# Didactical Phenomenology of Mathematical Structures by Hans Freudenthal

By Hans Freudenthal

The release ofa new e-book sequence is usually a not easy eventn ot just for the Editorial Board and the writer, but in addition, and extra relatively, for the 1st writer. either the Editorial Board and the writer are delightedt hat the 1st writer during this sequence isw ell in a position to meet the problem. Professor Freudenthal wishes no creation toanyone within the arithmetic schooling box and it really is rather becoming that his e-book may be the 1st during this new sequence since it was once in 1968 that he, and Reidel, produced the 1st factor oft he magazine Edu cational stories in arithmetic. Breakingfresh flooring is accordingly not anything new to Professor Freudenthal and this e-book illustrates good his excitement at this sort of job. To be strictly right the ‘ground’ which he has damaged here's now not new, yet aswith arithmetic as a tutorial job and Weeding and Sowing, it is very the newness oft he demeanour during which he has performed his research which supplies us with such a lot of clean views. it truly is our goal that this new e-book sequence may still supply those that paintings int he rising self-discipline of mathematicseducation with an important source, and at a time of substantial quandary in regards to the complete arithmetic cu rriculum this booklet represents simply such source. ALAN J. BISHOP coping with Editor vii a glance BACKWARD AND a glance ahead males die, platforms last.

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Bruner), Toward a Theory of Instruction, 1966, pp. 10–11. THE METHOD 31 Consider the number concept “three” and the geometrical concept “straight”. Before the child masters these words, he can be familiar with what they mean: clapping his hands thrice and running straight to a goal if it is suggested to him (the enactive phase); sorting out cards with three objects or straight lines pictured on them (the ikonic phase). Mastering the word three (or straight) means he is in the symbolic phase, since “three” as a word is a symbol for the concept three (or “straight” is for straight).

For a while, general set theory was a subject of profound axiomatic research; in particular, the continuum hypothesis was in the focus of attention. 2. Sets as Substrata of Structures Wherever sets in advanced mathematics are not an aim in themselves, they fulfill various tasks. For instance, they serve as substrata for structures – a metric space, a group, a category is a set with a number of properties. ) Structuring the substratum set can happen in various ways, for instance: a set becomes a metric space by putting a distance function upon its pairs of elements, a set becomes a group by prescribing a certain operation between its elements, and a set becomes a category by imposing certain mappings between the member sets.

If sets serve as substrata for certain structures, they are in general not subjected to drastic set theory operations. Subsets are formed, mainly to introduce substructures; substrata of structures are mapped on each other to define mappings of the imposed structures; set theory products are formed to get structure products of the imposed structures; and the substratum set is partitioned, for instance, as a set of equivalence classes, in order to derive new structures. 3. Sets as a Linguistic Tool In an even weaker way, sets occur virtually as only a linguistic tool where some 34 SETS 35 predicates are replaced by their extension.