IC-NRLF 
 
 17 Elt 
 
UNIVERSITY OF CALIFORNIA. 
 
 GIFT OF 
 
 U/*rxv/T 
 ^Accession 
 
 Class 
 
AN INQUIRY INTO THE TEACHING 
 OF ADDITION AND SUBTRACTION 
 
 THESIS 
 
 Presented to the Faculty of Philosophy of the University of 
 
 Pennsylvania in Partial Fulfillment of the 
 
 Requirements for the Degree of 
 
 Doctor of Philosophy 
 
 ALBERT DUNCAN YOCUM 
 
 PHILADELPHIA 
 AVIL PRINTING COMPANY 
 
 igoi 
 
AN INQUIRY INTO THE TEACHING 
 OF ADDITION AND SUBTRACTION 
 
 THESIS 
 
 Presented to the Faculty of Philosophy of the University of 
 
 Pennsylvania in Partial Fulfillment of the 
 
 Requirements for the Degree of 
 
 Doctor of Philosophy 
 
 ALBERT DUNCAN YOCUM 
 
 PHILADELPHIA 
 AVIL PRINTING COMPANY 
 
 IQOI 
 
OF 
 
 AVIL PRINTING COMPANY 
 
 MARKET AND FORTIETH STREETS 
 
 PHILADELPHIA 
 
INTRODUCTORY. 
 
 In the following discussion it is assumed that whatever 
 the numerical knowledge which children should ultimately 
 possess, and whatever the time when their formal instruc- 
 tion in number should begin, they must at the very outset 
 become thoroughly familiar with the fundamental facts of 
 addition and subtraction ; that painstaking inquiry should 
 be made into the numerical content of their minds on their 
 first entering school; that so far as possible this content 
 should become the basis of elementary arithmetical in- 
 struction; that aside from the order in which the funda- 
 mental number- facts can be most readily mastered by each 
 individual, there is a psychological order in which they 
 should be taught to pupils studying en masse i. e., that 
 there is an order in which they can be most readily 
 mastered by the great majority of individuals who are 
 under common instruction; that similarly there is a 
 psychological method by which they should be taught ; and 
 that such order and such method may be suggested by a 
 priori discussion and determined by practical investiga- 
 tion. 
 
 This inquiry therefore begins with a resume of the 
 results of all investigations into the numerical content of 
 children's minds on entering school which were accessible 
 to the writer, supplemented by the data obtained in a 
 somewhat restricted investigation of his own; attempts 
 to prove that there is no necessary antagonism between 
 logical order and psychological order; seeks to indicate 
 what on a priori grounds, should be the psychological 
 order and method of teaching the fundamental facts of 
 addition and subtraction; and finally reports the results 
 of the use of the order and method thus indicated, in the 
 schools of a large manufacturing town. 
 
 (3) 
 
While the conclusions resulting from this inquiry are 
 purely tentative, and may of course be disproved by 
 broader investigation, it must be none the less remem- 
 bered that they are based upon facts and have been suc- 
 cessfully subjected to the test of impartial practice. 
 
 Throughout the entire discussion effort has been made 
 to avoid all difficulties not necessarily involved in the 
 simple problem which it seeks to solve. It cheerfully leaves 
 to the mathematical philosopher the controversy over the 
 number 'concept, taking it for granted, however, that 
 such concept will have either as an important factor or a 
 necessary condition, an intelligent knowledge of the deci- 
 mal system, as well as of the fact that each number in 
 the scale is greater by one than that which immediately 
 precedes it. 
 
 With equal firmness it turns aside from seductive specu- 
 lations concerning the origin of number and its historic 
 development, assuming that even should the Culture 
 Epoch Theory be regarded as demonstrated, the study of 
 the child would be more likely to throw light upon the 
 history of number, than that of barbarous dialects and 
 ancient texts upon the order in which number should be 
 taught the child. 
 
 What number shall be taught to children, whether it 
 shall be taught them on their first entering school, what 
 proportion the time devoted to it shall bear to that given to 
 other branches of the curriculum, these and kindred 
 questions of pedagogic importance it fails to discuss, not 
 through a minimizing of their importance, but from the 
 fact that they are judged worthy of special investigation. 
 
 Certain fundamental sums and differences must be 
 mastered by the children. Intelligently derived, they 
 must be mechanically used in numerical operation. If 
 this investigation serves to indicate an order and a method 
 of instruction which will bring about their readier mastery 
 by the majority of the pupils when taught in common, it 
 will accomplish the end for which it was undertaken, 
 
not, it is hoped, with the result that more number-work 
 will be done in a given school year, but that less time need 
 be devoted to the number-work at present demanded by 
 school curricula, whatever that time may be. 
 
 It is only proper that these introductory remarks should 
 conclude with formal acknowledgment of the practical 
 assistance afforded by Dr. Lightner Witmer, Assistant 
 Professor of Psychology, and the suggestive discussions 
 which characterize his seminaries. To him and to Dr. 
 Brumbaugh, Professor of Pedagogy, is due in no small 
 measure such success as may have attended the inquiry. 
 
CHAPTER I. THE NUMERICAL CONTENT OF 
 
 CHILDREN'S MINDS UPON THEIR FIRST 
 
 ENTERING SCHOOL. 
 
 From 1869, when the Berlin Pedagogical Union initi- 
 ated the first general inquiry into the ideational range 
 (Vorstellungskreiz) of children entering school, 1 such 
 tests have been common in the pedagogic centres of Ger- 
 many. Nor has the least service which Dr. Hall has done 
 to education in America been the impetus given to simi- 
 lar investigation through the forceful demonstration of its 
 value and suggestive directions as to its method, contained 
 in his now classical report on the tests in Boston and 
 Kansas City schools. 2 Numerous, however, as these 
 tests have been, they have shed but little light upon the 
 numerical knowledge possessed by children at the begin- 
 ning of their school career. Among the 10,000 first 
 tested in Berlin, 74.35 per cent were found to be familiar 
 with the number 2 ; 73.99 per cent, with 3, and 72.65 per 
 cent, with 4 and even this sparse data loses much of 
 its value, when it is known that some of the returns from 
 which it was obtained were made after the children had 
 been several weeks in school. These are the only facts 
 concerning children's numerical knowledge which the 
 tests sought to obtain, and yet inquiry was made into their 
 familiarity with almost a hundred concepts. When one 
 takes into account the proportionate time given to number- 
 work in the curriculum of the elementary school, together 
 with the fact that a main object in these and succeeding 
 tests was to determine what knowledge the beginner could 
 be certainly assumed to possess, the neglect to further test 
 his numerical knowledge can be understood only by assum- 
 ing that investigators have taken it for granted perhaps 
 quite unconsciously that the child enters school prac- 
 
 (7) 
 
tically ignorant of even the simplest facts of number. Dr. 
 Lange fails to touch the subject of number at all in his 
 test of 500 beginners at Flauen, and 300 in the surround- 
 ing country. 3 Dr. Hall himself after a searching inquiry 
 as to how many among 200 Boston children fail to have 
 any proper concept of a hundred simple and familiar 
 things, only adds to the numerical data, resulting from the 
 Berlin test, that 8 per cent of his subjects did not know 
 the "number-name" 3; 17 per cent, 4, and 29.5 per 
 cent, 5. 4 This he does because "number cannot be 
 developed to any practical extent without knowledge of 
 the number-name," 5 that is, he assumes a minimum of 
 knowledge necessary to the teaching of number, then 
 seeks to discover how many children do not possess it. 
 
 Indeed all these inquiries into the general content of 
 children's minds have necessarily been negative in their 
 results. The knowledge which children may possess is 
 so extensive and so various, that a given test, even though 
 limited to that part of the possible content, likely to be 
 common to a majority of individuals, while showing much 
 that the individual does not know, can show but little of 
 that which he does. On the contrary, a special inquiry 
 into the numerical content of children's minds being very 
 limited in its extent, can be quite positive in its results. 
 
 Another characteristic of Dr. Hall's test was that his 
 200 children were carefully selected, being limited to 
 those of "average capacity," and care being taken to 
 "avoid schools where the children came from homes repre- 
 senting extremes of either culture or ignorance." 6 This 
 usual scientific precaution must of course be omitted in a 
 special inquiry, which seeks among other things to deter- 
 mine the numerical knowledge which can be counted upon 
 as common to the great majority of beginners, without 
 previous knowledge as to their capacity, and with no 
 regard to their environment. This objection does not hold 
 against the results of Hartmann's test at Annaberg on the 
 660 boys and the 652 girls, between the ages of six and 
 
three-fourth and five and three-fourth years, who began 
 school from 1881-84. Of these, 69 per cent of the boys 
 and 62 per cent of the girls could count from i to io. 7 
 In Superintendent Greenwood's tests in Kansas City, 
 identical with those of Dr. Hall and reported by him, the 
 "number-names" were omitted, probably owing to the 
 fact that at the time of the test the children had been some 
 seven months in school. 
 
 It would seem then that the teacher, discouraged by 
 the difficulties encountered in imparting the fundamental 
 numerical facts, has led the scientist to assume that the 
 beginner knows little or nothing of number. More than 
 this, it is already being urged on what may be or may 
 not be good physiological and psychological grounds, that 
 he ought to know nothing of it, and that he should not be 
 taught it at the time when he first enters school. "The 
 child," it is said, "is too young for abstractions ; he is still 
 in the imaginative epoch of his cultural development ; cer- 
 tain nerve fibers are not yet medullated ; nature must not 
 be forced," an argument which the assumption of his 
 numerical ignorance tends somewhat to reinforce. 
 
 But since aside from the sparse data already sum- 
 marized, the teacher has only dogma upon which to de- 
 pend, it becomes necessary for a stable pedagogy to deter- 
 mine by adequate investigation the numerical content of 
 children's minds on their first entering school. 
 
 The following test was made primarily to determine in 
 so far as was possible, what certain individual children 
 knew on entering the schools of a particular locality, and 
 secondarily as a step in the more general determination of 
 what numerical knowledge the great majority of Ameri- 
 can children may be assumed to have in common on enter- 
 ing the public school. For obvious reasons it was not made 
 by the regular teachers, but by experienced teachers' 
 assistants, who had been carefully instructed together in 
 a uniform mode of procedure. These assistants were 
 seven young women one for each school, accustomed 
 
10 
 
 to the management of children and selected with special 
 reference to their fitness for the task. The results obtained 
 by them were checked by the regular teachers as they 
 began the work of the year. 
 
 The test was made on the first day of school, in order 
 that its results might be reached before any of its sub- 
 jects had been in charge of a teacher, or had been given 
 any systematic instruction. Fifty boys and fifty girls 
 were subjected to it. The average age of the former 
 was 5.64 years; of the latter 5.48 years. The highest 
 age given was eight years, two months. Only thirty 
 children were older than six years. That a number of 
 children admitted were below school age there can be little 
 doubt, as many hardworking parents look upon the school 
 as a day-nursery, and feel that a misrepresentation is 
 excusable which will relieve them from serious care, and 
 at the same time admit their children a little earlier 
 to educational advantages which they must relinquish all 
 too soon. The test was made in a manufacturing town of 
 from 10,000 to 12,000 inhabitants, comprising an un- 
 usually small proportion of professional men, but by way 
 of compensation, very few foreigners. The children 
 tested, coming from every section of the town, are in all 
 probability fairly representative of the great mass of 
 native-born beginners in the country as a whole. 
 
 Great care was taken to so put the questions that the 
 results may be universally valid. Where a fact was put 
 concretely reference was strictly limited to such objects 
 as all American school children are likely to have counted 
 marbles for the boys, candies and pennies for both 
 boys and girls. A possible exception is to be found in 
 the use of apple, top, and orange in Questions XII to XV. 
 The second part of Question XIX is theoretically unfor- 
 tunate owing to the tendency of children to answer "yes," 
 when they think that an affirmative will please. Prac- 
 tically, however, this tendency would be hard to trace, 
 very few children answering the question at all. Ques- 
 
II 
 
 tions VII and VIII should have been supplemented else- 
 where by others in which the facts were not arranged in 
 series, in order to determine how far regular arrange- 
 ment was helpful to those who gave them correctly. The 
 later experience of teachers, however, showed conclusively 
 to what extent such was the case. As a possible source of 
 serious error any embarrassment that could be detected 
 was noted. Most children, however, were natural in 
 their manner, and seemed to respond freely to questions 
 which they were able to answer. 
 
 Suggestion was from the very nature of the questions 
 impossible. Prompting was permitted in counting, but 
 note was made wherever it occurred. No conditions of 
 the experiment, therefore, seem such as to seriously 
 modify its results. 
 
 As the recording of the answers given was so planned as 
 to involve a minimum amount of writing, each investiga- 
 tor filled out her own blanks. Although designed as they 
 are to be within certain limits fairly exhaustive, the ques- 
 tions seem quite numerous, it will be seen on investiga- 
 tion, that it was unnecessary to ask all of every individual. 
 The average time taken for each pupil was about ten 
 minutes. The following is a copy of the blank. 
 
 MATHEMATICAL CONTENT OF CHILDREN'S MINDS ON THEIR 
 FIRST ENTERING SCHOOL. 
 
 I. Age Years Months. . .... .... . . ...... 
 
 II. Boy or girl? 
 
 III. American or Foreign? 
 
 IV. Manner: Natural or embarrassed? (An X placed above a word 
 
 or figure will indicate it as the answer; above a fact in number, 
 that it has been correctly given; above a question, that it has 
 been correctly answered.) 
 
 V. Can he count? Yes or no. To what number without objects? 
 
 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. 
 (Cross out any numbers omitted by him, or on which while 
 hesitating he was prompted.) 
 
12 
 
 VI. To what number can he count actual objects? 2, 3, 4, 5, 6, 7, 
 
 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. (Cross out as 
 above.) 
 
 VII. Which of the following sums can he give correctly: ("One and 
 
 one more?" "Two and one more?" etc., is the form in which 
 these sums should be required; first, abstractly correct replies 
 being denoted by the usual X; second, concretely correct replies 
 being denoted by an O placed after the X). 1 & 1, 2 & 1, 3 & 1, 
 4 & 1, 5 & 1? (Only when the preceding sums are correctly 
 given should the following be required) : 6&1, 7&1, 8&1, 
 9&1, 10 &1, 11 &1, 12 &1, 13 &1, 14 &1, 15 &1, 16 &1, 
 17 & 1, 18 & 1, 19 & 1, 20 & 1? 
 
 VIII. Which of the following differences can he give correctly? ("If 
 you have one and I take away one, how many have you left?" 
 etc., is the form in which these differences should be required, 
 both abstractly and concretely as above) . 1 1,2 1,3 1, 
 4 1, 5 1, 6 1? (Only when the preceding differences 
 are correctly given should the following be required) : 7 1, 
 8 1, 9 1, 10 1, 11 1, 12 1, 13 1, 14 1, 15 1, 
 16 1, 17 1, 18 1, 19 1, 20 1? 
 
 IX. Particular Number Associations. Does he know without counting 
 
 how many hands he has? Yes, or How many fingers 
 
 on one hand? Yes, or The number of legs a horse has? 
 
 Yes, or (If his answer to one of the questions under IX 
 
 is a wrong number, put it in the blank; if so to one under X, put 
 it above each successive number in place of the X.) 
 
 X. Number Perception too Rapid for Conscious Counting. Can he 
 
 instantly tell the number of splints successively shown him in 
 groups of 2, 4, 6, 3 and 5, held distinctly apart in one hand? 
 The total number when 2 are thus held in each hand? When 
 
 3 are so held? 
 
 XI. Can he correctly give the following facts? (First abstractly, then 
 
 concretely, and indicated as above) : 1 1, 2 2, 3 3, 
 
 4 4, 5 5, 6 6? 2 & 2, 2 & 3, 3 & 2, 4 & 2, 2 & 4, 3 & 3? 
 3 _ 2, 4 2, 4 3, 5 2, 5 3, 5 4, 6 2, 6 4, 6 3, 
 6 5? 
 
 XII. If one apple costs lc., what will two apples cost? 
 
 XIII. If a ball costs 2c., what will three balls cost? 
 
 XIV. If a top costs 2c., how many can I buy for 4c.? 
 
 XV. If an orange costs 3c. how many can I buy for 6c.? 
 
 XVI. 2 ones? 2 twos? 2 threes? 3 ones? 3 twos? 
 
XVII. How many twos in 4? In 6? How many threes in 6? 
 
 XVIII. You have three sticks and I have two sticks Which has more? 
 ........ How many more? ........ Which less? ........ How 
 
 many Ies3? ........ 
 
 XIX. Into how many parts do you cut an apple to divide it in half? 
 
 ........ Should those parts be of the same size? ........ To 
 
 divide it into thirds? ........ Into quarter?? ........ 
 
 Name of Pupil . . . 
 Name of Teacher. 
 
 PERCENTAGES FOR FIFTY BOYS AND FIFTY GIRLS. 
 
 V. Can Not Count at All... 
 Without 
 With Obi 
 
 Can Count With and With 
 Without Objects 
 
 VI. With Objects From 
 
 Can Count Higher Withe 
 With 
 
 VII. Know Without Objects. 
 
 With Objects 
 
 VIII. Know Without Objects 
 
 With Objects... 
 
 IX. Know number Hands . . . 
 Fingers . . 
 Horse's Legs 
 
 
 Boys. 
 Per 
 No. Cent. 
 
 4 8 
 6 12 
 2 4 
 38 76 
 
 Girls. 
 Per 
 No. Cent. 
 
 2 4 
 10 20 
 2 4 
 36 72 
 
 Objects 
 ects 
 
 out Objects . . 
 
 From 2 to 5 . . 
 
 39 
 
 78 
 
 38 
 
 76 
 
 5 to 10.. 
 
 39 
 
 78 
 
 33 
 
 66 
 
 10 to 20.. 
 
 31 
 
 62 
 
 26 
 
 52 
 
 2 to 5 . . 
 
 43 
 
 86 
 
 46 
 
 92 
 
 5 to 10.. 
 
 37 
 
 74 
 
 36 
 
 72 
 
 10 to 20.. 
 
 24 
 
 48 
 
 28 
 
 56 
 
 ut Objects . . . 
 
 10 
 
 20 
 
 10 
 
 20 
 
 Objects 
 
 16 
 
 32 
 
 19 
 
 38 
 
 .... 1 + .. 
 
 14 
 
 28 
 
 19 
 
 38 
 
 2 + .. 
 
 20 
 
 40 
 
 19 
 
 38 
 
 to 10 + . . 
 
 11 
 
 22 
 
 13 
 
 26 
 
 20 + .. 
 
 5 
 
 10 
 
 8 
 
 16 
 
 .... 1 + .. 
 
 31 
 
 62 
 
 35 
 
 70 
 
 2 + .. 
 
 32 
 
 64 
 
 35 
 
 70 
 
 to 10 + 1 .. 
 
 10 
 
 20 
 
 17 
 
 34 
 
 20 + 1 . . 
 
 4 
 
 8 
 
 7 
 
 14 
 
 .... 2 .. 
 
 11 
 
 22 
 
 17 
 
 34 
 
 3 .. 
 
 11 
 
 22 
 
 14 
 
 28 
 
 to 10 .. 
 2 . . 
 
 4 
 18 
 17 
 
 8 
 36 
 34 
 
 9 
 25 
 22 
 
 18 
 50 
 44 
 
 3 .. 
 
 to 10 1. . 
 
 3 
 41 
 23 
 39 
 
 6 
 
 82 
 46 
 78 
 
 11 
 41 
 29 
 38 
 
 22 
 82 
 58 
 76 
 
 
 32S . 
 
: " : Boys. Girls. 
 
 Per Per 
 
 No. Cent. No. Cent. 
 
