Child's Conception Of Geometry

by Jean Piaget

Routledge is now re-issuing this prestigious series of 204 volumes originally published between 1910 and 1965. The titles include works by key figures such asC.G. Jung, Sigmund Freud, Jean Piaget, Otto Rank, James Hillman, Erich Fromm, Karen Horney and Susan Isaacs. Each volume is available on its own, as part of a themed mini-set, or as part of a specially-priced 204-volume set. A brochure listing each title in the International Library of Psychology series is available upon request.

Genres: Psychology, Education

Hardcover

First published January 28, 1960

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

[William Bies · Rating 4 out of 5]

Jean Piaget and Bärbel Inhelder’s The Child’s Conception of Space (previously reviewed by us here) is about space in general, culminating in Euclidean concepts (i.e., as a vector space equipped with a topology). In contrast, the two authors are joined by a third collaborator, Alina Szeminska, in the present book The Child’s Conception of Geometry, to study the development of a metrical structure in particular (i.e., measurement of lengths, angles, areas, volumes).

The format will be familiar to anyone who has seen earlier books by Piaget and company: a mixture of theoretical exposition interspersed with detailed reports from actual experimental observations on children, who are asked to solve various geometrical problems posed to them. The inclusion of the children’s actual words as they justify their thinking can be very illuminating as to their thought-processes, especially when accompanied by the psychologist’s running commentary. This reviewer has never seen a style of presentation like this elsewhere, but here it tends to work pretty well in support of the authors’ contentions.

What this reviewer appreciates in Piaget et al. is their ability to resolve a psychological competence into its elements and to study their genesis in the child, step by step. For instance:

To measure is to take a whole out of one element, taken as a unit, and to transpose this unit on the remainder of a whole: measurement is therefore a synthesis of sub-division and change of position….The idea of change of position is doubly difficult for the young child. In the first place the concept of measurement goes beyond the ability to carry out the necessary bodily movements or the ability to transpose things; it implies the representation of changes of position, and the ability to reconstruct a sequence of action come much later than the ability to carry it out. But secondly, to be able to image movements is not enough, for the subject must link movements to reference points. Some system of reference it implicit in the representation of any sequence of movement. An understanding of measurement demands that the several reference points be linked in a systematic whole, which implies ‘coordinate axes’. [pp. 3-4]

To us who are adults already in possession of the relevant competency, it can be a struggle to put oneself into the mind of a child who does not. That is why a psychological analysis is so impressive, for it means a recovery from a higher perspective of what one has forgotten. Where do the youngest children begin? Here is Piaget’s initial claim:

In several of the above mentioned researches it was shown that children begin by considering a change of position in terms of the end-position only. They make no attempt to link end-point with starting point, still less do they consider both in terms of a more embracing system of reference. [p.4]

Clearly, to fill such an inadequate concept with more adequate ones necessitates a process of development in which the child gradually becomes aware of how things fit together. Here, in brief, is the authors’ description of the intermediate stages:

Before this operational generalization, the child’s space was virtually split in two heterogeneous entities instead of forming a homogeneous common medium. On the one hand, there were objects, of filled space. These were characterized by form, by intuitive notions of length, etc., and by being subject to changes of position with respect to one another. On the other hand, there was empty space which formed the intervals between objects. This intuitive notion of empty space contained the germ of the operational notion of distance, but it entered neither into the representation of change of position nor into that of length. There could be no conservation of intervals so conceived because distances were modified whenever a solid object was interposed. Even the distance between stationary objects was so modified because these were not thought of as fixed reference points for changes of position. The generalizations of operations of order and change of position coordinates these heterogeneous entities, i.e. filled and empty space, not by simple fusion which would be absurd but by means of a tertium which comprehends both. This is the system of ‘sites’ which may be either occupied by objects or free, and these in turn are fixed by means of elements considered stationary. [p 88]

But to make progress something else figures as a necessary aid, viz., the idea of conservation – obvious enough to us adults but not at all to the child! Here is what the authors have to remark on this:

