Class _J^i_:_: GopghtN^ COP^-RIGHT DEPOSIT. PRACTICE IN THE CASE OF SCHOOL CHILDREN BY THOMAS JOSEPH KIRBY, Ph.D. PROFESSOR OP ELEMENTARY EDUCATION AT THE UNIVERSITY OF PITTSBURGH TEACHERS COLLEGE, COLUMBIA UNIVERSITY CONTRIBUTIONS TO EDUCATION, NO. 58 PUBLISHED BY ^tntijjtrB CHolUge, Qlolttmbia 3lntttprHttg NEW YORK CITY 1913 /!/) V xp^ y vP^« Copyright, l})i;{, by Thomas Joseph Kirby 4U i ACKNOWLEDGMENTS No acknowledgments I can make adequately show my obliga- tions to Dr. Edward L. Thorndike for assistance in connection with this study. To him I am indebted for suggesting and out- lining the study, for guidance during its progress, for assistance in computing and interpreting the results, and for reading and revising the manuscript. To Dr. J. McKeen Cattell and Dr. R. S. Woodworth I am indebted for valuable suggestions received during reports on the study in their seminar. To Dr. C. L. Robbins 1 am under obli- gations for careful checking of the statistical results, and to Dr. H. W. Reddick for a careful reading of the manuscript. To the children, teachers, principals, and superintendent of the Chil- dren's Aid Society of New York City, I am indebted for cheer- ful cooperation during the progress of the practice. To my sister, Miss Nelle Kirby, I am indebted for conducting the practice of one class reported in this study. T. J. K. CONTENTS Chapter Page I The Administration of the Exi-eiuments i Children Participating i Conductors of the Practice 2 Material Used 2 Plan of the Practice 3 Method of Conducting the Practice 6 Method of Scoring 9 Efforts for Uniformity 10 II Improvement in the Group as a Whole 12 Addition 12 Initial Ability i5 Accuracy 16 Gross Gain in Numl)er of Problems Correctly Added 18 Relative Gain in Addition from Sixty Minutes of Practice.. 20 Gross Gain in Accuracy in Addition 22 Summary 24 Division 24 Initial Ability 27 Accuracy in Division 28 Relative Gain in Division 31 Gross Gain in Accuracy 32 Summary 34 The Value of the Practice Experiment as a Method of Teaching 35 Factors Contributing to the Improvement 37 III The Effect of the Distribution and Length of Work Period UPON the Rate of Learning 45 Plan of Practice 45 Addition 47 Initial Ability of the Groups 47 Gross Gain 49 Percentile Gain Si Gain in Accuracy 52 Summary 54 Division 55 Initial Ability 55 Gross Gain 59 Percentile Gain 60 Gain in Accuracy 60 Summary oO Improvement in Relation to Initial Ability 68 General Summary 69 vi Contents Chapter Page IV The Permanence of the Practice Effect 71 Permanence of Association Normally Used in School Work... 72 Addition 72 Gross Gain 73 Gain in Accuracy 73 Division 75 Gross Gain 75 Gain in Accuracy 76 Summary jy Permanence of Associations Through Summer Vacation 78 Addition 78 Gross Loss 78 Loss Per Cent 78 Loss in Accuracy 79 Division 79 Gross Loss 80 Loss Per Cent 81 Loss in Accuracy 8i Permanence in Terms of Advance Over the Initial Practice- Period of the Original Experiment 81 Addition 81 Gross Gain 81 Change in Accuracy 82 Division 83 Gross Gain 83 Loss in Accuracy 84 Permanence of Associations in Addition and Division during Vacation as Shown by the Amount of Practice Required to Restore these Associations to their Previous Efficiency 86 Addition 86 Time of Practice 86 Gross Gain 87 Accuracy 87 Division 89 Time of Practice 89 Gross Gain 90 Accuracy 90 Appendix I. The Use of the Method of the Practice Experiment in Teaching Handwriting and Spelling 93 Appendix II. Sample of Addition and Division Sheets Used 97 PRACTICE IN THE CASE OF SCHOOL CHILDREN CHAPTER I THE ADMINISTRATION OF THE EXPERIMENTS Children Participating The practice on which this study is based was conducted dur- ing the years 191 1 and 1912 in the schools of the Children's Aid Society in the third- and fourth-year classes as a part of the regular grade work in arithmetic. In all, about 1350 children took part. It was the intention that the study should be con- ducted under school conditions normal for children, teachers, and supervisor, in order to meet, as far as possible, a current criticism that results obtained from studies with small groups of persons under laboratory conditions, are not applicable in school conditions ; and also in order to establish greater confi- dence in the validity of the results of the study and insure the applicability of its results to actual school-room problems. It is true that one meets a greater number of factors under such conditions that can not be controlled than in an experiment con- ducted under laboratory conditions, but they are the factors that enter into all school-room work and help to detemiine both its quality and quantity. There was approximately the same simi- larity of ability in the classes that took the practice that one ordinarily finds in a school system where a common course of study is followed and where all the teachers are under similar supervision, but where there is diversity of nationality and ex- treme individual variations among the children. Two different experiments were conducted, one in addition, the other in division. The addition was given to classes in the fourth year of the elementary school ; the division, to classes in the last half of the third year and the first half of the fourth year. These classes had done the work outlined in arithmetic in the New York City course of study. Hence the children had been taught the facts and processes involved in the experiment. I 2 Practice in the Case of School Children Their exact proficiency in these functions at the beginning of tlic experiment will be defined more accurately later by means of tiieir records in the first practice-period. Conductors of the Practice Thirty-nine classes took part in the practice. The writer con- ducted the entire practice in thirty-four of these classes. In four other classes (VT, X, XXV, and XXVI) he had charge of the initial and final practice-periods and of some others, but some of the intervening practice-periods were in charge of the principal of the school or the teacher of the class, who followed the same plan as the writer. They could do this because they were present during all the previous i)enods of practice and had been asked to note the exact method of procedure in order that they might be able to duplicate it exactly when the writer could not be present. Their interest and fine spirit of coopera- tion insured the most careful observance of all the requirements of the experiment. Since the writer usually gave the practice- period preceding and following one given by them, it was pos- sible to note any variations that might occur in results. No greater variations were noticeable in the performance of these classes at such times from one day to the next than when the writer conducted the tests continuously. In the case of class XX, the complete i)ractice was conducted by the class-room teacher in a school outside of New York City. All the details of the experiment were carefully observed. The results in this class conformed so closely to those in other classes that it seemed perfectly justifiable to use them in order to make the number of classes in the difl^erent groups equal. Anyone who wishes to do so, will find that the results are practically uninfluenced by omitting these five classes from the totals. Material Used In the addition practice, the Thorndike 'Addition Sheets* (which consist of seven different sheets, each containing 48 columns of one-place numbers, each column containing ten ad- dends, I's and o's omitted, so arranged that any successive five of the columns are of nearly equal difficulty) were used. By using the seven diflFerent sheets, the chance of remembering The Administration of the Experiments 3 answers to any column because of its position is reduced to a minimum. These sheets are printed in large clear type on un- glazed paper which obviates eye-strain with its attending fatigue, and gives the best possible conditions for rapid, concentrated, accurate work. A sample of one of these sheets may be found on page 97. For the practice in division, sheets were devised by the writer on the " Remainder Division Table " plan worked out by Thorn- dike in his " Exercises in Arithmetic," Nos. 2 and 3. These division combinations include the entire series from " 20 ^ — 3s and — remainder" up to "89 = — 9s and — remainder," thus involving not only the combinations which are the inverse of the multiplication tables through 9 times 9, but also the division of the intervening series of numbers which give a remainder in the answer. No dividends smaller than twenty were used be- cause they involve associations which are usually well established in children's minds and so need no further drill. A sample of these sheets may be seen on page 98. Three sheets of these combinations were arranged in such a way as to make any group of eight or ten successive combinations practically equal in diffi- culty to any other such group. This arrangement was the only means of obviating the inequality of difficulty in individual com- binations. An examination of these sheets will convince one that the difficulty involved in any continuous group of eight or ten combinations is, for all practical purposes, the same as that for any other similar group, even though the individual com- binations may differ widely in degree of difficulty. The records made by classes on these tests are further evidence of the equality of different groups of eight or ten successive combinations. Plan of the Practice In the addition experiment each class practiced adding these columns for 75 minutes ; but this time was distributed differently for different classes. Each class had an initial practice-period of fifteen minutes. These were identical in character with the intervening practice-periods ; but besides serving as practice- periods, they served as measures of ability at the beginning and end of practice. From them the change in ability for each indi- vidual was measured. The intervening practice of forty-five 4 Practice in the Case of School Children minutes between the initial practice-period and the final practice- period was broken up for different groups of classes in four different ways. For Group I it was divided into two practice- periods of 22^ minutes each ; for Group II, into three practice- periods of 15 minutes each; for Group III, into eight practice- periods, seven of six minutes each and one of three minutes ; for Group IV into twenty-two practice-periods, twenty-one of two minutes each and one of three minutes. The following statement of this plan in tabular form adds to clearness : Guoui's Initial rioiuou Inti' ;i{V10NING 45 MlNUTKS Final Piouiod I 15 min. •) 22 i min. 15 min. II 15 min. ii 15 min. 15 min. 111 15 min. 7 G min. and 1 .'} min. 15 min. IV 15 min. 21 2 min. and 1 ;i min. 15 min. It is of special importance for an understanding of the entire discussion to get clearly in mind that all the classes practiced for the same aggregate amount of time. It is further to be carefully noted that for all classes the initial period and final period of practice were of equal length. Since all four groups had the same initial and final practice-periods, but had the inter- vening forty-five minutes of practice distributed in different ways, any difference in the results for the groups will be a consequence of distributing a given amount of time in periods of dilTcrcnt length.' In the division experiment the aggregate time of practice was sixty minutes for each class. This was divided into periods of different lengths for different groups of classes. Each class had an initial practice-period of ten minutes and a final period of ten minutes ; and the intervening practice of 40 minutes was divided in three different ways. In the division, as in the addition, the initial and final practice-periods were identical in character with the intervening practice-periods ; but besides serving as practice- periods, they served as measures of ability at the beginning and end of practice from which any change in ability was measured. The first group had two intervening practice-periods of twenty minutes each ; a second group had four intervening practice- periods of ten minutes each; a third group had twenty inter- vening practice-periods of two minutes each. The following tabular statement helps to kcc]) the plan for each group in mind : ^ Except iov a factor to ho noted on page 62 ff. The Administration of the Experiments tours Initial Piomou iNTEllVENINa 40 MiNUTKH Final Period I II III 10 min. 10 min. 10 min. 2 20 min. 4 10 min. 20 2 min. 10 min. 10 min. 10 min. The plan of liaving each class drill for the same aggregate length of time, sixty minutes, makes it i)ossiblc to measure the gain in all classes for this amount of drill ; while breaking the forty minutes intervening between the initial test and the final test into periods of different lengths gives the opportunity, as in the addition experiment, to find the comparative effect of drill- periods of different lengths, which aggregate a constant amount. As nearly as normal school conditions would i)ermit, all prac- tice-periods were given one each on successive school days to any class. So in addition, in the classes of Group I, four days were required to complete the experiment; of Grouj) IT, five days; of Group J II, ten days; and of Group IV, twenty-four days. In division four days were required for the classes of Group I ; six days for Group II ; and twenty-two days for Group 111. In three classes it was found necessary to give a two- minute practice-period twice a day, one in the morning and one in the afternoon for the last five days in order to complete the experiment within a time made necessary by school conditions. No noticeable effect was produced by this change. The practice was given to any individual class as nearly at the same time of day as was possible. In general the time of day did not vary for any class more than half an hour, but sometimes it was necessary to give a practice-period to a class in the afternoon in- stead of in the morning. The teachers felt that children would not do so well in the afternoon, but the results showed little or no influence from such changes of time. Of course different classes did not have their practice at the same time of day. Some classes had most of their practice early in the morning, others later in the morning. Some classes had most of their practice early in the afternoon while others had theirs late in the afternoon. Some of the very best records were made late in the afternoon despite the general feeling on the ])art of the teachers that a class could not do as well late in the day. Some classes seemed to be affected by the loss of school work on vSaturday and Sunday, or Ijy the intervention of a holiday. Yet in other classes remarkaljle gains were made following such a 6 Practice in the Case of School Children holiday, which seems to obviate the need of carefully evaluating any such variable factors. In fact other studies have shown that where interest is keen children do as effective work late in the day as early. Hence this factor receives no consideration. The fact that the interval between the initial and final practice- periods was larger when the intervening practice-periods were shorter and so gave more opportunity for practice out of school and for the influence of regular school training outside the specified practice of the experiment, conditions all the results of this study. TJiere were three possibilities. The practice could be given as it was ; or it could be given with longer intervals when the practice-periods were long, so as to have the final practice-period always come, say, 24 school-days after the initial practice-period ; or it could be given daily as it was, but with a varying interval between the next to the last and the last prac- tice-period so that the final practice-period would always come, say, 24 school-days after the initial practice-period. Each method has its disadvantages. The method used here gives the pupils ])racticing in shorter periods greater chances of help from prac- tice outside the experiment. The second method mixes the effect of interval-length with the effect of practice-period-length. The third gives the shorter periods the advantage of less for- getting. In interpreting the results given in this monograph, the reader may allow for the greater chance of outside practice in the short-period groups as seems wise. The Method of Conducting the Practice The purpose of the practice was explained to the children as carefully as possible, and in the same way to all classes. The fact that much of arithmetic work both in school and out in- volved addition was impressed as well as the need for speed and accuracy in it. They were told that this work would help them improve their addition. The value to be derived from the prac- tice was indicated. In the case of addition, no preliminary work was given, because it could be assumed that any fourth- year child knew how to add columns of figures. Before the papers were passed out the children were shown one of the sheets and instructed to place their answers on the sheet which would be given to them, in the space provided. They were told The Administration of the Experiments 7 to add as rapidly as possible, but to make as few mistakes as possible, as all mistakes would count against their scores. The need of accuracy combined with speed was impressed further by showing them the method of scoring, no credit save for each column correctly added. When asked what they were to do, their most common response was, "Add as fast as you can and don't make mistakes." They were also told how many minutes they were to work, but they had no means of knowing the time, once the signal to begin was given, until the signal to stop was given. The papers were passed out by the class teacher in her regular way, with the printed side down. In all the practice each child was supplied with more problems than he could finish in the allotted time. The children placed their names, age, and grades on the back of the sheets. Then they were instructed again to add as rapidly and as accurately as possible and to place their answers on the sheets beneath the columns. They were told that the signal for beginning would be " Go," and for ending " Pencils down." The oft-repeated injunction, " Look only at your own paper " was given, and was followed quite conscien- tiously. After a practice-period was finished and pencils were laid down, the children were asked about the number of problems they had worked. This was done because the children were eager to tell their own scores and to learn the scores of others. It acted too as an immediate reward and so as an incentive. Care was taken to avoid undue excitement, but an attempt was made to inspire the children to their best individual efforts. Just before beginning the second and each following practice- period, the exact score of each child in the preceding day's practice was read, both the number of columns worked and the number correct.- High scores with few errors were commended. If some score showed that perhaps this child was adding beyond his best speed, as indicated by the number of errors, he was advised, not in a fault-finding way but for the good of his own record, to try for greater accuracy with the hope of improving his score. These facts were put in concrete form for the chil- dren as follows : " Which gives the better score, 20 with 4 wrong or 16 with i wrong?" Of course this warning was not With a very few exceptions. 8 Practice in the Case of School Children always pertinent, though usually a large number of mistakes indicated that the child was working beyond his own norm of speed. At all times during the practice the value of accuracy was stressed ; and the undesirability of working an excessive number of problems with many errors was pointed out. Further instructions calculated to maintain effort were given just before beginning the second practice-period and repeated always at the beginning of each subsequent one. The children were told that their individual improvement was to be measured and they were shown that no matter how low or high their pres- ent record their final standing would be determined by the amount of gain made. They were shown that it was not pri- marily a contest among the individuals of the class, but an effort on the part of each one to surpass his own previous record. The children v/ere encouraged to compare their last record with their own previous records, and at times the scores were read to them in such a way as to indicate gains made. A hypothetical curve was drawn upon the board to indicate the ascent that would result from supposed gains, as well as to show them how to keep their own individual curves. After a few periods of practice, actual scores were used on the board in the construction of these curves. No doubt, had each child been encouraged to draw his own curve from day to day, this concrete showing of his gain would have resulted in added effort that would have justified fully the time spent. The desire to surpass one's own previous record was the incen- tive most appealed to ; still the spirit of rivalry was strong. One little girl expressed it well for the class by saying : '* Why, you just try to beat yourself," but an ambitious boy, when asked what record he was trying to surpass, pointed to his neighbor behind him and said : " I'm trying to beat him." The average number of problems done correctly in the preceding practice- period by boys and girls as separate groups was also read to the class before beginning the day's practice. This group com- parison served as an additional incentive, but would not satisfy the children as a substitute for their individual scores. In division, preliminary work was given before the initial practice-period was begun. The need of knowing the multiplica- tion and division combinations, both for school and life, was explained. The children were told that this work would aid The Administration of the Experiments 9 them in both. Then many of the combinations were placed upon the blackboard to insure familiarity with the form of expression used in the sheets. Particular attention was given to assisting any child who appeared not to understand. After a seemingly general response was obtained in this way, in order to make doubly sure that each individual knew exactly what he was to do and how to record the result properly on the sheet, one-sixth of a sheet was given to each child on which to fill in the results. During this part of the preliminary trial very great care was taken by the writer and the grade teacher to see that all the children understood fully what was to be done. Despite this care the first papers of some few individuals could not be used because they showed clearly a failure to grasp the situation. In most cases these failures were from children who had recently entered the classes and were not yet properly classified, or who had not sufficient mastery of English to understand the instruc- tions. Yet some failed because they were backward in learning the process of short division. Almost without exception the chil- dren who failed to grasp the situation after the explanation mentioned above, were found to be unable to handle short division in any form. Later the teacher explained the matter