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 Experiments Involving Flowers, Beads, Vases.docx.

A Child’s Conception of Numbers by Jean Piaget

Part I. Conservation of Qualities and Invariance of Wholes

I.                    Conservation of Continuous Quantities

1.       Technique and general Results

2.       Stage I. Absence of Conservation

3.       Stage II. Intermediary Reactions

4.       Stage III. Necessary Conservation

II.                  Conservation of Discontinuous Quantities and It’s Relation to One-One Correspondence

1.       Stage I. Absence of Conservation

2.       Stage II. Beginnings of Construction of Permanent Sets

3.       Stage III. Conservation and Quantifying Co-Ordination

Part II Cardinal and Ordinal One-One Correspondence

III.                Provoked Correspondence and Equivalence of Corresponding Sets

I.                    One-One Correspondence between glasses and bottles

II.                  Correspondence between flowers and vases and between eggs and egg-cups

III.                One for one exchange of pennies and objects

IV.                One for one exchange in conjunction with counting aloud

IV Spontaneous Correspondence-Cardinal Value of Sets

I.                    Reproduction of Sets

II.                  Single Rows

III.                Conclusion

V Seriation, Qualitative Similarity And Ordinal Correspondence

I.                    Technique and General results

II.                  Construction of Serial Correspondence (Qualitative Similarity)

III.                From serial Correspondence to Ordinal Correspondence

IV.                Reconstruction of Cardinal Correspondence

VI Ordination and Cardination

I.                    The experiment with Sticks and the Problem of Seriation

II.                  The experiment with cards forming a staircase

III.                The experiment with mats and hurdles

IV.                Conclusions: Ordination and Cardination

Part III. Additive and Multiplicative Compositions

VII Additive Composition of Classes Relations Between Class And Number

I.                    Technique and Results

II.                  Stage I. Absence of Additive Composition

III.                Stage II and III. Progressive reversibility of operations

IV.                Number, and additive composition of classes

VIII Additive Composition of Numbers and Arithmetical Relations of Part to Whole

I.                    Technique and Results

II.                  Relations between parts and whole, and changes in composition of parts

III.                Equating of Quantities

IV.                Division into equal parts

V.                  Conclusion

IX Co-Ordination of Relations of Equivalence and Multiplicative Composition of Numbers

I.                    Constitution of One-One Correspondence and Composition of relations of equivalence

II.                  Stages of composition of relations of equivalence

III.                Multiple correspondence and numerical multiplication

IV.                Conclusion: Multiplication of Classes and Numbers

X. Additive and Multiplicative Composition of Relations and Equalization of Differences

I. Problems and Results

II. Development  of the notion of measure

III Composition of relations and of numerical units

V.                  Conclusions

Glossary

Index

The Child’s Conception of Numbers by Jean Piaget

Chapter II

Conservation of Discontinuous Quantities and It’s Relation to One-One Correspondence 

Stage II: Beginnings of Construction of Permanent Sets

As in the case of continuous quantities, we can distinguish here also a second stage characterized by intermediary solutions. In general, the situation is as follows. One the one hand, the child tends to think that there is conservations, either because the two sets are put into two identical containers (A1 and A2) or because the two sets have been built up by a one-one correspondence. But on the other hand, this tendency comes into conflict with the appearance of the sets, i.e. with the difference in level, cross-section, etc. Two new features are then observable, as distinct from the reactions of the first stage. In the first place, there is a real conflict; the factors of conservation do not immediately give way before the factors of change, and the struggle between the two is most instructive. Secondly, as a result of this conflict, the perceptual relationships are co-ordinated as relations, and are integrated in a system susceptible of justifying conservation while taking into account the concomitant variations.

Here we have examples in which the one-one correspondence does not intervene.

Marg (5;6): ‘Is there the same number of beads (In A1 and A2)?- Yes, the same.- And if we made necklace, etc.?- The same length- Why?-…-And if I pour this into that (A1 into L)? There are more there (A2).- Why?- Because it gets bigger here (pointing to the narrower column in L). Where is there more? In the big one (= the wide one, A2).- And if we make two necklaces (with L and A2)/- They’ll be the same length.- And if we pour these (L) into those (M1+M2)?- There’ll be more in the two little ones.- Why?-… - And if we make a necklace? It will be longer with the little once (than with A2). And before, when the beads were here (A1 and A2)/- The necklaces were the same length.- And if I put those (A2) here (E1+E2+E3+E4), will the two necklaces be the same (i.e. 2M and 4E)?- No, it’ll be longer from the little ones (4E).’

Ari (5:6): ‘They’re the same (A1 and A2).- And if we make two necklaces, etc.?- the same length.- And if we pour these (A2) into that (L)?- There’ll be more there (L).- Why?- Because it’s higher.- And if we make two necklaces?- They’ll be the same length.- And if we make a necklace?- It’ll be longer.    

The first point to notice here is that, as in the case of continuous quantities, the child at this level is capable of postulating conservation when there is only a slight change in pattern, but not when the change is more significant. Thus Marg and Ari think that the two necklaces will still be the same length if the beads in A are poured into l, but not if they are poured into 2M or 4E. But this is not all. By the very fact that he hesitates to assume conservation because of the change in shape, the child comes to distinguish between evaluations based on mere perception of the relationships of height and width and those which result from representation of the length of the necklaces. For instance although Marg and Ari think that the quantity changes in passing from A to L because the level rises, they still think that the necklace made from beads in L will be the same length as the one made from those in A. hence there is conservation when the child is thinking of the row of discontinuous elements, and non conservation when he is thinking of one or other of the dimensions of the set as a whole. Such distinctions between the various evaluations are extremely instructive. On the one hand they indicate how much and varied are the operations that the child has to co-ordinate in quantification. On the other hand, the fact that the evaluations based on representation of the necklace are more correct than the others suggest that the notion of conservation involves decomposition into elements. We shall now see whether this is borne out by facts.

Tis (5:1) put one bead into V1 each time the experiments put one into V2: ‘Are they the same?- Yes, because I put the same as you every time. –If we make two necklaces, etc? –They’ll be the same length, because there are a lot of beads, and you’ve got a lot as well.- (V1 was poured into L+M) Are they still the same?- In yours (L+M) there are a lot.- And in yours?-Not many.- And if we make two necklaces, etc.?-Yours will be longer, mine won’t be so long.- Why? Because there are more beads in yours.- But how did we put the beads in? two each time.- Then why have I got more?- There are two big lots in yours, look.’ So far then, Tis’s reaction is characteristic of the first stage, but we shall now see the transition from this reaction to the conflicts typical of the second stage. 

Tis put one bead into L every time the experimenter put one into P. He counted each bead as he put it in and reached the correct total of 12. L was then full, and Tis cried spontaneously: ‘I’ve got more.-Why? There are more in mine.- And if we make tow necklaces?- This one (L) will be longer.- Why? The glass is bigger, and that one (P) is smaller (pointing to the height).- But are there more beads? – In that one (L).- Why? – It’s bigger.- How did we put the beads in? We put two every time.- Have we got the same, or have you more or less?- Both the same.- Why?- Because we put every time.- What will the necklaces be like?- Yours will be long and mine will be the same length.- Why?- Because this one (L) is big, and mine (P) is little. You’ve got a lot of beads in yours.- And what about you?- Not as many, but a lot all the same.’ It is clear that as soon the child is reminded of the one-one correspondence there is a conflict between it and his perception of the dimensions, the former tending towards equality and the latter towards difference, with the results that no real synthesis is achieved.

Von (5;10) also failed to reconcile the correspondence with perceptual relationships. When 11 pink beads were put into E and 11 blue ones into P, one at a time, he said that they were the same, although E was full, ‘Why?- Because I counted, and I knew it’s right.- And if we make a pink necklace and a blue one? – There’ll be the same difference for both, the same size necklace.- How do you know?- I counted. There’s the same number of beads.- Then why are they like that here (pointing to the level in E)?- Yours (E) is round and thinner, mine (P) is round and bigger (with a gesture indicating the width)._ Well? They’re the same, because I counted. We put in the same number; I put the same as you every time.

Von then put one bead into G each time I put one into L. ‘it’s the same in both.- Why?- We put them in together.- And if we make two necklaces/- Both the same.- Well why id l full and not the other?- Because here (L) it’s round and long, and there (P) it’s round and bigger (=wider) and we put in the same amount.- (The beads in G were then poured into another glass (Gp) is smaller (=lower) and flatter, and this one (L) is longer; it’s bigger and there’s more.- More what (Gp and L were full to the brim and von was therefore not distinguishing between the volume of the glasses and the quality of beads)?- More beads There are more beads there (L).- And if we made two necklaces?- You (L) have more, the blue necklaces will be longer.- And what about the pink one (Gp)? – Shorter, because there are less beads.

These intermediary reactions are very interesting both from the point of view of quantification in general and of the significance of correspondence itself.

There is obviously a systematic conflict in the case of Tis and Von, and all the other cases typical of this stage, between a factor of equalization and conservation and a factor of difference . Every one of these children concluded that there is equality if the same number of elements is dropped, one at a time, into two containers, irrespective of the shape of the containers. But when the child afterwards considers the result obtained when the shapes are different, his belief in the equivalence is shaken by an evaluation based on the perceptual relationships. Although he himself has just made the one-one correspondence, he reacts like the child at the first stage and think that any variation in height or width entails a change in the quality as a whole. But whereas at the first stage his belief in the equivalence was destroyed, now there is merely conflict between the two tendencies, neither of which definitely triumphs. When the child considers the sets of beads he thinks that there is non-equivalence, and when he remembers the correspondence he again thinks that the sets are equivalent. Even when, as in the case of Tis, a final decision appears to have been reached, the words the child uses express his doubt (‘You have got a lot… I’ve not got so many, but a lot all the same’).

How is the child going to reconcile these two opposing tendencies? It is interesting to find that he solves the problem of the beads in exactly the same way as that of continuous quantities. The synthesis of the real equivalence and the apparent variations is achieved through co-ordination of the relations in question, and here again this co-ordination first takes the form of mere logical multiplication, which rapidly develops into true proportionality. This is begun at the second stage and completed during the third. Von, for example, who begins by accepting the equivalence because of the correspondence , explains the apparent variations in quantity by saying that the width of P compensates for the height of L. But his attempt at the operation of multiplication of relations (‘They’re the same… because here (L) it’s round and long, and there (P) it’s round and bigger’), is as yet so ill-founded that the second time he does it he forgets to correlate the height of l with the width of the other container and suddenly decides that L ‘is longer and bigger and so there’s more,!   

                                 

Number, and Additive Composition of Classes

The lesson to be learnt from the preceding facts is that class and number have a common mechanism, that of their additive and multiplicative operations. Let us now consider how classes must be group in order to function normally, how the ‘grouping’ of classes differ from ‘groups’ of numbers, and what are the relationship between these two kinds of systems.

The two equations which determine the solution of the problem of the beads examined in this chapter, i.e. A+A’=B and A=B-A’, constitute the elements of any additive ‘grouping’ of classes. Once these elements are acquired, the following sequence can be composed: if B is included in C, and C in D, then B+B=C; C+C=D; etc.., and the inverse operations will be D-C=C of D-C+C’; C-B=B of C-B=B’ etc. These equations are associative with respect to addition and subtraction. i.e. B+B +C=C+C.’ etc and D-C=B+B’ = A+A+B’, etc. But with respect to itself, each term is an ‘identical operation’, since A+A=A, and in the case of classes containing others A+B=B, if A is included in B1.

This peculiar relationship which distinguishes logical ‘groupings’ from ‘groups’ of integers, 1+1=2, 1+2+3, etc., indicates the fundamental difference between classes, in which iteration is disregarded, and numbers, of which iteration is characteristic.

In what does this difference consist from the psychological point of view? If we again take the example of the beads, class A is defined by the union of the elements having the common quality ‘brown’, but obviously the number of these elements like those of class A,’ is not fixed, unless by a convention  which as nothing to do with the logic of classes. If B=A+A’, and if A and A’ contain at least one element, all the we know is that class B will contain more elements than A or A’, and that the attribute characterizing B is common to all the elements of A and A’, while no element in A has the attribute of A, and conversely. Apart from the intensive quantifications A greater than B or B Less than A’, A+A=B, and the terms ‘one’, ‘none’, ‘some’ and ‘all’, class in extension ignores quantity and has nothing of the extensive quantification characteristic of number. Obviously, in order to be able to state that A+A+2A the two classes A and A must be quantitatively comparable, but we do not know whether  there are more, less or the same number of elements in A and A.’ When A+A represents the logical addition of two classes defined by the same attribute, they form a single class, therefore A+A=A.

How then are classes to be transformed into numbers? In order to simplify the problem, let us suppose the various classes A,A’, B’, C’,.. etc., to be singular, i.e. each containing a single element while B,C,D etc.. will contain several elements. Let us suppose, for instance, that A is round, brown, wooden bead; A is a round wooded bead, but is not brown; B’ is a round bead, but is not wooden; C’ is a square bead, D’ is a counter; E’ is a bean and so on. Then A+A=B will be the round, wooden beads; B+B+C will be the round beads; C+C+D will be the beads D+D=E will be the beads and the counter; E+E=F will be the things on the table, and so on. The question then is: what operations are necessary to obtain the number 1,2,3,…6 from this classification?

As we saw in Chapter III-VI, Russell’s solution to this problem is too simple. For him and his followers, two classes have the same number when there is one-one correspondence between their elements. Let us suppose we have another table on which there is an identical set of objects A,A’,B’,C’ etc… being defined in the same way. There will be an exact one-one correspondence between the two sets F and F1, but what do we mean here by the term correspondence? If we confine ourselves to the logic of classes, by which elements are united according to their attributes, it is obvious that A will correspond to A1, A’ to A1, B’ to B1, C to C1 etc., but it would be false to say that the square beads C will correspond to the counter D1 or that the round glass bead B’ will correspond to the bean E1. The qualitative correspondence between the two classes F and F1 merely means that these two classes have the same numbers. As we saw in Chapter III-IV these are various kinds of qualitative correspondence, which depend on the spatial position of the elements and have no numerical significance. When an anatomist compares the bones in the skeleton of mammals with those of the other classes of vertebrates, the correspondence he makes is qualitative, and not mathematical, whereas in our cases, if we assert that any element in F can correspond to any element if F1, we have the right to concluded that F and F1 correspond numerically term for term and that this correspondence defines the number six. This number is not a ‘Class of Classes’, but the results of a new operation brought in from outside, which is not contained in the logic of classes as such. In fact, this ‘quantifying’ correspondence is only achieved by disregarding all the attributes in question., i.e. by disregarding the classes.

In order to transform classes F and F1 into numbers, the first essential condition is that their terms shall be regarded as equivalent from all point of view simultaneously. This is however contrary to our earlier statement with respect to classes. If we assume that we disregard the difference between A and A’, then class B will not be equivalent to the number 2 but only to the union of ‘round, wooded beads’ irrespective of their color. If we then again take into account the difference between A and A’, A and A will no longer be equivalent to one another except with respect to B. In order that B shall be equivalent to the number 2, B must be the union of any pair of elements, A and A’, or A and E’, or B’ and C’, etc. In that case, A=A’ =B=c=d’, etc., and these objects, having lost their differences, will constitute a homogeneous class, ‘the things on the table’. In a word, if we say that A+A= two objects, A+A’+B=three objects, etc., we are regarding these elements as being so many units equivalent to one another and yet distinct, and this twofold prerequisite cannot be reduced to the schema of additive composition of classes without some further operation.

