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 OF CALIFORNIA 
 
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 LOS ANGELES 
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 LOS ANGELES 
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 Gollege 
 
 R.ECOR.D 
 
 A Journal Devoted to the Practical Problems 
 of Elementary and Secondary Education 
 and the Professional Training of Teachers 
 
 March, 1903 
 
 MATHEMATICS 
 
 IN THE 
 ELEMENTARY SCHOOL 
 
 
 IAY 1917 
 
 PUBLISHED BY 
 
 THE COLUMBIA UNIVERSITY PRESS 
 
 Columbia University, New York 
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 Entered as Second-Class Matter at the New York, N.Y., Post Office, June 12, 1900 
 (COPYRIGHT, 1900, BY TEACHERS COLLEGE)
 
 Teachers College Record 
 
 Edited by DEAN JAMES E. RUSSELL 
 
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 TEACHERS COLLEGE RECORD is a serial publication issued by 
 Teachers College, Columbia University, for the purpose of pre- 
 senting to students of Education, and to the public generally, a 
 comprehensive view of the history and principles of education, 
 of educational administration, and of the theory and practice of 
 teaching as advocated and followed by Teachers College and its 
 schools of observation and practice. 
 
 ANNOUNCEMENT OF VOLUME IV 1903 
 
 The May issue of the TEACHERS COLLEGE RECORD will deal 
 with " Methods of Teaching French and German," according to 
 the new or reform method, by Dr. Leopold Bahlsen, of Berlin, 
 Lecturer on Methods of Teaching Modern Languages, Teachers 
 College. 
 
 Other issues will discuss the principles which should govern 
 the curriculum of primary and secondary schools, illustrated from 
 outlines of courses as conducted in the Teachers College Schools, 
 and followed by special issues dealing with specific work in the 
 first, second, third, and fourth grades, and in the high school. 
 
 Numbers concerning Kindergarten work and Music in 
 Schools may also be expected. 
 
 Address all communications to 
 
 The Editor of Teachers College Record 
 
 Columbia University, New York City
 
 Q A 
 
 TEACHERS COLLEGE RECORD 
 
 VOL. IV MARCH, 1903 No. 2 
 
 1917 
 MATHEMATICS IN THE ELEMENTARY SCHOOL 
 
 By DAVID EUGENE ,SMITH, PH.D., Professor of Mathematics, and 
 FRANK M. McMURRY, PH.D., Professor of Elementary Education, in 
 Teachers College, Columbia University 
 
 The following outline of theory and of subject-matter is pro- 
 posed rather as a basis for discussion with students in profes- 
 sional courses than as a fixed body of thought for use in the 
 elementary school. Yet the course of work here outlined is fol- 
 lowed to -a very large extent in Teachers College in its Horace 
 Mann School of observation and its Speyer School of practice, 
 although the arrangement of topics is necessarily different in the 
 two schools. The particular order here suggested is expected to 
 apply more fully to the Horace Mann School than to the Speyer 
 School. 
 
 I. CONTROLLING IDEAS THROUGHOUT THE 
 CURRICULUM 
 
 The numerous changes in the curriculum of mathematics in 
 the grades during recent years have been principally due to two 
 causes, namely, a growing sympathy with children, and a more 
 enlightened interpretation of the needs of society. The former 
 has led, in many schools, to the omission or neglect of number 
 work as a regular study during the first year or two of the course, 
 on the plea that it is somewhat foreign to child nature ; it has 
 demanded less abstract problems, in order that truer interest 
 91] i
 
 2 Teachers College Record [92 
 
 might be aroused; and it has opposed especially long and diffi- 
 cult solutions, as in equation of payments and compound interest, 
 on the ground that they were an unnecessary physical tax, besides 
 giving a wrong idea of actual business life. The second cause 
 of change, namely, the better interpretation of the needs of 
 society, has likewise demanded less theoretical problems, and 
 has urged the entire omission of topics that are not called for 
 in the quantitative relations of daily life; as, for example, the 
 table of troy weight, partnership, and cube root. 
 
 The question arises: Should these two ideas constitute the 
 principal standard in the selection of a curriculum in mathe- 
 matics? And, if so, dare they have fuller control than they have 
 thus far obtained? Or, if neither of these should control, what 
 standard should take their place? 
 
 A full discussion of these questions is out of place here, 
 because it is a general educational problem that is involved, 
 applying alike to all branches of knowledge, and in the main 
 it should be assumed as settled, when an individual study is 
 under consideration. Nevertheless, a brief mention of the former 
 standard of selection of materials in mathematics for children, 
 and a comparison of the same with the generally accepted 
 standard in certain other studies, may suggest our answer to the 
 proposed questions, and an outline of our argument. 
 
 Twenty years ago, acquaintance with arithmetical processes, 
 and mental discipline, constituted the chief purposes of instruc- 
 tion in arithmetic. As to subject-matter, whatever problems 
 promised best to acquaint pupils with these useful processes, 
 and to furnish this mental training, were deemed acceptable. 
 This is probably still the prevalent view. 
 
 But these aims, or standards of worth, are far from accept- 
 able in certain other studies. Turn, for example, to literature, his- 
 tory, nature study, and art, as carried on in the grades. Who would 
 rank business utility and mental discipline foremost among their 
 purposes, and be guided primarily by these objects in the selec- 
 tion of topics in those subjects? Indeed, the whole point of view 
 has radically changed as to these branches. The teacher's first 
 ) aim is the excitement of a deep interest, possibly love, for these 
 fields of human experience. That is the important, immediate, 
 goal to be reached. When it comes to the subject-matter, those 
 topics must be chosen which are capable of arousing interest;
 
 93] Controlling Ideas Throughout the Curriculum 3 
 
 this is one of the controlling ideas, and the teacher must become 
 a close student of children in order to avoid serious error at this 
 point. 
 
 But not everything that is thoroughly interesting can be 
 accepted, and there is, therefore, a second controlling idea for 
 selection. The subject-matter in each of those studies must 
 reveal some side of life, and do it in such a way that the pupil 
 feels forcibly the relation between it and practical living. The 
 truths of literature must offer a guide for daily action; those of 
 history must throw light upon and awaken sympathy for present 
 social problems ; nature study must lead into agriculture ; and 
 art must end in the real enjoyment of such pictures as should 
 grace the home, as well as in the desire and ability to make home 
 more attractive. Whatever matter contributes only slightly to 
 this purpose of richer, more effective present living, must go; 
 hence, non-classics in literature, numerous names and dates in 
 history, mere sense-training object-lessons " of any sort " in 
 nature study, and the bare mechanical drawing of geometric 
 forms in art, are discarded. In other words, an actual weeding-; 
 out process is going on in these parts of our curriculum, with^ 
 the object of searching out and retaining only those materials 
 that (a) correspond with child nature, and (b) identify the child 
 with actual life. 
 
 If these large purposes are accomplished, minor objects are 
 also attained, including such as were above mentioned in con- 
 nection with arithmetic. For instance, when genuine interest is 
 assured, the prime condition for concentration of attention is ful-1 
 filled ; hence, good mental discipline is bound to follow, provided, j 
 of course, the method of presenting the subject is good. In' 
 brief, good mental discipline, as an aim, has little to do with the 
 choice of subject-matter; it is thought of only after this matter 
 has been chosen, and in that sense is a subsidiary object. Who, 
 for example, would allow the development of logical power to 
 be a controlling factor in selecting literature and history topics? 
 And even if it were so allowed, how could it be a clear guide? 
 
 Again, the business utility of a subject is of value; but if 
 history leads into the heart of present social problems, if nature 
 study opens to view many broad principles of agriculture, and if 
 art makes the home more attractive, a good degree of business 
 utility is attained, and much more. Utility taken in a narrow
 
 4 Teachers College Record [94 
 
 sense kills the spirit of any study, as we all know ; but when 
 each of these branches aims in this broad way to meet the needs 
 of life, not only useful knowledge is acquired, but spirit and 
 energy are aroused. 
 
 Enough has been said to indicate our answer to the query 
 raised at the beginning of this article. Business utility and 
 mental discipline should not rank as the primary aims in teaching 
 mathematics to children ; particularly in the selection of subject- 
 matter. This study should stand on the same plane, should be 
 
 controlled bv the same broad ideas, as other studies. Accord- 
 ' . / . 
 
 ingly,(the child's interest in the quantitative side of life should 
 
 be the highest immediate aim of the teacher of mathematics in 
 the gradesy just as his interest in the spiritual side is the highest 
 
 " immediate aim of the teacher of literature; and the nature of the 
 child, together with the needs of society, should constitute the 
 main standard in selecting subject-matter. This signifies that 
 these two ideas should be given a fuller control in mathematics 
 than has thus far been allowed. 
 
 What is there, however, in mathematics in the grades that 
 is capable of appealing to interest? Has it a body of thought 
 comparable in attractiveness to children to that in literature or 
 history or nature study? That is the most fundamental question 
 in this field at the present time. If there is no such body of 
 thought, then of course we must drop back upon useful knowl- 
 edge and mental discipline as our chief aims. To be sure, these 
 are purposes that have been " conceived by the adult and forced 
 upon the child," but what else would be left ? On the other hand, 
 if there is a rich body of thought here for the pupil, then it is 
 time that we were finding it out. 
 
 Manual training has faced this same question and is in 
 advance of arithmetic in its solution. It was not long ago that 
 this study, like arithmetic, aimed mainly at utility, by leading 
 
 " children to make useful objects, and at mental discipline, by 
 exercising the mind through the hand. But now there is a 
 marked tendency to regard handwork as a means of opening up 
 the industrial side of life. Through it, one is not only to develop 
 some skill in carpentry, blacksmithing, masonry, basketry, and 
 bent-iron work; but he is to be introduced into the constructive 
 experiences of mankind, making possibly many excursions to 
 factories, in order the more fully to comprehend and sympathize
 
 95] Controlling Ideas Throughout the Curriculum 5 
 
 with this phase of human activity. And why is this not reason- 
 able? Just as literature, relying upon words, tries to present 
 man's highest aspirations, why should not some other study, 
 relying largely upon observation and participation in manufac- 
 turing, try to present man's constructive experience? Little 
 seems to be lost, and much gain is promised, by this change; for 
 the thought-content of manual training is thereby increased far 
 beyond the principles involved in the use of tools. The fact that 
 we are distinguished as an industrial nation emphasizes the need 
 of this advance, on the social side, while it promises an especially 
 rich addition to the subject-matter of manual training. When 
 this aim is more fully realized in practice, as it can be, manual 
 training will stand on the same educational plane as other studies, 
 and the instructor in "that branch can then justly rely upon im- 
 planting a permanent interest in it, as do instructors of literature 
 and nature study in their respective fields. 
 
 Again the question is before us : What is there in arith- 
 metic that is capable of arousing the direct, immediate, interest 
 of children? Ordinarily the part of any study that is depended 
 upon to arouse interest, at least initial interest, is the part that' 
 is concrete, as the individual facts in history and nature study. 
 The mere operations in arithmetic are abstract, as we know, and 
 the principal other part of the study is the applied problems. 
 Hence, these applied problems must be the concrete part of the, 
 subject, and if its content is to be greatly enriched, the change 
 must be expected first of all in this particular. 
 
 But some persons, no doubt, will here raise the questions: 
 Are the problems of arithmetic really concrete? Has arithmetic, 
 in fact, any concrete part at all, as other studies have? 
 
 It becomes necessary to define what we mean by concrete. 
 Some years ago it was found that a very large percentage of 
 examples in our arithmetics were plainly abstract, consisting 
 of unnamed units, such as 5^2X3%- To make these concrete 
 they were made denominate; as, for example, If one barrel of 
 apples cost $3^, what will 5^2 barrels cost? But still there is 
 the common complaint that these are practically as abstract as 
 they were before. When, therefore, is an example concrete? 
 
 If one goes to a dictionary to learn the meaning of a list 
 of new words, he is engaged in an abstract kind of work, 
 because the words are then dissociated from their original, con-
 
 6 Teachers College Record [96 
 
 crete, connections. If, on the other hand, one learns their mean- 
 ing through the context, as most children do, the work is 
 concrete, because the words are then approached and viewed in 
 their individual relations. So, in general, a statement is truly 
 concrete only when it is seen in its original associations; the 
 
 moment it is " isolated from its origin and its goal, it becomes 
 abstract." 
 
 Applied to mathematics, this means that problems are 
 abstract when they are chosen at random, without the purpose 
 of clarifying the individual situations in which they arose, but 
 merely with the intent of giving exercises in processes. They 
 ^-are concrete only when they deal with real things and with actual, 
 significant, situations. In the former case, the answers reached 
 have no worth in themselves, and the work is unreal because 
 the processes, which were originally a mere means, become the 
 end. Motive for hard work on the part of pupils is no longer 
 provided in the subject matter, and for that reason competition 
 of child with child, an artificial incentive, must be largely relied 
 upon. In the latter case the answers are the end, because they 
 ' are valuable thoughts, like those in other studies. 
 
 For example, let us take for our topic a comparatively 
 insignificant industry, namely, the manufacture and consumption 
 of printers' ink in the United States : 
 
 1. At the present time about 30 million pounds of this ink are manu- 
 factured in the United States per year. How many tons is that? {Answer, 
 15,000.) 
 
 A single factory in New York makes i l /2 million pounds. What part 
 of the whole does it make? (V.) How many tons? (750.) 
 
 2. The worth of this one plant is about $125,000. At that rate, how 
 much money is invested in this industry in the United States? ($2,500,000.) 
 
 3. This one factory employs 20 workmen, and 35 men in all, including 
 salesmen and others. What, then, are the corresponding numbers employed 
 in the entire industry? (700.) 
 
 4. The workmen in this factory, on the average, receive about $2 per 
 day. At that rate what amount is paid per day to such workmen in the 
 United States? Per year, of 300 days? The 35 men average about $5 per 
 day. What would be the approximate pay roll per week, and per year, for 
 all such factories together? 
 
 5. This one factory, in making the ink, consumes per month 250 bbls. 
 of rosin brought mainly from the pine trees of Georgia and South Caro- 
 lina. It costs $6 per bbl. What facts can you figure out from these state- 
 ments? (3,000 bbls. are used in the one factory, and they cost $18,000. 
 The cost for all the factories is $360,000.)
 
 97] Controlling Ideas Throughout the Curriculum 7 
 
 6. This one factory consumes per month 10,000 gal. of linseed oil, 
 made from flax seed, and worth 75 cents per gal. What facts can you 
 learn from this statement? (It costs the one factory $7,500 per month, or 
 $90,000 per year, for such oil. It costs all the factories $1,800,000 per year, 
 the total amount being 2,400,000 gal. The acreage of flax might also be 
 considered.) 
 
 7. Four hundred tons of anthracite coal are burnt per year in tnis 
 factory. What is its price, at the present rate? 
 
 8. This factory consumes 600 bbls. of water per day, 300 days in the 
 year, mainly to keep cylinders cool. The city charges $i per thousand 
 cu. ft. of water. The barrels contain 50 gal. each, and I gal. of water weighs 
 8 Ibs. A cubic foot of water weighs 62% Ibs. The factory has recently 
 dug a well of its own. Estimate how much it saves by that means, sup- 
 posing its pumping to cost nothing. 
 
 9. One and one-fourth Ibs. of black ink will print 1000 eight-page 
 newspapers of ordinary size. How much would an eight-page weekly paper, 
 having a circulation of 12,000, consume per year? Find out the circulation 
 of your principal county paper, and compute the amount of ink it uses. 
 
 10. How much would be consumed by one issue of the New York 
 Morning Journal, a sixteen-page paper, having a circulation in 1898 of 
 309,472 ? One issue of the New York Sunday World, a fifty-six-page paper 
 with a circulation in 1898 of 500,000? (2 tons and 375 Ibs. for the World.) 
 
 11. The retail price of black ink varies from 4 to 10 cts. per Ib. At 
 6 cents per Ib., how much would one of the above named papers pay for 
 ink, per year? (World, $262.50 per Sunday.) 
 
 12. The black part of black ink is like lampblack, and is made from 
 gas. One pound of such black is worth 8 cents; and it takes 2000 cu. ft. 
 of gas to make it. Gas in New York City costs $i per thousand cu. ft. 
 Is this lampblack probably manufactured in the city? If not, can you 
 suggest where it might be made? 
 
 These examples, instead of being hypothetical, are actual 
 cases, and in that sense are concrete. Some of the answers 
 might well be worth remembering, like facts in history. Of 
 course, if pupils have not become interested in New York City 
 and its industries, this topic might be relatively unattractive to 
 them. But that is a danger encountered in all studies, and 
 suggests the desirable relationship in the grades between mathe- 
 matics and other branches. Teachers in composition and in 
 manual training try to select those topics to work upon, that 
 have already become of interest to a class. These are recognized 
 as branches of knowledge that are quite dependent upon others 
 for suggestions as to subject matter; and the recognition of this 
 dependence by no means detracts from their dignity. Indeed, 
 dignity is not concerned here; this arrangement merely insures
 
 8 Teachers College Record [98 
 
 a greater degree of interest. Since elementary arithmetic is only 
 the quantitative side of human experience, it is peculiarly de- 
 pendent upon other fields for the content of its problems, although 
 by no means dependent solely upon other school studies. The 
 more the problems deal with interesting topics, the better; hence, 
 correlation of the right kind is a matter of supreme importance 
 to mathematics in the grades. 
 
 Further than that, the correlation of mathematics is of very 
 great importance to other branches, and even necessary to the 
 effectiveness of the curriculum as a whole. For instance, geog- 
 raphy gives the causes for the location of factories in a certain 
 place; nature study shows how the earthworm grinds up soil; 
 and the United States history describes the campaigns of the 
 armies in Virginia during the Civil War, when it was necessary 
 to change the base of supplies. Now each of these topics has a 
 quantitative side, and the picture is very incomplete in each case 
 until that side has been presented. In fact the main part of the 
 definition of factory must be gotten through arithmetic, as shown 
 by the examples on ink manufacture, just given. The agricul- 
 tural value of earthworms begins to be appreciated only when 
 we measure the quantity of their castings on a given area; and 
 the difficulties attending an army's change of its base of supplies 
 are not clear until one knows how much bread, meat, flour, sugar 
 and water 100,000 men consume in a day ; and how many horses, 
 wagons and men it takes to haul that amount of supplies a given 
 distance in a given time. The single fact that it requires not 
 less than 40 bbls. of salt per day for such an army throws much 
 light on the matter. Figures give descriptions, just as words do, 
 and our appreciation of the quantitative side of life is lamentably 
 poor because we have made so little use of figures in school for 
 this purpose. Indeed, many of our most pressing problems are 
 largely quantitative, as for example, the maintenance of the Erie 
 Canal in the State of New York. Why should we not know 
 the relative cost and speed of transportation by rail and canal, 
 together with cost of maintenance, as well as the parts of speech 
 and the causes of our wars? And 'must we not possess such 
 quantitative knowledge before we can be intelligent as to public 
 policy in such matters? 
 