further to these children. They continued in the practice, but their records did not enter in the final computation. In all other respects, the same plan was followed in the division as has been described in the addition, though greater stress could be placed on speed, inasmuch as the percentage of accuracy showed a gradual increase along with the absolute gain in number of com- binations. Method of Scoring In scoring the addition, credit was given according to columns. The score for any paper was found by deducting one from the total number of columns added for each column incorrectly added. If a child added 16 columns with 2 wrong, his score was taken as 14. Thus efficiency was measured by the rapidity and accuracy of getting the sums. In division a combination was taken as a unit, correct quotient and remainder being required for any credit. The score for any paper was found by deducting one from the total number of combinations solved for each one lo Practice in the Case of School Children incorrectly solved. If a child did 30 combinations with 2 incor- rect, his score was 28. One might contend that some credit should be given for a correct quotient with an incorrect re- mainder ; but after all in the use of such combinations the result as a whole determines efficiency. Thus ability was measured by the rapidity and accuracy of getting the quotients and remain- ders. These methods of scoring whereby speed and accuracy are reduced to a single variable, are entirely arbitrary, but the author feels reasonably certain that any system of weighting errors which might be used would not materially change the results on which conclusions are based. The record of no child could be used if he was absent at a first or last practice, inasmuch as the change in his ability could not be measured. But records of those who missed one or more intervening tests were used.'' In all, about 1350 children were tested. The smallest number of ])apers scored for any child who was present every day during the time an experiment was in progress was four, and the greatest number twenty-four. This means that in all about 15,000 papers were graded and their scores recorded in the process of giving the two experi- ments. Efforts for Uniformity Special effort was made to keep the conditions uniform in all the classes. Each teacher was requested to give no drill during the experiment in the line of work being tested, since one purpose of the work was to measure as accurately as possible the amount of improvement her children could make with this special method of work in the exact time allotted for the practice. A fine spirit of cooperation prevailed which insured compliance with this request. No mention was made to the children about practicing for fear it might act as a suggestion. In a few cases children became so enthusiastic that they did some work at home. How- ever, they bad none of the sheets that were used in the experi- ment and so had to devise their own means for any practice car- ried on. This can only be taken as an indication of the interest ' This was perhaps an error of administration, but its influence on any result to be stated was very slight; hence, having begun in that way, I have not thought it worth while to undergo the labor of com- puting the results separately for those who were present at every single practice-period. The Administration of the Experiments ii displayed and must be considered, as I have already said, as a factor that can not be completely controlled in a school experi- ment. Perhaps the greatest difficulty in carrying on the tests uni- formly for all classes was met in trying to give the instructions and appeals to different classes in such a way as to cause them to react with an equal amount of enthusiasm for the work. At all times every incentive was used that would seem to cause the children to surpass their previous records. When one bears in mind that the writer was the regular supervisor equally ac- quainted with all the classes, controlled by no preconceived notions of results, and interested in the experiment as a piece of accurate school work, it will be seen that the conditions would naturally be kept as nearly uniform in all classes as is possible in such a piece of work as has been done. Some one may express surprise at not finding here a printed set of instructions given orally or read to all classes in the same way to insure uniformity. However, the author aimed not so much at uniformity in kind and amount of instruction as at uni- formity of understanding and enthusiasm on the part of the dif- ferent classes for the task in hand. While this plan introduced again the judgment of the writer to determine uniformity of un- derstanding and enthusiasm, he feels that, even so, less substan- tial variability resulted in the total reactions of the classes meas- ured, than if the same instructions had been uniformly read or given orally at all times. In making this effort for uniformity of understanding and enthusiasm, no great variations in instructions and appeals were necessary ; and, so far as could be judged, much the same intensity of effort obtained in the performance of any class from day to day, and in the performance of different classes. CHAPTER II IMPROVEMENT IN THE GROUP AS A WHOLE Addition This chapter consists of a consideration of the results of the practice already described, in so far as they pertain to the group as a whole. The large problem to be presented is the amount of improvement in the functions tested, made by these school children of the third and fourth year in about one hour of drill with the practice experiment as the method of class work. To measure the improvement in a group of persons subjected to a given amount of practice in any phase of activity, it is necessary to devise methods and materials which will exercise the function in question and to obtain exact and comparable measures of their ability at the beginning and end of the prac- tice. The methods and material have been discussed in a pre- vious chapter. We shall proceed to a presentation of the data which give the initial and final ability of the entire group and hence the basis for measuring any change which resulted from the practice. The clearest presentation of the results of this study would demand that all the class records be published in detail. But since these records alone would require more than half the space ordinarily allotted to such a report as this, only a sample record can be published. However, from this record the reader can obtain a thorough understanding of the exact sources from which all summaries presented later were obtained. The table is a complete record of one class in addition in which all the practice-periods were 15 minutes in length. Such a record is presented because it affords the best opportunity to see the progress from one period of practice to the next. Table I shows that in the initial practice-period boy A added 85 columns, 81 of which (or 95 per cent) were correct. In the second period he added 105 columns, 97 of which were correct. In the third period he added no columns, 102 of which were 12 Improvement in the Group as a Whole 13 TABLE I Record of Class I in Five Fifteen-minute Practice-periods IN Addition Indi- Initial 2nd 3rd 4th Final Gain vid- Practice- Practice - Practice - Practice - Practice- uals Period Period Period Period Period Per cent Boys S. C. %c. S. C. S. C. S. C. S. C. %c. Gross Ac. A 85 81 95 105 97 110 102 115 107 118 116 98 35 43 4- 3 B 38 30 79 39 27 44 32 46 38 61 43 70 13 43 — 9 C 83 81 98 80 53 85 61 98 74 92 77 83 —4 —5 —15 D 40 27 68 39 31 48 36 57 34 42 7 26 —26 E 49 42 89 57 51 64 59 76 64 80 67 83 20 43 — 6 F 80 74 93 90 89 101 99 119 117 112 112 100 38 51 + 7 G 51 49 96 68 67 75 71 89 86 98 97 99 48 98 -1- 3 H 58 51 88 49 41 54 45 54 46 57 43 75 8 16 —13 I 48 46 98 65 49 55 54 52 50 59 57 97 11 24 — 1 J 37 26 70 44 39 43 40 48 45 59 50 83 24 92 + 15 K 89 82 92 40 82 99 92 112 104 96 91 95 9 11 + 3 L 112 109 97 128 127 138 136 140 132 156 148 95 39 36 — 2 M 42 38 90 47 42 51 45 49 42 51 47 92 9 24 + 2 N 30 27 90 41 40 45 44 47 44 50 45 90 18 67 24 15 63 35 19 52 17 74 25 51 27 53 12 80 —10 P 50 49 98 60 57 76 75 80 77 95 91 95 42 86 — 3 Q 37 29 78 41 31 46 37 28 20 48 39 81 10 36 + 3 R 24 17 71 40 25 39 30 37 27 75 10 59 -1- 2 s 33 27 82 32 30 37 32 55 47 48 39 81 2 7 — 1 Girls A 19 5 74 24 4 29 6 31 5 34 15 47 10 200 —27 B 42 31 74 38 37 24 22 22 20 52 48 83 12 39 + 9 C 46 37 80 50 38 55 44 61 52 61 48 79 11 30 — 1 D 30 18 60 38 30 34 26 37 26 37 32 87 14 78 -1-27 E 32 31 97 44 41 56 48 67 64 73 69 95 38 123 — 2 F 19 16 84 43 41 50 45 55 47 78 77 99 61 381 -f-15 G 26 20 75 44 30 33 19 49 32 36 25 69 5 25 — 6 H 45 39 87 54 47 61 57 84 79 96 85 89 46 118 + 2 I 23 21 91 24 15 28 23 25 13 37 32 87 11 52 — 4 J 35 25 71 40 35 53 48 60 55 64 59 92 34 136 + 21 K 56 52 93 71 66 54 54 68 67 63 61 97 9 17 + 4 L 25 18 72 33 22 38 33 48 35 40 37 93 19 106 + 21 M 32 26 81 45 36 49 47 64 63 70 65 90 39 150 + 9 N 26 19 73 36 30 35 30 48 33 52 36 70 17 89 — 3 75 74 99 80 80 83 77 89 79 96 89 93 15 20 — 6 P 37 36 97 40 40 41 39 45 43 40 36 90 — 7 Q 37 35 95 30 26 35 34 37 30 49 44 90 9 26 — 5 R 45 37 82 56 54 59 55 62 58 71 65 92 28 76 + 10 S 32 30 94 40 35 40 38 40 39 48 41 85 11 37 — 9 T 34 31 91 38 35 48 42 56 47 56 41 75 10 32 —16 1 2 3 4 6 6 7 8 9 10 // IS IS 14 IS 16 S.=Problems solved. C.=Problenis correct. %C.=Per cent of problems correct. Ac.=Accuracy. correct. In the fourth period he added 115 columns, 107 of which were correct. In the final period he added 118 columns, 116 of which (or 98 per cent) were correct. His gross gain was 35 columns. That is, he added correctly 35 more columns in the final fifteen-minute period than in the initial fifteen-minute 14 Practice in the Case of School Children period. This gross gain of 35 columns was on an initial ability of 81 columns; and hence means a gain per cent of 43. In the initial fifteen-minute period he add 95 per cent of his columns correctly ; and in the final fifteen-minute period he added 98 per cent of his columns correctly which made a gain of 3 per cent in accuracy. Each of the columns of the table has been given a number (printed at the bottom) by which it will be referred to in the future discussion. Referring to these small figures in italics at the bottom of the columns, the reader can gain a complete interpretation of the entire record from the following explanation, bearing in mind also that the letter " S " at the top of a column designates the entire number of problems solved, " C " the number of prob- lems correct, " %C," the per cent of problems worked correctly, "Ac," accuracy, and that the word " Gain " at the head of the last three columns is to be taken with each. Column / gives the individuals, boys and girls separate. Column 2 gives the entire number of problems solved in the initial 15-minute period; column 5 gives the number of problems that were correct in the same period ; and column 4 gives the per cent of problems worked correctly, or the per cent of accuracy in the initial period. The percentages of accuracy in column 4 were found by dividing the number in column 5 by the corresponding number in column 2. Columns 5 and 6, 7 and 8, 9 and 10 give the entire number of problems worked and the number right in the three inter- vening 15-minute periods. These pairs of columns correspond to 2 and 5 in the initial practice-period. The per cent of accuracy for these intervening tests was not necessary for later discussion and so was omitted. Columns //, 12 and Jj of the final period correspond to columns 2, 5, and 4 in the initial period. Column 14 gives the gross gain in number of problems correct or the absolute gain, and is the algebraic difference between the num- ber in column j and the corresponding number in column 12. Column 75 gives the gain expressed in hundreths obtained from dividing the number in column 14 by the corresponding number in column j. Column 16 is the gross gain in accuracy expressed in per cent of problems correct and is the algebraic difference between the number in column 4 and the corresponding number in column jj. It is to be noted that this last column is not a gain per cent as is column 75, but the gross gain in accuracy Improvement in the Group as a Whole 15 expressed in a per cent. A minus sign indicates a loss wherever used. Every other class record in addition used in this study corre- sponds closely to this one in the categories of initial record, final record, per cent of accuracy, gross gain, percentile gain, and per cent gained in accuracy. The only difference in other class records results from the fact that the intervening periods of practice were of different lengths, some being 22^/4 minutes, some 6 minutes, and some 2 minutes in length. For the entire addition experiment, there are 21 class records. These complete records are placed on file in Teachers College where they are accessible to any one wishing to use them. Initial Ability The first group of facts to be considered in arriving at a solu- tion to the problem with which this chapter deals, provides data which give a measure of the ability of the entire group in addi- tion at the beginning of practice. All the individual records of " problems added correctly " in the initial period of which those in column j of Table I are samples, have been distributed, class by class, boys and girls separate. A summary of these facts affords the data necessary to show the distribution, central ten- dency, and variability of the group as a whole, as well as the reliability of the measures used. Table II gives a distribution of the number of problems worked correctly in the initial fifteen-minute period by 732 fourth-year children. The number of problems worked is given in groups of 5. The table shows that 3 children (or 0.4 per cent of the entire group) worked from o to 4 problems correctly in fifteen minutes; 50 children (or 6.8 per cent) worked from 5 to 9 problems correctly; 80 children (or 10.9 per cent) did from 10 to 14 problems, etc. The figures show that the distribution is de- cidedly skewed toward the high end. The table shows the range in number of problems added correctly by these fourth-year children to be from ' o to 4 ' problems up to ' 105 to 109 ' prob- lems. The median number of problems worked correctly is 23.3 ; that is, there are as many children who worked 23.3 problems or more as there are who worked 23.3 problems or less. The variability of the group is further shown by the P. E., 7.8, a i6 Practice in the Case of School Children TABLE II NtTMBER OF Problems Added Correctly in the Initial Fifteen- minute Period Problems added correctly .... to 4 5 to 9 10 to 14 15 to 19 20 to 24 25 to 29 30 to 34 35 to 39 40 to 44 45 to 49 50 to 54 55 to 59 Number of indi- viduals 3 50 80 133 130 111 81 57 33 14 13 7 Per cent .4 6.8 10.9 18.1 18 15.2 11 7.8 4.5 1.9 1.8 .9 TABLE 11— Continued Problems added correctly .... 60 to 64 65 to 69 70 to 74 75 to 79 80 to 84 85 to 89 90 to 94 95 to 99 100 to 104 105 to 109 Total Number of indi- viduals 4 3 7 3 1 1 1 732 Per cent .5 .4 .9 .4 .1 .1 .1 100 Median 25 Percentile 75 Percentile P.E. P-E- t.-obt. Av. 23.3 16.5 32.1 7.8 .29 problems number which when subtracted from and added to the median, gives the approximate Hmits within which the middle fifty per cent of the class falls. In other words, when the lowest twenty- five per cent and the highest twenty-five per cent of the class are cut out, the limits within which the remaining middle fifty per cent falls, are 16.5 columns and 32.1 columns. The per cents in the table show that 81 per cent of the children ranged from 10 to 39 problems; 51 per cent from 15 to 29 problems. Accuracy In order to define more exactly the initial ability of this group of children in addition it is necessary to know the degree of accuracy of their performance. The accuracy for each child was determined by dividing the number of problems worked cor- rectly by the entire number worked. Just what these per cents of accuracy are the reader may see from Table I, column 4. Improvement in the Group as a Whole 17 The following table is a distribution of the per cents of accuracy of the entire group of 732 children, of which those per cents in column 4 of Table I form a part. TABLE III Per Cent of Problems Added Correctly in the Initial Fifteen-minute Period Per cent of accuracy. . . 6 to 10 11 to 15 16 to 20 21 to 25 26 to 30 31 to 35 36 to 40 41 to 45 46 to 50 51 to 55 Individuals 1 1 2 3 9 13 16 22 17 Per cent of group .1 .1 .3 .4 1.2 1.8 2.2 3 2.3 TABLE III— Continued Per cent of accuracy. . . 56 to 60 61 to 65 66 to 70 71 to 75 76 to 80 81 to 85 86 to 90 91 to 95 96 to 100 Total Individuals 25 46 58 84 100 81 108 76 70 732 Per cent of group 3.4 6.3 8 11.5 13.7 11.1 14.8 10.4 9.6 100% Median 79 . per cent 25 Percentile 68. 75 Percentile 89. P.E. 10. P-^- t.-obt. Av. .38 Table III gives the per cents of accuracy in groups of five. It shows that 70 children (or 9.6 per cent of the entire group) added correctly from 96 to 100 per cent of the entire number of problems which they worked ; 76 added correctly from 91 to 95 per cent of the problems which they worked, etc. These figures show that the distribution is skewed toward the low end. The range in per cent of problems added correctly is from 6 to 100. The median per cent of accuracy is 79, which shows that there were as many children who added correctly four-fifths or more of their problems as there were who added correctly four- fifths or less. The upper 25 per cent of the class added with an accuracy of 89 per cent or more ; the upper 60 per cent of the class added with an accuracy of 68 per cent or less. The range iB Practice in the Case of School Children of accuracy for the middle 50 per cent of the class was from 68 per cent to 89 per cent. By combining now the data derived from Tables II and III, the initial ability of the group can be defined both in terms of amount and accuracy of performance. We can give the clearest idea of the initial performance of this group in addition by say- ing that the median ability in number of problems worked cor- rectly in fifteen minutes was 23.3 problems, which was 79 per cent of the entire number of problems worked. Or, taking as a basis the entire number of problems worked instead of those worked correctly, one can say that the chances are that one child in any two picked at random would be able to add 29.5 columns or more in 15 minutes with an accuracy of 79 per cent or more. Gross Gain in Number of Problems Correctly Added We are now ready to consider the gain made in problems cor- rectly added by this group of fourth-year children in the course of seventy-five minutes of drill with the practice experiment as the method of class work.* This gain will be considered in two ways, absolute gain and percentile gain. The present considera- tion has to do with the absolute gain. It will be recalled that the initial test and the final test for all classes were of equal length, 15 minutes, and that between the initial and final tests all classes practiced for the same length of time, 45 minutes. It is now our problem to measure the amount of improvement that the entire group showed in the final 15 minutes of practice over the initial 15 minutes. Referring again to Table I, columns 5, 12 and 14, the reader will understand clearly the exact measures that are involved in this part of the discussion. It will be seen that boy A in the initial test had 81 problems correct; in the final test he had 116 problems correct which gave him an absolute gain of 35 prob- lems in the final 15 minutes over the number of problems worked correctly in the initial 15 minutes. So, boy B gained 13 prob- lems, girl A 10 problems, etc. All of these gross gains were * The gain is measured for only 60 minutes of practice. The initial period is 15 minutes and the final period is 15 minutes. The record for each of these periods gives the adding rate at the middle of each period. Hence the time of practice that is measured is from the middle of the initial period to the middle of the final period, or (to minutes. Improvement in the Group as a Whole 19 distributed in a table, class by class, boys and girls separate. Only a summary of this distribution for the entire group is pre- sented here. TABLE IV Gross Gain, in Number op Columns Added Correctly in Fifteen Minutes, Resulting from Sixty Minutes of Practice Columns gained. . . —15 to —12 —11 to —8 —7 to —4 —3 to 1 to 4 5 to 8 9 to 12 13 to 16 17 to 20 21 to 24 25 to 28 Individuals 5 11 28 63 93 105 111 90 75 29 29 Per cents .7 1.5 3.8 8.6 12.7 14.3 15.2 12.3 10.2 4 4 TABLE lY— Continued 29 33 37 41 45 49 53 57 61 65 Columns gained. . . to to to to to to to to to to Total 32 36 40 44 48 52 56 60 64 68 Individuals 25 24 16 5 7 5 2 3 4 2 732 Per cents 3.4 3.3 2.2 .7 .9 .7 .3 .4 .5 .3 Median 10.7 columns 25 Percentile 3.6 75 Percentile 18.8 P.E. 7.6 ^•^- t.-obt. Av. .28 Table IV gives a summary of the distribution of the gross gain in number of columns added correctly in fifteen minutes resulting from 60 minutes of practice. The number of problems gained is given in groups of 4. The table reads : " Five children (or 0.7 per cent of the entire group) gained from — 15 to — 12 problems." This means, of course, that they lost. The other end of the table reads: "Two children (or 0.3 per cent of the entire group) gained from 65 to 68 problems ; four children (or 0.5 per cent of the group) gained from 61 to 64 problems, etc." The figures show that the distribution is skewed toward the high end. The range for the group is from 15 to 12 problems lost to 65 to 68 problems gained. The median gain is 10.7 columns, which means that the median increase in ability was such that 10.7 more columns were added correctly in 15 minutes at the end of practice than in 15 minutes at the beginning of practice. 