JUL 39 ]m A Contribution to the Pedagogy of Arithmetic BY ERNEST C. McDOUGLE Clark University A DISSERTATION SUBMITTED TO THE FACULTY OF CLARK UNIVERSITY, WORCESTER, MASS., IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DE- GREE OF DOCTOR OF PHILOSOPHY, AND ACCEPTED ON THE RECOMMENDATION OF WILLIAM H. BURNHAM. OP THE UNIVERSITY Reprinted from the Pedagogical Seminary June, 1914, Vol. XXI, pp. 161-218 A Contribution to the Pedagogy of Arithmetic BY ERNEST C. McDOUGLE Clark University A DISSERTATION SUBMITTED TO THE FACULTY OF CLARK UNIVERSITY, WORCESTER, MASS., IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DE- GREE OF DOCTOR OF PHILOSOPHY, AND ACCEPTED ON THE RECOMMENDATION OF WILLIAM H. BURNHAM. Reprinted from the Pedagogical Seminary June, 1914, Vol. XXI, pp. 161-218 r\ 0^* » • « •' « A CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC* By Ernest C. McDougle, Clark University I. Introduction Nearly three centuries ago Comenius undertook to give a method to teaching. At the present time there is a wide- spread movement through Experimental Education to estab- lish it upon a sound scientific basis. Much that now finds acceptance in both curriculum and didactic procedure has found a place through conservative respect to traditional philosophies and dogmas of the past, coupled with some later * It is fitting that I should here acknowledge my indebtedness to Dr. G. Stanley Hall, President of Gark University, for suggesting the field of investigation which led to this thesis, and for helpful direction in its development; to Dr. William H. Bumham, Professor of Pedagogy and School Hygiene, for his many favors in suggest- ing material to be embraced in the research upon the thesis and for valuable criticism of the treatment of the topics included; to Eh". Louis N. Wilson, Librarian of Clark University, for his un- sparing pains in procuring special documents for my use in gather- ing data; to Dr. S. A. Courtis, of Detroit, Mich., for the loan of some private materials; and to Miss Rose A. Carrigan, of the Boston Normal School, whose kindness made.it possible for me to witness the recent tests of the Boston school children and to be furnished with additional statistical facts. I wish also to thank the following persons for prompt response to my personal letters asking about material to be included in my survey: Dr. E. L. Thorndike, of Columbia University; Dr. G. Deutchler, Tubingen, Germany; Dr. A. W. Stamper, Chico, Cal.; Dr. W. H. Maxwell, Superintendent New York City schools; Mrs. Adelia R. Hornbrook, San Jose, Cal.; Prof. J. E. Calfee, Berea, Ky.; and Prof. C. H. Dietrich, Winchester, Ky. 162 CONTRIBX)frON tb*THE PEDAGOGY OF ARITHMETIC empirical and pragmatic considerations. The present critical studies and experiments are concerned with both the subjects in the curricula and the methods of instruction. In these re- searches many laboratories have been busy, many investiga- tors have been active, and much helpful work has been done. It is now necessary that the results should be brought to- gether and put into usable form. Experimental Psychology has been too busy in exploring many fertile fields, as yet, to give attention to the full bearing of its discoveries upon Didactics, while Experimental Pedagogy is frequently too empirical to be scientific. For these reasons it is essential that the synthesist should bring together the modern needs of Pedagogy and the contributions of Experimental Psychol- ogy germane to the processes of learning, and state in clear and simple language the norms of method so the average teacher, who possesses little or no technical nomenclature of the psychological laboratory, or even of the experimental pedagogist, may find assistance in the daily routine of school duties. It is the purpose of this research to bring together many of the recent tests and experiments in Arithmetic and, in connection with conclusions drawn from them bearing upon better texts and methods, to evaluate them for practical use in the regular school work. II. Brief Historic Sketch of Arithmetic Arithmetic is the oldest science developed by man. As an art it runs much farther back into antiquity. Its first use is so remote that it is difficult to separate the mythical from the real. Anthropological investigations have brought much helpful material to light and the historical genesis of Arith- metic is coming to be better understood. Callisthenes found in Babylon, in 331 B. C, when Alex- ander the Great captured the city, burned brick astronomical records running back to 2234 B. C. These were sent to Aristotle, according to Porphyry. In Egypt no uncivilized state of society has been found. Their oldest mathematical books date back to 3400 B. C, although Josephus (62, chap. 7j P- 50) gives Abraham credit for introducing Arithmetic into Egypt, when he says: " He communicated to them Arithmetic and delivered to them the science of Astronomy, for before Abram came into Egypt they were unacquainted with those parts of learning, and that science came from the Chaldeans into Egypt, and from thence to the Greeks also." Among the Greeks and Romans, as well as among most CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 163 primitive peoples, systems of counting were clumsy and im- perfect, and it became necessary to invent symbols for num- bers and systems of numeration for the practical use of these symbols. Counting-boards and the abacus v^ere early invented and have persisted down to the present time. We find the Chinese and Japanese (64, 179) using the swan-pan and the soroban very generally even to-day. It is of historic interest to note that the first printed Arithmetic, published at Treviso, near Venice, was entitled " The Art of the Abacus for Arithmetic." Because of their awkward number symbols the Romans used calculating-boards for computations and employed their symbols only in stating results. The process of calculation derives its name from the Latin, calculus, " pebble," since pebbles were used as counters by many people around the Mediterranean. The so-called Arabic characters are more properly Hindu. Leonardo of Pisa, in 1200 A. D., obtained them from the Moors and two years later he published a system of com- putation using them. As he had obtained the characters from the Arabs they received the name Arabic. They have been traced, however, to the Hindus. After their introduction into Europe, a long contest ensued between the abacists and the algorists, but the Hindu system gradually spread over the continent and was well known by 1400 A. D. Merchants discarded the Roman notation in 1550 and the monasteries and colleges followed a century later. With the new system, Florentine traders and writers developed double-entry book- keeping and worked out seven operations : Numeration, Addi- tion, Subtraction, Multiplication, Division, Involution, and Evolution, while Italian and English arithmeticians simplified the processes. The Arabs added from left to right. Garth, an Englishman, devised a plan to add from right to left. One has only to compare the solution of a problem in Division by Pacioli or Tartaglia with the work done even in the third grade of the schools to-day to note what simplifi- cation has taken place. The invention of printing with movable types, together with the great commercial activity carried on through the Han- seatic League and other agencies gave a remarkable stimulus to algoristic Arithmetic in Europe. In the sixteenth century sweeping transitions occurred. The manuscript Arithmetics were replaced by printed books; Roman symbols yielded to the Hindu characters ; the Arithmetic of the learned became the possession of the common people; and counters dis- appeared in favor of figures. Many authors were found in Italy, France, Germany, Holland, and England, and the art of reckoning with the pen rapidly replaced the art of calcu- 164 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC lating with the abacus. The Hanseatic League established Rechenschulen along its trade routes and these commercial schools had much to do in keeping Arithmetic out of the regular schools. The first German Arithmetic appeared in 1482 from the pen of Ulrich Wagner, a Rechenmeister of this League. Many topics, such as Partnership with Time, came into use then as a practical business subject, and have been retained in our modern text-books long after the business world has discarded the methods they present. It was through the influence of Pestalozzi that Arithmetic was given such a prominent place in the schools. Despite strenuous efforts to discredit its value and to minimize its standing among the branches of learning in recent years, it is still receiving from 12% to 26% of the entire time devoted to recitation in the Elementary Schools throughout the civilized countries of the world (90). III. Psychology of Arithmetic The big question among teachers is to know how to teach according to sound principles. As with other subjects in the curriculum, scientific methods in Arithmetic must ulti- mately be based upon genetic psychology. Until we approach from the lower side the many questions of material and method, there will be only an approximation to the real solution. Most methods have been superimposed, so to speak, upon the child from above and only in late years has there come any decided scientific tendency to study the genesis of number and the processes of computation from the child's point of view. These studies are yet confessedly few and do not warrant an attempt at a full statement of scientific method based upon them. What has been learned may assist in better pedagogical procedure and incite to further original research. Genesis of Number Ideas. — Major (75, 167) observed his son among other things for the rise of his ideas of number. He found him able to miss one ball out of his wagon, when three were in it, at the age of 21 months. While the child had a confused idea of 3, 4, and 5, at three years of age, Major received, many times, the correct number, i, or 2, or 3, when apples were used, by throwing them on the ground and asking for a certain number. Later, the child's interest declined. Preyer's boy missed one of his 10 toys (75, 166), at the age of only 18 months, and at 878 days of age counted his nine-pins by standing them in a row and saying : " Eins ! Eins ! Eins ! Noch Eins ! Noch Eins ! " and so on to the end of the row. On her 584th day, Dearborn's little daughter CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 165 (32a, 