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 THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 OF CALIFORNIA 
 
 LOS ANGELES
 
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 educational 
 
 EDITED BY HENRY SUZZALLO 
 
 PROFESSOR OF THE PHILOSOPHY OF EDUCATION 
 TEACHERS COLLEGE, COLUMBIA UNIVERSITY 
 
 THE TEACHING OF 
 PRIMARY ARITHMETIC 
 
 A critical study of recent tendencies in method 
 
 BY 
 HENRY SUZZALLO 
 
 WITH AN INTRODUCTION BY DAVID EUGENE SMITH 
 
 HOUGHTON MIFFLIN COMPANY 
 
 BOSTON, NEW YORK AND CHICAGO 
 
 fiitiettfbe ptcjsrf Cambridge 

 
 COPYRIGHT, 1911, BY TEACHERS COLLEGE, COLUMBIA UNIVERSITY 
 COPYRIGHT, 1913, BY HOUGHTON MIFFLIN COMPANY 
 
 ALL RIGHTS RESERVED 
 
 "V
 
 A 
 
 5 
 
 CONTENTS 
 
 INTRODUCTION . . David Eugene Smith v 
 
 THE TEACHING OF PRIMARY ARITH- 
 METIC ....... Henry Suzzallo 
 
 I. THE SCOPE OF THE STUDY ..... i 
 II. THE INFLUENCE OF AIMS ON TEACHING 9 
 
 III. THE EFFECT OF THE CHANGING STATUS 
 
 OF TEACHING METHOD ..... 21 
 
 IV. METHOD AS AFFECTED BY THE DISTRI- 
 
 BUTION AND ARRANGEMENT OF ARITH- 
 METICAL WORK ........ 32 
 
 V. THE DISTRIBUTION OF OBJECTIVE WORK 42 
 
 VI. THE MATERIALS OF OBJECTIVE TEACH- 
 
 ING ............ 47 
 
 VII. SOME RECENT INFLUENCES ON OBJEC- 
 
 TIVE TEACHING ........ 53 
 
 VIII. THE USE OF METHODS OF RATIONALI- 
 
 ZATION ........... 60 
 
 iii
 
 CONTENTS 
 
 IX. SPECIAL METHODS FOR OBTAINING AC- 
 CURACY, INDEPENDENCE, AND SPEED . 69 
 
 X. THE USE OF SPECIAL ALGORISMS, ORAL 
 
 FORMS, AND WRITTEN ARRANGEMENTS 83 
 
 XI. EXAMPLES AND PROBLEMS 96 
 
 XII. CHARACTERISTIC MODES OF PROGRESS 
 
 IN TEACHING METHOD no 
 
 OUTLINE .119
 
 INTRODUCTION 
 
 J" ' "' ' ' > - 
 
 BY DAVID EUGLiNb SivilTH 
 
 2^ ^. -2. a a 
 
 THE evolution of the teaching of primary arith- 
 metic extends over a period of about two hundred 
 years, although numerous sporadic efforts at 
 teaching the science of number to young child- 
 ren had been made long before the founding 
 of the Francke Institute at Halle. During the 
 eighteenth century not much progress was made 
 until there was established the Philanthropin at 
 Dessau, and perhaps it would be more just to 
 speak of primary arithmetic as having its real be- 
 ginning in this institution at about the time that 
 our country was establishing its independent ex- 
 istence. It is, however, to Pestalozzi, at the be- 
 ginning of the nineteenth century, that we 
 usually and rightly assign the first sympathetic 
 movement in this direction, and it is the period 
 from that time to the present that has seen the 
 real evolution of the teaching of arithmetic to 
 children in the first school years. 
 v
 
 INTRODUCTION 
 
 The evolution of this phase of education is one 
 of the most interesting and profitable studies 
 that a teacher of arithmetic can undertake. It re- 
 veals the experiments, many of them puerile but 
 a few of them virile, that have been made from 
 time to time ; it brings into light the failures 
 which should serve as warnings, and the suc- 
 cesses which will inspire the teacher to better 
 and more carefully considered effort ; it shows the 
 trend of primary instruction, and it makes the 
 student of education more sympathetic with the 
 great problem before him.yjln_particular, he will 
 see the danger of narrowness in matters of 
 method, the futility of expecting to create genu- 
 ine interest by any single line of devices, the 
 pmfnt-fesijks that came from over-emphasis of 
 the doctrine of formal discipline, and the errors 
 of judgment that have been made in deciding 
 upon what constitutes reality in an arithmetical 
 problem for children. If by such a study he 
 should see the childishness of confining one's 
 self to the use of cubes alone, or of some special 
 type of number chart, or of some particular form 
 of number cards, or of sticks of varied lengths or 
 vi
 
 INTRODUCTION 
 
 shapes, the result would be salutary. If by his 
 study of the absurd extreme to which formal dis- 
 cipline was carried a generation ago he is led to 
 see the equal absurdity to which so many teach- 
 ers are tending to-day the denying that any 
 such discipline exists at all his labor will bear 
 good fruit in the school room of the present. 
 
 It fortunately happens that after this century 
 of experiment we are getting about ready to take 
 some account of stock ; to weigh up values ; to 
 select from what the world has produced, and to 
 select with some approach to good judgment. It 
 is not probable that the time is entirely ripe for 
 this labor, because we are at present in the midst 
 of a period of agitation that seems certain to warp 
 our judgment, the period of agitation fora some- 
 what narrow phase of industrial education ; but 
 even with this danger we are better able to 
 weigh up the values in the teaching of arith- 
 metic than we have ever been before. One rea- 
 son for this is that we now have men and women 
 of sufficiently sound education and sufficiently 
 broad view to attack the problem. These men 
 appreciate the efforts of Pestalozzi, but they 
 vii
 
 INTRODUCTION 
 
 recognize that these are to the present what 
 the science of Franklin is to that of Kelvin and 
 Thompson. They appreciate what Tillich did for 
 number work, and the influence of Grube ; but 
 they know that these men were narrow in view 
 and dogmatic in statement, and that they stand 
 rather as warnings than as founders of any 
 worthy theory. 
 
 In looking for such a man to prepare a report 
 upon the teaching of arithmetic in the primary 
 grades, the American members of the Interna- 
 tional Commission on the Teaching of Mathe- 
 matics turned first of all to Professor Suzzallo. 1 
 They felt that his standing as a scholar, his ex- 
 perience as a practical school man, and his posi- 
 tion in the educational world fitted him perfectly 
 for a work of this importance. That their good 
 judgment did not fail them will be seen in read- 
 ing the following report. In it Professor Suzzallo 
 
 1 The material presented in this study was originally col- 
 lected and organized for the purposes of a special report to 
 a sub-committee of the International Commission on the 
 Teaching of Mathematics. This commission was created at the 
 International Congress of Mathematicians held in Rome in 
 1908. The report was first published in the Teachers College 
 Record, March, 1911. 
 
 viii
 
 INTRODUCTION 
 
 has set forth very clearly the aims of instruction 
 in primary arithmetic, rightly considering these 
 aims in their evolutiojg.a% rather than. in their 
 static aspec^s^. emphasizing the importance of 
 this phase of the study, 
 
 tendencies that seem makingjorji_more rational 
 
 view of teaching. He has discarded the narrow 
 and trivial concept of method that characterized 
 the educational work of a generation now passing 
 away, and has brought dignity to the term by 
 considering it from the modern scientific stand- 
 point. He has discussed both historically and 
 psychologically the important question of object 
 teaching, showing the failures that have resulted 
 from narrow views of the purpose of such an 
 aid and the success that may be expected from 
 a more rational use of number material. The 
 whole question of the role of reason, or rather of 
 the pupil's effort to express the rationalizing pro- 
 cess has been considered, the extreme danger 
 points have been indicated, and the bearing of 
 some of our saner forms of psychology upon the 
 subject has been set forth. The technique of num- 
 ber, including the question of accuracy and speed 
 ix
 
 INTRODUCTION 
 
 in the operations, has also been treated in a very 
 acceptable manner, Professor Suzzallo's experi- 
 ence in this phase of work having been unusual. 
 And finally, the vexed question of what consti- 
 tutes a genuinely concrete problem has been con- 
 sidered in its various bearings, and the present 
 tendencies in problem-making have been indi- 
 cated. It is needless to say that the last word has 
 not been spoken on this phase of the work, and 
 that it never will be. New generations produce 
 new lines of application of arithmetic, even for 
 children. But with respect to the general prin- 
 ciples of the selection of matter for the framing 
 of problems, Professor Suzzallo speaks with a con- 
 viction that will carry weight. 
 
 With all of the opinions expressed in such a 
 report probably no reader will agree. It would be 
 a poor discussion that would not provoke some 
 opposition. But with the general tenor of the 
 report it is certain that most thinking teachers 
 of to-day will find themselves in hearty accord. 
 More important still is the fact that the report 
 sets forth in clear language the present status of 
 primary arithmetic in the more thoughtful edu-
 
 INTRODUCTION 
 
 cational circles in this country, and that it will 
 state to teachers at home and abroad the tend- 
 encies as they now appear to the leaders of edu- 
 cational thought in the United States. 
 
 TEACHERS COLLEGE, COLUMBIA UNIVERSITY, 
 February, 1911.
 
 THE SCOPE OF THE STUDY 
 
 IT is the function of this study to convey some 
 notion of the methods employed in teaching math- 
 ematics in the first six grades of the American 
 elementary school. No attempt is made to give a 
 minute description of the endless details of 
 teaching procedure, nor even to enumerate all 
 the types of teaching method employed. Its 
 purpose is restricted to an analysis of the larger 
 tendencies in teaching practice which are repre- 
 sentative of the spirit of mathematical instruction 
 in the lower schools. 
 
 Function of the Study to Trace General Tendencies 
 
 We should have a much simpler task if it were 
 ours to sketch the purposes of mathematical iji- 
 struction, or to outline the nature and organiza- 
 tion of the various mathematical courses of study. 
 As it is, we have to describe something less con- 
 cr^tc, namely the method employed in the pre- 
 
 9 I
 
 TEACHING PRIMARY ARITHMETIC 
 
 sentation of mathematical subjects. Intimately 
 dependent upon the subject matter involved, 
 dominated by the special aims of mathematical 
 instruction in a non-technical school, adjusted to 
 the immaturities of childhood, and reflecting the 
 personal habits of mind of the teacher, teach- 
 ing method emerges a powerful, variable, and 
 subtle thing. In spite of subtlety and variability 
 there are, however, certain general practices 
 that can be described and analyzed. It is with 
 these that this study will deal 
 
 Teaching Method is a Mode of Presentation 
 
 Owing to the existing confusions, it is well at 
 the very outset to have in mind a clear definition 
 of the term " teaching methods." Teaching 
 methods are always methods of presentation. In 
 this respect the teaching art is like any other 
 art, literary, graphic, plastic, or what not. The 
 literary artist, for example, has a purpose, a sub- 
 ject matter, a particular audience, and a. special 
 style of presentation. All these factors are pre- 
 sent in the teaching art. The aims of instruction, 
 the particular facts to be taught, the immaturity 
 
 2
 
 THE SCOPE OF THE STUDY 
 
 of the child taught, and the inevitable personality 
 of the teacher determine the style of instruc- 
 tion, or, to use our own " trade word," a 
 method of teaching. Every teacher, then, has ' 
 a style or method conscious or unconscious, 
 good, bad, or indifferent. Unlike the literary 
 artist, he has many ends to serve rather than one. 
 His functions are general to life, and include 
 moral, social, and personal ends, as well as those 
 that are aesthetic. His methods of communica- 
 tion, too, are more than one. He presents his ex- 
 periences objectively and graphically, as well as \ 
 through the medium of written words and speech. 
 Always the teacher's end is to stimulate growth -~~ 
 through the presentation of experiences. When 
 that presentation takes a form and order differ- 
 ent from that usual to adult life for the precise 
 purpose of making the fact more readily compre- 
 hensible to the immature mind of the child, then , 
 that modification may be called a method of \ 
 teaching.^ Teaching methods are always special J 
 manners of readjusting adult wisdom to the 
 special psychological conditions of a student's 
 m'md.f 
 
 3
 
 \ 
 
 TEACHING PRIMARY ARITHMETIC 
 
 Distinct Uniformities Exist among its Variations 
 
 In the concrete, methods of teaching are always 
 specialized responses to situations, and as variable 
 as situations are variable. Life is never just the 
 same at any point. Yet certain essential similari- 
 ties exist and give us the opportunity to inter- 
 pret life in terms of law. The same may be said 
 of the teaching life. In a sense it never repeats 
 itself ; yet to the degree that the same end, the 
 same subject matter, and the same immaturity of 
 mind recur in class rooms, teachers will tend to 
 use similar modes of adjustment. In describing 
 mathematical teaching in the primary schools, it 
 is these similar modes of teaching adjustment, 
 these similar " general methods," that we shall 
 describe and analyze. 
 
 The Methods of Public Elementary Schools are 
 Representative 
 
 It will be unnecessary to have a separate 
 treatment of the "general methods "of mathe- 
 matical teaching for public schools and private 
 schools. Whatever may be said of the state-sup- 
 4
 
 THE SCOPE OF THE STUDY 
 
 ported schools will in general be true of private 
 institutions. It is true of elementary schools as 
 it is not of secondary and higher schools, that 
 private institutions hold a relatively minor place, 
 as compared with public or state schools. They 
 are in a sense mere adjuncts to the public school 
 system, claiming, in the generality of cases, no 
 real difference in their ideals and methods of in- 
 struction. In the larger number of cases they 
 draw their courses of study and methods from 
 the public school systems of. the immediate neigh- 
 borhood. The social jtatus of the parent or the 
 personal incapacity of the child, rather than 
 difference of school methods, is the cause of the 
 special clientage of the private primary schools in 
 the United States. Hence, a description of the 
 characteristic methods of the public schools will 
 be representative of the prevailing modes of in- 
 struction in American private elementary schools. 
 
 Elementary Mathematics is Mainly Arithmetic 
 
 Mathematical instruction in the first six years 
 of the elementary school concerns itself almost 
 exclusively with the teaching of arithmetic. A 
 5
 
 TEACHING PRIMARY ARITHMETIC 
 
 decade or so ago there was a vigorous movement 
 for the introduction of algebra and geometry 
 into the elementary school. As a result, these 
 subjects made their appearance in the seventh 
 and eighth grades seldom in the first six 
 school years. Where the influence of this move- 
 ment penetrated the lower grades or persisted 
 in the higher grades, the algebraic and geometric 
 elements involved were so restricted and simpli- 
 fied that they became part and parcel of the 
 subject of arithmetic, rather than the elemen- 
 tary phases of two more advanced subjects. This 
 is true of the simple algebraic equation or the 
 measurement of simple geometric figures where 
 introduced below the seventh year. 
 
 Elementary Arithmetic Emphasizes the Four 
 Fundamental Processes 
 
 By common practice, even the arithmetic 
 taught in the primary grades has been given a 
 restrictive emphasis. For the most part it is con- 
 cerned with the mastery of the fundamental pro- 
 cesses in manipulating integers and fractions. 
 The casual observer, in reading American courses 
 6
 
 THE SCOPE OF THE STUDY 
 
 of study, will note that in the lower grades the^ 
 mathematical subjects taught are named after ( 
 the abstract process involved rather than after \ 
 the particular business institution to which the ) 
 arithmetic is concretely applied. Thus in the~^ 
 lower grades we teach counting, subtraction, 
 fractions, decimals, percentage, etc. ; while in 
 the higher grades we teach interest, stocks and 
 bonds, commission, insurance, etc. There is of 
 course no hard and fast line of demarcation ; one 
 emphasis gradually passes over into the other, 
 an approximate balance being maintained in the 
 intermediate grades. It will perhaps simplify the 
 task of this study and make its treatment more 
 thoroughly representative of all conditions, if 
 the general methods described be restricted 
 to that field which is most characteristic of the 
 first five or six years of mathematical instruction, 
 namely, to the teaching of the fundamental pro- 
 cesses of manipulating integers and fractions 
 along with their simple applications to concrete 
 problems. This discussion, then, will be limited 
 to the period of school life in which the tools of 
 
 arithmetic are acquired. 
 i 
 
 7
 
 TEACHING PRIMARY ARITHMETIC 
 
 The Need for Studying Exceptional Reform 
 Tendencies 
 
 While the aspects of mathematical instruction 
 here studied and presented are selected because 
 of their representative nature, it would be un- 
 wise to restrict ourselves to a statement of the 
 commonly accepted procedures of school-room 
 practice. There are in America certain reform 
 tendencies which are as characteristic of condi- 
 tions as are the conservative practices. These 
 modifying forces need to be mentioned along 
 with the practices that they alter. Again, there 
 are certain scientific effects now well under 
 way to study the problem of methods in teach- 
 ing. While these have, as their immediate aim, 
 the acquisition of new knowledge rather than di- 
 rect educational reform, their ultimate effect will 
 be to change methods of teaching. For this reason 
 they are important, and have a proper place in 
 this presentation.
 
