V. THE WORK IN MATHEMATICS IN THE FOURTH SCHOOL

YEAR.

AUSTRIA.-Much attention is still given to speed and accuracy in the four fundamental operations. A large amount of oral drill is provided. The idea of fractions is extended, and decimal fractions are introduced. Numerous exercises involving easy operations with common and decimal fractions are given. The common units of weight, length, surface, capacity, and money receive a good deal of attention.

BELGIUM.-Common fractions are formally introduced, and the reduction of common fractions to decimal fractions is explained and practiced. Numerous practical problems involving the systems of weights and measures are given. The subject of decimal fractions is correlated with that of the metric system. The fundamental ideas of simple proportion are taught. Simple interest is begun.

DENMARK.-The four operations are extended to exercises involving large numbers. Multipliers with three and divisors with two figures are used. Addition, subtraction, multiplication, and division of denominate numbers are taught. The concept of fraction is extended to include all fractions with denominators not greater than 10. The pupils are required to find fractional parts of a number.

The fundamental ideas of simple proportion are taught. The metric units are used in many of the problems.

ENGLAND.—The tens, hundreds, and thousands groups are studied. Much emphasis is laid upon the factors of numbers less than 100. Addition is extended to numbers of four figures, and subtraction is extended to hundreds and thousands. Multiplication by easy hundreds is taught, and division by easy factors is studied. The four operations are employed in problems involving money, yards, feet, inches, gallons, quarts, pints, pounds, and ounces. Ascending and descending reductions are presented. The pupils are taught to use the fractions, 1, 1, 1, 3, 4 to 3, and to H. Numerous problems involving these fractions are given, and they are represented on lines and by means of paper folding. The use of the measuring tape is taught.

The pupils are made familiar with the idea of bisection by paper folding. A study is made of squares and rectangles. The pupils are required to measure and compare the opposite sides and the diagonals. The equilateral and isosceles triangles are briefly studied. Figures are drawn to easy scales and simple geometric patterns are constructed.

FINLAND.-Both common and decimal fractions are taught during the year. Most of the common fractions have small denominators, and most of the exercises do not involve decimals of more than the third order. The pupils are taught how to reduce a common to a decimal fraction. The fundamental ideas of ratio and proportion are taught. Some very elementary problems are given in simple interest.

FRANCE.-The ideas of common fractions are extended, and decimal fractions are introduced. The four operations are applied to decimal fractions. The fundamental ideas of proportion are studied, and a few problems are given involving simple interest. There is a good deal of attention given to problems involving the legal systems of weights and measures, and the four operations are involved in many mental problems. Pupils are required to represent at the board and on paper the simple geometric figures and to study from models the fundamental properties of the cube, prism, cylinder, and sphere. The texts include numerous problems relating to losses caused by alcoholic drinks.

GERMANY.-In Germany a pupil generally spends his fourth school year in the Volksschule, the Gymnasium, the Realgymnasium, or the Oberrealschule, although, as in other countries, there are other types of schools which he may attend.

Course in the Volksschule.—Oral multiplication by 12, 15, and 25 is presented. The four operations with denominate numbers are studied. Simple problems involving frac52452°-153

tions are given. Notation and numeration are extended up to seven places. Continual practice is given in the four operations. Multiplication involves multipliers of four digits, and division involves three place divisors.

Decimal fractions are taught to three decimal places, and the subject is closely correlated with the metric system. Denominate numbers are frequently written in decimal form. Ascending and descending reduction are applied to the metric system. Addition and subtraction of common fractions are taught, and aliquot parts are freely used. In some of the German Volksschulen formal written arithmetic begins with this grade. Applications are postponed until the processes are fixed.

Course in the Gymnasium.-The four fundamental operations with abstract and denominate numbers are extensively drilled upon. The simple measures of weight, length, capacity, surface, and volume are taught. Problems involving parenthesis are given. The ideas of common fractions are extended and decimals are given some attention.

Course in the Realgymnasium.—The course in this type of school is the same as for the Gymnasium.

Course in the Oberrealschule.-Four hours a week are given to arithmetic. The course includes drill in the four operations with abstract and denominate numbers; practice in decimal notation; the rules for divisibility by one place numbers; prime factors; multiples and divisors. The ideas of common fractions are extended. Oral arithmetic receives a good deal of emphasis.

HOLLAND.-The four operations with integers are thoroughly reviewed. Emphasis is placed upon denominate numbers and the various common units of measure. The concept of fractions is extended; decimal fractions are taught and pupils reduce common to decimal fractions. The rectangle, cube, and parallelopiped are studied. Four and a half hours a week are given to arithmetic.

HUNGARY.-The attempt is made to secure a high degree of accuracy and a fair degree of speed with the four operations. A thorough study is made of denominate numbers and the simple units of measure. The decimal notation is taught and the concept of common fraction is extended. The pupils learn how to reduce a common to a decimal fraction.

ITALY.-The written work includes the four operations with integers and decimal fractions; the reduction of a common to a decimal fraction; a study of the Roman numerals; and numerous practical exercises on the metric system.

