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factor, which has perhaps not received the attention in the reports that it merits, is that in general the instructors in normal schools are themselves not university trained, and the demand for universitytrained instructors can not become strong until the principles upon which the training of elementary-school teachers is based become broader and more liberal. Finally, it may be that in most countries, and in the United States in particular, the standards in mathematics have been greatly affected by the fact that the great majority of candidates entering the teaching profession are women. It is almost universally the case in European normal schools that in mathematics lower attainments are required from the women than from the men students.

The general problem is, however, receiving considerable attention both in practical administration and in theoretical discussions. At present the European countries are all passing through a transition stage, which finds expression in dissatisfaction with the prevailing arrangements. The tendency generally is in the direction of raising the standards of the academic or cultural training—a tendency which reaches its culmination in the admission of certain elementary-school teachers to some university courses in Germany; in the provision of facilities to attract graduates of secondary schools to the elementaryschool service, as in Hesse, England, Italy, and Switzerland; in the gradual separation of professional and academic training by such a provision as the introduction of a fourth year for students in the departments of education of universities in England; and in the United States in the continually increasing opportunities for the improvement of teachers in service. That the best thought in this country has not been backward in formulating the task that lies before those interested in the training of teachers is indicated in the recommendations of the American committees, which are quoted in the section on the United States.


The normal schools for men and women in Belgium are organized on the basis of a four-year course. The students are admitted at the age of 15, after an entrance examination on the subjects of the elementary school. Arithmetic is the only mathematical subject included in the examination, and consists of a written test (two questions of general arithmetic and two problems) and an oral test (two questions of general arithmetic and an exercise in mental arithmetic). The students must obtain 50 per cent of the marks to pass. The mathematical subjects in the normal schools consist of arithmetic, algebra (in first and second years), and plane geometry (in second and third years) for men, and only arithmetic for women. The time allotment per week is as follows:

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Algebra and geometry are dropped in the fourth year of the men's course, and the time is given to a review of arithmetic with special reference to the requirements of the elementary school and the method of instruction. Promotion examinations take place at the close of each year on the subjects of that year. The final examination includes only mental and general arithmetic and the metric system.

The following aims are prescribed in the regulations of 1896: (1) The subjects are to be limited to what is necessary with special reference to the needs of the different stages of the elementary school; (2) attention is to be confined to the purely professional aspect without digressions into theory, except that instruction in everyday arithmetic must emphasize the practical utility of the subject and its content with reference to partnership, insurance, savings banks, annuities, etc.; in mental arithmetic the students are to be drilled in short methods and rules rather than principles; (3) the students must be trained by frequent handling of concrete objects, by questions, oral work, and practical exercises to promote their own development. It is found, however, that as a general rule the method of instruction is didactic, and the students tend to be passive recipients.

The outline of the work in the men's normal schools is as follows:


Arithmetic.-Integers. Preliminary introduction and definitions. Decimal system. Fundamental operations, their number and character; methodical explanation; principles of multiplication and division.

Decimal numbers. Numeration and its properties. Fundamental operations. Approximations.

Concrete and practical study of the legal system of weights and measures. Divisibility and properties of numbers. Principles of divisibility, e. g., by 2 and 5, by 4 and 25, by 8 and 125, by 9 and 3, by 111. Remainders in divisions by these numbers. Test of multiplication and division by casting out the 9's.

Greatest common measure of two or more numbers by the method of successive divisions.

Common fractions; origin and definitions. Numeration. Simplification of fractions. Reduction of fractions. Reduction to the same denominator. Fundamental operations. Conversion into decimal fractions.

Exercises in mental arithmetic, chiefly with the aid of short method with whole numbers, decimals, and common fractions.


Solution (by the method of reduction to unity) of various problems bearing on everyday needs.

Discussion of the work in the lower stage of the elementary school.

Algebra.-Easy problems to illustrate the value of algebraic notation. Introduction and definitions. Fundamental operations. Factors. Fractions; definition; simplification by factors. Fundamental operations. Equations; definitions; general principles of the solution of equations. Solution of numerical equations of the first degree with one, two, and three unknowns. Methods of elimination by addition and subtraction, by substitution, by comparison of values. Problems. Principles of divisibility. Division of xn±am by x±a.


Arithmetic.-Properties of numbers. Least common multiple and greatest common measure of two or more numbers. Reduction of fractions to the same denominator. Divisibility by 6, 18, 15, 45, etc. Recurring decimals.

Ratio and proportion. Numbers directly and inversely proportional. Problems. Method of reduction to unity. Application to rule of three. Simple interest; simple discount at home and abroad; average maturity; revenues; contracts; savings banks and State annuities; insurance; proportional shares and partnership; rate, exchange, and commission; alligation; and alloys.

Mental arithmetic on these operations.

Discussion of the work of the intermediate stage of the elementary school, including the legal system of weights and measures.

Algebra.-Algebraic equations of the first degree with one unknown. Problems. Examples of indeterminate and impossible numerical equations.

Meaning of negative solutions of problems. Exercises on negative quantities.

