Piaget’s Cognitive Theory of Child Development

Read this article to learn about the Piaget’s Cognitive Theory of Child Development. After reading this article you will learn about: 1. Introduction to Piaget’s Cognitive Theory of Child Development 2. The Schema 3. Adaptation: Assimilation and Accommodation 4. The Period of Concrete Operations (Age Seven to Eleven) 5. The Period of Formal Operation.

Contents:

1. Introduction to Piaget’s Cognitive Theory of Child Development
2. The Schema
3. Adaptation: Assimilation and Accommodation
4. The Period of Concrete Operations (Age Seven to Eleven)
5. The Period of Formal Operation

1. Introduction to Piaget’s Cognitive Theory of Child Development:

Piaget traces the development of the child since its birth. He observes day-to-day changes that come about in the behaviour of the child.

His theory of child development is the result of his long and minute observations of the child’s behaviour that goes on changing from the day the child is born till such time when he attains adulthood or reaches to the maximum possible point of development—though, to some extent, the process of development ever remains dynamic.

Thus, the approach of Piaget was empirical; his findings are, mainly, based on case studies or cross-sectional studies. He observed closely how a neonate starts, apparently, with reflex reactions, and, in due course of time attains the stage of Formal Operations by the time he is 11+.

Jean Piaget was basically related to Biology. He received his doctoral degree in biological science in 1918 at the age of twenty-two. He was professor in the University of Geneva. But his first research was related to the development of reasoning in children, and it was published in 1921.

He was then offered the post of Director at the JJ Research Institute at Geneva. Ever since he held that post along with the post of Principal at the Center of Genetic Epistemology, Geneva; and came out with so much work on child development that he impressed as the most prolific writer and the theorist on the development of the child.

Three factors influenced the research and the theorising work of Piaget:

(i) Biological studies related to the adaptations of fauna and flora to their natural environment;

(ii) Philosophical study of the genesis of knowledge, that is, Epistemology and

(iii) Studies on developmental psychology. Though the entire work of Piaget falls into the category of developmental psychology, his approach throughout bears evidence to the impact of biological background that he had. Adaptiveness and the development of the biological structures have been made the basis of his showing the genesis of epistemology or the development of knowledge.

As mentioned at the outset, Piaget’s conclusions are the results of empirical researches conducted through the clinical device of observations and questionings. Besides those of the children of his own family, his were the cross-sectional studies widely conducted—had he conducted more longitudinal studies, the findings, related to cognitive development, would have been more coherent.

There are some terms which are very important for understanding the theory of Piaget, such as SCHEMA. It is not possible to follow the elaborated explanation of his theory without being clear what he means by the term ‘schema’.

2. The Schema:

The schema is “behavioural parallel of structure in biology.” A biological structure may be unitary and as simple as a finger, but it may also be a wide system of different structures, as complicated as the digestive system.

A finger reacts in response to a stimulus, overt or covert, and so does the digestive system, but the action of a unitary structure illustrates a single reaction to a single stimulus; that of the digestive system, illustrates how different structures work in correlation to one another in an integrated way.

Piaget starts with the ‘sucking reflex’ as one of the earliest schemas. Apparently, it appears to be a one-to-one response, that is, a single response to a single stimulus. But actually it is not so, many acts are involved in the process.

The mother touches the cheek the infant turns its face towards the mother, the touch on the lips, makes the baby open its mouth; sucking when the inside of the mouth is stimulated; and swallowing when the liquid reaches the throat.

The sucking structure is developed as a result of the child’s sucking behaviour. This behavioural structure is a ‘schema’ as Piaget calls it. Thus, each schema is a result of a unitary behaviour or a set of behaviours, that is, consisted of a series of actions enabling the organism to achieve some specific end.

Another simple schema is that of looking at an object. Even a neonate can see an object when its eyes are pointed to it, but it is not looking at an object. Looking involves the actions of moving the eyeballs, adaptive movements of the lens of the eyes, and paying attention to the object- it is not merely fixating the eyes upon the object.

Thus, looking at an object needs the development of a schema, the role of which is very important in the overall development of the child.

To be able to grasp an object indicates that the schema to do the same has developed. But a child’s being able to grasp objects both big and small, or thick or thin, illustrates how the schema of grasping has adapted itself to the demands of the situation.

