L B 
 
 UC-NRLF 
 
 I 
 
 BIENNIAL REPORT 
 
 President of the 
 
 OF CALIFORNIA 
 
 1893 
 
LIBRARY 
 
 OF THE 
 
 UNIVERSITY OF CALIFORNIA 
 
 GIFT OK 
 
 Accession 90259 Class 
 

 Spiral Plan 
 Teaching Arithmetic 
 
 BY 
 JOHN H. WALSH 
 

I. THE SPIRAL METHOD. 
 
 II. WHAT IS NEEDED IN AN IDEAL 
 TEXT BOOK. 
 
 III. PLAN AND SCOPE OF THE WALSH 
 BOOKS. 
 
 130- 
 
I. 
 
 THE SPIRAL METHOD, 
 
 A brief description of one or more typical text- The Old-time 
 books of the old style will enable the reader to realize Te3rt - book - 
 the fundamental defects of those made after that plan. 
 
 The author of a book of that class fails to under- 
 stand that the logical development of the subject mat- 
 ter produces almost invariably a faulty arrangement 
 of topics to be followed by the young pupil. The re- 
 sult of his labors may be a good book of reference, but 
 a bad teaching book. 
 
 In the old grammars, the first place was given to Grammar, 
 an explanation of the word ' 'grammar, " then one 
 of "English grammar." This was followed by the 
 enumeration of the four parts, and a definition of each. 
 Next in order would come the elaboration of the first 
 part, covering many 'pages. The very young pupil 
 was expected to learn that "orthography treats of let- 
 ters, syllables, separate words, and spelling;" after 
 which the classification of the letters was given. 
 Vowels and consonants (with the double-barreled w and 
 y), mutes and liquids, labials, dentals, etc., would help 
 to befog the youthful intellect. 
 
 However, the student's troubles were only begin- 
 ning. Although he might know the declensions, com- 
 parisons and conjugations almost as well as the 
 
THE SPIRAL METHOD. 
 
 Spelling- 
 book. 
 
 teacher, he didn't know that he knew them, nor was 
 he taught how to study them. Blindly he struggled 
 for many weary months, and with what result? Pos- 
 sibly he became able to repeat many meaningless 
 forms. 
 
 Geography. A geography text-book almost invariably began 
 
 with a definition of ''geography," then stated the 
 division of the subject into mathematical, physical, 
 political, etc. , with the definition of each. Although 
 mathematical geography is the least interesting, as 
 well as the most difficult, for the beginner, it had to be 
 completed before the next division was touched; and 
 so on with the remaining divisions. 
 
 The arrangement of the speller was frequently 
 according to the number of syllables in the word 
 monosyllables taking the first place, followed by 
 disyllables, trisyllables, etc., utterly regardless of the 
 needs of the pupil. If homonyms were included, they 
 came late in the book and were arranged alphabetically. 
 The child could not learn the difference between to t 
 two, and too, until he had worried through the un- 
 familiar words under the earlier letters cygnet and 
 signet, crewel and cruel, fane and feign, for instance. 
 
 Arithmetic. In arithmetic, after definitions of quantity, unit, 
 
 number, concrete, abstract, etc., the subject of numera- 
 tion and notation was reached. When the author 
 deemed this sufficiently exhausted some books go up 
 to twenty periods, vigintillions he took up addition, 
 not forgetting, however, to give several principles and 
 rules in the first topic. Addition had its definitions, 
 principles, and rules, and a more or less exhaustive 
 treatment. Then came subtraction, multiplication, 
 and division, each containing its set of definitions> 
 
THE SPIRAL METHOD. 5 
 
 principles, and rules, and each being practically com- 
 pleted before the next was begun. This work gener- 
 ally filled up four years, all through which many un- 
 fortunates were compelled for various reasons to leave 
 school. 
 
 Before the sacred ground of fractions could be trod- 
 den, a probationary period had to be spent among the 
 "properties of numbers," which gave the author an 
 opportunity to tax the child mind with a lot of things 
 about prime numbers and composite numbers, and 
 composite numbers prime to each other; with factors 
 and multiples, and common factors; with common 
 divisors and greatest common divisors; with common 
 multiples and least common multiples. Each new 
 name had its definition, which made it no clearer; each 
 subdivision had its principles and rules; each had its 
 set of examples with possibly a few heart-breaking 
 "problems. 
 
 The subject of fractions afforded rare ground for 
 the old-time author and his modern follower. Defini- 
 tions were given of fraction t fractional unit ', unit of the 
 fraction , numerator ', denominator, common fraction, pro- 
 per fraction, improper fraction, simple fraction, com- 
 pound fraction, complex fraction and mixed number. 
 Before the pupil was permitted to add y* and ^ , he 
 had to be taught how to reduce a whole number to an 
 improper fraction, and a mixed number to an improper 
 fraction; a simple fraction to lowest terms, to higher 
 terms, and to given higher terms; fractions to equiva- 
 lent fractions with the least common denominator; and 
 compound fractions to simple fractions. 
 
 Sufficient has been given to show the dreariness of 
 the old text-book, especially in the hands of a teacher 
 
THE SPIRAL METHOD 
 
 The Two- 
 book Series. 
 
 Reform in 
 Text-books. 
 
 who required the memorizing of each definition, prin- 
 ciple and rule; although the gradual development of 
 the subject in a "logical" order might delight an aged 
 philosopher. It is difficult, however, to defend a 
 scheme of instruction that would prevent a pupil from 
 seeing in a text-book how to find the cost of a half of 
 a 10 cent pie until he had been at school nearly six 
 years; or the method of calculating how much would 
 have to be paid for a pint of milk when it sold for 6 
 cents a quart, until he had spent still another year in 
 the study of arithmetic. 
 
 When the author of this style of text-book decided 
 that a two-book series was advisable, he did not change 
 the "logical" arrangement. All of the sins of the 
 higher book generally appeared in the lower one, and 
 in an aggravated form because of the condensation 
 necessary to make a smaller book. To obtain as many 
 purchasers as possible, the first book of arithmetic 
 contained nearly all the topics of the higher . one, but 
 with many fewer examples for practice. 
 
 The German educators were probably the first to 
 realize the defects of the old-time text-book, although 
 the German pupil suffered comparatively little from 
 its use, owing to the slight dependence of his teacher 
 upon the book. When, however, it was found that 
 the study of grammar did not result in any lessening 
 of the number of mistakes made by a pupil in speaking 
 or in writing, the intelligent teacher came to under- 
 stand that correctness in speaking and in writing comes 
 from long-continued practice in correct speaking and 
 writing, that many people are correct in these respects, 
 who have never studied technical grammar, and that 
 many others, able to repeat glibly all of the rules of 
 
THE SPIRAL METHOD. 7 
 
 syntax with their exceptions, habitually blunder. It 
 is now admitted that the science of grammar is of no 
 use in bringing pupils to correct habits of speech, that 
 all it can do is to help training in thought. In the 
 elementary school it has no place except in the high- 
 est classes, and there for its disciplinary rather than 
 for its practical value. 
 
 To give the required practice in correct talking and "Spiral" 
 writing, it became necessary to develop a systematic Lessons, 
 series of language lessons intended to lead pupils to 
 the employment of the proper forms. In these lessons 
 the "spiral" arrangement was necessarily adopted, 
 each year getting its share in drills in the correct use of 
 the common irregular verbs, for instance, and in speak- 
 ing and writing correctly the sentences likely to contain 
 mistakes in default of such practice. No fear of 
 disturbing the "logical" order of topics prevents the 
 maker of a course of study in language from prescrib- 
 ing such lessons for pupils of even the lowest grades 
 as will bring their ' ' Him and me done it " into something 
 more in accordance with the best usage. He considers 
 the arrangement of the subject matter from the stand- 
 point of the proper training of the child, and leaves 
 the "logical" arrangement for books of reference or 
 for text-books used by students of some maturity. 
 
 The modern geography is gradually dropping its "Spiral" 
 thought-depressing definitions. Children are led to 
 getting accurate notions of land and water forms with- 
 out being compelled to memorize set collections of 
 words, to them meaningless. The subject is led up 
 to, before the text-book is reached, in systematic oral 
 lessons through the lower classes. A new elementary 
 geography goes over the whole ground three times in 
 
8 
 
 THE SPIRAI, METHOD. 
 
 "Spiral" 
 Spelling. 
 
 "Spiral" 
 Arithmetic. 
 
 the one book, adding a few more details in the second 
 treatment, and still more in the third. 
 
 The winner of the prize at an old time spelling- 
 match was frequently unable to write a short letter 
 without making some orthographical blunders. The 
 new graded speller, besides providing for much prac- 
 tice in writing words correctly, has so changed the old 
 arrangement as to teach children to use "their" or 
 "there" properly before the letter / is reached in the 
 homonym subdivision. Each series of lessons now 
 has its proper share of the things that should be taught 
 a child likely to leave school before the middle of the 
 book is studied. 
 
 Text-books in arithmetic have been -somewhat 
 slow in responding to the demand for modern improve- 
 ments. Owing to the fact that books are not placed 
 in the hands of pupils of graded schools until they are 
 nearly half through the elementary course, superin- 
 tendents were able to effect many changes for the 
 better in the lower classes. Pupils of these grades 
 have been required during the first school year to solve 
 oral problems involving any of the four fundamental 
 operations, and even to find fractional parts of small 
 numbers. The more commonly used denominate units 
 were brought into the work of these grades, and many 
 other valuable reforms were made. The authors, in 
 self-defense, were compelled to re- write their first 
 books ; but the majority of them have left in the sec- 
 ond books all of the old, old faults. 
 
 The result has been to handicap the superinten- 
 dents of schools in a great measure. A good course 
 in the work of the first four years, with the details 
 carefully elaborated, is followed in the fifth year with 
 
THE SPIRAL METHOD. 9 
 
 the requirement that Blank's Arithmetic be studied 
 from page to page . A similar one is made for 
 the remaining years. 
 
 It is time for children to begin to use books, if they 
 have not had them before, and it is inadvisable to sug- 
 gest too much flitting about from one part of the book 
 to another. In this way, pupils that have multiplied 
 by a mixed number in the lower grades, and worked 
 simple problems involving pounds and ounces, pecks 
 and bushels, are compelled to drop these topics en- 
 tirely until the "logical" order brings them again into 
 view. The pupils are not even permitted, in this re- 
 spect, to "mark time;" they must retreat, through the 
 failure of the books to furnish ammunition. 
 
 As an illustration of the ' 'spiral" method in history, "Spiral" 
 a plan followed in good schools may be shown in a few Teaching 1 
 words. Before the regular text-book is reached, the History, 
 subject is taught orally, being commenced in the lower 
 grades with stories about historical personages. The 
 idea is to cover the whole period of American history 
 as frequently as possible. The first year's pupils are 
 told about, say, Columbus, John Smith, Washington, 
 Lincoln, and McKinley. The next class is told about 
 or reads about the same persons with additional 
 details, and other characters are introduced , say, 
 De Soto, Penn, Putnam, Grant, Dewey. In a city 
 system in which free books are supplied, a short bio- 
 graphical history is taken up and read in a year. The 
 next year another is read and discussed, all this being 
 done before history is taken up as a formal study. 
 
 The foregoing method, while offensive to the person 
 who would prefer to stick to one book divided chron- 
 ologically into as many periods as there were classes 
 
10 THE SPIRAL METHOD. 
 
 "Concentric studying history, is probably traceable to the French 
 Courses" in , r Zj ,, 
 
 France. plan of concentric courses. 
 
 After their defeat by the Prussians, which the 
 French attributed largely to the superior education of 
 their opponents, they resolved to spare neither money 
 nor effort to increase the efficiency of their schools of 
 all kinds to the greatest possible extent. They have 
 expended enormous sums of money in equipping the 
 schools and in obtaining the best obtainable talent to 
 teach and to direct the teaching. Under these cir- 
 cumstances, it is not strange that the French schools 
 during the past twenty-five years have forged ahead 
 more rapidly than those of any other country of the 
 world. 
 
 The striking feature of the French system is the 
 organization of the studies of the elementary schools 
 into three "concentric courses" of two years each, the 
 pupils of these six years corresponding approximately 
 to our classes from the third year to the eighth, inclu- 
 sive. The first two years are spent in the ' 'maternal 
 school," in which the teaching is chiefly oral; but 
 during the other six years text- books in reading, 
 language, geography, history and arithmetic are used. 
 In each of the last four subjects a different book is 
 taken up every second year, each book covering the 
 whole subject, but in a method adapted to the capacity 
 of the student. The * 'spiral' ' arrangement in France 
 is more properly called the ' 'concentric circle' ' method. 
 
II. 
 
 WHAT IS NEEDED IN AN IDEAL TEXT-BOOK. 
 
 To determine the requirements of a good teaching 
 book, or series of books, for elementary schools, two 
 things must be determined. One has been alluded to 
 previously the proper arrangement of the subject mat- 
 ter from the standpoint of the learner, which is almost 
 the opposite of the "logical" arrangement desirable in 
 a book of reference. 
 
 The other is the careful consideration of the 
 arrangement that will best take care of the proper de- Matter, 
 velopment and training of the too large number of un- 
 fortunate children driven from school at all stages of 
 the course through poverty or other misfortune. 
 Two thirds of the school children are found in the 
 classes of the first four years, and nearly one -third are Lif of j 
 in the classes of the second four years. The number Pupils, 
 in the high schools constitute fewer than one-fiftieth, 
 while the number in college constitutes but an inap- 
 preciable fraction. 
 
 The following figures taken from a recent report of 
 a city school system show the number per thousand 
 of pupils on register in classes of each school year 
 from the first to the twelfth. As tuition, text-books, 
 and supplies of every kind are furnished free of cost, 
 the showing is more favorable as to length of school 
 life than is likely to obtain in cities less favored and in 
 the rural districts. 
 
 (ii) 
 
12 AN IDEAI, TEXT BOOK, 
 
 Primary classes 1st year 185 
 
 2d " 173 
 
 3d " 159 
 
 4th " 144 661 total 
 
 Grammar classes 5th " 125 
 
 6th , " 93 
 
 7th " 58 
 
 8th " 35 311 " 
 
 High school classes 9th " 16 
 
 10th " 7 
 
 llth " 4 
 
 12th ' 1 28 " 
 
 1000 " 
 
 While it is impossible for the average person to 
 draw accurate conclusions from the foregoing figures 
 they show nevertheless a constant dropping-out of 
 those that need all our assistance. If a superinten- 
 dent could be sure of keeping all his pupils 1 2 years 
 or 8 years or even 4 years, he could so arrange his 
 course of study as to give a certain completeness to the 
 education received up to that time. The problem, 
 however, is somewhat more difficult, especially as the 
 short school life of too many is interrupted by fre- 
 quent absences from various causes. 
 
 The system that provides for only those that go to 
 the high school is doing a very small share of its pro- 
 per work. 
 
 The ideal text-book, therefore, must contain such 
 an arrangement of its subject matter as will give some- 
 thing substantial at as many points in the course as 
 possible. 
 
 The book that takes up each topic as frequently as 
 possible helps out the school life of its user by making 
 
AN IDEAL TEXT BOOK. 13 
 
 it easier to promote the boy or girl that is a little 
 " below grade" because of enforced absence or of the Ideal Book 
 possession of fewer brains than the other pupils. In Qn e Topical 
 the use of the old time book, a pupil that had failed to 
 master a topic had no chance to review it properly; 
 the user of the ideal book should have several op- 
 portunities. 
 
