ABSTRACT
The study of children’s spontaneous behaviour in a measuring situation (ch. II) revealed that the notion of a metric unit is evolved only at level IIIB and depends on the previous mastery of qualitative operational transitivity (level IIIA) and on the coordination of changes of position at the level of representation, itself a function of a system of references. The ensuing detailed analysis of that development showed that the coordination of changes of position involved in measuring and the elaboration of a reference system were alike impossible without conservation of distance and length. That conservation is therefore an essential condition of measurement, and in ch. III and IV it was shown that its achievement, both in respect of stationary distances and of the length of moving objects, was dependent on the simultaneous coordination of positional changes and sites. Ch. V then dealt with the relations between conservation of length, transitivity of relations of equality or inequality and the understanding of measurement. The method consisted in presenting problems of conservation of length and of measurement separately, using paired lines, some of which were straight and others bent. The aim was to lay bare the parts played by subdivision, or grouping of intervals, and relations of order and change of position, and to demonstrate their importance to the genesis of conservation and qualitative transitivity, and also to that of metrical units. The underlying unity of this development has been made apparent. On the one hand, the conservation of length was shown to be the product of coordination as between subdivision (or nesting interval relations) and order of position (i.e. relations of order and of changes in that order), (chs. III, IV and V). As a result of that coordination, these two qualitative groupings, which had previously been undifferentiated for want of external reference elements, now became complementary to one another, through a comprehensive reference-system. On the other hand, the measurement of length was shown to be the outcome, not of a complementary relation between subdivision and change of position, but of their operational fusion. Thus a metric unit was shown to be essentially a part which could be applied to the remaining parts of the same whole, through changes in the position of equivalent middle terms. The whole itself could then be expressed as a multiple of that unit.
