ABSTRACT
The operations considered so far in relation to areas and volumes were all capable of being handled on a purely logical plane, that is, without numerical multiplication. Frequently the elements and relations involved were such as could be expressed in quantitative terms, but the problem of calculating an area in such terms has not yet been studied. This in fact introduces a new element which goes beyond the purely logical operations envisaged up to now; a rectangle measuring 2 × 3 linear units gives six square units. Thus even where the conservation and measurement of areas was being expressly considered, the questions put to the children were such that they only needed to apply ready-made units of area (squares, rectangles, equilateral triangles, etc.); they did not have to calculate the area as a function of linear measurements. Similarly in the chapter which dealt with the subdivision of areas, the process consisted in dividing whole areas into partial ones, so that the units involved were all square units and the numerical relation between area and length of side was irrelevant.
