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yet fully appreciated. History shows how for a period the world neglected synthetic methods in its enthusiasm over the newly discovered analytic geometry, but later had to be brought back to a realization of the great beauty, power, and simplicity of synthetic geometry methods; and so also teachers-to-be after being introduced to the wonderful analytic geometry, in a later course need to be impressed with the value of a combination of the two methods.

Another important course is a unifying course generally offered under some such title as "Fundamental Concepts" or "Synoptic Course." Teachers particularly need to have perspectives, foundations, and essential conceptions impressed upon them. Teachers in high schools have spoken to me with considerable enthusiasm on the value of such a course. Young's book on Fundamental Concepts of Algebra and Geometry has ably made a start in this direction but in my own opinion we mathematicians are not yet appreciating and emphasizing this type of work as strongly as it deserves. Mitchell found only 8 such courses in the 100 institutions which he investigated (Bibliography 8, p. 396).

Teachers need work in the History of Elementary Mathematics since historical remarks in the class-room arouse interest; and teachers familiar with the historical background of their subject are not so easily misled by radical would-be reformers. History courses were offered in 24 of Mitchell's 100 institutions; 9 others devoted part of their teachers' course to history, making 33 institutions giving some formal teaching in the history of mathematics (Bibliography 8, p. 396). Columbia University and the University of Michigan offer graduate courses in the History of Mathematics.

These mathematics courses offered particularly for students preparing to teach-"Teachers' Course," "Fundamental Concepts" or "Synoptic Course," and "History of Elementary Mathematics"-are usually taken in the senior, occasionally in the junior year, and with few exceptions require as prerequisites courses at least through the calculus. Sometimes these courses are offered in the Schools of Education but generally by instructors specially trained in pure mathematics. Teachers College of Columbia University offers the widest variety of courses on the teaching of mathematics; their 1919-1920 special bulletin offers 11 different courses not counting practice teaching; the 1917-18 catalog of the University of Chicago School of Education offers four such courses and the University of Illinois gives three special teachers' courses.

One type of valuable teacher training seems to be nowhere explicitly offered, that is, the oral presentation of papers on special topics worked up in the libraries. Teachers certainly need training in the use of libraries, acquaintance with books other than text-books, as well as practice in the lucid and convincing presentation of well-organized material collected from the best sources our libraries afford. No doubt many colleges give some of this type of training implicitly in connection with other courses or in clubs. We raise the question whether this important type of training is not being neglected by being allowed to remain implicit? Is not a seminar for undergraduates aimed explicitly to train in the use of libraries and oral presentation worthy of emphasis? Is it not probable that high school teachers

thus definitely introduced to the riches of our libraries would show a greater interest in mathematics as a science and make greater efforts to increase their knowledge of the subject by more individual reading and study?

Besides the mathematics courses specially considered students preparing to teach frequently elect even up to a total of 40 hours from courses in pure mathematics such as differential equations, theory of functions of both real and complex variables, differential geometry, solid analytic geometry, higher algebra, limits and series. They sometimes elect applied courses in statistics, astronomy, physics and mechanics.

When so many colleges require twenty or more hours in mathematics before they will recommend graduates for high school positions, why is it that we still find so many of much less training teaching mathematics today? One great cause is the practice, which administrators believe is a necessity, of assigning to a teacher well prepared to teach one or two subjects classes in a third or fourth and sometimes a fifth subject. In our high schools we find many teaching mathematics as a third or fourth subject who make no pretense of either special training or interest in mathematics. One suggestion for meeting this so called necessity is to increase the number of junior, township and community high schools since these types of organizations make possible earlier and more complete departmentalization and make places for more special teachers who can devote their time to one or at most two departments of instruction.