 This quantitative knowledge, too, calls for more than arith- 
 metic alone, even in childhood. It requires mensuration, some
 
 99] Controlling Ideas Throughout the Curriculum 9 
 
 geometry, and even a little algebra. Why should we try to 
 maintain the barriers between these fields when they are so 
 intimately related in daily life, and so simple in their elements? 
 Why should the mediaeval " Rule of False Position " still find 
 place in our arithmetics under the form : " Let 100 per cent 
 represent the cost " ? It is in every way simpler to adopt the 
 symbolism that for nearly three centuries has been conventional 
 in mathematics, and say : " Let x represent the cost." In either 
 case we employ a symbolism, but the second one is the simpler, 
 the more easily handled in the solution, and hence the more 
 widely accepted. Indeed, if we had adopted the symbolism of 
 the simple equation long ago, in our own elementary arithmetic, 
 topics like proportion and the applications of percentage would 
 have been much more clearly understood, and the subjects of 
 cube root and the progressions might have developed enough 
 of value to have made it worth while to retain them. The fact 
 is, there is no well defined line between our common arithmetic 
 and algebra, and the examiner in the former who appends to a 
 question the statement, " Do not solve by algebra," merely reveals 
 his own ignorance. He might insist that 2 : 5=7 : ( ?) is a 
 
 oc c 
 problem in arithmetic, and that -= is one in algebra, but a 
 
 little thought shows that they are identical. The fact is that, 
 strictly speaking, algebra is a science treating of a certain class j 
 of functions, and that arithmetic is a science treating of numerical 1 
 values. But arithmetic and algebra, as we commonly speak of 
 them, are sciences each of which considerably overlaps the other,] 
 and rightly so. 
 
 The same overlapping is apparent for arithmetic and geome- 
 try. Geometry is not merely a modern form of Euclid, although, 
 in England and America especially, we are wont to speak of it 
 as such. The necessary mensuration of common life is a part 
 of it, and as such is deserving of a much more scientific treat- 
 ment than it has already had. This topic, however, is reserved 
 for a subsequent article. 
 
 From these statements it follows that the subjects to be 
 studied in mathematics should be selected with the same care 
 as to age and present interests of the children concerned, as 
 are classic poems. Answers would then be worth something, and 
 children would not need to be blamed for " working for them "
 
 io Teachers College Record [100 
 
 when they have nothing else for which to work. Then mathe- 
 matics in the grades would be recognized as revealing one very 
 interesting and large side of life, as does any other study worthy 
 of pursuit. 
 
 And is this not natural? Children have a constant call for 
 number in their play. In fact, one reason advanced for having 
 no regular study called number in the first two years of school is 
 that they learn nearly as much of this outside of school as in it. 
 But it is not for the processes that they care ; they wish quantitative 
 facts, and the answer, if the problem is real, is after all the true 
 goal. It is so with adults also; they are wanting the answer 
 when they figure at all; it is this that furnishes the motive for 
 accuracy. Why, then, should children in school be confined to 
 / processes, when the purpose of children outside of school, and 
 of adults, is quite apart from processes? 
 
 The whole history of printed arithmetics shows how periodi- 
 cally has arisen the question as to the nature of the work to be 
 demanded of pupils. While there has been a constant evolution 
 towards a better treatment of the subject, a better understanding 
 of the processes, a clearer presentation of the operations, and a 
 recognition of the more valuable symbols of mathematics, there 
 has not been a corresponding evolution in the nature of the 
 problems. This has resulted from the ever present struggle 
 between culture and utility; between" mental discipline and the 
 problem of daily bread; between aristocracy and trade. This 
 struggle was seen in the earliest printed books, when the first 
 / arithmetics printed in Italy (Treviso, 1478) and in Germany 
 (Bamberg, 1482) took the utility side of the case; while the 
 great Italian treatise of Paciuoli (1494), and the first English 
 arithmetic (Bishop Tonstall's, 1522), took the culture side. But 
 this is a general historic truth: No arithmetic has flourished 
 among a democratic people, and but two or three have had any 
 great hold in schools that stood for pure culture, that did not 
 minimize processes and magnify the study of those problems 
 that revealed the actual quantitative side of the life of their day. 
 Thus it was that Kobel and Riese in the i6th century in Germany 
 had an enormous influence; that Trenchant and Savonne at the 
 close of the same century in France dominated the teaching in 
 that country; that Recorde did the same for England then, as 
 Cocker did later; and that, with the great awakening of com-
 
 IQI] Controlling Ideas Throughout the Curriculum n 
 
 merce in Holland in the i/th century, a flood of books of this 
 class appeared, revealing in a most interesting way the actual life 
 of the people of that period. 
 
 The problems of present arithmetic might well be of two 
 kinds : f ir^t, those dealing with the quantitative side of matters 1 / 
 of local interest; as the cost of blasting rock for a cellar in New 
 York City, the quantity and cost of a mile of asphalt pavement 
 in Buffalo, the quantity of water necessary for irrigating for a 
 season an acre of land in Colorado, and the cost and quantity 
 of materials necessary for fattening a herd of 25 cattle on an 
 Ohio farm ; second, those dealing with the quantitative side of . 
 matters of general interest ; as the Pennsylvania Railroad system, 
 an ocean steamship, the comparative cost of transportation of ore 
 from Lake Superior to Pittsburg by rail and by water, the 
 amount of freight carried on the Mississippi compared with that 
 carried by the Illinois Central Railroad which parallels it, a great 
 department store; the labor and money saved by the cotton gin 
 and by other inventions concerned in cotton production; the 
 sugar-cane and beet-sugar industries compared; and dairying 
 and ranching. Since nearly all of the arithmetical processes 
 are mastered by the end of the fifth year at school, the last 
 three years of mathematics in the grades might be spent al- 
 most entirely in a study of important industries and other > 
 matters on the quantitative side: as mining, banking, invest- 
 ments, manufacture of clothing, government revenues, commis- 
 sion business, gardening and farming, the cost of paving, of 
 water and of gas in different cities, the comparative cost of gas 
 and electric light, a comparison of the steel industry in this 
 country and in England, or a comparison of the growth of cer- 
 tain cities at home and abroad. 
 
 Such study would allow mathematics to culminate in two 
 classes of generalizations : ( I ) the rules of arithmetic to which 
 we are already accustomed, and (2) another large class of truths 
 dealing with the material side of human affairs. 
 
 Furthermore; consider how such arithmetic would meet the 
 " needs of society," since the subject-matter leads so directly 
 into practical life. The relation, too, may be one of sympathy 
 as well as of knowledge. A New York street car conductor very 
 often rides 80 miles per day. When we recall this fact, together 
 with the number of fares collected, the number of stops made,
 
 12 Teachers College Record [102 
 
 the number of persons helped on and off, and the number of 
 hours consumed by all this, his life touches ours as it never did 
 before; and this closer union means deeper sympathy. When, 
 through actual data, we learn that the owner of city houses to 
 rent must count upon their being idle from one-tenth to one- 
 quarter of the time, we understand better the reason for his high 
 prices. And when again, from actual facts, we know that even 
 a careful grocer in a city expects to have 15 per cent of his 
 fresh fruit spoil in hot weather, and to lose one-tenth of all the 
 money that he credits, we are at least able to understand, if not 
 always to condone, some of his actions. 
 
 The fact that the answers to many of these actual, living, 
 problems can be only approximately correct, shows the necessity 
 for a class of examples that do not " come out even," a class that 
 really meet the " needs of society." A large percentage of busi- 
 ness is of necessity merely a matter of estimates ; while even in 
 /the most careful scientific experiments, the data can never be 
 exact, and the results of computations cannot be closer to the 
 truth than are these data. 
 
 It is by such a plan as the preceding that mathematics in 
 . the grades might accept the " interest of the child " and the " re- 
 quirements of society " as its controlling ideas in the selection 
 I \ 'of subject-matter. And, under such control, the utility of the 
 knowledge and the value of the mental discipline, using these 
 terms in the common sense, would be enhanced rather than 
 diminished ; while much higher purposes in the study would also 
 be accomplished. 
 
 But the above plan is far from practicable when present 
 conditions are considered. In mathematics there must be more 
 dependence upon text-books than in most other studies, and our 
 text-books have been prepared along traditional lines. We can 
 omit obsolete matter, but satisfactorily to add such topics as 
 some of those mentioned, and to rearrange the whole on newer 
 lines, is impossible without much time and effort. 
 
 The difficulty is particularly noticeable in the sixth, seventh, 
 and eighth grades. The children there know the fundamental 
 operations with both integers and fractions, and should be ready 
 ' to devote all their time to applications needed in modern life, 
 and the facts that arise from such application. But the arith- 
 metics, instead, omit this latter side and often offer applications
 
 103] Controlling Ideas Throughout the Curriculum 13 
 
 of past generations. For instance, they elaborate proportion, 
 which is a mediaeval method of solving 1 simple equations, and 
 neglect the geometric application to measuring figures of the * 
 same shape (see p. 62), which is really about the only application 
 of importance outside of physics. 
 
 Under the present conditions, however, it is impossible to 
 do more than keep clearly in mind the plan which we should like 
 to follow, and to struggle toward it as best we can, here and 
 there, in practice. It should be remembered, also, that it is the) 
 purpose of this publication to suggest suitable controlling ideas 
 and a corresponding outline of subject matter, rather than to be 
 a text-book in itself. 
 
 In conclusion, it is the mission of mathematics in the grades 
 to make a large contribution to the knowledge of the children, 
 and to appeal to their interest by the richness of its content. 
 Therefore processes can be only a subordinate part of the subject- 
 matter. Deep interest, surprise, and excitement should be pro- 
 duced by the valuable thoughts in this subject as in others. 
 There is less danger in the criticism that an arithmetic is too 
 much like an encyclopaedia, than in the criticism that it is too 
 much like a dictionary.
 
 II. OUTLINE FOR THE FIRST FIVE GRADES 
 
 Grade I 
 
 I. General Suggestions 
 
 (a) In the spirit of the foregoing article, the purpose of 
 mathematics for six-year-old children is to meet from their 
 point of view, their daily need of number as it arises in their 
 school studies and in their relations outside of school. And lest 
 it may be felt that we are emphasizing the abstract, in speaking 
 of number rather J:han measurement, of which latter we have 
 recently been hearing so much, some explanation should be given. 
 To a child, as to us, to measure anything is to count the number 
 of times some arbitrary unit is contained in that thing. To find 
 a ratio of one thing to another is to find how many times the 
 one contains the other, or what part it is of the other; or, 
 what is the same thing, to find how many times each contains 
 some common unit. Hence the somewhat persistent argument 
 that we should teach a child measurement rather than number, 
 or ratio as the basis of number, is simply to say that we should 
 teach number with due regard to its applications. To this we have 
 already agreed; but to confine the early study of number to the 
 measurement of lengths alone, and ratio to the consideration of 
 an uninteresting series of blocks, is entirely foreign to our belief 
 as expressed in the preceding article as to what arithmetic, at 
 any stage, should be. Having now explained our understanding 
 of the relation of number to measurement, we return to the 
 work of the first grade. 
 
 Systematic instruction in this subject at this age is extremely 
 difficult, owing to the danger of its being too formal. On that 
 account, and also because of their peculiar need of other kinds 
 of work, the children of the Speyer School should probably 
 have no regular study of mathematics during the first school 
 year. On the contrary, it should be quite incidental. But, 
 
 14 [104
 
 105] Outline for the First Five Grades 15 
 
 as they are using a large number of materials that require some 
 kind of measurement, they will incidentally acquire much knowl- 
 edge in this field, provided the teacher is fairly attentive to the 
 quantitative side of their experience. 
 
 The children of the Horace Mann School, on the other hand, 
 begin the course with more mature minds, owing to their home 
 training. They have heard more of business in a large way, 
 they have come more closely in contact with nature, they have 
 traveled and have listened to lively table conversation, and they 
 can more safely run the danger of formal work. For them some 
 regular study in this line is not only safe but valuable. But 
 while the recitation period should be regular, the body of thought 
 offered should make no attempt to constitute a system, such, for 
 example, as that outlined by the " Grube Method." Such a plan 
 is the logical arrangement of the adult mind, and ignores the need 
 of motive on the part of the child. The two schools, therefore, 
 cannot cover the same ground either during the first school year, 
 or later. The following outline suggests the quality and arrange- 
 ment of work; each of the schools should do as much or as 
 little as is fitting. And what is true of these schools is true of 
 others. It is always dangerous to lay down a strict programme 
 for several schools ; some latitude is always necessary. Even 
 more dangerous is a series of books definitely limited as to grade 
 work. 
 
 (&) Desirable Materials. The materials needed are: foot 
 rules, yard sticks, i-lb. weights with balance scales for weighing, 
 toy or real money, pint and quart measures, building blocks of 
 definite dimensions, including many inch cubes ; Speer blocks, 
 splints, materials for number games, and cardboard for making 
 furniture to a scale. The use of fingers for counting should be 
 discouraged, for the reason that they cannot later be removed 
 entirely from reach, when not wanted. The same objection 
 applies to the habit of making marks. But splints and other 
 objects can be so removed and hence are unobjectionable. While 
 materials should vary, in order to hold the interest, and should 
 not lack esthetic value, they should not prove so attractive as 
 to draw attention away from the number work. Much of the 
 gaudily colored material often sold is objectionable on this 
 ground. 
 
 (c) Variety is itself worthy of being regarded as a principle
 
 1 6 Teachers College Record [106 
 
 of education, and since very young children quickly tire of a 
 single kind of work, it is important, where mathematics is a 
 regular study, frequently to vary the materials and the nature 
 of the problems. 
 
 (d) Normally in mathematics, written work should supple- 
 ment oral work, by taking up those problems that are too diffi- 
 cult for oral treatment. Written numbers should receive very 
 little attention in this grade, and should be undertaken mainly 
 for the sake of variety and for exercise in penmanship. 
 
 (e) Accuracy, within a fraction of the large units used, is 
 especially important at this age; it not only impresses the basal 
 units of measurement on the mind, but is necessary for a proper 
 appreciation of the spirit of this branch of knowledge. First 
 impressions are peculiarly important in the attainment of this 
 latter aim. Accuracy is much more important than speed, 
 although it should be remembered that accuracy within small 
 fractions of the units used finally becomes impossible, and the 
 attempt to reach it in measuring is injurious physically. 
 
 (f) A text-book is unnecessary at this age, although it 
 might secure additional variety in the Horace Mann School and 
 prove helpful in assigning very simple kinds of seat work. On 
 the whole, however, it is likely to increase the danger that the 
 work will become too formal. 
 
 (g) A reasonable effort should be made to develop the num- 
 ber sense of each pupil; but such effort should stop short of 
 persecution both in this and in higher grades, even though some 
 children are thought to show " no ability to learn mathematics." 
 Excellence in every study is not a sine qua non in the Elemen- 
 tary School. 
 
 (h) One of the principles of education especially applicable 
 at this stage, though especially neglected until recently, is that 
 of motor activity. Its application is called for both by the 
 physical activity of the child, and by the nature of the subject, 
 which requires actual measurement. 
 
 2. The Mathematical Work. 
 
 (a) The number-space is limited, extending from I to 100. 
 The chief interest which children have in arithmetic, on entering 
 school, is in counting, and this within the number-space above 
 indicated.
 
 107] Outline for the First Five Grades 17 
 
 (b) Counting: Since much interest lies in counting, as 
 seen in children's " counting out," " keeping score " in games, 
 counting by 5's and by ID'S, in being " It " in other games, and 
 in referring to pages of their books, special attention is given 
 to the number series in the space I to 100. The children are 
 encouraged to count rapidly by units, and by the groups 2, 5, 
 10. The material objects of counting should vary so as to show 
 a wide range of application. The emphasis laid upon counting 
 appeals not only to the interest of the child, but it was historically 
 the first stage in the development of the world's mathematics; 
 and it is in harmony with the latest stage, which looks upon 
 mathematics as the science of order, rather than the science 
 of quantity. 
 
 In connection with writing, the common notation to 20 
 should be taught, with incidental use of written numbers to 100. 
 On account of the clock-face, the Roman numerals to XII are 
 taught in the second half-year. 
 
 The number idea precedes the symbol in the space I to 10. 
 When the value of symbols is somewhat appreciated, the symbol 
 properly becomes the more important of the two. 
 
 (c) Measuring: All number is, of course, the result of 
 measure; and as already suggested, there is no dividing line 
 between counting and measuring. When the child counts the 
 number of inches in a foot, he has measured the foot. Hence 
 measuring appeals to his interests and needs at this time, and 
 should form a considerable part of the work. The measures 
 chiefly used in this grade, and with which the child should 
 become familiar by frequent actual use, are the following: 
 
 Length, the inch, foot, yard. 
 
 Capacity, the pint, quart, gallon ; the quart, peck, bushel. 
 
 Weight, the ounce, pound. 
 
 Time, the day, week. 
 
 The rod and mile are omitted, in this grade, because they 
 are not within the child's range of interest. Similarly the gill, 
 ton, month, year, second, minute (as 1 / 60 of an hour). But of 
 course no teacher should feel limited to the terms given; ex- 
 pressions like " 5 minutes," " 2 miles," and " 6 square inches," 
 may well be used whenever the need arises. Geographical con- 
 siderations determine many questions of this kind ; a child in 
 the country being much more apt to know the width of a street
 
 i8 Teachers College Record [108 
 
 in rods than a city child, and similarly such measures as the peck 
 and bushel. 
 
 The metric system is not introduced in the primary grades, 
 because it is not the one that children of this generation will 
 chiefly use. It is needed for general information later in the 
 course, but it would be a mistake to have it interfere with the 
 common system here. 
 
 (d) Fractions: As a result of attempts to measure, the 
 fraction appears. It is seen in paper-folding, in the separating 
 of groups of objects as half the class, a quarter of the blocks, 
 and in such other comparisons as the length of one stick compared 
 with that of another. All this involves the idea of ratio, a fun- 
 damental notion in dealing with number, but one which it is 
 neither necessary nor advisable to make very prominent, per se, 
 with children. 
 
 The fractions with which children should become familiar 
 in this grade are ^2, Yz, %, Y- This does not, however, exclude 
 the incidental use of such other fractions as may naturally enter 
 into the work of the class; for if the child knows thirds, he 
 knows Y$ an d Vs as we N as Y$' 
 
 (e) Operations : The only operation to which much atten- 
 tion need be given in this grade is addition, and this only as 
 it is necessary in such problems as " 5 inches and 4 inches are 
 how many inches?" or "2 ft., 3 ft., and 5 ft. are how many 
 feet ? " In general, such problems should be arranged in columns, 
 as is done in ordinary business life ; the equation form, " 5 in. 
 -f- 4 in. = 9 in.," is relatively of less importance and should come 
 in the second semester. It is a mistake, for various reasons, 
 to attempt to treat the four fundamental processes simultaneously. 
 They are not of equal difficulty ; the child does not need them to 
 an equal degree; and the world of business does not use them 
 with equal frequency. Hence addition, the easiest, the most im- 
 portant, and the most interesting to the child, occupies the chief 
 attention in this year. Incidentally, as needed in the simple 
 problems proposed, the ideas of subtraction, multiplication, and 
 division, are introduced ; but the work, even in addition, is limited 
 to the number-space I to 20, and no tables are learned. 
 
 So far as subtraction is treated, the " making change " 
 method should be used. For example, " If you have 10 cents, 
 and you buy a pencil costing 3 cents, how many cents have you
 
 109] Outline for the First Five Grades 19 
 
 left ? " The child should see that he has 7 cents left, because 
 3 cents + 7 cents =. 10 cents. This is the oral subtraction of 
 business life, and it is the basis for the " Austrian subtraction " 
 recommended in Grade III. 
 
 No attempt should be made in the first year to cover system- 
 atically all number relations within any set number-space. 
 
 (/) Symbols : In Grade I there should not be any system- 
 atic attempt to have the symbols -}-, , X, -j- used by the 
 children. They may be used by the teacher, and explained, and 
 made part of the lessons in writing, but it is not wise to give 
 any considerable number of written exercises of the form 2 + 3 
 4 + i =? Still more objectionable are forms like 2X3 + 2, 
 and, particularly, 2 + 2X3- 
 
 The reasons for opposing such examples are as follows: 
 
 (1) These chains of operations enter very little into real 
 mathematics. In practical life we never meet a problem like 
 2X3 + 4X6-^-3 2, nor do we find these symbols much 
 used in algebra or the higher mathematics. Hence they should 
 play but a very small part, if any, in the education of children. 
 