 X. Recognize 2 37 74 35 70 
 
 3 27 54 30 60 
 
 4 23 46 23 46 
 
 5 11 22 11 22 
 
 6 7 14 6 12 
 
 2 Two Groups 21 42 19 38 
 
 2 Three Groups 6 12 4 8 
 
 XI. Know Without Objects 1 1 . . 6 12 16 32 
 
 2 2 . . 8 16 14 28 
 
 to 6 6 . . 9 18 14 28 
 
 With Objects 1 1 . . 13 26 21 42 
 
 2 2. . 17 34 21 42 
 
 to 6 6. . 14 28 21 42 
 
 Without Objects 2 + 2.. 12 24 6 12 
 
 2 + 3.. 12 24 5 10 
 
 to3 + 3.. 8 16 3 6 
 
 With Objects 2 + 2. . 10 20 14 28 
 
 2 + 3.. 6 12 18 36 
 
 to 3 + 3.. 5 10 6 12 
 
 Without Objects 3 2. . 11 22 6 12 
 
 4 2.. 5 10 4 8 
 
 to 6 5.. 4 8 4 8 
 
 With Objects 3 2. . 10 20 11 22 
 
 4 2.. 7 14 6 12 
 
 to6 5.. 4 8 6 12 
 
 XII. Know 1X2 Concretely 28 56 29 58 
 
 XIII. Know 2X3 Concretely 1 2 3 6 
 
 XIV. Know 2's in 4 Concretely 4 8 7 14 
 
 XV. Know 3's in 6 Concretely 3 6 3 6 
 
 XVI. Knowl X 2 Abstractly 7 14 8 16 
 
 2X2 Abstractly 4 8 3 6 
 
 3X2 Abstractly 2 4 2 4 
 
 1X3 Abstractly 3 6 3 6 
 
 2X3 Abstractly 1 2 2 4 
 
 XVII. Know 2's in 4 Abstractly 3 6 2 4 
 
 2's in 6 Abstractly 3 6 1 2 
 
 3's in 6 Abstractly 2 4 1 2 
 
 XVIII. Understand Term " More" 41 82 36 72 
 
 "How Many More" 8 16 9 18 
 
 Term "Less" 14 28 11 22 
 
 "How Many Less" 2436 
 
15 
 
 Bays. Girls. 
 
 Per Per 
 
 No. Cent. No. Cent. 
 
 XIX. Know That There Are Two Halves in 
 
 Whole 27 54 18 36 
 
 That These Parts Should Be 
 
 Equal 18 36 11 22 
 
 That There Are Three Thirds in 
 
 Whole 5 10 2 4 
 
 That There Are Four Quarters 
 
 in Whole 5 10 2 4 
 
 SUMMARY OF RESULTS. 
 
 While the results obtained from the testing of so small 
 a number of pupils are by no means conclusive, they are 
 none the less significant. No generalizations not purely 
 tentative can be based upon them alone ; scientifically, they 
 are merely suggestive, or at most but a step in the induc- 
 tion of positive knowledge; but pedagogically they are 
 final, in that definitely showing the numerical knowledge 
 of certain individual children who are to be taught 
 together, they certainly indicate the numerical known to 
 which the unknown facts of number are to be joined and 
 through which they are to be apperceived. 
 
 The first significant fact to note is that the girls knew 
 quite as much of number as the boys, if not somewhat 
 more. If they are at a disadvantage later on in the arith- 
 metical course, it is not due to a poor start as compared 
 with that of their brothers. 
 
 Again, as was to be expected, there was noticeable a 
 great variation in the character of the knowledge dis- 
 played by various individuals, ranging from the almost 
 absolute ignorance of the three or four who knew the 
 least, to the remarkable readiness with which two or three 
 gave the most difficult facts. In the case of the majority, 
 however, the knowledge of individual facts was so scat- 
 tered and unsystematic as to compel the teachers to ignore 
 it, if they taught in the usual logical order. There were 
 few children who did not have a considerable knowledge 
 
i6 
 
 of number, but in scarcely any two cases was that knowl- 
 edge the same. Eighteen per cent of those tested knew ab- 
 stractly that 2 and 2 are 4, and 17 per cent that 2 and 3 are 
 5, while concretely 24 per cent knew these same facts. 
 Eleven boys and six girls knew 3 2 as an abstract fact, 
 while ten boys and seven girls knew it in the concrete* 
 But all those who knew it in the concrete did not 
 know it in the abstract, nor did the same boys and girls 
 who knew the differences always know the sums. With 
 any method of teaching which took up in regular order the 
 combinations and separations of the numbers from 2 to 
 20, a classification which would from the start group 
 together in separate divisions pupils having approxi- 
 mately the same degree of development was plainly impos- 
 sible. 
 
 All this seems at first glance to justify the assumption 
 of practical ignorance to which reference has already been 
 made, and the consequent practice far too common by 
 which with few exceptions beginners are grouped together 
 at the start, and only separated into divisions if sepa- 
 rated at all when they are found to widely vary in the 
 facility with which they acquire the elementary facts. 
 
 The fact, however, none the less remains, that the ma- 
 jority of the children, thus tested on entering school, dis- 
 played a considerable knowledge of number. There were 
 only four boys and two girls who could not count at all, 
 approximately three-fourths of the whole number could 
 count both with and without objects. Of those who could 
 not, six boys and ten girls who could count actual objects, 
 could not count abstractly, while two boys and two girls 
 who could count abstractly, could not count actual objects, 
 that is to say, with but 4 per cent of the whole number 
 of pupils, was the counting merely mechanical. Ten boys 
 and ten girls could count higher without objects than 
 with; sixteen boys and nineteen girls higher with than 
 without. More than half of the pupils, however, could 
 count equally high in the abstract and in the concrete. As 
 
17 
 
 full comprehension of the counting process can call for 
 nothing in addition to this but the generalization that each 
 number is one greater than that immediately below it in 
 the scale, 13 it is surprising to learn that so many of the 
 children possessed the knowledge necessary to it. While 
 but 24 per cent of the whole number knew that 2 cents and 
 2 cents are 4 cents, 49 per cent knew even abstractly that 2 
 and i are 3, while 62 per cent knew that 2 cents and i cent 
 are 3 cents, thus beginning a generalization, which 24 per 
 cent had carried to 10 and i abstractly and 27 per cent 
 concretely, and respectively 13 per cent and n per cent 
 to 20 and i. Seventy-seven per cent knew that 2 is more 
 than i, and 17 per cent answered correctly what must have 
 been the unfamiliar question, "How many more?" It was 
 not remarkable that but 25 per cent could tell whether I 
 was "less" than 2 or 2 than i, and that but three or four 
 knew how many less. Fifty- four per cent of the boys and 
 36 per cent of the girls knew that a thing to be divided in 
 half must be cut into two parts, and 36 per cent of the 
 boys and 22 per cent of the girls understood that these 
 parts should be equal. Ten per cent of the boys and 4 
 per cent of the girls knew the number of parts into which 
 a whole must be cut to divide it into thirds or into fourths 
 when called quarters. 
 
 Although 1 8 per cent of the beginners could not tell how 
 many hands they had, 48 per cent the number of fingers on 
 one hand, and 23 per cent the number of a horse's legs, 
 the customary assumption that such lack of knowledge is 
 indicative of an ignorance of number was not proven. On 
 the contrary many thus ignorant knew as much as those 
 who were not, their ignorance being due to the fact that 
 their attention had not been directed to the numbering 
 of the particular concrete things in question. Many 
 adults would be found on investigation unable to give the 
 number of ribs in the human frame, or of legs normally 
 possessed by a fly. 
 
 It is significant, too, that 72 per cent of the children 
 
i8 
 
 recognized the number 2 with counting made practically 
 impossible; 57 per cent, the number 3; 46 per cent, 4; 
 22 per cent, 5 ; and 13 per cent, 6. When two splints were 
 held in each hand, 40 per cent recognized the total as 4, 
 and when three were so held, 10 per cent the total as 6. 
 With but very few exceptions, all children who knew the 
 number of hands they had, could recognize 2 without 
 counting, and all who knew a horse had four legs could 
 so recognize 4 something which scarce a child without 
 that knowledge could do. It is significant in this connec- 
 tion that while 77 per cent knew the number of legs a 
 horse has, though horses' legs are in pairs, only 24 per 
 cent knew that 2 and 2 are 4, and 7 per cent, two 2's. 
 
 Incidentally acquired though this knowledge is, scant 
 in the case of the few, but quite considerable in that of the 
 majority, if repeated tests in various localities confirm the 
 foregoing results, it will at least be certainly demonstrated 
 that the great majority of children know something of 
 number on entering school. Whether they have enough 
 in common for the public school teacher to be able to 
 utilize it, will be discussed later. 
 
 COMMENTS UPON THE FOREGOING RESULTS. 
 
 Whatever the facts that may be ultimately associated 
 with a given digit, however various the numerical relations 
 in which it plays a part, primarily it seems to stand for 
 aggregation. The child knows that i and i are 2 before 
 he learns that two I's are 2 ; that 5 and i are 6 before he 
 discovers or is led to discover that three 2's are 6. He 
 asks for "another" or "one more" rather than for two 
 times as many as he had before. He learns to count before 
 he masters the elementary products. Multiplication as 
 taught him in the school may never be to him "a short 
 method of addition," but nevertheless were he left to him- 
 self, 8 would be 4 and 4, before it would be 4 taken two 
 times, or multiplied by 2. It has been left for the philos- 
 
19 
 
 opher to assert the fact that number is ratio. The child 
 counts before he measures. To him addition is counting 
 on, subtraction is counting off. Counting, then, is the 
 fundamental process. 8 It begins in so far as it is not 
 merely mechanical with the child's plea for "just one 
 more/' his perception of one thing and another one, not, of 
 course, with the abstraction I and i. 9 Sixty-six per 
 cent of the 100 children tested on entering school knew that 
 i cent and another cent are 2 cents ; only 33 per cent that 
 i and i are 2. Half, however, of those who knew this 
 primal fact of number at all, had made the generaliza- 
 tion necessary to apply it to any one thing and another, 
 a fact which is of significance in a later connection. While 
 only four boys and two girls were unable to count at all, 
 thirty-four boys and girls did not know that i more than 
 i familiar thing, or i familiar object and another, are 
 2. Hence it is plainly obvious that many children who 
 are reported as being able to count were completely 
 ignorant of the fact that each successive digit is one more 
 than the one which immediately precedes it. It would be 
 erroneous, however, to assume that counting thus unintel- 
 ligent, is a purely mechanical repetition. Only 16 per 
 cent of the whole number of children were unable to count 
 actual objects, and thus to definitely number a group of 
 I's, of whose number as compared with other groups they 
 had no knowledge. For example, the great majority of 
 beginners were at least able to determine that there were 
 four objects in a given group, though ignorant oi the fact 
 that their number was i more than that of a group of 3 
 or i less than that of a group of 5, while a very consider- 
 able number had made the generalization necessary to 
 intelligent counting. These results tend to demonstrate 
 that upon entering school, a few children will be found 
 who cannot count at all (in this case 6 per cent) ; 10 a few 
 whose counting is merely mechanical repetition originat- 
 ing in the series concept, 8 conventionalized by imitation, 
 and stimulated by a love of rhythm ( 16 per cent) j 11 a 
 
2O 
 
 large number who having taken the next step can by 
 counting determine within narrow limits the number of 
 objects in a given group though ignorant that each digit 
 which they name is greater by I than that immediately 
 preceding it (34 per cent) ; 12 a still larger number (44 
 per cent), comprising almost half of the children tested, 
 who have at least begun to count intelligently. 13 
 
 Counting, then, it would seem, is at first to the great 
 majority of children a rhythmical series of names with 
 little or no meaning, sooner or later correctly applied to 
 particular groups of objects, and gradually perceived as 
 a series of number-names each one of which may be 
 applied to a group containing one more object than does 
 that to which the number-name last preceding it maybe ap- 
 plied. To some, counting may never be a mechanical pro- 
 cess. They may successively add by I's, may know that i 
 and i are 2, 2 and i, 3, etc., before they repeat i, 2, 3, as a 
 series. Indeed this addition of successive I's and the me- 
 chanical counting may go on side by side, without the 
 child's perceiving any connection between them. 
 
 The fact that "counting on" or "forwards" is so popular 
 a process with children is likely to make it a matter of 
 imitation before it becomes intelligent; not so, however, 
 with "counting off." Very few of the children tested 
 could count backwards from 10 to i, yet 28 per cent knew 
 in the abstract that 2 less i are i, 25 per cent that 3 less 
 i are 2, and 13 per cent the most of the remaining facts to 
 10 less i, while in the concrete the percentages were 
 severally 43 per cent, 39 per cent and 14 per cent. The 
 fact that few children knew these differences in any 
 systematic order strongly indicates that they were 
 mastered as separate and distinct facts. They did not 
 seem to know that 3 less i are 2, because 2 is next below 3 
 in the digit series. Unlike the sums formed by adding i, 
 there was nothing so far as could be seen to differentiate 
 the facts in which i was subtrahend from those in which 
 the subtrahend was 2 or 3. It is evident, then, that the 
 
21 
 
 only way in which these children became familiar with 
 certain differences was the process by which they came to 
 know such of the elementary sums as were not obtained by 
 adding i. Just as they counted out the number of objects 
 to be added and then counted out their sum, so they must 
 have counted out the number of objects in the minuend, 
 the number that was taken away and the number remain- 
 ing. They were almost without exception, ignorant of 
 the term "less ;" subtraction was to them a "taking away" 
 with a certain remainder, not the lessening of a number by 
 a certain amount. The number remaining does not seem 
 to have been inferred from the number to which the sub- 
 trahend is added to obtain the minuend. That is, they did 
 not know that 3 less 2 is i , because they already knew that 
 2 and i are 3, but because they met with it in their experi- 
 ence as a separate and distinct fact. Twenty-three per 
 cent abstractly and 35 per cent concretely seem to have 
 made the generalization that when a number is taken from 
 itself the remainder is "nothing." 
 
 However few the disconnected facts of number which 
 a child masters as distinct operations, and which repeated 
 in his experience with different objects become abstrac- 
 tions, it is probable that they would be fewer still did he 
 have to laboriously count out every number which they 
 involve. Three-fourths of the children tested could recog- 
 nize 2 without noticeable counting; more than half, 3; 
 and a little less than half, 4. Twenty-two per cent and 
 13 per cent respectively thus recognized 5 and 6. As the 
 objects used in this test were splints held fan-like and dis- 
 played but for an instant, the resulting forms were hardly 
 those that the child would already have associated with 
 particular number-names. This being so, while I fully 
 agree with Dr. Phillips that much which passes as number 
 recognition "is only the recognition of an individual 
 form/' 14 in the present case the children seem to have 
 recognized number. It may be more than a mere coinci- 
 dence, however, that with but very few exceptions, all 
 
22 
 
 children who knew the number of hands they had, could 
 recognize 2, and that all who knew a horse has four legs, 
 could recognize 4, something which scarce a child with- 
 out that knowledge could do. Two objects can not be 
 so placed that they will not be in a straight line, three and 
 four objects are usually so grouped that they may respec- 
 tively form vertices for triangles and quadrilaterals. The 
 relative position of the horse's legs, human hands or the 
 feet of a three-legged stool is not different from that of the 
 corresponding number of splints. 
 
 In the absence of recognition by the children of what is 
 thus in effect an imaginary general form, far-fetched 
 though its assumption may seem, whether we grant the 
 truth of Kulpe's hypothesis of a direct recognition, de- 
 pendent upon the "effectiveness" of a percept for central 
 excitation and the "mood" which it induces, 15 or assume 
 that the recognition is here due to the comparison of the 
 present percept with a memory image in part identical, 
 an adequate explanation of this instantaneous number 
 recognition is exceedingly difficult. In the first place, 
 whatever may be true in the case of the adult, in that of 
 the child just entering school, there is little likelihood of 
 unconscious counting, and no possibility whatever of 
 either conscious or unconscious estimation. Few of the 
 children tested were orally rapid counters, and the great 
 majority in counting objects assisted discrimination by 
 pointing to each successive one or by handling it. It is 
 altogether unlikely that a "mental counting" of which 
 there was no external sign would be rapid, if indeed in 
 the case of the majority it could even be possible with 
 any degree of accuracy. 
 
 Tests in which the pupils would be led to so count the 
 numbers certainly recognized would be indeterminate, in 
 that in the absence of introspection the observer would 
 not determine whether the result was reached through 
 counting or not, while if they were found able to 
 "mentally" count higher numbers, the results would not 
 
23 
 
 necessarily apply to the lesser numbers directly recog- 
 nized. 
 
 Unconscious counting being thus improbable, uncon- 
 scious estimation was plainly impossible, where a ma- 
 jority of the subjects were ignorant of the necessary facts. 
 The grouping of objects instead of helping to a more 
 accurate judgment, resulted in the reverse, only 40 per 
 cent of the pupils being able to recognize two 2 groups 
 as 4, and 10 per cent two 3 groups as 6. The separation 
 into two groups only made it more difficult to perceive the 
 objects as one whole. 
 
 While it may be well to recall in this connection the 
 famous experiments in which young children through 
 the display of a number of large and irregular objects side 
 by side with a larger number of smaller objects, have been 
 deceived into pronouncing the former number the greater, 
 they merely serve to demonstrate that children, like adults, 
 may be led to form a wrong judgment of comparative 
 number. 
 
 In conclusion, then, we may state as true of the particu- 
 lar children tested and tentatively assume as true of 
 American children in general: 
 
 1. That the majority of children have some knowledge 
 of number on first entering school. 
 
 2. That to them counting is the initial numerical pro- 
 cess, and the ability to count, the only numerical knowl- 
 edge which they have in common. 
 
 3. That they have little or no knowledge of ratio. 
 
 4. That to them the numbers i, 2, 3 and 4 have become 
 concepts which in most cases can be instantaneously and 
 correctly applied whenever the corresponding groups of 
 objects are distinctly seen. 
 
CHAPTER II. AN A PRIORI DETERMINATION 
 
 OF THE PSYCHOLOGICAL ORDER OF 
 
 TEACHING NUMBER. 
 
 I. DISCUSSION OF LOGICAL AND OF PSYCHOLOGICAL 
 
 ORDER. 
 
 As has been already asserted, it is impossible to utilize 
 the numerical knowledge possessed by children on enter- 
 ing school, if number is to be taught them in the usual 
 logical order, that is, in the order which exhausts the 
 combinations and separations of each successive number 
 from i to 20 before the teaching of any fact connected 
 with those above it. The question therefore naturally 
 arises, is this logical order the only order in which number 
 can be taught, and if not, is it all things considered 
 the order in which the fundamental facts of addition and 
 subtraction can be most readily mastered by beginners? 
 There is a tendency among disciples of "the new educa- 
 tion" to take for granted, an antagonism between "logical" 
 and "psychological" order. The knowledge of the 
 primary school, it is assumed, must not be systematic 
 knowledge, because the order in which facts are best 
 grouped by the adult mind is not that in which they can 
 be most readily apperceived by the mind of the child. 
 Now it has by no means been demonstrated that knowl- 
 edge logically arranged can not be more readily compre- 
 hended by the beginner than can any aggregation of 
 isolated facts, however simple those facts may be. It is 
 characteristic of a logical order that facts having some- 
 thing in common are grouped together. A logical classi- 
 fication, basing itself on an element common to a large 
 
 (24) 
 
25 
 
 number of things, will separate out from the mass two or 
 more groups which will together contain all things having 
 the common element, but which are differentiated from 
 each other in that the things in each group have an addi- 
 tional common element, which those in the others have 
 not. Similarly each group may then be separated into 
 additional groups, the partial identity of the things in each 
 sub-group growing greater and their number less, until it 
 is impossible for partial identity to be thus increased. 
 Facts are taught in a logical order, when beginning with 
 the things having least in common, instruction succes- 
 sively passes to those which have most, or when beginning 
 with those which have the most in common, it similarly 
 passes to those which have least. Teaching which thus 
 goes from the general to the particular is called deductive ; 
 that which taking the opposite course, passes from the 
 particular to the general, is known as inductive. As the 
 things included in a given class may be differently grouped 
 accordingly as this or that element common to many of 
 them, is made the basis of classification, the same things 
 may usually be classified in many different ways, with the 
 result that the young may be taught both deductively and 
 inductively in more than one logical order. Now aside 
 from such general conditions as attention and fatigue, 
 nothing is more necessary to the processes involved in 
 apperception and recall, than such arrangement of indi- 
 vidual facts as makes more readily perceivable the presence 
 of some common element. 
 