176) counted 6 cattle in a picture by saying: "One, two; one, two; one, two." He thinks she knew the number 2 as early as the 543rd day, and also says the same thing about I. Decroly thinks his little girl loiew 2 before she recognized what i is (33a, 119). The 2 seemed to remain for some time as the only number the child grasped. Binet's daughter at 30 months of age comprehended 2, and could get the idea of 4 at 51 months (33a). Lindner's son at 23 months had the number 2 (33a). Moore's three children had the idea of 2 at 22, 26, and 29 months; and a good idea of 3 at 32, and 53 months. Scupin's child (33a) knew 2 at 22 months. Clara and William Stern studied their children very care- fully and report that they could use numbers correctly in connection with apples, for instance, long before the abstract idea of number arose, or even before number could be rightly applied to other objects of less interest (33a). Ordinals were learned before cardinals. Major reports ordinals and cardinals as confusing to his child (75, 173), while Hilde Stern could get the fifth finger, but did not seem to under- stand the sum of her fingers and thumb on one hand (74a). Decroly made many careful observations upon his daughter and contributes a very interesting opinion that she knew the number 2 long before she had the idea of i. This does not seem to have been observed by others who have studied the genesis of number in infants and deserves to have more attention. While his child got the idea of two at 19 months of age, she was able to differentiate three, at 28 months, from two or one. In another month she picked out 2 objects by the aid of her fingers, and at 35 months did this without using the fingers. At 41 months, she seemed to have the idea of 3 quite well in mind and in another month the idea of 4 appeared to be somewhat clear. Still, as many other child- observers have found in their studies, there was a reversion of both interest and apparent. ability later, and at 46 months of age the little girl had a confused idea of 3 and 4, calling them simply, *' more." At 51 months, 3 arose again to clear- ness and soon was well comprehended. By her 57th month she was able to hand the correct number of objects, up to 5, indicated by having held up for her problem one or more fingers. This showed an ability to abstract the idea of num- ber, say from four fingers held up by her father, and of applying the number thus gained to the apples or other objects asked for. Sully found his child at 4 years 3 months calling big beads " 6," smaller ones, '* 5," and still smaller ones, "4." At 5 years old, however, he placed four crab- apples upon the sand, added two to them mentally, and begin- 166 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC ning with calling the first one '' 3/' counted them correctly up to 6. He could also count his four playthings, two dolls, a tin soldier, and a shell, from memory, after they had been put away. Sully says (121, 352) that the drawings of children 5 years of age show small regard for the five fingers of the hand. In the growth of infants in number ideas we find marked individual differences, and it can only be said yet that the idea of number seems to be forming intelligently about the fourth year (75, 165). Lietzmann {J2, 22) thinks schools often make a mistake in proceeding as if the child has no ideas of number when it enters school. " The beginnings of number lie much earlier. When the child comes to school, it will, in a majority of cases, already know the number words possibly up to 10, or 12." Of 1,217 children entering the Volksschulen in Breslau, 10% could count up to 5, 78% could count up to 10, 4% up to 100, and nearly 1% over 100. Ballard says (9, 58) : " It is a well known fact that chil- dren learn to count of their own accord. They do it at home, on the playground, and at their games. It is impossible to stop them." " I tested a school where no counting is especially taught and found that about 60% of the children could count up to 20, before they were 5 years old, and about 30% could count up to 30 before they were 6." Meumann (79, Bd. 2, 345) agrees with Preyer and the other few child psychologists that the child has a somewhat fully intelligible use of numbers from i to 10 at the earliest toward the close of the 4th year of its life. In his report of the Fielden School, Manchester, Eng., Harrison (54, 269) says the chil- dren play with dominoes and " the result is, that already the five-year olds are able to write correctly on the blackboard the result of 2 plus 7, 9 plus 5, etc." This agrees with Mon- tessori who has children at three to begin counting with but- tons, plates, or money, and later with sets of blocks. There are two very distinct schools of educators on the question of the origin of the child's notion of number. One party maintains that ideas of number are developed through the simultaneous perception of several objects, or stimuli, presented to the senses. With Newcomb (87) they hold that " our teaching of numbers is too abstract, — too much dissociated from objects of sense." Many experiments have been performed in recent years to determine how many objects one may be able to apprehend accurately without counting. For this purpose the time exposure is made as brief as possible so as to prevent counting. Nanu (86) used bright dots on a dark background and gave a time exposure of 33/1000 of a second. She ar- CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 167 ranged the dots in different figures and found that 5 could be perceived in a line, 10 in a parallelogram, and 8 in a hexagon in 75% of the cases. Lay (67) performed many- experiments and found a greater percentage of successes at- tended the arrangement of dots in quadrate form. He is a strong advocate of the objective method of teaching the early number work. He quotes the experiments of Goldscheider and Miiller, von Scheele, Schneider, Kiilpe, Cattell, Dietze, Warren and Messenger in support of his claims. However, his contentions are not without vigorous criticism from Walse- mann, Knilling, Knoche, and the whole school of Herbartians. McLellan and Dewey (76) would base the development of number upon measuring, — upon the ratio idea. They hold that there is no number without measurement (p. 242) nor measurement without the fraction implied. Tear reviewed the Speer arithmetic a half generation ago and quotes Newton (123, 631) : " Number is the abstract ratio of one quantity to another quantity of the same kind." So, also, the great Swiss mathematician, Euler, is cited : " Number is the ratio of one quantity to another quantity taken as a unit." In direct opposition to this notion of the origin of number in the child mind and to the consequent procedure in teach- ing, there is a strong party which maintains that the presen- tation of the objects in a series, or the stimuli in succession, is the proper method. Gilbert (43, no) believes that count- ing is " the first step in systematic thinking," while Phillips decides (95, 22'j') that " the first step is surely the forma- tion of the series-idea." He holds that counting is funda- mental, and that children forbidden to count on their fingers sometimes count by using their toes, or move an elbow, or press a muscle, or clear the throat slightly in order to follow the series. Among his reasons for rejecting the Grube method, Badanes says (6, 34) : " It is false from the point of view of Arithmetic as a science and as an art. It ignores the process of counting." The dispute really carries us back into the philosophical question as to whether number has time-relation only, or space-relation only, or neither, or both. It will suffice to present the writer's view on the pedagogical significance of the discussion to quote with approval the words of Meumann (79, 338) : From the psychological point of view number-concepts seem to possess both temporal and spatial qualities, and either of the views alone represents necessarily a one-sided notion, that by the series-method and by the simultaneous-method the child gains something of power not contained in the other. " The fact that by both methods good results can be obtained shows that a comple- ment of these methods, or the simultaneous employment of 168 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC both, must be the right way of complete comprehension of number for the child." On the side of anthropology, many systems of notation have been found among primitive peoples, and no definite correlation seems to exist between these systems and the civil development of the tribes. The highly civilized Peru- vians knew almost no arithmetic as an art and nothing of it as a science, according to Conant (24, 150), while the Yoru- bas, a very barbarous tribe in Africa could count quite ex- pertly. Many savage tribes count only up to two, and have number words only for " one," " two," and " many " or some other verbal device for distinguishing their first definite number ideas from the indefinite ones lying beyond. The lowest Brazilian tribes count to 3, and the Carribees, Galibi, Abipones and many others go up to 4. As a rule the South American and Australian tribes count seldom above 3 or 4. Among some of the Australians only binary systems prevail, while ternary and quaternary systems abound among the Indian tribes of South America. Yet, some Pacific island tribes have been found with ability to count up into millions in their trade in fish and breadfruit. Quinary scales are widely diffused throughout the world, and a few octonary systems are believed to have existed. Some scant traces of vigesimal systems have been found. All these facts show how slowly and imperfectly the concept of number has arisen among primitive peoples, who seem to have used numbers only as necessity forced the matter upon them. Need for counting in their barter with one another had more influence upon number than did their general intelligence, or any sub- jective interest in it. Conant (25, 31) says: "If the life of any tribe is such as to induce trade and barter with their neighbors, a considerable quickness in reckoning will be developed among them. Otherwise this power will remain dor- mant because there is but little in the life of primitive man to call for its exercise." More recently Boas (10, 65-66) has stated this as follows: " The fact that generalized forms of expression are not used does not prove inability to form them, but merely proves that the mode of life of the people is such that