Zusammenfassung
Während der letzten dreißig Jahre haben sich viele nordamerikanische Mathematikdidaktiker dafür eingesetzt, dem Beweisen im Mathematikunterricht der Sekundarstufen eine geringere Rolle als bisher zuzuweisen. Diese Auffassung wurde großenteils mit Entwicklungen in der Mathematik selbst, mit den Ideen von Lakatos und mit geänderten sozialen Werten begründet. Der folgende Artikel versucht zu zeigen, daß keines dieser Argumente einen solchen Auffassung swandel rechtfertigt und daß das Beweisen weiterhin seinen Wert für den Schulunterricht behält, wegen seiner zentralen Bedeutung für das mathematische Arbeiten und als ein wichtiges Mittel zur Beförderung von Verständnis.
Abstract
Over the past thirty years, many mathematics educators in North America have suggested that proof be relegated to a lesser role in the secondary mathematics curriculum. This view has been shaped in large part by developments in mathematics itself, by the thinking of Lakatos and by changing social values. This paper argues that none of these factors justifies such a move, and that proof continues to have value in the classroom, both as a reflection of its central role in mathematical practice and as an important tool for the promotion of understanding.
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References
Agassi, J. (1981). Lakatos on proof and on mathematics. Logique et Analyse, 24, 437–439.
Andrews, G. (1994). The death of proof? Semi-rigorous mathematics? You’ve got to be kidding! The Mathematical Intelligencer, 16(4), 16–18.
Atiyah, M., et al. (1994). Responses to “Theoretical mathematics”: Towards a cultural synthesis of mathematics and theoretical physics. Bulletin of the American Mathematical Society, 30(2), 178–207.
Babai, L. (1994). Probably true theorems, cry wolf? Notices of the American Mathematical Society, 41(5), 453–454.
Blum, M. (1986). How to prove a theorem so no one else can claim it. Proceedings of the International Congress of Mathematicians, 1444–1451.
Blum, W., & Kirsch, A. (1991) Preformal proving: Examples and reflexions. Educational Studies in Mathematics, 22(2), 183–203
Cipra, B. (1993). New computer insights from “transparent” proofs. What’s Happening in the Mathematical Sciences, 1, 7–12.
Confrey, J. (1994). A theory of intellectual development. For the Learning of Mathematics, 14(3), 2–8.
Davis, P. J. & Hersh, R. (1981). The mathematical experience. Boston: Houghton Mifilin.
Dawson, A. J. (1969). The implications of the work of Popper, Polya and Lakatos for a model of mathematics instruction. Unpublished doctoral dissertation, University of Alberta, Canada.
Dossey, J. (1992). The nature of mathematics: Its role and its influence. In D. Grows (Ed.), Handbook of research on mathematics teaching and learning, 39–48. New York: Macmillan.
Epstein, D., & Levy, S. (1995). Experimentation and proof in mathematics. Notices of the American Mathematical Society, 42(6), 670–674.
Ernest, P. (1991). The philosophy of mathematics education. London: Falmer.
Ernest, P. (1996). Popularization: Myths, massmedia and modernism. In Bishop, A. et al. (Eds.), International handbook of mathematical education, Part 2. Dordrecht: Kluwer Academic Publishers.
Feyeraband, P. (1975). Imre Lakatos. British Journal for the Philosophy of Science, 26, 1–18.
Greeno, J. (1994). Comments on Susanna Epp’s chapter. In Schoenfeld A. (ed.), Mathematical thinking and problem solving. Hillsdale, NJ: Lawrence Erlbaum Associates.
Goldwasser, S., Micali, S. & Rackoff, C. (1985). The knowledge complexity of interactive proof-systems. Proceedings of the 17th ACM Symposium on Theory of. Computing, 291–304.
Hacking, I. (1979). Imre Lakatos’ philosophy of science. British Journal for the Philosophy of Science, 30, 390–39
Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.
Hersh, R. (1986). Some proposals for reviving the philosophy of mathematics. In T. Tymoczko (Ed.), New directions in the philosophy of mathematics. Boston: Birkhäuser.
Horgan, J. (1993). The death of proof. Scientific American, 269(4), 93–103
Jaffe, A. & Quinn, F. (1993). “Theoretical mathematics”: Towards a cultural synthesis of mathematics and theoretical physics. Bulletin of the American Mathematical Society, 29(1), 1–13.
Kitcher, P. (1984). The nature of mathematical knowledge. New York: Oxford University Press.
Lakatos, I. (1976). Proofs and refutations. Cambridge. Cambridge University Press.
Lakatos, I. (1978). Mathematics, science and epistemology (Philosophical papers vol. 2). J. Worrall & G. Currie, (Eds.) Cambridge: Cambridge University Press.
Lampert, M. (1990). When the problem is not the question and the answer is not the solution. American Educational Research Journal, 27, 29–63.
Laudan, L. (1990). Science and relativism. Chicago: University of Chicago Press.
Lehman, H. (1980). An examination of Imre Lakatos’ philosophy of mathematics. The Philosophical Forum, (22), 1, 33–48
MacLane, S. (1996). Despite physicists, proof is essential in mathematics. Paper presented at the Boston Colloquium for the Philosophy of Science “Proof and Progress in Mathematics,” Boston University. National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. National Council of Teachers of Mathematics, Reston, VA.
National Council of Teachers of Mathematics (1991). Professional standards for teaching mathematics. National Council of Teachers of Mathematics, Reston, VA.
Nickson, M. (1994). The culture of the mathematics classroom: An unknown quantity. In S. Lerman (Ed.), Cultural perspectives on the mathematics classroom, 7–36. Dordrecht: Kluwer.
Putnam, H. (1987). Mathematics, matter and methods: Philosophical papers, Volume 1. Cambridge: Cambridge University Press.
Steiner, M. (1983). The philosophy of Imre Lakatos. The Journal of Philosophv, LXXX, 9, 502–521.
Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177.
Wittmann, E., & Müller, G. (1990). When is a proof a proof? Bull. Soc. Math. Belg., 1, 15–40.
Zeilberger, D. (1993). Theorems for a price: tomorrow’s semi-rigorous mathematical culture. Notices of the American Mathematical Society, 40(8), 978–981.
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Hanna, G. The Ongoing Value of Proof. JMD 18, 171–185 (1997). https://doi.org/10.1007/BF03338846
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DOI: https://doi.org/10.1007/BF03338846