Child's Conception Of Geometry

Jean Piaget, Barbel Inhelder, Alina Szeminska

• Publisher: Routledge
• Year: 2013
• Print ISBN: 9780415209991
• eBook ISBN: 9781136316739
• Topic: Medicine
• Subtopic: Health Care Delivery

This is Volume XIX of thirty-two in a series on Developmental Psychology. Originally published in 1960. There are certain problems of a general nature in regard to the development of spatial concepts in young children, the problem of spatial intuition as a whole is exceedingly complex, and this study discusses of a child's conception of geometry of measurement and metrical geometry.

Part One
Introduction

Change of Position and Spontaneous Measurement

COMPLEX operations often depend on others which are more elementary. This is certainly true of measurement, which is a synthesis of two operations, one of these being the coordination of changes of position. Since this first part is designed to form an introduction to our study, we have deliberately dealt with situations in which the child’s activity is as spontaneous as it can be. This should enable us to observe the way in which measuring behaviour as a whole is elaborated in the course of the child’s activity. Moreover, we should see clearly how measurement implies certain other operations, not only in the context of our experimental situations but in real life. The following two chapters are therefore inductive rather than deductive in approach. The first deals with the way in which children reconstruct their own movements or changes of position by drawing on their own conceptions of the spatial field, and how, as they grow in maturity, so the latter become increasingly coordinated. The second deals with the way in which they measure, or show those more primitive kinds of behaviour which eventually lead to measurement itself; the actual situation is a kind of game in which the child is asked to make a construction which he recognizes sooner or later as one which demands a degree of precision that only measuring can guarantee.

Chapter One
Change of Position¹

THE concept of change of position is fundamental for Euclidean metrics. These changes form a mathematical ‘group’, and can be represented in terms of a three-dimensional space structured by a coordinate system. The subject has already been touched on in two earlier investigations. The first of these was concerned with the organization of changes of position at the level of sensori-motor intelligence, during the first eighteen months of life.² The second was devoted to the study of spatial representation between the ages of 4 and 12³: we saw how younger children described changes of position in terms of end-positions only, while older children compared paths of movement. Dimensions and coordinate systems have also been investigated in an earlier work in which we showed how gradually children learn to structure their space in terms of horizontal and vertical axes and went on to trace the stages of development in the construction of a ground-plan like that of a village.⁴ We have yet to study how children come to reconstruct changes of position in terms of a comprehensive system of references, or coordinate system.

A good way to bridge the gap between these earlier studies and the present researches on problems of spatial measurement is to study how children learn to use reference systems in thinking of a group of movements. This investigation should prove a valuable introduction to the study of Euclidean metrics in general.

To measure is to take out of a whole 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. However, although this way of looking at it seems clear and self-evident, the process is far more intricate in fact. As often happens in psycho-genetic development, a mental operation is deceptively simple when it has reached its final equilibrium, but its genesis is very much more complex.

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 comes much later than the ability to carry it out correctly. But secondly, to be able to imagine movements is not enough, for the subject must link movements to reference points. Some system of reference is implicit in the representation of any sequence of movement. An understanding of measurement demands that the several reference points are linked in a systematic whole, which implies ‘coordinate axes’. This is especially true of measurements in an area or a space, ch. VII, where the subject must take account of two or more measurements simultaneously, and of angular measurement, ch. XI, where he has to note differences in direction over and above the changes of position involved.

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. We will meet this phenomemon again and again in the present series of enquiries, particularly in that which deals with the constancy of the length of an object when it is moved (ch. IV). Before analysing the way in which children build up the concept of metric space, we must return briefly to the study of movement, bearing in mind the difference between actually moving an object and describing change of position, and especially how closely linked the latter must be to a system of references (see C.C.S., chs. XIII and XIV).