 (2) The teacher's personal judgment as to how a chain of 
 operations ought to be treated has no validity unless supported 
 by the conventions of mathematicians. For example, a teacher 
 might say that he thinks that children should be taught that 
 2 + 2 X 3 = 12, taking the operations in the order stated; but 
 the mathematical convention is that 2 + 2X3 = 8, the multi- 
 plication being performed first. If the child gets the wrong 
 idea now, it will trouble him throughout his subsequent mathe- 
 matical work. If such chains are to be given, they should be 
 in forms that admit of no misunderstanding, as 2 X 3 + 2. But 
 even these are open to such serious objection that they should 
 not be recommended. 
 
 The symbol X is preferably read " times " when the multi- 
 plier comes first and " multiplied by " when it comes second, as 
 
 (a) 2 X $3, " 2 times $3." 
 
 (&) $3X2, "$3 multiplied by 2." 
 
 This enables the sentence to be read from the left to the right. 
 The reading 
 
 (c) $3X2, "2 times $3," 
 
 has good authority, particularly in England and France, and 
 among older American writers, but it is coming into disfavor
 
 2O Teachers College Record [no 
 
 because it is not in accord with the genius of our language. 
 The reading first suggested (a) has the sanction of a rapidly 
 growing number of our best writers, and is thus becoming con- 
 ventional. There is the added reason for it that, in algebra, the 
 numerical multiplier is generally put first, as in ^ab. Practically, 
 however, the method of reading the X is not so serious as teach- 
 ers often suppose, because in business the X is used more com- 
 monly to mean " by," as in the expression 2 ft. X 5 ft., 3X9, 
 where it does not indicate multiplication. Furthermore, it is well 
 to know that in mathematics the symbol has lost all standing, 
 except in the logical statement of an arithmetical solution. In 
 algebra, and even in the theory of numbers, it has long since 
 been nearly forgotten, the dot or the absence of sign having 
 taken its place, as in 2-4, and ^ab. Indeed, symbols -)-, , X, 
 -4-, and =, were invented for algebra rather than for arithmetic, 
 and their chief value is still there, although the X and -r- have 
 been generally discarded by algebraists. It is only in the written 
 analysis of problems in the grammar school grades that arith- 
 metic has much use for them. 
 
 3. Problems Suggestive of the Type Desired 
 
 As previously stated, arithmetic is merely the quantitative 
 side of our experience, and those problems will be of most 
 interest that are drawn from fields that have already become 
 attractive. Hence they should be taken from other school studies, 
 and from experiences of daily life outside of school. Since the 
 curricula of the Horace Mann and the Speyer Schools are 
 radically different in some respects, some of the following prob- 
 lems that are suitable for the one, will not prove fitting for the 
 other school. 
 
 1. With toy dishes, or blocks to represent them, set a table for four 
 persons. How many plates? knives? forks? napkins? How many spoons, 
 with two for each person? How many of each are necessary for a family 
 of four persons? For your family? 
 
 2. In your reader find the following pages as they are called: 5, 7, 21. 
 
 3. Count by s-cent pieces ; by dimes. 
 
 4. Tell the length and breadth of each of these blocks (building 
 blocks). Name each by its measurement. For example, 2 X 4-inch 
 block, 8-inch block. Use these names always in building, and, in general, 
 call denominate numbers by their full names.
 
 in] Outline for the First Five Grades 21 
 
 5. Make a plan of a room, the length being three 8-inch blocks, and 
 the width two 8-inch blocks. Make other plans. Measure the dimensions 
 of the school room, of the yard. Compare the school room in size with the 
 home of the Pilgrims, and with an Eskimo hut. 
 
 6. Show a foot-measure. Draw lines, and point out objects, i ft. long. 
 Do the same with the yard measure. Estimate the length, breadth, and 
 height of objects, and then test by measurement. For example, "Mary, 
 estimate John's height; now measure to see how nearly right you were, 
 making your answer correct within one inch." " Give what you think to 
 be the height of this table, and measure as before." Give similar problems 
 for chairs and doors. Draw the plan of a washcloth of suitable size; of 
 an iron holder. Give their dimensions. 
 
 7. Find objects that you think weigh I Ib. See if they do. Weigh 
 various objects on scales. Likewise use other measures until the units 
 of measurement are quite familiar. 
 
 8. A good milch cow averages about 10 quarts of milk per day. 
 Show with the measures how much that would be. How many families 
 could she supply with i quart for each? With 2 quarts? 
 
 9. Name things that the grocer sells by the quart, the pound, the bushel. 
 Give such orders as your mother gives to grocers. Learn the prices, and 
 pay for some of these things in toy or real money. 
 
 10. Do the same in connection with a bakery shop. 
 
 11. Using sand to represent sugar and other commodities, weigh out 
 what different children call for. 
 
 12. Make measurements on paper for seed envelopes, boxes, and toy 
 furniture of a certain size. 
 
 13. On the sand table lay out a garden according to some scale ; 
 include the walks and a garden plot for each child. Measure the growth of 
 plants from time to time. 
 
 Problems in making a dirt digger 
 
 1. The stick is 10 inches long. If we use 2 inches for the point, how 
 much is left for the handle and blade? 
 
 2. Half of the remaining part is for the handle ; half for the blade. 
 What is the length of each? 
 
 3. The stick is ij4 inches wide. The handle is to be half of that 
 width. How wide is it? 
 
 4. How much must be cut off from each side, in order to leave the 
 handle in the middle? 
 
 Seed marker 
 
 Cut strips 6 X i X l /& inch. Draw a line across the stick i l /2 inches 
 from the end. Find the center of the end. Draw a line from this point 
 to each end of the cross line. Cut the stick. 
 
 Wool carders 
 
 i. Our board is 4 inches wide. How many rows of nails will it hold 
 */2 inch from the edge and Yz inch apart?
 
 22 Teachers College Record [112 
 
 2. How many nails are needed for each row, when the board is 4^ 
 inches long and the nails are to be l / 2 inch from the edge and l /2 inch 
 apart ? 
 
 Games 
 
 1. Draw on the floor or table three concentric circles of radii 3, 5, 
 8 inches, giving each circle a certain value. Now roll a marble so as to 
 stop it in one of these circles, and count the value given in favor of pupil.- 
 Let each child keep his own score. 
 
 2. Bean-bag game. Have a board with a hole in it, and three bags 
 of different sizes, each having a certain value. Throw these bags through 
 the hole. Let each child keep his own score. 
 
 3. Ring toss. Same in principle as the preceding. 
 
 4. An exercise that children enjoy very much is counting by I, 2, 5, 
 or 10, while certain children are doing a given task. For example, see if 
 we can count to 20 by 2's while the pencils are being collected. Teachers 
 find this to be a valuable drill. 
 
 5. Shuffle board. Make bags out of heavy ticking. Fill them with 
 sand, weighing respectively J4, i, i l /2, 2 Ib. Vary from time to time the 
 values placed upon bags. Have the captain keep the score. 
 
 Some of this work can best be done in other recitation 
 periods than those for arithmetic, even though a separate period 
 for arithmetic is planned. At such times it is important to keep 
 in mind, as a pedagogical principle, that any one recitation should 
 aim primarily to do work of one kind only, be it literature, learn- 
 ing to read, nature study, or something else; and that only as 
 much apparently foreign subject matter should be allowed as is 
 directly necessary for the accomplishment of this one aim. If 
 this rule is not followed, a recitation period may be occupied 
 with such a medley of matter that neither children nor teacher 
 can tell what has been accomplished. The arithmetical games 
 suggested should soon become part of the out-door play. 
 
 Grade II 
 
 i. General Suggestions 
 
 (a) Review the work of Grade I in the same spirit in which 
 it is there taken. In general, the first three or four weeks of 
 each year should be devoted to a review of the work of the pre- 
 ceding year. When this is done, the common complaint of the 
 ignorance of children of the preceding work may cease. 
 
 It should also be remembered that the review should be a
 
 113] Outline for the First Five Grades 23 
 
 prominent factor in every lesson, occupying perhaps half the 
 time in the primary grades. Drills, of course, are included under 
 such reviews. Although much advance has been made in edu- 
 cation, drill in the fundamental operations, even to the point of 
 automatic work, has not yet been found unnecessary. In the 
 higher grades, it is less necessary to plan so directly for reviews, 
 because they are incidental to the solution of the problems. 
 
 Whatever part of the work outlined for Grade I has not 
 been covered by the beginning of the second grade in the Speyer 
 School will, of course, be taken up at the beginning of that year. 
 In general, there is likely to be from six months to one full 
 year's difference between the pupils of the two schools in mathe- 
 matical ability and knowledge. This outline is planned primarily 
 with reference to the Horace Mann School, when it comes to 
 division of work into certain grades. 
 
 (&) In this grade the work in arithmetic comes to be treated 
 more as a system, the child's interest having now been aroused 
 to such a point as to permit it. This means that numerical tables 
 are learned as such, and a more or less definite amount of work 
 is laid out in the domain of pure number. As in other studies, 
 however, this abstract part should follow a large amount of 
 concrete matter. 
 
 (c) There is no radical departure, however, from the kind 
 of work done in Grade I. There should be the same close rela- 
 tion to manual training, to other studies, and to the problems of 
 daily life. The children should be encouraged to invent problems ; 
 the quantities of food, fuel, and clothing used, with cost, forming 
 a basis for much of the work. 
 
 (d) Materials as before. Also cards with dots and figures. 
 Dominoes and other games, including the number games, of 
 the Cincinnati Game Company. Objective work in establishing 
 number relations is still necessary. There is, for example, an 
 advantage in counting by lo's, and in seeing bundles of splints. 
 But there is a danger in carrying the objective work beyond the 
 needs of the children, as there is in abolishing it too early. A 
 support is necessary at first to a child learning to walk ; it ceases 
 to be necessary later; to tell when it ceases to be necessary and 
 begins to be harmful, demands careful thought for each indi- 
 vidual, and so for objective teaching. But the children them- 
 selves may be relied upon for some aid in this matter, if a pride
 
 24 Teachers College Record [114 
 
 has been developed in them to abandon the use of objects as 
 soon as possible. 
 
 (e) If the oral work is rapid and accurate, the written work 
 will be so. But in quantity the oral work should greatly pre- 
 dominate. There is a temptation to have too much written arith- 
 metic at this stage, simply because it is easily assigned for seat 
 work. The great danger is that such young children, thus left 
 alone at their seats, will drop into careless habits involving divi- 
 sion of attention. No text-book is advisable, at least in the first 
 part of this year, for reasons previously given. 
 
 (/) In the written work the same attention should be given 
 to neatness and accuracy that is required in other written exer- 
 cises. Indeed, these qualities are particularly important objects 
 of written arithmetical work in this grade. 
 
 The equation form may now be introduced ; as 24-5=?, 
 2+? = 7, ? + 8 = 8, 2X? = 6, 6 + ? = 2; or with n (for 
 number) in place of the " ?." But it should be remembered that 
 these are not the forms of practical life, and more time should 
 be directed to the forms more common; namely, the work in 
 columns. Such symbols of operation are necessary later, when 
 the child conies to the explanation of problems of some length; 
 in this grade they enter only incidentally. If the teacher gives 
 chains of operations like 2 + 3 1+2, or 2X3 4, mathe- 
 matical conventionalities must be observed ; e.g., 2 -j- 3 X 4 = 
 14, not 20, and hence it would be better to write this 3X4 + 2, 
 so that the operations could be taken in their order. (See p. 19.) 
 Written work of this kind is not, however, very valuable; the 
 oral treatment is better. 
 
 (g) Bibliography : Smith : Teaching of Elementary Math- 
 ematics, Chaps, i-v ; McClellan and Dewey : Psychology of Number, 
 Chaps, i-ix; Phillips, in the Pedagogical Seminary, Oct., 1897. 
 
 2. The Mathematical Work 
 
 (a) The number-space, i to 10,000 for reading (as in larger 
 street numbers), I to 1,000 for counting and writing, I to 100 
 for operations. The Roman numerals may be limited, as in 
 Grade I, to the space I to XII, this being sufficient for the present 
 uses of the children; although many city courses carry this 
 system to C.
 
 115] Outline for the First Five Grades 25 
 
 (&) Counting: The counting of Grade I is carried for- 
 ward. Counting by twos, from 2 to 20 ; by threes, from 3 to 30 ; 
 by fours, from 4 to 40 ; the object being to lay the foundation for 
 the multiplication table to 4 X 10. This counting will naturally 
 and incidentally extend farther as an interesting exercise. 
 
 Also counting by twos from i to 9, by threes from I and 2 
 to 10, and by fours from i, 2, 3, to n ; the object being to form 
 the tables of addition. This work can easily be made an interest- 
 ing exercise in games, and rapid oral work will have an interest 
 per se. But for interest, much dependence should be placed on 
 the proper kind of problems. The meaning of " dozen." 
 
 (c) Tables: When the counting suggested in (&) has 
 been carried out, the children know the table of addition, and 
 also the table of multiplication through 4 X 10. The latter 
 should, however, be learned and drilled upon as such. Also the 
 table of ID'S and 5's, the former to 100, the latter to 50. The 
 subtraction table need not be developed and learned, since sub- 
 traction is made to depend upon addition as stated below. 
 
 (d) Measuring: The measures chiefly used in this grade 
 are those used in Grade I, with a few additions. They are: 
 
 Length, the inch, foot, yard. 
 Capacity, the gill, pint, quart, gallon ; 
 
 the quart, peck, bushel; 
 
 the cubic inch. 
 
 Weight, the ounce, pound. 
 Time, the day, week, month; 
 
 minutes in hour, hours in day, days in week. 
 Surface, the square inch, square foot. 
 Here, as in all grades, the children should become thoroughly 
 familiar with the measures by actual use. 
 
 (e) Operations: Addition of several one-figure num- 2 
 bers ; or of two- figure numbers, neither column at first ex- 3 
 ceeding 9. Accustom the children to say, " 5, 14, 21, 24, 26," 7 
 from below; and also, "2, 5, 12, 21, 26," from above. In 9 
 general the child should read a column as he reads a sentence, 5 
 never counting by units any more than he would spell by 
 letters. 26 
 
 Subtraction has from the earliest times been taught in 
 various ways. Some methods of treatment are more rapidly per- 
 formed, others are more easily explained. Since, however, the
 
 26 Teachers College Record [116 
 
 child is not expected to learn or to repeat any but very simple 
 explanations, and since the purpose of an explanation is to justify 
 a process that soon should be merely mechanical, it is reasonable 
 to expect that the shortest operation is the one that, in the long 
 run, will be accepted by the world. This is seen in many of the 
 other operations. For example, in the division of fractions, the 
 world now inverts the divisor and multiplies, although for 
 generations it divided by reducing the fractions to a common 
 denominator. 
 
 Multiplication is in this year limited to one-figure factors, 
 and to the domain I X i to 4 X 10. 
 
 Division is limited chiefly to " exact division," not involving 
 mental " carrying." It is introduced only as needed, in simple and 
 natural problems, as in taking y 2 , Y$, etc. 
 
 The terms addend, sum, subtrahend, minuend, difference, 
 multiplier, are used as necessary, but no formal definitions are 
 given. 
 
 Fractions: Halves, thirds, fourths, fifths, sixths, eighths, 
 tenths. The seventh and ninth are omitted, or are met only 
 incidentally, there being almost no use for them in the measure- 
 ments and the problems of this grade. In general, no inexact 
 fractional parts, like y$ of 7; such cases may, however, enter 
 into certain impromptu problems. 
 
 3- Suggestive Problems 
 
 i. Practice stepping off 2 feet at each step. Then, by stepping, find 
 the width of the street in front of your school-building. What is the 
 width of the widest streets near you ? What seems to be the average width 
 of streets in this city? In the same way find the width and depth of 
 near-by lots. Show the size of an ordinary yard in a village. 
 
 2. How many children can be accommodated at the blackboard in 
 your school-room at one time, counting 2^2 feet for each child ? 
 
 3. Measure your school-room. Make a floor plan of it, with blocks, 
 according to some scale. Likewise measure and make a floor plan of some 
 grocery store you have visited; also of a bakery. 
 
 4. Name the months in each of the four seasons. Each season is 
 what part of one year? 
 
 5. Measure the growth of plants, twigs, etc., correct to within ^ inch. 
 
 6. Make a floor plan of the apartment, or of one of the floors of the 
 house, in which you live. Use blocks or make a drawing to scale. 
 
 7. How many chairs are needed for a six-room apartment containing 
 4 persons? Explain your answer.
 
 117] Outline for the First Five Grades 27 
 
 8. Give the size of a fair-sized napkin. Of a towel. Show by draw- 
 ings. How many towels are needed to supply a family of four? Explain 
 your answer. 
 
 9. Read the thermometer from time to time and explain what it 
 signifies. 
 
 10. Count the number of grains on an ear of corn. One grain of corn 
 often produces a stalk with three such ears. 
 
 11. A family in this city, burning coal in the range, often spends 
 25 cents per week for kindling, receiving 3 bundles of sticks for 5 cents. 
 What is the cost per month? How many bundles are burned per week? 
 
 12. It takes zy 2 gallons of milk to make I Ib. of butter. One gallon 
 weighs 9 Ibs. How many pounds of milk make i Ib. of butter? Milk 
 costs us about 7 cents per quart. What is the worth of the milk necessary 
 for I Ib. of butter? What is the cost of one Ib. of butter? How does this 
 compare with the cost of the milk? Why this difference? 
 
 13. Give the dimensions of a pencil-box of suitable size for use. 
 
 14. Games, including dominoes, and the number games of the Cin- 
 cinnati Game Company. 
 
 Grade III 
 
 i. General Suggestions 
 
 (a) Review the work of Grade II, devoting three or four 
 weeks to it. See General Suggestions (a) under Grades I and II. 
 
 (b) The text-book is necessary in this grade. Some reasons 
 for the delay have already been given. (See page 16.) The use 
 of the text-book is that it may serve : ( i ) as a reference book 
 for facts, like a dictionary, as in the case of tables; (2) as a 
 collection of problems to save the time of dictation. The prob- 
 lems often fail to appeal to interest and must be supplemented 
 or omitted, but the choice of evils leads to using a book from 
 now on. There is a serious danger, however, in introducing a 
 book, of departing too radically from the child's daily interests. 
 
 (c) The work in this grade becomes still more systematic. 
 The object of the year, so far as operations are concerned, is to 
 become familiar with " the four processes " in the number-space 
 i to i ,000, division being limited as set forth below. The object, 
 so far as rich thought-content is concerned, is to acquaint the 
 pupils with numerous separate facts of interest, and thus to reveal 
 to them the quantitative side of a few large topics. 
 
 (d) Objective work becomes less and less needed in number 
 relations of integers. But it is just as necessary as ever in
 
 28 Teachers College Record [118 
 
 presenting radically new subjects, as in reducing fractions, in 
 studying new forms, and in taking up new measures. 
 
 (e) Written work is still subordinate, being confined largely 
 to computations too extensive for purely mental treatment. A 
 large amount of rapid oral analysis should be given in very 
 abbreviated form. Elaborate analysis should be avoided both 
 by the teacher and the pupil. 
 
 (/) There is a great advantage in the occasional use of oral 
 analysis, giving the main steps without performing the operations. 
 For example: If a book costs $2, how much will 36 such books 
 cost ? Answer : " 36 books will cost 36 times $2." 
 