 II 
 
 THE INFLUENCE OF AIMS ON TEACHING 
 
 . ^^ 
 Factors Influencing Teaching Methods 
 
 IT has been suggested that all teaching methods 
 represent adjustments to several variable factors 
 in the school-room situation. Teaching method 
 is never, or should not be, just one thing. It is as 
 variable as the factors that determine its situa- 
 tion. The purposes of mathematical instruction,--- 
 the nature of the fact to be taught, the imma- 
 turity of the child, the teacher's scholarly equip- 
 ment, his personality, his attitude toward the 
 very idea or institution of method, all these are 
 influences in determining the status of mathe- 
 matical teaching. Some of them are so important 
 that it will be necessary to discuss them in de- 
 tail at the very outset. 
 
 The Influence of a Scientific Aim 
 
 The purposes of mathematical instruction in 
 the elementary school must always be very in- 
 9
 
 TEACHING PRIMARY ARITHMETIC 
 
 fluential upon method. It makes a great differ- 
 ence whether one is merely teaching the elements 
 of mathematics or is teaching mathematics as a 
 tool for business life. It has not been long since 
 the aim of mathematical teaching was merely 
 scientific. The facts taught were the beginning 
 of a science, and the end was to obtain a foun- 
 dation for more advanced facts of the same kind, 
 which were dependent upon this foundation. As 
 the teacher had learned his mathematics, so he 
 taught the subject. To a considerable degree, as 
 the master's adult mind classified the facts of the 
 subject, so he presented it to the child. His 
 methods were logical rather than psychological. 
 He gave the finished product or process to the 
 child without special adaptation to the child's 
 immaturity; a roundabout method tha slowly 
 approximated and only finally achieved the full 
 result was with such a teacher exceptional. 
 
 Such a scientific aim, implicit rather than ex- 
 pressed, dominated the methods of teaching when 
 arithmetic was handed over to the elementary 
 schools by the higher institutions of education. 
 The first purpose to be rooted in the traditions 
 10
 
 INFLUENCE OF AIMS ON TEACHING 
 
 of mathematical teaching, it still persists with 
 all the rigidity of a conservative force. Teachers 
 still tend to teach future workmen in the lower 
 schools.as they themselves were taught by sci- 
 entific scholars in the universities. And high 
 school and college instructors still impose their 
 standards upon the lower schools so as to influ- 
 ence their methods of instruction. As higher in- 
 struction still remains largely scientific in purpose 
 and method, its effects reinforce the earliest tra- 
 dition in the elementary schools. Under such 
 
 an influenceithe worth of a mathematical fact is--' 
 
 vt'jr 
 
 measured by its place in a logical scheme, rather } 
 than by its significance and recurrence in every-' 
 day life.\ The mathematician may need to know 
 all abour the names of the places in notation and 
 numeration ; the layman cares only about the ac- 
 curate reading and writing of numbers, and not 
 at all about the verbal title of "units of thou- 
 sands " place. Again the rational needs of a stu- 
 dent of mathematics may require an understand- 
 ing of the reasons why we "carry" in column 
 addition, but the effective everyday use demands 
 an accurate habit of "carrying" rather than an 
 ii
 
 TEACHING PRIMARY ARITHMETIC 
 
 accurate explanation. Yet just such methods 
 persist in our schools because of the domination 
 of a scientific treatment of the subject. 
 
 The Influence of the Aim of Formal Discipline 
 
 The remoteness of such mathematical teach- 
 ing from the needs of common life constantly 
 threatens the loyalty and support of the public. 
 Some defense becomes necessary on other than 
 scientific grounds. Such a sanction could not be 
 found in utilitarianism, for the waste was evi- 
 dent. It remained for a psychological theory to 
 sketch a defense upon "disciplinary" grounds. 
 The doctrine of "formal discipline" says that 
 such mathematical teaching trains the powers of 
 the mind so that any mastery gained in mathe- 
 matics is a mastery operating in full elsewhere, 
 regardless of the remoteness of the new situa- 
 tions from those in connection with which the 
 power or ability was originally acquired^) The facts 
 and processes mastered may not be those most 
 needed in daily life, but they are good for every 
 nan inasmuch as they train his mind.) Such was 
 .he dictum of the doctrine of " f ormaTdiscipline." 
 
 12
 
 INFLUENCE OF AIMS ON TEACHING 
 
 The effect of such doctrine is to/defend and 
 perpetuate every obsolete, unimportant, and 
 wasteful practice)in the teaching of mathematics.^ 
 No matter that partnership as taught in the 
 schools had its original sanction in its close cor- 
 respondence to the reality of business practice ; 
 no matter that the old sanction has passed ; 
 teach it now for its ability to discipline the* mind ! 
 No matter that " life insurance " touches more 
 men than "cube root" ; the latter should be kept 
 because of its power to train the mind. In life, 
 where "approximation" of amounts suffices, the 
 teacher demands absolute accuracy, and the eth- 
 ical worth of such precise truth is the high law 
 for its defense. In life, "the butcher, the baker, 
 and the candlestick maker " figure out the total 
 of a bill mainly "in their heads,", with a few ac- 
 cessory pencil scribbles upon paper; the teacher 
 finds sanction in aesthetics for requiring a com- 
 plete statement written or re- written in exquisite 
 form. Regardless of the truth that is concealed 
 in the doctrine of " formal discipline," it must be 
 confessed by those who know the history of 
 teaching method in the United States that it is 
 13
 
 TEACHING PRIMARY ARITHMETIC 
 
 the main defense of conservatism and the larg- 
 est cause of waste in teaching methods. 
 
 The Shift in Emphasis from Academic to 
 Social Aims 
 
 Such has been the ground upon which recent 
 educational reform has operated. Slowly the 
 \ older scientific and disciplinary aims of instruc- 
 \ tion have given way to the newer purposes of 
 business utility and social insight. In that step 
 a large transition has been covered. Before, the 
 school measured the worth of its work by stand- 
 ards internal to educational institutions. The 
 schoolmaster and the scholar, rather than the 
 man on the street, had formulated the scientific 
 classifications of mathematics and expounded 
 the doctrine of "formal discipline." Thereafter, 
 the measure of efficient school instruction was 
 determined by standards external to the school, 
 the product of conditions outside of school life. 
 \J3usiness need and social situation determine 
 whether a fact or a process is worth comprehend- 
 ing, and whether the method of instruction has 
 been effective?\ 
 
 14
 
 INFLUENGE OF AIMS ON TEACHING 
 
 Business Utility as an End 
 
 The utilitarianism that first attacked the older 
 course of study and its methods was the utility of 
 the business world. The arithmetic of business life- 
 became the standard. The practices of the market 
 determined what matter, skill, and accuracy 
 should be demanded of the elementary school 
 pupil. Recently it became the habit to call upon 
 the business man to give his opinion as to what 
 constitutes good arithmetical training; and no 
 criticism was so feared as that of the business 
 leader who said that the boys that came to him 
 were incompetent. Committees on courses of 
 study have even investigated the relative fre- 
 quency and importance of specific arithmetical 
 processes in the business world with the idea of 
 utilizing the results as a basis for changes in the 
 mathematical curriculum. 
 
 This aim of business utility, coming at a time 
 when the elementary school course was felt to 
 be overcrowded, met with a ready reception. It 
 operated for the time being as the standard by 
 which materials and methods in arithmetic were 
 15
 
 TEACHING PRIMARY ARITHMETIC 
 
 to be eliminated, if not actually selected. Ma- 
 terials not general to the business world, such 
 as the table of Troy weight, were therefore 
 eliminated. Processes of computing interest in- 
 frequently used were supplemented by more 
 widespread and up-to-date methods. More doing 
 and less explaining characterized the instruction 
 in adding columns of figures, and such manipu- 
 lation mimicked the exact conditions of its use 
 in the world at large. If strings of figures are 
 usually added in vertical columns in the busi- 
 ness world, then they should be taught in ver- 
 tical columns more nearly exclusively than before. 
 The obsolete and the relatively infrequent, the 
 over-complex and the wasteful processes of the 
 old arithmetic tended to disappear. More than 
 any other influence, this aim of business utility 
 has combated the over-conservative influence of 
 scientific and disciplinary aims which dominated 
 previous decades. The newer methods of teach- 
 ing have kept the best of the old movements. 
 The work is still scientific in that it is accurate; 
 it is still disciplinary in that it trains ; but the 
 truth and the training which are given are 
 16
 
 INFLUENCE OF AIMS ON TEACHING 
 
 selected by and associated with actual business 
 situations common to every-day life. 
 
 Broad Social Utilitarianism as a Standard 
 
 There is evidence in the present thought of 
 teachers that a utility broader than that of the 
 business world is beginning to obtain in the 
 schools. Everywhere in these days the Ameri- 
 can teacher and the educational writer speak of 
 the social aims of education. More than ever be- 
 fore, the social consciousness of the teacher en- 
 larges. This general increase in the social con- 
 sciousness of the teacher is reflected in mathe- 
 matical instruction. 
 
 In spite of an increasing movement toward 
 specific vocational training, as seen in the in- 
 dustrial education movement, for example, 
 there has been a reactionary defense of the ele- 
 mentary schools as an institution for very gen- 
 eral training in the things that arc-socially fun- 
 damental and common. This movement toward 
 the preservation of the elementary school as a 
 place for giving a broadly socialized and modern 
 culture not only is checking the inroads of a 
 17
 
 TEACHING PRIMARY ARITHMETIC 
 
 narrow vocational education, but is broadening 
 the conception of the older studies, of which 
 arithmetic is but one. Arithmetic is not a sub- 
 ject in which only the skills of calculation are 
 cultivated ; it is one that contributes social in- 
 sight, just as history and geography do. 
 
 The influence of the social aim of instruction 
 upon mathematical instruction is subtle but ob- 
 vious. The business man's opinion with refer- 
 ence to arithmetical instruction is not always 
 taken as gospel. There are other standards. 
 " Why," says the schoolmaster, " should I train 
 people for your special needs, any more than for 
 the demands of other trades that men ply? To 
 be sure, our graduates do not fit perfectly into 
 your shop at once. But that precise and local 
 adjustment is the work of the business course 
 or of shop apprenticeship. My function is to 
 train men for the situations common to all men 
 and special to no class. The elementary school 
 is a school for general culture or social apprecia- 
 tion, not a business college or a trade school." 
 The sociologist usurps the place of the business 
 man as the school's proper critic. 
 18
 
 INFLUENCE OF AIMS ON TEACHING 
 
 Some Concrete Effects of the Change in Aim 
 
 The immediate effect is that arithmetical ap- 
 plications find a larger place in teaching. A 
 saner relation is established between abstract 
 examples and concrete problems. And the prob- 
 lems, in increasing extent, are real problems, 
 typical of life, if not actual. No more does arith- 
 metic, in the best schools, confine itself to fig- 
 ures alone. Figures are applied in concrete 
 problems. There may be days of teaching when 
 not a figure is used during the arithmetic period. 
 The social setting, the business institution, which 
 calls for the calculation, is studied as carefully 
 as the process of calculation. The students are 
 given a knowledge of banking as well as skill in 
 the computation of interest. They may even 
 visit a bank, a factory, a shop, as the case may 
 require. Instead of having fifteen problems that 
 .deal with fifteen different subjects all more or 
 less remote from one another, as was almost uni- 
 versally the case with older text-books and teach- 
 ing methods, the class hour may be given over to 
 fifteen problems related to one situation, such 
 19
 
 TEACHING PRIMARY ARITHMETIC 
 
 as might develop in the business of a bakery 
 shop or an apartment house. Thus arithmetic 
 gradually gains social setting and unity. 
 
 To-day teaching methods in arithmetic are in 
 a state of transition : old and new purposes min- 
 gle with unequal force and give us a mixed pro- 
 cess of instructing. Old materials and methods 
 still persist, for logical and disciplinary ideals 
 still hold ; but the newer regimen ushered in by 
 the demands of business utility and social under- 
 standing gains ground. The obsolete, the untrue, 
 the wasteful methods pass from arithmetic teach- 
 ing ; and the pressing, tfiodern, and useful activi- 
 ties and understandings enter. Arithmetic is less 
 abstract and formal as a subject than it was ; it 
 has become increasingly vital and concrete with 
 real interests, insights, and situations. The grind 
 of sheer mechanical drill decreases in teaching ; 
 and a reasoned understanding of relations, in 
 some degree at least, is substituted. Artificial mo- 
 tives and incentives are less frequently used to 
 get work done, while the quantitative needs of the 
 child's life and the intrinsic interest of children 
 in the institutional occupations of their elders pro- 
 vide a more vital motive for the use of arithmetic.
 
 Ill 
 
 THE EFFECT OF THE CHANGING STATUS OF 
 TEACHING METHOD 
 
 Method as a Psychological Adjustment to 
 the Child 
 
 AT the very beginning, it was suggested that 
 many factors enter into the nature of our teach- 
 ing methods. There was occasion to show the 
 effect of varying aims on the spirit and manner 
 of instruction, for the end in view inevitably in- 
 fluences any presentation of facts, in school or 
 out. The most significant factor, however, in 
 teaching method is the attempt to adjus^ meth- 
 ods of presentation to the psychological con- 
 ditions of childhood. Teaching method in the 
 school is primarily a mode of presentation de- 
 signed to stimulate the energies of children. As 
 long as the teacher was the most active person 
 in the classroom, method as such was not impor- 
 tant in pedagogical theory. The focusing of at- 
 tention on the child as an active human factor to 
 
 21
 
 TEACHING PRIMARY ARITHMETIC 
 
 be given careful consideration is responsible for 
 the extended development of teaching technique. 
 The growing importance of "method" in educa- 
 tional theory marks a growth in the teacher's 
 consciousness of psychological factors, precisely 
 as the appearance of the newer aims in teaching 
 has marked an increased regard for social factors. 
 
 The Effect of an Increased Reverence for 
 Childhood 
 
 Two important movements have been respon- 
 sible for the development of a psychological con- 
 sciousness of the pupil as a dominating factor 
 in teaching method. One is humanitarian ; the 
 other, scientific. 
 
 There has been a steady growth in reverence 
 and sympathy for childhood. As yet it has 
 scarcely expressed itself with fullness. Its pre- 
 sence is revealed by the widespread enactment of 
 laws designed to guarantee the rights of child- 
 hood laws against child labor and in favor of 
 compulsory education. The growth of special 
 courts for juvenile offenders, the development of 
 playgrounds, and the decreased brutality of dis- 
 22
 
 CHANGING STATUS OF METHOD 
 
 cipline at home and school, are other symptoms 
 of the public attitude toward childhood. The 
 wide acceptance of the " doctrine of interest " 
 in teaching ; the enrichment of the curriculum ; 
 specialized schools for truants and defectives; 
 individual instruction, these are the school- 
 master's recognition of the modern attitude to- 
 ward childhood. Under such conditions teach- 
 ing becomes less and less a ruthless external 
 imposition of adult views, and more a means 
 of sympathetic ministry to those inner needs of 
 child life which make for desirable qualities of 
 character. While it is true that teaching method 
 is a condescension to childhood, it is a socially 
 profitable condescension in that it is a guarantee 
 of more effective and enduring mastery of the 
 life that is revealed at school. Since the child's 
 acquisition tends the more to be part and parcel 
 of his own life under such sympathetic teaching, 
 the products of such instruction are enduring. 
 
 The Reconstruction of Method through Psychology 
 
 Such a humanitarian movement naturally 
 called for knowledge of the child the wisdom 
 23
 
 TEACHING PRIMARY ARITHMETIC 
 
 of common sense soon exhausts itself, and more 
 scientific data are demanded. Thus the " child 
 study movement" came into existence. The 
 movement was in some degree disappointing, for 
 frequently it busied itself in cataloguing the ob- 
 vious rather than in classifying new and hitherto 
 unexplained data. But one thing it did : it focused 
 attention upon the child as the crucial factor 
 in education, the prime conditioning force in all 
 methods of instruction. Since then, a saner psy- 
 chological foundation has been laid for educa- 
 tional procedure, one which is criticising and 
 reconstructing teaching method at every turn. 
 Hitherto, teaching methods had been improved 
 fitfully through a crude empiricism. As the 
 ablest teachers became dissatisfied with their 
 teaching and dared to vary their methods, they 
 selected the successful experiments, and other 
 teachers willingly adopted the methods that 
 seemed better than their own. Now a body of 
 general psychological knowledge, rich in its 
 criticism of old methods and in its suggestion 
 of new means of procedure, gives a scientific 
 basis to teaching method. Where additional psy- 
 24
 
 CHANGING STATUS OF METHOD 
 
 chological knowledge is needed, the educational 
 psychologists seek it through special investiga- 
 tions. And where the counter claims of com- 
 peting methods defy ordinary psychological 
 analysis and investigation, judgment is sought 
 through an experimental pedagogy which sub- 
 mits teaching processes to comparative tests 
 under normal classroom conditions. 
 