We need to examine the representation of changes of position in a situation with which the subject is thoroughly familiar. It will not do to face him with a concrete problem comprising movements which are quite new to him, asking him to describe them. Not only should the changes of position themselves be familiar; the reference system of which they form a part should be one with which the subject is familiar and which he can readily envisage (unlike the imaginary village of C.C.S., ch. XIV). Accordingly the preliminary investigation deals with large-scale changes of position as against the minute and accurate ones which figure in measuring. Moreover, we shall leave out the question of measurement altogether, and concentrate on (i) the conflict between representation and action, and (ii) the necessary relation between the representation of change of position and a system of references.

By studying the way children between the ages of 4 and 10 describe a familiar walk, such as the road from home to school, or that from school to various well-known places round about, we will see clearly how they learn to describe changes of position by using landmarks and eventually link these reference points in a comprehensive system. Thus the main buildings and squares, bridges, rivers or streams (in Geneva, the Rhone and the Arve), their own school and their homes are the landmarks which go together to form a reference system, and this in turn enters into their description of changes of position.

§1. Outline of method and results

Children cannot be questioned below the age of 4 or 5, which is when, in Switzerland, they first enter the Kindergarten, that is, at Stage II. Even between 4 and 7 children cannot be made to stay the length of the experiment unless they become interested in the questions asked.

The subject comes into the experimental room attached to the school and is taken to the window where he is asked to point out various buildings and well-known places. This is merely to ascertain the extent of his local knowledge and sense of direction. Next, he is made to sit at a table with his back to the window and given a sand-tray with wet sand, carefully levelled off. He is also given a number of little wooden houses of various sizes, representing the school and nearby buildings, little pieces of wood representing greens, recreation grounds, public squares and bridges, and a ribbon to represent the Arve. (The experiment was carried out at a school quite close to the river.) The experimenter takes the biggest house and puts it in the middle of the sand tray, saying: “Now this is the big school (meaning the primary school as against the kindergarten). There are plenty more houses, little ones and big ones. These little bits of wood are to make bridges with and this blue ribbon is the Arve. Now I want to know about everything near the school. You put the things in the right places.” (Children who are too nervous to use the models can be asked to draw it all on the sand for a start.) At the end of the first part of the experiment the subject is asked to draw a plan in the sand or on a piece of paper, showing how he would go home from school or, better still, how he would go to a place which they all know (the ‘Place Neuve’ which is quite close to the area shown on the sand tray). The drawing is a free drawing but the child is asked to show how it fits in with his general plan. Next the child is asked to make a drawing of the sand-model on a large sheet of paper. When he has done this, the experimenter turns the ‘school building’ through 180° and asks: “Now if I turn the school round like this, must we move everything else about as well or can we leave it just as it is?” The child is asked to make the necessary changes himself. With older children the entire experiment can be carried out with pencil and paper.

There are thus three related parts to the enquiry: (1) a plan of the school buildings and the principal features in the immediate vicinity; (2) a reconstruction of the route from school to a well-known landmark; and (3) changes in the location of features when the school building has been turned through 180°. The first of these has already been covered in C.C.S., ch. XIV, and may therefore be passed over in the present chapter. We shall concentrate instead on the representation of changes of position as shown by a child’s reconstruction of the route from school to a given destination or as necessitated by turning the original plan through 180°. However, it is clear that both these tasks demand a system of references which is in effect a topographical schema or plan of the neighbourhood.

The responses made by children to this enquiry show three levels of development. During stages I and II, that is, up to an average age of 7, children produce three kinds of reaction, all of which are interrelated. (1) When dealing with a route, they think of their own actions first, as though these were some kind of absolute, and the various landmarks are fixed in terms of them, instead of vice versa. (2) Their landmarks are not organized in terms of an objective spatial whole; the links between any two are conceived of as being independent of the system as a whole; subjective considerations determine the location of the various points. (3) Subjects cannot rotate a plan through 180°, nor can they reconstruct a route in the reverse direction, because both imply operational thinking.

The next level (IIIA) is marked by limited objective coordinations of space. Children reconstruct routes, using sub-systems of reference which are not linked together as a whole. Their plan of the district is made up of several portions which are correct in themselves but do not agree with one another. In rotating the plan they reverse some of the relations required but not all. Finally, stage IIIB shows complete spatial coordination alike in the representation of a route, in the topographical schema and in the correct reversal of a plan of the school.

