The fact that the college teacher has need of much more mathematical knowledge than he can possibly secure during the period of his preparation, especially if he expects to take an active part in research and in directing graduate work, has usually led to the a.s.sumption that the future teacher of college mathematics should devote all his energies to securing a deep mathematical insight and a wide range of mathematical knowledge.[10] On the other hand, students prepared in accord with this a.s.sumption have frequently found it very difficult to adapt themselves to the needs of large freshman cla.s.ses of engineering students entering upon the duties for which they were supposed to have been prepared.
The breadth of view and the sweep of abstraction needed for effective graduate work have little in common with accuracy in numerical work and emphasis on details which are so essential to the young engineering students. The difficulty of the situation is increased by the fact that the young instructor is often led to believe that his advancement and the appreciation of his services are directly proportional to his achievements in investigations of a high order.
This belief naturally leads many to begrudge the time and thought which their teaching duties should normally receive.
The young college teacher of mathematics is thus confronted with a much more complex situation than that which confronts the mathematics teachers in secondary school work. Here the success in the cla.s.sroom is the one great goal, and the mathematical knowledge required is comparatively very modest. Possibly the situation of the college teacher could be materially improved if it were understood that his first promotion would be mainly dependent upon his success as a teacher, but that later promotions involved the element of productive scholarship in an increasing ratio.
The schools of education which have in recent years been established in most of our leading universities have thus far had only a slight influence on the preparation of the college teachers, but it seems likely that this influence will increase as the needs of professional training become better known. It is probably true that the ratio of courses on methods to courses on knowledge of the subject will always be largest for the elementary teacher, in view of the great difference between the mental maturity of the student and the teacher, somewhat less for the secondary teacher and least for the college teacher; but this least should not be zero, as is so frequently the case at present, since there usually is even here a considerable difference between the mathematical maturity of the student and that of the teacher.
It may be argued that the future college teacher will probably profit more by noting the methods employed by his instructors than he would by the theoretic discussions relating to methods. This is doubtless true, but it does not prove that the latter discussions are without value. On the other hand, these discussions will often serve to fix more attention on the former methods and will lead the student to note more accurately their import and probable adaptability to the needs of the younger students.
Among the useful features for the training of the future mathematics teachers are the mathematical clubs which are connected with most of the active mathematical departments. In many cases, at least, two such clubs are maintained, the one being devoted largely to the presentation of research work while the other aims to provide opportunities for the presentation of papers of special interest to the students. The latter papers are often presented by graduate students or by advanced undergraduates, and they offer a splendid opportunity for such students to acquire effective and clear methods of presentation. The same desirable end is often promoted by reports given by students in seminars or in advanced courses.
Prominent factors in the training of the future college teachers are the teaching scholarships or fellowships and the a.s.sistantships. Many of the larger universities provide a number of positions of this type. It sometimes happens that the teaching duties connected with these positions are so heavy as to leave too little energy for vigorous graduate work. On the other hand, these positions have made it possible for many to continue their graduate studies longer than they could otherwise have done and at the same time to acquire sound habits of teaching while in close contact with men of proved ability along this line.
It should be emphasized that the ideal college teacher of mathematics is not the one who acquires a respectable fund of mathematical knowledge which he pa.s.ses along to his students, but the one imbued with an abiding interest in learning more and more about his subject as long as life lasts. This interest naturally soon forces him to conduct researches where progress usually is slow and uncertain.
Research work should be animated by the desire for more knowledge and not by the desire for publication. In fact, only those new results should be published which are likely to be helpful to others in starting at a more favorable point in their efforts to secure intellectual mastery over certain important problems.
Half a century ago it was commonly a.s.sumed that graduation from a good college implied enough training to enter upon the duties of a college teacher, but this view has been practically abandoned, at least as regards the college teacher of mathematics. The normal preparation is now commonly placed three years later, and the Ph.D. degree is usually regarded to be evidence of this normal preparation. This degree is supposed by many to imply that its possessor has reached a stage where he can do independent research work and direct students who seek similar degrees. In view of the fact that in America as well as in Germany the student often receives much direct a.s.sistance while working on his Ph.D. thesis, this supposition is frequently not in accord with the facts.[11]
The emphasis on the Ph.D. degree for college teachers has in many cases led to an improvement in ideals, but in some other cases it has had the opposite effect. Too many possessors of this degree have been able to count on it as accepted evidence of scientific attainments, while they allowed themselves to become absorbed in non-scientific matters, especially in administrative details. Professors of mathematics in our colleges have been called on to shoulder an unusual amount of the administrative work, and many men of fine ability and scholarship have thus been hindered from entering actively into research work. Conditions have, however, improved rapidly in recent years, and it is becoming better known that the productive college teacher needs all his energies for scientific work; and in no field is this more emphatically true than in mathematics. Some departmental administrative duties will doubtless always devolve upon the mathematics teachers. By a careful division of these duties they need not interfere seriously with the main work of the various teachers.