 It is only when the common element is too insignificant 
 for children to readily perceive it, or when the things to 
 which it is common are too complex or too various for it 
 to be readily perceived, that facts logically grouped can not 
 be readily mastered. The psychological order of teaching 
 a number of particular facts, is that through which a given 
 individual or group of individuals can most readily ap- 
 proximate their full apperception and prompt recall. 
 While the psychological order thus concerns itself with 
 
 v 
 
 OF THE 
 
26 
 
 the succession in which facts are presented to the indi- 
 vidual mind, and the logical with the manner in which 
 they are related to each other, it is readily to be seen that 
 the former is conditioned by the latter. 
 
 Not only, therefore, is there no necessary antagonism 
 between the logical and the psychological orders, but it 
 may be seriously questioned whether to the vast majority 
 of individuals any order of teaching the facts of the 
 school curriculum is fully psychological without being 
 logical. If the facts to be imparted to a given individual 
 have no element in common, that order of teaching will be 
 psychological, which so presents them that each separate 
 fact will precede all others more difficult for that individ- 
 ual to perceive. If on the contrary the facts to be thus im- 
 parted have anything in common, which mastered in one 
 will materially aid in the mastery of all, that order of 
 teaching will be psychological, which groups together the 
 facts possessing this common element, provided that it is 
 readily perceivable to the individual taught. That such 
 a grouping is logical is evident per se. Since the great 
 majority of facts included within the school curriculum, 
 have more or less in common with other facts, it neces- 
 sarily follows that wherever possible, they must be pre- 
 sented in a logical order whose common element readily 
 perceived by the majority of pupils, in the mastery of one 
 or more individual facts, will insure the ready mastery of 
 the remaining facts with which they are grouped. It will 
 be convenient to designate such a common element as an 
 "effective" element. 
 
 The logical order will therefore be psychological, when 
 its common element is one thus effective, and readily per- 
 ceivable to the individuals taught. Other things being 
 equal, the larger the number of the facts included in its 
 several groups, the more psychological will an order be, 
 for the larger the number of facts in the groups, the 
 smaller the number of groups, and hence the readier the 
 mastery of the whole body of facts. Moreover it is plain 
 
27 
 
 that the common element should not only be effective, but 
 that it should be that particular effective element which is 
 the basis of scientific classification, wherever such an ele- 
 ment is readily perceivable by the individuals taught. 
 
 To assert that without exception all psychological order 
 of grouping the facts included in the school curriculum 
 must be logical order would be to assume not only that 
 each of these facts has in common with other facts more 
 or less that is effective, but that every fact has among the 
 various elements thus common and effective at least one 
 which through psychological instruction can be readily 
 perceived by the great majority of individuals taught. 
 While this assumption is quite unnecessary to the present 
 argument, experience has abundantly demonstrated that 
 as a rule such an element can be found in the various facts 
 which go to make up the common school curriculum. 
 
 If a given logical order based upon an essential common 
 element seems too difficult for the child, before condemn- 
 ing it as unpsychological,it would be well to know whether 
 the attempt to impart the facts so grouped has been made 
 in accordance with a psychological method of instruction. 
 (See Chapter III.) And even should this prove to be the 
 case, the failure of the child to comprehend them is no 
 indication that logical order in general is unpsychological, 
 but rather that in a particular logical order, the common 
 element can not be readily perceived by the child mind, 
 the right course in such an event being to seek to arrange 
 the same facts in an order no less logical, but in which 
 the essential common element can be more readily per- 
 ceived. 
 
 The discussion thus far has made clear the conditions 
 necessary to the psychological sub-grouping of the facts 
 included in any general category. 
 
 In each psychological group, 
 
 i. The common element must be an effective one; one 
 whose perception in the mastery of one fact in the group 
 will insure the ready mastery of every other fact. 
 
28 
 
 2. It must be the most effective one which through 
 psychological instruction can be readily perceived by the 
 great majority of particular individuals to be taught, 
 unless some other effective element thus readily perceiv- 
 able is the basis of the accepted scientific classification. 
 
 3. It must be common to more facts than is any other 
 readily perceivable and equally effective. 
 
 All the facts of a general category that must be 
 mastered by the child, having been arranged so far as pos- 
 sible in psychological groups, it is evident that the psycho- 
 logical order of teaching these groups will depend upon 
 the relative effectiveness of their common elements, the 
 relative readiness with which their common elements are 
 perceived by particular individuals taught, and the rela- 
 tive number of facts which they include. When, as is 
 usually the case, mastery of the effective element of one 
 group is a necessary condition to that of the effective ele- 
 ment of another, or when, more rarely, the effectiveness, 
 perceivability and inclusiveness of the elements common to 
 the various groups increase or diminish together, psycho- 
 logical order is readily determinable. But when, as may 
 possibly occur, among a number of unrelated groups, the 
 common element of one has relatively the highest effec- 
 tiveness, of another the readiest perceivability, and of a 
 third the greatest inclusiveness, it is not so easy to deter- 
 mine which of the three should be taught first. Each 
 group, it may be, can be readily mastered by the child, 
 but should they vary considerably in the readiness with 
 which they can be mastered, and not too widely in the 
 number of facts which they include, it is probably more 
 psychological to teach first the one which, whether through 
 the higher effectiveness of its common element or through 
 the greater readiness with which that element is perceiv- 
 able, can be mastered most readily. If on the contrary 
 one should include many more facts than the others, being 
 effective and readily mastered, though not most readily 
 
29 
 
 mastered, it should be taught first, tending as it does to 
 the readier mastery of the whole. 
 
 Where, as will usually be the case with psychological 
 groups, the groups themselves have elements in common 
 they should be grouped together in their turn, under the 
 same conditions as the single facts which they themselves 
 include, it being remembered that here, the readier mastery 
 of some one group must be subordinated to the ready 
 mastery of the whole collection of groups of which it is 
 a part. 
 
 It is fortunate that this general discussion of psycho- 
 logical order has not involved the determination of what 
 knowledge is of most worth. Whether or not physiology 
 should be taught before number or aesthetics before read- 
 ing, it is not for psychology to determine. But whatever 
 the sequence of the general branches of study utilitarian, 
 ethical, or historic, there will be few among the facts 
 selected as necessary to the child, that can not be included 
 in some psychological group. Relative usefulness, char- 
 acter building, or culture epoch may demand that some 
 individual fact shall be taught before the psychological 
 group of which it would otherwise form a part, or that 
 some logical group shall be taught before the effective 
 element common to its members can be readily mastered 
 by the child, but after all such demands have been met, 
 there is an order in which the facts thus included from 
 year to year in the school curriculum can be most readily 
 mastered by a majority of the pupils taught, and this 
 order is psychological. 
 
 However great the variation in our courses of study, 
 the facts which are taught in school are all included in 
 some general classification. They are grouped together 
 into branches of study in each school curriculum, and 
 taught for the most part with the aid of separate text- 
 books. Even in the primary grades they are grouped as 
 number-work, nature-study, and the like. The common 
 element upon which this general classification is based 
 
30 
 
 soon comes to be recognized though perhaps somewhat 
 vaguely by even the youngest pupil. He is not able to 
 tell what it is, unless the teacher has pointed it out or de- 
 noted it by some suggestive term, but he will not confuse 
 a number-fact with his reading or a sentence with his 
 number, unless "correlation" has quite unified his mental 
 content. At the point, however, when on successive sub- 
 division, the common element, however effective, is not 
 readily perceivable or where being readily perceivable in- 
 struction does not lead the learner to perceive it, logical 
 order becomes of no avail, except in so far as it prevents 
 the instructor from omitting any of the necessary facts. 
 To the learner each fact stands by itself, unrelated to other 
 facts, and is perceived by him without being well apper- 
 ceived. It was this misuse of logical order that char- 
 acterized the so-called formal instruction which here and 
 there still exists in American schools. To the influence 
 of German pedagogy and more latterly to the populariza- 
 tion of psycho-genesis especially in the form of "child 
 study," has been largely due the reaction against formal 
 instruction, which is now almost everywhere triumphant. 
 Unfortunately, however, iconoclasm here as elsewhere the 
 companion of reform, has shattered the old idol into a 
 thousand fragments, and thereby too often substituted 
 for logical order in elementary instruction, the study of 
 isolated facts. 
 
 II. THE PSYCHOLOGICAL ORDER OF TEACHING THE 
 FUNDAMENTAL SUMS AND DIFFERENCES. 
 
 (I) THE REACTION AGAINST FORMALISM IN THE 
 TEACHING OF ELEMENTARY NUMBER. 
 
 The changes which have taken place in the study of 
 arithmetic during the past fifty years well illustrate this 
 swing of the pedagogic pendulum. But a generation or so 
 ago formalism reigned supreme. Beginners were taught 
 
notation and numeration to billions, before they began ad- 
 dition ; the addition of columns long and broad, before they 
 studied subtraction, and so on and on through a logical 
 order which they were rarely led to comprehend, counting 
 on fingers, learning by rote, and working by rule, they 
 plodded from multiplication table to the rule of three. No 
 one will claim that to the majority of those taught such 
 teaching was psychological, but notwithstanding its 
 theoretic faults, the reformer is occasionally non-plussed 
 by the undeniable fact that it turned out a host of strong 
 mathematicians. 
 
 The reason is not far to seek. All that was lacking, 
 where the element common to each group was an effective 
 common element, was that it should be comprehended by 
 the individuals taught. And many a plodder did compre- 
 hend even as he labored, while many another came with 
 the aid of a maturer mind, to ultimately comprehend what 
 had been mechanically but permanently drilled into his 
 memory. To such the old formal instruction was either 
 psychological or became so. Not so, however, with the 
 great majority, stupefied by a logic they could not under- 
 stand, and arrested in mental development by what was to 
 them for the most part the mechanical memorizing of dis- 
 connected facts. With the more or less individual instruc- 
 tion of the old district school, the formal grouping of the 
 facts of number was undoubtedly helpful to the few. But 
 as no educational plan, through its helpfulness to a minor- 
 ity can find justification for its application to all, educa- 
 tional leaders largely influenced by the German reform- 
 ers began to protest against the mechanical mastery of 
 fact and rule, and to urge in its place the teaching of 
 "intellectual arithmetic." In school journal and teachers' 
 meeting they persistently insisted that the pupil should not 
 recite rules "by heart without thinking or considering 
 thereon," that in the solution of problems he should 
 examine the "question," and not his stock of rules ; that, 
 in fact, there should be no rules at all, but in their stead 
 
32 
 
 a few general processes which would furnish the key to 
 all solutions. Thus arithmetical analysis and mental 
 arithmetic came to find a place in the curriculum, substi- 
 tuting for rigidity of rule a no less rigidity in demonstra- 
 tion, but necessitating the teaching of the subjectively 
 simple, before that of the subjectively complex. The 
 triumph of reform, even yet by no means complete, was 
 not won without a struggle. The embattled host of tradi- 
 tion long continued to hold its own against the cham- 
 pions of reason. The educational literature of the period 
 is rich in material for a history yet to be written, of this 
 stage in the old warfare between "Trojan" and "Greek." 
 But while here and there may be found a locality where 
 every rule has been hurled from its sacred niche, there has 
 hardly yet resulted, as feared by a grandiloquent conserva- 
 tism, a generation of pupils, who independent alike of 
 teacher and text-book, "never soar by faith to regions of 
 inconceivable glory, but ever plod on by the dim light of 
 reason." 16 
 
 The only innovation that need be here discussed is the 
 change which took place in the order of teaching the 
 elementary facts of number. Largely through the influ- 
 ence of Colburn and Emerson, the fundamental facts of 
 addition, subtraction, multiplication and division began to 
 be taught before the study of numeration and notation, or 
 operation with numbers greater than 10 or 20. Later 
 came the introduction of the Grube method, 17 involving 
 another order no less logical than the old but which has 
 for its end the exhaustion of the "simpler" of the funda- 
 mental facts before the study of those objectively more 
 complex. No effort was made to so group the facts that 
 the mastery of one might involve the ready mastery of 
 many. On the contrary ready mastery was to come 
 through the method by which each individual fact was to 
 be imparted. Even to-day while the Grube method is 
 rapidly being displaced by what are perhaps its betters, 
 the Grube order still reigns supreme in American 
 
33 
 
 schools. 18 Is this order psychological? If not, what is 
 the psychological order of teaching the elementary sums 
 and differences ? 
 
 (II) DESCRIPTION OF THE VARIOUS LOGICAL ORDERS 
 POSSIBLE IN TEACHING THE ELEMENTARY SUMS 
 AND DIFFERENCES. 
 
 The terms "sum" and "difference" are here used to 
 denote not merely the results of certain numerical opera- 
 tions, but in their popular signification, as denoting the 
 operations themselves, including their results. For 
 example, 3 + 2 = 5 is called a sum, and 5 3 = 2, a dif- 
 ference. An "elementary" sum is one formed by the 
 addition of one digit to another ; a "secondary" sum, one 
 in which one or both of the numbers added are greater 
 than nine. The "inversion" of a sum only differs from 
 the sum itself, in that the number to which addition is 
 made has changed place with the number to be added. 
 An "elementary" difference is one formed by the sub- 
 traction of one digit from another; a "secondary" differ- 
 ence, one in which either subtrahend or remainder is 
 greater than nine. An "alternation" is a difference in 
 which the remainder has changed places with the subtra- 
 hend. All sums and differences, whether elementary or 
 secondary, whose mastery is necessary to ready operation, 
 may be regarded as "fundamental." These fundamental 
 sums and differences together with the fundamental prod- 
 ucts, quotients, etc., are commonly known as "the number 
 facts." 
 
 As the secondary facts are readily derivable from those 
 that are elementary, and the identity of result in sum and 
 inversion, or the interchange of result in difference and 
 alternation, are as will be presently demonstrated 
 readily perceivable by the great majority of children, it is 
 unnecessary for the present to consider the arrangement 
 of all fundamental facts that are not elementary, or of 
 elementary facts that are inversions or alternations. 
 
34 
 
 Again, while it has not been taken for granted that the 
 elementary differences should not be taught contempor- 
 aneously with the elementary sums, it will better serve the 
 purposes of the discussion to fix the psychological order 
 for the teaching of the latter, before considering whether 
 the former should be taught thus contemporaneously, at a 
 later period, or not at all. 
 
 Can the elementary sums and differences be grouped in 
 more than one logical order? If so, what are these logi- 
 cal orders, and which among them is the psychological 
 order for the majority of children on first entering the 
 lowest school grade? It is not until this latter question 
 has been fully determined by a strict application of the test 
 already laid down that we can judge as to how far we can 
 utilize the knowledge of number which children possess 
 on first entering school. 
 
 In building up a group of different elementary sums it 
 is evident that but one term can remain constant. If the 
 result the "sum" in the stricter sense of the term is not 
 varied, both the number to be added and that to which 
 addition is to be made, become variables ; if either of the 
 latter is kept constant, the number added to it or to which 
 it is added, and their result, must be variables. Moreover, 
 it is evident that if a system of grouping is to include all of 
 the elementary sums, it must involve the addition of each 
 digit to every other digit, though not necessarily in the 
 usual digit order. Hence only three distinct sets of groups 
 of elementary sums are possible : ( i ) Those in which the 
 "result" is constant; (2) those in which the digit to which 
 addition is to be made, is constant, and (3) those in which 
 the number to be added is constant. In each set every 
 digit must be the constant in a group containing all the 
 elementary sums, which can be formed by the variation 
 of the other two terms. If the order of teaching the 
 elementary sums is to be psychological in the highest 
 degree it must be determined : 
 
 i. Whether one system of grouping is psychological 
 
35 
 
 throughout as compared with either of the others, or 
 whether psychological order necessitates the use of groups 
 from more than one system; 
 
 2. The psychological order of the groups selected. 
 
 3. The psychological order of the facts within each 
 group. 
 
 For convenience in the discussion, the forty-five 
 elementary sums not inversions are given below, grouped 
 according to each system and following the order of the 
 digits. 
 
 TABLE OF THE ELEMENTARY SUMS ARRANGED UNDER 
 EACH OF THE THREE LOGICAL SYSTEMS OF GROUPING. 
 
 A. The Grouping Together of the Sums Obtained by the Addition of 
 
 the Several Digits in Order to Each Successive Digit. 
 
 1. 1 & 1, 1 & 2, 1 & 3, 1 & 4, 1 & 5, 1 & 6, 1 & 7, 1 & 8, 1 & 9; 
 
 2. 2 & 2, 2 & 3, 2 & 4, 2 & 5, 2 & 6, 2 & 7, 2 & 8, 2 & 9; 
 
 3. 3 & 3, 3 & 4, 3 & 5, 3 & 6, 3 & 7, 3 & 8, 3 & 9; 
 
 4. 4 & 4, 4 & 5, 4 & 6, 4 & 7, 4 & 8, 4 & 9; 
 
 5. 5 & 5, 5 & 6, 5 & 7, 5 & 8, 5 & 9; 
 
 6. 6&6, 6&7, 6&8, 6&9; 
 
 7. 7&7, 7&8, 7&9; 
 
 8. 8&8, 8&9; 
 
 9. 9 & 9. 
 
 B. The Grouping Under Each Successive Digit of the Sums Whose 
 
 Results Are Equal to It. 
 
 1. 1& 
 
 2. 2& 
 
 3. 3& 
 
 4. 4& 
 
 5. 5& 
 
 6. 6& 
 
 7. 7& 
 
 8. 8& 
 
 ; 10. 9 & 2,8 & 3,7 & 4,6 & 5; 
 
 ; 11. 9 & 3,8 & 4,7 & 5,6 & 6; 
 
 , 2 & 2; 12. 9 & 4, 8 & y, 7 & 6; 
 
 ,3&2; 13. 9 & 5,8 & 6,7 & 7; 
 
 ,4&2, 3&3; 14. 9 & 6,8 & 7; 
 
 , 5&2, 4&3; 15. 9 & 7,8 & 8; 
 
 ,6&2,5&3,4&4; 16. 9&8; 
 
 , 7&2, 6&3, 5&4; 17. 9 & 9. 
 
 9. 9 & 1, 8 & 2, 7 & 3, 6 & 4, 5 & 5; 
 
 C. The Grouping Together of the Sums Obtained by the Successive 
 Addition of Each Digit, First to Itself, Then to Each Inferior 
 Digit. 
 
 1. 1 & 1, 2 & 1, 3 & 1, 4 & 1, 5 & 1, 6 & 1, 7 & 1, 8 & 1, 9 & 1; 
 
 2. (1)2&2,4&2,6&2,8&2; 5. (1) 5&5; 6. (6) 8. (3) 9. (4) 
 
 (2)3&2,5&2,7&2,9&2; (2) 6&5; 7. (1) 7&7; (4) (5) 
 
36 
 
 3.(1)3&3,6&3,9&3; (3) 7&5; (2)8&7; (5) (6) 
 
 (2)4&3,7&3; (4) 8&5; (3)9&7; (6) (7) 
 
 (3)5&3,8&3; (5) 9&5; (4)- (7) (8) 
 
 4 . (1) 4&4, 8&4 ; 6. (1) 6&6 ; (5) (8) (9) 
 
 (2)5&4,9&4; (2) 7&6; (6) 9. (1) 9&9; 
 
 (3)6&4; (3)8&6; (7) (2) 
 
 (4) 7&4; (4) 9&6; 8. (1) 8&8; (3) 
 
 (5) (2) 9&8; 
 
 (III) COMPARISON OF THE THREE SYSTEMS WITH A 
 VIEW TO THE DETERMINATION OF THE PSYCHO- 
 LOGICAL ORDER. 
 
 Is one system of grouping more psychological through- 
 out than is either of the others ? 
 
 To determine this, the usual test must be applied to the 
 various groups resulting from the completed development 
 of each system. Have the groups of any one system, as 
 compared with those of the other two, common elements : 
 
 1. More effective? 
 
 2. More readily perceivable by those to be taught ? 
 
 3. Common to a greater number of the elementary 
 sums? 
 
 It must be steadily held in mind that among these tests 
 that of effective identity is of the highest comparative 
 importance, that, if the mastery of one individual of the 
 group is not the key to the mastery of all, the grouping 
 can be psychological only in so far as it insures the teach- 
 ing of facts subjectively simple before those relatively 
 complex. 
 
