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leaving school spent a little time at business "colleges" and then drifted into unskilled positions which paid but a pittance. To compete with the business-school interest, he organized courses in "business arithmetic," "business English," put typewriting in the seventh, eighth, and ninth grades, bookkeeping and stenography in the eighth and ninth, and commercial law and "elementary banking" in the ninth. Along with these special subjects, work in penmanship, in history, in geography, in manual training or domestic science, in drawing, in music was continued. The effect upon attendance of this change in the character of the work offered was immediate, and for the first time in the history of the school there developed a large ninth-grade class with many individuals declaring their intention of going on into the upper high school for more advanced work along similar lines.

In this secondary period it is important that a survey of the chief departments of human knowledge be made before the individual settles down to an intensive study of lines which are intended to converge toward his future specialty. The work of the first cycle of this period, then, can well comprise the giving of courses in general science, general mathematics, general history, literature, courses affording a start in the languages for those desiring language study, music, art, and, finally, that special knowledge which science contributes relating to personal and sex hygiene, without which neither physical nor moral health can long be conserved. Thus landmarks in the chief fields of knowledge will be established which will serve to orient the pupil to a degree in the totality of race experience and culture. Furthermore, such a survey, extensive and popular rather than intensive and narrowly scholarly, harmonizes completely with the natural impulses of those entering the period of adolescence which demands change, variety, and human interest rather than completeness and logical arrangement. Again, by passing in procession before the student of this age the salient features of the important departments of knowledge, opportunity will be given for the determination of individual aptitudes and the forming of interests which may prove permanent in their enduring, and which also may fundamentally and completely modify the future course of the individual's development. Courses such as can be formulated from this point of view will provide an excellent "topping off" for those who find it impossible to continue their schooling beyond the end of the ninth or tenth year, and for those who are able to remain throughout the last cycle of the period such courses will give an excellent introduction to the more intensive work which can and should be expected in the advanced years of secondary school training.

To divide the field of science, for example, into water-tight compartments, labeling each with the terms physics, chemistry, biology,

botany, physiography, or astronomy, is a procedure that for specialists has a meaning and use, but for young students, indifferent to the reasons which have led scholars to form these cleavages, but tremendously interested in the facts and eager for a bird's-eye view of the whole, such a device is confusing, misleading, and deadening, and accounts for the distaste which many young people form for a field of knowledge that should have for them an interest dramatic in its intensity. It is not a difficult matter for the broadly prepared teacher to develop a course in science that will start with an explanation of the common phenomena of one's environment and progress from point to point, guided largely by the class interest of the moment, so that upon its completion the important conclusions in each of the special sciences will have been examined and their value and significance in the familiar things of everyday experience noted.

So, too, with mathematics. Because there is a special body of mathematical truths called algebra, and another body called geometry, and still a third called trigonometry; and because it is the custom of publishing houses to issue texts dealing with each separately; and because among teachers some like to think that they are specialists in one and some in another of these branches, we find our courses of study almost universally following the same plan. So in our high schools young people are plunged at once into "algebra," into "geometry," and later still are offered "trigonometry," and perhaps some of the branches of "higher" mathematics. Except for the inertia of teachers, it would be possible to provide a course, extending, say, over a year, that would comprise the important, yet simple, elements of each, arranged in an orderly whole and applied to the concrete situations falling within the experiences and intellectual grasp of the youth of this period. Such a course would accomplish three things: It would, in the pupil's mind, be related to his experience, and hence he would see in mathematics an instrument of practical rather than of theoretical value; it would serve as an excellent introduction to the more highly specialized branches if his interest in mathematical study developed; and it would give all the mathematics needed by one not wishing to enter technical vocations. Such a course is not an easy one to organize, it must be admitted, and few steps have as yet been taken in this direction; nevertheless, such a general approach to the specialized branches of mathematics is possible and highly desirable. Miss Thirmuthis Brookman, head of the department of mathematics of the Berkeley schools, has made a hopeful beginning. Her analysis of the problem is as follows:

The study of mathematics in the public schools of Berkeley is undertaken by two classes of pupils-those who need mathematics as part of their equipment as efficient citizens and home makers and those needing it as direct preparation for the earning of their livelihood. The arrangement of courses offered in

mathematics is therefore guided by two principles: First, in the earliest grades courses are offered essential to self-preservation, in order that those who leave school early may not be unduly handicapped; second, in order to avoid requiring pupils to specialize at too early an age, general courses in mathematics aim to include the essential principles required by the specialist. By this means the average pupil enlarges his horizon by studying mathematics in terms of life; at the same time the future specialist is unconsciously emphasizing the principles he will need when he chooses his vocation.