For picking up a big doll, and for picking up a small thread, different sorts of muscular movements are required; the child’s skill of lifting both of them, indicates his adaptability to the situation, and the equilibrium that comes about in his behaviour because of the development of the different variants of the schema of grasping. And, the development of the different variants of a schema, illustrates his mobility.

Until the time these different variants of the schema have developed, there would be a state of un-equilibrium, and efforts would continue till equilibrium is attained. But equilibrium is not finally attained with the development of a schema a change in situation demands a new schema for the organism to perform the now needful function.

Again, there would develop a state of un-equilibrium; the present state of equilibrium is disturbed to be restored again only when a new schema or a new variant of a schema, which enables the organism to perform the new function, is fully developed.

An adaptation does not take place in isolation. There is interrelatedness in the process, and a complex, coherent pattern is formed when changes take place for adaptation to the environment.

The new adaptations are rooted in the old ones to suit the new demands of the environment; and thus, there is continuity in the process “the development of new structures to fill old functions under changed circumstances” (AL Baldwin).

The biological principle of teleological adaptation is based upon the mechanism governed by causal laws. Each adaptation is to serve some specific purpose, and each is related to other adaptations, and to the demands of the environment. Thus, there is interrelatedness, continuity and complexity in the process leading to the development of new structures, new schemas or schemata.

A child learns to walk—the development of a new sensorimotor schema enables the child to perform the function of walking, which in turn enables the child to perform still other such functions which cannot be possible unless the child can walk. This is an example of teleological adaptation.

The development of the structure or schema of walking is quite essential to enable the child to explore its widening environment; which again, is so essential for enabling the child to perform so many other functions which the continuously changing circumstances, as the child develops, will be demanding from it.

Walking and holding some object in the hand simultaneously, requires still a new variant of the schema of walking; a change in the nature of the function of walking also depends upon the shape, weight and so on, of the object being held while walking. And, the development of so many variants of a schema indicates its mobility.

So far, we have been referring to the development of sensorimotor schemas only. By the time the child moves from the stage of infancy to the preoperational stage, generally beginning by the end of second year, the development of the cognitive schemas becomes apparent though this development is rooted in the sensorimotor schemas themselves.

The development of cognitive schemas, marks the beginning of the internalisation of the sensorimotor functions.

For example, looking is a sensorimotor function, and the visual image that is created in the mind of the child, is the internalisation of the same function, and, is a result of the development of a cognitive schema. Interpreting Piaget’s theory, AL Baldwin writes: “cognitive processes are best conceptualized as actions.”

Sensorimotor schemas involve overt actions as cognitive schemas involve the internal function of the mind. The child’s knowing of the number system, and his concept of space, and the laws of logic, all mark that, there must have developed the cognitive schemas for the same. The development of cognitive schemas enables the child to think in abstract.

Thinking is the internalised use of language. Different cognitive schemas rapidly develop, enabling the growing child to perform mental functions beginning from very simple, such as, integrating two different pieces of experience to perform a sensorimotor function, to integrating a whole series of logical thinking to make conceptual conclusions.

During the stage of formal operations, the adolescent child can perform very complex mental functions after going through a long exercise based on the logico- mathematical principles of groups and groupings, and cause and effect relationship.

3. Adaptation: Assimilation and Accommodation:

Adaptation is an important term which Piaget has borrowed from Biology. Adaptation is a process consisted of assimilation and accommodation of something into the structure; a schema is an element of a structure as Piaget explains it.

‘Assimilation’ is ‘taking in’ something from the situation. But an organism would not ‘take in’ or ‘assimilate’ anything unless the situation can activate the schema to do the same; the situation must be motivational for the organism for the initiation of assimilation. And, the situation becomes motivational only when there is some challenge in it.

When challenge ceases to be there in the situation because of the fact that the organism has adapted to that particular aspect of the situation, it would now endeavour to adapt to some other aspect of the situation which demands for the same; it is such a demand of the situation which activates a schema for assimilation.

The next development, after assimilation, in the process, is ‘accommoda­tion’. Accommodation implies changes that take place in the structure or in the schema so that the organism may adapt to the demands of the situation or to the new situation. It was the biological background of Piaget which led him to describe the development of the child in terms of Adaptation, Assimilation and Accommodation.

