In the Realschule four hours a week are usually devoted to arithmetic and two to geometry. The course is practically the same as in the Oberrealschule.

Oberrealschule. The time devoted to mathematics varies, but it is usually five or six hours a week. The course in arithmetic is practically the same as in the Gymnasium. Special attention is devoted to discount and business practice. Some practice is given in using letters in the solution of arithmetical problems.

Geometry. The pupils are made familiar with the fundamental concepts of plane and solid forms. The course provides abundant practice with the ruler, compasses, and protractor. Simple problems involving surfaces and volumes are given. The pupils learn how to construct triangles from given parts. The congruency theorems are discovered intuitively. Special attention is devoted to the study of the right, isosceles, and equilateral triangles and to the volumes of the parallelogram and trapezoid. The properties of the circle are also considered, and elementary ideas of loci are presented. There is no sharp distinction between plane and solid geometry because of the propedeutic instruction in the subject. Two hours a week are usually devoted to geometry.

HOLLAND. The sixth year is the last year of the primary school. A comprehensive review of the arithmetic of previous school years is given, and numerous practical applications are made. The pupils learn how to compute the surfaces and volumes of the simple solids. Unitary analysis is emphasized. The content of the course is very similar to that of the corresponding school year in Belgium.

HUNGARY.-The sixth school year is the last school year of the Volksschule. The course in the Volksschule and the Bürgerschule includes the study of the fundamental operations, with common and decimal fractions. Short cuts and numerous problems involving statistics of the home, the city, agriculture, and the industries are taught. Two hours a week are usually devoted to the subject.

Gymnasium.-Abridged multiplication and division are taught, and short cuts are given considerable attention. The subject of ratio and proportion is presented, and numerous problems involving percentage are based on economic conditions. Four hours a week are usually devoted to the study of arithmetic.

The course in geometry includes the study of cubes, prisms, pyramids, cylinders, cones, and spheres. The areas and surfaces of cubes, pyramids, cones, and spheres are determined, and pupils make models of all of these solids.

Realschule.-Four hours a week are usually devoted to arithmetic, and four hours to geometric drawing. The courses in both subjects are practically the same as in the Gymnasium.

ITALY. The sixth school year is the last year of the elementary school. Unitary analysis, and simple and compound proportion are taught. Commercial arithmetic. is especially emphasized. Numerous problems involving the various money systems are given. A comprehensive review of rules and principles learned in previous years is given. The pupils learn how to use the ruler and compasses, and many theorems of geometry are discovered intuitively.

JAPAN.—The sixth year is the last of the ordinary primary. The chief aim of the course is to give simple computations in common and decimal fractions, ratio and percentage, and to afford a comprehensive review of the arithmetic of the first five school years. The review is given the latter part of the year.

NORWAY.—(No report is available.)

ROUMANIA.—The course in arithmetic includes the four operations with integers, common and decimal fractions. The metric system, powers and factors, and the reasons underlying the processes are explained to the pupils. Applications of the metric system are especially emphasized during the sixth school year.

The course in geometry includes the intuitive study of angles, triangles, quadrilaterals, circles, parallels, and perpendiculars. The heuristic method is used, and most of the important theorems preceding the subject of similar figures in the Euclidian geometry are discovered. Simple constructions are given a good deal of attention.

RUSSIA. Ratio, simple and compound proportion, interest and partial payments are taught. Four hours a week are usually devoted to the subject of arithmetic. The course in algebra includes the solution of simple arithmetical problems by means of letters, the rules for signs, the four operations with monomials and easy exercises with polynomials.

Realschule. Direct and inverse proportion, compound proportion, percentage and its practical applications, and alligation are taught. A comprehensive review of arithmetic is given.

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The course in algebra includes all that is offered in the Gymnasium, and in addition to this, the squares and cubes of binomials of the types (a+b),2 (a+b) 3 and easy equations of the first degree.

SWEDEN.-The sixth school year is the last year of the primary school. Arithmetic and geometry are closely correlated. Emphasis in arithmetic is placed chiefly on rapid and accurate computation with integers, common and decimal fractions, and their direct applications. Five or six hours a week are usually devoted to the subjcet. Realschule.-The course in arithmetic includes the four operations with integers, common and decimal fractions, ratio and proportion, percentage and its application to interest, and the metric system. Proportion is usually introduced in connection with simple geometric problems.

The course in algebra includes the removal of parentheses, easy factoring, and simple reductions.

The course in geometry is of a propedeutic nature and includes mensuration and simple constructions. Five hours a week are devoted to the study of mathematics. SWITZERLAND.-The sixth school year is the last of the primary school. A pupil must complete the work of the primary school before he can enter the Gymnasium.

The course in arithmetic includes the review of the four operations with integers, common and decimal fractions, the application of percentage to business, and applications of proportion to easy geometrical problems.

The course in geometry includes the intuitive study of the triangle, rhombus, rhomboid, and trapezium.

UNITED STATES.-(New York State course.) The subject of common fractions is reviewed, especial emphasis being placed upon the three problems:

(1) To find a fractional part of a number.

