STATE OF NEW JERSEY DEPARTMENT OF PUBLIC INSTRUCTION TRENTON The Teaching of Elementary Arithmetic December, 1912 CONTENTS J'l n't Pages. Foreword. 3 Influences afl'ecting the Teaching of Elementary Mathematics. 5 The Field of PJlementary Mathematics . 5 Mathematics of Common Business . 6 Concrete Mathematics . 8 The Text-Book . 9 Training for Skill . 9 Accuracy and Speed. 9 Use or Application . 10 Interpretation of Problems. 10 Calculation . 11 Addition . 12 Subtraction. 13 Multiplication . 14 Division. 16 Division and Partition . 17 Economy in Calculation . 19 In Oral and Written Work . 19 Estimating Besults . 20 Checking Work . 20 Use of the Pencil . 20 Oral Arithmetic .‘. 20 The Class Time . 21 Time Given to Arithmetic . 22 Home Work . 22 General Methods. 22 Inductive Teaching and the Use of Objects . 22 Drill . 24 Quality, Appearance and Form . 25 Solving Problems . 26 Diagrams . 26 Graphs. 27 Testing . 27 Marks . 29 The Individual . 29 Geometry . 29 Algebra . 30 Informational or Community Arithmetic . 30 Books for Supervisors and Teachers . 33 Note on the Assignment of Work by Grades. 34 Elementary Mathematics by Grades . 35 First Year (First Half) . 35 “ (Second “ ) . 36 Second “ (First “ ) . 37 “ “ (Second “ ) . 39 Third “ (First “ ) . 41 “ ‘‘ (Second “ ) . 42 Fourth “ (First “ ) 44 “ “ (Second “ ) . 45 . Fifth “ (First “ ) . 47 “ (Second « ) . 49 Sixth “ (First “ ) 50 “ “ (Second “ ) . 51 Seventh “ (First “ ) 65 “ “ ( Second “ ) . 56 Eighth “ (First “ ) 62 “ “ (Second “ ) . 65 Foreword. 3 /> This pamphlet consists of two parts. The first is a monograph on the teaching of elementary mathematics; the second is a course of study. In the monograph an attempt is made to set forth some of the principles which seem to be widely accepted at present in the teaching of elementary mathematics. Two of the most important of these are, first, that there should be taught in the elementary schools only that kind of mathematics which is use¬ ful in common life; secondly, that in this restricted field of the useful, training be directed toi the cultivation of skill in the ap¬ plication or use of mathematical knowledge. Accordingly, some of the traditional topics are omitted. The course of study is a minimum course of study. It is not proposed to make the use of it obligatory in the schools of the State; on the contrary, its use is optional. Those using the course, who feel that other subjects or topics should be included, may, of course, add them. It is believed that subject material, at least in the higher grades, should be co-ordinated to a degree and as far as prac¬ ticable, with the other activities of the school, such as the school shop, the kitchen, the garden or the farm; that the material ^j>^hould also be related to the civic or .industrial life of the com¬ munity. Such co-ordination or relation adds interest to the study and gives motive to pupils to do their best work. If it is not practicable to teach this so-called applied arithmetic in the higher grades, traditional subjects of the text book may, of course, be substituted therefor. Mathematics is a subject in which pupils are supposed to ac¬ quire skill in the application of mathematical knowledge to the problems and exercises most useful in large ranges of human experience, and a subject in which a premium is put upon accur¬ acy and exactness. r / .i '6 CL IS 3 4 The pamphlet will in part serve its purpose, if it furnishes aid to teachers. The main purpose, however, is to bring about, by means of good teaching, more intelligent action on the part of pupils in this particular field of elementary school study. Courses of study or monographs upon teaching are valueless in proportion as they fail to affect the conduct of pupils in school and out of school. To prepare such a monograph, and to outline a course of study in a subject concerning which almost every educated person has an opinion, is a difficult task. The judgment of business men, as well as teachers, has been sought. The latter have included many of the leading directors and teachers in the State, in the normal schools, in city schools, and in rural schools. A few teachers outside the State have also been consulted. An attempt has been made to harmonize conflicting opinions. It may be said with some confidence that the monograph and course of study reflect the consensus of opinion of the persons who so generously assisted. To them grateful acknowledgement is made. Special mention should be made, too, of the assistance ren¬ dered by the Assistant Commissioner in charge of Elementary Schools, Mr. George A. Mirick. December, 1912. CALVIN N. KENDALL, Commissioner of Education. The Teaching of Elementary Mathematics The influences that are now affecting the teaching of elemen¬ tary mathematics are not different from those influences that have affected it in the past. These influences have brought the instruc¬ tion more into harmony with the nature of children and more into conformity with the demands of society. The child has a mind that must be taught in accordance with its nature. As knowledge of this developing mind increases, it is inevitable that changes will be made in methods of instruction and in readjustments of the material of instruction. Inasmuch as the child is being trained for present and future usefulness in society, the training that is given must be modified from time to time to meet the new demands and conditions of an ever changing social order. Therefore it has come to pass that many changes are now being made in the entire program of school work, and particularly in the teaching of elementary mathematics. THE FIELD OF ELEMENTARY MATHEMATICS. Among the very evident demands that are being made upon the schools are two in relation to the teaching of mathematics in the elementary grades. 1. That there be taught in these grades only that kind and amount of mathematics that is useful in common life. 2. That in this narrow field of the useful, training be di¬ rected to the cultivation of skill in the application or use of mathematical knowledge. In this process it is undoubt¬ edly true that the best mental training or discipline will be given. 5 6 Mathematics If all subjects and parts of subjects in arith- metic, that are not now found in common busi¬ ness practice, are dropped from the elementary course of study, a number of subjects will be eliminated that have in the past consumed much time. Among these subjects may be mentioned: 1. Cube Root. 2. Square Root. (The square and cube roots of numbers, whose roots are dis¬ coverable by inspection, should be learned—the squares of num¬ bers at least to 12, some may think to 25, with the corresponding square roots; the cubes of numbers at least to 5, some may think to 10, with the corresponding cube roots.) 3. Greatest Common Divisor. 4. Least Common Multiple. (Fractions with denominators larger than 16 are seldom found in business. Fractions with denominators to 64 are often used in machine shops as units of measure, but there is no call to calculate with these fractions except in the office of the expert. Those with larger denominators are expressed in the decimal form. The common denominators of the fractions of common business and all the factoring needed in using such fractions can be calculated mentally. It is true also in the use of fractions that practical business finds no place for such problems as “If ^ of my money is $725, how much money have I?”) 5. Uncommon applications of Percentage. (In business it is seldom, if ever, required to find “the whole” when a part is given. The business man knows “the whole,” i. e. his capital and his costs. He needs to find “the part,” i. e. gains and losses, as related to capital and costs, expressed in common and decimal form.) (In interest problems, also, for the reason just given, it is unnecessary to teach “finding the principal when time, rate and interest are given,” or “finding the interest” for unusual rates or for periods not common in business practice.) 6. True Discount, Partnership, Compound Proportion, Tables of Surveyors’ Measure, Troy and Apothecaries’ Weight, 7 Paper Measure and obsolete units in all other tables should be omitted. 7. Problems in Taxes, Insurance, Bonds, Stocks, Partial Payments, Bank Discount, Compound Interest, Longitude and Time, Ratio and Proportion, Lumber Measure should be of the simplest kind and they should be used for their informational as well as for their mathematical value. 8. Mensuration. (It is desirable that the study of mensuration should be baseu on real, objective experience. The kind of problems should be determined by circumstances. Farm children may profitably calculate the capacity of bins and silos. City children will get more profit from solving problems relating to streets, side¬ walks, etc. The problems of the paper hanger, carpet dealer, plasterer, floor layer and brick layer, are specialized trade problems subject to trade practices and should have little attention in the elementary grades. Omit mensuration of spheres and frustums of pyramids and cones.) 9. The Metric System. (The metric system in this country is a specialized form of mathematics used in the sciences only. It is, at present, like simplified, spelling, a form, theoretically desirable, that has no like¬ lihood of being immediately adopted for general use. A few lessons may profitably be given for information.) There will then remain as the legitimate field of elementary mathematics: f. Counting numbers ir. Reading numbers 1. Integers—Arabic and Roman 2. Common Fractions 3 . Decimal Fractions [II. Writing numbers 1. Integers—Arabic and Roman 2. Common Fractions 3. Decimal Fractions IV. The Processes 1. Addition 2. Subtraction 3. Multiplication 4. Division (a) Integers (b) Common Fractions (c) Decimal Fractions to three places 8 \'. l^erceiitage applications. (a) Trade or Commercial Discount (b) Profit or Loss (c) Commission (d) Simple Interest VI. The following subjects should be treated largely for information purposes. See p. 30. (a) Taxes (b) Insurance (c) Stocks (d) Partial Payments (e) Bonds (f) Bank Discount (g) Compound Interest VII. Denominate numbers in useful problems of community value To bring mathematics to complete accord with the standard of the “useful” not only must the work be made as simple as is the mathematics of business, but it must deal with the material of actual business. To this end text books have been generally remade in recent years and the material and data found in the business world have been used for the making of the problems. Concrete However, while it is true that modern busi- Mathematics. ness fumishes better material for problems than does ancient business, it cannot be denied that modern business terms and data, when met with in' books, may be and in general are as abstract for the school boy or girl as those of ancient business. Real mathematics is found only in real things and in real activities. Book mathematics is real and con¬ crete only in proportion as the mathematical foundations have been laid in concrete experience and as there is adequate oppor¬ tunity for concrete, real application. There are likely to be progressive changes in the teaching of this subject until in all grades, not simply in the kindergarten and primary grades, there are appropriate activities in which material for real, concrete problems may be found. Some ad¬ vance is being made in this direction. 1. A larger use is being made of dramatized “occupa¬ tions” and games in the lower grades. 2. In the upper grades industrial or manual work of various kinds for boys and girls is being more closely studied for its mathematical content. 3- A first hand study of the life of the community by the pupils is revealing- real, concrete material for school mathematics. It is evident that a new attitud'e must be The Text Book. t^ken towards the text-book. If concrete mathematics cannot be found in a book, but only in real things, the book can be a guide only and a reference help. It may con¬ tain suggestions as to the method and illustrate types of problems, hut the reservoir, which holds the material for real mathematical problems, is the life of the community; and this reservoir must be drawn upon by the children themselves in all grades. A teacher should feel free to omit problorns and topics from a text-book and daily to supplement the work therein laid down, as long as this is in harmony with the curriculum. The text-book may be expected to furnish a large part of the abstract work required, but no book can supply all of the concrete work suited to the various localities of the State. TRAINING FOR SKILL. The long time complaint of business men, that their young- employes are not able to use skillfully the mathematics they have studied in school, constitutes a demand that should not be over¬ looked. The public school, as a matter of fact, has never under¬ taken the task of training for skill in any field, the assumption being that this belongs to special training, not general, and that there is not time for special training. If, however, the field of elementary mathematics is to be limited by the needs of common business, there will be ample time for training for skill. It may also prove to be true that this same training for skill is the best general or disciplinary training. Besides a knowledge of what and how, skill Accuracy and Speed • , ^ « , , , t involves accuracy and reasonable speed. In mathematics this accuracy and reasonable speed is developed by 1. Using (applying) mathematical knowledge in a large variety of ways and in a variety of fields of experience. 2. Training in the interpretation of problems. 3. Training in calculation. lO 4. Training in estimating results. 