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are suggested for bringing about this last relationship: By more systematic observation work in the grades, by more responsibility for the course of study and the methods of teaching in the grades, and by the closer supervision of practice teaching in arithmetic by the department of mathematics.

From inferences based upon the foregoing, and from reflection upon the general problem, the committee believe that the best theory and practice of to-day point to the following conclusions: That a foundation in subject matter as a basis for the professional study of mathematics for teaching the subject in the first six grades of the elementary school should include a minimum of one-half year in high-school arithmetic, one year of algebra, and one year of geometry;

, that, exclusive of all courses in psychology, pedagogy, principles of teaching, general method, or history of education, a minimum of one-half year's professional study of arithmetic should be required to include the following: A course in “special method,” the teaching of elementary mathematics which should consider the more elementary phases of the psychology of number; principles of general method in their application to arithmetic; educational values of arithmetic and the place of arithmetic in the general educational scheme; the organization of the elementary school curriculum in arithmetic; the organization of typical units of subject matter for presentation to appropriate grades; the development and writing of typical plans for teaching; the utilization of local and general economic studies for number application; the observation and discussion of typical lessons in the grades showing concrete applications of principles developed; the place of games and other recreational devices in lower grade work in number, and the historical development of the teaching of arithmetic, showing the place and value of certain "methods,” as the Grube, Speer, etc.

The committee believe, further, that every school engaged in the preparation of teachers of mathematics should develop a museum or teaching collection of materials—apparatus, books, pamphlets, papers, etc.—which will aid in interpreting the historic development of the subject, present-day practice, textbooks, etc.; that the head of the department of mathematics should be largely responsible for the organization of the course of study in mathematics in the training school in cooperation with the department of education and the supervisors in the training school; that the head of the department of mathematics should aid in the supervision of the teaching of mathematics in the training school; and that he, as well as the supervisors or critic teachers, should be able to give demonstration lessons in the training school illustrative of principles of teaching developed in the “methods” class, and that the points of emphasis in all observations, discussions, plans, and criticisms should be upon the basis of fundamental principles rather than upon petty details.

The aim in the whole professional consideration of mathematics for teachers of these grades is, broadly, to give acquaintance with the fundamental principles of teaching arithmetic, of the organization of its subject matter, of its place in the educational scheme, and of its historical development. The teacher should be given the pedagogical outlook and perspective of arithmetic, as well as the ways and means of teaching its details. He should know enough of the psychology of number to enable him to secure healthy interest and adequate drill and to sacrifice neither at the expense of the other.

The greatest problem of all at the present time would seem to be to find teachers for departments of mathematics in normal schools who themselves have the wide pedagogic outlook desired for such work. When teachers can be found who have this larger perspective and who will regard the training school as the laboratory for developing insight, intelligence, and a minimum of skill in the teacher to be sent out into the field at large, this vital, daily union of theory and practice will do much to increase the efficiency of the prospective teacher in elementary mathematics.

To this may be added the recommendations received by the subcommittee that inquired into the conditions which a rational preparation of teachers for the seventh and eighth grades should fulfill. The replies received were as follows: Massachusetts—(a) normal or college diploma, (b) summer school course at least every third year, (c) membership in local mathematics club, (d) membership in New England Mathematical Association; Virginia—(a) high-school course, (b) good normal course, with practice work; Georgia—(a) teaching knowledge of subject, (b) practice course, (c) rational course in pedagogy; Indiana-inexperienced teachers should not be allowed in seventh or eighth grade work; Iowa-good knowledge of arithmetic, algebra, and geometry; Missouri-more attention to subject matter, but method not neglected.


The agencies for the further training of teachers in service are far more numerous in the United States than in any other country. Among these may be mentioned teachers' institutes, summer schools, reading circles, and the ordinary routine supervision. Teachers' institutes are classified into State, county, and city, the county being the most usual unit. The meetings are generally held during the school year, and the teachers are compelled to attend. The length of the sessions varies from half a day, when the institutes are held several times during the year, to five days or longer, when the institute assumes more the character of a regular training school at which the regular lessons are assigned and recitations are conducted. There is at present a widespread demand for reform in the method of conducting the institutes to provide for genuine professional growth instead of being what has been termed “inspirational.” It is proposed to replace the discussion of general topics with a class of pupils and discussions by the teachers. In such a program arithmetic often finds a place, from two to three hours in a five-day session being devoted to the subject. In this time the instructor may give his time wholly to the detail of special method, or, perhaps, set before the institute the latest reforms and tendencies in the teaching of arithmetic. The topics of a more general nature discussed by institute instructors may be those concerned with the enriching of the course of study, the making of arithmetic practical, the elimination of wasto material, the relative value of the topical and spiral plans in the arrangement of subject matter, the place of algebra and geometry in the elementary school, and others. In presenting special method the instructor may relate it to devices in the lower grades, the teaching of the fundamental operations, explanations of problems, special topics like percentage, etc.