According to Piaget, it is the schema which assimilates, provided, the situation can activate it for the same; that is, the situation must be motivational for the process to start. A change in the covert or overt behaviour is the result of accommodation and it testifies that the process of adaptation has been completed. Now, we shall take one or two examples by way of illustrations.

First, we are taking an example of general nature, related to the adaptation of system of schemata. A young one of an animal, and so also that of a human being, changes the items of food as it grows older; for the digestion of those new items of food, the system concerned has to accommodate itself.

And, when such an accommodation for the needful changes in the functioning of the system has occurred, the organism may be considered to have covertly adapted to the situation, and a state of equilibrium ensues.

Let us take another example. A child, say of eight months or so, sees a lot of things in the situation. Some are big items, like a table, a pitcher, a stone; and, some are tiny things—a doll, a small piece of paper and a small piece of thread.

Now, which of these items attracts the attention of the child for an attempt to pick up, depends on two factors; first, the thing must be graspable to the child; second, it must contain the element of challenge; that is, the child must not be already perfectly equipped with the skill of picking up the same.

These two factors are essential for the activation of a schema. And, when a schema or a whole system of schema is activated, the existing schema assimilates the ailment or food, which is the experience that the organism has while being active in reference to the new function that the situation demands to be performed.

The performance of that new function entails some changes in the schema or the system of the schemas; the changes thus occurred constitute the Accommodation. And, the result is Adaptation to the new situation.

Adaptation alone can lead to a state of equilibrium. The organism continues to be in a state of un-equilibration until it has developed the skill to perform the new function as demanded by the situation present before it.

This process of adaptation, which includes both assimilation and accommodation, serves motivational as well as learning purposes. One is motivated to be active when the situation poses some challenge, and it contains the element of feasibility with which one can meet the challenge. And, the skill to perform a new function is developed as a result of the action done.

In the Piagetian terms, a schema develops when some action is done through the process of adaptation. Development of each schema is a measure of new learning. Each schema serves as a tool in the process of future development of a new schema.

As regards other terms that are important for understanding the Piagetian theory of child development, we shall be explaining the same while describing the main characteristics of the different periods that the child develops through.

Piaget has divided the child’s development into four periods:

1. Infancy,

2. Pre-operational,

3. Concrete Operational, and

4. Formal Operational. Piaget measures development cognitively.

Each of the four periods of development is characterised by distinct schemas.

Development of the different schemas synchronizes with biological maturity. Piaget has quantified development on the basis of various schemas that distinguish one period from the other. Though time-scheduled development of each Stage has been quantified, nevertheless, it is not the case that there is a water-tight compartment separating one period from the other.  

4. The Period of Concrete Operations (Age Seven to Eleven):

It is during this period that the child’s thought processes become much more stable. The child gives indications of having started thinking reasonably. His placing things in order of size, is an indication of the kind. He can compare two sets of objects, thinking similarities between each object of the one to the corresponding object of the other.

Now, he has developed the perceptual constancy, thereby is able to grasp that spatial rearrangement of objects, does not change the number of objects in the set. He can follow simpler relationships between different classes.

His cognitive schema tells him that the number of members in a class is equal to the total number of members in its sub-classes; or, at least, the number of members in the class cannot be fewer than what it is in its sub-classes altogether.

He also knows the results of reversal of the order of objects, and can tell in advance about the result of the second reversal. He has achieved the schema which can guide him regarding distance between two objects; he can also realise that the length of the path between two objects may be more than the distance that is there between two objects.

For a proper understanding of events and objects, sedimentary conceptions regarding time, space and number must have been achieved, and the period of Concrete Operations, is the time when the same are achieved. The child now starts thinking with some logic. All this, helps in a better understanding of events and objects.

None the less, there are some fields of conceptual limitations for the child during this period. His conception of volume does not happen to be adequately developed, with the result that he fails to realise the relationship between quantity and density in a correct way. Still, the child has no schemas to enable him to arrange situation in his mind before he has actually to confront the same in his environment.

The period of Concrete Operations (between age seven and eleven) is characterised by the following factors:

1. Operational thinking

2. Consistent thinking

3. The groups and the groupings (classes and classifications)

4. Conservation of quantity.

Operational Thinking:

It is during the period of Concrete Operations that the child’s thinking assumes operational character in the sense that an event or object in the environment, not merely evokes some thought but it sets in motion a series of thinking; and it is what is meant here by the term “operational thinking”. The operational thinking is the basic factor, for want of which the remaining three factors, mentioned above, cannot be possible.

