mathematical sciences, an introduction might be given to elementary trigonometry, which is at present neglected entirely. The standards of attainment for women are in all cases lower than for men students.


The regulations recommend that the Socratic method of instruction be used in the normal schools. It is conceded that this recommendation has much in its favor, but it neglects the time element, for much more ground could be covered by the didactic method. Instruction of this character, however, is in agreement with the general aim of the teaching of mathematics. It also enables the teachers to emphasize the characteristic methods of mathematics, of deduction, analysis and synthesis, and induces the students to consider the theory underlying the methods employed and so to discern for themselves the appropriate method in each case. It is admitted, however, that in spite of the prescription of the regulations to emphasize principles and theory, the students are not sufficiently mature to look upon arithmetic as anything but an accumulation of facts and a collection of operations or to recognize the value, use, or connection of these. Hence, while they can perform the four operations, they fail to arrive at their true meaning or at exact definitions and rigorous proofs. The charge is as usual laid at the door of the elementary schools, for too often the instructor is compelled to repeat the work of the lower stage. It is, however, suggested that such repetition can be made valuable if it is done in such a way as to throw new light on the general theory and principles of the subject, the aim suggested in all the regulations since 1881.

On the practical side the regulations recommend an emphasis on mental arithmetic in order to secure readiness and flexibility in working with numbers. Problems are selected from the operations of everyday life-commerce, industry, manufacture, and agriculture. The impossible or improbable exercises of former years are excluded.

The same spirit permeates the suggestions on the teaching of algebra. Long and complicated exercises are to be avoided and those parts of the subjects are to receive attention which can be of some service in other subjects of the normal-school course. At the same time from six to seven hours are to be devoted to the theory of algebra.

Geometry is, so far as possible, made practical, and little attention is given to theory. The subject is correlated with drawing, and the figures that are studied in the latter connection are made the objects of study in the geometry lessons.

The instructor is not limited in any way in the selection of a textbook, but the textbook is only to be used for the study of details. The work of the classroom is intended to bring out the essential points and to develop the most difficult parts.

At the close of the second year of the normal-school course the students are given an examination in the general academic work of the first two years, leading to the brevet supérieur. This examination is required as a qualification for appointment as a director of a cours complémentaire or as teacher in such a course or in the higher elementary school. It is also one of the preliminary requirements for the certificat d'aptitude au professorat des écoles normales and certificates for teaching special subjects. Candidates for the brevet supérieur must be over 18 and hold the brevet élémentaire. The examination consists of two parts, written and oral. In the written part, mathematics is grouped with science and the examination consists of (a) a problem in arithmetic or geometry, applied to practical operations, (b) a question on the theory of arithmetic, and, for men only, (c) a question on physics and natural science as applied to hygiene, industry, and agriculture. In the oral part, questions are given on arithmetic and problems to be performed mentally, and for men, on algebra and geometry. It will easily be seen that little can be expected in an examination of this character, and the importance of the subject is still further minimized by the small proportion of the marks allotted to it.

The third year of the course is devoted to the professional preparation of the students. The work includes a study of the principles of education and methods of instruction, with observations of model lessons and practice in an elementary school of the locality or attached to the normal school. The topics studied in connection with the special methods of teaching arithmetic include the following:

General principles governing the teaching of elementary arithmetic.

Study of courses in arithmetic in elementary schools with reference to such special problems as the teaching of fractions or the introduction of decimals.

Knowledge of numbers. Importance of the first 10 numbers. Arithmetical operations and their introduction at different stages of the elementary school course.

Solution of problems. Development of reasoning ability. Principles underlying mental arithmetic. Critical study of the textbook.

Model and criticism lessons are conducted in connection with this study. The students themselves are expected to do two months of practice work during the year in two periods of a month each. There is no provision for the time to be allotted to the teaching of arithmetic. The practice work is not regarded with much seriousness, since teachers are expected to have at least two years' actual experience before they receive their permanent appointments as titulaires.

1 General method is included under pedagogy and principles of education.

The final examination for the certificat de fin d'études normales consists of three parts: (1) A written theme on a pedagogical topic selected from a list, for which two months are allowed; (2) a practice lesson after one hour's preparation; (3) an oral examination with questions on the organization of a class, school programs, methods of instruction, with special reference to the practice lesson.

In general the work of the normal schools is criticized on three grounds, over the first of which the normal school has obviously no control: (1) The poor preparation of the students; (2) too much subject matter is to be covered and too many class periods are required to enable the students to do any genuine independent study; and with reference to mathematics, (3) the proportion of the marks allotted to mathematics in the various examinations is small.