 Since children often fail to get the exact condition of a 
 problem, it is well to have them restate it, in their own words, 
 if it admits of such a change. It is well also to have them state 
 the number of steps involved in the solution, showing very briefly 
 what they are, and giving the approximate answer. The actual 
 solution may then follow, or may be omitted entirely for the 
 time. Such work forces children into the " thought-side " of 
 arithmetic, and away from the impression that it is all " mere 
 figuring" ; in other words, it overcomes the tendency to become too 
 formal. The oral solution of problems with only approximate 
 answers may well increase in prominence as the work advances. 
 One argument in its support is the fact that many of the problems 
 of adults are actually solved only approximately; also, a very 
 large percentage of adults use the approximate answer as a guide 
 to correctness in the actual calculation. 
 
 (g) Geographical considerations should influence the subject 
 here as in the other grades. (See Grade I, 2, (c).) For 
 example, such words as penny, bit, shilling, nickel, are common 
 as names of coins, in some parts of the country, but not in others. 
 
 (h) It is important to consider how far induction is legiti- 
 mate in elementary arithmetic. The world induced before it 
 deduced; and the child does the same. Even in the highest 
 mathematics induction plays a great part, in discovery; then 
 deduction comes to play its part in proving that the induction is 
 correct. So in the primary grades induction is not only allow- 
 able, but it is the natural method. The child comes, however, 
 in the later grades, to the period of more or less rigid deduction, 
 because he is largely applying principles already familiar. Then, 
 however, another kind of induction should be going on, since
 
 119] Outline for the First Five Grades 29 
 
 (see page 50) he should be reaching new generalizations about 
 industries and other phases of our quantitative experience. Truly 
 inductive work in arithmetic requires that the rule be put in the 
 background at first; concrete problems should first occupy the 
 attention, and only after several of these have been solved, and 
 the methods compared, should the rule itself be broached and 
 worded. Even then there is a great danger in making altogether 
 too much of memorizing rules, particularly those of operation. 
 Teachers should, however, distinguish between a rule that is 
 originated by the pupil, and one which is dogmatically given to 
 him. The former has high value; the latter is dangerous. 
 
 For example, take this problem: Your garden is 24 yards 
 long and 3 yards wide. How many feet of wire will it take to 
 enclose it? A rectangle represents the plot of ground, and we 
 have this column for addition: 24 
 
 24 
 
 3 
 
 3 
 
 Different devices may be employed to explain the process, 
 as shown on page 31. 
 
 Another example might be : A soldier on a long march must 
 carry a knapsack weighing 2 Ibs; food in it weighing 4 Ibs; 
 a gun weighing 9 Ibs ; and a blanket weighing 14 Ibs. How 
 much must he carry in all? 
 
 Similar problems should be solved before an attempt is made, 
 if ever, to phrase any rule of addition. 
 
 The following might be suitable problems in subtraction : 
 (i) A good carriage horse often weighs 1,025 Ibs. A good 
 dray horse often weighs 1,850 Ibs. What is the number of 
 pounds of difference? Why need there be such a difference 
 in weight? (2) In May, 1902, anthracite coal sold at $5.25 
 per ton; in August, owing to the strike, at $10.00 per ton. 
 How much advance was made in price? (3) For children who 
 have been studying history: How many years is it since Henry 
 Hudson sailed up the Hudson River? 
 
 In truly inductive work there is no feeling of hurry to reach 
 the rule ; the latter is gradually brought to light through numerous 
 positively concrete examples. 
 
 (i) Types : The question arises, How should the rule be
 
 3O Teachers College Record [120 
 
 recalled by the child ? Shall he learn the teacher's rule verbatim ? 
 Or shall he memorize a wording of his own? Or shall he recall 
 some practical problem, and reproduce the process there fol- 
 lowed? Many teachers prefer the last plan. For example, the 
 garden mentioned on page 29 is 24 yards long and 3 yards wide. 
 
 20+4 24 24 
 
 20 -j- 4 24 24 
 
 3 or 3 or 3 
 
 > 3 33 
 
 40 + 14 = 54 14 54 
 40 
 
 54 
 In subtraction, taking the example about the horses, we have: 
 
 1,850 1,850 = i, 800 -(-40+10 
 less 1,025 1,025=1,000+20+ 5 
 
 800 + 20+ 5 = 
 
 For a discussion of the various methods of solution in sub- 
 traction see page 31. Whatever the method followed, the par- 
 ticular example is held in memory for some time as an aid to 
 the solution of new problems. Being concrete, it can easily be 
 remembered; and since it must be recalled many times for help, 
 the rule itself gradually comes to light and may finally be care- 
 fully worded, preferably in the pupil's own way. This plan 
 avoids the premature learning of the generalization. 
 
 (/) Bibliography : Smith : Teaching of Elementary Mathe- 
 matics; and McClellan and Dewey: Psychology of Number, as 
 before ; McMurry : Method of Recitation, Chap. 2, p. 14. 
 
 2. The Mathematical Work 
 
 (a) The number-space: i to 1,000,000 for reading and 
 writing, i to 1,000 for operations. The Roman numerals I to D, 
 but not to work out every numeral, the reading of chapter num- 
 bers being the important feature. 
 
 (&) Counting: The counting of Grades I and II is con- 
 tinued sufficiently to complete the development of the multiplica- 
 tion table to 10 X 10, which is as far as it need be carried.
 
 I2i ] Outline for the First Five Grades 31 
 
 (c) Tables of abstract number: The multiplication table 
 as above, thoroughly learned and made the object of continued 
 and rapid oral drill. The addition table reviewed daily, and also 
 made the object of continued oral drill. 
 
 (d) Measuring: In addition to the measures introduced 
 in Grades I and II, the following are studied : 
 
 Length, the rod and mile. Number of New York blocks 
 to the mile, north and south, also east and west. 
 Weight, the ton. 
 Time, months in year ; 
 
 days in months, " Thirty days hath September," 
 
 etc.; 
 
 days in year; 
 
 weeks (approximately) in the month and year. 
 Meaning of " quire " and " score." 
 
 Particular attention to the table of United States money, but 
 always omitting the " eagle," and omitting the " mill " in the 
 primary grades. 
 
 (e) Operations: the four fundamental processes in the 
 number-space I to 1,000, division being limited to divisors of 
 two figures. 
 
 In addition, teachers are not limited to any one device. In 
 explaining a process involving some mental retention of number, 
 as in " carrying," it may be necessary to consider sticks in bundles 
 of 10, as is often done, and to adopt also such devices as 
 
 26 = 20 + 6 26 
 
 39 = 30+9 39 
 
 50+15-65 15 
 50 
 
 65 
 
 in order to bring out the idea. Since no explanation should be 
 required from the child at this stage, further than the answer 
 to simple questions, all that is necessary is that he should (i) 
 know how to perform the operation, and (2) feel that he has 
 taken nothing on mere authority, beyond the conventionalities of 
 the subject. 
 
 In subtraction, of the various methods suggested, the one 
 that is now having the greatest favor is the so-called " Austrian
 
 32 Teachers College Record [122 
 
 method," which has the advantages of brevity and of requiring 
 no subtraction table to be memorized. The general steps neces- 
 sary to its development are : 
 
 ( i ) Meaning of difference, that number which added to the 
 smaller number produces the larger, an idea sufficiently com- 
 plete for children. For example ; 9 5 = 4 because 5 -f- 4 = 9. 
 By recalling the addition table, such differences are at once 49 
 brought to mind. Apply to cases like 49 25 : 25 
 
 Any subtraction of two-place numbers where no figure 
 in the subtrahend signifies a larger amount than the corre- 24 
 sponding figure in the minuend, follows this plan. 
 
 (2) Adding 10, 100, or 1,000 to each number does not 
 change the difference. Inductively shown from 
 
 9 19 29 
 
 5 15 25 
 
 444 
 
 (3) Consider the case of 52. 
 
 27 
 
 . 
 Excluding the advanced idea of negative numbers from this 
 
 discussion, we may say, there is no number which added . to 7 
 makes 2, but 5 added to 7 makes 12. We have now increased 
 the 52 by 10, and we must add 10 to 27 so as not to change the 
 difference. 3 (tens) and 2 (tens) are 5 (tens). Hence the dif- 
 ference is 25. 
 
 Like all innovations in our own long established mental pro- 
 cesses, this may seem difficult; but looked at from the stand- 
 point of the child it saves memorizing many facts, and it reduces 
 subtraction to the much simpler operation of addition. The ob- 
 jection that it does not give the notion of " taking away," is 
 hardly worth notice; it gives the result with a minimum of 
 mental friction, and this is the practical end in view. The details 
 of presentation are, of course, capable of great variation, in- 
 volving the use of materials, and such devices as : 
 
 52 == 50 -\- 2 the difference being the same 50+12 
 27 20 -f- 7 as between 30 -f- 7 
 
 20+5
 
 123] 
 
 Outline for the First Five Grades 
 
 33 
 
 In short division, it may frequently be necessary to use the 
 " long division " form, showing that the former is only an abbre- 
 viation of the latter. And, in general, the abridgment naturally 
 follows the more complete form in all operations. As in addi- 
 tion, an introduction to " carrying " often requires such a device 
 as this : 
 
 27 
 
 36 
 
 13 
 50 
 
 63 
 
 and as in subtraction we have to resort to complete forms like this : 
 The difference between 52 and 27 equals that of 
 
 40 -f- 12 or 50 -f- 12 
 and 20+7 and 30 -f- 7 
 
 20+5 
 so in multiplication we lead to 
 
 12 12 
 
 7 through 7 
 
 84 
 
 and in division we lead to 
 5)134 
 
 14 
 70 
 
 84 
 
 through 
 
 26, and 4 remainder, 
 26 
 
 5)134 
 100 
 
 34 
 30 
 
 4 remainder, 
 or through 5 1 100 4-30 
 
 "- * 
 
 I \J~ I 
 
 20 -j- 6, and 4 remainder.
 
 34 Teachers College Record [124 
 
 No definite rule can be laid down for these cases. The 
 teacher should use the more complete form whenever necessary 
 for the class in hand, and come to the abridged form as soon 
 as the pupils are ready for it. 
 
 We shall find that it is generally better in " long division " 
 to put the quotient above the dividend. (See page 39.) In " short 
 division " this could also be done were it worth while to change 
 the well established custom of writing it underneath. 
 
 (/) Geometric forms. The square and rectangle, as met 
 in manual training, basketry, paper-work, and elsewhere, are 
 studied with reference to perimeter and area. The cubic inch 
 and cubic foot. Study of the circle, including the diameter and 
 the radius, but without mensuration. The details of the work in 
 geometry are reserved for a subsequent article. 
 
 (g) Fractions and ratios are incidentally studied, carrying on 
 the work of Grade II. There is no formal treatment of the sub- 
 ject. Such simple reductions as f = , f i> an d such operations 
 as + i, \ of , \ \ are made part of the oral work. 
 
 The decimal form is used in writing United States money, 
 but without any explanation of the decimal system. For the sake 
 of practice, papers may be marked in per cent, and the term should 
 be explained and used in other work. 
 
 3. Suggestive Problems 
 
 1. One gas jet burns about 5 cubic feet of gas per hour. How much 
 gas is consumed per week in a house that burns 3 jets each evening from 
 7 to 10.30 o'clock? Estimate the amount that a street lamp burns in one 
 month. 
 
 2. Gas costs $i per thousand cubic feet. Estimate the cost of the 
 different amounts of gas above mentioned. Estimate the cost of gas for 
 street lights for a distance of one-quarter of a mile on one street. 
 
 3. A faucet actually leaked in a New York house at the rate of i pint 
 of water every thirty seconds. How much would that amount to in one day 
 of twenty- four hours? In one month? The water was allowed to run 
 six months; how much was the waste? 
 
 4. Make out an actual bill for ice for your family for one month. 
 
 5. Live cattle sell at 7 l / 2 cents per Ib. Good beef-steak now costs 
 28 cents per Ib. What is the difference? Why is this difference? 
 
 6. Make yourself acquainted with some of the more common fish, 
 and the average weight of each. Tell what one fish of each kind will cost 
 at the customary price per pound.
 
 125] Outline for the First Five Grades 35 
 
 7. Estimate the number of persons living in some apartment house 
 near you, and explain how you do it. 
 
 8. Tell time by the clock, to minutes and fractions of a minute. 
 
 9. What distance do you walk each day in going to and from school 
 once? Show how you find it. 
 
 10. In order to run a single ordinary rock drill, used for blasting, one 
 engineer and two men are necessary. The leading man with the drill often 
 receives $2.75 per day; his assistant, $1.50; the engineer, $3. One drill will 
 keep a blacksmith busy about one-fourth of the time, sharpening the drills ; 
 he receives $3 per day. How much does the labor cost to run a single drill 
 one day? 
 
 11. A business man often sends 18 letters per day, 6 days in the 
 week, each requiring a 2-cent stamp. At that rate, what is his postage bill 
 per month of four weeks? 
 
 12. Talk with a gardener to find out how large a lettuce bed it would 
 take to supply all the lettuce your family would need for a season. Show 
 how you estimate it. 
 
 13. Young chickens are now worth 20 cents per Ib. What is the cost 
 of one weighing 2^2 Ibs. ? Cost per dozen? 
 
 14. The chicken industry : 
 
 (a) Of the eggs that a farmer "sets" for raising chickens, seldom 
 more than 4 out of 5 hatch out. If he sets 600, how many chickens does 
 he get? 
 
 (fe) Of those that are hatched out, about i out of 3 is drowned, killed 
 by wild animals, or sickens and dies, before the age of 3 months is reached. 
 How many grow up from the 600 eggs? 
 
 It would be well to continue a story like this, referring to the number 
 eaten by a family, the number sold, the weight and price per pound, and 
 the amount and cost of feed. Similar series of problems in raising a herd 
 of sheep. Estimate the cost of feeding a horse in this city. 
 
 15. A motorman on a New York trolley car very frequently runs his 
 car 80 miles per day. How many times does he stop if he averages one 
 stop every third block, counting 20 blocks to one mile? 
 
 16. The conductor often collects 75 fares on one trip " up town," 
 covering about n miles. How many dollars would this amount to, count- 
 ing fares at 5 cents each? 
 
 Grade IV 
 
 i. General Suggestions 
 
 (a) Review the work of Grade III, devoting three or four 
 weeks to it. See General Suggestion (a) under Grades I and II. 
 
 (&) On the use of the text-book, see General Suggestion (&) 
 under Grade III.
 
 36 Teachers College Record [126 
 
 (c) Touching processes, the work of the year is chiefly given 
 to the ability to handle the four operations in general problems 
 and to the addition and subtraction of common fractions, the essen- 
 tially new work being the completion of " long division." As to 
 quantitative facts of value, it is now possible to present the prin- 
 cipal sides of many leading industries, and thus to build up as 
 full and vivid a picture of many matters as is done in other 
 studies, though always through figures rather than through 
 words. 
 
 (d) This is the grade in which children often begin to lose 
 interest in the subject, and well they may if processes only are 
 depended upon to arouse life. It is the teacher's duty to plan much 
 variety in method, and to make fair use of competition ; but more 
 than this should be done. Really valuable subjects should be se- 
 lected; interest from any other source is likely to be largely arti- 
 ficial. The text-book, of course, contains but few such topics, 
 and the teacher will simply have to do the best she can, under 
 present circumstances, to find them herself. 
 
 (e) On the proper use of diagrams. These can be exten- 
 sively used to advantage, and should be drawn and studied before 
 the computation begins ; otherwise, the child wanders about in the 
 dark and gets no good from them. For example : In finding the 
 area of a floor, the diagram, if needed at all, is needed before the 
 computation. In general, the children should become accustomed 
 to thinking out the main steps to be taken and the approximate 
 answer, before beginning the computation. 
 
 (/) For further suggestions, see Grades I, II, III. 
 (g) Bibliography : Smith, McClelland and Dewey, and 
 McMurry, as noted in the preceding grades. 
 
 2. The Mathematical Work 
 
 (a) The number space. No limits need be observed hereafter, 
 except that number names beyond billions are of little value. 
 Roman numerals up to MM being sufficient for dates, form the 
 limit for this system in the grades. 
 
 (&) Tables. The addition and multiplication tables have 
 already been learned, and are to be made the subject of constant 
 review. 
 
 The following table of aliquot parts should become known
 
 127] Outline for the First Five Grades 37 
 
 this year, in anticipation of the work in decimal fractions for 
 next year: 
 
 y 2 = 0.50 = $0% 
 
 y 4 = 0.25 = 25% 
 
 = 0.2O = 20% 
 
 = O.IO== 10% 
 
 Such facts, like the rules of arithmetic, should be reached 
 through study of concrete examples. The other plan, deductive 
 as it is called, though not truly deductive, is to study each fact 
 and the rules first ; and, after having learned them, to apply them 
 to examples. 
 
 (c) Measures. This year should see the tables of denominate 
 numbers systematized, and the great basal units of arithmetic 
 emphasized. 
 
 The following tables are given as the ones to be memorized, 
 the chief units being italicized : 
 
 Length. 12 in. (12") = i //. (:') 
 
 3 ft. = i yd. 
 
 S l /2 yds. or i6 l / 2 ft. = i rd. 
 320 rds. or 5280 ft. = i mi. 
 
 The surveyor's table is omitted, as being part of the technical 
 education of a surveyor, and unnecessary in common life. 
 Surface. 144 sq. in. = i sq. ft. 
 
 9 sq. ft. = i sq. yd. 
 30^4 sq. yds. = i sq. rd. 
 1 60 sq. rds. = i acre 
 640 acres = i sq. mi. 
 
 In certain communities the study of township sections would 
 have place. 
 Capacity. 
 
 Liquid. Dry. Cubic. 
 
 4 gi. = i pt. 2 pts. = i qt. 1728 en. in. = i cu. ft. 
 
 2 pts. = i qt. 8 qts. = i pk. 27 cu. ft. = i cu. yd. 
 
 4 qts. = i gal. 4 pks. = i bu. A "cord of wood" ex- 
 
 plained. 
 Weight. 1 6 oz. = i Ib. 
 
 2000 Ibs. = i ton
 
 38 Teachers College Record [128 
 
 This table of avoirdupois weight is the one which the children 
 will practically need. The long ton should be mentioned, but troy 
 weight and apothecary's weight are technicalities of trade and are 
 not needed in common life. 
 
 The child's original motive for studying these facts should be 
 found largely in the needs arising in manual training and other 
 studies, and in home experience; but the teacher should now col- 
 lect the various facts into " tables," and drill upon them until they 
 are properly fixed. 
 
 (d) Compound numbers are involved in the measures men- 
 tioned. In this grade the operations with them are confined chiefly 
 to such simple reductions as are necessary in the applied problems 
 mentioned ; namely, feet to inches, pounds to ounces, days to weeks, 
 and similar cases. 
 
 (e) Operations with integers. The essentially new feature 
 of the year is the completion of " long division." The early printed 
 arithmetics gave several methods of division, and the one now 
 commonly used was finally adopted nearly two centuries after the 
 beginning of printing, showing that it was by no means universally 
 accepted as the best. Now a slight improvement on the common 
 method has been suggested, and is rapidly meeting with favor. 
 It is one of the forms often called by the name of " Austrian 
 Method." The successive steps for explaining " long division " 
 vary with the class. In a general way, they cover the following 
 ground : 
 
 Write Think 
 
 78 
 
 3)234 
 
 210 
 
 24 3 210 + 24 
 
 24 70+8 
 
 The quotient of 200 -r- 3 is not any number of loo's. The 
 quotient of 23 tens -r- 3 is 7 tens. And since the product of 3 and 
 7 tens is 21 tens, there remains 24 to be divided. 24 -=- 3 = 8. 
 And since the product of 3 and 8 is 24, there remains nothing to 
 be divided.
 