they are not required; that they would, however, develop just as soon as needed. This point of view is also corroborated by a study of the numeral systems of primitive languages. As is well known, many languages exist in which the numerals do not exceed two or three. It has been inferred from this that the people speaking these languages are not capable of forming the concept of higher numbers. I think this interpretation of the existing conditions is highly erroneous. Peoples like the South American Indians (among whom these defective numeral systems are found), or like the Eskimo (whose old system of num- bers probably did not exceed ten), are presumably not in need of CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 169 higher numerical expressions, because there are not many objects that they have to count. On the other hand, just as soon as these same people find themselves in contact with civilization, and when they acquire standards of value that have to be counted, they adopt with perfect ease higher numerals from other languages and develop a more or less perfect system of counting. ... It must be borne in mind that counting does not become necessary until objects are considered in such generalized form that their individualities are entirely lost sight of. For this reason it is possible that even a person who has a flock of domesticated animals may know them by name and by their characteristics without ever desiring to count them. Members of a war expedition may be known by name and may not be counted. In short, there is no proof that the use of numerals is in any way connected with the inability to form con- cepts of higher numbers." It may not be an unfair or unwarranted deduction from all these studies and views to believe that children at school entrance in America may have ample ability to delight in numbers although they may show little interest in them up to that time. The child-mind no doubt expands intelligently with its growth in experience with objects of multitude, much as the development of the primitive mind does, as described by Boas. Its early work with these numbers can be moti- vated and made attractive in a manner paralleling the race expansion. " The child is a natural symbolist," says Mary R. Ailing- Aber (3, 171). "A corn-cob with a dress on it will do for a baby and a stick with no additions, for a horse. To let one thing stand for another is as easy to a child as to breathe." There is an easy transition from the objects, too, to numbers and then from numbers to symbols at an age corresponding to school entrance. Time of Beginning Arithmetic. — With regard to the time when Arithmetic should be introduced into the schools and when it should be completed there is some difference of opinion, and the matter is just now in the polemical stage. One group of educators holds that it should not be taught, except incidentally, in the first grade, or first and second, or the first three grades. The majority report of the Committee of Fifteen of the National Educational Association, in 1895, urged the beginning of the subject in the second and its completion in the sixth grade. Burnham (17, 65) believes there is " ample reason for postponing the work of Arithmetic until the age of 10, or, more accurately, to that stage of devel- opment which is likely to be found in normal children at this age. While it is greatly to be desired that more investi- gations be made in regard to this subject, with our present experience this seems to be a wise rule." Chartres (22, 278) says that " separate Arithmetic classes should not be taught in the first grade; it is better to defer them to the second 170 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC grade, and probably it would be better to begin Arithmetic in the fourth grade if our text-books were built with that in view." So also writes Stamper (113, 258): "The general tendency in this country is to refrain from all drill work in the first year. In fact, some schools defer such work until the middle of the second year, or the beginning of the third." On the other side of the controversy are to be found many ardent protagonists. Smith (105, 128) says: "Not to put Arithmetic as a topic in the first grade is to make sure that it will not be seriously or systematically taught there in nine- tenths of the schools of the country. The average teacher, not in the cities merely, but throughout the country generally, will simply touch upon it in the most perfunctory way. What- ever of scientific statistics we have show that this is true, and that children so taught are not, when they enter the intermediate grades, as well prepared in Arithmetic as those who have studied the subject as a topic from the first grade on." In a very recent article (106, 95) he further argues: "All the talk about having no Arithmetic in grades i and 2, or leaving it to the whim of the teacher, has not shaken the belief of the great schools of the world in the wisdom of Pestalozzi's judgment." "Arithmetic is a game and all boys and girls are mere players. We have not learned this very thoroughly yet, but we are making progress." Montessori (83, 326) claims to achieve some wonderful results even with children of pre-school age. Greenwood (47) dissented from the report of the Committee of Fifteen and furnished a verbatim report of some actual teaching and results in the Kansas City Schools as proof of his contention that Arith- metic is eminently successful in the lower grades. " No greater difficulty to get small children to grasp the nature of a fraction as such than in getting them to grasp the simple whole numbers, . . . Children get the idea of half, third, quarter, long before they enter school." Hence, he advocates teaching them to add, subtract, multiply and divide fractions in the first grade. Cook (7) says: "I visited the Kansas City schools and testify that Mr. Greenwood has not overestimated conditions. I took some third-grade work home and tried it on Normal students, and they couldn't do it as rapidly as those children did it." In Germany a little more than 20% of the time in the first two grades is devoted to Arithmetic. In the United States, according to the report formulated in 191 1 by the American Committee of the Inter- national Commission on the Teaching of Mathematics (56, 16-65, 75-7^) cities reporting about one-tenth of the school population of the country show that Arithmetic is taught as a topic in the first grade in 71.5% ; in the second grade it CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 171 is introduced in 22% ; and in the third grade in 6.5%. (See Table E for further details.) In country, or rural schools, it may be said to be taught in the first grade in practically all the schools. The writer personally believes that there should be sys- tematic teaching of numbers in the first grades. The child's natural interest in numbers and the rich opportunities for pre- senting numbers in concrete objects should be utilized. Mere rote work must be avoided, but these early years are of inestimable value in furnishing a substantial foundation in the child's individual growth in the comprehension of num- bers. Counting may also be used in these grades to diversify the work and add to the useful results. Utility and Discipline. — ^Another question of much interest and one upon which there is considerable controversy yet, is that of the so-called disciplinary value of Arithmetic. The matter is at least as old as Plato, who says in his Republic: "And you have further remarked that those who have a natural talent for calculation are generally quick at every other kind of knowledge, and even the dull, if they have arithmetical training, gain in quickness, if not in any other way." In mathematics, perhaps, more than in any other subject, the doctrine of formal discipline, or transfer of train- ing, has been most successfully maintained. If Arithmetic has not been kept in the curriculum as a practical subject it has staid there as a disciplinarian of the intellect. John Stuart Mill attributed his success in speculation to his mathe- matical training, — " the habit of never accepting half solu- tions of difficulties as complete, never abandoning a puzzle, but again and again returning to it until it was cleared up; never allowing obscurities in a subject to remain unexplained because they did not seem important; never thinking I per- fectly understood any part of a subject until I understood the whole." This is not the place to discuss the general ques- tion of mental discipline, or transfer of training, upon which an extensive and varied literature has been produced within the past decade. It is not well to accept either of the two extreme positions noted in the literature, but at present the value of arithmetic in the school course may be defended upon both practical and disciplinary grounds. In its utilitarian aspects the demands of every-day commercial life are suf- ficient proof. And on the subjective side, if it should be shown that one intellectual trait does not and cannot assist any other, still the Einstellung toward matters under con- sideration trained into children in Arithmetic may be ad- vanced as evidence of the subjective discipline of Arithmetic. 