 The Increased Professional Respectability of 
 Conscious Method 
 
 Increased sympathy with childhood and in- 
 creased scientific knowledge of human nature 
 together give teaching method a new j ustification. 
 The result is that the era of complete depend- 
 ence upon teaching genius and mere common 
 sense in methods of instruction has passed out 
 of the American elementary school. We are now 
 in a period where a specific professional tech- 
 nique in teaching is demanded, a technique partly 
 developed out of crude personal and professional 
 experience, and partly founded upon scientific 
 criticism and experiment. A new humanitarian 
 and scientific attitude toward the mental life of 
 25
 
 TEACHING PRIMARY ARITHMETIC 
 
 children elevates teaching method to a position 
 it has never before enjoyed. 
 
 The public elementary-school teacher is con- 
 servative indeed who will deny that there is any- 
 thing worthy in the notion of " method." As a 
 class, teachers have faith in the special profes- 
 sional technique which is included under the 
 term. They are critical of the many abuses which 
 have been committed in the name of method. 
 Method cannot be a substitute for scholarship. 
 It cannot be a "cut and dried" procedure indis- 
 criminately or uniformly applied to class-room 
 instruction. Like every other technical means, 
 teach ing method is subject to its own limitations 
 and strengths, a fact which the average teacher 
 recognizes. 
 
 The Prevalence of Methods Emphasizing a 
 
 Single Idea 
 
 *-. 
 
 In spite of the^act that the majority of ele- 
 mentary teachers Keep reasonably sane on the 
 problem of method in teaching, it must be ad- 
 mitted that a considerable proportion of teachers 
 are inclined to be attracted by systems of method 
 26
 
 CHANGING STATUS OF METHOD 
 
 that greatly over-emphasize a single element of 
 procedure. The hold which the " Grube method," 
 with its unnatural logical thoroughness and 
 progression, gained in this country two or 
 three decades ago is scarcely explicable to-day. 
 Scarcely less baffling is the very large appeal 
 made by a series of textbooks which lays the 
 stress upon the acquisition of arithmetic through 
 the idea of ratio and by means of measuring. 
 Manual work as the source of arithmetical ex- 
 periences is another special emphasis, which, like 
 the others, has had its enthusiastic adherents. 
 Again it is "arithmetic without a pencil" or 
 some other over-extension of a legitimate local 
 method into a "panacea "or "cure-all," which con- 
 fronts us. The promulgation and acceptance of 
 such unversatile and one-sided systems of teach- 
 ing method are indicative of two defects in the 
 professional equipment of teachers : (i) the lack 
 of a clear, scientific notion as to the nature and 
 function of teaching method, and (2) the lack of 
 psychological insight into the varied nature of 
 class-room situations. Untrained teachers we still 
 have among us, and others, too, to whom a little 
 27
 
 TEACHING PRIMARY ARITHMETIC 
 
 knowledge is a dangerous thing. These are fre- 
 quently carried away by the enthusiastic appeals 
 of the reformer with a system far too simple to 
 meet the complex needs of human nature. Our 
 experiences seem to have sobered us somewhat, 
 the increase of supervision has made responsible 
 officers cautious, and increased professional in- 
 telligence has put a wholesome damper upon 
 na'fve and futile proposals to make teaching 
 easy. 
 
 The Tendency toward Over-Uniformity 
 in Method 
 
 A more serious evil than that just mentioned 
 is the tendency of the supervising staff to over- 
 prescribe specific methods for class-room teach- 
 ers. Recently there has developed, more par- 
 ticularly in large city systems, a tendency to 
 demand a uniform mode of teaching the same 
 school subject throughout the city. The prime 
 causes of this tendency are to be found (i) in the 
 specialization of grade teaching, and the conse- 
 quent interdependence of one teacher on another; 
 and (2) in the mobility of the school population, 
 28
 
 CHANGING STATUS OF METHOD 
 
 which involves considerable lost energy if teachers 
 do not operate along similar lines. 
 
 The result of such imposed uniformity is a 
 reduction of spontaneity in teaching. The pro- 
 cess of instruction proceeds in a more or less 
 mechanical fashion, the teacher working for bulk 
 results by a persistent and general application of 
 the methods laid down. That teaching which at 
 every moment tends to adjust itself skillfully to 
 the changes of human doubt and interest, diffi- 
 culty and success, discouragement and insight, 
 now taking care of a whole group at once, now 
 aiding an individual straggler, now resolutely 
 following a prescribed lead, now pursuing a line 
 of least resistance previously unsuspected, can- 
 not thrive under such conditions. The demand 
 for an excessive uniformity stifles teaching as 
 a fine art, and makes of it a mechanical busi- 
 ness ; only those activities that fit the machine 
 can go on. Thus it happens that we memorize, 
 cram, drill, and review ; and soon the subtler pro- 
 cesses of thinking and evaluating, which are 
 the best fruit of education, cease to exist.
 
 TEACHING PRIMARY ARITHMETIC 
 
 Method as a Series of Varied, Particular 
 Adjustments 
 
 Fortunately the one-method system of teach- 
 ing will soon belong to the past ; and fortunately, 
 too, the imposition of uniform methods is begin- 
 ning to lose ground, even in our cities. For the 
 most part, the common sense of teachers and 
 the positive statements of our better theorists 
 keep teaching methods in a sane and useful 
 status. Teaching methods should be as infinitely 
 variable as the conditions calling for their use are 
 endlessly changeable. Not one method but many 
 are necessary, for methods are supplementary 
 rather than competitive. No one method should 
 be used with a pre-established rigidity ; each must 
 be flexible in its uses, so as to accomplish the 
 varied work to be done. The teacher, with his 
 everyday contact with the problems of child- 
 hood, is the best interpreter of conditions and 
 the best chooser of the tools of instruction. 
 The supervisor may criticize, suggest, and ad- 
 vise ; he may call attention to fundamental prin- 
 ciples involved ; but the teacher himself must
 
 CHANGING STATUS OF METHOD 
 
 finally choose his own methods. He is the only 
 one who can know conditions well enough to ad- 
 just teaching methods to the needs of his own 
 children. 
 
 Arithmetic teaching has suffered from false 
 uses of teaching method. In this respect it has 
 shared the common professional lot. But in 
 addition it has had special difficulties and adven- 
 tures of its own. We have now to note those 
 special phases of teaching method which are 
 peculiar and local to mathematical instruction. 
 
 J f '
 
 IV 
 
 METHOD AS AFFECTED BY THE DISTRIBIK 
 TION AND ARRANGEMENT OF ARITH- 
 METICAL WORK 
 
 The Tendency toward Shortening the Time 
 Distribution 
 
 SEVERAL decades ago arithmetic, as a formal 
 subject, was begun in the first school year and 
 continued throughout the grades to the last 
 school year. This is no longer a characteristic 
 condition, much less a uniform one. There have 
 been forces operating to complete the subject of 
 arithmetic prior to the eighth year, and to delay 
 its first systematic presentation in the primary 
 grades for a period varying from six months to 
 two years. The report of the "Committee of 
 Fifteen " of the National Education Association 
 summarizes the tendency existing in 1895 when 
 it states that, "with the right methods, and a 
 wise use of time in preparing the arithmetic 
 lesson in and out of school, five years are surfi- 
 32
 
 DISTRIBUTION AND ARRANGEMENT 
 
 cient for the study of mere arithmetic the five 
 years beginning with the second school year and 
 ending with the close of the sixth year. 
 
 The Attempt to Eliminate Waste 
 
 The attempt to shorten the period of formal 
 instruction in arithmetic has had its effects upon 
 the methods of teaching as well as upon the ar- 
 rangement of the course of study. The presence 
 of a large number of children who leave school 
 by the seventh year, the example of a varied 
 European practice, the overcrowded curriculum, 
 all these have combined to suggest a short- 
 ened treatment of arithmetic. Hence economy, 
 through the elimination of obsolete and unim- 
 portant topics in the course of study and through 
 better methods of instruction, has become a 
 pressing matter. Its influence on method is ob- 
 vious. 
 
 It has focused attention upon "teaching 
 method" and given it an increasing importance 
 in the eyes of mathematics teachers. Specifically, 
 it has tended to reduce the amount of objective 
 work, to eliminate the explanation or rationaliza- 
 33
 
 TEACHING PRIMARY ARITHMETIC 
 
 tion of processes which in life are done auto- 
 matically; it has made teachers satisfied with 
 teaching one manner of solution where, before, 
 two or three were given ; it has laid the emphasis 
 upon utilizing old knowledge in new places, rather 
 than on acquiring new means. 
 
 Delay in Beginning Formal Arithmetic 
 Teaching 
 
 The tendency toward delay in beginning for- 
 mal arithmetic instruction is to be explained in 
 terms of several causes. Under a regimen where 
 complicated and obsolete problems, difficult 
 of comprehension, were common in elementary 
 school tests, it was natural that teachers should 
 believe that arithmetic is too difficult a subject 
 for young children and that better results could 
 be obtained if the subject were not commenced 
 till the children were more mature. 
 
 This belief persists even after the curriculum 
 is purged of all obsolete and over-complex ma- 
 terials, and has become a modern course of 
 study with materials well within the compre- 
 hension and interest of primary children. 
 34
 
 DISTRIBUTION AND ARRANGEMENT 
 
 The Incidental Method of Teaching 
 
 The by-product of this belief is that any arith- 
 metic taught during these first few years should 
 be taught "incidentally," as a chance accompani- 
 ment of other studies. Only after one or two 
 years of incidental work should the formal arith- 
 metic instruction be given. This " incidental " 
 method of teaching beginners is difficult to es- 
 timate. It has been so variously treated that a 
 comparative measure of its worth is difficult to 
 obtain. The contention that children who are 
 taught incidentally for two years and systemati- 
 cally for two years more have at the end of 
 four years of school life as good a command of 
 arithmetic as those who have had a systematic 
 course through four school years, is difficult 
 to substantiate or deny on scientific grounds. 
 Sometimes "incidental" teaching required by 
 the course of study becomes " systematic " in 
 the hands of the teacher. Sometimes the two 
 years of " systematic " teaching which follow the 
 incidental teaching mean far more than two 
 years, since the teachers, in order to catch up, 
 35
 
 TEACHING PRIMARY ARITHMETIC 
 
 give more time and emphasis to the subject than 
 the relative time-allotment of any general sched- 
 ule would seem to warrant. Such have been the 
 facts frequently revealed by a class-room inspec- 
 tion that penetrates beyond the course of study, 
 the time schedule, and regulations of the school 
 board. 
 
 Reactions against the Plan of Incidental Teaching 
 In the lack of specific comparative measures of 
 the worth of such methods of instruction, there 
 is a growing conviction (i) that beginning school 
 children are mature enough for the systematic 
 study of all the arithmetic that the modern 
 course of study would assign to these grades ; (2) 
 that, considering the quantity and quality of their 
 experiences, they can think or reason quite as 
 well as memorize ; and (3) that what the school re- 
 quires of the child can be better done in a re- 
 sponsible, systematic manner than by any hap- 
 hazard system of " incidental " instruction. 
 
 These reactionary attitudes by no means imply 
 a return to " systematic " teaching of arithmetic 
 in the first two school years, nor to such formal 
 36
 
 DISTRIBUTION AND ARRANGEMENT 
 
 methods as were previously employed. Other 
 grounds forbid. The crude, uninteresting memori- 
 ter methods of the past have gone for good. Ob- 
 jective work, plays, games, manual activities make 
 arithmetical study easier and more efficient. In- 
 deed, these newer methods have been a large fac- 
 tor in convincing teachers that children have the 
 ability to master the first steps in arithmetic 
 during the first two years. Regardless of this 
 change in prof essional belief, it is a fairly general 
 opinion that arithmetic should not be thrown 
 upon the school-beginner along with the other 
 heavy burden of learning to read. Learning the 
 mechanics of reading is quite the most important 
 part of the first school year, and the addition of 
 the difficulties of another language for such 
 number is would overburden and distract the 
 child. Hence a common-sense distribution of 
 burdens and tasks, regardless of questions of 
 child maturity, would delay the formal and sys- 
 tematic study of arithmetic a half or whole school 
 year, little reliance being placed upon previous 
 " incidental " acquisitions. 
 
 37
 
 TEACHING PRIMARY ARITHMETIC 
 
 Logical and Psychological Types of Arrangement 
 
 There are other problems of method less con- 
 cerned with the time for beginning the study, or 
 with the span of school life to be given to it. 
 These deal with the arrangement of sub-topics 
 within the course of study, or with the manner 
 of progression from one aspect of arithmetical 
 experience to another. I refer now to the vari- 
 ous methods of planning the work in arithmetic 
 from grade to grade, of which the "Grube 
 method " and the " spiral " methods are types. 
 
 The methods that have been employed in the 
 United States for the arrangement or ordering 
 of topics within the course of study have varied 
 considerably from time to time, but all these 
 variations may be grouped around two types: 
 (l) The "logical" types of arrangement, and (2) 
 the "psychological" types of arrangement. If 
 the course of study proceeds primarily by units 
 that are characteristic of the mathematics of a 
 mature adult mind, the type may be said to be 
 "logical." If the 'course of study proceeds pri- 
 marily by units that are characteristic of the 
 38
 
 DISTRIBUTION AND ARRANGEMENT 
 
 manner in which an immature child's mind ap- 
 proaches the subject, then the type may be said 
 to be "psychological." The dominant arrange- 
 ments have been " logical " up to within the past 
 two decades. The older text-books taught " nota- 
 tion and numeration "rather thoroughly, then pro- 
 ceeded to a fairly adequate mastery of "addi- 
 tion," then to " subtraction," and so on. Such 
 an arrangement is distinctly "logical." So also 
 was the later " Grube method," which progressed 
 by numbers rather than by processes. 
 
 The courses of study which have been most 
 familiar to us in the past decade have used the 
 "concentric circle" or "spiral" methods of ar- 
 ranging the sub-topics of arithmetic. These ar- 
 rangements are "psychological" in type. They 
 are attempts to give a systematic order of mas- 
 tery which shall approximate the child's order of 
 need in knowing. Here the first mathematical 
 facts and skills taught are those that the child 
 first requires, regardless as to whether they em- 
 ploy integers or fractions, additions or divisions. 
 A little later, he deals with the same subjects 
 and the same numbers in more complicated 
 39
 
 TEACHING PRIMARY ARITHMETIC 
 
 manipulation and in more extended application. 
 The field is re-covered, as it were, by ever widen- 
 ing circles or by an enlarged swing of the " spiral " 
 progression. 
 
 Estimates of Worth 
 
 The older "logical" plans are thorough and 
 definite in their demands ; the teacher always 
 knows just what he is about. But such a system 
 of procedure is unnatural and remote from the 
 child ; it lacks appeal and motive. The child pur- 
 sues the subject as a task laid down for him, not 
 as an answer to his own curiosities or necessities. 
 The newer psychological plans meet the different 
 levels of child-maturity effectively; they are 
 nearer the natural order of acquiring knowledge. 
 The difficulty with all psychological arrange- 
 ments is that the teacher cannot readily re- 
 member what the child has and has not been. 
 The supervisor, too, finds it hard to locate re- 
 sponsibility for the teaching of definite arith- 
 metic sub-topics. As orders of teaching they are 
 psychologically natural but administratively in- 
 effective. 
 
 40
 
 DISTRIBUTION AND ARRANGEMENT 
 
 The Present Mixed Metho&of Procedure 
 
 The result is that, to-day, the two types of 
 arrangement are modifying each other and giv- 
 ing a mixed method, partly " logical " and partly 
 "psychological." That line, of least resistance in 
 which the children study arithmetical facts and 
 processes with greatest success is modified by 
 definite demands that topics e.g., addition 
 be mastered thoroughly " then and there." The 
 method is partly " topical " and partly " spiral." 
 The child in the second grade may have a little 
 of all the fundamental processes, a few simple 
 fractions, antf United States money; but just 
 there he will be held definitely responsible for a 
 very considerable number of the addition com 
 binations. The pupil may have had fractions in 
 every grade, but the fifth grade will be respon- 
 sible for a thorough and systematic mastery of 
 the same. Such is the mixed method of arrange- 
 ment which is to-day prevalent in American 
 schools.
 