This brings us to the second condition; the equivalent terms must remain distinct. In saying that A+A =two beads, we are stating that A is any bead, and A’ and other bead, but essentially different. In what does this difference consist? We can no longer have recourse to the difference in color, or to any other qualitative difference, for in so doing we should merely have the schema of classification described earlier, that of addition of classes and not of numbers. By “any other bead’, we shall therefore simply mean a bead ‘placed next to’, ‘coming after’, ‘selected after’ the first one. That is to say that in addition to the inclusion A+A=B characteristic of classes, a principle of seriation is involved, and as we saw in Chapter V-VI seriation is merely an addition of differences, as distinct from addition of classes which is an addition of elements that are equivalent from one given point of view. 

These two condition are necessary and sufficient to give rise to number. Number is at the same time a class and an asymmetrical relation, the units of which it is composed being simultaneously added because they are equivalent, and seriated because they are different one from another. In qualitative logic, the operational fusion of these two characteristics is impossible, since addition of classes is commutative because the terms are not equivalent. Number, on the contrary, is the outcome of generalization of equivalence and generalization of seriation. For instance, the first unit in two is equivalent to the second, and if their order of enumeration is changed, the second becomes the first and vice versa.

To sum up, we can now see why the additive hierarchy of classes, seriation of relationship, and operational generalization of number (i.e the construction of the numbers above, 1,2,…5 which are intuitive) appear at approximately the same time, above the age 6 or 7, when the child’s reasoning is beginning to go beyond the initial pre-logical level. The reason is that class, asymmetrical relation, and number are three complementary manifestations of the same operational construction applied either to equivalences, differences or both together. It is, in fact, when the child’s intuitive evaluations have become mobile and he has therefore reached the level of the reversible operations, that he becomes capable of inclusions, seriation and counting.

It is obvious that if number is the fusion of class and asymmetrical relation into a single operational whole, this synchronism has its logical explanation, but it can also be explained psychologically. Since each number is a whole, born of the union of equivalent and distinct terms, it cannot be constituted without inclusion and seriation. Moreover, although intensive quantification does not presuppose individual numbers, it does require that the child shall be capable of construction these numbers, otherwise the relationship which determine the extension of classes would lose all meaning. It is because this is so that all the facts contained in this chapter show that classes and number are mutually dependent, in that while number involves class, class in its return relies implicitly on number.2

1Cf. in Compte rendu des’ senances de la Societe de Physique et d’Historie naurelle de geneve, vol lviii (April 18, 1941), our article on the relationship between logical class and number.

2 Madame Refis mehmed Semin who repeated the same experiments with Turkish children, principally in Istamboul, obtained results very similar to our.

Conclusion: Ordination and Cardination

The Experiment With Cards Forming a Staircase.

We have as material a square of cardboard A, representing one unit, a rectangle B the same width but twice as high (thus representing two units), a rectangle C representing three units (same width as A but three times as high), etc Thus A=1; B=2A; C=3A D=4A E=5A; F=6A; G=7A; H=8A; J=9A; and K=10A. The cards therefore form a staircase, but one which is based on a construction of units and not on arbitrary relations like the staircase in $1.

We begin by asking the child to make the series himself so that he may grasp the principle of this ordination, and then get him to count the cards as far as he can without hesitating. He is then asked; ‘How many cards like A could we make with (B) or (C), etc.’, until he understands that the second card can be cut up into 2A, the third into 3A, and so on. Once this is grasped, a card is picked out at random (F) for instance), the staircase remaining in position, and the child is asked how many units this card represents. It is the child’s solution to this third type of question that interests us here. If he can at once make F’s cardinal value six correspond with its position (sixth), it is clear that he has grasped the relation between ordination and cardination. If, however, he needs to measure each time how many times A goes into E, F, etc., we shall be justified in stating that this correspondence has not yet been grasped. We find three stages, corresponding to those of $1. In the first, seriation is still global and the relation between order and cardination is not understood beyond three or four, even when the order A K is followed. In the second stage, intuitive seriation gives a correct result after trial and error, and the relation between ordination and cardinal value is understood provided that the order is followed, but not when the cards are mixed. In the third stage, the last question also is successfully solved. Here are some examples of the first stage.

Tes (4:6) seriated the cards correctly as far as D, then made the mistakes typical of this level. He could count up to 15. ‘How many cards like that ONE (A) could we make with this one (B)?- Two.-And with that one (C)?-Four… No, it’s the same as those (B+A). (Counting.) One, two, three, it makes three.- And with that one (D)?- (He counted with his finger.) Four.-etc.- And with that one (J=9)?- (He counted again, following with his finger imaginary sections of the card) 1,2…8.- And with that one (C)?-1,2…5.- And with that one (C again)?- That makes 4 cards.’

Fiv (5;0) also merely made a global seriation, which we then corrected with him. He counted them correctly. ‘How many cards like this one (A) could we make with that one (B)?- Two.- And with that one (C)?- Three.- And with that (D) is bigger that this one (C)?- By that (pointing to the difference in height).- How many more are there each time?- then how many cards like this one (A) can we make with that one (D)?-Five, no two.- How many?- (he counted the whole series from A to K.) 1,2…10.- then how many cards can we make with this one 9A)?-One.- And with this one (B)?-Two- And with this one 9C)?-Three.- And with this one (D)?-Five.’

These children show that they can count the terms of the series without hesitation, and that they can even understood, when two successive terms are compared, that the difference is A or 1. Tes, for instances, sees that C is identical with B+A, and Fiv states that there is ‘one more each time’. And yet, when asked to say how many times unit A goes into N, they are unable to find the solution merely by reference to the position of N, and to say, for instance, that since the card is the forth it contains four units. More important still, they do not even grasp this correspondence between position and cardinal value when the various cards are taken in order. They can see I by direct perception up to three, but after that they either count the possible divisions each time, like Tes, or else they judge by appearance, like Fiv. In a word, they are in possession of all the empirical elements necessary for understanding the law, but they do not understand it.

The paradoxical situation characteristic of the second stage illustrates even better what happens since children we shall now quote as examples appear to discover the law when the progressive order is followed, but are baffled when the order is reversed, when cards are picked out at random, and even more when the visual seriation is destroyed.

Bet (5;0) first made the series as follows: A,B,C,D,F,G, then inserted E, etc., thus arriving unaided at the correct results, though after some trial and error. ‘If we cut up this one (B), how many little ones like that (A) can we make? -3.- And out of (D)/-4, etc., correct up to 10.- And out of this one (Pointing to J again)?-9.- And out of (H)?-.10.-And out of (G)?-11.-And out of (F)?-12/’ Thus when the series was reversed, Bet began correctly by going down to 9, but went on with 10, 11, 12,.. and only saw the absurdity of it when he came to B. When the cards were disarranged, Bet said 4 for D (Judging by its appearance), but 5 for G, then again 4 for D, then 6 for G, and 6 also for h (tracing with his finger the possible division on the card itself).

Mic (5;0) successfully seriated and counted the ten cards. ‘How many of those (A0 can we make out of that one (B)?-2.-And our of that one (c)/-4.NO3.-And out of (D)/-5, etc.’ But when, with the series still in place, F was indicated, he did not think of using the system he had discovered and said that F was 4A (Counting on the card itself), G was 6A (same method), E was 5 and F ‘8, no12’ (same method).

Bru (5;0): ‘Which is the first one?-(he pointed to A.)-And the second?-(B).- If we cut up that one (B), how many little cards like this one (A) can we make?-2-And out of this one (C)/-3.-And out of this one (D)?-4;etc., up to 10.’ But when, without disturbing the series, we pointed to cards at random, he thought there were 4 in E,6 in H, 3 in D, 7 in E, and so on.

Dit (5;0) succeeded after trial and error in arranging the 10 cards as a staircase and at once understood the law; ‘That one (A) is 1, that one (B) makes 2; that one (C) makes 3; that one (D) is 4; that (E) makes 5; that (F) 6… and that (K) makes 10 little cards.’ The row was then disarranged and F (6) was indicated. ‘How many does this one make/- 4 little ones.- Are you sure?- No, but I think so.- What could we do to make sure ?-Make a staircase.(He remade it and counted.) 6 Little ones.’ But in the case of D (4), Dit failed because he reversed the order. He began by seriating the first 6 cards: ‘Can we tell from those how many it makes?-No, we must put the others as well.’ He made the complete row of 10, then counted from the 10th to the 4th and said: ‘that one (D) makes 7, we can make 7 little ones.- Why/- Because there are 7 places.- How’s that?- (He put card A seven times on card D and said): That’s 7, that’s how it is.

The significance of these examples is clear, and they throw considerable light on what occurred at the first stage. These is no doubt that at this stage the child understands the law of the series and adds one to the cardinal number for each new ordinal when the order A to K is followed. And yet, even with the staircase still in place, when a card larger then three or four is picked out at random he can no longer estimate the cardinal value of its units by reference to its position, and at first tries only direct evaluation, tracing with his finger the hypothetical division on the card itself, as did bet, Mic and Bru. Dit, who was more advance, having reached the upper limit of stage II, understood that once the cards were disarranged he had to reconstruct the series and find the position of each element in order to determine its value. And yet, on the one hand, he thought he had to reconstruct the whole series from A to K and could not tell from the first four cards that the forth represented four units, and on the other, when he had  reconstructed the whole series he calculated the position of D from the wrong end and thus made it seven instead of four.

These strange behaviors make it reasonable to suppose that children at this level have systematic difficulty in grasping the relations between ordination and cardination. What is the explanation, since in this experiment with cards arranged as a staircase the material represents with the maximum of intuitive clarity the law governing the formation of the first ten finite numbers, each ordinal corresponding to each cardinal and vice versa? Obviously the most natural hypothesis is to relate this difficulty to the lack of understanding we found in our analysis of cardination. It will be remembered that at a certain stage of development, in which the average age corresponds to that of the present stage, the child who is successful in making a one-one correspondence between two sets of six objects no longer grasps their cardinal correspondence once the elements of the one set are displaced. What appeared to be a one-one correspondence leading to stable colligation thus proves to be only a system without conservation and therefore without true cardination. In this system the colligation is still only intuitive, i.e. dependent on perception of the figure or of the space occupied by the sets in question, and although the child can use numbers to enumerate the elements, this enumeration does not as yet involve true cardinal numbers. Similarly, in the present case it may well be that what appears to be ordinal understanding of the series is dependent on the intuitive act of passing term by term from the beginning to the end, and ceases to exit when this is replaced by consideration of any given term.

This comparison between the difficulties of cardination and ordination is justified by the fact that once the two ideas are grasped they appear to be essentially linked one with the other. If the child fails to grasp the notion of the constant value of a set regardless of the arrangement of its elements, it is because at this level he has not yet acquired the notion  of stable ordination. To put it more simply, this hypothesis amount to a statement that intuitive seriation becomes true seriation only when it becomes operational, and that it becomes operational precisely when there is co-ordination with cardination. Conversely, colligation and intuitive correspondence becomes true cardination only when they become operational, and this occurs only when there is co-ordination with ordination. Viewed thus, the child’s seriation at this stage merely produces as it were ‘rigid series’ such that the position of the elements depend on the whole set of seriating, and cannot serve for detailed operations once they are separated from the whole. Just as (until he has acquired the notion of lasting equivalence) the child can only evaluate corresponding sets by a kind of rigid colligation, i.e without reconciling the conservation of the whole with the mobility of the elements, so, during this stage, he can only base cardinal value on position as long as the positions are linked one with another in a whole, continuous series. Once the series is broken up and it becomes a matter of considering the relationship between a given element and the others, with out following the progressive order step by step, ‘before’ and ‘after’ lose their cardinal significance, or the cardinal values cease to represent ‘before’ and ‘after’.

As a final test of our thesis let us consider some examples of the third stage, characterized by complete understanding of the problem, i.e. by operational ordination and cardination.

Let (6;0): ‘How many cards are there?-10.- How many like this one (A) can we make with that one (B)?-2.- And with (C)?-3. (he counted on the card).-And with that one (D)?- (He began by counting on the card and cried: We can make 4.-And that (E)?- 5, because I know how the figures go!-(I passed over the next element.) And that one (G)?-7,’ etc. The cards were then disarranged and G was indicated: ‘How many with that one?-(he counted and then at once began to seriate from A to G and said); ‘Yes, it’s 7.’

Ald (6;6) made the series without hesitating and was first asked ‘How many like that one (A) can we make out of (C). He replied: 3 I knew I had to count that one (A) as well.’ He had therefore at once counted the positions. ‘And with that one (F)?-6, because there are 3 there (Pointing to K, J, H), that makes 10, 9, 8, then that one (G) 7,’ When the series was disarranged, he reconstructed the staircase as far as the element in question and thus found: 5 for E, etc.

These children were immediately able to find the value of one of the cards, whether it was pointed out at random in the staircase or when the series was disarranged. In other words, the series had ceased to be rigid and had become mobile or operational, each term being susceptible of consideration in itself and in its relation to the others, whatever the order, Thus Ald reconstructed the cardinal value of F by determining its position in the decreasing order and Let found the question so easy in the increasing order that he said, ‘I know how the figures go’

In conclusion, we find that the gradual co-ordination of cardination and ordination confirms, on this essentially numerical plane, our earlier findings as to qualitative seriaton and serial correspondence. 

Part II. Cardinal and Ordinal One-One Correspondence

Chapter III

Provoked Correspondence And Equivalence of Corresponding Sets.

One for One Exchange of Pennies and Objects.

In this section, we shall study the dynamic correspondence which one for one exchange provides, and  we shall being with a technique which is merely an extension of those used in the earlier experiments. The child is told that we are going to have a shopping game, and he is given some pennies to buy flowers, sweets, etc., the price of each object being one penny. He can first be asked to estimate how many things he will be able to buy(which will show whether he is using global comparison, one-one correspondence, or numeration). The actual exchange    then takes place, after which we try to discover whether or not there is equivalence, for the child, between the number of pennies and the things he has bought. As however these methods of correspondence are essentially the same as those we shall be examining in the next chapter, it is more particularly the question of the equivalence of corresponding sets with which we shall be concerned here.

I Stage I: Global comparison and Lack of Equivalence

Naturally, all children at the first stage can make the correct one for one exchange of pennies for objects. But on the one hand they are incapable of estimating, through correspondence, how many elements they will have to exchange, and on the other they do not conclude that the two sets exchanged are equivalent.

Gui (4:4) put 5 flowers opposite 6 pennies, then made  a one for one exchange of 6 pennies for 6 flowers (taking the extra flowers from the reserve supply). The pennies were in a row and the flowers bunched together: ‘What have we done?- We’ve exchanged them.- Then is there the same number of flowers and pennies?.-No.-Are there more on one side?-Yes.- Where?- There )Pennies). (The exchange was again made, but this time the pennies were put in a pile and the flowers in a row.)-Is there the same number of flowers and pennies?-No.- Where are there more?-Here (Flowers).-And here (Pennies)?- Less;

Mic (4:4) was also unable to make a correct estimate of the number of flowers required for the pennies.-A one for one exchange of 6 flowers for 6 pennies was made, the flowers being put in a row and the pennies close together: “Is there the same number of both?- No, there are more flowers.-Why?-Because the flowers are more spread out.’

Duc (4:6) also succeeded only in making a global estimate of the number of flowers required. Then 6 flowers were exchanged for 6 pennies, the pennies being spread out: ‘Have we got the same number of flowers and pennies?-No, there are more pennies.- (The money was returned and the exchanged again made, the pennies being put in a pile.) And now?- No, there are more flowers.

It is unnecessary to comment on these examples before examining those of the second stage.

