Child's Conception Of Number

by Jean Piaget

Jean Piaget was one of the most salient & inspirational figures in psychological & educational research in the 20th century. He was prolific, authoring or editing over eighty books & numerous journal papers which have spawned a huge, fertile continuation of his research over the decades. A major component of any course on children's psychological development & a research tradition that is expanding, scholars need access to the original texts rather than relying on secondhand accounts. "Jean Piaget: Selected Works" is a chance to acquire key original texts, most of which have been previously unavailable for years.

Genres: Psychology, Education, Nonfiction

260 pages, Paperback

Published October 17, 1965

About the author

Jean Piaget (1896 - 1980) was a Swiss philosopher, natural scientist and developmental theorist, well known for his work studying children, his theory of cognitive development, and his epistemological view called "genetic epistemology." In 1955, he created the International Centre for Genetic Epistemology in Geneva and directed it until his death in 1980. According to Ernst von Glasersfeld, Jean Piaget was "the great pioneer of the constructivist theory of knowing."

Reviews

[Steven H · Rating 5 out of 5]

THE BOOK THAT RECOUNTS PIAGET’S MOST FAMOUS EXPERIMENT

Jean Piaget (1896-1980) was a Swiss developmental psychologist known for his epistemological studies with children. His theory of cognitive development and epistemological view are known as "genetic epistemology.”

He wrote in the Foreword to this 1941 book, “In our earlier books… we analyzed various verbal and conceptual aspects of the child’s thought. Later on, we examined the beginnings of thought on the practical and sensory-motor planes… It now remains, in order to discover the mechanisms that determine thought, to investigate how the sensory-motor schemata of assimilating intelligence are organized in operational systems on the plane of thought. Beyond the child’s verbal constructions, and in line with his practical activity, we now have to trace the development of the operations which give rise to number and continuous quantities, to space, time, speed, etc., operations which, in these essential fields, lead from intuitive and egocentric pre-logic to rational co-ordination that is both deductive and inductive.

“In dealing with these new problems, appropriate methods must be used. We shall still keep our original procedure of free conversation with the child, conversation which is governed by the questions put, but which is compelled to follow the direction indicated by the child’s spontaneous answers. Our investigation of sensory-motor intelligence has, however, shown us the necessity for actual manipulation of objects…

“In the present volume, it has not been possible to include all that we should have wished to say on the subject of the evolution of number. In particular, there is an inexhaustible mine of information, on which we have deliberately not drawn… Our hypothesis is that the construction of number goes hand-in-hand with the development of logic, and that a pre-numerical period corresponds to the pre-logical level…. In our view, logical and arithmetical operations therefore constitute a single system that is psychologically natural, the second resulting from generalization and fusion of the first, under the two complementary headings of inclusion of classes and seriations of relations, quality being disregarded… when the same system is applied to sets irrespective of their qualities, the fusion of inclusion and seriation of the elements into a single operational totality takes place, and this totality constitutes the sequence of whole numbers, which are indissociably cardinal and ordinal…

“Discussion as to the relationship between number and logic has… been endless. The logisticians, with Russell, have tried to reduce cardinal number to the notion of ‘class of classes,’ and ordinal number, dissociated from cardinal number, to the notion of ‘class of relationships,' while their opponents maintained… that the whole number is essentially synthetic and irreducible. Our hypothesis seems to obviate the necessity for this alternative, for if number is at the same time both class and asymmetrical relation, it does not derive from one or other of the logical operations, but from their union, continuity thus being reconciled with irreducibility, and the relationships between logic and arithmetic being regarded not as unilateral but as reciprocal. Nevertheless, the connections established in the field of experimental psychology needed to be verified in the field of logistics, and we proceeded to attempt this verification.

“In studying the literature on the subject, we were surprised to find to what extent the usual point of view was ‘realist’ rather than ‘operational’… This fact accounts for the connections, many of them artificial, established by Russell, which forcibly separated logistic investigation from psychological analysis, whereas each should be a support for the other in the same way as mathematics and experimental physics. If, on the contrary, we construct a logistics based on the reality of operations as such, in accordance with, and not in opposition to, the psychogenetic processes, we discover that the natural psychological systems of thought, such as simple and multiple classifications… correspond from the logistic point of view to operational structures closely akin to mathematical ‘groups,’ and which we have called ‘groupings.’”

