# Objectives

Chapter

## Abstract

I have been concerned ever since I began to teach mathematics that a great deal of what is commonly taught and what is offered in textbooks provides a comparatively meaningless form of activity for pupils. They learn to perform certain computational processes, they perform algebraic manipulations, but those activities which would show them the purpose of mathematics are scarce.

## Keywords

Mathematics Teaching Axiomatic System Reflective Thinking Deductive Argument Axiomatic Method
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## References

- ATM (1978): Maths for Sixth Formers. Association of Teachers of Mathematics, Nelson, Lancs.Google Scholar
- Bell, A.W., Wigley, A.R. and Rooke, D.J. (1978): Journey into Maths, The South Nottinghamshire Project. Teachers Guides I and 2, Number Skills I and 2, Pupils worksheets Packs 1 - 10. Blackie, Glasgow.Google Scholar
- Horton, B. (1979): Tests of Mathematical Process Shell Centre for Mathematical Education, University of Nottingham.Google Scholar
- Leapfrogs group: Leaflet from Coldharbour, Newton St. Cyres, Exeter.Google Scholar
- 1.Joseph Agassi, “On mathematics education: the Lakatosian revolution”, For the learning of mathematics, Vol. I, No. I, Montreal, 1980. pp. 27 - 31.Google Scholar
- 2.A.J. (Sandy) Dawson. The Implication of the Work of Popper, Polya, and Lakatos for a Model of Inquiry in Mathematics. Unpublished doctoral dissertation, University of Alberta, 1969.Google Scholar
- 3.The journal, For the learning of mathematics is available by writing to the editor, David Wheeler, at the Department of Mathematics, Concordia University, 7141 Sherbrooke Street West, Montreal, Quebec, Canada, W4B IR6.Google Scholar
- 4.Caleb Gattegno, “The foundations of geometry,” For the learning of Mathematics, Vol. I, No. I, Montreal, 1980. pp. 10 - 15.Google Scholar
- 5.Caleb Gattegno has written many books most of which are available from Educational Solutions, Inc., 80 Fifth Avenue, New York, New York 10011. Some which may be of most interest to mathematics educators are: The Universe of Babies Google Scholar
- b) Caleb Gattegno has written many books most of which are available from Educational Solutions, Inc., 80 Fifth Avenue, New York, New York 10011. Some which may be of most interest to mathematics educators are: The Mind Teaches the Brain Google Scholar
- c)Caleb Gattegno has written many books most of which are available from Educational Solutions, Inc., 80 Fifth Avenue, New York, New York 10011. Some which may be of most interest to mathematics educators are: Evolution and Memory Google Scholar
- d)Caleb Gattegno has written many books most of which are available from Educational Solutions, Inc., 80 Fifth Avenue, New York, New York 10011. Some which may be of most interest to mathematics educators are: The Common Sense of Teaching Mathematics Google Scholar
- e)Caleb Gattegno has written many books most of which are available from Educational Solutions, Inc., 80 Fifth Avenue, New York, New York 10011. Some which may be of most interest to mathematics educators are: What We Owe Children Google Scholar
- 6.This activity was suggested by an item in the Newsletter published by Educational Solutions, lnc., “Mathematics: visible and tangibl”, Vol. IX, No. 3, February, 1980, New York.Google Scholar
- 7.The geometry films developed by Gattegno are also available from Educational Solutions, Inc. They are computer animated films, relatively short in duration, but rich in their potential for helping learners to “image” in mathematics.Google Scholar
- 8.Imre Lakatos, Proofs and Refutations, Cambridge University Press, New York, 1976.MATHCrossRefGoogle Scholar
- Armitage, J.V., The Relation Between Abstract and ‘Concrete’ Mathematics at School, Proceedings of the First International Congress on Mathematical Education, Dordrecht-Holland: Reidel 1969, 48 - 55.Google Scholar