 THE DISTRIBUTION OF OBJECTIVE WORK 
 
 Objective Teaching is Generally Current 
 
 THE use of objects in teaching arithmetic is cur- 
 rent in the elementary school. Particularly is 
 this true in the lowest grades of the school, in 
 primary work. It may be said that there is a very 
 large quantity of objective teaching in the first 
 year of schooling and that it decreases more or 
 less gradually as the higher grades are ap- 
 proached. By the time the highest grammar 
 grades are reached, the use of objects has reached 
 its minimum. 
 
 The teaching of arithmetic prior to the middle 
 of the nineteenth century was little associated 
 with object teaching. That is to say, the general 
 practice of instruction was non-objective. The 
 use of objects in giving a concrete basis for ab- 
 stract arithmetical concepts and for memoriter 
 manipulations, seems to have gained its initial 
 42
 
 DISTRIBUTION OF OBJECTIVE WORK 
 
 hold on the schools through the introduction of 
 Pestalozzian methods of teaching. The later 
 introduction of school subjects requiring objec- 
 tive treatment, such as elementary science, na- 
 ture study and manual training, fortified the 
 previous movement and gave it considerable 
 enlargement. Together these two movements 
 established the respectability of objective teach- 
 ing in arithmetic. School-room experience quickly 
 gave it an empirical sanction. It remained for 
 the modern psychological movement in educa- 
 tion to give it a scientific sanction, and to refine 
 its uses. 
 
 Its Distribution is Crudefe* Gauged 
 
 It is quite fair to say that the use of objective 
 work decreases more or less gradually from the 
 first to the last year, the underlying assumption 
 being that the use of objects has a teaching 
 value that decreases as the maturity of the pupils 
 increases. Current practice does not proceed far 
 beyond the application of the simple and some- 
 what crude psychological statement that the 
 youngest children must have much objective 
 43
 
 TEACHING PRIMARY ARITHMETIC 
 
 teaching, the older less, the oldest least of all. 
 The lack of a more refined analysis of the worth 
 of object teaching necessarily leads to some 
 neglect and waste. 
 
 If a new topic enters late into the course of 
 study, as in the case of square root, the subject 
 is not so well taught because of the current pre- 
 judice or tradition against the use of object 
 teaching in the higher grades. On the other 
 hand it is also probable that the teaching of ad- 
 dition is often accompanied by wasted time and 
 energy simply because lingering over objects 
 in the lower classes is the current fashion. 
 
 Tendency toward a More Refined Correlation of 
 Object Teaching with Particular Immaturity 
 
 Reform in the direction of a more refined 
 and exact use of object teaching has already 
 appeared in the treatment of fractions and men- 
 suration, where, regardless of the increased 
 maturity of the children studying these topics, 
 a large amount of objective method is utilized. 
 This is a considerable departure from the slight 
 objective treatment of other arithmetic topics 
 44
 
 DISTRIBUTION OF OBJECTIVE WORK 
 
 taught in the same grades. Such exceptional 
 practices suggest that the novelty of an arith- 
 metic topic is the condition calling for objec- 
 tive work in instruction. I It is immaturity in 
 a special subject or situation which determines 
 the amount of basal objective work.\ The jxr- 
 relation is not with the age of the pupil, but 
 with his experience with the social problem or 
 subject in hand. It is of course true that the 
 younger the student is, the greater the likelihood 
 that any subject presented will be novel and 
 strange. Only in this indirect manner does the 
 novelty of subject matter coincide with mere 
 youth as an essential principle in determining 
 the need of objective presentation. The naive 
 assumption of the older enthusiastic reformers 
 that objective work is a good thing psycho- 
 logically, one of which the pupil cannot have too 
 much, is by no means the accepted view of the/ 
 new reformer. With the latter, objective pre^j 
 
 "V ~~ ~ "^ 
 
 sentation is an excellent method at a given stage 
 of immaturity in the special topic involved ; but 
 it may be uneconomical, even an oBstacle^to) 
 efficiency, if pushed beyond. 
 45
 
 TEACHING PRIMARY ARITHMETIC 
 
 The Movement Supported by both Scientific and 
 Common- Sense Criticism 
 
 There is, then, a certain coincidence of the sci- 
 entific criticism of the psychologist and of the 
 common-sense criticism of the conservative 
 teacher, who look suspiciously upon a highly ex- 
 tended object teaching. The teacher, on grounds 
 of experience, says that too much objective 
 teaching is confusing and delays teaching. The 
 psychological critic says it is unnecessary and 
 wasteful. The result is that, in these later days, 
 the distribution of objective work has changed 
 somewhat. More subjects-are developed in the 
 higher grades through an objective instruction 
 than before. Perhaps no fewer subjects in the 
 lower grades are presented objectively, but the 
 extent of objective treatment of each of these 
 has undergone considerable curtailment.
 
 VI 
 
 THE MATERIALS OF OBJECTIVE TEACHING 
 
 The Indiscriminate Use of Objects 
 
 THE existing defects in objective teaching are 
 not restricted to a false placing or distribution. 
 The quality of the teaching use of objects is 
 likewise open to serious criticism. Object teach- 
 
 aoykA*. 
 
 ing is a device, so successful, as against prior 
 non-objective teaching, that it has come to be a 
 standard of instruction as well as a means. As 
 long as objects any convenient objects are 
 used, the teaching is regarded as good. Given 
 such a sanction, the inevitable result is an un- 
 discriminating use of objects. The process of 
 objectifying tends not to be regulated by the 
 needs of the child's thinking life ; it is determined 
 by the -enthusiasm of the teacher and the ma- 
 terials convenient for school use. 
 
 The Artificiality of Materials Utilized 
 The first fact which is noted in observing ob- 
 jective teaching is the artificiality of the materials 
 47
 
 TEACHING PRIMARY ARITHMETIC 
 
 employed. Primary children count, add, etc., with 
 things they will never be concerned with in life. 
 Lentils, sticks, tablets, and the like are the stock 
 objective stuff of the schools, and to a consider- 
 able degree this will always be the case. Cheap 
 and convenient material suitable for individual 
 manipulation on the top of a school desk is not 
 plentiful. But instances where better and more 
 normal material has been used are frequent 
 enough in the best schools to warrant the belief 
 that more could be done in this direction in the 
 average classroom. The " playing at store," the 
 use of actual applications of the tables of weights 
 and measures are cases that might be cited. 
 
 Narrowness in the Range of Materials 
 
 The materials used are not only more artificial 
 than they need be, but too restricted in range. 
 As has already been said, the types of material 
 capable of convenient and efficient use in a 
 schoolroom are not numerous. But the series 
 can and ought to be extended. More forms of 
 even the artificial material should be used, thus 
 minimizing the danger of monotony. The blame 
 48
 
 MATERIALS OF OBJECTIVE TEACHING 
 
 for the narrow range of materials used falls partly 
 on school boards who do not vote a sufficient 
 allowance for teaching materials to primary 
 teachers ; partly on teachers who do not exercise 
 sufficient ingenuity in devising new forms of 
 objects, or who do not show the vigor requisite 
 to a shift from one material to another; and 
 partly on the supervisory staff which has neither 
 been insistent upon, nor sensitive to, the need 
 of a more interesting range of objective stuffs. 
 
 Inadequate Variation of Traditional Materials 
 
 Even the narrow range of materials in general 
 use might be better employed than it is. There 
 is, of course, a distinct tendency to vary the ob- 
 jects, merely because a child gets tired of one 
 kind as a material. But a different quality of 
 variation is required when the pupil is to derive 
 abstract notions from concrete materials. It is 
 too frequently the case that the teacher will 
 treat the fundamental addition combinations with 
 one set of objects, e.g., lentils. In all the child's 
 objective experience within that field there are 
 two persistent associations " lentils " and " the 
 49
 
 TEACHING PRIMARY ARITHMETIC 
 
 relation of addition." The accidental element is 
 thus emphasized as frequently as the essential 
 one and, being concrete, has even a better chance 
 to impress itself. A wide variation in the objective 
 material used would make teaching more effective, 
 particularly with young children. 
 
 The Restricted Use of Diagrams and Pictures 
 
 The nature of the materials proper to objective 
 teaching has likewise been too narrowly inter- 
 preted. Objective teaching has meant, almost 
 exclusively, instructing or developing through 
 three-dimensional presentations. There is a wide 
 range of two-dimensional representations which 
 have been neglected, but which for all the psy- 
 chological purposes of education have as much 
 worth as so-called objects. I refer here to the 
 use of such material as pictures. Such quasi-ob- 
 jective material has been little used by teachers 
 save as it appears in textbooks. Even the text- 
 book writers have not used pictures with a deep 
 sense of their intrinsic worth. They are printed 
 as a mere substitute for objects in a period when 
 objects are popular pedagogical materials. The 
 50
 
 MATERIALS OF OBJECTIVE TEACHING 
 
 geometric figure or diagram has had a slight use 
 with both the teacher and the textbook writer. 
 Its most frequent use has been in treatments of 
 mensuration. There are, of course, obvious dis- 
 advantages to pictures and diagrams. The things 
 represented in them are not capable of personal 
 manipulation by the child in the ordinary sense. 
 But. they have a superiority all their own. They 
 offer a wider, more natural, and more interesting 
 range of concrete experiences. 
 
 Plays and Games in Object Teaching 
 
 There are other curious phases of narrowness 
 in the current pedagogical interpretation of what 
 constitutes a concrete or objective experience. 
 It will be noted that visual objects are the ones 
 generally employed and that they are generally 
 inanimate objects. Of late there has been some 
 tendency to use hearing and touch in giving a 
 concrete basis to teaching. Advantage is taken 
 of the social plays of children, and their games 
 with things. Here the children themselves, and 
 
 their relations and acts are the experiences from 
 
 
 
 which the numerical units are obtained. With 
 Si
 
 TEACHING PRIMARY ARITHMETIC 
 
 some of the best teachers in the lowest grades 
 it is no longer unusual to see children moving 
 about in all sorts of play designed to add reality 
 to, and increase interest in, number facts. 
 
 The Lack of Unity in the Use of Objects 
 
 The conservative teacher's use of objects is 
 hopelessly artificial and lacking in unity. If he 
 brings a series of objects into the development 
 of a single topic, they have little relation to each 
 other, and they represent no actual grouping. 
 Their sole connection with one another is that 
 they exemplify the same abstract arithmetical 
 truth. Beans, cardboard squares, and shoe-pegs 
 may all be employed in the same lesson. The 
 progressive teachers offer more logical unity in 
 their materials. To "play at store," to utilize 
 games, to deal with things within a picture, is to 
 bring the concrete materials into the classroom 
 with a more nearly normal setting. It is in no 
 small measure due to this better use of material 
 that the progressive teacher is gaining power 
 throughout the elementary grades.
 
 VII y 3 
 
 SOME RECENT INFLUENCES 'ON OBJECTIVE 
 TEACHING 
 
 THE wasteful use of objective teaching in the 
 lowest grades has undergone some correction. 
 The sheer enthusiasm of the modern reformer is 
 partly responsible for this modification of con- 
 servative practice. When did single-minded men 
 ever keep within bounds ? In our social economy 
 the defense of the radical is found in the fact 
 that other single-minded men are conservatives. 
 Men at one extreme need to be overcome by like 
 men at the other. But the check of one enthu- 
 siast on another is not always perfect. Other 
 contributory causes make it easy to go to unfor- 
 tunate extremes not easily corrected. 
 
 The Influence of Inductive Teaching 
 
 Inductive teaching has been one of several 
 movements affecting objective teaching. The 
 effort of teachers to escape the slavishness of 
 53
 
 TEACHING PRIMARY ARITHMETIC 
 
 mere memoriter methods and to approximate 
 real thinking led to the introduction of inductive 
 teaching. Necessarily objective teaching became 
 more or less identified with the new movement 
 and was influenced by it So, it has been said 
 of objective work in arithmetic as it has been 
 said of laboratory work in the sciences, that 
 such instruction is a method of " discovery " or 
 "rediscovery." Such an alliance has had its ben- 
 eficial effects upon objective teaching ; it has re- 
 deemed it from the aimless "observational work " 
 of an earlier " objective study." But in the teach- 
 ing of arithmetic, at any rate, it has confused an 
 objective mode of presentation with a scientific 
 method of learning truth, two activities having 
 a common logical basis, but not at all the same. 
 Under the assumption that the " development " 
 method is one of "rediscovery," the tendency is 
 to give the child as complete a range of con- 
 crete evidences as would be necessary on the 
 part of the scientist in substantiating a new fact. 
 The result is, that long after the child is con- 
 vinced of the truth, say that 4 and 2 are 6, the 
 teacher persists in giving further objective illus- 
 54
 
 INFLUENCES ON OBJECTIVE TEACHING 
 
 trations of the fact. The child loses interest in 
 the somewhat monotonous continuance of ob- 
 jective manipulations, and the teacher has natu- 
 rally wasted time and energy. If the fact or the 
 process that the teacher wishes to convey can 
 be transmitted with fewer objective treatments 
 (the authoritative treatment of the teacher count- 
 ing for something in school, as authority counts 
 everywhere), then it is unnecessary to exhaust 
 the objective treatments of a numerical fact. In- 
 ductive teaching and learning are not equivalent 
 to inductive discovery ; and to hold them iden- 
 tical is necessarily to overdo the use of objects in 
 teaching. 
 
 The Movement for Active Modes of Learning 
 
 Another modern movement in teaching method 
 which has had a conspicuous effect on objective 
 teaching is the Froebelian demand for "self -ac- 
 tivity ".on the part of the child. The recent favor 
 enjoyed by manual training, nature study, self- 
 government, and other active phases of school 
 life is indicative of the sway of this doctrine. Its 
 influence has forced the introduction of new sub- 
 55
 
 TEACHING PRIMARY ARITHMETIC 
 
 jects and changed the manner of presenting the 
 older subjects of the elementary curriculum. 
 Arithmetic has responded along with the other 
 studies, and an active use of objects by the chil- 
 dren themselves is, found in increased degree. 
 
 There was a time when objective work in the 
 schools was a passive matter so far as the child 
 was concerned. Any active manipulation of the 
 objects that might be required was cared for by 
 the teacher, the child being merely a passive 
 observer. This is much less the case than for- 
 merly, the influence of "self-activity" having 
 entered with contemporaneous pedagogy. The 
 present situation is one where the child some- 
 times merely observes objects and sometimes 
 actually handles them. 
 
 At present, then, we have about the same 
 range of methods employed in teaching arith- 
 metic as in teaching science. At one extreme 
 the teacher himself demonstrates by the help of 
 objects in the presence of the class, and records 
 the relations in appropriate arithmetical symbols, 
 the class being in the position of interested spec- 
 tators of a process. At the other extreme the 
 56
 
 INFLUENCES ON OBJECTIVE TEACHING 
 
 teacher puts the material on the desks of the 
 children and, with a minimum of instruction in 
 advance, directs them toward the desired expe- 
 riences and conclusions. 
 
 The Abbreviated Use of Objects 
 
 As might be expected, there has been some 
 reaction against the influences emerging from 
 inductive or developmental teaching and active 
 modes of learning. To a considerable degree the 
 reactionary influence expresses itself in the ab- 
 breviated use of objects in presenting a mathe- 
 matical relation, process, or manipulation. One 
 mode of abbreviation will suffice as an example. 
 
 There are two methods of relating the sym- 
 bols and processes of arithmetic to the actual 
 relations among objects. For convenience these 
 maybe called the methods of "parallel corre- 
 spondence" and of "final correspondence." 
 
 The Method of Parallel Correspondence 
 
 The method of " parallel correspondence " is 
 generally used in the development of all the 
 simpler combinations or processes of arithmetic. 
 57
 
 TEACHING PRIMARY ARITHMETIC 
 
 In learning to count, the child sees the first ob- 
 ject and says the symbolic " one," sees the sec- 
 ond object, and says the symbolic "two." Again 
 in addition, he sees " ten," and writes the sym- 
 bol 10 ; sees " six," and writes 6 ; sees the whole 
 as sixteen, and writes 16 ; then summarizes the 
 work in the form 10 + 6= 16. Each stage in the 
 symbolic process is noted in connection with ob- 
 jects. This, the method of "parallel correspond- 
 ence," is the more current method of using 
 objects. 
 
 The Method of Final Correspondence 
 
 The method of showing a "final correspond- 
 ence" of result between objective manipulation 
 and symbolic manipulation is much less fre- 
 quently used. It is used with more complex pro- 
 cesses than those mentioned above, in connec- 
 tion with column addition or "borrowing" in 
 subtraction. It is a mode of object teaching used 
 in place of the usual "explanation" or "ration- 
 alization " which attempts to explain what is sim- 
 ply a correspondence between the manipulation 
 of a series of facts and the manipulation of a se- 
 58
 
 INFLUENCES ON OBJECTIVE TEACHING 
 
 ries of symbols. Under this method the teacher 
 usually tells the child directly how to perform 
 the process in the conventional manner, no spe- 
 cial explanation being given. Then a case in- 
 volving the actual use of objects is considered, 
 and this result is compared with the result ob- 
 tained by the symbolicv manipulation. One or 
 two such cases suffice to convince the pupil that 
 the authoritative mode is true to nature. This 
 method of " final correspondence " in the use of 
 objects represents a new and restricted, but in- 
 creasing, tendency.
 
