UNITED STATES.-In many of the schools of the United States only incidental number work is given during the first school year. In some schools regular number work is begun the latter half of the first year. In a few schools incidental number work is given during the first two school years.

A great variety of courses exists in those schools which provide special periods for number work during all or a part of the first school year. The following outline is from the New York State course of study. It represents one of the most advanced

courses.

Pupils are taught to count, read, and write numbers to 100 and to memorize the 45 addition combinations. The drill in these combinations is given in such a way as to prepare for subtraction as well as addition. Pupils learn to count 100 by twos, fives, and tens. The children are taught to carry in addition. No attempt is made to teach the science of numbers; the art of computation is emphasized. Oral work greatly predominates, but a good deal of seat and blackboard work is given.

GENERAL SUMMARY OF THE FIRST YEAR'S WORK.

There is not much divergence between the courses in arithmetic in the most progressive schools of the various countries. In general it may be said that the aim is to teach the children to count and to read and write the numbers to 100; to perform easy additions and subtractions within these limits; to know the fractions,, and †; and to make a few easy multiplications and divisions involving numbers less than 20. Practically all the work is oral, and objects are freely used.

III. THE WORK IN MATHEMATICS IN THE SECOND SCHOOL YEAR.

AUSTRIA.-Drill upon addition, subtraction, multiplication, and division is continued, and the exercises involve numbers from 1 to 100. The reading and writing of numbers is extended to 1,000. The pupils are taught the use of the common meas

ures.

BELGIUM.—Knowledge of the number scale is extended to 100, and pupils frequently learn to write numbers to 1,000. The four operations involving numbers not greater than 100 are taught, and place value is emphasized. The multiplication tables through 10 times 10 are built up and learned. Considerable attention is devoted to the small fractions in common use. Both oral and written exercises are given, but the oral work largely predominates.

In boys' schools four hours a week and in girls' schools three hours are devoted to the study of arithmetic.

DENMARK. The four fundamental operations are taught simultaneously for numbers up to 100. The pupils usually learn to read numbers to 1,000. The work in division includes divisors of only one figure.

Danish money is studied, and the following terms are learned and used: Meter, decimeter, centimeter, kilogram, liter, dozen, year, month, week, and day. Simple reductions in the metric system are taught. The fractions, 1, 1, and are taught objectively and are used in simple computations.

ENGLAND. The year that most nearly corresponds to our second school year is called "Standard I." This is usually preceded by at least two years in the infant department. During this period a good deal of incidental number work is given.

The general practice is to teach the reading and writing of numbers from 1 to 100. The course includes addition and subtraction of two numbers of one digit each, but no formal multiplication or division is given. Rapid addition and subtraction are

emphasized. These processes are taught on the "completion of the ten" system. The fractions,, and are taught.

A good deal of attention is given to the drawing of the simplest geometrical figures and to estimates of length. Such objects as cubes, bricks, and coins are extensively used.

FINLAND.-The course includes the four fundamental operations involving numbers not greater than 100, and the reading and writing of numbers to 1,000.

The common measures of length, weight, volume, and capacity are also taught. FRANCE.—The pupils are taught to count and write numbers to 1,000. The four fundamental operations involving easy combinations with numbers less than 100 are taught. Division is limited to divisors of one figure.

The terms meter, liter, gram, franc, and their easy multiples and submultiples are learned. Pupils learn to construct squares, rectangles, and triangles. A good deal of emphasis is put upon simple estimates and measurement.

Forty minutes a day are usually devoted to the study of arithmetic. GERMANY.—The reading and writing of numbers is extended to 1,000, and numerous oral exercises involving the four operations with small numbers are given. The operations are proved by reversing. Special emphasis is put upon the multiplication and division tables up to five. The fractions,, and are taught and used in easy problems. The terms mark, pound, meter, and centimeter are taught. The pupils are required to do a good deal of measuring and estimating. Rhymes and games are used to fix the sequence of numbers and their combinations.

In all the States oral arithmetic greatly predominates. In a few States special emphasis is put upon the fractional parts of small numbers, and upon counting to 100 by fives and tens..

HOLLAND.—The pupils are taught to count, read, and write numbers to 1,000. The four fundamental operations are quite thoroughly taught for numbers less than 100. A good deal of emphasis is placed upon the common denominate numbers. In Amsterdam four and one-half hours a week are devoted to arithmetic. HUNGARY. (See the course in Austria.)

ITALY.—The course includes the reading and writing of numbers to 1,000, and oral exercises involving the four operations on numbers less than 100. In multiplication and division the multiplier and divisor are limited to one-digit numbers. The fractions,, and are taught objectively. The elementary notions of units of weight, length, and capacity are taught. Numerous easy practical problems are given.

JAPAN.-Notation and numeration are taught to 1,000. Pupils are taught to count by tens to 1,000. Oral addition and subtraction involving numbers less than 100 and multiplication and division by easy two-figure numbers are taught. The use of the abacus makes the learning of the tables beyond the nines useless. The tables and the inverse operations are begun in this grade.

