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Developments in Mathematical Education
Proceedings of the Second International Congress on Mathematical Education
Edited by A. G. Howson
Publisher: Cambridge University Press
Print Publication Year: 1973
Online Publication Date:September 2011
Online ISBN:9781139013536
Hardback ISBN:9780521201902
Paperback ISBN:9780521098038
Chapter DOI: http://dx.doi.org/10.1017/CBO9781139013536.017
Subjects: Education - Maths, Educ - Mathematics
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What do we mean by ‘Axiomatics’?
Modern mathematics, as it has developed during the last century and is still progressing, is characterised both by the axiomatic method and a tendency to abstractness. There are two types of mathematics constructed axiomatically, namely, categorical and non-categorical theories. Examples of the former are Euclidean geometry and the theory of the natural numbers, and of the latter, group, ring, field, vector space and metric space. Roughly speaking, the former is an axiomatisation of a substance – an entity – and the latter is an abstraction of a structure.
A categorical theory can be constructed in two different ways. One way is to discover the essential basis, look for the foundation of a theory, and axiomatise the theory. This is in the spirit of Euclid and Hilbert and may be called the historical development. Historical Euclidean geometry, Hilbert's rigorous theory and Peano's theory of the naturals are examples of this historical development.
The other way is as follows. First, we break down a theory into essential pieces, observe some similarity in different theories, and abstract a common structure. Thus we obtain a non-categorical theory. Next, we organise several non-categorical theories to characterise a categorical theory. This is in the spirit of modern mathematics and may be called the modern development. Thus, an approach in which the Euclidean plane is characterised as a two-dimensional linear space with an inner product typifies this development.
pp. i-vi
pp. vii-viii
Editor's Acknowledgements : Read PDF
pp. ix-x
PART I - A CONGRESS SURVEY : Read PDF
pp. 1-3
pp. 4-74
PART II - THE INVITED PAPERS : Read PDF
pp. 75-76
pp. 77-78
Comments on mathematical education : Read PDF
pp. 79-87
The Presidential Address : Read PDF
pp. 88-100
What groups mean in mathematics and what they should mean in mathematical education : Read PDF
pp. 101-114
Nature, man and mathematics : Read PDF
pp. 115-135
Some anthropological observations on number, time and common-sense : Read PDF
pp. 136-153
Mathematical education in developing countries – some problems of teaching and learning : Read PDF
pp. 154-180
Some questions of mathematical education in the USSR : Read PDF
pp. 181-193
Modern mathematics: does it exist? : Read PDF
pp. 194-210
PART III - A SELECTION OF CONGRESS PAPERS : Read PDF
pp. 211-212
Investigation and problem-solving in mathematical education : Read PDF
pp. 213-221
Intuition, structure and heuristic methods in the teaching of mathematics : Read PDF
pp. 222-232
Mathematics and science in the secondary school : Read PDF
pp. 233-240
Geometry as a gateway to mathematics : Read PDF
pp. 241-253
The International Baccalaureate : Read PDF
pp. 254-261
The role of axioms in contemporary mathematics and in mathematical education : Read PDF
pp. 262-271
Implications of the work of Piaget in the training of students to teach primary mathematics : Read PDF
pp. 272-282
Are we off the track in teaching mathematical concepts? : Read PDF
pp. 283-296
pp. 297-298
1 - The congress committees and officers : Read PDF
pp. 299-299
2 - The working groups : Read PDF
pp. 300-304
3 - ICMI and congress recommendations : Read PDF
pp. 305-306
4 - Films and videotapes on mathematics and its teaching : Read PDF
pp. 307-314
pp. 315-318