Child's Conception of Number

Selected Works vol 2

Jean Piaget

• Publisher: Routledge
• Year: 2013
• ISBN:
  9781136220517
• Edition: 1
• Topic: Medicine
• Subtopic: Health Care Delivery

First published in 1997. This is Volume II of selected works of Jean Piaget which looks at his thinking on children and how their interaction and perception of number, touching on the areas of development in thinking on conversation of quantities and invariance of wholes; cardinal and ordinal one-one correspondence; as well as additive and multicatitive compositions. These explorations look into tracing the development of the operations which give rise to number and continuous quantities, to space, time, speed, etc., operations which, in these essential fields, lead from intuitive and egocentric pre-logic to rational co-ordination that is both deductive and inductive.

Part One

Conservation of Quantities and Invariance of Wholes

Chapter I

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 object¹ 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 does this mean that arithmetical notions acquire their structure because of this conservation, or are we to conclude that conservation precedes any numerical or quantifying activities, and is not only a function, but also an a priori structure, a kind of innate idea present from the first awareness of the intellect and the first contact with experience? It is experiment that will provide the answer, and we shall try to show that the first alternative is the only one that is in agreement with the facts.

§1. Technique and general results

This chapter and the one that follows will be devoted to experiments made simultaneously with continuous and discontinuous quantities. It seemed to us essential to deal with the two questions at the same time, although the former are not arithmetical and were to be treated separately in a special volume,² since it was desirable to ascertain that the results obtained in the case of discontinuous sets were general.

The child is first given two cylindrical containers of equal dimensions (A₁ and A₂) containing the same quantity of liquid (as is shown by the levels). The contents of A₂ are then poured into two smaller containers of equal dimensions (B₁ and B₂) and the child is asked whether the quantity of liquid poured from A₂ into (B₁ + B₂) is still equal to that in A₁. If necessary, the liquid in B₁ can then be poured into two smaller, equal containers (C₁ and C₂), and in case of need, the liquid in B₂ can be poured into two other containers C₃ and C₄ identical with C₁ and C₂. Questions as to the equality between (C₁ + C₂) and B₂, or between (C₁ + C₂ + C₃ + C₄) and A₁, etc., are then put. In this way, the liquids are subdivided in a variety of ways, and each time the problem of conservation is put in the form of a question as to equality or non-equality with one of the original containers. Conversely, as a check on his answers, the child can be asked to pour into a glass of a different shape a quantity of liquid approximately the same as that in a given glass, but the main problem is still that of conservation as such.

The results obtained seem to prove that continuous quantities are not at once considered to be constant, and that the notion of conservation is gradually constructed by means of an intellectual mechanism which it is our purpose to explain. By grouping the answers to the various questions, it is possible to distinguish three stages. In the first, the child considers it natural for the quantity of liquid to vary according to the form and dimensions of the containers into which it is poured. Perception of the apparent changes is therefore not corrected by a system of relations that ensures invariance of quantity. In the second stage, which is a period of transition, conservation gradually emerges, but although it is recognized in some cases, of which we shall attempt to discover the characteristics, it is not so in all. When he reaches the third stage, the child at once postulates conservation of the quantities in each of the transformations to which they are subjected. Naturally this does not mean that this generalization of constancy extends at this stage beyond the limits of the field studied here.

In our interpretation of these facts, we can start from the following hypotheses, some of which directed the research of this chapter while others arose in the course of our experiments. The question to be considered is whether the development of the notion of conservation of quantity is not one and the same as the development of the notion of quantity. The child does not first acquire the notion of quantity and then attribute constancy to it; he discovers true quantification only when he is capable of constructing wholes that are preserved. At the level of the first stage, quantity is therefore no more than the asymmetrical relations between qualities, i.e., comparisons of the type ‘more’ or ‘less’ contained in judgements such as ‘it’s higher’, ‘not so wide’, etc. These relations depend on perception, and are not as yet relations in the true sense, since they cannot be co-ordinated one with another in additive or multiplicative operations. This co-ordination begins at the second stage and results in the notion of ‘intensive’ quantity, i.e., without units, but susceptible of logical coherence. As soon as this intensive quantification exists, the child can grasp, before any other measurement, the proportionality of differences, and therefore the notion of extensive quantity. This discovery, which alone makes possible the development of number, thus results from the child’s progress in logic during these stages.