II Stage II: Correct Correspondence but not lasting equivalence

The only progress shown at this stage is that the child is capable of making a correct estimate, through visual correspondence, of  the number of elements to be exchanged. In spite of this, and of the fact that the estimate is empirically confirmed by the exchange, he still does not believe that the two sets are equivalent:

Nic (4:1) counted 10 flowers and 10 pennies, but did not total them into a single cardinal number; ‘How many pennies are there, then?-One, Two, Three, Four…ten (Counting from memory)._ Well then, buy.- (He gave one penny for each flower, up to 10, but the pennies were put in a row while he kept the flowers in his hand.)- Is there the same number of flowers and pennies?- There are more pennies (But he then spontaneously made the correspondence by putting one flower opposite each penny). Oh yes! They’re the same.- (The flowers were then bunched together.) And now?- there are more pennies.- 9The pennies were put in a pile the flowers in a row.) And now?- No, because there are a lot of flowers.

Lib (4;5) put 4 pennies opposite 4 flowers; ‘Is there the same number of pennies and flowers?-Yes, they’re the same.-All right, now you’re going to buy some flowers. Here are your pennies (Six).- You must give one penny for each flower.- (We exchanged 6 flowers for 6 pennies the pennies being arranged in a row while the flowers were in his hand.) is there the same number of flowers and pennies?-yes, they’re the same…No, they’re not. Here there are more (Pointing to the flowers).-Could we put one flowers opposite each penny?-No, there are two many flowers (he tried it and found that they corresponded exactly.) yes they’re the same.- Let’s begin again, shall we? (The exchange was again made, with the pennies spaced out and the flowers bunched together.)-Will that be all right?-There’ll be too many flowers bunched together.)-Will that be all right?- There’’ be too many flowers. You’ll see. (He made the correspondence and was very much surprised at the results.

Par (5;2); ‘You must pay a penny for each flower. How many flowers can you buy with that (1)?-One.- And with those (3)?- Three flowers for three, because there are three pennies.-All right, now you’re going to buy all those (he exchanged 6 pennies in a pile). Is there the same number of both?-No.- Why?- because there are more flowers.- If I wanted to buy those flowers with these pennies (i.e. the 6 pennies that the child himself had paid out one by one), could I do it?- No, Yes.- they’re the same then?-No, there are more flowers.-And if I put a penny opposite each flower (Doing so opposite the first two flowers to make sure he understood)?- No, there’ll be some flowers left over.

Fur (5;9) exchanged 7 pennies for 7 flowers after correctly making the correspondence 5 to 5. The flowers were still in his hand and the pennies were in a row; ‘Is there the same number of both?-No, there are a lot of pennies and not many flowers.- (The flowers were then arranged in a row but rather closer together the pennies .) Is there the same number?.-No, there are more pennies. There’s one past the end.- Count the flowers.- Seven.-And count the pennies.- One … seven.- So they’re the same?- No, there are some past the end.- Let’s see (The one for one exchange was given made.) –So there’s the same number?-(He remained silent, and was obviously shaken.)- If we counted the pennies and the flowers (the latter were now further apart) Would we have to count longer, or would it be the same?-We’d have to count longer for the flowers.

Aud (6;7); ‘We’re going to play at selling flowers. Here are your pennies.-(He counted the correctly.) Eight pennies.- Each flower costs a penny. How many will you be able to buy?-Eight.-(The one for one exchange was made, and the pennies were put in a row while the child kept the flowers in his hand.) is there the same number of flowers and pennies?._ No, There (Pennies)there are more.-Why?- They’re spread out.- Could we put one flower on each penny?-Yes.- then is there the same number of both?-No. There are more there (Pennies), Because they’re spread out.’

The reaction show quite clearly that one for one exchange is no guarantee that the child sees two equivalent sets as being bound by the same cardinal number.

Children of the second stage do not differ from those of the first stage as far as equivalence is concerned.

It is in no way surprising that Gui, Mic, Duc (Stage I), who, before the exchange takes place, cannot make the correspondence between two sets of objects, and estimate quantities by the space they occupy, are also unable to see after the exchange that the sets are necessarily equivalent. What is extraordinary, however, is that Nic, who makes the correspondence spontaneously in order to see if the two sets are equivalent, Lid, who before the exchange makes the correct correspondence between four pennies and four sweets, and Par, who uses numbers in estimating that he will get three flowers for three pennies, are incapable of assuming the equivalence of the sets they themselves have exchanges.

The most curious cases are those of Par, Fur and Aud, who use numeration. Par states , before he makes the experiment, that he can buy three flowers with three pennies, but as soon as the flowers are spread out, he no longer sees the equivalence. Fur when the seven pennies are piled together and the seven flowers spread out, counts both sets, discovers that they have the same cardinal number, but refuses to accept that they are equivalent; ‘No, there are more pennies. There’s one past the end. ‘Similarly Aud counts eight pennies, says that he will be able to buy eight flowers, makes the exchange, and then cannot see that the sets are equivalent; ‘There are more, because they’re spread out.’ These cases clearly show that perception of spatial properties carries more weight than even verbal numeration.

As to the question of the possibility of restoring the original situation, the children were asked, when they thought that the flowers and pennies exchanged were not equivalent, whether one flower could be put on each penny. Most children at this stage think that this could not be done. Lid says, ‘There’ll be too many flowers, you’ll see’, and Par, ‘no, there’ll be some flowers left over’. Only Aud thinks it could be done, but does not deduce from the fact that there is equivalence, although he is very near it.

III Stage III: Lasting Equivalence

To begin with, we have two reactions that are intermediary between stage II and III.

Pit (6;11). Ten flowers were exchanged for 10 pennies. Pit kept the flowers in his hand and the pennies were put in a row; ‘is there the same number of flowers and pennies?- (he made the flowers into a row, putting one opposite each penny, in order to check the number and then said0; yes, there are as many flowers as pennies.-(And now (flowers spaced out, pennies in a pile)?-They’re not the same. there are more flowers (He re-made the correspondence of his own accord). Oh Yes! They’re the same.-But before, you said there were more flowers, didn’t you?-Yes, but they were like that (with a gesture to indicate that the pennies were close together).’

Fran;(6;3) at once counted the 10 pennies he was given, ‘How many flowers can you buy if each one costs a penny?- Ten (The exchange was made, the flowers remaining in his hand and the pennies spaced out on the table.) Is there the same number of flowers and pennies?- Yes.- Why?- Because they’re the same.- (The exchange was again made, and the pennies were spread out.) Are they the same? yes._ (The pennies were out together and the flowers spaced out.) And now? –No.- Why? There are more there (Pointing to the pennies). –Could we cover each penny with a flower?-Yes.-Well then? – they’re the same.’

These two cases, in which the correct answer is arrived at, are extremely interesting, particularly the spontaneous verification of Pit, who was obviously using abstract operations that were as yet barely mastered, in order to combat perceptual appearance. Fran finally succeeded in arriving at this abstraction, i.e. at the operation proper.

Here finally, are some examples of correct reactions. 

Gin (4;11) counted his 10 pennies and estimated that he would have 10 flowers. After the exchanged, he said ‘They’re the same’ whatever the configuration, But without giving any reasons.

Du (5;8) exchanged 10 for 10, the flowers remaining in his hand and the pennies spread out on the table. ‘Is there the same number of pennies and flowers?.-Yes.- Why?- Because we’ve used them all.-(the flowers were spaced out and the pennies put together.) Are they the same now?-Yes.- why?- Because we’ve used them all.’

Ler (5;8) kept the flowers in his hand after the exchange: ‘Is there the same number of both?-Yes- Why?- Because they go like that (spontaneously putting one flower opposite each penny).’

Clav (5;8). Same situation: ‘ Are they the same?-Yes.-Exactly?- Yes.- Why?- Because I gave you my pennies.’

For these children, it is clear that the equivalence has become obvious and logically necessary . The reason they give to justify their assumption are interesting because of their operational character. For Ler, it is the fact that there is one-one correspondence, and hence the possibility of returning to the visible correspondence after the exchange; for Du and Clav, it the exchange itself, because by it, the two sets were exhausted simultaneously: ‘I gave you pennies’ and ‘We’ve used them all.

In conclusion, the results of the test using one for one exchange are exactly the same as these found in the case of static, visible correspondence between objects. This constitutes a valuable aid to the understanding of the notion of correspondence. It indicates that the importance given to the procedure of one for one exchange, in which so many authors have attempted to see the beginnings of cardination, is unjustified, since this procedure does not in itself result in the notion of necessary equivalence of exchange sets. Before this results can be achieved, one for one exchange, like intuitive correspondence, must become operational, i.e. it must be grasped as a reversible system of displacement and relations.

One for one exchange in Conjunction with counting aloud.

we have just seen, from the reaction of Par, Fur and Aud, that counting aloud appears to have little influence on the belief in the equivalence of two sets as a result of one-one correspondence. We have already frequently had occasion between the acquired ability to count and the actual operation of which the child is capable.

We have now reached the point at which this needs to be examined systematically. We first of all find out how far the child can count without difficulty. We then repeat the above experiment with one for one exchange, choosing a number of pairs of objects smaller that the highest number to which the child can count. We then ask him to count the objects he has been given, and we cover with out hand the pennies he has given us in exchange (so that he cannot count them). We then ask him merely to guess how many pennies are hidden.

The results show that counting aloud makes no difference, and we find the same stages as those we found with the earlier techniques.

I Stage I Global Comparison and Lack of Equivalence.

Ras (3;6) could only count up to 4 or 5. He was given 2 pennies and asked to give in exchange ‘the same number of sweets’. He gave 5 and then 2 For 3 pennies, he gave 4 sweets, etc. I then exchanged 4 pennies for 4 sweets, one at a time. When I hid the sweets with my hand, he thought that there were none left when 3 had been taken out, and then thought that there was still one left after I had taken out the fourth.

Ber (3:11) could count correctly up to 5, but had difficulty in making the correspondence between two sets containing more than 2 or 3 elements. I made the one for one exchange of 3 sweets for 3 pennies and covered up the pennies. I then brought one out and said; ‘Are there ant left/-Yes- How many?-…-And now (There was 1 left)? –No.- And now (When the last had been brought out)?-Yes.- How many? –There’s one penny left.’ When 2 pennies were exchanged for 2 sweets, Ber answered correctly, but as soon as there were 3 or 4 his answer were again given quite at random. Finally I made sweets have I given you? –One, two, three, four,- And how many pennies have I in my hand?-… - How many do you think?- I don’t know.’

II Stage II: Correct Correspondence, Without Lasting Equivalence

The only difference between this and the preceding stage is that the child can make the correspondence before the actual exchange is made. 

Mard (5;6); ‘Look, I’m going to buy some sweets from you. Here are my pennies (7, in a row). Give me as many sweets as there are pennies.- (He counted) 1,2,3,…7.- And how many pennies?- 1,2,3…7- Good. And how many sweets have you given me (having covered them with my hand)?-…-How many did you give me for one penny?- One.- Yes. And for two pennies?-two.- that’s right And for three pennies?- Three.-That right. And how many pennies have you there?-1,2,3…7.-Good. And how many sweets did you give me? How many sweets are there here (Uncovering them for a moment and then hiding them again with my hand)?-I,2,3,4,5,’ I tried it again: ‘Look. Here are some pennies (5, in a row). How many are there? -1,2,3,4,5.- That’s right. (I picked them up). Now, when I give you a penny, you’ll give me a sweet (we made a one for one exchange up to 3). How many pennies have you? -1,2,3.- (two more were exchanged.) And now how many pennies have you?- 1…5.- Good. And how many sweets have I got (covered them)?-..-9

Cauch (5.6) was also able to make the one-one correspondence between sweets and pennies, when the pennies were in a row, even up to 15 or 17. He could count the pennies he was given up to and beyond 10. But before a one for exchange of 8 pennies for 8 sweets he did not conclude that there was necessary equivalence: ‘How many sweets have I given you?- (He counted.) 8.- That’s right. How many pennies have you given me ( They were hidden under my hand)/-10

Per (6;0) made 7 sweets correspondence to 7 pennies, and then made the exchange: ‘How many pennies have you?- (Counting)7.- And how many sweets have you given me (Hidden under my hand)?-…’

I began again with 5; ‘How many pennies have you? – 5.- And how many sweets have you given me?-…7.’ At a tird attempt, he counted 10 pennies and though he had had 9 sweets; etc.

It is not difficult to interpret these facts. When the child makes the one for one exchange he knows very well that there is equivalence. Mard, for instance, knows that he has to give one sweets for one penny, two for two pennies, three for three, and so on. But once the exchange has been made, and one of the sets is no longer visible, the child no longer considers it to equivalent to the one he can see. These reactions, therefore, are exactly the same as those of the corresponding stage studied earlier. If the child has not yet reached a certain level of understanding which characterized the beginning of the third stage, counting aloud has no effect on the mechanism of numerical thought.

III. Stage III: Lasting Equivalence

When, as here, one for one exchange is accompanied by counting aloud, interesting cases sometimes occur in which the child who has reached the stage of being able to give the correct answer, checks the equivalence by counting the number of exchanges, but does not as a result assign a definite number to the equivalent sets.

Mad (5;6) made a one for one exchange of 7 pennies for 7 sweets counting them as he did so: ‘How many sweets have you ? 1,2,…7.- And how many pennies have you given me?- 1,2,….7.’ When, how-ever, she did not count the elements as they were being exchanged, she showed that she was still as the earlier level. When 5 sweets were exchanged for 5 pennies, she said; ‘There are 5 sweets.- And how many pennies under my hand?-4,’ Etc.

Fred (6;0) also exchanged 5 sweets for 5 pennies and evaluated the two sets correctly by repeating the series of numbers; 1…5.’ But afterwards, when he was asked how many pennies were hidden, it did nor occur to him to count the 5 sweets in a row in front of him.

Such reactions clearly indicate progress over the earlier stage in the direction of the construction of true equivalence between the sets in question. But the equivalence recognized by Mad and Ferd is in reality only the equivalence of the operations they have just performed, i.e. the action of the moving a sweet and the action of moving a penny. As long as the child confines himself to the enumeration of these exchange, he can accept the notion that the correspondence is lasting, but when he tries to abstract the cardinal number from the operations by means of which he arrived at it, he does not yet see that there is necessary equivalence.1

Finally, we have two examples of genuine cases of stage III, Children for whom the one for one exchange involves the notion of lasting equivalence.

Sim (6:6) We made a one for one exchange of 6 pennies for 6 sweets. ‘How many pennies have you? -6.- And how many sweets have I?-6.- Are you sure? Quite sure.- Why?-..’

Far (6:6) exchange 8 pennies for 8 sweets; ‘ How many sweets are there?-8.- And how many pennies here (raising my hand and showing the pile of pennies)?-8.- Are you sure?- Yes.’ Same results with 11 etc.

This, then, is the evolution of estimates of equivalence.

It is in this way, then, that the child’s capacity to estimate equivalence develops in cases where he makes use of numeration. It is therefore no exaggeration to say that the verbal factor plays little part in the development of correspondence and equivalence. We find here the same stages, correspondence to approximately the same ages, as we found in the experiments of $$$1-3. At the point at which correspondence becomes quantifying, thereby giving rise to the beginnings of equivalence, counting aloud may, no doubt hasten the process of evolution. Our only contention is that the process is not begun by numerals as such.

It remains to explain the relationship between correspondence and equivalence. Before we can do so, we must first examine the evolution of correspondence itself, i.e., of its mechanism in its spontaneous form, not when it is provoked. We shall attempt to do this in the following chapter, in which we shall investigate how the child estimates quantities, how he thereby discovers one-one correspondence, and how he makes use of it in the cases of correspondence between similar objects that are not qualitatively complementary.

1 A. Rey also observed children who counted the action they were doing, and found that they did not all, as a result, acquire the notion of equivalence (L’ Educateur, May 1931, p, 151.)