Up to this point, with the possible exception of the "Teachers' Course," we have considered only that part of teacher training which aims at knowledge of subject matter. But mathematics in our high schools is a means of educating boys and girls; student psychology is the second and a very important side of teacher training. "Professional Training" is the term usually employed to characterize that part of teacher training which aims to give knowledge of adolescent interests, motives, and learning processes. A teacher may be saturated with mathematical knowledge and yet find it impossible to induce lively youngsters to increase their knowledge of the subject. The fluent character and great variations in adolescent mental life make teaching an arduous task. Too many pass by this side of teacher training with the dogmatism, "Oh, teachers are born, not made," or "Actual experience is the only way of learning how purposefully to influence students." Is it not as reasonable to say that physicians and surgeons must get all their knowledge of the human body from practice? Doctors are required to spend long periods in the study of human physiology and anatomy with all the generalizations from the experiences of previous successful doctors before they are permitted by law to practice medicine themselves. Why should it be thought unreasonable to expect teachers to study the experience of successful teachers and get their generalizations and suggestions before they are permitted to undertake the important work of directing the mental life of our young people? While granting all possible importance to natural abilities yet in no work at all comparable with education in importance do we rely entirely on methods of trial and error in experience with no preliminary training in theory. Notwithstanding

some objectors the requirement of some professional training for teachers is already a fact. The North Central Association of Colleges and Secondary Schools, including over one thousand high schools, now requires that "the minimum professional training of teachers of any academic subject shall be at least eleven semester hours in education" (Bibliography 7). The Illinois school law now requires an examination in educational psychology, and the principles and methods of teaching for a state high school teacher's certificate (1919 Illinois School Law, Circular 138, p. 6).

Schools and Colleges of Education are meeting this demand by developing courses in educational psychology, principles of secondary education, theory of teaching, special methods in particular subjects, observation and practice teaching, history of education, and educational sociology. While we who are workers in an old, exact and well-developed science may have felt that some of the past work in education has been unscientific in method and contains hasty and unwarranted generalizations, yet is not blundering always common in a new field? Education is very very much younger than mathematics and yet educational investigators of really scientific ideals have already appeared. Cannot we mathematicians be constructive rather than merely destructive critics; can we not be sympathetic and suggestive in our attitude toward this comparatively new, important and complex field of investigation? A body of educational facts and guiding principles is slowly being accumulated to help the teacher. The wide use of mental tests during the World War is a tribute to the painstaking workers in education. We should therefore require prospective teachers to take good courses in educational psychology, technique and theory of teaching, history of education, and, in case they have never taught, some practice teaching as their work in Professional Training.

The work in practice teaching is being constantly improved in quality and there is a noticeable tendency to raise the standards of scholarship of supervisors of practice teaching in our demonstration high schools. For example, at the University of Illinois where a new high school is about to be opened, the ideal set for the supervisor of mathematics work is one with a Ph.D. degree in both mathematics and education, if such cannot be found then there must be a Ph.D. degree in mathematics and as much training in education as possible. Likewise in each department they plan to require a doctor's degree in the field to be supervised together with specialization in education.

The Wisconsin plan of "directed teaching" in contrast to practice teaching deserves special mention. This plan is fully described in the Eighteenth Year Book of the National Society for the Study of Education. Limited time makes possible here only a hasty sketch of this unique plan. Each class is at all times in charge of an expert regular staff teacher; from one to three college seniors in training are assigned to a particular class; they become students in this class and participate in all class activities by preparing lessons and reciting just as regular students do, and this participation gives them the exact student viewpoint; at the same time these seniors discuss methods of organization and technique with the

expert teacher in charge to get the teacher's viewpoint; by consistent unfailing excellence a senior may win the right to assist the staff teacher and finally becomes the class leader for short or even long periods, but always under the direction of the staff expert; preparation through participation is the key note of this plan. The purpose of this special mention of the Wisconsin directed teaching scheme is its suggestions. Why cannot this plan be adapted to teacher training in our college classes as now organized? Why not assign one prospective teacher to each expert member of our present college faculties; by participation, assisting and personal contact with this expert teacher cannot the student get the best possible teacher training? Would not a plan of associating inexperienced beginners with older experts for careful direction and help be much better than our present plan of putting assistants in sole charge of sections to sink or swin unaided? Can we get practice teaching in college classes without waiting for the formation of practice schools which are often long delayed by the budget difficulties? It is not suggested that experimental schools be abolished or discouraged but rather that teacher training possibilities may be extended to college classes as at present organized.