 VIII 
 
 THE USE OF METHODS OF RATIONALIZATION 
 
 The Tendency toward Rational Methods 
 SOME of the marked changes which have occurred 
 in the methods employed for the presentation of 
 number to children have already been mentioned 
 in connection with the objective teaching of 
 arithmetic. The main tendency to be noted is 
 that objective instruction, which has been used 
 as a mere device of illustration, becomes the 
 first step in inductive or developmental teaching. 
 It is subsumed under a more inclusive method. 
 The change is significant, for it is a symp- 
 tom indicating that mathematical teaching is 
 becoming less dominantly memoriter and more 
 rational. 
 
 The Era of Direct Instruction and Drill 
 
 Several decades ago it was not at all unusual 
 for the bare facts of arithmetic to be given to 
 the child by the teacher without much attempt at 
 60
 
 METHODS OF RATIONALIZATION 
 
 providing a basis in the pupil's own experience. 
 The teaching was "direct," the teacher's atten4 
 tion being focused on getting the fact from his) 
 own mind into the child's mind, -the whole env 
 phasis being placed upon the problem of trans- 
 mission and the subsequent difficulty of retention. 
 In so far as objects and illustrations were used, 
 they were merely incidental and reinforcive. They 
 did not constitute any basic body of personal ex- 
 perience by which the child was to seize a con- 
 cept, relation, or process to be handled thereafter 
 through symbols and conventional forms. Under 
 such a system of instruction, still too widely prev- 
 alent, the child had to memorize outright, with- 
 out any concrete basis for his belief, tables of 
 addition, multiplication, etc., and the rules for 
 
 solving various types of problems. It-was outright 
 
 * 
 
 memorization for which little vital motivation 
 was provided. In insuring retention the teacher 
 
 therefore relied, not upon interested a'nd varied 
 
 '.*.' 
 impressions, but upon the number of verbal re- 
 
 petitions. " Drill " was characteristic of this era 
 in teaching. 
 
 61 
 
 \,
 
 TEACHING PRIMARY ARITHMETIC 
 
 Indirect Teaching as a Rational Method 
 
 Under the influence of the inductive or de- 
 velopmental method of teaching, the emphasis 
 on the repetitional memorization of number facts 
 and processes is reduced. Teaching now becomes 
 "indirect" rather than "direct" ; the child learns 
 through his own experience rather than through 
 the statement of book or teacher. Here the child's 
 own thought and activity, not the teacher's, are 
 conspicuously central in the teaching situation. 
 The teacher stimulates the child into action ; he 
 suggests, guides, corrects, does everything in fact 
 save obtrude his authority and opinion into the 
 child's interpretation. The child's activity gives 
 him many vivid and varied impressions of the sub- 
 traction combination or other relation. When he 
 has found the fact, he has already learned it ! 
 Further drill or review is not primary, but simply 
 supplementary a further guarantee of the per- 
 sistence of the impression. Even the spirit of 
 such supplementary drill and review is, in these 
 days, something different from a monotonous re- 
 petition of the same words ; it is a reimpression 
 62
 
 METHODS OF RATIONALIZATION 
 
 or review of the essential fact in many varied and 
 interesting forms. 
 
 Interest as a Factor in Methods of Rational- 
 ization 
 
 It is perfectly natural that, in shifting the 
 teacher's attention from his own activities to those 
 of the children, the interest of the child should 
 be considered in increasing degree. If the child 
 is to learn directly, with a maximum use of his 
 initiative, it is absolutely essential that the 
 teacher should provide some motive. This im- 
 plies that the child is to be interested in some 
 fundamental way in the activities in which he is 
 to engage. Instead of thumbing the fundamental 
 facts with his memory, in an artificial and effort- 
 ful manner, "sing-songing" the tables rhythmi- 
 cally, so as to make dull business less dull, the 
 child studies the arithmetic involved in his own 
 life, for the modern teacher brings the two to- 
 gether. The number story, the arithmetical game, 
 playing at adult activities, constructive work, 
 measuring, and other vital interests of the child 
 and community life become increasingly the basis 
 63
 
 TEACHING PRIMARY ARITHMETIC 
 
 of instruction in number. Such is the pronounced 
 tendency wherever the movement is away from 
 the traditional rote-learning or drill. 
 
 Of course there is a slight tendency in Amer- 
 ican elementary schools, where a soft and false 
 interpretation of the "doctrine of interest" is 
 gospel, to teach only those things that can be 
 taught in an interesting fashion. But this tend- 
 ency is less operative in arithmetic than in 
 other subjects. Here the logical interdependence 
 of one arithmetical skill on another has quickly 
 pointed the failure of such a haphazard mode of 
 instruction. 
 
 The Reaction against Rationalization 
 
 There is, however, in " advanced " as well as 
 in reactionary quarters, a revolt against the tend- 
 ency to objectify, explain, or rationalize every- 
 thing taught in arithmetic. On the whole it is a 
 discriminating movement, for this opposition to 
 "rationalization" in arithmetical teaching, and 
 in favor of "memorization" or " habituation," 
 bases its plea on rational grounds, mainly derived 
 from the facts of modern psychology. 
 64
 
 METHODS OF RATIONALIZATION 
 
 It is specifically opposed to explaining why 
 "carrying" in addition, and "borrowing" in sub- 
 traction are right modes of procedure. These 
 acts are to be taught as memory or habit, inas- 
 much as they are to be performed by that method 
 forever after. To develop such processes ration- 
 ally or to demand a reason for the procedure once 
 it is acquired, is merely to stir up unnecessary 
 trouble, trouble unprompted by any demands of 
 actual efficiency. 
 
 Four Principles for the Use of Rationalization 
 
 A study of the actual arithmetical facts upon 
 which this opposition expresses itself suggests 
 the four following general principles as to the 
 use of "rationalization" and " habituation," as 
 methods of mastery: (i) Any fact or process 
 which always recurs in an identical manner, and 
 occurs with sufficient frequency to be remem- 
 bered, ought not to be "rationalized" for the 
 pupil, but "habituated." The correct placing 
 of partial products in the multiplication of two 
 numbers of two or more figures is a specific case. 
 (2) If a process does recur in the same manner, 
 65
 
 TEACHING PRIMARY ARITHMETIC 
 
 but is so little used in after life that any formal 
 method of solution would be forgotten, then the 
 teacher should "rationalize" it. The process of 
 finding the square root of a number illustrates 
 this series of facts. (3) If the process always does 
 occur in the same manner, but with the frequency 
 of its recurrence in doubt, the teacher should 
 both "habituate" and " rationalize." The division 
 of a fraction by a fraction is frequently taught 
 both "mechanically" and "by thinking it out." 
 (4) When a process or relation is likely to be ex- 
 pressed in a variable form, then the child must 
 be taught to think through the relations involved, 
 and should not be permitted to treat it mechan- 
 ically, through a mere act of habit or memory. 
 All applied examples are to be dealt with in this, 
 manner, for such problems are of many types, 
 and no two problems of the same type are ever 
 quite alike. These laws will, of course, not be 
 interpreted to mean that no reason is to be given 
 a child in a process like " carrying " in addition. 
 The reason is not essential to efficient mastery, 
 but it may be given to add interest or to satisfy 
 the specially curious. 
 
 66
 
 METHODS OF RATIONALIZATION 
 
 The Substantiating Psychology 
 The theoretic basis which seems to underlie 
 such a statement of general principles is derived 
 from functional psychology. Memory and reason- 
 ing are not separate functions ; they are interde- 
 pendent ; but we mean differently by these terms 
 because they have distinguishable emphases. It 
 may be said that memory as a function is effi- 
 cient only in the face of familiar situations 
 where, if the association is present at all, the 
 response is quick and precise. In the face of 
 new situations it is incapable of accurate re- 
 sponse. Reason is slow and uneconomical in 
 action, but it is the only efficient method of 
 arriving at the essential nature of a problem 
 largely unfamiliar. It would be wasteful to meet 
 many of the necessary events of life with a 
 purely reasoned reaction. It would be too de- 
 vious and deliberate in reaching conclusions. 
 
 Rationalization as a Substititte for Object 
 
 Teaching 
 
 There is a sense in which all proof through 
 objects is a type of rationalization, but we do 
 
 67
 
 TEACHING PRIMARY ARITHMETIC 
 
 not ordinarily so consider it. Such a mere "cor- 
 respondence with the objective facts " is suffi- 
 ciently different from the method of " explaining 
 a new fact in terms of previously acquired facts" 
 to warrant a separate classification. Were it 
 not that the methods are sometimes inter- 
 changeable in developing arithmetical truths, 
 they would not need to be mentioned here. A 
 citation will make the point clear. In teaching 
 the multiplication tables, the combination " six 
 twos are twelve " may be taught as a direct ob- 
 jective fact, as when six pupils with two hands 
 each are shown. On the other hand the same 
 fact of multiplication may be taught as a " de- 
 rived fact," as when six twos are added in a 
 column and make twelve. They are both rational 
 methods of proving that six twos are twelve. 
 One method shows it objectively, the other 
 through the use of well established addition 
 combinations viewed as multiplication. Such a 
 rational mode of "deriving" multiplication is 
 used more frequently perhaps than the objective 
 method.
 
 IX / 
 
 SPECIAL METHODS FOR OBTAINING ACCU- 
 RACY, INDEPENDENCE, AND SPEED 
 
 Supervision of Learning after First Development 
 
 IT is not alone the first stages in the acquisition 
 of an arithmetical process which have received 
 attention in the re-organization of teaching 
 methods, though, to be sure, the problem of 
 first presentations has in recent decades been 
 given the most attention. More and more, the 
 American tendency is to watch every step in 
 the learning process, to provide for all necessary 
 transitions, and to safeguard against avoidable 
 confusions. It might be suggested that the in- 
 termediation of the teacher at every step in the 
 child's work might destroy the pupil's initiative 
 and independence. Apparently, however, those 
 who are deeply interested that the child should 
 not be permitted to fall into the errors which 
 unsupervised drill would convert into habits, are 
 fully as cautious to provide steps for forcing the 
 69
 
 TEACHING PRIMARY ARITHMETIC 
 
 child to assume an increasing responsibility for 
 his own work. The distinction made is that an 
 over-early independence is as fatal to accurate 
 and rapid mathematical work as an over-delayed 
 dependence. 
 
 The Use of Steps, or Stages, in Teaching 
 
 Some of these serial treatments or related 
 stages of method to which reference has been 
 made may be cited. Necessarily only the more 
 important are mentioned. In indicating certain 
 clean-cut steps or processes, there is no intent to 
 convey the idea that these stages are fixed or 
 conscious matters. The statement is merely in- 
 dicative of the habitual tendency of the average 
 practitioner with an implied theory. As will be 
 readily evident, there is no assumption that such 
 a formal, classified, theoretic statement of stages 
 is a conscious possession of the teaching staff in 
 general. Again, in actual school work there are 
 many types of variation from the characteristic 
 modes here suggested. Always the steps pverlap ; 
 frequently they are extended, abbreviated, or 
 omitted. But the statement represents, in a very 
 70
 
 SPECIAL METHODS 
 
 real way, the trend of underlying theory, whether 
 conscious or merely implied. 
 
 Stages in the Presentation of Problems 
 
 One of the specific controversies much argued 
 in the primary school concerns the medium 
 through which arithmetical examples and prob- 
 lems shall be transmitted to young children. 
 There are three typical ways in which a situa- 
 tion demanding arithmetical solution may be 
 brought to the child's mind : (i) The situation 
 when visible may be presented through itself, 
 that is, objectively. (2) The situation may be 
 described through the medium of spoken lan- 
 guage, the teacher usually giving the dictation. 
 (3) The situation may be conveyed through 
 written language, as when the child reads from 
 blackboard or text. Inasmuch as objects are a 
 universal language, no difficulty arises through 
 this basic method of presentation. It is when a 
 language description of a situation is substituted 
 for the situation itself that difficulty occurs. The 
 child might be able to solve the problem if he 
 really understood the situation the language was 
 71
 
 TEACHING PRIMARY ARITHMETIC 
 
 meant to convey. Owing to the difficulty that 
 primary children have in getting the thought out 
 of language, it has been urged that problems 
 in any unfamiliar field should be presented in 
 the following order : (i) Objectively or graphi- 
 cally ; (2) when the fundamental idea is grasped, 
 through spoken language ; and (3), after the type 
 of situation is fairly familiar, through written or 
 printed language. It is seriously urged by some 
 teachers that no written presentation should be 
 used in the first four grades. Such an extreme 
 tendency would practically abolish the use of 
 primary text-books. There are many exceptional 
 teachers who do not put a primary text in the 
 hands of children at all. Such a tendency is in- 
 creasing. Particularly is this true among primary 
 teachers in the schools of the foreign quarters of 
 large cities. Accurate communication through 
 the English language is always more difficult here. 
 Hence, the period of objective teaching is neces- 
 sarily prolonged, dependence on the "number 
 stories " told by the teacher extended, and the 
 solution of written problems much longer de- 
 layed than elsewhere. 
 
 72
 
 SPECIAL METHODS 
 
 An Opposite Method in Presenting Examples 
 
 The situation is somewhat different, almost 
 the opposite in fact, when " examples " rather 
 than "problems " are presented, meaning by " ex- 
 ample" a "problem " expressed through the use 
 of mathematical signs. It is easier to present 
 " examples " in written form on blackboard or in 
 text than it is to dictate them orally. This ob- 
 viates the necessity of holding the examples in 
 mind during solution. The permanence of the 
 visual presentation saves the re-statement fre- 
 quently necessary in oral presentation. Hence it 
 is a common practice to supply the youngest chil- 
 dren with mimeographed or written sheets of "ex- 
 amples." It is with older children, or with younger 
 children at a later stage in the mastery of a typical 
 difficulty, that oral presentation of examples is 
 stressed. Then we have that type of work which 
 is called " mental " or " silent " arithmetic. 
 
 Better Transitions from Concrete to Abstract 
 
 Work 
 
 There is some tendency toward the provision 
 of better transitions from the objective presenta- 
 73
 
 TEACHING PRIMARY ARITHMETIC 
 
 tion of applied problems to the symbolic pre- 
 sentation of abstract examples. Behind all uses 
 of objective work is the belief that it is a mere 
 foundation for more rapid and efficient abstract 
 work. Objective teaching is fundamental, but 
 purely preparatory. The child ought to pass 
 from objects and sense-impressions, through 
 images of various degrees of abbreviation, to 
 symbols and the abstract concepts for which 
 they stand. But in American practice a sharp 
 jump is usually made from concrete objects to 
 abstract symbols. The transition through ade- 
 quate transitional imagery is not made. Wher- 
 ever psychological influences are directly at work 
 in the schools, there is a minority movement 
 favoring a better transition. The nature of such 
 transition is scarcely reasoned out as so much 
 psychological science, but is the accompaniment 
 of a widening professional movement for the 
 enlarged use of pictures, diagrams, number 
 stories, and the like. A critical examination of 
 the various means of presenting arithmetical 
 situations would order them as follows in making 
 the transitions from objective concreteness to 
 74
 
 SPECIAL METHODS 
 
 symbolic abstraction: (i) Objects, (2) pictures, 
 (3) graphs, (4) the concrete imagery of words, 
 (5) more abstract verbal presentations, (6) pre- 
 sentations through mathematical symbols. No 
 such minuteness of adjustment is apparent in 
 existing methods, though it might seem desir- 
 able in teaching young children. At any rate, it 
 would be more effective than an unreasoned tra- 
 ditional procedure full of over-emphasis and 
 omission. 
 
 The Child's Four Modes of Work 
 
 We have thus far discussed merely the teach- 
 er's activity in instruction. We have to note the 
 graded requirement made in the child's own ac- 
 tivity. What is the existing custom with refer- 
 ence to the manner !n which children are required 
 to solve the problems or examples presented to 
 them ? There are four typical ways in which the 
 child does his work, the names of which are de- 
 rived from the differentiating element : (i) The 
 " silent " method, otherwise spoken of as " mental 
 arithmetic," "arithmetic without a pencil," etc. 
 (2) The "oral" method where the child works 
 75
 
 TEACHING PRIMARY ARITHMETIC 
 
 aloud, that is, expresses his procedure step by 
 step in speech. (3) The " written " method where 
 the child writes out in full his analysis and cal- 
 culation. (4) The "mixed" method where the 
 child uses all three of the previously mentioned 
 methods, in alternation, as ease and efficiency 
 may require. 
 