5. Training in checking work as it progresses. A person who learns a fact from a book may Use or Application. recognize that fact when it is tnet outside the book. A person who learns the theory of farming does not therefore become a successful farmer. A man who learns to handle tools on a farm, is thereby trained to take up similar work at once and skillfi>'ty in a highly organized factory. For the same reason that it is necessary for a mature' person to learn anew each dif¥erent application of knowledge, it is necessary for children to have a large variety of uses in school for their mathematics, if they are to enter the business world as skillful mathematicians. If, as seems to be true, the elementary field of mathematics can be surveyed and even lightly worked, in the first six grades, the seventh and eighth grades become years in which there is imperative need, as there is ample time, for varied application. The applications have been before referred to. They are in problems drawn fresh from the reservoir of commun¬ ity life, and in problems found in the school shop, school printer)^ school kitchen and sewing room, and in the school agricultural studies. Interpretation of skillful ill any department of of Problems. life, he must know how to attack his problems, how to analyze them so that a correct and reasonably speedy determination may be made of what the problems call for and the data that they give ^for the solution. Two modifications have been made in recent years in the method of training in the interpretation or “explanation” of problems in arithmetic. Tst. The “explanations” are now less complex in their state¬ ments. Children are naturally logical but they seldom are logicians. Much practice is now given in stating what a problem calls for, and how the results are to be obtained; but little is re¬ quired in elaborate or “complete explanations” that were formerly much insisted upon. 2d. Pupils are now trained, even in primary grades, to think through a problem to determine approximately the probable II answer, before they begin to solve it on paper. This training in “estimating results” cultivates a kind of judgment most useful in business. To illustrate these considerations, the following problem may be used— “A dealer buys 250 Ib^ of twine at 19^ a pound. He sells it • at 25^ a pound. How much is the profit on the whole lot?” The pupil may proceed somewhat as follows: Statement —The problem calls for the profit on the whole lot. Explanation —The profit on one pound is the difference between the cost and selling price of a pound. The profit on the whole is the profit on one pound multiplied by the number of pounds. Estimate —$.19 is approximately $.20. Then $.05 is approxi¬ mately the profit on one pound. If the profit were $.10 per pound, the profit on the whole would be $25. As the profit per pound IS half of $.10, the total profit would be Half of $25.00 or $12.50. The total profit is somewhat more than $12.50. Solution —The profit on one pound is $.06. The profit on 100 lb. is $6.00; on 200 lb. is $12.00; on 50 lb. is $3.00; on 250 lb. is $iq.oo. $15.00 is somewhat more than $12.50. Skill in solving problems requires not only ready judgment in interpretating them but ac¬ curacy and reasonable speed in calculation. In general the methods of calculation learned in the early grades persist through the high school. Now, however, good text-books are suggesting more economical methods of calculation as the work progresses. To habituate the mind to the primar}^ grade methods is to habit¬ uate it to activity on a low plane, and this is no more desirable in mathematics than in reading or in penmanship, or in any other subject . Skill in calculation seems to depend upon (a) Familiarity with the addition, subtraction, multipli¬ cation and division tables, (b) Ability to see relations among numbers that will simplify computation. 12 A few illustrations may help to call the attention of teachers t( this important phase of their work in teaching arithmetic. There are forty-five so-called primary com- Addition. binations in addition. These are the possible addition combinations of any two digits through eighteen. Of these, twenty-five combinations make ten or less. The twenty- five combinations should become so familiar, before a study of the others is taken up, that the sums and corresponding differences are recognized instantly without calculation, e. g. 7 -|- 7 should be seen as 14 without thinking 7 and 7 are 14: 3 6 is thought 9 without adding. It is as easy for pupils to see the sum or difference in these related numbers, as it is for them to see the word, at, without spelling it. The pupils and teacher should become conscious of the fact that some of these combinations are more difficult to remember than others. The difficult ones should be found and special stress should be laid on them. When working with the primary combinations above ten, the pupils may be led to see that in adding ten and three, they make three and ten or “three-teen”; four and ten makes “four-teen”; etc. When nine and four are added, the sum is one less than ten and four; and when eight and four are added, the sum is two less than ten and four; etc. By studying with the pupils the forty-five possible combinations i^- will be discovered that those that cause trouble are very few and they should have special drill. As the work in addition progresses, if the pupils are shown how, they will rapidly learn to see the sum in several numbers at a glance and will learn to add more nearly as they read. In reading, after the words are well known, the mind grasps words and groups of words as wholes; it does not stop to spell the words. In the same way, after learning to see five and three as eight zi’ithout adding, it will be just as easy to- see five and three and * two as ten without adding. In this way naturally and without pressure pupils will learn how to add by combining familiar groups of numbers rather than by combining two numbers only 13 at a time. Later it will not be difficult to acquire the habit of i6 . i6 4 24 seeing — as twenty and — as forty without resorting to the slower process. After the forty-five primary combinations are learned pupils should be led to build up other combinations for themselves. ^ 19 + 9=28, 29 -{- 9=38, 9 -j- 7—16, 19 -|- 7=26, etc. This is sometimes called adding “by endings.” This should be a preparation for column addition. The early drill exercises during the first three years should be principally with numbers 4 5 , arranged vertically — rather than 4 -|- 5=9. The purpose of this is not to make “lightning calculators.” There is a normal speed beyond which it is not desirable to train pupils. This normal speed varies with individuals. There is need of a speed standard for each grade. No such stand’ard has yet been established. To meet the demands of business there should be some practice in adding numbers placed horizontally as w.ell as vertically. This is illustrated in the following tabulated items of expense in a school city: Coal Books Salaries Total District i. | $2,000 | $200 | $10,000 | $12,200 District 2. | 500 | 50 | 2,000 | 2,550 District 3. j 550 j 150 | 6,000 | 6,700 District 4. | 450 | 100 j 5,050 | 5,600 Total . I $3,500 I $500 i $23,050 I $27,050 Subtraction is the reverse of addition. Pupils Subtraction. should see the relation between the two pro¬ cesses. Training to see the difference, where calculation is not . necessary, is as desirable as training to see the sum. Subtraction “by endings” should be taught, e. g., 8—3=5, 18—3=15, 28—3=25, etc. ■ 14 The “addition” (Austrian) method of finding the difference is preferred by some. e. g. 9 -5 4 The pupil would say 5 and four are 9, placing 4 as the answer. This method has evident advantages. This method of finding the difference between numbers, some¬ times called the “making change” method, should be taught and used in all problems which involve transactions in which that method would be used in business, e. g. If a purchase is made for $.75 and a dollar bill is given in payment, the change would be calculated by the trader not by subtracting the $.75 from $1.00, but by adding to $.75 enough to make a dollar, using such change as was most convenient. He might say “$.75 and $.25 is a dollar” passing to the purchaser a twenty-five cent piece; or he might say “$.75 and $.05 are $.80 and $.10 are $.90 and $.10 are $1.00,” passing the purchaser a five cent piece and two dimes. Very early in the grades pupils learn that Multiplication. multiplying a number by ten is the same as an¬ nexing a zero to the number, e. g., loXi —10, 10X10=100. After this illustration they may easily be led to adopt the method of annexing one zero when multiplying by ten, two zeros when multiplying by one hundred, etc. When they have come to use the decimal point, they should be led to see that moving the decimal point in a number to the right multiplies the number by ten; moving it two places to the right muliplies the number by one hundred, etc. Afterward it will not be difficult for pupils to see the economy that often results from multiplying by one hundred and taking a third rather than by multiplying by 33 1-3, multiplying by one hundred and taking a fourth rather than by multiplying by 25, etc., etc. If pupils are to be reasonable and intelligent in their mathe¬ matical study, they must understand the nature of the processes which they learn to perform. In addition, they are putting together a number of like things. In subtraction, they are finding the dif¬ ference between numbers of like things. In multiplying they* are finding the result after taking one or more like things a certain 15 number of times, the multiplier indicating “the number of times.” Mathematically then the multiplier must always be an abstract number, i. e., the number of times that the multiplicand is taken. It is evident that the answer in a multiplication example must always be the same kind as the multiplicand. From this it follows that in problems in mensuration statements like the following represent extensions of the concept of multi¬ plication that are not only unnecessary but also undesirable in the elementary school: 6 ft. X 3 ft. = i8 sq. ft. Here there are two mathematical errors—ist the multiplier 6ft. is not abstract; 2d the answer, i8 sq. ft., is not like the multipli¬ cand. In all examples of this and a similar nature pupils should be led to realize that if they are to find the area (square measure) from data given in length, they must translate their data into square measure. For example—“Find the area of one side of a board 6in. x 3in.” 6 X I sq. in.—6 sq. in. 3 X 6 sq. in. = i8 sq. in. After the pupils are familiar with the fact that if they are to have square inches in the answer they must start with a multipli¬ cand in square inches and that the multiplier represents the num¬ ber of times the multiplicand is to be taken (as addend), it will be possible to unite the two processes above into one, as 3 X 6 sq. in.=i8 sq. in. Of course this principle in multiplication applies in all problems. It would be incorrect to state ^qts.^2pts.—6 pints, or 2')<^^qts.= 6pts. Pupils should not only think but state, that if one quart contains 2 pints, 3 quarts will contain 3 times 2 pints, i. e., 3X2 pints = 6 pints. This is all very simple for children even in the primary grades, except possibly in problems in which the multiplier is the larger number. They have, however, learned (it may need reteaching at this point) that the answer is the same, so far as the numbers are concerned, whether one multiplies 2 by 3 or 3 by 2. There¬ fore, while it is necesary to always think and express, (orally i6 and in written form) a multiplication example in its proper or¬ der, the actual multiplication may be done abstractly in the most economical way. Pupils should learn to use the economical Division. dividing by ten, one hundred, etc., by crossing out or dropping one or more right hand zeros, or by moving the decimal point to the left. If they wish to divide by 33 1-3, they should learn that it is often easier to divide the number by 100 and then multiply by 3, etc. These economical ways of ' calculating should become the natural and easy w^ays both in oral and written work. Division and multiplication bear a relation to each other similar to that that addition and subtraction bear to each other. The mul¬ tiplication drills may be given in division examples, e. g., i6-f-4=4, 4X4=16; 16 may be divided by 2, by 4, by 8, by 16: 16=4X4 or 2X8 or 1X16; the factors of 16 are 2 and 8 or 4 and 4 or 2 and 2 and 2 and 2. There are tw'o traditional positions which the quotient may occupy in a long division example. It may be placed at the right cf the dividend, e. g., 27)476(17 27 206 189 It may be placed over the dividend, e. g., 17 27)476 27 206 189' All examples in w^hich the decimal point is used seem to be better solved by placing the quotient above, e. g., 3.14)72.50 The troublesome part of division of decimals is the location of the decimal point in the quotient. If this point were fixed at the outset, a cause of mistake would disappear. By placing the 17 quotient above the dividend the point in the quotient is first es¬ tablished, and easih-’ established. The proper location of the decimal point in the quotient may be fixed in several ways. These ways may be indicated as follows: 72.50-^3.14 (i) 314.)7250. (2) 3.14)72.50 (3) 3.14)72.50. In (i) both divisor and dividend are multiplied by 100 in order to make the divisor a whole number. In (2) this multiplication i< implied and in (3) it is indicated. As an introduction to division of decimals pupils should be led to see, by proving it to be true in a number of simple ex¬ amples, that the divisor and dividend may be multiplied by any (the same) number without afifecting the relation or quotient. Thus 2)6 4)^ 93=3; 18-f-6=3. 3 ~ 3 ~ It would seem to be wise to teach the method that is likely to produce the most accurate results. The method of placing the quotient above, and of fixing at once the quotient decimal point, appears to give the best results. Whether the muliplication is actually done as in (i) or only implied or indicated as in (2) and (3) is not material. It is important, however, that the figures of the quotient be properly placed in relation to the figures of the dividend. The first figure of the quotient should be exactly over the last figure of the group in the dividend that is used as a trial dividend, e. g., 2 . '! : 3.14) 72.50 Division and There are two types of problems solved by Partition. the process of division. Type I. A man sold land for $15.00 per acre, receiving $750.00 for it. How many acres did he sell? Type 2. A man sold 50 acres of land for $750.00. What was the price per acre ? Formerly it was the practice to think through the problem, determine the ‘‘name” of the answer, and perform the division as if it were a pure abstract computation. For instance, in the problem indicated Type i—the problem calls for “acres.” The i8 problem evidently calls for the use of division process. It may, therefore, be stated as follows: 5 Q 15)750 Answer—50 acres. In problem indicated as Type 2—the problem calls for price per acre. Having determined that the answer is to be price per acre, the calculation may again be performed without refer¬ ence to the ‘"names” in the problem. As— 15 50)750 50_ 250 250 - Answer—$15 price per acre. This abstract way of performing problems is the business way. It is now, however, the common school practice to carry the logical thinking through the calculation process. Problems like Type I are performed thus: 50 times $i5)$75o 75 Answer—50 acres. Problems like Type 2 are performed thus: $15 50)$750 50_ 250 250 Or Po of $750=$! 