20 Practice in the Case of School Children It also means there were just as many children who gained 10.7 columns or more, as there were who gained 10.7 columns or less. The members of the upper 25 per cent of the group gained 18.8 columns or more while the members of the lower 25 per cent of the group gained 3.6 columns or less. This leaves the range for the middle fifty per cent of the class from 3.6 columns to 18.8 columns gained. The percentages in the table show that 65 per cent of the entire group gained from i to 20 problems and that a larger per cent gained from 9 to 12 columns, than any other group of four columns. Relative Gain in Addition From Sixty Minutes of Practice Gain may be expressed in absolute terms, as in the last section ; or it may be expressed in the per cent which the gross gain is of the original quantity on which it was gained. We are so accustomed to think of gain or improvement in relative terms that such terms often become a more satisfying measure of the change in question than the absolute terms. We found in the last section that the median gain in number of problems added correctly in 15 minutes was 10.7 columns. It is now the problem to find what per cent of the initial ability this gross improve- ment was. Referring again to Table I, column i^, the reader will clearly understand the source from which the present data are obtained. Boy A made a gross gain of 35 problems ; that is, he worked correctly 35 more problems in the last fifteen minutes than in the initial fifteen minutes. Or he made a gain of 35 problems on an initial ability of 81 problems, or 43 per cent. So boy B gained 43 per cent, girl A gained 200 per cent and girl B, 39 per cent, etc. In almost every case these per cents are of a dif- ferent initial ability, and to have their proper significance they need to be considered in relation both to the initial ability and the gross gain of the individual whose performance they repre- sent. That is, the same gain per cent for two individuals does not necessarily indicate that their gain was equally meritorious. Taking the cases of boys A, B and E, who have gained the same relative amount, we would be justified in saying that A's 43 per cent gain represents a much higher efficiency of performance that that of B or E. Likewise we feel certain that L's 36 per Improvement in the Group as a Whole 2 1 cent of gain is far more noteworthy than Q's 36 per cent, and it is plainly evident that G's 98 per cent does not mean almost twice as meritorious a gain as that of F. We are badly in need of a more equitable means of measuring relative gain than is afforded by per cents, but until such a means is perfected we shall have to continue using the one now in vogue. Such consid- erations, however, have to do especially with individual differ- ences which will be considered in a later portion of this study. All the per cents which correspond to those of column 15, Table I, were distributed class by class, boys and girls separate, but only a summary of this table is presented here, from which an adequate notion of the central tendency and variability of the group in relative gain can be explained. TABLE V Gain Per Cent in Columns Added Correctly in Fifteen Minutes Resulting from Sixty Minutes of Practice Gain, per cent . . . -89 to -75 -74 to -60 -59 to -45 -44 to -30 -29 to -15 -14 to 1 to 15 16 to 30 31 to 45 46 to 60 61 to 75 76 to 90 54 91 to 105 40 106 to 120 Individuals 3 1 6 11 20 61 66 97 85 92 80 20 Per cents .4 .1 .8 1.5 2.7 8 3 .9 13.2 11.6 12.6 10.9 7.4 5.5 4 TABLE V— Continued Gain, per cent.. . 121 to 135 136 to 150 151 to 165 166 to 180 181 to 195 196 to 210 211 to 225 226 241 to to 240 255 256 to 270 271 to 285 286 to 300 301 to Total Individuals 14 21 9 1 10 732 Per cents. 1.9 2.9 1.2 .9 .3 1.1 .1 1.4 100 Median 48 per cent 25 Percentile 18 75 Percentile 83 P.E. 32.5 In Table V the relative gains are distributed in groups of 15. The range is from a loss of 89 per cent to a gain of 300 per cent or more. The mode is in the group 61 to 75 per cent. The median gain per cent is 48, which means that there were as 3 2 Practice in the Case of School Children many children who gained more than 48 per cent as there were who gained less than 48 per cent. The upper 25 per cent of the group gained 83 per cent or more while the lower 25 per cent gained 18 per cent or less, which leaves the range for the middle 50 per cent of the group from a gain per cent of 18 to 83. The percentages in the table show that 14 per cent of the entire group lost as a result of the practice, and that 56 per cent of the entire group gained from 16 to 90 per cent. Again bear- ing in mind the wide variability of the class, we can say that the central tendency of the relative gain of the group is a gain of 48 per cent. Gross Cain in Accuracy in Addition To determine the complete effect of the practice upon this group one other factor remains to be considered, viz., the influ- ence of the practice upon the accuracy of making the associations involved. It will be recalled from the instructions given to the classes just before each i)ractice-period that two factors were emphasized at all times during the practice as constituent ele- ments of the efficiency of the performance, speed and accuracy. We have seen that the group as a whole worked with enough increase of speed to add correctly 10.7 more columns in the final fifteen-minute period than in the initial period, which meant an increase in speed of about fifty per cent. The question now arises, was this increase in speed accompanied by an increase in accuracy, or was it at the expense of accuracy, or did the group as a whole add with as good a chance of getting a correct answer at the end of the practice as at the beginning? Vox an understanding of the data on which the answer to this question is based, the reader is referred again to Table 1. columns 4, 75, and 16. The degree of accuracy of boy A at the beginning of practice was 95 per cent ; at the end of practice his percentage of accuracy was 98 per cent. Hence his gross gain in accuracy expressed in percentage of answers that were correct was 3. It is to be noted that this is a gross gain and not a gain per cent. So boy B lost 9 per cent, girl A lost 2y per cent, and girl ?> gained 9 per cent. All these per cents were distributed class by class, boys and girls separate. A summary of this table will suffice to give sufficient data to determine the change in the group as a whole, in control tendency and variability. Improvement in the Group as a Whole 23 TABLE VI Gross Gain in Accuracy in Addition Expressed in Per Cent OF Answers that Were Correct Resulting from THE Sixty Minutes of Practice Per cent gained. -55 to -51 -50 to -46 -45 to -41 -40 to -36 -35 to -31 -30 to -26 23 -25 to -21 -20 to -16 -15 to -11 -10 to -6 -5 to -1 to 4 5 to 9 Individuals 8 2 2 6 19 17 54 63 68 101 111 79 Per cents 1.1 .3 .3 .8 2.6 3.1 2.3 7.4 8.6 9.3 13.8 15.2 10.8 TABLE V — Continued Per cent gained. 10 to 14 15 to 19 20 to 24 25 to 29 30 to 34 35 to 39 40 to 44 45 to 49 50 to 54 55 to 59 60 to 64 65 to 69 Total Individuals 61 47 28 17 6 8 2 5 1 3 1 732 Per cents 8.3 6.4 3.8 2.3 .8 1.1 .3 .7 .1 .4 .1 Median -.4 25 Percentile -12.4 75 Percentile 9.2 P.E. 10.8 per cent In Table VI, the gross gains in accuracy expressed in per cents are given in groups of 5. Negative signs indicate loss. The range in the change in accuracy effected by the practice is from a loss of 55 per cent to a gain of 69 per cent. The table reads, " Eight children (or i.i per cent) of the entire group lost from 55 to 51 per cent, etc." The figures indicate a curve which approaches close to the normal curve of distribution. The mode is at o to 4 per cent gain. The median is .4 per cent loss, which shows that there were as many children who lost .4 per cent or more in accuracy as there were who lost .4 per cent or less or gained in accuracy. The figures following '' Individuals " show that 363 of the children lost in accuracy while 369 children either maintained their initial proficiency for accuracy or worked with greater chance of getting a correct answer. The upper 25 per cent of the group gained 9 per cent or more in accuracy while the lower 25 per cent lost 12 per cent or more, which gives the range for the middle 50 per cent of the group from a loss of 12 per cent to a gain of 9 per cent. 24 Practice in the Case of School Children Summary The initial performance in addition of the group of 732 fourth- year children showed a median ability on their part to add cor- rectly in fifteen minutes 23.3 columns such as were used in the practice. Their median accuracy in this initial performance was found to be about 80 per cent, which shows that the median number of problems added regardless of the answer obtained was about 30 columns. From these two central tendencies we can say that the median initial ability of this group of 732 fourth- year children in addition was such that they could add correctly 23.3 columns, such as were used in the test, with an accuracy of 80 per cent. The change effected by the practice of 60 minutes m the group as a whole was an improvement in speed such that they could add 10.7 more columns correctly in the final 15 minutes of the practice than in the initial 15 minutes, while the change in accur- acy was so small as not to need consideration. Division The experiment in division, with children in the second half of the third year and the first half of the fourth year, was con- ducted on the same general plan as the one in addition with fourth-year children, except that the initial and final periods of practice were 10 minutes each instead of 15, and that the entire time spent in the practice was 60 minutes instead of 75. The intervening 40 minutes between the initial and final practice- periods were divided for different groups of classes into periods of 20 minutes, 10 minutes and 2 minutes. This plan of time re- quired 4 school days to complete the experiment when the inter- vening time was divided into 20-minute periods. 6 consecutive school days wiicn the intervening time was divided into lo-minute periods, and 22 consecutive school days when the intervening 40 minutes were divided into periods of 2 minutes each. The method of conducting the practice was described in Chapter I, and a sample sheet of the material used may be found on page <)8. It is necessary to remember that in scoring these division combinations a credit of one was given for each combination if the quotient and remainder were both correct. If either was wrong, no credit was given. In other words, the score was Improvement in the Group as a Whole 25 found by deducting one from the entire number of combina- tions worked for each one that was incorrectly worked. In Table VII, an exact class record is given of a class for which the intervening 40 minutes were divided into periods of ten minutes each. This arrangement made all the practice- periods of this class of equal length. Such a record is presented because it gives samples of all data to be used in this discussion and affords the best opportunity to note the change in ability from day to day. In all, eighteen classes took part in the division experiment, which means that there were eighteen such records as the one here presented, from which the data to be presented in the following discussion were obtained. Six of these 18 records involve exactly the same number of practice-periods as the one given in Table VII. Six others have but two practice- periods between the initial and the final periods and so arc more brief. The other 6 have 20 practice-periods between the initial and final periods and hence occupy more than three times the space occupied by the one given in Table VII. Six hundred six third- and fourth-year children took part in the experiment, in the course of which about 6500 papers were scored and en- tered in these 18 records. These can not be printed here, but they are placed on record in Teachers College where any one may use them. The following table gives sufficient data to enable the reader to understand the exact sources from which the summaries that are to be presented later were ol)tained. Table VII reads as follows, " Boy B in the initial practice- period worked 70 combinations, 63 of which were correct, or 90 per cent of them. In the second practice-period he worked 72 combinations with 72 correct. In the third practice he worked 80 combinations with 80 correct. In the fourth prac- tice he worked 87 combinations with 83 correct. In the fifth practice-period he worked 99 combinations with 92 correct. In the final practice-period he worked 119 combinations with 112 correct, or 94 per cent of them. His gain in number of com- binations worked correctly was 49, his gain per cent 78, and his gain in accuracy expressed in per cent was 4 j^er cent." So for any other individual. In the following discussion the quantities mentioned above will be referred to by the numbers in italics at the bottom of the columns. The mctliod of finding the per cent of accuracy, the gain per cent in number of problems 26 Practice in the Case of School Children TABLE VII Record of Class VII in Six Ten-minute Practice-periods in Division Indi- Initia 2nd 3 -d 4th 5 th Final Gain vid- Practice- Practice- Practice - Practice - Practice - Practice- uals Tt^^inA -parir^A •Po^-.r^A Po^;^,4 p«,.i«^i p««;,^^ IT r^er xer -L er x^er " i: Cll-Jl Pai- x^er Boys S. C. %C. s. C. s. C. s. C. S. C. s. C. %C. Gross cent Ac. A 55 54 98 60 58 84 82 77 75 76 75 74 71 96 17 30 — 2 B 70 63 90 72 72 80 80 87 83 99 92 119 112 94 49 78 + 4 C 27 25 93 45 38 55 51 59 56 66 60 70 66 94 41 168 + 1 D 16 12 75 14 14 25 24 25 24 18 17 95 5 63 + 20 E 77 77 100 99 98 106 106 112 112 133 133 153 151 99 74 96 — 1 F 27 21 78 28 18 19 13 30 23 36 32 45 42 93 21 100 + 15 G 31 25 81 36 31 46 43 41 35 38 31 49 42 86 17 68 + 5 H 50 49 98 51 48 61 61 56 53 55 53 65 63 97 14 29 — 1 I 62 62 100 62 62 69 69 69 67 60 60 68 68 100 6 10 J 34 30 88 48 44 58 57 54 52 52 50 53 52 98 22 73 + 10 K 18 17 95 36 33 54 50 49 44 77 70 63 59 94 42 40 — 1 L 30 19 63 35 25 46 33 64 58 40 36 37 35 95 16 84 + 32 M 90 86 96 104 98 114 113 128 127 99 41 48 + 3 N 69 68 99 62 59 83 81 78 75 77 77 92 89 98 21 31 — 1 43 38 89 53 45 69 65 61 58 56 53 63 60 95 22 58 + 6 P 27 27 100 27 27 33 32 31 27 28 28 30 27 90 —10 Q 78 74 95 89 86 109 99 104 104 108 98 107 98 91 24 32 — 4 K 37 37 100 58 58 67 67 62 60 69 67 77 77 100 40 105 S 48 44 92 63 57 55 52 66 66 77 77 100 33 75 + 8 T 67 66 99 88 86 103 101 102 101 99 95 112 111 99 45 68 U 54 51 94 57 54 92 68 69 67 62 62 64 64 100 13 25 + 6 V 35 32 92 40 37 35 32 44 41 56 54 97 22 69 + 5 w 27 25 92 27 25 43 39 38 35 36 36 39 39 100 14 56 + 8 Girls A 14 11 79 29 21 37 29 31 19 38 31 82 20 55 + 3 B 24 23 96 37 36 30 29 35 32 45 45 50 50 100 27 117 + 4 C 39 36 93 39 38 52 51 53 49 50 48 57 53 93 17 47 D 50 46 92 59 56 69 68 62 62 51 51 81 80 99 34 74 + 7 E 56 54 97 61 58 61 61 65 65 55 53 66 62 94 8 15 — 3 F 52 49 94 54 52 71 71 66 63 50 48 76 73 96 24 49 + 2 G 46 41 89 49 46 51 49 50 47 49 46 66 61 91 10 24 + 2 H 42 39 93 53 51 68 68 65 64 79 77 98 38 97 + 5 I 76 74 98 20 19 83 83 78 77 81 79 110 107 97 33 45 — 1 J 47 46 98 50 48 69 68 55 55 55 55 66 61 97 18 37 — 1 K 10 8 81 7 2 25 24 14 7 28 26 20 13 65 5 63 —23 L 72 69 96 71 69 93 92 93 93 90 87 98 96 98 27 39 + 2 M 45 45 100 55 54 66 62 77 76 63 54 71 67 95 22 19 — 5 N 6f) 57 95 59 59 72 72 57 56 70 70 84 83 99 26 46 + 4 O 24 16 67 37 22 36 27 45 36 41 32 52 45 87 29 181 + 20 P 45 45 100 39 38 59 59 64 58 62 61 71 70 99 25 56 — 1 Q 24 20 83 16 15 40 39 32 30 34 31 29 26 90 6 30 + 7 R 42 40 95 43 42 50 50 50 49 45 44 52 49 94 9 23 — 1 S 23 20 87 29 29 36 25 33 32 37 36 98 16 80 + 1 T 26 23 89 44 41 56 53 51 50 43 42 56 56 100 33 143 + 11 U 30 16 46 30 16 37 29 32 23 37 32 38 36 95 30 125 + 49 / 2 S 4 5 6 7 8 9 10 11 12 IS H IB 16 17 18 S.=Problenis solved. C.=Problems correct. %C.=Per cent of problems correct. Ac.=Accuracy. Improvement in the Group as a Whole 27 worked correctly, and the gain in accuracy expressed in per cent is the same as that used in addition, Table I, to which the reader may refer for the method if the meaning of these figures is not clear. " S " indicates problems solved, " C " problems solved correctly, and "Ac." accuracy. Initial Ability To determine the initial ability of this group of 606 children of last half of third and first half of fourth year in giving the results for the division combinations, two factors must be con- sidered, — first the number of such combinations they worked cor- rectly, and second the accuracy of their performance. The data to determine the first factor are the number of combinations worked correctly in the initial ten-minute period, a sample of which is given in Table VII, column 5. The numbers of com- binations answered correctly by the 606 children were distributed class by class, boys and girls separate, but only the following summary of this distribution table can be presented here. (Table VIII.) TABLE VIII Number of Division Combinations Answered Correctly Initial Ten-minute Period IN THE Number of combinations to 4 5 to 9 10 to 14 15 to 19 20 to 24 25 to 29 30 to 34 35 to 39 40 to 44 45 to 49 44 50 to 54 38 55 to 59 42 60 to 64 28 4.6 65 to 69 Individuals 23 33 65 56 52 51 57 45 15 Per cent 3.8 5.4 10.7 9.2 8.6 8.4 9.4 7.4 7.3 6.3 6.9 2.5 TABLE VIII- —Continued Number of combinations .... 70 to 74 75 to 79 80 to 84 85 to 89 90 to 94 95 to 99 100 to 104 105 to 109 110 to 114 115 to 119 120 to 124 125 to 129 Total Individuals 13 16 3 9 5 5 3 1 1 1 606 Per cent 2.1 2.6 .5 1.5 .8 .8 .5 .2 .2 .2 100 Median 34 . 5 combinations 25 Percentile 21.7 75 Percentile 52.5 P.E. 15.4 P-E- t.-obt. Av. .63 28 Practice in the Case of School Children Table VI 11 shows a distribution (in groups of 5) of the num- ber of combinations done correctly in the initial 10 minutes. It shows that 23 children (or 3.8 per cent of the entire group) did from 5 to 9 combinations, 33 children (or 5.4 per cent of the entire group) did from 10 to 14 com])inations, etc. The range in number of ccjmjjinations done correctly is from 5 to 126. The median number is 34.5 combinations. This means that there were just as many children who worked 34.5 combinations or more correctly in 10 minutes as there were who worked 34.5 coml)inations or less. The upper 25 per cent of the group did 53 combinations or more while the lower 25 per cent did 22 com- binations or less. The percentages in the table show that 74 per cent of the group did from 15 to 59 combinations. The variability of the group is also shown by the P. E., 15.4. The (lata presented here give 34.5 combinations as the most likely true median. Acctiracy in Dnnsion With what degree of accuracy did this group work the com- binations which they attempted in the initial to minutes, is the next (|uestion for consideration. The source of the data used in answering this question can best be seen in Table VII, column ./. The per cents of accuracy of which those in column ./ are I)art were distributed class by class, boys and girls separate. The following table (Table IX) shows a summary from this larger tabic. Table IX reads as follows from the right side: " Two hundred and thirty-seven children (or 39.1 per cent of the entire group) worked 96 to lOO per cent of their combinations correctly, etc." The ]>er cents are given in groups of 5. The range is from 26 per cent of accuracy to 100 per cent. The figures show almost a right-angle distribution. However, had the per cents been scaled more finely at the upper end, the distribution would then have shown a very decided skewness toward the lower end. The median i)cr cent of accuracy is 93, that is, just as many children worked with an accuracy of 93 to 100 per cent as with an accuracy of 26 to 93 per cent. The upper 25 per cent of the children worked with :m accuracy of 97 per cent or more. The lower 25 per cent worked with an accuracy of 85 per cent or less. Improvement in the Group as a Whole 29 TABLE IX The Per Cent of Coukect Answeks to the Division Combina- tions IN THE Initial Ten-minute Pehiods Per cent of correct answers 26 to 30 31 to 35 36 to 40 41 to 45 46 to 50 51 to 55 56 to 60 61 to 65 Individuals 1 1 3 7 5 7 12 Per cent .2 .2 .5 1.2 .8 1.2 2 TABLE IX- —Continued Per cent of correct answers 66 to 70 71 to 75 76 to 80 81 to 85 86 to 90 91 to 95 96 to 100 Total Individuals 11 22 31 53 70 146 237 606 Per cent 1.8 3.6 5.1 8.7 11.6 24.1 39.1 100 Median 93 per cent 25 l^ercentile 85 75 Percentile 97 P.E. 6 The upper 63 per cent of these children worked with an accuracy of 91 per cent or more. With the accuracy of the group determined, we can now define the initial ability of the group more exactly by saying that it was such that the median number of columns worked correctly by the group as a whole was 34.5 combinations, and according to the median per cent of accuracy 34.5 combinations were 93 per cent of the median number of combinations at- tempted. The chances are that one child out of two in the group taken at random would do 34.5 combinations correctly in 10 minutes with an accuracy of 93 per cent. Gross Gain in Number of Combinations Worked Correctly What gain in ability did this group of 606 third- and fourth- year children make in the course of 60 minutes of practice, is the next question to be answered.^ This gain has been measured " While there were 60 minutes of practice the gain was measured for only so minutes. The initial practice-period and the final practice-period were 10 minutes each. But the record for each of these periods gives the adding rate at the middle of each period. Hence the amount of practice whose eflfect is measured is from the middle of the initial period to the middle of the final period, or 50 minutes. 3° Practice in the Case of School Children both absolutely and relatively. (Xir ])rc.sent consideration is the gnjss fjain, wliicii was found for each individual by subtracting the number (^f combinations worked correctly in the first lo min- utes of practice from the number worked correctly in the final lo minutes of practice. Jveferring to Table VII, column 16, the reader will sec that boy A J^^•lined 17 combinations, boy 15, 49 combinations, etc. The j:,Mins made by the OoC) cliildren were distributed class by class, boys and girls separate, but for our discussion a summary of this large distribution table must suffice. 'J'his summary appears in Table X. 'IWIilJO X Thbi Guumh Gain in Numiiku ok (Jomiiinationh Anhwkiikd ('ourkctlv, TBOM Firrv Minutkh of J*itA(,"ncio Conil)in;i(i()riH gniiiod . -19 to -14 10 -9 to 5 -4 lo U 1 to 5 to 10 11 to ir, 16 to 20 21 to 25 26 to 30 31 to 35 36 to 40; 41 to 45 IndividuiilH 2 5 3 15 20 49 55 GO G9 ()0 53 41 39 Per ceuta .3 .8 .6 2.5 3,3 8.1 9.1 9.9 11.4 9.9 8.7 0.8 6.4 TABLE 'X.—Continwd Cuiitbinutiuuu gained. to 50 51 to 55 50 to (50 Gl to G5 G() to 70 71 to 75 7() to 80 81 to 85 8G to 90 91 to 95 to 100 Totiil IndividualB 31 18 25 17 16 10 5 5 4 2 2 606 Per centB 6.1 3 4.1 2.8 2.6 1.7 .8 .8 .7 .3 .3 100 Median 27.6 combinations 25 I'ercontilo 10.7 75 I'crcoiitilo 43.4 iM<;. IG 3 iM':-,.,.,.< A„ .t>6 In I'ablo X the gioss gains are distributed in groups of 5. beginning at the end of the tabic one sees that 2 children (or .3 per cent of the entire group) worked correctly from 96 to 100 cond)inations more in the final ten-minute i)eriod than in the initial ten-minute i)eriod ; 2 children (or .3 per cent) worked from ()i to ()5 more, etc. The distribution is somewhat skewed Jmprovcmenl, in the (irouj^ (is a Whole 3 r toward the Iii^di end. The range is from a loss of 19 com- l)inations to a j^^ain of 100 combinations. 'J'hc median j^ain is 27.5 combinations. That is, the g'roui) as a whole, measured by the median gain, i)rorited by the practice of fifty minutes to the extent that it worked correctly in 10 minutes at the end of practice 27.5 more cf)mbinations than it worked in 10 minutes at the beginning of the ])ractice. This median gain means that there were as many children who gained 27.5 combinations or more as there were who gained 27.5 combinations or less. The U])per 25 ])er cent of the class profited by the i)ractice to the extent that they worked c(jrrectly in the final 10 minutes 43.4 combinations (jr more in excess of the number wf>rked in the im'tial 10 minutes. 