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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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(36) J. Piaget, "Apprentissage et Connaissance," P. Greco et J. Piaget, Apprentissage et Connai­ssance, Klaus Reprint, 1974, pp.21-67.

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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.

68 Ilrfi * ~ ~ (40)

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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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2i.4 {! %-zJ..£.£. 0]3f-cl.A1 OF ~ .j! 014. (59)

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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.'