(2) To find what fractional part one number is of another.

(3) Given a fractional part of a number and its relation to the whole, to find the whole.

Denominate numbers are reviewed, and drill is given on industrial problems demanding their use. The idea of percentage is introduced, and percentage is applied to profit and loss, trade and cash discount, commission, simple interest, and the making of promissory notes. Some problems are given in simple interest in which three of the elements principal, rate, time, and interest are given to find the fourth. The simple equation is introduced and used in the solution of some of the problems.

In some schools of the country the course includes, in addition to the above, the keeping of simple accounts, the making out and receipting of bills, and some simple measurements. These measurements are usually made in connection with the study of denominate numbers.

In many schools special provision is made for daily drill in oral computation.

GENERAL SUMMARY OF THE SIXTH YEAR'S WORK.

The course in mathematics in practically all of the European countries is decidedly more advanced than in the United States.

The sixth school year is the last year of primary instruction in many of the countries.

The courses abroad include all that is given during the corresponding school year in the United States and also many subjects that are not included in the course in this country.

One marked contrast to the work in this country is found in the emphasis that is put upon the function concept. This feature of the work is especially emphasized in the schools of Germany, Austria, and Hungary, but it is also given some attention in several other countries. In some countries the function concept is introduced in connection with the work in mensuration or graphs; in others, it is first presented with direct and inverse proportion. The tendency abroad is to increase still further the emphasis already placed upon the idea of function.

Another marked contrast to the work in the United States is found in the emphasis that is put upon the propedeutic study of geometry. The courses abroad, almost without exception, provide for the study of intuitive or observational geometry. The amount of time given to this work varies, but the general prevalence of such work is indicative of the importance attached to it. In several of the countries provision is made for the study of geometric drawing. The pupils learn to use the ruler, protractor, compasses, and triangle, and to make the simple geometric constructions. This work is closely correlated with the work in intuitive geometry and the classes are usually taught by the same teacher. In Germany the systematic and serious study of geometry begins with the sixth school year. In Germany and England easy loci problems are introduced.

Short methods and abbreviated processes receive more emphasis abroad than in this country. This is especially true in Austria, Belgium, Germany, and Hungary. Alligation is taught in several of the countries; for example, in Germany and Russia. The subject is seldom taught in the United States.

The elementary ideas of algebra and of algebraic computation are introduced during the sixth school year in a few of the European countries; for example, in France, Russia, and Sweden.

The time devoted to the study of mathematics abroad is about the same, on the average, as in this country. In some of the countries the time devoted to mathematics is somewhat in excess of that in the United States, if we consider the course in drawing as a part of the course in mathematics.

VIII. THE WORK IN MATHEMATICS IN THE SEVENTH SCHOOL YEAR.

AUSTRIA.-The seventh school year is the second year of the Bürgerschule and the third year of the Gymnasium, Realgymnasium, and Realschule.

In the course in the Bürgerschule five hours a week are devoted to mathematics, three hours being given to arithmetic, one to observational geometry, and one to geometric drawing and constructions.

The pupils are taught how to find the greatest common divisor and the least common multiple of two or more numbers. The study of common fractions is completed, and the method of reducing common to decimal fractions is taught. The four operations with recurring decimals are taught. Fractions are applied to direct and inverse . proportion, and the functional idea is emphasized. Some simple problems in interest are given. The fundamental ideas of planimetry are presented. The pupils learn how to bisect angles and certain regular figures. The idea of the symmetry of certain plane figures is presented, and a special study is made of triangles, quadrilaterals, polygons, and circles. Formulas for the surface and volume of a pyramid, prism, cone, cylinder, and sphere are learned. The study of the sphere is correlated with the work in geometry. The work in constructions includes the drawing of angles, triangles, quadrilaterals, polygons, and a few simple solids.

Gymnasium and Realgymnasium.-A comprehensive survey of the arithmetic of previous years is made. Rules are studied both in words and in letters, and simple transformations of formulas are made. The pupils are taught the use of parenthesis, and how to make substitutions in formulas and in equations. The ideas of negative number are presented.

The course in geometry includes the computation of simple surfaces, and the volume of the right prism, cylinder, pyramid, and cone. The pupils measure numerous objects both in and out of doors. The Pythagorean theorem is presented and is applied to plane and solid figures. Formulas for the surface and volume of the sphere are taught without proof. A good deal of attention is given to simular figures, and the idea is emphasized by means of reductions and enlargements. The pupils draw many figures to scale. The instruction in arithmetic and geometry is very closely related. The pupils are taught the graphic representation of the four fundamental operations and of (a+b)2, (a−b)2, (a+b), (a−b), (a+b)3, and (a-b)3. Abbreviated and approxi mate computations are given a good deal of attention, and square and cube root are presented. The pupils are taught to estimate their results and to check these estimates by measuring and weighing. Functional thinking receives continual emphasis. A few of the simplest equations are studied. Geometric drawing may be studied, but it is not obligatory. In some of the Realgymnasia it is required.

Realschule. Three hours a week are usually devoted to mathematics and two to geometric drawing. The drawing is usually taught by the mathematics teacher, and this is considered a decided advantage. Arithmetic is always taught in close connection with geometry.