 The maker of a good text-book must not be too 
 radical. The method given in the previous chapter, Selection of 
 of reading history by covering the whole ground each Topics, 
 year or two, is not applicable, at least in all of its de- 
 tails, in a text-book of arithmetic. While a boy 
 can easily understand all about Dewey, although the 
 latter belongs in the last chapter of a chronological 
 history, and while he is likely to be as much interested 
 in the hero of Manila Bay as in the Northmen of the 
 first chapter, the same is not true of cube root as com- 
 pared with addition. The early curves of the arithmetic 
 "spiral" should not include too many topics, nor ones 
 too advanced. Some authors, finding the "spiral" of some 
 method a good one, have carried it to too great Authors, 
 extremes. Having convinced themselves that they 
 have discovered an ingenious method of simplifying 
 an advanced topic, they work it into an early page of 
 their books. They forget, however, that the impor- 
 tant thing to do for every pupil of the common school 
 is to give him, at the earliest possible moment, a work- 
 ing familiarity with the fundamental processes, the abil- 
 ity to use simple fractions in their commoner applica- 
 tions, and some acquaintance with the solution of pro- 
 blems involving the most commonly used denominate 
 units. To permit the child to fritter away valuable 
 
14 
 
 AN IDEAL TEXT BOOK. 
 
 Measure- 
 ments, Per- 
 centage and 
 Interest 
 
 Omission of 
 Non-essen- 
 tials. 
 
 time on less important matters, at the risk of failing to 
 obtain the essentials, is an educational crime. 
 
 On the other hand, it is unwise to allow the scholar 
 of the fifth and sixth years to give his whole time to 
 tiresome drills in fractions, decimals and denominate 
 numbers to the exclusion of even elementary lessons 
 in measurements, percentage, and interest. It is possi- 
 ble to give these by the end of the sixth year, or a lit- 
 tle earlier; and no child should be deprived of this 
 much arithmetic, who is forced through poverty to 
 give up attendance at school before taking up the work 
 of the seventh year. The boy or girl able to remain 
 longer will obtain a better knowledge of these topics; 
 but the others should receive at least some instruction 
 therein. 
 
 In considering what should be contained in the 
 ideal book, it must not be forgotten that judicious 
 omissions constitute a source of strength in a teaching 
 arithmetic. There was a time when the author of a 
 geography prided himself upon the multitude of details 
 that were crowded into his maps; to-day he calls 
 attention to their small number. Arithmetic is studied 
 to develop mathematical power in the learner, and not 
 to give him a mass of isolated facts; and the more the 
 pupil's attention is distracted by the latter, the less 
 likelihood there is that he will obtain the full benefit 
 to be derived from the study. A boy that is "good at 
 figures" can readily adapt himself to the arithmetical 
 requirements of almost any calling, as soon as he 
 learns the few facts peculiar to his position. 
 
 Ability to work examples involving denominate 
 numbers can be very much better obtained from the 
 use of a few tables containing familiar units, than from 
 
AN IDEAL TEXT BOOK. 15 
 
 the introduction of the other tables. The absence of 
 the "related predicates" in the case of the latter, tends 
 very much to the confusion of the youthful learner. 
 Meeting a new table a few years later in a new business 
 gives him no trouble, because the daily routine fur- 
 nishes the "related predicates" that are not present in 
 his school days. 
 
 Many books prevent the pupil from testing his Unnecessary 
 powers. Each topic or subdivision of a topic is treated 
 as if it were something entirely new; and explanations, 
 principles and rules are furnished where their intro- 
 duction is a positive injury to the learner's develop- 
 ment. The older authors were content to give only 
 four sets of rules, after they had reached addition of 
 .fractions; some newer ones work in at least two more: 
 "To find what fractional part one number is of an- 
 other," and "To find the whole when the fractional 
 part is given." 
 
 Probably the worst teaching done in our schools 
 occurs in the arithmetic classes in the seventh year. 
 At this stage, percentage is generally reached; and at 
 a time when some mathematical power should have 
 been gained, the author and teacher endeavor to pre- Bad Teach- 
 vent its display. The only new thing in percentage 
 that the thirteen-year-old pupil needs to know before 
 being set to work at the exercises, is the mean- 
 ing of the term "per cent." Being told that, he 
 should be able to solve every question that does not 
 contain any strange technical words. The pupil has 
 solved similar problems in the fifth year in fractions, 
 and in the sixth year in decimals; and it would seem 
 a pedagogical blunder to force unnecessary assistance 
 
16 
 
 AN IDEAL TEXT BOOK. 
 
 Unnecessary 
 Subdivisions. 
 
 New-fangled 
 Systems of 
 Arithmetic. 
 
 upon him, were it not an offence committed by authors 
 of high repute. 
 
 Not content with stifling all growth at the outset of 
 the new topic by their wrong treatment of it, these 
 authors present the same thing again and again under 
 such new names as insurance, commission, brokerage, 
 profit and loss, taxes, duties, and apparently endeavor 
 to prevent a scholar from ascertaining that he is not 
 taking up something strange by giving him a set of 
 rules, principles, and cases, with each subdivision. A 
 boy or girl would be positively benefited by the omis- 
 sion from the text book of every one of these sub- 
 topics. This would not prevent a teacher from giving 
 problems usually placed under one or other subdivis- 
 ions if she were careful to avoid the introduction of 
 unfamiliar words, whose meaning could not readily be 
 determined from the context. Any other examples 
 are unnecessary. The well-taught pupil can handle 
 any he meets in his particular business. 
 
 Because of the meagerness of the results obtained 
 from the study of arithmetic, some well-meaning peda- 
 gogues have assumed that the present methods are 
 wrong, root and branch. Instead of endeavoring to 
 improve the present system by the needed reforms, 
 they begin with the assumption that nothing is right. 
 They wish to kill the rats by burning the barn, a 
 rather wasteful procedure. The Grube method was 
 the first, in recent years, to obtain any wide-spread ac- 
 ceptance. The good things in this method are still 
 employed in a modified way; but the interminable 
 grind prescribed by its author is not now carried out 
 by any sane teacher. 
 
 Two new methods, each guaranteed to be a specific 
 
AN IDEAL TEXT BOOK. 17 
 
 for all the mathematical ills we suffer, have recently The Rational 
 been proposed in all seriousness. The first assumes Method - 
 that failure to teach arithmetic properly is due to the 
 adherence of teachers to the "fixed unit" of the Grube 
 system. If the current practice of the schools were to 
 begin work in number with an elaborate drill on the 
 number one, there might be some reason for writing an 
 article to show the absurdity of such procedure; but as 
 no teacher does anything of the kind, the author of 
 the "movable unit" method is threshing the air. If 
 teachers were bound to inflict upon babes tiresome 
 drills on the "unit," it would be well to suggest the 
 employment of the "movable" one; but as teachers, in 
 the main, are rational beings, they will not worry 
 about either variety of unit. How ridiculous it is to 
 waste energy in teaching children what they know 
 already; or what they will inevitably learn without 
 teaching, and learn better. What need to drill elabo- 
 rately on the facts that 1 cent is different from 1 nickel, 
 that 1 hour is not 1 day, etc., etc. The only new 
 things introduced into the teaching of arithmetic by the 
 author of the "Rational method" are the worse than 
 useless ones of obscuring the terms of a straight-for- 
 ward problem by the introduction of references to his 
 "movable" units. The text-book written to exploit 
 the method would be vastly improved by the omission 
 of every reference thereto. It contains many good 
 things, but they are all old; what is new in it is bad 
 (for teaching purposes). 
 
 The zealous introducer of the "ratio" method The "Ratio' 
 points with pride to the wonderful results produced by 
 the use of his system, with its elaborate outfit of 
 splints, squares, rectangles, triangles, cubes, prisms, 
 
18 AN IDEAL TEXT BOOK. 
 
 cones, and what not. The same remarkable things 
 are claimed by each inventor of something new in 
 education. Some of the lookers-on are deluded into 
 the belief that the philosopher's stone has been found 
 at last. 'Others, more experienced, understand that 
 children can be taught anything; but they realize that 
 the price paid may be the irreparable injury of the 
 children. Bright teachers, proud of the advertising 
 they get from streams of visitors, and desirous of 
 obtaining the commendation of their superiors, will 
 fail to appreciate the harm they are doing their young 
 charges by prematurely forcing them into tasks beyond 
 their years. The complicated ratios introduced by 
 this system into the number work of the lowest years, 
 will inevitably do much harm to the development of 
 immature minds. What is sought of the babe can be 
 obtained much better if postponed until the proper time. 
 Too much One thing that will inevitably militate against the 
 
 Machinery, success of a method requiring too much machinery in 
 its operations is the failure of the over-burdened 
 teacher to employ the necessary material, even when 
 it is supplied, and the unwillingness of the school 
 authorities to continue to furnish supplies that are 
 kept unused in dark closets. While teachers in city 
 schools, having only pupils of a single grade, may 
 struggle for a time with the method, it cannot possibly 
 meet with any favor at the hands of teachers having 
 pupils of more than one grade. The very best results 
 in objective teaching are generally found in the classes 
 in which the objective material is obtained through 
 efforts of the teacher or pupils. Home-made apparatus 
 of nearly every kind is much more effective in class 
 work than that supplied by school boards. 
 
AN IDEAL TEXT BOOK. 19 
 
 It is not so much a new method of teaching arith- Science of 
 metic that is needed, as a modification of existing ^jJrtoT 
 methods. In the first place, teachers must understand Computing, 
 that babes have no business with the science of number; 
 the main object during the early school years should 
 be to teach the art of computing. This makes un- 
 necessary all definitions and principles. When notation 
 and numeration are taught beginners, there should be 
 no wasteful elaboration of method of showing the value 
 of the figure in the tens' place or of the one in the 
 hundreds' place, the bundles of ten splints and the 
 larger ones of a hundred splints. Young children can 
 learn to read and write 39 without much trouble; all 
 the time spent in endeavoring to give them an adequate 
 notion of 39 is thrown away. The little parrots can 
 be made to say anything the teacher wishes them to 
 say; they can learn to manipulate the single splints and 
 the bundles of ten; but the speeches and the per- 
 formance are meaningless to the baby actors, although 
 their teacher does not always realize it. 
 
 When written addition is to be taught, the teacher The "how" : 
 should show his pupils the "how" of the carrying, not 
 the "why." This is the opinion of all the soundest 
 educators of our country, even if it disagrees with the 
 practice of the girl whose normal-school diploma is a 
 few months old. While she may think the children 
 know what they are talking about, when they glibly 
 rattle off the unmeaning formulas prescribed by her, 
 she is seriously deluding herself. It is not claimed, of 
 course, that children are much injured by this simple 
 teacher's methods; a kind Providence has ordained 
 that the infantile mind does not waste much effort in~ 
 the endeavor to grasp matters entirely beyond its 
 
20 
 
 AN IDEAL TEXT BOOK. 
 
 Lines of 
 Work. 
 
 Oral 
 Problems. 
 
 Drills. 
 
 range. Still, the teacher is wasting time; and, when 
 she ascertains that her pupils' previous repetitions of 
 things she wished them to say, mean nothing to them, 
 she may do harm by endeavoring to force them to 
 understand. 
 
 Of course, oral problems are to occupy a fair por- 
 tion of the time given to number work and even some 
 written problems should be given; but a large part of 
 the energy of the beginning pupils should be given to 
 abstract written addition and to the necessary oral 
 drills. The following are the lines of work that 
 should be followed throughout the arithmetical course: 
 1st, oral problems, to develop proper ideas of number, 
 and to strengthen the reasoning powers; 2d, oral drills 
 in rapidly combining abstract numbers, to give the 
 necessary facility in performing written operations; 
 3d, written addition, subtraction, etc., of abstract 
 numbers; 4th, written solution of problems. 
 
 The oral problems will be of two kinds: First, 
 those in which the child will have to determine for 
 himself the operation involved; and these should con- 
 tain small numbers, so that the size of the numbers 
 will not make the problem unnecessarily difficult, 
 while failing to help the pupil's mathematical develop- 
 ment. In those of the second kind, which hardly 
 deserve to be called "problems," except that they are 
 "concrete," the chief object is to drill the pupil in 
 making the needed combination rapidly without a 
 pencil; they should involve, therefore, as a rule, but 
 one operation. 
 
 The oral drills should be given very frequently, but 
 for very short periods; and they should demand rapid 
 answers on the part of the pupils. These help the 
 
AN IDEAL TEXT BOOK. 21 
 
 scholars in their subsequent abstract written work. 
 Sight drills must not be overlooked. 
 
 The written problems show the pupils the applica- 
 tions of the fundamental processes to the ordinary 
 affairs of life. 
 
 The ideal arithmetic should contain each of the Careful 
 foregoing lines of work, carefully graded f and in the Gradation 
 proper proportion. The book should be divided into 
 sections, each containing the work of a year or a half 
 year. The attempt to divide by daily lessons is un- 
 wise; as nobody but the teacher can determine just 
 what subject requires special attention at a given time, 
 with a corresponding number of examples. 
 
 The abstract work should be developed very slowly Abstract 
 and carefully, with a very, very large number of ex- 
 amples. Difficulties should be encountered one at a 
 time. As facility in calculation is obtained only by 
 very much practice, it is almost impossible to furnish 
 too many abstract examples. The task of selecting 
 them from other arithmetics, or of making them, 
 should not be thrown upon the over-burdened teacher; 
 and it is better for the pupils to have them in their 
 books than to be compelled to copy them from the 
 blackboard. Reviews of processes already learned 
 should be continued to the end of school life. The 
 complaints of college professors and of business men 
 that graduates of high schools cannot even add or 
 multiply correctly is only too well founded. 
 
 As the chief object of the written problems is to written 
 develop reasoning power, they must, if possible, be Problems, 
 even more carefully graded than the abstract work. 
 A problem should generally require no explanation 
 from the teacher; the conditions, therefore, should be 
 
22 
 
 AN IDEAL TEXT BOOK. 
 
 "Miscellane- 
 ous" 
 Problems. 
 
 Analysis of 
 Problems* 
 
 such as are reasonably familiar to the pupil. This le- 
 quires an author to skillfully adapt the wording of a 
 problem, and the matters contained in it, to the age of 
 the average pupil that will be required to work it. The 
 numbers employed should not be so large as to daze 
 the learner; since the latter, as a rule, stumbles over a 
 written problem expressed in the same words and con- 
 taining the same conditions as an oral one with smaller 
 figures that gives him no trouble whatever. This lat- 
 ter difficulty can frequently be overcome by using 
 written problems as "sight" ones, the numbers being 
 changed by the pupil to those small enough to be han- 
 dled without a pencil, and the pupil then working the 
 problem again with the figures given in the book. 
 
 Problems, both oral and written, should always be 
 "miscellaneous;" that is, there should be no heading 
 to indicate the character of the operation required to 
 solve one. If the scholar knows that all the so-called 
 problems on a given page involve multiplication, he 
 need give no attention to the conditions. No two 
 consecutive problems should involve the same opera- 
 tions, except in the few cases where two or more suc- 
 cessive ones are introduced to lead up to a more com- 
 plicated one. 
 
 While problems constitute a very important part of 
 a child's arithmetical training, progress in abstract 
 work should not be retarded by the teacher, in the 
 vain attempt to make each pupil work out and under - 
 ,jgtand every problem. The wise teacher will not re- 
 quire every member of the class to keep pace with his 
 mates in problems, nor will she require all to work the 
 same problems at the same time, nor will she insist 
 upon any pupil solving all of them. She will also in 
 
AN IDEAL TEXT BOOK. 23 
 
 problem work, both oral and written, assume that the 
 child has reasoned correctly when he gets the correct 
 answer; and, therefore, she will not exact an unmean- 
 ing analysis, oral or written. If she wishes the pupil 
 to give a reason, she will take any one suited to the 
 child's years, even if it does not contain the conven- 
 tional "wherefore" or "hence." She will not expect 
 written solutions to follow a set plan, nor will she 
 spoil a^ bright boy by requiring him to place upon his 
 paper any more figures than are necessary to aid him 
 in obtaining the result. An occasional written analy- 
 sis, as an exercise in composition, she will probably 
 ask. 
 