The school year is 10 months, and 6 periods a week are devoted to arithmetic. NORWAY. (No data are available for this year.)

ROUMANIA. (No data are available for this year.)

RUSSIA.—(See the report for the third school year.)

SWEDEN.-Pupils learn to count, read, and write numbers to 1,000; to perform the four operations orally on numbers less than 50, and to perform them in writing on numbers less than 100. The multipliers and divisors are always single digits or multiples of 10. The simplest common measures are studied and the easiest fractions are taught.

SWITZERLAND.-Details of the course for the second year are not given in the reports. Oral arithmetic is the basis of the work throughout the primary grades, and great emphasis is put upon easy practical problems within the experience of the child.

UNITED STATES.-In the New York State course there is continued drill on the use of the 45 combinations in addition and subtraction. There is also drill on series in addition and counting by twos, threes, fours, and fives. The addition method is used in subtraction. There is continued drill in rapid additions. The pupils memorize the 45 combinations in multiplication. These are so taught as to prepare for division at the same time. The process of carrying in multiplication is taught. Good model forms are extensively used. No explanation of the processes is attempted.

GENERAL SUMMARY OF THE SECOND YEAR'S WORK.

The course of the second school year varies more than that of the first year. In general, the aim of the work may be said to be to teach the children to count, read, and write numbers to 1,000; to perform the fundamental operations on numbers less than 100; and to learn the simple units of measure. In several countries, multipliers and divisors are limited to one figure. The pupils are taught to count to 100 by twos, fives, and tens. The 45 addition combinations are learned in this year and the multiplication tables involving products up to 10 times 10 are usually studied.

Subtraction is usually taught by the addition method and is studied at the same time as addition. The fractions,, and are taught, and objects are very extensively used. The simple denominate numbers are studied and much attention is devoted to measures and estimates. Oral work predominates. Numerous concrete problems involving the experiences of the pupils are given.

The course in the most progressive schools of the United States compares favorably with the most advanced courses in Europe. In the great majority of the schools of the United States, however, not so much is attempted in arithmetic as in the best schools of Europe during the second school year. The longer school year and the longer school day enable the European teachers to devote more time to drill in fixing the number facts, and the pupils leaving the second grade there are probably more thoroughly grounded in the fundamentals than is the case in this country.

IV. THE WORK IN MATHEMATICS IN THE THIRD SCHOOL YEAR.

AUSTRIA. A great deal of emphasis is put upon speed and accuracy in the four fundamental operations. A large amount of oral drill is given, and numerous exercises are solved at the blackboard or on paper.

The fractions include all those with denominators less than 10. Simple exercises involving these fractions are given. Exercises involving simple estimates and measurements are frequently given.

BELGIUM. The pupils are given an intuitive and practical knowledge of the terms meter, liter, gram, and franc, and of their multiples and submultiples. The units of measure are put into the hands of the pupils whenever practicable. The four operations are explained, and computation is extended to large numbers. The decimal notation is introduced, and the various operations are involved in problems. Emphasis is put upon easy, practical problems. The textbook is first used in this grade. In the boys' schools four hours a week and in the girls' schools three hours are devoted to the study of arithmetic.

DENMARK.-Notation and numeration are extended to 10,000. Place value, in the reading and writing of numbers, is emphasized. The four operations are taught, using both abstract and concrete numbers. Multipliers and divisors not exceeding two figures are used, except in the case of powers of 10. The terms day, week, hour, minute, second, meter, kilometer, millimeter, hectometer, decileter, gram, kilogram, and ton are taught, and are used in numerous problems. Simple exercises

involving easy fractions are given. There is daily oral and written drill on the multiplication tables.

Five hours a week are devoted to arithmetic.

ENGLAND.-In Standard II the course includes the reading and writing of numbers from 1 to 1,000. The four operations are extended to include addition of hundreds, multiplication of tens by units and of hundreds by easy tens; subtraction of tens and easy hundreds; division of tens and easy hundreds by units.

The fractions, }, and are used in easy additions, subtractions, multiplications, and divisions. The tape measure and ruler are extensively used, and the pupils are taught the following terms: Yard, foot, inch, one-half inch, pound, and one-half pound. They are also taught the values of the various English coins.

A good deal of attention is devoted to the drawing of the simplest geometrical figures. The pupils are required to draw rectangles and squares of given dimensions. Squared paper is rather extensively used in these drawings. The pupils are taught the right angle and how to divide angles by means of paper folding.

FINLAND.-Place value is carefully explained. The four operations are extended to 1,000, for both abstract and concrete numbers. The common units of length, weight, and capacity are taught, and numerous exercises involving these measures are given.

Four hours a week are given to arithmetic.