 Certain numbers are to be firmly associated together as 
 individual number facts. If any considerable number of 
 these facts can be so grouped together that the mastery of 
 one involves the mastery of all, their partial identity may 
 be regarded as an effective identity. The group is psycho- 
 logical, however, only when this effective identity is 
 readily perceivable by a majority of the individuals who 
 are to thus firmly associate the digits together as facts. 
 The more effective this identity, the more readily it is per- 
 
37 
 
 ceivable, and the greater the number of facts which possess 
 it, the more psychological the resulting group. 
 
 i. Why the Grouping Together of the Sums Obtained 
 by the Addition of the Several Digits in Order to Each 
 Successive Digit is Not the Psychological System of 
 Grouping. 
 
 This grouping which results from the addition of all the 
 digits to each digit as a constant, prevailed in American 
 schools from the time of Colburn until the Grube method 
 became popular. It is found for example in such well- 
 known text-books as those of Emerson (1838), Green- 
 leaf (1851), and Davies (1862), as well as in many 
 others. The constant addend to which addition is made 
 and the fact that when the sums are taken in the order of 
 the digits, each result is greater by one than that immedi- 
 ately preceding it, are the elements common to all the 
 facts of a given group, not considered in relation to other 
 groups. There is nothing, however, in the fact that the 
 pupil has added one number to 6, to aid him in giving 
 the result obtained by the addition of another number to 
 6, unless he successively gives all the facts which, when 
 taken in order, intervene between them. Thus, knowing 
 that 6 and 2 are 8, he can know that 6 and 5 are n, by 
 repeating to himself, 6 and 2 are 8, 6 and 3 are 9, 6 and 4 
 are 10, 6 and 5 are n, an operation which was usually 
 performed with the aid of finger-counting. 
 
 But this knowledge is not due to the grouping. The 
 result may be just as readily obtained for the isolated 
 fact. The method employed is merely the counting out 
 of the sum, either mentally or with the aid of objects. It 
 is evident then that the elements common to the various 
 groups of this system, considered within themselves, are 
 not effective. When, however, the group formed by the 
 addition of the digits to 5 has been thoroughly mastered 
 before the study of the 6 group begins, the facts included 
 in the latter have an effective common element, in that 
 
any fact in the 6 group will have a result greater by one 
 than that of the corresponding fact in the 5 group. If 5 
 and 5 are 10, 6 and 5, or i and 5 and 5 will be 1 1. This, 
 however, is not so readily perceivable by the child as is the 
 fact that if 5 and 5 are 10, 5 and 5 and i, or 5 and 6 will be 
 1 1 which is, as will be seen, all that it is necessary for him 
 to perceive in the third system of grouping. In the former 
 case he must be led to see that the addition of i to the 5, 
 is the addition of I to the 5 and 5 ; in the latter, that the 
 addition of the i to the 5 and 5 involves the addition of 
 the i to the 5. As the result can be obtained inductively 
 only by first adding the i to the 5 and 5, or the 5 and 5 to 
 the i, it is evident that the latter method of derivation will 
 be more readily mastered by the child. It is the algebraic 
 5 and 5 and i compared with i and (5 and 5). The 5 
 must be added to the 5 before the child perceives that the 
 addition of the i results in the n. It is after this that he 
 must perceive that the addition of I and 5, or of 5 and I, 
 is the addition of 6. This, too, is more readily seen by the 
 latter plan, where it has not been necessary to overempha- 
 size the addition of the 5 to the 5. Under this system of 
 grouping, which it may be convenient to designate as the 
 Colburn, there are nine groups containing from nine to 
 one elementary facts. 
 
 2. Why the Grouping Under Each Successive Digit of 
 the Sums Which Have It for Their Result is Not the 
 Psychological System of Grouping. 
 
 This is of course the Grube system of grouping, limited 
 to a certain class of fundamental facts. For thirty years 
 or more it has been gradually superseding the Colburn 
 plan, until to-day there are few if any elementary arithme- 
 tics that do not follow it. The element common to the 
 sums composing one of its groups is their result, and that 
 alone, unless, as is not done in the Grube method of teach- 
 ing, the result is considered in its relation to that of the 
 
39 
 
 group immediately below. In this latter case, it may be 
 regarded as one greater for each sum than the result for 
 the corresponding sum in the lower group, a common 
 element readily enough perceived after it has been ob- 
 tained, but hardly effective, in that, for example, 4 and 
 2 are known to be 6 because 5 and i are 6. The child 
 could only recognize an unknown sum as belonging to the 
 6 group through the fact that another sum, one of whose 
 addends was identical with one addend of the unknown, 
 and the other less by one than the other addend of the un- 
 known, had 5 for its result, a generalization which once 
 mastered would be applicable to all groups, but whose 
 mastery is at least sufficiently difficult to have prevented 
 any attempt to utilize it by the disciples of Grube. Under 
 the Grube order each elementary sum has been taught as 
 a fact separate and distinct from other elementary sums, 
 except in so far as with each sum has been taught its 
 inversion, which practice with that of simultaneously 
 teaching the corresponding elementary differences, prod- 
 ucts and quotients, need not now be considered. As a 
 grouping is possible in which the mastery of one elemen- 
 tary sum in a group will insure the ready mastery of all, 
 the Grube order is obviously unpsychological. 
 
 It has been continually urged in its favor that it in- 
 sures the teaching of the "simpler" facts before that of the 
 "more complex." By it, it is claimed the children are not 
 confused by the introduction of the more difficult numbers, 
 before the easier ones are mastered. Unfortunately for 
 this argument, however, the great majority of children on 
 entering school, are already familiar with the names of all 
 the digits. In fact many a teacher of the Grube system 
 has found herself forced into the absurd position of wish- 
 ing that her children knew less. But granting that as 
 far as number is concerned, the mind of the beginner is a 
 tabla rasa, it by no means follows that by teaching the 
 sums as isolated facts, the Grube order presents first 
 those that are the "easiest" for the child. No fact has 
 
40 
 
 been better demonstrated by psychologists than that the 
 objectively simple may be subjectively complex, and the 
 objectively complex, the subjectively simple. It has not, 
 for example, been satisfactorily shown, that the associa- 
 tion 5 and 4 is more difficult for the child than is that of 
 2 and 3. Even aside from this, objectively considered, it 
 would be hard to express in mathematical terms the differ^ 
 ence in "simplicity" between 4 and 2 and 8 and i. And 
 yet, it is only in so far as through it the beginner may be 
 first taught those individual facts, which he can most 
 readily master, that the Grube order is in any respect an 
 improvement upon the most arbitrary order of teaching 
 the elementary sums which will include them all, especially 
 as it divides them into seventeen groups, where the Col- 
 burn groups them in nine. 
 
 3. Why the Grouping Together of the Sums Obtained 
 by the Successive Addition of Each Digit, First to Itself 
 Wid Then to Each Inferior Digit, is Psychological. 
 
 There remains to be considered the system in whose suc- 
 cessive groups the constant term is the digit added. While 
 it has never been used so generally as the two already dis- 
 cussed, it has occasionally found its champions, as, for 
 example, when in the early sixties, Felter and Eaton used 
 it in their elementary arithmetics. In it, the element com- 
 mon to the facts included in a given group is the number 
 to be added combined with the fact that the result is 
 always one greater than the corresponding result in the 
 group immediately below. Unlike that just considered, 
 this common element is highly effective. For example, 
 the pupil who is once led to see that 5 and 3 are 8 because 
 knowing that 5 + 2 = 7, he will know that 5 + 2+1=8, 
 will readily come to derive for himself the result of every 
 other elementary sum involving the addition of 3. 19 That 
 the element thus effective was readily perceivable by the 
 great majority of the pupils in the first school grade of 
 one representative locality, will be presently demonstrated. 
 
Meanwhile, being effective, as the common element in 
 the Grube group is not, being, if not more effective, at 
 least far more readily perceivable, than that of the Colburn 
 series, it is plain that the system of which it is the basis, 
 with its nine groups, is the psychological system. 
 
 The pupil who has once made the simple generaliza- 
 tion that one added to a number always results in the 
 number next above it in the scale, will readily derive for 
 himself with the aid of proper instruction which in 
 its primal sense means, after all, the right arrangement 
 of subject matter group by group, all of the elementary 
 sums. The thorough mastery of one group insures the 
 ready derivation of the next. Knowing all the elementary 
 sums obtainable by the addition of four, he will readily 
 derive and for the most part, quite abstractly, all of 
 those obtained by the addition of 4 and I or 5. 
 
 As this is the case with all the groups and the deriva- 
 tion of each group depends upon the thorough mastery of 
 that immediately preceding, it follows that the system 
 under discussion, being psychological throughout as com- 
 pared with either of the others, it is unnecessary to use 
 groups from more than one system. 
 
 (IV) THE DETERMINATION OF THE PSYCHOLOGICAL 
 ORDER OF THE GROUPS SELECTED AS A RESULT OF 
 THIS COMPARISON. 
 
 One group being dependent for its derivation upon the 
 thorough mastery of the group immediately below, the 
 psychological order of the groups within the system is 
 naturally determined : First, the elementary sums formed 
 by the addition of i ; second, those formed by the addition 
 of 2, etc., to those formed by the addition of 9. 
 
 (V) THE DETERMINATION OF THE PSYCHOLOGICAL 
 ORDER FOR THE FACTS WITHIN EACH GROUP. 
 
 The elementary sums are not only to be independently 
 derived by the pupils, but so thoroughly memorized that 
 the result will be instantly called to mind, when the 
 
42 
 
 addends are seen or heard. This ready recall can be con- 
 sequent only upon repetition, a repetition either abstract 
 and mechanical, or varied and concrete, of the isolated 
 fact or of the facts in relation to each other. For 
 example, 4 and 2 are 6, once derived, can be memorized 
 either by repeating concrete applications such as 4 cats 
 and 2 cats, or 4 cents and 2 cents, or by a mechanical repe- 
 tition of the abstract 4 + 2 = 6. Neither plan is wholly 
 satisfactory. The first is uneconomical, in that too much 
 time is taken in getting the necessary amount of repeti- 
 tion. Though properly used, it appeals to the child's 
 interest, and though applying the facts, it is a necessary 
 form of the number drill, it is not the form of repetition 
 which will result in the memorizing of the facts with a 
 minimum expenditure of time and effort. The second is 
 economical until the attention of the pupil is lost through 
 his lack of interest and the fatigue arising from the un- 
 varied repetition. It can be occasionally used, to fix in 
 the pupil's memory some individual fact, but is manifestly 
 unfitted for the regular drill upon the facts of a given 
 group. 
 
 Again the 4 and 2 are 6 can be memorized in an abstract 
 series where it is associated with other facts, as in the 
 "addition table." In the case of the tables as usually 
 arranged, the pupil mechanically commits to memory a 
 series of facts, which he may or may not have derived 
 for himself, in such a way that when a fact is demanded 
 out of its usual order, it often can be recalled only through 
 a running over of that part of the table immediately pre- 
 ceding it, to the end that a false and unnecessary associa- 
 tion may be utilized. Nevertheless the table has always been 
 popular with the pupils, and much of the disrepute into 
 which it has fallen among the "more advanced" teachers, 
 is due to their naturally but unnecessarily associating it 
 with the irrational instruction of which it so often formed 
 a part, and their erroneously assuming that the child finds 
 the monotonous repetition as unpleasant as does the adult. 
 
43 
 
 It is this very rhythmical, sing-song movement that gives 
 pleasure to the child; it is this combined with the joy of 
 achievement that makes him love to count. 21 If the 
 fundamental sums could be so arranged that the associa- 
 tion formed by repeating them in succession would be an 
 essential association of result with its addends, rather 
 than the false one of sum with sum, the repetition of these 
 facts in tables would be the psychological way of memo- 
 rizing them, because it would be the readiest way, and the 
 way most pleasant to the child. Now it is just such an 
 arrangement that results from the sub-grouping of tha 
 sums formed by the successive addition of a given digit, 
 into those in which it is successively added beginning 
 with itself, and those in which it is successively added be- 
 ginning with each inferior digit, that is, the arrangement 
 which is based upon counting by 2's from 2 and from I ; 
 by 3's, from 3, from i and from 2, etc. 20 Of course this 
 drill includes not only the elementary sums not inversions, 
 but all the fundamental addition facts. Two and 4, though 
 identical in its result with 4 and 2, and 14 and 2, though 
 but 4 and 2 and 10, must be as readily recalled as the 4 
 and 2 itself. 
 
 Knowing as the children do, that they are counting say 
 by 4's from 2, as they repeat "6, 10, 14, 18," while the 
 teacher interjects "and 4?" they are forming the associa- 
 tion 6 and 4 are 10, 10 and 4 are 14, etc., in such a way 
 that the series once mastered, all of its individual facts 
 are mastered. The drill is therefore economical. While 
 abstract it is varied, that is, it is not the mere repeti- 
 tion of one fact again and again. It is interesting to the 
 children from their love of counting, due not alone to its 
 pleasurable rhythm, but as they master successive series, 
 to the joy of achievement as well the satisfaction arising 
 from the knowledge that they can count by 3's and 4's 
 instead of merely by I's or by 2's. 21 Where a digit is 
 successively added to itself, the way is opened for the 
 ready derivation of the fundamental products and quo- 
 
44 
 
 tients. 22 Finally, the only numerical knowledge which 
 the great majority of children have in common on enter- 
 ing school, the ability to more or less mechanically count 
 by i's, within narrow limits, is utilised from the very 
 start. It is with counting by I's that the work begins. 
 
 It is of course admitted that the counting is "mechani- 
 cal memory work" but what of that, if, though abstract, 
 it is interesting to the child, and, through the very fact 
 that it is mechanical the readiest way of fixing firmly in 
 his mind facts that he has already intelligently derived. 
 
 This sub-grouping, then, of the psychological groups 
 is psychological, since the facts are so arranged that as 
 each group is derived, it may be memorized by the child 
 with a maximum of readiness and interest. 
 
 So far, it is only those of the fundamental sums which 
 are less than 24, that have been psychologically grouped, 
 There remains a number of groups no less fundamental, 
 whose psychological order must be determined. The 
 sums obtained by the successive addition of lo's from 10; 
 the sums obtained by the addition of the various digits to 
 the multiples of 10 thus derived; the sums obtained by the 
 additions of i to every number from 24 to 99, necessary 
 to intelligent counting to 100, while so related to each 
 other that they must be taught in the order in which they 
 were just named, should have their place in the general 
 psychological order as certainly established, as that of 
 the sums already discussed. To do this it is necessary to 
 assume, what later will be convincingly proven, that it is 
 as easy for the majority of children to add ID'S as to add 
 apples, and far easier to add a digit to 30 or 40, or i to 25 
 or 48, than to add 2 to 6 or 9. This being the case, the 
 psychological order for these groups of fundamental 
 sums is immediately after the i sums from 3 and i to 9 
 and i, and on account of the difficult terms n and 12 
 before the i sums from 10 and i to 19 and i. Thus 
 the decimal concept is formed from the start, and a large 
 number of fundamental facts mastered through the ready 
 
45 
 
 mastery of a common element necessary to the compre- 
 hension of the decimal system. 
 
 (VI) THE GENERAL PSYCHOLOGICAL ORDER OF TEACH- 
 ING THE FUNDAMENTAL SUMS, AS THUS DETER- 
 MINED ON A PRIORI GROUNDS. 
 
 1. Intelligent counting by I's to 10; the "one sums" 
 and their inversions. 
 
 2. Intelligent counting by lo's to 90; the "ten mul- 
 tiples" and the "ten sums." Place value to lo's. 
 
 3. Intelligent counting both by the addition of the 
 successive digits, to the 10 multiples and by i's (i) 
 from 20 to 100; (2) from 10 to 20. 
 
 4. Intelligent counting first to 12 and then to 24, by 
 2's, 3's, 4's, 5's, etc., successively (i) from 2, 3, 4, 5, 
 etc., respectively; e. g. "4, 8, 12," or "2, 4, 6, 8, 10, 12." 
 (2) Successively from all other digits less than the one by 
 which the counting is being done; e. g. "i, 5, 9, 13; 
 2, 6, 10; 3, 7, ii." All the elementary sums formed by 
 such successive addition of each digit, being with their 
 inversions, thoroughly mastered, before the beginning of 
 the drill on counting by the digit next succeeding. 
 
 (VII) THE DETERMINATION OF THE PSYCHOLOGICAL 
 ORDER OF TEACHING THE FUNDAMENTAL DIFFER- 
 ENCES. 
 
 In determining the psychological order of teaching the 
 fundamental differences, it is necessary to consider not 
 only the system of grouping in which they can be most 
 readily mastered by the individuals taught, as a series 
 distinct from all other facts, but also to fix the relation 
 which their teaching should hold, to that of the funda- 
 mental sums it being assumed for the present that the 
 teaching of the fundamental products and quotients 
 should follow after, except those whose derivation is inci- 
 dental to elementary addition. 
 
4 6 
 
 Should the fundamental differences be taught at all? 
 If so, should they be taught contemporaneously with the 
 fundamental sums, or after the fundamental sums have 
 been thoroughly mastered? Whether they should be 
 taught as distinct from the elementary sums, or in other 
 words, whether subtraction should be taught other than 
 as a particular form of addition, is a question which has 
 not as yet been experimentally settled. A priori, there 
 is little doubt that a difference can be just as accurately 
 and as quickly obtained by considering its units, tens, etc., 
 as the numbers to be added to the corresponding digits 
 of the subtrahend in order that they shall equal those of 
 the minuend, as by taking them to be the direct result of 
 the successive subtraction of the corresponding digits of 
 the subtrahend from those of the minuend. But while as 
 a mechanical operation, the former may be as satisfactory 
 as the latter, it is more difficult of application. For 
 example, an individual who wishes to determine the re^ 
 mainder which will result from taking 9 from 17, must 
 know that it is the number which added to 9 will give 17. 
 This he can learn either unintelligently or intelligently ; he 
 can either blindly accept the fact that the difference is the 
 number which added to the subtrahend will give the minu-< 
 end, or accept it because he has made the generalization 
 that the minuend being the sum of the subtrahend and 
 some other number, if the subtrahend is taken away the 
 other number must remain. As intelligent operation is 
 here assumed, this generalization becomes a necessary 
 condition to subtraction "by addition." But an individual 
 who has made it can derive at will all of the fundamental 
 differences. The customary drill in the rapid subtrac- 
 tion of small numbers, would therefore be coincident with 
 drill upon the fundamental differences. In view of this 
 fact, it is probably better to consider the making of the 
 generalization, where it can be made, as the psychological 
 method of deriving them, rather than as a necessary step 
 in a mode of subtraction, which with them once mastered, 
 presents no advantages over that in general use. 
 
47 
 
 If the fundamental differences are to be taught, 
 another question arises shall they be taught simultane- 
 ously with the fundamental sums, or after the funda- 
 mental sums in whole or in part have been thoroughly 
 mastered. Against the former alternative is the fact, 
 that at least where number is taught to children on their 
 first entering school, many are so immature in their 
 mental development, as to be sadly confused by the intro- 
 duction of the "less facts" while they are striving to 
 master the sums. In favor of it is the practice existing 
 in many schools of teaching sums and differences, inver- 
 sions and alternations, together the only psychological 
 grouping possible under the Grube system. For example, 
 with 5 and i and i and 5 are taught 6 i and 6 5, all 
 four of the facts having the 5 and the i variously associ- 
 ated with the 6. But this is, after all, merely the objective 
 derivation of each fundamental difference, aided by the 
 generalization already discussed. It is manifestly far 
 more psychological to thus derive no more of the differ- 
 ences than are necessary to the making of the generaliza- 
 tion, by which, once made, all can be derived without the 
 aid of objects. The postponement of this derivation until 
 after the first groups of fundamental sums are thoroughly 
 mastered, will give abundant opportunity for all who are 
 capable of making the generalization to be led to make it, 
 and at the same time will allow those not capable of 
 making it, and therefore in the stage of mental 
 development where confusion of sum and difference will 
 be most puzzling, to devote their undivided attention 
 to the mastery of the fundamental sums. If after those 
 failing to make the generalization had thus mastered the 
 sums, they would be compelled to objectively derive the 
 individual differences with no aid from the grouping, 
 especially, if the number thus failing was, as it proved 
 to be, a considerable fraction of the whole, the teaching 
 of the fundamental differences after the sums, might not 
 be psychological. But aside from the possibility of hav- 
 
4 8 
 
 ing such pupils wait for the study of subtraction until they 
 are more mature and are able with their "brighter" com- 
 panions, to derive the fundamental differences at will, or 
 to dispense with them altogether, the differences, like 
 the sums, can be taught in psychological order, and, by the 
 same simple mental process, one group after another 
 readily derived. 
 