 While the science of arithmetic has no place in the 
 elementary school, and while the "how" should gen- O f Power 
 erally precede the "why," the thoughtful instructor 
 will give her brighter pupils an opportunity to develop 
 power by reasoning out the "how," before she shows 
 it to the slower ones. After pupils have added num- 
 bers of two figures that need no "carrying," she 
 might lead up to the latter, by asking on the board 
 the sum of 12 and 12, then 13 and 13, then 14 and 14, 
 then 15 and 15. Some members of her class would 
 know that the sum of the last set is 30, even if the 
 oral combinations had been confined to numbers of one 
 figure; and they would appreciate the method that 
 produces this result. Subtraction as the reverse of 
 addition should be led up to; but no time should 
 be spent in explaining the reason for the method em- 
 ployed. Those that can get any benefit from the per- 
 liminary work, should be permitted to get it; but as 
 the "why" will come anyhow, although later, the 
 progress of the class should not be delayed. The 
 
24 
 
 AN IDEAL TEXT BOOK. 
 
 Development 
 of "Rules." 
 
 Topical 
 Work. 
 
 Arithmetic 
 vs. Algebra. 
 
 addition of 15 and 15 will show pupils how to get the 
 product of 15 by 2, after they have worked multiplica- 
 tion examples in which there is no "carrying." 
 
 As the child grows older, and as his reasoning 
 power increases, he should be given more frequent 
 opportunities to develop for himself the so-called 
 "rules." All that is necessary in the case of most 
 children is that the teacher should furnish examples 
 in which the successive steps occur in the proper order. 
 She need scarcely say a word in explanation; in fact, 
 the better the teacher the fewer her words, if she takes 
 care to present difficulties no more rapidly than they 
 can be surmounted. 
 
 The working-out of this kind of work is hardly 
 possible in the old-time book. Bach topic has to be 
 contained in a chapter, of which the padding consti- 
 tutes a large part. Work that is best spread over a 
 year or so, and in which results are to be obtained 
 only after working hundreds of examples, has to be 
 dismissed in a few pages. 
 
 The erroneous notion formerly prevalent, that 
 pupils were somehow improved by over-difficult tasks, 
 prevented the adoption in this country of the use of 
 algebraic methods by students of arithmetic. While 
 the old-timer was ready to admit that some of the so- 
 called arithmetic problems could be solved in a quar- 
 ter of the time by the use of the letter x, he pretended 
 to believe that it was a good thing for the child to 
 make him attempt the almost impossible. Although 
 Spartan methods of killing off the weaklings have long 
 been discontinued on the physical side, the school life 
 of many has been made unnecessarily severe by requir- 
 ing them to attempt to solve problems in the hardest 
 
AN IDEAL TEXT BOOK. 25 
 
 way, because of the supposed mental growth thereby 
 induced. It signifies nothing to the believers in this 
 plan that much the larger number of pupils get nothing 
 out of it; they are content at the success of the for- 
 tunate few. 
 
 The opponents of the introduction of algebraic Formal 
 methods into the elementary school claim that this 
 subject should be confined exclusively to those able to 
 go to a high school. While it may be true that the 
 formal study of algebra as a science should not be 
 commenced until about the ninth school year, no one 
 will have the hardihood to assert that other pupils 
 should be deprived of the use of anything that would 
 tend to lighten their burdens. 
 
 Even the fiercest opponent of the use of x- in a so- 
 called arithmetical problem, will himself, in solving Arithmetical 
 such a problem, make use of an algebraic method; Circumlocu- 
 but to conceal the latter, he will explain his solution * 
 by a tedious circumlocution in which the tale-telling 
 x is carefully kept from view. 
 
 There is nothing in the application of algebraic The use O f ^ 
 methods to the solution of problems that is beyond 
 the capacity of the average child or teacher. If in 
 a first-year class-room, you put on the blackboard 
 3+?= 5, the correct answer is given by nearly every 
 pupil. While no one would object to such a problem 
 as an arithmetical one, it is called algebra when an x 
 is substituted for the interrogation point. Is this fair? 
 
 The mind of the average child works forward, so Going 
 to speak, better than it works backward. A beginner Forward ' 
 in subtraction can tell the answer to "7 and what makes 
 10?" more readily than to ' ' 1 less 7 equals what?' ' The 
 former is algebraic in a way; the latter, arithmetical. 
 
26 AN IDEAI, TEXT BOOK. 
 
 The following problem, frequently found in the 
 arithmetics, cannot be solved arithmetically without 
 help, by any but the most phenomenal scholars of the 
 age at which they reach it in the books: "The votes 
 cast for A and B number 6836, of which A gets a 
 majority of 748. How many does each receive?" 
 An author (who has studied algebra) furnishes a neat 
 explanation. His excuse for inserting this problem and 
 many other similar ones is that children may meet 
 them later in life. Instead of furnishing the general 
 (algebraic) method, he gives a different, pretendedly 
 arithmetical one for each, and the scholar can solve 
 only those whose types he has already had, instead 
 of being enabled by the algebraic master-key to open 
 any lock, whether he has seen it before or not. 
 
 Problems in Let us see how the "problems" in interest are 
 
 Interest. handled by arithmetic makers that are unwilling to 
 give their clients the benefit of the simple methods. 
 To find what principal will produce $64 in 3yr. 6mo. 
 9da. at 6%, they assume a principal of $1, on which 
 the interest is calculated at the given rate for the 
 given time. The result is found to be $0.2115. The 
 pupil is then told that the required principal is as 
 many times $1 as $64 is times $0.2115. 
 
 While the bright boy that had never seen algebra, 
 could not of himself evolve the foregoing method, the 
 bright student of algebra could solve the problem 
 without having encountered it previously. Instead of 
 finding the interest on $1 (which is almost as much 
 algebra as arithmetic) he would calculate it on x dollars, 
 and call this result .2115:*: equal to 64, the equation 
 being, without decimals, 2115^=640,000. His prac- 
 tice in solving equations suggests the remaining step. 
 
AN IDEAL TEXT BOOK. 27 
 
 These interest problems are at least six in number: 
 
 1 . Given interest, rate, time, to find principal. 
 
 2. Given principal, rate, interest, to find time. 
 
 3. Given principal, time, interest, to find rate. 
 
 4. Given amount, rate, time, to find principal. 
 
 5. Given amount, rate, principal, to find time. 
 
 6. Given amount, time, principal, to find rate. 
 While others might be enumerated, the foregoing 
 
 are sufficient to show the advisability of applying 
 algebraic methods when they are available. These 
 six "problems" really should constitute no "problem" 
 of special note from the algebraic standpoint. Any- 
 one that knows a little of algebraic ways, and able to 
 calculate interest, needs no "rule". He works out the 
 interest (or amount) , using x to represent the missing 
 factor, and he then forms the necessary equation. By vs. 
 following the old way, the learner is required to mem- 
 orize six rules; especially as the text-book maker fails 
 to make clear the connection between them. Not 
 knowing the underlying reasons, he forgets the rules 
 shortly after leaving school, and he becomes helpless. 
 The algebra boy gets no "rule" for solving any prob- 
 lem, except to treat the x just as he would the number 
 it represents. 
 
 As most teachers are from high or normal schools, Easy to 
 or both, but very few will be compelled to make any 
 special study of the algebraic method. These few, if 
 competent to teach arithmetic, can make themselves 
 familiar with the method after very little practice. 
 
 The writer of these lines deprecates the attempts of 
 some misguided people who seek to introduce formal High school 
 algebra into the grammar school course, reasoning that A1 g ebra - 
 it will save time later in the high school. The 
 
28 
 
 AN 
 
 TEXT BOOK. 
 
 Constructive 
 Work. 
 
 elementary course is intended primarily for those whose 
 education ends there. Nothing should be introduced 
 that needs high school work to give it any value. A 
 boy that gives a year or so to addition, subtraction, etc. 
 of algebraic quantities gets no benefit whatever from 
 the study, if he does not go to high school. This time 
 should be given almost entirely to the solution of the 
 equation. The pupils that go to the high school should 
 begin the study of algebra at the beginning, just the 
 same as those who had never seen it, except that they 
 may progress much more rapidly from the insight 
 given them of the utility of the new study. Diluted 
 high school algebra should have no place in an eight 
 years' course of study for the elementary schools. 
 Those who propose it do not appreciate the real prov- 
 ince of "algebra below the high school." 
 
 While a few French and German schools postpone 
 algebra until a late period of the elementary school 
 course, they all introduce geometry very early in the 
 child's school life, the French schools beginning it 
 during the first year. 
 
 As we do not teach high school algebra when we use 
 x in the arithmetic work of the lower schools, the word 
 "geometry" as employed in European courses of study 
 does not mean that demonstrative geometry is inflicted 
 upon infants. Geometry is begun in France, as in the 
 United States, by clay modeling, drawing squares and 
 circles, constructing paper prisms and cones. It in- 
 cludes mensuration of surfaces and of solids, as well as 
 the solution of simple problems in construction. It 
 does not mean the study of the high school text-book. 
 
 For every pupil of the elementary school that will 
 be called upon to work out an example in partial pay- 
 
AN IDEAL TEXT BOOK. 29 
 
 ments, there will be a thousand likely to require some 
 knowledge of geometrical facts. These facts should 
 be commenced in the lower school at as early an age as 
 possible. The drawing courses in many schools, 
 urban and rural, provide the best possible instruction 
 for the smaller children. By the end of the fourth utility of 
 year systematic work should come into the arithmetic Geometl T- 
 work, a little at a time. Mensuration of rectangles, 
 formerly left for the end of the eighth year, and later 
 pushed forward a year or so, should commence at the 
 beginning of the fifth year. Each year of the last four 
 should contain its proper share of calculating surfaces 
 and volumes; and the last year or two should contain 
 a well worked-out set of. construction problems, the 
 working of which would put the pupil in possession of 
 the most important facts of geometry. 
 
 As in algebra, the injudicious should not be per- 
 mitted to push demonstrative geometry into elementary 
 schools. The latter is studied in the high school for 
 its disciplinary value. The constructive work of the 
 other, besides being extremely valuable to those quit- D emonstra . 
 ting school at the end of the eighth year, is also useful tive Geom- 
 to the later student of Euclid or of Legendre. Know- etry ' 
 ing the facts, he is better able to appreciate the chain 
 of reasoning employed in the text-book. 
 
 It is hardly necessary to enumerate the persons to 
 whom some knowledge of geometrical facts is useful. 
 The farmer, mason, plasterer, carpenter, tinsmith, 
 painter, have all to deal with mensuration to a greater 
 or less extent, and also with the construction of some 
 of the geometrical forms. 
 
 The work in constructive geometry can be done by Easily 
 comparatively young children. What the latter Tau 2 ht - 
 
30 
 
 AN IDEAL TEXT BOOK. 
 
 How to Find 
 Time. 
 
 Cutting out 
 
 Useless 
 
 Topics. 
 
 can readily do, should present no difficulty to the 
 teacher. 
 
 The question of time is always an important one in 
 the present over-crowded course. Friends (or enemies) 
 of the schools are continually coming to the front with 
 additional studies to be inflicted upon the system, and 
 those teachers are hardly blamable who look with 
 disfavor upon the attempt to add algebra and geometry 
 to the over-loaded curriculum. If, however, they 
 realize that there is no desire to introduce the high- 
 school subjects known by these names, that all that is 
 intended is to suggest the employment of a few simple 
 algebraic expedients in solving arithmetic problems, 
 and the use of drawing as an aid in mensuration work, 
 their objections will be less vigorous. 
 
 The benefits to teachers and scholars will be in- 
 creased if the adoption of the new method leads to bet- 
 ter arrangement of the old topics and the elimination 
 of every unnecessary one. By cutting out the things 
 that now overload the books, the time at present given 
 to arithmetic may be lessened, the subject will be bet- 
 ter taught, and plenty of time will be found to use the 
 equation and to work out problems in construction. 
 
III. 
 
 PLAN AND SCOPE OF THE WALSH BOOKS. 
 
 The Walsh arithmetics constitute a one-book series 
 bound for convenience in two or in three parts. The 
 first page of one book follows immediately after the 
 last page of the preceding one, without a break. The 
 purchaser of the second book does not buy a number 
 of useless pages, as he must frequently do in the 
 case of other series. 
 
 Each arithmetic chapter after the first contains Half-year 
 work for a half year. Besides the appropriate advance ap er8 ' 
 work in all the lines, oral and written, drill work and 
 problems, it contains the necessary reviews. 
 
 Children will not willingly turn backward to get mat- R ev i ews 
 ter for reviews. The advisability of taking new lessons 
 for this purpose is especially appreciated by teachers of 
 language, English and -foreign, ancient and modern. 
 Besides, the need of constant review is likely to be 
 overlooked unless matter for the purpose is brought 
 directly to the teacher's attention. 
 
 Another strong feature is the careful grading. In Large Num- 
 the abstract work, the examples are so numerous that 
 the difficulties are introduced as slowly as is compati- 
 ble with good work. The very great number of the 
 abstract examples gives the needed facility in rapid 
 and accurate calculation. The examples are so graded 
 that the child can begin written work early in his 
 school life, and continue it without interruption. He 
 can begin to add as soon as he knows* a few sums, and 
 
 (31) 
 
32 THE WAIH ARITHMETICS. 
 
 the successive examples grow difficult by slow degrees. 
 In the first nine pages, for instance, there are 258 ex- 
 amples in addition, oral and written, with the total of 
 each column less than 10. The children learn to do 
 by doing. Then follow 121 examples, problems, etc., 
 leading to written subtraction, without ''borrowing,' 
 then 65 examples in written subtraction and 10 * 'mis- 
 cellaneous" problems, 454 examples in all before 10 is 
 used as a sum of any column or as a separate minuend. 
 In addition to the foregoing there are 93 exercises in 
 numeration and notation of numbers to 99, or 547 ex- 
 ercises of all kinds in the first fourteen pages. This 
 shows the absence of padding and the care taken in 
 the development of the work. The remaining 21 
 pages of the first chapter, devoted to addition and sub- 
 traction of easy numbers, contain 879 exercises of 
 all kinds, including drills, etc., making a total of 1426 
 for 35 pages. These will be none too many, as the 
 work of this chapter is intended to cover what is 
 usually done by the end of the second year. 
 
 The next chapter is arranged for pupils of the first 
 
 half of the third school year. It extends the previous 
 Multiplica- 
 tion and work in addition and subtraction, and takes up multi- 
 Division, plication and division, beginning with the new work. 
 Multiplication is commenced at once on the supposi- 
 tion that the child has learned from hi-s addition work 
 the products by 2 up to twice 4. Ten exercises and 
 60 examples are given with 2 as a multiplier, and 
 without carrying, to fasten the child's knowledge of 
 the early table. The last 10 examples introduce 
 larger products, but the examples still involve no car- 
 rying. Division by two is next taken up, with each 
 figure of the dividend a multiple of the divisor, the 
 
THE WALSH ARITHMETICS. 33 
 
 same number of exercises being employed as in multi- 
 plication. 
 
 After nearly a half dozen pages of review work 
 and "miscellaneous" problems, also some extension of Development 
 numeration and notation, pupils are led to "carrying" of Process by 
 in multiplication by being asked to find products of up 
 12, 13, 14, 15, 16 and 17 by 2. Then come the quo- 
 tients of 24, 26, 28, 30, 34 and 38 divided by 2. It is 
 left to the teacher to decide whether or not she needs 
 to show the "how." When the pupil has used 2 as a 
 multiplier and a divisor in a number of examples, he 
 is plunged into a number of others in which the mul- Commtita- 
 tipliers (or divisors) include numbers to 9. The ^on. 
 work, however, is kept within his range by using in 
 the multiplicand only (or quotient) numbers composed 
 of O's, 1's and 2's. From this he learns that he al- 
 ready knows a portion of the "3 times" table, also a 
 portion of each of the others. 
 
 The child learns the "3 times" table by working 
 numerous examples in multiplication and division with 
 this number as the multiplier or the divisor. He then, 
 as before, uses the other numbers to 9 as multipliers 
 or divisors, with multiplicands or quotients limited to 
 numbers made up of O's, 1's, 2's and 3's. Working 
 in this way, he not only learns each table easily and 
 thoroughly, but he begins to understand the law of 
 commutation, and to realize that as he goes towards 9 
 times, he has fewer facts to memorize in each table. 
 
