stated in the every day terms of the working world and which require
the student to go through the successive mental steps in the same way
that he would if he were working in a shop. The problem referred to
above is one of division of fractions. If we state it thus: "81/2÷5,"
the pupil takes pencil and paper, performs the operation and announces
the result. If we say, "A bar 81/2 feet long is to be cut into five
pieces of equal length; how long should each piece be?", the problem
calls for the exercise of greater intelligence, as the pupil must
determine which process to use in order to obtain the correct result.
It becomes still more difficult if we merely show him the bar and say:
"This bar must be cut into five pieces of equal length; how long will
each piece be?" Several additional preliminary steps are required,
none of which was involved in the problem in its original form. Before
the length of the pieces can be computed he must find out the length
of the bar. He must know what to measure it with, and in what terms,
whether feet or inches, the problem should be stated. Again, if we
say: "Lay this bar out to be cut in five equal lengths," another
step--the measurement and marking for each cut--is added. Many
variations might be introduced, each involving additional
opportunities for the exercise of thought.

It is through practice in solving problems of this kind that the pupil
acquires what the employer called mathematical intelligence. It
consists in the ability to note what elements are involved in the
problems and to decide which process of arithmetic should be used in
dealing with them. Once these decisions are made the succeeding
arithmetical calculations are simple and easy. In technical terms the
ability that is needed is the ability to generalize one's experiences.
In every-day terms it is the ability to use what one knows.