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By Hans Freudenthal

This publication is a made of love and appreciate. If that sounds fairly ordinary I in the beginning apologise, yet allow me clarify why i exploit these phrases. the unique manuscript used to be in fact Freudenthal’s, yet his colleagues have carried the undertaking via to its end with love for the guy, and his principles, and with a recognize built over years of communal attempt. Their invitation to me to jot down this Preface e- bles me to pay my respects to the nice guy, even if i'm most likely incurring his wrath for writing a Preface for his publication with no his permission! I simply desire he is familiar with the sentiments of all colleagues engaged during this specific venture. Hans Freudenthal died on October thirteenth, 1990 whilst this publication undertaking was once good in hand. in truth he wrote to me in April 1988, announcing “I am wondering a brand new publication. i've got acquired the sub-title (China Lectures) notwithstanding I nonetheless lack a title”. i used to be astonished. He had retired in 1975, yet in fact he saved operating. Then in 1985 we were assisting him rejoice his eightieth birthday, and even if I stated in an article assertion in academic reports in arithmetic (ESM) on the time “we look ahead to him having fun with many extra years of non-retirement” i didn't anticipate to work out one other long manuscript.

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Extra info for Revisiting Mathematics Education: China Lectures (Mathematics Education Library)

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V=w. Such systems are called algebraic structures. There are plenty of them, with two or even more operations, which may satisfy widely different conditions. 4 Structures ---from smaller to larger The best known among the algebraic structures are the number structures of the natural numbers 0,1,2, ... , of the integers. ,-2,-1,0, 1,2,. 33 of the real numbers, among which, for instance, e, , of the complex numbers a + b i where i = It is a sequence of structures of increasing size, although not primarily -- and not even necessarily -- of increasing richness.

Some of these models, with a large domain of transfer, are certainly worth acquiring by students who MATHEMATICS PHENOMENOLOGICALLY 35 are expected to put them to good use. On the other hand, however vaguely I defined modelling -- as idealising and simplifying -- it did hit the nail on the head: to grasp the essentials of a (static or dynamic) situation and to focus on them within what I earlier called a rich context and, as things progress, within ever richer ones. This, then, is the viewpoint from which I will continue to look for aspects of mathematising.

On the scale of concreteness, however, one can progress even further, by stepping out of pure mathematics, and consider physical, or merely experienced space as a model of its axiomatically defined pre-image. Only for the sake of completeness did I mention this use of the term “model”, which is actually the inverse of the term we started with. As a matter of fact, in 34 CHAPTER 1 the present context we are not concerned with models of axiomatic systems, which are lavishly used in foundational research, but rather with modelling in the sense of idealising.

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