 The Worth of these Modes 
 
 The worth of these four methods of work is 
 necessarily variable. The rapidity of the " silent 
 method" with simple figures is obvious. The 
 "silent method "and the "mixed method" (which 
 is more slow but more manageable with complex 
 processes and calculations) are the two methods 
 normally employed in social and business life. 
 The purely " oral " and "written " methods, with 
 their tendency toward analysis and calculation 
 fully expressed in oral or written language, are 
 highly artificial. They are valuable as school 
 devices for revealing the action of the child's 
 mind to the teacher so that it may be corrected, 
 guided, and generally controlled. The present 
 tendency is toward an over-use of these methods 
 76
 
 SPECIAL METHODS 
 
 and toward an under-use of the other two, more 
 particularly the " mixed " method. It would seem 
 that there is little conscious attempt to make 
 certain that the child moves from full oral or 
 written statement to the judicious application of 
 the more natural " silent" and " mixed " methods. 
 It may be that full oral and written statements 
 of work have seriously hampered the right use of 
 the more natural methods of statement. 
 
 The Traditiottal Quarrel between " Mental " and 
 " Written " Arithmetic 
 
 One conspicuous traditional quarrel in the 
 schools is between the " silent " and the " writ- 
 ten " methods. Up to within a decade or so ago, 
 "silent" or "mental " arithmetic was much over- 
 emphasized, being carried to absurd extremes. 
 The reaction that followed was equally extreme 
 in its emphasis on the " written " method. There 
 are signs now of a more rational use of the two 
 as supplements of each other. 
 
 The order in which different statements of 
 arithmetical work should come has also been a 
 subject for pedagogical argument. The usual 
 77
 
 TEACHING PRIMARY ARITHMETIC 
 
 order, due to the fact that first treatments of a 
 topic are simple both in the steps of reasoning 
 and the calculations involved, has been " silent " 
 arithmetic followed by " written " arithmetic. A 
 more recent order has been: (i) "silent" (2) 
 " written " (3) " silent " a much superior serial 
 order, though by no means an accurate state- 
 ment of a perfect procedure. The fixed treat- 
 ments necessitated by text-books have made 
 teaching method arbitrary in its steps, here as 
 elsewhere. It is altogether probable that many 
 simple calculations or analyses can be done 
 "silently" ("mentally") from the beginning; 
 that others require visual demonstration, but once 
 mastered can thereafter be done without visual 
 aids ; that still others will always have to be per- 
 formed, partially at least, with some written work. 
 It is purely a matter for concrete judgment 
 in each special case, but the existing practice 
 scarcely recognizes this truth. The result is that 
 many problems are arbitrarily done in one way, 
 and it is too frequently the uneconomical and in- 
 efficient way that is used.
 
 SPECIAL METHODS 
 
 The Transition from Development by Teacher to 
 Independent Work by Pupil 
 
 It is well to recall that in all these efforts to 
 control the child's activity, there is a tendency 
 to leave the child over-dependent upon the 
 teacher. It is vitally important that a child 
 should be kept free of any error which unsuper- 
 vised drill would fix into a stubborn habit ; but 
 it is likewise important that the child should 
 acquire some self-reliance. While not always 
 clearly defined, there is a distinct tendency in 
 the direction of releasing the teacher's control 
 of the child. A characteristic practice would be 
 one in which the teacher's work with the child 
 would pass through various stages, each one of 
 which would mark a decrease in the control of 
 the process by the teacher and an increase in 
 the freedom of the child to do his example, or 
 problem, by himself. 
 
 Four Characteristic Stages of the Transition 
 
 One characteristic series of stages quite fre- 
 quently used in the presentation of a single topic 
 79
 
 TEACHING PRIMARY ARITHMETIC 
 
 in arithmetic, say "carrying" in addition, is the 
 following: (i) The teacher performs the pro- 
 cess on the blackboard in the presence of the 
 class, the children not being allowed to attempt 
 the process by themselves until after the process 
 is clearly understood from the teacher's develop- 
 ment. (2) The children are then allowed to per- 
 form the process upon the blackboard, where it 
 is exceedingly easy for the teacher to keep the 
 work of every child under his eye. An error is 
 caught by a quick glance at the board and im- 
 mediately corrected before the child can reiterate 
 a false impression. (3) Other cases of the same 
 type of example are assigned to the children at 
 their seats where they work upon paper, still 
 under the supervision of the teacher a super- 
 vision which is less adequate, however. (4) The 
 same difficulty, after the careful safeguarding of 
 the previous stages, is then assigned for "home 
 work," where the child relies almost completely 
 upon himself. Once more it is necessary to sug- 
 gest that these stages are merely roughly im- 
 plied in the variations of existing practice. 
 
 80
 
 SPECIAL METHODS 
 
 Special Methods of Attaining Speed 
 
 Most of the methods discussed in this chapter 
 have had as their sanction the attainment of ac- 
 curacy in thinking and calculating. Some efforts 
 to insure independent power on the part of the 
 child have already been noted. But nothing has 
 been said of the effort to add speed to accuracy 
 in getting efficient results. Such special efforts 
 have been made. These efforts may be classified 
 into two groups : (i) Those aiming to quicken 
 the rate of mental response. (2) Those aiming at 
 short-cut processes of calculation. 
 
 Typical of the first are (a) the use of an es- 
 tablished rhythm as the child attacks a column 
 of additions ; (b} the device of having children 
 race for quick answers, letting them raise their 
 hands or stand when they have gotten the an- 
 swer ; (c) the assignment of a series of problems 
 for written work under the pressure of a restricted 
 time allotment for the performance of each. 
 These and similar devices are much used in the 
 schools. They are open to the objection that they 
 quicken the rate of the better students, but foster 
 81
 
 TEACHING PRIMARY ARITHMETIC 
 
 confusion, error, and discouragement among the 
 less able children, thus actually hindering speed. 
 The various shorter methods which represent 
 an effort to reduce the number of mental pro- 
 cesses required are usually not of general appli- 
 cability, and consequently have not attained gen- 
 eral currency in the elementary schools which aim 
 to teach merely one generally available and ef- 
 fective method even though it requires more 
 time, special expertness being left to later de- 
 velopment in the special school or business which 
 requires it. 
 
 The Relation of Accuracy to Speed 
 
 It has come to be quite a common opinion 
 among teachers that the fundamental element in 
 rapid arithmetical work is certain and accurate 
 calculation. If pupils know their tables of com- 
 binations and are sure of each detail of calcula- 
 tion, there is no confusion or hesitancy ; speed 
 then follows as a matter of course. This belief, 
 as much as anything else, explains why the lower 
 schools have developed few special means for at- 
 taining speed other than those mentioned.
 
 THE USE OF SPECIAL ALGORISMS, ORAL 
 FORMS, AND WRITTEN ARRANGEMENTS 
 
 THE methods of teaching arithmetic are influ- 
 enced not only by the aims of such instruction, 
 but by the peculiar nature of the matter taught. 
 The use of special algorisms, temporary algor- 
 istic aids or teaching " crutches," oral and written 
 forms of analysis, are of considerable moment 
 in determining the difficulties and therefore the 
 methods of teaching. Their condition and influ- 
 ence will need to be given some slight notice. 
 
 The Traditional Nature of Algorisms and Forms 
 
 The algorisms and forms used in the American 
 schools are those that have been determined by 
 social and educational traditions. It is probable 
 that wide social practice has largely determined 
 the traditions, though it must be admitted that 
 the traditions of text-book makers have also 
 given it form. In consequence the ruling school 
 83
 
 TEACHING PRIMARY ARITHMETIC 
 
 tradition in the matter of algorisms does not al- 
 ways coincide with current community practice. 
 It is probable that the various modes of comput- 
 ing interest, given by the average arithmetic of 
 ten years ago, were once current in the business 
 world. These methods have changed somewhat, 
 and the school form of computation has not al- 
 ways been changed to accord. Such misadjust- 
 ments between the forms used in school and 
 those used in daily life are not numerous, but 
 they are more frequent than they ought to be. 
 
 The Number of Algorisms Used 
 
 The use of various algorisms for a single pro- 
 cess is not very frequent. There is a fairly gen- 
 eral prevalence of a single algorism for a single 
 process among American teachers. Certain strik- 
 ing exceptions are to be noted in connection with 
 subtraction and division, where the so-called 
 " Austrian " methods are being brought into 
 competition with the traditional modes of the 
 American school. It may be said, however, that 
 even when two distinct algorisms are in contem- 
 poraneous use in a school system, the teachers 
 84
 
 USE OF SPECIAL FORMS 
 
 are usually careful to employ only one algorism 
 with a given child. Even when a child moves from 
 a school using one kind of algorism to a school 
 using another, the tendency is to allow him to 
 follow his own method. 
 
 Reform in the Use of Algorisms 
 
 The tendency toward the substitution, dupli- 
 cation, and modification of existing algorisms is 
 inconsiderable, and very recent. The introduction 
 of the "Austrian" algorisms, already mentioned, 
 is perhaps the most conspicuous movement, hav- 
 ing a very wide group of adherents. There are, 
 however, a group of teachers and educational 
 psychologists who are attempting to refine teach- 
 ing methods so as to attain a greater economy 
 and efficiency in the learning process. These are 
 responsible for a movement toward the reform 
 of existing algorisms. The movement expresses 
 itself in a number of ways, it offers new forms, 
 modifies others, and aims to bring a larger sim- 
 ilarity and consistency into algorisms employed 
 in the various stages of the same process. Its 
 function is always to simplify for the child and 
 85
 
 TEACHING PRIMARY ARITHMETIC 
 
 thus to increase the practical efficiency and the 
 mental economy of his methods. 
 
 The Standard of Social Usage 
 
 One of the standards set, is, that as far as is 
 consistent with economy, the algorism employed 
 with greatest frequency in social life is to be 
 preferred. If people add, subtract, and multiply 
 with their figures arranged above and below each 
 other in vertical form 3 , then the vertical form 
 
 9 
 
 is to be preferred to the horizontal method 
 6+3=9 so largely used and imposed by text- 
 book writers. 
 
 The Extended Use of Acquired Forms 
 
 To learn two forms for one thing, particularly 
 when one has no sanction either in current use 
 or on general grounds of psychological efficiency, 
 is a waste. Hence there is an increasing disposi- 
 tion both in general practice and among the 
 more critical to utilize a single form in as many 
 places as possible. " Subtracting by adding " is 
 merely using the same association and word form 
 for both addition and subtraction. Hence only 
 86
 
 USE OF SPECIAL FORMS 
 
 one set of tables, instead of two, has to be learned. 
 The meaning, the applicability, and the visual 
 form of addition and subtraction are still different. 
 Only the process of remembering and using the 
 fundamental combinations is the same. But this 
 is a large saving : " Dividing by multiplying " is 
 an analogous situation, though not so much em- 
 ployed in American schools as " subtracting by 
 adding." The most radical suggestion for utilizing 
 a simplified common form is one in which these 
 forms of division, as used in the tables 18-7-6=3, 
 in short division 6)1832, and in long division 
 
 62)i8325(, are reduced to one consistent form 
 in all three cases, as 6 [78, 6| 1832, 62 | 18325. 
 Such a simplification is urged in other situations. 
 The movement has not passed far beyond theo- 
 retic acceptance, though several city school sys- 
 tems are trying the experiment, San Francisco 
 being a notable instance. 
 
 The Use of " Crutches" or Temporary Algorisms 
 
 The use of special and temporary algoristic 
 aids or learning " crutches " in mathematical cal- 
 culation is one of the problems of method under 
 87
 
 r TEACHING PRIMARY ARITHMETIC 
 
 constant controversy. Teachers seem fairly evenly 
 divided upon the question. Typical situations in 
 which such " crutches " are used may be noted 
 as follows : Changing the figures of the upper 
 number in " borrowing " in subtraction ; rewrit- 
 ing figures in adding and subtracting fractions. 
 In the broad sense any algorism which is used 
 during the teaching or learning process tempor- 
 arily, to be abandoned completely later, is an 
 "accessory algorism" or "crutch." The objec- 
 tions to their use lie in the facts : (i) that skill in 
 manipulation is learned in connection with stages 
 and forms not characteristic of final practical 
 use ; (2) that this implies, psychologically at any 
 rate, the waste of learning two forms or usages 
 instead of one ; and, (3) that it decreases the 
 speed with which mathematical calculation is 
 done. If there is a drift in any direction, it is prob- 
 ably toward the abandonment of " crutches." 
 
 Full and Short Forms of Calculation 
 
 The division of opinion, which exists in con- 
 nection with the temporary use of special algor- 
 isms or "crutches," likewise exists with reference 
 88
 
 USE OF SPECIAL FORMS 
 
 to the use of " full forms " and " short forms " 
 of manipulation and statement. The temporary 
 use of a " full form," in a case where a " short 
 form " will finally be used, is similar to the em- 
 ployment of a "crutch." There is one important 
 difference, however, which explains the relatively 
 larger presence of temporary " full forms " than 
 of "crutches." The "full form " is an accurate 
 form which is used somewhere, in a more com- 
 plex stage of the same process or in some other 
 process ; the " crutch " is not. Thus : a " full 
 form " in column addition with partial totals and 
 a final total of partial totals, will be utilized in 
 column multiplication, the " long division form " 
 of doing " short division " (which is the fully ex- 
 pressed form of dividing by a number of one 
 figure) will be utilized in division by numbers of 
 more than one figure. 
 
 Forms of Analysis or Reasoning 
 
 The problem of form applies not alone to the 
 
 algorism or special method of computation, but 
 
 likewise to the special methods of reasoning used 
 
 in determining the specific series of steps to be 
 
 89
 
 TEACHING PRIMARY ARITHMETIC 
 
 taken in achieving the answer. In every problem 
 the child solves, he must not only decide what is 
 to be done (reason), but he must do it (calculate). 
 There are forms of reasoning as there are forms 
 of calculation. As any calculation may have 
 several algorisms, the solution of a problem may 
 be expressed in several forms. It is the latter dif- 
 ficulty which appears in the teacher's demands 
 for " formal analysis" of problems. The analysis 
 is usually required in full statement. 
 
 The Traditional Requirement of Full Formal 
 Analysis 
 
 It has been a very general requirement in 
 American schools that the child give a full oral or 
 written statement of his analysis and computation. 
 Formal statements have been demanded of the 
 child as much on the side of reasoning as on that 
 of calculation. One of the causes of this demand 
 is found in the tendency of the teacher to en- 
 courage full statement by the child, merely as 
 a revelation of his inner processes so that the 
 source of error in results might be detected and 
 the error eliminated. We have already noted this 
 90
 
 USE OF SPECIAL FORMS 
 
 predisposition of the teacher to call for full oral 
 and written statements for purposes of control in 
 the various methods designed to achieve and safe- 
 guard accuracy. 
 
 The Limitations of Full Formal Analysis 
 
 There is, however, a marked tendency away 
 from formal analysis of arithmetic problems in 
 the elementary school, just as there is a move- 
 ment away from a formal deductive logic in 
 the higher schools. Natural, genetic modes of 
 thought are supplanting the unnatural, formal 
 statement of steps. It is felt that while such full 
 formal statements of reasoning and calculation 
 may assist in the teacher's control, they may 
 actually interfere with accuracy and rapidity on 
 the child's part. 
 
 To write out each step in full often means 
 giving an enlarged attention to factors that are 
 merely touched and assumed in actual thinking. 
 To delay the thought process, with attention held 
 on a fully developed linguistic statement, whether 
 oral or written, may be to distract from the chain 
 of essential ideas or meanings that really solve 
 
 91
 
 TEACHING PRIMARY ARITHMETIC 
 
 the problem. Frequently children lose the thread 
 of thought midway of the process because of the 
 necessity of dealing with the form side, and have 
 to begin anew. 
 
 " Labeling " the Steps of Calculation 
 
 A conservative protest against the old formal 
 expression of reasoned steps is found in omitting 
 for the most part the linguistic statements deal- 
 ing with the logic of the problem and merely 
 " labeling " the numbers that occur in the calcu- 
 lation. This is a more restricted form of state- 
 ment, much more used at the present time than 
 hitherto. But it is still open to psychological 
 objections that make the more scientific critics 
 protest. There are many stages in a calculation 
 where there is no association whatever with the 
 concrete problem in hand. The concrete problem 
 is studied, the decision is made that all the fac- 
 tors named are to be added. They are added, 
 purely abstractly, and a number is given as the 
 total. The result is then thought of in terms 
 of the concrete problem in hand. A disposition 
 to label each item in the addition may be ne- 
 92
 
 USE OF SPECIAL FORMS 
 
 cessary in the rendering of a bill, but it is a 
 false and obstructing activity in the actual solv- 
 ing of the problem. The same situation exists 
 where there are two or three processes to be 
 utilized in series. Once the child has grasped his 
 concrete situation and reasoned what to do he 
 may proceed to mechanical manipulation, never 
 thinking of the concrete applications till he has 
 finished. 
 
