Intelligence as Arising from Cognitive Structures and Processes 145reverse of addition and division is the reverse of multiplica-
tion. The child can go to the store and back home again or
trace out a route on a map and see the way back.
The period is labeled as one of “concrete” operations be-
cause operations are performed for objects that are physically
present. A major acquisition of the period is conservation,
which involves a child’s recognizing that objects or quanti-
ties can remain the same despite changes in their physical ap-
pearance. Suppose, for example, that a child is shown two
glasses, one of which is short and fat and the other of which
is tall and thin. If a preoperational child watches water poured
from the short, fat glass to the tall, thin one, he or she will say
that the tall, thin glass has more water than the short, fat one
had. But the concrete-operational child will recognize that
the quantity of water is the same in the new glass as in the old
glass, despite the change in physical appearance.
The period of formal operations begins to evolve at
around 11 years of age and usually will be fairly fully devel-
oped by 16 years of age, although some adults never com-
pletely develop formal operations. In the period of formal
operations, the child comes to be able to think abstractly and
hypothetically, not just concretely. The individual can view a
problem from multiple points of view and can think much
more systematically than in the past. For example, if asked
to provide all possible permutations of the numbers 1, 2, 3,
and 4, the child can now implement a systematic strategy for
listing all of these permutations. In contrast, the concrete-
operational child will have essentially listed permutations at
random, without a systematic strategy for generating all of
the possible permutations. The child can now think scientifi-
cally and use the hypothetico-deductive method to generate
and test hypotheses.
Vygotsky and Feuerstein’s Theories
Whereas Piaget has emphasized primarily biological matura-
tion in the development of intelligence, other theorists inter-
ested in structures, such as Vygotsky (1978) and Feuerstein
(1979), have emphasized more the role of interactions of in-
dividuals with the environment. Vygotsky suggested that
basic to intelligence is internalization,which is the internal
reconstruction of an external operation. The basic notion is
that we observe those in the social environment around us
acting in certain ways and we internalize their actions so that
they become a part of ourselves.
Vygotsky (1978) gave as an example of internalization the
development of pointing. He suggested that, initially, point-
ing is nothing more than an unsuccessful attempt to grasp
something. The child attempts to grasp an object beyond his
reach and, initially, is likely to fail. When the mother sees the
child attempting to grasp an object, she comes to his aid and
is likely to point to the object. He thereby learns to do the
same. Thus, the child’s unsuccessful attempt engenders a re-
action from the mother or some other individual, which leads
to his being able to perform that action. Note that it is the so-
cial mediation rather than the object itself that provides the
basis for the child’s learning to point.
Vygotsky also proposed the important notion of azone of
proximal development,which refers to functions that have not
yet matured but are in the process of maturation. The basic
idea is to look not only at developed abilities but also at abil-
ities that are developing. This zone is often measured as the
difference between performance before and after instruction.
Thus, instruction is given at the time of testing to measure the
individual’s ability to learn in the testing environment
(Brown & French, 1979; Feuerstein, 1980; Grigorenko &
Sternberg, 1998). The research suggests that tests of the zone
of proximal development tap abilities not measured by con-
ventional tests.
Related ideas have been proposed by Feuerstein (1979,
1980). Feuerstein has suggested that much of intellectual de-
velopment derives from the mediation of the environment by
the mother or other adults. From Feuerstein’s point of view,
parents serve an important role in development not only for
the experiences with which they provide children but also
for the way they help children understand these experiences.
For example, what would be important would be not so much
encouraging children to watch educational television or tak-
ing children to museums but rather helping children interpret
what they see on television or in museums.EvaluationBy any standard, Piaget’s contribution to the study of intelli-
gence was profound. First, his theory stands alone in terms of
its comprehensiveness in accounting for intellectual develop-
ment. There is no competition in this respect. Second, even
the many individuals who have critiqued Piaget’s work have
honored the work by deeming it worthy of criticism. To the
extent that a theory’s value is heuristic, in its giving way to
subsequent theories, Piaget’s work is almost without peer.
And much research today, especially in Europe, continues in
the tradition of Piaget. Neo-Piagetians, although they have
changed many of the details, still build upon many Piagetian
theoretical ideas and tasks for studying development. Third,
even the most ardent critics of Piaget would concede that
many of his ideas were correct. Many of those ideas, such as
of centration, conservation, and equilibration, remain alive
today in a wide variety of forms. Fourth, Piaget provided an
enormous database for developmental psychologists to deal