if (^ Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924002999849 INDIANA UNIVERSITY STUDIES Study No. 32 STUDIES IN ARITHMETIC. Edited by Melvin E. Hag-- GEKTY, Ph.D., formerly Associate Professor of Psychology and Education in Indiana University. For Sale by the University Bookstore, Bloomington, Ind. Price, 25 ets. The Indiana University Studies are intended to furnish a means for ptiblishing some of the contributions to knowledge made by instructors and advanced students of the University. The Studies are continuously numbered; each number is paged 4iidependently. Entered as second class matter, March 27, 1916, at the post-ofiSce at Bloomlugton, Ind., under the Act of August 24, 1912. The Indiana Univehbitt Stitdies are pub- lished six times a year, in January, March, May, July, September, and November, by Indiana University, from the University OiHce, Bloomlngton, Ind. Indiana University StudiesI, Vol. Ill ) September, 1916 Study No. 32 STUDIES IN ARITHMETIC. Edited by Mblvin E. Hag-- GEBTY, Ph.D., formerly Associate Professor of Psychology and Education in Indiana University. Gontents PAGE Prefatory Note 3 I. Second Report on the Measurement of Arithmetic in In- diana CiTT Schools By Mblvin E; Haggertt Introduction '5 Causes of Variability in Scores 8 Speed and Accuracy 9 Standard Scores and Charts 9 Comparison of 1915 Scores and 1914 Scores from Nine Cities. 12 Tables and Charts 13 II. Arithmetic Tests in Rural Schools of Five Indiana Coun- ties By Paul R. Mort Introduction 59 Results from the District and Graded Schools of Five County Systems 60 Factors Afifecting Achievement in the Fundamentals of Arith- metic , 68 The Achievement of the Retarded 78 III. Effects of Six Weeks' Daily Drill in Addition. . , .By Mary A. Kerr 79 IV. An Experimental Study op the Effects op Drill in Arith- metical Processes under Varying Conditions. . . .By Herman Wimmer. Introduction 96 Normal Growth without DriU during Six Weeks 98 Comparative Progress with Drill and without 99 Comparison of Progress with DriU on Fundamentals and Progress with Drill on Reasoning 99 Drill for Speed and Drill for Accuracy 100 The Best Distribution of Time for Drills 101 Comparison of the First Six Weeks' Gain under Drill in each Grade 102 V. Experiments with. Courtis Practice Pads. .By Flora Wilbbr Nature of the Study. 103 Speed : . , 106 Accuracy. • i 107 Colnparison of Speed and Accuracy , 109 Summary , 110 Prefatory Note This Study contains five separate investigations in the learning of arithmetic in Indiana schools. In each instance the Courtis standard tests were used to measure the efficiency of the children whose work was tested. The first two investigations are con- cerned with the measurement of the actual efficiency of the schools in question near the close of the second semester, 1915. The material for these studies was gathered for the most part thru the Indiana University Bureau of Cooperative Research. For the purposes of this report this material is divided into two parts, one dealing with the city schools, the other with the rural schools. The first, covering 22 city systems, is reported by Mr. M. E. Haggerty. Mr. Paul Mort is responsible for the report on the rural schools of five counties. In addition to these measurement results there are three in- vestigations on the effects of drill in arithmetic, as follows: (a) An investigation of the effects of six weeks' daily drill in addition, by Mary A. Kerr, principal of the Department school, Bloomington, Ind. (&) An experimental investigation of the effects of drill in arithmetical processes under varying conditions, by Herman Wimmer, superintendent of schools in Rochelle, 111. (c) An experiment with Courtis Practice Pads, by Flora Wilbur, principal of Fort Wayne training school. Acknowledgement is due to the numerous school superin- tendents, principals, and teachers for their services in giving tests and doing the initial work in scoring the results. No less credit is due the University authorities for their cooperation, to President William L. Bryan and the Board of Trustees for the funds with which the work of re-scoring and tabulation was carried, on, to the graduate students and assistants who assisted in the work, and to the ever-patient Office of University Publica- tions in whose hands a somewhat chaotic manuscript has taken final shape. M. E. Haggerty. June 5, 1916. I. SECOND REPORT ON THE MEASUREMENT OF ARITHMETIC IN INDIANA CITY SCHOOLS By Mblvin E. Haggbety, Ph.D., formerly Associate Professor of Psychology and Education in Indiana University. Inthoduction The second report on the measurement of arithmetic in Indiana city 'schools is based upon tests — Courtis, Series B — which were given in May, 1915. As in the case of the 