172 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC The certainties found in Arithmetic, too, have a moral value to children as they approach so much of uncertainty in other studies. IV. Experimental Studies Many experimental studies of the teaching of Arithmetic and the various processes of learning the different operations in handling numbers have been made, notably within the past two years. While they have been concerned principally with only the fundamental operations, a synthetic study of their results should throw some added light upon our problem of finding a scientific basis for Arithmetical Methods. Thous- ands of children have been tested and drilled under more or less controlled conditions, and the results are now becom- ing available for comparative pedagogical purposes. Much yet remains to be learned by further experimentation, but a consideration of those studies that have been made and pub- lished will assist materially in determining more clearly than has been done heretofore the weakness of the present methods, their strong features, and suggest the next step in the search for a sound pedagogy of Arithmetic. Influence of Puberty. — Voigt made some studies upon chil- dren from ten to fourteen years of age in the Volksschulen (80, 117). Instead of using the ordinary decimal system of notation, he employed systems in which 8 and 6 were the bases. By this means he reduced to a minimum the use of knowledge already possessed by the children. Among his results are these: i. The learning of new systems does not progress gradually, but by " leaps ;" 2, Between the ages of 13 and 14 the boys showed a marked increase in ability; 3,. Between the ages of 12 and 13 the girls showed this rise in ability. It was noticed that the onset of puberty gave a decided increase in the ability to work independently in numbers. Prior to this time the children as a rule work mechanically, according to " copy." The boys reached the period of independent work from i to i^ years in physiolog- ical age later than the girls ; hence, the girls of the same age as the boys after the beginning of puberty are usually more than one year ahead of them in number work. That is, problems which boys can solve independently in their eighth year in school can be as easily solved by girls in their seventh year. Rice and Courtis both found the 6th grade especially troublesome, as there were disturbances in the scores of that grade in the many records which they gathered. In the recent tests in Boston (38, 23) this errancy was found in the 7th grade. Since Boston admits children at five years CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 173 of age, the 7th grade there would correspond to the 6th grade in physiological age elsewhere. It appears from the tests of Voigt, Rice and Courtis that the dawn of adolescence, affect- ing as it does markedly the physiological nature of the child, has also a great effect upon its ability to do accurate and independent work with numbers. Before this time, mechan- ical work is done; after this period the rise of independence and self-reliance changes the emphasis to the reasoning phases of arithmetical solutions. Value of Drill. — ^A number of experimental studies have been made to arrive at some fundamental facts concerning the value of drill. Brown (13, 8iff) gave the Stone tests to 6th, 7th, and 8th grades of the practice school of the Eastern Illinois State Normal School, consisting of 51 pupils, 18 boys si^d 33 girls. Two sections were made of them. One was drilled upon fundamentals for five minutes each day for thirty days, while the other pursued regular work. A test at the end showed that the drilled pupils made a much larger ad- vancement than the others. His results also showed that the 6th grade children profited more from drill than did the next higher grades. Later, he carried his experiments into three school systems (13, 488), confining the tests to the 6th grade and to the four fundamental operations. In all, he tested 222 pupils, no boys and 112 girls. Only twenty days were allowed between the first and the second tests, during which time one group in each school was given five minutes extra drill in the fundamentals besides the regular work which the other section followed. The results showed a gain on the part of the non-drill group in problems solved of 6.4% ; in Addition, of 6.8%; in Subtraction, of 11.9%; in Multipli- cation, of 10.9%; in Division, of 15.4%. The drill group gained on these same items respectively: 16.9%, 18.5%, 32%, 24.1%, and 34.2%. These are only the aggregates. Of the 112 cases of drilled pupils, 95 gained, 5 did not advance and 12 lost. Of the no cases of non-drilled pupils, 50 gained, 7 had the same score as at the beginning, and 53 lost. In the individual studies and aggregates submitted it appears that the drilled groups gained from two to three times as much as the non-drilled groups. "All teachers of the drill classes reported an improvement also in the text-book work." The drill excited them to keener interest in the regular lessons (13, 489). In March, 1912, Phillips (93) gave the Stone tests to 33 boys and 36 girls of the 6th, 7th, and 8th grades. Each grade was divided into two groups, one pursuing regular work and 174 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC the other receiving in addition a ten-minute drill daily upon fundamental operations and upon reasoning upon mental problems. At the end of two months the two groups were again submitted to the Stone test with the following results: 6th grade non-drill, gain in Fundamentals, 55% ; drill group, 45% ; 7th grade, in Fundamentals, non-drill, 10% ; drill group, 22% ; 8th grade, in Fundamentals, non-drill, 16% ; drill, 25% ; all grades, gain in Fundamentals, non-drill, 27%; drill, 31%. In Reasoning : the 6th grade gain was : non-drill, 36% ; drill, 55%; 7th grade, non-drill, 17%; drill, 29%; 8th grade, non- drill, 12%; drill, 15%; all, non-drill, 22%; drill, 33%. Starch (114) gave 15 observers eight preliminary tests, six in arithmetical operations and two in auditory memory span for numbers. Of these observers, 8 were then given fourteen days' practice in mental multiplication of 50 prob- lems each day, totaling 700 problems. The other 7 observers were given no practice. On the second test the practiced ob- servers showed from twenty to forty per cent improvement more than the others in the arithmetical operations, while there was little change in the memory span in either group. Thorndike (125) experimented with 33 adults to learn the increase in efficiency in mental multiplication, judged by the reduction in the time required. He used no figure below 3 and none was repeated. All his subjects showed improve- ment through drill, and he says: " The fact that these mature and competent minds improved in the course of so short a training so much as to be able to do an equal task in two-fifths of the time first taken is worthy of atten- tion." "The most ardent advocate of the general influence of specific practice would not, I judge, claim that ten hours drill in any one thing could improve an already well-educated adult 50%, or 5%, or even 1% in the average of all kis intellectual processes." He found a rapid rise in the rate of improvement during the early practice, an observation generally confirmed by all ex- perimenters. At another time he used 19 adult subjects, 8 men and 11 women students in Columbia University, giving them each day for seven days 48 columns of ten figures each to add, in all 2,592 additions. These subjects were able on the second test to reduce the time 31% and the errors 29%, with a total improvement of 29%. Only fifty-three minutes' practice was actually given. Thorndike says of this test: " That the practice represented by only 2,592 additions made by an educated adult whose addition associations have been long estab- lished and often used should produce an improvement of three- tenths bears witness to the continued plasticity and educability of the synapses involved." CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 175 In another connection he has stated (127, 290) : " So apparently simple an ability as ordinary addition of integers can be shown to require analysis into at least nine separate abilities, each of which probably requires further analysis, in one case, into perhaps ninety component ability-atoms." Similarly, Donovan and Thomdike (128, 134) used 29 fourth- grade pupils and found in a practice series, given two periods of two minutes each per day for three school weeks, or a total of thirty two-minute periods, the average rose from 294 to 4^/2 examples per minute. Kirby (65, 24) studied experimentally 732 children in the fourth grade, testing them before and after sixty minutes of practice, after the method just mentioned. He found the average score changed from 31 columns with 24 correct per minute to 50 columns with 37 correct. The children, therefore, gained 61% in attempts and 54% in correct additions, and maintained the same rate in their accuracy and in their speed, almost. Hahn (128, 134) has obtained similar results. Kirby also gave a series of drills for fifty minutes to 606 third and fourth grade children. At the beginning the children averaged 40 simple divisions per minute, with 37 correct, an accuracy of almost 93%. At the close of the drill series they performed 73 divisions per minute with 70 correct, an accuracy of almost 97%. They had gained 83% in speed and almost 90% in their accurate results. At McLean Hospital, Wells found that ten nurses, five men and five women, in oral addition of digits printed one above the other, in five weeks, practic- ing five minutes per day, six days in the week, — a total of 150 minutes of practice, increased their speed nearly 100% and maintained about the same rate of gain in accurate work. The five women on the first day performed 1,115 additions in five minutes, while the five men reached 1,120. On the thirtieth day the women aggregated 2,210, and the men 2,178 additions. The lowest score at first was made by a woman who got 150, and at the end she reached 280. The highest at first was by a man who got 290 and on the thirtieth day he went to 540. It is observed that the lowest and the highest made practically the same rate of gain. Whitley (134, 129) tested nine subjects in mental multiplication, giving a prac- tice series of three examples per day for twenty days, omitting Sundays. These subjects averaged 2.8 minutes practice per day, or a total of fifty-six minutes. The results show more than 100% gain in speed, with no ill effect upon accuracy. In the Dumfries schools, Jeffrey (61, 392) selected 9 boys and 9 girls, and placed them in three groups according to their mental ability, as disclosed in previous school work. 176 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC Each group consisted of three boys and three girls, the groups being classified as bright, average and dull. They were all given fifteen-minute drills for five consecutive days and then were tested with the following results: The 9 boys made 19,717 additions and the 9 girls made 18,304, with 126 errors by the boys and 134 errors by the girls. The group results were: bright boys gained from first to second day, 49.1%, girls, 32.7% ; average boys, 28.2%, girls, 32.3% ; dull boys, 25.7% ; girls, 44.6%. From first to fifth day the gains were: bright boys, 90.9%, girls, 58.9% ; average boys, 62.9%, girls, 67.2%; dull boys, 47.4%, girls, 91.1%. Short and Long Periods of Drill. — Kirby tried the effect of dividing the total practice time into periods of different lengths. He used 1,338 children (65, 63) and his results have considerable pedagogic value. Using 100 as a standard of comparison his data may be expressed as