 This lengthy explanation of the work of the first 
 two chapters is intended to show how much attention 
 has been paid to making the child's path in numbers 
 as smooth and as interesting as possible. The short- child? 8 
 ness of the examples and the care to avoid introducing Interest. 
 
34 THE WALSH ARITHMETICS. 
 
 difficulties too rapidly, tend to give the pupil a sense 
 of power. This obtained, his interest is secured, and 
 everything goes smoothly. When a child that has 
 used only 2 as a multiplier is asked to multiply 
 121 by 3, 222 by 4, 201 by 5, 121 by 6, 202 by 7, 
 112 by 8, and 212 by 9, he is delighted to find that the 
 new multipliers present no new difficulties, and he is 
 encouraged in his onward course. 
 
 In discussing the matter of the second chapter, no 
 reference has been made to the drills, oral and sight, 
 opics. p roD i ems ora i an( j written; nor to the new matters in- 
 troduced United States money, fractional parts of 
 numbers, Roman notation, liquid measure. Each turn 
 of the ''spiral" brings in its new matter, besides am- 
 plifying and extending the old. It must not be sup- 
 posed, either, that abstract work constitutes the sole 
 important feature of the Walsh books. 
 
 These books lay as much stress on the reasoning 
 Accuracy and s ^ e as ^ an ^ ther good books; but they also recog- 
 Rapidity. nize the important fact that the ability to reason cor- 
 rectly in mathematics is useless if not accompanied by 
 the ability to compute accurately. The early school 
 years are the ones to be given to the endless examples 
 needed to secure accuracy and rapidity in performing 
 operations, as children at this period are ready and 
 willing to give themselves up to the grind necessary to 
 secure these results. If they haven't mastered the 
 fundamental processes before they are 11 or 12, the 
 chances are that they will always be slow and inaccur- 
 ate. 
 
 The care shown in the gradual development of the 
 
 work contained in the first and second chapters extends 
 
 Development, to all the others. In long division, for instance, Chap- 
 
THE WALSH ARITHMETICS. 35 
 
 ter IV, none of the first 300 examples has a dividend 
 of over four figures, although the multiplication re- 
 sults in the same chapter generally contain five figures. 
 As long division is rather difficult, the likelihood of 
 the pupil becoming discouraged is diminished by the 
 shortness of the examples, especially as the earlier 
 quotients consist of numbers containing small figures; 
 such as, 13, 21, 22, 23, 12, 211, 123, 222, 11, 12, etc. 
 The divisors, too, are carefully chosen; 21, 31, 41, 22, 
 32, 42, etc., being used before 16. The limit of four 
 figures in the dividends of the early examples permits 
 of the early introduction of large divisors without real- 
 ly increasing the difficulty of the example, since the 
 longer the divisor the fewer figures there will be in the 
 quotient; the answers to 8199-i-911 and 9872-^2468 
 consisting of a single figure. A word will be said later 
 about the long-division drills. 
 
 A form of fraction work is begun very early. As Fractions, 
 soon as children learn to divide by 2, they find one- 
 half of a number. Later, they find fourths and thirds, 
 without, however, hearing of "fraction," "numera- 
 tor,'* "denominator," or the like. The sum of ^ and 
 YV, and of 1% and 1^, etc., begins addition of frac- 
 tions in Chapter III, although the formal work is not 
 reached until Chapter VII. Each' turn of the * 'spiral' ' 
 brings in its appropriate work in Chapters IV, V, and 
 VI, while each chapter after the seventh has the need- 
 ed reviews. 
 
 Chapter II marks the commencement of work in Denominate 
 denominate numbers, with problems involving pints Numbers, 
 and quarts. The intervening chapters to the ninth 
 extend the child's knowledge of this important topic, 
 while the ninth summarizes and completes the subject, 
 
36 THE WALSH ARITHMETICS. 
 
 except for the subsequent inevitable reviews. I/ong, 
 tedious examples are avoided by limiting the number 
 of denominate units in any example to two successive 
 units before the ninth chapter, and to three successive 
 
 Examples. units in the ninth and remaining chapters. This fea- 
 ture of short examples is a prominent one in the 
 Walsh books, and it appeals to every one interested in 
 education. The old time teacher that covered the 
 blackboard with a single example in addition, for in- 
 stance, did much to kill the pupils' interest in mathe- 
 matics. A child that is given ten or a dozen short 
 examples during an arithmetic lesson has a chance of 
 getting the correct answer to a large proportion of 
 them, while unable to continue the strain needed to 
 work out a single very long one. 
 
 From the beginning of the sixth chapter, the point 
 at which pupils generally take up a second book, the 
 superiority of the (< spiral" method becomes more 
 apparent. In the lower grades, many teachers do 
 
 The Old-time good work because they are not hampered by * 'logical' ' 
 books in the pupils' hands. When the children of 
 the 5th year get a book of this kind, they are fettered. 
 The early pages, devoted to the fundamental processes, 
 appear too elementary, and are not touched; while the 
 topical arrangement prevents the extension, until it is 
 regularly reached, of some work already begun. In 
 the first chapter of the second Walsh book (Chapter VI) 
 all the previous work is continued and extended, while 
 new ground is broken in decimals and mensuration, 
 each treated in such a way as to be readily understood 
 by the 5th year scholar. A year later, Chapter VIII, 
 marks the point at which are introduced percentage 
 and interest. 
 
THE WALSH ARITHMETICS. 37 
 
 The employment of the "spiral" method does not 
 prevent the author of the Walsh books from adopting 
 the good features of other books. The early intro- Treatment^ 
 duction of an advanced topic is always accompanied by Topic*. 
 its systematic treatment in the chapter especially de- 
 voted to that topic, which chapter is reached in the 
 Walsh books at just the same time it is reached in the 
 old-line texts. Thus, Chapter VII of Walsh is the 
 fraction one; Chapter VIII, the decimal one; Chapter 
 IX, the denominate number one; Chapter XI, the next 
 arithmetical one, being given to percentage; etc., etc. 
 
 Besides being strong in its general features, the 
 Walsh books are particularly useful for teaching pur- Special 
 poses because of their special features. One marked I 
 characteristic is the space devoted to "drills," oral and 
 sight, each chapter containing its share. One kind of 
 drills is intended to make children masters of all the 
 combinations needed in their work in the fundamental 
 processes, including two sets, never before used in this 
 country, to enable pupils to obtain rapidly each figure 
 of the quotient in a long division example. (See Art. 
 321 and Arts. 397 401.) These preliminary drills are 
 furnished in great variety, to prevent the weariness to 
 children that comes from tiresome repetitions of the 
 same exercise. 
 
 As children brought up on the old books advanced 
 into more advanced topics, they seemed to lose their 
 earlier facility in computation, because of lack of re- 
 views, discontinuance of drills, etc. The Walsh books 
 aim not only to keep up the skill obtained in the lower 
 classes in adding, multiplying, etc. , mentally and on 
 paper, but to increase it as far as possible. A business 
 man should not need to hunt up a pencil every time he 
 
38 
 
 THE WALSH ARITHMETICS. 
 
 Special 
 Drills. 
 
 wishes to make a simple calculation. f To enable a boy 
 or a girl to readily combine large numbers, each half- 
 yearly chapter has a page (or more) devoted to "spec- 
 ial drills," which gradually increase in difficulty, as 
 will be seen from an examination of the following 
 selections: 
 
 FROM CHAPTER III. 
 
 50+30 
 
 20+60 
 
 50+40 
 
 40+50 
 
 30+60 
 
 9050 
 
 5020 
 
 8040 
 
 5030 
 
 9070 
 
 20X2 
 
 3X30 
 
 20X4 
 
 ysX90 
 
 20X3 
 
 40-5-2 
 
 90-^-30 
 
 y>> of 60 
 
 80-5-4 
 
 40-5-2 
 
 FROM CHAPTER IV. 
 
 13+13 19+30 43+46 51+37 22+23 
 
 2513 3120 6511 8775 4626 
 
 13X2 32X3 21X4 23X3 41X2 
 
 88-r-4 39-^13 26-5-2 63-5-21 86-^-2 
 
 FROM CHAPTER V. 
 
 56 + 17 13+78 25+16 18+45 34+19 
 
 6619 5639 6012 7657 4318 
 
 13X4 5X15 28X3 7X13 47X2 
 
 42-5-3 42-5-14 78n-6 x 78-5-13 90-^6 
 
 The foregoing types indicate the gradual develop- 
 ment of the drills; the next set, from Chapter XIV, 
 shows what pupils of the eighth school year should 
 be able to do without using a pencil: 
 
 112+91+85 
 15023 + 48 
 
 63X28 
 676-T-13 
 
 84Xlf 
 
 43+131+61 
 
 172+ 1966 
 
 54X42 
 
 527^-17 
 
 211^X13 
 
 95 + 144+79 
 183 (72 37) 
 
 26X58 
 704-f-22 
 36X49| 
 
THE WALSH ARITHMETICS. 
 
 68+56+174 
 161 + 7912 
 71X82 
 837-^-27 
 
 After each set of drills, there are given many oral 
 (or sight) problems involving similar combinations, a 
 feature found in no other arithmetic, written or mental, problems 
 
 The advanced mental examples, even in books devoted Involving the 
 
 "SDecieil 
 exclusively to this form of arithmetic, use smaller Drills." 
 
 numbers as the work progresses, the authors consid- 
 ering, apparently, that facility in computation is of no 
 value as compared with the ability to crack mathematic 
 chestnuts. While the mental arithmetic work in the 
 Walsh books includes examples and problems such as 
 are described above, at the beginning of this paragraph, 
 it also includes the customary exercises bearing upon 
 the topic under immediate treatment. 
 
 Other sets of drills have for their object the devel- 
 opment of skill in the use of short methods by pupils. 
 They include mental multiplication and division by commutation 
 fractional parts of 100; the use of 99, 24, 49, also of Drills. 
 99^, 24^, 49^, etc., as multipliers in mental exam- 
 ples, etc. The problems under this head, and, in fact, 
 problems all through the books, show, where possible, 
 the law of "commutation": for instance, that 25 yards 
 at 48c. per yard can be solved mentally in the same 
 way as 48 yards at 25c. ; that in subtraction examples 
 the child can take 43 from 50 when he can take 7 from 
 50; that, in multiplication, he knows 9 fours when he 
 has learned 4 nines; that, in division, if 35 contains 5 
 sevens, it contains 7 fives. 
 
40 THE; WALSH ARITHMETICS. 
 
 "Approxima- The "approximation" drills are new to text books 
 tion" Drills. j n arithmetic. Their value is appreciated on sight by 
 all teachers anxious to prevent their pupils from offer- 
 ing absurd answers. The use of the method of 
 ~ approximation before attempting to solve a problem, 
 will frequently lead a pupil to discover the operations 
 necessary to its solution, whereas, the too frequent 
 practice among average children is to begin to work 
 without a full appreciation of the conditions involved. 
 With the purpose of making the arithmetical jour- 
 ney as smooth as possible for young learners, the Walsh 
 arithmetics suggest some improvements upon the 
 method now in vogue. Teachers that believe children 
 should not do anything without a knowledge of the 
 Method. underlying reasons, have made subtraction unneces- 
 
 sarily difficult by requiring pupils to work examples 
 in the ' 'logical' ' way. In finding the difference between 
 835 and 398, the little learner is supposed to make a 
 speech in some such fashion as this: "Eight units 
 from 5 units I cannot take, so I borrow 1 ten from the 
 3 tens. Adding this ten, which equals 10 units, to 
 the 5 units, I have 15 units. Then I take 8 units from 
 15 units which leaves me 7 units, and this I write in 
 the units column. Since I took 1 ten from the 3 tens, 
 I have two tens remaining. As I cannot take 9 tens 
 from 2 tens I must borrow 1 hundred from the 8 hun- 
 dreds. Adding this hundred, which equals ten tens," 
 etc., etc.; but why continue this rigamarole? And 
 how the difficulty is increased when the minuend 
 contains a few ciphers, say 1000 473, where the 
 1 'next higher order" has nothing to lend. The old 
 way is just as "logical": 8 from 15 leaves 7, 10 from 
 13 leaves 3, 4 from 8 leaves 4; but it is more difficult 
 
THE WALSH ARITHMETICS. 41 
 
 to "explain". Is there, however, any likelihood that 
 infants understand the explanation they profess to give 
 of the other method? Should school children be re- 
 quifed to repeat an unmeaning formula they will never 
 use after leaving school? An accountant does not 
 think of units, tens, or hundreds as he makes his daily 
 calculations. 
 
 In the Walsh book, the "building up" method is 
 advised; or, as it is sometimes called, the "computer's Method in 
 method." Instead of being told to take 8 from 15, Subtraction, 
 the child is asked "8 and what make 15?" as expe- 
 rience shows that the mind travels forward more easily 
 than it goes backward, especially after giving all its 
 attention previously to addition. By this method the 
 operation resembles addition so much as to make it 
 less difficult for beginners. The use of this method 
 enables the pupil to shorten many other operations. 
 He can, for instance, ascertain the result of the follow- 
 ing: 1000 (643+287 + 25) without first finding the 
 sum of these numbers to be added, (Art. 384); or 
 4832 (456 X 8) . Long division can be performed with- 
 out writing the partial products (Art. 616); or the mixed 
 
 1 1 223 
 number equivalent to can be written at once. 
 
 In long division, the children are advised to write 
 each quotient figure above the corresponding figure of Quotient, 
 the dividend, to prevent the omission of one or more 
 ciphers in the quotient, or the introduction in the quo- 
 tient of a figure too many (Art. 282). This plan is 
 similar to the one used in short division; and it makes 
 the necessary multiplication more easy to the young 
 student by bringing the multiplying quotient figure 
 nearer to the divisor an important trifle. 
 
42 
 
 THE WALSH ARITHMETICS. 
 
 Division of 
 Decimals. 
 
 Short 
 Method. 
 
 Omission of 
 Unnecessary 
 Figures. 
 
 The method given for "pointing-off" in division of 
 decimals (Art. 663) is a mechanical one, but it pre- 
 vents the pupil from getting the decimal point in the 
 wrong place. Another good method, which helps by 
 mechanical means in getting the correct answer in 
 multiplication, is given in Art. 344. The intelligent 
 teacher will not despise anything that will aid her 
 pupils to secure accurate results, even if it is sneered 
 at as ''mechanical" by the user of "logical" methods 
 whose pupils frequently blunder. 
 
 Besides giving much attention to special short 
 methods, the Walsh books offer suggestions in every 
 chapter as to the disuse of unnecessary figures. 
 Children learn just as readily to cipher without these 
 aids (?) as with them, and their written work becomes 
 more accurate by not being too long drawn out. For 
 instance, in finding the least common multiple of 3, 
 9, 7, 14, 6, 14, 2, 12, some teachers permit pupils to 
 retain all these numbers, instead of using only the 
 necessary ones 9, 14, 12 (Art. 595). In reducing 
 28^ to an improper fraction or in multiplying 16^ by 
 8, scholars are not required, as they should be, to write 
 the answer directly (Art. 653). 
 
 To reduce 15 gal. 3 qt. to quarts, the average boy 
 or girl will use several lines of figures, when one is 
 sufficient (Art. 766); see also under interest (Art. 936), 
 discount (Art. 937), commercial discount (Art. 944), 
 compound interest (Art. 983). To enumerate all 
 the places in which suggestions are made as to omit- 
 ting unneccessary figures, would be to make too long 
 a list. The index to the Grammar School Arithmetic 
 gives 56 pages on which are found "short methods," 
 
THE WALSH ARITHMETICS. 43 
 
 and these do not include suggestions given in the 
 Teachers' Manual. 
 