5. Answer—$15. It is readily seen that problems like Type 2 are logically pro¬ blems calling for the division of the dividend into parts. From this logical standpoint therefore this type of problem is called a “partition” problem although so far as the mathematical division is concerned it is no different from the calculation involved in problems of Type i. It is very essential that teachers understand clearly the logical difference in these two types of problems. It is essential that the children be trained in thinking through all problems determining 19 what kind of an answer they are to have and how they are to ar¬ rive at the answer, but teachers should not confuse them by the term, partition. Problems in denominate numbers, reducing from low^er to higher denomination fall under Type i, Economy in Calculation. Ability to see readily the relations between numbers and to choose the economical way to use these relations in calculation is gained only by training. Only a few of these economical uses of number relations have been mentioned for purposes of illustration. If pupils are shown the way, if they are trained to keep an open mind, (mathematically) and if they are given opportunities for abundant practice they will develop skill, varying in degree with the individual, without pressure. Again it should be emphasized that no encouragement is here given to train “lightning calculators.” Such specialized training as that term implies, has no place in school. The pur¬ pose is to so teach elementary arithmetic that the pupils may grow in power to understand and interpret the problems that belong properly to the elementary field of arithmetic and that they may have such a command of number combinations and of number relations that they may with accuracy, readiness and economy of labor perform the processes necessary for the solution of such problems, e. g.. Problem—“A lady bought cloth at per yard. How much does she pay for 20 yards?” Evidently the 20 yards will cost 20 times I2l^^. At this point the pupil should be taught to pause, and consider the best way to make the cal¬ culation. He may multiply 123/2^ by 20; he may think $.125, mul¬ tiply by 10 by moving the decimal point:=$i.25 and then think twice that; or he may think, 1254^ is one-eighth of a dollar and then think an eighth of $20, which is $2 f, f being Training will cause the mind instinctively to picture calcu¬ lation possibilities and it will give skill in the selection of the economical plan of procedure. In Oral and Often pupils are taught to calculate pro- Written Work. perly in oral work but they use the long, prim¬ itive, laborious methods in written work. The economical methods should be used in both written and oral calculations. Pupils should be encouraged to find and use economical ways of doing all problems. However, here as elsewhere, good judgment 20 should control, so that their calculations may be in reality eco¬ nomical. Pupils should be encouraged to use the pencil as little as possible in the solution of all problems. Closely allied to training for skill in calcu- Estimating Results, -g training in “estimating” what the reasonable answers to problems are, as a preliminary to their solution. This training may begin early in school life in esti¬ mating lengths, areas, capacities, weights, etc. Much oral and written work of this character with all kinds of problems may profitably be given in all grades until pupils have established the habit of doing their mathematical work with an expectation of arriving at a reasonable result. Good text books give fre¬ quent suggestions and opportunities for this training. Checking Work. This “checkine” The habit of “ gresses should be checking” work as it pro- fixed in the early grades. consists in repeating one process, before the next process is begun. In adding, the calculation should be re-- peated in the reverse order. In substracting, the remainder and subtrahend may be added. The repetition should be made after each part of the problem has been solved, rather than after the entire problem has been solved. This is the business custom. USE OF THE PENCIL. Children in school rely too much on the pencil. What can be done mentally, should not be done wfith the pencil. In mental and oral exercises the pencil may often be used to record data and answers. In written exercises the pencil should not take the place of the mind. All pupils will not be alike in their abil¬ ity to free themselves from slavery to the pencil, but all may be taught how to do it and they may be encouraged and rewarded for successful efifort in this direction. ORAL ARITHMETIC Mental and oral work should be given daily, if possible, for about one third of the time of the recitation, usually preceding the other w^ork. In five minutes two or three examples in ab¬ stract arithmetic, and two or three problems may be solved. The problems may be dictated to the class, the calculations may 21 be made mentally, and the answers may be recorded on a slip of paper. The pupils making mistakes should solve the problems orally. The problems not understood should be noted and the principles involved should be introduced at another time. A teacher may dictate to the class in two or three minutes ten or twelve problems. Each pupil may indicate by a letter the process to be used in solving each problem, as i. m. (multi¬ plication), 2. d, (division), etc. This exercise centers on the process apart from calculation. These problems, whether dictated by the teacher from her own lists or taken from the text book, should be simpler in state- m.ent and in calculation than problems assigned for written work. They should be planned by the teacher in advance to emphasize the points that need review and drill. They should be of two kinds—(a) abstract, leading to skill in calculation— (b) concrete, leading to skill in interpretation, and in using calculation in applied problems. These applied problems should often correlate with local interests, and with other school work. The habit of mental calculation should be established in all work, both written and oral. THE CLASS TIME. The class period may consist of four parts: 1st. The mental and oral review, occupying about one-third of the time. 2nd. Marking the written work for the day by the pupils—occupying about three to five minutes. 3rd. The discussion of questions regarding the writ¬ ten work of the day—emphasizing the principles in¬ volved—occupying a third of the recitation time. 4th. The discussion of new work, involving often the next day’s written problems. No one class period will perhaps be divisible in just this way, but the general proportion may be maintained. Sometimes an entire period may be given profitably to any one of these phases. The thought to be kept in mind is that the “recitation” should not be a time of testing and examination only, but also a time in which pupils and teacher will work together to increase the pupils’ mathematical knowledge and power. 22 TIME GIVEN TO ARITHMETIC. Beginning in the second and third grades, with ten minutes with the teacher per day, this time should gradually be extended to 125 or 150 minutes a week of actual class time, not counting time taken in going to and from class. HOME WORK. Home work should be given but sparingly and should con¬ sist chiefly in performing the mechanical calculations involved after the ‘"what to do” has been discussed in the classroom. When pupils are assigned home work that they do not under¬ stand, either they waste a great deal of time or they are obliged to depend upon help given by parents. In the latter case there is no uniformity of instruction given the various members of the class. GENERAL METHODS Inductive This subject is well discussed in McMurry’s Teaching and “The Method of the Recitation.” The im- The Use of ^ , . , . , , • • j Objects. portance of the inductive method is pointed out in the following quotations: “There has been only one possible way by which the race has arrived at its store of general truths, i. e., through con¬ crete or individual experiences.” “The progress of the race has been necessarily inductive.” “It is the first duty of the teacher then to direct attention to the past related experiences.” A full discussion of inductive teaching cannot be given here, but an illustration of its application in arithmetic is quoted from “The Method of the Recitation.” This quotation describes a lesson (inductively taught) on the question raised by the teacher: “How shall we add fractions whose denominators are un¬ like?” “What fractions have you already learned to add? Try these: I and f. I and |. j\ and Can you add J and |? What change is necessary before adding them? Why not add them in the same way as the others? How can you add one bushel 23 and one peck? Change the bushel into pecks. Add two yards and one foot. What change was necessary in both examples? Add i and i=|. | + i==^. Illustrate this with a square divided into fourths and eighths. Add I to -1%. Add J and J. What was done in all these cases before adding?” “How shall we add and J ? How can you change the two fractions so that they will be alike, that is, have the same fractional unit? Change them to twelfths. One-third equals how many twelfths? One-fourth equals how many twelfths? } — tV* Illustrate this with a sheet of paper folded into thirds, fourths and twelfths.” “Add J and What is the common fractional unit? | -|- i _ 4 ! 3- 7 3anH 3 -? ill 3 1516 -21- t1 ” -¥ T "6^- S'- ¥ TO- 4 "T T0 = 2 0 -r ^0-2^— “Notice now, what was done in each of these problems; T + 1+4 J + T 7 - The fractions in the finst were changed to twelfths, in the second to sixths, and in the third to twentieths. Was the value of the fractions changed? But in each example the fractions were changed to a common fractional unit, or a common denominator. What was done to the numerators? In each fraction they were changed to cor¬ respond with the change in the denominator. Then in adding, the numerators were added.” “Make a rule for adding fractions that will cover all the cases so far.” “To add these fractions change the fractions to equivalent fractions having a common denominator. Add the numerators for a new numerator and use the common denominator for the new denominator.” “To acquire skill and accuracy in this kind of addition: 1. Add oral proldems as follows: 4+4- 4 + f. i +#. I + 2. “For written work add such as these: ‘•3 -1 + 1 + S=? A + i “3. Add mixed numbers as follows: “32f + I7A + i8f = ?” This illustrative lesson will repay careful study. From it may be drawn the follow-ing suggestions: 24 1. In teaching a new fact or process, begin with the fact or process that the pupils know. 2. Have them use these known facts in familiar ways. 3. Introduce the new fact or process in its simplest form. 4. Suggest the new by means of illustrations of simi¬ lar facts or processes with which they are familiar. 5. Apply the principle of the illustrations by the use of objects which the pupils themselves manipulate. 6. Have pupils apply the principles employed in their objective work to the fact or process that they have set out to learn. 7. Make several such applications. 8. Formulate a statement of what was done in each instance and the results that followed. 9. Formulate a general statement covering all the instances or cases. 10. Make numerous applications of the new fact or process in oral and written exercises. Independent work upon a new process should not be assigned until by class exercises, with the teacher, the new point has been made reasonably clear. Illustrations and objects should be employed only in so far as they are needed to establish the foundation of thinking in experience and in real things. Their use beyond this will tend to confuse the pupils’ minds and distract attention from the real purpose of the lesson. There is a natural tendency on the part of ” some teachers to neglect the drill side of arith¬ metic. After a principle has been made clear to the pupil it must be fixed by persistent, thoughtful drill. Eflfectiveness in drilling depends upon (a) The interest of the teacher in it, (b) The interest of the pupil in it, (c) The variety of ways employed in using the fact to be drilled. . (d) The concentration of drill exercises on the parts of a problem or points in the tables that need drill. 25 For instance, it may be that there is a class or individual weakness in “interpreting” problems. In this case, drill not on solving the problems, but give abundance of practice in having pupils state only what problems ask to ha-ve done. Give at other times abundance of practice in having pupils state only what the problem gives as data to be used in working out the desired result. In drilling on “tables” and other elements of abstract com¬ putations, children should become conscious of the combina¬ tions which they need to drill upon, as contrasted with those on which they are certain. Lay the stress of drill upon the difficult parts. e. g. A teacher is drilling a class on substraction. She finds that some of the pupils have difficulty with the “borrowing” or “taking” from a higher place figure. She may use an example somewhat like the following: 75084692 34296893 Let the pupil who has trouble with the “borrowing” do all the “borrowing” while another pupil does the subtracting. The first pupil says “12”, the other, subtracting, says “9”. The first one “18”, the other “9”. The first one “15”, the other “7”, etc. By this “division of labor” in the exercise each pupil receives the drill that he in particular needs. (e) The subordination of the teacher, and the activity of the pupil. The drilling exercises need careful planning to the end that they may not be monotonous, that the emphasis and repeti¬ tion be at the difficult points and withi the weak pupils and that the teacher be largely a director—the pupils being responsible for the talking and the doing. Aim in all grades to secure first, accuracy, Appearance, and then reasonable rapidity. The first ob- ject is by far the more essential, but in order to develop power of application, so necessary to success, pupils should be required to prepare their written work in arithmetic within a specified time, usually a study period. It should be 26 impressed upon the pupils constantly that nothing but absolute accuracy will suffice in mathematics. If the work is so planned as to be well adapted to the pupils, they will take delight in seek¬ ing to reach and to maintain this and the other high standards, alluded to, providing the school, because of the teacher’s influ¬ ence is pervaded with the spirit of ambition and the determina¬ tion to achieve. This standard, however, should not be used in a way to discourage the slow pupils. Train children to make neat figures and to arrange all their seat and blackboard work in an orderly manner. Lines should be parallel with the edges of the paper or blackboard and be drawn with precision. Unnecessary lines should be omitted. The solution of a problem consists of four Solving Problems. , * parts: 1. Reading, analyzing, understanding the problem— what it gives and what it requires. 2. Planning how to solve it—determining the pro¬ cesses to be used and choosing the most economical way of proceeding. 3. Executing the plan—forecasting the probable re¬ sult, using the shortest roads to the answer by taking advantage of the relations existing between numbers. 