoff, G.; Johnson, M., Philosophy in the flesh (1999), New York: Basic Books, New York |
| [41] | Lakoff, G.; Núñez, R., Where mathematics comes from: How the embodied mind brings mathematics into being (2000), New York: Basic Books, New York · Zbl 0987.00003 |
| [42] | Lawler, RW, The progressive construction of mind, Cognitive Science, 5, 1-34 (1980) |
| [43] | Lénárt, I., Non-Euclidean adventures on the Lénárt sphere (2003), Emeryville: Key Curriculum Press, Emeryville |
| [44] | Leron, U., A direct approach to indirect proofs, Educational Studies in Mathematics, 16, 3, 321-325 (1985) |
| [45] | 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. |
| [46] | MacLane, S., Responses to theoretical mathematics, Bulletin (new series) of the American Mathematical Society, 30, 2, 190-191 (1994) |
| [47] | Maher, CA; Martino, AM, The development of the idea of mathematical proof: A 5-year case study, Journal for Research in Mathematics Education, 27, 2, 194-214 (1996) |
| [48] | Mason, J.; Burton, L.; Stacey, K., Thinking mathematically (1982), London: Addison Wesley, London |
| [49] | Meltzoff, AN; Kuhl, PK; Movellan, J.; Sejnowski, TJ, Foundations for a new science of learning, Science, 325, 284-288 (2009) |
| [50] | National Council of Teachers of Mathematics, Curriculum and evaluation standards for school mathematics (1989), Reston: NCTM, Reston |
| [51] | National Council of Teachers of Mathematics, Principles and standards for school mathematics (2000), Reston: NCTM, Reston |
| [52] | 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).* |
| [53] | Núñez, R.; Edwards, LD; Matos, JF, Embodied cognition as grounding for situatedness and context in mathematics education, Educational Studies in Mathematics, 39, 45-65 (1999) |
| [54] | Paivio, A., Dual coding theory: Retrospect and current status, Canadian Journal of Psychology, 45, 255-287 (1991) |
| [55] | Papert, S., An exploration in the space of mathematics educations, International Journal of Computers for Mathematical Learning, 1, 1, 95-123 (1996) |
| [56] | Parker, J., R. L. Moore: Mathematician and teacher, vii (2005), Washington, DC: Mathematical Association of America, Washington, DC · Zbl 1069.01007 |
| [57] | Pinto, M. M. F. (1998). Students’ understanding of real analysis. PhD thesis, University of Warwick, Coventry. |
| [58] | 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. |
| [59] | Pirie, S.; Kieren, T., Growth in mathematical understanding: How can we characterize it and how we can represent it?, Educational Studies in Mathematics, 26, 2-3, 165-190 (1994) |
| [60] | Poincaré, H. (1913/1982). The foundations of science (G. B. Halsted, Trans.). The Science Press (Reprinted: Washington, DC: University Press of America). · JFM 44.0086.16 |
| [61] | Rav, Y., Why do we prove theorems?, Philosophia Mathematica, 7, 1, 5-41 (1999) · Zbl 0941.03003 |
| [62] | 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. |
| [63] | 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. |
| [64] | Rodd, MM, On mathematical warrants, Mathematical Thinking and Learning, 2, 3, 221-244 (2000) |
| [65] | Rota, GC, Indiscrete thoughts (1997), Boston: Birkhauser, Boston · Zbl 0862.00005 |
| [66] | Senk, SL, How well do students write geometry proofs?, Mathematics Teacher, 78, 6, 448-456 (1985) |
| [67] | Sfard, A., 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 (1991) |
| [68] | Shapiro, S., Foundations without foundationalism: A case for second-order logic (1991), Oxford: Clarendon, Oxford · Zbl 0732.03002 |
| [69] | 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. |
| [70] | 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. |
| [71] | Tall, DO, The transition to formal thinking in mathematics, Mathematics Education Research Journal, 20, 2, 5-24 (2008) |
| [72] | Tall, D. O. (2012, under review). How humans learn to think mathematically. |
| [73] | Tall, DO; Vinner, S., Concept image and concept definition in mathematics with particular reference to limits and continuity, Educational Studies in Mathematics, 12, 2, 151-169 (1981) |
| [74] | Van Hiele, PM, Structure and insight (1986), New York: Academic, New York |
| [75] | Van Hiele, PM; Geldof, D.; Fuys, D.; Geddes, D.; Tischler, R., The didactics of geometry in the lowest class of secondary school, English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele (1984), Brooklyn: Brooklyn College, Brooklyn |
| [76] | Weber, K., Student difficulty in constructing proofs: The need for strategic knowledge, Educational Studies in Mathematics, 48, 1, 101-119 (2001) |
| [77] | Weber, K., 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 (2004) |
| [78] | 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. |
| [79] | Yevdokimov, O. (in preparation). Mathematical concepts in early childhood. |
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