 The non-progressive teacher hesitates to do things 
 in a strange way ; but she will soon realize the advis- 
 ability of saving time as indicated in the Walsh books. 
 The methods given are the straightforward ones that 
 can be understood by the dullest pupil, and which, by 
 being applicable in the daily work, are readily appre- 
 ciated. They do not include such as 9} X9}^, 85 X 
 85, 64X66, found in some old magazine under the 
 head of "Mathematical Recreations." 
 
 The algebra work is contained in chapters X and 
 XV; of these, every pupil should study the former. 
 It contains only 1 1 pages, and is readily understood Algebra, 
 by a very young pupil. To enable a teacher unfami- 
 liar with this work to do it successfully, she has only 
 to follow the lines laid down in the Manual. While 
 chapter XV is intended more particularly for schools 
 that have a nine-years' course, its study is advised 
 even if time has to be obtained therefor by the omis- 
 sion of some of the work in arithmetic: bonds and 
 stocks, for instance, compound interest, exchange, 
 partial payments, proportion, equation of payments, 
 etc. 
 
 While the construction exercises and problems have 
 been placed in chapter XVI, they should be commenced 
 about the time chapter XII is reached, and carried 
 along with the arithmetic work, even if portions of the 
 latter receive less attention in consequence. The ex- 
 ercises in calculating heights and distances are very in- 
 teresting to scholars and are very useful to many of them 
 later in life. They are likely to be employed by more 
 pupils than the problems in equation of payments. 
 
44 THE WAI^SH ARITHMETICS. 
 
 The teacher will find in the Manual minute directions 
 as to the best way to conduct the geometry lessons. 
 
 The Walsh books furnish problems in greater num- 
 ber and variety than any other series. Each problem, 
 being unlike the previous one, will require the pupil 
 to read it carefully; he cannot work it by referring to 
 a ' 'sample" one at the head of the page. The absurd 
 types are all omitted, such as the far-fetched ones in 
 some books under the headings of greatest common 
 divisor and least common multiple. 
 
 These books are offered to the teaching profession 
 in the belief that they contain more strong features 
 and are better teaching books than any now before the 
 public. It is not, however, claimed that they are per- 
 fect as books of reference. Although an index is 
 really unnecessary in a teaching book, a good one is 
 furnished for such teachers as desire to use the book 
 topically. 
 
The Walsh Arithmetics 
 
 C9ntain abundant, varied, and practical problems. 
 Omit nothing essential, yet contain only the essentials. 
 Are fresh, original, and well graded. 
 Secure constant review without actual repetition. 
 Are arranged on the " spiral " plan. 
 
 Three=Book Series. 
 
 Elementary Arithmetic. For third and fourth grades. 
 
 Cloth. 2 18 pages. 30 cents 
 
 Intermediate Arithmetic. For fifth and sixth grades. 
 
 Cloth. 252 pages. 35 cents. 
 
 Higher Arithmetic. For upper grades. 
 
 Half Leather. 387 pages. 65 cents. 
 
 Two-Book Series. 
 
 Primary Arithmetic. For third, fourth and fifth grades. 
 
 doth. 198 pages. 30 cents. 
 
 Grammar School Arithmetic. For upper grades. 
 
 Half Leather. 433 pages. 65 cents. 
 
 Each series is provided with Teacher's Manuals in parts. 
 Correspondence is cordially invited. 
 
 D. C. HEATH & CO., Publishers, Boston, New York, Chicago. 
 
22 MATHEMATICS. 
 
 Mathematics for Common Schools. 
 
 A graded course in arithmetic, with simple problems in algebra and geometry. 
 By JOHN H. WALSH, Associate Superintendent of Public Instruction, Brooklyn. 
 Two-Book Series. Primary Arithmetic. Cloth, 206 pages. Introduction 
 price, 30 cents. Grammar School Arithmetic. Half leather, 458 pages. Intro- 
 duction price, 65 cents. 
 
 Three-Book Series. Elementary Arithmetic. Cloth, 220 pages. Introduc- 
 tion price, 30 cents. Intermediate Arithmetic. Cloth, 255 pages. Introduction 
 price, 35 cents. Higher Arithmetic. Half leather, 403 pages. Introduction 
 price, 65 cents. 
 
 IN several important particulars the Walsh Arithmetics mark a de- 
 parture from the traditional method and arrangement. 
 
 1. By the "spiral advancement plan" the elements of all the im- 
 portant topics are taken up early in the course, adding to the interest 
 and practical worth of the study. 
 
 2 . In each case the subject taken up is not exhausted at once, but 
 practice in it is carried on with problems of gradually increasing diffi- 
 culty throughout the course. 
 
 3. Drills in addition, subtraction, multiplication and division of ab- 
 stract numbers are given at intervals throughout the books of the 
 series, thus insuring in pupils of the upper grades, accuracy and speed 
 in the fundamental processes. This is an important and unique feature. 
 
 4. The series contains a larger number of varied and practical con- 
 crete problems than any other. 
 
 5. It is the only series containing drills in securing "approximate 
 answers," work of great advantage in calling the pupil's attention 
 to the condition of a problem, and thus giving the power to detect at 
 once the absurdity of any result greatly wide of the mark. 
 
 Such obvious merits of the lower books as the alternation of oral, 
 sight and written work, the early introduction of United States 
 currency (leading to decimals), the easy beginnings with fractions and 
 denominate numbers, and the freshness and interest insured by the 
 great variety of means used to secure perfect mastery of simple 
 number combinations, cannot be too strongly emphasized. 
 
 In the higher book are to be noted the wide range of subjects treated 
 in their simple elements, the great variety of practical problems, the 
 
MA THEM A TICS. 23 
 
 early introduction of percentage and simple interest, of bills and re- 
 ceipts, and all the matters connected with simple commercial arithmetic. 
 
 Unique features are : the many short methods noted, the use of ap- 
 proximate answers, the abundant drills in the four fundamental 
 processes, and the introduction of algebra in a way so natural and sim- 
 ple that children of ten may easily grasp enough of it to shorten many 
 of the longer arithmetical processes. 
 
 The Walsh Arithmetics constitute a one-book series bound for con- 
 venience in two or in three parts. The first page of one book follows 
 immediately after the last page of the preceding one, without a break. 
 The purchaser of the second book does not buy a number of useless 
 pages, as he must frequently do in the case of other series. 
 
 The Walsh books illustrate most admirably what every teacher 
 knows so well, that many things that are complex in their completeness, 
 are in their elements simply and easily comprehended by young chil- 
 dren. 
 
 The series thoroughly satisfies demands of modem pedagogy ; it is 
 inductive in method, practical and varied in treatment, makes clear 
 thought and accurate computation matters of habit, and lays the foun- 
 dation for the intelligent use of mathematical principles. 
 
 The Walsh Arithmetics anticipated the recommendations of the Com- 
 mittee of Ten and of the Committee of Fifteen. 
 
 Full descriptive circular ', and valuable pamphlets upon * ' The Spiral 
 Method" and "Suggestions to Teachers and Courses of Study in 
 Arithmetic" sent free on request. 
 
 Teachers Manuals to Mathematics for Common Scttools. 
 
 By JOHN H. WALSH, Associate Sup't of Public Instruction, Brooklyn, N. Y. 
 
 Manual to Three-Book edition. Complete. 385 pages. Cloth. Retail price, $1.50. 
 Manual to Elementary Arithmetic. 63 pages. Paper. Retail price, 15 cents. 
 Manual to Intermediate Arithmetic. 124 pages. Paper. Retail price, 20 cents. 
 Manual to Higher Arithmetic. 343 pages. Paper. Retail price, 40 cents. 
 Manual to Primary Arithmetic. 67 pages. Paper. Retail price, 15 cents. 
 Manual to Grammar School Arithmetic. 342 pages. Paper. Retail price, 50 cents. 
 
Elementary Mathematics 
 
 AtWOOd'S Complete Graded Arithmetic. Presents a carefully graded course, to 
 begin with the fourth year and continue through the eighth year. Part I, 30 cts.; Part 
 II, 65 cts. 
 
 Badlam's Aids tO Number. Teacher's edition First series, Nos. i to 10, 40 cts.; 
 Second series, Nos. 10 to 20, 40 cts. Pupil's edition First series, 25 cts.; Second 
 series, 25 cts. 
 
 Branson's Methods in Teaching Arithmetic. 15 cts. 
 
 Hanus's Geometry in the Grammar Schools. An essay, with outline of work for 
 
 the last three years of the grammar school. 25 cts. 
 
 HOWland's Drill Cards. For middle grades in arithmetic. Each, 3 cts.; per hun- 
 dred, $2.40. 
 
 Hunt's Geometry for Grammar Schools. The definitions and elementary con- 
 cepts are to be taught concretely, by much measuring, and by the making of models 
 and diagrams by the pupils. 30 cts. 
 
 Pierce's Review Number Cards. Two cards, for second and third year pupils. 
 
 Each, 3 cts.; per hundred, $2.40. 
 Safford'S Mathematical Teaching. A monograph, with applications. 25 cts. 
 
 Sloane's Practical Lessons in Fractions. 25 cts. Set of six fraction cards, for 
 
 pupils to cut. 10 cts. 
 
 Sutton and Kimbrough's Pupils' Series of Arithmetics. Lower Book, for 
 
 primary and intermediate grades, 35 cts. Higher Book, 65 cts. 
 
 The New Arithmetic. By 300 teachers. Little theory and much practice. An excel- 
 lent review book. 65 cts. 
 
 Walsh's Arithmetics. On the "spiral advancement" plan, and perfectly graded. 
 Special features of this series are its division into half-yearly chapters instead of the 
 arrangement by topics; the great number and variety of the problems ; the use of the 
 equation in solution of arithmetical problems; and the introduction of the elements of 
 algebra and geometry. Its use shortens and enriches the course in common school 
 mathematics. In two series: 
 
 Three Book Series Elementary, 30 cts.; Intermediate, 35 cts.; Higher, 65 cts. 
 Two Book Series Primary, 30 cts.; Grammar school, 65 cts. 
 
 Walsh's Algebra and Geometry for Grammar Grades. Three chapters from 
 
 Walsh's Arithmetic printed separately. 15 cts. 
 
 White's TWO Years With Numbers. For second and third year classes. 35.0*5. 
 White's Junior Arithmetic. For fourth and fifth years. 45 cts. 
 
 White's Senior Arithmetic. 6 5 cts. 
 
 For advanced -works see our list of books in Mathematics. 
 
 D.C. HEATH & CO., Publishers, Boston, New York, Chicago 
 

THE WALSH ARITHMETICS 
 
 SUGGESTIONS TO TEACHERS 
 
 AND 
 
 OUTLINES OF COURSES OF STUDY 
 
 THREE-BOOK f E " ementar y Arithmetic . . . Introduction price,. 30 cents 
 SERIES l Interme( ^ iate Arithmetic . . ** " 35 " 
 
 I Higher Arithmetic * " 65 " 
 
 TWO-BOOK [ Primary Arithmetic Introduction price, 30 cents 
 
 SERIES 1 Grammar School Arithmetic, " * 65 u 
 
 D. C. HEATH & CO., PUBLISHERS 
 
 BOSTON NEW YORK CHICAGO LONDON 
 
CONTENTS. PART I. 
 
 CHAPTER I. 
 
 PAOB 
 
 ADDITION AND SUBTRACTION 1 
 
 NOTATION AND NUMERATION TO 99 ..-.*, 3 
 
 Addition 3 
 
 Subtraction 9 
 
 Addition ... . 15 
 
 NOTATION AND NUMERATION TO 999 18 
 
 Subtraction 25 
 
 Drills 28 
 
 CHAPTER II. 
 
 MULTIPLICATION AND DIVISION 36 
 
 Multiplication by 2 36 
 
 Division by 2 38 
 
 NOTATION AND NUMERATION TO 9,999 42 
 
 Multiplication by 3 50 
 
 Drills 51 
 
 Division by 3 ,..-.... 54 
 
 Multiplication by 4 57 
 
 Division by 4 59 
 
 Multiplication by 5 . 62 
 
 Division by 5 63 
 
 United States Money 64 
 
 Addition and Subtraction 65 
 
 Fractional Parts of Numbers 67 
 
 Roman Notation . 68 
 
 Liquid Measure *.,.,. 69 
 
VI CONTENTS. PART I. 
 
 CHAPTER III. 
 
 PAGE 
 
 MULTIPLICATION AND DIVISION OUNCE AND POUND Two OPERA- 
 TIONS HALVES, THIRDS, FOURTHS MULTIPLICATION BY A 
 
 MIXED NUMBER 72 
 
 Multiplication by 6 72 
 
 Division by 6 75 
 
 Quotients and Remainders 80 
 
 Multiplication by a Mixed Number 81 
 
 NOTATION AND NUMERATION TO 99,999 83 
 
 Multiplication by 7 85 
 
 Division by 7 85 
 
 Multiplication by 8 90 
 
 Division by 8 90 
 
 Multiplication by 9 94 
 
 Division by 9 . . . , 94 
 
 Multiplication by 10 ...... 98 
 
 Division by 10 98 
 
 Special Drills 99 
 
 Halves (Addition and Subtraction) 105 
 
 Fourths (Addition and Subtraction) . 106 
 
 Thirds (Addition and Subtraction) 107 
 
 CHAPTER IV. 
 
 MULTIPLIERS AND DIVISORS or Two OR MORE FIGURES MULTI- 
 PLIERS CONTAINING FRACTIONS ADDITION AND SUBTRACTION 
 
 OF EASY MIXED NUMBERS INCH, FOOT. AND YARD . . . 110 
 
 Halves and Fourths (Addition and Subtraction) 110 
 
 Multiplication by 11 and 12 113 
 
 Division by 11 and 12 114 
 
 Multipliers ending in 118 
 
 Divisors ending in 118 
 
 Long Measure 122 
 
 Multipliers of Two Digits , 122 
 
CONTENTS. PART I. vii 
 
 FAGB 
 
 Long Division 124 
 
 Special Drills 127 
 
 Halves, Fourths, and Eighths 129 
 
 Multipliers ending with Ciphers 144 
 
 Halves, Thirds, and Sixths 145 
 
 Long Division Drills . . 148 
 
 Divisors ending with Ciphers 150 
 
 Thirds and Ninths . ,....' 154 
 
 Special Drills . . ". . . . . . . ... 156 
 
 Multipliers of More than Two Figures ..-...... . . 157 
 
 CHAPTER V. 
 
 MULTIPLIERS AND DIVISORS OP THREE OR MORE FIGURES ADDI- 
 TION AND SUBTRACTION OF EASY FRACTIONS MULTIPLICATION 
 
 BY A MIXED NUMBER EASY DENOMINATE NUMBERS . . . 160 
 
 Multiplication by a Mixed Number . . . 160 
 
 Long Division 161 
 
 Special Drills 162 
 
 Mixed Numbers (Addition and Subtraction) ........ 167 
 
 Dry Measure 170 
 
 NOTATION AND NUMERATION TO 999,999 172 
 
 More than One Operation 177 
 
 Easy Fractions (Addition and Subtraction) . . , 179 
 
 Short Methods ..'.% / 181 
 
 Halves and Fifths .' ." 184 
 
 Fourths and Fifths 184 
 
 Long Division, Drills 186 
 
 Thirds and Fourths 192 
 
 Denominate Numbers 194 
 
 Thirds and Fifths ....... , 201 
 
 Special DrUls . . . . v 203 
 
 Roman Notation . . 211 
 
CONTENTS. PART II. 
 
 CHAPTER VI. 
 