=The mathematical textbook=
The American teachers of mathematics follow the textbook more closely than is customary in Germany, for instance. Among college teachers there is a wide difference of view in regard to the suitable use of the textbook. While some use it simply for the purpose of providing ill.u.s.trative examples and do not expect the student to begin any subject by a study of the presentation found in the textbook, there are others who expect the normal student to secure all the needed a.s.sistance from the textbook and who employ the cla.s.s periods mainly for the purpose of teaching the students how to use the textbook most effectively. The practice of most teachers falls between these two extremes, and, as a rule, the textbook is followed less and less closely as the student advances in his work. In fact, in many advanced courses no particular textbook is followed. In such courses the princ.i.p.al results and the exercises are often dictated by the teacher or furnished by means of mimeographed notes.
The close adherence to the textbook is apt to cultivate the habit on the part of the student of trying to understand what the author meant instead of confining his attention to trying to understand the subject. In view of the fact that the American secondary mathematics teachers usually follow textbooks so slavishly, the college teacher of mathematics who believes in emphasizing the subject rather than the textbook often meets with considerable difficulty with the beginning cla.s.ses. On the other hand, it is clear that as the student advances he should be encouraged to seek information from all available sources instead of from one particular book only. The rapid improvement in our library facilities makes this att.i.tude especially desirable.
An advantage of the textbook is that it is limited in all directions, while the subject itself is of indefinite extent. In the textbook the subject has been pressed into a linear sequence, while its natural form usually exhibits various dimensions. The textbook presents those phases about which there is usually no doubt, while the subject itself exhibits limitations of knowledge in many directions. From these few characteristics it is evident that the study of textbooks is apt to cultivate a different att.i.tude and a different point of view from those cultivated by the unhampered study of subjects. The latter are, however, the ones which correspond to the actual world and which therefore should receive more and more emphasis as the mental vision of the student can be enlarged.
The number of different available college mathematical textbooks on the subjects usually studied by the large cla.s.ses of engineering students has increased rapidly in recent years. On the other hand, the number of suitable textbooks for the more advanced cla.s.ses is often very limited. In fact, it is often found desirable to use textbooks written in some foreign language, especially in French, German, or Italian, for such courses. This procedure has the advantage that it helps to cultivate a better reading knowledge of these languages, which is in itself a very worthy end for the advanced student of mathematics. This procedure has, however, become less necessary in recent years in view of the publication of various excellent advanced works in the English language.
The greatest mathematical treasure is const.i.tuted by the periodic literatures, and the larger colleges and universities aim to have complete sets of the leading mathematical periodicals available for their students. This literature has been made more accessible by the publication of various catalogues, such as the _Subject Index_, Volume I, published by the Royal Society of London in 1908, and the volumes "A" of the annual publications ent.i.tled _International Catalogue of Scientific Literature_. All students who have access to large libraries should learn how to utilize this great store of mathematical lore whenever mathematical questions present themselves to them in their scientific work. This is especially true as regards those who specialize along mathematical lines.
In some of the colleges and universities general informational courses along mathematical lines have been organized under different names, such as history of mathematics, synoptic course, fundamental concepts, cultural course, etc. Several books have recently been prepared with a view to meeting the needs of textbooks for such courses. College teachers of mathematics usually find it difficult to interest their students sufficiently in the current periodic literature, and one of the greatest problems of the college teacher is to instill such a broad interest in mathematics that the student will seek mathematical knowledge in all available sources instead of confining himself to the study of a few textbooks or the work of a particular school.
