course, be worked out by trial and experiment.
They will probably vary in different schools and from year to year.

INDUSTRIAL MATHEMATICS

Of the hundreds of employers who were interviewed by members of the
Survey Staff as to the technical equipment needed by beginners in the
various trades, nearly all emphasized the ability to apply the
principles of simple arithmetic quickly, correctly, and accurately to
industrial problems. Many employers criticized the present methods of
teaching this subject in the public schools. In the main their
criticisms were to the effect that the teaching was not "practical."
"The boys I get may know arithmetic," said one, "but they haven't any
mathematical sense." Another cited his experience with an apprentice
who was told to cut a bar eight and one-half feet long into five
pieces of equal length. He was not told the length of the bar, but was
given the direct order: "Cut that bar into five pieces all of the same
size." The boy was unable to lay out the work, although when asked by
the foreman, "Don't you know how to divide 81/2 by 5?", he performed the
arithmetical operation without difficulty. The employer gave this
instance as an illustration of what to his mind constituted one of the
principal defects of public school teaching. "Mere knowledge of
mathematical principles and the ability to solve abstract problems is
not enough," he said. "What the boys get in the schools is
mathematical skill, but what they need in their work is mathematical
intelligence. The first does not necessarily imply the second."

This mathematical intelligence can be developed only through practice
in the solution of practical problems, that is, problems which are
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 lon