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 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.