Approximate computation of decimals is presented and applied to the finding of surfaces and volumes. The pupils make approximate measurements of various objects, distances, and heights, and use the data in problems. General arithmetic is taught, and a summary of the work of previous school years is given. Rules are studied both in words and in letters, and the pupils are taught to generalize rules whenever it is possible. Simple transformations and checks are given a good deal of attention. The terms "coefficient," "powers," and "exponent" are taught. The pupils learn the rules for the square and cube of a binomial, and they represent graphically (a+b)2 and (a+b)3. Graphic representations of roots are also made. The idea of the negative number is presented, and the four operations with algebraic numbers are taught. The course in geometry is quite similar to the course in the Gymnasium. The pupils learn the fundamental laws regarding areas and volumes. The functional idea is especially emphasized during this school year. Figures are usually drawn to scale, and results are frequently checked by drawing. When solids are involved, results are usually checked by weighing. The course in geometric drawing relates principally to the mensuration and transformation of areas. Some attention is given to decorative forms using circles or arcs.

BELGIUM.-The seventh school year is the first year of the middle school and of the Athénée Royal.

Course for the middle school.-The four fundamental operations with integers, common and decimal fractions are explained and extensively drilled upon. The pupils are taught the tests of divisibility for 2, 3, 4, 5, 8, 9, 11, 25, and 125. The casting out of nines is used as a check for multiplication and division. The greatest common divisor of two numbers is found by successive divisions. The changes effected by adding, subtracting, multiplying, or dividing both terms of a common fraction by the same number are studied. The subject of legal weights and measures is given a good deal of attention. Numerous problems involving the rule of three, simple interest, profit and loss, and commission are solved by means of unitary analysis.

Geometry. The fundamental concepts of geometry are presented and the pupils are led to discover the conditions under which triangles are congruent. The theorem

for the sum of the angles of a triangle is also developed. The principal properties of triangles and quadrilaterals are studied.

Athénée Royal.-Practically the same course is offered in both the classical and the modern-language schools. The fundamental operations with integers, common and decimal fractions are explained and drilled upon. Tests of divisibility for 2, 3, 4, 5, 6, 9, and 11 are taught without proof. The method of reducing a common to a decimal fraction and the various principles of common fractions are taught without proof. Only small denominators are used. The study of the metric system is continued. DENMARK.-The seventh school year is the last year of the Folkeskole and the second year of the intermediate or Mellemskole.

In the Folkeskole four hours a week are devoted to mathematics. The study of proportion is continued, and percentage is applied to profit and loss. A comparative study of foreign coins is made.

Geometry. The course in geometry includes the study of triangles, quadrilaterals, and circles. The pupils learn to compute the volume of right prisms and cylinders with given altitudes and bases. Drawings and models of plane and solid figures are made. Instead of making these drawings, girls are taught the method of keeping simple accounts.

Provision is

Intermediate school.-Five hours a week are devoted to mathematics. made for almost daily drill in oral arithmetic. Simple and compound proportion, percentage, simple interest, partnership, simple bookkeeping, and easy mensuration are studied. Geometry is usually taken up before algebra. One or two hours a week are devoted to the study of intuitional or observational geometry. The fundamental properties of straight lines and planes, the measurement of angles, the congruency and similarity of triangles, and the principal properties of quadrilaterals and circles are taught.

ENGLAND. The four fundamental operations for common and decimal fractions are reviewed, and in a few schools some facts relating to recurring decimals are presented. Numerous problems are worked by means of unitary analysis. The pupils receive a good deal of practice in the drawing up of invoices and the making out of bills. The subject of simple interest is taught, most of the problems being worked by means of formulas. The applications of percentage include profit and loss, discount, and taxes. Some easy problems are given in generalized arithmetic. Many rules are expressed with letters. The algebraic equation is introduced, and easy equations are solved. Simple substitution is also taught, and the meaning and use of negative numbers are briefly presented. Pupils are taught to use rulers marked in centimeters and millimeters. Volumes are found by the use of cubic blocks, by graduated vessels, and by emptying or displacement. The rules for the mensuration of various kinds of triangles, of the rhombus and rhomboid are derived. Simple volumes are studied. Squared paper is extensively used for finding the area of irregular figures. The relation between the length of the circumference and the diameter is determined. Pupils graph simple statistics, such as the school attendance, temperature, and prices.

The work in geometry includes the finding of the locus of points equidistant from a given point, the locus of points at a given distance from a given line, and the locus of points equidistant from three given points. The pupils prove in several ways that the sum of the angles of a triangle is 180 degrees. The relation of interior and exterior angles of triangles is discovered, and the method of dividing a line into equal parts or into parts having a given ratio is taught. The equality of triangles is shown by means of superposition. Some attention is given to the study of circles and to inscribed and circumscribed squares.

FINLAND.-The seventh school year is the last year of the primary school. From one to two hours are devoted to arithmetic. The work of previous years is reviewed, and proportion and discount are studied. The study of algebra is begun in the lycée.