Now, thinking relates itself not only to different perspectives but it also integrates the past experiences to the present ones. And thus, the cognitive development moves towards a better perception of objects and events.

But still, the thought process gets started as a result of some concrete object- some object or event in the environment working as a stimulus. The child cannot still start thinking with some idea coming to mind without there being any concrete stimulus in the external world to evoke the same.

Consistent Thinking:

During the Pre-operational Period, sometimes, a child may give the evidence of the presence of logic in his thought process but it is not generally the case. It is only during the Period of Concrete Operations that the child achieves some logic to give direction to his thinking process, though intuitive solution of problems would continue even during the period of adult cognitive development.

Nevertheless, there is consistency in the child’s thinking now. He thinks with some logical sequence, and the thought process becomes organised. The system of thinking becomes stabilized. It happens to be the result of equilibration. All the logical implications of a belief are available to a child whose thought process has thus got organised.

The child gets prepared for the formal operations of the next period as a result of the development of interrelated systems in the organisation of operational thoughts. The behaviour of the child indicates that now, he has started searching for some logic for what he does or says but whatever he says or does, is in reference to something concretely present in the environ­ment.

Notwithstanding, his behaviour is not a mere response of some stimulus in the environment, it is a result, on the one hand, of the unfolding of a maturational process, and, on the other hand, it is a result of the con­catenation of experiences—all aspects of the child’s behaviour are thus active.

His experiences are acquired through his active involvement in responding to the stimuli received from the environment, and, a simult­aneously going on thought process leads to the organisation of operational groupings which Piaget would like to call well-organised cognitive schemas.

Consistency in thinking is the evidence that the thinking process is well-organised now. Every piece of thought is the result either of some new experience or is the natural consequent of the piece of thought preceding the present one—consistent thinking process is the achievement of the third period of cognitive development, that is, the period of concrete operations.

The Groups and the Groupings (Classes and Classifications):

According to Piaget, the organisation of mental actions in operational thinking “can be described in terms of mathematical groups and groupings.” The interrelated intellectual actions are organised into certain sets which have properties of mathematical groups or groupings.

Groups:

In Mathematics, a group is a set of certain elements having relationships with one another based on certain properties. These elements may be numbers, points, people, classes of objects operations or anything. These elements combine together through certain operations.

These operations are numerical or logical operations. The result of an operation may be the combination of two classes into a single one, or, may be the formation of a third number when two separate numbers are added together.

Only such a set of elements can become a group where the elements have been combined under some law. This operation may be multiplication, though in character it may not exactly be an arithmetical multiplication. If the two elements of a group are two events or actions, the two actions or events must follow one to the other in succession, so as to make a combination.

Let us consider an example to illustrate how non-quantitative elements may be treated to form different permutations of the same set. Let us suppose the set to be that of three beads—red, yellow and green in colour respectively. Now, six different permutations are possible on this set: 1. (RGY) 2. (RYG) 3. (RG) (Y) 4. (RY) (G) 5. (R) (GY) 6. (R) (G) (Y).

In each case R goes to G, G goes to Y, and Y goes to R. The elements in the set, are the six rearrangements as shown above. These rearrangements have been possible through a succession of operations conducted on elements.

Here, the elements are not the beads of three different colours but the rearrangements of the same. The set becomes a group where its elements are combined under some law. The operation conducted here should be labelled as “multiplication”.

Many other operations besides thus per-mutating the order of beads are possible. If the path is combining the two regions A and B; and if B is combined to C, then to combine A to C is also possible. But the path from M to N cannot be joined to that from A to B.

Another example of operations may be the rotations of a wheel to 120 degrees, 240 degrees and 360 degrees—the three positions making three elements of the set. There is a relationship among the three elements of the set governed by some law of combination, and thus these three elements of the set make a group.

To assess whether a set is a group or not we should see three things:

1. Is the law of combination applicable to every pair of elements in the set?

2. Is it possible to combine an element with every other element in the set?

3. Is every such combination making an element of itself of the set?

If answers to all these three questions are found to be positive, then the set should be taken to be a group. The set of elements would be closed when the possibility of all the rearrangements has been answered: In the example of three beads, given above, when the six rearrangements have been made, the set is said to have “closed” under that operation.