 129] 
 
 Outline for the First Five Grades 
 
 39 
 
 Similarly for 625 -=- 25 : 
 Write 
 
 25 
 
 25)625 
 500 
 
 125 
 
 125 
 
 Think 
 
 25 [500+ 125 
 20+ 5 
 
 After this is understood, the abridgment as shown in the 
 common " Austrian Method " should be used. For example : 
 
 25 
 
 25)625 
 50 
 
 125 
 125 
 
 While the decimal fraction is not taken up in this grade, an 
 example involving such fractions is given to show more fully 
 the reason for adopting this particular algorism. 
 
 Required to divide 6.275 by 2.5 : 
 
 OLD METHOD 
 2.5)6.275(2.51 
 50 
 
 I 27 
 I 25 
 
 COMMON AUSTRIAN METHOD 
 
 2.51 
 25)62.75 
 
 50 
 
 25 
 25 
 
 " Point off as 
 many places in the 
 quotient as the num- 
 ber of decimal places 
 in the dividend ex- 
 ceeds that in the di- 
 visor." 
 
 127 
 125 
 
 25 
 25 
 
 Dividend and divisor having 
 been multiplied by such a power 
 of i o as makes the divisor a whole 
 number, the decimal point in the 
 quotient simply goes above that in 
 the dividend.
 
 4O Teachers College Record [130 
 
 The following method is recommended for the early work: 
 
 2-51 
 25)62.75 
 
 50 
 
 12-75 
 12.5 
 
 0.25 
 0.25 
 
 The entire remainder is brought down each time, and the 
 decimal point is preserved throughout. 
 
 In this grade the child is ready to consider the twofold notion 
 of division, not very philosophically, but sufficiently to insure 
 accuracy of statement. Division is the inverse of a simpler pro- 
 cess, multiplication, just as subtraction is the inverse of a simpler 
 process, addition. A complication arises, however, in division, 
 which does not arise in subtraction, as is seen in the following : 
 
 Addition and subtraction : 
 
 $3 + $2 = $5 ; therefore, (a) $5 $3 = $2 
 or, (b) $5 $2 = $3. 
 
 These two inverses are essentially the same, because the num- 
 bers all represent dollars. 
 
 Multiplication and division : 
 
 2 X $3 = $6 ; therefore, (c) $6 -^ $3 = 2 
 or, (d) $6 -H2 = $3. 
 
 These two inverses are different, because the numbers do not 
 all represent dollars, as in the case of addition and subtraction. 
 
 These two cases have been dignified by different names, the 
 first (c) being called " measuring " and the second (d) " par- 
 tition." But it is unwise to use these names with children. The 
 two cases are easily understood by children sufficiently to lead 
 them to the use of correct forms, if only the teacher will use them. 
 
 (f) Fractions. The year's work includes the addition and 
 subtraction of common fractions, generally with one figure in 
 each term. Simple cases of multiplying fractions by integers are 
 introduced. In many schools children in Grade IV go beyond 
 this limit, but the eventual gain by so doing is not sufficient to 
 overcome the loss in interest by carrying the child beyond his
 
 131] Outline for the First Five Grades 41 
 
 ordinary needs in this year. Improper fractions and mixed num- 
 bers are also introduced in this grade. 
 
 Sufficient work in factoring is introduced to allow the 
 reduction of relatively simple fractions to lowest terms. This 
 includes the tests of divisibility by 2, 3, 5, inductively presented. 
 The idea of cancellation is also introduced. 
 
 The question as to the precedence of decimal and common 
 fractions demands consideration at this point, although the former 
 is met only incidentally in this grade. It is often argued that 
 the decimal fraction, being merely an elaboration of the deci- 
 mal notation with integers, should be taken up along with the 
 latter; that since United States money is taken up in Grades I 
 and II, decimal fractions also belong in those grades. But this 
 is on the old theory that when a topic is first met it must then 
 be mastered. It is one thing to know how to indicate values in 
 United States money, and quite another thing to know how to 
 divide one decimal fraction by another. 
 
 The needs of the child in Grade III require that he be shown 
 how to write $1.25, and what the symbol means. They do not 
 require that he know how to operate with decimal fractions, 
 although he should know that y 2 of */ 2 is % ; y 2 of % is l /^. The 
 idea of the decimal fraction is relatively so abstract that is was not 
 until well along in the i8th century, fully one hundred fifty years 
 after the first general use of the decimal point, that this form of 
 the fraction found much favor in the business world or in the 
 schools. Hence, it cannot be expected that children will readily 
 grasp the idea of decimal fractions before the idea of common 
 fractions is fairly well fixed in mind. Both present needs and 
 propaedeutic considerations demand, however, that there be con- 
 tinued practice in the writing of numbers in dollars and cents, 
 and in the incidental use of such forms as 0.50 and $0%, as set 
 forth under (&) "Tables." 
 
 (g) Business forms. The simple form of bill, as used in 
 retail trade, is introduced in this grade, for the reason that children 
 at about this period begin practically to meet it in their reading, 
 and in purchasing articles at stores. 
 
 (h) Oral Drill. There should be in this, as in every other 
 grade, frequent drill in the addition of columns of figures, with 
 a view to accuracy and fair rapidity; and in subtraction by the 
 " making change " method. The three parts of arithmetic that
 
 42 Teachers College Record [132 
 
 people most often use are (i) addition of columns of figures, 
 (2) making change, (3) the multiplication table; and a school 
 day should seldom pass without some brisk exercise in these 
 subjects. Above all, children should be urged to read columns 
 of figures as they read words and sentences; they do not spell 
 the words, neither should they slowly count up a column. 
 
 (*) Particular attention is given to the simple problems 
 involving the use of United States money, manual training, the 
 measurement of rooms for papering and carpeting (but without 
 complications involving fractional widths and patterns, which prac- 
 tically are left to dealers), the common daily purchases of a 
 family, and the common occupations. Maps are drawn to a scale. 
 Much attention is given to accuracy, to a fair degree of speed, 
 and to neatness of work. 
 
 Grade V 
 
 i. General Suggestions 
 
 (a) Review the work of Grade IV, as suggested in the notes 
 for that grade. 
 
 (b) On the use of the text-book, see General Suggestion 
 (b) under Grade III. 
 
 (c) The work of the year is chiefly directed to the funda- 
 mental operations with compound numbers and with fractions, 
 and to the natural extension of the applications of arithmetic to 
 various occupations, and to other quantitative phases of life. 
 
 (d) For further suggestions see Grades I-IV. 
 
 (e) Bibliography : Smith's Teaching of Elementary Math- 
 ematics, as noted in preceding grades. On the geometry : Sundara 
 Row's Geometric Paper Folding, for the spirit of the work ; also, 
 Beman and Smith: Higher Arithmetic, p. 66. On the formal 
 statement of the work: Beman and Smith: Higher Arithmetic, 
 p. 41. 
 
 2. The Mathematical Work 
 
 (a) Compound numbers. All of the necessary tables have 
 now been taken. The reasons for omitting certain of those that 
 formerly had place, have been stated. The kilowat and horse- 
 power, as units of measure, are spoken of by more people in 
 New York City to-day than are the scruple and the link ; so that if
 
 133] Outline for the First Five Grades 43 
 
 we were to add to the number of the tables, it would not be in 
 the direction of the apothecary's weight or the surveyor's measure, 
 but rather in the direction of the newer units and the metric 
 system. These, however, if taken at all, should enter into the 
 work of the later grades, where the pupil's interest demands 
 them. 
 
 The reduction of compound numbers to units of higher and 
 lower denominations, " ascending " and " descending," is con- 
 fined to numbers of not more than three denominations. The 
 reasons for this are that in practical life we rarely use more than 
 two, as feet and inches, yards and inches, pounds and ounces ; 
 and that one who can perform reductions with three denomina- 
 tions can easily perform those with more, if occasion ever demands. 
 The recent change in custom, with respect to compound numbers, 
 is quite marked. But a relatively short time ago it was not 
 uncommon to see the area of a field stated in acres and rods, 
 while now it is in acres and decimal parts of an acre ; lengths were 
 stated in rods and feet, but now in feet and decimals; and, in 
 general, compound numbers were far more extensively used a 
 generation ago than now. 
 
 The operations with compound numbers, generally involving 
 only two denominations, and limited to three, are taken in this 
 grade as suggested. The reasons for this limitation have just been 
 given. When one considers the rarity of occasions for the use 
 of such numbers, by himself or in ordinary business, he will be 
 convinced that the time formerly devoted to the subject might 
 better be spent on other portions of arithmetic, or on other 
 subjects. 
 
 (b) Reduction of common fractions. The growth in the use 
 of the decimal fraction has been so great during the past century, 
 that much of the work formerly necessary in common fractions 
 has now become almost obsolete. Text-books have been rather 
 slow in recognizing these changes, usually being followers rather 
 than leaders in any movement of this nature. It therefore becomes 
 necessary for the teacher to orientate himself somewhat, before 
 undertaking the work in fractions per se. 
 
 When arithmetics began to be printed, the sexagesimal frac- 
 tion was used for all scientific purposes, as we use it now in 
 degrees, minutes, and seconds. The common needs of trade 
 demanded another fraction, the form of which was brought from
 
 44 Teachers College Record [134 
 
 the Arabs, and this was known as the common fraction. In 
 cases where we, for example, now would use fractions like 0.432 
 and 0.9375, they were compelled to use jYs and || . If these frac- 
 tions should appear as |^f and $ it was desirable to reduce them 
 to lowest terms, and, if they were to be added, it was necessary 
 to reduce them to common denominators, and preferably to the 
 least common denominators. Hence arose the necessity for a 
 study of fractions, and for finding the greatest common divisor 
 and the least common multiple of two or more numbers. Now 
 that the common fraction is used in only a few denominations, 
 and those very small, the decimal fraction becoming more com- 
 mon in general, and nearly universal in scientific and monetary 
 computations, there is no longer the practical necessity for any 
 extensive treatment of factoring, and of divisors and multiples, 
 on the part of children. 
 
 Hence the subject of factoring is limited to the ability to 
 detect the factors 2, 3, 5, these being sufficient for any ordinary 
 reduction of fractions to lowest terms. The subject of greatest 
 common divisor is omitted, as a topic, for the reason that frac- 
 tions formerly requiring the use of such divisors, are now reduced 
 to the decimal form before operating, and the " practical (applied) 
 problems " are always so palpably artificial as to be unworthy 
 of attention. The subject of greatest common divisor should 
 receive some attention, however, for the reason that the general 
 information of pupils may still demand it for a time, but it is 
 limited to the treatment by factoring as stated. There is a little 
 more demand for the least common multiple, since this is necessary 
 for finding the least common denominator of several fractions, 
 and the addition and subtraction of common fractions is still 
 important, within certain limits. For this, however, the factoring 
 method is again ample. 
 
 (c) Operations with common fractions. While the conven- 
 tional order of treatment is possibly justified on the ground of 
 difficulty, and should probably be followed ; it should not be felt 
 to be the order of importance, and it is a fair question as to 
 whether it is the order of difficulty. Other things being equal, 
 addition and subtraction of common fractions are simpler than 
 multiplication, when the denominators are alike, but otherwise 
 they are usually not so. The operation 2 / 3 of 6 / 7 is easier to 
 perform than that of 5 / 7 2 / 3 , and it is more important; while
 
 135] Outline for the First Five Grades 45 
 
 the explanation of one is about as difficult as that of the other. 
 The natural order of presentation, however, would seem to be: 
 Addition and subtraction of fractions having the same denomi- 
 nator; addition and subtraction of fractions not having the same 
 denominator, and hence the necessity for a common denominator, 
 and in particular for the least common denominator; multipli- 
 cation; division. 
 
 It should be remembered, however, that the most important 
 of these operations is that of multiplication, the others being 
 used comparatively seldom. That is, we use " 4 times i2 l / 2 cents," 
 and expressions similar to this, more often than we add common 
 fractions. It should also be remembered that however much we 
 may be tempted to use other devices for division, like that of 
 reducing to a common denominator and then dividing numerators, 
 the practical plan is that of multiplying by the inverted divisor. 
 This, therefore, should be the one with which children should 
 become familiar; sufficient explanation being given to justify the 
 process to their minds, but no elaborate explanation being 
 required. 
 
 (d) Decimal fractions. As already stated, these are carried 
 along with common fractions, so far as notation is concerned. 
 But it now becomes necessary to study them per se, with respect 
 chiefly to the operations. 
 
 Having become more or less familiar with the notation, 
 it is not difficult for children to see that 1 / 10 , 2 / 20 , 3 / 30 , 10 /ioo> 
 o.io, 10%, ten hundredths, ten per cent, all have identically the 
 same value, and all mean from this standpoint, exactly the same 
 thing. Sometimes it is more convenient to write 1 / 10 , sometimes 
 10%, sometimes o.io, as in the case of $2.10. 
 
 The addition and subtraction of decimal fractions usually 
 offer no difficulties unless the teacher suggests them by some 
 attempt at over-explanation. 
 
 The difficulty in multiplication has often been made the 
 greater by the attempt to refer too much to the common- fraction 
 notation. The most approved forms are the following : 
 
 (i) The decimal points are, as is natural, arranged in a 
 column. Since 5 times hundredths is hundredths, the right hand 
 number of the product is placed under hundredths. The rest of 
 the work is identical with that of integers, the decimal point 
 going under the others.
 
 46 Teachers College Record [136 
 
 6.25 
 
 5- 
 
 31-25 
 
 (2) In this case, since hundredths multiplied by tenths is 
 thousandths, the right-hand figure of the product goes in the 
 thousandths place. 
 
 6.25 
 0.5 
 
 3-125 
 
 (3) In this case, since hundredths multiplied by hundredths 
 is ten-thousandths, the right-hand figure of the product goes 
 in the ten-thousandths place. 
 
 6.25 
 0.25 
 
 0.3125 
 1.250 
 
 1-5625 
 
 Not only is this arrangement the simplest and most natural, 
 but there is an ulterior value that will be appreciated by computers. 
 Since a large amount of computation in scientific matters need 
 be carried to only two or three decimal places, this arrangement 
 is the best for the approximate multiplications demanded. Thus 
 in the approximate multiplication of 10.48 by 3.1416 we have, 
 multiplying first by the units: 
 
 10.48 
 3.1416 
 
 31-44 
 1.048 
 0.419 
 
 O.OIO 
 
 0.006 
 
 32-92 
 
 The result is correct to two places, and the unnecessary 
 work is eliminated. The saving is more pronounced in more
 
 137] Outline for the First Five Grades 47 
 
 elaborate problems. Of course this latter work is not intended 
 for this grade. 
 
 The division of decimal fractions has already been men- 
 tioned in Grade IV, 2, (e). 
 
 The operations with decimals are generally limited, for rea- 
 sons already stated, to fractions having not more than three places. 
 
 Such simple aliquot parts (the name is not used) as .33^, 
 .66%, .20, .25, are taken up in this grade. 
 
 (e) Business forms. Carrying forward the work of Grade 
 IV, attention is given to the bill, the receipt, and the meaning of 
 debit and credit. 
 
 (/) Oral work. See Grade IV, 2, (h). 
 
 (g) Geometry. The subject of geometry in the grades 
 has of late attracted much attention. The nature of the work 
 depends upon the answer to the question, Why should it be there ? 
 Its presence might be justified, (i) because the child's interests" 
 demand it, (2) because the child needs the logical training which 
 it offers, (3) because general information demands it, or for 
 similar reasons, all more or less connected, and all summed up 
 in the statement that the child is ready for it. But for what is 
 the child ready? For a knowledge of geometric forms certainly, 
 and this he has with his manual training all through the grades, 
 learning the names of simple geometric figures as a part of 
 that work. It may be doubted if he needs to know of any other 
 forms of plane geometry than those met in that way. 
 
 Furthermore he has interest in knowing how to measure 
 certain of these forms, such as the parallelepiped and prism, which 
 may be met even in sewing. In other words, mensuration should 
 be undertaken rationally, such concrete problems being solved as 
 are actually met in working with material things, this being the 
 " laboratory method " in its truest form. 
 
 But he does not need at present, and he is not yet prepared 
 for, the deductive proof of geometric propositions. Proof by 
 things. As before stated, however, the subject of geometry is 
 actual trial, and proof by logical reasoning, are two very different 
 reserved for a later article. 
 
 So much being premised, the work in this grade is limited 
 to the proper naming of the common geometric forms, particu- 
 larly as met in the manual training, and to the mensuration of 
 certain of these forms, as follows :
 
 4 8 
 
 Length : circumference of circle compared with the diameter. 
 
 Area: square, rectangle, parallelogram, triangle, circle. 
 
 Volume: rectangular parallelepiped and prism. 
 
 Much interest may, however, be awakened by extending 
 this work to the construction of regular polygons; tiles or hard- 
 wood floors offering an example of the hexagon, and the bee's 
 cell showing its economic value. 
 
 It is very desirable to hold to the unity of mathematics, not 
 letting the child feel that his study of geometry is separate from 
 the study of arithmetic. Both are mathematics and are inter- 
 related in life; the school too often separates such close inter- 
 relations. 
 
 (h) Formal work. Children now approach the stage of 
 more formal reasoning about the work. As a consequence, the 
 solution of applied problems should now appear in steps, neatly 
 arranged and numbered. Accuracy of form is necessary to 
 accuracy of reasoning. The following will illustrate what is 
 meant : 
 
 COMMON INACCURACIES 
 
 1. 2 X 25 $50. 
 
 2. 2 X $25 = $50. + $2. = $52. 
 
 3. 2 ft. X 4 ft. = 8 ft., or 8 sq. ft. 
 
 4. 100 -4- 4 = $25. 
 
 5. $100-1-4 = 25. 
 
 6. 100 -H $4 = $25. 
 
 7. V 144 sq. ft. 12 ft. 
 
 8. " As many times as 4 is con- 
 
 tained in $100." 
 
 9. " Tens o' thousandths." 
 
 THE ACCURATE FORM 
 
 1. 2 X $25 $50. 
 
 2. 2 X $25 = $50 
 
 $50 4- $2 = $52. 
 
 3. 2 X 4 sq. ft. = 8 sq. ft., or 
 
 2 X 4 X I sq. ft. = 8 sq. ft. 
 
 4. $100 4 = $25. 
 
 5. $100 $4 = 25, or 
 $100 4 = $25. 
 
 6. Impossible. See 5. 
 
 7. V*44 ft. = 12 ft., or 
 
 VH4X i ft. = 12 ft. 
 
 8. " As many times as $4 is con- 
 
 tained in $100 " or " as 4 
 is contained in $100." 
 
 9. " Ten thousandths."
 
 III. GENERAL DISCUSSION OF THE WORK OF THE 
 LAST THREE GRADES 
 
 (a) MODERN SUBJECT-MATTER 
 
 The work of the last three grades is chiefly devoted to the 
 application of arithmetic to the affairs of life. It is, therefore, 
 desirable to know the nature of the problems demanded by the 
 business of to-day, by the science, manual training, and other 
 subjects in related courses in the school. This matter needs there- 
 fore to be considered at some length from the point of view of the 
 teacher. 
 
 Knowing thus, in general, the kind of subjects that should 
 constitute the principal topics for study in the sixth, seventh and 
 eighth grades of school, we have a standard for judging the 
 worth of the traditional topics and processes customarily dealt 
 with in these years. In other words, we can decide as to what is 
 really important in social life, and what is really used; and we 
 can also decide what is not so useful, and what may even be cast 
 aside as having slight value. 
 
 Thus far in the grades, the mathematics has had two promi- 
 nent purposes, so far as knowledge is concerned; (a) acquaint- 
 ance with certain processes; and (b) acquaintance with valuable 
 facts of a quantitative nature. To the teacher the former may 
 rightly have been a prominent object; just as in teaching litera- 
 ture the inculcation of the underlying truth may be the great 
 purpose. But to the child, the answers to problems, if the problems 
 have been well chosen, may have been the attractive aim ; just as 
 in literature it is the incidents with their " outcome," in the narra- 
 tive that he reads. In other words, the teacher's object may be 
 quite different from that of the child; and very often it is so. 
 