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Discussion of problems of moving bodies, etc., especially of geometric problems. Algebraic equations of the first degree with two or three unknowns. Problems. Discussions of the general principles for solving two equations with two unknowns. Geometry.-Definitions. Axioms. Properties of triangles. Equality of triangles. Properties of perpendiculars and oblique lines. Equality of right triangles. Parallels. The sum of the angles of a triangle and of some polygons. Properties of a parallelogram.

Properties of the circle and figures resulting from combinations of circles and straight lines. Relative positions of two circles.

Incommensurables in general. Measurement of angles. Inscribed and circumscribed quadrilaterals.

Noteworthy points of triangle; circumscribed, inscribed, and escribed circles; center of gravity, common point of the altitudes.

Problems of construction. Loci. Analysis and synthesis in geometric solutions. Applications.

Measurement of plane areas. Principal relations between the parts of a triangle.


Powers. Extraction of square and cube roots. Determination of these roots approximately.

Arithmetical and geometric progressions.

Logarithms. Use of tables. Application to compound interest and annuities. Discussion of the work of the upper stage of the elementary school.

Geometry.-Proportional lines. Similarity of figures. Proportional lines in a circle. Area of the triangle as a function of the three sides.

Problems of construction based on the previous course.

Properties of regular polygons. Construction of regular polygons with 3, 4, 5, 6, 8, 10, 12, and 15 sides. Area of a regular polygon. Measurement of circumference, circle, and sector. Discussion of a method of determining the relation of the circumference to the diameter. Problems of construction.

Surveying. Description of use of surveying instruments. Measurement of the size of plats. Drafting of plans with the aid of the instruments and the plane table. Practical applications. Leveling.



Arithmetic.-Thorough review of the work of the first two years. Review of the program of the elementary school as a whole with model lessons, especially on those parts less thoroughly studied in the normal schools.

Various problems and exercises in rapid calculation.

In addition to this program the students are given a course in special method by the professor of method. This course includes a discussion of the methods of the mathematical sciences-analysis and synthesis, induction and deduction, and of the theory and method to be followed in teaching arithmetic, the system of weights and measures, and the prescribed program. This course is supplemented by the professor of mathematics, who discusses the development and correlation of the programs of the elementary and adult school, and directs the model lessons in the subject and the subsequent criticisms. In the different classes of a practice school the students give practice lessons in computation, theoretic arithmetic, and the metric system.

The program of the normal schools is admittedly inadequate, but the tendency to adopt the suggestions of the modern reform movement is not yet appreciably strong. All that the reformers propose at present is an extension of the arithmetic program in the norma] schools for women by one year and the introduction of two books of plane geometry, and in the normal schools for men the continuation of algebra up to the solution of equations of the second degree with one unknown and of geometry up to the chief principles of solid geometry.


The normal schools of Denmark are public and private, but all are under State supervision; a few are coeducational; students, who must be over 18 years of age on entering, must have had one year's teaching experience before they are admitted. The minimum entrance requirements in mathematics consist of ability to handle the four fundamental operations with integral numbers and fractions (including decimals), to solve easy problems, and to work the four fundamental operations in algebra. In the normal schools the courses of the first two years are of a general cultural character, while the

third year is devoted to professional studies and practice teaching. Mathematics is taught only in the first two years, 8 hours a week in the first year and 7 in the second out of a total of 36 hours.

The standards attained in mathematics in the normal school are indicated in the scope of the final examination conducted by the State board of education:

Arithmetic.-Rule of three. Compound rule of three with application to problems including foreign coinage, measures, and weights; divisibility; alligation; compound interest; common business processes; decimal fractions. Extraction of roots; equations of the first and second degrees; calculation of surfaces and solids.

General arithmetic and algebra.-The four fundamental operations; powers and roots; factors; divisions of polynomials; greatest common measure and least common multiple of numbers and polynomials; proportion; equations of first and second degrees with one or more unknowns; simple equations of the second degree with several unknowns; decimal fractions; logarithms; compound interest and annuities; arithmetical and geometric progressions.

Geometry. The chief principles of plane geometry; straight lines and circles; equality and similarity; calculation of areas and volumes; application of these principles to simple constructions.

It is probable that some reform will be introduced. Those who are especially interested-the normal-school instructors-demand some change in the content of the subjects and more attention to the special method of teaching arithmetic.

The State offers a one-year course and several shorter vacation courses which afford opportunities for the further education of elementary-school teachers. Here the students may select groups of special subjects. In mathematics 10 hours a week are given to arithmetic, algebra, plane and solid geometry, analytic geometry, trigonometry, and the elements of differential and integral calculus. While the underlying aim is to give the students a more thoroughgoing perspective of the school requirements, efforts are made also to impart some training in scientific methods and to develop independence.


A study of the training of teachers in England is somewhat complicated by the fact that the teachers may be recruited in several different ways. The central authority still recognizes uncertificated teachers and certificated teachers. The former may have received their preparation in pupil teacher centers, which are fast disappearing, or in secondary schools, with preliminary education in elementary schools. But in both cases the candidates to be recognized as uncertificated teachers must pass the preliminary examination for the elementary school teachers' certificate or some equivalent examination, usually the entrance or matriculation examination of one of the universities. The certificated teachers again are of two classes-those who have passed through a training college with a two years' course

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