 Where pupils, once having mastered the fundamental 
 sums, can be readily led to make and to apply the gen- 
 eralization, that whenever one of two numbers is sub- 
 tracted from their sum, the other remains, so far as the 
 mere derivation of the fundamental differences is con- 
 cerned, the order may be purely arbitrary. But it is 
 necessary, not only that a difference shall be derived but 
 that its minuend and subtrahend shall be so firmly associ- 
 ated with it that is, with the "difference," in the nar- 
 rower sense, that perception of the former shall instan- 
 taneously call up the latter. Here, as in the case of the 
 fundamental sums, counting seems to furnish the most 
 psychological form of the repetition without which this 
 firm association would be impossible. 
 
 Where pupils who have mastered the fundamental 
 sums are unable thus to derive the fundamental differ- 
 ences, the psychological order of teaching the latter can 
 be shown by precisely the same line of reasoning, to be 
 essentially the same though in inverse order, as that in 
 which the former should be taught. The pupils will first 
 learn to count backwards by I's say from 24. This 
 will, with proper instruction, give them the fundamental 
 differences, resulting from the subtraction of i. Those 
 resulting from the subtraction of 2 will then be obtained, 
 by the successive subtraction of I, thus 18 i i or 
 1 8 2 ; those resulting from the subtraction of 3, from 
 the subtraction of i from the corresponding differences of 
 the "2 group," etc. 
 
 The sub-grouping, as before, will be such as to make 
 possible counting backwards by 2's from 20 and from 19, 
 
49 
 
 by 3's from 21, 20, and 19, etc., the rhythmical repe- 
 tition resulting in the readiest memorizing of the funda- 
 mental differences, however they may have been derived. 
 As when a series has been once thoroughly mastered, it is 
 far easier to learn to repeat it backwards, than it was to 
 originally master it, the fundamental differences will 
 be more readily mastered than were the fundamental 
 sums, and at the same time, the fundamental sums will 
 be more thoroughly mastered than before. Hence the 
 grouping together of the differences obtained by the suc- 
 cessive subtraction of each digit first from its multiple 
 which is next above 20 and then from the corresponding 
 number for each inferior digit, is psychological: 
 
 1. For those pupils who failing to make the generali- 
 zation find in the mastery of the effective element com- 
 mon to successive groups the readiest method of deriving 
 the fundamental differences. 
 
 2. For all pupils who through love of rhythm find 
 counting the form of repetition by which they can most 
 readily memorize the differences, whatever the method 
 by which they have been derived. 
 
 And this grouping is psychological, whether the funda- 
 mental differences are taught simultaneously with the 
 fundamental sums, or after the fundamental sums have 
 been thoroughly mastered. 24 
 
 Granting then, that a priori the psychological order of 
 teaching the fundamental sums and differences has been 
 satisfactorily determined, it yet remains to be demon- 
 strated that these a priori determinations are correct. In 
 the schools which had among their pupils the hundred 
 children whose numerical knowledge was tested on their 
 first entering the lowest school grade this demonstration 
 has been attempted, 23 and the results of six months' work 
 subjected to a rigid and impartial test. While only a 
 wide adoption of the order recommended, in schools in 
 which the conditions greatly vary, will make possible a 
 series of general tests, which will be universally accepted 
 
50 
 
 as demonstrative, the results so far attained at least 
 serve to indicate that such adoption would be justifiable, 
 and that such experimental demonstration is not unlikely 
 to prove the determinations to be indisputably correct. 
 
 Since so far at least as economical derivation is con- 
 cerned, the results of teaching in psychological order are 
 dependent upon the employment of psychological method, 
 the report of the test will be preceded by a discussion of 
 method, whose determinations like those concerning 
 psychological order, the results of the test will tend to 
 confirm. 
 
CHAPTER III. THE A PRIORI DETERMINA- 
 TION OF THE PSYCHOLOGICAL METHOD 
 OF TEACHING THE FUNDAMENTAL 
 SUMS AND DIFFERENCES. 
 
 I. A DEFINITION OF PSYCHOLOGICAL METHOD. 
 
 That method of teaching a particular branch of study 
 is psychological, which while insuring the readiest 
 mastery of its subject-matter by the individual taught, 
 gives a maximum of mental training along the lines of 
 development for the furtherance of which the mastery of 
 that subject-matter is peculiarly adapted. That no one 
 method will be psychological for all individuals must of 
 course be admitted. Since, however, most individuals 
 are taught en masse, the method which in the case of the 
 great majority meets the conditions just specified, is the 
 psychological method for the school. In the absence of 
 any satisfactory demonstration of the common assump- 
 tion that all special mind training is general mind training, 
 it is safer to judge what training should result from the 
 study of a particular branch, both in the light of that 
 assumption, and in that of its negation. Every subject 
 involves in its mastery the predominance of some particu- 
 lar mental activity. All other activities of the mind are 
 more or less involved in it as they are more or less in- 
 volved in every mental state, but one predominates in the 
 study as one predominates in any phenomenon of mind. 
 Thus imagination is the ruling activity in the study of 
 descriptive history or geography, and representation in the 
 use of the copy-book. If all special training is general 
 training, it follows that so far as the school curriculum is 
 concerned, the general training say of the imagination 
 should be given through those branches in whose mastery 
 
52 
 
 it is the predominant activity. Otherwise much time will 
 be wasted in overemphasizing the importance of some 
 mental activity, which is only incidentally involved in the 
 study of a given branch. If all special training is not 
 general training, it follows that the training of the imag- 
 ination should be limited so far as it is not purely inci- 
 dental, to those fields of activity in which it will be 
 naturally exercised, or to those which most closely 
 resemble them. In either case the method of teaching 
 a particular branch should be such as to train the mental 
 activity necessary to its mastery. Other activities should 
 be incidentally trained, but the teaching, in addition to 
 mastery of subject-matter, should involve training con- 
 centrated upon the activity which predominates in the 
 science or art which is being taught. Pupils who are 
 mastering descriptive history, should reason about the 
 facts with which they deal, but as the main object of their 
 study is to make those facts real, the method of instruc- 
 tion should be planned to train the imagination, rather 
 than the reason. In other words the intensive study of 
 history is premature wherever it precedes or is taking the 
 place of the realistic presentation of important events. 
 
 II. CONDITIONS NECESSARY TO A PSYCHOLOGICAL 
 
 METHOD OF TEACHING THE FUNDAMENTAL 
 
 FACTS OF NUMBER. 
 
 The important place that mathematics occupies in the 
 curriculum has always been justified, and must still be 
 justified, if it is to be justified at all, by the fact that of all 
 the so-called mental disciplines, it more directly and more 
 fully than any other, involves the pure reasoning activity 
 of the human mind. If the special training in pure 
 reason, popularly supposed to be involved in its study, is 
 a general training in pure reason, it is the duty of peda- 
 gogy to make good the popular belief. If not, only such 
 mathematics should be taught as can be permanently 
 
53 
 
 mastered, and in such a way that the pupils will be thor- 
 oughly trained in such mathematical reasoning as they 
 are likely to exercise in everyday life. 
 
 In either case, such training as is given should be 
 mainly a training of reason ; in neither should it be mainly 
 a training of representation or imagination. The im- 
 agination can be better and more economically trained in 
 some study in which it is the dominant mental activity. 
 The visualizing of lines and angles, and plane surfaces 
 in varying perspective all the trigonometrical training 
 recommended by Herbart in his "A. B. C. of Sense Per- 
 ception/' may or may not be a necessary part of the train- 
 ing of the child's imagination. It certainly, however, is 
 no necessary factor in the study of number, even though it 
 be a preparation for the visualizing of the forms of 
 geometry. Much less necessary is special training in the 
 visualizing of complex columns and rows of figures. In 
 the teaching of mathematics, including the teaching of 
 the fundamental facts of number, psychological method is 
 that which while insuring the readiest mastery of the facts 
 by the individual, will give a maximum of training to the 
 reasoning activity of that individual's mind. It should 
 assuredly carry with it training of other mental powers, 
 but just as assuredly such training should be incidental to 
 the training of reason. 
 
 This discussion must therefore submit method to the 
 following tests : 
 
 1. Is a given method better adapted than any other to 
 insure the readiest derivation by the great majority of 
 the pupils, of the various groups of fundamental facts ? 
 
 2. Is it better adapted than any other to insure the 
 readiest memorizing by the great majority of the pupils 
 of the individual facts when once derived ? 
 
 3. Does it involve a maximum of mental training, 
 chiefly through the exercise of the reasoning activity? 
 
 The method which stands these tests can be safely char- 
 acterized as psychological. 
 
54 
 
 III. SPECIFICATION OF THE GENERAL METHODS OF 
 TEACHING ELEMENTARY NUMBER. 
 
 However ambiguous the term "method" may be, used 
 now by Comenius to apply to his general analogy between 
 nature and instruction, or by the pedagogic inventor of 
 to-day to designate some useful but restricted device, no 
 one will deny that there are certain general methods of 
 instruction call them what you will based upon the 
 various ways in which the human mind acquires knowl- 
 edge. The subject-matter of arithmetic may be acquired 
 inductively or deductively, synthetically or analytically, 
 concretely or abstractly. It has been shown that to teach 
 the number-facts in psychological order is for the most 
 part to teach inductively, while it is evident that to so 
 teach the fundamental sums is to teach synthetically. 
 That this teaching is here psychological, no one will at- 
 tempt to disprove who remembers that it has for its object 
 the firm association in the mind of certain number-facts, 
 leaving each philosopher to his own devices for filling 
 the child mind with his number "concept," though being 
 based none the less upon the fact that to the child number 
 is an expression of aggregation, a concept with which 
 philosophers are beginning to agree. While it is per- 
 haps equally evident that the psychological order of 
 teaching the sums and differences involves a method 
 which is largely abstract, the existing prejudice in favor 
 of the objective method, the general assumption that the 
 concrete basis admittedly necessary to the comprehension 
 of the abstract, involves the objective teaching of each in- 
 dividual fact of number, makes highly desirable a thor- 
 ough exposition of all that which goes to prove the 
 abstract method psychological. 
 
 With this end in view, the various "methods" of deriv- 
 ing and memorizing the fundamental facts, may all be 
 included in the following classification : 
 
55 
 
 1. The Objective Method, in which each individual 
 fact is derived by the direct use of objects. 
 
 2. The Memorial Method in which so far as possible 
 each individual fact is derived by the summoning up of the 
 memorial images of objects. 
 
 3. The Abstract Method by which so far as possible the 
 facts of each group are derived by abstract reason from 
 those of the group immediately preceding. 
 
 IV. A COMPARISON OF CERTAIN OF THESE METHODS 
 
 WITH A VIEW TO THE DETERMINATION OF 
 
 PSYCHOLOGICAL METHOD. 
 
 (I) THE PESTALOZZIAN OR GRUBE METHOD, IN WHICH 
 EACH INDIVIDUAL FACT is DERIVED WITH THE 
 DIRECT AID OF OBJECTS. 
 
 i. Does the Grube Method Insure the Readiest Deriva- 
 tion of the Fundamental Facts of Number? 
 
 However necessary the use of objects to the formation 
 of the number "concept," however essential that such use 
 should result in the discrimination, abstraction and group- 
 ing which has number for its product, it by no means fol- 
 lows that each fact of number should be objectively 
 taught. 25 In order that an individual who has formed 
 adequate concepts of the numbers involved, shall fully 
 comprehend a fundamental sum or difference, it is only 
 necessary that he shall comprehend the simple and obvious 
 fact of addition or subtraction, and perceive for himself 
 the invariable result. The objective derivation of indi- 
 vidual facts is justifiable, if in order to perceive that the 
 result is invariable, in order to intelligently accept it as 
 an abstract fact, it must first be perceived to be true in 
 various concrete forms, and then the generalization made 
 that being true of these, it is true of all, that 3 cubes 
 and 2 cubes being 5 cubes, 3 splints and 2 splints, 5 splints, 
 
56 
 
 etc., 3 and 2 must invariably be 5. It is a serious mistake 
 however, to assume that it is necessary to repeat this 
 mental process in deriving every fundamental sum or 
 difference. Sooner or later, the generalization is made 
 that every fact of number, however derived, is universal 
 in its application, that if 2 and i are invariably 3, and 
 3 and 2 invariably 5, 3 and 4, found to be 7 in one in- 
 stance, whether by a derivation that is concrete or one that 
 is abstract, will be known to be 7, no matter what the 
 concrete form of the addition may be. No one would 
 dream of teaching 36 and 2 objectively, in order to lead 
 pupils to perceive by the repeated addition of various 
 objects that whatever things may be added, 36 and 2 will 
 always equal 38. They already know that if through 
 abstract addition, 36 and 2 be found to equal 38, the result 
 will be invariable, whether the objects concerned are 
 splints or trilobites. It remains for advocates of concrete 
 number derivation to show that this is not equally true 
 of 6 and 2. Generalization requires but few particulars. 
 After the child has found that 2 and i 3 with one set of 
 objects, are still 3 though the objects be varied, and 
 being 3 though the objects be varied, are therefore 3 
 whatever the objects may be, he will not require 
 many repetitions of this process of abstraction and gen- 
 eralization in the derivation of other facts, before he will 
 assume that each fact which he derives will likewise apply 
 to all things. In cases where the abstract fact seems to 
 be derived by the summoning up of one that is concrete, 
 as, for example, where a pupil failing to give the ab- 
 stract 3 and 5 can be led to recall the fact that 3 marbles 
 and 5 marbles are 8 marbles, would seem to indicate that 
 abstraction and generalization had not yet taken place, if 
 it were not that the recall is often followed by the im- 
 mediate generalization 3 and 5 are 8. It is not that 
 3 and 5 had not been generalized, but that not having 
 been thoroughly memorized, the familiar concrete associa- 
 tion helped in its recall. 
 
57 
 
 As then, the use of objects is not necessary to intelli- 
 gent generalization in the case of each fundamental fact, 
 it having already been clearly shown in the discussion of 
 psychological order, that the fundamental sums and dif- 
 ferences can be more readily derived by a simple pro- 
 cess of abstract reason, it necessarily follows that 
 objects should be used only in general drill in num- 
 ber perception and in the occasional verification of facts al- 
 ready derived, except in so far as their use is necessary 
 to the mastery of the effective element common to each 
 group. 26 The Grube method, therefore, objectively de- 
 riving each individual fact, would seem at least as regards 
 derivation, to be uneconomical and unpsychological. 
 
 2. Does it Insure the Readiest Memorising of the Facts 
 When Once Derived? 
 
 When a number-fact has been once derived, it must be 
 memorized by repetition, now as in the days of Richter, 
 and as it must ever be, "the mother not only of study but 
 also of education." Just so far as the Grube method 
 limits its memory drill to repetition with actual objects or 
 of concrete facts, it is unpsychological. There is a ten- 
 dency among the friends of the "new education" to con- 
 fuse the mechanical with the irrational. Word for word 
 memorizing of the subject-matter of education is mechani- 
 cal, but it is irrational not on account of its being thus 
 mechanical, but first, because the words of few text-books 
 are worth memorizing, and second, because thoughts are 
 more certain of recall through their manifold associa- 
 tion with other thoughts. If words are to be thoroughly 
 memorized, no matter how full the apperception of the 
 thoughts which they express, the necessary repetition 
 should be "mechanical." So with the abstract fact of 
 number. Five is to be permanently associated with 3 
 and 2, an association of symbols and not of thoughts. 
 This association is assisted by the common association of 
 
58 
 
 the 5 and the 3 and 2 with cubes, splints and geometrical 
 forms, only through the repetition involved in the process. 
 At least no individual is likely to recall a number- fact 
 because of its association with the same series of objects 
 with which all other number- facts have been associated. 
 But such repetition is plainly uneconomical when com- 
 pared with the mechanical repetition of the abstract fact, 
 unless, to be sure, the former is, as is assumed, more in- 
 teresting to the child, and so holds his attention as the 
 latter can not. It is, however, at least very questionable, 
 whether this assumption of interest will stand the test of 
 impartial investigation. The use of a great variety of 
 attractive objects interests the child, but in the objects, 
 not in the fact which they are intended to exemplify, 
 while the use of the same set of objects again and again, 
 no matter how finely polished the cubes or how brightly 
 colored the splints, is no stimulus to interest, especially 
 as in the end it must come to be associated with failure 
 to recall and sometimes with more or less merited rebuke. 
 Such interest as is manifested in concrete repetition may 
 after all be traceable to a love of counting combined with 
 the joy of mastering a new fact or recalling one that was 
 forgotten, supplemented in many cases by the sympathy, 
 enthusiasm and dramatic ability of a Pestalozzian 
 teacher, rather than by the inert concreteness of Festaloz- 
 zian matter. 
 
 There is possible, however, another form of concrete 
 repetition in which a given number-fact is firmly associ- 
 ated with a particular concrete thing, which often seen, 
 will repeatedly recall the fact to mind. 2 + 2 are 4 may 
 thus be associated with the legs of a chair, or 2 and i are 3 
 with the vertices of a triangle. While granting that the 
 numerical association would often arise when the objects 
 were seen, and that this incidental and natural repetition 
 would firmly fix the facts in mind, it would be mani- 
 festly impossible to select a sufficient number of familiar 
 objects adapted to this system of repetition. 
 
59 
 
 A priori, then it would seem that any form of abstract 
 repetition of a number-fact once it has been intelligently 
 derived, so limited in time as not to become wearisome, 
 will result in a readier memorizing than will repetition 
 in the concrete. Especially will this be the case, where 
 the repetition can be made rhythmical and, so, pleasing to 
 the child. 
 
 The Grube method therefore can not be psychological 
 in so far as it involves repeated objective illustration in 
 the memorizing of fundamental facts. Its use of 
 memorial images or of concrete associations to the same 
 end, will be considered later on in this discussion. 
 
 3. Does it Involve a Maximum of Mental Training, 
 ChieHy Through the Exercise of the Reasoning Activity? 
 
 The most enthusiastic advocates of objective operation 
 are hardly likely to urge it as a means to mental training. 
 They rather assume it as a necessary condition. The 
 discrimination into separate wholes of objects so similar 
 that they can be perceived as a united whole, however 
 necessary to "number perception/' is not a mode of train- 
 ing necessary to supplement the child's experience. If 
 the similarity were to be sought out, it would be valuable 
 training of an essential factor of the reasoning process, 
 but this would be merely adding difficulty to the formation 
 of the "number concept." This analysis of wholes into 
 component wholes easily apparent, far from training 
 sense perception, serves but to strengthen the every-day 
 experience, which discriminates wholes without perceiv- 
 ing them in detail. But even were training in sense per- 
 ception and in the discovery of concrete similarities in- 
 volved in the teaching of the fundamental sums and differ- 
 ences, it should be purely incidental to the training of 
 abstract reason. Only in systematic nature-study and 
 the experimental study of the natural sciences should 
 such training become a dominant factor. 
 
6o 
 
 The only remaining form of mental training- involved 
 in the objective method is exhaustive drill in generaliza- 
 tion. It has already been demonstrated that continual 
 repetition of this process is unnecessary in so far as it is 
 intended to lead the child to comprehend that a fact true 
 for one collection of objects will hold for all. Generali- 
 zation, and, too often, generalization from insufficient 
 particulars, is a natural activity of the human mind. 3 
 and 2 5 in the one case actually tested will be 5 in all, 
 unless it is proven to be 6. It is only so far as. the ob- 
 jective method acts as a check upon this dangerous ten- 
 dency that it can be useful as training in generalization. 
 As the fact that 3 and 2 will always prove to be 5 only 
 serves to confirm the usual assumption, training which 
 looks toward a surer judgment should be reserved for 
 studies which demonstrate that one Emerald Island does 
 not make all islands green, or that a broken rule does not 
 infallibly indicate a malicious child. 
 