 From the beginning of the sixth grade, the mathematical 
 
 processes as objects of thought should grow less prominent in 
 
 the minds of both teacher and pupil, for the reason that most of 
 
 these processes have already been mastered ; the other phase of 
 
 139] 49
 
 50 Teachers College Record [140 
 
 the work, meanwhile, should increase in prominence. In other 
 words, the primary aim of teacher and pupil should now be 
 increasingly a study of social and business life. Just as literature 
 and history teach ethical and political principles of conduct; so 
 arithmetic should now teach quantitative principles of business, 
 and problems are the means for arriving at this knowledge. 
 
 In outlining the work for the next three grades, therefore, 
 it is our duty to name not primarily the arithmetical processes 
 to be treated, but the large topics that must be investigated on 
 the quantitative side. Some of these, for example, are transporta- 
 tion by rail, by canal and ocean, with comparisons ; corporations ; 
 government revenues and expenditures; manufacturing; farming. 
 Modern arithmetics already show a tendency in this direction; 
 for, while they discuss topics like ratio and proportion, discount, 
 interest, profit and loss, which are mainly concerned with proc- 
 esses, they also take up topics like banking, stocks and bonds, 
 and commissions, which are definite kinds of business. In fact, 
 a modern arithmetic is a confused mixture of processes, and of 
 the occupations of men, although it is evident all the time that 
 there are very few new processes that need be taught. Our 
 proposal amounts only to the suggestion that this confusion be 
 remedied by a fuller acceptance of great business enterprises as 
 topics of study. 
 
 And if this were done, the leading defect of our present 
 study of arithmetic would be largely overcome. For every one 
 now has the feeling that children begin to figure on a given 
 problem too quickly, although it is certain that the work is 
 none too accurate; they are at work with the pencil with 
 altogether too little thought of the general conditions under 
 which the actual business is conducted. This is because our 
 very definition of arithmetic has been " figuring." If we 
 adopted the other conception, namely, that arithmetic is a study 
 aiming to bring one into an understanding and appreciation of 
 leading business undertakings, the emphasis would naturally fall 
 on the nature of each undertaking. This would insure more 
 intelligence from the start, and give motive for the mechan- 
 ical work in the solution of problems. The effect, too, would 
 be that the processes themselves would be better understood ; just 
 as the conventional forms in written language are better learned 
 when children have really interesting subjects to write about.
 
 141] Work of the Last Three Grades 51 
 
 Aside from this argument, while many of these topics are 
 among the most important themes of daily conversation, they are 
 not and cannot be provided for in other studies, because an under- 
 standing of them is reached mainly by a study of quantity. It 
 seems strange, under such circumstances, that arithmetic has been 
 so long willing to confine itself exclusively to processes. With 
 the examples of reform set by other studies along the same line, 
 notably language work in its endeavor to enliven form through 
 a richer content, it is fitting that a forward movement be made 
 in arithmetic. Interest, therefore, may be as great an aim in 
 arithmetic as in literature or in nature study; for the field of 
 thought may be as inspiring. 
 
 Hence it is well to begin by enumerating some of the principal 
 business undertakings that should constitute the course of study 
 in arithmetic in the sixth, seventh, and eighth grades. A discus- 
 sion of the various obsolete and of the acceptable methods or 
 processes will next be in order. Then the special work of each 
 year will follow. 
 
 Topics for study from which selections might well be made 
 for the sixth, seventh, and eighth grades are as follows : 
 
 Farm life: Dairy farm in New England; dimensions of a 
 typical farm ; cattle ; milk ; prices ; butter ; feed. Size of typical 
 farm in Illinois ; typical space given to each product ; quantity of 
 this product. Irrigated farms in Colorado; size; quantity and 
 price of water; variety and quantity of products (compare with 
 the two former) ; effect of irrigation on the price of land. Wheat 
 farm in Washington ; area ; quantity ; cost of production ; prices ; 
 manner of renting land ; variation from year to year. Fruit farms 
 in New York ; in Southern California ; in New Jersey ; in Florida. 
 Cotton, rice, and sugar plantations. 
 
 Rainfall : Variations in different places and from year to 
 year. 
 
 Mining: Nature of a mine; coal mine; dimensions; number 
 of men ; amount of coal ; bituminous and anthracite mines com- 
 pared. Iron ore in our own states ; comparison with Great Britain. 
 Copper output of Calumet and Hecla Mines ; general comparisons. 
 Gold and silver mining. 
 
 The Oil Industry : variety and quantity of products ; com- 
 parison with Russia. 
 
 Lumbering.
 
 52 Teachers College Record [142 
 
 Fishing. 
 
 Ranching: Cattle ranch; sheep ranch; dimensions; losses; 
 prices. 
 
 Transportation : Wagon roads ; extent and cost of various 
 kinds of roads and pavement in various states and cities. Canal ; 
 extent ; history of the Erie Canal on the quantitative side ; amount 
 of freight carried. Pennsylvania Railroad system ; freight on the 
 N. Y. Central Railroad compared with that on the Erie Canal; 
 that on the Illinois Central Railroad compared with that on the 
 Mississippi River. Ocean traffic ; size of steamships ; freight 
 capacity ; coal consumption. Elevators for grain. 
 
 Manufacture: Cotton; comparison of factories in the South 
 and in Massachusetts; wages; prices. Wool; herd of sheep in 
 Ohio; in Nebraska; factories; wages; prices. Leather. Clothing 
 in New York ; the sweat-shops. Sugar ; cane and beet sugar 
 compared. The Linen industry. The Silk industry. Gas com- 
 pared with electricity as to cost; price of each in different 
 cities. 
 
 Sale of Commodities: Grocery Store. Bakery. Department 
 Store. Wholesale Store; dimensions; stock; traveling men. 
 
 Banking business : deposits ; loans ; exchange. 
 
 Corporations : stocks ; bonds. 
 
 Rents : How farms are rented ; amounts. Tenement houses 
 in New York, with reference to recent reforms. Apartments in 
 New York. Houses. Office buildings. 
 
 Insurance: fire; accident; life. 
 
 Revenues and expenses of New York City : taxes of various 
 kinds; departments; employees; supplies; cost of elections; com- 
 parisons with other cities at home and abroad. 
 
 Revenues and expenses of the United States Government: 
 Patent Office Department. Treasury Department; internal 
 revenue ; tariff ; imports and exports ; custom houses. War 
 Department; its size; the cost of maintaining. Pension Depart- 
 ment. Navy Department; warships; cost of navy. The capitol, 
 and the various other expenses, such as those of the White House 
 and the Department of State. 
 
 Immigration : Nationalities ; restrictions. 
 
 Power for manufacturing; horse-power required in various 
 factories; electricity and steam; Niagara Falls as a generator of 
 power.
 
 143] Work of the Last Three Grades 53 
 
 Labor : The cost at home and abroad ; in various occupations. 
 
 Education : Cost of our schools ; a great library ; a university. 
 
 The cost of printing: The extent of the industry; consump- 
 tion of ink and paper. 
 
 Public health : The effect of improved sewerage, of good 
 water, of clean streets, of better ventilation. 
 
 The investigation of almost any one of these topics on the 
 quantitative side calls into use many, and perhaps all, of the 
 arithmetical processes that have thus far become familiar to the 
 pupil. In arithmetic, as in geography, children are in the habit 
 of forgetting one subject while learning another; but the above 
 arrangement renders such forgetting impossible in arithmetic. 
 Aside from that, however, text-books in arithmetic have too few 
 " promiscuous examples." The examples and problems ordinarily 
 bear so exclusively on one process that the pupils drop into a 
 mechanical application of a single new rule before the series of 
 examples is half completed, and the remainder of the problems 
 are solved with the minimum amount of thought. A greater 
 variety of work would prevent the application of the new rule, 
 or of any particular new thought, to every example, and would 
 thus compel the pupil to think at every step of his progress. 
 
 (&) TRADITIONAL SUBJECT-MATTER 
 
 The preceding discussion suggests an examination into the 
 traditional topics of the more advanced arithmetics, with a view 
 to determining their value. This examination must, in this case, 
 be very brief and limited to the more prominent topics. 
 
 (i) Percentage. This subject, formerly a topic by itself, is 
 merely one phase of decimal fractions, and should be so treated. 
 A large part of business arithmetic involves the finding of per 
 cents, so that the method is continually applied after it is once 
 presented. The treatment of the subject by " cases," and the 
 learning of definitions of terms like " amount," " difference," or 
 even " percentage," may be considered obsolete. There is need 
 to know what " per cent " means, namely, " hundredths " (" hun- 
 dredth," or "of a hundredth," as in 6%, i%, l / 2 %), and there 
 is occasionally some value in using the term " base." But the 
 two leading problems of the subject are illustrated by two 
 examples not requiring any elaborate vocabulary, namely:
 
 54 Teachers College Record [144 
 
 1. 6% of $250 is how much? 
 
 2. If 104% of x = $7.28, what does x equal? 
 Practical problems in percentage rarely require any other 
 
 forms. 
 
 (2) Discounts, (a) Commercial: Of high value, since it is 
 used in all business, from extensive wholesale transactions down 
 to so-called bargain sales. 
 
 (b) Bank: Generally used by those having occasion to bor- 
 row money at banks. Of high value. See (8) below. 
 
 (c) So-called " true " discount : The very name condemns it. 
 It gives no true idea of business and should be omitted. It should, 
 of course, be recognized that this form of discount is more advan- 
 tageous to the borrower than bank discount, and on this account 
 it has had many champions on theoretic grounds. For us, how- 
 ever, it is merely a question, in the grades, as to which is practi- 
 cally used. 
 
 (3) Interest, (a) Simple: Necessary in all business life. 
 But in teaching this, as other similar subjects, the teacher needs 
 to look at the matter in perspective. That is, one class of 
 problems is very important; several others are relatively unim- 
 portant, and the teacher should present the subject with this in 
 mind. The important point is that the pupils should know what 
 simple interest is and how it is computed, with tables and with- 
 out. It is unimportant, from the business standpoint, that the 
 pupil should be able to find the time, given the principal, rate, and 
 interest. Hence, while this latter case is valuable as fixing the 
 former in mind, it deserves little emphasis. 
 
 (b) Compound: It is valuable for a child to know what 
 compound interest is, and that it is allowed by savings banks in 
 most states, and that the rate of interest is relatively low. It is 
 well that there should be enough exercise in computing com- 
 pound interest, with and without tables, to fix the idea in mind. 
 This being accomplished, the subject has little practical value. 
 To those who, either here or in other subjects of arithmetic, plead 
 for mental gymnastics, it should again be said that arithmetic 
 offers ample facilities for this valuable work without giving a 
 false idea of business customs and values. Hence the value of 
 this subject is relatively low. 
 
 (c) Annual: This is rarely used and hence is of low value. 
 When met, however, in coupon notes, it involves no new principle.
 
 I 45] Work of the Last Three Grades 55 
 
 (d) Partial Payments : This was formerly of high value, but 
 the increase of banking facilities, and the simplification of banking 
 methods, have diminished this value. In rural communities it still 
 has considerable importance, the " United States Rule " being the 
 only one generally recognized in most states. In order to secure 
 the value of the subject, problems should involve common amounts, 
 as a principal of $100 with payments like $10 and $25, rather than 
 amounts like $251.42 and $19.79. If complicated multiplications 
 are desired, let them enter where they belong, in problems involv- 
 ing scientific measurements. To introduce them here is to make 
 the problem both unreal and uninteresting. Taught as suggested, 
 the subject has a moderate value. 
 
 (4) Insurance. This is important in three general lines : fire, 
 accident, and life insurance. As to the first, it is important that 
 one class of problems should be emphasized, the usual one-year arid 
 three-year policies, at the prevailing rates in this part of the 
 country, the premium being required. As to the second, it is 
 important that the system be explained and a few simple problems 
 solved. As to the third, life insurance, the classes of policies are 
 now so numerous that only two or three standard cases need be 
 considered. The " straight life," and the endowment policies are 
 the most frequently found, and these should be considered in the 
 light of the premiums and the returns. (See Beman & Smith's 
 Higher Arithmetic, p. 177.) The value of this subject is 
 high. 
 
 (5) Stocks and bonds. Since, in these days, most business 
 requiring large capital is conducted by corporations, and the ques- 
 tion of the rights and duties of capital are vital in economics, it 
 is very necessary to relate arithmetic to this phase of modern life. 
 The following matter should be, with due regard to sequence, 
 explained to the class : Corporations, stock companies being special 
 forms ; directors and officers of stock companies ; stock, preferred 
 and common ; dividends ; certificate of stock ; bonds, coupon and 
 registered; interest; par, above and below par; usual par value 
 of a share of stock and of a bond ; how stocks and bonds are pur- 
 chased ; stock exchange ; broker's commissions ; newspaper quota- 
 tions used in class ; distinction between legitimate and illegitimate 
 dealings in stock; the percentage against the gambler. 
 
 The problems should be confined to the usual cases, and 
 may best be assigned with reference to the newspaper quotations.
 
 56 Teachers College Record [146 
 
 The following errors are not uncommon in text-books, and should 
 be avoided : Buying a fractional part of a share ; paying a com- 
 mission different from y% per cent; sending a certain sum to a 
 broker to be entirely invested, instead of ordering the broker to 
 buy a certain number of shares and then paying him; quotations 
 differing from the prevailing prices of standard stocks. 
 
 Taken up in the modern spirit, with blank forms of stock 
 certificates and of bonds to be seen by the class, the subject has 
 high economic value, although mathematically it offers nothing 
 new. (For suggestions, see Beman & Smith's Higher Arith- 
 metic, p. 171.) 
 
 (6) Commission and brokerage. In this is involved the 
 element of the question of food supply for a city like New York, 
 and from this practical standpoint it should be approached. The 
 practical mathematical side of the case is simple, and the non- 
 practical side, involving unusual business problems, should not 
 be considered. (See Beman & Smith's Higher Arithmetic, p. 168.) 
 The economic value is therefore high, although no new mathe- 
 matical processes are involved. 
 
 (7) Government revenues. Instead of isolating the chapter 
 on taxes, it is much better to consider this detail in connection 
 with government revenues in general. There should then be 
 explained to the class the following: The expenses of the United 
 States Government, and where the money goes; the same for the 
 state and the city; the sources of income to meet the expenses of 
 the United States Government ; internal revenues, including post- 
 age, and customs duties ; the revenues for the state and city ; taxes 
 on real and personal property ; inheritance taxes ; prevailing tax 
 rates ; construction of a tax-table, and explanation of its use. 
 
 The problems should then relate to the common cases of 
 direct taxation, of internal revenue, and of customs. To manu- 
 facture unreal problems of a complicated nature is to put out of 
 relief the practical problems. Mathematical difficulties are better 
 presented in other ways. From the standpoint suggested the 
 subject has high economic and rather high mathematical value. 
 (Beman & Smith's Higher Arithmetic, p. 163.) 
 
 (8) Banking business. Arithmetics quite generally separate 
 bank discount from exchange, and each of these topics from 
 numerous other features of banking, with which pupils should 
 become acquainted. The ordinary citizen needs to know some-
 
 147] Work of the Last Three Grades 57 
 
 thing of the following subjects: Savings banks versus general 
 commercial banks ; how to open a bank account ; deposit slips and 
 checks; certificates of deposit; interest on deposits; the banking 
 business of express companies and the post office, as well as of 
 banks, as shown in money orders and drafts. 
 
 If one is in business he will need to know : The process of 
 borrowing money from a bank ; bank discount ; security ; discount- 
 ing notes that he may hold ; collecting by means of drafts ; 
 sending money by means of drafts. 
 
 His general stock of information should also include a 
 knowledge of the clearing houses in various cities, and particu- 
 larly in New York, together with the volume of business trans- 
 acted. 
 
 From this standpoint the economic and mathematical value 
 of the subject is high. (Beman & Smith's Higher Arithmetic, 
 
 P- 148.) 
 
 (9) Profit and loss. A very old name for a special chapter. 
 At present the business expression " Profit and loss " has an 
 entirely different meaning from that given in arithmetic. Since 
 the chapter involves no new principles, pertains to no particular 
 line of business, and involves no problems not naturally to be 
 placed in miscellaneous business exercises, it may well be omitted 
 as a special topic. 
 
 (10) Equation of payments. An old chapter, once of much 
 practical value, but nearly obsolete in America. Improved bank- 
 ing facilities have made these long-standing accounts unnec- 
 essary. 
 
 (n) Partnership. This was formerly a very valuable sub- 
 ject, particularly before the invention of stock companies. Part- 
 nerships still exist; they are, indeed, more numerous than ever, 
 but the style of problem set forth in the arithmetics, under this 
 topic, has long been obsolete. 
 
 (12) Longitude and Time. This subject, formerly of much 
 practical importance, ceased to have some of this value with the 
 general adoption, at the close of the I9th century, of standard 
 time in a large part of the civilized world. Practically, it is at 
 present rather a part of geography than of arithmetic. Problems 
 based on the 15 scheme should predominate. Other problems 
 should generally refer to ships at sea and to observatories, 
 becoming thus too technical for the grades.
 
 58 Teachers College Record [148 
 
 (c) SUMMARY CONCERNING SUBJECT-MATTER 
 
 These conventional subjects have been mentioned by some- 
 what the same names found in the ordinary arithmetics, so that 
 the suggestions may be of service, no matter what book may be 
 in use. The change in the standard chapters has recently been 
 very marked. When the old " Rule of False," or " Rule of False 
 Position," disappeared, many conservative teachers felt that there 
 had been a great loss. When alligation disappeared, a few years 
 ago, there was a similar protest. Now that equation of payments, 
 and compound proportion, and unitary analysis, and profit and 
 loss, and other chapters, are going, many very good teachers feel 
 that nothing will be left of arithmetic. But in place of every 
 business custom or mechanical process that becomes antiquated, 
 new customs and new processes appear, and for these we need to 
 be prepared. As the opportunity offers the following newer topics 
 are introduced, the problems taken from current literature: 
 
 1. The question of the agricultural interests of the country, 
 connecting with commissions, banking, taxes, and transportation. 
 
 2. The questions of fishing, lumbering, mining, manufac- 
 turing, and trade. 
 
 3. The question of transportation, connecting with the study 
 of corporations, taxes, agriculture, mining, and banking. 
 
 4. The questions of labor and of labor organizations, and the 
 relation of each to the questions above mentioned. 
 
 5. Modern treatment of the question of government revenues 
 and expenditures. 
 
 6. Modern treatment of the problems of banking. 
 
 7. Modern treatment of problems involving corporations, as 
 suggested above under stocks and bonds. 
 
 These topics are alive and are valuable for every citizen. It 
 will be a better day for the schools when they replace the obsolete 
 subjects to which reference has been made. 
 
 It is important to note that the topics named suggest our 
 broader commercial, industrial, and social life as the field for 
 arithmetic in the higher grades. Thereby, elementary mathe- 
 matics is made to stand on the same plane as literature and other 
 studies, for all these now culminate in rich generalizations. We 
 should expect in each grade, therefore, a crude formulation of 
 rules of business rather than rules for arithmetical processes.
 
