Developmental psychology was described above as the link between the two apparently unrelated subjects of biology and epistemology. Developmental psychology is concerned with the structure of the nervous system, whereas biology is concerned with its function and epistemology with its content. Let us look in turn at function, structure and content to see how structure links function to content.

Piaget, like Darwin, considers the function of the nervous system to be the adaptation of the organism to its environment. Adaptation involves the complementary processes of assimilation and accommodation. The nervous system assimilates information from the environment and, if necessary, accommodates to that information.

The process of cognitive development is like the progress of a worm, as it stretches its front forward and then pulls its back up. The alternating stretches and pulls of the worm correspond to the alternating assimilations and accommodations of the person. But what moves us to stretch and pull? The spate of evidence for a need for stimulation and a need for consistency, as presented in Section 3.2, suggests the organic basis for assimilation and accommodation, respectively. We assimilate information from our environment because we have an need for stimulation and we change to accommodate that information, if it is not consistent with the information already stored, because we have a need for consistency.

Because of the need for consistency, the stored information is organized into a structure. Input information, which is inconsistent with this stored information, forces changes in this structure if this new information is to be accommodated. When the structure changes qualitatively, the child is said to move into a new stage. There are three broad stages corresponding to three basic structures - the sensorimotor stage (up to 2 years), the concrete operations stage (2 to 11 years) and the formal operations stage (after 11 years). This is, of course, a very bold, bald statement - there are substages and sub-substages within each stage, the transition from one stage to another is not abrupt, and the stages are not tied rigidly to ages.

The content of your nervous system is the product of this process of alternating assimilations to and accommodations of those evolving structures. Your content is all the information you have assimilated from your environment. This information is not, however, simply poured in - it is assimilated into an organized structure of previous information and that structure may have to change to accommodate it. Thus, structure serves to mediate between function (adaptation through alternating assimilations of and accommodations to information from the environment) and content (information assimilated by the person from the environment).

Now that we have the developmental psychology of Jean Piaget firmly anchored, on the one side, to biology, through the function-structure relationship, and, on the other side, to epistemology, through the structure-content relationship, let us take a closer look at it. We have learned so far that, as a different structure emerges, the child is said to move into a different stage, and that the three major stages are sensorimotor, concrete operations, and formal operations. Here is a description of you during each of those stages.

In the sensorimotor stage, you live in the here and now. Not because your guru recommended it but because you have no option. Living in the there and then requires some internal representation of things which are not here and now, and you have as yet no such internal representation of your environment. You are interested in your environment only insofar as it is doing things to you (sensory) and you are doing things to it (motor) - that is, you are in your sensorimotor stage. The behaviorists are right. You are an S-R organism - but only up to the age of 2. Your task during your first two years is to co-ordinate stimuli and responses - stimulus with stimulus, stimulus with response, and response with response. Let us look at each in turn.

Co-ordinating stimulus with stimulus involves the acquisition of the object concept and of object constancy. You now perceive your world as composed of objects, which continue to exist even when you are no longer looking at them (object concept) and which remain the same despite the different ways you look at them - that is, they remain the same size despite variation in your distance from them, the same brightness despite variation in the illumination on them, and the same shape despite variation in your orientation to them (object constancy). That is, you believe that the Statue of Liberty is still in New York harbor, where you last saw it, even though you are now back home in Klamath Falls, Oregon, and you believe that it will continue to be 150 feet high, even if it were shipped back to France.

It is difficult to imagine how the world could be perceived otherwise. Yet children do, and you as a child did. When Piaget presented his daughter Jacqueline with a bottle, she made the appropriate reaching motions and gurgling sounds as long as she could see it, but lost interest in it when he put it behind his back. Out of sight, out of mind. This faith that an object exists independently of you (object concept) is a prerequisite to the faith that it conserves its size, brightness and shape despite variation in distance, illumination and orientation (object constancy). It was one of your accomplishments during your sensorimotor stage.

