lowed closely by Wiglaf, his faithful friend, he rode to the lair of the dragon. It was in a fearful region. Great pillars of rock stood as sentinels on all sides; the very sands seemed to threaten them as they drew near. Soon they saw the firebreathing dragon ahead of them. As he lashed his great tail and cast fire from his nostrils the Geatmen turned. Cowards, in time of need they deserted their master. Wiglaf only, remained at his side. On he advanced steadily, though the breath burned him, though he panted and struggled at each step. He lifted his mighty sword and attacked the monster, whose vicious strength was far the greater. Wiglaf aided all he could, calling meanwhile to the cowardly thanes to assist him. No voice answered and he waited, fearing Beowulf's defeat, which he knew would come only by death. But Beowulf was not defeated, for with one last effort he made a terrible thrust with his sword and killed the fire-breathing monster. With victory came the consciousness of suffering and the smarting of the wounds made by the poisonous claws of the dragon warned him that his end was near. He looked around at the treasure which now belonged to the Geatmen and felt joy that even in his old age he had been able to defend and enrich his people. Then death came to Beowulf.

His liegeman carried him back to where the cowardly thanes were delaying, who grieved now right sorely at the loss of their ruler. With lamentation they built the funeral pyre and as the flames ascended they cried aloud in their sorrow, for they knew they had lost the bravest of Geatmen, the noblest of men and the staunchest of kings.

Students in College Mathematics

H. A. FOERING, Bethlehem PREPARATORY SCHOOL, BETHLEHEM, PA.

Paper read before the Sixth Meeting of the Association of Teachers in Mathematics in the Middle States and Maryland, April 14, 1906

T

I

HE causes for the failure of students in college are largely poor teaching of elementary mathematics in preparatory schools and the manner in which the subject-matter is presented in the text-books. The indications are, at present, that we are about to make certain changes for improvement in the teaching of elementary mathematics. What form this improvement will take, it is impossible to forecast exactly. But it will mean the better preparation of teachers of mathematics for their work, and I hope the establishment of a standard of professional skill as a requisite qualification for those who wish to become teachers of mathematics, better definitions of the subject-matter of elementary mathematics and better methods of teaching.

A successful teacher is one fitted by nature, attainments, and training for his work. Besides possessing a natural gift for teaching, he must be endowed with a talent and a love for his subject. He naturally anticipates the boy's future, and by his experience is enabled to foresee the difficulties to be met by the student in his college work and then lays the foundation accordingly. He knows intuitively that this foundation consists not only in the acquisition of knowledge, but also and more largely in the acquisition of correct methods of study and reasoning.

The teacher must next be master of his subject, not master merely of his text-book. He must also have a knowledge of mathematics beyond that which he teaches, otherwise he is evidently unfitted to undertake the preparation of students for such work. He should also be brought into closer relations with those who have charge of mathematics in colleges.

You will find in a thousand high schools and other secondary schools throughout the country, teachers whose only qualifications are a certificate or a college diploma and the experience

of a year, or two, or none in teaching. It is most likely, too, that no part of this experience was obtained under a competent master. Others are teaching whose only qualification for the position is exceptional intellectual attainment in the subject they are supposed to teach, and who in no other way are adapted to their work. It is not necessary to refer to those who make teaching a stepping stone to some other profession or business.

While we have been discussing curricula, definitions of elementary mathematics, reforms of various kinds, we have been neglecting a matter that transcends them all in importance—the training of the teacher. We are apt at times to speak disparagingly of low standards and low requirements at some of our smaller colleges, but we should remember that with more uniform courses and methods in secondary schools, and above all, with better fitted secondary school teachers in mathematics, there will of necessity be an advance in standards, and conditions will tend to correct themselves. We have schools of medicine, schools of law, schools of music; now let our colleges arrange for the proper training of the teacher of mathematics. The German law requires that the teacher of mathematics in the gymnasium shall have taken in the university the special three years' course in mathematics established for that purpose. Even this has been found insufficient to provide teachers who are properly prepared for the work. Our physicians must have four years of special training and then must pass a state examination. There is no such provision for the teacher. The care of the intellect is apparently of less concern than the care of the body. It should be regarded equally as necessary for the teacher also to obtain professional experience under an able master, just as the earnest physician finds it necessary to have hospital experience before practicing his profession. Klein, one of the leaders in Germany of the movement for the improvement of the teaching of secondary school mathematics, and a member of the commission appointed there for that purpose, urges that the prospective teacher of mathematics should spend a part of the college year in traveling and observing conditions relating to the teacher's future work.

In the case of my own school I have for several years been considering the advisability of sending members of the teaching force on travels for a short period each year to observe what is being done in other schools, and I cannot help believing that it would be very profitable.

However, we are just awakening to a clearer understanding of the present day needs, though we seem more tardy in rising to meet the situation in this country than educators in Germany and other foreign countries. It is to be hoped that the present movement, though scarcely well started, will not end until the training of teachers of mathematics in secondary schools shall have come to be an essential part of our scheme of education. We must have skillful, methodical teachers not only in a few schools but throughout the length and breadth of the land, if any uniformly high grade of work is to be done. It is perfectly evident that this is a task of great proportions, requiring the co-operation and organization of those mutually concerned, but the time is admittedly ripe for the inauguration of the work.

In discussing further some elements of weakness in preparation of students, it is to be borne in mind that my remarks have reference especially to the requirements of our technological colleges. Nor do I wish to be construed as opposed to making mathematics more practical and as upholding the teaching of only the logical side. Elementary mathematics can to great advantage be made more concrete, and emphasis laid upon applications. But while the beginner does not enjoy or profit by fine spun logic, the last year's work in the secondary school cannot avoid being devoted more to development of the logical side. The time has come for it. The fact of the matter is, that this is now being too much neglected and it is this neglect that causes many failures.

When it occured to me to undertake a discussion of this question I felt that my view would tend to be one-sided and that a great deal of light might be thrown upon the subject by those in charge of mathematics in our colleges. A circular was therefore sent to the head of the department of mathematics at every college east of the Mississippi, and with very few ex

ceptions the questions were answered. These answers furnished some valuable information.

In spite of the cry in Germany, England and America for concrete adaptations, the weakness of our students according to these replies would seem to be the insufficiently developed methods of thought, and that failures are due to poor preparation as a result of poor teaching rather than to mental deficiencies on the part of the student.

I shall take up several of these subjects, beginning with geometry, and discuss the replies with my views and suggestions.

There seems to be less fault found with the preparation in geometry than with the preparation in algebra. This seems rational. Advanced mathematics is more algebraic in its methods and very apparently depends far more largely upon algebra than upon geometry. I agree with those who think that geometry as taught in our best secondary schools is overdone, that too much time is devoted to needless theorems. It is generally agreed that the first object in teaching geometry is to teach the boy to reason logically, and the second to teach him to express his demonstration in good English and with some elegance. The replies to my circular indicate that while we may say the first is moderately well done, the second is in general so badly done as to call for universal complaint. It appears to me that it would be better to curtail the amount to be taught in geometry, to lay more stress on teaching the student correct methods of procedure in geometrical analysis and upon fittingly expressed demonstrations. In requiring frequent written exercises the teacher should be as scrupulous in requiring good English as in requiring good logic. There is no reason why the mathematics master should be less careful as to language than the English master.

I cannot help feeling, though I know the general opinion seems to be the contrary, that in geometries pictures of figures are bad, and that the mechanical methods of arranging proofs as exemplified in most of our popular text-books do not tend to the above ends. Teaching is easier with these books, but I cannot believe it gives the boy equally efficient training