- Athen, H., Kunle, H. (ed.) Proceedings of the Third International Congress on Mathematical Education, Karlsruhe 1977.Google Scholar
- Bell, A.W., A Study of Pupil’s Proof Explanations in Mathematical Situations. Educ. Stud. in Math. 7 (1976), 23 - 40.CrossRefGoogle Scholar
- Beth, E.W., Piaget, J., Epistemoloqie Mathematique et Psychologie. Etudes d’epistemologie genetique XIV. Presses Universitaires de France 1961.Google Scholar
- Christiansen, B., Induction and Deduction in the Learning of Mathematics and in Mathematical Instruction, Proceedings of the First International Congress on Mathematical Education, Dordrecht-Holland: Reidel 1969, 7 - 27.Google Scholar
- Dieudonne, J., L’abstraction et L’intuition Mathematique, NICO 20 (1976), 45 - 65.Google Scholar
- Fischbein, E., Intuition, Structure and Heuristic Methods in the Teaching of Mathematics, In: Howson 1973, 222 - 232.Google Scholar
- Fischbein, E., The Intuitive Sources of Probabilistic Thinking in Children. Dordrecht-Holland: Reidel 1975.MATHCrossRefGoogle Scholar
- Freudenthal, H., Mathematics as an Educational Task. Dordrecht-Holland: Reidel, 1973.MATHGoogle Scholar
- Giles, G., Does Teaching Inhibit Learning? Mathematics Teaching 65 (1973), 33 - 38.Google Scholar
- Griffiths, H.B., Mathematical Insight and Mathematical Curricula, Educ. Stud. in Math. 4 (1971), 153 - 165.CrossRefGoogle Scholar
- Griffiths, H.B., Surfaces. London: CUP 1976.Google Scholar
- Griffiths, H.B., Some Comments on MacDonald’s Paper, Educ. Stud. in Math. 9 (1978), 421 - 427.MATHCrossRefGoogle Scholar
- Hilbert, D., Cohn-Vossen, S., Geometry and the Imagination. New York: Chelsea 1952.MATHGoogle Scholar
- Howson, A.G., (ed.) Developments in Mathematical Education, Proceedings of the Second International Congress on Mathematical Education, Cambridge 1973.Google Scholar
- Lakatos, I., Proofs and Refutations. London: CUP 1976.Google Scholar
- MacDonald, I., Insight and Intuition in Mathematics, Educ. Stud. in Math. 9 (1978), 411 - 420.MATHCrossRefGoogle Scholar
- Manin, Yu., How Convincing is a Proof?, The Mathematical Intelligencer 2 (1976), No. I, 17 - 18.MathSciNetGoogle Scholar
- Shibata, T., The Role of Axioms in Contemporary Mathematics and in Mathematical Education. In: Howson 1973, 262 - 271.Google Scholar
- Thom, R., Modern Mathematics: Does it Exist? In: Howson 1973, 194 - 209.Google Scholar
- Walther, G., Illuminating Examples - An Aspect of Simplification in Mathematics Teaching. Inaugural Lecture, University of Dortmund 1979.Google Scholar
- Wittman, E., Grundfragen des Mathematikunterrichts, Braunschweig 1978Google Scholar
- Wittmann, E., Beziehungen zwischen operativen, Beziehungen zwischen operativen “Programmen” in Mathematik, Psychologie und Mathematikdidaktik, to appear in Journal fur Mathematikdidaktik 1980.Google Scholar
- A version of this paper has appeared in Educational Studies in Mathematics, Vol. 12, No.3, August 1981, pp. 389-397, Copyright 1981 by D. Reidel Publishing Co., Dordrecht, England.Google Scholar
- Cohen, P.J. Set theory and the continuum hypothesis. New York, W.A. Benjamin, 1966.Google Scholar
- Courant, R., & John, F. Introduction to calculus and analysis. Wiley, International Edition, New York,Google Scholar
- Kline, M. Logic versus pedagogy. American Mathematical Monthly, March 1970, pp. 264 - 281.Google Scholar
- Piaget, J. Le possible, L’impossible et le necessaire. Archives de Psychologie, 1976, 44 (No. 172), pp. 281 - 299.Google Scholar
- Suppes, P. The axiomatic method in high school mathematics. In The role of axiomatics and problem solving in mathematics. Conference Board of the Mathematical Sciences, Washington, D.C., Ginn & Co., 1966, pp. 69 - 76.Google Scholar
- Wilder, R.L. Introduction to the foundations of mathematics. Wiley, New York, 1952, 1955.MATHGoogle Scholar

## Copyright information

© Birkhäuser Boston, Inc. 1983