Stimulus and response get co-ordinated through a series of circular reactions. Between 1 and 4 months, you develop primary circular reactions. You happen to make the sound "ah" (response), you hear the sound (stimulus), you imitate the sound (response), you hear the sound again (stimulus), and so on as you imitate yourself over and over again.

Between 4 and 8 months, you develop secondary circular reactions. You happen to make the sound "dah" while doting daddy is present (response), you see daddy jumping up and down with excitement at being recognized (stimulus), you repeat the sound (response), daddy jumps up and down some more (stimulus), and so on in that eternal process by which two generations continue to condition one another. Whereas your primary circular reaction was centered on yourself, this secondary circular reaction is centered on your environment. It is, in Piaget's words, "behavior designed to make interesting sights and sounds last ". You have started to act on your environment. However, your behavior is not yet intentional, because you have not yet disentangled your means (your response) from your ends (to make interesting sights and sounds last). You shake your rattle in order to produce those interesting sounds but, if your rattle were taken away, you would continue to shake your arm.

Between 12 and 18 months, you develop tertiary circular reactions. As before, you repeat a behavior which has an interesting effect, but you vary that behavior to discover what changes there will be in that effect. Thus, your babbling is not just "ah, ah, ah, ah, ah" as you imitate yourself in the primary circular reaction, nor just "dah, dah, dah, dah, dah" as you perpetuate an interesting effect in the secondary circular reaction, but "ah, dab, mah, maaah, damaa, daaa" as you vary your response to test their effect on your environment. You are conducting experiments; you are beginning your career as a scientist. You are not simply behaving in order to behave but are behaving in order to check out your environment. Behavior was an end in itself but is now a means to another end.

One President of the United States is reported to have said of another that he couldn't both walk and chew gum at the same time. This is true of all of us at birth. You learn to co-ordinate your responses during the sensorimotor stage. You are born with a small repertoire of inborn responses or reflexes - you can look at things, you can reach for things, you can grasp things, you can suck on things. During your first month, you exercise each of those reflexes individually - you look at, reach for, grasp, or suck on things. However, you can't simultaneously look at and reach for a thing or successively look at, reach for, grasp, and suck on a thing. Your set of isolated responses are not organized into systems of responses until about 8 months. You know things in your environment purely in terms of your organized systems of responses to them. By 18 months, you begin to represent those things by a organized system of internal responses. Such an internal representation of your external environment heralds the beginning of mental life and the end of the sensorimotor stage.

As you move from the sensorimotor to the concrete operations stage, your behavior is determined not only by your external environment but also by your internal representation of your external environment. This internal representation is made possible by the development of the symbolic function. That is, you learn to represent an object in your environment by something else. An object (for example, a gun) may be represented by another object (a stick), a gesture (pointing a finger), a sound (bang! bang!), an image (a drawing of a gun), or a word ("gun"). Note that the word is only one of many possible representations of an object. The development of the symbolic function is a necessary condition for the development of language and not vice versa, as some theorists have argued.

The concrete operations stage can be subdivided into pre-operational and operational substages. Sometimes, the pre-operational substage is presented as a separate stage. However, I prefer to include it within the concrete operations stage, since what is essentially a pre-sensorimotor substage is included within the sensorimotor stage. Just as the sensorimotor child must practice the isolated responses before organizing them into a system, so the concrete operations child must practice the isolated internal representations before organizing them into a system. The pre-operational substage may perhaps be best considered as this preparatory phase of practice.

The pre-operational substage can be described in terms of the "mistakes" you make as you first explore your new, confusing world of symbols. Piaget illustrates two such mistakes in the following anecdotes about his children [PIAGET 1951]:

When walking with his son, they passed a snail.
"There is a snail", said his son.
Later, they happened to pass another snail.
"There is the snail again" said his son.