 PAGE 
 
 MIXED NUMBERS FEDERAL MONEY BILLS DENOMINATE NUM- 
 BERS DECIMALS MEASUREMENTS. . . . .; 213 
 
 MIXED NUMBERS . .. . . .... 213 
 
 Addition of Mixed Numbers , . 217 
 
 Multiples and Factors 218 
 
 Subtraction of Mixed Numbers 220 
 
 NOTATION AND NUMERATION 224 
 
 Multiplication of Mixed Numbers 231 
 
 Division of Mixed Numbers . 234 
 
 FEDERAL MONEY 236 
 
 Fractional Parts of a Dollar . . . . . . . . . . . . . . 239 
 
 Division of Federal Money 245 
 
 Sight Approximations i . . 247 
 
 DENOMINATE NUMBERS 250 
 
 Time Measure .. . 250 
 
 Dry Measure . 251 
 
 Avoirdupois Weight .,... 251 
 
 Liquid Measure 251 
 
 Special Drills . 254 
 
 BILLS . . . | . . 263 
 
 DECIMALS .....* . . 264 
 
 Notation and Numeration . . . . . . .... . ... . 265 
 
 Addition of Decimals . . . ..... . . . ... . . . 267 
 
 Subtraction of Decimals 268 
 
 Multiplication of a Decimal by an Integer 268 
 
 Division of a Decimal by an Integer 270 
 
 MEASUREMENTS 277 
 
 v 
 
VI CONTENTS. PART II. 
 
 CHAPTER VII. 
 
 PAGS 
 
 FRACTIONS DECIMALS BILLS DENOMINATE NUMBERS MEAS- 
 UREMENTS 280 
 
 ADDITION or FRACTIONS . * 280 
 
 SUBTRACTION OF FRACTIONS 281 
 
 Factors and Multiples . 281 
 
 Prime Numbers 282 
 
 Greatest Common Divisor 283 
 
 Lowest Terms . . V . . 283 
 
 Least Common Multiple 285 
 
 ADDITION AND SUBTRACTION OF FRACTIONS 286 
 
 Special Drills ...... , . 291 
 
 Cancellation 296 
 
 MULTIPLICATION OF FRACTIONS 298 
 
 DIVISION OF FRACTIONS 300 
 
 Fractional Parts of a Dollar . . 309 
 
 BILLS 311 
 
 Short Methods 316 
 
 MULTIPLICATION OF DECIMALS 318 
 
 DIVISION OF DECIMALS - . . . 319 
 
 Sight Approximations 322 
 
 DENOMINATE NUMBERS 327 
 
 Long Measure 327 
 
 MEASUREMENTS 330 
 
 CHAPTER VIII. 
 DECIMALS BILLS DENOMINATE NUMBERS MEASUREMENTS 
 
 PERCENTAGE INTEREST ...;... 337 
 
 DECIMALS 337 
 
 Reduction '.......... 337 
 
 Addition ..."..-.. 338 
 
 Subtraction 339 
 
 Multiplication . . . . . ; ; . 339 
 
 Division . , 339 
 
CONTENTS. PART II. Vll 
 
 PAGE 
 
 MEASUREMENTS 345 
 
 Special Drills 346 
 
 Short Methods 352 
 
 Approximations . 355 
 
 DENOMINATE NUMBERS . . . ; / 357 
 
 Reduction Descending 364 
 
 Reduction Ascending 366 
 
 Addition and Subtraction > : . <" "." . . . 367 
 
 Multiplication and Division . . . . ... . . . . . . . 370 
 
 PERCENTAGE . .' ; . 372 
 
 BILLS > ''. . . . \ . . 376 
 
 INTEREST ; . , . 377 
 
 AREAS OF RIGHT-ANGLED TRIANGLES - . .^ . . . 379 
 
 Short Methods , 381 
 
 CHAPTER IX. 
 
 DENOMINATE NUMBERS SURFACES AND VOLUMES PERCENTAGE 
 
 INTEREST 389 
 
 DENOMINATE NUMBERS 389 
 
 Reduction Ascending and Descending 389 
 
 Compound Addition 392 
 
 Compound Subtraction 394 
 
 Compound Multiplication 395 
 
 Compound Division 396 
 
 Special Drills . . . ." ; =. , .' .'. . ,-.v, ""-.-. . 400 
 
 Short Methods ... . .' , . '. . . . . . v ': . . ' . 403 
 
 Avoirdupois Weight (Long Ton) 404 
 
 MEASUREMENTS r !J '. . 408 
 
 Time between Dates . . . . . . 411 
 
 PERCENTAGE . . . .." . . . . . . . . 415 
 
 INTEREST. 415 
 
 Approximations 419 
 
Viii CONTENTS. PART II. 
 
 PAGE 
 
 SURFACES 419 
 
 Square Measure 420 
 
 VOLUMES 424 
 
 Approximations .,... 426 
 
 Cubic Measure 427 
 
 Troy Weight 427 
 
 ANGLES, TRIANGLES, QUADRILATERALS 443 
 
 Areas 445 
 
 CHAPTER X. 
 
 ALGEBRAIC EQUATIONS 447 
 
 ONE UNKNOWN QUANTITY 447 
 
 Clearing of Fractions 451 
 
 Transposing 455 
 
CONTENTS. PART III. 
 
 CHAPTER XI. 
 
 PAGE 
 
 PERCENTAGE INTEREST DISCOUNT SURFACES AND VOLUMES . . 459 
 
 PERCENTAGE 459 
 
 To find the Base or the Rate 461 
 
 Profit and Loss 464 
 
 MEASUREMENTS 467 
 
 INTEREST 471 
 
 Interest-bearing Notes . 472 
 
 Special Drills 481 
 
 Approximations 483 
 
 Short Methods . 484 
 
 BANK DISCOUNT 489 
 
 Discount of Interest bearing Notes 503 
 
 English Money 504 
 
 COMMERCIAL DISCOUNT 509 
 
 SURFACES AND VOLUMES . . 517 
 
 CHAPTER XII. 
 
 SIMPLE AND COMPOUND INTEREST DISCOUNT CAUSE AND EFFECT 
 PARTNERSHIP BONDS AND STOCKS EXCHANGE LONGITUDE 
 
 AND TIME SURFACES AND VOLUMES 519 
 
 SIMPLE INTEREST 519 
 
 To find Principal, Rate, or Time 519 
 
 Interest by Aliquot Parts 523 
 
 V 
 
Vi CONTENTS. PART III. 
 
 PAGE 
 
 COMMERCIAL DISCOUNT 529 
 
 BANK DISCOUNT 534 
 
 To find Face of Note, Rate of Discount, or Time 534 
 
 Special Drills 537 
 
 Short Methods 542 
 
 MEASUREMENTS 550 
 
 CAUSE AND EFFECT 553 
 
 PARTNERSHIP 558 
 
 Approximations 562 
 
 BONDS AND STOCKS 562 
 
 COMPOUND INTEREST 565 
 
 EXCHANGE 575 
 
 Domestic Sight Exchange 576 
 
 Circular Measure 578 
 
 Time Drafts 579 
 
 LONGITUDE AND TIME 580 
 
 Bills of Exchange (Foreign) 585 
 
 CHAPTER XIII. 
 
 PARTIAL PAYMENTS RATIO AND PROPORTION SQUARE ROOT 
 
 SURFACES AND VOLUMES 588 
 
 PARTIAL PAYMENTS U. S. RULE 588 
 
 Present Worth and True Discount 592 
 
 SURFACES AND VOLUMES 601 
 
 SQUARE ROOT 607 
 
 RATIO 610 
 
 Special Drills 620 
 
 PROPORTION 624 
 
 Applications of Square Root 634 
 
 MEASUREMENTS 643 
 
 Exact Interest 648 
 
 PARTIAL PAYMENTS MERCHANTS' RULE . . 653 
 
CONTENTS. PART III. Vll 
 CHAPTER XIV. 
 
 PAGE 
 
 EQUATION OF PAYMENTS MENSURATION OF SURFACES AND VOL- 
 UMES BOARD MEASURE ANNUAL INTEREST GOVERNMENT 
 
 LANDS METRIC SYSTEM 657 
 
 EQUATION OF PAYMENTS 657 
 
 MENSURATION OF PLANE SURFACES 667 
 
 Special Drills '. .... . . . . 674 
 
 SURFACES OF SOLIDS 681 
 
 Prisms and Cylinders 681 
 
 Pyramids and Cones 682 
 
 VOLUMES . . . . . : .... ....... 688 
 
 Lumber Measure 692 
 
 Surface of Sphere 697 
 
 CUBE ROOT '. . . . . 699 
 
 Volume of Sphere 701 
 
 ANNUAL INTEREST . , . 719 
 
 Government Lands . 720 
 
 METRIC SYSTEM . 721 
 
 CHAPTER XV. 
 
 ALGEBRAIC EQUATIONS Two UNKNOWN QUANTITIES THREE UN- 
 KNOWN QUANTITIES PURE QUADRATICS AFFECTED QUAD- 
 RATICS . .... . ...... . . ... . v. . . 728 
 
 ADDITION OF ALGEBRAIC QUANTITIES , . - . 728 
 
 SUBTRACTION OF ALGEBRAIC QUANTITIES . 730 
 
 Removing Parentheses ' - . . . 732 
 
 Two UNKNOWN QUANTITIES . . 736 
 
 
 
 THREE UNKNOWN QUANTITIES .- ., . . . < . . . 742 
 
 MULTIPLICATION OF ALGEBRAIC QUANTITIES 745 
 
 PURE QUADRATICS 747 
 
 AFFECTED QUADRATICS 749 
 
Vlll CONTENTS. PART III. 
 
 CHAPTER XVI. 
 
 PAQI 
 
 ELEMENTARY GEOMETRT PROBLEMS IN CONSTRUCTION PRACTICAL 
 APPLICATIONS CALCULATION OF HEIGHTS AND DISTANCES 
 
 MENSURATION 755 
 
 ELEMENTARY GEOMETRY 755 
 
 Exercises in Construction 757 
 
 Problems in Construction 773 
 
 Equal Triangles Equivalent Triangles 782 
 
 Similar Triangles - 784 
 
 CALCULATION OF HEIGHTS AND DISTANCES. ......... 785 
 
 MENSURATION OF SURFACES 790 
 
 Prisms, Cylinders, Pyramids, Cones 792 
 
 Frustum of Pyramid or Cone .............. 794 
 
 Sphere . . 796 
 
 VOLUMES 797 
 
 Prisms and Cylinders 797 
 
 Pyramids and Cones ........... 798 
 
 Frustums of Pyramids and Cones 799 
 
 Oblique Prisms 802 
 
 Sphere 802 
 
SUGGESTIONS TO TEACHERS 
 
 INTRODUCTORY 
 
 Plan and Scope of the Work, In addition to the subjects 
 generally included in text-books in arithmetic, The Walsh Arith- 
 metics contain such simple work in algebraic equations and con- 
 structive geometry as can be studied to advantage by pupils of 
 the elementary schools. The arithmetical portion is divided 
 into thirteen chapters, each of which, except the first, contains 
 a half-year's work. The following nine-year and eight-year 
 courses will show the arrangement of topics : 
 
 NINE-YEAR COURSE 
 FIRST AND SECOND YEARS 
 Chapter I, Numbers of Three Figures. Simple Processes. 
 
 THIRD YEAR 
 
 Chapters II, and IIL Numbers of Five Figures. Multipliers 
 and Divisors of One Figure. Addition and Subtraction of Halves, of 
 Fourths, of Thirds. Multiplication by Mixed Numbers. Pint, Quart, 
 and Gallon; Ounce and Pound. Roman Notation. 
 
 FOURTH YEAR 
 
 Chapters IV, and V, Numbers of Six Figures. Multipliers and 
 Divisors of Two or More Figures. Addition and Subtraction of Easy 
 Fractions. Multiplication by Mixed Numbers. Simple Denominate 
 Numbers. Roman Notation. 
 
 FIFTH YKAK 
 
 Chapters VI, and VII, Fractions. Decimals of Three Places. 
 Bills. Denominate Numbers. Simple Measurements. 
 
 SIXTH YKAU 
 
 Chapters VIII. and IX, Decimals. Bills. Denominate Num- 
 bers. Surfaces and Volumes. Percentage and Interest. 
 
 1 
 
2 SUGGESTIONS TO TEACHERS 
 
 SEVENTH YEAR 
 
 Chapters X, and XI, and Articles 931 to 963 in Chapter XII,, 
 and Articles 1251 to 1269 in Chapter XVI, Percentage. Meas- 
 urements. Interest. Discount. Surfaces and Volumes. Elementary 
 Algebra and Geometry. Exercises and Problems. 
 
 EIGHTH YEAR 
 
 Chapter XII,, Articles 964 to 1007, Chapter XIII,, and Article 
 1270 of Chapter XVI, Partnership. Bonds and Stocks. Com- 
 pound Interest. Exchange. Longitude and Time. Partial Pay- 
 ments. Surfaces and Volumes. Square Root. Ratio. Proportion. 
 Measurements. Elementary Geometry. Problems in Construction. 
 
 NINTH YEAR 
 
 Chapters XIV, and XV,, and Chapter XVI, completed, Equa- 
 tion of Payments. Mensuration of Plane Surfaces and Volumes. 
 Cube Root. Annual Interest. Metric System. Elementary Algebra. 
 Elementary Geometry. Calculation of Heights and Distances. 
 
 EIGHT- YEAR COURSE 
 FIRST, SECOND, THIRD, AND FOURTH YEARS 
 
 As in nine-year course. 
 
 FIFTH YEAR 
 
 Chapters VI, and VII, Fractions. Decimals of Three Places. 
 Bills. Denominate Numbers. Simple Measurements. 
 
 SIXTH YEAR 
 
 Chapters VIII, and IX, Decimals. Bills. Denominate Num- 
 bers. Surfaces and Volumes. Percentage and Interest. 
 
 SEVENTH YEAR 
 
 Chapters XI, and XII, Percentage and Interest. Commercial 
 and Bank Discount. Cause and Effect. Partnership. Bonds and 
 Stocks. Exchange. Longitude and Time. Surfaces and Volumes. 
 
 EIGHTH YEAR 
 
 Chapters XIII, and XIV, Partial Payments. Equation of Pay- 
 ments. Annual Interest. Metric System. Evolution and Involution. 
 Surfaces and Volumes. 
 
INTRODUCTORY 3 
 
 While all of the above topics are generally included in an 
 eight years' course, it may be considered advisable to omit some 
 of them, and to take up, instead, during the seventh and eighth 
 years, the constructive geometry work of Chapter XVI. Among 
 the topics that may be dropped without injury to the pupil are 
 Bonds and Stocks, Exchange, Partial Payments, and Equation 
 of Payments. 
 
 Grammar School Algebra. Chapter X., consisting of a dozen 
 pages, is devoted to the subject of easy equations of one unknown 
 quantity, as a preliminary to the employment of the equation in 
 so much of the subsequent work in arithmetic as is rendered 
 more simple by this mode of treatment. To teachers desirous 
 of dispensing with rules, sample solutions of type examples, etc., 
 the algebraic method of solving the so-called " problems " in per- 
 centage, interest, discount, etc., is strongly recommended. 
 
 In Chapter XV., intended chiefly for schools having a nine 
 years' course, the algebraic work is extended to cover simple 
 equations containing two or more unknown quantities, and pure 
 and affected quadratic equations of one unknown quantity. 
 
 No attempt has been made in these two chapters to treat 
 algebra as a science ; the aim has been to make grammar-school 
 pupils acquainted, to some slight extent, with the great instru- 
 ment of mathematical investigation, the equation. 
 
 Constructive Geometry, Progressive teachers will appreciate the 
 importance of supplementing the concrete geometrical instruction 
 now given in the drawing and mensuration work. Chapter XVI. 
 contains a series of problems in construction so arranged as to 
 enable pupils to obtain for themselves a working knowledge of 
 all the most important facts of geometry. Applications of the 
 facts thus ascertained, are made to the mensuration of surfaces 
 and volumes, the calculation of heights and distances, etc. No 
 attempt is made to anticipate the work of the high-school by 
 teaching geometry as a science. 
 
4 MANUAL FOR TEACHERS 
 
 While the construction problems are brought together into a 
 single chapter at the end of the book, it is not intended that 
 instruction in geometry should be delayed until the preceding 
 work is completed. Chapter XVI. should be commenced not later 
 than the seventh year, and should be continued throughout the 
 remainder of the grammar-school course. For the earlier years, 
 suitable exercises in the mensuration of the surfaces of triangles 
 and quadrilaterals, and of the volumes of right parallelopipedons 
 have been incorporated with the arithmetic work. 
 
