Wage Earning and Education

by R. R. Lutz

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 8½ 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: "8½÷5," the pupil takes pencil and paper, performs the operation and announces the result. If we say, "A bar 8½ 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.

The work in applied mathematics should cover a wide range of problems worded in the language of the trades and constantly varied in order to establish as many points of contact as possible between the pupil's knowledge of mathematics and the use of mathematics in industrial life. Practical shop work is one of the best means to this end. The trouble with much of the shop work given in the schools is that it runs to hand craftmanship in which the object is to "make something" by methods long ago discarded in the industrial world, rather than to give the pupil exercise in the sort of thinking he will need to do after he goes to work. Successful teaching does not depend so much on the use of tools and materials as on the teacher's knowledge of the conditions surrounding industrial work and his ability to originate methods for vitalizing the instruction in its relation to industrial needs.

Mechanical Drawing

At the present time the junior high school course provides for one hour a week of mechanical drawing. All the boys who may be expected to elect the industrial course can well afford to devote more time to drawing. For such boys no other subject in the curriculum, except perhaps applied mathematics, is of greater importance. In many of the trades the ability to work from drawings is indispensable and the man who does not possess it is not likely to rise above purely routine work.

In a drawing course for future industrial workers the emphasis should be placed on giving the pupil an understanding of the uses of drawing for industrial purposes, rather than on fine workmanship in making drawings. Seventh grade boys can't be made into draftsmen in three years and if they leave school at 15 they are not likely to become draftsmen. The ordinary skilled workman seldom has any need to make drawings or designs, beyond an occasional rough sketch, but he often has to work from drawings. To put it in another way, drawing to the average workman is like an additional language of which he needs a reading but not a writing knowledge. No doubt it would be well to teach him to write and read with equal skill, but in the two or three years most of these boys will remain in school there is not time enough to do both.

Industrial Science

In many of the trades an introductory knowledge of physics and chemistry is of considerable advantage. Boys in the junior high school cannot be expected to take formal courses in these subjects, but they should not leave school without some acquaintance with them and a knowledge of their relations to industrial processes. A fair equipment should be provided for demonstrational and illustrative purposes. The subject matter should be correlated as closely as possible with the shop work, and the principal mechanical and chemical laws explained as the shop problems furnish examples of their application.