The second requirement for a group is that it be associative. That is, each element in the set must be so related to each other element of it that the result of any two elements of the set may be associated with the remaining elements making no change in the total value of the set.

To make the point clear, let us take the example of an arithmetical group consisted of three elements as 1 +7 + 8, the total value of which is ’16’; now, it makes no difference whether ‘1’ and ‘7’ are added first, or the result of ‘7’ and ‘8’ is added to’ 1′, the total value of the group will remain unchanged; that is, there is no difference between (1 + 7) + 8 and 1 + (7 + 8).

The third requirement of a group is that one of its elements be called the identity element; adding of it to any other element in the group makes no difference. Zero is such an element, the addition of which to any other element of the group will cause no change. In the set of permutations (R) (G) (Y) is the identity element. It leaves the arrangement just as it was with no effect of a change in the order of beads.

The fourth and the last criterion of a group is that every element of it, has an inverse. The entire set forms a group when a negative integer and an identity element are added to it, the positive integers alone cannot form a group, “they are closed and associative but contain neither an identity element nor inverses”.

As mentioned above, ‘zero’ is an identity element, and if +5 is a positive integer, the negative integer which works as its inverse is -5, all together they make a set, a group of elements. In the set of permutations, there is an inverse for every element. (RGY) and (RYG) are inverses of each other. The combination of the element and its inverse, is the identity element.

Groupings:

Piaget has coined the word ‘grouping’ for a set of operations that has many of the properties of a group but which does not form a ‘group’. When a larger class is divided into such classes, there occurs a sequence of legitimate operations which can be undone through another sequence of operations.

Thus, there is closure as well as reversibility, and the system of operations resembles a group in some respects. But actually, the system of operations does not form a group because only certain classes may be legitimately combined.

As for example, the child during the period of Concrete Operations, can understand that setters can be combined with dogs of other breeds to form the class—domestic dogs; these dogs can be combined with various wild dogs, and they together make the genus called canis, but there can be no class representing the combination of domestic dogs and non-carnivorous mammals.

Thus, because of lack of certain things, and because of certain resemblances with the characteristics of groups, Piaget has used the term ‘grouping’ for the system of operations leading to it.

One can realise the characteristics of closure, completeness and reversibility in the system leading to such groupings, by the time one is in the range of the second half of the period of Concrete Operations. Piaget distinguishes eight types of groupings and he considers their role to be very significant in the cognitive development of the child during the period of Concrete Operations commencing from the age of 7 years.

Conservation of Quantity:

The child’s perception regarding size, volume or mass happens to be illusionary till he attains the stage of Concrete Operations. The example of famous Piaget-Taponier experiment or Muller-Lyer illusion, has already been given previously which illustrates how the two sticks appeared to be of unequal size when the place or the physical position of one of them was changed.

Likewise, the child declared the liquids to be unequal when the liquid from one of the two containers was poured into a container narrower in size but one having more height.

Though the changes were made in the presence of the child even then he could not perceive or have the conception of the constancy or conservation of size, mass or volume because a child up to that period of age cannot integrate his past experience with that of the present one.

For, the development of the conception of conservation of quantity, size, mass or volume or of weight without the same being affected by the change of place, or due to the passage of time, is possible when the child is able to integrate his past experiences with that of the present ones.

Mass or volume of an object remains unchanged even when its shape is changed. The example given is that of two chapatti-like structures of clay, which are equal in mass. But when one of the two structures, is reshaped as a roller, the child declares that now its mass has grown more in quantity.

This conception is illusionary, illusion being caused because of this structure’s length being much more than that of the other. Even when this reshaping is done in the presence of the child, he will not be able to perceive the conservation of mass.

When an object is broken into parts or pieces, the weight is conserved, that is, the weight of the parts, generally, remains the same as was that of its previous intact position. It is after many years of maturity that the child obtains the conception that an object loses its weight in proportion as it is carried to a point higher and higher, away from the centre of the gravitational force.

An object becomes almost weightless when it is on the moon, or high in the space. Nevertheless, conservation of quantity as a perception, is an important cognitive achievement of the period of Concrete Operations.

