By Hans Freudenthal
The release ofa new ebook sequence is usually a difficult eventn ot just for the Editorial Board and the writer, but in addition, and extra fairly, 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 advent toanyone within the arithmetic schooling box and it really is quite becoming that his publication could 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 reports in arithmetic. Breakingfresh flooring is hence not anything new to Professor Freudenthal and this ebook illustrates good his excitement at any such activity. To be strictly right the ‘ground’ which he has damaged here's no longer new, yet aswith arithmetic as an academic job and Weeding and Sowing, it is vitally the newness oft he demeanour within which he has conducted his research which gives us with such a lot of clean views. it's our purpose that this new e-book sequence may still offer those that paintings int he rising self-discipline of mathematicseducation with a necessary source, and at a time of substantial drawback concerning the entire arithmetic cu rriculum this publication 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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Of course, if need be, they would be able to draw them. Does this mean that they are in the enactive or in the ikonic phase? Of course not. For the concept in question they possess a symbol, the words “spiral stairs”, though if a definition is to be produced, one would have more or less difficulty in passing from the enactive or ikonic to the symbolic representation. * Studies in Cognitive Growth (Edited by J. S. Bruner), Toward a Theory of Instruction, 1966, pp. 10–11. THE METHOD 31 Consider the number concept “three” and the geometrical concept “straight”.
Set as a purely linguistic phenomenon expresses itself in this kind of school mathematics in particular by the cult of creating sets in extension. Every predicate can be transformed into a set – indeed, the set of things sharing the predicate. From “... is red” one can form the set of red things; from “... has long hair and wears spectacles” the set of long-haired spectacle-wearers, which in turn is the intersection of the long-haired people and the spectaclewearers. Or closer to mathematics: from “...
Bastiaan (4; 10) spontaneously measured the width of a path by steps. “This is six further”. I show him I can do it in one step. He does the same with two steps. He continues measuring by pacing. Bastiaan (6; 5) has made a large construction of roads, bridges, walls and tunnels in a sandpit. In order to make a drawing of the construction he measures distances with his two forefingers parallel at a fixed distance (about a decimeter), proceeding with the left forefinger in the hole made by the right one.