FRANCE.-A thorough review of the work of the previous grade is given, and the operations are extended to large numbers. The general ideas of fractions are introduced. Simple reasoning problems are given, the data being selected from matters within the experience of the child. The pupils make numerous estimates and comparisons of distances, the metric units being used. Exercises involving the most elementary plane figures are given, and the various kinds of angles are studied. The simplest solids are studied from models.

Forty minutes a day are devoted to the study of arithmetic.

GERMANY.-The number scale is extended to 1,000 and the pupils are taught to count by 10, 50, and 100. Oral addition and subtraction are extended to numbers of two digits, and written addition and subtraction to numbers of six digits. Multiplication up to four-place numbers by two-place numbers is taught. Division of numbers containing from two to six figures by numbers of one figure and by easy tens is taught. The terms mark, pound, meter, centimeter, liter, hectoliter, gram, kilogram, and kilometer are taught, and are used in numerous exercises. The four operations involving units of length and weight are given a good deal of attention.

There are in Germany numerous collections of problems designed to give facility in oral computation.

In most of the German States, five hours a week are devoted to the study of arithmetic.

HOLLAND.—The course is practically the same as in Belgium. Exercises involving the four operations with all numbers are given. Much emphasis is placed upon the various weights and measures.

Five hours a week are given to arithmetic.

HUNGARY.—(For details, see the course in Austria.) The problems are chosen almost entirely from practical life. There is a good deal of freedom and variation in courses.

ITALY.-Notation and numeration are extended to 10,000. Numerous oral exercises on numbers less than 100 are given. The written exercises involve numbers up to 1,000. Multipliers and divisors are usually small or else a multiple of 10. The decimal notation is introduced and easy common fractions are reduced to decimals. A good deal of attention is given to practical exercises involving the metric system. Some of the elementary notions of geometry are derived by intuition. Free-hand drawing is emphasized.

JAPAN.-Notation and numeration are extended to 10,000. Oral and written exercises involve all four operations. The chief aim of the year is to teach written addition, subtraction, multiplication, and division with numbers less than 10,000. From five to ten minutes' oral drill is given daily to help fix the number facts firmly in mind.

NORWAY.-(No data are available on the work of this year.)

ROUMANIA. (No data are available on the work of this year.)

RUSSIA. The course for the first three school years includes the four operations for abstract and concrete numbers and the easiest fractions. The tables of weights and measures and of Russian money are taught. There are numerous collections of good problems for use in the primary schools. The fundamental ideas of geometry are taught by means of paper folding and cutting, using squares, rectangles, and simple solids. The pupils do not use a textbook.

Five hours a week are given to arithmetic.

SWEDEN.—(The course is not given in detail. See the sixth school year.) SWITZERLAND.-Notation and numeration are taught to 1,000. Mental arithmetic is the basis for all work. Calculation, and not reasoning, is emphasized. All the operations are taught.

UNITED STATES.-(New York State course.) Drill in counting is continued. The pupils are taught to count by fives to 100, beginning with 0, 1, 2, 3, or 4, and by sixes, beginning with each of the numbers from 0 to 5, inclusive. In short division, 2, 3, 4, 5, 6, 7, 8, and 9 are used as divisors. Multiplication, with two or more figures in the multiplier, is taught. Definitions of the terms addend, sum, minuend, subtrahend, remainder, multiplicand, multiplier, product, dividend, divisor, and quotient are learned. Pupils are taught to measure, using the inch and the foot. Square inch and square foot are also taught. The fractions and are applied to the use of the linear unit in measuring.

During the second half of the year the following topics are taught: Long division, multiplication tables of the tens, elevens, and twelves, and their use as divisors in short division; tests for divisibility by 2, 3, 5, 9, and 10; the definition of factor and prime factor. The pupils memorize the prime factors up to 25; linear and square measurement of objects in the school room, and liquid measure are taught.

A great deal of attention is given to oral drill and written work for accuracy and rapidity in the four operations. At the close of the year the pupil is expected to be able to add, subtract, multiply, and divide integers with accuracy and facility.

GENERAL SUMMARY OF THE THIRD YEAR'S WORK.

There is greater variety in the third-year courses than in those of the first and second school years. In a few of the countries-for example, Belgium and Italy— the notation of decimal fractions is introduced. This is usually not done in the United States until the latter part of the fourth or the early part of the fifth year. It is a common practice abroad to introduce fractions with denominate numbers. In all of the European countries and in Japan oral arithmetic greatly predominates. In Japan a special part of the recitation is set apart for this oral drill. The textbooks in several of the countries, notably Germany, Austria, and Italy, are collections of problems rather than expositions of number.

In most of the schools subtraction is taught by the addition method.

At the end of his third school year the German boy may either (1) continue in the Volksschule, (2) pass to the Bürgerschule, or (3) enter a secondary school (Gymnasium, Realgymnasium, or Oberrealschule). The Russian pupil may enter either the Realschule or the Gymnasium. If he prefers he may continue in the primary school a year or two more. In Sweden the primary school lasts six years, but a pupil may enter the Realschule after he has completed the third school year.