follows : Gains made with 22-minute practice periods, 100; 15-minute periods, 121 ; 6-minute periods, loi ; 2-minute periods, 146.5. These were in addition. The results in division were: from 20- minute periods taken as a basis, 100; lo-minute periods, 1 10.5; 2-minute periods, 177. That is, the short periods of practice scattered over more days give a higher rate of gain. These are subject, however, to discount since the children during the longer time elapsing from the first to the last test would gain more from their regular work than those taking the longer single periods of practice. In the Whitley tests already referred to it should be noted that after twenty days, with practice upon only three examples each day, the subjects were able to reduce the time from 338 seconds, with 1.7 errors per example to 135 seconds with 1.4 errors per example. Thorndike drilled sixteen subjects continuously on sixty examples, the number used by Whitley, but employing only one period of practice, varying from 2 to 12 hours. His adult subjects took an average of 352 seconds per example with 1.2 errors on the first test, and 160 seconds with 0.8 errors on their final test. Their total time of practice averaged higher than did those tested by Whitley, while their gain was not so marked. However, it is to be noted that Thorndike's subjects took their final test at the close of three or four hours of unrelenting practice, and some allowance is to be made for their jaded condition. Permanency of Improvement through Drill. — Wells tested for the permanence of improvement in the adult nurses who had taken the drill from January to April, 1910, giving six of them two tests in December, 191 2, after a lapse of 2 years and 8 months (128, 323). In January, 1910, they had CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 177 averaged 234 additions in their first test and 274 on the second, while their average score in the final test in April, 1910, had been 447. In December, 1912, these same adults on their first test scored 343 additions and the next day- raised it to 375. Kirby (65, 7 iff) tested 258 of his fourth- grade children from three to twelve weeks after they had relinquished their practice and found them able to do as well as they had done on their previous final tests. He again submitted 152 of them to tests in September, at the opening of school after the summer vacation, and they showed a decided loss in speed with some decrease in accuracy. Prac- TABLE A Twenty Leading "Type-Errors," Made by Two Hundred and Thirty- eight Children in Addition of Digits, Arranged in Order of Frequency. Compiled from Phelps (92) Rank Combination Total number of errors Percent- age of the attempts Number of children making the error Percentage of the children making the error Number of children making the error more than once Percentage of children making the error two or more times 1 9 plus 7 395 3.32 120 50.42 63 26.47 2 8 plus 5 369 3.10 109 45.80 59 24.79 3 9 plus 6 320 2.69 94 39.49 49 20.59 4 9 plus 3 304 2.55 91 38.23 47 19.75 5 9 plus 5 298 2.50 86 36.13 38 15.96 6 9 plus 8 145 2.44 79 33.19 37 15.55 7 8 plus 4 142 2.38 79 33.19 33 13.86 8 8 plus 7 137 2.30 74 31.09 32 13.44 9 6 plus 5 135 2.27 72 30.25 31 13.02 10 7 plus 5 134 2.25 71 29.83 31 13.02 11 7 plus 3 120 2.02 67 28.15 26 10.92 12 8 plus 8 118 1.98 67 28.15 25 10.50 13 7 plus 4 232 1.95 60 25.21 25 10.50 14 8 plus 3 231 1.94 59 24.75 24 10.08 15 7 plus 6 186 1.56 58 24.37 23 9.66 16 3 plus 3 87 1.46 52 21.85 21 8.82 17 7 plus 2 158 1.33 52 21.85 21 8.82 18 5 plus 3 78 1.31 48 20.17 20 8.40 19 6 plus 3 71 1.19 48 20.17 17 7.14 20 9 plus 2 71 1.19 44 18.48 15 6.30 178 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC tice drills of fifteen to forty-five minutes were sufficient to restore the speed of the former tests. Brown (13) tested his practice school class, already discussed, after the summer vacation of twelve weeks and found that the drilled section in September was able to raise the averages made in June, while the undrilled section either showed no gain or had retrograded. Type-Errors. — Phelps (92) reviews the Otis-Davidson tests upon 270 children in the eighth grade and uses their data in the study of errors. These tests were given in the grammar school at San Jose, Cal., and 238 sets of papers, 5,950 separate tests, were obtained. In Table A I have arranged the chief mistakes in addition in the order of their frequency. There is a remarkably consistent showing in the table, — the combinations, which were the more difficult as shown by the number of times they occur, are also those which are made by the largest number of children and are more often repeated by the same child than the other combinations. Of the eight schools reported, those showing the highest speed had the lowest percentage of accuracy. Phillips (95, 245), out of 440 returns made to him, gives the following list of difficult combinations of the digits in addition: 157 find 9 plus 7 the hardest; 88, 7 plus 8; 34, 6 plus 7; 42 find 7 alone troublesome; 18, 9 only; 26 are bothered in using 3, 6, and 8; 327 mention 7, and 204 give 9 in the list of digits they find hard to handle correctly. In 1905 in Budapest, Ranschburg (97) tested 153 children, to whom he gave 65 tests, 20 each in Addition and Subtrac- tion, 15 in Multiplication, and 10 in Division, in an attempt to determine which fundamental operation is the most diffi- cult to school children. If we consider accuracy alone, his order, placing the easiest first, is : Addition, Multiplication, Subtraction and Division; but if they are arranged according to speed they are: Multiplication, Addition, Division and Sub- traction. If both speed and accuracy be combined the order is: Multiplication, 68.75%; Addition, 60.25%; Division, 50.65% ; and Subtraction, 46.82%. Manner of Adding. — Amett (5, 327fT) tested eight adults in their habits of adding, using 15 columns of 2y figures each. He had them add for thirty minutes, rest a few min- utes and then add for thirty minutes more. Nearly 200 col- umns were added. It was found that some of the subjects employed straight addition, following the columns and making as many additions as there were intervals between figures, while others used combinations of digits. Out of a possible CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 179 use of 840 combinations one announced 840 results, while another gave only 519 results for a possible 810, having made 171 combinations of two figures and 43 combinations of three figures each. Cole gave three tests to 35 persons selected at random to determine their habits of adding upward and downward. The first test consisted of 20 columns of 40 figures each, the subjects adding the odd columns upward and the even ones downward. The downward adding required 15.3% more time than the upward adding, but there were fewer errors in it, — 54% of the errors being made in the upward and 46% in the downward adding. The second test consisted of 10 col- umns, identical for the upward and the downward adding, and a third tested the reading of numbers from left to right and from right to left in a horizontal line. It was found that the established habit of adding upward gave more speed but resulted in greater liability to error. In the reading of numbers the average time from left to right was 34.4 seconds with 62 errors, while the reading to the left averaged 37.1 seconds with a total of 36 errors. The usual reading habit afforded greater speed to the right, but the additional atten- tion that was demanded in the reversed reading resulted in a higher accuracy (23, 83!?). Socialisation of Arithmetic. — Paine reports a recent experi- ment in Boston with some sixth-grade children who were slow and indolent in Arithmetic, but not mentally defective. They were chronic ** failers " and were particularly deficient in Arithmetic. A " grocery store " was fitted out for them with enough of the real supplies to make the experiment more than symbolic. It was found that the children took on new life not only in Arithmetic but in their Language work as well. The results in Arithmetic are given as follows: 1. Increased accuracy and speed in computations. 2. Confidence was established in independent solution of problems. 3. A good drill was afforded in making up original prob- lems. 4.Some valuable training was gained in business methods. Dooley (37) also reports some interesting work in the suc- cessful motivation of Arithmetic in the Massachusetts Indus- trial School. The Wording of Problems. — The effect upon results occa- sioned by a change in the wording of problems is reported by Phillips (95, 268) in the case of 224 teachers or those preparing to teach. When 40 problems involving Gain or 180 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC Loss, expressed in common fractions, was given them 8i.6% of them solved all correctly while the others averaged 5 prob- lems missed. Later the same problems were given to 212 of the same group, with a change only in the wording from fractions to per cent and 62% solved all correctly, while the others averaged 3 missed. During the past year Courtis (31, 4) turned his attention to the question of the wording of problems. He was able to construct twenty-one varieties of problems, based upon a single situation, by changing the form of the question and the relative position of the phrases em- ployed, and his tests showed that one of these problems, measured by the errors made by children in solving them, was nineteen times as difficult as another. That the mere rearrangement of the words and phrases in problems causes such wide difference in results should have careful consid- eration from both authors of Arithmetics and teachers of that subject. Correlation of Abilities in Arithmetic. — Lobsien (73) con- cludes upon some experimental studies of arithmetical abilities with the following: 1. There is no correlation between ability to remember num- ber images visually and the ability to write numbers. 2. The greater the ability for solving problems in the head, the weaker is the memory of numbers gained through the eye. 3. The highest correlation exists between acoustic number- memory and ability to write numbers. 4. The good head-reckoner generally performs written work well, and znce versa. 