Part III Additive and Multiplicative Compositions

IX. Co Ordination of Relations of Equivalence and Multiplicative Composition of Numbers

3. multiple correspondence and numerical multiplication

We now have to see how composition of equivalence can be generalized in the form of one-one correspondence between n sets-which we shall call ‘multiple correspondence’, a rather unsatisfactory term, but which we shall only use in this connection- and numerical multiplication.

During the first stage, we find the following reactions:

Dal (5;1) did not think set X (10 Flowers) and set Z (10 Flowers) were equivalent, although he himself had put first one and then the other into the same 10 vases Y. ‘Now we’er going to pull all the flowers into these little tubes (the opening of which was only large enough to contain one flower). Take enough tubes for all the flowers: you can see that we can only put one flower in each tubes.- (He made a row of 10 tubes opposite the 10 vases.) have you got enough for all the flowers? –(He added 4.)- Is is right now?-…-Try them.- (He put one flower in each tube, and when he came to the twelfth he added two more, but did not grasp that he needed two tubes for each vase.) he was then told to put all the flowers in the 10 vases: ‘How many will there be in each vase?- (He tried them one by one).

\in the cases of the egg-cups, he put in 10 eggs, one by one, and they were then taken out and out close together in a pan. He did the same with 10 other eggs, which were then spaced out in a second pan. ‘Are they the same?- No, there are more there (Z).- Well now, if we give all the eggs to these children (The 10 dolls in front of which the 10 egg-cup were standing), how many can we give to each of them?-One.- Are you sure?- (he began putting one egg in each egg-cup, then after 3 or 4 he exclaimed): No, a lot, 6 the children will have a lot to eat.’

Com (4;11) after first flowers X and then flowers Z had been taken out of the same 10 vases Y, did not think X was equivalent to Z. ‘If we want to put all the flowers (pointing to X and Z) in these vases, how many can we put in each vase?- (He put one in each vase, then began putting two, and then, looking at those still to come, three, after which by successive trial and error, he finally put two in each vase. That makes 2 for each vase.’

Immediately afterwards: ‘All right then. Now we’re going to put them in these little tubes, one flower in each tube.-Yes.- (He put 10 tubes opposite the vases.) –Have you got enough for all these flowers?-yes.- And will they all go in?-Yes.- try them.-(He began, and then said); there’ll be more tubes vases. They’re be a long line (Adding 5 or 6).’

Blu (5;6) had constructed sets X and Z corresponding to 10 vases. ‘And now, if I want to pull all the flowers back in these vases, how many shall I have to put in each one?-You’ll have to put one.- Do you think they’ll all go in?- (he tried it, but having reached the 5th or 6th he exclaimed); Oh1 well have to put more.-(He put 2 and obtained the correct result, and then asked spontaneously): Why did we have to put 2?- What had we done before?-Oh Yes1 1,1,1, (Set X), then we took them out, and then again, 1,1,1,(Set Z).- That’s right.

Now look. We’re going to take some little tubes, and we shall only put in one flower because there’s only a very small hole. Take enough tubes for all flowers.- (He put one tube in front of each vase.)- How many flowers were there in each vase? -6, no,2.- And how many are we going to put in each tube?-1.- Do you think you’ve got enough tubes?-Yes, it’s the same number as the vases.-try them then.- (he put one flower in each tube, but when he was about half-way through the series he exclaimed); Oh! There aren’t enough.- (He added 4 and said); I think that’ll be enough (and went on putting in the flowers).

No, there are some flowers left (Adding 3 tubes to the other end and putting 3 flowers in them.) No, there are still 3 flowers left (Adding 3 more tubes). They’re right now, but why are they more than the vases?- Look, in one vases there were two flowers. How many tubes did you take for two flowers? (He indicated one.)- Yes, for one flowers, but what about the other?- Oh Yes! (putting 2 tubes opposite each vase). That’s exactly right!’

The reaction of this first stage are extremely interesting. These children, who, at, least at the beginning of the test, cannot make a one-one correspondence between two sets of objects (except when the elements of one set are actually placed inside the elements of the other), are obviously equally incapable of deducing that because X=Y and Y=Z, then X=Z. Thus, when they are required to make the two sets X+Z correspond simultaneously with the vases Y, i.e. two flowers to each vase or to find as many tubes as there are elements in X+Z, i.e. two tubes to each vase, their behavior shows that they are quite incapable of multiplicative composition.

The most primitive reaction consists in merely identifying the new correspondence required with one of the earlier correspondences, without understanding the necessity for a correspondence of two to one, or duplication. Thus, in the test with tubes and vases, all the children began by putting out ten tubes because there were ten vases, for, as Blu said, ‘It’s the same number as the vases.’ And yet every child clearly understood that only one flower was to be put in each tube, as they showed when they tried to verify the correspondence empirically. In the case of the questions, each child also began by giving only one egg  to each doll, and by putting only one flower in each vase.

When however the child finds that in fact there are too many flowers for the vases, and too many eggs for the egg-cups, he quickly ceases to assimilate the new situation to the former one. But he does not succeed in making the hypotheses of a definite relationship between X+ Z and Y, i.e. he does not understand that if X and Z correspond simultaneously to Y, it means that there are two elements, and not one, corresponding to each element Y. It does not therefore occur to children at this stage to double the number; they merely recognize the necessity for a global increase, and choose any number at random. This constitutes the characteristic feature of this level, at which there is absence of exact correspondence and also of composition of the relations of equivalence. Dal, for instance, merely adds four tubes to his original series of ten, Conn adds five or six, Blu four, then three, and then three more, without understanding why: ‘But why are they more that the vases?’ There is the same lack of understanding in the problem of the eggs, and the number of flowers to be put in each vase. 

In a word, these children merely make an arbitrary estimate of the increase and are unaware of the duplication. Although they understand that n blue flowers correspond to n vases, and that n pink flowers also correspond to the same vases, they do not understand that the n vases correspond to n pairs. And although they understand that all the flowers together correspond to the tubes they do not understand that therefore each vase correspond to two tubes. It should be noted that Blu, after his failure, becomes aware of this relationship, and is thus on the threshold of the second stage.We now come to some examples of this second stage, the first of which shows the transition from the first stage to the second:

Rys (4;9) did not think flowers X and Z were equivalent (cf.42). ‘I wonder how many we’d have to put in each vase so that they’ll all go in?- I don’t know (he put them in one by one, then towards the end two by two and said):2’

‘Now we’re going to take the tubes. How many flowers did we put in one vases?-2._ And how many are we going to put in the tubes ?- 1.- Well then, get the tubes ready.- (He put one tube opposite each vase, put one flower into each tube, then looked at the remaining flowers, without counting them, and put another row of 10 tubes opposite the vases. He put in the flowers and said): That’s exactly right.’ 

A few days later, he put 10 eggs into egg-cups, then 10 more. ‘How many eggs will each child eat?-2.- Why?- One before and one after.- And if we give them these as well ( A new set of 10 that were put into the egg-cups and then placed by the other set of 10) How many will that make for each one?-2.- Try it. (He put 2 in front of each, and left the last 0 on one side.)- And what about these?- They’ll be for tomorrow (Obviously brushing the question aside.)’

Rol (5;4) knew that X and Z were equal to Y, but not that X and Z were equal. ‘Now if I want to put all the flowers in the vases, how many shall I have to put in each one?-I (he began doing so, then exclaimed): Oh! It’ll be 2 then.’

With the eggs: ‘How many eggs will that make for each child?-2.-And if we add these (10 more)?-2.- Why? (He counted) 3.’

Uld (5;8); ‘If I put all these flowers (X+Z) in the vases (Y), how many will that make in each vase?-2,3,or more.- Try it.- (He tried putting 2 and went on to the end of the row.) That’s just right.’

‘Now take just enough tubes for us to be able to put one flower in each.-(He put 1 tube in front of the 1st vase, 2 in front of the 2nd, 3rd, and 4th , 1 in front of the 5th, 2 in front of the 6th and the 7th, 3 in front of the 8th and 9th , and 2 in front of the 10th. He then equalized them.)’

With 10 eggs X and 10 eggs Z: ‘How many eggs will each child be able to eat?- That one (the 1st) will have 2 (He went on putting 2 eggs in front of each egg-cup). I didn’t think there’ll be enough (he went on). Yes, that’s right.- And suppose we gave them these as well (10 eggs that were also put in correspondence with the egg-cups), How many would that make for each one?-4, no 5.- Why?-Because they’ve got more.’

Hoeg (5;11) after he replies of $2; ‘And now, if we wanted to put all the flowers (X+Z) back in the vases (Y), how many would there be in each vases?-3,4.-Why?- Because we can put a lot in.-Yes, but there must be the same number in each vase, and we must use them all.-Yes. Then I’ll put that number (6).-Try it.-(After 3 vase he stopped.)  Well then, I’ll have to put 3.-Why?-Because it’s less (He tried it, but stopped at about the middle of the series). It’s not right yet then. I must only put 2 (doing so). That’s right.-Why only2?_ Because.-How many heaps did we have?-Oh! 2, and they were both the same, and the vases too, so that makes 2 for each vase.- That’s right.’

‘You see these little tubes? We’re going to put just one floor in each tubes. Take enough of them for all the flowers.- (He put one opposite each vase.)- Will all the flowers go in?-Yes.- Why?- Because there are .. I’ll count them (He counted) 10, like the vases, because the flowers went into the vases. Oh! I know, I’ll have to put some more everywhere, like this (putting a second tube opposite each vase).-Try it.- (He put in the flowers.) It’s exactly right! (surprised at the exactness of the results).’ 

The next day after being questioned about the eggs (cf.$2): ‘Now how many eggs will each child have eaten?-2.- Why?- Because we measured 2 twice (i.e. we twice put into eggs-cups a set of corresponding eggs).- That’s right. And suppose we gave them these as well (10 more eggs)?- That would make 3._makAnd if they had these as well (10 more eggs)?- That would make 4.’

The difference between these reactions and those of the earlier stage is obvious. When these children make a one-one correspondence between tubes and vases, and then find that the tubes will not correspond to all the flowers, they pass straight from the system ‘One-One’ to the system ‘Two to One’. This marks an important progress towards multiplication. 

Let us compare, for example, the case of Rys with that of Blu. The latter, although he finally discovers the two-one correspondence is still at the first stage because he proceeds by arbitrary addition. Rys, on the other hand, begins by putting ten tubes for ten vases, and when he finds that there are some flowers left, does not, like Blu, attempt to estimate the number, but at once adds ten tubes, puts in the flowers without any hesitation, and says ‘That’s exactly right.’ Similarly, Rol adds a row of ten to his original row, and Hoeg says ‘Oh! I know, I’ll have to put some more everywhere, like this.

In the case of the problem of discovering how many flowers will go into each vase if X+Z is to correspond to Y, Rys, who is still at the first stage as far as this problem is concerned, begins by putting one flower in each vase, and goes on to two after the first few. Rol also begins by putting one and goes on to the end of the row, the, seeing that there are some flowers left, reacts as in the case of the tubes, and without counting says ‘Oh! It’ll be two then.’ Uld and Hoeg, who are typical of the majority of children at this level, at once state that there will be ‘Two, Three, or more’, i.e. they think in terms of n to one correspondence, and then reduce n to two.

As regards the problems with the eggs, Uld is still at the first stage and tries two eggs for each egg-cups, but without being certain, whereas the other children are certain from the start. Obviously some ‘training takes place between the first test and the present one, which may also be easier in itself. Nevertheless, it is interesting that the reasons given by Rol (‘I must give two eggs each time’) and Hoeg (‘We measured twice’) belong to the third stage, since they express awareness of the existence of two One-One correspondence that have to be multiplied.

Each of these children understand, then, that if two sets containing an equal number of elements correspond respectively to a training an equal number of elements correspond respectively to a third set ‘One to One’, then the first two combined will correspond to the third in the relationship ‘Two to One’. But is it true to say that these children can already grasp the relationship n+n as a true multiplication, i.e. as the transition from ‘I’ times ‘n’ to ‘2’ times ‘n’? To our mind there are three reasons for not accepting this view. The first is that, as we saw in$2, these children have not yet mastered composition of the relation of equivalence, and it would be difficult to accept that there is understanding of arithmetical multiplication without mastery of the logical relationships involved in the composition of these equivalences.

The second reason is that these children do not immediately succeed with the multiple correspondence. It is only when they discover that there is a reminder, after attempting the One-One correspondence that they go from n to n + n. It is certainly a great advance on the arbitrary procedure of the first stage, but there is still some hesitation, and not yet immediate understanding such as we find at the first stage.

The third and most important reason is that if the child at once interpreted multiple correspondence as a multiplicative relationship, he would certainly be able to generalize if from 2n to 3, 4 or 5n, since 3,4, and 5, are familiar to him as 2, as we shall see at the third stage. Here, however, the multiple correspondence is not generalizable. Rys, for instance, thinks that if three sets of eggs correspond one to one with the same ten eggs-cups, each child will have two eggs, and when on making the distribution he finds that he has ten eggs left, he puts off the solution to the next day. In the same situation, Uld is inclined to think that each child will have four or five eggs, finally deciding on five because ‘they’ve got more’. Only Hoeg is able to generalize the multiplication to 3 and 4 without hesitation, and in so doing he shows that he has reached the third stage.

We come now to some examples of the third stage.

Gros (5;10) was convinced that X=Z if X=Y and Z=Y. ‘If I put all the flowers (X+Z) in these vases (Y), how many will there be in each? –I blue and I pink.- How many is that?-2.- And if I added these (anew set of 10), how many would there be in each vases?-3.- Why?-I’d put one, one, one,-And now suppose we wanted to put them in these tubes that will only hold one flower?-(He took 10+10+10 tubes.)’

Thi (6;10); ‘We’ll have to put 2 flowers in each vase.’ He afterwards took 10+10 tubes. In the case of the eggs, he at once saw that when there were 10+10+10 they would have 3, etc.

Bora (6;0) also understood at once that there would be 2 flowers in each vase ‘because there are 2 things (i.e. 2 sets of 10)’. He also took 2 or 3 tubes for each vase according as there were 2 or 3 equivalent sets of flowers.

Same reaction in the case of 2 and 3 sets of eggs. ‘And if I have these (10+10+10+10)?-4 for each doll.- And if I add these (10)?-9, oh no!5.’

It is clear that these children, all of whom are capable of composing the equivalence ($@), are equally able to understand, by combining the relations and no longer by intuitive trial and error, the relationships of multiple correspondence involved in the problem put to them: two flowers to one vase, two eggs to one doll, and a double series of tubes for the row of vases. Moreover, and this is the interesting feature of this stage, as soon as the ‘Two to One” relationship is grasped, it becomes generalizable to three, four and five. From this fact, two conclusion can be drawn. Firstly, the transition from the intuitive to the operational method pr procedure entails the possibility of generalization, of which we have just seen some almost instantaneous examples in the case of the small numbers familiar to the child. Secondly, the operation of correspondence is revealed in its true light, as being a multiplicative3 composition. In the various correspondence, one to one, two to one, three to one, etc., the value of each new set is no longer regarded only as an addition, but as a multiplication, ‘I’ times ‘n’, ‘2’ times ‘n’, ‘3’ times ‘n’, etc.

Conclusion: Multiplication of classes and numbers

Our examination, I n chapter III and iV, of the various types of one-one correspondence showed that the equivalence reached through one-one correspondence is multiplicative in character.

