Conservació de la
superfície amb l'obbliq':
Conservación de la superfície con 'obbliq':
Conservation of surface with 'obbliq':
|Hi ha la mateixa quantitat de líquid als dos recipients? -> Si. Es pot comprovar amb l'obbliq:|
|Is there the same amount of liquid in both containers? -> Yes. You can demonstrate it with the obbliq:|
|Hay la misma cantidad de líquido en los dos recipientes? -> Si. Lo pueden comprobar con el obbliq:|
CONSERVATION OF CONTINUOUS QUANTITIES
Every notion, whether it be scientific or merely a matter of common sense, presupposes a set of principles of conservation, either explicit or implicit. It is a matter of common knowledge that in the field of the empirical sciences the introduction of the principle of inertia (conservation of rectilinear and uniform motion) made possible the development of modern physics, and that the principle of conservation of matter made modern chemistry possible. It is unnecessary to stress the importance in every-day life of the principle of identity; any attempt by thought to build up a system of notions requires a certain permanence in their definitions. In the field of perception, the schema of the permanent object 1 presupposes the elaboration of what is no doubt the most primitive of all these principles of conservation. Obviously conservation, which is a necessary condition of all experience and all reasoning, by no means exhausts the representation of reality or the dynamism of the intellectual processes, but that is another matter. Our contention is merely that conservation is a necessary condition for all rational activity, and we are not concerned with whether it is sufficient to account for this activity or to explain the nature of reality.
This being so, arithmetical thought is no exception to the rule. A set or collection is only conceivable if it remains unchanged irrespective of the changes occurring in the relationship between the elements. For instance, the permutations of the elements in a given set do not change its value. A number is only intelligible if it remains identical with itself, whatever the distribution of the units of which it is composed. A continuous quantity such as a length or a volume can only be used in reasoning if it is a permanent whole, irrespective of the possible arrangements of its parts. In a word, whether it be a matter of continuous or discontinuous qualities, of quantitative relations perceived in the sensible universe, or of sets and numbers conceived by thought, whether it be a matter of the child's earliest contacts with number or of the most refined axiomatizations of any intuitive system, in each and every case the conservation of something is postulated as a necessary condition for any mathematical understanding.From the psychological point of view, the need for conservation appears then to be a kind of functional a priori of thought. But...
____________________ 1 La Construction du Réel chez l'Enfant, chapter i.
CONSERVATION OF DISCONTINUOUS QUANTITIES AND ITS RELATION TO ONE-ONE CORRESPONDENCE 1
The experiments described in the previous chapter can all be repeated with discontinuous quantities that can be evaluated globally when the elements are massed and counted when they are separated. Sets of beads, for instance, can be used. If they are put into the containers used in Chapter I, they can serve for the same evaluations as the liquids (level, crosssection, etc.), and in addition they are material for a further global quantification with which children are familiar: that of the length of necklaces made from the beads. The evaluation of this length can thus be used in each case to check the quantification of the contents of the various containers used. On the other hand, when the beads are considered as separate units they can be used in operations of correspondence. If the child is told, for instance, to put beads into a container, one by one, at the same time as the experimenter is putting beads one by one into another container, he can then be asked whether the total quantities are the same, with or without identity in the shape of the two containers.
In going on from the analysis of continuous quantities to that of discontinuous quantities we are therefore not merely checking our earlier findings. We are also making a preliminary study of the relationship between conservation of quantities and the development of one-one correspondence, which is, as is well known, one of the origins of number. We shall then be in a better position to approach the question of cardinal and ordinal correspondence as such.
It should be noted that the stages we shall find here correspond exactly to those of the previous chapter.
1. Stage I: Absence of conservation