(b) In contrast to this schedule there are State universities and privately endowed institutions, like Leland Stanford Junior University, which permit a wide range of electives. The University of Minnesota, for example,' accepts the following:

3

(c) Finally, it may be of interest to give a composite picture of class work in the 15 States of the North Central Association (Colorado, Illinois, Indiana, Iowa, Kansas, Michigan, Minnesota, Missouri, Montana, Nebraska, North Dakota, Ohio, Oklahoma, South Dakota, and Wisconsin). The percentage of units given in the different subjects of the high schools of these States in 1913-14 was as follows: English (13.1), Latin (11.5), history (9.5), commercial (9.2), German (8.5), algebra (5.2), geometry (4.8), manual training (4.7), physics (3.3), domestic science (3), chemistry (2.6), cooking (2.5), drawing (2.5), sewing (2.4), normal subjects (2.4), agriculture (2.0), botany (2.0), French (1.7), physical geography (1.7), music (1.6), civics (1.5), physiology (1.2), zoology (0.9), education (0.8), other subjects (1.4).

After completing his high-school course our future teacher in one of the secondary schools must of necessity take a college course. There are upward of 800 so-called "colleges" and "universities" in the United States, but many of these are really secondary schools. In 1908-9 there were only 261 colleges which had 100 collegiate students enrolled in the four regular college classes, or which had an endowment to the amount of $100,000. There are about 40 endowed and 40 State universities in the United States. The student who wishes to go to one of the chief university mathematical centers would

1 Illustration given by Capen (1. c.).

In each case the number of units given is the maximum number which can be taken in the corresponding subject.

G. S. Counts, A Study of the Colleges and High Schools in the North Central Association. (Bu. of Educ.. Bul., 1915, No. 6.) Washington, 1915, p. 115. See also the association's "investigation into the status of the teaching of almost all subjects appearing in the secondary schools, programs of study," by L. V. Koos: The Administration of Secondary-School Units, Supplementary Educational Monographs, University of Chicago, vol. 1, No. 3, July, 1917.

Monroe's Cyclopedia of Education, vol. 2, p. 74.

There are also about 50 State colleges. See Statistics of State Universities and State Colleges (Bu. of Educ., Bul., 1917, No. 55), Washington, 1918.

probably select one of the following: Harvard, Chicago, Illinois, Princeton, or Columbia.

The college course leads to the degree of A. B., Sc. B., Litt. B., Ph. B., or some other degree of similar rank, but the title alone does not convey a very definite idea of achievement.1

As to work leading to the degree of A. B., some colleges do not require any courses in mathematics; others require higher algebra, solid geometry, plane and spherical trigonometry, plane and solid analytic geometry, calculus and differential equations. But the normal requirement is a three-hour semester course in each of the subjects: (1) Higher algebra, or (2) solid geometry, and (3) plane trigonometry. If our undergraduate student wishes to proceed in mathematics beyond the required work, the opportunities for doing so vary widely with different colleges. In one none but required courses in mathematics are offered; in another so many courses in mathematics are offered that under the "elective system" there in vogue it would be possible for the student to elect the major part of the courses he takes for his degree from those given by the department of mathematics.

THE TRAINING OF TEACHERS FOR SECONDARY SCHOOLS.

But before going into this question more particularly let us consider how a student who is taking a college course may definitely prepare himself to be a high-school teacher. To get a broad view of the question, we should recall the "joint recommendations of the committee of seventeen on the professional preparation of high-school teachers," adopted by the National Education Association.

The committee on the preparation of high-school teachers recommend: I. That the academic preparation include the following elements:

A. A detailed and specialized study of the subjects to be taught. The program of studies selected by each student should include work in subjects outside of those in which he is making special preparation, sufficient to give some insight into different fields of knowledge and to avoid the dangers of overspecialization.

B. One or more subjects from a group including history, economics, and sociology, which will give the teacher a proper outlook upon the social aspects of education.

C. A course in general psychology and at least one from a group of subjects including history of philosophy, logic, and ethics, which will give the teacher a proper outlook upon education as the development of the individual.