 Except in so far as all mental activities are involved 
 in each, this is the only mental training involved in the 
 Grube method. It not only fails to train the abstract 
 reason, but involves the very minimum of mental training 
 which can result in the mastery of the fundamental sums 
 and differences. 
 
 (II) THE MEMORIAL METHOD, IN WHICH, so FAR AS 
 POSSIBLE, EACH INDIVIDUAL FACT is DERIVED AND 
 MEMORIZED BY THE SUMMONING UP OF MEMORIAL 
 IMAGES OR OF CONCRETE ASSOCIATIONS. 
 
 i. Does the Memorial Method Insure the Readiest 
 Derivation of the Fundamental Number-Facts? 
 
 All that has been urged against the excessive use of 
 objects in teaching the fundamental facts of number 
 can be urged with more than equal force against such 
 teaching as is dependent upon the summoning up of 
 
6i 
 
 memorial images. Indeed no one would seriously advo- 
 cate the deriving of 7 as 3 + 4 through the recall of the 
 visual images of 3 objects and 4 objects and their com- 
 bination into 7. The memorial method, however, has 
 been very generally used in the memorizing of facts when 
 once they have been objectively derived, or objectively ap- 
 plied, whether as the result of instruction or of numerical 
 experience outside of the school. 
 
 2. Does it Insure the Readiest Memorizing of the 
 Facts When Once Derived? 
 
 It has already been shown that the very general use of 
 "number stories" however valuable as a drill in lan- 
 guage, is unnecessary as a drill in the application of a 
 generalized fact. If the child knows that 4 and 2 are 6, 
 he knows that 4 oranges and 2 oranges are 6 oranges 
 without ever having applied that particular abstract sum 
 to the limited stock of objects which he has come to use 
 in his number stories. Unnecessary so far as the sums 
 and differences are concerned, as a means to application, 
 the story even more than the use of actual objects, is 
 uneconomical as a mode of repetition. If the child really 
 visualizes as numbered groups the objects that he names 
 which will presently be disputed, all that has been urged 
 against objective repetition, is fully applicable, with the 
 additional argument that the visualizing is unnecessary. 
 If he merely visualizes unnumbered objects or associates 
 the unfamiliar fact with the name of some familiar con- 
 crete thing, his repetition is plainly uneconomical. The 
 argument that he will be more interested in the concrete 
 has been already met. That the association of the un- 
 familiar with the familiar will result in its recall and so 
 in its frequent repetition has also been shown to be 
 improbable. It only remains to be demonstrated that 
 even though it is assumed that the child visualizes the 
 objects used as addends or as subtrahend and difference, 
 
62 
 
 he can not visualize sum or minuend as a numbered 
 whole. 
 
 It is a generally accepted fact that but five or six objects 
 can be simultaneously perceived, if indeed it has been 
 satisfactorily demonstrated that even this number can be 
 perceived simultaneously. It is possible through training 
 to perceive more than five or six objects instantaneously, 
 but in all cases where this has been successfully accom- 
 plished, it seems to be as the result not only of a well- 
 trained observation, but of an estimation so ready as in 
 many cases to be quite unconscious. That is, the percep- 
 tion is in all probability successive and not simultaneous. 
 Where the objects to be perceived have been so placed as 
 to prevent so far as possible their separation into more 
 or less definite groups, patient and persistent training has 
 failed to increase the number of objects "simultaneously" 
 perceivable. 27 Hence it is obvious that the individual, 
 looking at the group of objects which for the first time he 
 has formed by the addition of say 8 to 9, can have no 
 numerical concept directly arising from his perception 
 of the group as a whole. It is not until he perceives it as 
 four 4's and i, three 5's and 2, or some other combination 
 with which he is familiar, or until he has counted the 
 objects composing it, that he knows it to be 17. That is, 
 he infers its equality with the group resulting from the 
 addition of two other groups, by perceiving that it may 
 be exactly separated into those groups, or by counting 
 proves it to be equal to 16 and i. The only visual image, 
 then, that can arise in his mind in response to the oral or 
 visual "17," is not that of one group containing seventeen 
 objects, but of a group composed of at least three other 
 groups as 6 and 6 and 5. 
 
 While it has yet to be determined to what percentage 
 of those versed in elementary number such a visual image 
 does actually present itself, it is evident that in the case 
 of larger numbers, the increasing complexity of an image 
 no group of which can be greater than 6, precludes 
 
63 
 
 its presentation when the number which it would visually 
 represent is seen or named. Even, for example, so small 
 a number as 57 could not be visually represented by fewer 
 than nine groups of 6 each and one of 3. It is perhaps 
 safe to say that under ordinary conditions, in the case of 
 the great majority of individuals, no visual, tactile, audi- 
 tory, or kinsesthetic image of a group of objects or sensa- 
 tions, rises in response to a number-name. And it may 
 be asserted with still more certainty that no such repro- 
 duction is necessary to the number concept. 28 
 
 There remains for consideration that form of the me- 
 morial method in which the concrete association is with 
 the written number symbol. Figures are objects. Should 
 the abstract fact after it has been intelligently derived 
 be memorized through the repetition of its written form ? 
 So far, the discussion has concerned itself with the oral 
 memorizing of the abstract fact, the desirability for which 
 no one is likely to deny. But much of the addition and 
 subtraction which the majority of individuals are called 
 upon to perform is not oral but written. More than this, 
 there are those who favor such drill in the visualizing of 
 the written process, that long and difficult problems can 
 be solved mentally. While the memorizing of the oral 
 fact is all that is necessary to written addition or written 
 subtraction, while either operation may be made a purely 
 oral one by the oral repetition of the numbers written, 
 it is obvious that operation will be more rapid where a 
 visual or a semi-visual association has been formed. In 
 written subtraction the sight of the digits to be subtracted 
 should at once call up the oral or the visual differences, 
 without the intermediation of the oral digit names, a 
 fact that is equally true of addition not in column. To 
 this end, then, concrete association is necessary. The 
 pupil should be drilled upon saying or writing the result, 
 when the numbers operated upon are written one above 
 the other, it mattering not whether the necessary kin- 
 sesthetic chain is recalled by its visual or its auditory 
 
6 4 
 
 image. The algebraic form, i + 3 = 4 O r 5 2 = 3, 
 so common in the written work of the school is no help 
 to this written association. It is only justifiable in so far 
 as it is desirable to familiarize the pupils with algebraic 
 expression, and has in all probability resulted not from 
 such justification, but as a survival of the old addition and 
 subtraction tables scattered by the Grube method into indi- 
 vidual facts. 
 
 A different form of concrete association the "semi- 
 visual" is necessary to the highest degree of facility in 
 addition in column. After the addition of the two digits 
 at the top of the column, the numbers to be added are not 
 seen together, nor are they successively pronounced. The 
 addition is not of two numbers written one above the 
 other, but of a spoken number or its auditory image, with 
 a number written in column, each oral result or its 
 auditory image being added to the number next above or 
 below the last number added. It is evident that to insure 
 rapid addition in column, drill on neither the oral nor the 
 written abstract facts alone, is all that is needed. The 
 drill should consist of the addition of a written digit 
 to one that has been spoken. 
 
 The memorial method, then, has to be utilized in the 
 drill preliminary to rapid operation. 
 
 3. Does it Involve a Maximum of Mental Training 
 Chiefly Through the Exercise of the Reasoning Activity? 
 
 It is at once apparent that teaching of number by the 
 memorial method involves more mental training than does 
 that which is limited to the objective method alone. Here 
 the power of the mind to recall the concrete, especially the 
 visualizing power, is more or less exercised. But 
 whether or not such training is general training, but little 
 of it is necessary in the teaching of number, while it can 
 be made far more effective in the study of drawing, geo- 
 graphy or history in which it is an essential factor. More 
 
65 
 
 than likely the power to visualize unnumbered objects and 
 geometrical forms is all the visualizing power desirable 
 in mathematics. Whether this power is developed by the 
 memorial method is to say the least very doubtful. The 
 representation which it usually involves is in all likeli- 
 hood, that of vague and indistinct wholes, only resulting 
 in images clear and distinct in all their parts when visual- 
 izing is made a main end of the number study, as in the 
 case of the mental solution of lengthy problems. It is 
 very questionable whether this power to visualize a com- 
 plex numerical operation is a useful accomplishment. 
 Occasions are rare when ready written operation will not 
 fully serve the same purpose. It is necessary to discrimi- 
 nate sharply between fundamental number fully compre- 
 hended, and other distinct ends with which it has been 
 confounded. Even correlation does not demand that 
 the economical teaching of one branch shall be sacrificed, 
 and the main end of the consequent training minified, in 
 order that something else more or less related to its sub- 
 ject-matter or more or less involved in the training shall 
 be taught with it. 
 
 (Ill) THE ABSTRACT METHOD, IN WHICH, so FAR AS 
 POSSIBLE, EACH OF THE FACTS IN EACH GROUP 
 ARE DERIVED BY ABSTRACT REASON FROM THOSE 
 OF THE GROUP IMMEDIATELY PRECEDING, AND 
 LARGELY MEMORIZED BY ABSTRACT COUNTING. 
 
 i. Conclusion that the Abstract Method Insures the 
 Readiest Derivation of the Fundamental Number-Facts. 
 
 As it is only through the use of this method that the 
 psychological order of teaching number can be fully util- 
 ized in the derivation of the fundamental sums and dif- 
 ferences, it is plainly the psychological method, wherever 
 it is adapted to the majority of the children taught. 
 Objects are used only in so far as they are necessary to the 
 
66 
 
 comprehension of the effective element common to each 
 group, to the occasional verification of a fact abstractly 
 derived, and to general drill in number perception. The 
 intelligent derivation of one fact in a group insures the 
 ready derivation of all. Indeed with many pupils, the 
 intelligent derivation of the facts in the first two or three 
 groups insures the ready derivation of all the remaining 
 facts. By no other method can the fundamental facts 
 be so readily derived. 
 
 2. Conclusion That It Insures the Readiest Memoriz- 
 ing of the Facts When Once Derived. 
 
 That the memorizing of the fundamental sums and dif- 
 ferences by forward and backward counting by 2's, 3's, 
 4's, etc., is economical, has been plainly demonstrated. 
 It will become unpsychological only, if as the series 
 multiply, the child who enjoys counting by I's and by 2's 
 should become weary of counting by 3's, 4's and 5's. This 
 constitutes the main method of oral drill. It should of 
 course be supplemented by the abstract written drill in 
 which with the two addends or the minuend and subtra- 
 hend written, the pupil calls out or writes the proper 
 result, and the semi-visual drill preliminary to addition 
 in column, in which the pupil gives the sum of the number 
 named by the teacher and that indicated by a pointer in 
 a column containing all the digits. 
 
 By means of these drills, not only may the facts be 
 readily and economically memorized, but if for the first 
 year or so, they are made preliminary to all working of 
 examples there need never be any slow operation. From 
 the first, pupils should not be allowed to engage in opera- 
 tion involving facts which they can not give with the 
 highest degree of facility. Examples should be devised 
 made up wholly of those sums or differences upon which 
 thorough drill has been given immediately before the 
 addition or the subtraction is to be performed. Thus 
 
6 7 
 
 operation being invariably rapid, rapid operation will be- 
 come a habit the degree of rapidity being dependent 
 upon the reaction time of the individual, and finding its 
 maximum at the point where rapidity threatens to result 
 in nervousness. 29 
 
 3. Conclusion That It Involves a Maximum of Mental 
 Training Chiefly in the Exercise of the Reasoning Ac- 
 tivity. 
 
 The abstract method not only makes possible the readi- 
 est derivation and the most economical memorizing, but 
 involves a maximum of mental training chiefly in the 
 exercise of the reasoning activity. If adapted to the 
 individual taught, it is therefore not only the mathemati- 
 cal, but unquestionably the psychological method of 
 teaching the fundamental sums and differences. 
 
CHAPTER IV. RESULTS OF A HALF YEAR'S 
 TEACHING IN THE ORDER AND ACCORD- 
 ING TO THE METHOD JUST 
 DETERMINED. 
 
 I. CONDITIONS UNDER WHICH THE ATTEMPT WAS 
 
 MADE. 
 
 These a priori determinations were put to the test in the 
 same schools in which effort had been made to ascertain 
 the numerical knowledge possessed by the child on enter- 
 ing school. On the whole, existing conditions were dis- 
 tinctly unfavorable to success. In two of the schools the 
 teachers were new and with practically no experience in 
 teaching young children, while a third, after being taught 
 for about three weeks by a teacher in the early stages 
 of typhoid fever, and for two months by a substitute, was 
 then placed in charge of a young girl, well-trained, but 
 who had never taught before. The number of pupils 
 under each teacher varied from twenty-eight to fifty- four, 
 and averaged forty-five. In two out of the seven rooms, 
 the great majority of pupils were beginners; in the 
 remainder, most of the children had entered during the 
 last school year, while many had been in school for a still 
 longer period. No special attention was given to begin- 
 ners. On the contrary, according to a long-established, 
 though unfortunate custom, far less time was devoted to 
 those classes which largely consisted of children just 
 entering school, than to those in which the great majority 
 were hold-overs. More than this, the same amount of 
 time was not allowed to corresponding classes in the dif- 
 ferent schools, one class of fourteen, for example, being 
 given on an average twenty minutes of oral drill each day, 
 while another of twelve was given twenty minutes of simi- 
 lar instruction. The more advanced of the beginners 
 were not taught with others who had just entered, but in 
 
 (68) 
 
69 
 
 classes in which hold-overs largely predominated. As the 
 schools did not open until September 5, and the classes 
 were tested during the first two weeks in March, the 
 results on the average represent the work of about one 
 hundred and twenty days which for half of those first 
 entering school could not have meant an approximate 
 total of thirty hours of actual instruction and for the 
 remainder from forty to fifty hours. As it was believed 
 that the experience of teachers during a short period, 
 unrestricted by a priori suggestions as to method of pro- 
 cedure, would be the only stable basis for detailed direc- 
 tions in the future, they were given no special instructions 
 other than that they should take up the elementary facts 
 in the order just described, and in such a way that the 
 pupils should so far as possible be led to perceive each 
 common element. 
 
 In view of the variability in teaching power, in length 
 of drill period and in manner of teaching, taken in con- 
 nection with the inexperience of some of the teachers, the 
 few hours spent in actual instruction in number, the 
 absence of detailed directions, and finally the fact that 
 even the few and simple directions that were given were 
 for a time in some cases misunderstood, the highly satis- 
 factory progress of the work, from the very first, appears 
 to justify the a priori determination of psychological 
 order and method. 
 
 Before proceeding to discuss the searching and im- 
 partial test given at the close of the first half year, with 
 its significant though of course inconclusive results, it 
 will be profitable to consider the suggestive data inci- 
 dentally furnished by the work that was subjected to it. 
 
 II. SOME OBSERVATIONS UPON FACTS WHICH IT INCI- 
 DENTALLY FURNISHED. 
 
 While no children were reported as having any serious 
 difficulty with mechanical counting, it was soon dis- 
 covered that probably from 10 per cent to 20 per cent of 
 
yo 
 
 the whole number of beginners would find it hard to 
 master the element common to the facts of even the first 
 group although such mastery was necessary to intelli- 
 gent counting. 30 The same children almost without ex- 
 ception also had trouble in their word-study, in the asso- 
 ciation of the word and its symbol. Indeed at this point 
 it may be well to state that among the children tested, it 
 was a very exceptional thing to find those who were in 
 more advanced reading classes than number-classes or 
 vice versa, only three being classified higher in number 
 and two in reading. The "slowness," then, of these 
 beginners, who with some "backward" hold-overs from 
 the year before, composed the lowest number section of 
 each first grade school, was plainly due to some condition 
 common to both number-work and word-study, a con- 
 dition which could be no other than the complexity of the 
 association required to memorize word-symbol or number- 
 fact. Children whose associations had so far been largely 
 confined to those uniting a name with a thing or one 
 name with another, found it well nigh impossible to asso- 
 ciate a name with a complex written symbol, or the "3," 
 "and," "2," "are," and "5," into a fundamental sum. As 
 they could readily associate a single letter with the cor- 
 responding sound, and a number with that adjacent to it 
 in the number scale, the mechanical counting to 10 or 20 
 already begun on their entering school was readily mas- 
 tered. But how were they to be led from that as a point 
 of departure, to the derivation and memorizing of the 
 essential facts ? All that was necessary to the derivation 
 of the first group was their comprehension of the fact that 
 the addition of one to a number, results in the number 
 next above it in the scale. But at the start, the pupils 
 comprising these lower classes, utterly failed to compre- 
 hend it. Hence the effort was made to simplify the 
 process of instruction. After the children could mechani- 
 cally count with a high degree of readiness from i to 9, 
 the teacher began to interject the expression "and one," 
 
between the terms of the series as given by the pupils in 
 counting the objects as she added them one by one. Thus 
 after they had become accustomed to the interruption, the 
 "and one" gradually came to carry with it its proper 
 meaning, and to be firmly associated with each number 
 and that which immediately followed it. The second 
 step in the process was to drill them upon the ready giving 
 of the number that came next in the scale after any other 
 which the teacher called out. Finally they were led to 
 observe that the addition of one denoted by "and one" 
 always resulted in that now familiar "next" number. 
 Within six weeks all but six of the "slow" ones had 
 mastered the facts of this group, together with the inver- 
 sions. Under the old plan they would have been vainly 
 struggling to remember the two or three "simpler" facts 
 which they had counted out with the objects every day. 
 During the first half year, even the remaining six yielded 
 to treatment and mastered the nineteen facts with their 
 inversions. A little less than one- fourth of the beginners, 
 however, in the corresponding time, failed to associate 
 20 with two IQ'S, 30 with three ID'S, etc. As they 
 were led to think of the "ty" as a short way of saying 
 "tens," and permitted to say "twoty," "threety," and 
 "fivety," in all cases where the corresponding contractions 
 did not seem to be sufficiently suggestive, and as those 
 classes who to the last pupil mastered these facts with but 
 little difficulty, were just as "slow" as those who to the 
 last pupil failed, the true reason for their failure to master 
 this and the two following groups appears to lie not so 
 much in their inability to learn, as in the false assump- 
 tion on the part of their teachers that having had difficulty 
 with the numbers from i to 10, there was little use in 
 attempting to teach them the "larger" and "more diffi- 
 cult" numbers from 20 to zoo. 31 
 
 While at the close of the first half year 34 per cent of 
 the beginners had not yet succeeded in firmly memorizing 
 all the fundamental sums resulting from the addition of 
 
7 2 
 
 2) very few of them had failed to comprehend, that the 
 addition of 2 was equivalent to the addition of i and i, 
 Great care, however, had to be taken to avoid plunging 
 those who were more immature into inextricable mental 
 confusion the result being brought about rather through 
 suggestion than by syllogism. For example, the series 
 "4, 6, 8," etc., was taught as follows: The teacher plac- 
 ing two cubes in plain view of the class, "How many 
 cubes have I ?" adding one and then another "Two and 
 i ? and i ?" putting the two new blocks together "What 
 are 2 and 2?" etc. Of course for a time some of these 
 pupils answered "4" to the 2 and 2, not because they per- 
 ceived the identity of the 2 with the i and i, but because 
 the "4" was the last answer that they had heard. It was 
 not long, however, before almost unconsciously the per- 
 ception came and they were able to apply the resulting 
 principle in finding other sums. 
 
 Very persistent repetition was necessary in the case of 
 these "slow pupils," to insure the thorough memorizing of 
 the facts thus derived. Again and again did the teacher 
 judge that the association had been firmly made, only 
 to discover when she somewhat relaxed the drill that her 
 faith had not been well founded. Had the counting 
 proved dull and tiresome to the child, opponents of num- 
 ber teaching during the first year, would have found addi- 
 tional argument in the case of this lowest third of the 
 beginners. On the contrary, the counting continued to 
 be for all pupils, from first to last, a continual pleasure. 
 They loved the rhythm, they gloried in conquering one 
 series after another, as they glory in achievements in their 
 play. The satisfaction with which they counted by 5's or 
 6's, and the contempt but half assumed with which they 
 who were thus advanced, would rush through I's and 2's, 
 settled once and for all the fear that the interest shown in 
 counting by ones might not continue in the higher series. 
 