II 
 
 GENERAL HINTS 
 
 Division of the Work, The five chapters constituting Part I. 
 of Mathematics for Common Schools should be completed by the 
 end of the fourth school year. The remaining eight arithmetic 
 chapters constitute half-yearly divisions for the second four years 
 of school. Chapter L, with the additional oral work needed in 
 the case of young pupils, will occupy about two years ; the re- 
 maining four chapters should not take more than half a year each. 
 When the Grube system is used, and the work of the first two 
 years is exclusively oral, it will be possible, by omitting much of 
 the easier portions of the first two chapters, to cover, during the 
 third year, the ground contained in Chapters I., II., and III. 
 
 Additions and Omissions. The teacher should freely supple- 
 ment the work of the text-book when she finds it necessary to do 
 so ; and she should not hesitate to leave a topic that her pupils 
 fully understand, even though they may not have worked all the 
 examples given in connection therewith. A very large number 
 of exercises is necessary for such pupils as can devote a half-year 
 to the study of the matter furnished in each chapter. In the 
 case of pupils of greater maturity, it will be possible to make 
 more rapid progress by passing to the next topic as soon as the 
 previous work is fairly well understood. 
 
 Oral and Written Work. The heading "Slate Problems" is 
 merely a general direction, and it should be disregarded by the 
 teacher when the pupils are able to do the work " mentally." 
 The use of the pencil should be demanded only so far as it mny 
 
 5 
 
6 MANUAL FOR TEACHERS 
 
 be required. It is a pedagogical mistake to insist that all of tlie 
 pupils of a class should set down a number of figures that are 
 not needed by the brighter ones. As an occasional exercise, it 
 may be advisable to have scholars give all the work required to 
 solve a problem, and to make a written explanation of each step 
 in the solution ; but it should be the teacher's aim to have the 
 majority of the examples done with as great rapidity as is con- 
 sistent with absolute correctness. It will be found that, as a 
 rule, the quickest workers are the most accurate. 
 
 Many of the slate problems can be treated by some classes as 
 " sight " examples, each pupil reading the question for himself 
 from the book, and writing the answer at a given signal without 
 putting down any of the work. 
 
 Use of Books, It is generally recommended that books be 
 placed in pupils' hands as early as the third school year. Since 
 many children are unable at this stage to read with sufficient 
 intelligence to understand the terms of a problem, this work 
 should be done under the teacher's direction, the latter reading 
 the questions while the pupils follow from their books. In later 
 years, the problems should be solved by the pupils from the 
 books with practically no assistance whatever from the teacher. 
 
 Conduct of the Eecitation, Many thoughtful educators consider 
 it advisable to divide an arithmetic class into two sections, for 
 some purposes, even where its members are nearly equal in 
 attainments. The members of one division of such a class may 
 work examples from their books while the others write the 
 answers to oral problems given by the teacher, etc. 
 
 Where a class is thus taught in two divisions, the members of 
 each should sit in alternate rows, extending from the front 
 of the room to the rear. Seated in this way, a pupil is doing a 
 different kind of work from those on the right and the left, and 
 he would not have the temptation of a neighbor's slate to lead 
 him to compare answers. 
 
GENERAL HINTS 7 
 
 As an economy of time, explanations of new subjects might be 
 given to the whole class; but much of the arithmetic work 
 should be done in "sections," one of which is under the im- 
 mediate direction of the teacher, the other being employed 
 in "seat" work. In the case of pupils of the more advanced 
 classes, "seat" work should consist largely of "problems" solved 
 without assistance. Especial pains have been taken to so grade 
 the problems as to have none beyond the capacity of the average 
 pupil that is willing to try to understand its terms. It is not 
 necessary that all the members of a division should work the 
 same problems at a given time, nor the same number of prob- 
 lems, nor that a new topic should be postponed until all of the 
 previous problems have been solved. 
 
 Whenever it is possible, all of the members of the division 
 working under the teacher's immediate direction should take 
 part in all the work done. In mental arithmetic, for instance, 
 while only a few may be called upon for explanations, all of the 
 pupils should write the answers to each question. The same is 
 true of much of the sight work, the approximations, some of the 
 special drills, etc. 
 
 Drills and Sight Work. To secure reasonable rapidity, it is 
 necessary to have regular systematic drills. They should be 
 employed daily, if possible, in the earlier years, but should never 
 last longer than five or ten minutes. Various kinds are sug- 
 gested, such as sight addition drills, in Arts. 3, 11, 24, 26, etc. ; 
 subtraction, in Arts. 19, 50, 53, etc. ; multiplication, in Arts. 71, 
 109, etc. ; division, in Arts. 199, 202, etc. ; counting by 2's, 3's, 
 etc., in Art. 61 ; carrying, in Art. 53, etc. For the young pupil, 
 those are the most valuable in which the figures are in his sight, 
 and in the position they occupy in an example ; see Arts. 3, 34, 
 164, etc. 
 
 Many teachers prepare cards, each of which contains one of 
 the combinations taught in their respective grades. Showing 
 one of these cards, the teacher requires an immediate answer 
 
8 MANUAL FOR TEACHERS 
 
 from a pupil. If his reply is correct, a new card is shown to 
 the next pupil, and so on. Other teachers write a number of 
 combinations on the blackboard, and point to them at random, 
 requiring prompt answers. When drills remain on the board 
 for any considerable time, some children learn to know the 
 results of a combination by its location on the board, so that 
 frequent changes in the arrangement of the drills are, therefore, 
 advisable. The drills in Arts. Ill, 112, and 115 furnish a great 
 deal of work with the occasional change of a single figure. 
 
 For the higher classes, each chapter contains appropriate 
 drills, which are subsequently used in oral problems. It happens 
 only too frequently that as children go forward in school they 
 lose much of the readiness in oral and written work they 
 possessed in the lower grades, owing to the neglect of their 
 teachers to continue to require quick, accurate review work in 
 the operations previously taught. These special drills follow 
 the plan of the combinations of the earlier chapters, but gradu- 
 ally grow more difficult. They should first be used as sight 
 exercises, either from the books or from the blackboard. 
 
 To secure valuable results from drill exercises, the utmost 
 possible promptness in answers should be insisted upon. 
 
 Definitions, Principles, and Eules, Young children should not 
 memorize rules or definitions. They should learn to add by 
 adding, after being first shown by the teacher how to perform 
 the operation. Those not previously taught by the Grube 
 method should be given no reason for " carrying." In teaching 
 such children to write numbers of two or three figures, there is 
 nothing gained by discussing the local value of the digits. Dur- 
 ing the earlier years, instruction in the art of arithmetic should 
 be given with the least possible amount of science. While prin- 
 ciples may be incidentally brought to the view of the children 
 at times, there should be no cross-examination thereon. It may 
 be shown, for instance, that subtraction is the reverse of addition, 
 and that multiplication is a short method of combining equal 
 
GENERAL HINTS 9 
 
 numbers, etc. ; but care should be taken in the case of pupils 
 below about the fifth school year not to dwell long on this side 
 of the instruction. By that time, pupils should be able to add, 
 subtract, multiply, and divide whole numbers ; to add and sub- 
 tract simple mixed numbers, and to use a mixed number as a 
 multiplier or a multiplicand ; to solve easy problems, with small 
 numbers, involving the foregoing operations and others contain- 
 ing the more commonly used denominate units. Whether or not 
 they can explain the principles underlying the operations is of 
 next to no importance, if they can do the work with reasonable 
 accuracy and rapidity. 
 
 When decimal fractions are taken up, the principles of Arabic 
 notation should be developed ; and about the same time, or some- 
 what later, the principles upon which are founded the operations 
 in the fundamental processes, can be briefly discussed. 
 
 Definitions should in all cases be made by the pupils, their 
 mistakes being brought out by the teacher through appropriate 
 questions, criticisms, etc. Systematic work under this head 
 should be deferred until at least the seventh year. 
 
 The use of unnecessary rules in the higher grades is to be 
 deprecated. When, for instance, a pupil understands that per 
 cent means hundredths, that seven per cent means seven hun- 
 dredths, it should ot be necessary to tell him that 7 per cent of 
 143 is obtained by multiplying 143 by .07. It should be a fair 
 assumption that his previous work in the multiplication of 
 common and of decimal fractions has enabled him to see that 
 7 per cent of 143 is -j-J^ of 143 or 143 X .07, without information 
 other than the meaning of the term " per cent." 
 
 When a pupil is able to calculate that 15% of 120 is 18, he 
 should be allowed to try to work out for himself, without a rule, 
 the solution of this problem : 18 is what per cent of 120 ? or of 
 this: 18 is 15% of what number? These questions should 
 present no more difficulty in the seventh year than the following 
 examples in the fifth : (a) Find the cost of -fo ton of hay at $12 
 per ton. (b) When hay is worth $12 per ton, what part of a 
 
10 MANUAL FOR TEACHERS 
 
 ton can be bought for $ 1.80 ? (c) If ^ ton of hay costs $1.80, 
 what is the value of a ton ? 
 
 When, however, it becomes necessary to assist pupils in the 
 solution of problems of this class, it is more profitable to furnish 
 them with a general method by the use of the equation, than 
 with any special plan suited only to the type under immediate 
 discussion. 
 
 In the supplement to the Manual will be found the usual defini- 
 tions, principles, and rules, for the teacher to use in such a way 
 as her experience shows to be best for her pupils. The rules 
 given are based somewhat on the older methods, rather than on 
 those recommended by the author. He would prefer to omit 
 entirely those relating to percentage, interest, and the like as 
 being unnecessary, but that they are called for by many success- 
 ful teachers, who prefer to continue the use of methods which 
 they have found to produce satisfactory results. 
 
 Language, While the use of correct language should be 
 insisted upon in all lessons, children should not be required in 
 arithmetic to give all answers in " complete sentences." Espe- 
 cially in the drills, it is important that the results be expressed 
 in the fewest possible words. 
 
 Analyses, Sparing use of analyses is recommended for begin- 
 ners. If a pupil solves a problem correctly, the natural inference 
 should be that his method is correct, even if he be unable to state 
 it in words. When a pupil gives the analysis of a problem, he 
 should be permitted to express himself in his own way. Set 
 forms should not be used under any circumstances. 
 
 Objective Illustrations, The chief reason for the use of objects 
 in the study of arithmetic is to enable pupils to work without 
 them. While counters, weights and measures, diagrams, or the 
 like are necessary at the beginning of some topics, it is important 
 to discontinue their use as soon as the scholar is able to proceed 
 without their aid. 
 
GENERAL HINTS 11 
 
 Approximate Answers. An important drill is furnished in 
 the "approximations." (See Arts. 521, 669, 719, etc.) Pupils 
 should be required in much of their written work to estimate 
 the result before beginning to solve a problem with the pencil. 
 Besides preventing an absurd answer, this practice will also have 
 the effect of causing a pupil to see what processes are necessary. 
 In too many instances, work is commenced upon a problem before 
 the conditions are grasped by the youthful scholar ; which will 
 be less likely to occur in the case of one who has carefully 
 "estimated" the answer. The pupil will frequently find, also, 
 that he can obtain the correct result without using his pencil 
 at all. 
 
 Indicating Operations, It is a good practice to require pupils 
 to indicate by signs all of the processes necessary to the solution 
 of a problem, before performing any of the operations. This fre- 
 quently enables a scholar to shorten his work by cancellation, etc. 
 In the case of problems whose solution requires tedious processes, 
 some teachers do not require their pupils to do more than to 
 indicate the operations. It is to be feared that much of the lack 
 of facility in adding, multiplying, etc., found in the pupils of 
 the higher classes is due to this desire to make work pleasant. 
 Instead of becoming more expert in the fundamental operations, 
 scholars in their eighth year frequently add, subtract, multiply, 
 and divide more slowly and less accurately than in their fourth 
 year of school. 
 
 Paper vs. Slates, To the use of slates may be traced very much 
 of the poor work now done in arithmetic. A child that finds the 
 sum of two or more numbers by drawing on his slate the number 
 of strokes represented by each, and then counting the total, will 
 have to adopt some other method if his work is done on material 
 that does not permit the easy obliteration of the tell-tale marks. 
 When the teacher has an opportunity to see the number of 
 attempts made by some of her pupils to obtain the correct quo- 
 
12 MANUAL FOR TEACHERS 
 
 tient figures in a long division example, she may realize the 
 importance of such drills as will enable them to arrive more 
 readily at the correct result. 
 
 The unnecessary work now done by many pupils will be very 
 much lessened if they find themselves compelled to dispense with 
 the " rubbing out " they have an opportunity to indulge in when 
 slates are employed. The additional expense caused by the 
 introduction of paper will almost inevitably lead to better results 
 in arithmetic. The arrangement of the work will be looked 
 after ; pupils will not be required, nor will they be permitted, to 
 waste material in writing out the operations that can be per- 
 formed mentally ; the least common denominator will be deter- 
 mined by inspection ; problems will be shortened by the greater 
 use of cancellation, etc., etc. Better writing of figures and neater 
 arrangement of problems will be likely to accompany the use of 
 material that will be kept by the teacher for the inspection of 
 the school authorities. The endless writing of tables and the 
 long, tedious examples now given to keep troublesome pupils 
 from bothering a teacher that wishes to write up her records, 
 will, to some extent, be discontinued when slates are no longer 
 used. 
 
The Walsh Arithmetics. 
 
 IN several important particulars the Walsh Arithmetics mark a 
 departure from the traditional method and arrangement. 
 
 1. By the "spiral plan" the elements of all the important 
 topics are taken up early in the course, adding to the interest 
 and practical worth of the study. 
 
 2. In each case the subject taken up is not exhausted at 
 once, but practice in it is carried on with problems of gradually 
 increasing difficulty throughout the course. 
 
 3. Drills in addition, subtraction, multiplication and division 
 of abstract numbers are given at intervals throughout the books 
 of the series, thus insuring in pupils of the upper grades, ac- 
 curacy and speed in the fundamental processes. This is an im- 
 portant and unique feature. 
 
 4. The series contains a larger number of varied arid practi- 
 cal concrete problems than any other. 
 
 5. It is the only series containing drills in securing " approxi- 
 mate answers," work of great advantage in calling the pupil's 
 attention to the conditions of a problem, and thus giving the 
 power to detect at once the absurdity of any result greatly wide 
 of the mark. 
 
 Such obvious merits of the lower book as the alternation of 
 oral, sight and slate work, the early introduction of United States 
 currency (leading to decimals), the easy beginnings with frac- 
 tions and denominate numbers, and the freshness and interest 
 
insured by the great variety of means used to secure perfect 
 mastery of simple number combinations, cannot be too strongly 
 emphasized. 
 
 In the higher book we note the wide range of subjects treated 
 in their simple elements, the great variety of practical problems, 
 the early introduction of percentage and simple interest, of bills 
 and receipts, and all the matters connected with simple commer- 
 cial arithmetic. 
 
 Unique features are : the many short methods noted, the use of 
 approximate answers, the abundant drills in the four fundamental 
 processes, and the introduction of algebra in a way so natural 
 and simple that children of ten may easily grasp enough of it to 
 shorten many of the longer arithmetical processes. 
 
 The Walsh books illustrate most admirably what every teacher 
 knows so well, that many things that are complex in their com- 
 pleteness, are in their elements simple and easily comprehended 
 by young children. 
 
 The series is thoroughly up to the demands of modern peda- 
 gogy; it is inductive in method, practical and varied in .treat- 
 ment, and pursues one object from start to finish, i. e., to make 
 clear thought and accurate computation matters of habit, and to 
 lay the foundation for the intelligent use of mathematical princi- 
 ples. 
 