§2. Stage I: Absence of conservation

For children at the first stage, the quantity of liquid increases or diminishes according to the size or number of the containers. The reasons given for this non-conservation vary from child to child, and from one moment to the next, but in every case the child thinks that the change he sees involves a change in the total value of the liquid. Here we have some examples:

Bias (4;0). ‘Have you got a friend?—Yes, Odette.—Well look, we’re giving you, Clairette, a glass of orangeade (A₁,

full), and we’re giving Odette a glass of lemonade (A₂, also

full). Has one of you more to drink than the other?—The same.—This is what Clairette does: she pours her drink into two other glasses (B₁ and B₂, which are thus half filled). Has Clairette the same amount as Odette?—Odette has more.—Why?—Because wi’ve put less in (She pointed to the levels in B₁ and B₂, without taking into account the fact that there were two glasses).—(Odette’s drink was then poured into B₃ and B₄.) It’s the same.—And now (pouring Clairette’s drink from B₁ + B₂ into L, a long thin tube, which is then almost full)?—I’ve got more.—Why?—We’ve poured it into that glass (pointing to the level in L), and here (B₃ and B₄) we haven’t.—But were they the same before?—Yes.—And now?—I’ve got more.’ Clairette’s orangeade was then poured back from L into B₁ and B₂: ‘Look, Clairette has poured hers like Odette. So, is all the lemonade (B₃ + B₄) and all the orangeade (B₁ + B₂) the same?—It’s the same (with conviction).—Now Clairette does this (pouring B₁ into C₁ which is then full, while B₂ remains half full). Have you both the same amount to drink?—I’ve got more.—But where does the extra come from?—From in there (B₁).—What must we do so that Odette has the same?—We must take that little glass (pouring part of B₃ into C₂).—And is it the same now, or has one of you got more?—Odette has more.—Why?—Because we’ve poured it into that little glass (C₂).—But is there the same amount to drink, or has one got more than the other?—Odette has more to drink.—Why?—Because she has three glasses (B₃ almost empty, B₄ and C₂, while Clairette has C₁ full and B₂).’

A moment later, a new experiment. Clairette was again shown glasses A₁ and A₂,

full, one with orangeade for herself and the other with lemonade for Odette. ‘Are they exactly the same?—Yes (verifying the levels).—Well, Odette is going to pour hers (A₂) into all those (C₁, C₂, C₃, C₄, which were thus about half full). Have you both the same amount?—I’ve got more. She has less. In the glasses there’s less (looking carefully at the levels).—But before, you both had the same?—Yes.—And now?—Here (pointing to the level in A₁) it’s more, and here (indicating the 4 glasses C) it’s less.’

Finally she was given only the big glass A₁ almost full of orangeade: ‘Look, Clairette does this: she pours it like that (into B₁ and B₂, which are then

full). Is there more to drink now than before, or less, or the same?—There’s less (very definitely).—Explain to me why.—When you poured it out, it made less.—But don’t the little glasses together make the same?—It makes less.’

Sim (5;0). She was shown A₁ and A₂ half full. ‘There’s the same amount in the glasses, isn’t there?—(She verified it) Yes.—Look, Renée, who has the lemonade, pours it out like this (pouring A₁ into B₁ and B₂, which were thus about

full). Have you both still the same amount to drink?—No. Renée has more because she has two glasses.—What could you do to have the same amount?—Pour mine into two glasses. (She poured A₂ into B₃ and B₄.)—Have you both got the same now?—(She looked for a long time at the 4 glasses Yes.—Now Madeleine (herself) is going to pour her two glasses into three (B₃ and B₄ into C₁, C₂ and C₃). Are they the same now?—No.—Who has more to drink?—Madeleine, because she has three glasses. Renée must pour hers too into three glasses. (Renée’s B₁ and B₂ were poured into C₅, C₆ and C₇). There.—It’s the same.—But now Madeleine pours hers into a fourth glass (C₄, which was filled with a little from C₁, C₂ and C₃). Have you both the same amount?—I’ve got more.—Is there more of the lemonade (C₅, C₆ and C₇) or of the orangeade (C₁, C₂, C₃ and C₄)?—The orangeade.—(The two big glasses A₁ and A₂ were then put before her.) Look, we’re going to pour back all the lemonade into this one (A₁) as it was before, and all the orangeade into that one. Where will the lemonade com...

Table of contents

• Cover Page
• Jean Piaget: Selected Works
• Title Page
• Copyright Page
• Contents
• Foreword
• Part I. Conservation of Quantities and Invariance of Wholes
• Part II. Cardinal and Ordinal One-One Correspondence
• Part III. Additive and Multiplicative Compositions
• Glossary
• Index