4. Testing the work—checking the calculation—com¬ paring the result with the statements of the problem. Beginning in the early grades pupils should Diagrams taught to illustrate their problems in mensuration with diagrams. These diagrams should be care¬ fully drawn with the ruler. If two or more lengths, areas or solids are represented, the diagrams should show accurately their mathematical relations. Sketch diagrams,—those drawn without the ruler—however quickly and roughly they may be drawn, should be made thoughtfully and should indicate the proper relations of lengths, areas and solids that are implied in the data of the problems, e. g.. If a field is to be represented 20 rd. long and 15 rd. wide, the pupils should be led to see that the relation between the length and the width is the same as the relation between 4 and 3. The diagram, therefore, should 27 show four units of length and three units of width, each unit representing in this problem 5 rd. This kind of diagramming is called “drawing to scale.” The expression of mathematical facts in a Graphs. graphical way is becoming common. In the daily newspapers and in many magazines and books comparisons of quantitative facts are shown by diagrams and by pictures of bales of cotton, barrels of flour, etc. In the elementary schools the pupils should be made intelligent in the reading of them and in making them when this kind of representation would be of use in illustrating their work. Very early in the grades the daily temperatures may be recorded graphically. Some teach¬ ers have found it very stimulating to have each pupil keep his own daily or weekly record in spelling or in some other subject, graphically. The class averages in a study or in attendance may be kept graphically on the board or bulletin. Charts with numbers arranged in columns and lines, either upon the blackboard or on large sheets of paper, “flash” cards devised for drill in addition, subtraction, multiplication and di¬ vision, aliquot parts and their per cent equivalents, etc., are very useful in all grades. Their value, however, depends upon their variety and the freshness of appeal, A good text-book will give suggestions in this fleld. Success in teaching any subject, and partic- Testing larly arithmetic, is dependent upon the ability of the teacher to ascertain what the child’s previous preparation for a topic about to be presented has been and whether lessons that have been taught or facts that have been drilled upon have really been grasped by the pupils and retained. Every success¬ ful teacher is constantly testing, frequently by means of writ¬ ten tests, but commonly by careful questioning, day by day, in order to reveal any existing weaknesses. This careful testing will enable the teacher to give the individuals just the help that each needs, and to plan the work of the following days so that the work demanded of the pupil will not present too many difficulties. It will also prevent the equally harmful practice of making assignments which, being too easy, do not call for effort by the pupils. 28 A test should reveal to the teacher the particular needs of the class and of the individual pupil. The failures in a test should be carefully studied by the teacher to discover the par¬ ticular point at which the failure occurred. e. g-. ‘‘Of a flock of sheep 420 were sold. This was 35% of the whole number. How many were left?” Nineteen pupils in a certain class failed on this problem. Of this nineteen pupils, twelve took 420 as the total number of sheep; seven said that 65% =420. It is evident that it would be a waste of time to assign to this class additional problems of this type until there had been further instruction and a great deal of oral drill upon the in¬ terpretation of this type of problem, and upon the relation of the term, per cent, to the equivalent part of the.whole which is in¬ volved in the problem, i. e. A per cent is a part, a hundreth, of some particular quantity. Therefore the pupil must see that he has said that 65% of all the sheep = 420 sheep, and that this statement contradicts the statement in the problem. If this particular class consisted of thirty-five pupils, only half failed on this example and therefore only half need this addi¬ tional help and drill. It is doubtless true that it would be profit¬ able to divide this class, for a time at least, into two sections. It is important to give a variety of tests, oral and written, at the beginning of the term, so that the teacher may commence the term’s work intelligently. Another kind of test, not intended for use in determing pro¬ motion, has been successfully worked out in recent years. It is called “standard” or “standardized test.” These tests are somewhat fully discussed by Dr. C. W. Stone, in “Arithmetic Abilities,” a book published by Teachers College, Columbia Imiversity. Dr. S. A. Courtis has devised a plan foi* such tests, which has been somewhat widely used. Dr. Courtis states, “The Courtis Standard Tests are not ‘examinations,’ but scientific measures of fundamental abilities of arithmetic involved in simple work with whole numbers. Their purpose is to show how efficiently the work of the entire school is conducted.” It is agreed that these “tests” are a contribution to the scien- tilic teaching of elementary arithmetic. 29 For samples address Dr. S. A. Courtis, 441 John R. St., Detroit, Michigan. Tests, especially of the more formal kind,^ should not call for the most difficult work that pupils might be expected to do, neither should they form the exclusive or most important con¬ sideration in determining standing or promotion. The mark given a test or a recitation or the work of a week, a month, or a term, should be considered not a rating indicating absolute value, nor should it be an end in itself, but rather an indication of mathematical proficiency or need. Pupils should realize that in arithmetic absolute accuracy is essential. Business cannot prosper on in¬ accurate arithmetic. 90% may therefore properly become the standard for daily work. No such standard should, however, be used in determining promotion. The Individual. If the instruction in mathematics is from the beginning to be given from the stand¬ point of the pupil, the teacher’s first consideration is to find out what each individual knows and also how he knows it. All defects cannot be made up at once, but they should be tabulated and definite plans should be made to remedy them. The more fundamental these defects the more important it is to remove them. It may be that a pupil has reached the eighth grade without learning to make good figures or to make them in a proper manner. The arrangement of work may be unbusinesslike. It may be that a pupil'has not learned to illustrate his work with properly constructed diagrams. It may be that a pupil has not learned to calculate economi- callv. These and other individual and class defects should be looked for at the beginning of each term and a thoughtful effort should be made to meet each difficulty. GEOMETRY. All mensuration is geometry. As much geometry, therefore, should be taught as is needed for the proper construction and reading of diagrams in mensuration and of working drawings 30 in the shop. The working drawings are best made on the draw¬ ing board with the drawing tools that go with it. The proper use of these tools should be taught. It is probable that in time there will be an extension of this work in the form of a course in geometrical construction of a useful nature adapted to the. seventh and eighth grades. This course will doubtless include instruction in the use of the simple tools of the architect’s office, referred to above, i. e., the drawing board, the T-square, the triangles and the compass. Such a course has not been planned that meets general acceptance, although a promising beginning has been made in some text¬ books. ALGEBRA. In schools that are large enough to have two eighth grade classes, it sometimes happens that pupils may be found strong in arithmetic, who would be profited by beginning algebra. Such pupils may complete half a year of high school algebra during the eighth year and at the same time continue the study of arithmetic, giving it, however, less time than the other class gives. For most pupils it is best to continue without interrup¬ tion the study of arithmetic. All pupils should, however, become familiar with the use of letters as symbols of number and quantity and should learn to solve problems by the equation method. In algebra as in geometry there is a probability that in the future there will be a somewhat more extended study than is here suggested. The practical uses of the algebraic symbol and the simple algebraic formula would seem to warrant such de¬ velopment. Some advance has been made in formulating a course for the seventh and eighth grades, but it seems at present undesirable to do more than call attention to this tendency. INFORMATIONAL OR COMMUNITY ARITHMETIC. To relate the arithmetic of the schools more closely and syste¬ matically with that in the world ouside the schools, part of the time of the seventh and eighth years should be given to an in- vestigation of the activities of the community to discover their mathematical content. Of necessity a considerable amount of reporting on investigations and of discussion must be allowed in the arithmetic time, although the exercise is informational rather than mathematical. Some of the time for this discussion may be taken from the English composition and civics time, if it is found to encroach too much upon the mathematics. There are a number of subjects that should be treated largely for their informational or community value. The 'mathematics of these subjects is in the field of the specialist and of the expert and therefore problems in them should be of the simplest kind, for the purpose of illustrating the principles involved. Many of these topics have been noted on page 8. The examinations given in these subjects should be in har¬ mony with the emphasis that has been placed upon them. (From Dr. Suzzallo—“The Teaching of Primary Arith¬ metic.’') “More doing and less explaining should characterize instruc¬ tion in arithmetic. Problems, in an increasing extent, are real problems typical of life, if not actual. There may be days of teaching when not a figure is used during the arithmetic period. The business institution, the social setting, is studied as care¬ fully as the processes of calculation. The pupils acquire a knowledge of banking as well as skill in the computation of in¬ terest. They may even visit a bank, a factory, a shop, as the case may require. Instead of fifteen problems that deal with fifteen different subjects, all more or less remote from one an¬ other, the class hour may be given over to fifteen problems re¬ lated to one situation, such as might develop in the business of a bakery or a farm. Advantage is taken of the social plays of the children and their games. “To ‘play at store,’ to utilize games, to deal with things within a picture, is to bring the concrete materials in the class room into more nearly normal setting.” (From David Eugene Smith—“The Teaching of Arith¬ metic.”) “Thus a country whose business was chiefly farming would need to emphasize agricultural problems; one that derived its wealth from metals or its coal would emphasize mining; a man- 32 iifacturing nation would find certain lines or problems of the factory peculiarly suited to its needs, etc.” “The earnest teacher, awake to the needs of the business community in which the school is located, will hardly fail to introduce genuine problems with local color to enliven the work in arithmetic—number games, problems of simple building, of mechanical effort, of national resources.” This leads to the conclusion that the supervisor and the the teacher need to acquaint themselves with the mathematics in the life immediately about them. If the school is in the city, mathematics will be found in the factory, in the department store, in the regular income and outgo of family life. If the school is in a country district, mathematics will be found in the field, forest, barn and home'as well as in the country store and in the school and its surrounding grounds. The coun¬ try school offers probably the very best conditions for effective mathematical teaching since in the country the uses of mathe¬ matics are so numerous, so near at hand and so vital to the children. BOOKS FOR SUPERVISORS AND TEACHERS. “The Teaching of Arithmetic,” by David Eugene Smith, Ginn & Co. (A short, readable, authoritative discussion of elernentary school mathematics. Price $i.oo.) “The Teaching of Mathematics,” by J. W. A. Young, Long¬ mans, Green & Co. (This is a book of 346 pages and is a standard book for supervisors’ use. The bibliographies on the various topics discussed are very full. Price $1.50.) “The Teaching of Primary Arithmetic,” by Henry Suzzallo, Houghton, Mifflin & Co. (An interesting and suggestive study of the teaching of arithmetic in the first six grades of the American public schools. Price $.60.) “The Method of the Recitation,” by Charles A. and Frank M. McMurry, Public School Publishing Co. (Valuable to the teacher of any subject. Price $1.00.) NOTES ON THE ASSIGNMENT OF WORK BY GRADES. Modifications of the following plan of distribution of work may be preferred by some. (1) There are some who believe that formal instruction in arithmetic should be delayed as long as possible, until the third or fourth year of school. Others believe that this formal in¬ struction should begin at once in the first year. There does not appear to be sufficient data to discredit either opinion or to support either opinion. The plan suggested here follows a middle course. It is believed that this plan is in ac¬ cordance with the prevailing practice in this country. (2) There are those who prefer to place the objective work in the fundamental processes with fractions in the fourth year rather than in the fifth. While it is doubtless true that in general in this country this work is assigned to the fifth grade, the quality of the particular class may *determine this matter. (3) It is recognized that the work outlined for the seventh and eighth grades is new for many schools. Not many text-books at present adequately treat the practical application of elementary mathematics, and text-books at best can be but guides in this field. All schools have not introduced the shop and domestic science activities in such a way as to make them immediately available for the stud}^ of their mathematical content. While making the transition from the study of less useful arithmetic, to the study of more useful, applied and community arithmetic, it may not be feasible to discard at once the work with which teachers are familiar. However, progress will be lapld if supervisors and teachers believe that the transition ought to be made, and with judgment set themselves the task of making it. There may be a feeling on the part of some that too much of the ordinary school arithmetic is here eliminated. The recog¬ nition of the principle of elimination, namely, that the schools should teach the kind of arithmetic which is useful outside of school, is much more important than the elimination or retention cd any particular topic. If the principle is accepted, the practice in the schools will be brought more and more into harmony with it. 34 ELEMENTARY ARITHMETIC BY GRADES. FIRST GRADE (First Half). The study of arithmetic during this year should be some¬ what informal and the kind should be determined by the needs of the children. These cannot be set down definitely. Children who have not had the advantage of kindergarten training or of somewhat exceptional home environment, come to school at the age of five or six years with the knowledge of a few number names which they apply indefinitely to groups of things; but with a very limited consciousness or conception of the mathematical significance of these names. I St. They may not be able to apply these names correctly. They may call a group of seven objects, five; a group of ten objects seven, etc. 2d. They may use the term 'Tour” when they mean the fourth one; "five” when they mean the fifth one, etc. 3d. They may be able to recognize a group of objects and may say correctly "This group is four,” and "That group is five,” etc.; but without realizing the relation of the group to one of its members as a unit; e. g. that the group of four ob¬ jects is the repetition of those objects four times, etc. This enumeration is not intended to be complete, but sug¬ gestive only, to indicate the point at which instruction should begin. A true conception of number, of number relations, and a knowledge of the fundamental processes have their origin in counting. It is, therefore, important in counting objects by ones, twos, fives, etc., that the learner realize the relation of the group counted to the one, two or five assumed as the unit. These rcJations can be made only through repeated experiments in counting objects. The kind of material and the way it is used have much to do in making children conscious of these relations. 