The Early Growth of Logic in the Child

Classification and Seriation

Jean Piaget, Inhelder, Brbel & Piaget, Jean, Barbel Inhelder

• Publisher: Routledge
• Year: 2013
• Print ISBN:
  9780415210010
• eBook ISBN:
  9781136317293
• Topic: Medicine
• Subtopic: Health Care Delivery

This is Volume XXI of thirty-two in a series of Developmental Psychology. Originally published in 1946, it looks the classification and seriation and the growth of logic in young children and this study incorporates the results obtained from the examination of a total of 2,159 children.

Chapter Six
Multiplicative Classification (Matrices)¹

SO far this study has been confined to additive classification. We will now turn to double and triple multiplicative classifications, these being frequently represented by matrices, i.e. two or more entry tables.

As well as having a more complex logical structure, these pose an interesting psychological problem: while additive classification improves as the subject develops beyond graphic collections (in the sense of ch. I), multiplicative classification lends itself very readily to a certain form of spatial representation. No doubt such an arrangement eventually becomes purely symbolic, but to begin with it could easily be the equivalent of a graphic collection which happens to show an intrinsic correspondence with the relevant logical structure.

The situation is somewhat paradoxical. From a logical point of view multiplicative classifications are more complicated than additive ones. On the other hand, they are supported by a graphic representation which is very much in tune with the kind of thinking found in younger children (graphic collections). Now multiplicative and additive classifications are mastered at the same time, about the age of 7-8 years. The question we have to decide is whether the graphic factor compensates for the added logical complexity, or whether the fact that both systems emerge at the same time is due mainly to the interdependence of the two kinds of operation. In the latter event, we would regard the graphic properties of multiplicative structures as subordinate: initially, and up to a point, they help to make the classification possible, but the help soon becomes more apparent than real, and in time the representation becomes nothing more than a form of symbolism.

1. Statement of the Problem

Suppose that we have a set of elements (e.g. red and blue squares and circles) which can be divided into two classes A₁ and A₁ according to one criterion (e.g. A₁ = squares and A₁ = circles), as well as into two different classes A₂ and A₂ on the basis of a second criterion (e.g. A₂ = red elements and A₂ blue = elements). We may use the term B₁ to denote the union of the first two classes, i.e. B₁ = A₁ + A₁ and B₂ for the union of the second two classes, i.e. B₂ = A₂ - A₂ Multiplicative classification consists of classing each element simultaneously in terms of the two additive orders B₁ and B_.2 Such a classification yields four sub-classes:

If we wish to divide the elements into these four classes in such a way that the elements belonging to any one of the original sub-classes (e.g. A₁ are next to each other, then the only possible spatial form is that of a two-by-two matrix.

In the diagram given here, A₁ and A₁ correspond to the two vertical columns and A₂ and A₂ the two horizontal rows. Of course, multiplicative classification does not have to be presented spatially, and could be described in a purely abstract way. But multiplicative inclusions can be symbolized by matrices, and only by matrices, just as class-inclusion can be symbolized topologically by Euler’s circles.

It is easy to check the two statements made at the beginning of this chapter: that a structure of this kind is more complicated than one of additive classification, but corresponds to a spatial configuration which subjects at stage I can interpret as a “graphic collection”.

As far as the first statement is concerned, we recall that there are 10 criteria for additive classification (ch. II, §1), that all of them are observed by stage III, and that, with the single exception of class-inclusion (criterion 7), all of them begin to be applied during stage II. Now every one of these criteria applies equally to multiplicative classification (since it is a composite out of two or more additive classifications). But two new criteria have to be added. These, together with their consequences, will be numbered 11 to 14.

(11) All the elements of B x belong to B² as well, and vice versa. Thus all the elements of B² are multiplied by B². If there were elements of B¹ not belonging to B² (e.g. if there were black squares and circles as well as red and blue ones), then a new class B² (the black elements) would have to be added to complete the classification, and there would be six sub-classes.

(12) All the elements of A¹ must also belong either to A² or A² (but not to both, because A² X A² = 0). Similar statements may be made for the classes A¹ A² and A².

(13) A¹ and A² contain only elements belonging either to A² or to A². Similarly, A² and A contain only elements belonging to A¹ and A¹.

(14) Each of the basic associations A¹ A₂, A₁ A₂, etc. constitutes one, and only one, multiplicative class.

On the other hand, it is obvious that a matrix is the sort of spatial configuration that makes a special appeal to perception by virtue of symmetry. If A¹ and A¹ are squares and circles, and A¹ and A² are red and blue elements, then the squares in A¹ A₂ are balanced by those in A¹ A₂, while the red elements in A¹ A₂ balance those in A₁ A₂, etc. The symmetry is twofold, corresponding as it does to the horizontal and vertical axes of the diagram, but it also corresponds to the two kinds of complementarity (by negation) in the logical structure.