 A comparison of the number of subjects treated in any given 
 one hundred pages of Walsh with a corresponding one hundred 
 pages in any other series makes evident Walsh's superiority both 
 in variety and freshness, and in drill and review upon essentials. 
 
The Heart of Oak Books. 
 
 A collection of traditional rhymes and stories for children, and of mas- 
 terpieces of poetry and prose, for use at school and at home, chosen with 
 special reference to the cultivation of the imagination and a taste for good 
 reading. By 
 
 CHARLES ELIOT NORTON. 
 
 These six volumes provide an unrivaled means of making good reading 
 more attractive than bad, and of giving right direction to uncritical choice, 
 by offering to the young, without comment or lesson-book apparatus, 
 
 SELECTED PORTIONS OF THE BEST LITERATURE, THE VIRTUE OK 
 WHICH HAS BEEN APPROVED BY IX>NG CONSENT. 
 
 The selections are of unusual length, completeness and variety, compris- 
 ing a very large proportion of poetry, and are adapted to the progressive 
 needs of childhood and youth by a unique principle of selection, grading and 
 arrangement, which makes each volume a unit, and makes the series the first 
 permanent contribution to the body of school reading by a man of letters 
 which children will love and cherish after school-days are over. 
 
 THE FINE TASTE AND RARE LITERARY EXPERIENCE AND RESOURCES 
 
 of the editor are a guarantee that the series contains nothing but the very 
 best. No author's name or reputation has been potent enough to save from 
 rejection any selection that did not meet the editor's exacting standard in at 
 least three particulars : First, absolute truth to nature (especially nature in 
 America); second, wide, healthy, human interest; third, the highest possible 
 merit in point of literary form. The result, therefore, is a body of reading of 
 extraordinarily trustworthy character. The youth who shall become acquaint- 
 ed with the contents of these volumes will share in the common stock of the 
 intellectual life of the race, and will have the door opened to him of all the 
 vast and noble resources of that life. 
 
 FOR HOME USE, 
 
 even by children most favored by circumstance, these volumes provide the 
 richest store of thought and music to grow up with and to learn by heart. 
 No happier birthday or Christmas gift can be conceived, especially for children 
 in the country, or remote from libraries and other means of culture, than a set 
 of the Heart of Oak Books. They are a veritable possession forever, and 
 their price puts them within the reach of all. 
 
 Descriptive pamphlet giving prefaces, tables of contents, specimen pages, 
 and indexes of authors sent on application. 
 
 D. C. HEATH & CO., PUBLISHERS 
 
 BOSTON NEW YORK ATLANTA CHICAGO 
 
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THE WALSH 
 ARITHMETICS 
 
 WHAT THEY ARE AND 
 WHAT THEY WILL DO 
 
 ALSO 
 
 WHAT THEY HAVE DONE 
 AND ARE DOING, TOLD BY 
 THOSE WHO USE THEM 
 
 BOSTON 
 ATLANTA 
 
 PUBLISHED BY 
 D. C. HEATH & COMPANY 
 
 NEW YORK 
 SAN FRANCISCO 
 
 CHICAGO 
 LONDON 
 
MATHEMATICS 
 FOR COMMON SCHOOLS 
 
 A Graded Course in Arithmetic with 
 
 Simple Problems in Algebra 
 
 and Geometry 
 
 By JOHN H. WALSH 
 
 Associate Superintendent of Schools, Brooklyn 
 
 Arranged in Three=Part or Two=Part Series 
 
 The Three-Part Series 
 
 Elementary Arithmetic. Cloth. 218 pages. 30 cents 
 
 Intermediate Arithmetic. Cloth. 252 pages. 35 cents 
 
 Higher Arithmetic. Half leather. 365 pages. 65 cents 
 
 The Two-Part Series 
 
 Primary Arithmetic. Cloth. 198 pages. 30 cents 
 
 Grammar School Arithmetic. Half leather. 411 pages. 65 cents 
 
 In the Two-Part Series the examination papers 
 are omitted, considerably reducing the bulk, but in 
 no way interfering with the completeness of the course. 
 
 The Walsh Arithmetics 
 
 INSURE 
 
 Rapid and Accurate Computation 
 
 GIVE 
 
 Constant Review without Repetition 
 
 Abundant and Varied Problems 
 
 OMIT 
 
 Nothing Essential, Everything Else 
 
 ARE 
 
 Fresh, Well Graded, and Teachable 
 
 EMBODY 
 
 the Recommendations of the Committee of Ten 
 
 AS WELIv AS 
 
 of the Committee of Fifteen 
 
The Walsh Arithmetics 
 
 over twenty years before his series of arith- 
 metics were published John H. Walsh had been 
 a deep and philosophic student, both in and out of the 
 classroom, of the faults of the old system of presenting 
 arithmetic and of the features which should characterize 
 the modern and effective text-book. 
 
 How far he had advanced on the right road was 
 shown by the fact that, when the Report of the Com- 
 mittee of Ten was issued, his arithmetics, then in press, 
 were found to have anticipated and practically embodied 
 all the important recommendations of that report. 
 Again, with the recommendations of the Committee 
 of Fifteen, the agreement is quite as close. 
 
 These facts show that sound principles underlie 
 the Walsh Arithmetics. Further evidence is that 
 several series of arithmetics have appeared since, built 
 on the same principles, but it is worth noting that no 
 one of these imitators has produced books as faultless 
 as those of Mr. Walsh. 
 
Features of Especial Merit 
 
 Division of Work 
 
 'T^HE work of this arithmetic is divided into 
 sixteen chapters, each containing a half-year 
 course. The elements of all the important topics are 
 taken up early, and treated more fully in each succeed- 
 ing chapter. Advanced work in all the lines with 
 oral and written drill is given in each chapter with 
 special reviews. 
 
 No Unnecessary Rules 
 
 /CHILDREN learn the quick and accurate work- 
 ing of processes long before they can compre- 
 hend the underlying principles, and it is a mistake to 
 misuse valuable time upon the unnecessary memorizing 
 of rules and definitions. True education will stimu- 
 late the mental activity of the child by helping 
 him to work out his own rules inductively. He 
 will then stand on his feet securely, and what he 
 knows he will know well. The real educator should 
 be the living teacher ; the text-book but an instrument. 
 
 4 
 
The Method of Grading 
 
 'T'HE old method was to take up one topic 
 after another, beginning with addition, and 
 to exhaust each one before going on to the next. 
 Under this system many pupils who left school 
 before the end of the course had no knowledge at all 
 of certain subjects which would be of great practical 
 value to them as percentage, denominate numbers, 
 etc. On the other hand, experience has shown that 
 many graduates cannot add, subtract, multiply, and 
 divide with facility and accuracy, from too little practice 
 during the later years of school life. 
 
 Mr. Walsh has taken what may be called " the 
 spiral plan," or, as the French express it, the "concen- 
 tric circle method." The book is divided, not by 
 topics but by half-year courses. Practice in each 
 subject is carried on with problems of gradually 
 increasing difficulty through the whole course. Drill 
 in the four fundamental processes is found in all the 
 chapters. On the other hand, certain subjects, as 
 mensuration and denominate numbers, are begun much 
 earlier, in simple form, of course, than in other 
 books, so that pupils who do not finish the course 
 will have a much better grasp of the whole subject 
 of arithmetic and be better fitted to apply its 
 principles in the practical business of life. 
 
Great Number of Problems 
 
 TNCLUDING drill exercises and oral work, there 
 are over 7600 exercises and problems in the first 
 book alone. The second book contains a propor- 
 tionate amount of practice work. No other 
 arithmetic has nearly as many exercises and practical 
 problems. Teachers who use the Walsh Arithmetics 
 do not need to search for extra material to give their 
 classes sufficient practice. 
 
 Variety of Problems 
 
 UACH problem is unlike the previous one, and 
 will require the pupil to read it carefully. 
 He cannot work it by referring to a "sample" at 
 the head of the page. Variety is also given by 
 the problems in a large number of examination 
 papers from many sources, which are included in 
 the Three-Book Course. Papers are given which 
 have been used in national, state, and municipal 
 civil service examinations, as well as in school and 
 college examinations. 
 
Oral and Sight Work 
 
 A GREAT amount of oral work is given all 
 through these books, both under the several 
 topics discussed and in the continuous reviews. 
 The amount and variety of this oral work covers 
 all the ground of a mental arithmetic and answers 
 all the demands for a manual of this subject. 
 
 In addition to the usual oral work, practice in 
 sight work has been introduced. Problems are put 
 upon the blackboard by the teacher, the pupils 
 perform the necessary processes mentally and write 
 the answers as quickly as possible. This is a 
 helpful form of drill. 
 
 Approximate Answers 
 
 pRACTICE in approximate answers will prove 
 to be of the greatest practical value, for, in 
 business life, one cannot always stop to reckon 
 with paper and pencil, but must be ready to 
 estimate quickly the approximate result in order 
 to make a decision. By such practice in rapid 
 computation pupils learn to note all conditions of a 
 problem and soon detect at once the absurdity of 
 any result wide of the mark. This useful drill is 
 found in no other series of arithmetics. 
 
 7 
 
Continuous Reviews 
 
 HE RE are not only frequent but continuous 
 reviews, which constantly apply all that has 
 been learned without actual repetition of material 
 previously used. Pupils thus cannot lose their 
 readiness in the application of the simpler and 
 more fundamental processes. Each half-year's 
 work contains its own review chapter and the 
 lessons arranged for this purpose will be especially 
 appreciated by teachers. 
 
 Algebra Work 
 
 JV/FODERN educational methods, as recommended 
 by the various committees who have made 
 a special study of elementary school problems, all 
 require the introduction of simple algebra work in 
 the grammar grades. The algebra work included 
 in the Walsh Arithmetics, in Chapters X and XV, 
 is within the capacity of the average pupil and 
 sufficiently full to give him all the training which 
 it is desirable for him to have in this grade of work. 
 It perfectly meets the recommendations of the 
 Committee of Ten and the Committee of Fifteen. 
 To enable a teacher unfamiliar with this work to 
 do it successfully, he has only to follow the lines 
 laid down in the manual. 
 
 8 
 
Elementary Geometry 
 
 'pLEMENTARY Geometry was recommended by 
 the Committee of Ten above referred to and is 
 now generally taught in the upper grammar grades. 
 The chapter on geometry in the Walsh books con- 
 tains sufficient material for two years' work. Pupils 
 who have mastered the work presented in this chapter 
 will find little difficulty in solving the questions in 
 inventional geometry commonly offered in examina- 
 tion papers. 
 
 General Statement 
 
 TN the lower book the alternation of oral, sight, and 
 written work, the early introduction of United 
 States currency (leading to decimals), the easy begin- 
 nings with fractions and denominate numbers, are 
 advantages which cannot be too strongly emphasized. 
 
 In the higher book the range of subjects, the prac- 
 tical problems, the early introduction of percentage 
 and simple interest, of bills and receipts, and all the 
 matter connected with simple commercial arithmetic, 
 together with the short methods for approximate 
 answers, are notable points of superiority. 
 
 The series is thoroughly up to the demands of 
 modern pedagogy. It is inductive in method and 
 aims to develop clear thought and the power of 
 accurate computation and to lay the foundation for the 
 intelligent use of mathematical principles. 
 
 9 
 
Weighty Words of Educators 
 
 ALBERT LEONARD, Pres. Mich. Nor. Schools, Tpsilanti, Mich. 
 Walsh's Arithmetics embody the best ideas of modern educational 
 philosophy and are a distinct improvement upon the older text-books. 
 I do not know of any better books for school use than this series. 
 
 W. V. HAILMAN, Superintendent, Dayton, Ohio. 
 
 The Walsh Arithmetics are highly satisfactory to me in every 
 respect. They are eminently practical and free from pernicious 
 puzzles. I appreciate the attention which the books pay to the 
 place of arithmetic in mensuration and in industrial pursuits, and 
 the effective manner in which, in the advanced grades, they make 
 the transition to considerations of general arithmetic or algebra. 
 
 C. W. CRUIKSHANK, Superintendent, Fort Madison, Iowa. 
 
 The books are standing the test of the classroom admirably. I find 
 them well graded, am well pleased with the amount of work furnished, 
 and believe that the " Spiral Method" is the correct method. The 
 books require independent work on the part of the pupils. I am 
 growing to like them better as I see the results of their use in our 
 schools. 
 B. E. JACKSON, Superintendent, West Superior, Wis. 
 
 The Walsh Arithmetics have been in use in this city for a number 
 of years, and they stand higher in the estimation of the Board, 
 teachers, and patrons of the school than ever before. I consider 
 them the best series published, and heartily endorse the "Spiral 
 Plan ' ' as sound pedagogical ly. 
 W. McK. VANCE, Superintendent, Urbana, Ohio. 
 
 My teachers without exception are earnest admirers and hearty 
 supporters of the spiral plan of teaching arithmetic, as it appears 
 in the Walsh series. The problems are graded with such nicety 
 that the pupil meets every difficulty with an increasing sense of 
 power. I commend the books heartily and without reservation. 
 
 10 
 
I. C. PHILLIPS, Superintendent, Lewiston, Maine. 
 
 We have used the Walsh Arithmetics for three years. Teachers 
 and pupils are well pleased with them, and the results are better 
 than I have ever obtained with any other series of arithmetics. 
 
 W. N. LISTER, County Com. Schools, Ann Arbor, Mich. 
 
 The Walsh Arithmetics are now in very general use throughout 
 the county and I have yet to hear the first unfavorable criticism 
 from my teachers. On the contrary, they are expressing them- 
 selves gratified with the results obtained from the new plan. 
 
 S. A. FARNSWORTH, Principal, St. Paul, Minn. 
 
 Four years ago the Walsh Arithmetics were adopted for the 
 St. Paul schools. When the period of adoption expired last 
 summer, the principals of the forty-five schools of the city were 
 practically unanimous in recommending their continuance. They 
 are eminently adapted to foster careful and accurate reasoning 
 along mathematical lines. A marked improvement has been 
 shown in the subject since we began the use of the books. 
 
 C. D. GRAWN, Prin. Tpsilanti Normal Training School. 
 
 During the twenty years of my experience as a teacher and a 
 superintendent I have never before made use of a series of arithmetics 
 that have enabled us to reach such gratifying results. 
 
 The books are well graded, afford frequent opportunities for 
 review, have well-selected and practical exercises, and in the second 
 book of the grammar course give the pupil a good working knowledge 
 of the algebraic equation and an intelligent understanding of the 
 concepts of elementary geometry, both of which features fully 
 comply with the recommendations of the Committee of Ten. 
 
 F. S. SUTCLIFFE, Superintendent, Arlington, Mass. 
 
 I have for two years supervised the work of classes using 
 Walsh Arithmetics, and for definite, reliable results I count them 
 the best books I ever used. 
 
 ii 
 
" FULLY A YEAR IN ADVANCE " 
 
 ELGIN, ILL. 
 D. C. HEATH & CO. : 
 
 We have been using the Walsh Arithmetics in our 
 schools for several years, and are pleased with the 
 books because of what they have done in making our 
 arithmetic work more efficient. Our pupils are fully 
 a year in advance of what they were when the books 
 were introduced, grade for grade, I mean. 
 
 Permit me to suggest a few of the strong points of 
 these books from the standpoint of teachers who are 
 using them : 
 
 1. A large number of simple problems. 
 
 2. Frequent reviews. 
 
 * 
 
 3. An abundance of oral work, thus doing away with the neces- 
 sity for a separate book in mental arithmetic. 
 
 4. Early introduction of easy work in fractions, denominate 
 numbers, percentage, and interest. 
 
 5. Simple explanation. 
 
 6. Distribution of elementary work in algebra and geometrical 
 measurements and constructions so that it may supplement and 
 elucidate the work in arithmetic. 
 
 7. Beginning algebra work with equation. 
 
 Our teachers find that this work very greatly aids 
 clear thinking and accuracy of statement. I unhesi- 
 tatingly recommend the books. 
 
 M. A. WHITNEY, Supt. of Schools. 
 
YB 051 12 
 
 y