'i1ie lower 25 ])er cent i)rofited to the extent of 10.7 combinations or more, f^nly 25 of the 606 children failed to ])rorit by the i)ractice. Relative Cain in Division What relation did the gross gain from 50 nn'nutes of i)ractice bear to the initial ability is our next question for consideration. Or, putting it in other terms, what gain per cent in ability to make the division associations involved in this experiment re- sulted from 50 minutes of practice? This gain for the entire grou]) was found from the individual gains made by the 606 children. Table VII, column //, gives these gain per cents for one class. P)oy A gained 30 per cent, boy I> gained 78 per cent, boy C, 168 per cent, etc. The same points that were made in discussing the gain ])er cents in addition obtain here. These individual gain per cents were distributed class by class, boys and girls separate, but only the smnmary of this large table is given in Table XI, which shows the form of the distribution, the central tendency, and variability of the group. In Table XI, the gain ])er cents in division combinations cor- rectly answered resulting from 50 mimites of ])ractice are dis- tributed in groui)s of 15. The figures show that the distribution is skewed t(jward the high end. A wide range, from a loss of 74 ])er cent to a gain of 400 per cent, is seen. The mode is in the grou]) 6t to 75 per cent. The median is 75 ])er cent. That is, there were just as many children who gained 75 per cent or 32 Practice in the Case of School Children more as there were who gained 75 per cent or less. The upper 25 per cent of the class gained 116 per cent or more, while the lower 25 per cent gained 47 per cent or less. The figures in the table show that 96 per cent of the children profited by the practice. TABLE XI Gain Pek Cent in Division Combinations Correctly Answered FROM Fifty Minutes of Puactice -74 -59 -44 -29 -14 1 16 31 46 61 76 Gain, per cent .... to to to to to to to to to to to -60 -45 -30 -15 15 30 45 60 75 90 Individuals 1 3 2 5 14 16 42 59 77 85 76 Per cents .2 .5 .3 .8 2.3 2.6 6.9 9.7 12.7 14 12.5 Gain, per cent .... 91 to 105 106 to 120 121 to 135 136 to 150 137 to 165 166 to 180 181 to 195 196 to 210 211 to 225 226 to 240 241 to 255 Individuals 50 33 30 22 9 10 8 8 10 11 4 Per cents 8.3 5.4 5 3.6 1.5 1.7 1.3 1.3 1.7 1.8 .7 Gain, per cent .... 256 to 270 271 to 285 286 to 300 301 to 315 316 to 330 331 to 345 340 to 360 361 to 375 376 to 340 341 to 405 Total Individuals 4 5 2 2 3 2 3 4 2 4 606 Per cents .7 .8 .3 .3 .5 .3 .5 .7 .3 .7 100 Median 75 per cent 25 Percentile 47 75 Percentile 116 P.E. 34.5 Gross Gain in Accuracy To know the complete effect of the practice we must measure the change in accuracy of making the associations as well as the gain in speed in making them. The gain in accuracy is given as a gross amount, but is expressed in per cents. Table VII, column 18, gives these gross gains for one class. They were found by subtracting the numbers in column 4 from the corre- sponding numbers in. column 75, Boy A lost 2 per cent, boy B gained 4 per cent, etc. The gains made by the 606 children were distributed class by class, girls and boys separate, but only a summary of the large table is given here, in Table XII, in Improvement in the Group as a Whole ?,2> which the form of the distribution, the central tendency and the variabiHty of the group are clearly shown. TABLE XII Gross Gain in Accuracy in Division, Expressed in Per Cents of Answers that Were Correct, from Fifty Minutes of Practice Per cent gained -45 to -41 -40 to -36 -35 to -31 -30 to -26 -25 to -21 -20 to -16 -15 to -11 -10 to -6 -5 to -1 to 4 Individuals 1 1 2 3 3 11 37 113 211 Per cents .2 .2 .3 .5 .5 1.8 6.1 18.6 34 8 TABLE 'Xll— Continued Per cent gained 5 to 9 10 to 14 15 to 19 20 to 24 25 to 29 30 to 34 35 to 39 40 to 44 45 to 49 50 to 54 Individuals 102 52 26 13 11 8 2 5 4 1 Per cents 16.8 8.6 4.3 2.1 1.8 1.2 .3 .8 .7 2 Median 25 Percentile 75 Percentile P.E. 2.6 per cent 1.4 8. 4.5 Table XII shows a distribution of the gross gains in accuracy in division expressed in per cents of answers that were correct resulting from the 6o minutes of practice. The gains are given in groups of 5. The figures show a distribution conforming closely to the normal distribution curve. The range is from 45 per cent lost to 54 per cent gained. The median gain is 2.6 per cent which shows that there were as many children who gained 2.6 per cent or more in accuracy as there were who gained 2.6 per cent or less. The figures show that 435 children made the associations with equal or greater accuracy at the end of the practice than at the beginning, while 171 made them with less accuracy. Combining now the evidence afforded by Tables X, XI, and XII, we can say that the 50 minutes of practice effected a very remarkable gain in the ability of these children to make the division associations concerned in the tests. Not only did the group as a whole increase its median ability to the extent that 24 Practice in the Ca^c of School Children in lo minutes at the end of practice it could do correctly 27.5 (^or 75 per cent) more combinations than in 10 minutes at the beginning of the practice, but it also increased by 2.0 per cent its median accuracy in giving the answers. Summary ADDITION 1. 732 fourth-year children practiced for 75 minutes adding such columns as are shown on page 97. The gain from 60 minutes of the practice was measured. 2. Initial ability: In the initial 15 minutes of practice the median number of columns added correctly was 33.3. 3. Initial accuracy: In this initial 15-minutc period, the median per cent of accuracy was 79 per cent. 4. Initial al)ility and accuracy: Expressing the initial ability by number of jiroblems worked rather than by number of prob- lems worked correctly, we can say that in the initial 15 minutes the group added a median number of 29.5 with a median accur- acy of 79 per cent. 5. Absolute gain : In the final 15 minutes of practice the group added correctly a median of 10.7 more columns than in the initial 15-minute period. This gain resulted from 60 minutes of practice. 6. Relative gain : This gain of 10.7 columns meant a median percentile gain of 48 per cent, from 60 minutes of practice. 7. Change in accuracy : The change in accuracy was a median loss of .4 per cent, which was so small that it hardly needs to be considered in measuring the effect of the ])ractice. 8. The central tendencies given are valid group measures only when considered with the wide deviations from them. DIVISION 1. 606 children of last half of third year and first half of fourth year practiced for 60 minutes making such division com- binations as are shown on page 98. The gain from 50 minutes of the practice was measured. 2. Initial ability: In the initial 10 minutes of practice the median number of combinations done correctly was 34.5. 3. Initial accuracy: In this initial lO-minute period, the median per cent of accuracy was 93 per cent. Improvement in the Group as a Whole 35 4. Initial aljility and accuracy : The initial al^lity may be ex- pressed in number of combinations worked rather than in number worked correctly. The median number of combinations worked or attempted in 10 minutes was 37 with a median accuracy of 93 per cent. 5. Absolute gain : In the final 10 minutes of practice, the group worked correctly a median of 27.5 more combinations than in the initial 10 minutes of practice. 6. Relative gain : This gain of 27.5 combinations meant a median percentile gain of 75 per cent from 50 minutes of practice. 7. Change in accuracy : Along with the median gain of 75 per cent in speed went a median gain in accuracy of 2.6 per cent. TiiFi Value of the Practici-: ILxi'kkimi':nt as a MF/riiOD of Teaching It would be easier to estimate the value of the practice ex- periment as a method for increasing skill if we had norms resulting from other methods of work with which to compare the results of this study. So far as the author knows, there has been no previous attempt to measure in a group of school children the progress resulting from the application of any par- ticular method of school work. Mr. Courtis measured the change in a fifth-year grade from September to June in addi- tion and division. However, the material used, the period of time for which the change was measured, and the method of work were all different, hence a comparison would confuse in- stead of clarify. Dr. Thorndike" has made a study of practice with university students in addition in which the same method and material used in this experiment were used. He found a median saving of time of 29 per cent, or improvement of 41 per cent in amount done per unit of time, the score used here. The time spent in practice amounted to about 53 minutes. He says : " The amount of improvement in this experiment may also add to our confidence that the method of the practice experi- ment wherein one works at one's limit and competes with one's own past record may well be made a regular feature in many school drills. Even if the same length of time produced in chil- " Practice in the Case of KcXAiiion, American Journal of Psychology, Vol. XXI, pp. 4«3-4««- 36 Practice in the Case of School Children drcn u ])t'iccntilc iiuprovcmcnt only half as j^reat as here, the gain would still probably be far greater than the j^ain by any of the customary forms of drill." This is the nearest norm for a comparison known to the author. Dr. Thorndikc's students practiced about 53 minutes; the children in the author's e.\])erimcnt practiced 60 minutes or about one and a fifth times as lonJ^^ I lis students showed a median gain of 41 per cent; the children in this experiment, 48 per cent. In 50 minutes of practice in division the children in this experiment made a median gain of 75 per cent. These gains, taken with Dr. Thorndike's statement that if children should make half as j;ieat a gain in the same time as was made by the university students in his ex])eriment, they would gain more than by methods ordinarily employed in school, inspire conlidence in the practice e.\])eriment as a most efiicient method for conducting many of the drills necessary in regular school work. J. C. l>rovvn'^ in a study on the value of drill in arithmetic says, " b'ive-minute drill periods upon the fundamental number fads, ])receding the daily lesson in arithmetic, were found to be beneficial in the sixth, seventh and eighth grades. The benefit was not limited to im])rove(l mastery of the number habits, but included increased efficiency in arithmetical reason- ing. The improvement was still in evidence after the lapse of the twelve weeks summer vacation." Air. Rrown does not (U'sciibe (lie drill given except as follows: " 'f he first five min- utes of each period was devoted to drill work in addition, sub- traction, multiplication and division. About one-half of the drill work was written and the other half oral." Mr. Hrown provided for a control group, lie ftumd that the classes having this drill improved from 10 to 20 per cent more in different l)hases of arithmetic in the course of 20 recitations than the classes that did not have the practice. It is c|uite probable that even greater impnnement than was found would have resulted had the drill been conducted by the use of practice experiment whereby each indiviilual would have been competing with his own past record, thus having greater incentive to master thor- ouj/liU' the facts involved. ^Journal of Educalioiuil rsyclio!ot;y, l'\-l>. i()ii, pp. 81 to 88, and Nov. 1012. pp. 485-492. Improvement in the Group as a Whole 37 Factors Contributing to the Improvement That a surprising improvement was made by these children working with the practice experiment as a method, has been conclusively proved by quantitative data. To try to determine some of the factors which contributed to this improvement is the present problem, a solution of which must rest upon con- clusions more subjective in their nature. But the conclusions presented here are products of impressions made on the author while observing the reactions of the children to the work of this experiment ; or of impressions from previous experience with children who studied by different methods the same facts in- volved in the experiment; or of impressions of the enthusiasm of children engaged in games where a perseverance in under- going drudgery was displayed, that is seldom seen in the ])erform- ance of school tasks, but whose presence in many phases of drill work would guarantee a most efficient and willing mastery of the facts involved. Any one who has watched children jump the rope, has observed their persistent efforts to outdo a rival, or to surpass their own previous performance. Boys will bounce a ball from the sidewalk against a wall and catch it with untiring zeal so long as there is a contest or a desire to surpass a pre- vious record. It is quite noticeable in these games where the greatest persistence is displayed that children keep their record from time to time. It would be interesting to know if they would continue jumping the rope and tossing the balls with any- thing near the same enthusiasm if they knew nothing of their results from one trial to the next. It was with the hope of finding a method for mastering some of the fundamental requirements of school work, that would at least make a partial appeal to this enthusiastic exuberance in children, that the practice experiment was tried, and the author feels that this appeal in the method used has been the greatest contributing factor to the gains made. However, there are many factors which may have entered into the gains made, some of which are here presented. To what extent was the improvement due : 1. To increased power of concentrating on the task in hand? 2. To mastery of the technique of the experiment? 38 Practice in the Case of School Children 3. To greater incentive to effort in later tests than in the first one? 4. To work done outside of the time inckided in the test? 5. The lack of previous use of the functions involved ? 6. To reinforcement of the mental activity by the motor activi- ties involved in writing the results and by the visual per- cepts formed? 7. To maximum of opportunity afforded for exercising the function tested? 8. To the concerted effort of the group, each individual con- tributing a share to the gain ? 9. To acquiring " higher order" habits of work? 10, To the freedom of the children to regulate their efforts for quantity and quality in the way best suited to their indi- vidual capacities for each? 11. To the stimulus resulting from knowing the exact score in quantity and quality of the previous test and to the desire to surpass this previous record? The first five factors, if present to any great extent, would tend to produce an exaggerated measure of the gain in adding power and dividing power, or the gain found would be made up of the improvement in the functions tested plus the gain resulting from these five factors. The remaining six factors, it seems, would affect the measure of gain only as they directly influenced the ability to add and divide. An effort will be made to evaluate the influence of each of these eleven factors. To what extent was the improvement due to increased power of concentrating on the task in hand? In most experiments there is an increase in the ability to disregard conditions that make against the result sought. The children in this experiment ac- quired greater power of concentration and showed greater power to withstand distractions. How much this factor influenced can not be determined, but whatever the gain was it should be regarded as one of the valuable features of the method. To what extent was the improvement due to mastering the technique of the experiment? In the addition experiment this factor certainly did not enter at all, since the children had been accustomed to add columns and place their answers beneath them. In the division the children had not been accustomed to writing short division in the form given on the sheets, but before Improvement in the Group as a Whole 39 the first recorded practice was made they were given enough practice in writing the results in this form to insure an under- standing of its meaning. For a complete discussion of the care that was taken to insure this understanding on their part the reader is referred to Chapter I, page 6. There were a few children who could not get the results required by these sheets, but almost without exception these same children were unable to get the results any better by the customary form of division. Hence their failure or extreme slowness was due not to the form in which the division was expressed, but to their lack of knowledge of the process of division. Moreover, such records were not used in determining the gain. Further, the class rec- ords show no indication of misunderstanding this form of ex- pressing division, since there is on the whole as great a gain between the performances of any two days as between those of the first and second. So the author feels certain that master- ing the technique of this experiment produced no greater influence on the results, than this same factor ordinarily pro- duces in an experiment. To what extent was the improvement due to greater incentive to effort in later tests than in the first one? The children had greater incentive in the later tests because they knew their own previous records and were trying to surpass them. They had no such record to go against in the first test. Just how much this factor contributed to the results can not be told. Every other incentive was as great in the first test as in later ones. To what extent was the improvement due to work outside of the time included in the test? As has been said before, some children worked at home to improve. This could not be pre- vented. Suggestions not to do so, might have resulted in more doing so. Very few did this ; and these had to devise their own material, as no sheets were at their disposal. This factor, so far as it operated, tended to exaggerate the gains. How much can not be judged. While this uncontrolled factor detracts from the psychological value of the results, it indicates the value of the practice-experiment from the standpoint of education. To what extent was the improvement due to lack of previous use of the function involved? The amount of improvement that may be expected of a group largely depends upon the place it has reached in the practice curve. Greater improvement may 40 Practice in the Case of School Children (at least in many cases) be expected in the early stages of the curve than in the later stages. One basis is available to show the comparative initial ability of the children in this group with that of other children in the same school grades. Three fourth- year classes containing ii6 children in a school of another system using the same addition material worked a median of 21 columns with a median accuracy of about 75 per cent in 15 minutes. The children in the author's experiment worked a median of 30 columns with an accuracy of 79 per cent in the initial 15-minute period. While the initial ability for the three classes mentioned represents but a single school and so can not be considered as a norm of ability for fourth-year classes in general, it does indi- cate that the children in the present experiment had at least a fair degree of ability at the beginning of the practice and that there is no good reason for believing that the gains were unduly influenced by lack of previous use of the functions tested. The five factors thus far discussed would tend in general to exaggerate the improvement made to the degree that they oper- ated in the group tested. However, the author feels that their influence was very slight. The following six factors are the ones which best account for the improvement made. To what extent was the improvement due to the reinforce- ment of the mental activity by the motor activity involved in writing the results and by the visual percepts formed of the results? This opportunity for improvement was present at least in the division to a much greater extent than is true in most practice given in school. Formulating the results so that they could be written required a complete decision which might be shirked in a method not demanding a written answer. Writ- ing the results required motor responses which in turn left their impression upon the mind. The results when written were ob- jective forms to make further lasting impression through the sense of sight. The author attributes much of the gain in division to the writing of the results which demanded sharper thinking, induced motor activity, and made possible more acute visual images. To what extent was the improvement due to the maximum of opportunity afforded by the practice for exercising the function tested? Any one who has observed classes drilling by the meth- ods commonly used in school knows that much of the time is Improvement in the Group as a Whole 41 consumed in the manipulation of cards, in writing numbers pre- paratory for the drill, or in mere dawdling. In the method of drill used in the experiments of this study, almost all of the energy expended was continually directed toward making the associations whose perfecting the drill expected to accomplish. There was a minimum of writing to be done, which demanded a maximum of thinking, and which also tended to keep the mind on the work to be done. One is quite certain that in ten minutes of drill with this method, children had the opportunity to think and express many more results than with methods ordinarily used in school. Greater opportunity for concentrating on the facts to be fixed in mind could hardly be devised. So we may be certain that much of the gain was due to the children's making very many associations within a short time, which pro- vided for frequent repetition of associations, one of the funda- mental requisites for perfect habit formation. To what extent was the improvement due to the concerted effort of the group, each individual contributing a share to the gain? A prominent element in determining the amount of im- provement in any group is the part of the group that contributes to the improvement. Effort distributed throughout the group gives expectation of greatest improvement. The author has never seen children work when there was as great an effort on as great a part of a group. To what extent was the improvement due to acquiring " higher order" habits of work? Very many children in the third and fourth years of school form their combinations, especially in addition, by counting. No effort was made to get children to change their methods. It is not easy to tell with certainty just what children use the lower habits and which ones add by use of the combinations. No effort was made to keep a record of individuals' methods. This could hardly be done because even children who are most addicted to adding by ones, add on the smaller numbers, like two and three, at one step. So, to make an exact classification of the children on this basis would be impossible even if one could tell just what method a child were using at any instant. However, if children do acquire the higher order habits of adding by gradually taking up new combinations, it is reasonable to assume that, with the desire these children had to advance, many of them acquired higher order habits of 4 2 Practice in the Case of School Children work which contrihutcd a share to the gain that was made. On the other hand, this changing to a new method, in some in- stances that were very noticeable, caused an increase of errors that resulted in a loss. It is believed by the author that many of the individual gross losses were due to an unusual increase of errors resulting from some such change in method of work. Still for the