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.
Bishop, A.J.: 1985, 'The Social Construction of Meaning: A Significant Development for Mathematics Education?', For the Learning of Mathematics 5, 1, 24–28.
Bloor, D.: 1983, Wittgenstein: A Social Theory of Knowledge, London, Macmillan.
Cobb, P. and Steffe, L.P.: 1983, 'The constructivist researcher as teacher and model-builder', Journal for Research in Mathematics Education 14, 83–94.
D'Ambrosio, U.: 1985, 'Ethnomathematics and its place in the history and pedagogy of Mathematics', For the Learning of Mathematics 5, 1, 44–48.
Davis, P.J.: 1972, 'Fidelity in mathematical discourse: Is one and one really two?', American Mathematical Monthly 79, 252–263.
Ernest, P.: 1991, The Philosophy of Mathematics Education, London: The Falmer Press.
Hadamard, J.: 1954, The psychology of invention in the mathematical field, New York: Dover.
Hamming, R.W.: 1980, 'The unreasonable effectiveness of mathematics', American Mathematical Monthly 87, 81–90.
Hanna, G.: 1980, 'More than Formal Proof', For the Learning of Mathematics 9, 1, 20–23.
Hardy, G.H., Seshu Aiyar, P.V. and Wilson, B.M. (Eds): 1962, Srinivasa Ramanujan (1887- 1920): Collected papers, New York: Chelsea.
Hersh, R.: 1979, 'Some Proposals for Reviving the Philosophy of Mathematics', Advances in Mathematics 31, 1, 31–50.
Heyting, A.: 1956, Intuitionism: An Introduction, Amsterdam, North-Holland.
Kitcher, P.: 1984, The nature of mathematical knowledge, Oxford University Press.
Kline, M.: 1980, Mathematics the Loss of Certainty, Oxford University Press.
Körner, S.: 1968, The philosophy of mathematics, New York: Dover.
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.
Lakatos, L.: 1976, Proofs and Refutations: The Logic of Mathematical Discovery, Cambridge: Cambridge University Press.
Lakatos, I.: 1978, Mathematics, Science and Epistemology (Philosophical papers Vol. 2), Cambridge, Cambridge University Press.
National Commission on Excellence in Education (NCEE): 1983, A nation at risk: The imperative for education reform, Washington, DC: US Government Printing Office.
National Council of Teachers of Mathematics (NCTM): 1980a An agenda for action, Reston, VA: The Council.
National Council of Teachers of Mathematics (NCTM): 1980b, Problem solving in school mathematics, Reston, VA: The Council.
Poincaré, H.: 1952, Science and Method, New York: Dover.
Poincaré, H.: 1956, 'Mathematical creation'. In J.R. Newman (ed.), The world of mathematics (Vol. 4), New York: Simon & Shuster.
Polya, G.: 1954, Mathematics and plausible reasoning (2 Vols.), Princeton: Princeton University Press.
Polya, G.: 1962a (Vol. 1) and 1965 (Vol. 2), Mathematical discovery, New York: Wiley.
Polya. G.: 1962b, 'The teaching of mathematics and the biogenetic law'. In I.J. Good (ed.), The Scientist Speculates, London: Heinemann, 352–356.
Polya, G.: 1963, 'On learning, teaching and learning teaching', American Mathematical Monthly 70, 605–619.
Popper, K.: 1979, Objective knowledge. An Evolutionary Approach, Oxford: Oxford University Press.
Resnick, L.B.: 1983, 'Mathematics and science learning: A new conception', Science 29, 477.
Restivo, S.: 1983, The social relations of physics, mysticism and mathematics, Dordrecht: Reider.
Robitaille, D. & Dirks, M.: 1982, 'Models for the Mathematics Curriculum', For the Learning of Mathematics 2, 3, 3–21.
Simon, M.: 1986, 'The teacher's role in increasing student understanding of mathematics', Educational Leadership 43, 40–43.
Sinclair, H.: 1987, 'Constructivism and the psychology of mathematics', Proceedings of the Eleventh Annual Psychology of Mathematics Education Conference, 28–41.
Steffe, L., Cobb, P. and Glasersfeld, E.: 1988, Construction of Arithmetical Meanings and Strategies, New York: Springer-Verlag.
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.
Thom, R.: 1971, 'Modern Mathematics: An educational and philosophical error?', American Scientist 59, 695–699.
Thom, R.: 1973, 'Modern Mathematics: Does it Exist?', In A.G. Howson (ed): Developments in Mathematics Education, Cambridge, 194–209.
Thompson, A.G.: 1984, 'The Relationship of Teachers' Conceptions of Mathematics and Mathematics Teaching to Instructional Practice', Educational Studies in Mathematics 15, 105–127.
Toumasis, C.: 1987, 'Mathematics Education during the last 200 years around the world', The Greek Mathematical Society, Euclid III 16, 17–60 (In Greek).
Tymoczko, T. (Ed): 1986, New Directions in the Philosophy of Mathematics, Boston: Birkhauser.
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.
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.
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Toumasis, C. The Nctm Standards and the Philosophy of Mathematics. Studies in Philosophy and Education 16, 317–330 (1997). https://doi.org/10.1023/A:1004909220965
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DOI: https://doi.org/10.1023/A:1004909220965