Jean Piaget, Children and the Class-Inclusion Problem

Dolph Kohnstamm

• Publisher: Routledge
• Year: 2021
• ISBN:
  9781000364958
• Edition: 1
• Topic: Éducation
• Subtopic: Psychologie de l'éducation

The Classic Edition of Dolph Kohnstamm's Jean Piaget, Children and the Class-Inclusion Problem, first published in 1967, includes a new introduction by the author, describing for readers the original context for his work, how the field has moved forward and the ongoing relevance of this volume.

This enduring text offers a critical study of a cornerstone of Piaget's theory that a child's ability to solve problems of class-inclusion marks the beginning of the period of concrete (logical) operations at about 7 or 8 years of age. Kohnstamm's experiments show, however, that, with a teaching method that provokes children's authentic logical thinking processes, most children of 5 can already learn to solve a variety of class-inclusion problems, up to a level where they can even invent similar but new problems themselves. These results question the basic assumption of Piaget's theory that logical operations can only develop in firmly connected groupings of operations. Kohnstamm argues that experimenters must, therefore, show that children who come to master one kind of operation should also show transference to other operations of the same grouping. This insightful volume questions the real existence in brain functioning of Piaget's families of logical operations. No experimental proof of such families has ever been demonstrated, and thus is solely an assumption in Piaget's theory.

This challenge to Piaget's theory is an invaluable resource for students and scholars of cognitive, developmental and educational psychology.

1
The Class Inclusion Problem in Piaget’s Theory

The problem central to this study was for the first time put before children systematically by Piaget and Szeminska during their study of the development of the child’s conception of number (Piaget and Szeminska, 1941).

The seventh chapter in that book consists of a report of an investigation in which children of five to eight years of age were confronted with a box of wooden beads, most of which were brown and only two white. The key question posed to the children was: “Are there more wooden beads or more brown ones?”

With this problem the authors wished to study the development of the additive composition of classes because they thought there was a close connection between certain aspects of the development of number, notably the development of conservation of number and quantity and certain aspects of the development of class, namely insight into the quantitative relations between classes within a hierarchy of containing and contained classes. As is well known, Piaget was always keenly interested in the connection between the development of number and class concepts in children, it being his epistemological conviction that both develop in close connection with one another in the human mind. It is for this reason that much attention is also paid to certain aspects of the development of class concepts in his book on the development of number.

Now, why was the problem calling for a quantitative comparison of two classes, one of which being a subclass of the other, of interest for the study of problems associated with the study of number? Quantitative comparison may take several forms, but what made this particular form of interest for Piaget was the fact that here a comparison of a whole and a part of the whole was required, of which most children seemed to be incapable before seven to eight years of age. The answer they gave to the question: “More wooden beads or more brown ones?” when confronted with the collection of wooden beads was a firm, “More brown beads.” An analysis of the incorrect answers revealed that the children did not regard the larger part as more numerous than the whole, but that they simply did not see the whole any more after having paid attention to the parts. Consequently, they took the comparison between part and whole asked for by the question to mean a comparison of both parts.

Naturally, Piaget saw a connection between the childish reactions to this problem and the problems of conservation and quantity that had been discussed earlier in the same book.

Here, too, the great difficulty appeared to lie in the mental retention of the whole whilst operating with the parts, just as it had been difficult for these children to understand the possibility of a certain quantitative whole remaining the same despite changes in spatial composition. But apart from this conservation analogy, the nature of which I shall consider further presently, Piaget perceived other developmental points of resemblance between the additive composition of classes and the additive composition of numbers. I shall not go into those similarities here but refer the reader to the above-mentioned book by Piaget and Szeminska.

In this section I shall confine myself to a consideration of the inclusion problem as such, thus independent of any connection with problems of number development. This is quite permissible, since the inclusion problem has received independent treatment in Piaget’s later publications to an increasing extent, especially in the study of the early growth of the logic of classes, and, therefore, ceased to be inextricably bound up with his study of the development of number.

Nevertheless, I must pause a moment by the above-mentioned chapter from The Child’s Conception of Number. Neither Piaget’s psychological analysis of the inclusion problem nor his theoretical interpretation of it have undergone much change since that chapter was written. I can, therefore, do no better than study his analysis and interpretation at the source.

Several variations of the problem were examined in the next series of experiments. One of the next variations was to ask the child which string of beads would be longer: the one threaded from wooden beads or the one threaded from brown beads. And as an introduction to this bead problem, the child’s attention was drawn to the difference between whole and part in the following manner:

Two empty boxes were placed alongside the box of beads and the child was asked: “If I take the brown beads out of the (full) box and put them in this one (one of any beads left in the (full) box?” and: “If I take away the wooden beads, will there be any beads left over then?”

The authors reported that the comprehension of this latter problem did not necessarily cause insight into the problem with the two strings of beads—a fact confirmed in subsequent studies.