Summer schools are conducted by universities and colleges and normal schools, or are specially organized by State educational authorities. Their duration varies from 2 to 12 weeks. The work is, in general, intended for teachers in service. Courses in mathematics have been regarded in most cases as of special importance in these summer curricula. In fact, in several instances the work began with courses in mathematics and perhaps one other subject, and was gradually extended to include courses in all the regular departments as the number of students increased and the demand became apparent.

The courses offered in mathematics may be roughly divided into two classes: (1) Those intended primarily to emphasize the pedagogic aspect of the subject, and (2) those intended primarily to develop the subject matter for its own sake or as a prerequisite to other courses. The pedagogic courses include critical studies of the various elementary and secondary branches with reference to scientific interpretation and methods of presentation and also studies in the history of mathematics, with special reference to the needs of teachers in elementary and secondary schools.

In general, it may be said that a large number of those in attendance at the summer sessions are teachers spending a part or all their vacation in study, but it does not follow that all are seeking the strictly pedagogic courses. In fact, very many are pursuing courses for degrees, and are therefore filling out requirements or choosing elective work in subject matter in which they are interested. It is true, however, that in most cases where special courses of a pedagogic character are given they are well attended and fully appreciated. On the other hand, there is a wide difference of opinion among institutions offering summer work as to the usefulness of the pedagogic courses as compared with the content courses. It is believed by many that the best pedagogic training comes through careful and diligent study of the subject matter under the guidance of an inspiring teacher who knows how to exhibit good methods and to impress them by example, rather than precept, upon the students.


The account of the training of mathematical teachers here presented is based on the following reports made to the International Commission on the Teaching of Mathematics, and a few other works of a general character.


Commission Internationale de l'Enseignement Mathématique. Sous-Commission

Belge. Sur l'enseignement des mathématiques . . . dans les écoles primaires, les écoles normales primaires, etc. Rapports. Brussels, 1911.


Heegaard, Poul. Der Mathematikunterricht in Dänemark. Basle & Geneva, 1912.


Board of Education. Suggestions for the consideration of teachers. London, 1912.

The teaching of mathematics in the United Kingdom. Special reports on educational subjects. Vol. 26, part II. London, 1912. The red code, 1912.


Bioche, Ch. Enseignement primaire. In Commission Internationale de l'Enseigne

ment Mathématique. Sous-Commission Française. Rapports. Paris, 1911.


Deutscher Ausschuss für den mathematischen und naturwissenschaftlichen Unter

richt. Vorschläge für den mathematischen, naturwissenschaftlichen und

erdkundlichen Unterricht an Lehrerseminaren. Leipzig & Berlin, 1912. Dressler, H., and Körner, K. Der mathematische Unterricht an den Volksschulen

und Lehrerbildungsanstalten in Sachsen, Thüringen und Anhalt. Leipzig &

Berlin, 1914. Körner, K., and Lietzmann, W. Die Organisation des mathematischen Unterrichts

in den Lehrerbildungsanstalten in Preussen. (In preparation.) Lietzmann, W. Die Organisation des mathematischen Unterrichtes in der Preus

sischen Volks- und Mittelschule. Berlin, 1914. Treutlein, P. Der mathematische Unterricht an den Volksschulen und Lehrer- und

Lehrerinnenbildungsanstalten in Süddeutschland. Baden, Hesse, Württemberg, Bavaria.


den Lehrerbildungsanstalten.

Goldziher, K. Der mathematische Unterricht an

Budapest, 1912.


Conti, A. L'ensegnamento della matematica nelle scuole normali. Rome, 1912.


Commission Internationale de l'Enseignment Mathématique. Sous-Commission Russe. Rapports présentés à la Délégation Russe. St. Petersburg, 1911.

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