 Although after a careful attempt had been made in 
 every school, the pupils, including the great majority of 
 
73 
 
 beginners, had experienced little difficulty in learning to 
 count backwards less difficulty of course than they 
 would have had in learning an entirely new series, when 
 it came to drill on individual facts, the number who per- 
 sistently confused the sums and differences, was, as had 
 been anticipated too considerable to be ignored. It was 
 plainly unpsychological to have a child who found some 
 difficulty in so simple a verbal association as that between 
 a given number, "and" and the digit next succeeding, 
 judge between such association and that formed between 
 the same number, "less," and the digit immediately preced- 
 ing it. For many the judgment was too complex. Some 
 first giving the sum and seeing disapproval in the teacher's 
 eye, would give the difference. Others of different temp- 
 erament, or more immature, familiar with both facts but 
 confused by the sudden demand for a judgment, would re- 
 main in a state of uncertainty, undoubtedly painful, until 
 relieved by kindly suggestion or stunned by sharp rebuke. 
 
 While a fair majority of the beginners and a very con- 
 siderable majority of the other pupils made the neces- 
 sary discrimination without apparent difficulty, as has 
 been already asserted there seemed to be no reason why 
 the minority should find a serious obstacle to progress 
 in an operation which was by no means a necessary condi- 
 tion to the more rapid progress of the more mature. As 
 it still appeared highly probable that the former might be 
 more able to judge between the two sets of facts later on 
 in the school year, after many of the sums had been cer- 
 tainly mastered, the teaching of the elementary differences 
 was abandoned in every school but one. 
 
 This action found additional justification from another 
 consideration already brought out in the a priori discus- 
 sion. Early in the school year, every class had been 
 asked and had readily answered the following question: 
 "If I had an apple and a peach, and took away the peach, 
 what would I have left? If I had the apple and the peach 
 and took away the apple? All but an inconsiderable 
 
74 
 
 minority were equally successful in answering the follow- 
 ing: "I have some fruit, an apple and a peach. If I take 
 the apple away from the fruit, what will I have left?" 
 etc. When numbers were substituted for apples and 
 peaches in the first question, the answer was given quite 
 as readily. When, however, the sum of the two numbers 
 was substituted for the term "fruit" in the second ques- 
 tion, a majority of the answers were confused, it being 
 of course quite natural for the "slower" children to give 
 the sum the number last named in place of the re- 
 mainder. Nevertheless, with the aid of concrete illustra- 
 tion and tactful questioning, even this difficulty was over- 
 come, by so large a percentage of the pupils, that it seemed 
 quite possible that with proper training, the great ma- 
 jority might be led to form the generalization, that when 
 one of two numbers is taken from their sum, the remainder 
 will always be the other. As this generalization is 
 "effective," in that through it all fundamental differences 
 can be readily derived, the steps necessary to it were from 
 this time on made a part of the regular drill. The gen- 
 eral test which will be presently described, having been 
 taken too early in the year for the results of this post- 
 ponement and preliminary drill to be statistically reported, 
 it may be well to state that the experiment was highly 
 successful. Half, for example, of the pupils in one 
 advanced class at the close of the first half year gave 
 with absolute correctness and with very considerable 
 readiness any fundamental differences corresponding to 
 fundamental sums which they had mastered. This in- 
 formal test was as much of a surprise to the teacher as to 
 the pupils, and so far as was known, involved no fact 
 which the class had previously derived. A majority of 
 the pupils still failing, however, to master the generaliza- 
 tion, it was found necessary for them to derive the differ- 
 ences as they had derived the sums, the backward count- 
 ing proving as popular and successful a process as count- 
 ing forwards. 
 
75 
 
 The abstract derivation of each new sum, involving 
 as it does, the oral addition of three numbers, was the 
 beginning of a thorough preparation for rapid operation. 
 Wherever the teachers strictly excluded all examples 
 involving facts insufficiently mastered and persistently 
 carried on the various forms of preliminary drill, the 
 habit of rapid operation was formed by the pupils. While 
 there is at present no way by which this can be satisfac- 
 torily demonstrated to those who have not visited the 
 schools, it is none the less a fact that the pupils thus taught 
 do not know what it is to add or to subtract slowly. 32 
 
 III. NUMBER TEST GIVEN AT THE CLOSE OF THE FIRST 
 HALF YEAR TO 237 FIRST GRADE PUPILS. 
 
 (I) OBJECT OF THE TEST. 
 
 The test was made to determine: 
 
 1. The percentage of the whole number of pupils as 
 well as of the beginners, who had thoroughly mastered the 
 facts of each series, as shown in the percentage of correct 
 answers immediately given. 
 
 2. The nature of the deficiency existing among the re- 
 maining pupils, as shown by the percentage : 
 
 (1) Of correct answers deliberately given. 
 
 (2) Of wrong answers immediately corrected. 
 
 (3) Of correct answers immediately given and un- 
 corrected. 
 
 (4) Of incorrect answers immediately given and cor- 
 rected wrongly. 
 
 ( 5 ) Of incorrect answers deliberately given. 
 
 (6) Of failures to answer. 
 
 3. The comparative readiness with which the facts 
 were given, as shown by the average time it took the 
 pupils of each class to give a fact in each series, deduc- 
 tion being made for all deliberations. 
 
 That is, the tests sought to discover how thoroughly 
 
7 6 
 
 the fundamental sums had been mastered and how readily 
 they could be called to mind and expressed. 
 
 Immediate answers deliberately given, indicated a 
 mastery, mechanical to a higher or lesser degree, as shown 
 in the average time it took to give a fact in each group. 
 The comparison of the time thus recorded with the records 
 resulting from future tests given under the same general 
 conditions, will afford the measure of the gain or loss in 
 mechanical facility. 33 
 
 Correct answers deliberately given indicate the power 
 to rationally derive an unknown or forgotten fact from 
 one that is known. Wrong answers immediately cor- 
 rected are more likely the result of nervous strain due to 
 the test, or of psychological or physiological conditions 
 peculiar to an individual, than of ignorance of the fact 
 required. Incorrect answers immediately given and un- 
 corrected while not necessarily indicative of individual 
 ignorance, make collectively a fair index of general inac- 
 curacy. While incorrect answers immediately given and 
 corrected wrongly, may be due to nervousness, as is prob- 
 ably the case with those which are corrected rightly, they 
 are more likely, perhaps, to result from uncertain knowl- 
 edge. Incorrect answers deliberately given, probably 
 for the most part result from some error in the process 
 of derivation or from the basing of that process upon an 
 incorrect sum or difference, although they may originate 
 in an effort to supply by a guess a forgotten fact that can 
 not be rationally derived. Failures to answer, though 
 frequently due to embarrassment, are here assumed to be 
 the result of ignorance. 
 
 (II) DESCRIPTION OF TEST AND OF PRECAUTIONS TAKEN 
 TO INSURE TRUSTWORTHY RESULTS. 
 
 Both in the classification just given, as well as in the 
 method of conducting the test, every effort was made to 
 prevent the more or less unconscious modification of con- 
 
77 
 
 ditions, so often due to the conducting of an investigation 
 by those interested in its results. Errors due to partiality 
 in questioning, that is, to the teacher's consciously or un- 
 consciously asking the easier facts of a series from the less 
 proficient pupils and the more difficult from those who are 
 brighter, were avoided by the adoption of a fixed order for 
 the facts of each series. This order was purely arbitrary, 
 and included all sums in each series, or their inversions. 
 In order that the test might measure the actual progress 
 of the schools, the facts in each series were given together. 
 Test was also made of the facts miscellaneously grouped. 
 
 The lack of uniformity in the rate of questioning, 
 destructive to the value of the time tests, was reduced to 
 a minimum by all facts being demanded by the same 
 individual in as quick succession as distinctness of enunci- 
 ation would allow, and for a period so limited as to prac- 
 tically eliminate variations due to fatigue. Whenever 
 a pupil hesitated, deduction was made in the time all 
 answers appreciably longer in coming than the others 
 being counted hesitations. In order to insure accuracy 
 and to prevent interference with the recording of results, 
 the time was kept by an assistant. The teacher took no 
 part in the test except to note the names of the pupils that 
 failed, and the probable explanation of their failure, 
 where explanation could be given. The use of an exceed- 
 ingly simple system of recording made it possible for the 
 individual who did the questioning to note each failure in 
 the proper class without loss of time, his pencil being 
 ever ready to mark a tally under "immediate answers," 
 while "deliberations" afforded time for the necessary 
 record. 
 
 All children present in the schools on the day when the 
 test was made, with the exception of one who had just 
 entered school, were subjected to it 237 in all. Of 
 these but 68 were beginners who had been in- 
 cluded in the original 100 tested, although many of the 
 others had entered school since the beginning of the 
 
78 
 
 school year. Each child was tested on one fact in every 
 group which his class was thought to have mastered. The 
 total number of facts demanded from the whole number of 
 children was 1,838. 
 
 (Ill) ANALYSIS OF THE RESULTS. 
 
 i. More Facts Were Mastered by the Pupils Than in 
 Any Corresponding Period in Previous Years. 
 
 Though the test was severe, its results fully justified 
 the order and methods that had been followed in the work 
 of the first half year. 
 
 In the past it was only the most advanced classes that 
 had mastered the twenty-five elementary sums and the 
 twenty-six elementary differences, involved in the study 
 of the numbers from i to 10, fifty-one facts in all, to- 
 gether with their inversions and alternations. A few 
 products and quotients and the differences resulting from 
 the subtraction of each digit from itself are not included in 
 this comparison, their mastery having been incidental to 
 both plans of work. 
 
 Under the new system, 16 per cent of all the pupils 
 and 15 per cent of the beginners had mastered 112 
 fundamental sums with their inversions over twice as 
 many facts as any class had previously mastered in the 
 same time, and in addition could count by lo's to 100, 
 could instantaneously add any number from i to 10 to any 
 multiple of 10 below 100, could add i to any number 
 from i to 100, and in many cases could immediately derive 
 all facts in subtraction corresponding to the sums which 
 they had mastered. Twenty-seven per cent of all pupils 
 and 21 per cent of the beginners, were as far advanced as 
 those just reported, except that they had mastered but 
 ninety-nine fundamental sums, and that it was a smaller 
 percentage that could readily derive the differences. Re- 
 
79 
 
 spectively 39 per cent and 31 per cent had mastered 85 
 facts ; 52 per cent and 34 per cent, 70 facts ; 56 per cent and 
 41 per cent, 54 facts ; and 79 per cent and 66 per cent had 
 mastered 37 facts. Seventy-two per cent of all pupils 
 and 52 per cent of the beginners could add i to any num- 
 ber from i to i oo; 76 per cent and 63 per cent respectively 
 could add any digit to any multiple of 10 below 100, and 
 68 per cent and 44 per cent could similarly add 10. 
 Eighty-five per cent and 78 per cent respectively could 
 count intelligently by ID'S to 100, and all could add i to 
 any number from i to 20. No facts have been included 
 in these figures upon which the pupils were not thor- 
 oughly tested. The majority of the classes, at the time 
 the test was made, had been more or less thoroughly 
 drilled on the series next above the highest upon which 
 they were tested, while several classes had begun the 
 systematic derivation of the fundamental differences. 
 
 It should be noted that the contrast between the knowl- 
 edge possessed by the most advanced classes in this and 
 former years, was no greater than that existing between 
 the attainments of the slowest groups. In the past it was 
 a source of gratification to the teacher if the unfortunates 
 of which these groups were composed had mastered more 
 or less thoroughly the first eight or ten facts. Under 
 the new plan no group had mastered less than nineteen 
 sums, while a large proportion of "slow"ones had mas- 
 tered the multiples of 10, could add the digits to them, 
 and could add i to any number below 100. 
 
 Such results, attained under unsatisfactory conditions, 
 and in spite of the blunders ever involved in a new at- 
 tempt, would seem to indicate that the plan thus tested, 
 properly carried out, will make possible either the mas- 
 tery of more number-facts during the first school years, 
 or what is perhaps more desirable, the mastery of the 
 customary number of facts with less time devoted to the 
 study of number. 
 
8o 
 
 2. The Test Showed a Remarkably High Percentage 
 of Correct Answers. 
 
 Notwithstanding the fact that every child present in 
 the various schools, with the exception already noted 
 were subjected to the test, 82.1 per cent of the 1,838 facts 
 called for were given correctly. As in the 17.9 per cent 
 of wrong answers are included those of feeble-minded, 
 backward, or nervous children, and of pupils who had 
 been but a few weeks in school some of whom failed on 
 most of the questions and a few on all the true per- 
 centage of correct answers should be much higher. For 
 example, in one school of forty pupils, four children made 
 thirty-four out of the eighty mistakes. With these four 
 eliminated, the percentage of correct answers for the 
 school increases from 80.5 per cent to 87.2 per cent. In 
 the same ratio the 82.1 per cent of correct answers for all 
 schools would be increased to 89 per cent. 
 
 3. The Pupils Displayed a High Degree of Certainty 
 in Their Answers. 
 
 Only 1.3 per cent of the answers were changed, when 
 once they had been given i per cent of these being 
 wrong ones that were corrected, and .3 per cent correct 
 ones that were made wrong. In most cases these errors 
 were the result of nervousness. 
 
 4. The Answers Were Given with Highly Satisfactory 
 Promptness. 
 
 As the average time taken in asking the questions dis- 
 tinctly is included in the average number of seconds given 
 in the following table, it has been thought best to give 
 for comparison the corresponding averages resulting, 
 when the individual who asked the questions in the test, 
 asked them aloud to himself and himself gave the an- 
 swers with maximum readiness. In each case this average, 
 
8i 
 
 which for convenience will be called the "standard," is 
 the lowest resulting from three of these self-tests. The 
 results are given for each of the three or four sub-groups 
 into which the number of classes was divided. 
 
 Stand. IstGr. 2d Gr. 3d Gr. 4th Gr. 
 
 I. 5 & 1, 1 & 2, etc 2.25 2.26 2.62 2.99 2.59 
 
 II. 6 tens, 2 tens, etc 2.20 2.15 2.18 2.18 2.59 
 
 III. 60 & 10, 10 & 10, etc. 2.55 2.46 2.43 2.48 2.77 
 
 IV. 20 & 4, 40 & 3, etc. ... 2.48 2.48 2.60 3.13 5.50 
 V. 26 & 1, 52 & 1, etc. . . . 2.60 2.58 2.75 2.89 
 
 VI. 6 & 2, 2 & 2, etc 2.50 2.50 2.85 2.96 
 
 VII. 12 & 3, 3 & 3, etc 2.55 3.33 3.43 3.61 
 
 VIII. Miscellaneous Sums . . 2.40 3.13 3.39 2.50 
 
 IX. Miscellaneous Inversions 2.45 3.38 3.94 3.75 
 
 X. 12 & 4, 4 & 4, etc 2.50 3.48 4.04 3.25 
 
 XI. 10 & 5, 5 & 5, etc 2.50 3.02 3.56 4.37 
 
 XII. 12 & 6, 6 & 6, etc 2.50 4.28 4.38 4.12 
 
 XIII. 14 & 7, 7 & 7, etc 2.50 3.91 3.57 
 
 It will be noted that in most cases the variation is not 
 great, much of it, notwithstanding all precautions, 
 being in all probability due to failure to question at a uni- 
 form rate, although the number of seconds for the first 
 group is uniformly lower than those of the second, and 
 those of the second than those of the third. For the 
 first six series, the answers are given with a high degree 
 of readiness. From the seventh series on, there is a very 
 marked decrease in the readiness with which the answers 
 are given, due to the fact that the pupils in most of the 
 classes had been given less drill upon the higher sums. 
 
 The averages for series VIII, in which the "2" and the 
 "3 sums" are mingled, are higher than those of series VI, 
 but lower than those of series VII, the indication being 
 that the pupils could give the miscellaneous facts as readily 
 as those of the groups. The fact that the average number 
 of seconds in series IX, where the facts of series VIII 
 are inverted, is higher than that in series VIII, is probably 
 due not to any less readiness in giving the ordinary inver- 
 sions, but to the unfortunate fact that it contained such 
 
82 
 
 inversions as 3 and 12, 2 and 16, 3 and 13, and 2 and 14, 
 which are rarely called for either in addition or in drill. 
 
 On the whole, the facts were given with a highly satis- 
 factory degree of readiness, especially when it is taken 
 into account that the main reason for thus timing the 
 pupils at a stage of their work when but comparative 
 little drill had been given them in operation, was to make 
 it possible to gauge the improvement that should result 
 by the close of the year. It is altogether likely, however, 
 that the variation due to oral questioning will in the end 
 prevent a satisfactory comparison. 83 
 
 5. Demonstration of the Ability of the Pupils to 
 Promptly Derive for Themselves Facts Upon Which 
 They Had Been Insufficiently Drilled. 
 
 This evidence of independence in derivation was one 
 of the most significant of the results. When pupils 
 hesitated, 3.2 per cent of the questions received no reply 
 at all owing to nervousness, dullness, or mental con- 
 fusion; 4 per cent after deliberation were answered 
 wrongly, either through their derivation being based 
 upon an error, or through some mistake in the process of 
 derivation, unless the children thus answering waited a 
 moment or so to give a guess that might just as well have 
 been given at once. Nine and four-tenths per cent, how- 
 ever, were answered correctly after more or less delibera- 
 tion. If the facts had been taught in Grubean isolation 
 such recall would have been impossible. The significance 
 of the foregoing percentage is largely due to the fact that 
 the pupils who thus hesitated, were for the most part 
 those, who being behind the majority of their fellows, 
 might be supposed to be less likely to abstractly derive 
 the facts that they had forgotten. 
 
 It should be remarked in conclusion that throughout 
 the test, the beginners showed as great accuracy, readi- 
 ness and independence in giving such facts as they had 
 
83 
 
 mastered, as did the pupils who had attended school 
 before. 
 
 IV. SUMMARY OF THE CONCLUSIONS WHICH THE 
 INQUIRY TENDS TO ESTABLISH. 
 
 At the close, then, of the first half year of work, the 
 results attained all tend to demonstrate the correctness of 
 the a priori determinations regarding psychological order 
 and method. While both are psychological only in so far 
 as they are adapted to the majority of individuals taught; 
 while it is altogether likely that there are individuals to 
 which they are not thus adapted ; they have proved them- 
 selves psychological in the only schools where thus far 
 they have been tried. Nor is it unreasonable to assume 
 that based as they are upon psychological fact, should they 
 be adopted by other schools, they may in course of time 
 be as certainly confirmed by general experience and 
 proven by scientific test, as they have been indicated by 
 individual reason and justified by individual success. For 
 the present, a single investigator tentatively asserts as the 
 result of a patient and impartial examination of limited 
 data, that the only numerical knowledge common to a 
 majority of children on entering school is the ability to 
 instantaneously perceive three or four objects as 3 or 4, 
 and to count more or less mechanically to 10 or 12; that 
 the majority of such children love to count, and as their 
 work progresses to count by 2's, 3's, etc., as well as by 
 I's; that this knowledge and this interest form the natural 
 basis for the study of number ; that there is no necessary 
 antagonism between the logical and the psychological 
 orders of teaching the fundamental sums and differences ; 
 that the order in which they can be most readily taught is 
 that logical order which results from the successive addi- 
 tion or subtraction of each digit from all other digits, 
 in that each of the resulting groups contains an element 
 which readily mastered in one or two of the facts to which 
 
8 4 
 
 it is common, readily effects the derivation of all ; that this 
 method of derivation involves the maximum of abstract 
 reason possible to the majority of children on first enter- 
 ing school and the minimum of concrete illustration neces- 
 sary to intelligent work ; that it is based upon the numeri- 
 cal knowledge possessed by beginners, insures from the 
 start a maximum amount of the mental training peculiar 
 to mathematics, and makes possible the use of abstract 
 counting by 2's, 3*5, etc., the readiest means of memoriz- 
 ing the sums and differences when once they are derived ; 
 that the order and the method which thus result in the 
 readiest mastery and the highest training are mathemati- 
 cal, and, being adapted to the great majority of the pupils 
 of even the first school grade, psychological; that since 
 their proper use insures the mastery of more funda- 
 mental sums and differences in the usual time or of the 
 usual number of sums and differences in less time, and 
 gives the pupils the power to readily derive forgotten 
 facts, they are economical ; that rapid operation in greater 
 or less degree, can be made invariable from the first, and 
 so from the first can become a habit. 
 