Moreover, a variety of materials were already employed during this first study. The whole could also be formed by blue beads, most of which were square and only 2 or 3 were round. Or else it consisted of a collection of flowers, consisting of 20 poppies and 2 or 3 cornflowers. (“Which bunch of flowers will be bigger, that consisting of all flowers or that of all the poppies?”) These variations did not bring about any change in the results.

The authors described three stages observable in their subjects. In order to facilitate that description, they introduced a system of notation that has also been applied throughout this book; the whole class is indicated by the capital letter B, and the larger of the two subclasses with the letter A. The smaller of the two subclasses is the complement of A, namely A′. The authors wrote:

During the first stage the child remains incapable of understanding that the B classes will always contain more elements than the A classes, and the reason for this lies in the fact that, psychologically, the child is unable to think of the whole B and the parts A and A′ simultaneously. Logically this amounts to saying that the child does not yet conceive the class B as the outcome of the addition B = A + A′ and the class A as resulting from the subtraction A = B – A′. (p. 134)¹

I draw attention to the difference made here between the “psychological” and the “logical” point of view because these different descriptions play an important role in this book. And since, as far as I know, this quotation is the first occasion in the Piaget literature concerning the inclusion problem upon which these two different descriptions are set against one another, I may do well to link up my first comments to it.

I understand and support the “psychological” explanation, although, as I shall substantiate below, it is not the complete explanation. But the real difficulties begin with the transition to the “logical” description. For whose logic is meant? That of the child or that of the scientific observer? Of course, the authors do not mean to say that the child will ultimately think in terms of the symbols A, A′, “is equal to,” etc. But the second possibility is that they might well mean that the child will finally understand that what he or she perceives as wooden beads (for us, B) is the consequence of a mental combination of brown and white beads and, similarly, that he or she comes to see the brown beads as the consequence of the mental subtraction of the white beads from the wooden beads. A third possibility, namely that the formulas B = A + A′ and A = B – A′ are intended by the authors purely and simply as a (logical) model of the mental event without implying that they are giving a description of it, must be rejected in the light of the quotation concerned (“does not yet conceive”). The second possibility is, therefore, the most probable. The child does not yet regard the whole as a conjunction of the two parts, nor yet either one of the parts as the result of the subtraction of the other part from the whole. But is that not then also a psychological explanation, just as the explanation that the child cannot yet simultaneously think about the whole and the parts is? On the one hand, the quotation tells me that the one—the psychological—really amounts to the other—the logical—which would mean that they could not both be psychologically true and complementary. On the other hand, the logical explanation is worded in such a way as to clearly attribute lack of comprehension of those logical activities—later termed “operations” by Piaget—to the child him- or herself.

Yet, the difference between psychological and logical is not caused by the fact that an abreviated written form closely associated with a certain kind of logic is conceivable for the latter and not for the former. For we can, for instance, imagine a mode of notation for the former as B → A → B → A′ → B, thereby indicating that the child is able to direct his attention to the whole and the parts without the whole disintegrating and disappearing from the field of attention because of the shift of attention to the parts.² The latter could then be written as B → A → A′ → A → A′ etc.

Is it then because Piaget does not include ability to think simultaneously of the whole and parts without the whole thereby ceasing to exist—thought being directed to the parts only—under “logical” thinking? Whereas he does include insight into the whole as sum of the parts, combined with seeing A as the result of B − A′?

This is a possible answer.

I look out into a large car park.³ A large number of the cars is white. If I direct my attention to the white cars, I do not forget that they form part of all the cars parked there. I do not experience that not forgetting and the ability to redirect attention back to “all cars” as a mental act calling for particular mental effort on my part. It takes place of its own accord; I need do no more than shift my attention and change the verbal organizers a little: the white ones, the others, all cars, etc. I am not conscious of any other mental activity being involved. It all happens so effortlessly that I can well imagine that Piaget does not see any “logical operation” in it, but only a mental prerequisite for logical operations.

But let me return to the perception of all the cars as a conjunction of the white and the non-white ones and the white cars as the result of the dissociation of the non-white from all the cars. That does cost me mental effort. I feel something of a process of mental activity in this, and I can imagine Piaget’s terming it an operation or combination of operations. It is, of course, the question w...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Contents
• Preface to the Classic Edition
• Preface to the 2014 Edition
• 1 The Class Inclusion Problem in Piaget’s Theory
• 2 Teaching the Inclusion Problem to Children in Geneva
• 3 Teaching the Inclusion Problem to Children in Amsterdam
• 4 A Genevan Comment on My 1963 Publication
• 5 Teaching the Inclusion Problem to Children in Montreal
• 6 Teaching the Inclusion Problem to Children in Utrecht (Netherlands)
• 7 Categorizing Children According to Their Degree of Understanding
• 8 On the Methodology of Learning Studies Relevant to Piaget’s Theory
• Summary
• Epilogue A Personal History
• Bibliography
• Author Index