It is essential, both psychologically and logically, to distinguish the various forms of equivalence. Additive and multiplicative equivalence differ by the fact that in the first, the equivalence is in respect to one attribute (e.g., blue and pink flowers are equivalent in being flowers), while in the second, two attributes are involved (e.g. the fact of being flowers, and having a certain position in a series). Such multiplicative equivalence between classes occur frequently in the comparative sciences, zoology, psychology, etc., and the whole of this volume is an illustration of them, since the stage of development of one notion have been related to those of another.

Construction of equivalences through qualitative correspondence, co-ordination of these equivalences, are already multiplicative operations, although number as such does not yet intervene. In order that multiplication of classes shall become multiplication of numbers, all that is necessary is that the classes in question shall be ‘singular’, as we saw in the transition from addition of classes to addition of numbers in Chapter VIII.

We need only add, therefore, that in the case of multiplicative operations, as in that of additive operations, qualitative composition of classes is not achieved on the operational plane before that of numbers, but at the same time. There is not one stage of logical multiplication and another stage of arithmetical multiplication. During the first stage of development, neither of these composition is possible; during the second, there is a beginning of both on the intuitive plane, and during the third, both of them become operational in true sense. Hence the child’s simultaneous success in the various test studied in this chapter, and his immediate generalization of multiplication once it has been discovered.

Part II Cardinal and Ordinal One-One Correspondence

IV. Spontaneous Correspondence –Cardinal Value of Sets

In the previous chapter, we tried to show that there are various types of correspondence, which can be distinguished by their relationship to the idea of equivalence they entail. While the highest level can be termed ‘quantifying correspondence’ because it results in the notion of necessary, lasting equivalence, the lower levels are intuitive, because the equivalence of the sets is recognized only when the correspondence is actually perceived, and ceases to exist when it is no longer in the field of perception.

Before going any further, we must analyse the mechanism of correspondence, no longer considering it results, but examining it spontaneous development, i.e. in situation required, therefore, is one in which the child spontaneously attempts to estimate the cardinal value of a set in such a way that we can discover what types of correspondence he uses, and what methods he adopts before, and immediately after, one-one correspondence.

For this purpose, what is most suitable is correspondence between like objects, the child being required to find a quantity equal to that contained in a given model. A problem of this kind is of course, similar to those we examined in the previous chapter, when the child was told, before being asked about the equivalence, to pick out the required number of correspondence elements. But in the first place, here our material will not be of such a kind that the correspondence is suggested by the fact that the objects are qualitatively complementary; the objects will be similar, and this may make some difference. In the second place, and more important, the child will not be told to ‘put one A opposite (or into) each B,’ or to ‘exchange one A for each B’. We shall say to him: ‘There is a number of objects: pick out the same number’, without suggesting any method. In other words, whereas the earlier problems imposed the correspondence so that we could examine its results, the problem to be studied here is merely one of evaluation, requiring no suggestion of the method to be used,  but enabling us to see what the child’s procedure actually is.

There were the techniques used. In the first place, we presented the child with a succession of figures made with counter, and asked him to take the same number of counters as each figure contained. If, as we were led to suspect in the previous chapter, correspondence is based on qualitative compassion, out first task must be to analyse how the child compares quantitatively two sets with different configurations. There were five types of figures: I, ‘Badly structured’ figures, e.g. a collection of counters distributed at random, but neither touching nor overlapping: II, open series, e.g., tow parallel rows of counters :III, closed figures, the shape of which did not depend on the number  of elements used, e.g. a circle, a house, a right angle: IV, closed figures of which the shape depended on the number of counters, e.g. a square, a cross, etc. V, more complex closed figures, less familiar to the child, e.g., a rhombus, etc.

With one of these figures in front of him, the child was told to look at it, and to pick out of the box the same number of counters.

In the second place, the child was shown a row of six beans, representing sweets or pennies, one to tow centimeters apart, and he was asked to pick out the same number. Obviously this second technique, which is similar to those of the previous chapter, is only  a special case of the first.

The results obtained with these two kinds of test can be classified into three types, corresponding to the three stage found in the earlier chapter. During the first stage, the child uses only global comparison, imitating the configuration of the model without attempting exact quantification. When he is dealing with rows, he makes a row of the same length as the model, but of different density. During the second stage, there is one-one correspondence, but ceases to exist when the figure is distorted. Finally, in the third stage, there is exact correspondence and lasting equivalence. 

Reproduction of Figures

W.A. Lay made a detailed study of the way in which various figures made with three, four, five, etc., objects arranged as triangles, squares, etc., are distinguished by the child, from the point of view of perception of n umber. The number four, for instance, is more easily recognized when the objects are placed at the four corners of a square than when they are placed at random. A Descoeudres and O. Decroly made use of these investigations in their interesting research into the development of  number. Our point of view here will be different, for while these authors examined what has come to be called perception of numbers, i.e/. the application of already existing numerical schemata to discrete objects perceived in the same field, we shall examine what might be called quantifying operations i.e., the elementary operations of correspondence, equating, etc., which constitute the logic of number. In a word, we shall ignore the problems of perception and shall concentrate on the problem of genesis of operation as such. This being so, our analysis of the reproduction of figures will serve merely as an introduction to the study of the mechanism of correspondence. We shall therefore not consider each of the figures separately, but shall describe simultaneously the whole set of reactions obtained by the first technique.

Stage I: Global Comparison

The characteristic feature of children at the first stage is that they do not as yet feel the need for a quantitative evaluation, since they have no precise notion of the cardinal number. They therefore confine themselves, in evaluating the given sets, to global qualitative comparisons, without co-ordination of the qualities that are compared. Our first examples are cases of non-structured figures.

Pa (4.6). To find the required number of counters to correspond to a collection of 15 elements, he took small handfuls of them and tried to arrange them so that they looked the same: ‘Are they the same?-No. _Why?- There are more here (The collection he had just made did in fact contain 2 extra elements).-well then?-(He did not remove any of the counters, but moved the ones that were too close together, so as to produce a configuration more like that of the model.)-Is there the same number of counters?-No, yes, I’ve put the same amount.’

Hug (5.0), for a collection of 15 counters, said: ‘I don’t know how many there are. I don’t know how to to do it (to find the same number). –Try.- (He collected a few counters and then spaced them out so that they resembled the model.)-Are they the same?-Yes.- How many are there there (Model)/ I don’t know.- Then how do you know they’re the same?-I looked twice (Model and copy). It’s right.’

In the case of the series, children at this stage attempts to reproduce both the configuration and the dimensions of the model, but are not concerned with detail:

Mul (4;1) reproduced the model by a series of the same shape and approximately the same length, bit with the elements closer together and though ‘there the same number of counters.’

Li (4;9), in order to reproduce the model first put 5 counters in a row, then made 4 of them into pairs by adding 4 more: ‘Are they the same?-Yes.-Why?- (She made a movement of the hand to indicate the direction of the row.) Are there as many?-yes. (She looked at the model and saw that her copy was a little shorter. She added 2 counters her series was then composed of 11 counters instead of 8 as in the model, but it was the same length.) – Where are there more?-There (pointing to her copy). I want them to be the same.- (She removed the 2 counters, leaving 9 as against 8, but the model series was longer, so she spaced out her counters to make here longer.)- Is there the same number? Yes.

In the case of closed figures, children at this stage can reproduce correctly those which require a definite number of elements, provided that the form is familiar (category IV), but if the shape is unfamiliar (V), or does not involve a definite number of elements (III), the copy is no longer numerically correct.

Mul (4;1), in attempting to find the same number of elements as there were in a circle of 10 counters, made a circle of 14. Similarly, for a circle made by 6 matches arranged as radii, he made one with 12 matches; ‘Is there exactly the same number of match-sticks?-Yes._ where are those (indicating on the copy some that were particularly close together)/- there (pointing at random to those of the model)’

For a right angle with 6 counters (sides of 4 and 3 elements), Mul made, three times in succession, angles with 4 counters (sides of 3 and 2); ‘Is there the same number of counters?-No. I don’t know.-Try again.-(He then made an angle with 8 counters.)- is there exactly the same number?-Yes.-Who has more?-Me.- Well then, take away the extra ones.- (He removed two and altered the spacing of the counters to make them like the model.- Are they the same?-Yes.- Who has more? –Me (which was incorrect since there were 6 in both. He then put back the 2 he had removed.)’ Similarly, for a house of 6 counters, Mul made one of 13, then removed 1 ‘so that it should be just right,’ etc.

In the case of figures whose shape depended on the number of elements, Mul successfully reproduced a square of 4 counters and a triangle of 6, but made a cross with 6 instead of 5. He also failed with a square of 9 counters, being careful to make the elements at the 4 corners correspond, but putting 1 counter too many in between. The rhombus of 13 counters was copied as a vague quadrilateral containing 15 elements.

Mar (4;6), in the case of figures whose shape depended on the number of elements successfully copied a cross of 5 counters, a square of 4 and even a triangle of 6, but failed with a square of 9, his square containing 15 elements. The figures in category III, circle, right angle etc., were copied, but with more elements.

The reaction of this first stage are of great importance to the psychology of numbers. It might at first seem that these children feel no need for a quantitative evaluation, and that they merely make a more or less correct copy of the model. Such an assumption would be too sweeping. Even if the child sometimes forgets, in making his copy, that he was told to ‘put the same number of counters,’ he shows by his answers to the question put to him that he understood perfectly what he had to do. Par, for instance, says, ‘there are more here’, and Li recognizes the differences and attempts to correct them. But the expressions ‘more counters’ and ‘less counters’ have a quite different meaning for the child of this stage than for us, and do not as yet represent a cardinal evaluation.

For the adult, cardinal evaluation always presuppose that units have been added or merged in order to form the whole obtained by correspondence with the set to be evaluated. For the child of this stage, on the contrary, the evaluation is based merely on the global qualities of the sets in question, these qualities being quantified by comparison as ‘more’ or ‘less’, without co-ordination of the comparisons. In other words, this experiment, like those in Chapter I and II, shows that the only quantification of which the child at this stage is capable takes place through the relationships ‘more’ or ‘less’. The child does not, however, co-ordinate the comparing his copy with the model. Thus Mul thinks that his series of pairs is equal to the model because it is the same length, and he disregards the distribution of the elements, while Par thinks there are more counters in his copy because they are closer together, and merely puts them further apart instead of removing some. In the case of the figure with matches, Mul thinks that a circle of twelve has the same number as one of six, because the diameter is the same.

This experiment, unlike the earlier ones, provides us with evidence that this failure to co-ordinate the quantitative relationships only makes its appearance when there are explicit judgments of comparison. The child’s initial procedure is to co-ordinate the perceived qualities, but this co-ordination has not yet become operational or logical. It is still purely intuitive, and consists only in the attempt to produce a general resemblance between the model and the copy. This is why, when the child tries to find the same number of counter as he has been shown, he contents himself with a rough reproduction of the configuration of the model. The copy is only approximate precisely because the child cannot analyse the figure, i.e., he cannot find the relationships which, when logically composed, constitute its qualities. In a word, it is general shape, i.e., the total surface covered accompanied by a vaguely similar structure, that the child of this level takes as the criterion for cardinal evaluation. When this general shape involves number, and is fairly familiar to the child, there is in addition one-one correspondence, but as a result of the qualitative similarity, not as its cause. When the general shape does not depend on number (types I-III) or is unfamiliar (type V), the child fails to make the correspondence.

To sum up, the most general characteristic of this stage is the irreversibility of the reactions. The evaluations of children at this level are purely perceptual, with the results that qualities that are not comparable are related and the synthesis can only be global. This means, in other wards, that the child’s intuitive judgment is not yet combined with operations through which the elements isolated by his analysis can be put together again. It is precisely because the judgments of the child at this stage lack mobility, that his thought is essentially irreversible.

Stage II: Intuitive Correspondence.

We shall apply the term qualitative to correspondence that is based only on the qualities of the corresponding elements, as for example, when the child puts angles in his copy when there are angles in the model, irrespective of whether he knows how many there are, or whether there is the same number as in the model. Numerical correspondence, on the other hand, will be that in which each elements is considered as a unit, irrespective of the qualities, e.g., n blue counters corresponding to n red counters, whatever their distribution. We shall call intuitive any correspondence that is entirely based on perception (or possibly on representative images0 and is consequently not preserved outside the actual field of perception (or of clear recollection). We shall say, on the other hand, that correspondence is operational when it is based on relationships of an intellectual nature, its distinctive characteristics therefore being the fact that it is preserved independently of actual perception, and it’s ‘reversibility’. Qualitative correspondence man therefore be either intuitive or operational, whereas numerical correspondence is essentially operational (except perhaps in the case of the first three or four numbers).

Having clearly defined our terms, we shall find that the second stage, that of intuitive qualitative correspondence, is merely a continuation of the first. As the copying of the models becomes more exact, it leads to one-one correspondence susceptible of greater precision. But by the very fact that this correspondence is based on perceptual comparison, it is not numerical from the start, in spite of appearances, but remains qualitative and intuitive. This can easily be proved by altering the configuration of the corresponding sets. As soon as this is done, the child no longer accepts the equivalence. This intuitive correspondence without lasting equivalence justifies our assertion that there is a second stage, which differs from the first in that there is systematic use of correspondence, and from the third in that the correspondence does not yet involve necessary equivalence. We must point out also that, while children at the first stage usually begin by putting a pile of counters on the table and then arrange them in imitate the model (adding or removing elements where they think necessary), children at the second stage usually begin by making the correspondence, taking the counters one by one to reproduce the various parts of the model.

First have some examples related to category I:

Ha (4;5) first looked carefully at the pile of 15 counters, then put down 16 elements one at time, copying the configuration of the model bit by bit, looking to see that he was making the correspondence (his one error being due to the fact that he counted one element twice): ‘Are they the same?-That one (Copy) is bigger. I’ll take some away (removing the extra counter).- Are they the same?-Yes.- Are you apart) Is there the same number of counters?-Yes…no (he added some counters to the model in order to imitate the new configuration of the copy)

We then distributed 13 counter in front of Ha, some in groups of 3 or 2, but making the configuration a recognizable whole. He was then asked to find as many matches as there were counters. He arranged 11 matches in groups of 3 and 2 so as to reproduce certain details of the model.- ‘Are there as many?- There (Counters), there aren’t many, here (matches), - there are a lot.- Make it so that there’s the same number in both. (Ha then spaced out the counters.)

Finally, 8 matches were distributed in front of him: ‘Find the same number of counters.-I don’t know how to make the pattern.- Never mind, try.’ He spaced out the matches a little, then took 14 counters one by one and tried to reproduce the pattern of the model. The correspondence thus broke down when the difference in shape of the objects prevented an exact copy of the model, whereas the copy was an exact reproduction of the model when counters only were used.   

Ba (4;9) put down his counters one by one, looking at each of the 15 in the model in turn: ‘ Are they the same? –Yes.- Are you sure?- Yes.- Show me how you know.- This one and that one, etc., (pointing to the correspondence elements). He also succeeded in making the correspondence between counters and matches, by imitating the configuration. But when the configuration was changed, he was no longer certain of the equivalence, even without the matches.

The next examples are of category II:

Min (5;0) succeeded at once in reproducing series of four or more pairs, but ceased to assume the equivalence when the counters were spaced out. When he had to make the correspondence between matches and counters, he could only make an approximation, e.g. 10 matches for 8 counters: ‘Are they the same?-Yes.- Count them.- (He counted correctly 8 and 10.)- Are they the same then? – Yes.’ There was thus no precise relationship between his counting and the quantitative evaluation. (When the terms to be compared are qualitatively equivalent, the evaluation is made by correspondence; when they are different, e.g., counters and matches, it is made by combination of correspondence and the global relationships of the space occupied.)