 149] Work of the Last Three Grades 59 
 
 For example, a study of farming- would result in a knowledge 
 of many principles, touching quantity, that guide the farmer; 
 such as the proportionate division of his land for certain crops, 
 the customary amount of stock, the variation in prices that may 
 be expected. A study of rents would acquaint the pupil with 
 methods of renting farms and tenements, the per cent of profit 
 than can be expected, the dangers, and the customary losses. 
 A study of insurance would likewise lead to some knowledge of 
 the way in which risks are calculated, what rates are paid, and 
 the provisions a man should make for the support of his family 
 after his death. A text-book should word these principles with 
 the same care with which rules for processes have heretofore 
 been worded. Thus, mathematics in the grades would cease to 
 be purely theoretical; but, on the contrary, would lead to an 
 understanding of practical affairs. 
 
 But arithmetic in the grades can easily go farther than this. 
 Why is it not a fair task to require a seventh or eighth grade 
 pupil to discover by himself how much it costs to keep a horse 
 or cow in a given community, or to estimate how much butter 
 a cow might make, averaging 2 l / 2 gallons of milk per day, or to 
 find the cost of fuel for the school, or to investigate other similar 
 matters? Such a lesson might be assigned a week or more in 
 advance, with the understanding that each pupil should collect 
 the most reliable data he could get, and make a full report, 
 including the answer to the problem, to the class. After some 
 experience, the children might be left to their own resources as 
 to where to go, how to make an appointment with a man, what 
 questions to ask, how to know when all necessary questions have 
 been put, and how to use the data collected. If such work were 
 definitely required, the children would soon learn to do it 
 properly. 
 
 Of course, the principal question here is whether or not such 
 employment comes properly within the range of school duties. 
 Waiving the fact that nature study and geography rightly call 
 for excursions; is it not the purpose of the school to bring the 
 pupil into an understanding and appreciation of community life, 
 and, also, to identify him with that life as much as possible, in- 
 cluding even the motor action? In other words, is not action, 
 " doing " of one kind or another, the end-point of school instruc- 
 tion? Certainly this is the desired end-point in literature, in
 
 60 Teachers College Record [150 
 
 history, and in manual training; and if it is likewise true of 
 arithmetic, might not this be one of the most effective ways for 
 bringing the pupil into direct touch with the occupations and the 
 people about him? Often such a task as that suggested above 
 could be assigned to committees of the class, and some co- 
 operative, administrative ability would need to be exercised 
 before proper data could be collected. Why not? Even the 
 kindergarten accepts the development of just such ability as one 
 of its prominent objects; why not the grades? It takes time, but 
 it is worth time. If the school is to cease being too theoretical, 
 it must find in some manner an outlet in action. If data for live 
 problems must be collected with care, it is only natural that pupils 
 do some of this work. It is an artificial situation when nothing 
 is left the pupil but to think and to figure. The fact that the 
 teacher would need to be in close touch with the environment and 
 reasonably well posted as to the facts, in order to carry on such 
 work successfully, makes it all the more desirable. Whatever 
 tends to make our teachers less " bookish," and to identify them 
 with their surroundings, particularly in a quantitative way, is 
 welcome. 
 
 (d) GENERAL METHODS OF SOLVING PROBLEMS 
 
 (1) General Analysis. By this we mean merely common 
 sense applied to any ordinary problems. Of course this method 
 is of the highest value. It excludes the memorizing of long and 
 ultra-scientific forms: of analysis, but includes a brief statement 
 of the main steps, with reasons. 
 
 (2) Unitary Analysis. A special form of analysis, in which 
 a series of terms is reduced to unity. For example : If two men 
 do a piece of work in 4 days of 10 hours each, how many men will 
 it require to do it in 5 days of 8 hours each? If the 4 days were 
 each i hour long, it would take 20 men; if there were only I day, 
 it would take 80 men; but since there are 5 days it will take 
 16 men, and since the days are 8 hours long, it will take 
 2 men. 
 
 This is a good plan for solving problems which, as given in 
 arithmetics, are usually of a bad type. For example the anti- 
 quated problems of compound proportion easily yield to this 
 method. These problems, however, are bad in that they give a
 
 151] Work of the Last Three Grades 61 
 
 false idea of business; for in business such extensive series are 
 very rarely met. A single problem here corresponds to a series 
 of problems in actual life. 
 
 (3) The equation method. This is the latest development 
 in the science of arithmetic, as applied to daily problems. 
 
 The equation, as needed in arithmetic, rarely involves cases 
 more difficult than the following, where a, b, and c stand for 
 known numbers : 
 
 a x -\- b = c, 
 
 consequently children should become familiar with problems like 
 the following: 
 
 Twice a certain number is 58 ; what is the number ? 
 
 Five times a certain number, together with 6, equals 41 ; 
 what is the number? 
 
 Ten plus a sixth of a certain number equals 12 ; what is 
 the number? 
 
 One-sixth of a certain number equals ^ ; what is the 
 number ? 
 
 Think of a number ; multiply it by 5 ; add 4 ; tell me the result 
 and I will tell you the number. 
 
 Problems like the above, while applying to no related sub- 
 ject, have in themselves a good deal of interest for children. 
 They like to make up problems like the last one, giving them to 
 one another as puzzle games. 
 
 This subject has not been sufficiently appreciated by teachers, 
 but it is of highest value; for example, If 105^/2% of a certain 
 sum is $2,1 10, what is the sum? Here, 1.055^ = $2,110, .'.x 
 = $2,000, a solution much simpler than the conventional one. It 
 is evident that this problem, with little value itself, is suggestive 
 of a long line of very real cases. 
 
 (4) Simple proportion. A favorite arithmetic method, under 
 the name " Rule of Three," before the equation was generally 
 known. It is merely a fractional equation, this fact being concealed 
 from the pupil by a distinctive symbolism. As an instrument of 
 business arithmetic it is antiquated and cumbersome. Its value 
 in applied arithmetic lies in problems in physics, as in the pressure 
 of gases, and in problems involving similar figures. Hence, in 
 ordinary arithmetic it has slight value, although in geometry its
 
 62 Teachers College Record [152 
 
 value is very great. It is generally better to translate it into the 
 simple equation. It is easier to solve 
 
 2 20 
 
 Or 2.QX = IO, 
 
 than to solve 
 
 20 : 5 = 2 : x, 
 
 although all these expressions are equivalent. 
 In other words, a proportion like 
 
 3:4= 15:* 
 is merely equivalent to 
 
 *:i5 = 4:3> 
 or to 
 
 x 4 
 
 15 3 
 
 which is solved by multiplying by 15, thus avoiding the old and 
 ill-understood proportion rule. 
 
 As a preparation for this work it is not necessary to give any 
 elaborate treatment of ratio. The concept, ratio, is valuable and 
 necessary; the topic, ratio, is not, at least as usually treated. 
 
 The valuable application of proportion in these grades is 
 very limited. The subject might be applied to problems like this: 
 If $2 draw $0.06 interest in i year, how many dollars will draw 
 $3.50 interest in the same time? But in the first place, the problem 
 is not genuine; and furthermore, if it were, it could better be 
 solved by simple analysis. And so with most of the conventional 
 problems of proportion. There is one subject, however, to which 
 proportion necessarily applies, the subject of similar figures. In 
 as far as these have an interest for children, the subject of pro- 
 portion is valuable in the grades. In particular, problems like the 
 following are recommended, the figure always being drawn in 
 advance of the solution: 
 
 If a yard-stick, standing on the level pavement, casts a 
 shadaw 5 ft. long, and the shadow of the gable of the school 
 building at the same time is 145 ft. long, how high is the gable 
 above the pavement ?
 
 153] Work of the Last Three Grades 63 
 
 The following form of solution is recommended: 
 
 1 . Let x = the number of feet in height. 
 
 2. Then because the ratio of heights equals the ratio of shadow 
 lengths, 
 
 x 145 
 
 3. Therefore, 
 
 3 X 145 
 x = 
 
 = 3 x 29 
 = 87, the number of feet required. 
 
 (5) Compound proportion. A method for solving certain 
 obsolete problems, which were solved in the i6th and I7th cen- 
 turies under a rule known by various names; as the double rule, 
 the rule of five, the rule of seven. Practically of no value. 
 
 The unreal problems commonly found in text-books, and the 
 entire subject of compound proportion, should be omitted. 
 
 (e) THE QUESTION OF GEOMETRY IN THE GRADES 
 
 With the exclusion from arithmetic of a large amount of 
 obsolete matter, which had gradually been accumulating up to 
 within a few years, educators looked about for mathematical sub- 
 jects to take its place. The most convenient material at hand was 
 algebra and geometry, and since much of this was simpler than 
 what had been excluded, it naturally found many advocates. As 
 is usual in such cases, the movement to include algebra and 
 geometry in the grades went to an extreme, with the result that 
 children received a smattering of each and entered the high school 
 with an exaggerated idea of their knowledge of these subjects 
 or with a preconceived dislike for them. The question now begins 
 to assume a more rational form : What is the value of these subjects 
 in the grades, and, as a consequence, how should they be presented ? 
 The question as it relates to algebra has already been briefly 
 answered. It remains to consider it as regards geometry. 
 
 Considering only the ability of children to demonstrate theo- 
 rems or to solve problems, there is no question but that a consider-
 
 64 Teachers College Record [154 
 
 able amount of demonstrative geometry can be introduced into 
 the last three of the eight grades. Considering only the question 
 of valuable mental gymnastics, the same answer would be valid. 
 But if the subject is to be presented in the same way as in the 
 high school, and if the child is there to go over the same ground, 
 the effect upon him is not altogether happy. It is a common 
 remark of teachers of geometry that they would prefer a pupil 
 who had never studied geometry to one who had taken it in the 
 grammar grades. The edge of interest is worn off, because 
 nothing new is being mastered. 
 
 If, then, the subject is to be taken in the grades, it should 
 be in such way as not only to appeal to the pupil's interests then, 
 but to leave that impression of only partial mastery which acts as 
 a stimulus to further consideration of the subject. What, then, 
 are these interests, and how shall they be met? 
 
 In connection with the pupil's mathematical work it is con- 
 venient for him to solve an equation. Therefore the equation 
 is introduced into his mathematics ; not algebra as a separate topic, 
 but such features of algebra as bear upon the work in hand. 
 Likewise it is convenient for him to measure things. Therefore 
 portions of geometry should come into his mathematics ; not 
 geometry as a separate science, but such features of it as bear 
 upon the work in hand, and with the increase of a rational manual 
 training this work is daily becoming more important. 
 
 This work in mensuration requires that the pupil should know 
 the names of the common surfaces and solids. This part of 
 geometry, therefore, is introduced in connection with mensuration. 
 It is also necessary that he should know how to measure these 
 common forms ; hence, this work is also taken up, and in a manner 
 quite scientific enough to merit the name geometric. As to the 
 actual geometric facts absorbed by the pupil, this has been done 
 for years. The new feature, developed of late years, is the dis- 
 covery of the facts. Formerly it was merely to memorize a set 
 of rules dogmatically given; now it is to work these rules out. 
 
 Bibliography: Smith's Teaching of Elementary Mathematics, 
 as already noted; Beman & Smith's Higher Arithmetic, much of 
 which is suggestive for grade work, both in business problems 
 and in mensuration ; on the geometry, consult Sundara Row, Geo- 
 metric Paper Folding, much of which is suggestive for grade 
 teachers, although the book is not entirely elementary.
 
 IV. OUTLINE FOR THE LAST THREE GRADES 
 
 Grade VI 
 
 In this grade the reduction of common fractions to decimals, 
 and vice versa, is the only topic in pure arithmetic demanding 
 attention; except as the others enter into reviews. The year is 
 given to applications, chiefly in percentage and denominate num- 
 bers. 
 
 At the beginning of the year the work of Grade V should 
 be reviewed as suggested in the notes for that grade. On the 
 subject of reviews in general, and the use of the text-book, con- 
 sult the general suggestions of the preceding grades. It is par- 
 ticularly desirable to carry out the general policies as laid down 
 in the notes on the work of all preceding grades, and hence 
 teachers and those carrying on professional observation work 
 should become familiar with those notes. 
 
 The following are some of the topics especially suitable for 
 the sixth school grade: I. Gardening; 2. Farming; 3. Rainfall, 
 variations in different places and from year to year; 4. Mining; 
 5. Lumbering; 6. Fishing; 7. Ranching; 8. Transportation; 9. 
 Manufacture and trade, introducing the metric system as con- 
 nected with our rapidly growing foreign market. 
 
 Aside from the English-speaking world, the metric system 
 is generally used in highly civilized countries. Our exports of 
 guns, machinery, and all manufactured articles are dependent to 
 quite a degree upon the use of metric measures in all descriptions 
 of our goods. The foreign trade cannot be expected to sur- 
 render its extremely simple system to our cumbersome one; and 
 whether we adopt the metric measures for our own use or not, 
 we must adopt them for our export trade, if we wish to see it 
 developed. 
 
 The traditional topics especially considered this year are : ( I ) 
 Simple commercial transactions; bills, receipts, commercial dis- 
 counts 5(2) Simple investments ; notes, mortgages, simple interest. 
 '55] 65
 
 66 Teachers College Record [156 
 
 The drill with abstract numbers should always hereafter 
 include the important problem of the addition of columns of 
 figures, checking by adding in reverse order. It should also 
 include subtraction, generally of numbers not exceeding thou- 
 sands; multiplication of numbers like 13 X I2j^, all these being 
 cases frequently met in ordinary business ; and division with one- 
 figure or two-figure divisors. In general all this work should 
 relate to the problems of daily life, rather than to tedious and 
 artificially difficult problems intended merely as tasks. 
 
 In geometry, the work is confined to the measurement of 
 surfaces and of solids, as indicated below; only commensurable 
 magnitudes (for example, lines having a common measure) 
 being considered. Paper- folding (Sundara Row, Geometric Exer- 
 cises in Paper Folding, pages 1-7, 14-15), paper-cutting, and draw- 
 ing, should be employed to illustrate mensuration propositions as 
 in the earlier grades. The proofs of the following propositions, 
 for commensurable lines, are entirely within the powers of pupils 
 of this grade: 
 
 The area of a square a units on a side, is a 2 square units. 
 
 The area of a rectangle, a units high, b units long, is ab square 
 units. 
 
 The area of a parallelogram a units high, b units long, is ab 
 square units. 
 
 The area of a triangle, a units high, b units long, is #ab 
 square units. 
 
 This work is a proper part of mathematics for children of this 
 age. The mathematical unity is preserved; the small amount of 
 geometry not being looked upon as a separate study. For the 
 sixth grade, besides the work above specified, the following should 
 be included: Mensuration of the circle, cylinder, and the forms 
 above mentioned; board measure (foot and thousand) and cord 
 measure ; application to manual training and to building. 
 
 The following problems are suggested as a type of a series 
 devoted to a single subject: 
 
 1. A township in the western farming states is usually 6 miles square. 
 How many square miles does it contain? Show by a drawing. 
 
 2. One square mile contains 640 acres of land and is often divided 
 into four equal parts, called quarter-sections, for four farms. What is the 
 size of each farm? What is the distance around each farm? Show by 
 a drawing. 
 
 3. Divide 160 acres into 4 equal squares. How many acres in each?
 
 157] Outline for the Last Three Grades 67 
 
 How far is it around each? Show by a drawing. How long would it 
 take you to walk around each? 
 
 4. Divide 40 acres into 4 equal squares. How many acres in each? 
 What is the length, in feet, of one of its sides? The distance around it 
 is what part of I mile? 
 
 5. Compare the area of some city block near you, with a 10 acre plot 
 of ground. Make drawings to a scale. 
 
 6. One acre contains 160 square rods, or is equal to a square with 
 each side 12.65 rods long. Mark out such an area somewhere near the 
 school. How many feet around it? 
 
 7. How many acres in the block in which your school building 
 stands? Pace it off, taking 2 ft. to a pace, in order to measure it. 
 
 8. The average farm in the United States contains about 140 acres. 
 That is what part of one square mile? What part of one quarter section? 
 
 9. An ordinary city lot, 25 by 100 feet, is what part of an acre? 
 
 10. The land on which your school-building stands, including the 
 yard, is what part of an acre? Estimate the area of other portions of 
 land, and test by measurement to see how nearly right you are. 
 
 11. Good farm land in New York State can often be bought for about 
 $60 per acre. According to that, what would be the price of the block of 
 land in which your school building stands? Find out the approximate 
 worth of this plot of land? Give some reasons for the difference in 
 price, if there is a difference. 
 
 12. Plot out a 40 acre piece of land in your vicinity. Also one of 
 160 acres. Show that you are approximately right. 
 
 13. A certain farmer in Central Ohio has a farm of 160 acres. Show 
 a plot of this ground, drawn on the scale of Vie of one inch to one rod. 
 
 14. Mark off a suitable area for this man's orchard, yard and garden, 
 barn and barn-yard, field for corn, field for clover and hay, field for wheat, 
 one for pasture, and a timber lot. 
 
 15. It takes him about 54 of a day to prepare an acre of land for 
 planting corn. How long would it take to prepare the field that you have 
 marked out for corn? 
 
 16. I takes him about J4 of a day to plant an acre of corn, and the same 
 to sow an acre of wheat. How long would it take to plant and sow these 
 two fields? 
 
 17. It takes a peck of corn to plant an acre, and 1^4 bushel of wheat 
 to sow an acre. How much corn and wheat are necessary for your fields? 
 
 18. The average yield of corn per acre on this man's farm is about 
 40 bushels. How much corn would the above field yield? 
 
 19. The average yield of wheat per acre is about 18 bushels. How 
 much wheat would the above field yield? 
 
 20. The highest prices that this farmer has received are 50 cents a 
 bushel for corn and 80 cents a bushel for wheat. What facts can you dis- 
 cover about the possible income of the farmer? 
 
 Such a series suggests how arithmetic in the higher grades 
 might give an extensive knowledge concerning an occupation,
 
 68 Teachers College Record [158 
 
 and at the same time furnish abundant drill in pure arithmetical 
 work. 
 
 Grade VII 
 
 Review the work of Grade VI as suggested by a similar note 
 under that grade. 
 
 The following topics in business arithmetic constitute a large 
 part of this year's work: Commercial discount; per cent of gain 
 or loss; commission and brokerage; taxes with customs, treated 
 as suggested on p. 56 ; simple interest ; compound interest as used 
 in savings banks. 
 
 The spirit in which this work is treated is suggested in the 
 notes on Grade VI. See Beman and Smith's Higher Arithmetic, 
 the portion relating to business arithmetic being entirely suited 
 to seventh and eighth-grade work. 
 
 The nature of the oral drill is set forth on p. 41. 
 
 Mensuration and geometry : Review mensuration of the circle 
 and cylinder ; add the pyramid and cone ; in connection with this, 
 review the geometric knowledge thus far secured, covering the 
 ground laid down in Hanus, Geometry in the Grammar School, 
 pages 37-46. 
 
 Ratio and proportion: See p. 61. 
 
 Grade VIII 
 
 i. General Suggestions 
 
 See p. 65. Children have now become mature enough to 
 appreciate difficult business transactions, and these, together with 
 certain geometry work, constitute this year's work. 
 
 2. The Mathematical Work 
 
 (a) Corporations : Organization, stocks and dividends, bonds 
 and interest. 
 
 For this work, the best text-book is the financial page of a 
 daily newspaper. It is not necessary to enter into all of the cus- 
 toms of the stock exchange; indeed only a few technical ex- 
 pressions are of any use to the average citizen. But a dozen 
 problems involving the cost of ten shares of various specified 
 stocks, and another dozen involving the amounts to be received
 
 159] Outline for the Last Three Grades 69 
 
 from the sale of certain stocks, and a few relating to the pur- 
 chase of bonds, all of the problems being based on the news- 
 paper quotations and all involving the usual brokerage, will be 
 interesting and valuable. Different pupils will have different 
 answers, depending on the quotations used, which brings up the 
 questions of the meaning of the expressions " highest," " lowest," 
 " opening," and " closing," and of the causes of fluctuation from 
 day to day. (See also p. 55.) 
 