No country is making greater progress in the training of its teachers than Germany, and in no subject is this more true than in mathemetics. Wedded as Germany, and more particularly Prussia, has been to the immediate demands of professional training, nothing less than a revolution is taking place in the whole conception of the training of teachers. In place of the narrow restrictions which the traditions of 50 years have imposed on the normal school, the realization is gaining ground that a good teacher must be broadly educated, with a liberal grasp of subject matter, and not merely the master of a few tricks of the trade. But the influences of tradition are not to be swept away in a few years. All that can be said for the present is that the tendencies are liberal and the recent reforms in mathematical study are gradually being introduced. It is not intended, however, that the mathematical curriculum of the normal schools shall be more extensive than that of the secondary school, nor again that the students of the normal schools shall be encouraged to proceed to the universities. But, as will be indicated later, while there is progress universally throughout Germany, it is not equally marked in all the States. The smaller States of central Germany, for example, are far more liberal than Prussia or Bavaria; while Saxony may be said to stand midway.

There are, in general, two main types or systems for training teachers—that of Prussia and that of Saxony, the remaining States following more or less closely the one or the other. Hesse and Bavaria until recently stood alone in having a five-year course, but in 1912 the Bavarian course was changed to the usual six years. The chief differ

1 The present account deals solely with the training of men; the women who intend to teach in elementary schools in the majority of cases pass through the secondary schools for girls (Höhere Mädchenschulen), but where this is not the case, as in Wurttemberg, the requirements in mathematics are always lower for women than for men.

ences of organization between these two systems are that the Prussian course is divided into two periods—the preparatory course of three years in the Präparandenanstalt and the three-year course in the normal school (Volksschullehrerseminar). The other system provides a course of six years in the same institution (Volksschullehrerseminar). This distinction is slight, however, for the standard of work does not vary, although in Prussia the qualifications of teachers in the preparatory institutions are not as high, for example, as those of the instructors in the normal schools of Saxony. The first five years of the whole course whether given in one or two institutions, are devoted mainly to general academic training, the professional preparation being introduced in the fourth year and given special emphasis in the final year.

The training career of the students begins at the age of 14, when they enter the preparatory course, which, like the normal-school course, is intended solely for future teachers. The majority enter from the elementary schools, the few exceptions coming from the "Real Schools” (Realschulen) and occasionally from a secondary school with full nine-year course.

Hesse alone is trying the interesting experiment of providing a one-year professional course for graduates of the nine-year secondary schools.

The students entering the training course are selected on the basis of an entrance examination on the subjects of the elementary schools. The outline prescribed in mathematics by the Prussian Government will serve to indicate the scope of the work:

Arithmetic.-Lower stage: Operations with concrete and abstract numbers from 1 to 100. Middle stage: Similar operations without any limit; averages; factors and reductions; simple rule of three. Upper stage: Fractions and applications to arithmetic of everyday life, and decimals.

So far as possible, it is required that the work in the lower stage should be mental and preparatory with work on the blackboard, and that the problems should be practical and related to the needs of everyday life. At all stages it is desired that emphasis should be placed on clear thinking and correct expression, with ability to do independent, accurate, and rapid work as the ultimate goal.

Geometry.-Lines, angles, triangles, quadrilaterals, regular figures, the circle and its parts, regular solids. To this are added, in graded schools, principles of lines and angles, and equality and similarity of plane figures.

The work in drawing is closely related both to geometry and arithmetic. Algebra has practically disappeared entirely from the elementary school program.

The elementary school programs in arithmetic are very similar in the other States. The larger cities are offering somewhat more extensive courses, plane and solid geometry, for example, being added to the descriptive geometry.


It will serve for the present purpose to disregard the division between the preparatory institutions and the normal schools which prevails in Prussia, and to consider the six-year course as a unit. The work of the first few years is devoted to a review of the elementary school course and the introduction of algebra. The aim of the whole course may be indicated by several quotations:

Prussia.—The aim is to secure clear comprehension, a readiness in solving problems and ability to impart instruction in arithmetic and geometry in the elementary school. The preparatory institutions serve to lay a sound foundation by reviewing and extending the work of the upper grades of the elementary school.

Saxony.—The aim is to secure accuracy and ability to solve problems in arithmetic and geometry that are appropriate to the elementary and continuation schools, with clear insight into the essence and principles of the methods necessary for professional work; and to secure the possession of such mathematical knowledge as will furnish a grasp of the problems of daily life beyond the standards of the elementary and continuation schools, or as may be essential for more thorough work in other subjects, e. g., physics and geography.

Bavaria.—The chief aims of instruction are the thorough grasp of the relations of number and space, correct understanding of operations and methods in calculations, accuracy, and rapidity in estimating and in solving problems.

Wurttemberg.The aim is to secure clear insight into the most important principles of elementary mathematics, their relation, application, and significance to other sciences; readiness in the solution of problems.

The time allotted to mathematics (arithmetic, algebra, geometry, and, in some States, trigonometry) shows considerable variation. In some States


little is done in the last, or sixth, year of the course; in general, no new work is then taken up, except where there are elective courses. The following table represents the number of hours per week given to the mathematical subjects in each year of the course:

Number of hours per week in mathematical subjects.

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In presenting an outline of the mathematical studies, it will be convenient to give in parallel columns the work of the normal schools in Prussia and Saxony, as the chief representatives of the two types, and to indicate the variations found in other States.

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