history for reference; mistakes by the profession are avoided; the present profits by the past; experiments and results are a matter of record. Not so with pedagogics; no vast indexed and convenient libraries are at hand: experiment is repeated with the same necessary conclusion.

In the common schools, one remembers departmental instruction; the Grübe method; the misapplication of Herbart's philosophy, whereby the pupil is to make effort only when interested; freedom in the selection of tasks with voluntary application only required; abolition of formal movements about the schoolhouse, the outgoing and incoming to be like that at concert and church; increasing time appropriated to rehearsing and preparing for public displays; encouraging the press in publicity of individual school exercises; coddling the vanity of parents by pupils' spectacular public performances; decrease of memory training, forgetting Comenius, who wrote "the acquisition of knowledge depends upon memory many such practises, each one of which depends, not on principle, but upon the skill of the teacher, measures which, like the Quincy Plan, tho approved, to be profitable must be accompanied by the spirit and skill and inspiration of the master, and be a part of the innovation.

A picture of the whole field today gives pleasure, encouragement, and enthusiasm to him who looks. The best American manhood and womanhood is found in our schools. Impelled by desire to great accomplishment-the greatest and noblest in life that of making virtuous and cultured men and women of the boys and girls in their care, thousands are active in the schoolrooms. Their opportunities are better than were ever before, with such equipment as our fathers never imagined; with pure hearts and cultured brains and vigorous characters and an intelligent and God-fearing Nation for promoter, one can predict for the future of our Country the highest prosperity and an enduring Government.

AARON GOVE

DENVER, COLO.

VIII

THE INTERNATIONAL COMMISSION ON THE TEACHING OF MATHEMATICS

At the fourth International Congress of Mathematicians, held at Rome in 1908, the following resolution was adopted: "The Congress, recognizing the importance of a comparative examination of the methods and plans of study of the instruction in mathematics in the secondary schools of the different nations, empowers Messrs. Klein, Greenhill, and Fehr to form an International Commission, to study these questions and present a general report to the next Congress.”

The committee was in due time organized as follows: President, Geheimrath Professor F. Klein of Göttingen; Vice-president, Professor Sir George Greenhill of London; General Secretary, Professor H. Fehr of Geneva. These gentlemen began work at once and appointed as members of the Commission three representatives for each of the countries usually having delegates at the congresses, and one for each of the other countries of importance.

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On account of the varied meanings assigned to the term secondary schools" it was evident from the beginning that the investigation could not be limited to the courses given in institutions that would be called by this name in America. Indeed, such limitation would be undesirable from every point of view. It was therefore decided that the Commission should extend its work so as to include the whole field of mathematical instruction, from the lowest steps to the highest, or at least to the point where individual research definitely begins. This includes the work of the common schools, or indeed, in America, those from the primary grades thru the college, and it also includes the preparation of teachers for these schools, which runs at once into the work of the universities. Furthermore it includes the mathematical courses in the increasingly

large number of technical schools, trade schools, private schools, and professional institutions of various types, schools in which the sexes are more apt to be taught separately than is the case in the American public schools, and where traditional mathematics is giving place to something entirely new in most parts of the world.

The results of such an international investigation can not fail to be stimulating to the best that is in mathematics, and indeed to the best that is in education in general. It is not a body of professional reformers that is at work, but a body composed of practical teachers and of some of the best known of the world's mathematicians. It is the desire of this body to know exactly what is done today in the teaching of mathematics in all of the important countries of the world; to know if any countries are accomplishing exceptional results in a given. time and under relatively the same conditions as obtain in other countries; to know how teachers are prepared to give instruction in mathematics in the high schools, for example, in France, and Italy, and Denmark; to know what general means are taken to give instruction in the subject in the universities of England, of Germany, and of Russia; to know the progress that is being made in answering the vext question of the best mathematics for the special types of school that are springing up all over this country as well as elsewhere, and to learn from others the effect of abandoning the rigid mathematics of the old school in favor of what many feel to be the sentimental mathematics that is now so often advocated. There is also the question of examinations that needs to be considered. What are the results of England's rigid system? of the examinations which France imposes upon those who would enter schools like the Ecole Normale Supérieure? of the college entrance examinations in our eastern states, as compared with the certification system? It is true that there is no unfailing unit of measure for results of this character, but after all some scientific results are obtainable, and even if they do not settle the issue, the world of mathematics will be benefited by knowing what they are. We have also certain influences tending to make better teachers of mathematics

that are somewhat peculiar to this country, and these it will be of interest to other countries to know. Such are some of our teachers' associations, our teachers' institutes, our systems of state inspection and supervision, some of our scientific societies and publications, and the often unappreciated but none the less valuable influence of publishers and their agents, an influence that is unknown in many parts of the world.

In order to set forth more clearly the exact purpose of the Commission, the following extract is made from the first preliminary report: The general plan of the work consists of two parts. Of these the first relates to the present state of the work and to the methods of mathematical instruction. Under this general topic are the following sections:

Section I. THE VARIOUS TYPES OF SCHOOLS.-In this first chapter will be given a concise exposition of the various public institutions of learning in which mathematical instruction is given and the aim of each school will be indicated. Schools for girls will be included.

The institutions will be distributed according to the following classification:

(a) Primary schools, lower and higher.

(b) Middle schools or higher secondary (lycées, German Gymnasiums and Realschulen, etc.).

(c) Middle professional schools (Technicums, etc.).

(d) Normal schools of the various grades (seminaries for teachers, "teachers' colleges," etc.).

(e) Higher institutions (universities and technical schools). It is desirable that this exposition be accompanied by a schematic table giving a general view and making evident the succession and correspondence between the diverse establishments and indicating also the average age of the students.

Section II. AIM OF THE MATHEMATICAL INSTRUCTION AND OF THE SEPARate Branches.-This question will be studied for the various types of institutions mentioned above, taking into account, wherever necessary, applied mathematics, notably mechanics.

Not only does the aim of mathematical instruction vary necessarily in different institutions, but it has undergone some

transformations in the course of the last decade. It may be purely formal, or formal but taking account of intuition; it may also lay stress on logical development and the utilitarian side simultaneously, or else regard only the practical. On the other hand, the development of the memory may be the principal aim, or contrariwise the development of the mathematical faculties.

What are the branches of mathematics taught in the different types of schools? The time allotted to the branches should be indicated and the extent of the program. To what extent is attention paid to correlation among the mathematical branches and, if there is occasion, the correlation between these branches and applied mathematics (including mechanics) and physics?

Section III. THE EXAMINATIONS.-It is unquestionable that the system of examinations has a great influence on the method of instruction. The characteristics of the examinations in each category of schools should be concisely indicated, and particularly those which lead to "certificates of maturity," to "degrees," etc., and the examinations of candidates for teaching.

Section IV. THE METHODS OF TEACHING.-What are the methods used in the various institutions, from the primary schools to the higher institutions? Material of instruction; mathematical models; use of manuals, textbooks, collections of problems. Theoretical problems; problems taken from the applied sciences. Practical work.

Section V. PREPARATION OF CANDIDATES FOR TEACHING. -Here again are to be included the diverse types of schools, and there are to be indicated the requirements demanded by the school authorities: (a) with regard to theoretical preparation; (b) to professional preparation.

The second part relates to the modern tendencies in the teaching of mathematics, and is subdivided into the following sections:

Section I. MODERN IDEAS CONCERNING SCHOOL ORGANIZATION.-Reforms in studies. New types of schools. The question of coeducation of the two sexes.