1 In 1911 the Bureau of Education published an interesting Classification of Universities and Colleges with Reference to Bachelor's Degrees. The author, Dr. K. C. Babcock, a specialist in the bureau, divided the institutions into four classes. The first and highest class he defined as "Institutions whose graduates would ordinarily be able to take master's degree at any of the larger graduate schools in one year after receiving their bachelor's degree, without necessarily doing more than the amount of work regularly prescribed for such higher degree." In the first class were only 15 State and 44 endowed and private institutions. See also Rep. of U. S. Commis. of Educ. for 1914, vol. 1, Washington, 1915, p. 168.

2 Compare Undergraduate Work in Mathematics in Colleges of Liberal Arts and Universities. International Commission on the Teaching of Mathematics. (Bu. of Educ., Bul., 1911, No. 7.) Washington, 1911.

II. That definite study be given to each of the following subjects, either in separate courses or in such combinations as convenience or necessity demands:

A. History of education.

1. History of general education.

2. History of secondary education.

B. Educational psychology with emphasis on adolescence.

C. The principles of education, including the study of educational aims, values, and processes. Courses in general method are included under this heading.

D. Special methods in the secondary-school subjects that the students expect to teach.

E. Organization and management of schools and school systems.

F. School hygiene.

III. That opportunity for observation and practice teaching with secondary pupils be given.

The committee recognizes the difficulties involved in this recommendation, but believes that they are not insurmountable. Each of the following plans has proved successful in some instances:

A. The maintenance of a school of secondary-school grade that may be used for observation and practice.

B. Affiliation with public or private high schools so situated geographically that practice teaching can be done without interfering with other work of the college course.

In addition to the above, the committee suggests that where competent critical supervision is possible, cadet teaching, in schools more remotely situated, may be attempted. In such cases, a teacher's diploma might be granted after a year's successful work as a cadet teacher.

IV. That the minimum requirement for a secondary-school teacher be graduation from a college maintaining a four-year course and requiring four years of highschool work for admission, or from an institution having equivalent requirements for admission and giving equivalent academic scholarship.

A year of graduate work divided between academic and professional subjects is desirable. Discussions of the relative value of college and normal schools for secondary-school teachers are to be found in the references below.1

V. That the study of subjects mentioned under II be distributed through the last two years of the college course.

The proportional amount of time given to these subjects will vary with local conditions, but an irreducible minimum is one-eighth of the college course. They should be preceded or accompanied by the subjects mentioned in I B, C. Recommendations as to the amount of time given to particular courses will be found in several of the accompanying papers.2

It will now be illuminating to consider in detail a definite scheme involving practice teaching in a manner which has won high praise from prominent authorities in recent writings." I refer to the pioneer system at Brown University, in Providence, R. I., where it has been in operation for over 20 years. The fundamental principles of the

1 Given on p. 538, Proc. Nat. Ed. Assoc., 1907.

See the report in Proc. Nat. Ed. Assoc., 1907, pp. 521-668.

J. F. Brown, The training of teachers for secondary schools in Germany and the United States, New York 1911, p. 242f. W. S. Learned, The Oberlehrer, A study of the social and professional evolution of the German schoolmaster. (Harvard Studies in Education, Vol. I). Cambridge, Mass., 1914, p. 131f.

scheme which has been advocated as worthy of wide acceptance have been summarized by Prof. Jacobs as follows:1

1. Practice teaching should be open only to graduate students; that is, students who hold a bachelor's degree. This rule is inflexible and has never been broken. Brown University looks askance upon the custom which has developed of including practice teaching and extended professional preparation as a part of the four years' work for the first college degree and regards it as a lowering of standards. It holds that a fifth year of college work is necessary to the proper preparation of secondaryschool teachers.

2. Practice teaching should be under actual schoolroom conditions. Hence, Brown University uses for its practice teaching the public and private secondary schools of the city of Providence and neighboring cities and towns.

3. Practice teaching should include the continuous instruction and control of a class for a long period. At Brown University the amount of practice teaching required varies from a minimum of 5 periods a week to 15 periods a week for one year. Student teachers teach very few classes, but they teach them continuously for a semester or a year. After many trials it has been found best to limit the student teachers to one or two subjects. To give a few sporadic lessons before a class is one thing; to teach a subject continuously is quite another matter.