5. The Period of Formal Operation:

The period of formal operations is the period of early adolescence. It is the period when cognitive developments reach to their adult or highest levels. Now, the adolescent has the fundamental grasp that “underlies the logical thought.”

His mode of drinking appears to have undergone a major change from accidental thinking to a sequential causal thinking which enables him for scientific experimentation, and for deducing proper implications.

During the period of Concrete Operations, the child sorts objects of the concrete world into groupings but he fails to understand the relationships which are there among different groupings. It is during the period of Formal Operation that the adolescent starts understanding such relationships.

The main characteristics of this period are:

1. Understanding logical relations among classes;

2. Combinatorial thinking;

3. Achieving some additional concrete constructions.

Understanding Logical Relations among Classes:

During the period of Concrete Operations, the child is able to divide the numerous objects of the concrete world into classes. This is his way of perceiving objects in reality and with greater details. But the relationships that exist among classes can be grasped only during the fourth period of cognitive development, which Piaget has named as the period of Formal Operations.

Formal means “of constructs”; the child has now entered into the stage of adolescence, and it is during this period that constructs or conceptions develop and become the bases for abstract thinking or logical thinking. An adolescent, by the period of 11+, starts thinking of relationships that exist among the various classes that he distributes the objects into.

Thus, his thinking assumes more and more abstract forms, based on concepts related to concrete objects: dealing with mental constructs rather than only with the concrete forms of the objects. Now, one starts surveying the logical relationships among classes that he has distributed the objects of the concrete world into.

During the period of Formal Operations, the adolescent becomes cognitively so developed that he can visualize the possible relationships that may develop in future among different objects; and so, can design future plans accordingly.

Such operations of the mind, are the formal operations, are of indispensable importance in scientific surveys or experimentations. Importance of a factor in the causation of an event may be established in advance when one can conduct Formal Operations.

So far, the child was able to realise simple relationships that existed among the events that presented themselves to him but now, he can design or create situations required to establish the needful relationships among the events and thus, now he is able to confirm or disconfirm a hypothesis.

Through a number of examples, AL Baldwin has tried to make clear how logical operations enable one to obtain the matrix of possibilities through multiplication of two characteristics together. The Fig. 19.3, given here, illustrates this matrix of possibilities.

In this figure, two classes of objects A and B are intersecting each other in such a way that four divisions have been made:

One area is both A and B; the area where A cross-hatches in one direction, and B in the other

(The area where one circle overlaps the other).

Then there is area which represents only the object A.

There is another area which represents only B.

And, there is one more area, the fourth one, which is neither A nor B.

“These are the four cells in the two-by-two matrix that emerge when two classifications (A versus non-A) and (B versus non-B) are multiplied together.”

Now, if we think these four as elements of statements, the four positions are:

p represents where the statement is true.

P̅ represents examples where the statement is not applicable. q represents another statement about the same class that the statement p is related to; and is true.

q̅ represents examples where the q̅ is true and q is false. The same situation has been expressed through another (Fig. 19.4).

Piaget has given such logico-mathematical examples to illustrate how an adolescent during the 11+ period has his faculty developing with age and enrichment of environment, so far formal cognitive operations are concerned. One grows continually more able to understand his environment, and can perform better in relation to his environment, his mental faculty for formal operations being more developed.

It is, thus, that an adolescent grows proficient in solving very complicated problems through a consistent sequential thinking in an abstract way. AL Baldwin in his book Theories of Child Development has given the truth table for 16 binary operations (Fig. 19.5)—This shows how in due course an individual cognitively develops so much so that he can perform formal operations of such a complicated nature.

Combinatorial Thinking:

Combinatorial thinking is the second important characteristic of this period. It is this ability of combinatorial thinking that enables the adolescent to go through all the choice sequentially or systematically. One, during this period is cognitively so developed that he can cover all the different aspects of an issue, and, can exhaust all possibilities relating to the development of an event.

One can obtain the needful information using the entire system of formal logic; sometimes even intuitively. Formal Operations always happen to be based on formal logic which enables one to survey exhaustively the possible relationships that may be there among classes of different objects.

The truth table for sixteen binary operations (Fig. 19.5) is the result of logical thinking which illustrates how combinatorial thinking can enable one to understand the relationships between two statements of a fact exhaustively. It is not enough for the child to understand the relationship among the sixteen binary operations.