 So-called Accuracies of Statement 
 
 The protest mentioned above even goes so far 
 as to attack the teacher's insistence upon certain 
 so-called accuracies of statement. The case of 3 
 pencils at 5 cents would be expressed 3 times 5 
 cents ==15 cents. 3x5 = 15 would be demanded, 
 and 5x3 not allowed at all. The protest is not 
 against insistence on a proper order where " la- 
 beled " statements are used. The objection is 
 made against the demand for the so-called proper 
 order when abstract figures are related merely 
 by signs. Where the child calculates ^symbol- 
 ically, he sees the situation as one to be worked 
 out by a purely conventional relation between 
 93
 
 TEACHING PRIMARY ARITHMETIC 
 
 two numbers. For all practical purposes 5x3 
 will solve the situation quite as accurately as 
 3x5. The insistence on one, as opposed to the 
 other, is a useless effort that cannot affect the 
 result. 
 
 Increased Use of Mathematical Symbols 
 
 The same tendency which is making for a re- 
 duction of verbal forms is increasing the use of 
 mathematical symbols. As logical relations are 
 less frequently written out, a simple sign such 
 as + or -f- is used. The algebraic x is supplied 
 in place of a whole roundabout series of awkward 
 preliminary statements or assumptions. With it, 
 of course, come changed methods of manipula- 
 tion, as in the use of the algebraic equation. 
 
 It is doubtless true that the rigidity of full 
 logical forms is giving way to a more flexible and 
 natural mode of expressing the child's thoughts. 
 Fixed oral and written forms of exposition may 
 assist the child, much as the acquisition of a 
 definite symbol fixes an abstract meaning, which 
 remains unwieldy till it attaches itself to a word 
 by which it is to be recalled. But increasing care 
 94
 
 USE OF SPECIAL FORMS 
 
 is manifested that children shall use only those 
 forms that will conform to practical need upon 
 the one hand, and to natural, efficient, and eco- 
 nomical mastery on the other.
 
 XI 
 
 EXAMPLES AND PROBLEMS 
 
 Formal and Applied Arithmetic 
 
 THE teaching of arithmetic is usually classified 
 under two aspects, formal work and applied work. 
 The formal work deals mainly with the memori- 
 zation of fundamental facts, processes, and other 
 details of manipulation. The applied work, as 
 the name implies, is the formal work utilized in 
 the setting of a concrete situation demanding a 
 solution. These two aspects of arithmetical in- 
 struction are very frequently sharply separated, 
 the child working alternately with one or the 
 other. The characteristic practice is to deal with 
 them without relating them as closely as the 
 highest efficiency would demand. 
 
 The Example and the Problem 
 
 Formal exercises in arithmetic are usually pre- 
 sented through the " example " ; the exercises in 
 application through the " problem " ; the distinc- 
 96
 
 EXAMPLES AND PROBLEMS 
 
 tion being that one is an abstract and symbolical 
 statement of numerical facts and the other a con- 
 crete and descriptive statement. 1 In the first 
 case the mathematical sign tells the child what 
 to do, whether to add, subtract, multiply, or di- 
 vide ; the example being a kind of pre-reasoned 
 problem, the child has only to manipulate ac- 
 cording to the sign, his whole attention through- 
 out being focused on the formal calculation. In 
 the second case the child has two distinct func- 
 tions ; he must, from the description of the sit- 
 uation presented, decide, through the process of 
 reasoning, what he is to do (add, subtract, divide, 
 or multiply), and having rendered his judgment, 
 he must proceed through the formal calculation. 
 
 The Traditional Precedence of Formal Work 
 
 As the problem involves two types of mental 
 processes in a single exercise, and the example 
 
 1 While this distinction is not general, it has sufficient cur- 
 rency to warrant its use here for the convenience of discus- 
 sion. The expression '* clothed problem " (from the German) 
 is occasionally used to mean what is here designated as 
 " problem," and " abstract problem " is used to mean what is 
 here designated as " example." 
 
 97
 
 TEACHING PRIMARY ARITHMETIC 
 
 but one, the usual procedure in arithmetic is to 
 take up the formal side through examples first 
 and, later on, the applied side through the use of 
 problems. This means that the first emphasis is 
 on formal and abstract work rather than on a 
 treatment of natural, concrete situations, an em- 
 phasis not wholly sanctioned by modern psy- 
 chology and the better teaching procedure of 
 other subjects. 
 
 Objective and Narrative Presentation as a 
 Reform Tendency 
 
 The reform tendency is found mainly in the 
 primary grades where the beginnings of new 
 processes are made through objective presenta- 
 tions of the problem. But the transition from 
 objectified problems to formal work is not imme- 
 diate. The children pass from objectified situa- 
 tions to "number stories," which are only de- 
 scriptions or narratives of a situation. This is 
 the interesting primary-school equivalent of that 
 more businesslike language description found in 
 the higher grades, the arithmetical problem. But 
 it precedes formal work and succeeds it, the 
 98
 
 EXAMPLES AND PROBLEMS 
 
 formal drill being a mere intermediate drill. 
 Here concrete presentations and formal work are 
 more closely related and more naturally ordered. 
 This reform tendency, which began in the pri- 
 mary school, is extending to the higher grades, 
 where it is no longer rare to find the attack upon a 
 process preceded by careful studies of the con- 
 crete circumstances in which the process is util- 
 ized. In the case of interest, several days might 
 be utilized in studying the institution of banking 
 in all its more important facts and relations. 
 Such an approach not only provides motive for 
 the formal and mechanical work, but gives a 
 necessary logical basis in fact. Hence the un- 
 derstanding of practical business life makes ac- 
 curate reasoning possible for the child when he 
 is called upon to solve actual problems in appli- 
 cation of the formal work. 
 
 The Over-Emphasis of Formal Work 
 
 It is perfectly natural under the general tradi- 
 tional practice of putting the first emphasis on 
 mastery of formal work that the largest amount 
 of attention should be given to the mechanical 
 99
 
 TEACHING PRIMARY ARITHMETIC 
 
 and technical side of arithmetic, and that the 
 concrete uses and applications should be slighted, 
 and this is generally true of the practice of 
 American teachers. Much more ingenuity has 
 been used in the careful training of the child on 
 the formal side than in teaching him to think out 
 his problems. There is no such careful arrang- 
 ing and ordering of types in teaching a child 
 to reason, as there is in teaching him to cal- 
 culate. 
 
 The Need for More Systematic Teaching of 
 Reasoning 
 
 Here and there a few thoroughly systematic 
 attempts are made to carry the pupil through 
 the simple types of one-step reasoning, to two- 
 step and three-step problems with their possible 
 variations. Just as the example isolates the diffi- 
 culties of calculation, by letting the sign + or 
 stand for the logic of the situation, there is a 
 tendency to give problems without requiring the 
 calculations. This affords a means of isolating 
 and treating the special difficulties of reasoning. 
 The child is merely required to tell what he 
 100
 
 EXAMPLES AND PROBLEMS 
 
 would do, without doing it ; the answer being 
 checked by the gross facts. A little later, as a 
 transition, he is permitted to give a rapid, rough 
 approximation of what the answer is likely to be. 
 With further command he tells what he would 
 do and does it accurately. But such a program 
 of teaching is still rare among teachers. 
 
 Existing Devices for Testing Reasoning 
 
 The care of the child's reasoning is largely 
 restricted to testing his comprehension of the 
 problem, (i) by having him restate the problem 
 to be sure he understands it, or (2) by having 
 him give a formal oral or written analysis of the 
 way in which he solved the problem. The first 
 requirement may not be thoroughgoing, as the 
 child may give a verbal repetition of the problem 
 without really knowing its meaning. The second 
 is a formal analysis of the finished result and 
 does not represent the genetic method of the 
 child's thinking. Consequently its formulas do 
 not, in any considerable degree, assist him in his 
 actual struggle with the complex of facts. 
 
 101
 
 TEACHING PRIMARY ARITHMETIC 
 
 Sources of Failure in the Solution of Problems 
 
 This lack of a systematic teaching of the tech- 
 nique of reasoning is manifest in the unrelia- 
 bility of children's thinking. When a child fails 
 in a problem assigned from the text-book, the 
 source of the error may be in one or more of 
 three phases : (i) In failing to get the meaning 
 of the language used to describe the details of 
 the situation ; (2) in failing to reason out what 
 needs to be done to solve the situation ; (3) in 
 failing to make an accurate calculation. The 
 first is a matter of language ; the second, one of 
 reasoning ; the third, one of memorization. The 
 elimination of errors, due to the first and third 
 sources, leaves a considerable proportion to be 
 accounted for by the second. Such informal in- 
 vestigations as have been made seem to show 
 that the children who fail in reasoning do not 
 make any real effort to penetrate into the essen- 
 tial relations of the situation. They depend on 
 their association of processes with specific words 
 of relation used in the description of the prob- 
 lem, an association determined of course by their 
 1 02
 
 EXAMPLES AND PROBLEMS 
 
 past experiences. As long as these familar " cue " 
 words are used, they succeed. Let unfamiliar 
 words or phrases be utilized in their stead or let 
 the relation be implied, and, like as not, the 
 children will fail to do the right thing. Practical 
 school people are familiar with the fact that 
 children solve the problems given in the lan- 
 guage of their own teachers and fail when the 
 problems are set by^grincipals ^r_superintend- 
 ents, whose language is' strange to them. 
 
 / 
 The Need of Varied Presentations of Problems 
 
 A varied use of materials in the objective 
 presentation of problems, and a more constantly 
 varied use of language in the descriptive pre- 
 sentation of problems would prevent the child 
 from making such superficial and unthought- 
 ful associations, and force him to think out 
 connections between what is essential in a 
 typical problem and the appropriate process of 
 manipulating it. But such a deliberate applica- 
 tion of modern psychology is far from being a 
 conspicuous minority movement. The subject- 
 matter of the problems given to children has, 
 103
 
 TEACHING PRIMARY ARITHMETIC 
 
 however, improved greatly. Obsolete, puzzling, 
 and unreal situations, which only hinder the 
 child's attempt to reason, are less and less used 
 in problem work. 
 
 Improvement in the Subject- Matter of Problems 
 
 Daily it becomes recognized with greater clear- 
 ness that right reasoning depends upon a com- 
 prehension of the facts of the case, and the 
 facts of the case in point must be within the 
 experience of the child. This is the only way 
 in which a problem can be real and concrete 
 to him. 
 
 Real and Concrete Problems Taken from the 
 Larger Social World 
 
 The recent effort on the part of text-book 
 writers and teachers to make arithmetical prob- 
 lems "real" and "concrete" has not always 
 recognized the above-mentioned psychological 
 principle. The terms "real" and "concrete" 
 have been interpreted in many ways besides in 
 terms of the child's consciousness. With some, 
 " real" has meant " material" ; and the problems, 
 104
 
 EXAMPLES AND PROBLEMS 
 
 more particularly with primary children, have, 
 in increasing degree, been presented by objects 
 or words connoting very vivid images. Others 
 have defined this quality in terms of actual exist- 
 ence or use in the larger social world. If these 
 problems actually occur at the grocer's, the 
 banker's, or the wholesaler's, it is said that they 
 "are indeed concrete." And much effort has 
 been expended in carrying these current prob- 
 lems into the classroom, in spite of the fact that 
 they may be neither comprehensible nor interest- 
 ing to the pupil. 
 
 Real and Concrete Problems Taken from the 
 Child's Own Life 
 
 There is another social world, nearer home to 
 the child, from which a more vital borrowing can 
 be made. There is an opportunity to use the 
 child's life in its quantitative aspects, to take his 
 plays, games, and occupations, and introduce 
 their situations into his mathematical teaching. 
 As his world expands from year to year, he will 
 be carried by degrees from personal and local 
 situations to those of general interest. The 
 105
 
 TEACHING PRIMARY ARITHMETIC 
 
 teacher can provide this progression without de- 
 vitalizing the facts presented. 
 
 The Imaginative or Hypothetical Problem 
 
 There is another error into which both the 
 socially-minded radical and the specialist in child 
 study fall. In their eagerness to improve the 
 arithmetical problem, they assume that problems 
 taken from the larger sbciaDworld or from the 
 child's experience are necessarily superior to hy- 
 pothetical, imaginative, or " made-up " problems. 
 The psychological fact that needs to be forced 
 upon the attention of the reformers is that, with 
 proper artfulness, an imagined problem may be 
 even more vital and real to the child than one 
 taken from life as a situation in a drama may 
 be more appealing and real to a child than one on 
 the street. This has some recognition, but not 
 enough. Those who stand upon the side of the 
 " made-up " problems are more likely to be re- 
 actionaries who tolerate the traditional type of 
 problem even though its stupid artificiality is ob- 
 vious to both the teacher and the child. They 
 might better be dealing with dull problems bor- 
 106
 
 EXAMPLES AND PROBLEMS 
 
 rowed from real life than with specially invented 
 dullness. 
 
 Valid Arguments for Actual Problems 
 
 Of course there is another argument for the 
 use of actual social materials. The child must 
 ultimately come into command of precisely these 
 facts, since their mastery will be demanded by 
 the business world. But must a primary school 
 child study his arithmetic through problems 
 taken from the dreary statistics of imports and 
 exports merely because tariff reform is a polit- 
 ical issue which every citizen ought finally to 
 comprehend ? There is a time for this, and, as 
 is the case with most of such borrowed busi- 
 ness problems, the time is later. In so far as 
 these are current situations within the con- 
 tacts of child life, let them enter. A quan- 
 titative revelation of life is important ; and it 
 is good teaching economy to gain knowledge 
 by the way, provided it does not distract 
 attention from whatever main business is at 
 hand. 
 
 107
 
 TEACHING PRIMARY ARITHMETIC 
 
 Unity in the Subject-Matter of Problems 
 
 The socializing of arithmetical problems has 
 one other additional good effect. It has tended 
 to bring some topical unity into the problems 
 constituting the assignment for a given lesson, 
 or group of lessons. Hitherto, a series of prob- 
 lems was almost always composed of a hetero- 
 geneous lot of situations. There was no unity 
 save that some one process was involved in each. 
 The movement is now in the direction of attain- 
 ing a more approximate unity within the subject- 
 matter of the problems themselves. The difficul- 
 ties of attainment have restricted this movement 
 to more progressive circles. 
 
 The Eclectic Source of Problems 
 
 The eclectic source of arithmetic problems is 
 apparent from the foregoing discussion. It would 
 seem that some better texts would naturally be 
 evolved through the implied criticism of each 
 movement upon the other. Such is the case. 
 Problems from child life emphasize the begin- 
 ning condition to which adjustment must be 
 1 08
 
 EXAMPLES AND PROBLEMS 
 
 made in all good teaching. Those from the 
 greater world suggest the final goals of instruc- 
 tion. Those "made up" by the teacher call 
 attention to what is too often forgotten, that the 
 educative process in school may be artful with- 
 out becoming artificial. Teaching is art, and 
 when well done is not less effective for the fact.
 
 XII 
 
 CHARACTERISTIC MODES OF PROGRESS IN 
 TEACHING METHOD 
 
 Variation in Method, and its Causes 
 
 THE existing methods of teaching arithmetic in 
 the American elementary schools are exceed- 
 ingly varied. This is due to many causes. The 
 democratic system of local control, as opposed 
 to a centralized supervision of schools, has in- 
 creased both the possibility and the probability of 
 variation. Even within the units of supervision 
 (state, county, and municipal) the opportunity 
 for reducing variation in the direction of a more 
 efficient uniformity is lost. This is partly due to 
 the lack of a thoroughly trained staff of super- 
 visors of the teaching process. Uniformity be- 
 yond the legal units of supervision has been 
 restricted by the lack of organized professional 
 means for investigation of and experimentation 
 in controversial matters. Even such crude ex- 
 no
 
 PROGRESS IN TEACHING METHOD 
 
 periments as are being tried in more than one 
 class room, school, or system are unknown, un- 
 reported, unestimated, because no competent 
 professional body gathers, evaluates, and diffuses 
 such knowledge. In this respect the teaching 
 profession is far below the efficient organization 
 of the legal and medical professions. 
 
 Characteristic Traditions and Reforms 
 
 It is exceedingly difficult therefore to analyze 
 the characteristic aspects of teaching method 
 except as these are interpreted in movements of 
 general significance. These may be actual or 
 potential, traditional or reformatory, general or 
 local in present acceptance. The situation is one 
 wherein tradition is mixed with radicalism, and 
 radicalism modified by reaction. In this medley 
 of movements there are dominant tendencies 
 both traditional and progressive. 
 
