# Popular Science Monthly/Volume 68/February 1906/Some Recent Tendencies in Mathematical Instruction

By Professor G. A. MILLER

LELAND STANFORD JUNIOR UNIVERSITY

SEVERAL prominent writers have suggested that 'pure mathematics' should be called 'free mathematics' in view of the great latitude of freedom both in subject matter and in methods of work.^{[1]} This view is diametrically opposed to the one commonly held. The student of elementary text-books on mathematics can not fail to be impressed by the close similarity in subject matter and in methods. In our arithmetics and algebras we find problems which are quite similar to those in the work written by Ahmes, an Egyptian priest living seventeen hundred years before Christ. Our geometries bear such a close resemblance to Euclid's, written three hundred years before Christ, that the terms Euclid and elementary geometry are still practically synonymous in the minds of many teachers.

Moreover, there are numerous *a priori* reasons to regard mathematics as a slave rather than as a free, living, pulsating being, exhibiting the many exhilarating changes characteristic of youthful development. The reasonableness of the main postulates of mathematics has never been questioned. In most cases contrary hypotheses would appear perfectly absurd to those who have not been trained to mistrust their intuitions. The main function of mathematics has been, and probably always will be, to draw necessary conclusions from such postulates. Whether the postulates have been explicitly stated or not is a secondary matter. Hence mathematics has become a vast structure which is perfectly invulnerable except possibly at its foundation, and here the attack seems merely a matter of words. The most that has been done in an effectual manner is to erect structures on other sets of postulates. It should be emphasized that the main duty of the mathematician is to build upon given postulates. In these operations he will always be free from attack, since he can not arrive at any conclusion without proving that it is the only possible one.^{[2]}

As mathematics enters into questions which are common to all society, from the most primitive to the most civilized, it is clear that some of its elements must always enter into the education of every one. Within these narrow limits there is little freedom of choice. In rising from this low view of mathematics to the one which recognizes its value in developing thought power and thought caution, one soon arrives in regions of the greatest freedom. From such a standpoint one can readily comprehend why so profound a thinker as Simon Newcomb should say, 'The mathematics of the twenty-first century may be very different from our own: perhaps the schoolboy will begin algebra with the theory of substitution-groups as he might now but for inherited habits.'^{[3]}

During the last few decades Germany has wielded a predominating influence on the development of mathematics in this country. Hence it is natural that the tendencies of German mathematical development should be strongly felt in this country. Two of these tendencies are especially prominent: viz., the uniting of pure and applied mathematics, and the encyclopedic character of pure mathematics. Definite evidence of the former tendency is furnished by the rapid increase in the number of courses on the mathematics of insurance, mathematics for students of physics and chemistry, and especially by the enactment of 1898 which made applied mathematics a distinct requirement of those who expected to become teachers of mathematics. While the American universities have been more conservative in these directions, yet there are many evidences that these tendencies are strongly reflected in the mathematical courses offered by our higher institutions.

Jacobi was perhaps the first eminent German mathematician who made special efforts to lead his students to the boundary between the known and unknown as rapidly as possible, and then to make them coworkers with him in investigating new problems. His methods were imitated very largely by others so that German mathematical instruction became, to an unusually large extent, instruction in research, or, at least, instruction in regions which had been very inadequately explored. These methods have been employed in other countries. In our own country their introduction was hastened by the teaching of Sylvester and Cayley, who employed similar methods while they were connected with Johns Hopkins University.

While the discovery of new truths gives an interest and charm which can scarcely be attained in any other way, yet the early attack on research problems has not been free from undesirable results. If the mind is centered on one line of thought it is less apt to be in condition to receive deep impressions of other fields which may be equally important. Only the greatest minds have been able to attain to those broad and impressive views which comprehend the true correlation of the different lines of mathematical activity, in addition to making important contributions along any one line. In recent years there has been a tendency to encourage breadth of scholarship even at the expense of research in early years,—a tendency which Klein has aptly named encyclopedic.^{[4]}

In American mathematical research activity was very limited until recent years. Within the last decade the mathematical productivity has more than doubled, both as to quality and as to quantity. This has been largely due to foreign training, as only very few of our larger institutions have a sufficient number of research men on their faculties to afford their students opportunities to enter upon fields of research which are best suited to their tastes and ability. Hence the encyclopedic tendency of German mathematics, and possibly also of that of France, should not affect us for a number of years.

The tendencies which have been mentioned relate principally to university instruction. During the last few years there has been an unprecedented activity along lines which relate principally to secondary schools. This movement is sometimes called the Perry movement, in view of the great activity of Professor John Perry, of the Royal College of Science, London. Perry's paper at the recent Glasgow meeting of the British Association for the Advancement of Science provoked a great deal of discussion, and was followed by the appointment of a committee with Professor Forsyth, of Cambridge, as chairman, 'to report upon improvements that might be effected in the teaching of mathematics, in the first instance in the teaching of elementary mathematics, and upon such means as they think likely to effect such improvements.'