35 36 In selecting material, objects of uniform size, as cubes, small cardboard squares, splints of definite length, are more desirable as they aid in giving clear images. These objects may be used in building, designing or construct¬ ing familiar forms, thus appealing to motor activity. Interest in the construction is more sure to lead to the desire to compare, count and measure than if objects are counted with no purpose. Observations made by the children, such as “My tower is higher than Mary’s,” “My house has more blocks in it than John’s,” “I have put six blocks in the steps,” “This stick is two inches longer than that,” “This card is two times that,” are all evidences of the growing sense of relation which is character¬ istic of numbers. In extending the series, objects other than the kind's indicated should be used. Exercises which necessitate assorting, classifying and count¬ ing objects and placing them in groups, piles or boxes, fre¬ quent tests in counting balls or numbers, counting frame houses in the block, boxes in the post office, etc., should be given. No special period need be set aside for this instruction. An occasional “teaching” exercise, supplemented by individual in¬ struction to meet the dififerent needs of children; by “seat work,” planned to give the desired kind of arithmetical experience; and by a frequent use of mathematical material in oral language exercises, may be more efifective than a regular daily lesson in arithemetic. FIRST GRADE (Second Half). The instruction of the first half year should be continued and carried to 1. Counting by twos and tens to lOO. 2. Reading numbers through joo. 3. Making figures through 10. 4. Using one-half, one-third, one-fourth, as they may need these terms in connection with their various kinds of class and seat work. No regular period need be set aside for instruction in arith¬ metic, but a special .period should be given to it when there is need in order to accomplish the results indicated. 37 SECOND GRADE (First Half). Pupils may be given a text-book at this time, or it may be withheld for another half year or even until the beginning of the third grade, according to the judgment of those in authority. The teacher should have two or three good primary arithmetics for her own use. Whether the pupils have books or do not have them, the spirit and method of instruction will be the same. Instruction should be more formal than in the First Grade and a special period should be set aside for it. The principal w^ork of the term is the teaching of the 25 primary number facts of addition whose sum does not exceed 10. These 25 sums should be taught by counting various kinds of objects and by the use of diagrams (dots, lines, figures, pictures, etc., on the blackboard). The objects are used so that the children may find for them¬ selves the sums and may see the results that come from com¬ bining the groups of 2 and 3, 4 and 3, etc. These results, they de¬ termine by counting, by this time a familiar process. There should be some variety in the objects to maintain in¬ terest and to avoid associating the number ideas exclusivelv with one set of objects. The variety should not be so great as to cause mental confusion. After the objects have fulfilled their purpose, which is to make clear a fact or a process, their use should be discontinued. It is not always easy to determine w^hen this time has arrived, as it is as much an individual as a class matter. If a pupil can readily and accurately arrange objects, as, 5 blocks and 3 blocks, and by counting determine the sum, he is ready, turning his back upon the blocks, to make the oral statement, 5 and 3 are 8; to make the statement with figures, 5-I-3—8; to read these numbers and to memorize the combination. While learning that 5 blocks and 3 blocks are 8 blocks, it is a simple matter to learn that 3 blocks and 5 blocks are 8 blocks, that 8 blocks less 5 blocks are 3 blocks, and thatl 8 blocks less 3 blocks are 5 blocks. After several combinations have been learned, it remains to fix them in the mind. This can be done best not by returning constantly to objects, or by mere repetition, but by using the 38 combinations in ways interesting to the children. Much thought has been given to devising occupations and games in which the number relations may be expressed repeatedly. Sight work must precede pure mental work in all drills. Pupils may “play store’' in a variety of ways. It is not neces¬ sary to have money or imitation money. Children use tooth picks, pebbles, nails, etc., for money in their play out of school. They can use the same things for money in school. Often in play out of school one marble is equivalent to the value of two, three, five or more marbles. This same kind of imagina-' lion may be utilized in school. “Playing Dominoes’” is an interesting game. Children may make their own dominoes for use at their seats. The teacher may have a similar set of large card board dominoes for class use. Of course the number of 'dots on the dominoes should be governed by the numbers on which the teacher wishes to drill. “Playing Soldiers” can be used in a variety of ways. This and games of a similar nature give opportunities for action. They should be played with “snap” and “energ}'.” “Playing Housekeeper” has many possibilities for number ex¬ perience. “Spinning the Arrow.” Make a large circle of cardboard. Place different numbers at regular intervals around the circum¬ ference. Fasten an arrow loosely in the center. Each child spins the arrow and announces the number to which it points and adds or subtracts an agreed on number. Such drills should not take the place of drills with figures written in columns. These games and dramatized occupations may be of real value if the play side does not crowd out the mathematical side. The play factor should be kept simple and subordinate and should involve a relatively large amount of serious mathematical drill. It is questionable whether a teacher is justified in infusing into the recreation periods an element of arithmetic or of any other study. The recreation periods should be periods of mental re¬ laxation. Beyond the making of figures and pictures of objects, no written work should be required. In addition to learning the 25 primary combinations men¬ tioned above the following subjects should be given careful attention: 39 1. Reading numbers and counting by twos, fives and tens to loo beginning with zero. 2. Writing in figures and spelling the names of numbers through 20. 3. Roman numerals as they are met with in books. 4. Halves and thirds of numbers giving exact divisions through products that are learned. 5. Measures used—inch and foot. Use the ruler naturally and teach the parts as used. It is desirable to have the pupils in this grade use a ruler divided into inches, halves and quarters only. 6. Teach dozen and half dozen. SECOND GRADE (Second Half). Read carefully the outline for the grades preceding. If the children have come from another teacher, time should be taken to review the work of the previous grades before beginning the new work. What has been noted regarding use of object games and dramatized occupations applies to this grade also. There are 45 so-called primary number facts of addition, i. e., there are 45 different groups of two numbers each whose sum is 18 or less. Twenty-five of these primary addition facts, those representing sums less than 10, were learned in the first half of this grade. It is believed that the addition facts between 10 and and 18 are better learned by calculation than by objects. Lead pupils to see that 13 is three and ten, that 14 is four and ten; etc. Eleven and twelve need to be learned by themselves. Realiz¬ ing that 9 is one less than 10, pupils will readily see that nine and four must be one less than ten and four; that nine and five is one less than ten and five, etc. The use of objects beyond ten or twelve involves too many objects for easy manipulation, in¬ volves groups of objects too large for mental imaging, and habituates the mind to thinking in terms of objects, a positive detriment to development of power in mathematical thinking. Pupils during this grade should begin to recognize groups of numbers as wholes, as they recognize groups of letters (words) as wholes. 40 2 4 7 Expressions like 123 etc., should be thought at once as three, six and ten, rather than as two and one are three, etc. The best test of a pupil’s knowledge of his addition and sub¬ traction facts is ability to see these facts, without conscious cal¬ culation just as words are recognized without conscious spelling. The chief emphasis of this half year’s work is on the 45 primary addition facts. Much waste of effort will be avoided if these facts are separated into groups according to their difficulty. It should be made evident to the children that all combinations of 1 and some other number are so easy that they need no attention. It is a profitable exercise for the pupils to help make these “diffi¬ cult” or “troublesome” groups. The “troublesome” groups should receive attention until they are as familiar as the “easy” groups. Drill by the use of games, dramatized occupations, various kinds of mechanical devices such as diagrams, “flash cards,” etc., etc., where the figures are written in column form. The following subjects should also receive attention: 1. Adding and subtracting by tens and fives to 100, beginning with zero. 2. Reading and writing, spelling names of numbers through 100. 3. Roman numerals as they are met with in books and in learning to tell time of day. 4. Addition of single columns of two to four, one-figure numbers. 5. Multiplication and division tables of 2, and if the class is able, the table of 3. 6. Halves, thirds, fourths of numbers, through 20 and of single objects. 7. The units of measures, one inch, one square inch, pint. Use also foot, yard, square foot and quart. 8. Coins up to $1.00. The children should be given much experience in using real measures. They should begin also to exercise their judgment in estimating lengths and surfaces. The ruler divided into iiiches, half inches and quarter inches is desirable for this grade. THIRD GRADE (First Half;. Pupils should have a text-book. Review thoroughly the 45 primary addition facts, giving special drill upon those facts that are troublesome. All these facts should be known instantly, with the corresponding differ¬ ences ; the multiplication and division tables of 2 and 3; the exact halves and thirds of products learned. Give much sight drill. More written work may be given in this review than was de¬ sirable in the Second Grade, although the greater 1 emphasis should still be on the oral form of expression. The new work of this half year is: 1. Adding one-figure numbers to two-figure numbers, orally and in written form. Subtraction examples with the same kind of numbers and taught in connection with addition. Add one- figure numbers in columns of two, three and four numbers. Add numbers arranged in horizontal lines. Pupils should use readily the terms, sum, difference, remainder. Addition and subtraction by ‘‘endings,’’ e. g., ^-{-2—^: 13-4-2=15; 6—3=3; t 6—3=13, etc. 2. Multiplication tables to 4X10. 3. Division tables to 40-4-4. 4. Multiplication and division by one-figure numbers. 5. Reading and writing numbers through 1,000. Roman numerals through XX. Dollars and cents. 6. Halves, thirds and fourths of single objects and of num- b(‘rs which allow of exact division to 40. 7. Use the measures of previous grades. 8. Introduce simple one step problems keeping them as near as possible to the children’s experience. The use of the ruler should be carefully taught so that it may be used skillfully, like any other tool, in drawing and card board construction work. Pupils may begin drawing to scale. Give ocasional time tests for speed and accuracy. Try not to discourage the slower members of the class. 41 42 THIRD GRADE (Second Half). Two or three weeks may be profitably spent in a careful review of the work of the previous half year. In this review emphasize ad¬ dition and subtraction by “endings,” e. g., 9-|-2=ii; 19+2 =- 21; 29-4-2=31; etc. During the review make a memoran¬ dum of the troublesome places and devise interesting exercises for drill. Oral expression should predominate over written ex¬ pression. The new work of this half year is: 1. Addition of one-figure numbers and one-figure and two- figure numbers at sight. Subtraction examples with the same kind of numbers. Add columns of one-figure numbers consist¬ ing of two, three, four, five and six numbers. Lead pupils to see the sum of four number columns at a glance, without laborious addition. 7 3 2 I — This sum may be seen as 13. It is not necessary to add one number at a time. If any pupil has diffi¬ culty in seeing the 13 at a glance, he may at least see two groups—the sum of one group being 10 and of the other 3. Have some addition in horizontal lines. Addition and subtrac¬ tion of U .S. money orally and in written form. 2. Tables of multiplication and division to 9X10 and 90-r-9. 3. Multiplication and division by one-figure numbers. Avoid examples in division involving remainders. 4. Halves, thirds, etc., to ninth of all numbers involved in the tables which will be divisible without a remainder. 5. Reading and writing numbers to 10,000. Roman numerals to L. 6. Use the measures, pints, quarts, gallons; quarts, pecks, bushels; ounces, pounds; inches, feet, yards; square inches, square feet, square yards; dozen, one-half dozen, one-third and one-fourth of a dozen. Teach also the table of minutes, hours, days, months and years. Study the relations of the parts of each table, one to the other; give abundance of practice in “estimating” lengths, areas, vol- 43 limes, weights, so that all terms in tables may have a reasonably definite meaning. 7. Simple one step problems of a practical sort. 8. Train pupils to a skillful use of the ruler in all work in which it is used. Continue simple drawing to scale. Use games and dramatized occupations in drills and reviews. Give occasional time tests for accuracy and speed. These tests are for inspiration and should not result in discouragement to faithful pupils who may be a little slow^ FOURTH GRADE (First Half). This grade has little work that may be called new. The children are led farther along lines they have already traveled. They will meet difficulties, but they will be difficulties involved in using larger numbers, in calculations and in application. It is most important that the Fourth Grade teacher shall be familiar with the work that the teachers in previous grades have been trying to do—the quantity, the variety and the method. It should be recognized that pupils have not learned all that teachers have tried to teach. Many points will need reteaching and time should be taken for it. * 1. In addition much drill should be given to adding “by endings.” The difficulties in addition are largely because a few simple combinations are troublesome such as g-j-y, 8-|-5, 8-[-7, 7-j-4, 5+2, 5+3, 8+5. The stress of drill should be placed upon these and other points of difficulty. However, the teacher must call out the active, interested effort of the children, if the drill is to make an impression. Pupils should become skillful in recognizing at sight sums of groups of two and three figures. The addition may be carried to three and four-figure numbers, and to six and eight numbers in column. Have some work in horizontal addition. Have pupils fix the habit of checking their addition by adding a second time in the reverse order. See page 20. 