He reports what is perhaps his most famous experiment: “The child is first given two cylindrical containers of equal dimensions (A1 and A2) containing the same quantity of liquid. The contents of A2 are then poured into two smaller containers of equal dimensions (B1 and B2) and the child is asked whether the quantity of liquid poured from A2 into B1+B2 is still equal to that of A1… [The children deny this.] The results seem to prove that continuous quantities are not at once considered to be constant, and that the notion of conservation is gradually constructed by means of an intellectual mechanism which it is our purpose to explain… it is possible to distinguish three stages. In the first, the child considers it natural for the quantity of liquid to vary according to the form and dimensions of the containers into which it is poured… In the second stage… conservation … is recognized in some cases… it is not so in all. When he reaches the third stage, the child at once postulates conservation of the quantities in each of the transformations.” (Pg. 4-5)

He asserts, “we are not suggesting that number is reduced to classes and relations; we are merely indicating their mutual relationship … the concept of class does not precede that of number, but is acquired simultaneously, the two concepts being interdependent.” (Pg. 157)

He argues, “How then are classes to be transformed into numbers?... Russell’s solution to this problem is too simple. For him and his followers, two classes have the same number when there is a one-to-one correspondence between their elements.” (Pg. 182)

He continues, “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… class and number are mutually dependent, in that while number involves class, class in its turn relies implicitly on number.” (Pg. 184)

He summarizes, “There is not one stage of logical multiplication and another stage of arithmetical multiplication. During the first stage of development, neither of these compositions 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 the true sense. Hence, the child’s simultaneous success in the various tests … and his immediate generalization of multiplication once it has been discovered.” (Pg. 220)

Piaget’s description of his famous experiment alone would make this book ‘must reading’ for those studying Piaget.

[William Bies · Rating 4 out of 5]

Einstein once praised the Swiss psychologist Jean Piaget for having had the audacity of discover a most elementary insight: that children think differently than adults do. They are not just immature adults, but have their own distinctive way of approaching the world that can yield to the sedulous investigations of the psychologist, if he is imaginative enough to forget his own accustomed manner of thought and to observe children with an attentive, open mind. No doubt, Piaget excelled at this very thing and, what is more, had the patience and determination to see his investigations through to a grand synthetic conclusion, which he has outlined in several of his works.

The present volume, The Child’s Conception of Number, turns Piaget’s fruitful approach to the origin and derivation of mathematical concepts, in particular, number and arithmetic. Now a child of three years can count, of course, but it does not mean, by a long shot, that he has a real conception of the numbers he enumerates, as opposed merely to having memorized a sequence of names (with certain built-in patterns, which vary from language to language). In fact, Piaget concludes from his researches that children do not arrive at an adequate conception of what a number is until as late as the age of eight. In the work under review, he reconstructs the stages through which children typically go, at which he has arrived through induction based on experimental study of young children over a range of ages. His technique is to interview the child and to ask him questions about toys provided for the purpose (of course, this presupposes the child to be old enough to have command of speech; earlier stages can be postulated hypothetically). The written record of these interviews can be sorted, organized and analyzed in order to infer what the child at a given age is actually capable of understanding. A nice feature of the present work is that the author has reproduced a representative selection of these interviews, so that the reader can retrace Piaget’s argument from the children’s very words themselves. The investigations recounted here may be seen as an application of Piaget’s general theory of intelligence (cf. his foundational Psychology of Intelligence), where the author’s views on accommodation, assimilation and the operationalist approach to intelligence come to the fore.

Piaget summarizes his findings in compressed form in the Foreward: ‘Our hypothesis is that the construction of number goes hand-in-hand with the development of logic, and that a pre-numerical period corresponds to the pre-logical level. Our results do, in fact, show that number is organized, stage after stage, in close connection with the gradual elaboration of systems of asymmetrical relations (qualitative seriations), the sequence of numbers thus resulting from an operational synthesis of classification and seriation. In our view, the logical and arithmetical operations therefore constitute a single system that is psychologically natural, the second resulting from generalization and fusion of the first, under the two complementary headings of inclusion of classes and seriation of relations, quality being disregarded.’

Let us unpack what he says here. The point is that, to begin with, the child has no concept of quantity at all. At the early, pre-numerical stage, all he knows is how to perform qualitative comparisons of two things presented immediately in his visual field: the level of water in this glass is higher than that in that glass etc. A lot of conceptual work is needed before the child can begin to form an adequate concept of number. Piaget conveniently demarcates the properties of number into two: ordinality and cardinality. Ordinality, as the term suggests, has to do with the property of the real line of being an ordered field: given two numbers, we can say whether one is greater than, less than or equal to the other. The child has to acquire the concept of ordinality gradually by becoming proficient at seriation, by which Piaget means to place a collection of various-sized items into sequential order. The youngest children cannot do this at all, even when the procedure is demonstrated to them before their eyes. A number of concepts have to be mastered first, such as that of conservation and one-to-one correspondence. At four to five years, for instance, a child will be convinced, when one pours liquid from a wide glass into a narrow glass, that the higher level in the latter means that the quantity of water has increased. Likewise, given a collection of pennies placed on a table, he will judge their number changes when one permutes them. Only when he recounts them by hand will he believe that the number has stayed the same.

