# The Nctm Standards and the Philosophy of Mathematics

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## Abstract

It is argued that the philosophical and epistemological beliefs about the nature of mathematics have a significant influence on the way mathematics is taught at school. In this paper, the philosophy of mathematics of the NCTM's Standards is investigated by examining is explicit assumptions regarding the teaching and learning of school mathematics. The main conceptual tool used for this purpose is the model of two dichotomous philosophies of mathematics-absolutist versus- fallibilist and their relation to mathematics pedagogy. The main conclusion is that a fallibilist view of mathematics is assumed in the Standards and that most of its pedagogical assumptions and approaches are based on this philosophy.

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### REFERENCES

- Benacerraf, P. & Putnam, H: 1964.
*Philosophy of mathematics: Selected readings*, Englewood Cliffs, N.J.: Prentice-Hall.Google Scholar - Bishop, A.J.: 1985, 'The Social Construction of Meaning: A Significant Development for Mathematics Education?',
*For the Learning of Mathematics***5**, 1, 24–28.Google Scholar - Bloor, D.: 1983,
*Wittgenstein: A Social Theory of Knowledge*, London, Macmillan.Google Scholar - Cobb, P. and Steffe, L.P.: 1983, 'The constructivist researcher as teacher and model-builder',
*Journal for Research in Mathematics Education***14**, 83–94.Google Scholar - D'Ambrosio, U.: 1985, 'Ethnomathematics and its place in the history and pedagogy of Mathematics',
*For the Learning of Mathematics***5**, 1, 44–48.Google Scholar - Davis, P.J.: 1972, 'Fidelity in mathematical discourse: Is one and one really two?',
*American Mathematical Monthly***79**, 252–263.Google Scholar - Ernest, P.: 1991,
*The Philosophy of Mathematics Education*, London: The Falmer Press.Google Scholar - Hadamard, J.: 1954,
*The psychology of invention in the mathematical field*, New York: Dover.Google Scholar - Hamming, R.W.: 1980, 'The unreasonable effectiveness of mathematics',
*American Mathematical Monthly***87**, 81–90.Google Scholar - Hanna, G.: 1980, 'More than Formal Proof',
*For the Learning of Mathematics***9**, 1, 20–23.Google Scholar - Hardy, G.H., Seshu Aiyar, P.V. and Wilson, B.M. (Eds): 1962,
*Srinivasa Ramanujan (1887- 1920): Collected papers*, New York: Chelsea.Google Scholar - Hersh, R.: 1979, 'Some Proposals for Reviving the Philosophy of Mathematics',
*Advances in Mathematics***31**, 1, 31–50.Google Scholar - Heyting, A.: 1956,
*Intuitionism: An Introduction*, Amsterdam, North-Holland.Google Scholar - Kitcher, P.: 1984,
*The nature of mathematical knowledge*, Oxford University Press.Google Scholar - Kline, M.: 1980,
*Mathematics the Loss of Certainty*, Oxford University Press.Google Scholar - Körner, S.: 1968,
*The philosophy of mathematics*, New York: Dover.Google Scholar - Leman., S.: 1983, 'Problem-solving or knowledge-centered: the influence of philosophy on mathematics teaching',
*International Journal of Mathematical Education in Science and Technology***14**, 1, 59–66.Google Scholar - Lakatos, L.: 1976,
*Proofs and Refutations: The Logic of Mathematical Discovery*, Cambridge: Cambridge University Press.Google Scholar - Lakatos, I.: 1978,
*Mathematics, Science and Epistemology (Philosophical papers Vol. 2)*, Cambridge, Cambridge University Press.Google Scholar - National Commission on Excellence in Education (NCEE): 1983,
*A nation at risk: The imperative for education reform*, Washington, DC: US Government Printing Office.Google Scholar - National Council of Teachers of Mathematics (NCTM): 1980a
*An agenda for action*, Reston, VA: The Council.Google Scholar - National Council of Teachers of Mathematics (NCTM): 1980b,
*Problem solving in school mathematics*, Reston, VA: The Council.Google Scholar - Poincaré, H.: 1952,
*Science and Method*, New York: Dover.Google Scholar - Poincaré, H.: 1956, 'Mathematical creation'. In J.R. Newman (ed.),
*The world of mathematics*(Vol. 4), New York: Simon & Shuster.Google Scholar - Polya, G.: 1954,
*Mathematics and plausible reasoning (2 Vols.)*, Princeton: Princeton University Press.Google Scholar - Polya, G.: 1962a (Vol. 1) and 1965 (Vol. 2),
*Mathematical discovery*, New York: Wiley.Google Scholar - Polya. G.: 1962b, 'The teaching of mathematics and the biogenetic law'. In I.J. Good (ed.),
*The Scientist Speculates*, London: Heinemann, 352–356.Google Scholar - Polya, G.: 1963, 'On learning, teaching and learning teaching',
*American Mathematical Monthly***70**, 605–619.Google Scholar - Popper, K.: 1979,
*Objective knowledge. An Evolutionary Approach*, Oxford: Oxford University Press.Google Scholar - Resnick, L.B.: 1983, 'Mathematics and science learning: A new conception',
*Science***29**, 477.Google Scholar - Restivo, S.: 1983,
*The social relations of physics, mysticism and mathematics*, Dordrecht: Reider.Google Scholar - Robitaille, D. & Dirks, M.: 1982, 'Models for the Mathematics Curriculum',
*For the Learning of Mathematics***2**, 3, 3–21.Google Scholar - Simon, M.: 1986, 'The teacher's role in increasing student understanding of mathematics',
*Educational Leadership***43**, 40–43.Google Scholar - Sinclair, H.: 1987, 'Constructivism and the psychology of mathematics', Proceedings of the Eleventh Annual Psychology of Mathematics Education Conference, 28–41.Google Scholar
- Steffe, L., Cobb, P. and Glasersfeld, E.: 1988,
*Construction of Arithmetical Meanings and Strategies*, New York: Springer-Verlag.Google Scholar - Steiner, H.G.: 1987, 'Philosophical and Epistemological Aspects of Mathematics and their Interaction with Theory and Practice in Mathematics Education',
*For the Learning of Mathematics***7**, 1, 7–13.Google Scholar - Thom, R.: 1971, 'Modern Mathematics: An educational and philosophical error?',
*American Scientist***59**, 695–699.Google Scholar - Thom, R.: 1973, 'Modern Mathematics: Does it Exist?', In A.G. Howson (ed):
*Developments in Mathematics Education*, Cambridge, 194–209.Google Scholar - Thompson, A.G.: 1984, 'The Relationship of Teachers' Conceptions of Mathematics and Mathematics Teaching to Instructional Practice',
*Educational Studies in Mathematics***15**, 105–127.Google Scholar - Toumasis, C.: 1987, 'Mathematics Education during the last 200 years around the world',
*The Greek Mathematical Society*, Euclid III 16, 17–60 (In Greek).Google Scholar - Tymoczko, T. (Ed): 1986,
*New Directions in the Philosophy of Mathematics*, Boston: Birkhauser.Google Scholar - Van der Blij, F., Hilding, S. and Weinzweig, A.I.: 1981, 'New Golds for Old: An Analysis of Reactions to Recent Reforms in Several Countries'. In R. Morris (ed.)
*Studies in mathematics education*, Unesco, Vol. 2, 105–118.Google Scholar - Von Glasersfeld, E.: 1983, 'Learning as a constructive activity'. Proceedings of the Fifth Annual Meeting of the North American Chapter of Psychology of Mathematics Education, 41–49.Google Scholar