Besides furnishing college students with first class teacher training our colleges and universities should also help teachers already in service. College instructors should coöperate heartily in high school teacher associations. The annual University of Illinois High School Conference has been a very potent factor in teacher contact and mutual inspiration; it has given high school teachers the points of view and advice of more expert mathematicians and it has given college instructors more sympathetic understanding of the high school teachers particular problems. The teachers' book-shelf should also receive the careful attention of college men. There is need for books in English such as Klein's Elementarmathematik vom höheren Standpunkte aus. High school teachers need more books like Young's Fundamental Concepts of Algebra and Geometry, Carson's Mathematical Education, Nunn's Teaching of Algebra, Whitehead's Introduction to Mathematics and Organization of Thought, Keyser's Human Worth of Rigorous Thinking, Monographs on Mathematics edited by Young, and specially little contributions similar to Teubner's "Mathematische Bibliothek" on topics like Der Begriff der Zahl in seiner logischen und historischen Entwicklung, Der Pythagoreische Lehrsatz, Konstruktionen in begrenzter Ebenen, etc. The writing or translating of books which will find places on the secondary teachers' book shelf would be a distinct service in raising standards of scholarship among teachers of mathematics.

In order that the important work of secondary teacher training should not be neglected each college department of mathematics, specially the larger ones, should assign to one member of its staff the problem of elementary teacher training. This instructor should visit high schools as often as possible and keep in touch with actual high school conditions; he should take as his field of research the problem of devising ways and means of improving instruction in mathematics. Summing up the suggestions as points for your discussion, we college teachers

can help high school teacher training (1) by emphasizing the desirability of the greatest possible scholarship in mathematics, (2) by developing strong courses in theory of equations, advanced geometry, fundamental concepts and history, (3) by emphasizing in all our courses perspectives and big ideas, (4) by developing the use of libraries, knowledge of best literature and ability to prepare and forcefully present papers on mathematical topics, (5) by making one member of the staff responsible for investigation in the field of teacher training, (6) by encouraging professional training specially in adolescent psychology, (7) by taking constructive and not destructive attitudes toward scholarly investigation in education, (8) by demanding that supervisors of mathematics teacher training work have a Ph.D. or equivalent in mathematics as well as special training in education, (9) by studying the Wisconsin directed teaching plan with the purpose of modifying it to apply to college classes, and (10) by writing and translating books or articles which appeal to the present interests and attainments of high school teachers.

American standards of mathematics teacher preparation, both in scholarship and professional training, are below those of some other countries, specially those of France and Germany, and we are still far from the ideal set up by our American Commission on Mathematics Teaching (Bibliography 2, pp. 13–14). However American standards are slowly but surely rising; let us help this upward movement.


1. The Training of Teachers of Mathematics, by R. C. Archibald. Bulletin, 1917, No. 27, U.S. Bureau of Education.

2. Training of Teachers of Elementary and Secondary Mathematics. Bulletin, 1911, No. 12, U. S. Bureau of Education.

3. Influences Tending to Improve the Work of the Teacher of Mathematics. Bulletin, 1912, No. 13, U. S. Bureau of Education.

4. "The Training of Mathematics Teachers in the Secondary Schools of the United States" by A. W. Stamper, School Science and Mathematics, April, 1910.

5. "The Professional Preparation of High School Teachers." Eighteenth Yearbook of the National Society for the Study of Education, 1919. This has a bibliography of 77 titles on teacher training.

6. "Report of the Committee of Seventeen on the Professional Preparation of High School Teachers." National Educational Assoc. Proceedings for 1907.

7. 1919 Report of North Central Assoc. of Colleges and Secondary Schools.

8. "What are the actual courses now offered in colleges and universities in this country for the preparation of teachers?" by U. G. Mitchell and R. C. Archibald. AM. MATHEMATICAL MONTHLY, Dec., 1916, pp. 395–399.


Geometry for Juniors and Seniors" by E. B. Stouffer, Am. Math. Monthly, April, 1918. 10. "Means for Scientific Development of Mathematics Teachers," by G. A. Miller, Science, Dec. 6, 1918.

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