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156               THE DISCOVERY OF THE CHILD

mathematical analysis of the thing. One could have grasped the
idea of a square form without knowing how to count up to four,
.and therefore without finding out the number of sides and angles.
Sides and angles are abstractions which do not exist of themselves;
what does exist is a piece of wood of a definite shape. Again,
the lengthy explanations of the teacher not only confused the
child's mind, but crossed that abyss which separates the concrete
from the abstract, the shape of an object from mathematics.

Suppose, I said to the teacher, that an architect was showing
you a cupola, the form of which interested you. He might give
you two illustrations. He might point out to you the beauty of
the surroundings, the harmony of the parts; might make you
.ascend and climb round the dome itself in order to appreciate the
relative proportions of its parts, so that the appearance of the
whole should be realized, and then recognized and believed in.
Or he might make you count the windows, the wide and the
narrow cornices and finally make a drawing of the structure, to
illustrate the laws of stability and to teach you the algebraic
formulae necessary to be resolved for the calculations relative to
these laws. In the first case you would visualize the form of the
cupola; in the second, you would understand nothing and instead
of an impression of the cupola you would get one of this architect
who imagined that he was talking to engineering colleagues instead
of to a lady who was travelling for amusement. The case is just
the same. Instead of saying to the child, " This is a square,"
and simply make him touch it and ascertain its material outlines,
we proceed to the geometrical analysis of it. We believe that
it is premature to teach plane geometrical forms to the child,
just because we associate them with the mathematical concept.
But the child is not incapable of appreciating simple form; in fact,
he can see square windows and tables without making any effort;
his eye rests on all the forms round about him. To direct his
-attention to one particular form is to make it stand out clearly
.and to fix an idea of it. In the same way, we ourselves may
"be standing on the margin of a lake, looking at its shores