group as a whole it is reasonable to believe that some share of the gain was due to the accjuisition of higher order habits of work. To what extent was the improvement due to the freedom of the children to regulate their efforts for quantity and quality in the way best suited to their individual capacities for each? In most school work such stress is placed on quality of work that a child not only is not rewarded for ([uantity of work, but is often made uncomfortable for excessive production regardless of whether he has maintained his standard of quality. Indi- viduals vary greatly in speed and accuracy of work. Rapid calculators are as a rule more accurate than slow ones. How- ever, this docs not mean that by speeding the slow one up to the fast one's norm he would become more accurate ; but there is considerable evidence to show that decreasing the speed of the fast one to the norm of the slow one would improve his accuracy little or not at all. In these tests children were left free within certain limits to regulate their effort on (juantity and (juality in such a way as to produce the highest score when the penalties for inaccuracy had been assessed. Most of the children soon saw that they could work more rapidly without increasing their errors and regulated their efforts accordingly. The fact that the group decreased only .4 of one per cent in median accuracy in addition and increased 2.6 per cent in median accuracy in division while there was a median gain per cent in speed of 50 per cent, shows conclusively that (luantity of work may be increased to a vast extent without deterioration in its quality. To just what extent the imi)rovcment was due to the way the children regulated their effort for quantity and qual- ity can not be told, but one may say with assurance that the imiM-ovcment would have been much less had the children fol- lowed the usual maxim, " Not how much but how well." To what extent was the improvement due to the stimulus re- sulting from knowing the exact score in quantity and quality Improvement in the Group as a Whole 43 of the previous test and to a desire to surpass this previous record? The author feels that the children's knowledge of the record of their previous performance, united with the desire each one had to surpass his previous record, was the greatest con- tributing factor to the improvement. Concerning the value of this factor I.add and Woodworth"* say : " Experimental condi- tions are stimulating largely because one has a measure of one's success and progress ; and the habit of checking up one's work can scarcely fail to prove of benefit wherever measures of suc- cess and failure are practicable." Not only were the children anxious to know the results of their previous tests, but they exerted such a determined effort each to surpass his own pre- vious record, that one felt progress must necessarily result. At the end of a test there were frequent expressions of regret that the time was up, but almost no indication of indifference or ennui. The children were deriving immediate satisfaction from winning in a game where each was striving to be a victor, not under the enervating condition that he try to surpass someone who is natively his superior, but under the energizing condition that he surpass his own previous day's performance. This factor provided incentive for the slow child who ordinarily realized the futility of striving to do as well as the exceptional members of the class ; it provided an incentive also for these excei)tional members to exert their best effort, and thus precluded the dawd- ling that many methods cultivate among the more gifted members of a class. So, under the conditions of this game of practice, most of the children were successful, and " nothing succeeds like success." Ladd and Woodworth" appraise the emotional tone attending successful and baffled effort as follows : " Exactly how these emotions act to strengthen one association and to weaken or counteract another cannot readily be seen ; but it is safe to assume that they corresponded to some genuine dynamic process of great efficacy." So, much of the gain in efficiency must have been due to the children's knowing their previous success, which in turn induced a pleasurable emotional tone to inspire continued intense effort and strengthen associations formed. The intense effort that was displayed, the improvement that was made, and the emotional attitude that was aroused, seem to "Elements of Physiological Psychology, p. 571. "Ibid., p. 552. 44 Practice in the Case of School Children be features that should recommend the practice experiment as a method for school work. IJryan and llarter^" say: "A school method must be judged by the moods and tempers which it cul- tivates not simply by what is learned, still less by the momentary interest it arouses. ]f one forces mastery of the multiplication tables by a method which keeps one half the school cowed and the other half rebellious, one has obtained a useful result but at disastrous cost. Better not know the multiplication tables Ih.'iii be thus morally maimed. There are many schools and homes where hard tasks are performed in a good temper; where thorough (bill does not arrest, but prepares the way for higher development, where childien begin to do what they must later do to succeed in any business — pass cheerfully from interest in desired ends to a resolute dnidgory necessary for the attainment of lliese ends." That tlio practice experiment, wherever applic- able, meets the demands made here of a method for school work, is confidently asserted. ^^ Psych. Rev., Vol. I, p. 370. CiiAi''n':i>: iii Till': i<:i'Fi":cT oi^ tiiI': DisrkiiuriMON and length OF WORK I'FRiOD Ul'UN Till': KATE OF LEARNING Tlic proMcin of lliis (li.iplcr is lo iiicasnix- (he cU'vci of ;;iviii;^ al)oii( oiu- iiotir of (liill ill arilliiiulic in periods of (IKVcrcnt Unt;(li. If we have an iiour lo devole to (h'ill in the division lahles. or inulliplicalion tables or in addition, is it better io ^ivc this (bill ill short periods extendiii}^ over a eorrespondinjjly }j;^realer iiuiiiber of days, or is it better to give the drill in loiif^^er periods thns eoneentraliiifj it into fewer days? Ti.AN oi" 'III!': rKAc'iicr; To understand the maimer in whieb the solution to this prob- lem was attempted, it is necessary to recall the plan of j^iving' the tests that have been discussed in the prcccdin^^ chapters. In the addition experiment with fourth-year children, there were four f^roups (;f classes on the basis of the distribution of the j)iaeti(e. In all the j^'-roups there was ,an initial practice |)eriod and a linal practice-period of fifteen minutes each, but the in- terveninj^' forty-live mintites were distributed dilTcrently for the different f^roups. In the first j^Moiip (he iiilerveninj;' forty live minutes were divided into two ])ractice-perio(ls of twenty-two and a half minutes each; in the second ^Mvnip into three practice- jjcriods of (ifteeii iiiiiiutcs each; in the third jMoiip into seven practice periods of six minutes each and one period of three minutes; in the fourth ^n'oup into twenty one two-minute periods and one three minute period. The followint^ plan makes this distribution clear: CinoiirH Initial Picuioi) Intiouvicnino ■1.') MrNnriiH I'^inai, I'lMdoi) 1 IT) min. 2 22i mill. IT) mill II ir> min. .'{ 15 mill. 1 Ti mill III If) mill. 7 n mill. .•IIKJ 1 '■'< niin, l.'^i mill IV l.'> mill. 21 2 mill, iind I '.'< min. I.'i min. 45 46 Practice in the Case of School Children Each group of classes practiced once a day on successive school days, as far as possible," which of course resulted in the experiment extending over a different number of days for the different groups. For Group 1, four successive days were required; for Group II, five successive days; for Group III, ten successive days; and for (n-oup IV, twenty-four successive days. TA15LK XI 11 NUMHKIl OK I'liOHI.lOMS AdDIOI) CoRItlOCTLY IN TIIK INITIAL FlFTEEN- MINUTK I'lOUIOl) HY THH FoUll (JUOUPS (Each DiaainNATKD by thb Numumk of Minutes in Its Intehvkninq Practich-Periodb) Coliiiiuis Group I Group II Group III Group IV ad.k'd CDiTC'ct ly 22 .^ mill. 15 min. G min. 2 min. Iiidi- Per Tndi- Per Iii.U- rer In.li- I'er viiluiils cents viiluiils cont.s viiliKils cent a viiiuals cents to 4 1 .5 2 .9 5 to 9 IG 8.2 (') 5.8 10 4.9 18 7.9 10 to 14 23 11.9 8.7 .35 17.1 13 5.7 15 to 1!) :v2 10.5 l(i 15 I 45 22 40 17.4 20 to 24 ;i8 19. G 19 18, :{ 3(> 17.0 37 10.2 25 to 2!) 27 i;{.9 11 10,0 33 10. 1 40 17.4 ;{0 to ;m 19 9.8 II 13.5 19 9.2 29 12.7 ;{5 to ;w i;{ G.7 12 11.5 11 5,4 21 9.2 40 to 44 7 ;i.G 4 3,8 8 3,9 14 0.1 45 to 49 •) 1 4 3,8 4 • ) 4 1,8 60 to 54 5 2,G • > 1,9 •> i. 4 1.8 55 to 59 4 2,1 3 1.3 GO to 04 • ) 1. 1 1. 1 .5 05 to 09 T .5 1 .5 1 .5 70 to 74 4 2.1 • > 1,9 1 .5 75 to 79 SO to 84 ;{ 2 , 9 85 to 89 1 .5 90 to 94 1 .5 95 to 99 100 to 104 105 to 109 1 1 Total 194 UK) 104 100 205 100 229 100 Average 25.9 29 . 2 22.7 20. 3 Median 22.9 25.4 21 1 25 . 1 25 P. IG. 17.9 15, 1 17.5 75 P. 82. 35.8 28, t) 33.3 P.E. S. 8.9 G.8 7.9 •P-1'^t.-oht. Av. G .9 .5 .5 " For exceptions and further details see Chapter I, " Plan of the Practice." Distribution and Length of Work Period 47 In Group T, there were 6 classes, containing 194 children ; in Group 11/- 3 classes, containing 104 children; in Group III, 6 classes, containing 205 children; and in Grouj) IV, 6 classes, containing 229 children. Addition Initial Ability of the Groups The classes to compose any group were chosen at random, or sometimes on the basis of convenience in reaching them. Since nothing was known of the exact ability of the classes in addition before beginning the tests, this plan of grouping seemed justifiable. Had it been possible to have the tests for all these classes in progress at one time, one could have grouped them on basis of ability, after the first practice-period, in such a way as to secure equality of initial ability in the four groups. Table XIII shows a summary of the distribution of the initial scores of all these classes grouped, as has been shown above, on the basis of the length of the practice-periods for the 45 minutes of practice between the initial practice-period and the final practice-period. The data composing this table are the same as those composing Table II, but they are distributed here to show the initial ability of the four groups that are being compared. The table shows the number of problems worked correctly in the initial 15-minute practice-period by the members of each group. Samples of the scores distributed here may be seen in Table I, column j. Table XIII reads as follows : In Group I, 16 children (or 8.2 per cent of the group) added from 5 to 9 columns correctly in the initial 15-minute period; in Group II, 6 children (or 5.8 per cent of the group) showed the same ability; in Group III, 10 children (or 4.9 per cent of the group) showed the same ability; and in Group IV, 18 children (or 7.9 per cent of the group) showed the same ability. The range of the four groups was slightly different, but not enough to deserve much weight. The central tendencies as shown by the averages of the groups "After the experiment had progressed for some time it seemed more profitable to discontinue giving the 15-minute practice-periods, since they differed so little from the 22^-minute periods, and to concentrate effort on the extreme and middle periods. This accounts for the smaller number of classes and children in Group II, which in turn causes less weight to be given to the measures of Group II in the conclusions. 48 Practice in the Case of School Children were 25.9, 29.2, 22.7 and 26.3. By medians they were 22.9, 25.4, 2 1. 1 and 25.1. The individuals comprising the middle 50 per cent of the groups had the following ranges: Group I, 16 col- umns to 32 columns; Group II, 18 columns to 36 columns; Group III, 15 columns to 29 columns; Group IV, 18 columns to 32 columns. According to the percentages given in the table, about 44 per cent of Group I, 52 per cent of Group II, 38 per cent of Group III, and 52 per cent of Group IV, added correctly 25 columns or more. TABLE XIV Per Cent of Pkoblems Added Correctly in the Initial Fifteen- minute Periods by the Four Groups (Each Designated by the Number op Minutrs in Its Intervening Practice-period) Group I Group II Group III Group IV Per cent, of problems 22 .^ min. 15 min. 6 min. 2 min. correct Indi- Per Indi- Per Indi- Per Indi- Per viduals cents viduals cents viduals cents viduals cents 6 to 10 11 to 15 1 1 1 .5 16 to 20 21 to 25 1 .5 1 .5 26 to 30 2 1. 1 .5 31 to 35 2 1. 1 1. 2 1. 4 1.8 36 to 40 6 3.1 1 1. 2 1. 4 1.8 41 to 45 5 2.0 2 1.9 5 2.5 4 1.8 46 to 50 4 2.1 2 1.9 7 3.4 9 3.9 51 to 55 6 3.1 4 2. 7 3.1 56 to 60 7 3.6 1 1. 10 4.9 7 3.1 61 to 65 21 10.9 5 4.8 9 4.4 11 4.8 66 to 70 14 7.2 7 0.7 16 7.8 21 9.2 71 to 75 13 6.7 17 16.4 28 13.2 26 11.3 76 to 80 22 11.4 12 11.5 28 13.2 38 16.7 81 to 85 31 16. 11 10.6 10 7.8 23 10.1 86 to 90 27 13.9 17 16.4 27 13.2 37 16.2 91 to 95 19 9.8 14 13.5 21 10.3 22 9.6 96 to 100 14 7.2 13 12.5 30 14.6 13 5.7 Total 194 100 104 100 205 100 229 100 Median 79 82 79 78 25 P. B4 72 59 38 75 P. S8 91 c)0 B8 P.E. 12 9.5 10.5 10 In order to give a more exact statement of the relative initial ability of these four groups in addition, it is necessary to show the accuracy with which each group did its work. In finding Distribution and Length of Work Period 49 the accuracy of the groups, the same data are used as were used in Table III, but now the summaries show the distributions for the classes grouped according to the length of the practice- periods which comprised the intervening 45 minutes of practice. This distribution is found in Table XIV. Samples of the indi- vidual records which comprise this table may be found in Table I, column 4. The medians for accuracy are : for Group I, 79 per cent ; for Group II, 82 per cent; for Group III, 79 per cent; and for Group IV, 78 per cent. This means that 50 per cent of Group 1 added with an accuracy of 79 per cent or more; 50 per cent of Group II added with an accuracy of 82 per cent or more; 50 per cent of Group III added with an accuracy of 79 per cent or more, etc. In terms of number of problems worked and accuracy the following is the initial status of each group; Group I added 29 columns with an accuracy of 79 per cent. Group II added 31 columns with a median accuracy of 82 per cent. Group III added 27 columns with a median accuracy of 79 per cent. Group IV added 32 columns with an accuracy of 78 per cent. The median initial ability of the groups summarized is as follows: Median Number of Median Number of Columns Marked Columns Correct Accuracy Group I 29 columns 22.9 79 per cent Group II 31 columns 25.4 82 per cent Group III 27 columns 21.1 79 per cent Group IV 32 columns 25.1 78 per cent Gross Gain Having defined the initial ability of the four groups in terms of quality and quantity of performance, we shall now deter- mine the absolute gain made by each group in the course of 75 minutes of practice, 30 minutes of which were occupied by the initial 15-minute practice-period and the final 15-minute prac- tice-period for all groups alike; the remaining 45 minutes of which were divided into practice-periods of 22^/2 minutes for Group I, 15 minutes for Group II, 6 minutes for Group III, and 2 minutes for Group IV. The data to be used in reaching a conclusion to this problem are the same as those used in Table IV, the source of which was described minutely in connection with that table. In Table 5° I'ractkc in the Case of School Children XV the gross gains made by the 732 children, samples of which may be seen in Table I, column 7./, are distributed for the four groups. TAliLE XV Cliio.sH (Jain in Niimiikii ok Columns Added Couukotly in Fifteen MiNii'i'KH, Madio in the Couusio ov Seventy-five MiNUTEH OF J'uacti(;e hy the Fouh (Juoui'S Group I Group II Group III Group IV ColiimiiH 22i min. 15 min. 6 min. 2 min. In.li- Por Indi- Per Indi- I'or Indi- Per vidiialN corit.H vidiiulH cents vid uulH cents viduals cents -15 to -12 2 1. 2 1.9 1 .6 -11 to -8 4 3 . H 5 2 . 5 2 .9 -7 to -4 H 1.1 4 3.8 11 5.4 5 2.3 ~.i to 12 (5.2 H 7.7 29 14.1 14 () . 2 1 to I ;',5 J8. <» S.7 21 10.3 28 12.2 r> to s ;m 17.5 10 9.(5 28 13.2 33 14.4 <) to 12 25 12.9 21 23. 1 31 15.1 31 13.6 i;{ to i() 27 13. 9 12 1 1 . 5 22 10.8 29 12.7 17 to 20 22 11.3 7 (5.7 25 12.2 21 9.2 21 (0 21 4 2. I I 3 . S 9 4.4 12 6 4 25 to 28 iO 5.2 3 2.9 7 3.4 9 2.9 29 to 32 (> 3.1 3 2 . 9 9 4 . 4 7 3.1 33 to M (i 3.1 3 2 . 9 2 1 13 5.7 37 to 40 1 .5 5 4.8 3 1.5 7 3 1 41 to 44 1 J. 1 .5 3 1.3 45 to 48 1 .5 2 1.9 1 .5 3 1.3 4«) (0 52 1 .5 J 1. 3 1.3 53 to 5<) 2 .9 67 to (iO 1 .5 2 .9 61 to (54 2 1.9 2 .9 05 to ()8 2 .9 Total 194 100 104 100 205 100 229 100 Average 11. 13.0 10.7 IG.l Median 9.5 11. 9.G 12.6 26 P. 3.5 4 1 1.7 6.4 76 P. 17. 19. 1 17. G 23.1 P.E. (5.8 7.7 8. 8.9 P-E-t.-obt. Av. .6 .8 .0 .0 Tabic XV reads as follows in the bflh lino: " In Group I, 35 children (or 18 per cent of the group) gained from i to 4 prob- lems ; in (Ironp If, <> children (or 8.7 per cent of the group) gained fioin 1 to 4 problems; in Group III, 21 children (or 10.3 per cent of the group) gained from i to 4 problems; in Group IV, 28 children (or 12.2 per cent of the group) gained Distribution and Length of Work Period 5 1 from I to 4 problems." Tlic range of distribution is somewhat wider for the two-minute grouj) than for the other three groups, being from 15 columns lost to 68 gained. The four distribu- tions have the same general form, being slightly skewed at the high end. The median gain for the 22r/^-minute group is 9.5 columns; for the 15-minute group, 11 columns; for the 6-minute group, 9.6 columns; and for the 2-minute group, 12.6 columns. The respective averages were ii.o, 13.6, 10.7 and 16.1. These medians and averages show an advantage for the group which had the forty-five minutes of practice in 2 minute periods. The 22^ -minute group and the 6-minute groups made about the same gain, while the i5-minute group made a larger gain. The percentages in the table show that : 40 per cent of Group I gained 13 columns or more; 41 per cent of Group II gained 13 columns or more; 39 per cent of Group III gained 13 columns or more; 50 per cent of Group IV gained 13 columns or more. Percentile Gain To get a measure of the gain of each group in terms of its initial performance is our next problem. To find the gain per cent of each group of classes, the gain per cents made, by all the individuals, samples of which may be seen in Table I, column ij, have been distributed in four groups, formed on the basis pre- viously described. These data are given in Table XVI. Table XVI reads as follows in the seventh line: "In Group I, 23 children (or 11.9 per cent of the group) gained from i to 15 per cent in columns added correctly; in Group II, 10 chil- dren (or 9.6 per cent of the group) gained the same; in Group III, 15 children (or 7.5 per cent of the group) gained the same; in Group IV, 18 children (or 7.9 per cent of the group) gained the same." The range of the groups is from 89 per cent lost to more than 300 per cent gained, that of each group being about the same. The form of the distribution for each group is simi- lar, being skewed toward the high end. The median gain per cents for the groups made in the course of 75 minutes of practice are as follows : Group I, 45 per cent. Group II, 43 i)er cent; Group III, 42 per cent, and Group IV, 56 per cent. 52 Practice in the Case of School Children TABLE XVI Gain Per Cent in Columns Added Correctly Made in the Course OP Seventy-five Minutes Practice by the Four Groups Group I Group II Group III Group IV Gain, per cent 22^ min. 15 min. 6 min. 2 min. Indi- viduals Per cents Indi- viduals Per cents Indi- viduals Per cents Indi- viduals Per cents -89 to -75 -74 to -GO -59 to -45 -44 to -30 -29 to -15 -14 to 1 to 15 16 to 30 31 to 45 46 to 60 61 to 75 76 to 90 91 to 105 106 to 120 121 to 135 136 to 150 151 to 165 166 to 180 181 to 195 196 to 210 211 to 225 226 to 240 241 to 255 256 to 270 271 to 285 286 to 300 301 to 305 1 2 4 11 23 29 28 29 24 15 8 5 5 3 2 1 1 1 1 1 .5 1. 2.1 5.7 11.9 14.9 14.2 14.9 12.4 7.8 4 1 2.6 2.6 1.5 1. .5 .5 .5 .5 .5 2 1 1 2 4 8 10 12 14 9 11 G 5 5 3 5 1 2 1 1 1 1.9 1. 1. 1.9 3.8 7.7 9.6 11.6 13.4 8.7 10.0 5.8 4 8 4.8 2.9 4.8 I 1.9 1. 1. 1. 3 7 5 29 15 27 21 20 15 16 9 10 3 8 2 1 1 4 1 3 1 2 2 1.5 3.4 2.5 13.7 7.5 13.2 10.3 9.8 7.5 8. 4.4 4.9 1.5 3.9 1. .5 .5 2. .5 1.5 .5 1. 1. 2 7 13 18 29 22 34 30 17 18 9 3 5 4 5 1 2 3 1 6 .9 3.1 5.7 7.9 12.7 9.6 14.9 12.1 7.4 7.9 3.9 1.3 2.3 1.8 2.3 .5 .9 1.3 .5 2.7 Total 194 100 104 100 205 100 229 100 Median 25 P. 75 P. P.E. 45 19 72 26.5 43 13 86 26.5 42 8 87 39.5 56 24 90 33 Gain in Accuracy One other factor remains to be considered in measuring the effect of the practice on the four groups, the change in accuracy wrought by the practice. The data to determine this result have been used before in Table VI. The source of these data may be seen by referring to Table I, column i6. A distribution of the gains in accuracy for the four groups is given in Table XVII. Distribution and Length of Work Period 53 TABLE XVII Gross Gain in Accuracy in Addition, Expressed in Per Cents OF Answers that Were Correct, R'Iade in the Course of Seventy-five Minutes of Practice by the Four Groups Group I Group II Group III Group IV Per cents 22* min. 15 min. 6 min. 2 min. gained Indi- Per Indi- Per Indi- j Per Indi- Per viduals cents viduals cents viduals cents viduals cents -55 to -51 1 1. 2 1. 5 2.3 -50 to -46 2 1.9 -45 to -41 2 1. -40 to -36 2 1.9 4 2. -35 to -31 2 1. 12 5.9 5 2.3 -30 to -26 2 1. 5 4.8 5 2.5 11 4.8 -25 to -21 3 1.5 3 2.9 4 2. 7 3.1 -20 to -16 9 4.5 5 4.8 20 9.8 20 8.7 -15 to -11 12 6.2 7 6.7 21 10.3 23 10.1 -10 to -6 17 8.7 12 11.6 19 9.2 20 8.7 -5 to -1 23 11.9 19 18.3 17 8.2 42 18.4 to 4 36 18.5 19 18.3 25 12.2 31 13.6 5 to 9 26 13.4 8 7.7 22 10.8 23 10.1 10 to 14 23 11.9 7 6.7 16 7.8 15 6.6 15 to 19 15 7.8 6 5.8 15 7.3 11 4.8 20 to 24 10 5.2 3 2.9 10 4.9 5 2.3 25 to 29 4 2.1 3 2.9 5 2.5 5 2.3 30 to 34 3 1.5 2 1. 1 .5 35 to 39 5 2.6 1 .5 2 .9 40 to 44 1 .5 1 1 45 to 49 2 1. 1 1. 2 1. 50 to 54 1 .5 55 to 59 1 .5 2 .9 60 to 64 65 to 69 1 .5 Total 194 100 104 100 205 100 229 100 Median 3.5 -1.6 -1.5 -2.7 25 P. -4.7 -10.1 -15 -13.5 75 P. 12.9 6.4 9.7 6.2 P.E. 8.8 8.3 12.4 9.8 Table XVII reads as follows near the middle : " In Group I, 36 children (or 18.5 per cent of the group) gained from o to 4 per cent in accuracy; in Group II, 19 children (or 18.3 per cent of the group) gained the same; in Group III, 25 children (or 12.2 per cent of the group) gained the same; in Group IV, 31 children (or 13.6 per cent of the group) gained the same." The median change in accuracy in the four groups is as fol- lows : Group I gained 3.5 per cent; Group II lost 1.6 per cent; Group III lost 1.5 per cent; Group IV lost 2.7 per cent. 54 Practice in the Case of School Children Summary The following summarized statement gives the data necessary for comparing the four groups and for interpreting the results. Medians of the Groups of Individuals Median Initial Ability: Average Median Median Median Examples Gross Gross Reliability: Gain Gain in Correct Gain Gain P.E t.-obt. Av. Per cent Accuracy Group I 22 . 