Gis (5;5) provides another examples of the distinction between verbal numerations and actual  operations. She counted correctly up to 27, pointing to each counter in the row in turn. After 27, there ceased to be any co-ordination between the counters to which she pointed and the number she said, but she counted from memory up to 54. This verbal enumeration did not however correspond to any systematic colligation except in the case of the first few numbers, because although she said that 12 is greater than 8 and 10 is greater than 7, she also said that 9 is greater then 13 and 19 is greater than 21: ‘where are there more (with the counters she had just counted in front of her)?- 19.- Why?- Because there are a lot.- And there?-21.-Well then?- That’s less because there aren’t many.

It is interesting to note that, on the plane of actual operations Gis was at the second stage: correct correspondence, bur without lasting equivalence. For example, she made the correspondence between four or more pairs correctly, but no longer thought they were equivalent when one set was spaced out.

We find the same reaction for categories III-V:

Nil (5;0) began by putting 2 extra elements in coping a cross of 9 counters, but corrected himself by pointing to the corresponding terms. He at once correctly reproduced the square of 9 counters, a house of 11 and a circle of 10. In coping the circle, he kept the same diameter. When he was asked if it was exactly the same, he pointed with his finger to the one-one correspondence. One counter was then put opposite each element of the model so that the one-one correspondence would produce a concentric circle larger in diameter: ‘Will there be enough counters to put opposite each one?- Yes.- Why?- They’re the same.’ But once the bigger circle was completed, he no longer though they were equivalent: ‘Is there the same number of counters?-No.- Why?- Because it’s bigger.’

Ba (4;9) Succeeded in reproducing, after one or two slight temporary errors, figures such as the circle of 9 counters, a right angle of 11, the square of 9, etc., and was also able to make the correspondence between matches and counters, in various combination, by always copying the perceived figure. For the rhombus of 13 counters, he put in the centre a row of 5 (correct) and below a triangle of 4 ( correct). But above, he put only 2 elements instead of 4: ‘Are there as many?- yes.- How do you know?- (He pointed to each element in the model and each corresponding counter in the copy, and when he got to the top exclaimed): I made a mistake, I did it wrong. ( He at once corrected himself)’

In spite of this, however,, Ba did not assume necessary equivalence when the distribution of one of the sets was changed, although he himself had just made the correspondence. For example, when a rectangle of twelve counters was turned so that the long side became the base, he no longer thought it was equivalent to the model.

These then are the reaction of the second stage. There is one-one correspondence, but it is always based on the particular properties of the figure, for without the figure, the child no longer thinks the two sets are equivalent.

Stage III: Operational Correspondence

During the third stage, the correspondence no longer depends on the intuitive figure, and the child begins to use spontaneous operations as a check, breaking up the figures himself and arranging them in series. The correspondence thus becomes either qualitatively or numerically operational. Here we have some examples of categories I and II:

Hen (5;0) was shown a collection of 11 counters: ‘Take the same number.-(he took 14, one at a time or in twos). Are they the same?- (He made the correspondence in order to find out.) No (he removed 3.)-And now (His 11 counters were distributed at random, and bore no resemblance to the model.)?-Yes.- (The elements of the model were then spread out.) And now?- Still the same.’

Cha (6;0), when he was shown a collection of 12 counters, took 11, one at a time, without making them into a figure, looking at the corresponding elements in the model, then of his own accord added one more. When the elements of the model were spread out, he continued to recognize the equivalence. Same reaction in the case of the series of 4 pairs: he at once took 8 counters and merely made the rows without copying the figure.

And now some examples of categories III-V:

Fav (5;6) had no difficulty in reproducing the figures in categories III and IV. He copied the figure, and also assumed the equivalence of the sets when the configuration was changed. For a figure of type V, he began by copying the model, and when counted aloud: ‘I MUST PUT 3 MORE’, ETC., THEN, GETTING CONFUSED, HE STOPPED BOTH COPYING THE FIGURE AND COUNTING AND ADPOTED A procedure very typical of this stage, i.e., he made an ‘arbitrary correspondence. He separated the elements of the model, arranged then 2 by 2 in two vertical rows, then did the same with the counters of his own set, but placed these horizontally. He then saw at once that there was one missing and added it.

Maw (6;0) also correctly reproduced complex figures like those of type V, but in order to check them, replied only on the correspondence: ‘Are they the same/-(he counted 12 and 13.) There’s one too many (removing 1, which was wrong as he had merely made a mistake in counting).-Then why is there an empty space here? He then behaved in the same way as Fav. He destroyed his own figure and put the counters in a row, then, with his finger made the correspondence with the counters in the model, which had remained in position. he then saw that there was one counter missing and added it.

Similarly, when he took 22 matches to correspond to 22 counters arranged in a complex figure, he counted to himself as he did so, ‘Are there as many?-Yes.- How many?- I don’t know (having forgotten the last cardinal number). –Then how do you know they’re the same?- Every time I put a match, I pointed to a counter.- And how do you know you didn’t make a mistake?- (He then arranged the counters in arrow and put a match opposite each one.)’

It is obvious that Fav and Maw believed in the equivalence of corresponding sets, since they themselves destroyed the figure in order to check their numerical equality.

No examples could indicate more clearly that it is only when the correspondence becomes operational the child feels certain of his counting. We can therefore distinguish a stage characterized by operational correspondence, with recognition of the equivalence of corresponding sets, and conservation of quantities. This stage is intermediary between mere intuitive correspondence and correspondence between the objects and the numerical used to count them. This last type of correspondence, which is characteristic of a fourth stage, in which practical correspondence is replaced by the ability to use numeration correctly, does not concern us here, since the object of this volume is the study of the genesis of number, and it is only when operations are logically established on the practical plane that counting becomes truly numerical.

 Single Rows

Before proceeding to discuss the reactions given above, it will be useful to analyse correspondence between single rows. This study will not be a repetition of that in Chapter III, since then we were concerned only with investigating why one-one correspondence, even between objects that are qualitatively complementary, does not entail the notion of necessary and lasting equivalence of corresponding sets . Here, on the contrary, our object will be to place correspondence among the various processes of cardinal evaluation, i.e., to study, making use of like objects, how the child makes two sets of equal value. Moreover, although in the preceding section we have already seen these processes at work, we feel that it is desirable to simplify the problem, in order to discover whether the same results are achieved when single rows are used instead of complex figures.

Stage I: Global Comparison

In the case of single rows, the reaction are very similar to those observed earlier. Instead of making the one-one correspondence, the child bases his evaluations on only or other of the two global qualities of the row, its length or the density of the elements, without co-ordinating the. Here we have some examples.

Don 4;1) had a sister, Myriam: Mummy is given Myriam all those pennies to go to the fair. Now you take the same number of pennies as Myriam.- (He took a handful at random (5), but arranged them so that his row was longer than the model.) It’s bigger, it’s not right!- Why? Is one of you richer, or have you both got the same?-Yes, I’m richer.- Well then, make it so that it’s right.- (He put them all back into the box, then took out 4 which he put close together, then 1 more which he put close to the others.) But this will be smaller (His row.) I’ll have to put some more. (He added 1 at each end, thus obtaining a row of 7, the same length as the model).- Are they the same like that, or will one of you be richer?- They’re exactly the same.’

Char (4;4) also began by making a compact row of 11 buttons to equal the 6 spaced out buttons of the model, then, as his row was longer than the other, he removed 3 from the end, thus obtaining the same length: ‘Are they the same? –Yes.- Exactly?-yes.- (The 6 elements of the model were then put further apart, and the 8 in his copy were put closer together).  And now?- There are more there (the 6).’

Boq (4;7); ‘Put as many sweets here as there are there. Those (6) are for Roger. You are to take as many as he has.- (He made a compact row of about ten, which was shorter then the model.)-Are they the same?- Not yet (adding some).- And now?- Yes.- Why?- Because they’re like that (indicating the length). –(The 6 in the model were then spread out.) Who has more?-Roger.-Why?-Because they go right up the there.- What can we do to make them the same?-Put some more (adding1).- (The 6 were then closed up and his were spread out.)- Now I’ve got more.’

Finally Boq was shown two rows of sweets, one containing 3 that were spaced out, and the other 4 that were close together, the first row thus being longer: ‘Where are there more?- There (the 3).- Why?- It’s bigger line.’

Arc (4;9): ‘Mummy is giving these sweets (8) to Luc. You’re going to have some too, exactly the same number.- (He put 13 in a compact row, taking great care to make the row the same length as the model.)- Have you got the same number?- Yes._ And how (Spacing Luc’s 8 rather further apart)?-No, Luc will have more.’

Ril (5;2) put 9 pennies to equal a row of 6: ‘There.- Are you as rich as Daniel, or have you got more?- We’ve both the same.- (His 9 were then put closer together and Daniel’s 6 were put further apart.) Who will be able to buy more?- Daniel.’

Ler (5;3) put 8 pennies close tighter opposite a row of 6, and then, when the 6 were spread further apart, he thought they were more than the 8 ‘because it’s bigger here.’

More rarely, the evaluation is based on the density of the elements, as in the following examples:

Don (4;1), just after the experiments quoted above, put 7 red counters close together under 6 blue ones: ‘Are they the same?- yes.- (The model row was then closed up and his copy spaced further apart.) – And now?- I’ve got more (7 red ones) Because it’s bigger. No, Myriam’s got  more (6 blue ones).- Why?- because they’re close together: there are a lot.’

Lin (5;3) also, although most of his estimates were based on the length of the row, said more than once that 6 elements close together were more then n elements spread out ‘because it’s bigger.’

Although the results of these estimates based on density is the converse of those based on the length of the row, the principle is obviously the same: the criterion of the evaluation is not the number of the elements or these one-one- correspondence, but perception of global quality.

This brings us back to the question raised in$1. What is the nature of this quantification prior to one-one correspondence, and why is one-one correspondence impossible at the level of the first stage?

With regard to the first of these, our conclusion are the same as before: elementary or ‘gross’ quantities are nothing else than the relationships expressing ‘more’, ‘equal to’ or ‘less’ immediately perceived in the given qualities, but not yet co-ordinated. The two qualities inherent in any row of objects, irrespective of the nature of the objects, are the total length and the density of the elements. Now it is impossible to compare any two rows without relating the qualities of one to those of the other, i.e., without one of the rows being seen as longer, shorter or the same length as the other, or as more compact, less compact or of the same density. It is therefore these relationship, which are as elementary as the qualities themselves, that are used by children as this level in their pre-cardinal evaluations. When Don, for instance, says ‘I’ve got more because it’s bigger’, and Ler thinks that six pennies spread out are more than eight close together ‘because it’s bigger,’ they are directly expressing the length of the rows in terms of quantity. If it were a question of comparing a larger number of rows, or of adding the relationship abstractly, new difficulties would doubtless appear sooner or later. (cf. Chapter V and VI), but as long as it is only a matter of immediately perceptible relationships, these two criteria suffice for an elementary evaluation. It is clear that the child is also capable of comparing two rows from the point of view of the intervals separating the elements. He recognizes that in one row the elements are more (or less) compact then in the other, and he can translate this perception into elementary quantitative relationships. For Don, for instance, ‘because they’re close together, there are a lot’, and for Lin, there are more when ‘it’s bigger’ (Chapter III, in which Mou, Lid and Fran used density as the criterion for estimating the larger quantity). Each of these relationships of total length and density thus constitutes a beginning of what will later be cardinal evaluation, both of them being, from the start, inseparable from the quantity.

For the adult, a row of n spaced-out elements keeps its cardinal value n if its length is diminished by closing up the elements. It is therefore the relation between the length of the row and the intervals between the elements which determines the conservation of the whole, whereas the relationships of length and density are variable. It is precisely this co-ordination, or logical composition of the two relationship in question, that the child of this level cannot make, and that is why the notion of conservation of the sets and even of one-one correspondence is still impossible. The elementary quantitative relationships are merely practical schemata which are still pre-logical because there is not as yet any operation properly so called.

Now why is it that one-one correspondence is not possible during the first stage? We must, of course, distinguish between intuitive and operational correspondence. Operational correspondence presuppose the intervention of special operations which we shall discuss later, but intuitive correspondence, when it is qualitatively correct, is the results of elementary multiplication of the relations, and we have seen that the relationship of total length and density are not susceptible of composition at this first level. It is obvious, for instance, that when Boq prefers three spaced-out sweets to four close together because ‘it’s a bigger line’, he is disregarding both the elements as such and the intervals between them, and that when Don, reasoning in a similar way, uses the other criterion and says ‘because they’re close together there are a lot’, he is disregarding the length of the rows and is therefore incapable of comparing the elements as such. It is only when two rows are of equal length and density that their equivalence implies their correspondence and goes beyond global evaluation. It can be seen, therefore, that correspondence is the expression of a construction in the true sense, even at the level of intuitive qualitative correspondence (i.e., before there is any question of numerical correspondence), the operation then being partially replaced by perception of the figures.

This construction can be analyze as follows. First, there must be a decomposition, so that composition may become possible. If the global relationships of length and density of a row are to be merged in a single whole, it must first be understood that the total length is formed by the sum of the intervals separating the elements, and that consequently, in a compact row the intervals will be shorter and  more numerous, while in a scattered row of the same length the intervals will be longer and less numerous. Secondly, when the two rows are compared, the construction on which the correspondence rests will presuppose a multiplicative composition of relationships: the two rows will correspond to one another spatially if they are at the same time equal in length and density, i.e., if each elements of the one can be placed opposite an element of the other.

In this connection, the hesitations of the most advanced children  of this stage, and the means they finally use to discover the correspondence, are most revealing. Having made their evaluation, like the other children, form the point of view of the length only, they begin to notice the density, hesitate for a moment between the two possible points of view, then, disconcerted by the contradiction involved, attempt to take account of both at once. They dictions involved, attempt to take account of both at once. They are thus led to make a spatial correspondence between both the elements and the intervals in the two sets. Here is an example:

Stu (5:11) was trying to make a row equal to one containing 6 sweets. She took8, one at a time, and put them in a row: ‘That’s more. – How can you tell?- You can see by the line.-(The 8 row were then closed up and the 6 spaced out.)- No, there are more there (6).- Why?-There (the 8) its smaller. They’re close together.- But are there less?-Yes.- ( She was then shown two rows, one of 6 sweets close together and the other of 4 spread out.) Where are there more?-There (6).- Why?- Because there’s more (=close together). – ( She was then shown two rows of 6, one compact, the other spread out.)And there? – There (spaced out), because it’s longer, there are more sweets.- (She was again shown two rows of which the shorter contained more elements.) And now?- There (the shorter one), because there are a lot.- (The original 6 were then put down again.) Put down the same number.- (Stu then made the correct correspondence and thus passed beyond the first stage.)

Thus for Stu, it was the discovery that a compact row of six elements may contain more than a longer row of four elements, that made her try to combine the relationships of length and density and consequently to decompose the sets, thereby discovering the one-one correspondence.

Stage II: Intuitive Correspondence, Without Lasting Equivalence

When children at the second stage are asked to pick out a number of elements equal to those in a model row of six, they react immediately (or almost immediately)by making an optical spatial correspondence with the model, but they no longer accept the equivalence of the two rows when the correspondence cannot actually be perceived.

Jon (4;5); ‘Take the same number as there are there (6 counters).- (He put 7 counters close together, and then made the correct correspondence.) – Are they the same?-Yes._ (His row was then spread out.)-Are they the same?-No.- Has one of us got more?-Me.- Make it so that you have the same number as I have.- (He closed his up.)- Are they the same?-Yes.-Why?- Because I pushed mine together.’ 

Pret (4;11) succeeded in making the correct correspondence after first putting one too many, but when the elements in the model were closed up, he said: ‘There are more there, because there’s a bigger line. It has to be the same line, then they’re the same.- What must we do then? – (He rearranged them so that the rows were the same length.’ (Same reactions in the case of the reverse operation.

