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school, the method teacher in a brief course of 15 hours gives a historical outline of the special method and discusses the aim and problems of mathematical instruction and the arrangement and distribution of the material for the elementary schools. On this subject the committee already referred to recommends that the course on method should include the methods of teaching in all grades of the elementary schools, a critical study of methods and theories, and an examination of the most usual apparatus and local textbooks. To give a comprehensive view of the subject, the course should not omit reference to the methods both of the kindergarten and of the secondary schools, while some assistance and advice should be given to the students for their own further study.

The permanent appointment of German teachers can only be secured by passing a second examination for which candidates become eligible as a rule two years after they have passed the first examination at the close of their normal school career. The second examination, however, is rather of a practical or professional character and is intended to test the ability of the candidate as a teacher rather than as a student. While a test is, as a rule, given in a special elective subject, it is not of so much importance as professional skill. Since the first appointments are in most cases in rural districts, the teachers are thrown on their own resources and any study that they undertake must necessarily be done independently and without guidance. Bavaria has instituted district extension courses to prepare for the second examination, but this practice appears to be isolated. A stimulus is afforded to further study by the examinations for promotion to principalships or positions in the middle schools and normal schools. Increasing opportunities are being offered, as in Saxony, Hesse, Wurttemberg, and Bavaria, to students who acquit themselves excellently in the final examination at the normal schools to proceed to the universities, but as students who do so rarely return to the elementary schools, a discussion of these facilities and examinations is not appropriate here.


The Hungarian normal schools are maintained by the State or the different religious denominations, schools of the latter type being in the majority. Separate schools exist for the training of men and women. All the schools follow more or less closely the program prescribed by the State. These have been recently revised (1911) and brought into closer agreement with modern reform movements. The normal schools furnish a four-year course to candidates who are admitted at the age of 14 from the lower classes of the secondary schools and the intermediate schools (Bürgerschulen).

The aim of instruction in mathematics is declared to bethe study of algebraic foundations of the common arithmetical operations, the knowledge and accurate application of practical arithmetical problems taken from daily life, and the study of the most important principles and the simplest practical application of elementary geometry, with particular attention to the needs of elementary school instruction.

The following outline indicates the scope of the course in mathematics:

FIRST YEAR (four hours). Arithmetic and elementary algebra: Introduction to the nomenclature of algebra; fundamental operations of algebra. Negative numbers, the number system; the fundamental operations in the decimal system. Divisibility of integral numbers. Fundamental operations with common fractions and decimals. Simple equations with one unknown; graphic equations with two or three unknowns.

Geometry: Measurement of length and angles. Angles. Parallel straight lines. The chief properties of the triangle, quadrilateral, polygon, and circle.

SECOND YEAR (three hours). Arithmetic and elementary algebra: Powers and roots. The second and third power of algebraic and decimal expressions. Fractional numbers; pure and mixed quadratic equations with one unknown; imaginary and complex numbers.

Geometry: Principles of equality. Problems of construction with reference to the triangle, quadrilateral, and regular polygons. Similarity of figures; similar triangles; the theorem of Pythagoras. Inscribed and circumscribed triangles and quadrilaterals. Calculation of the side of an equilateral triangle and the regular hexagon. The circumference. Calculation of area of these figures.

THIRD YEAR (two hours). Arithmetic and elementary algebra: Proportion; rule of three and proportional division. Percentage in commercial and statistical applications; coinage; alloys; national and foreign money systems.

Geometry: Analytic geometry of the point, straight line, and triangle, with corresponding development of the prerequisite algebraic knowledge, based upon the function concept and graphic methods.

Method: Discussion and explanations of the syllabus and suggestions for teaching arithmetic and geometry in the elementary schools, including a consideration of manuals, textbooks, and apparatus.

FOURTH YEAR (three hours). Arithmetic and elementary algebra: Simple interest and discount. Arithmetical and geometric progressions. Principles of compound interest (savings banks, annuities, redemption of loans); compound-interest tables. The most important features of commercial practice and exchange.

Geometry: Relation of lines and planes in space. Simple exercises in leveling. Lines of equal height. Reading of topographic, especially military, charts; simple exercises in surveying. Construction, surface, and volume of the prisms, cylinder, pyramid, cone, and conic sections, and the sphere. Definition and construction of the ellipse, parabola, and hyperbola.

The chief criticism that is made against this syllabus is on the ground of arrangement. While it is thought that the addition of logarithms and the elements of plane trigonometry would be a great gain, it is felt that this is the only criticism that can be made on the ground of subject matter. Considerable improvement could, however, be made if all the arithmetic were completed as a foundation for the study of principles and of algebra, and if the algebra and geometry were more closely correlated.

The professional study in the normal schools begins with the observation of instructions in the model school from the third year on. Practice teaching is commenced in the third year, but is not fully developed until the fourth year, when each student teaches six hours a week.



The training of teachers for Italian elementary schools is provided in normal schools, which are classed with secondary schools. The normal schools are organized on a three-year basis, and until recently boys and girls were taught in separate institutions. Coeducational schools have, however, sprung up within the last few years. Since the normal schools can not supply the demand for teachers created by the establishment of new elementary schools and the extension of the elementary-school period, special two-year courses (corsi magistrali) have been established for students who have passed through the ginnasi, which give a five-year secondaryschool course. The corsi magistrali are mainly of a professional character. Students are admitted to the normal schools at about the

age 15 from higher elementary or intermediate schools with three-year courses beyond the elementary schools—scuole tecniche for boys and scuole complementari for girls. Admission is by certificate from these schools; entrance examinations are given only in special cases. Each institution is equipped with a complete elementary school for practice and observation, while the girls' normal schools have, in addition, kindergartens and complementary schools attached to them.