NOTES AND REFERENCES. 
 
 1 " Vorstellungkreis der Berliner Kinder beim Eintritt in die Schule." 
 Berlin Stadisches Jahrbuch. 1870. Pp. 59-77. 
 
 2 "Contents of Children's Minds on Entering School," by G. Stanley 
 Hall, Ph. D., LL. D. 
 
 3 Ibid., p. 12. Dr. Hall here refers to " Der Vorstellungskreis unserer 
 sechsjahrigen Kleinen." Allg. Schul-Zeitung. Jena, 1879. P. 327 
 et seq. Tliis article I have been unable to consult. 
 
 4 "Contents of Children's Minds," p. 19. 
 
 5 Ibid., p. 34. 
 9 Ibid., p. 17. 
 
 7 Ibid., p. 49. 
 
 8 This conclusion, while merely tentative in so far as it is based upon 
 the foregoing results, is in accord both with common experience and 
 the general trend of philosophical determination. The Committee of 
 Fifteen asserts that "counting is the fundamental operation of arith- 
 metic, and all other arithmetical operations simply devices for speed 
 by using remembered countings instead of going through the detailed 
 work again each time." ("Report of the Committee of Fifteen," pub- 
 lished for the National Educational Association by the American Book 
 Company. New York, Cincinnati, Chicago, 1895, p. 53.) 
 
 Believers in the Culture Epoch Theory can not but accept this asser- 
 tion, on the ground that in the historical development of the race, 
 counting preceded all other forms of numerical operation. ("The Teach- 
 ing of Elementary Mathematics," D. E. Smith. Macmillan,1900, p. 49.) 
 It finds its fullest justification, however, in Dr. D. E. Phillips' article 
 on "Number and Its Application, Psychologically Considered," in 
 "Pedagogical Seminary," Vol. V, No. 2, pp. 221-281. (J. H. Orpha, 
 Worcester, Mass.) "The earliest and most rudimentary form of 
 knowledge in the cognitive sense," says Dr. Phillips, "is a knowledge 
 of a series of changes" (p. 228). This "series-idea," established by 
 a multitude of successive and rhythmical sensations conveyed through 
 the different senses, and embodying the rudiments of the number 
 concept, finally becomes abstract a general idea applicable to any 
 series of successions. At this stage symbols of representation are neces- 
 sary. Hence results counting, first the naming of the series without 
 reference to objects of any kind, then the counting of objects inde- 
 pendently of the order of number-names, and finally the counting of 
 objects with the number-names applied in proper order. 
 
 "Counting is fundamental, and counting that is spontaneous, free 
 from sensible observation, and from the strain of reason" (p. 238). 
 
 (85) 
 
86 
 
 9 Such records as investigators have made of the mental develop- 
 ment of individual children fully confirm this statement. Perez reports 
 the case of a child two and a half years old who could count to 12, but 
 did not understand the expression "3 days" until "another and 
 another and another" made it clear. To the same child a year was- 
 "many, many to-morrows." ("The First Three Years of Childhood," 
 Bernard Perez. Edited and translated by Alice M. Christie. E. L. 
 Kellogg & Co., 1894, p. 186.) 
 
 Similarly in a case reported by Preyer, by the twenty-ninth month 
 numbering begins to be active to a noteworthy degree. "The child 
 began, viz, on the 878th day, suddenly of his own accord entirely, ta 
 count with his nine-pins, putting them in a row, saying with each one, 
 eins (one) eins! eins! eins! eins! Afterward saying, eins! noch eins 
 (one more) ! noch eins ! noch eins ! The process of adding is thus 
 performed without the naming of the sums." ("The Development 
 of the Intellect," W. Preyer. Appleton, New York, 1895, p. 172.) 
 
 10 Among 453 children first entering the schools of Chester, Pa., in 
 September, 1900, but 56 (i. e., 12 per cent) were found unable to 
 count to 8 or 10. 
 
 11 While Preyer's generalization that " the boy of four years counts 
 objects with effort up to six" and that "numbers remain for a long 
 time merely empty words" (Dev. of Int.," p. 271), may be based upon- 
 too few particulars, the latter statement at least is undoubtedly correct. 
 (See Dr. Phillips' "No. and Its Ap.," p. 234.) Cases where "counting 
 proceeds independently of the number-names" (Phillips, p. 234), as 
 where a child of nineteen months counts "2, 3, 5, 6, 7, 8, 9" (Preyer), 
 while probably characteristic of "at least nearly all children" when 
 they first learn to count, are by no means common among the children 
 whom I have tested on their first entering school probably less than 
 half a dozen in several hundred. The inference is plain; most children 
 in all probability begin to count long before they reach school age. 
 
 12 The fact that in counting objects, children often run the number 
 series ahead, counting "seven" before the teacher has placed the 
 seventh object with the five or six, should not be taken to indicate 
 inability to number objects. The error is due rather to a lapse in 
 attention, and is not so likely to occur when the pupils are themselves 
 handling the objects which they count. 
 
 13 By "intelligent" counting, I mean nothing more than the num- 
 bering of things successively perceived, by a parallel repetition of the 
 accepted number series, accompanied by the realization that each 
 number-name applies to one thing more than that which immediately 
 precedes it in the scale. 
 
 Of course this realization of the inclusiveness of each number-name 
 involves the discrimination, abstraction and grouping justly empha- 
 sized by Professor Dewey (McLennan and Dewey's "Psychology of 
 
87 
 
 Number," Appleton, 1898, pp. 31 and 32), the number-name denoting 
 not the one more, but the number of ones in the group of which the 
 one more forms a part. But when children who have mastered the 
 number series abstractly, apply it to objects in order to determine " how 
 many" it is under conditions which insure the presence of each factor 
 in the intellectual process. 
 
 Professor Dewey asserts that in order " to promote the natural action 
 of the mind in constructing number, the starting point should be not 
 a single thing or an unmeasured whole, but a group of things or a 
 measured whole." So far as the "single thing" is concerned, he might 
 have gone a step farther. The starting point in counting not only 
 should not be a single thing, but can not be so. Before an individual 
 attempts to determine "how many," there must be in his mind the 
 thought of the unified group that he is to number. Its unity need not 
 be that of qualitative likeness. It may consist in the mere fact of 
 possession or location, as when he numbers the Christmas presents 
 that he has received, or the various objects on a table. Ignoring quali- 
 tative differences with ease, or failing to observe them, abstraction 
 presents less difficulties to him than to the philosopher. He unhesi- 
 tatingly counts things that the philosopher would tend to subgroup. 
 But the thought of some constant unity such as the number of legs 
 possessed by a horse, or of a constantly increasing unity such as the 
 telegraph poles seen from the car window, always precedes his count- 
 ing. There is never a " how many" without a supplementary " what?" 
 the thought of which precedes it, and the "what" is never a single 
 thing, but always a unified plurality. 
 
 Just as generalization, abstraction and grouping thus precede the 
 counting of things, discrimination acccompanies it. As each thing, 
 observed in its proper sequence, serves as a stimulus to the correspond- 
 ing number-name, it must be clearly perceived as a one of right belong- 
 ing to the group. If things can not be discriminated they can not be 
 counted. It therefore follows that if children succeed in applying the 
 number series to things at all, they are grouping, abstracting and dis- 
 criminating like so many philosophers. Professor Dewey does not 
 give too philosophical a meaning to children's counting (Phillips, p. 243) ; 
 he merely describes in philosophical terms an experience which children 
 necessarily have whenever they apply the number series to things, but 
 of which they are and may be permitted to remain unconscious. 
 Before they attempt to determine how many, they generalize; if they 
 count things at all they discriminate them. 
 
 If he would go on to assert that the naming of each successive number 
 as one thing more is added to the group already counted, carries with 
 it a recollection and discrimination of the individual things already 
 numbered and their regrouping with the additional thing into the new 
 unity to which the number-name is applied, I should venture to dispute 
 
88 
 
 it on what appears to me fully adequate grounds. (See p. 61 of present 
 discussion.) 
 
 That there is a vague recollection of the mass of number-names that 
 have gone before, made possible in the case of the larger numbers, by 
 the decimal terminology, I believe to be true. When in counting we 
 say "twenty-six," there is no clear recollection, visual or otherwise, 
 of 26 things (see p. 62 of present discussion), but rather the abstract 
 thought of 26 as 25 and 1, and the vague knowledge perhaps not 
 present in every instance, but always possible that many successive 
 additions of 1 were necessary before the first 1 was increased to 25. 
 When the decimal system has been mastered, added to this, if not 
 entirely displacing it, there may be the abstract recall of two tens and 
 six, a concept fully adequate to the demands of every-day life, if not 
 to mathematical philosophy. 
 
 A vague content may be given the number-name long before mechani- 
 cal counting becomes intelligent. Indeed, Professor LeFevre asserts 
 that from the first there is a fuller recall of number-names, and hence 
 a more exact notion of number than I can venture to claim. On p. 23 
 of his "Number and Its Algebra" (Heath, 1896), he states that "the 
 order being learned by rote, any word numeral by suggesting its definite 
 place in the fixed series of words, recalls all those gone before; and 
 from this comparison the mind conveys or receives an exact notion of 
 the number of individuals in the group of objects numerically charac- 
 terized by any such number-name." 
 
 14 P. 249, " Number and Its Application." 
 
 15 See pp. 171-176 in " Outlines of Psychology," translated by Dr. E. 
 B. Titchener from the German of Professor Oswald Kiilpe. (The 
 Macmillan Company, New York, 1895.) 
 
 18 See early files of the "Pennsylvania School Journal" and other 
 educational periodicals. 
 
 17 See "Elements of the Grube Method," Levi Seeley, M. A., Ph. D. 
 (E. L. Kellogg & Co., New York.) 
 
 18 Occasionally the general order is varied, as where 2, 4 and 8 are 
 analyzed before 3, 5 and 7, but it may be safely asserted that in almost 
 all American schools the various combinations and separatione of a 
 particular number are exhausted before the teaching of any fact involv- 
 ing the numbers above it. A useful piece of work yet to be accom- 
 plished is the certain determination of the number of schools using 
 each of the possible orders of presenting the fundamental number-facts 
 and of the various general methods of teaching them. 
 
 19 The naturalness of such derivation as well as its rationality is seen 
 in the following abstract from Louise E. Hogan's "Study of a Child." 
 (Harpers, 1898, New York, p. 168.) "To-day we heard him say 'six 
 and three are nine; six and four are ten.' His aunt asked him how 
 he knew it. He replied, 'I know that six and three are nine, and four 
 
8 9 
 
 is one more than three, and ten is next to nine, so it must be so.' " (See 
 Note 26.) 
 
 30 A summary of methods given in David Eugene Smith's admirable 
 book, "The Teaching of Elementary Mathematics" (Macmillan, 1900, 
 pp. 94-97), leads me to infer that this plan has been long in use in some 
 of the elementary schools of Germany. He refers to Knilling's "Zur 
 Reform des Rechenunterrichtes " and "Die Naturgemasse Methode des 
 Rechen-Unterrichts in der Deutschen Volksschule," and to Tanck's 
 "Rechnen auf der Unterstufe," "Der Zahlenkreis von 1 bis 20," and 
 " Betrachtungen uber das Zahlen." To these works I unfortunately 
 have not had access. 
 
 31 Some who read this statement may prefer to agree with Dr. Dewey, 
 when, after asserting that " the concept two involves the act of putting 
 together and holding together the two discriminated ones," he goes 
 on to say, "It is this tension between opposites which is largely the 
 basis of the childish delight in counting." ("Psychology of Number," 
 p. 31.) It seems to me, however, that the delight is in mechanical 
 counting which children love without any thought of the fact that 
 one and one are two. 
 
 While Dr. Phillips finds that the "passion for counting" objects, 
 most common in children between the ages of seven and ten, had its 
 beginning in a "passion for following the abstract series" in but 57 
 cases out of 131 (Ped. Sem., V, p. 252), I have failed to discover any 
 children of school age who do not heartily enjoy the rhythmical flow 
 of mechanical counting even when "sing-song" is prohibited. 
 
 33 See Phillips, p. 265. "Counting is fundamental and even com- 
 binations furnish the first step, hence counting by 2's, 3's, 4's, etc., 
 furnish the first steps in multiplication." 
 
 38 Encouraged by the results of this first attempt at demonstration, 
 during the past school year (1900-1901), I followed the same plan in 
 teaching the elementary sums and differences to 1,200 children con- 
 stituting the first grade in the public schools of Chester, Pa. While I 
 shall from time to time refer to the results there attained, it is perhaps 
 well at this point to make the general statement that this second 
 attempt was even more successful than was the first. 
 
 34 See Phillips, p. 265 : " The difficulties of subtraction are greater 
 and more lasting than those of addition, hence its introduction on 
 beginning addition must at first be incidental. Care must be taken 
 not to confuse the child by introducing too many processes at once." 
 While the results of my own work show the wisdom of Dr. Phillips' 
 caution (see present article, p. 48), I find that after pupils have thor- 
 oughly mastered the fundamental sums formed by the addition of two, 
 they can take up the corresponding differences without any serious 
 confusion of the two sets of facts. 
 
 38 An early protest against the "Bitter End of the Object Method" 
 
9 o 
 
 was made by Ida F. Foster in Educational Review, March, 1894. " It 
 is well that a child should see that four straws and three straws are 
 seven straws, before he learns to say that 4 + 3 = 7, but it is not neces- 
 sary that four splints and three splints should be set before his eye 
 every time he is asked how many 4 + 3 make." Surely someone, 
 though unknown to me, must have long ago pointed out the equal 
 absurdity of objectively deriving each new fact. 
 
 38 Hensleigh Wedgwood, in his article on " The Foundation of Arith- 
 metic" ("Mind," Vol. Ill, p. 572), declares that the Grube Method 
 assumes with Mill that the number-facts are mastered only "by con- 
 stant observation of the result when groups of actual objects are com- 
 bined or separated, i. e., by experience." Not so, he goes on to state, 
 with the derivation of numbers by mental association "from conscious- 
 ness on the contemplation of our own thoughts, that what we mean 
 by two is nothing else than the aggregation of 1 + 1." When with 
 the aid of objects, children have learned that 2 + 1 are 3, and that 
 3 + 1 are 4, they can be quite readily led without other aid than their 
 own powers of generalization unconsciously exercised, to the conscious- 
 ness of the fact that 8 + 1 are 9. This statement I make not as a 
 reasonable assumption, but as an assertion based upon the experience 
 of 1,500 children. 
 
 37 A prolonged effort which I made a year or so ago to train twenty 
 young men and women to instantaneous recognition of the number 
 8, with counting and estimation both conscious and unconscious 
 practically eliminated, through the use of slightly curved lines of 
 equidistant and uniform black dots, resulted in their ready recognition 
 of the particular 7's, 8's and 9's on which they had been drilled, 
 but in no appreciable improvement when they were called upon to dis- 
 tinguish from each other, 8's, 7's and 9's arranged in curves slightly 
 different from those used in their training. Since with irregular 
 arrangement I found training in recognition quite easy, I infer that 
 results such as those which have been achieved by Miss Aiken (see 
 " Mind Training" Catherine Aiken) are due to training in instantaneous 
 and perhaps unconscious estimation rather than to simultaneous 
 number perception. 
 
 38 Children often distinctly visualize the smaller numbers when count- 
 ing. Of this Sully gives us a good example his child saying, "One, 
 two, three, four two dollies, a tin soldier and a shell," as he succes- 
 sively pointed to their images on his coverlid. (P. 484, " Studies of 
 Children," Appleton, New York, 1896.) 
 
 To what limit we can appreciate the number of units involved in any 
 written or spoken quantity is problematical. Dr. Conant thinks 100 
 somewhat high for most of us, and 1,000 a rather certain maximum. 
 (P. 33, " The Number Concept," Macmillan, New York, 1896.) Accord- 
 ing to Kay, in his "Memory, What It Is and How to Improve It," 
 
91 
 
 (p. 320, Appleton, New York, 1895), "G. P. Bidder, before he was six 
 years old, was accustomed to amuse himself by arranging peas, marbles 
 and the like in rows and squares of different numbers, and by counting 
 them over to ascertain the results of the combinations. Hence the 
 figures did not present themselves to him merely as symbols, but they 
 represented to his mind an equal number of definite objects. He never 
 went beyond 100." As a result, however, "he could multiply a row 
 of fifteen figures by another row of the same number and give the actual 
 result in a few minutes without seeing or writing down a single figure." 
 Such use of imaged objects by a mathematical prodigy probably as 
 a mnemonic system furnishes no justification for the excessive use 
 of splints and cubes in teaching the fundamental number-facts to normal 
 children. 
 
 In the Educational Review for 1893 (p. 467), Adelia R. Hornbrook 
 claims pedagogical value for "number forms," and suggests a system 
 of instruction based upon imaged columns of the numbers written in 
 convenient forms a device which in many cases might be practi- 
 cable, but which in most cases would be unnecessary, and hence uneco- 
 nomical. 
 
 28 Kay asserts ("Memory," p. 321) that if the fundamental sums are 
 so fully associated together in the mind, "that afterwards, whenever 
 the two forms occur, the third being their sum, will at once come up 
 one may sum up a whole column of figures almost at sight. This is 
 best done by simply using the eye without naming the figures." That 
 is, the column 3, 2, 1, 4, etc., should be read " 5, 6, 10, " without naming 
 the 3 + 2, the 5 + 1, etc. 
 
 But, Kay goes on to say, he will never thoroughly learn the combina- 
 tion 9 + 8 = 17, "if he immediately adds, and 6 = 23, and 7 = 30. 
 The mind must dwell upon one set of figures, and thoroughly master 
 it before proceeding to the next." 
 
 When the number-facts, as here suggested, are taught in groups 
 formed by the successive addition of the same digit as, for example, 
 "2 and 2, 4; 4 and 2, 6; 6 and 2, 8," etc. it is the introduction by the 
 teacher of the "and 2" into the series, which thus causes the mind to 
 dwell upon the individual facts, and so completes the necessary associa- 
 tion. This introduction of the common element should therefore 
 invariably supplement the simple counting, in order that six will come 
 after four in the children's minds only when they are counting by two's, 
 and after five when they are counting by one's. Then, as the result of 
 the mechanical drill, 4 and 2 will always be 6, and not 5 or 7. 
 
 30 Of the 453 beginners in Chester, 12 per cent after learning to count 
 had difficulty in telling what number came next after each digit, and 
 15 per cent were slow to learn the " one sums" to ten. 
 
 31 This inference has been borne out by my later experience. 
 
 33 In order to achieve this end, it was found necessary to specially 
 
92 
 
 prepare for the teachers the examples needed for proper drill and to 
 insist upon their exclusive use in seat work as well as in class exercises. 
 The semi-oral drill upon the two or three facts other than the "one 
 sums" involved in each set of examples was invariably insisted upon 
 before the pupils were permitted to add, and the corresponding written 
 drill before they were allowed to subtract. 
 
 After examples were copied for desk-work, the pupils added or sub- 
 tracted them simultaneously. Dawdling was thus eliminated, and 
 time economized. 
 
 83 No such comparison has been made. In a test necessarily oral 
 on account of the main objects it sought to accomplish and the inability 
 of beginners to write with sufficient ease, it was impracticable to allow 
 each class an equal period of time, and to make comparison on the basis 
 of the number of facts written or of examples solved the only mode 
 of procedure by which can be obtained the uniform results necessary 
 to an exact comparison. 
 
 BRA FTp 
 OF THE 
 
 (( UNIVERSITY jj 
 
 ^LIFOR]^ 
 
UNIVEESITY OF CALIFORNIA LIBRARY, 
 BERKELEY 
 
 THIS BOOK IS DUE ON THE LAST DATE 
 STAMPED BELOW 
 
 Books not returned on time are subject to a 'fine of 
 50c per volume after the third day overdue, increasing 
 to $1.00 per volume after the sixth day. Books not in 
 demand may be renewed if application is made before 
 expiration of loan period. 
 
 
 
 w 
 JUN 5 1947 
 
 lOm-4,'23