Jacqueline, seeing her sister Lucienne in a new bathing suit with a cap, says to her mother:
"What's the baby's name?"
Her mother explained that it was a bathing costume.
However, Jacqueline pointed to Lucienne's face and asked: "But what's the name of that?" repeating the question many times.
As soon as Lucienne had her dress on again, Jacqueline said very seriously:
"It's Lucienne again" as if her sister had changed her identity by changing her clothes.

In the following experiment, Piaget illustrates another competence you gain as you move from the pre-operational to the operational phase of the concrete operations stage [PIAGET 1952]. Children are shown 10 dolls of differing heights and 10 miniature walking sticks also of differing lengths. They are asked to arrange dolls and sticks "so that each doll can easily find the stick that belongs to it ". Pre-operational children could not place the dolls or the sticks in order. They seemed to lack any organizing principle for ordering objects - for example, finding the tallest doll, then the next tallest doll, and so on until the series is complete. They did not understand the principle underlying this procedure that, if doll A is taller than doll B and doll B is taller than doll C, then doll A is taller than doll C.

There was a transitional stage during which the child could order the dolls and the sticks but could not assign the sticks to the dolls unless they were lined up evenly with them. If Piaget squeezed the row of sticks closer together or spread them further out, then they could no longer tell which stick belonged to which doll. Finally, in the operational substage, the child could order both the sticks and the dolls and could assign the correct stick to each doll. At this point, you can place objects along dimensions and consider the relationship between objects along those dimensions. You are capable of ordinal reasoning.

In summary, at the end of the concrete operations stage, you are capable of both class reasoning and ordinal reasoning. That is, you can place objects in your environment into classes and along dimensions. On this firm foundation of logic, you are now ready to build the superstructure of mathematics. Now that you know "doll A is taller than doll B" and "doll B is taller than doll C" imply that "doll A is taller than doll C", you may be interested in making the more precise statement than "doll A is x units taller than doll B" and "doll B is y units taller than doll C" implies that "doll A is x plus y units taller than doll C". The natural numbers (1, 2, 3, --) make this possible. By placing those numbers evenly along your dimension, you can now see that dolls A, B, C are, let us say, 9, 7, and 4 units tall, respectively. Thus, "doll A is 2 units taller than doll B" and "doll B is 3 units taller than doll C" which implies that "doll A is 5 units taller than doll C".

Within this system of natural numbers, you can add and always get an answer. However, when you reverse this operation and subtract, you can't always get an answer. Mathematicians invented zero to provide an answer when you subtract a number from itself and negative numbers to provide an answer when you subtract a number from a smaller number. When you multiply within this enlarged system of numbers, you can always get an answer. However, when you reverse this operation and divide, you can't always get an answer. Mathematicians invented fractions to provide an answer when you divide a number by another number which does not divide evenly. When you square within this system of numbers, you can always get an answer. However, when you reverse this operation and take the square root, you can't always get an answer. Mathematicians invented irrational numbers to provide an answer when you take the square root of a number which is not a perfect square.

In this way, more and more systems of numbers are invented to permit closure under more and more sophisticated operations and thus the superstructure of mathematics is erected. However, no matter how esoteric it becomes, it still rests on the foundation of class and ordinal reasoning and cannot be assimilated by you until you have mastered those basic logics. Having mastered concrete operations, you now have the basis for learning mathematical operations. You cannot yet deal, however, with some of the products of those operations - with negative numbers, irrational numbers, and imaginary numbers (the square root of irrational numbers), because there is no concrete equivalent to them. You can deal with 5 oranges but not with -3 oranges or Ö7 oranges or Ö-2 oranges. You can see, smell, touch, and taste an orange but not a negative, irrational, or imaginary one.