9 11.0 9.5 .5 45 3.5 Group II 25.4 13.6 11.0 .8 43 -1.6 Group III 21.1 10.7 9.6 .6 42 -1.5 Group IV 25.1 16.1 12.6 .6 56 -2.7 In computing all averages and medians thus far, the indi- vidual in the group has been considered as the unit. We shall now present the same data computed, with the class as the unit. The median ability of each class was computed ; then the average of these class-medians for each group was found. The reliability of this average was computed on the basis of the number of classes. The following figures show the result. Average op Class Medians Average Initial Average Average Median Ability Reliability Gross Median Reliability of Median of Classes: of Average: Gain of of Average: Percentile Examples Correct t.-obt. Av, Classes P.E. ^ . ^ . Gains t.-obt. Av. Group I 23.7 1.7 10.2 1.1 42 Group II 25.7 2.3 9.6 1.2 41 Group III 21.3 .9 9.4 .8 42 Group IV 25.5 1.5 13.9 1.9 58 Using now the best measure of efficiency — the number of examples correct, which includes credit for both speed and ac- curacy — and the three methods of computing the gain, we have : Average Gross Gain OF Individuals Median Gross Gain OF Individuals Average of Median Gross Gains OF Classes Group I Group II Group III Group IV 11.0 13.6 10.7 16.1 9.5 11.0 9.6 12.6 10.2 9.6 9.4 13.9 It appears that, taking the results at their face value, the 2-minute practice was the most favorable. Part of its superiority to the 225^ -minute and 6-minute practice was probably due to the greater initial ability of those taking it. This will be studied later. Part of its superiority may have been due also to chance factors, but these are as likely to have worked against as for it. Distribution and Length of Work Period 55 Part of its superiority may have been due to the greater length of time covered, and so the greater amount of opportunity for training in and out of school, apart from the special practice. We shall return to this after surveying the facts concerning division. Division Initial Ability In the division experiment, with children in the last half of the third year and the first half of fourth year, there were three groups of classes on the basis of the distribution of the 60 minutes of practice. In all of the groups there was an initial practice-period and a final practice-period of 10 minutes each, but the intervening 40 minutes were distributed differently for the different groups. In Group I, the intervening 40 minutes of practice were divided into 2 practice-periods of 20 minutes each ; in Group II, into 4 practice-periods of 10 minutes each ; and in Group III, into 20 practice-periods of 2 minutes each. The following plan shows clearly this distribution of the practice. Initial Intervening 45 Minutes Final Groups Practice-Period op Practice Practice-Period I 10 min. 2 20 min. periods 10 min. II 10 min. 4 10 min. periods 10 min. Ill 10 min. 20 2 min. periods 10 min. The classes of each of the groups practiced once a day on successive school days as far as possible,^^ which of course re- sulted in the experiment extending over a different number of days for the different groups. For Group I, 4 successive days ; for Group II, 6 successive days ; and for Group III, 22 suc- cessive days were required to complete the experiment. In Group I, there were 6 classes containing 204 children ; in Group II, 6 classes containing 209 children ; and in Group III, 6 classes containing 193 children. The classes to compose the different groups were chosen at random or on the basis of accessibility for the author, since nothing definite was known of their ability in division before beginning the tests to serve as a basis for grouping whereby equality of initial ability might be secured in the dift'erent groups. Table XVIII shows a distribution of the number of combina- ^' For exceptions and further details see Chapter I, " Plan of the Practice." S6 Practice in the Case of School Children tions worked correctly in the initial lO-minute period of prac- tice by the 606 children, arranged in three groups made on the basis of the length of the intervening periods of practice. Sam- ples of these scores may be seen in Table VII, column j. TABLE XVIII Thk NrTMiiKH OF Division Combinations Answered Cohrectly in Tuio Initial Ten-minute Period by Each of the Three Groups (irouD I ( jioup II G roup III Numhor of combina- 20 min. 10 min. 2 min. tions Iruli- Per Indi- Per Indi- Per viilualH cents vidui! .« cents vidual s cents 5 to 9 10 4.9 10 4.8 3 1.6 10 to 14 15 7.4 12 5.7 6 3.1 15 to 19 19 9.3 22 10.5 24 12.4 20 to 24 13 6.4 24 11.5 19 9.8 25 to 29 15 7.4 26 12 . 4 11 5.7 30 to 34 15 7.4 21 10. 15 7.8 35 to 39 20 9.8 21 10. 16 8.3 40 to 44 17 8.3 12 5,7 1() 8.3 45 to 49 12 5.9 19 9.1 12 6.7 50 to 54 IS s.s 12 5 7 S 4.1 55 to 59 14 6 . 9 10 4.8 IS 9.3 60 to 64 16 7.8 5. 2.4 7 3.6 65 to 69 4 2 S 3,8 :>, 1.6 70 to 74 2 T. 3 1.4 s 4.7 75 to 74 S 3.9 3 1.4 5 2.6 80 to 84 3 1.6 85 to 89 2 1. 1 .5 6 3.1 90 1,0 94 1 .5 4 2.1 95 to 99 3 1.5 ') 1. 100 to 101 105 to 109 3 1.6 110 to 114 1 .5 115 to 119 120 to 124 1 .5 125 to 129 1 .5 Total 204 100 209 100 193 100 Avenifie 3 9.1 34.6 44.6 Median ;, S.3 32. 40.3 25 P. 9 2.2 21.2 23.5 75 P. 5 4.2 46.8 58.3 P.E. 1 6 12.8 17.4 ^ •^'•t.-obt. Av. 1.1 .9 1.3 The average number of combinations worked correctly was : for Group I, 39.1; for Group II, 34.6; for Group III, 44.6. The median number of combinations worked correctly was : for Distribution and Length of Work Period 57 Group I, 38.3; for Group II, 32; for Group III, 40.3. The per cents in the table show that 47 per cent of Group I, and 35 per cent of Group II, reached the median of Group III. In evaluating the gains made by the three groups, this difference in initial ability must be taken into consideration. The accuracy of the three groups in doing the division com- binations is another factor which must be determined in order to show exactly the comparative ability of the groups in their initial performance. The data to be used are the per cents of accuracy for the 606 children, samples of which may be seen in Table III, column 4. These data are distributed in Table XIX for the three groups. TABLE XIX Per Cent of Correct Answers to the Division Combinations in THE Initial Ten Minutes by the Three Groups Per cent Group I Group II Group III of correct answers 20 min. 10 min. 2 min. Indi- viduals Per cents Indi- viduals Per cents Indi- Per viduals cents 26 to 30 31 to 35 36 to 40 41 to 45 46 to 50 51 to 55 56 to 60 61 to 65 66 to 70 71 to 75 76 to 80 81 to 85 86 to 90 91 to 95 96 to 100 2 2 3 2 6 5 6 6 11 22 47 92 1. 1. 1.5 1. a. 2.5 3. 3. 5.4 10.8 23. 45.1 1 1 1 3 2 4 3 3 11 15 15 27 57 66 .5 .5 .5 1.4 1. 2 1^4 1.4 5.3 7.2 7.2 12.9 27.2 31.6 2 1 3 3 5 10 27 21 42 79 1. .5 1.6 1.6 2.6 5.2 14. 10.9 21.7 40.9 Total 204 100 209 100 193 100 Median 25 P. 75 P. P.E. 94 87 98 5.5 92 83 97 7 93 85 98 6.5 Table XIX reads as follows at the bottom: In Group I, 92 children (or 45.1 per cent of the group) worked the combina- tions in the initial practice-period with an accuracy of 96 to 100 per cent; in Group II, 66 children (or 31.6 per cent of the 58 Practice in ike Case of School Children group) worked with the same accuracy; and in Group III, 79 children (or 40.9 per cent of the group) worked with the same accuracy. The three distributions are very similar, being de- cidedly skewed toward the low end. The median accuracy for the three groups is almost equal. Group I worked with a median initial accuracy of 94 per cent, Group II 92 per cent, and Group III 93 per cent. So in the factor of accuracy there is not enough difference to demand evaluation. TABLE XX Number of Division Combinations Correctly Answered Gained in the Course of Sixty Minutes of Practice by the Three Groups Number of Group I Group II Group III combina- lions 20 min. 10 min. 2 min. Indi- viduals Per cents Indi- viduals Per cents Indi- viduals Per cents -19 to -15 -14 to -10 -9 to -5 -4 to 1 to 5 6 to 10 11 to 15 16 to 20 21 to 25 26 to 30 31 to 35 36 to 40 41 to 45 46 to 50 51 to 55 56 to 60 61 to 65 66 to 70 71 to 75 76 to 80 81 to 85 86 to 90 91 to 95 96 to 100 1 2 8 10 17 27 25 29 23 18 8 8 8 2 7 4 3 3 1 .5 1. 3.9 4.9 8.3 13.2 12.3 14.2 11.3 8.8 3.9 3.9 3.9 1. 3.5 2 i;5 1.5 .5 1 3 2 6 5 25 18 29 26 20 19 13 15 12 3 2 4 4 2 .5 1.4 1. 2.8 2.5 11.9 8.6 13.9 12.4 9.5 9.1 6.2 7.2 5.6 1.4 1. 2. 2. 1. 1 1 5 7 10 6 14 17 16 20 16 11 13 16 9 9 5 5 5 I 2 .5 .5 2.6 3.6 5.2 3.1 7.3 8.8 8.3 10.4 8.3 5.7 6.7 8.3 4.7 4.7 2.6 2.6 2.6 1.6 1. 1. Total 204 100 209 100 193 100 Average Median 25 P. 75 P. P.E. ^•■^•t.obt.- Av. 1 1 5.1 2.6 2.9 3.5 0.3 .7 1 1 5.5 3.5 3.4 6.6 1.6 .8 A A 2 I 1 2.6 0.4 6,8 7.9 5.6 1.1 Distribution and Length of Work Period 59 TABLE XXI Per Cent of Division Combinations Correctly Answered Gained in the Course op Sixty Minutes of Practice by the Three Groups Gain, per cent Group I Group II Group III 20 min. 10 min. 2 min. ImJi- \'iduals Per cents Indi- viduals Per cents Indi- viduals Per cents -74 to -60 -59 to -45 -44 to -30 -29 to -15 -14 to 1 to 15 16 to 30 31 to 45 46 to 60 61 to 75 76 to 90 91 to 105 106 to 120 121 to 135 136 to 150 151 to 165 166 to 180 181 to 195 196 to 210 211 to 225 226 to 240 241 to 255 256 to 270 271 to 285 286 to 300 301 to 315 316 to 330 331 to 345 346 to 360 361 to 375 376 to 390 391 to 405 1 3 7 8 16 26 42 27 22 12 9 12 4 6 1 1 1 1 1 1 1 1 1 .5 1.5 3.5 3.9 7.8 12.8 20.6 13.2 10.8 5.9 4.4 6. 2. 3. .5 .5 .5 .5 .5 .5 .5 .5 .5 3 2 2 5 5 18 24 20 31 27 14 14 6 8 3 5 3 3 8 1 1 1 1 1 1 2 1.4 1. 1. 2.5 2.5 8.6 11.2 9.6 14.8 12.9 6.6 6.6 2.8 3.8 1.4 2.5 1.4 1.4 3.8 .5 .5 .5 .5 .5 .5 1. 2 3 8 9 15 27 27 24 10 12 10 3 6 2 4 6 2 2 3 5 2 1 2 1 2 2 2 1 1. 1.6 4.1 4.7 7.8 14. 14. 12.5 5.2 6.2 5.2 1.6 3.1 1. 2.1 3.1 1. 1. 1.6 2.6 1. .6 1. .5 1. 1. 1. .5 Total 204 100 209 100 193 100 Median 25 P. 75 P. P.E. P-^-t.-obt. Av. 60 40 92 26 1.8 73 41 112 35.3 2.4 94 67 147 40 2.9 Gross Gain To determine the gross gain of the three groups in division made in the course of 6o minutes of practice, the gross gains in number of combinations worked correctly by each of the 6o Practice in the Case of School Children 606 children, samples of which may be seen in Table VII, column 16, were distributed for the three groups in Table XX, and the measures for each group calculated. The average number of combinations gained in the course of 60 minutes by Group I was 25.1; by Group II, 25.5; and by Group III, 42.6. The corresponding medians were 22.6, 23.5 and 40.4, At their face value these facts show a very decided advantage for the group which did the intervening 45 minutes of practice in 2-minute periods over the groups which did it in lo-minute periods and 20-minute periods. Only 18 per cent of the 20-minute group and 21 per cent of the lo-minute group reached the median of the 2-minute group. Percentile Gain The data used in determining the gain per cent for the groups are the individual records of the 606 children in gain per cent, samples of which may be seen in Table VII, column 17. These gain per cents are distributed (separately) in Table XXI, for the three groups. Gain in Accuracy The gross gains in accuracy for the 606 individuals, samples of which may be seen in Table VII, column 18, were distributed for the three groups in Table XXII and the measures for each group found. The median gains in accuracy were: for Group I, 2.1 per cent; for Group II, 3.5 per cent; for Group III, 2.3 per cent. These gains are so nearly the same that we can say that in division no advantage was shown in accuracy from one length of practice period over the others. All periods resulted in about an equal absolute gain in accuracy. Sitmmary There were 18 3B and 4A classes, containing 606 children in the division experiment. Group I consisted of 6 classes con- taining 204 children ; Group II consisted of 6 classes containing 209 children; and Group III consisted of 6 classes containing 193 children. The groups were made on the basis of length of practice-period in the 40 minutes of practice between the initial lo-minute period and the final lo-minute period, which were the Distribution and Length of Work Period 6i TABLE XXII Gross Gain in Accuracy in Division, Expressed in Per Cents op Answers that were Correct, Made in the Course of Sixty Minutes of Practice by the Groups Group I Group II Group III Gain, per cent 20 min. 10 min. 2 min. Indi- viduals Per cents Indi- viduals Per cents Indi- viduals Per cents -45 to -41 -40 to -36 -35 to -31 -30 to -26 -25 to -21 -30 to -16 -15 to -11 -10 to - 6 - 5 to - 1 Oto 4 5 to 9 10 to 14 15 to 19 20 to 24 25 to 29 30 to 34 35 to 39 40 to 44 45 to 49 50 to 54 1 1 6 15 36 82 28 17 3 4 5 4 2 .5 .5 .3 7.4 17.6 40.2 13.7 8.3 1.5 2 2.5 2 1 1 2 1 1 6 38 70 43 13 10 8 5 3 4 4 .5 1 .5 .5 2.8 18.1 33.5 20.6 6.2 4.8 3.8 2.4 1.4 2 2 1 3 4 16 39 59 31 22 13 1 1 1 1 1 .5 1.6 2.1 8.3 20.2 30.6 16.1 11.4 6.7 .5 .5 .5 .5 .5 Total 204 100 209 100 193 100 Median 25 Percentile 75 Percentile P.E. 2.1 -1.6 G.6 4.2 3.5 -.3 8.9 4.6 2.3 -2.4 8.2 5.3 same for all three groups. In Group I, these intervening practice periods were 20 minutes each; in Group II, 10 minutes each; in Group III, 2 minutes each. The following is a summary of the important facts concerning the differences in improvement. Group I Group II Group III Medians of the Groups op Individuals Median Initial Ability 38.3 32. 40.3 Average Gross Gain 25.1 25.5 42.6 Median Gross Gain 22,6 23.5 40.4 Reliability of Average t.-obt. Av. 1.1 Median Median Gain Gain in Per cent Accuracy 60 73 94 2.1 3.5 2.3 62 Practice in the Case of School Children 1. The initial ability of Group II was enough below that of the other two groups to demand careful consideration in evalu- ating the relative gain. 2. The initial accuracy of the groups was so nearly equal that it may be considered a constant factor in the discussion. 3. Group III gained almost twice as many combinations done correctly in the course of the 60 minutes of practice as did the other two groups, whose gains were practically the same. 4. In gross gain in accuracy the three groups were so nearly equal that this too may be considered as almost a constant factor. The measures given have been computed from the scores of the individuals comprising these groups. The following measures are computed from the scores of the classes in the groups. The median for each class was found, then the average of these medians was computed. Average op Class Medians AvernKe _ _ Average Average Initial Median Reliability Gross Reliability of Medinn Ability of of Average: Median of of Average: Percentile Classes P.E. . , . . Classes P.E. . , , Gains t.-obt Av. t.-obt. av. Group I 38.4 2.2 20.6 2. 58 Group II 33.4 1.7 25.1 1.5 77 Group III 41.4 4.5 44.7 2.7 114 Using the three methods of computing gross gain, we have: Average Median Average op Median Gross Gain Gross Gain Gross Gains OP Individuals op Individuals op Classes Group I 25.1 22.6 20.6 Group II 25.5 23.5 25.1 Group III 42.6 40.4 44.7 Considering the facts for both addition and division, it appears that, subject to discounts for the inequalities of the groups in initial ability, there is considerable advantage in the short period lengths, when the length is two minutes. The advantage there is noteworthy, since in addition the gain is greater than in the longer-period groups even when their ability was greater in the longer period (Group II) ; and since in division the gain is so very much greater than in the 20- or lo-minute group. The facts can be freed from the influence of inequalities in ability at the beginning of practice by comparing only those of Distribution and Length of Work Period 63 equal initial ability. For example, we find in the case of division that those of initial ability 15 averaged in gain, 18.0, 26.3 and 23.0 according as they had practiced in 20-, 10-, or 2-minute periods; those of initial ability 16 averaged in gain 16.4, 15.3 and 41.0 according as they had practiced in 20-, 10-, or 2-minute periods. Making such calculations for those of each initial ability in division from 5 to 64 and allowing equal weight to each succes- sive set of five successive groups, it appears that on the average the 20-minute, lo-minute, and 2-minute period varieties of prac- tice brought to those of equal initial ability gains in the relation of 100, 1105^ and 177. In the case of addition the same procedure, carried out with those of initial abilities 5, 6, 7, and so on up through 49 gives the following results : According as the practice was in 22^ , 15, 6, or 2-minute divisions, it brought to those of equal initial ability gains in the relation of 100, 121, loi and 146^. It appears, then, that the superiority of the shortest practice- period length remains when inequalities of initial ability are eliminated. It appears further that the periods of intermediate length have really a greater superiority over the longest period than the results irrespective of differences in initial ability showed. There is a positive relation between initial ability and gross gain. Consequently Group III in addition and Group II in division, which happened to be groups of low initial ability, suffered in the comparison. The detailed facts for the gains of each variety of initial ability according to the nature of the practice are shown in Tables XXIII and XXIV. 64 Practice in the Case of School Children TABLE XXIII The Relation of Length op Practice-Periods to Gross Gain IN Addition Initial 22^ min. 15 min. 6 min. 2 min. ability. Exam- practice- periods practice- periods practice- periods practice- Average Gross Gain periods ples correct Average Gross Gain Num- ber of Cases Average Gross Gain Num- ber of Cases Average Gross Gain Num- ber of Cases Num- ber of Cases 1 2 3 2 1 4 5 1 14 1 5 4 2 8 2 4 1 4 1 6 30 4 7 9.3 3 8. 2 2.5 2 14 2 4 8 7.6 3 9.5 2 11.2 5 9.8 5 9 13.4 7 -1.5 2 4.7 4 10 6.6 3 25 1 14.7 11 13.3 3 11 19 1 2 2 13 4 12 7 9 4 3 17.7 4 9.2 4 13 10.5 4 9.3 6 8.8 5 14 7.3 6 11 3 12.8 8 23.3 3 15 7.6 3 -6 2 14.6 9 19.1 12 16 5.2 6 21.3 3 6.2 5 16.7 8 17 7.5 9 5 3 5 8 14.2 4 18 9.3 3 13.3 4 9 21 8 9 19 7.6 11 15 3 4 5 8.2 8 20 7.5 10 6.1 7 10.8 16 11.1 7 21 9.7 9 10.2 5 6 2 17.8 6 22 6.7 4 7 1 12.7 4 9.4 12 23 6.3 7 14 2 6.9 10 4.3 7 24 12.6 8 7.7 4. 9.6 5 20.8 6 25 10 6 21 2 8.1 8 25.8 5 26 7.2 4 28.3 3 7.3 7 11.5 11 27 8.6 8 9.6 3 10.6 7 17.2 5 28 7.5 6 9 1 8.6 8 15 13 29 10.8 4 -3 1 2 11.3 6 30 23.7 6 8.4 5 1.4 5 10.3 4 31 12.4 5 16.3 4 6.8 4 11.9 10 32 14.5 2 22.6 3 3.5 2 25.4 5 33 15 3 37 1 16 3 27.3 6 34 5.6 3 2 1 14.4 5 11.8 4 35 4.8 4 3 1 13.4 7 14.3 3 36 11.8 4 8.8 4 19 1 34.5 2 37 12.8 5 29 1 10.6 8 38 8 1 14 1 22 2 17.6 5 Distribution and Length of Work Period TABLE XXIII— Continued 6S Initial 22* min. 15 min. 6 min. 2 min. ability. Exam- practice- periods practice- periods practice- periods practice- periods ples correct Average Gross Gain Num- ber of Cases Average Gross Gain Num- ber of Cases Average Gross Gain Num- ber of Cases Average Gross Gain Num- ber of Cases 39 20 3 61 1 35 2 40 15 1 24 1 10.7 6 33.6 5 41 7 1 3 1 38 3 42 23.5 2 21.5 2 37 1 28.5 2 43 20 1 15 3 44 4.5 2 28 1 8 1 45 27 1 16 1 46 23 1 14 1 47 16.5 2 48 33 1 26.5 2 49 49 2 26.3 3 50 26 1 4 1 34 3 51 39 1 -6 1 19 1 52 30 5 2 18 1 53 30 1 29.5 2 54 53 1 55 56 57 1 57 29 1 58 59 1 59 17.5 2 33 1 60- 64 20.5 2 31 1. 12 1 65- 69 28 1 46 1 24 1 70- 74 30.3 3 37 2 43 1 80- 84 23 3 85- 89 22 1 90- 94 IS 1 18 1 105-109 48 1 Total 194 102 207 231 66 Practice in the Case of School Children TABLE XXIV The Relation of Length of PnACTiCE-PERiODS to Gross Gain IN Division Initial ability. 20 min. 10 min. 2 min. practice-periods practice-periods practice-periods Examples correct Average Number Averaee Number Averocte Number Gross Gain of Cases Gross Gain of Case.s Gross Gain of Cases 5 20 1 15.5 2 6 13 2 12 3 7 9 1 10.5 2 8 15.3 3 5 1 24 1 9 4.6 3 11.5 2 12 2 10 10 3 8 1 11 9.6 3 21 4 12 12 2 5 3 24 2 13 13.8 5 13.5 2 35.5 2 14 39 2 14.5 2 38 2 15 18 4 26.3 4 23 7 16 16.4 5 15.3 3 41 4 17 15.5 6 13 4 38.2 5 18 2 5 2 40.5 6 44.7 6 19 15 2 11.8 5 41 1 20 -15 1 26.2 9 34.5 2 21 28 2 38.5 2 38.8 5 22 23 2 27.5 8 43.6 5 23 30,6 3 20.5 4 17.7 3 24 14.2 5 30.5 4 74.7 3 25 21.3 3 21.2 6 57 1 26 25 1 13 2 40.2 5 27 10.5 4 17.1 7 20.5 2 28 22.3 4 25 4 s 23 . 5 2 29 25.3 3 41.5 2 19.5 2 30 23.6 5 19.5 6 38.4 5 31 18.5 4 20 4 40.5 2 32 16.5 2 28.6 3 33.5 2 33 28 2 31.6 3 39.3 4 34 23 2 29 4 33.5 2 35 21 3 44 5 36 19.7 7 31.8 5 44 1 37 22.3 4 31.1 8 46.8 4 38 27.3 3 48.5 5 39 25.3 6 20.8 5 17 2 40 19 4 7.5 2 43 1 41 46 1 30.2 5 46.3 3 42 32 2 39.3 4 43 25.4 8 34 8 44 22.5 2 13 5 45 1 Distribution and Length of Work Period 87 TABLE XXIY— Continued Initial 20 min. 10 min. 2 min. ability. practice- periods practice- ;)eriods practice-periods Examples correct Average Number Average Number Average Number Gross Gain of Case.s Gross Gain of Cases Gross Gain of Cases 45 10.5 2 18 4 64 1 46 6 1 38.3 6 18 3 47 26 1 16 6 3 41.6 3 48 32 4 15.5 2 47.5 2 49 37.5 4 25.8 4 44.3 4 50 17.5 2 44 1 27 1 51 16,6 7 29 7 42.8 4 52 24.3 3 6 1 37.5 2 53 32 3 54 28.3 3 22 3 67 1 55 19.5 4 19.5 2 31.8 6 56 49 1 44.5 2 57 37 4 27.8 6 40.5 2 58 18.3 3 50 1 27.8 4 59 23.6 3 37.3 3 60 26 2 27 1 83 1 61 25 3 47 1 45.3 3 62 31 3 34 3 63 41.4 5 49 1 47 1 64 42 3 48 1 65 51 1 63 1 62 1 66 35 1 48.5 2 92 1 67 31 1 48 1 68 40 1 34 . 