MATHEMATICS

To arithmetic, the Cleveland schools are devoting a somewhat larger proportion of time than the average of cities.

TABLE 7.—TIME GIVEN TO ARITHMETIC =========================================================== | Hours per year | Per cent of grade time| Grade |———————————————————————- | Cleveland | 50 cities| Cleveland | 50 cities | —————————————————————————————- 1 | 38 | 60 | 5.2 | 6.9 | 2 | 136 | 96 | 15.5 | 10.7 | 3 | 142 | 131 | 16.3 | 14.4 | 4 | 152 | 149 | 17.2 | 15.4 | 5 | 142 | 144 | 17.1 | 14.9 | 6 | 155 | 146 | 17.5 | 15.0 | 7 | 142 | 140 | 16.1 | 14.4 | 8 | 158 | 142 | 17.9 | 14.1 | =========================================================== Total | 1065 | 1008 | 15.5 | 13.3 | —————————————————————————————-

That everybody should be well grounded in the fundamental operations of arithmetic is so obvious as to require no discussion. Beyond this point, however, difficult problems arise. The probabilities are that the social and vocational conditions of the coming generation will require that everybody be more mathematical-minded than at present.

The content of mathematics courses is to be determined by human needs. One of the fundamental needs of the age upon which we are now entering is accurate quantitative thinking in the fields of one's vocation, in the supervision of our many co-operative governmental labors, in our economic thinking with reference to taxation, expenditures, insurance, public utilities, civic improvements, pensions, corporations, and the multitude of other civic and vocational matters.

Just as the thought involved in physics, astronomy, or engineering needs to be put in mathematical terms in order that it may be used effectively, so must it be with effective vocational, civic, and economic thinking in general. Our chief need is not so much the ability to do calculations as it is the ability to think in figures and the habit of thinking in figures. Calculations, while indispensable, are incidental to more important matters.

Naturally before one is prepared to use mathematical forms of thought in considering the many social and vocational problems, he must have mastered the fundamentals. The elementary school, at as early an age as practicable, should certainly give the necessary preliminary knowledge of and practice in the fundamental operations of arithmetic. This should be done with a high degree of thoroughness, but it should always be kept in mind that this is only a preliminary mastery of the alphabet of mathematical thinking. The other part of our problem is a development of the quantitative aspects of the vocational, economic, and civic subjects. One finds clear recognition of this in Cleveland in the new arithmetic manual. The following quotations are typical:

"The important problem of the seventh and eighth grades is to enable the pupils to understand and deal intelligently with the most important social institutions with which arithmetical processes are associated."

In discussing the teaching of the mathematical aspect of insurance, we find this statement: "Owing to the important place this subject holds in life, we should emphasize its informational value rather than its mathematical content."

Under taxation and revenue: "If the general features of this subject are presented from the standpoint of civics, the pupils should have no difficulty in solving the problems as no new principle is introduced."

Under stocks and bonds: "Pupils should be taught to know what a corporation is, its chief officers, how it is organized, what stocks and bonds are, and how dividends are declared and paid, in so far as such knowledge is needed by the general public."

These statements indicate a recognition of the most important principle that should control in the development of all of the mathematics, elementary and secondary, beyond the preliminary training needed for accuracy and rapidity in the fundamental operations.

When this principle is carried through to its logical conclusion, it will be observed that most of these developments will not take place within the arithmetic class, but in the various other subjects. Arithmetic teaching, like the teaching of penmanship, etc., is for the purpose of giving tools that are to be used in matters that lie beyond. The full development will take place within these various other fields. For the present, it probably will be well for the schools to develop the matters both within the arithmetic classes and in the other classes. Neither being complete at present, each will tend to complete the other.

On the side of the preliminary training in the fundamental operations, the present arithmetic course of study is on the whole of a superior character. It provides for much drill, and for a great variety of drill. It emphasizes rapidity, accuracy, and the confidence that comes to pupils from checking up their results. It holds fast to fundamentals, dispensing with most of the things of little practical use. It provides easy advances from the simple to the complicated. The field of number is explored in a great variety of directions so that pupils are made to feel at home in the subject. One large defect is the lack of printed exercise materials, the use of which would result in greatly increased effectiveness. Such printed materials ought to be furnished in great abundance.

ALGEBRA

In the report of the Educational Commission of Cleveland, 1906, we find the following very significant sentences relative to the course of study for the proposed high school of commerce:

"An entirely new course of study should be made out for this school. Subjects which have been considered necessary in a high school, because they tend to develop the mind, should not for this reason only be placed in a commercial course. Subjects should not be given because they strengthen the mind, but the subjects which are necessary in this course should be given in such a way as to strengthen the mind. The mathematics in this school should consist of business arithmetic and mensuration. We can see no reason for giving these students either algebra or geometry. But they should be taught short and practical methods of working business problems."

We find here a recommendation since carried out that indicates a clear recognition of the principle of adaptation of the course of study to actual needs. Carried out to its logical conclusion, and applied to the entire city system, it raises questions as to the advisability of requiring algebra of girls in any of the high school courses; or of requiring it of that large number of boys looking forward to vocations that do not involve the generalized mathematics of algebra. Now either the commercial students do need algebra or a large proportion of these others do not need it. It seems advisable here to do nothing more than to present the question as one which the city needs to investigate. The present practice, in Cleveland as elsewhere, reveals inconsistency. In one or the other of the schools a wrong course is probably being followed. The current tendency in public education is toward agreement with the principle enunciated by the Cleveland Educational Commission, and toward a growing and consistent application of it.

Differentiation in the mathematics of different classes of pupils is necessary. The public schools ought to give the same mathematics to all up to that level where the need is common to all. Beyond that point, mathematics needs to be adapted to the probable future activities of the individual. There are those who will need to reach the higher levels of mathematical ability. Others will have no such need.

There is a growing belief that even for those who are in need of algebra the subject is not at present organized in desirable ways. It is thought that, on the one hand, it should be knit up in far larger measure with practical matters, and on the other, it should be developed in connection with geometry and trigonometry. The technical high schools of Cleveland have adopted this form of organization. Their mathematics is probably greatly in advance of that of the academic schools.

GEOMETRY

Form study should begin in the kindergarten, and it should develop through the grades and high school in ways similar to the arithmetic, and in conjunction with the arithmetic, drawing, and construction work. Since geometrical forms involve numerical relations, they supply good materials to use in making number relations concrete and clear. This is now done in developing ideas of fractions, multiplication, division, ratio, per cent, etc. It should be done much more fully and variously than at present and for the double purpose of practising the form-ideas as well as the number-ideas. Arithmetic study and form-study can well grow up together, gradually merging into the combined algebra and geometry so far as students need to reach the higher levels of mathematical generalization.

At the same time that this is being developed in the mathematics classes, development should also be going on in the classes of drawing, design, and construction. The alphabet of form-study will thus be taught in several of the studies. The application will be made in practical design, in mechanical and free-hand drawing, in constructive labor, in the graphical representation of social, economic, and other facts of life. The application comes not so much in the development of practical problems in the mathematics classes as in the development of the form aspect of those other activities that involve form.

We have here pointed to what appears to be in progressive schools a growing program of work. Everywhere it is yet somewhat vague and inchoate. In connection with the arithmetic, the drawing, the construction and art work, and the mathematics of the technical high schools, it appears to be developing in Cleveland in a vigorous and healthy manner.