If I were to hand you three dolls called Mary, Sue and Jane with Mary being taller than Sue and shorter than Jane, then you would be able to line them up in order of height and tell me that Jane is the tallest of the three. However, if I were to tell you that "Mary is taller than Jane and shorter than Sue" and asked you "Who is the tallest of the three?", you would have great difficulty in answering. You can solve the problem in your hands but not yet in your head, in much the same way as, at one point, you could count on your fingers but not in your head. That is, you can perform operations on concrete objects but not yet on propositions about those objects. Your internal representation of your environment helps you deal more effectively with it but you are not yet emancipated from it. You are capable of class reasoning and ordinal reasoning but not yet of propositional reasoning.

The following one-minute course in propositional reasoning will serve to show what you gained when you moved from the concrete operations stage to the formal operations stage. First, try the short test in propositional reasoning provided in Figure 6-1.

Let us imagine an empty universe and let us introduce into it the proposition. In the beginning, there was the proposition. At the risk of appearing unduly familiar at such early acquaintance, let us call it simply p. Propositions come in many guises - today is Tuesday, Now is the time for all good men to come to the aid of Jennifer 343-7020, Kafka is a kvetch, E equals M.C squared, and so on - but they all have in common the fact that they may be said to be either true or false. Let us represent this fact, in shorthand, as p or -p, where p means proposition p is true and -p means proposition p is false. Our proposition looks lonely all alone in its empty universe. Let us then introduce another proposition, q, which, like all propositions may be said to be either true or false. If those two propositions get together, as inevitably they must all alone in their empty universe, they will generate four possible states of affairs:

p is true and q is true
p is true and q is false
p is false and q is true
p is false and q is false

p.q or p.-q or -p.q or -p.-q

A certain subset of propositions (for example, "This is a typewriter", "I am a Scotsman") state that a particular object is a member of a set. Because such propositions occur so often, the cumbersome "If X is a member of set A, then X is a member of set B" is compressed to "All As are Bs". The relations "Some As are Bs" and "No As are Bs" can be generated in the same way. Class reasoning, involving those relations "all", "some" and "no", permits you to place the things in your environment into sets and consider the relationships among those sets. The subset of propositions stating the position of a thing on a dimension generates, in a similar way, the relations "is greater then", "is equal to", and "is less than". Ordinal reasoning, involving those relations, permits you to place the things in your environment along dimensions and consider the relationships between their positions on those dimensions. In your concrete operations stage, you could deal only with those two limited subareas of propositional reasoning - class reasoning and ordinal reasoning - because they refer directly to things in your environment. In your formal relations stage, you learn how to operate with propositions as well as with things and can thus deal with all propositional reasoning.

You may now use the following strategy to check your answers to the short test in propositional reasoning in Figure 6-1: The premises (the propositions after SUPPOSE YOU KNOW THAT) eliminate certain alternatives from the four possible alternatives, and the status of the conclusion (the proposition after THEN WOULD THIS BE TRUE?) is determined by the remaining alternatives.

If it is contained in all the remaining alternatives, then it must be true (YES);
if it is contained in some of the remaining alternatives, then it may be true (MAYBE);
if it is contained in none of the remaining alternatives, then it can't be true (NO).

1 YES 2 MAYBE 3 MAYBE 4 NO.

The Jean Piaget Archives, published in 37 volumes, contains over 50 books and over 500 articles [JEAN PIAGET FOUNDATION]. It is presumptuous to try to compress this huge body of data based on a lifetime of work into a chapter and it is very presumptuous to try to summarize it in a paragraph. Here is a very presumptuous paragraph.

Your cognitive development was not, as it appears when you look back on it, a gradual accumulation of information and skills. It was a series of revolutions in which you moved from one stage to another. "Each stage is qualitatively different from every other, but each results from the one that preceded it, and prepares (you) for the one that follows it" [LEFRANCOIS]. First, you deal directly with your environment (sensorimotor stage), then with propositions about your environment (concrete operations stage), and then with propositions about propositions (formal operations stage), You free yourself from the tyranny of your environment by acting on it and, thereby, building up an internal representation of it. Your behavior is subsequently determined not only by your objective environment but also by your subjective map.