An analysis of the place of mathematics in life reveals the following classification:

I. Arithmetic of investment and of expenditure: For the masses.-The arithmetic of the home and of the business world and of the city, as each affects the home. How to increase the earning capacity, insurance, taxes, etc. For the specialist. The arithmetic of trade, of the financial world, and of city, State, and Government revenue; bookkeeping, banking, tariffs, etc.

II. Arithmetic of measurement: For the masses.-The arithmetic of the home and of the business and scientific world as they affect the home; house plans, division of real estate, map reading, etc. For the specialist.-The arithmetic of trade and of the workshop; freight transportation, furniture making, etc.

III. Elementary algebra (formulas and equations): For the masses.-The algebra underlying the general literature dealing with mathematical subjects. High and low gears in bicycles, automobiles, etc. For the specialist.-Ability to use algebra in the science laboratory and in mechanical construction and computation. Intensity of light on city streets, revolutions per minute of an electric motor, etc.

IV. Elementary geometry and trigonometry: For the masses.-The principles of geometry and trigonometry underlying the general literature concerning measurement. Amount of surface in an aeroplane; elementary surveys in reclamation work. For the specialist.-1. The principles of geometry and trigonometry useful in the mechanical arts and in advanced mathematics. 2. Practice in proving geometrical principles as training in logical thinking and in power to grasp abstractions.

V. Advanced algebra: For the specialist.-1. The laws of numbers as an introduction to higher mathematics. 2. The mastery of advanced algebra as training for teachers of elementary mathematics.

From the foregoing analysis two facts stand out clearly. First, that mathematical training affects two classes-the masses and the specialist. As far as possible this distinction is recognized by making mathematics for the masses more or less compulsory and that for the specialist wholly elective.

The second fact revealed by the analysis is that as the mathematics for the masses progresses down the list it influences smaller and smaller classes of people. This fact has been recognized by drawing a dividing line between the years of study in the curriculum. At the end of the eighth grade sufficient mathematics has been completed for graduation from the lower high school, and also from the upper high school, although an additional year in the latter school is strongly recommended. At the end of the eleventh year pupils who have studied the subject each term have completed the requirements for a degree from the State university in all colleges which do not specialize in mathematics. As far as possible the mathematics work in the Berkeley high schools has been adapted to the varying stages of the child's development. In the seventh grade children have mastered the operations of numbers through fractions and decimals, but have not learned to attach a meaning to the same. At this age they are frequently sent on errands to the stores and are familiar with the prices of sugar, but rarely handle pocket money over 25 cents a week. The boys may have paper routes or may earn money cutting lawns or running errands, but they are usually not old enough to work in stores on Saturdays. The work, therefore, consists in handling money in ways that

are connected with the home. The children learn to keep personal accounts and make out bills. They are also required to keep the home accounts of a family on an income not to exceed $100 a month. Rent, living expenses, food and clothing are at prices set by the children from their own observations. If their accounts fail to provide reasonably well for a family and to purchase a lot at Berkeley prices on the installment plan besides, the work must be repeated. In this connection, the amount devoted to philanthropy is considered equal to the amount devoted to luxuries. During this work the instinct to save is fostered by the desire to build a house when the lot is paid for. Insurance, taxes, and commission fees appear incidentally in the problem of providing for the family. The children keep strict personal accounts, including their clothing and personal expenses, which reveal the wonderful prominence of the moving picture show. At the end of the year the children have a working knowledge of the arithmetic of money from the viewpoint of the home and have incidentally received lessons in the value of thrift versus extravagance. The problem of providing for the home lighting, furnishing, etc., is adapted to the stage of development of the children. If estimates are too high for the income, the girls appeal to the sewing and cooking teachers to plan their prices in order to get them inside the required limits.1