5 2 43 1 69 70- 74 69.5 2 28 . S 4 57.9 9 75- 79 45.4 8 46.6 3 49.6 5 80- 84 63 3 85- 89 64 2 41 1 43 6 90- 94 90 1 74 4 95- 99 59.3 3 77.5 2 100-104 67 1 105-109 87 2 110-114 62 1 115-119 120-124 90 1 125-129 72 1 Total 205 210 192 68 Practice in the Case of School Children Distribution and Length of Work Period 69 General Summary While the results of both the addition and division experiment show that the greatest gains were made by the groups which had their practice in the shortest periods, we can not conclude that all of this excess of gain was due to the difference in the length of the practice-period. Other factors probably contributed a share to this extra gain, (i) The groups, working in shorter periods, because of the number of days over which the experi- ments ran, had greater opportunity during the experiment to profit from the regular school work than other classes. This difference might have been overcome by giving the experiment to all groups in the same number of days, which would necessitate giving more than one of the shorter practice-periods within a day or allowing days to lapse between the longer practice-periods. This is a problem for another study. (2) The groups working in shorter periods had a longer time in which to catch the spirit of the experiment and to become enthusiastic over surpassing their previous performance. They had their records read to them more times and had the incentives to intense effort repeated more often. (3) They also had greater opportunity and incentive to do work outside of the time given to the experiment. On the other hand the 2-minute group in addition was handi- capped by the difficulty of making an absolute gain from one 2-minute period to the next. The median gross gain for this group was 12.6 columns from 23 periods of practice which shows that only those who gained most rapidly could have had the pleasur- able effect from day to day of surpassing their previous day's record. Then, too, if they did surpass their previous days record it was by a small amount which did not give the same feeling of success and incentive to effort that a larger gain would have given. This condition did not operate much in the 2-minute period in division where the median gain was 40.4 combinations in 20 periods of practice. In all other periods of practice the gain from day to day was larger and resulted in a more intense feeling of success and greater incentive to effort. This handicap on the 2-minute group, it seems, would offset the extra gains from conditions previously mentioned. From the administrative side of school-room work, the longer periods of practice can be much more economically managed in 70 Practice in the Case of School Children that preparation needs be made for them much less often. Per- haps this factor would compensate in a large measure for the disadvantage which the long period seems to have in comparison with short ones. CHAPTER IV THE PERMANENCE OF THE PRACTICE EFFECT In order to measure the permanence of the abiHty acquired in the course of the experiments that have been described, retention tests were given at the end of the school year in June, 1912, to those classes that had taken their practice during the preceding four months and finished it long enough before the end of June to make a retention test seem profitable. No one knew that these later tests were to be given after the regular experiment was finished. This makes it reasonable to suppose that about the regular amount of work was done in addition and division from the close of the experiment until the tests for retention were given and that no extra drill was given in anticipation of them. These tests afford data by which to determine to what extent the efficiency acquired during the experi- ments persisted from the close of the practice in the experiment to the end of June under the normal exercise of performing regular school work. Retention tests were also given at the beginning of the follow- ing school year in September to as many of the same classes as the author had access to. These tests afford data by which to determine to what extent the efficiency shown at the end of June persisted through vacation, during which practice was substan- tially zero. Immediately following these retention tests in September, the classes were given sufficient practice to restore them approximately to the same efficiency they had attained at the end of their practice the preceding year. These retention tests correspond in every way to the initial and final practice-periods of the experiment. In addition there were practice-periods of 15 minutes, and in division practice- periods of 10 minutes. Hence, the scores made in them were directly comparable to the scores in both the initial and final periods of the experiment. In the practice conducted to restore the ability lost during 71 7 a Practice in the Case of Scliool Children vacation some fivc-niinutc practice-periods were used, but the last ])racl ice-period jj;iven was a 15-iiiinute ])eriod in addition and a lo-niiiuite period in division. The retention tests and the practice f^iven in Septciuljer were conducted by the author in exactly the same manner as the practice of the experiment had been conducted. During some of the 15-minute tests in September some of the children showed signs of weariness that were not noticeable during the original practice. The author flit that this was due to the children's inability to apply them- selves to mental work immediately after vacation with the same jiersistence that they had shown during the experiment, which was conducted at a time when they were accustomed to do con- tinuous mental work. There is no doubt in the author's mind that shorter periods of practice would have been more effective in this practice given during the first two weeks of school, and that the time required for these classes to regain their former cfliciency would have been considerably lessened if given in shorter periods. The present experiment affords no data to prove this opinion, but it would be i)rofitable to iind out if practice-periods should be shorter in the early weeks of the school term than in the later ones. This chapter treats, then, three phases of the problem of the permanence of associations: (i) The permanence of well- practiced addition and division associations during a period of time when they were normally used in regular school work. (2) The permanence of the same associations after almost com- plete disuse during the summer vacation. (3) The permanence of these associations as shown by the amount of practice required to restore them to their former efficiency. Pl'-RMANKNt'l': OK ASSOCIATIONS NORMALLY ITsiCD IN SciIOOL Work AnniTioN Nine classes that had taken the addition practice finished the experiment and took the retention test in June at the time indi- cated below : The Permanence of the Practice Effect 73 Class Final Pkactice Teuiou Retention Teht XIV March 28 June 21 XVI March 28 June 20 XIII March 2U Juno 21 XVII April 18 June 20 VIII April I'J June 19 IX April I'J June 18 X May 9 June 19 XI May 9 Juno 19 V May 27 June 20 The question which wc are considering is, What change in ability to add took place in the children of these nine classes in the time intervening between the final practice-period and the retention test near the end of June, a time varying for the different classes from 12 weeks to 3 weeks, in which no si)ecial drill was given in arithmetic, but in which regular work in arithmetic was in progress? The scores made in the retention tests were not compared directly with the scores made in the final i)ractice-period of the experiment, but with the scores made in the initial practice-period. So, to answer the question ])ro- posed above, we shall first find what gain was made in the reten- tion test near the end of June over the initial test and then compare this gain with that made during the original practice- experiment by these same classes. Any difference found l)etween these two gains will be the change that took place in the interval between the close of practice and the test near the end of June. Gross Gain: Table XXV gives under (a) the gross gain of these nine classes in the retention test at the end of June over the initial period of the experiment and under (b) the gross gain of these same classes during the original experiment. The median gain in columns added correctly in the retention test near the end of June over the initial ])ractice-i)eriod of the experiment was 11 columns, while the median gain of these same classes during the experiment was 10.4 columns. This means that, in the time elapsing between the end of the regular practice of the experiment and the end of June in which only the regular school work in addition was carried on, these classes not only did not lose any of the ability acquired during the experiment, but gained the difference between these two medians or .6 column. The 25 percentile and the 75 percentile together with the median indicate that this gain was distributed all along the curve. Gaifi in Accuracy: Table XXVI gives under (a) the gain of 74 Practice in the Case of School Children these nine classes in accuracy in the retention test at the end of June over the initial period of the experiment, and under (b) the gain of the same classes in accuracy during the experiment. Between the initial practice-period and the retention test there was a gross gain in accuracy of .2 per cent while during the experiment there was a loss in accuracy of .4 per cent. The difference between these two medians, .6 per cent, is the median gain in accuracy from the end of the original practice to the end of June. This indicates a slight gain in accuracy of performance after the intense practice ceased and normal use of the associa- tions was continued. However, this gain is so small that little weight should be given to it. TABLE XXV (a) Gross Gain in Addition FROM THE Initial Period OF THE Experiment to THE End of June (b) Gross Gain of the Same Individuals During the Experiment Columns gained Individuals Per cents Individuals Per cents -15 to-12 4 1.6 1 .4 -11 to - 8 3 1.5 6 2.6 - 7 to - 4 7 2.7 7 2.7 - 3 to 14 5.4 25 9.7 1 to 4 35 13.5 40 15.5 5 to 8 39 15.1 35 13.5 9 to 12 44 17. 31 11.9 13 to 16 38 14.3 34 13.1 17 to 20 24 9.3 29 11.2 21 to 24 13 5.1 14 5.4 25 to 28 4 1.6 13 5.1 29 to 32 12 4.7 9 3.5 33 to 36 5 1.9 6 2.6 37 to 40 4 1.6 1 .4 41 to 44 6 2.6 1 .4 45 to 48 2 .8 3 1.5 49 to 52 1 .4 53 to 56 1 .4 57 to 60 1 .4 61 to 64 1 .4 1 .4 65 to 68 2 .8 Total 258 100. 258 100. Median 11 Median 10.4 25 Percentile 4.6 25 Percentile 3 . 1 75 Percentile 18.1 75 Percentile 17.1 P.E. 6.8 P.E. 7. The Permanence of the Practice Effect 75 TABLE XXVI (a) Gain in Accuracy in Addi- tion FROM Initial Score TO THE End of June (b) Gain in Accuracy of Same Individuals During the Experiment Per cent gained Individuals Per cents Individuals Per cents in accuracy -55 to -51 4 1.7 -50 to -46 1 .5 -45 to -41 3 1.2 1 .5 -40 to -36 1 .5 1 .5 -35 to -31 3 1.2 6 2.4 -30 to -26 5 1.9 9 3.6 -25 to -21 16 6.1 7 ' 2.6 -20 to -16 20 7.8 20 7.8 -15 to -11 22 8.5 25 9.7 -10 to - 6 21 8.2 23 9. - 5 to -1 32 12.4 32 12.4 to 4 36 13.9 42 16.3 5 to 9 37 14.4 28 10.9 10 to 14 20 7.S 22 8.5 15 to 19 12 4.7 17 6.6 20 to 24 16 6.1 8 3.1 25 to 29 8 3.1 2 .8 30 to 34 2 .8 2 .8 35 to 39 1 .5 3 1.2 40 to 44 1 .5 2 .8 45 to 49 3 1.2 50 to 54 1 .5 55 to 59 1 .5 Total 258 100. 258 100. Median .2 Median -.4 25 Percer itile -12. 25 Percentile -12.2 75 Percer itile 9. 75 Percentile 8.7 P.E. 10.5 P.E. 10.5 Division Gross Gain: Only two classes in the division experiment were given the retention test in June. The other classes did not finish the experiment until the closing days of the term. One o£ these classes finished the experiment June y, the other June lo. Both took the retention test June 20. Table XXVII gives under (a) the gross gain of these two classes in number of combina- tions worked correctly in the retention test on June 20 over the initial practice-period of the experiment, and under (b) the gross gain of the same individuals during the experiment. The median gain in the retention test over the initial period was 30.7 com- 76 Practice in the Case of School Children binations while the gain during the experiment was 28.5 com- binations. This means that these two classes in an interval of about 12 days not only did not lose any of the proficiency acquired during the intense practice of the experiment but actually gained a median of 2.2 combinations. Gain in Accuracy: The change in accuracy may be seen in Table XXVIII. From the initial period to the retention test, the median gain was 2.7 per cent, while the median gain during the experiment was 2 per cent. These figures show a median loss of .7 per cent in accuracy in the interval from the close of the experiment to the retention test. TABLE XXVII (a) Gross Gain of Two Classes in Division from Initial Test to End of June (b) Gross Gain of Same Classes During Experi- ment Combinations Individuals Per cents Individuals Per cents gained -4 to 1 1.2 1 1.2 1 to 5 1 1.2 2 2.4 6 to 10 3 3.6 6 7.1 11 to 15 2 2.4 3 3.5 16 to 20 7 8.4 10 12.9 21 to 25 16 19.3 13 15.3 26 to 30 11 13.3 11 12.9 31 to 35 15 18.1 13 15.3 36 to 40 10 12.1 4 4.7 41 to 45 6 7.2 9 10.6 46 to 50 3 3.6 5 5.9 51 to 55 2 2.4 1 1.2 56 to 60 1.2 1 2.4 61 to 65 12 1 1.2 66 to 70 1 1.2 71 to 75 1.2 2 2.4 76 to 80 1.2 81 to 85 86 to 90 1.2 91 to 95 1.2 96 to 100 Total S3 100 83 1 100 Median 30.7 Median 28.5 25 Percer itile 22.6 25 Percer itile 19.7 75 Percer itile 38.6 75 Percer itile 39 . 6 P.E. 8. P.E. 9.9 The Permanence of the Practice Effect 77 Summary: For two classes in division, in an interval of about 12 days between the close of the experiment and the reten- tion test, the median gross gain was 2.2 combinations done cor- rectly, and the median loss in accuracy was .7 per cent. Here, as in addition, these classes not only did not recede from the high efficiency reached during the experiment when practice ceased, but they made a slight gain while using these same associations in the performance of regular school work. Both the addition experiment and the division experiment indi- cate that these third- and fourth-year children who had improved their proficiency in adding and in doing the division combina- tions 50 per cent or more by intense practice in a brief space of time, lost none of this recently acquired proficiency so long as they continued to exercise these same functions to the extent demanded by the performance of regular school work. TABLE XXVIII (a) Gross Gain in Accuracy OF Two Classes in Divi- sion FROM Initial Test TO End of June (b) Gross Gain of Same Classes During Experi- ment Per cent gained in accuracy Individuals Per cents Individuals Per cents -25 to -21 1 1.2 -20 to -16 -15 to -11 1 1.2 -10 to -6 4 4.8 2 2.4 -5 to -1 21 25.3 17 20. to 4 31 37.1 35 41.1 5 to 9 16 19.3 14 16.5 10 to 14 4 4.8 8 9.4 15 to 19 2 2.4 3 3.5 20 to 24 3 3.6 1 1.2 25 to 29 1 1.2 2 2.4 30 to 34 1 1.2 45 to 49 1 1.2 Total 83 100. 85 100. Median 2.0 Median 2.7 25 Percentile -2.8 25 Percer itile -.4 75 Percentile 6.1 75 Percer itile 7.6 P.E. 4.5 P.E. 4. 78 Practice in the Case of School Children Permanence of Associations Through Summer Vacation Our next problem is to measure the change in ability in addi- tion and division which occurred during the summer vacation, a time when the associations concerned fell into almost complete disuse. This change will be shown by comparing the scores in the retention tests given at the beginning of September with the scores made at the end of June. Addition Gross Loss: Five classes containing 152 children that had taken the practice during the latter half of the school year 191 1- 1912 and the retention test at the end of June 1912 were accessible to the author the following September. School opened September 9. These five classes were given the retention tests September 10 and 11 with one exception. This class had been promoted into three or four other schools and could not be assembled for the test until September 16. Many children did not return in September. Others, of course, had taken their places, but only the records of children who were present at the end of June and on the day of the retention test in September could be used. Table XXIX gives under (a) the gross gain of these five classes in the September retention tests over their standing in the retention tests in June. The median loss in columns added correctly was 7.5 columns. The lower 25 per cent of the group lost 14. i columns or more and the upper 25 per cent of the group lost 2. columns or less. Only the upper 15 per cent of the group could add as many columns correctly in 15 minutes at the beginning of September as at the end of June. Loss Per Cent: The loss per cent used in this portion of the discussion was computed by finding what per cent the gross loss was of the number of problems worked correctly in the retention test at the end of June. If a child added 40 columns correctly in the retention test in June and 32 in September his gross loss was 8 columns and his loss per cent was 20, which would appear in the table as — 20. These per cents were dis- tributed as shown under (b) in Table XXIX. The median change was a loss of 17 per cent; that is, the group lost during vacation, as shown by the median, 17 per cent of the ability which it had at the end of June. The lower 25 per cent of the indi- The Permanence of the Practice Effect 79 viduals lost 32 per cent or more, and the upper 25 per cent lost 6. per cent or less. Loss in Accuracy: Anyone to whom the method of com- puting the change in accuracy is not clear may find this discussed under Table I. In Table XXIX, under (c), the gross gains in accuracy are distributed. The median loss was 4.2 per cent. The upper 35 per cent of the class did not lose in accuracy. TABLE XXIX Gain op Five Classes in Addition from End of June to Beginning OF September (a) (b) ^0 G ROSS Gain Gain Per Cent Gain in Accuract Columns Indi- Per Gain Indi- Per Per cent Indi- Per Gained viduals cent Per cent viduals cent Gained viduals cent -47 to -44 2 1.3 -65 to -61 2 1.3 -55 to -51 2 1.3 -43 to -40 1 .7 -60 to -56 5 3.3 -50 to -46 1 .7 -39 to -36 2 1.3 -55 to -51 -45 to -41 3 2. -35 to -32 -50 to -46 7 4.6 -40 to -36 3 2. -31 to -28 4 2.6 -45 to -41 4 2.6 -35 to -31 3 2. -27 to -24 7 4.6 -40 to -36 10 6.6 -30 to -26 2 1.3 -23 to -20 5 3.3 -35 to -31 15 10. -25 to -21 7 4.6 -19 to -16 12 7.9 -30 to -26 9 5.9 -20 to -16 10 6.6 -15 to -12 14 9.2 -25 to -21 13 8.6 -15 to -11 21 13.8 -11 to -8 29 19.1 -20 to -16 16 10.5 -10 to -6 17 11.2 -7 to -4 29 19.1 -15 to -11 17 11.2 -5 to -1 28 18.4 -3 to -0 24 15.8 -10 to -6 16 10.5 Oto 4 26 16.5 Ito 4 19 12.5 -5 to -1 12 7.9 5 to 9 16 10.5 5 to 8 2 1.3 Oto 4 16 10.5 10 to 14 6 3.9 9 to 12 2 1.3 5 to 9 5 3.3 15 to 19 2 1.3 10 to 14 1 .7 20 to 24 2 1.3 15 to 19 2 1.3 25 to 29 20 to 24 1 .7 30 to 34 3 2. 25 to 29 1 .7 Total 152 100. 152 100. 152 100. Median -7.5 Median -1 7. Median - 4.2 25 Perce ntile - 14.1 25 Percen tile -3 2. 25 Percentile -1 S.8 75 Perce ntile -2. 75 Percen tile - 6. 75 Percentile 2.8 P.E. 6.1 P.E. 1 3. P.E. 3.3 Division Seven of the classes that had taken the practice during the last half of the preceding school year were accessible in Septem- ber. These classes were given the retention test on September 10 and II except one class which could not be reassembled until September 16. Since all of these classes finished their practice So Practice in the Case of School Children in June, and since only two of them could be given the retention test in June, the September retention score is compared with the final practice-period of the practice in the experiment. Gross Loss: The gross losses were found by taking the differ- ence between the number of combinations done correctly in the retention test in September and the number done correctly in the final practice-period of the experiment in June. These losses for the 221 individuals comprising these seven classes were distributed in Table XXX under (a). The median loss for the group was 18.3 combinations. About 90 per cent of the group lost during vacation. The upper 25 per cent of the group lost 10 or fewer combinations. TABLE XXX Gain in Seven Classes in Division Combinations from the End OF June to September (a) Gross Gain (b) Gain Per Cent (c) Gain in Accuracy Combina- tions Gained -64 to -60 -59 to -55 -54 to -50 -49 to -45 -44 to -40 -39 to -35 -34 to -30 -29 to -25 -24 to -20 -19 to -15 -14 to -10 -9 to -5 -4 to 1 to 5 6 to 10 11 to 15 Total Indi- viduals 5 1 3 4 7 9 22 2S 23 35 34 16 17 Per cent 221 2.3 .5 1.4 1.8 3.2 4.1 10. 12.7 10.4 15.8 15.4 7.2 7.7 3.7 3.7 .5 100. Gain Per cent -70 to -65 to -60 to -55 to -50 to -45 to -40 to -35 to -30 to -25 to -20 to -15 to -10 to -5 to Oto 5 to 10 to 15 to 20 to 25 to 30 to 35 to 40 to 45 to -66 -61 -56 -51 -46 -41 -36 -31 -26 -21 -16 -11 -6 -1 4 9 14 19 24 29 34 39 44 49 Indi viduals 1 2 3 2 9 13 11 18 19 36 25 25 15 14 11 3 4 5 1 1 221 Per cent .5 1. 1.4 .9 4.1 5.9 5. 8.2 8.6 16.3 11.3 11.3 6.8 6.3 5. 1.4 1.8 2.3 .5 .5 1. 100. Per cent gained in accuracy -55 to -51 -50 to -46 -45 to -41 -40 to -36 -35 to -31 -30 to -26 -25 to -21 -20 to -16 -15 to -11 -10 to -6 -5 to Oto 5 to 10 to 15 to 20 to 25 to 30 to Indi- viduals 1 1 2 1 3 6 8 12 19 42 67 44 11 3 221 Per cent .5 .5 .9 .5 1.4 2.7 3.7 5.4 8 6 19. 30.3 19.9 5. 1.4 100. Median -18.3 25 Tercentile -28.7 75 Percentile -10.3 P.E. 9.2 Median -21 . 25 Percentile -32. 75 Percentile -10. P.E. 11. Median -4 . 3 25 Percentile -10.2 75 Percentile -. 1 P.E. 5. The Permanence of the Practice Effect 8 1 Loss Per Cent: The loss per cent was computed by finding what per cent the gross loss was of the number of combinations worked correctly in the final practice-period of the experiment. If a child did 75 combinations correctly in the final practice- period in June and 60 combinations in the retention test in September the gross loss was 15 combinations, or 20 per cent. These per cents are distributed in Table XXX under (b). The median loss was 21 per cent. Loss in Accuracy: The losses in accuracy are given in Table XXX under (c). The median loss was 4.3 per cent. Both in addition and division there was a great loss in ability during the summer vacation. The loss in division was greater than that in addition. This difference may have been due to the fact that younger children took part in the division work than in the addition or it may have been due to the fact that the children had been working in addition longer than in division.* Both the difference in age of the children and the fact that they had been working in addition longer than in division would afford conditions for causing the addition associa- tions to be more firmly set than the division combinations and hence more likely to persist. Permanence in Terms of Advance Over the Initial Practice-period of the Original Experiment The permanence of the improvement may also be measured by the superiority of the scores attained in September to the scores made at the beginning of the original experiment. In presenting these facts I shall present also the gain made from the initial to the final period of the experiment itself. The amount of permanence can then be viewed as the superiority of the September scores to the initial or their superiority to the final scores of the original experiment. Addition Gross Gain: The records of five classes of 148 children are available. One of these classes finished the experiment April 14, two May 4, one May 27, and one June 18. The questions to be *There was not only the excess of 15 minutes of practice in the orig- inal experiment, but also the extra June 15 minute retention test. Only two-sevenths of the classes had this June retention test. 82 Practice in the Case of School Children answered are, What gain on their initial ability did the pupils of these five classes make from the initial practice-period to the beginning of September? and, How much of their initial ability did they lose from the end of the experiment to the beginning of September ? The data for answering these questions are given in Table XXXI. The table shows that in September the indi- viduals in question were still 10.4 columns ahead of their standing at the beginning of the experiment but that from the close of the practice to the beginning of September they lost a median of 6.4 columns. Change in Accuracy: Similar data concerning accuracy are found in Table XXXII. The median loss in accuracy from TABLE XXXI (a) Gross Gain in Addition FROM Initial Practice- Period TO THE Reten- tion Test in September (b) Gross Gain op the Same Individuals During the Experiment Columns gained Individuals Per cents Individuals Per cents -19 to -12 1 .7 -11 to -8 1 4.7 2 1.4 -7 to -4 9 6.1 1 .7 -3 to -0 12 8.1 12 8.1 1 to 4 18 12.2 19 12.8 5 to 8 19 12.8 16 10.8 9 to 12 17 11.5 10 6.8 13 to 16 17 11.5 13 8.8 17 to 20 13 8.8 16 10.8 21 to 24 13 8.8 11 7.4 25 to 28 6 4.1 15 10.1 29 to 32 6 4.1 8 5.4 33 to 36 4 2.7 5 3.4 37 to 40 3 2. 7 4.7 41 to 44 1 .7 45 to 48 3 2. 49 to 52 4 2.7 53 to 56 57 to 60 1 .7 2 1.4 61 to 64 2 1.4 1 .7 65 to 68 2 1.4 Total 148 100. 148 100. Median 10.4 Median 16.8 25 Percentile 2. 25 Percentile 5.3 75 Percentile 19.9 25 Percentile 27.2 P.E. 8.8 P.E. 11. The Permanence of the Practice Effect 83 the initial period to the beginning of September was 4.1 per cent, while the loss in accuracy from the beginning to the end of the original practice was 1.5. TABLE XXXII (a) Gain in Accuracy in Addi- tion FROM Initial Prac- tice-Period TO THE Re- tention Test in Septem- ber (b) Gain of Same Individuals IN Accuracy During THE Experiment Per cent gained Individuals Per cents Individuals Per cents in accuracy -55 to -51 2 1.4 2 1.4 -50 to -46 2 1.4 -45 to -41 2 1.4 1 .7 -40 to -36 1 .7 -35 to -31 2 1.4 3 2. -30 to -26 8 5.4 4. 