This perceptual factor is so important that it facilitates, and even produces, solutions which appear to be operational but which are based on the methods of “graphic collections”. This is true of the sort of test usually referred to as a matrix test, e.g. Raven’s “Progressive Matrices”. In tests such as these, the subject is given a multiplicative table with all the spaces but one already filled, and asked to complete it by filling in the last space. (A2 x 2 matrix corresponds in our notation to the multiplication: B₁ x B₂; a 3 x 2 matrix is simply an extended matrix: B₁ X C₂.) Using our terminology, if A₁ A₂, A₁ A₂ and A₁ A₂ are given, the subject must find A₁ A₂. This means that the first ten criteria have been met in advance by the experimenter, while even criteria 11-13 have been met in part. Thus the three elements given are already classified simultaneously in B₁ and B₂ the two elements of A₁ already belong to A₂ or A₂; the given element of A₁ belongs to A₂, and all that remains is to find an element of A₁ which belongs to A₂; the sub-class A₁ contains only elements of A₂ and A₂, and the sub-class A₂ contains only elements of A₁ and A₁. In short, for the elements already given, the conditions of operational multiplicative classification are met by the perceptual configuration of the matrix. To find the fourth element, the subject need only extend these graphic properties by following the vertical and horizontal symmetries in the matrix arrangement.

Thus, the conditions for operational multiplicative classification are already contained in the spatial lay-out of matrices. This means that such problems can be solved without one single logical operation, by following through the similarities and differences which are thrown into relief by the twofold symmetry of these diagrams.

What makes a psychological analysis so complicated is the fact that the subject can also complete these graphic structures by using more or less operational relations, i.e. pre-logical and logical relations, which arise as he progresses from stage I to stage III. It is very difficult to separate operational and perceptual factors, and their relative importance is likely to vary with the particular situation involved. We know that a child is dealing with his problem at an operational level, and not on the level of graphic collections, if he is reasoning in terms of classes and not in perceptual terms, and this in turn is simply a matter of attributing similarities and differences to the elements as such, without considering their spatial position. But it is very difficult to know when he is doing this and when he is not. One obvious solution is to ask subjects to construct their own classifications, and this we have done. Nevertheless, here again the subject may use multiplicative operations, or graphic collections, or intermediate methods.

However, although interpretation is bound to be difficult, the problem we have to solve is simple enough. We have to decide among the following three hypotheses:

(1) Operational structures are not developed out of graphic structures. This would imply that multiplicative operations appear quite independently of spatial configuration. We might still find occasions when the spatial configuration triggers off the operational insight, others where it blocks it for a time, and some where it renders the operations superfluous.

(2) Operational structures are foreshadowed in spatial configurations, and are directly derived from activity related to them.

(3) There is a phase in the development of operational classification, whether multiplicative or additive, when graphic collections play a predominant part. Nevertheless, their final form owes a great deal more to the co-ordination of such inadequate data as these may yield. The coordination is a matter of assimilating, and structuring, and generalizing whatever experience is relevant to classification as a whole. This would mean that we can expect a gradual progress in multiplicative classification exactly parallel to that in additive classification.

Hypothesis (1) implies a sharp discontinuity between the initial and final stages; (2) implies complete continuity, and (3) implies a relative discontinuity since the effects of spatial arrangements would be gradually being replaced by those of operational logical coherence. Comparing subjects’ reactions to matrix tests with their spontaneous multiplicative classifications, hypothesis (1) would predict discontinuity in both situations, hypothesis (2) predi...

Table of contents

• Cover
• Title
• Copyright
• CONTENTS
• PREFACE
• TRANSLATOR’S INTRODUCTION
• INTRODUCTION
• I. GRAPHIC COLLECTIONS
• II. NON-GRAPHIC COLLECTIONS
• III. “ALL” AND “SOME”: CONDITIONS OF CLASS-INCLUSION
• IV. CLASS INCLUSION AND HIERARCHICAL CLASSIFICATIONS
• V. COMPLEMENTARY CLASSES
• VI. MULTIPLICATIVE CLASSIFICATION (MATRICES)
• VII. FLEXIBILITY IN HINDSIGHT AND FORESIGHT
• VIII. THE CLASSIFICATION OF ELEMENTS PERCEIVED BY TOUCH
• IX. SERIATION
• X. MULTIPLE SERIATION
• CONCLUSIONS
• INDEX