Development of the power to reconstruct changes of position proceeds along similar lines to that which enables young children to handle these relations at the perceptual level. We have shown in an earlier work how this is achieved by the building-up of a group at the perceptuo-motor level.¹ The present development, although many years later in point of time, conforms to the same important principle. In both, there is development away from what might be termed egocentric coordination towards objective coordination, that is, coordination which no longer hinges on the subject’s own actions. Whereas earlier the subject’s actions were the only reference system available for the representation of spatial reality, these actions themselves are later seen within the context of an objective system. He can therefore describe his own movements in terms of objective references which are systematically interrelated. At the level of representation, it is when children pass from an egocentric attitude to objective coordination that they build up a group of positional changes² and work out a coordinate system. A similar move away from an egocentric attitude and an equally remarkable achievement occurs in the first eighteen months of life when children build practical groups of changes of position in dealing with concrete objects. The youngest children in the present study cannot describe changes of position as an objective group because they lack a coherent network of reference points (see C.C.S., chs. XIV and XV, sec. 5, op. vi). They are as incapable of the operation of inversion as of associative and transitive composition, as is clear from their attempts to describe changes of position, and from the landmarks they use. From their own point of view these are coordinated, but this is in terms of their own actions. When dealing with sensori-motor space we called this a ‘subjective group’, since from the point of view of the objects there is no group. Later, when changes of position and landmarks are coordinated and allow of inversion, there is a true group.

§2. Stages I and II: Landmarks are uncoordinated and changes of position cannot be described

We have said that the above experiment cannot be carried out with children under 4. However, by noting spontaneous remarks made by children aged between 18 months and 4 years as we take them for a walk, we may observe the earliest spatial reactions which lead into stage II.

JAC (1; 7), when some 100 yards away from a certain mountain chalet, shows in what direction it lies and also points out the road taken by his grandfather when he left three days previously. At the age of 1; 11 he is able to show the direction of his own home although it is half a mile away and he has his back to it. But when he starts on his way back he begins by thinking his home is still behind him, till he discovers his mistake. At 3; 3 he walks along the right hand side of a road to avoid oncoming traffic, but on the way back he goes to the left, mistaking it for his right.

L, even at 3; 11, when travelling in a car, thinks the mountains are moving.¹

Their sense of direction is quite good, for the first child knew where his own home was as well as the road along which he accompanied his grandfather a few yards. On the other hand they are still inclined to hesitate and make mistakes when dealing with an about face which switches ‘behind’ to ‘in front’ and left to right, and even the framework of distant landmarks is for them not a stationary one.

This type of thinking along two distinct planes continues throughout stage II, and it is not until stage IIIA, when their powers of representation have become greater, that children are able to link the two together. The first plane is that of practical spatial orientation or effective action. When children are taken to the window of the experimental room they can show where they live, where the Arve is, and various other landmarks they know, and if we walk home with them they manage to show the way quite well, although they can only explain each portion as they come to it. The second plane, however, which is the one we are studying, is that of true representation, which is temporarily divorced from action. Such representation occurs when children merely think out a route and the spatial relations between various well-known landmarks. Quite clearly, the step from action to representation is still quite a large one at stage II. Children at this stage can orientate themselves in a practical way, that is, in action, which means that up to a point they can even anticipate the spatial relations between one landmark and the next as they come to them; but instead of building up a picture of their environment in general as an ordered whole, they stop short at a more or less incoherent juxtaposition. Their arrangement of landmarks draws on their own personal recollections of actual journeys and on their main interests, for these helped to fix in their minds the ...

Table of contents

• Cover
• Title
• Copyright
• CONTENTS
• Preface
• PART ONE INTRODUCTION
• PART TWO CONSERVATION AND MEASUREMENT OF LENGTH
• PART THREE RECTANGULAR COORDINATES, ANGLES AND CURVES
• PART FOUR AREAS AND SOLIDS
• PART FIVE CONCLUSIONS
• Index