Why do these subjects not regard conservation as logically necessary? They are willing to compare differences of length resulting from a given set of positional changes with those produced by others. It is these comparisons which lead to increasing comprehension of an intuitive nature. In the end they guess at conservation, without basing this notion on an exact composition of the spaces left empty by the change in position of the test objects and the corresponding spaces which are occupied: they do not realize that in every change of position these two factors are mutually compensating. Their thought does not yet embrace a system of fixed sites and deals only with the transformation of objects. That limitation precludes the operational construction of length. It does however, admit of an intuitive conservation of relations of equality, which anticipates operation and may even come near it. [p. 101]

The goal towards which children at the intermediate stage tend is the adult concept of length, which implies the ability to measure it. How does the child get there? Here is what the authors determine:

The construction of a metric unit thus involves a twofold generalization. True measurement of distance and lengths begins when the subject recognizes that any length may be decomposed into a series of intervals which are known to be equal because one of them may be applied to each of the others in turn. Generalized subdivision therefore gives rise to measurement because it enables the subject to think of a unit as forming part of any number of wholes, i.e. as an elementary common part. But generalized subdivision cannot be achieved without generalized change of position because the former implies that the unit-part may be applied hypothetically to an indefinite number of wholes. Thus the notion of an elementary unit which may be applied indefinitely in a continuous and contiguous series of changes of position involves the operational synthesis of subdivision and change of position. [p. 149]

This concludes Part II. Let us skip over Part III on the representation of space in terms of rectangular coordinates and the emergence of a sense of how curves or other geometrical figures fit into such a scheme, and turn to Part IV, which concerns the special problems that arise when one has to perform a measurement of a higher-dimensional object, viz., areas and volumes. For areas and volumes scale differently with size than do lengths, something the child does not initially realize. Nevertheless, upon analysis the authors conclude that, from a psychological point of view,

Thus the development of conservation and measurement runs exactly parallel whether the objects are lengths or whether they are areas and the level at which they are finally grasped is the same for both. Conservation is always the outcome of complementarity between the two groupings: that of additive subdivision and that of ordered positions and changes of position. [p. 300]

Towards the end, the authors draw together what they have learned in order to form some conclusions about the psychology of the child of a wider nature. Here are a couple key passages:

We are apt to be misled by our constant inclination to think of the successive levels of development as monolithic because they are also hierarchical strata of mental functioning in the adult. We therefore think of intelligence as a unique entity which ranges from perception and sensori-motor adaptation to verbal and formal thinking. But the unity of intelligence is only functional while from the structural point of view it entails a progressive series of different and heterogeneous structures. [pp. 307-308]

It is obvious that the difficulty to which we point arises only because there is no anticipatory schema in which the desired fraction or part is recognized beforehand as something bound up with a divisible whole equal to the sum of its parts. In this field, as in every other domain which we have studied, the facts are clear: thought alone cannot give rise to anticipatory schema, because representation cannot foresee the possible unless it is guided by action. Only when the child has reached the level where his actions are coordinated and form reversible operational groupings do we find operational anticipatory schemata. [p. 334]

The final stage studied in this book is the acquisition of the concept of a Euclidean continuum, by which the authors understand space as equipped not merely with topological notions of togetherness or apartness, but with a properly metrical notion amenable to measurement. The challenge faced by the child may be summed up thus:

Hence the systematic difficulties found by children at earlier levels when trying to relate areas and volumes with linear quantities. The child thinks of the area as a space bounded by a line which is why he cannot understand how lines produce areas. We know that the area of a square is given by the length of its sides, but such as statement is intelligible only if it is understood that the area itself is reducible to lines, because a two-dimensional continuum amounts to an uninterrupted matrix of one-dimensional continua. [p. 350]

Yet somehow everyone manages to surmount these difficulties so as to arrive at the mature geometrical concept of space, which process the authors describe thus:

Euclidean notions of space are completed in the course of stage IV and it is then that the duality of thinking in relation to area and volume is finally overcome. The first new discovery of stage IV is the conservation of volume in the sense of the physicist. Conservation of volume is no longer limited to ‘interior’ volume and divorced from the spatial context….Continuity is therefore achieved by adding an infinite number of lines to the existing matrix. Having elaborated the notion of a continuum, children can now use mathematical multiplication to arrive at areas and volumes, and by so doing they reconstruct the notion in metric terms. Here we see the final phase in the reconstruction of Euclidean space. The level at which it is completed is also that at which topological notions themselves achieve an equilibrium of their own by virtue of formal operational thinking. [pp. 407-408]

The descriptions of the process of development of a competence rely, needless to say, on a general perspective on the psychology of intelligence that Piaget has worked out elsewhere. Thus, read the present work to solidify one’s understanding in the context of a significant application but not necessarily to learn Piagetian theory in the first place.

Can we apply the principles disclosed in this book to non-Euclidean or Riemannian geometry? Clearly these advanced notions correspond to a stage of conceptual development that goes beyond any that the authors study here, but can one extrapolate their ideas? A whole new order of reflection is called for since one has to call into question the adult equilibration that for most everyone works perfectly well in any everyday problem he is likely to encounter. What, then, were Bolyai resp. Lobachevsky’s guiding insights that prompted them to go beyond the conventional Euclidean geometry? Their key realization was that parallelism might be only an approximate concept applicable locally but not globally. What about same question for Riemann? In his case, the immediate spur to his investigations seems to have been his abstraction of the notion of measurement from its concrete context and its extension to purely imaginary spaces of any dimensions (mehrfach ausgedehnte Größen). But Riemann’s supreme power of intuition reveals itself in what he does next, to undertake an analysis of the elementary notion of curvature so as to abstract and formalize its core content in the definition of the curvature tensor named after him. What Riemann perceives is that the intrinsic curvature of space around a given point reveals itself to our inspection when we consider how the infinitesimal surface spanned by two linearly independent tangent vectors bends when we prolong it outwards by projecting along geodesic rays, and his curvature tensor may be seen as a convenient device for organizing all these data (viz., the ½ n(n–1) Flächenrichtungen) in a way that further proves to have the desirable property of being independent of any choice of coordinates.

From a Piagetian point of view, what Bolyai, Lobachevsky and Riemann do is to become aware of a novel type of a conservation law that emerges at a higher level of reflection (following the lead of Carl Friedrich Gauss’ celebrated ‘theorema egregium’ which states that the integral of the Gaussian curvature over a surface forms a geometrical invariant, proportional in fact to its genus). Having once surmised that such geometrical invariants might exist, they then deploy all their mature formal operational competence in order systematically to explore them and to characterize their properties. Thus, at a higher level, these mathematicians go about their research in much the same manner as do the children in many of Piaget’s experiments, with the difference that there is no psychologist superintending the experiment who already knows what they are about to learn from it!

The present work is also very interesting from point of view of a prospective quantum theory of gravity, in that the authors stress that the rise of formal operational competence is possible only subsequent to the child’s reflection on the actions he can perform with given concrete problems – but these latter are altogether lacking as far as quantum gravity goes. The ladder of spatial concepts from topological to differentiable to metrical recalls the problem of identifying the correct algebra of observables in a generally covariant theory, i.e., what can one gauge away and what can one not gauge away? Sustained progress on the thorny problem of quantum gravity is likelier to result if, rather than to continue down the popular path of mindless spinning out of toy models (which offer, indeed, plenty of scope for an impressive display of mere technical skill and cleverness but whose usefulness in furthering insight into the original problem approaches the vanishing point), one were to get elementary and, like the children in Piaget’s experiments, try to figure out what the necessary concepts and physical principles may be (which, compared to deriving ever more properties of toy models, is far harder work, of a lonely and exploratory nature, and requires the patience to persist for a long time in a state of confusion, without being able to generate many publishable papers along the way; this latter circumstance explains why, despite the expenditure of prodigious energy over the decades by hundreds and thousands of brilliant theorists, almost nobody is willing to attempt anything original).