The pupils are taught the fundamental properties of the simple geometric figures. Free-hand drawing forms an important part of the work. The rules for the mensuration of the common plane figures and the names of the solids are taught.

JAPAN.—The chief aim of the work of the fourth school year is to secure a high degree of accuracy and speed in computations involving numbers less than 100,000,000 and to teach computation with compound and decimal numbers.

Multipliers are usually of two or three digits, and divisors and quotients are usually of not more than three digits. The pupils are taught to find a fractional part of a number.

In compound numbers the units of length, distance, weight, capacity, area, time, and money are taught.

Decimal notation is taught only to thousandths in this school year.

NORWAY. (No report.)

ROUMANIA. The four operations are extended to larger numbers; the decimal notation is introduced, and the relation between common and decimal fractions is emphasized. The various metric units are studied.

RUSSIA. The Gymnasium and the Realschule begin with the fourth school year. A pupil may continue his course in the primary school. All courses for the fourth year are much alike. The four operations are emphasized. Common and decimal fractions are introduced. In the Gymnasium four hours a week are devoted to

arithmetic. Measurements and easy drawing to scale form an important part of the

course.

SWEDEN.-The Realschule begins the fourth school year and lasts for six or seven years. Four hours a week are devoted to mathematics.

There is continued attention given to the four operations. Common and decimal fractions are introduced, and easy examples involving them are solved. Only inte gers are used as multipliers and divisors, and no remainders occur in the divisions. The study of the metric system is continued.

SWITZERLAND.-The attempt is made to develop speed and accuracy in the four operations with integers and denominate numbers.

The decimal notation is introduced, and the ideas of common fractions are extended, The four operations are taught for both integers and common fractions; only easy problems are considered. A good deal of attention is put upon estimates and drawing to scale.

UNITED STATES.—(New York State course). Roman numerals are taught from 1 to 100 and by hundreds to 1,000. The pupils learn to read and write United States money; to use cancellation, when possible, in the solution of problems; and to use the terms pint, quart, peck, and bushel. Common fractions are developed objec< tively. The pupils are taught to change fractions to equivalent fractions of higher and lower denominations; to add and subtract fractions the denominators of which do not contain more than two digits; to multiply a fraction by an integer and by a fraction, and to multiply an integer by a fraction; to divide a fraction by an integer and by a fraction, and to divide an integer by a fraction. The principles for multiplying or dividing a fraction by the proper operation upon its numerator or denominator are taught; also the effect of multiplying both terms of a fraction by the same number or dividing both by the same number.

There is continued drill throughout the year on the four operations with integers. Addition and subtraction of mixed numbers are taught. The pupils learn how to factor and to find the least common multiple of numbers to 100. Problems are carefully stated before being solved. Cubic measure is taught. Volumes studied include cubic inches, cubic feet, and cubic yards. Simple problems are given in bills and

accounts.

GENERAL SUMMARY OF THE FOURTH YEAR'S WORK.

At the close of the fourth school year (in Germany, the third school year) the pupil may enter a "higher" school of some sort; in Austria the Gymnasium, the Realgym nasium, or the Realschule; in France, the higher primary or lycée; in Germany, the Gymnasium, the Realgymnasium, or the Oberrealschule; in Hungary, the Bürgerschule, the Gymnasium, or the Realschule; in Italy, the Ginnasio, or the modern school; in Roumania, the Gymnasium; and in Switzerland, the middle school. Conditions are such that pupils can not well enter a "higher" school except at the begin ning; so that the fourth (or the third) school year is the last year of the primary for many European pupils.

The attempt is made to fix the four operations for abstract and denominate numbers firmly in mind by the time the pupil has completed his fourth school year. In all of the countries a large amount of both oral and written drill is provided. Speed and accuracy in the fundamentals are watchwords everywhere. In addition, the con cept of common fractions is much extended, and the decimal notation is introduced. In several of the countries, only addition and subtraction of common and decimal fractions are taught, but in others multiplication and division are also included in the course. When this is done, only the easy cases are usually considered.

The general use of the metric system in the countries of Continental Europe makes the introduction of decimals practicable at an earlier date than in the United States. Usually the subject of decimal fractions is closely correlated with the metric system,

Most courses in the United States are not so advanced as the New York State course. In general, it may be said that the courses in European countries include all that is offered in the United States during the fourth school year and a good deal of the work of the fifth school year. The formal study of common or decimal fractions is seldom begun in the United States before the fifth school year.

In most of the European countries emphasis is put upon computation rather than upon reasoning during the first four school years.

VI. THE WORK IN MATHEMATICS IN THE FIFTH SCHOOL YEAR.

AUSTRIA. The fifth school year is the last year of the Volksschule and the first of the Gymnasium, Realgymnasium, and Realschule.

Course in the Gymnasium.-The four operations with abstract and denominate numbers are continued. Roman notation is introduced. Austrian money, weights, and measures are studied. Common fractions are studied in connection with denominate numbers, most of the fractions being of small denomination; and the study of decimal fractions is continued.