G. A. MILLER _University of Illinois_
REFERENCES
For articles on the teaching of mathematics which appeared during the nineteenth century, consult 0050 _Pedagogy_ in the _Royal Society Index_, Vol. I, Pure Mathematics, 1908. For literature appearing during the first twelve years of the present century the reader may consult the _Bibliography of the Teaching of Mathematics_, 1900-1912, by D. E. Smith and Charles Goldziher, published by the United States Bureau of Education, Bulletin, 1912, No. 29. More recent literature may be found by consulting annual indexes, such as the _International Catalogue of Scientific Literature_, A, Mathematics, under 0050, and _Revue Semestrielle des Publications Mathematiques_, under V 1. The volumes of the international review ent.i.tled _L"Enseignement Mathematique_, founded in 1899, contain a large number of articles relating to college teaching. This subject will be treated in the closing volumes of the large French and German mathematical encyclopedias in course of publication.
Footnotes:
[3] P. Zuhlke. _Zeitschrift fur Mathematischen und Naturwissenschuftlichen Unterricht_, Vol. 45 (1915), page 483.
[4] Committee No. XII, American Report of the International Commission on the Teaching of Mathematics, 1912, page 9.
[5] _Internationale Mathematische Unterrichtskomission_, Vol. 3, No. 6 (1912), page 2.
[6] _Journal de l"Ecole Polytechnique_, Vol. 1 (1896), part 4, page lx.
[7] F. Cajori, _Teaching and History of Mathematics in the United States_, 1890, page 22.
[8] A. E. H. Love, _Proceedings of the London Mathematical Society_, Vol. 14 (1915), page 183.
[9] V. V. Bobynin, _L"Enseignement Mathematique_, Vol. 1 (1899), page 78.
[10] The Training of Teachers of Mathematics, 1917, by R. C.
Archibald. Bulletin No. 27, 1917, United States Bureau of Education.
[11] Cf. M. Bocher, _Science_, Vol. 38 (1913), page 546.
IX
PHYSICAL EDUCATION IN THE COLLEGE
=Lessons for physical education from the world war=
The events of the four years between the summer of 1914 and the winter of 1918 have brought us to a full realization of the real significance of physical education in the training of youth. America and her allies have had very dramatic reasons for regretting their careless indifference to the welfare of childhood and youth in former years.
Only yesterday, we were told that the great war would be won by the country that could furnish the last man or fight for the last quarter of an hour. America and her allies looked with a new and fearful concern upon the army of young men who were found physically unfit for military service.
With the danger of war past, there is no lack of evidence that we and our allies will make practical application of this particular lesson.
It will be fortunate indeed if the enlightened people of the earth are really permanently awake to the importance of the physical education of their citizens-in-the-making.
Governmental agencies have already started the movement to guarantee to the coming generation more extensive and more scientific physical education. Public and private inst.i.tutions are joining forces so that the advantages of this extended program of physical education will be enjoyed by the young men and young women in industry and commerce as well as by those in schools and colleges.
It is to be hoped that the American college will do its full share and neglect no reasonable measure whereby the college graduate may be developed into the vigorous and healthy human being that the mentally trained ought to be. It must be admitted that our findings by the military draft boards, as well as other evidences secured through physical examinations, are not such as to make the American college proud of the quality or the extent of physical education which it has given in the past. We must express our keen disappointment at the prevalence of under-development, remediable defects, and unachieved physical and functional possibilities in our college graduates.
=Aims of physical education=
Physical training is concerned with the achievement and the conservation of human health. It has to do with conditioning the human being for the exigencies of life in peace or in war. Its standards are not set by a degree of health which merely enables the individual to keep out of bed, eat three meals a day, and run no abnormal temperature. Physical training is concerned with developing vigorous, enduring health that is based upon the perfect function, coordination, and integration of every organ of the human body; health that is not found wanting at the military draft; health that meets all its community obligations; health that is not affected by diseases of decay; and health that resists infection and postpones preventable death.
=Formulations of aims and scope of physical education in official doc.u.ments--By Regents of the State of New York=
Official statements and information from reliable sources indicate that physical education and hygiene and physical training are regarded by authorities as covering about the same general field. The general plan and syllabus for physical training adopted by the Regents of the University of the State of New York in 1916 interprets physical training as covering "(1) Individual health examinations and personal health instruction (medical inspection); (2) instruction concerning the care of the body and the important facts of hygiene (recitations in hygiene); (3) physical examinations as a health habit, including gymnastics, elementary marching, and organized, supervised play, recreation, and athletics."
=By national committee on physical education=
In March of 1918 a National Committee on Physical Education, formed of representatives from twenty or more national organizations, adopted the following resolutions:
I. That a comprehensive, thoroughgoing program of health education and physical education is absolutely needed for all boys and girls of elementary and secondary school age, both rural and urban, in every state in the Union.