By Hans Freudenthal
The release ofa new ebook sequence is often a tough eventn ot just for the Editorial Board and the writer, but additionally, and extra really, for the 1st writer. either the Editorial Board and the writer are delightedt hat the 1st writer during this sequence isw ell capable of meet the problem. Professor Freudenthal wishes no creation toanyone within the arithmetic schooling box and it truly is rather becoming that his booklet may be the 1st during this new sequence since it used to be in 1968 that he, and Reidel, produced the 1st factor oft he magazine Edu cational experiences in arithmetic. Breakingfresh floor is as a result not anything new to Professor Freudenthal and this booklet illustrates good his excitement at this kind of activity. To be strictly right the ‘ground’ which he has damaged here's now not new, yet aswith arithmetic as a tutorial activity and Weeding and Sowing, it is very the newness oft he demeanour during which he has performed his research which gives us with such a lot of clean views. it truly is our purpose that this new ebook sequence should still offer those that paintings int he rising self-discipline of mathematicseducation with a vital source, and at a time of substantial challenge concerning the complete arithmetic cu rriculum this e-book represents simply such source. ALAN J. BISHOP coping with Editor vii a glance BACKWARD AND a glance ahead males die, structures final.
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Extra info for Didactical Phenomenology of Mathematical Structures
So there are many more questions I would like to have answered. The most urgent question, I think, is about the significance of the break–make transformations for socalled conservation (not only of length). If I may trust my own unsystematic experience, I would consider them as crucial. Yet in order to answer such questions, a quite different design of experiments is required than that of snapshots, registering which percentage of subjects at a certain age do “conserve”. Also required is a more positive mentality than that of tricking children into making mistakes.
For instance I would like to know whether constituting rigidity mentally precedes length, whether length invariance under congruence mapping and length invariance under flexions help or impede each other, what role is played by similarities in the mental constitution of length, and how the equivalence of “long” and “far” is acquired. So there are many more questions I would like to have answered. The most urgent question, I think, is about the significance of the break–make transformations for socalled conservation (not only of length).
Thirdly, the uncountability of R guarantees in a simple way the existence of non-algebraic numbers and in a more general way that by the difference of cardinality one set can be distinguished as a true extension of another. Fourthly, the unexpected phenomenon that forming the product of an infinite set by itself does not increase its cardinality and, as a consequence, that line segment, square, cube, and so on have the same cardinality, is the source of the problem of how to distinguish dimensions, which has been solved by paying attention to more structure, namely topological structure.