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(1) H. Freudenthal, Mathematics as an Educational Task, D. Reidel Publishing Company, 1973,pp.114-119.G. Polya, Mathematics and Plausible Reasoning, vol. I, Induction and Analogy in Mathe·matics, Princeton University Press, 1973, pp. v-vi.
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(2) R. Descartes, Rules for the Direction of the Mind, translated by L.]. Lafleur, The BobbsMerrill Company, Inc., 1961, pp.15-16.
(3) I. Lakatos, Proofs and Refutations, The Logic of Mathematical Discovery, Edited by].Worrel and E. Zahar, Cambridge University Press, 1976, pp.142-143.
(4) J.S. Bruner, The Process of Education, Vintage Books, 1963, pp.17-32.(5) ]. Agassi, "On Mathematics Education: the Lakatosian Revolution," For the Learning of
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Mathematics, vol. 1, num. 1, 1980, pp.27-3l.(6) 1. Lakatos, Proofs and Refutations, op. cit.(7) Ibid., p.2.(8) Ibid., p.5.(9) Ibid. pp.142-154.
(10) 1. Lakatos, Mathematics, science and epistemology, edited by J. Worral arid G. Currie,Cambridge University Press, 1978, p.70.
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1. Lakatos, Proofs and Refutations, op. cit., p. xii,(ll) Ibid., p.31.(2) Ibid., p.139.(3) Ibid., pp.144-145.(4) iu«, pp.145-146.(15) 1. Lakatos, Mathematics, science and epistemology, op, cit., pp.3-10.
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(16) K.R. Popper, Conjectures and Refutations, The Growth of Scientific Knowledge, Routledgeand Kegan Paul, 1963, p. vii.
(17) Ibid., p.46.(18) Ibid., p.37.
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(19) I. Lakatos, "A renaissance of empiricism in the recent philosophy of mathematics ?,"Mathematics, science and epistemology, op. cit., pp.24-42.
(20) Ibid., p.29.(21) Ibid., p.42.(22) P.]. Davis, R. Hersh, The Mathematical Experience, Penguin Books, 1981, p.346.
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(23) Ibid.(24) 1. Lakatos, Proofs and Refutations, op. cit.(25) tua.. p.38.(26) tus.. p. 67.(27) Ibid., p.1l5.(28) G. Polya, Mathematical Discovery, vol. II, John Wiley & sons, Inc. 1965, p.157.(29) G. Polya, Mathematics and Plausible Reasoning, vol. II, Patterns of Plausible Inference,
Princeton University Press, 1968, pp. 3-37.
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(33) Ibid., p.37.(34) G. Polya, Mathematical Discovery, II, op. cit., pp.1l8-1l9.
G. Polya, ~iEal!i ~, oj W/l] .;t-;;J] ~ ~ ?;( 'il7}, '7(:1' tzN, 1986, p.6.(35) 1. Lakatos, Proofs and Refutations, op. cit., p.140.
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(37) J. Agassi, "On Mathematics Education: the Lakatosian Revolution," op. cit.(38) Z.P. Dienes, Building up Mathematics, Hutchinson Educational, 1960, p.31.(39) H. Freudenthal, Weeding and Sowing, D.· Reidel Publishing Company, 1978, pp.192-210.
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(40) L Lakatos, Proofs and Refutations, op. cit., pp.50-56.(41) R. Thorn, "Modern Mathematics: An Educational and Philosophic Error?" The American
Scientist, vol. 59, no. 6, 1971, pp.695-699.(42) P.M. van Hiele, Structure and Insight, A Theory of Mathematics Education, Academic
Press, 1986.(43) H-G. Steiner, "Two kinds of 'elements' and the dialectic between synthetic-deductive and
analytic-genetic approaches in mathematics," For the Learning of Mathematics, vol. 8,num. 3, 1988, pp.7-15.
(44) J. Dieudonne, "Should We Teach 'Modern Mathematics'?" American Scientist, 61, 1973,pp.16-19.
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(45) H-G. Steiner, op. cit. p.9.(46) G. Schubring, Das genetische Prinzip in der Mathematik-Didaktik, Klett-Cotta Typoscript,
1978.(47) 1. Lakatos, Proofs and Refutations, op. cit., p.2.(48) Ibid., p.142.(49) Ibid.,p.144.
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(50) D. Hawkins, "The edge of platonism," For the Learning of Mathematics, vol. 5, num. 2,1985, pp.2-6.
(51) H. Freudenthal, op. cit., pp.99-108.(52) J.S. Bruner, "On Learning Mathematics," edited by J.A. McIntosh, Perspectives on Secondary
Mathematics Education, Prentice-Hall, Inc., 1971,pp. 63-73.
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(53) Conference Board of the Mathematical Sciences, National Advisory Committee on Mathematical Education, Overview and Analysis of School Mathematics, Grades K-12, 1975, pp.136137.
(54) tm~. J. Piagets] Jl!j(~ ~J!f~S1 'I§"~J3'9 1r~, Bulletin of the Korean Mathematical Society,vol.' 20, no. 2, 1983, pp.1l1-122.
(55) Gattegno, C., What We Owe Children, Routledge & Kegan Paul, 1971.
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(56) I. Lakatos, Proofs and Refutations, op. cit., p.127.(57) H. Wussing, The Genesis of the Abstract Group Concept, The MIT Press, 1984.
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73
An Educational Study on 1. Lakatos' Philosophy of Mathematics
Cheong Ho Woo
Abstract
The philosophical views about the problem of how mathematical knowledges grow have
exerted the deepest influence upon mathematical education. Recently, according to the
Zeitgeist regarding "teaching to think" or problem solving as a primary aim of school
mathematics, the fallibilist philosophy of mathematics and the logic of mathematical
discovery, which were developed by I. Lakatos under the influence of Hegel's dialectic,
Popper's critical philosophy, and Polya's mathematical heuristics, have became a matter
of deep concern to mathematics educators.
The present study was undertakened to analyse the fallibilist position of the works of
Lakatos and discuss the implications of it for the teaching of mathematics.
Lakatos challenged to the mathematical formalism striking at the traditional heartland
of infallibilsm, mathematical proofs, and characterized mathematics as a quasi-empirical
science. According to his views, informal mathematics is so called a thought experimental
science, which doesn't grow through a monotonous increase of the number of indubitably
established theorem, but through improvement of conjectures by the logic of "proofs and
refutations," and informal mathematical knowledges are no more than conjectures.
Lakatos' philosophy of mathematics and logic of mathematical discovery suggest it as a
major goal of mathematical education to develop the students' abilities and attitudes of
critical and reasonable thinking and to learn how to do mathematics through genuine
bona fide experience of mathematical rediscovery consisting of guessing, checking, proving,
refuting, improving conjectures, and proof-generating concepts.
Lakatos' philosophical position also rejects the traditional Euclidean deductive approach
with the Parmenldes-Platonic philosopy and the logic of inductive generalization, and
suggests a way of humanizing mathematical education, realizing the idea of 'activism' in
mathematical education, through critical fallibilist approach, that is, genetic-heuristic
Socratic-problematic situational- apprehensive approach.
Lakatos' views challenge the formalist tradition of mathematics teaching, but this does't
mean disregarding the logical construction of mathematics, rather require harmonizing
the systematic deductive approach to ready-made mathematics and the heuristic approach
to mathematics 'in statu nascendi.'