2.7 -25 to -21 6 4. 5 3.4 -20 to -16 11 7.4 12 8.1 -15 to -11 17 11,5 13 8.8 -10 to -6 18 12.2 14 9.5 -5 to -1 18 12.2 25 16.9 to 4 19 12.8 22 14.9 5 to 9 17 11.5 15 10.1 10 to 14 11 7.4 13 8.8 15 to 19 5 3.4 11 7.4 20 to 24 3 2. 2 1.4 25 to 29 1 .7 4 2.7 30 to 34 2 1.4 1 .7 35 to 39 1 .7 40 to 44 1 .7 45 to 49 50 to 54 1 .7 1 .7 Total 148 100. 148 100. Median -4.1 Median -1.5 25 Percen tile -14.6 25 Percen tile -11.7 75 Percen tile 6. 75 Percen tile 7.8 P.E. 10.3 P.E. 9.8 Division Gross Gain: The gross gain in number of combinations worked correctly from the initial period of the experiment to the beginning of September, and the gross gain of the same indi- viduals during the original experiment, are shown together in 84 Practice in the Case of School Children Table XXXIII. The median gain in the test at the beginning of September over the initial period was 17.5 combinations, while the median gain during the experiment was 37.5 combinations. At the beginning of September the group was thus still a median of 17.5 combinations ahead of the median ability shown in the initial period, but had lost a median of 20.0 combinations from the end of practice to the beginning of September. TABLE XXXIII (a) Gross Gain in Division FROM Initial Practice Period to the Reten- tion Test in September (b) Gross Gain of the Same Individuals During the Experiment ColuTiins Intli\'idual.s Per cents Individuals Per cents gained -14 to -10 3 1.4 1. .5 -9 to -5 3 1.4 1 .5 -5 to 12 5.7 2 .9 1 to 5 28 13.3 2 .9 6 to 10 16 7.6 9 4.3 11 to 15 32 15.2 9 4.3 16 to 20 28 13.3 17 S. 21 to 25 16 7.6 19 9. 26 to 30 21 10 18 8.6 31 to 35 17 8. 20 9.5 36 to 40 11 5.2 17 8. 41 to 45 8 3.9 18 8.6 46 to 50 8 3.9 14 6.6 51 to 55 2 .9 12 5.7 56 to 60 3 1.5 17 8. 61 to 65 1 .5 10 4.8 66 to 70 9 4.3 71 to 75 1 .5 5 2.4 76 to 80 2 .9 81 to 85 4 1.9 86 to 90 2 .9 91 to 95 1 .5 96 to 100 1 .5 Total 210 100. 210 100. Median 17.5 Median 37.5 25 Percer itile 7.5 25 Percer itile 23.5 75 Percer itile 30.1 75 Percer tile 54.9 P.E. 11.3 P.E. 15.7 Loss in Accuracy: Table XXXIV shows the facts. The median loss in accuracy from the beginning of the experiment to The Permanence of the Practice Effect 85 the beginning of September was .3 per cent while the median gain in accuracy during the experiment was 2.7 per cent. Hence during vacation there was a median loss of 3.0 per cent in accuracy. It must be remembered that these measures are from one 15-minute test to another, and so do not measure tlie effect of the disuse of the associations pure and simple. The loss from the last half-minute of the original practice to the first half- minute of the test in September might well be greater. But for practical purposes we wish to know just the fact which is measured here; namely, the change in the ability as shown in a test of ordinary length. Whatever loss is recovered in a minute or two of practice is educationally of no moment. TABLE XXXIV (a) Gain in Accuracy from Initial Practice-Period TO THE Retention Iest in September (b) Gain op the Same Indi- viduals IN Accuracy During Experiment Per cent gained Individuals Per cents Individuals Per cents -40 to -36 1 .5 -35 to -31 -30 to -26 2 .9 1 5 -25 to -21 4 1.9 -20 to -16 7 3.3 -15 to -11 9 4.3 4 1.9 -10 to -6 19 9. 7 3.3 -5 to -1 61 29. 40 19. to 4 67 31.9 82 39 5 to 9 20 9.6 39 18.5 10 to 14 6 2.9 17 8 15 to 19 10 4.8 7 3.3 20 to 24 2 .9 4 1.9 25 to 29 3 1.4 30 to 34 1 .5 2 .9 35 to 39 2 .9 40 to 44 2 .9 45 to 49 1 .5 Total 210 100. 210 100. Median -.3 Median 2.7 25 Percentile -4.6 25 Percentile -.5 75 Percentile 3.6 75 Percentile 7.4 P.E. 4.1 P.E. 4. 86 Practice in the Case of School Children Permanence of Association in Addition and Division During Vacation as Shown by the Amount of Practice Required to Restore These Associations to Their Pre- vious Efficiency. As soon as possible after the retention tests in September, each class was given sufficient practice in addition and division to restore it approximately to the same efficiency it had attained at the end of its practice the preceding year. Here " approxi- mately " means that the standing of a class, as shown by an average that could be calculated quickly, was within a few units of its standing at the end of the practice experiment. This practice was conducted under the same conditions as the practice of the experiment except that a practice-period could not always be given to each class on successive school days since there were four classes in addition and six in division to be re-practiced. Addition Only four classes containing 114 of the children who had taken the practice the preceding year were given practice to make good the loss of vacation. Time of Practice: These four classes had their retention test and their practice as indicated below : Retention Final Class Practice Test Practice Practice IX Sept. 10—15 min. Sept. 16 — 5 min. Sept. 18-15 min. X Sept. 12—15 min. Sept. 17 — 15 min, XI Sept. 12—15 min. ■ Sept. 17—5 min. 1 ■ Sept. 18 — 5 min. \ Sept. 17—15 min, VI Sept. 11 — 15 min. Sept. 25—15 min. . Sept. 19—15 min. J A wide difference in the amount of time required by different classes to regain their former efficiency is apparent. Class VI, which required the greatest amount of time, made a very great gain during the experiment of the previous year and so had a high standard to reach. This may partly account for the extra time required by this class to reach its former efficiency. The time required to regain this former efficiency was measured in the same way as was the time in the original experi- ment. While Class IX practiced 35 minutes, its gain was measured for only 273^ minutes. Its rate of adding during the The Permanence of the Practice Effect 87 last 15-ininute period was the rate of the mid-point of that period, or at the end of 73^ minutes; hence the actual time required for this class to regain its former efficiency was the 15 minutes of practice in the retention test of September 10, plus the 5 minutes on September 16, plus the 7^ minutes of the final practice-period, or 27^^ minutes. The time for the other classes was calculated in the same way. Accordingly Class X practiced 225^2 minutes; Class XI, 22^ minutes; and Class VI, 473^ minutes. The average time required to regain approximately the efficiency shown at the end of the practice of the preceding year was 30 minutes for the four classes. In other words, 30 minutes of practice were required to bring this group approximately to the efficiency which it reached as a result of 60 minutes of practice the preceding year. COMPARATIVE STANDING OF THESE FOUR CLASSES AT THE END OF THE EXPERIMENT AND AT THE END OF SEPTEMBER PRACTICE Gross Gain: Table XXXV gives under (a) the gross gain of the pupils of these four classes at the end of practice in September over their standing in the initial period of the original experiment, and under (b) the gross gain of these same individuals during the original experiment. A comparison of these two gains will show how nearly the 30 minutes of practice restored the group to its former efficiency. The median gain at the end of the 30 minutes of practice was 12.5 columns, while the median gain for the same group during the experiment was 14.3 columns. This means that 30 minutes of practice restored the median ability of the group to within 1.8 columns of the median ability of the group at the end of the experiment of the previous year. Accuracy: Table XXXVI shows that the median gain in accuracy for this group at the end of 30 minutes of practice over its accuracy in the initial practice-period was — 2 per cent. The loss of the group during the experiment was 3.1 per cent. Hence, so far as accuracy of performance is concerned, the condition of the group was practically the same at the end of the 30 minutes of practice in September as at the end of the experiment. 88 Practice in the Case of School Children The facts of Tables XXXV and XXXVI, in other words, are: 114 individuals who had taken part in the addition practice of the preceding year practiced again in September until they reached approximately the ability they had reached at the end of the experiment the preceding year. The average duration time of practice was 30 minutes, at the end of which time the group was within 1.8 columns of the standing which it had reached at the end of the 60 minutes of practice in the experi- ment of the previous year. The accuracy of performance was practically equal. Hence more than one-half as much practice was required in September to regain the standing reached in the experiment before vacation as was required at that time to reach it. TABLE XXXV (a) Gross Gain in Addition FROM THE Initial Prac- tice-Period OF the Ex- periment TO THE Last Practice in September (b) Gross Gain of the Same Individuals During the Experiment Columns Individuals Fer cents Individuals Per cents giiinetl -11 to - 8 4 3.5 2 1.7 -7 to -4 4 3.5 2 1 7 -3 to 7 6.1 9 7.9 1 to 4 14 12.3 16 14. 5 to 8 [16 14. 13 11.4 9 to 12 12 10.5 10 8.7 13 to 16 14 12.2 11 9.6 17 to 20 10 8.7 10 8.7 21 to 24 5 4.4 8 7 25 to 28 6 5.3 8 7 29 to 32 7 6.1 4 3 5 33 to 36 3 2.6 4 3.5 37 to 40 3 2 6 6 5.3 41 to 44 1 .9 45 to 48 5 4.4 2 1.7 49 to 52 3 2.6 53 to 56 1 .9 57 to 60 1 .9 2 1.7 61 to 64 1 .9 1 .9 65 to 68 1 .9 2 1.7 Total 114 100. 114 100. Median 12.5 Median 14.3 25 Percer tile 4.4 25 Percen tile 4.4 75 Percei tile 24.1 75 Percei tile 26.8 P.E. 9.9 P.E. 10.2 The Permanence of the Practice Effect TABLE XXXVI 89 (a) Gain in Accuracy in Addi- tion FROM THE Initial Practick-Pf:riod of the Experiment to the Last Period in September (b) Gain in Accuracy of the Same Individuals Dur- ing the Experiment Per cent gained Individuals Per cents Individuals Per cents -55 to -51 1 .9 3 2.0 -50 to -46 1 .9 -45 to -41 2 1.8 1 .9 -40 to -36 3 2.6 -35 to -31 1 .9 4 3.5 -30 to -26 5 4.4 4 3.5 -25 to -21 3 2.6 5 4.4 -20 to -16 8 7 10 8.7 -15 to -11 10 8.7 11 9.6 -10 to -6 16 14 10 8.7 -5 to -1 10 8.7 19 16.6 to 4 15 12.1 16 14 5 to 9 10 8.7 10 8.7 10 to 14 6 5.3 7 6.1 15 to 19 9 7.9 7 6.1 20 to 24 7 0.1 2 1.8 25 to 29 4 3.5 3 2.6 30 to 34 2 1.7 1 .9 35 to 39 1 .9 1 .9 40 to 44 Total 114 100. 114 100. Median -2. Median -3.1 25 Percen itile -13.3 25 Percen tile -14.8 75 Percei] itile 9.9 75 Percen tile 5.8 P.E. 11.6 P.E. 10.3 Division Time of Practice: Only six classes, containing 163 children who had taken the practice the preceding year, could be reprac- ticed in September. These six classes had their retention test and their practice as follows : 90 Practice in the Case of School Children Retention Class Practice-test XII Sept. 10—10 min. X Sept, 10—10 min. XI Sept. 10—10 min. VII Sept. 10—10 min. VI Sept. 11—10 min, XIX Sept. 11—10 min. Practice Sept. Sept. Sept. Sept. Sept. Sept. Sept. Sept. Sept. Sept. Sept. sept. 16— 5 m 18—10 m 16— 5 m 18—10 m 23—10 m 16— 5 m 16— 5 m 18—10 m 17 — 5 m 18—10 m 16— 5 m 20—10 m n. [■ Sept. 25—10 min, 40 n. J n Final Practice- period Amount Sept. 19—10 min. SO* 1 i n.J -I Sept. 18—10 min I Sor>+ 1Q in TTiin Sept. 19—10 min. Sept. 19—10 min. Sept. 25—10 min. 20 30 30 30 The average time required for these classes to regain approxi- mately their former ability was 30 minutes.* In order to give an exact basis for comparing the standing of this group at the end of 30 minutes of practice in September with its standing after 50 minutes of practice in the experiment of the previous year, the facts of Tables XXXVII and XXXVIII are presented. COMPAIL\TIVE STANDING OF THESE SIX CLASSES AT THE END OF THE EXPERIMENT AND AT THE END OF PRACTICE IN SEPTEMBER Gross Gain: Table XXXVII shows that the median gain from the initial period of the original experiment to the end of the 30 minutes of practice in September was 30.3 combinations, while, during the original experiment itself, the median gain was 34.2 combinations. The 30 minutes of practice lacked 3.9 com- binations of restoring the ability of this group to the ability it reached at the end of 50 minutes of practice the preceding spring. Accuracy: Table XXXVIII shows that the median gain in accuracy at the end of the 30 minutes of practice over the accuracy of the initial performance in the original experiment was 2.6 per cent; while during the experiment of the previous year a gain of 3.0 per cent was made. In other words Tables XXXVII and XXXVIII report as follows : 163 individuals who had taken part in the division practice of the preceding year practiced again in September until they reached approximately the ability reached at the end * Here, too, the rate for the last 10 minutes was the rate at the mid- point of the period or at the end of 5 minutes. Hence only 5 minutes of the last 10 minutes are included in the time required to restore the former ability. The Permanence of the Practice Effect 91 of the experiment the preceding year. The average duration of practice was 30 minutes, at the end of which time the group was within 3.9 combinations of the standing which it had reached at the end of the 50 minutes of practice in the experiment of the previous year. There was also a difference of .4 per cent in accuracy of performance. Hence more than three-fifths as much practice was required in September to regain the standing reached in the experiment of the previous year as was required at that time to reach it. TABLE XXXVII (a) Gross Gain in Division FROM THE Initial Period OF THE Experiment to THE Last Practice in September (b) Gross Gain of the Same Individuals During the Experiment Combinations gained Individuals Per cents Individuals Per cents -19 to -15 1 .6 -14 to -10 1 .6 1 .6 -9 to -5 2 1.2 -4 to 4 2.5 2 1.2 1 to 5 5 3. 2 1.2 6 to 10 8 4.9 8 4.9 11 to 15 11 6.8 7 4.3 16 to 20 16 9.8 17 10.4 21 to 25 18 11. 19 11.6 26 to 30 16 9.8 13 8. 31 to 35 11 6.8 17 10.4 36 to 40 15 9.2 14 8.6 41 to 45 13 8. 15 9.2 46 to 50 8 4.9 11 6.8 51 to 55 10 6.1 9 5.5 56 to 60 5 3. 11 6.8 61 to 65 6 3.7 5 3. 66 to 70 3 1.9 4 2.5 71 to 75 2 1.2 4 2 5 76 to 80 4 2.5 81 to 85 1 .6 2 1.2 86 to 90 91 to 95 2 1.2 1 .6 96 to 100 1 .6 1 .6 Total 163 100. 163 100. Median 30.3 Median 34.2 25 Percen tile 18.2 25 Percen tile 21.5 75 Percen tile 46.3 75 Percen tile 48.8 P.E. 14. P.E. 13.7 92 practice in the Case of School Children TABLE XXXVIII (a) Gain in Accuracy in Divi- sion FROM THE Initial Period of the Experi- ment TO THE Last Peri- od in September (b) Gain in Accuracy of the Same Individuals Dur- ing the Experiment Per cent gained Individuals Per cents Individuals Per cents -45 to -41 1 .6 -40 to -35 2 1.2 1 .6 -33 to -31 -30 to -26 1 .6 2 1.2 -25 to -21 -20 to -16 1 .6 -15 to- 11 2 1.2 2 1.2 -10 to S 10 6.1 6 3.7 -5 to -1 30 18.4 30 18.4 to 4 50 34.3 58 35.5 5 to 9 26 16. 31 19. 10 to 14 14 8.6 14 8.6 15 to 19 9 5.5 7 4.3 20 to 24 3 1.9 4 2.5 25 to 29 4 2.5 3 1.9 30 to 34 1 .6 2 1.2 35 to 39 2 1.2 40 to 44 2 1.2 45 to 49 1 .6 1 .6 Total 163 100. 163 100. Median 2.6 Median 3. 25 Percer itile -1 . 5 25 Percer Itile -.5 76 Percer itile 8.2 75 Percer Itile 8.2 P.E. 4.9 P.E. 4.4 APPENDIX I The Use of the Method of the Practice Experiment in Teaching Handwriting and Spelling Handivriting: The method of the practice experiment was applied by the author to writing in five fourth-year classes. Pages of Robinson Crusoe were printed to serve as material for the classes to copy. Uniform penholders and pens were supplied in each class. The regular school paper, of good quality, white and ruled, was used for the writing. The children were supplied with more material than they could copy in the time allowed. They were told to write as rapidly and as well as they could. About equal stress was put upon quantity and quality. Proper position was encouraged. Class I practiced 3 minutes a day for 16 successive school days. Class II " 3 " "9 alternate " Class III " 1 " " 12 successive " " Class IV " 1 « "9 alternate Class V " 6 " " 12 successive " The papers were not scored during the practice so the children had no definite measure of the quantity and quality from one day to the next as they had in arithmetic. This omission made the method radically different, since the incentive from knowing their exact progress was lacking. However, the children were always asked if they had written more and better than on the previous day. The author and the grade teacher observed the writing and urged that improvement in quality was as desirable as improvement in quantity. The first and last days' papers were scored for the number of letters written. They were also scored for quality by six different judges, by means of Thorndike's Scale for Handwrit- ing, and the average judgment for individuals and class was found. There was a decided gain in quantity in all the classes and a gain in quality in two classes. The following table gives (i) the average number of letters written per minute in the 93 94 Practice in the Case of School Children initial performance, (2) the average gain per minute in number of letters in the last day's performance over the first, and (3) the average change of quality expressed in steps of Thorndike's scale. Average Number of Letters Written per Minute in Initial Performance Average Number OP Letters Gained per Minute Average Change IN Steps op Thorndike's Scale Class I 35 7 + .02 Class II 24 22 —1.16 Class III 22 32 + .17 Class IV 43 21 —1.53 Class V 17 16 — .9 It is easy to interpret the results where there was a gain both in quantity and in quality as was the case in Classes I and III, the first of which gained 20 per cent in quantity and slightly in quality, the second of which gained 145 per cent in quantity and slightly in quality. Just what merit attaches to the per- formance where there was a very great gain in quantity, but a loss in quality, remains for future interpretation. The results suggest that it is possible by means of the practice experiment, to increase speed in writing in the case of fourth- year school children very much and at the same time increase the quality. Had there been a constant measure of quality to act as a check upon quantity, no doubt the classes that lost in quality might have transferred part of the effort from speed to quality, thus maintaining or improving the quality and at the same time increasing their speed. An interesting problem for future experiment would be one similar to that outlined above with the added feature of having copies of Thorndike's scale hanging in the room readily accessible to children by which they might measure at any time the quality of their own writing. This would serve as an incen- tive to improvement in quality along with speed. Spelling: Five fourth-year classes were given work in spelling with the practice experiment as a method. A list of words was used, each one of which 60 per cent of the fourth-year children in a large school of high grade had missed in a certain test. There were about 100 words in the list. These words were placed on cards, one on each card, which were shuffled until Appendix 95 they were promiscuously arranged. Then these words were dealt out in three piles until 30 words were placed in each pile. These groups of 30 words were then recorded for use. Then these 90 words were shuffled as before. Next they were dealt out in six piles until there were 12 words in each pile. These lists were recorded for use. Then these 90 words were shuffled again as before. This time they were dealt out in fifteen piles of 6 words each. These lists of 6 words were recorded. The groups of 6 words and 12 words were printed on large cards to be hung in the front of the room during the study period. It was found impracticable to put the list of 30 words on such a card, so they were typewritten, a copy for each child. Class I had 15 minutes to study 30 words on each of 3 successive school days Class II had 6 minutes to study 12 words on each of 6 successive school days Class III had 6 minutes to study 12 words on each of 6 alternate school days Class IV had 3 minutes to study 6 words on each of 13 successive school days Class V had 3 minutes to study 6 words on each of 9 alternate school days The children were told that there were words on the cards which they should study when the cards were turned at a given signal. They were told how much time they would have for study. They were told to study in any way that they choose, except that they should not take up their pencils during the study. This was done to prevent any attempt at writing words so that they could be used later. They were given no further instructions as to how they should study, except that they should try to learn every word. Before the study was begun, papers and pencils were passed out to be ready for use immediately after the study ended. At a given signal the chart or typewritten paper was turned for study. They studied diligently until the chart was turned or the signal to turn their typewritten papers face down was given. Then the words on the chart were pronounced by the experimenter in a promiscuous order at the rate of about 4 words per minute, and written by the pupils. The papers were marked before the next day's practice. Before beginning the following day's study the scores were read, 96 Practice in tJte Case of School Children and good records commended. All were urged to try to beat their previous day's record. This was continued for the number of days indicated above. The following is the record comprising (i") the percentage of words spelled correctly during the entire experiment. (2) the percentage of words spelled correctly in the tirst two tests, and (3) the percentage of words spelled correctly in the last two tests. Per Cknt of Wokds Per Cent of Words Per Cent of Words Sfelled Correctly Spelled Correctly Spelled Correctly IX Entire List in First IVo Tests in L.\st Two Tests Class I S2 84 S2 Chiss II 91 87 92 Class III S7 81 93 Class IV 77 88 70 Class V 93 95 90 The results indicate, as have other studies of spelling by Corn- man and Rice, that spelling is not amenable to method. Class II and Class III showed improvement in the last two tests over the first two. but the other three classes showed a loss. APPENDIX II Samit-ic ok AnniTioN Siii:i:is Reduced to ; 7s and r. 22= 6s and 56= 9s and 83= 93 and r. 53= 6s and 30= 7s and 21 = 7s 33= 4s and 89= 9s and 54- 8? and r. 77= 8s and 20= 8s and 32= 4s 22= 9s and 56= 6s and 80= 9s and r. 52= 7s and 31 = 4s and 22= 3s and r. 33= 7s and 86= 9s and 53= 9s and r. 75= 9s and 21 = 4s and 32= 7s and r. 23= 5s and 55= 7s and 78= 9s and r. 51 = 8s and 34= 4s and 24= 7s and r. 36= 7s and 74= 8s and 50= 6s and r. 67= 9s and 23= 8s and 35= 7s 25= 9s and 45= 9s 69= 9s and r. 48= 6s 34= 7s and r. 25= 3s and r. 38= 7s and 72= 9s' 49= 4 7s 66= 9s and 24= 4s 36= 4s 26= 5s and 50= 9s and 68= 9s and 47= 8s and 35= 4s and 25= 6s and 39= 4s and 71 = 8s and 48= 9s and 65= 9s and 26= 8s and 62= 9s and 38= 4s and 47= 5s and 28= 3s and 59= 9s and 39= 7s and 44= 9s and 29= 5s and 64= 9s and 40= 5s 42= 7s 27= 4s and 61 = 9s and 41 = 8s and 46= 7s and 28= 6s and 59= 6s and 37= 4s and 44= 6s and 29= 8s and 63= 9s 41 = 5s and 43= 5s and 27= 7s and 60= 9s and 57= 8s and 45= 7s and 28= 9s and 58= 7s and 37= 7s and 43= 8s and !• 31= 4s and LIBRARY OF CONGRESS; 021 350 393 5