In the eighth grade emphasis is placed upon measurements. In general, we find that pupils in this grade travel to a distance from Berkeley, to San Francisco, or to Oakland, etc., much more frequently than those in the seventh grade; so the measurements concerning the San Francisco Bay are full of meaning to them. Railroad time tables are used extensively for plotting the distances of various towns with reference to the Diablo Base Line and meridian. Distances on maps are computed according to scale. Latitude and longitude of places in the vicinity, as Mount Diablo, Grizzly, and Tamalpais, are computed. These are estimated from the maps and are used as the basis for problems in measurements. In every case pupils are required to obtain these dimensions directly from the maps. Lumber in house building, which is constantly going on in Berkeley, is computed, using Berkeley prices. Plans and elevations for a five-roomed cottage, durable, artistic, and economical, are used as the basis, and, wherever possible, pupils measure the lumber in their own basements, stair cases, roofs, etc. Squared paper is used for laying out triangles, parallelograms, etc., and the township map of California is made the basis for computing its areas. The children lay out baseball or basket-ball fields with the tape line, measure side walks, lots, etc. This work is extended in the high eighth grade to include the simple formulas for the measurement of solids, which lead naturally to the introduction of simple algebra in so far as it concerns understanding simple equations based on measurement. At the end of the eighth grade there is given the last of the compulsory work in mathematics. This is a six weeks' course in the arithmetic underlying civic finances, the bonding of a city, city and county taxes, State and Government revenues. This work is deferred as late as possible that the pupil may have broadened his experience to get the most from the course.

The ninth grade mathematics is elective. This is accordingly chosen by pupils who expect to graduate from the upper high school. This motive lends dignity to the work and eliminates those who have no particular interest therein. It includes some who are expecting to use their algebra in the high-school shops and who become the authorities of the class in matters pertaining to machinery. The main work of the class is the mastery of simple and quadratic equations as

1 See Brookman, Family Expense Account, for problems in investment and expenditure arranged for pupils of the lower high school.

they appear in the formulæ of the shop and of the physics and chemistry laboratories. Emphasis is placed upon the ability to understand simple formulæ as they appear in technical magazines, etc. Since so much of the working use of mathematics depends upon proportion and variation, these are carried through the year and appear in a variety of forms, arithmetic, algebraic, geometric, so that the pupil who does not continue his mathematics is equipped with the simplest elements of algebraic manipulation in such terms as he may need later.

The tenth year is the first year of the cycle of the upper high school. The course outlined plans to give a survey of the facts of geometry, including the ability to construct accurately with drawing instruments, to compute, using geometric formulæ and trigonometric functions, and to read blue prints of machinery, house plans, etc., with intelligence. When a thorough groundwork of the facts of geometry in their relations to life has been established, the work is followed by training in logic. The usual theorems of plane geometry are reduced to a minimum and used as the basis for more original work involving careful consideration throughout. This work is deferred as late as possible in the year on account of the immaturity of the average pupil. The work in the eleventh year concludes the required mathematics demanded by the University of California of such students as do not specialize in mathematics. The course aims, therefore, to give such knowledge of algebra and trigonometry as will give meaning to them as they occur in the reading of the average man, and such training as will lay a firm foundation for the specialists who will continue the subject. Algebra lays a strong emphasis upon the graphs of equations, statistics, and the laws of science. It lays stress upon the ability to use formulæ such as are found in the handbooks of mechanics and engineers. Trigonometry emphasizes measurement for simple triangulation and surveying rather than complex manipulations of formulæ. The purpose of this work is to give exercise in the essentials of the algebraic equation, its graph, and its application in surveying, for the benefit of those now stopping the subject and also as an introduction to higher mathematics.

Twelfth-year mathematics marks a distinct advance in the difficulty of the subject. So one is advised to enter the course only upon a display of marked mathematical ability during the first six weeks of the work. The class, is composed chiefly of those planning to become engineers, teachers of mathematics, or who expect to follow some other specialty. At this stage it becomes necessary to rescue the boy who is handy with tools and has therefore hoped to become an engineer, but who has displayed no power in mathematics. The large number of such who have previously failed in college classes in engineering are here given a final test and are advised to take up some work in which they have a better chance of success. With the class in senior mathematics highly specialized, it becomes possible to do careful, rigorous work in algebraic theory, induction, progression, logarithms, etc. In solid geometry the emphasis is placed on deductive logic and is kept at a high standard because the class is composed of those pupils who have elected mathematics as part of their life work.

No special comment needs to be passed on the courses provided in the Berkeley schools for the primary purpose of giving a survey of the field of history. The work is clearly summarized by the department head, William J. Cooper:

The work of the history department concerns itself with two blocks: First, that of the lower high school; second, that of the upper high school. In the first

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