the street and points at it and the parent says “dog” and the child learns
that the furry four-footed animals are called “dogs.” The child sees a horse
and cries out “dog” and the parent says “horse” and the child learns to
differentiate different categories of mammal.
In Piagetian terms, the early stages of intellectual element occur in
babyhood and are related to the growing capacity of the child to control
its body. This is the sensory-motor stage when the child learns to move
and to distinguish its own body from the rest of the physical universe. Its
actions are repeated until they are perfected and, after this stage, the child
transitions to what Piaget calls the concrete operational stage, when the
child manipulates physical objects and builds up an understanding of their
interrelations. For instance, fi ve toy bicycles are put on the desk with fi ve
toy riders nearby. The child begins to match up the riders with the bicycles
to appreciate that there is the same number of each set. The child takes a
tall glass of water and pours the water into a short fat glass. The preop-
erational child assumes that there is more water in the tall glass because
the water level is higher but, in the concrete operational stage, the child
can imaginatively reverse the operation and pour the water back from
the short fat glass into the tall thin glass and so learns “conservation of
volume.” These intellectual processes are not rationalistic in the sense of
being spun out by ratiocination but emerge through interactivity between
the child and the environment.
When a child sees a vehicle, the parent says “car” and eventually the
child learns that a whole group of objects can be placed into the category
“car.” This is a Ford car but that is a Chevrolet. The child, aged between
about 7 and 11, who thinks in a concrete operational way, is able to classify
objects on the basis of their appearance and to subdivide classes of objects.
If the class of cars contains the class of Fords and the class of Chevrolets
and the class of cars which are neither Fords nor Chevrolets, the child
is able to understand that the class of Chevrolets equals the class of cars
minus the class of Fords and the class of cars which are neither Fords nor
Chevrolets. This sort of logic can also work with simple relations: A is big-
ger than B and B is bigger than C so A is bigger than C, but this is really
another way of expressing the logic of classes because B and C may be
thought of as subclasses of A.
At the beginning of secondary education, at about the age of twelve,
the child is expected to enter the stage of formal operational thinking.
There are two main differences here. First, formal operational thinking is
formal in the sense that it deals with the shape of arguments. It can see
270 W.K. KAY