5. Acoustic memory of numbers and ability to perform opera- tions well have a smaller correlation. Stone (117, 43) believes that ability in any fundamental, with the exception of Addition, implies ability in an equal degree in the other fundamentals, nearly, and he found that many factors influence individual abilities. In the extensive tests in the New York City schools, Courtis reports (30, 79) that speed and accuracy have no correlation and (p. 84) " it was found that a child with good reasoning ability did not make mistakes in the abstract work." So far as his analysis of the results goes, accuracy is dependent upon reasoning and simple reasoning is directly related to ability in abstract work, high scores in "test 6" (a simple reasoning test) being associated with high scores in "test 7" (a test in the fundamentals). Up to a certain critical point there appears to be a definite correlation between a good knowledge of the tables and ability to work speedily in the abstract examples, and a lower corre- CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 181 lation with accuracy. The curves given by Courtis for the 13,629 boys and the 13,542 girls agree quite closely. The evidence which he submits warrants the deduction that, in general, a knowledge of the tables makes for speed and accuracy up to a certain point, beyond which other factors play such an important part that further knowledge is of no benefit (92-96). Winch (137) tested four schools in 1909 and one in 1910 in London, and reports his findings on this question, as the writer has gathered them together, as follows: TABLE B Correlation Tests in London Schools Tests Grade Number of pupils Average age Correlation between accuracy and reasoning Eflfect of practice 1 7 and 8 32 girls 13 .68 Improvement. No "transfer" was observed 2 3 43 girls 10 .79 Accuracy gained 3 4 38 girls 10.5 .69 21% gain in accuracy 4 4 35 boys 10 .85 20% gain in accuracy 5 4 72 boys 10.25 .736 40% gain in accuracy From his series of tests it appears that he found accuracy in computation to accompany good reasoning ability, but im- provement in computation did not affect perceptibly accuracy in reasoning. Starch (114, 310) gave special attention to the question of transfer of training in his investigations, and concludes : " The improvement in the end was due to the identical elements acquired in the training series and directly utilized in the other arithmetical operations." Comparison of Adults and Children. — Freeman (40a) sought in some experiments with 14 adults and 14 children, ranging in age between 6 and 14 years, to determine, if pos- sible, how children differ from adults in the elementary scope of attention, and also what differences there are in the num- ber of objects which may be grasped in a single act of atten- tion by adults and children. He used spots of light thrown upon a screen as stimuli, and varied the time exposure from .018 to .040 of a second for the adults to more than a second for some of the children. He found the range of attention 182 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC of adults and children to be nearer together than is generally supposed (p. 309). In adults he thinks the range may aver- age 6, although his observers varied from 4 to 7, while chil- dren between 12 and 14 years of age will average 5 and younger children, 4 (p. 309). He gave 1,806 exposures to his adults, while Nanu gave only 100, and he also figured out very carefully the optimum distance at which to place his observers from the screen. For these reasons he thinks his results are better. Nanu found with her observers a decided tendency to underestimate the number of spots shown; only I of her 5 observers overestimated the number, while 11 of the 14 sitting for Freeman overestimated them. He reports one pure analytic type, four mixed types with strong analytic inclination, two mixed with inclination to the synthetic type, and seven of the pure mixed variety. He differs strikingly from the findings of Nanu, who reported that she found the synthetic thinkers always inclined to underestimate the num- bers and the analytic type to overestimate them. Of Free- man's observers, the three who underestimated the numbers showed no tendency toward synthetic thinking, and the two who were at all inclined to synthesis overestimated the spots, — one in 94% of the erroneous judgments, and the other in 86% of the cases, — while four of the five who gave evidence of analytic thought underestimated them. Some of the introspections seemed to show that the ob- servers could image the groups of objects and describe them without having a grasp of the correct number, and he con- cludes that the number name is not essential to a compre- hension of a group of objects. " Neither the word nor the name is necessary for the number-concept." In his experiments with children he found less satisfactory results, as they were unable to give reliable introspections and only a small number of children of any age was used. Of the 14, two were 6 years of age; two were 7; two were 8; two were 10; four, 12; one, 13; and one, 14. As these were scattered through six of the eight grades, with no repre- sentative in the fourth and sixth grades, it gives small results for each grade. These children had been taught numbers upon the Russian reckoning-machine, so they were not en- tirely in new experiences. He found it necessary to exclude from his final results the four younger children's reactions, as they were unreliable. Of the remaining 10, he made two groups of 5 each according to ages. The 5 children between 8 and 10 years of age showed marked differences from the 5 who were 12 to 14 years old, while the older group resem- bled very much the adults, both in range of attention and in its behavior. CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 183 As a general result of his experiments Freeman concludes that: (i) Children as a rule comprehend a number of objects less correctly than adults. (2) Children prefer a horizontal arrangement of objects to be seen. (3) Children show a more rapid decline in correct answers as the number of objects is increased. (4) The range of attention in children is from i to 2 less than in adults. (5) Children underestimate more frequently than adults. (6) Definite arrangements in groups is less favorable for children than for adults. (7) Attention in young children is very irregular. (8) No correlation was found between school-talents and cor- rect answers. (9) Groups of 5 were better for the children and groups of 4 for adults. This last finding agrees with the conten- tion of Lay that the quadrate form is the best; but Freeman finds it only true for adults, rather than for school-children. It must be confessed that we do not yet have adequate results to justify any conclusions upon the perception of simultane- ously presented objects as a basis for early number training. Too few observers have been used so far. From such experiments as those of Freeman, one may be led to infer that the basis for the difficulties encountered by children and adults in mastering the multiplication table lies in the inability to handle numbers in groups of more than 5 or 6 readily. To master the table of 7's or 8's or 9's, one has to group the numbers in bundles of 7, or 8 or 9. The pupil usually finds the numbers below 6 rather easy in com- parison with numbers between 6 and 10. To say the 9's, one has to group the numbers up to 90 in bunches of 9's, and the attention has to pass rapidly over the groups if the learner is at all visually minded. Individual Differences. — There is to be seen from the vari- ous experiments reported a very wide divergence in number ability among school children. These differences are shown in the aversion of some to the subject as a school topic, while others choose it as their favorite branch. This is often caused by wrong motivation at some previous time, or to attitudes of parents toward the subject, or to poor teaching in a lower grade, in the case of those who dislike it; and to home en- couragement, proper motivation, or good teaching, or per- 184 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC haps, to all these influences combined in the case of those who prefer it. Frequently, it has been found that differences in habits of thinking cause variations in school interest in Arithmetic. No doubt, the varying degrees of interest may sometimes lie far back of school experience, in the child's opportunities to satisfy his inclinations for number in the nursery. Provisions that are made for the natural growth in number in young children in various materials afforded in the home, coupled with an active interest on the part of the mother, often determines the future bent of the child. While no other subject has been as much taught in the modem school, it is equally true that in no other subject has there been so much bad teaching. So to-day, partly through im- perfect teaching and a variety of pre-school inclinations, we find in the school grades children of almost every degree of advancement in the same grade. Some in the first or second grades have as good ability in numbers as others in the sixth, seventh, or eighth grades. Sex Differences. — In addition to the effect of puberty al- ready cited from Voigt, a number of observations have been recorded upon the differences in arithmetical abilities depend- ent upon sex. Ballard (9, 18) says that in a series of tests in the London schools the girls showed better mechanical skill in the solutions, but the boys did the problems better, and ** on the whole the boys were considerably ahead of the girls." Phillips (93, 163) tested 69 pupils in the Granite Falls, Minn., schools from March to May, 191 2, and found in the progress made in drill work that ** the girls did better than the boy« in tests in fundamentals " and the boys " did better work in reasoning," while the boys made a greater gain between the tests than did the girls, their gain being about 24% over that of the girls. In his extensive tests in New York City, Courtis (30, 136) found also that " the girls exceeded the boys in the speed tests in multiplication, but they fell below them in accuracy in reasoning." From a comparison of all the scores made by the boys and the girls it is seen that the girls excel the boys in mechanical work in the fundamentals and the boys excel in simple reasoning. He concludes : " Dif- ferences between the abilities there undoubtedly are, but whether due to sex or to environmental influences the differ- ences are too slight to be of any significance so far as present knowledge goes." Smith (109) reports on 3,869 students tested in the Normal School at Cortlandt, N. Y., as follows: Of 1,265 ^^^ put on examination, 58.7% passed with an average of 84.1%. CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 185 Of 2,603 women put on examination, 50.6% passed with an average of 83.4%. He also gives the records of 1,462 men and 1,564 women at Ypsilanti, Mich., which show practically no distinctions are to be made in Arithmetic on account of sex. The Rice Tests. — In the autumn of 1902, Dr. J. M. Rice gave a test of eight problems to 6,000 school children in seven cities (98, 100). The children were chosen from grades four to eight inclusive. In all, 18 schools, some in the slum districts, some in the better districts and some in aristocratic neighborhoods, were included. He studied the effect of home environment, size of classes, total time per day given to Arith- metic, average age of pupils, forenoon and afternoon periods of recitation, methods of instruction, teaching ability of the instructors, and concluded that none of these was the deter- mining factor in securing good results in the subject. Rather surprisingly he puts the whole responsibility ultimately upon the supervision. " This means (p. 136) in other words that the controlling factor in the accomplishment of results is to be found in the systems of examinations employed, some systems leading to better results than others." He found wide variations in the upper grades, mechanical errors increasing in them, with a decided deterioration in the 5th and 6th grades. Cities usually ranked with their individual schools; that is, good work in one school usually signified good work through- out the whole city. The Stone Tests.— Dr. C. W. Stone (117) gives detailed data from 26 school systems scattered well over the United States and comprising tests in Arithmetic which he personally gave to 152 classes of pupils in the 6A grade. Of the sys- tems tested, 6 were located in New England, 1 1 in the Middle East, and 9 in the Middle West. The tests were given under controlled conditions, and covered the fundamental opera- tions and the children's ability to reason upon the solutions. With 6,000 sets of papers to study he draws the following conclusions : 1. The net result of arithmetic work in the first six years is several products, rather than a product. The^ study called Arithmetic makes demand upon a plurality of abilities (p. 43). 2. There is a great variability in the products of different systems, and greater still among individuals in any sys- tem. The variability among boys does not appreciably differ from that among girls. 186 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 3. The possession of a certain amount of ability by a system is a better guarantee of the same amount of another ability than the possession of a certain amount of ability by an individual is that he will have the same amount of another ability. He agrees with Rice upon the eifect of good supervision, but found different results on the tests in reasoning (p. 45). He had the courses in Arithmetic in these 26 systems rated for him by 21 professors and graduate students in education (p. 71) and compares the results of his tests with these ratings. He says: "The situation seems to be that the course of study is not at present the factor that it ought to be in producing abilities. In certain systems it is evidently working well, but in others there is a wide-spread disparagement between ex'cellence in abilities and excellence in the course of study." "The course of study may be the most important single factor but it does not produce abilities unless taught. The other essential features for successful teaching are teachers and children of usual abilities, a reasonable time allot- ment, intelligent supervision and adequate measurement of results by tests (p. 91). The Courtis Standard Tests. — Following immediately upon the results obtained by Stone, Dr. S. A. Courtis began by giving the Stone tests to 317 girls in the Liggett school at Detroit, in 1908 and 1909. These girls were scattered through the grades from the 3rd to the 13th. Using his results as a basis, in September, 1909, he devised a new set of tests covering speed in each of the four fundamental operations, one in copying figures, two in reasoning and one general test in all four fundamentals. Under controlled conditions these were given to the same school in September, 1909, and in June, 1910. Among other results he found as did Rice that the 6th grade was a " notoriously difficult " one, although it had ranked high as a 5th grade the previous year (29, p. 361). From this beginning, Courtis came to believe that a uniform standard test could be devised, so he sought during the school year, 1910-11, to establish such a standard. He gathered papers from near 9,000 children in from 60 to 70 schools scattered in 10 states. These children had been given his 8 tests and a " standard " table was constructed from the results, which is given here after corrections have been made in it from later facts gathered altogether from 66,837 children and revised to August, 1913. From Table C it will be seen that a 6th grade child should be able to give correctly 50 combinations in addition of digits, 38 in subtraction, 37 in multiplication and division, CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 187 and copy figures at the rate of 92 per minute. The scores from grades 3 to 8 inclusive are revised to suit the figures from 66,837 tests, while the others, I understand, are those made out from the first studies gathered from 19 cities. TABLE c Courtis Standard Scores, Showing What a Should be Able to Accomplish in Child in Each Grade One Minute Grade Number of simple additions Number of simple subtractions Number of simple multiplications Number of simple divisions Number of figures copied 1 (6) (6) (29) 2 (21) (12) (10) (12) (51) 3 26 19 16 16 63 4 34 25 23 23 75 5 42 31 30 30 84 6 " 50 38 37 37 92 7 58 44 41 44 100 8 63 49 45 49 108 9 (65) (50) (50) (50) (120) 10 (57) (45) (43) (46) (112) 11 (59) (47) (44) (48) (114) 12 (61) (48) (44) (49) (112) 13 (71) (56) (50) (56) (116) 14 (74) (51) (58) (59) (124) The New York City Tests. — From March 15 to April 26, 1912, Courtis applied his tests in New York City to a list of 33,350 pupils, representing a school register of 40,000 pupils, or about one-tenth of the city school population from the 4th up to the 8th grade. Representative schools were selected in various parts of the city; 21 schools furnishing 380 classes with 12,147 pupils, were in Manhattan borough; 9 schools, with 148 classes and 4,488 pupils, were in the Bronx; 18 schools, with 315 classes and 10,243 pupils, were in Brooklyn ; 2 schools with 2^ classes and 646 pupils were in Richmond borough ; and 2 schools with 37 classes and 1,145 pupils in Queens. From these, 28,669 complete returns were received, but only 27,171 were tabulated because of apparent irregularities in the remainder. Courtis went to New York and personally took charge of the tests at the request of the " Hanus Committee in Charge of Educational 188 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC Aspects of School Inquiry." His purpose was to determine, if possible, the following: 1. The standard of achievement in fundamental operations with whole numbers, and in simple reasoning. 2. The relative achievement of the schools tested, as measured by standards. 3. The relative achievement of grades and individuals, as measured by standards, so far as is necessary to indicate to teachers, principals, and superintendents how such knowledge could be used to make their work more efficient. 4. The relative achievement of New York City schools as a whole, as measured by standards derived from tests in other cities. In the following table will be given the score of the various grades in each one of the eight tests used and at the same time the highest score made by any pupil, with some other data that may assist in one's understanding the gross results of the experimental study. It is not possible for us to enter upon any comprehensive consideration of the minor details reported by Courtis in his 158-page booklet. A few findings have already been mentioned and a few others remain to be noted after the table is studied. TABLE D Correct Score Averages Made by the Different Grades in the New York City Schools Grade 4th 5th 6th 7th 8th 9th 10th nth 12th Numb 5396 5386 5670 4771 4502 440 257 179 120 Avera ge age of the pupils . . 10.5 11.3 12.4 13.9 14.6 No. Kind of test Max. Score 1 Addition,— speed... 125 41.9 50.2 56.9 62.2 69.5 71.6 71.7 73.9 74.2 2 Subtraction, — speed 125 29.5 36.8 41 45.8 52.2 52.2 55.3 55.2 54.1 3 Multiplication, — 125 28.7 35 38.3 40.9 45.8 46.5 46.8 48.6 46.6 4 Division, — speed . . , 125 26.6 34.7 39.7 44.6 50.9 52.5 52.7 54.2 55 5 Copying figures, — speed 205 75.4 85.5 92.5 100 106.8 98.8 104.5 109.4 105 6 6 Reasoning, — one step 16 1.8 2.3 3 3.7 4.4 5.1 5.1 5.1 5 7 7 Fundamentals 19 4.2 5.8 7 8.5 10.1 10.9 11.5 10.5 11 8 Reasoning, — two steps 8 .^« 1.3 1.7 2.1 2.5 2.6 2.7 2.7 ' CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 189 It must be borne in mind that the Courtis tests are exactly the same for all the grades and that he holds that a score of forty answers per minute means double the ability that gives only twenty per minute (p. 17), and to change from 20 to 25 answers corresponds to a change from 40 to 45 per minute. The errors that were shown in the papers he classifies as follows: 1. Carelessness in bringing down the wrong figure in division or placing partial products in multiplication under the wrong figure, 12.5%. 2. Copying incorrectly, reversing figures, as 639 for 693, 8%. 3. In fundamental combinations, 50%. 