Hab (5;3) began by putting 9 sweets opposite the 6 of the model, but made the row the same length. ‘That’s it.- Are they the same?-I’m not sure.- Where are there more?- there (row of 9, close together).- What must we do then?- (She put 6 opposite 6 of the model and removed the rest.)- (The 6 of the model were then closed up.) Are they the same?-No.- Are there as many here (model) as there?- No, there (copy) there are more.- Is there more to eat on one side than on the other, or are they both the same?- I shall have more to eat.- make them both the same, then.- (She removed 2, then made the one-one correspondence, and finally put the 2 back when she found they were missing!).’

Per (5;7) had no difficulty in making a row of 6 sweets corresponding to the model. The model was then closed up: I’ve got more.- Why?- Because it’s longer line.- (The process was reversed.)- Now there are more there, Because it’s a  big line.’ But a moment later, Per said the opposite: ‘Are there more here (spaced out)?-No-Why not?- Because it’s long.- And there (Closed up)?- there are more there, Because there’s a little bundle (=close together). – then are there more in a little bundle than in a big line? –Yes.’ After this Per went back to using length as the criterion, made the two rows the same length again, and said: ‘Now they’re both the same.’

Plus (5;7) also used correspondence in order to find a set equal to the model, but chose 3 spaced out sweets in preference to 4 close together, Because ‘there are more’. In order to make 6 closed up elements again equal to 6 that were spaced out, he merely spread out the first set.

These examples clearly confirm our hypothesis as to the origin of qualitative correspondence. Elementary though this form of correspondence is on the purely intuitive plane, it nevertheless constitutes a complex relation involving a system of comparisons, i.e. of logical multiplication of division. If, for instance, the child is to compare the sides or angels of two rhombuses, he must decompose the figures in order to abstract the similarities in the position of the sides alone and the angles alone. Such relationships obviously belong to a more advances level than the undifferentiated, globally perceived relationships of the previous stage. In the case of our rows, it is the co-ordination of the relationships of length and density of the rows that leads to correspondence. As long as the child judges the quantity by length or density alone, there is no possibility of correspondence. The children quoted above are now able to make a copy of the same length and density as the model, the twofold equality being ensured by the fact that each element of the copy is placed opposite an element of the model. Thus Hub (like Job, Pret, etc.,). Who begins, like the children of the first stage, by putting too many sweets under the model, says that she is ‘not sure’ of the equivalence, and then discovers the correspondence by spacing out her own sweets and thus co-ordinating the relationships od density with that of length. Moreover, when one of the rows is closed up or spaced out, and the child is asked to restore the equivalence in which he no longer believes the reaction at this stage is different. Children at the first stage usually merely add or remove elements in order to have the same length or density again, whereas Jon, for instance, closes up the elements and gives as his reason for thinking the sets are equivalent: ‘because I pushed mine together’. Pret, Per, Plus and hub alos restore the equivalence by remaking the optical correspondence, i.e. by co-ordinating the density and length of the rows.

And yet, although the child begins to make the one-one correspondence as soon as he can think simultaneously of the length and density of the rows, this correspondence does not at once lead to the notion of lasting equivalence of corresponding sets, nor to that of their quantitative constancy. The question then arises.’ How is it that the equivalence lasts only as long as the correspondence is perceived and ceases to exit when it is no longer perceived? Why is it that co-ordination of the two relationship does not lead to necessary lasting equivalence? In order to find the answer, we must distinguish two aspects of the matter: that of the general co-ordination, and that of the nature of the operations involved.

As regard the first, it is clear that, while there is here a continuous process of co-ordination, its stage indicate the existence of successive planes of structuration. First there are the elementary, global, perceptual relationships that are inherent in perception of the length and density of the rows. The, when these perceptual relationships begin to be co-ordinated by means of seriations and logical multiplications, the co-r=ordination takes place first of all on an intuitive and still perceptual plane, and is only semi-operational. This is precisely what happens during the present stage. The child certainly considers the relationship of length and density simultaneously, since he can produce a copy equal in length and density to the model, but the co-ordination does not go beyond the plane of perception, i.e. as soon as the perceived figure, which made the correspondence possible, is altered, not only does the correspondence vanish, but also the co-ordination between length and density.

Indeed, when one of the rows is contracted or extended, the child does not say: ‘it’s shorter, but closer together, so we can’t tell’, but it alone. Per provides an excellent illustration. Although this child could make a one-one correspondence between the model and his copy, he was at a loss as soon as the distribution of the elements in one row was changed. He then hesitated again between two relationships, saying first that the spaced-out elements were more numerous ‘because it’s longer line’ and then the opposite, because there was a ‘bundle’. This clearly proves  and although Per could co-ordinate the relationship on the perceptual plane, and thus make a one-one correspondence between the rows when the elements were visually in contact, he was not capable of linking them sufficiently closely for the co-ordination to go beyond the intuitive level and to constitute a system of true operations.

Even at this second level, however, there are signs of a beginning of operational co-ordination. For if, as it is the purpose of this book to prove, and operation is indeed a reversible action, it is clear that the reactions of Jon, Pret, Per, etc., who space out or close up the elements I n order to restore the equivalences, are forerunners of the construction of true operations.

Coming now to the second aspect, we find that the global relations ‘more or less long’, ‘more or less close together’, which characterize the rows themselves and not the detail of the relationships between the elements, correspond to the first level, that of unco-ordinated perceptual relationships. This co-ordination between these two kinds of relationships. The co-ordination between these tow kinds of relationships, which takes place during the second stage on the intuitive plane only, is both additive (seriation) and multiplicative (Correspondence), and thus corresponds exactly to the beginning of operations which we described in Chapter I and II in connection with the second stage of conservation of quantities. Indeed, on the one hand, the density of a row is nothing more than the succession (perceived or grasped) of the intervals separating the elements, and the sum of these lengths is identical with the total length of the row. Hence, to co-ordinate total length and density is merely to decompose the former into segments which define the latter, and this constitutes an additive seriation. On the other hand, to make the one-one correspondence between two rows is to construct two series whose elements are exactly opposite one another, i.e. to multiply the relations ‘situated at a certain distance along’ by the relations ‘situated above’. What the child discovers in making the perceptual one-one correspondence is the beginning of serration and of multiplication of the qualitative relations of position, but nothing more. When the density or total length of one of the rows is altered, he ceases to believe in the correspondence, because he cannot understand that the displacements compensate one another and that the differences can be equated, an understanding which requires a higher level than that of mere qualitative groupings. All that the child can do is to restore the equality of the two rows, but he does not as yet deduce from this empirical return to the initial position that operational reversibility is always possible.

Stage III: Operational Correspondence and Lasting Equivalence

At the third stage, on the contrary, we find correspondence freed from perceptual or spatial limitations, and persisting in spite of any displacement of the elements. One-one correspondence thus becomes really quantifying, and from now on, expresses numerical equality and not merely qualitative equivalence. Here are some examples.

Fet(5;5); ‘take the same number of pennies as there are there (6).’ He made a row of 6 under the model, but put his much closer together so that there was no spatial correspondence between the rows. Both ends of the model extended beyond those of the copy. ‘Have you got the same number?-yes.- Are you and that boy just as rich as one another?-yes.- (The pennies of the model were then closed up and his were spaced out.)- And now?- the same.- Exactly?-Yes.- Why are they the same?- Because you’ve put them closer together.’

Cran (5;8) was shown 6 beans spaced out, and began by putting one opposite each one of the model. His copy was then closed up; ‘Are they the same?- Yes.- The same amount to eat?-yes.- How do you know?- I can see.’ Yet when he was asked to evaluate quantities numerically, he was never sure of his answer above 6.

Lan (6;2),  to reproduce a row of 6 matches, picked up 4, without counting, but looking at the correspondence elements. Having reached this point, he put his finger on the fourth match of the model, took 2 more, then put his 6 matches in front of the model row, but in a heap, so that there was no spatial contact. We then spread out his matches in a row and made the others into a bundle: ‘Are they the same?-Of course.-Why?-Because before, those (his own) were in a bundle, and now you’re put them like that (spread out), and these (the model) were spread out before, and now you’ve made them into a bundle.

The contrast between these children and those of the preceding stage is obvious. In the first place, they do not rely on the perceptual contact between the elements, even when they make the one-one correspondence. Fet, for instance, at once made a series that was more compact than the model, and Lan put his six matches in a heap opposite the model row. What is more important, however, is that these children can relate the successive configuration of the corresponding sets by correctly co-ordinating their relationships. For instance, in order to prove that the pennies in the model are still equivalent to those in his copy, Fet uses the same argument that the younger children used to prove the contrary: ‘because you’ve put them closer together.’ This justification can have one meaning only; when the pennies are put closer together, if none are added or removed the total length decreases but the density increases. The child is therefore now capable of considering simultaneously the relationship of length and density, not only when the series to be compared are similar, but also (and this is the progress over the second stage) when they differ in both length and density. In other words, if we consider the total length and density of the rows as two distinct relationships, as the child does before he co-ordinates them, or when he determines the density by the length of the intervals between the elements, we can say that the third stage indicates the completion of the multiplication of these two relationships. Even at the second stage, there is already a beginning of this multiplication, but at that level the child either compares two corresponding rows correctly because they are equal in length and density respectively, or, when one of these qualities is the same in both, concludes correctly as the other. When the two qualities vary together, at the second stage the child’s estimate is wrong, while at the first stage, by multiplication of the two relationships, he grasps for the first time that a row which is both shorter and more compact than another can be equal to it.

It is by the inclusion of each perceptual situation in the system of all the possible perceptual situation that from now on multiplication of the relationships is independent of actual perception of the rows. There is therefore, in reality, a freeing from perception in general, since each perception of the configuration takes its place in a coherent system of transformation governed by the logic of relationships, each composition corresponding to a possible perception of sets.

It is freeing from perception that marks the beginning of operations properly so called, which are thus seen to be the results of the progressive reversibility of thought. The terms used by Lan to explain the relationships between the two corresponding sets might be translated as flows; heap becomes row=row becomes heap. In other words, the two sets are still equivalent because their transformations are only reversible changes of position, i.e. they are the results of operations that can be reversed.

We must, however, point out that logical multiplication of qualitative relations is not in itself sufficient to account for numerical correspondence, with lasting equivalence, just as it was not sufficient to lead the child to the notion of conservation of continuous and discontinuous quantities (Chapter I and II). In the case of correspondence, as in that of the relationships inherent in conservation of quantities, the transition from the qualitative to the arithmetical operations is explained by the equating of the differences, i.e. by the implicit or explicit introduction of the notion of unit.

The assumptions that a short, compact row corresponds term for term with a longer and less compact row, involves the understanding of two facts essential to the construction of numbers. Hitherto, the elements of the corresponding rows have been considered by the child as being determined by their intuitive spatial qualities; the first counter on the left in the upper row, the counter next to it on the right, and so on, but once the correspondence is accepted as lasting irrespective of these positions, the elements merely becomes units, each equivalent to the others, and the correspondence then rests only on the notion of equal units differing one from the other only by their relative order of enumeration. In other words, the correspondence is now the application of the same order of enumerations to two sets of equal units. From the point of view of the intervals between the units we find the same mechanism. Up to the third stage, the child accepted the one-one correspondence between two rows of elements only if the total lengths and also the density, i.e. the intervals between the elements, were both equal. From the third stage onwards, however, he recognizes that the correspondence persists when one of the rows is more compact, the difference in length being compensated by the difference in the intervals. The discovery of truly arithmetical correspondence thus always presupposes a new operation as compared with ore qualitative logic, this operation being the equating of the difference, or to put it more concretely, the seriation of units seen as equal in all respects except the temporary relative position of each one in the series.

Conclusion: 

 The facts contained in this chapter are remarkably in agreement one with another and with those of the previous chapter. They provide us with the general picture of the stage through which correspondence passes, and thereby enable us to attempt a general explanation of the successive methods of quantification.  We have seen, in the course of our analysis, that the problem has gradually becomes more and more precise. In Chapter I and II we saw that the child does not at first assume conservation of either continuous or discontinuous quantities when the perceptual configuration is changed. We then examined the question of whether one-one correspondence, in its more familiar forms (Correspondence between container and content, and one for one exchange) was sufficient to ensure conservation, i.e. necessary lasting equivalence, of corresponding sets. Chapter III led us to give a negative answer to this question. There is a level of perceptual correspondence which is characterized by the disappearance of the equivalence as soon as the corresponding  elements are no longer in contact. Reactions of this kind are a relic of the first stage during which the one-one correspondence is not grasped even when it is imposed by the external situation, the equivalence resulting from a global estimate of the space occupied. We found it necessary, therefore, to study the spontaneous processes of quantification used by the child in order to determine the cardinal value of sets, and we therefore devoted Chapter IV to the question of spontaneous correspondence.

The tests used in Chapter IV involved the children in an activity which was a direct continuation of their everyday experience, and this activity yielded a rich crop of spontaneous procedure following a regular order: global evaluation, correspondence without lasting equivalence, and numerical correspondence with lasting equivalence. Three question then arose. Why does the child not feel the need, from the beginning, to decompose the global totalities that he thinks he can evaluate?  How does the first form of decomposition or intuitive qualitative correspondence make it appearance? In what condition does operational qualitative correspondence become numerical correspondence? In $2 we outlined the answer to these problems in the case of rows, but we must now generalize it. In so doing, we shall distinguish between (I) psychological analysis, which is causal and genetic, and (II) analysis of the logical construction of operations though, as will eventually appear, the two kinds of interpretations are parallel.

I. All the preceding experiments have shown that at a first level (usually at about the age of 4 1/2 – 5), the child evaluations discontinuous quantities as if they were continuous, i.e., extended, quantitative. His quantitative judgment are thus based only on the general shape of the set and on global relationships such as ‘more or less long’, ‘more or less wide’, etc.

There are two possible explanations of this initial reaction. Either the child does not feel the need to decompose the wholes he can perceive and is trying to evaluate, or he is incapable of such decomposition. Psychologically, the need and the capacity are of course closely linked, but for the moment we shall assume that they are distinct. That the child should begin by considering non-analysed wholes, without feeling the need to decompose them as long as experience does not compel him to do so, is perfect consistent with what we know of the psychology of thought at this level.

Hence the child who is asked to pick out ‘as many’ counters as there are in given set is in no way equipped by his intellectual structure to consider the set as begin a union of units, i.e., 1+1+1.. etc., which would imply that he already possessed the notion of whole numbers. For the child, therefore, ‘as many’ merely means a set similar to the model with respect to its overall qualities. But if he does not feel the need for decomposition, does this mean that he is incapable of it. To your mind, the preceding reactions have given a decisive answer to this question.

The only principle of synthesis at the disposal of the child at the first level is the intuitive global perception of the general configuration itself, in the absence of ‘operations’ which would enable him to put together again the various parts of the perceptual intuition if it were broken up. That is why, as soon as there is an alteration in the data, children of the first stage apparently base their evaluations on one criterion only: length of the rows, width of the figures, density, etc.

But might it not be said that in the very action of copying there is co-ordination of the global qualities, and therefore at least a suggestion of decomposition, since the model is roughly reproduced? There precisely is the point: the copy is only roughly correct. Closed figures depending on a given number of elements are well reproduced, because they involve a ‘good’ configuration, but collection, rows, open figures, and even closed figures with an arbitrary number of elements, are not correctly copied. Linear series, in particular, are evaluated by their total length only, irrespective of their density.