 (&) Lending money: Notes, bonds, and mortgages; partial 
 payments of a practical nature; writing notes and receipts. 
 
 (c) Banking : Starting accounts ; bank books, deposit-slips, 
 checks, the blanks being in actual use; bank discount; savings 
 banks and compound interest. 
 
 It is now possible to procure school sets of blanks, including 
 bank book, deposit slips, checks, notes, and drafts. The use of 
 such sets adds greatly to the interest and the value of the work. 
 
 (d) Exchange treated from the modern standpoint; drafts, 
 checks, money-orders. 
 
 The exchange rates of banks may be found on the financial 
 page of any of the leading daily papers. 
 
 (e) Our foreign trade; its growth, customs, duties; the 
 metric system reviewed, with the story of its origin, the extent 
 of its use, its advantages ; foreign exchange and the money systems 
 of England, France and Germany; longitude and time reviewed 
 in connection with study of geography of the countries involved. 
 
 (/) Insurance. 
 
 (g) Mensuration and geometry: Review mensuration of im- 
 portant figures, and cover the work laid down in Hanus, Geometry 
 in the Grammar Grades, part 2, page 47-52. In connection with 
 this work, consider square root as stated below. 
 
 (h) Square root approached through practical problems: 
 Relation between the geometric and the algebraic forms ; the latter 
 used in explaining the process. 
 
 Since this article treats mainly of arithmetic in the grades, 
 merely suggesting the use of algebra and geometry ; and since in 
 the eighth grade it is intended to review the arithmetic with special 
 reference to its business applications, and to devote some time to 
 elementary algebra and geometry, further consideration of the 
 latter work is reserved for a subsequent article. 
 
 In general it may be said, however, that there is a manifest
 
 /o Teachers College Record [160 
 
 danger of overdoing the work in algebra and geometry in the 
 eighth grade. Except so far as these subjects have practical 
 application to problems of the life or the science that the pupils 
 of this grade meet or have interest in, the work may better be 
 left for the high school. Therefore the work in geometry may 
 well be limited to mensuration and constructions, and that in 
 algebra may well be directed to the applications of simple equa- 
 tions. Any elaborate treatment of algebraic functions, such as 
 complicated fractions, or any considerable amount of work in 
 literal equations, is of doubtful value at this time. Teachers who 
 contribute to the stock of genuine applications of the simple 
 equation, especially in the problems of business or of simple 
 science, will perform a real service to education, for it is here 
 that the greatest value of algebra for the eighth grade is to be 
 found.
 
 Can Card Games be Used to 
 Advantage in the Schoolroom ? 
 
 THIS is a question which teachers everywhere are asking. Card Games, prepared 
 by practical schoolmen, are being advertised in this and other educational jour- 
 nals. They are constructed and edited by educators of note. The claim is made 
 by the publishers, and with obviously good reason, that these games interest the pupil ; 
 stimulate him to greater efforts and secure better results than are possible with usual 
 and routine methods alone. 
 
 Should these claims be justified by actual schoolroom use, then teachers everywhere 
 who wish to keep abreast of the times will be glad to know it, and will be interested in 
 the following reports from teachers who are using these games. 
 
 Dr. A. E. WINSHIP, Editor " Journal of Education," Boston, Mass. 
 
 " It is in every way interesting, is easily learned, and makes the children quick in their 
 combinations. It is ingenious and attractive. " 
 JONATHON RIGDOV, President Central Normal College, Danville, Ind. 
 
 " I am glad to assure you that I tried your game of Addition and Subtraction and most 
 heartily recommend it. It seems to me that it is not possible to add more pleasure to a game. I 
 shall be glad to speak to our students concerning each of these games that I have an opportunity 
 to test. If your Multiplication and Division game is ready, please send it." 
 Mr. E. J. LLEWELYN, City Supt., Arcadia, Ind. 
 
 " While we have not had time to put the games to a very severe test I can truthfully say 
 that I believe them to be all and more than represented to be. In the short time that we have had 
 the game, we have had it in almost constant use in our 3rd and 4th grades, and are much pleased 
 with the results. I believe that such games as these will have a tendency to awaken an interest 
 among students in the study of Arithmetic. It quickens their ability to add and subtract readily 
 and accurately and exerts a good moral influence over the pupils. I have no criticism to offer. 
 They are just the thing." 
 W. A. COLLINGS, City Supt., Charlestown, Ind. 
 
 " We are greatly pleased with the results of the Mathematical games. They present a dry 
 subject in an attractive manner and hold the child's attention without effort." 
 J. B. JORDAN, Principal, Carbon, Texas. 
 
 " I find the Fraction games not only interesting and fascinating, but instructive, and am 
 sure they will be a means to good results in the way of mental development and individual 
 improvement. I have found the Addition game very interesting for the children in class use. 
 It is of special benefit to those who have little interest in arithmetic generally, in the way of 
 arousing an interest in class work." 
 Mr. HORACE ELLIS, President Idaho State Normal School, Allison, Idaho. 
 
 " We tried this game with some beginning preparatory students. Results were interesting 
 and highly valuable. We believe the game strictly pedagogic and valuable." 
 Mr. PAUL A. COWGILL. City Supt, Michigan City, Ind. 
 
 " I have tried the addition and subtraction game with pupils selected from the 4th, sth and 
 6th grades, and find they are intensely interested in it. It is undoubtedly of great educational 
 value. I am sure that it is just what we need." 
 Mr. W. E. STIFF, County Supt., Mitchell, Ind. 
 
 " I have tested the game of addition and subtraction and find it to be very interesting. It 
 is a game that gives the child a valuable drill in learning the necessary combinations of numbers." 
 Mr. E. C. SHIELDS, County Supt., Clearfield, Pa. 
 
 " I find the game very instructive and beneficial in rapid and accurate operations in addition 
 and subtraction. They would be a good thing in any school or home." 
 Mr. I. V. BUSBY, Ex. Supt., City Schools, Alexandria, Ind. 
 
 " It is highly _interesting, and helpful in a marked degree in stimulating rapid combinations 
 of numbers. I believe the games will prove to be a happy device for strengthening pupils in their 
 number operations." 
 
 Mr. H. F. GALLIMORE, City Supt, Zionsville, Ind. 
 
 " I placed the game of addition and subtraction in the hands of one of my teachers and 
 instructed her to try it with the children. She reports that the results are very gratifying. 
 I feel that you are making a valuable addition to the equipment of our schools." 
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 First Steps in Arithmetic. 36 cents. 
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 75
 
 A CANADIAN LANDMARK! Columbia Universit * 
 
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 A wide range of courses will be 
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 the various University degrees and 
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 The following departments will 
 be represented : 
 
 Anthropology, i course 
 Chemistry, 5 courses 
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 Education, 7 courses 
 English, 6 courses 
 Fine Arts, i course 
 Geology, 2 courses 
 German, 8 courses 
 History, 4 courses 
 Latin, 4 courses 
 Manual Training, 3 courses 
 Mathematics, 5 courses 
 Music, 3 courses 
 Nature Study, i course 
 Philosophy, 4 courses 
 Physics, 5 courses 
 Physical Training, 5 courses 
 Physiology, i course 
 Psychology, 4 courses 
 Romance Languages, 6 courses 
 
 Tuition Fee, $25.00. For further 
 information concerning the courses 
 of instruction and concerning Board 
 and Lodging, address 
 
 Secretary Columbia TJnirersity 
 New York, N. Y. 
 
 A NOTE ON COLOR 
 
 For Teachers of Elementary ScHools 
 By CAROLINE: -WEST VAN HELDEN 
 
 Illustrated with Twelve Colored Plates 
 Price, in boards, 5O cents 
 
 Madame van Helden, having had wide experience with teachers, 
 is able to put into small compass many valuable suggestions 
 for practical work in water colors. 
 
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 Send for a special circular about it, and for a full price-list of 
 Water Colors and all material for color instruction. 
 
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 77
 
 Seacbers College 
 Columbia 
 
 Teachers College is the professional school of Columbia University for the study 
 of education and the training of teachers. The purpose of the College is to afford 
 opportunity, both theoretical and practical, for the training of teachers of both sexes 
 for elementary, secondary, and normal schools, of specialists in various branches of 
 school work, and of principals, supervisors, and superintendents of schools. 
 
 The College offers 56 courses in Education, including 6 courses on the History 
 and Principles of Education, 4 courses on Educational Administration, 7 courses on 
 
 Educational Psychology, and 25 courses on the theory and practice 
 Courses of teaching Biology, Domestic Art, Domestic Science, English, Fine 
 
 of Arts, French, Geography, German, Greek, History, Kindergarten, 
 
 Instruction Latin, Manual Training, Mathematics, Music, Physical Science and 
 
 Physical Education. Other courses of instruction supplementary 
 to those above are as follows : Biology, 6 courses ; Domestic Art, 5 courses ; Domestic 
 Science, 10 courses ; English, 6 courses ; Fine Arts, 14 courses ; French, 3 courses ; 
 German, 2 courses ; Geography, 4 courses ; History, 4 courses ; Kindergarten, 4 courses ; 
 Manual Training, 9 courses ; Mathematics, 3 courses ; Music, 5 courses ; Physical 
 Science, 4 courses, and Physical Education, 6 courses. Qualified students of Teachers 
 College may also pursue University courses in History, Language and Literature, 
 Natural Science, Mathematics, Philosophy, Psychology, Ethics, Anthropology, Music, 
 
 Economics and Social Science. Teachers College maintains two 
 Teachers schools of observation and practice: one, the Horace Mann School, 
 
 College the other known as the Speyer School. The Horace Mann School 
 
 Schools comprises three departments a kindergarten for children of three 
 
 to six years of age, an elementary school of eight grades, and a 
 high school of four grades. The Speyer School consists of a kindergarten, elementary 
 school, and special classes in sewing, cooking and manual training. 
 
 Courses of Study are as follows: (i) A two-year Collegiate 
 Courses Course which if followed by a two-year professional course leads 
 
 of to the degree of B.S. ; (2) Two-year professional courses lead- 
 
 Study ing to the Bachelor's diploma in (a) Secondary Teaching, (b) 
 
 Elementary Teaching, (c) Kindergarten, (d) Domestic Art, (e) 
 Domestic Science, (f) Fine Arts, (g) Music, and (h) Manual Training; (3) Gradu- 
 ate courses of one and two years, respectively, leading to the Master's and Doctor's 
 diplomas in the several departments of the College. Students holding the degree of 
 B.S. or A.B. may become candidates for A.M. and Ph.D. 
 
 The requirements for admission are as follows: (i) To the Col- 
 Admission legiate Course completion of a high-school course; (2) to the 
 Require- two-year courses (a, b, and c above) completion of the Collegiate 
 ments Course or its equivalent in an approved college or graduation from 
 
 an approved normal school; (d, e, f, g, h) same as for (a) and (fc) 
 or two years of technical training or experience in teaching; (3) to the graduate courses 
 college graduation or its equivalent. 
 p .. . . Tuition in graduate courses and courses leading to a degree, $150; 
 
 in other courses, $100. The faculty annually awards 5 Fellowships 
 
 of $650 each, i Scholarship of $400, 12 Scholarships of $150 each, 
 Scholarships and 4 Scholarships of $75 each. 
 
 For circulars and further information, address the Secretary. 
 
 JAMES E. RUSSELL, Ph.D., Dean. 
 
 78
 
 Columbia "University 
 in tbe City of IRew li>orfc 
 
 Columbia University includes both a college and a university in the strict sense 
 of the words. The college is Columbia College, founded in 1754 as King's College. 
 The University consists of the Faculties of Law, Medicine, Philosophy, Political Science, 
 Pure Science, and Applied Science. 
 
 The point of contact between the college and the university is the senior year 
 of the college, during which year students in the college pursue their studies, with 
 the consent of the college faculty, under one or more of the faculties of the university. 
 
 Barnard College, a college for women, is financially a separate corporation; but, 
 educationally, is a part of the system of Columbia University. 
 
 Teachers College, a professional school for teachers, is also, financially, a sep- 
 arate corporation ; and also, educationally, a part of the system of Columbia University. 
 
 Each college and school is under the charge of its own faculty, except that the 
 Schools of Mines, Chemistry, Engineering, and Architecture are all under the charge 
 of the Faculty of Applied Science. 
 
 For the care and advancement of the general interests of the university educa- 
 tional system, as a whole, a Council has been established, which is representative of 
 all the corporations concerned. 
 
 I. THE COLLEGES. 
 
 Columbia College offers for men a course 
 of four years, leading to the degree of 
 Bachelor of Arts. Candidates for admission 
 to the college must be at least fifteen years 
 of age, and pass an examination on pre- 
 scribed subjects, the particulars concerning 
 which may be found in the annual Circular 
 of Information. 
 
 Barnard College, founded in 1889, offers 
 for women a course of four years, leading 
 to the degree of Bachelor of Arts. Candi- 
 dates for admission to the college must be 
 at least fifteen years of age, and pass an 
 examination on prescribed subjects, the 
 particulars concerning which may be found 
 in the annual Circular of Information. 
 
 II. THE UNIVERSITY. 
 
 In a technical sense, the Faculties of Law, 
 Medicine, Philosophy, Political Science, 
 Pure Science, and Applied Science, taken 
 together constitute the university. These 
 faculties offer advanced courses of study 
 and investigation, respectively, in (a) pri- 
 vate or municipal law, (b) medicine, (c) 
 philosophy, philology, and letters, (d) his- 
 tory, economics, and public law, (e) mathe- 
 matics and natural science, and (/) applied 
 science. Courses of study under all of 
 these faculties are open to members of the 
 senior class in Columbia College. Certain 
 courses under the non-professional faculties 
 are open to women who have taken the first 
 degree. ^ These courses lead, through the 
 Bachelor's degree, to the university degrees 
 of Master of Arts and Doctor of Philosophy. 
 The degree of Master of Laws is also con- 
 ferred for advanced work in law done under 
 the Faculties of Law and Political Science 
 together. 
 
 III. THE PROFESSIONAL SCHOOLS. 
 
 The Faculties of Law, Medicine, and Ap- 
 plied Science conduct respectively the pro- 
 fessional schools of Law, Medicine, and 
 Mines, Chemistry, Engineering, and Archi- 
 
 tecture, to which students are admitted as 
 candidates for professional degrees on terms 
 prescribed by the faculties concerned. The 
 faculty of Teachers College conducts profes- 
 sional courses for teachers, that lead to a 
 diploma of the university. 
 
 1. THE SCHOOL OF LAW, established in 
 1858, offers a course of three years, in the 
 principles and practice of private and public 
 law, leading to the degree of Bachelor of 
 Laws. 
 
 2. THE COLLEGE OF PHYSICIANS AND 
 SURGEONS, founded in 1807, offers a course 
 of four years, in the principles and practice 
 of medicine and surgery, leading to the de- 
 gree of Doctor of Medicine. 
 
 3. THE SCHOOL OF MINES, established in 
 1864, offers courses of study, each of four 
 years, leading to a professional degree in 
 mining engineering and in metallurgy. 
 
 4. THE SCHOOLS OF CHEMISTRY, ENGI- 
 NEERING, AND ARCHITECTURE, set off from 
 the School of Mines in 1896, offer respect- 
 ively, courses of study, each of four years, 
 leading to an appropriate professional de- 
 gree, in analytical and applied chemistry ; in 
 civil, sanitary, electrical, and mechanical en- 
 gineering; and in architecture. 
 
 5. TEACHERS COLLEGE, founded in 1888 
 and chartered in 1889, was included in the 
 university in 1898. It offers the following 
 courses of study : (a) graduate courses lead- 
 ing to the Master's and Doctor's diplomas 
 in the several departments of the College ; 
 (b) professional courses, each of two years, 
 leading to the Bachelor's diploma for Sec- 
 ondary Teaching, Elementary Teaching, 
 Kindergarten, Domestic Art, Domestic 
 Science, Fine Arts, Music, and Manual 
 Training; (c) a collegiate course of two 
 years, which, if followed by a two-year 
 professional course, leads to the degree of 
 Bachelor of Science. Certain of its courses 
 may be taken, without extra charge, by 
 students of the university in partial fulfil- 
 ment of the requirements for the degrees 
 of Bachelor of Arts, Master of Arts, and 
 Doctor of Philosophy. 
 
 NICHOLAS MURRAY BUTLER, LL.D., 
 
 President. 
 
 79]
 
 Teachers College Record 
 
 No. x 
 January 
 
 No. a 
 March 
 
 No. 3 
 May 
 
 No. 4 
 September 
 
 No. 5 
 
 November 
 
 CONTENTS OF VOLUME I 1900 
 
 The Function of the University in the Training of Teachers. JAMES 
 
 E. RUSSELL. 
 Historical Sketch of Teachers College from its Foundation to 1897. 
 
 WALTER L. HERVEY. 
 The Organisation and Administration of Teachers College. JAMES 
 
 E. RUSSELL. 
 
 Aims of Nature Study. FRANCIS E. LLOYD. 
 
 Outline of Course in Nature Study in the Horace Mann School, etc. 
 ELIZABETH CARSS. 
 
 Outline of Course in English in the Horace Mann School. FRANKLIN 
 
 T. BAKER. 
 English Composition and Topical Studies in Literature. HERBERT V. 
 
 ABBOTT, ELLEN Y. STEVENS, and EMILY BRINCKERHOFF. 
 
 Syllabi for Teachers College Courses: 
 
 History of Education. PAUL MONROE. Principles of Education. 
 NICHOLAS MURRAY BUTLER. School Administration. SAMUEL T. 
 BUTTON. National Educational Systems. JAMES E. RUSSELL. 
 
 Outlines of Courses in Hand Work in the Horace Mann School: 
 Fine Arts. ALFRED V. CHURCHILL. Domestic Art. MARY S. 
 WOOLMAN. Domestic Science. HELEN KINNE. Manual Train- 
 ing. CHARLES R. RICHARDS. 
 
 CONTENTS OF VOLUME II 1901 
 
 No. i Biology in the Horace Mann High School. FRANCIS E. LLOYD and 
 
 January MAURICE A. BIGELOW. 
 
 No. a Geography in the Horace Mann Schools. RICHARD E. DODGE and Miss 
 
 March C. B. KIRCHWEY. 
 
 No. 3 Child Study. Sources of Material and Syllabi of College Courses. 
 
 May EDWARD L. THORNDIKE. 
 
 No. 4 Syllabi of Courses in Elementary and Applied Psychology. EDWARD 
 
 September L. THORNDIKE. 
 
 No. 5 The Economics of Manual Training {dealing with the cost of the equip- 
 
 November ment and the maintenance of Manual Training, Domestic Art and 
 
 Domestic Science in each of the grades and the high school). 
 
 Louis ROUILLION. 
 
 CONTENTS OF VOLUME III 1902 
 
 No. i Horace Mann School: Dedication Number. 
 
 January Papers by PRESIDENT DANIEL C. OILMAN, SUPERINTENDENT S. T. 
 
 DUTTON, and others, on Present-Day Problems in Education. 
 
 No. a Chemistry and Physics in the Horace Mann High School. PROFESSOR 
 
 March JOHN F. WOODHULL. 
 
 Nos. 3 and 4 Helps for the Teaching of Casar. PROFESSOR GONZALEZ LODGE, and 
 May, September MESSRS. H. H. HUBBELL and WILLIAM F. LITTLE. 
 
 No. 5 The Speyer School. Part I : Its history and purpose. DEAN RUSSELL, 
 
 November PROFESSOR F. M. McMuRRY and MR. JESSE D. BURKS. 
 
 Back numbers for sale singly or by the hundred at regular rates. 
 
 80
 
 UNIVERSITY OF CALIFORNIA LIBRARY 
 
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