4. Scattered, such as errors in carrying, etc., all the rest. Many errors were found in handling zero. It is worthy of mention here that in Columbia University, in the summer of 1910, 41 adult graduate students, teachers and superintendents were tested and 18 made 104 mistakes in zero combinations particularly when zero occurred in the multiplier. He found in the New York schools, as has been found wherever tests have been applied, that the work in funda- mentals is very low both in speed and in accuracy in com- putation and simple reasoning. The critical period for the mastery of these fundamentals seems to lie down in the lower grades. In the introduction to the Courtis Report, Dr. Hanus calls attention to the low degree of efficiency in the schools and says : " Children of every level of ability are found in every grade and differences between individuals greatly ex- ceed the difference between grades." For this reason the fundamentals should be adapted to the individuals and they should be taught in the light of individual capabilities. " This condition is universal and is not due to lack of effort or other conditions that could be easily removed, but to a neglect of one basic factor, — the difference in the powers and capabilities of children," (p. ']6'), because children are already at school entrance highly specialized in their mental habits. Courtis thinks (p. 130) that a more simple and practical course in Arithmetic, based directly upon the social needs of the chil- dren, would influence for good a greater number of both children and teachers. The Boston Tests. — The same eight tests were given by Courtis to 29 of the schools in Boston, in October, 191 2, and again in March, 1913. More than 500 classes and about 25,000 children were included. These tests were designed to determine the following facts: 190 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 1. The standard of arithmetical work in the Boston schools, and their comparative standing with other schools already- tested. 2. The nature and degree of change produced by six months regular work in Arithmetic. 3. The effect of certain special methods of individual in- struction. Supt. Dyer devotes only thirteen pages of his recent report (38) to these tests, yet some valuable facts are disclosed: 1. Boston is lower in abstract work and higher in reasoning than New York. 2. A comparison of the tests in March with those six months earlier shows that 53% of the pupils made improvement, 30% stood still and 17% lost. 3. Four special methods were employed in the experimental study : (a) In one group, each teacher took one period per week to work with individual pupils who seemed from the October tests to need particular strengthening upon certain points. This was continued for twelve weeks, and the group as a whole gained 14%. (b) A second group had the assistance of an able specialist, one to each school being assigned to the task of giving individual help to children sent by the regular teachers. This also ran for twelve weeks, and 2,187 children out of 3,443 received individual assistance, in 60,000 interviews lasting 15 minutes each. The report says this method did not yield the results expected. (c) A third group pursued regular work with no reme- dial help, and were used as a " control " group in measuring the others. (d) A fourth group was formed in March, of pupils who had not taken the October tests, and they were given special daily drills on the fundamentals for ten minutes each day during the regular recitation period. In the March tests this group attempted fewer problems and examples than the others and "this method was the least effective." A comparison of the three drilled groups with the " control " group shows that: I. Group one, in which the regular teachers devoted one period per week to a class drill upon fundamentals, exceeded the " control " group by 8% in accuracy. CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 191 2. Group two, in which a special teacher gave individual help, exceeded it by 5%. 3. Group four, in which the ten-minute drill upon funda- mentals preceded the test, exceeded it by only 3%. These facts seem to emphasize the value of systematic work under the regular teacher and to discount the common prac- tice of " cramming " for tests and examinations. Personal Observation of the Recent Boston Tests. — On Thursday, January 8, 1914, the writer went to Boston to observe the giving of the new Courtis tests which were to be given to more than 20,000 children, beginning that day under the supervision of Miss Rose A. Carrigan, of the Boston Normal School. Three schools were selected, one mixed and one each in which the boys and girls are taught separately. Miss Carrigan had furnished me a list of all the schools to be tested each day, so I went on Thursday without her knowledge to schools of my own selection. In all, I saw the tests given to 8 rooms, 5th, 6th, and 7th grades, 350 children being present and taking the tests. I saw 15 of the young lady " cadets " doing the testing at the three schools, two working together in each room. One would explain the sig- nals to be observed while her assistant manipulated the signal- box and announced the time to start and quit. There was splendid management in handling the tests and I have no criti- cism to offer on the fairness of the application of the tests. The children seemed to enjoy the game and so far as I could see took no notice of my presence or that of the master who accompanied me to their room. 53 cadets were busy in these tests after being trained by Miss Carrigan for three days. Another test was given the second week in April, 1914, but it will be several weeks before results can be known. These tests are to (i) set definite standards for each grade, (2) measure the results of each teacher's work, (3) assist in motivating the children's work, and (4) to furnish some studies in the grading. Criticism of Standardisation. — It is proper that I give some discussion upon the general proposition to attempt a " stand- ardization " of the achievements in Arithmetic, which is, after all, only a subordinate notion of " standardization " in all studies. There appears a strong tendency to-day to try to standardize everything, products and producers alike, and educational circles seem to have been caught in the current. The trades are said to have a definite number of doors for a carpenter to hang in a day, or a specific product to be turned out, beyond which a good workman will not endeavor to go. With discriminating accuracy there are attempts to 192 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC " standardize " in this world-movement not only Arithmetic, but also penmanship, compositions, cattle, peanuts and parents ! Courtis has selected a certain line of level from the thous- ands who have been tested by his method and sets these levels up as a standard for the schools everywhere. This would not seem so serious did we not find him saying (31, 14) : " It should be noted that standards will not produce uniform products unless they are treated both as goals to be reached, amd as limits not to be exceeded." He points out that the average score, for instance, of 11,059 8th grade pupils on test No. 7 was 9.5 examples correctly done, but 38% of these pupils had been so overtrained that they exceeded this score by 10% to 100%. " These high scores of school children represent waste effort." " When standard ability has been attained, additional degrees of me- chanical skill are products of the least importance." While the writer is in complete sympathy with every at- tempt to place Pedagogy upon a sound and genuine scientific basis, he does not consent willingly to the effort to " Pro- crusteanize " the schools by requiring them to be measured and directed by semi-arbitrary standards. Abilities of chil- dren are too widely variant and future callings are too diverse for us to agree that each child in the great public school system shall be moulded into the same set form in Arith- metic, or, indeed, in any school subject, by accepting stand- ards which are to be reached but not exceeded. The present movement is awakening much interest among the school public and must result in good. Its chief temptation lies in the extravagant application of some of its obviously useful features until the practical educator will be led to reject even what help it should be able to offer. The selection of a semi-arbitrary standard in Arithmetic has gone no farther than the four fundamental operations and simple problems in reasoning, and has not approached fractions, denominate numbers, ratio, percentage, mensura- tion, — yet to me it seems much like taking the average per- acre corn crop of the country, or more correctly, of a few sections, and setting this up as a " standard " to be reached but not exceeded by farmers everywhere! Boys' Corn Clubs have shown the com raisers the fallacy of taking " aver- ages " as standards. It is a little similar in educational work. As land differs in fertility, so children differ in Begabungen or talents. As farmers differ in their methods of cultivation, so parents and teachers differ in Erziehung. We may have to leave the Begabungen with the biologist and the eugenist; but with more fertile methods of instruction, with a more- socialized curriculum, it is possible for these experimental CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC 193 studies to become " stepping-stones " upon which we may rise to higher things. At all events we should recognize and respect, nay more, develop and encourage the individuality in every child. TABLE E Distribution of Arithmetical Topics in the Grades in American Cities* All figures are to be read as percentages Topic Cities laving it ^ 1 CM i II . 1 .c 00 ^ujulj^rs introduced all 71.5 22 6.5 all 5 78 1 7 .. all 14 21 17 21 2 7 .. all 2 10 6 3 23 2 * * T>prim5il frartion^ taiipht ... all 9 3 2 40 13 5 all 5 8 11 20 S6 Pprrpntapf ppnpral cases all 5 3 45 20 ' ' ^imnl«» intprfst all 5 20 45 30 64.7 5 40 55 Commission all . 15 55 30 all . 10 50 40 all 40 60 42.8 23 77 Stocks and bonds 71.4 -■-I, __ 5 23 20 77 60.7 75 57.1 28 7? Simple proportion all 12 21 67 42.8 . 24 76 Mensuration, — of plane figures all 10 1 2 20 30 28 ^Mensuration — of solids all .. 12 18 25 45 Square root . . all 15 85 28.5 . 25 75 all 12 2 \3 25 18 12 Least common multiple all 15 : \7 34 10 4 Greatest common divisor all 12 : J7 37 10 4 35.7 •• •• 28 72 * In the above table the distribution of the topics is given for 28 American cities reporting to the American Committee of the International Commission on the Teach- ing of Mathematics, and published in the U. S. Bureau of Education Bulletin No. 13, 1911, pp. 16-65, 75-78. These cities represent about one-tenth of the scholastic popu- lation in the United States. 194 CONTRIBUTION TO THE PEDAGOGY OF ARITHMETIC TABLE F Important Facts About the Experimental Studies in Arithmetic Investigator Country Subjects Date* Purpose of experiments 1. Phillips. D. E. America 260 children 1897 Popularity of arithmetic 2. Lewis England 8,000 1913 " 3. Messenger America 6 adults 1903 Kinds of thinkers 4. Nanu Germany 7 " 1904 .. 5. Freeman " 14 " 14 children 1910 Differences in adults and chil- dren 6. Lobsien " 1913 Sex differences 7 Ballard England 1912 4< ^- "itfCA-ti * 2.^3 Y/7 }r?^ V) A/ - UNIVERSITY OP CAUFORNIA LIBRARY