There is no need to reconstruct Piaget’s entire argument here. The reader will have the pleasure of watching how Piaget proceeds step by step to recover the development by which the ingredients of the concept of number are built up in the child, basing himself everywhere on the children’s self-reports during his experiments. The critical stage begins at seven to eight years, when the child masters concrete operations and applies the understanding thereby gained to the concept of number. The child learns to seriate and to perform reversible operations, such as permuting a set of objects and seeing that their number is conserved. The final stage in acquiring an adequate concept of number is the mastery of elementary arithmetical operations of addition, subtraction and multiplication. As Piaget relates, a lot has to go on here behind the scenes, such as the comprehension of division into equal parts and the composition of parts back into the whole. To compare two sets of objects for their cardinality, mentally one abstracts units one at a time, where it has to be understood that the total quantity in each set remains invariant during this process. The mature concept of number emerges as the conviction of reversibility of operations grows and the child begins to coordinate operations into a coherent unity.

We leave it to the reader to follow the flow of Piaget’s logic in detail and append some reflections on the significance of what Piaget has accomplished in this work. These reflections have a bearing on the controversy between the analytic (Frege, Russell) versus synthetic (Kant) philosophies of mathematics. In the view of Russell’s logical formalism, arithmetic involves nothing but concepts subject to logical operations, while for Kant, intuition (Anschauung) has to intervene, for to form the sum of two natural numbers is to construct (in the imagination) the series of successive units until one arrives at their sum, somewhat as one draws lines or circles in the course of a geometrical demonstration, which for Kant, of course, involves spatial intuition (outer sense), as against the temporal intuition (inner sense) underlying arithmetic. For this reviewer, what Piaget has done is to show that number is derived from a synthesis of perception and intelligence applied to operation. As such, the axioms of arithmetic represent a condensation of thought; we could not have arrived at them without first carrying out the synthesis in intuition.

The preceding observation invites another speculation: perhaps we could go back a step and rederive another larger concept of number than the standard one. The numbers with which we are familiar are useful to us because of the interaction with the world made possible by calculation by means of them, but that does not necessarily mean that we have exhausted, up to now, the range of possibilities. What the reviewer has in mind is something like a decategorification, as it is understood in modern mathematics. Indeed, it is possible that we could apply to number something analogous to the idea of equivalence up to homotopy in higher category theory. Just a suggestion for enlarging and enriching our experience of the world and corresponding mathematical formalism! Then, what Russell regards as analytic truths about number could be relative only to a certain formalization of them, just as Kant falsely imagines Euclidean geometry to be analytically true of spatial intuition. Now, all we need is for a young genius to discover the non-Peano arithmetic that would parallel non-Euclidean geometry!

P.S. For a graphical illustration of what the last-mentioned point could imply, consult the not-otherwise-distinguished Hollywood movie from a few years ago, Lucy, which is notable for the implicit philosophy of mathematics it espouses. This reviewer would like to contradict those who criticize the movie for being mathematically unrealistic, not to say impossible.

[Tom Schulte · Rating 3 out of 5]

I don't know what I expected from this, but I really didn't enjoy it. There was too much detail on the details of experiments and dialogue with the child subjects. Well, I am more into numbers than children, so that about figures.

[Ilib4kids · Rating 4 out of 5]

ILL (BF 723 N8 P53)
It is very hard to read and grasp author's idea,But still it is very good book to understand children's psyche about numbers.


Book summary based on my poor understanding:
Stage of understanding number:
Stage 1: intuition, perceptual, no understanding of number concept (conservation law law hidden in the number)
For example, two bunch of flowers same number, identical configuration 摆的一样
3-4 year old children judge them same, if one bundled up, or spread out, that is spatial perception is changed, they judge different. Even some children can count 1, 2, 3 and count them, when spatial perception is changed, they judge different, either one more or less than another, prove no understanding of number.
Stage 2: still influenced by perception
Stage 3: old children, doing right. understand number concept, its inherent conservation law and more.
My own comment:The children thought is different from adult, when we teacher, I think we should try to understand children cognitive psyche. Sometime, they think differently than adults.

Number is at the same time both class and asymmetrical relation. Page ix
Therefore, number is seen to be synthesis of class and asymmetrical relation. p243
meaning of this sentence:
1 apple, 1 orange, 1 pear on the table, you count them, 1, 2, 3. In this process, you conscientiously identify those as 3 things,not apple, not orange, not pear, but 3 things which they are belong to (that is, class means, identical things group together), at the same time you give them position, 1st is the apple, 2nd is orange, 3rd is pear is asymmetrical relation which you include the number, member of number are identical but different in position.
You can count 3 apple, but class identical meaning is not so obvious.
More example,
1 apple + 1 orange = 2 fruit
1 apple + 1 orange = 2 food
1 apple + 1 orange = 2 thing .....
In this addition process, you look for common attribute, disregard difference, apple, orange is fruit, food, or thing, etc, but not 2 oranges, 2 apples. This process give rise to number concept, which is what author means class.
My own comment:I've never thought number in this angle.

[Doni]

This is one time when reading the original material does not enhance the concepts. Terribly boring... consists mostly of direct dialogue with test subjects. Kept wishing he would just synthesize the material.