1914 Study^, the tests were given and scored by the local school oflftcers and the results, tabulated for class and school scores, were sent to Indiana University. Twenty-two city school systems reported tests of all or a part of their pupils. Below is given the list of cities and the name of the superintendent or principal thru whom the work was carried on: Akron Supt. H. Gf. Knight, Alexandria Supt. A. L. Trester, Bloomington Supt. H. L. Smith, Blufifton Supt. P. A. AUen, Columbia City Supt. J. C. Sanders, Crown Point Supt. W. S. Painter, Decatur Supt. C. E. Spaulding, Elwood Supt. J. L. Clauser, Port Wayne Prin. Flora Wilber, Huntington Supt. J. M. Scudder, KendallviUe Supt. P. C. Emmons, Mishawaka Supt. John F. Nuner, Montezuma Supt. J. G. Hirshbruner, Noblesville Supt. E. C. Stopher, Orleans Supt. Lewis Hoover, Poseyville Supt. O. H. Horrall, Princeton Supt. J. W. Stott, Richmond Supt. Joe Giles, Roachdale Supt. L. E. Michael, Seymour Supt. T. A. Mott, Sheridan Supt. James W. Kirk, South Bend Supt. L. J. Montgomery. iM. E. Haggerty. Arithmetic: A Co-operative Study in Educational Measurements {Indiana University Studies, No. 27). 6 Indiana University Studies In the following pages these cities are indicated by number but the order is not alphabetical. At the University the tabulated results were re-examined and corrected. The amount of this corrective work was considerable and could not be trusted to careless hands. Where so many thousands of computations are involved the chances for error are very great. Even graduate students require the most careful supervision both as to method and in accuracy of work. What- ever dependability is to be placed in the accuracy of school medians and variabilities is largely due to Miss Mary Kerr, principal of the Department school in Bloomington, who gave her summer vacation chiefly to the examination of the returns from the cities. Assistance was also rendered by Miss Cecile White, fellow in philosophy, and by Mr. Earl Moore, technical assistant in the department of philosophy. The results of the study are presented in the distribution tables, Tables VI-IX. The median scores are collected in Table I, which gives the scores in attempts, rights, and 'dependability for grades 5 to 8 in all of the 22 cities, the medians for Indiana and similar scores for 9 cities and towns in Iowa (1915), and 16 cities and towns in Kansas (1915). Reading across this table for grade 5, the medians for city 1 are: addition — attempts, 7.7 problems; rights, 4.2 problems; dependability 55 per cent. Then follow in order to the right the scores in subtraction, multipHcation, and division. The ranking of the several cities is shown in Table II. The most striking fact about these scores is the wide variation which they show from city to city. City 22 attempts 8.5 problems with 66 per cent of the work correct, while city 2 attempts but 4.4 problems and scores but 35 per cent on this meager effort. Fourteen schools fail to equal the fifth grade score of city 22 in their sixth grades, 9 fail in their seventh grades, and 5 fail in their eighth grades. In these upper grades city 22 likewise excels the median of the entire group, altho it is excelled by city 4 in the sixth grade, by cities 4 and 20 in the seventh grade, and by city 4 in the eighth grade. Compared to the Courtis standard, city 22 is in all grades below the level of efficient work. Much more are also the cities which are inferior to city 22. The difference in median scores is so great as to suggest not the distribution of schools more or less closely about a common center but the existence of actually different types of schools. Haggerty: Studies in Arithmetic 7 Note the sixth grade division scores in cities 15 and 16. The one is 8 problems correctly solved; the other is 2, or one-fourth the score of the first. The Hiultiphcation rights for city 17 is 3.1 ; for city 12 it is more than double. The number of subtraction rights of city 2 is 5.3; for city 21 it is 9. The minimum score in addition is 3.4 rights for cities 1 and 7, and for city 4 it is 7.5. It is the same story for all the grades. Think of such ranges of efficiency as 4.4 to 9, 7 to 11.2, 6.5 to 12, and 7.0 to 12.1 for the eighth grade; and 4.1 to 9, 4.2 to 8.4, and 4 to 9 for the seventh grade. There are 8 eighth grades as low as the median of all seventh grades in addition, and 2 as low as the median of all sixth grades, and 1 whose median is but .2 above the median of all fifth grades, while there are 1 sixth grade and 4 seventh grades higher than the median of all eighth grades. For a correct appre- ciation of these figures 'it must be remembered that they do not represent individual children. There are in all groups half the individuals below