2. In subtraction the work should parallel that in addition. If it has not before been taught, familiarize the pupils with the business way of making change, i. e., by adding to the amount of the purchase money, what is needed to complete the amount of money passed to the dealer in payment. 3. In multiplication and division continue previous work to use of multipliers and divisors of three figures. Tables through 12X12 and 144 -M2. Pupils should use readily the terms multiplier, multiplicand, product, dividend, divisor, quotient. 4. Reading and writing of numbers to 1,000,000. Roman numerals to C. 44 45 5- Complete the various tables of measure by teaching rods, and miles. Make sure that pupils have a definite idiea of the value of each item in the various tables and fix these ideas of values by frequent reviews and drills in which they are used concretely. Use the black board, the school room floor, the school yard to illustrate distances and areas. Accustom pupils to ‘‘pace” yards and rods. Determine the distance of a mile from the school to some well known point as a “standard mile.” The repetition of tables is of little use, if there is not this conscious¬ ness of concrete values. Give much practice in “estimating.” 6. One step problems and possibly simple two step problems. Begin to train pupils to think through the problem and to es¬ timate an approximate answer before they begin the written solution. Pupils should at times be asked to make problems for the class, using problem material in the home or school life. 7. In fractions continue work of previous grades. 8. Continue use of the ruler and drawing to scale. Drills and reviews by the use of games and dramatized oc¬ cupations are valuable in this grade. Keep these simple and keep the mathematics prominent. The pupils will be able to suggest new games if they are taken into the confidence of the teacher. Children may find it interesting to compete with themselves, as a person often does in playing golf or other games, trying to “beat his own record.” Each pupil may keep his or her own “score card.” FOURTH GRADE (Second Half). The outline and suggestions given for the First Half of this grade apply also to the Second Half. This is a time of complet¬ ing the foundation. The Fourth Grade teacher has a right to assume that pupils have been trained to perform skillfully the abstract processes as follows: 1. Addition and subtraction as previously outlined. 2. Multiplication and division by three-figure numbers. 3. Addition, subtraction, multiplication and division with U. S. money. 46 4. The simple use of such fractions as i, J when considered as parts of a unit, or of numbers which are divisible without a remainder. 5. The use of the various tables of measure in length, area, volume, bulk, time, w^eight, founded on a large body of practical, concrete experiences with these measures. 6. Reading and writing numbers to 1,000,000. Fifth Grade teachers have also a right to expect that the pupils coming to them from the Fourth Grade have been trained to the habit (a) of “checking” their w^ork by going over each pro¬ cess a second time before passing to the next process, (b) of “estimating” the probable approximate result before beginning the real solution of a problem, (c) of calculating with some thought of economy of effort. If the pupils are not proficient they should not necessarily for that reason be kept from promotion. Promotion should be determined by general fitness, and should not be delayed by a particular failure or weakness. FIFTH GRADE (First Half). The new work of the Fifth Grade is the teaching of addition, subtraction, multiplication and division of fractions—common and decimal. This should be divided about equally between the two halves of the year. The first half of the year common frac¬ tions (addition and subtraction thoroughly, multiplication and division less thoroughly) should be taught. The Fifth Grade teacher should study carefully the work of the preceding grades and she may profitably spend tw^o weeks at the beginning of the term in reviewing carefully the fourth grade work, reteaching those parts upon which the pupils are not clear and devising new drills for those parts that need em¬ phasis. This review may continue while the new work in frac¬ tions is being introduced. Pupils have had much experience with simple fractions, etc., of single objects and the equal parts of small num¬ bers. They can add -J and J and' I-, etc., and dou{btless will have no trouble with a problem requiring that J be taken from I or |. They have, however, had no experience in add¬ ing ^ and ^ or, in general in calculating with fractions not having a common denominator. 1. Teach reduction of fractions to higher and lower terms. 2. Addition and subtraction of fractions should be taught exactly as addition and subtraction of integers were taught in the second grade—objectively. Paper cutting, diagrams of circles, rectangles and lines should precede the use of figures and these should be used to ilustrate the first problems in which figures are used. Have much oral work. A detailed discussion of this objective teaching is not given here because the principles have been somwhat fully set forth in the introduction under “Inductive Teaching and the Use of Objects” and under “Second Grade,” and also because full sug¬ gestions will be found in any good text book. It may be said, however, that the use of objects and diagrams should be with small fractions, and that objective teaching should not be con- 47 tinned after its purpose has been realized, i. e., after pupils see the validity of the operations with fractions in real things. To continue the use of objects with large fractions will confuse. To dwell on the objects too long will cause pupils to think habit¬ ually in terms of objects and this will limit their mathematical development. 3. The terms numerator, denominator, common denominator and least common denominator should be taught. As only those fractions with small denominators (not larger than 16) will be used, all common denominators can be found by inspection. Teach also reduction of improper fractions to mixed numbers, of mixed numbers to improper fractions, and the addition and sub¬ traction of mixed numbers. 4. Teach multiplication of fractions and the use of the ‘‘of’' when the multiplier is a fraction. 5. Division of fractions. 6. Reading and writing numerals through 100,000,000. Reading and writing common and decimal fractions (through three places) and mixed numbers. In reading mixed numbers use the word, “and”, only between the whole number and the fraction. Read and write Roman numerals through C also the numerals D and M. 7. Continue drill in addition by “endings” and by groups of numbers. Have pupils find and learn the aliquot parts of $1.00 and 100 and apply these in the solution of problems. Continue practice in interpretation of problems, in “estimating’ results and in application of short methods in the solution of problems. Pupils in this grade should appreciate the importance of recog¬ nizing relations of numbers, should habitually look for relations and untilize them in calculations. e. g. “If 5 lbs. of butter costs $1.50 how much will 15 lbs. cost ?” Pupils should see that 15 is 3 times 5 and therefore that 15 lb. will cost 3 times $1.50. They should also realize that the easiest way to multiply $1.50 by 3 is to multiply $1.00 by 3, adding $.50 multiplied by 3. 8. Problems in denominate numbers. Two step problems. 9. Measuring and drawing to scale. 49 FIFTH GRADE (Second Half.) Review carefully and systematically addition, subtraction, multiplication and division of common fractions. Have much practice with mixed numbers. Continue reviews of all subjects treated in previous grades with special attention to training for skill in interpretation of problems, in estimating results, and in economy of calculation. The work of this year will be much simplified if the numbers are kept small and the problems are kept simple. The emphasis should be placed on familiarizing pupils with the processes. 1. Teach cancellation. Cancellation should be taug^ht as a case of “reduction” of fractions to lower terms. This reduc¬ tion is not a process of dividing numerator by denominator or denominator by numerator, but is the process of dividing both numerator and denominator by a common divisor. Thus f is reduced by dividing 4 and 8 by the common divisor 4, with the result I 2. Instruction in decimals should be introduced by illustra¬ tions with U. S. money. Teach addition, subtraction, multipli¬ cation and division of decimal fractions, using decimals to three places. In reading mixed numbers use the word “and” only at the decimal point. Keep the work simple. In multiplication limit the work to examples having the decimal point in the multiplier only and in division to examples having the decimal point in the dividend only. 3. Change common fractions to decimal form. At this point, if it has not been done before, teach that: I St. A fraction is an expression of a part—(a part of a unit, e. g., J of an orange; or of a group of units, ^ of 16 oranges. 2nd. A fraction is an expression of division in which the line separating numerator and denominator is a sign of division. Therefore, the numerator may be divided by the denominator and an improper fraction may be reduced to a whole or a mixed number; and a common fraction may be changed to the decimal form. (From this point of view a fraction is the ratio of two magnitudes of the same kind.) 4. Problems relating to the common affairs of life, market¬ ing, travelling, etc., should be made by the pupils. 5. Drawing to scale. SIXTH GRADE (First Half). Teachers of this grade should not expect pupils at the be¬ ginning of the term to be proficient in the work of the preceding grade. Two or three weeks or even a longer time should be spent in a careful and systematic review of the work in frac¬ tions. The sixth grade teacher should be “at home” in the work of the fifth grade teacher. During the time of this review much practice may be given in interpretation of problems, in estimating results, in becoming skillful in economical calculation,^ in choosing short methods of solving problems by taking advantage of number relations, using the aliquot parts of $i.oo or loo, etc. Continue drawing to scale. 1. Teach in decimals multiplication and division with the decimal point in both terms of the example, using also the more troublesome combinations of numbers, such as numbers with one or more zeros, etc. 2. Change common fractions to decimals and decimals to common fractions. 3. Teach the reading of decimals to six places, but limit their common use to three places. 4. About the only new work that should be attempted in this half year is simple work in per cent, the decimal fraction limited to two places or the common fraction with 100 as de¬ nominator, sometimes indicated by sign %. Limit this work to finding simple per cents of numbers. Avoid such per cents as 4 i%, 6i %, etc. Pupils should realize that they are not doing anything new \v hen they multiply a number by .04 or yf 0 • They should relate this work to their former work in common and decimal fractions. The notation is the only new thing in percentage. 5. Besides problems in the book, have many problems in mensuration, and denominate numbers drawn from the pupils’ ex¬ perience and formulated by them. 50 51 Ask the pupils to bring to school bills that have been used in business transactions. Talk them over with the class showing the advantage of the conventional forms of ruling, heading, receipting, etc. The pupils should be asked to make out bills first on blank forms, if they can be obtained, afterwards on paper which they themselves have ruled. Let their imagination be used to make these business transactions represent actual transactions as far as possible. Let it be imagined or represent¬ ed that the teacher or one of the members of the class has sold the rest of the class a bill of goods or done a piece of work for them or rented a house to them. The resulting bill should con¬ tain the real names of the parties to the transaction. The pupils should learn how to receipt a bill properly and also how to give a receipt for any obligation; e. g., an account settled, rent paid, wages received, and part-payment receipt. SIXTH GRADE (Second Half). The purpose of the review work in this grade is to establish a strong foundation for future mathematical study by making pupils sure and skillful in the use of the essential processes previously studied. At the outset the teacher should make sure that every pupil can add and subtract accurately and use the multiplication combinations automatically. All cases of “arrested develop¬ ment’'—such as counting in adding, adding instead of using the multiplication combinations, and writing down numbers to be added to the next higher order in multiplication—should be ear¬ nestly sought out and corrected. The teacher, particularly of this grade, should feel that unless this fundamental work is thoroughly accomplished her work in arithmetic is, to a con¬ siderable extent, a failure. Pupils should now be able to read and write the quantities, appearing under various forms, which they will need to use. They will probably have no difficulty with common fractions and denominate numbers, but they may need considerable drill in integers, Arabic and Roman, and in decimals. Their knowl¬ edge of the Roman system wHl be used almost exclusively in connection with the numbering of chapters in books, and it 52 should be taught with this in view. Drill only upon numbers below C, Sufficient drill should be given to enable pupils to reduce denominate numbers to higher and lower denominations and to give them proficiency in changing fractions to other fractional forms; common and decimal fractions to higher and lower forms; improper fractions to mixed numbers, etc. 1. Much practice, oral as well as written, should be given this half year in common and decimal fractions. 2. The new work of this half year is an extension of the use of per cent in simple problems. Pupils should realize that work in percentage is the same as work in common and decimal frac¬ tions. The terms, “base’', “rate” and “percentage” are seldom, if ever used, outside the school room. Their use in the school room should therefore rapidly disappear. The terms per cent, interest, (infrequently rate of interest), commission, etc., etc., should be used. The term, percentage, is often used to desig¬ nate that part of arithmetic in which per cents are used; e. g. “The pupil studies percentage.” “These are examples in per¬ centage.” Only two kinds of problems involving per cents require at¬ tention— (a) Those in which the part is to be found, when the per cent is given; e. g. “Find 35% of 600.” (b) Those in which the per cent is to be found when the part is given: e. g. “200 is what part of 800?” 