Although the method of global comparison allows of a rough comparison between two sets of the same form, covering the same area, and having approximately the same density, it becomes inadequate as soon as these properties differ substantially. Tow sets which the child has stated to be identical by global comparison, are no longer considered equivalent when the elements of set are spaced out. He cannot understand that when there is a change in the shape, and therefore in the distribution of the parts, something remains invariant, namely, the number of elements. The reason is that he has not yet acquired the notion of number, but only the perceptual wholes. Hence, in our view, the notion of conservation of the set is lacking because the elementary relationships inherent in the global perception are merely juxtaposed instead of being co-ordinated.

The method of global comparison is therefore not only vague, but lacking in mobility, since there is no operational mechanism by which the various successive states can be lined into a dynamic whole or system of relationships. Hence the characteristic feature of this first stage is the almost complete irreversibility of though. It is true that each of the global qualitative relationships established by the child, e.g., ‘longer’, etc., is capable of giving rise to an inverse relationship, ‘less long’, etc., but as these relationships are not decomposable into units and are not co-ordinated one with the other, but merely put together in a non-structured whole, they cannot yet form a reversible system, and perceptual intuition therefore holds sway, operations being as yet impossible.

The second method, which is characteristic of the second stage, and which consists in comparison of figures by intuitive qualitative correspondence, indicates some progress, but this is less than might at first appear. The progress lies in the precision brought to the analysis of shapes and qualities, which results in deeper understanding of the intuitive data. At the previous level, the only details noticed are those necessary for the construction of the overall shapes: angles, ends of rows, etc., but from now on there are no privileged details. Each part of the whole is perceived and taken into account, and the child begins to co-ordinate the various criteria. It is because he is puzzled at finding that he arrives at different evaluation according as he stresses one or other of the criteria, that he eventually comes to co-ordinate them.

The child’s success in thus reproducing the various geometrical configuration of the set is achieved through the development of the schema which already existed within global comparison, but which comes to the fore at this level, i.e., intuitive qualitative correspondence. Psychologically, to make a correspondence is merely to systematize judgments of resemblance and difference. In fact, the child perceives the details of one figure only through their resemblance to, or difference from, those of the other to which he is comparing it. He therefore works out the correspondence between all the analogous parts, and no longer concentrates-only on certain outstanding details and the general configuration. This qualitative correspondence, or ‘comparison of the parts’ is the tool used by the child for the reproduction of all the figures whole shape depends on the number of elements, and although it is not always successful in other cases, it constitutes a new and much more efficient method of reproducing the sets.

Yet this second method, although more precise than the first, is not much more mobile. It enables the child to compare only certain privileged static states of the sets in question, i.e. those which give the figure their particular shape. Ba, for instance, was doubtful as to the equivalence of two sets of twelve counters arranged as similar rectangles, because in one, the longer side was horizontal, and in the other, vertical. Similarly, Nil, was not prepared to accept the equivalence of the counters in two concentric circles in spite of the fact that each element in the outer one was opposite an element in the inner one. While it is true that precise comparison of figures has more possibilities than global comparison, since there is an infinite variety of figures, from the point of view of co-ordination of relationships and conservation of quantity, there is only a slight process, since displacement of the elements, resulting in a change in the figure, is sufficient to make the child think that the total number of elements has changed.

There is, indeed, a beginning of co-ordination of the relationship of length, width, density, etc., but it takes place only on the purely practical or intuitive plane. The tool used for the co-ordination of these relationships is not yet the operation as such, but the configuration itself. In order to make his own figure correspond to the model, the child must of course take into account simultaneously dimensions, density, shape. Etc., As compared with the first stage, the progress lies in the fact that in the actual construction of the figure these relationships are co-ordinated, but as soon as the figure is changed, the child is incapable of abstract operational co-ordination of the relationships involved, and has recourse to the only available principle of unification: in figure he perceives. This explains why the child at this stage ceases to believe in the constancy or correspondence of the sets when the evaluation on one criterion only, forgetting the others.

To sum up, the second method is merely a continuation of the first. It is more precise, slightly more mobile, but still restricted to sensory intuition, and not yet capable of truly logical and operational dissociations and compositions. It can be described as semi-operational, since on the practical plane it allows of qualitative correspondence, which implies intuitive co-ordination of relationships. With this semi-operational charter goes progress in reversibility of thought, reversibility being the psychological expression of the operation. This can be seen from the fact that although children at this level think that the number of elements changes with a change in the configuration, they nevertheless assume that a return to the original figure is possible. Thus in order to restore the equality between two rows, they merely space out or close up the elements of one of them, and do not add or remove any elements. But obviously this reversibility is incomplete and the relationships still do not constitute a reversible system, in other words, they are not yet independent of perception.

The third method, on the other hand, represents a definite progress: correspondence leads to necessary and lasting equivalence. The sets are now assumed to be equivalent, whatever the configuration or the distribution of the elements. The progress takes place gradually as a continuous process, the correspondence depending less and less on the figure itself, i.e., on perceptual intuition. As soon as qualitative correspondence is to any extent independent of the precise form of the respective parts of the two figures, the elements become interchangeable units, and the correspondence thus acquires a numerical character. But this freedom presupposes lasting co-ordination between the relationships, with out recourse to the figure, i.e., correlation of successive perceptions hitherto assumed to be irreducible.

The fundamental factor of this development is, in our view, the complete reversibility of the action involved in the child’s procedure. The operation he performs is no longer immediately absorbed in the intuitive results obtained. It frees itself, as it were, and becomes capable of moving in reverse. Each transformation can be compensated by it inverse, so that any arrangement many give rise to any other, and conversely. Thus instead of relying on an all-important figure, the child proceeds exclusively by reference to the one-one correspondence, and thereby succeeds for the first time in decomposing the wholes and co-ordinating the relationships. From now on, therefore, his actions constitute a reversible system involving constancy of the set. This system serves as the principle for generalization of qualitative correspondence or logical co-ordinations of relationships into numerical correspondence, in which each element, irrespective of its qualities, is considered as a unit equal to the others and differing from them only by its temporary position in the series.

II. We shall now consider, in its broad lines, the logical structuration of the child’s judgments, as it develops, by a series of transformations, from the elementary, unanalyzed relationship to general one-one correspondence.

The logical construction of correspondence which takes place during the three stages analysed in Chapter III and IV finds a natural place within the framework of quantification which we described in relation to conservation of continuous and discontinuous quantities. Global evaluation corresponds to ‘gross quantity’, qualitative correspondence to ‘intensive quantity’, and numerical correspondence to ‘extensive quantity’, it being understood that the logical multiplications which give rise to qualitative correspondence merely begin at the second stage and only on the intuitive plane, and that they are generalized during the third stage when they combine with those that give rise to numerical correspondence.

At the level of global quantification, the child is incapable of multiplying and relationships logically. He is also incapable of decomposing and re-combining them. Since therefore, they can neither be added nor multiplied logically, they are not as yet relations. They merely express qualities perceived in the comparisons made, as ‘more’, ‘less’, or ‘equal’, and are therefore what we called in our analysis in Chapter I and II ‘gross quantities’ or relationships between ‘gross qualities’.

The transformation of these global relationships into relations proper, and the construction of qualitative correspondence, are characterized by the following operations (semi-operational or intuitive at the second stage, and abstract at the third); additive seriation, and multiplication of additive series, this multiplication constituting the correspondence.

In the present case, this additive seritaion and multiplication of relationships are the complementary operations of decomposition into parts and reproductive composition, as we should in $2 in the case of linear series. Additive serition of relationships here consists in establishing that the total length of a row is formed by the sum of the intervals separating one element from the next,  while multiplication of relationships, which makes qualitative correspondence possible, consists in the construction of a new row having exactly the same length and the same intervals as the first.

In terms of classes rather then relationships, the correspondence of two sets will consist in the distribution of the second set in such as way that the elements of two sets are similarly placed, and that they can also be paired to form similar sub-sets which are also similarly placed.

When two rows of elements are in qualitative correspondence, the second series has the same total length and the same intervals as the first. From logical multiplication of the relationships it follows that in the case of

1. Equal length and equal density, the sets are the same;

2. Greater length and greater density, or smaller length and smaller density, one set has more, or less, elements than the other.

3. Equal length and greater density, or equal length and smaller density, one row contains more, or less, elements than the other;

4. Inverse lengths densities in (3), the conclusion is the same as in (3);

5. Greater length and smaller density, or greater density and smaller length, the number of elements in one series is larger and smaller than in the other, and therefore equal.

The child at the second stage has no difficulty in grasping intuitively the first four of these compositions. The first merely expresses qualitative correspondence; the second states that if a row is both longer and denser than  another it contains more elements; the third states that equal length and greater density implies more elements; the forth states that equal density and greater length also implies more elements. None of these multiplications requires an abstract operation, since perception is sufficient. The child at the second stage is capable, however, of grasping the fifth composition, although it is implied by the earlier ones: if a row is both shorter and denser than another, it may contain more, less, or as many elements. It might be suggested that it is the uncertainty which is the obstacle to understanding, but the relation (5) could be expresses as follows, in which case there is no longer uncertainty:

(6a) Equal number multiplied by greater length implies smaller density, and equal number multiplied by smaller length implies greater density, or

(6b) Equal number multiplied by greater density implies smaller length, and equal number multiplied by smaller density implies greater length.

How is it then, that the child at the second stage who has just established the equivalence by one-one correspondence, and who understands the compositions (1) to (4), does not grasp the multiplications of (6a) and (6b)?

At the second stage, the child is still on the intuitive plane. He sees that the length and density of the rows are variable and he is perfectly coherent from the point of view of perception in assuming that this variation involves variation in the number, and this prevents him from attempting composition (6). Where he ceases to be logical, however, is in failing to grasp that in a contracted series the decrease in length carries with it increase in density. Instead of deducing that the result of composition (5) in uncertain, he then dissociates length and density and makes the mistake of assuming that the number of elements depends on one of the two.

How then does the child at the third stage succeed in overcoming this difficulty and discovering the constancy of number? Here there is a distinction to be made. On the one hand, if the comparison of two sets is merely a matter of ‘more or less numerous’ (e.g. all the blue counters that correspond to the red counters), it is obvious that generalization of logical multiplications will suffice to establish the constancy of the sets, because of the operational reversibility of qualitative correspondence. The correspondence can be made and unmade, i.e. the two sets, which are two series of relationships, can be multiplied by another relationships which links them together and which can then be disregarded. Through this relationships, the two sets become, and remain equivalent, which constitutes a certain constancy. When, for instance, a set of blue counters has been placed term for term below a set of red counters, the two series remain linked by the relationship ‘below’ or ‘above’, which ensures their equivalence from this particular point of view.

But this constancy of classes or series of relations is not as yet constancy of number, and the equivalence to which we have just referred is not numerical equality. For example, there will be qualitative correspondence between three blue counters forming a triangle and three red counters also arranged in a triangle. If the red counters are then arranged in a row, the possibility of arranging them in a triangle still remains, and from that point of view, and that only, they are still equivalent to the blue ones. This equivalence is therefore of a special kind, depending on the same configuration. This is understood by the child in the third stage when, like Lan, he says that two sets which formerly corresponded are still ‘the same’ because the heap can be made into a row as easily as the row can be made into a heap. A further step involves the assumption that the three counters in a row still correspond to the three in the triangle, and to any other set of three counters, whatever their distribution. This also is grasped by children at the third stage, since they frequently succeed in making the correspondence without troubling about the shape of the sets.

1. This brings us to the third stage in the logical construction of correspondence, that of numerical correspondence proper, which is parallel to what we called ‘extensive quantities’ in Chapter I and II and reaches its apex at the same time as the discovery of constancy of classes and series of relations takes place.

The explanation of the transition from qualitative to numerical correspondence can perhaps best be given in terms of classes. Let us again consider the case of the three blue counters in a triangle with a corresponding set of three red counters also in triangle. Obviously any one of the counters can be used to start the correspondence, but one it is in the figure, its position is defined absolutely, and the correspondence is defined by the spatial properties of the counter chosen. Order in no way intervenes in the definition of the total classes formed by the corresponding elements. Such is the qualitative correspondence used by the child (and also frequently by adults). If, instead of making a triangle with red counters, the child puts them in a heap or a row, or distributes them at random, the one-one correspondence he makes is of a new type, since each counter on longer carries with it qualities that distinguish it from the others, but becomes a unit equal to the others. Each red counter can then correspond to any one of the blue counters, but to one only. Hence the set of red counters acquires its quality of number three, and is no longer the class of counters forming a triangle, just as a pair od=f counters acquires the number two and is no longer the class of counters forming the ends of one side of the triangle. Moreover, an union of pair counters will produce the same class containing two elements irrespective of their qualities.

In this numerical correspondence, how are the elements distinguished one from another? It os merely by the other in which they appear in the correspondence, an order which is relative, and varies from one operation to another, and which we shall therefore call ‘vicariant’. For instance, in order to find the number of red counters corresponding to the blue ones in a complex figure, one child will point to each blue counter in turn, in any order (provided he does not count any element twice and does not omit any) and each time he  will add red counter to his row; another child will make a pile of red counters, and third will merely take one each time from his reserve pile and put them aside at random, and so on. The only order common at all is in the act of pointing, but this order is essential if the correspondence is to be correct.

The correspondence therefore ceases to be qualitative and becomes numerical when the elements are grasped as being equivalent in all aspects, and when the differential properties that distinguished them within the set are replaced by only difference compatible with their equality, i.e. their relative position in the order of the correspondence. Once gain, therefore, we find that it is the equating of difference that is the origin of the unit, and consequently of number.

We find the same mechanism of arithmetization when we examine the evolution of relationships. Children at the third stage accepts the fact that there is equivalence in spite of the distortion of the one of the corresponding rows, without feeling the necessity for a return to the initial state. This means that they replace qualitative correspondence by mathematical correspondence, and that they understand that the decrease in length is exactly compensated by the increase in density. The multiplication of qualitative relationship is thus from now on completed by a higher operation, that of the equating of differences. This being so, the length of the row and the size of the intervals lose their significance, and it is only the constant relationship between the two which counts for the child. Each interval becomes a unit equivalent to the others and the relation that represents the shift caused by the interval is then equivalent to ‘plus one’.

It is clear therefore, that whether it be the elements as such or the relations that are being considered, in order that there shall be the transition from qualitative to numerical correspondence, a process of reasoning that goes beyond mere qualitative logic is necessary. It is the construction of units which are at the same time equal to one another and susceptible of seriation, and this construction takes place through the equating of differences. A class is, in fact, a union of terms seen as equivalent irrespective of their differences. For instance, all the counters of a series from a class to which those at the ends belongs on the same terms as the others; red and blue counters are counters irrespective of their color, i.e. tow classes are united to form a third by disregarding their difference, in this case the color. An asymmetrical relation, on the other hand, is the expression of a difference, e.g. when counter B is on the right of A and a certain distance away, A. and B are conceived as different. To unite two asymmetrical relationships in one is to add the differences. Symmetrical relationships are those which units the elements of the same class, without thereby adding anything. None of these qualitative compositions, therefore, would make it possible  to define units properly so called. The un ion of two classes in a third does not constitute tow units, because their union results from their common qualities, and in the new class their difference are disregarded. The union of two asymmetrical relationships also does not produce two units, since the partial differences which are composed are not necessarily equivalent. The construction of number, however, consists in the equating of difference, i.e. in uniting in a single operation the class and the asymmetrical relationship. The elements in question are then both equivalent to one another, thus participating of the class, and different from one another by their position in the enumeration, thus participating of the asymmetrical relationships. Moreover, since these differences are difference only of order, they are all equivalent one to the other. It therefore follows that the criterion for the transition from an arbitrary qualitative series to the corresponding number series is that each elementary relationship shall be considered as equivalent to the others.