The Growth Of Logical Thinking From Childhood To Adolescence

AN ESSAY ON THE CONSTRUCTION OF FORMAL OPERATIONAL STRUCTURES

Jean & Inhelder Piaget

• Publisher: Routledge
• Year: 2013
• ISBN:
  9781136317545
• Edition: 1
• Topic: Medicine
• Subtopic: Health Care Delivery

This is Volume XXII of thirty-two on a series of Developmental Psychology. Originally published in 1958, this text offers a translation from French of an essay on the construction of formal operational structures to explain part of the growth of logic in a child's brain and development. It looks at propositional logic, the integration of formal thought and the operational schemata of formal logic.

Part I

The Development of Propositional Logic

IF WE are to explain the transition from the concrete thought of the child to the formal thought of the adolescent, we must first describe the development of propositional logic, which the child at the concrete level (stage II: from 7-8 to 11-12 years) cannot yet handle. Experimentation shows that after a long period during which only operations appropriate to class and relational groupings and to the numerical and spatiotemporal structures which resulted from them are used, the beginnings of stage III (substage III-A, from 11-12 years to 14-15 years; and substage III-B, from 14-15 years onward) are distinguished by the organization of new operations performed on the propositions themselves and no longer only on the classes and relations that make up their content.

To study the questions raised by this development, we must analyze how children or adolescents at stage III go about solving problems which appear purely concrete but which experiments indicate can be resolved only at stage III and which actually presuppose the use of interpropositional operations. Part I of the present work will be devoted to this analysis.

1

The Equality of Angles of Incidence and Reflection and the Operations of Reciprocal Implication¹

OUR AIM in this chapter, and in the remainder of Part I, is not a systematic study of the concept of the equality of two angles. Actually, we already know how the concept is constructed: that it is first acquired at the level of concrete operations.² But it is precisely the fact that the concept is already so well known by the time the formal level (stage III) is reached that makes the reasoning process involved in the discovery of the equality between the angles of incidence and reflection so instructive. One of the aims of this study, then, is to isolate the operational mechanisms involved in the formal reasoning process itself, when this reasoning rests on notions already constructed at the concrete level.

The experimental apparatus consists of a kind of billiard game. Balls are launched with a tubular spring device that can be pivoted and aimed in various directions around a fixed point. The ball is shot against a projection wall³ and rebounds to the interior of the apparatus. A target is placed successively at different points, and subjects are simply asked to aim at it. Afterwards, they report what they observed.

But the equality between the angles of incidence and reflection is discovered only at stage III-A (11-12 to 14 years) and is often not formulated until stage III-B (14-15 years). Our problem is to understand why a concept as familiar after 7-9 years as that of the equality of two angles is utilized in the induction of an elementary law only at this late date and, especially, why formal operations are necessary for its use. We shall try to answer this question by retracing briefly the ground covered by the child before his arrival at the formal level, then by examining the latter more closely.

§Stage I

In the course of stage I (up to about 7-8 years) subjects are most concerned with their practical success or failure, without consideration of means; often even the role of rebounds is overlooked. The result is that, except toward the end of the stage, the trajectories are not generally conceived of as formed of rectilinear segments but rather as describing a sort of curve:

DAN (5 ; 2)^*succeeds at first: “I think it works because it's in the same direction” He adjusts the plunger by himself, but proceeds by empirical trial-and-error. Then he asks spontaneously: “Why do you have to turn the plunger sometimes?… No, you have to put it there [he fails]. If it could be pushed a little further” [he does this and succeeds]. But, although he knows how to control the rebounds successfully, DAN has no idea that they are made up of angles: the curve he describes with his finger is not tangent to the wall; he takes into account the starting point and the goal but not the rebound points.

WIRT (5 ; 5): “It came out here and it went over there…. I’m sure to make it,” etc. He succeeds occasionally but describes the trajectories with his finger only in the form of curves not touching the walls of the apparatus; he considers only the goal as if there were no rebounds.

FIG. 1. The principle of the billiard game is used to demonstrate the angles of incidence and reflection. The tubular spring plunger can be pivoted and aimed. Balls are launched from this plunger against the projection wall and rebound to the interior of the apparatus. The circled drawings represent targets which are placed successively at different points.

NAN (5 ; 5), on the other hand, is astonished by the detour made by the ball which first touches the walls. “It always goes over there ” But he does not succeed in adjusting his aim: “Oh, it always goes there. . . . it will work later.”

PIT (5 ; 5) notes about one of his tries [a failure]: “It was straight [as if this were an exception].— Why didn’t it hit it?—I thought I hit it”[no comprehension].

ANT (6 ; 6) becomes aware of the existence of rebounds at the same time that he notices the rectilinear character of the trajectory segments: “It [the ball] hits there, then goes over there ” [his gesture indicates straight lines].

PER (6 ; 6), in contrast, in spite of his age, resorts to the curvilinear model: “It goes there and it turns the other way” [gesture indicating a curve].

The reactions of this stage are extremely interesting, for although the children demonstrate by their behavior that they know how to act in the experimental situation, sometimes successfully, they never internalize their actions as operations, even as concrete operations. In a general sense, by concrete operations we mean actions which are not only internalized but are also integrated with other actions to form general reversible systems. Secondly, as a result of their internalized and integrated nature, concrete operations are actions accompanied by an awareness on the part of the subject of the techniques and coordinations of his own behavior. These characteristics distinguish operations from simple goal-directed behavior, and they are precisely those characteristics not found at this first stage: the subject acts only with a view toward achieving the goal; he does not ask himself why he succeeds. In the experiment under consideration he is not aware of either the rectilinear nature of the trajectory segments or the existence of rebounds except toward the end of the stage (toward 6 or 6-7 years); consequently he cannot take note of the presence of angles at the rebound point.

§Stage II (Substages II-A and II-B)

Substage II-A is distinguished by the appearance of concrete operations in the sense just defined:

VIR (7 ; 7 ) succeeds after several attempts. He points out and then draws trajectories with two distinct rectilinear segments, saying: “To aim more to the left, you have to turn [the plunger] to the left.”

TRUF (7 ; 10 ): “I know about where it will go”; in fact, he shows by his gestures that he realizes that the angle of rebound is extremely acute when the plunger is raised and extremely obtuse when it is lowered. Thus, he shows us that he has a vague global intuition of the equality between the angles of incidence and reflection. But he does not make it explicit, since he fails to divide the total angle indicated by his gesture into two equal angles.

BEND (8 ; o): “It's the corner [the angle of rebound] that makes it turn; you change the contour [the size of the angle] when you change the plunger” [inclination of the plunger]. He demonstrates as did the preceding subject that the angle is extremely acute when the plunger is slightly inclined and extremely obtuse when it is sharply inclined. We ask him what he means by the contour, and he points to the opening of the angle with a gesture indicating that he is thinking of the very generation of the angle by the progressive rotation of the plunger and of the rebound of increasing amplitude which results.

DESI (8 ; 2): “The ball always goes higher when the plunger is higher.”Then: “The ball will go there [further] because the plunger is tilted more; I put my eyes high up [ = I pinpoint the rebound point] and from the rubber [ = the rubber band attached to the wall on which the ball rebounds] I look at the round pieces” [ = the disks serving as goals].

At substage II-B the preceding operations, which give rise to a model that includes straight lines and angles, are complemented by an increasingl...

Table of contents

• front cover
• title page
• Table
• half title
• copyright page
• Introduction
• Preface
• contents
• PART I The Development of Propositional Logic
• PART II The Operational Schemata of Formal Logic
• PART III The Structural Integration of Formal Thought
• Index