the median score and half above. How much wider then must be the range between the lowest 10 per cent of the lower group and the upper 10 per cent of the upper group! The facts represented by these numbers appear more emphatic when shown in graphic form as in Figures 1 to 4. Here you see at a glance the enormous amount of overlapping existing from grade to grade. In general the fifth grade scores (triangles) are grouped at the lower lefthand corner of the' figure, and the eighth grade scores (stars) are found at the upper righthand part, with the sixth grade scores (squares) and the seventh grade scores (circles) sprinkled between. In the center of the figure the symbols of the several grades mingle in a most disorderly fashion. The amount of overlapping would be simply astonishing if we had not learned to expect such results from the application of standard tests. In Figure 5 is shown the overlapping of the fifth and seventh grades in the several schools within a single city. The seventh grade medians for both attempts and rights are clearly above those of the fifth grade, thus showing in general a, superior achieve- ment for the upper grade. On the other hand, in a large number of schools the fifth grade work is superior to that of many seventh grade classes in the same system and in one case the more ad- vanced class is inferior to the fifth grade class within the same school. 8 Indiana University Studies Causes of Variability in Scoeeb There are but two possible causes for the variability of scores. One is the variability in original endowment of children. The other is the effectiveness of environmental conditions on the growth and development of children. If we should accept the first hypothesis as an adequate explanation of the variability found in these scores we must argue that in those schools with low scores we have children of less than average ability and that other cities attain high scores because their children are unusually gifted. Two things are against such an explanation. First, there is no conclusive evidence from the experimental studies of in- dividual differences that such widely separated types of individuals are to be found among such fairly homogeneous populations as live in these Indiana cities. The second obstacle to such a theory is that you get widely different results within the same cities and within the same school. Thus city 2 scores 6.5 in the eighth grade addition and but 1.5 in the fifth; city 4 is 40 per cent above the median in sixth grade addition and 14 per cent below the median in multiplication in the same grade. Such facts as these do not indicate differences of mental endowment on the part of the children. They rather indicate ill-adapted environmental conditions. These environmental conditions may be divided into two groups, the school and the non-school. The latter comprise such facts as conditions of family life, feeding, recreation, and outside work. Of the great importance of such factors as bearing upon school work there can be no doubt. There is, however, no reason to believe that such conditions are peculiar to any city in a manner adequately to account for the variability of scores. In fact, some cities and individual schools in the poorer sections of other cities show approximately as good work as do the better- cared-for children in the more favored localities. With these possible explanations eliminated we must recur to the differences- in the school conditions to account for the differences in the scores. Where one city scores higher than another, the difference is probably due to the fact that the school conditions in the one city are superior to those of the city with the lower score. When city 20 scores 100 per cent in dependability in division in three grades it means that the administrative and teaching force in that city have learned how to teach division effectively, and when city 7 makes but 58 per cent in eighth grade addition, while scoring much higher in the other processes Haggerty: Studies in Arithmetic 9 it means that city 7 has not learned how to teach addition as effectively as the other subjects. The variability in median scores probably indicates more or less accurately the differences in teaching methods, and if any city is to secure a change in score it will do so by changing its methods of instruction. Speed and Accuracy One of the common causes urged for a low dependability is that the children work too rapidly. The evidence in this table of medians is directly contrary to that argument. Twenty-four grades which fall below 60 per cent in dependabihty have a median score in attempts of 7.1, while 11 grades which scored from 70 to 75 have a median score of 7.7, and the 7 cities which score above 