4. Practice teaching must be under the continuous direct or indirect supervision of an experienced teacher who knows the school, the class, and the detailed progress of the subject taught. Hence, all supervising teachers at Brown University are selected from the experienced teachers of the schools. Each student teacher has one supervising teacher, and each supervising teacher one student teacher. The work is strictly individual.

5. Practice teaching must be closely correlated with the university work. At Brown University supervising teachers are selected by the university and paid a small remuneration. Each student teacher is visited once a week by the professor in charge of the practice teaching, and private conferences are held. He also meets once a week in a general conference all the student teachers. Plan books for the past week are presented and discussed and later returned to the student teachers. Student teachers are at the same time pursuing other courses (1) in education, a course in secondary education and a seminary in current educational problems, and (2) in departments allied with the subjects they are teaching.

Student teachers who teach more than five periods a week usually receive some remuneration for their work from the school. In the case of the city of Providence this is provided for by an agreement between the university and the city of Providence. In other cases it is arranged as the cases arise. Student teachers who have shown themselves efficient are assigned other classes under supervision. For this work they receive remuneration. The work then becomes closely similar to what is known as "part-time work" in vocational education. The work at Brown University, however, long antedates the vocational "part-time work.”

Graduate students who are admitted to the practice teaching at Brown University usually have taken as undergraduates four semester courses in education. These courses are: History of education, principles of education, educational psychology, and general method. In the last course there is some systematic work in observation and some teaching of the class by members of the class. For this last purpose the class is divided into sections of about 10 each. The student, then, who is admitted to the graduate practice teaching, is not a mere novice, but one who has already had some experience.

6. The last principle is one which is fundamental and appears in all of the work. Practice teaching must not be an injury to the school or to the pupil, but rather a

1 School and Society, Apr. 8, 1916, vol. 3, p. 534.

benefit. Hence the student teacher is called upon freely to assist the supervising teacher or the principal of the school in doing a limited amount of clerical work, work with individual pupils, or other work which can be assigned with profit to the student teacher and to the school alike. The student teacher becomes, to all intents and purposes, a part of the school staff, subject to regulations as other teachers and working as the other teachers are, in harmony with the general purposes and spirit of the school. The work of the student teacher is frequently superior to that of many of the regular teachers. He has more time for preparation and individual work with pupils and frequently more enthusiasm.

But, quite apart from these general requirements for all teachers, the department of mathematics at Brown University lays down a minimum course of study which it requires students who are prospective teachers of mathematics to take if they wish the backing of the department in starting on their careers. In outline the course is as follows: Plane trigonometry (3 semester hours), higher algebra (3), solid geometry (3), plane analytic geometry (4), differential and integral calculus (8), teachers' course in algebra (6), and teachers' course in geometry (6). The teachers' course in algebra constitutes an introduction to some of the concepts of modern analysis. Among the topics treated are: The number system with special reference to irrational numbers, limits, infinite series, the fundamental operations, and determinants.

In the teachers' course in elementary geometry the student is taught: Methods for attacking Euclidean problems; discussions of famous problems; the existence of transcendental numbers and the proofs of the transcendence of e and ; means of rigorous discussion of the more delicate and difficult parts of the subject, such as the systems of axioms; something about (a) the history and literature of elementary geometry; (b) the most important French, German, and Italian texts; and (c) non-Euclidean geometry.

The students are also urged to take a year each (three hours a week) in physics and chemistry, in addition to the courses in education and psychology referred to above. . For the A. B. degree they are required to take, also, 12 hours in rhetoric, composition, and English literature; 6 hours in European history; 6 hours in economics and either social or political science. Other courses are elective.

Students preparing to teach in secondary schools constantly elect further mathematical work, such as (1) the two-hour course in differential equations; (2) the six-hour course in theory of functions of a real variable (text, first part of Goursat-Hedrick's Mathematical Analysis); (3) the six-hour course in functions of a complex variable (text, Pierpont's or Burkhardt-Rasor's work); (4) six-hour course in differential geometry (text, latter part of Goursat-Hedrick's work and Gauss's memoir); (5) six-hour course in projective geometry; (6) three-hour course in solid analytic geometry; (7) three-hour course