First, each of the four classes has to be taken systematically, one at a time. Then, every possible pair of classes has to be described. Then, every possible triplet of classes is considered, and finally comes the combination of the quadruplet of classes. Such a combination is possible only because of a certain kind of thought process.

Each binary operation represents a kind of possible combination. The thought process enabling one for one, such an operation is possible only during the period of formal operations. An adolescent during this period only can be cognitively developed enough to appreciate the difference between all the combinations that occur; and he can also acquire ability to process the entire range of possibility.

I shall describe here the experiment devised by Piaget and Inhelder to illustrate how formal operations of combinatorial thinking lead one to certain conclusions.

There were four bottles containing some colourless liquid in each; and they were labelled as 1,2,3 and 4. The first bottle contained sulphuric acid; the second water; the third oxygenated water; and fourth, thiosulfate. There was a smaller vial which contained potassium iodide, it was labelled as g.

The subject was presented two glasses; in one of them there were liquids from bottles first and third; and in the other it was from the second. While the subject was watching, the experimenter dropped liquid from the vial ‘g’ into each of these two glasses. The liquid in the first glass (which contained liquids) both from bottles 1 and 3 turned yellow, while in that of the other glass, no change in colour occurred.

Later, the child was asked to describe how each liquid had functioned in effecting change in colour. The child could tell the function of each liquid only on the basis of trial—watching the effect of each liquid separately, and in combination with some other liquid, upon some other liquid. The child by no way, can tell of such a change ahead of a concrete operation.

The trial and error is an important method in developing conceptual schemas in such a situation. The combinatorial thinking is an important characteristic of the period of Formal Operations. As here, the subject enhances his conceptual knowledge through the function of combining liquids.

Likewise, adolescent during the period of Formal Operations, thrusts ahead cognitively by way of combining different pieces of information to come to new conclusions. This is combinatorial thinking— pieces of a wide range of knowledge related to far off places and times are combined together with the result of the development of new/advanced cognitive forms/schemas during this period.

To achieve the result, all possible combinations may have to be tried. Ability to form systematic combinations is a characteristic of this period. If a subject belongs to this period of cognitive development, then only he can have the ability to cover systematically all the possibilities; in the example given above, he will try g with each.

Achieving of Some Additional Constructs:

It is during this period that the child is able to understand the problem of equilibrium. In such a problem there happens to be two different inverses leading to the development of the 4 groups. In Fig. 19.6 there is a beam, on both sides of its fulcrum, the weights are equal; the weights are at equal distance, too. But with the addition of weight on any one of the two sides, the beam is thrown out of balance.

There are several ways for the restoration of balance. The balance may be restored by inverting the previous function; that is, by subtracting the weight previously added. The other way for the restoration of balance or equilibration is to add weight to the side where it is less.

To resolve the problems of equilibration, the same double inverse is resorted to in a number of fields. By way of another example, the velocity of an object depends upon the distance it covers over a period of time.

The velocity is said to have increased in the proportion as the time taken remains the same in spite of there being increase in the distance covered. To bring the object to its original velocity either the time factor is increased or the distance covered is decreased.

The problem of equilibration may likewise be resolved in cases of relationships in the fields of gravity, volume and weight of a liquid. The simple law of proportion works everywhere—one variable equals the quotient of two other variables. We can restore the value of fraction to its original one either by changing the numerator or the denominator.

Two kinds of operations are involved in the solution of an equilibration problem. These two operations are inverses of two kinds—a negation of the original change or a reciprocation which changes some compensating factor. An actual mathematical group is the result of such a system of negation or reciprocation. The Fig. 19.7, illustrates this relationship.

The total operations involved here are—negation, reciprocation, correlation and identity, these four have already been explained in some other relevant part of the book. Negation attains a balance through subtraction; reciprocation through addition; the relationship between the opposite comers is that of correlation—the comers are the equivalent results of negation and reciprocation.

A clear cognitive improvement during the period of Formal Operations is witnessed over that of the Concrete Operations in the sense that during the period of Formal Operations, the child acquired the ability to understand relations of different groupings to one another. He can also have a preview of possibilities that may develop in future.

It is because of combinatorial thinking, during the period of Formal Operations that one can have a full understanding of the full implication of prepositional thinking as well as that of the equilibrium problems. His understanding of the variety of relations that exist among different classes of elements also becomes complete during this period.