Normal-school instructors must be graduates of universities and are appointed on the basis of a competitive examination. In the boys' normal school the same instructor has charge of mathematics, physics, and natural sciences; in the girls' schools these subjects are in different hands, but the instructor of mathematics is also required to teach in the complementary school.

A striking feature of the mathematical program, which is given below, is the omission of arithmetic in the first year of the course and the introduction of algebra in its place. Special attention is also given to method in the second year, a survival probably from

the time when a teacher's lower certificate could be obtained at the close of that year.

The program is as follows:

FIRST YEAR (three hours boys, two hours girls). Algebra: Introductory ideas. The four operations with integral quantities; equations of the first degree with one unknown. Square and cube roots with approximations.

Geometry: Definitions and introduction to plane geometry. Angles, triangles, and quadrilaterals. Regular and irregular polygons. Circle. Equality of polygons. Measurement of straight lines, angles, polygons, and circles. Equality of plane figures and principal theorems concerning plane figures.

SECOND YEAR (two hours). Arithmetic: Magnitudes; numbers; numeration; analysis of the four operations. Methods of teaching numbers and the four operations in the elementary schools. Ratio and proportion.

Geometry: Proportional lines and similar polygons. Methods of teaching the notions of plane geometry in the elementary schools. Bookkeeping: Inventory; trial balance; accounts rendered.

THIRD YEAR (two hours). Arithmetic: Magnitudes in direct and inverse ratio; rule of three, simple and compound; solution of related problems by proportion and reduction to unity. Methods of teaching the rule of three in the elementary schools.

Geometry: Straight lines and surfaces and their relative position in space. Dihedral and polyhedral angles; polyhedrons; prisms, cylinder, pyramid, cone, sphere. Fundamental notions of congruence and similarity. Methods of teaching solid geometry and the metric system in elementary schools. Bookkeeping: Daybook; ledger; cashbook.

This is by no means an ambitious program. Its critics object that the time allotted to it is insufficient, and that the arrangement is illogical, since much of the early work depends on a knowledge of matter which is at present postponed to a later year. Furthermore, the students under the present arrangement fail to obtain a comprehensive and systematic view of the methods of instruction and of the program of the elementary schools. The regulations prescribe that arithmetic must be taught with scientific rigor, while instruction in geometry is to be deductive in the first year and inductive in the second and third—a distinction which the reformers regard as vague and meaningless.

The normal schools have been under fire for some time. It is felt that the course of three years is too short a time in which to train cultured and efficient teachers. The practice of crowding the academic and cultural subjects and the professional subjects together is not only dangerous, but results in inefficiency. Demands are now being made for a longer course; for a separation of the two main purposes of the normal schools——the general and professional training; for better coordination between the various subjects; and for a reorganization of the program to the exclusion of the superfluous and an emphasis on the new elements demanded by modern culture.


The Russian public elementary school system provides for three types of schools: The ungraded elementary school with a course of three years, gradually being extended to four years, receiving pupils at the age of 7; the two-class elementary school which gives a fiveyear course, divided into two sections of three and two years, and which receives pupils at the age of 7; and the municipal elementary school, which in general gives a four-year course to pupils coming from the elementary school at the age of 10 or 11. The first two types of school are taught by a single teacher. The mathematical work of these schools is necessarily simple. In the elementary schools arithmetic only is taught, with the briefest outlines of geometry. The arithmetic here covers the four operations with abstract and concrete numbers and simple fractions, and geometry includes simple ideas of form and measurement. Five hours a week for the whole school are allotted to the subject. In the elementary schools with two classes the program is somewhat extended, but there is much variation both in selection of subject matter and in the time allotment, which ranges from three to six hours for arithmetic and from one to three hours for geometry. Arithmetic is carried up to common fractions, decimals, problems in the rule of three, alligation, alloys, partnership, and interest. Instruction in geometry is limited to a concrete basis.

The mathematical work in the municipal elementary schools is still organized on the basis of regulations issued in 1877, and includes arithmetic, geometry, and algebra. Arithmetic, to which are devoted from 8 to 16 hours a week in all the classes together, covers recurring decimals and proportions. Geometry, from 6 to 9 hours a week, is restricted to a fairly complete study of the problems occurring in plane geometry. The work in algebra, from 2 to 4 hours a week, is carried as far as simple equations.

It is obvious from the above that the preparation of students who enter the normal schools is very slight. Two types of normal schools are maintained, corresponding to the two main types of elementary schools. The normal seminaries are intended for the training of teachers (men and women) for the lower elementary schools—the ungraded and the two-class schools—and the normal institutes for the training of teachers (men only) for the municipal schools. Students are admitted to these institutions on the basis of a competitive examination. Candidates for the normal seminaries are drawn from among the more able pupils of the two-class schools, who, on completing their course, usually remain for review and practical work under the direction of their teachers until they reach the eligible age of 14. Candidates for admission to the normal institutes must have

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