The work in geometry is of a propedeutic nature. The cube and sphere are studied. Exercises with ruler, compasses, triangle, and protractor are given. A good deal of emphasis is placed upon the measuring and drawing of easy geometrical figures. The pupils are made familiar with the common solids. The constructions include angles of 30, 60, and 90 degrees; isosceles, equilateral, and right triangles; parallels; and perpendiculars. The pupils find the areas of squares and rectangles, and the volumes of cubes and right prisms.

Course in Realschule and Volksschule.-The course is practically the same as in the Gymnasium. Even more attention is put upon the securing of speed and accuracy in the four operations with integers and fractions.

Arithmetic is taught in very close connection with geometry in all types of schools. The two subjects complement each other and form one instruction unit.

The pupils make numerous simple models of pasteboard and sticks. Planes and solids are taken up in close connection, since the study is made from models.

Three hours a week are usually devoted to the study of arithmetic and one to geometry.

Course in Realgymnasium.—The course is the same as in the Realschule, except that there is no special period for the study of geometry.

BELGIUM.-The four processes with integers, common and decimal fractions are explained. Quotients are obtained correct to one-thousandth. Cancellation is extensively used. Pupils are taught the tests of divisibility for 2, 3, 4, 5, 8, 9, 25, and 125. Prime numbers are studied and applied to tests of divisibility by 6, 12, 15, 18, 21, and 35. Unitary analysis is taught. The problems are based largely on local industries, trades, and agriculture. The study of the metric system is continued. The mensuration of the rectangle, parallelogram, triangle, trapezoid, circle, and polygon is studied. Oral computation is emphasized, and short processes are encouraged.

Most of the teaching of geometric forms is done in connection with the metric system, manual training, and drawing.

DENMARK.—The pupils are taught to resolve numbers into prime factors. Addition and subtraction of common fractions are studied, and the study of decimal fractions is extended. Proportion is taken up, the fractional form of writing the proportion being used. The metric system is studied as a practical application of decimal fractions, and addition and subtraction of decimal fractions and multiplication and division of a decimal by an integer are taught.

Special periods are not devoted to the subject of geometric forms, but much is done in the drawing classes. Mensuration is taught in connection with arithmetic. The compasses are but little used.

ENGLAND.-Notation and numeration are extended to tens of thousands. Factors and easy long division are taught. Common fractions are studied, the denominators being restricted in most cases to numbers less than 12. Addition and subtraction of fractions are taught. Decimal fractions up to three decimal places are introduced, and the four operations involving decimals are taught by means of divided squares, rectangles, and rulers. Numerous problems involving distances and heights expressed in yards, feet, and inches are solved. The terms ton, hundredweight, chain, pound, ounce, gallon, quart, pint, month, day, hour, minute, and second are used in exercises. The method of unitary analysis receives a good deal of attention. The area of the square and the rectangle is taught, and the term perimeter is used. An attempt is made to give the pupil the idea of a standard pound and yard. Metric rulers are used, and pupils learn some of the equivalents. Calipers and wedges are used to determine internal and external diameters.

The work in geometry includes the bisection of lines, angles, and triangles by folding; the superposition of triangles; the use of compasses; folding circles to get angles of 180, 90, 45, and 22 degrees; the drawing of parallels by means of the set square and by the eye; the drawing of parallelograms, and drawing to scale.

FINLAND. The course in arithmetic includes decimal and common fractions, tests of divisibility, simple proportion, introductory work in percentage, and simple interest.

One hour a week is given to the study of geometry. The course includes the study of lines, angles, rectilinear figures and their areas, curvilinear figures and their areas. FRANCE. The fifth school year is the first year of the higher primary and the first of the lycée. The higher primary is not found in all schools.

The course in the higher primary includes the following subjects: The study of prime numbers and tests for divisibility by 2, 3, 4, 5, 9, and 25; the resolution of numbers into factors; greatest common divisor and least common multiple; unitary analysis; the metric system; the four operations with common and decimal fractions. Letters are used in the solution of some simple problems.

The work in geometry is given in connection with drawing, and is frequently taught by the same teacher. The following figures are studied: Square, rectangle, triangle, circle, perpendiculars, obliques, parallels, and parallelograms. The pupils learn the relation of circumference and radius, the terms chord, arc, tangent, and secant. Angles are measured and there is a good deal of graphic construction. Very elementary projections are introduced.

In the first year of the lycée of France the pupils may enter either the classical or the scientific section.

In the classical lycée two hours a week are devoted to mathematics, and the course includes the study of integers, common and decimal fractions.

In the scientific lycée three hours a week are given to arithmetic and one to geometric drawing. The work in arithmetic includes the study of common and decimal fractions, the rule of three, problems in interest, and alligation.

In the work in geometrical drawing the pupils are taught to use the ruler and compasses, and to make simple designs.

GERMANY.-Volksschule.-The subjects of denominate numbers and common fractions are closely related, and pupils are given a good deal of practice in computation in both topics. All four processes are studied with both common and decimal fractions, and the pupils learn how to reduce a common to a decimal fraction. In some schools decimal fractions are studied before common fractions, and in others this order is reversed. The problems involve a good deal of computation with time. In a few of the States abridged multiplication and division are being experimented with.