 Forces for Progress in Method 
 
 It is quite impossible to indicate the progres- 
 sive tendencies with clearness save in connection 
 with the discussions of concrete difficulties in 
 in
 
 TEACHING PRIMARY ARITHMETIC 
 
 mathematical teaching. The forces that are be- 
 hind these tendencies may, however, be summa- 
 rized here. For convenience, they may be classi- 
 fied into eight types of influence, extending from 
 more or less vague and general movements to 
 very particular, scientific contributions. No at- 
 tempt is made to indicate the achievement of 
 each ; the form of each influence is only roughly 
 defined, and illustrative movements or studies 
 are suggested : 
 
 General Pedagogical Movements 
 
 (i) It is obvious that any general pedagogical 
 movement that influences the professional atti- 
 tude of teachers will influence the special meth- 
 ods of mathematical teaching. The appearance 
 of the doctrine of interest made mathematical 
 instruction less formal. The growing enthusi- 
 asm for objective work enlarged the use of ob- 
 jects in the arithmetic period. The child study 
 movement laid emphasis upon the child's own 
 plays and games as a source of problems and ex- 
 amples. 
 
 112
 
 PROGRESS IN TEACHING METHOD 
 
 Special Pedagogical Movements 
 
 (2) Certain special movements in methods of 
 teaching, local to the subject of mathematics, 
 have also been effective. Here one has only to 
 recall the "Grube" method, with its influence 
 on the order and thoroughness with which the 
 elements of arithmetic are taught. 
 
 Daily Trial and Error 
 
 (3) The tendency of every teacher, who is at 
 all sensitive to the defects of his methods, is to 
 vary his daily practice. Constant trial, with error 
 eliminating and success justifying a departure, is 
 thus a source of progress. The new devices of 
 one teacher are taken up by the eager profes- 
 sional witness, and innovation is thus diffused. 
 We have no ability to measure how much pro- 
 fessional progress is due to individual variation 
 in teaching and its conscious and unconscious 
 imitation. The disposition of school systems to 
 send their teachers on tours of visitation without 
 loss of salary is a recognition of the value of 
 this method of advance.
 
 TEACHING PRIMARY ARITHMETIC 
 
 Experimentation of Progressive Teachers 
 
 (4) A far more efficient and radical source of 
 change than that just mentioned is the delib- 
 erate, conscious, experimental teaching of pro- 
 gressive individuals. Some new idea or device 
 occurs to the teacher of original mind, and it is 
 tried out with a fair proportion of resulting suc- 
 cesses. An illustration of such a contribution is 
 found in one conspicuous effort to get more rapid 
 column addition. The first columns to be added 
 were allowed to determine the selection and 
 order of addition combinations learned. Thus if 
 6 + 7 + 9 + 6 + 7=35 is the first column to be used, 
 then the first combinations mastered will be 
 6 + 7=13, 3 + 9=12, 2 + 6 = 8, 8 + 7=15. Arising 
 as a fruitful idea and seeming to give a measure 
 of success, it has been carried, in the particular 
 locality in mind, from school to school, and from 
 system to system. 
 
 Reconstruction through Psychological Criticism 
 
 (5) A prolific source of radical change is found 
 in the critical application of modern psychology 
 
 114
 
 PROGRESS IN TEACHING METHOD 
 
 to teaching methods. Algorisms, types of diffi- 
 culty, the order and gradation of these, as well 
 as many other factors in method have been rad- 
 ically reorganized on psychological grounds. Ex- 
 amples of such psychological modifications of 
 method are found in the " Courses of Study for 
 the Day Elementary Schools of the City of San 
 Francisco." Still more extensive critical applica- 
 tions are found in the " Exercises in Arithmetic " 
 devised by Dr. E. L. Thorndike, Professor of 
 Educational Psychology in Teachers College, 
 Columbia University. 
 
 Studies in the Special Psychology of 
 Mathematics 
 
 (6) Attempts have been made to inquire into 
 the special psychology of arithmetical processes 
 through careful experimentation and control. 
 They have not been numerous, nor have they 
 been influential on current practice. Such a field 
 needs development. A typical attempt to investi- 
 gate and formulate the special psychology of 
 number is found in a Clark University study of
 
 TEACHING PRIMARY ARITHMETIC 
 
 "Number and its Application Psychologically 
 Considered." 1 
 
 Investigations of Existing Methods 
 
 (7) Educational investigations as to the effi- 
 ciency of existing arithmetical teaching among 
 school systems, sufficiently varied to be repre- 
 sentative of American practice, have also been 
 conducted. These have usually gone beyond the 
 field of the special methods of presentation em- 
 ployed in the classroom, and have inquired into 
 the conditions of administration and supervision, 
 the arrangement of the courses of study, and 
 other similar factors. Dr. J. M. Rice's studies into 
 "The Causes of Success and Failure in Arith- 
 metic" 2 investigated such specific factors as: 
 The environment from which children come, their 
 age, time allotment of the subject, period of school 
 day given to arithmetic, arrangement of home 
 work, standards, examinations, etc. A subsequent 
 
 1 Phillips, D. E., " Number and its Application Psycholog- 
 ically Considered," Pedagogical Seminary, 1897-8, voL 5, 
 pp. 221-281. 
 
 2 Rice, J. M., " Educational Research : Causes of Success 
 and Failure in Schools," Forum, 1902-03, vol. 34, pp. 281-97, 
 437-S 2 - 
 
 116
 
 PROGRESS IN TEACHING METHOD 
 
 study of similar type, but employing more refined 
 methods, is that of Dr. C. W. Stone on "Arith- 
 metical Abilities and some Factors determining 
 them." 1 The main problem of this study was to 
 find the correlation between types of arithmet- 
 ical ability and different time expenditures and 
 courses of study. These two studies have prob- 
 ably attracted more general notice than any other 
 studies of arithmetical instruction. While they 
 have largely dealt with administrative conditions 
 that limit teaching method, rather than with the 
 details of teaching method itself, they have stim- 
 ulated the impulse to investigate conditions and 
 practices of every type. 
 
 Special Experiments in Controlled Comparative 
 
 Teaching 
 
 (8) The latest source of progress in teaching 
 method is found in the movement for compara- 
 tive experimental teaching under normal but care- 
 fully controlled conditions. Several such exper- 
 iments are being conducted in the Horace Mann 
 
 1 Stone, C. W., " Arithmetical Abilities and Some Factors 
 determining them," Columbia University Contributions to Ed- 
 ucation, Teachers College, N. Y. City, 1909, p. 101. 
 
 II/
 
 TEACHING PRIMARY ARITHM TIC 
 
 Elementary School of Teachers College, Colum- 
 bia University, under the direction of Principal 
 Henry C. Pearson, with the co-operation of the 
 staff of Teachers College. This experimental 
 work is designed to determine primarily the 
 relative value of competing methods in actual 
 use throughout the country, the assumption being 
 that every substantial difference in practice im- 
 plies a difference of theory and consequently a 
 controversy that can be resolved only on the 
 basis of careful comparative tests. Two parallel 
 series of classes of about the same age, ability, 
 teacher equipment, etc., are selected for this 
 work. One series is taught by one method ; the 
 other series by the other method. The abilities 
 of these children are measured both before and 
 after the teaching, and the growth compared. 
 The standards and methods of this type of com- 
 parative experimentation, together with a list of 
 current competitive methods requiring investiga- 
 tion, are given in Dr. David Eugene Smith's 
 monograph on " The Teaching of Arithmetic." 1 
 
 1 Smith, D. E., " The Teaching of Arithmetic," chap, xvi, 
 Teachers College, January, 1909.
 
 OUTLINE 
 
 I. THE SCOPE OF THE STUDY 
 
 1. Function of the Study to Trace General Tenden- 
 
 cies I 
 
 2. Teaching Method is a Mode of Presentation . . 2 
 
 3. Distinct Uniformities Exist among its Variations 4 
 
 4. The Methods of Public Elementary Schools are 
 
 Representative 4 
 
 5. Elementary Mathematics is Mainly Arithmetic . 5 
 
 6. Elementary Arithmetic Emphasizes the Four 
 
 Fundamental Processes 6 
 
 7. The Need for Studying Exceptional Reform 
 
 Tendencies 8 
 
 I. THE INFLUENCE OF AIMS ON TEACHING 
 
 1. Factors Influencing Teaching Methods .... 9 
 
 2. The Influence of a Scientific Aim 9 
 
 3. The Influence of the Aim of Formal Discipline . 12 
 
 4. The Shift in Emphasis from Academic to Social 
 
 Aims 14 
 
 5. Business Utility as an End 15 
 
 6. Broad Social Utilitarianism as a Standard . . . 17 
 
 7. Some Concrete Effects of the Change in Aim . 19 
 
 119
 
 OUTLINE 
 
 m. THE EFFECT OF THE CHANGING STATUS 
 OF TEACHING METHOD 
 
 1. Method as Psychological Adjustment to the Child 21 
 
 2. The Effect of an Increased Reverence for Child- 
 
 hood 22 
 
 3. The Reconstruction of Method through Psychol- 
 
 ogy 23 
 
 4. The Increased Professional Respectability of Con- 
 
 scious Method 25 
 
 5. The Prevalence of Methods Emphasizing a Single 
 
 Idea 26 
 
 6. The Tendency toward Over-Uniformity in Method 28 
 
 7. Method as a Series of Varied, Particular Adjust- 
 
 ments 30 
 
 IV. METHOD AS AFFECTED BY THE DIS- 
 TRIBUTION AND ARRANGEMENT OF 
 ARITHMETICAL WORK 
 
 1. The Tendency toward Shortening the Time Dis- 
 
 tribution 32 
 
 2. The Attempt to Eliminate Waste 33 
 
 3. Delay in Beginning Formal Arithmetic Teaching 34 
 
 4. The Incidental Method of Teaching 35 
 
 5. Reactions against the Plan of Incidental Teaching 36 
 j 6. Logical and Psychological Types of Arrangement 38 
 
 7. Estimates of Worth 40 
 
 8. The Present Mixed Method of Procedure ... 41 
 
 120
 
 OUTLINE 
 
 V. THE DISTRIBUTION OF OBJECTIVE WORK 
 
 1. Objective Teaching is Generally Current ... 42 
 
 2. Its Distribution is Crudely Gauged 43 
 
 3. Tendency toward a More Refined Correlation 
 
 of Object-Teaching with Particular Immatu- 
 rity 44 
 
 4. The Movement Supported by both Scientific and 
 
 Common-Sense Criticism 46 
 
 VI. THE MATERIALS OF OBJECTIVE TEACHING 
 
 ; I. The Indiscriminate Use of Objects . . . . .47 
 
 2. The Artificiality of Materials Utilized .... 47 
 
 3. Narrowness in the Range of Materials .... 48 
 
 4. Inadequate Variation of Traditional Materials . 49 
 
 5. The Restricted Use of Diagrams and Pictures . 50 
 
 6. Plays and Games in Object Teaching . . . .51 
 
 7. The Lack of Unity in the Use of Objects . . .52 
 
 VII. SOME RECENT INFLUENCES ON OBJEC- 
 
 TIVE TEACHING 
 
 1. The Influence of Inductive Teaching 53 
 
 2. The Movement for Active Modes of Learning . 55 
 
 3. The Abbreviated Use of Objects 57 
 
 4. The Method of Parallel Correspondence ... 57 
 
 5. The Method of Final Correspondence .... 58 
 
 121
 
 OUTLINE 
 
 Vffl. THE USE OF METHODS OF RATIONALI- 
 ZATION 
 
 1. The Tendency toward Rational Methods ... 60 
 
 2. The Era of Direct Instruction and Drill ... 60 
 
 3. Indirect Teaching as a Rational Method ... 62 
 
 4. Interest as a Factor in Methods of Rationalization 63 
 
 5. The Reaction against Rationalization .... 64 
 
 6. Four Principles for the Use of Rationalization . 65 
 
 7. The Substantiating Psychology 67 
 
 8. Rationalization as a Substitute for Object Teach- 
 
 ing 67 
 
 IX. SPECIAL METHODS FOR OBTAINING AC- 
 CURACY, INDEPENDENCE, AND SPEED 
 
 1. Supervision of Learning after First Development 69 
 
 2. The Use of Steps, or Stages, in Teaching ... 70 
 
 3. Stages in the Presentation of Problems . . . .71 
 
 4. An Opposite Method in Presenting Examples . 73 
 
 5. Better Transitions from Concrete to Abstract 
 
 Work -73 
 
 6. The Child's Four Modes of Work 75 
 
 7. The Worth of these Modes 76 
 
 8. The Traditional Quarrel between " Mental " and 
 
 "Written" Arithmetic 77 
 
 9. The Transition from Development by Teacher to 
 
 Independent Work by Pupil 79 
 
 10. Four Characteristic Stages of the Transition . . 79 
 122
 
 OUTLINE 
 
 11. Special Methods of Attaining Speed 81 
 
 12. The Relation of Accuracy to Speed 82 
 
 X. THE USE OF SPECIAL ALGORISMS, ORAL 
 FORMS, AND WRITTEN ARRANGEMENTS 
 
 1. The Traditional Nature of Algorisms and Forms 83 
 
 2. The Number of Algorisms Used 84 
 
 3. Reform in the Use of Algorisms 85 
 
 4. The Standard of Social Usage 86 
 
 5. The Extended Use of Acquired Forms .... 86 
 
 6. The Use of "Crutches " or Temporary Algorisms 87 
 
 7. Full and Short Forms of Calculation 88 
 
 8. Forms of Analysis or Reasoning 89 
 
 9. The Traditional Requirement of Full Formal 
 
 Analysis 90 
 
 10. The Limitations of Full Formal Analysis ... 91 
 ir. " Labeling " the Steps of Calculation 92 
 
 12. So-called Accuracies of Statement 93 
 
 13. Increased Use of Mathematical Symbols ... 94 
 
 XI. EXAMPLES AND PROBLEMS 
 
 1. Formal and Applied Arithmetic 96 
 
 2. The Example and the Problem 96 
 
 3. The Traditional Precedence of Formal Work . . 97 
 
 4. Objective and Narrative Presentation as a Re- 
 
 form Tendency 98 
 
 5. The Over-Emphasis of Formal Work .... 99 
 
 6. The Need for More Systematic Teaching of Rea- 
 
 soning ico 
 
 123
 
 OUTLINE 
 
 7. Existing Devices for Testing Reasoning . . . 101 
 
 8. Sources of Failure in the Solution of Problems . 102 
 
 9. The Need of Varied Presentations of Problem . 103 
 
 10. Improvement in the Subject-Matter of Problems 104 
 
 11. Real and Concrete Problems Taken from the 
 
 Larger Social World 104 
 
 12. Real and Concrete Problems Taken from the 
 
 Child's Own Life 105 
 
 13. The Imaginative or Hypothetical Problem . . 106 
 
 14. Valid Arguments for Actual Problems .... 107 
 
 15. Unity in the Subject-Matter of Problems . . . 108 
 
 16. The Eclectic Source of Problems 108 
 
 XH. CHARACTERISTIC MODES OF PROGRESS 
 IN TEACHING METHOD 
 
 1. Variation in Method and its Causes no 
 
 2. Characteristic Traditions and Reforms . . . . ill 
 
 3. Forces for Progress in Method 112 
 
 4. General Pedagogical Movements 112 
 
 5. Special Pedagogical Movements 113 
 
 6. Daily Trial and Error 113 
 
 7. Experimentation of Progressive Teachers . .114 
 
 8. Reconstruction through Psychological Criticism 114 
 
 9. Studies in the Special Psychology of Mathe- 
 
 matics 115 
 
 10. Investigations of Existing Methods 1 16 
 
 n. Special Experiments in Controlled Comparative 
 
 Teaching 117
 
 OUTLINE 
 
 7. Existing Devices for Testing Reasoning . . . 101 
 
 8. Sources of Failure in the Solution of Problems . 102 
 
 9. The Need of Varied Presentations of Problem .103 
 
 10. Improvement in the Subject-Matter of Problems 104 
 
 11. Real and Concrete Problems Taken from the 
 
 Larger Social World ......... 104 
 
 12. Real and Concrete Problems Taken from the 
 
 Child's Own Life .......... 105 
 
 13. The Imaginative or Hypothetical Problem . . 106 
 
 14. Valid Arguments for Actual Problems .... 107 
 
 15. Unity in thf Suhiprt-MaH-Ar nf Pr/->Klemc T 
 
 16. TheE 
 
 xn. CH. 
 
 1. Variati 
 
 2. Charac 
 
 3. Forces 
 
 4. Gener; 
 
 5. Specia 
 
 6. Daily ' 
 
 7. Experi 
 
 8. Recons 
 
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 Work and materials furnished are guaranteed to wear 
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 9. Studie: 
 
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 10. Investigations of Existing Methods 116 
 
 11. Special Experiments in Controlled Comparative 
 
 Teaching 117
 
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