3. Problems in simple commercial discount and in interest for integral per cents involving years and months only. Teach the simplest form of interest calculation, i. e., find the in¬ terest for one year and multiply by the number of years. 4. Learn the per cent equivalents for i,, tV 1 2 3 3 A 7 1_ These per cent equivalents should become so familiar that when the per cent form is thought, the common fraction form is thought simultaneously and vice versa, and they should be used interchangeably in oral and written work. 5. In measurements, review ’ the work with the rectangle, square, triangle, cube, distinguishing them from each other and associating these geometrical names with the figures. 53 The problems under this head should be practical. Require pupils frequently to make their own measurements in order to obtain the conditions of the problems they are to solve. Illustration: “Find the dimensions of your schoolroom.” “What is the total area of the walls and ceiling?” “How much window surface?” “How much blackboard surface?” “Deter¬ mine the cost of sodding the yard of your school and inclosing the school yard with a fence.’’ One good way of treating this topic is to work out problems in connection with building and equipping a house or the school building. If a building is being erected in the neighborhood, it should be used for this work in order that the problems may be real. 6 . In order to train pupils to keep simple accounts—an ac¬ complishment that every one should possess—teachers are re¬ quested to have at least one exercise a week in which the chil¬ dren make a record of their income and expenses. This exer¬ cise need not occupy an entire period. Except when class in¬ struction is needed, the records may be made before school, or at odd times. The accounts should be of actual moneys belonging to the pupils, if that is possible. It is better to have a briefer account and have it a real one than to fill it with imaginery transac¬ tions. The accounts should indicate briefly, but adequately, the sources and dispositions of income: not. Rec. from Aunt Mary,—but; Gift from Aunt Mary: not. Paid out— but; Paid for Candy: not. Earned—but. Shoveling Snow: etc. The headings at the top of the pages should be uniform through each book—either, Received—Paid ; or. Income—Outgo; or, Re¬ ceipts—Payments. Balances should be computed at the close of each month. These balances should be properly brought down by the use of red lines, if possible, and carriec^ forward to the next month’s account. Begin each month on two pages, with proper statements of accounts brought forward from previous month. Have a uniform use of capitals and punctua¬ tion. Pupils may keep memorandum slips, but entries in the account books should be made in school, with ink, and accounts should not be copied. Accounts through vacations may be kept ii. memorandum form and entered in the books in school after vacation. 54 Sample page of Account Book: . 1910. RECEIPTS. Mar. 1. On hand. “ 6. From Father .... “ 14. Errand . “ 20. Paper Route .... “ 29. Errand . . .. $1.18 .25 .10 .75 .15 1910. PAYMENTS. Mar. 2. Papers . “ 3. Car Fare . “ 4. Pencil . “ 31. On hand . . . . $ .25 .10 .05 ... 2 03 $2.43 $2.43 Ask the pupils to bring to school bills that have been used in business transactions. This work is review and pupils should become proficient in making and receipting bills of various sorts that are used in the common business of life. Make the work practical and frequently during the term give practice in this work. SEVENTH GRADE (First Half). Devote the first tzvo or three ivceks—more or less according io the needs of the class—to a review of the essential processes. Jt is ONLY when a good foundation is established in this review work that the best results in the advance can be secured. Make sure that all pupils can use the fundamental processes skillfully. Resort to various mechanical devices for drill upon the points of weakness. Pupils themselves may often suggest excellent devices, if they are made partly responsible. Give much attention to oral arithmetic. The work of this year consists in applying in practical ways the principles and practices learned in previous grades. These applications are in the fields of denominate numbers, of mensura¬ tion and of percentage. 1. Do not allow pupils to become confused by the new terminology found in the applications of percentage to various kinds of business. Cause them to see that these involve the identical principles and processes which they used in their study in the preceding grades. Keep the work simple, practical, and in acordance with the present business methods. Train pupils to investigate by polite and pointed inquiries the different kinds of business studied. The information thus gained should be re¬ ported to the class in an interesting and profitable manner. 2. The subject of commercial discount is of great value be¬ cause of its extensive use “from wholesale transactions down to bargain sales.” Pupils should understand some of the reasons for allowing discounts, buying in large quantities, paying '‘cash down” or within a specified time, the usual deductions from list prices, etc. It would be well, for obvious reasons, to review bills and receipts in connection with the study of this topic. Let the pupils interview commission merchants and obtain data for some practical problems, and, at the same time, a social interest in this important business. All problems should state whether or not the sum of money includes the commission and the price of the goods bought. It is contrary to business prac¬ tice to include the commission in this way. 55 56 3- The subject of taxes should be treated briefly. It should be shown that there is expense involved in conducting the af¬ fairs of a city. To make this concrete, use as an illustration the fact that the public schools are dependent upon the tax¬ payers for their support. Point out the necessity of each citizen [ aying his rightful share of the public expenses of government. The simple mathematical work in the subject may be developed by the analogy between levying a city tax and taxing the mem¬ bers of a ball team to meet its expenses. The attempt in the former case to apportion the taxes, according to the individuars property, may be brought out by the contrast in this respect be¬ tween these two illustrations of taxation. Confine the work in this grade to the subject of municipal taxes, discussing the subject only in a simiple way. Use the actual or current tax rate in problems. Pupils should be shown a ‘‘tax notice” form. 4. Mathematics of the school shop, of agriculture and .of domestic science or sewing in those schools where these activi¬ ties are caried on. SEVENTH GRADE (Second Half). Continue review and drills to increase skill in interpretation of problems, in estimating results, in choice of method in solu¬ tion of problems and in calculation. The following subjects may be studied this half year. 1. Mathematics of the school shop, of agriculture and of do¬ mestic science in those schools where these activities are car- • ^ ^ ned on, 2. Problems of the home, of the grocery store, meat market, department store. Making change. Cost of heating and light¬ ing the home. Cost of furnishings for the home. Other prob¬ lems relating to the cost of food, heat, light, clothing, etc., for the family. 3. Saving money, investing money, banking, interest, real estate, loans. 4. Making and reading working drawings and drawing to scale. 57 Problems of These problems are designed to give drill The Home. Upon the essential processes through the use of problems that are met with by individuals in their daily life. They are to be solved upon the basis of current prices in every case. For this reason it is important that the pupils should get in advance a list of the items, together with the current prices. These problems are the kind all of us as purchasers should be ])repared to solve mentally, or, if not so, then in writing wdth very brief notes. They are the kind that salesmen and clerks aie called upon to solve in their daily work. The problems will be found useful for both mental drill and for written lessons. Begin with a brief discussion of the need Savmg. value of saving money. There are many people who never learn to save money wisely. Many people pre¬ fer to gratify their immediate desires rather than provide for the future. Why is this a bad plan? On the other hand there are people that, in order to save for the future, deny themselves the things which it would be real economy to buy. Give illus¬ trations. There is always the temptation to live extravagantly, b-xtravagance includes not only living beyond one’s means, but also spending money foolishly. Every boy and girl should be¬ gin to save early, no matter if it be but a few dollars, or cents a year. It will be found a help to wise saving to keep a little account book, showing the amounts received and the amounts paid out, giving the date of each item. Froni such an account it should be possible to determine the amount of money on hand at any time. See suggestions for keeping these accounts under Sixth Grade (Second Half.) Having saved money by wise economy, it Investing. essential that we know how to take care of it. Formerly people hoarded their savings in hidden places about their homes. It is customary now to invest money, that is not needed, in various ways so that it will not only be safer than when kept at home but will also bring in some income. Unfortunately, some of the investments offered to the public are offered by dishonest men, but many others are perfectly safe, being made by honest business men of long experience. It is zvell to remember that the larger the income offered by any form- of investment the greater is likely to be the risk of losing the money put into it. Would it be better to put money into an in¬ vestment which promised a larger income, but which might not be safe, or into something which promised a very small income, but which was perfectly safe, like United States Government bonds? We should learn how to distinguish between good and bad investments, so that we shall be able to invest wisely what¬ ever money we may be able to save, avoiding any form of in¬ vestment that is in the least fraudulent or unsafe. Namie the various ways of investing money. Which are the safest? Which is the most common form of investm.ent? Consider the various ways of investing money; U. S. Postal Banks, l>anks and trust companies; co-operative banks; real estate; loaning money. Establish the economic principle that the element of safety in investments is of far greater importance than the amount of income from them. ^ ^ The bank is most commonly used for the safe keeping of money. Certain kinds of banks—for instance, savings banks and trust companies—not only guarantee to take care of all money left with them by depositors but they also pay a certain per cent of interest. Na¬ tional banks also generally allow interest on what are called “inactive” accounts, that is, deposits that remain undisturbed for some time. When practical, pupils should go to a bank and ask some official to show thern about the bank. A report should be made to the class. xA.scertain what officials there are in the hank, and the duties of each. Many schools have found it interesting and profitable to or¬ ganize school banks, electing the officers and carrying on a regular banking business, either with small amounts of real money placed on deposit by pupils and transferred by the teach¬ er to some bank or trust company, or with imitation money, which can be used by cutting out rectangular pieces of paper for bills and circular pieces for coins, marking on each piece its denomination. Teach the method of depositing money, of drawing checks and of borrowing money. 59 When one puts money in a bank for safe keeping, he is called a depositor. The first deposit made with a book is termed the opening of an account. It is usually necessary for the depositor to be introduced by some one known to the bank. Why is this advisable? The de¬ positor is asked to write his name for filing, so that his signa¬ ture may be verified at any time in the future. He is then given a deposit book, in which his account with the bank is kept. Let the pupils examine a deposit book and find the amount on deposit by subtracting the sum of the checks (amounts drawn out) from the sum of the deposits. Whenever a deposit is made, the depositor, or the bank of¬ ficial, makes out a deposit slip, on which the amount of deposit, the date and the depositor’s name are written. (Deposit slips may be obtained at a bank for the asking and practice may be given in filling them out.) The common form of withdrawing money from the bank is to issue checks against the account. This is now the most com¬ mon method of paying bills. Show the pupils the customary form of a check and teach them how to fill out a check, and how to keep the “stub.” In 1910 Congress passed a law establishing a Postal Savings Rank in connection with the postoffice. Any one may now de¬ posit at any postoffice authorized to receive it, a sum of money not exceeding $500. Interest is paid at the rate of 2% per an¬ num, compounded quarterly. The government of the United States guarantees the safe keeping of these deposits. It is the policy of the state and national governments, more and more, to keep a close supervision over the banks of the country, so as to protect the savings of the people from losses caused by dishonesty and incompetence. Periodically “state¬ ments” must be made by the banks to the State and National Bank Departments. Bank officials are thus compelled to conform to wise, conservative plans for handling and investing the funds entrusted to them. As a result, few people lose money deposited ir banks. Interest problems should be kept simple. Interest for unusual per cents or for days except as they may be considered as frac¬ tional parts of months are unusual in business. 