We may pursue this example further as typical of the complications which ultimately grow up around any course of study. The original purpose of Harvard was expanded with the passing years. A demand arose for lawyers and doctors; in the effort to meet this demand the institution was divided into separate schools. Still later students came to college seeking a general training not leading to any profession. Through all these changes in the demands of the student body the original courses of study have persistently battled their way down to the present. No clearer evidence can be found than this, that courses of study once created become vital factors in all the later life of the school. The college courses of study were in the first place the product of a particular professional demand. While satisfying this particular demand they became strong enough so that at a later period they have often dominated educational policies.
It is too flippant a remark to say that the classical education of the clerical period became the fashion and that later generations were afraid to be out of fashion, but something of this sort is what really happened. The traditions of a generation are hard to break. The father who took Greek as a part of his education hesitates to see his son enter upon life without the same equipment. Courses of study thus come to have an intellectual sanction which it is extraordinarily difficult to break down.
Traditional Character of Mathematics Courses in High Schools
Another example of no less impressive a type can be drawn from the high-school curriculum of the present time. There is hardly a tradition of high schools which is more fixed than that of requiring algebra in the first year and geometry later. This practice persists even though it is a well-known fact that in many schools failures in high-school algebra are more numerous than in any other high-school course. Also, there is a clear recognition of the fact that by being required of all students in the first year algebra is in effect made the prerequisite of admission to the courses in science and literature which are open only to students who have reached the later years of the high school. The question which the student of education must raise is this: How did algebra secure this position of commanding importance, and how does it hold this position when experience shows that so many students cannot take it with success? The answer to these questions throws a strong light on the nature of the curriculum.
Mathematics in general gained a preëminent position in the educational scheme of the Western World as far back as the fifth century before Christ, in the days of Pythagoras. The branches of mathematics which were chiefly cultivated in those days were geometry and arithmetic. Geometry flourished as an experimental science, and arithmetic consisted in the most elaborate speculations about prime numbers and the properties of odd and even numbers. After these sciences had reached a certain maturity they were transferred to the University of Alexandria, where, in the third century before Christ, Euclid formulated the principles of geometry into the logical form which has persisted to our own time. If one asks why the same service was not rendered for arithmetic at the University of Alexandria, the answer is to be found in the fact that the Greeks had no adequate method of expressing number. They used a system of letters even more clumsy than the system employed by the Romans after them. If one needs further demonstration of the reason why arithmetic did not develop in the classical world, let him try to multiply DCCLXXVII by XCIX. Arithmetic was very little cultivated, therefore, while geometry was put into perfect logical form. Since arithmetic was so little developed in the ancient world, algebra never succeeded in getting a real start.
Geometry, thus launched as a systematized branch of learning superior to arithmetic, has held its place through all generations. In the medieval institutions the perfect logical form of geometry was fully recognized. Geometry was used to sharpen the logic of many a mind. Arithmetic developed only so far as it was needed for the practical purposes of daily life.
In due time there came into Europe oriental scholars who brought with them that marvelous invention—the Arabic numerals. They brought also the science of algebra with its profound abstractions. The Arabic numerals soon superseded the clumsy Roman numerals, and the common man found that he could easily deal with the practical matters of life by means of this number system which rendered all calculations simple. With arithmetic of the new type came algebra. The scientists of Europe found that the algebraic methods opened up possibilities of mathematical reasoning which were of the first importance to science. Algebra and arithmetic flourished. But did these two newcomers in any degree disturb the position of geometry? Not at all. Algebra may be as abstract as any subject in the curriculum, but its historical relations were from the first with arithmetic, while geometry was related to logic and the higher subjects. Geometry has continued since 300 B.C. to be a higher course. The situation in the high schools of to-day is in no sense due to a careful study of the degree of abstraction involved in geometry and algebra. It is in no sense a recognition of the fact that geometry was the first of the two subjects to develop. The present situation can be understood only by recognizing the strength of tradition and the persistence of a practice when once it gets itself established.
The situation is the more impressive because even a superficial study of the intellectual needs of pupils shows that there ought to be instruction in the lower grades in the discrimination of forms and designs. One does not master the forms even of common things until his attention has been turned to them again and again. The consequences to the curriculum of the elevation of geometry to the upper school are far-reaching in a negative as well as in a positive way. Space study has been kept out of the lower schools because the only orthodox form of space study is the geometry of the higher schools. Space study ought to have a place in the curriculum of every grade.
In the case of algebra, on the other hand, tradition has operated to keep the subject in the lower classes of the high school. That it would be better to change this situation appears in the fact that textbooks in algebra have in recent years been made much easier in the effort to fit the subject to pupils’ needs, in the fact that some high schools have made it elective, and in the fact that some high schools have rearranged the whole subject-matter of mathematics, breaking up the historical lines of division.