In our own country the movement has been brought into prominence largely through the efforts of Professor Moore, who devoted a part of his presidential address before the American Mathematical Society to questions related to this movement. That the time was ripe for such a movement seems evidenced by the numerous organizations of teachers of mathematics with a view to the discussion of questions related to the improvements in teaching and in the selection of subject matter.^{[5]}

As might be expected these agitations have led to the most severe attacks on the present state of mathematical instruction. Professor Perry seems to have especial gifts along this line, as may be inferred from the following quotation:^{[6]} "I would rather be utterly ignorant of all the wonderful literature and science of the last twenty-four centuries, even of the wonderful achievements of the last fifty years, than not to have the sense that our whole system of so-called education is as degrading to literature and philosophy as it is to English boys and men." This is the view of a man who, as chairman of the Board of Examiners of the Board of Education of London in engineering, applied mechanics and practical mathematics, has charge of about a hundred thousand apprentices in English night schools.

One of the main contentions of the agitators is that our mathematical instruction should be more concrete and inductive. This is frequently expressed by the term 'the laboratory method of teaching mathematics' and several of our leading universities have announced courses to be taught by this method. This gives evidence of a profound movement in methods of mathematical instruction, which will doubtless effect many reforms even if it can not be expected that the extreme views will find general adoption.

The teaching of elementary geometry has perhaps been most severely attacked. This attack has been supported, if not directed, by some of the very foremost mathematicians. Klein has recently said that the methods adopted in Euclid's geometry are unsuited for boys.^{[7]} It has become the fashion of text-book writers to call especial attention to the rigor of their presentation. Fortunately these claims are generally unsubstantiated. There are few things that would give more definite proof of the perfect unsuitableness of an elementary text-book than the fact that every step in the presentation was rigorous. The history of mathematics shows that periods of discovery are followed rather than preceded by examination into the rigor of methods, and the same general principle holds in reference to the training of students.

In France C. Méray has perhaps done the most effective work towards reform in elementary instruction in geometry. At the last meeting of the French Association the section of mathematical sciences passed a resolution to request the association to address the minister of public instruction with a view to encouraging the introduction of 'Mèray's method' in the teaching of geometry. During last year this method was employed in at least thirty of the French schools, and very enthusiastic reports of improvement were received. The method is expounded in 'Nouveaux Eléments de Géometrie par Méray,' as well as in several of the French journals. One of its important features is the introduction of the notions of displacements and the group of displacements into elementry geometry.

While reform movements in mathematical instruction are not new, yet the present movement is without a parallel both with respect to its extent and with respect to its vigor. The question naturally arises whether the mental inclinations of civilized races are undergoing gradual changes so that concepts which at one time give pleasure and power to the youthful mind, at other times are beyond the capacity of the average mind, or whether we are becoming more sympathetic with the difficulties which have to be overcome in mental development and take greater precautions to make the steps natural. Whatever may be the cause, it seems probable that readjustment will be effected without departing very far from the older methods. The most likely changes in elementary instruction are those which Klein has been so actively advocating; viz., the early introduction and frequent use of the concept of a function and the teaching of the elements of analytic geometry and the differential and integral calculus at a much earlier period, before the student begins to specialize.

- ↑ Cf. Liebmann,
*Jahresbericht der Deutschen Mathematiker-Vereinigung*, Vol. 14 (1905), p. 231. - ↑ No attempt is here made to define the term mathematics. The views expressed are equally true whether mathematics is defined as a method, or whether the objects which it considers should enter into the definition. From the former standpoint, geometry, as commonly understood, is not mathematics. For a splendid exposition of the definitions of mathematics we refer to Bôcher, 'The Fundamental Conceptions and Methods of Mathematics,'
*Bulletin of the American Mathematical Society*, Vol. 11 (1905), p. 115. - ↑ Newcomb,
*Bulletin of the American Mathematical Society*, Vol. 3 (1894), p. 107. - ↑ Klein,
*Jahresbericht der Deutschen Mathematiker-Vereinigung*, Vol. 13 (1904), p. 473. - ↑ The following associations have been organized within a few years: Central Association of Science and Mathematics Teachers, Association of Teachers of Mathematics in the Middle States and Maryland, Association of Ohio Teachers of Mathematics and Science, Mathematical Section of the California Teachers' Association, Missouri Society of Teachers of Mathematics, Mathematical Association of Washington, Association of Mathematical Teachers in New England, Kansas Association of Mathematics Teachers.
- ↑
*Nature*, Vol. 65 (1902), p. 484. - ↑
*Jahresbericht der Deutscken Mathematiker-Vereinigung*, Vol. 14 (1904), p. 41.