6o Reed Estate. Another common form of investment is buying real estate, that is, land and houses. This makes a particularly desirable form of investment because of its comparative safety, of the tendency of wisely selected proj>- erty to increase in value and the possibility of using such prop¬ erty for a home for the owner, thus saving the expense of renting. The disadvantages of this form of investment are, first, the expense of maintaining the property—insurance, taxes, repairs, etc., together with the loss of interest on the investment; sec¬ ondly, the liability to depreciation in value, which is, of course, more often greater with houses than with land; and, thirdly, the difficulty of selling readily at its full value. Let each pupil in the class, if practicable, select some piece of property and ascertain the price at which it was held five years ago, and its present price. What has been the gain or loss per cent during the five years? If it can be ascertained, the amount expended for insurance, taxes, repairs, etc., and the interest on the original investment for the five years at 4% or 5%, should be deducted from the gain or added to the loss. Loans. All forms of investment that we have already studied are in reality different ways of loaning money, the income being the amount paid for the use of the money loaned. Show how this is true of interest. In addition to these indirect methods of loaning money, people frequently loan money as a matter of accommodation, or for interest re¬ ceived, usually at a somewhat higher rate than can be obtained at banks. When money is loaned a note should be obtained from the borrower, promising to pay the amount at a specified time, or “on demand.” (Show a “time” and a “demand” note.) Usually, also, the loaner insists upon some form of security from the bor¬ rower, which will secure the loaner from loss should the borrower be unable to repay the loan. This security may take the form of having the note endorsed (the name of the endorser written across the back of the note) by some responsible person, who thus agrees to pay the amount of the note should the borrower fail to do so; or, of a- mortgage on real estate or other valuable |-roperty owned by the borrower; or. of certificates of stocks or 6i bonds placed in his keeping as a guarantee of payment. A good business man will not loan money without satisfactory security, especially if the loan is made as an investment. It would be well to have the pupils see a mortgage deed, and, in a general way understand it. When a note is taken for a loan or in paym.ent for some obli¬ gation, it can be presented to the maker for payment when due, or it can be sold to a third party before it becomes due, for its face value, less a certain per cent of that amount for the length of time that the note has to run, which is deducted and retained by the one taking the note as payment for the interest. This is called discounting a note, and, as it is most commonly done by banks, it is called Bank Discount. When the note becomes due the person holding it will collect the amount indicated by the face value of the note. A person wishing to borrow money from a bank will often make a note payable to the bank after a specified number of days and have it discounted at the bank, thus securing the loan of money for the intervening time. The charges for “discounting” a note should be learned at a local bank. In New Jersey this charge is limitecf by law to 6%. Pupils should have practice in making out and discounting the different kinds of notes—demand, time and non-interest bearing. Pupils may imagine that $10,000 was placed in their hands for investment. Each one should select a list of investments which he believes would, at the same time, insure the safety of the prin¬ cipal and bring in a satisfactory income, using as far as possible current prices, and compute the total amount of annual income from its capital. The pupils should criticize one another’s invest¬ ments from the point of view of safety and desirability. This may be tried on successive days until pupils have learned to apply their knowledge to the different forms of investment that have been discussed. Try, also, larger and smaller sums of money as the amount of capital to be invested. This work, of course, must be very elementary and simple, but it may be made a valuable summing up of the term’s work. EIGHTH GRADE (First Half). Some pupils reach the eighth grade as ready to take up high school mathematics as others are a year later. In schools where two divisions are made those who are thus ready may be put together in one division and may be allowed to begin their high school algebra. Considering the shorter time devoted to this subject in the grammar grades they may not be able to complete the work assigned for the year in the high school. A review of essentials both in ’oral and written exercises should be taken by those who begin algebra as well as by the others, See suggestions for previous grades. 1. Mathematics of the school shop, of agriculture and of do¬ mestic science in those schools where these activities are car¬ ried on. 2. Problems relating to the industries of the locality and of the state: (a) Agricultural. (b) Manufacturing. (c) . Transportation. (d) Mercantile. Data for these problems must be secured first hand from the industries themselves. Some statistical data may be found in Year Books or State Reports. 3. Paying and collecting money by checks, drafts, and money orders. At one time there was no such thing as money. Instead of buying what was needed or wanted, people “bartered '' i. e., exchanged one commodity for another. In the early history of our country there was but little money to be had, and consequently, much “bartering” was carried on. The doctor and the minister were paid largely in farm products and other necessities. As trade increased, it was found more and more inconvenient to exchange produce, for it was difficult often to make a just ex¬ change. A horse would be worth more than a cow and not so much as two cows, and it was necessary to “throw in” some 62 63 thing, often not needed, to complete the bargain. The use of metals as a means of exchange was gradually introduced, as they were staple in value, could easily be made into convenient shapes and sizes, and would not wear out easily with handling. In time, the precious metals, gold and silver, came to be used for the higher denominations. As a matter of farther convenience the government began to make paper money which, of course, would be valueless except for the government’s willingness to ex¬ change the bills at any time for their value in gold or silver. ‘ Nearly all retail business is on a “cash” basis; in fact, the two and a half billion dollars’ worth of money in this country is largely used for the numberless forms of retail trade. But other means of paying bills have now been devised for the large amount of business that requires the sending of money from one place to another. For, while money is sometimes sent by mail (which is unsafe and often inconvenient) and occasionally by express (which is expensive), these methods of sending money can not be largely used. The most common method of paying bills is Checks. 1^^^ bank checks, over 90% of the total amount of indebtedness in the country being cancelled by their use. They are used so largely because of their convenience and safety. They can readily be transferred to another person by endorsing them, and their endorsement by the payee insures a receipt that the law will always recognize. Pupils in the Seventh Grade have had much practice in making out checks, and have become familiar with the conventional form and the method of using them in pay¬ ing bills. Blank checks for practice may be had, possibly, on ap¬ plication to a local bank, or the school board may furnish them, but it is desirable to have much practice in writing the entire form and in using checks instead of money in the transaction of school business. United States post-offices and the express Money Orders, Etc. . • , i • i r companies issue money orders, which are a sate and convenient means of transmitting money. A small fee, proportionate to the amount sent, is charged for making out the order and cashing it at its destination. The pcst-offir's alone, in a recent year, issued over fifty-two million money orders, amount- 64 ing in all to $420,000,000. To send money by money order, one obtains at the money order window at the post office an application blank and fills it out. (Show pupils money order blanks.) On receipt of this application, with the amount that it is desired to send by mail, plus the charge for making out the order, the clerk will prepare an order to the payee, which can be enclosed in a letter to him, who on receiving it will have it cashed at the specified money order post-office by signing the order. Money may also be sent by telegraph from some of the larger telegraph stations, but at a much larger cost than the methods of transmitting money already referred to. The charge is one per cent of the amount, plus double the rate of telegraphing between the two places. Such an expensive method is, of course, used only in emergencies. People about to travel in foreign countries commonly secure foreign express checks or letters of credit, which will be ac¬ cepted by specified banking houses and other places for their value in the currency of the country visited. They make a safe and convenient method of carrying money when traveling abroad. 4. In mensuration. (a) Pythagorean Theorem, demonstrated by diagrams and paper cutting. (b) Areas and perimeters of circles, rectangular quad¬ rilaterals, triangles, (right, isosceles, and equilateral) and irregular surfaces whose areas are found by the use of the triangle. (c) Volumes, capacities and surfaces of rectangular solids and of cylinders when the pupils need to find these in solving concrete problems. Classes that are studying the silo will need to determine the area of its surface and its capacity. It may be desirable to make an approximate reckoning of the capacity of barrels and baskets. Slight emphasis on the volumes of pyramids and cones, largely for informational purposes. (d) Lumber if the class can measure real lumber and learn the business practice of the lumber trade. 6S EIGHTH GRADE (Second Half). All pupils, whether they are taking algebra or not, should be given thorough drills in computation, in addition, subtraction, multiplication and division of integers, com^mon and decimal fractions, using the short methods taught in previous grades. Much practice should also be given in interpreting problems, in estimating results, in choosing economical methods of solving problems, and in checking work as it progresses. Review also applications in denominate numbers, in percentage and in men¬ suration. 1. Make use of all opportunities for the application of mathe¬ matics, including drawing to scale and the making and reading of working drawings in shop, in agriculture and in domestic activities. 2. Teach the use of letters as symbols of quantity and number in simple problems that may be solved by the use of the equa¬ tion. Teach the use of the equation in solving simple problems. 3. Teach the simple use of the graph to record variable tem¬ peratures and other statistical data. 4. Stocks and bonds. In treating of stocks and bonds show that a great deal of the wealth of the country is invested in the various kinds of business which supply our wants and minister to our welfare. Name several business enterprises in or near your community which are owned and controlled by single individuals; name several in which a number of persons have invested their money. Show that much of the business of the country is carried on by these corporations—for instance, railroads, trolley systems, and many of the large manufactories. The money contributed to carry on these industries is represented by bonds or certificates of stock, which are bought and sold like land or merchandise. Explain some of the causes for fluctuation of stock prices. Pupils can find the market price of the more common stocks and bonds quoted in the daily newspapers. Certain stocks are given twice in the lists, with entirely different prices quoted. After one or the other of these names of the same stock, the abbreviation '‘pfd” (preferred) is written; the stock not thus designated is “common stock.’’ The same name 66 will be found also in the list of bonds, with still another quotation given. Most railroads and many other business enterprises issue these three forms of securities, bonds, preferred and common Itock Pupils should study these different forms of securities as issued by some local corporation, as a traction company or a railroad. Bonds are evidence of indebtedness secured by mortgage on all property owned by the corporation. They promise to pay a fixed rate of interest on the par value of the bond. Although they usually do not bring so high a rate of income as good stocks, they are preferred to stocks by most investors on account of their greater safety. Why are they a safer investment than most stocks ? When the interest on a bond becomes due—usually semi-annually—a coupon is detached and may be cashed for the amount of interest due, indicated on the coupon, at any bank. In the case of “registered’’ bonds, which have no coupons, a check is sent the owner, by the treasurer of the corporation which issued the bond, for the amount of the interest. The United States Government has from time to time issued bonds, which, though they pay as low a rate of interest as 2%, are in great demand by investors because of their absolute security. State and city governments also frequently obtain money by issuing bonds. (Show a bond certificate if one can be secured.) Preferred stock, as its name indicates, has preference over the common stock in the property and profits of the company; it is entitled to dividends at a fixed rate, before the common stock can receive any dividends. How would this arrangement enable the organizers of a company to secure outside capital? (Show share of preferred stock.) After the running expenses of the corporation, the interest on its bonds, and the dividends upon the preferred stock have been paid, the balance of the profits of the business belongs to the holders of the common stock and is frequently distributed as dividends upon the common stock. (Show share of common stock.) A board of directors is generally elected by the stockholders to manage the business of the company. In their election each 67 holder of stock is entitled to as many votes as he has shares of stock. In some cases both the owners of preferred and common stock are entitled to vote; frequently, however, only holders of common stock are allowed to vote, the preferred stockholders, like the bondholders, being considered merely lenders of money to the company. The advantage of stocks and bonds as a form of investment is the ease with which they can be bought and sold. Their chief disadvantage is their liability to fluctuation in value, although this is not true of many bonds. One needs to be familiar with the '‘market” (the stocks and bonds offered for sale), in order to invest wisely in these securities, Men have found it wise to make their purchases and sales of stocks and bonds only through reliable, trustworthy brokers. The work in stocks and bonds will be limited to: (a) Finding the cost of a given number of shares of stock, or of a given number of bonds, at a given market price. (b) Finding the income from a given number of shares of stock or a given number of bonds. In reckoning the cost of stocks or bonds omit brokerage. Brokerage is generally not reckoned on the amount of money handled or invested, but it is subject to trade customs. 5. Insurance—Fire. Pupils should understand the underlying principle in insurance; paying a relatively small, but carefully estimated sum of money to a company for its guarantee to pay a much larger sum of money in case of a specified contingency. 6. Cost and support of local, state and national governments. The problems in the cost and support of the government should be designed to maintain and increase efficiency in the es¬ sential processes of arithmetic through constant application and to convey information concerning matters of local or national interest, of which all citizens should have some knowledge. This aspect of the work is designed to correlate with the work of the civics class. The problems should afford opportunity for the use of the four fundamental operations of integ-ers, of percentage, and of some of their applications. It includes such topics as, cost and income of our government, cost of city or town schools, cost and income of city or town government, money raised by taxes^ inoney raised by sale of bonds, money raised by special assess¬ ment. 7. Some study may be given to the subject “Longitude and Time.”