75 per cent in accuracy have a median of 9.8. These figures would seem to justify the statement that the more rapid the work the more accurate it is. Whether this extreme state- ment could be justified may well be doubted. The ratio between accuracy and speed is probably an individual matter, and there are certain limits of speed within which an individual does his best work. If he is pushed beyond this limit he loses in accuracy; likewise if he is slowed down too much he tends to become in- accurate. No one has yet determined for children in general or for any individiial child in particular the limits within which he will do his best work in any subject. The value of speed in securing greater accuracy is doubtless a function of attention. When the mental processes are rapid their inhibitory power is greater and, hence, the subject is less open to distraction from interfering stimuli. He has concentrated attention. To slow down these processes weakens their resistance, and contending neural impulses get the right of way with con- sequent dispersed attention, providing the condition for in- accurate work. For this reason many children would be rendered more accurate if they were speeded up within certain limits. Standard Scores and Charts One cannot from the median score of a class judge the efficiency of that class except in reference to a standard of achievement for a class of that grade. In the 1914 study^ we pubUshed a chart which we called the Indiana Standard. The form of the chart was "M. E. Haggerty. Arithmetic: A Co-operative Study in Educational Measurements {Indiana University Studies, No. 27). 10 Indiana University Studies determined by the median scores in attempts and rights for each of the processes for grades 5 to 8. The assumption was that the score attained or exceeded by one-half of all the pupils was a minimum ideal for the grade in question. This assumption was, of course, quite arbitrary, and its chief justification was that it estabhshed a point of reference for judgments in specific cases. It seems hardly worth while to make a chart from the 1915 median scores, inasmuch as the alterations would not be great and would serve no very useful purpose. Two charts made on another basis are presented in Figures 5 and 6. Indiana Standard B is based on those scores which were made or exceeded by the children in at least three cities. The several scores do not all come from the same cities but were taken from Table I wherever they occurred, selection of any score being based on the number of rights. In a similar fashion Indiana Standard C was made from the highest scores in rights occurring in Table I. The value of these standards, while still arbitrarily chosen, is that they offer an ideal that is beyond mediocrity and yet is within the limits of whs^t has actually been accomplished. Chart C is the most exacting standard and is attained by less than 5 per cent of the classes. More than 15 per cent of the cities attained Standard B in one or more classes. In the charts the scores used as standards are represented by short horizontal lines across the page so drawn as to appear at mid-section of the vertical dimension. These lines are the' Standards. Each vertical line represents the scale for the test in question. The first, third, fifth, etc., Hnes represent the number of problems attempted in the several tests in the several grades. The second, fourth, sixth, etc., lines in a corresponding manner scale the examples right. The portion of each Une below the Standard is proportional to the difference between zero and the median score. It is accordingly scaled into the proportional number of parts. The part of the line above the Standard is similarly scaled. Each vertical Hne is therefore a different scale from every other vertical hne, since the score is different in every case. It is possible on this form to graph the results from any class, school, or city and to see quickly its relative standing. To do this you locate the proper score on its appropriate vertical line. If you join the points so located on the attempts scale by a solid Une with the similar point on the rights scale, you have represented the dependability of the work. If the hne so drawn is parallel Haggerty: Studies in Arithmetic 11 to the Standard the per cent of dependabiUty is the same as the Standard. If the line slants upward to the right, the dependabiUty is greater; if the line slants downward to the right, the dependa- bility is less. Continued investigation of achievement in arithmetical sub- jects should lead some time to a standard much less arbitrary than the ones here presented. To escape the charge of arbitrari- ness a standard must have due regard to the following conditions : (a) The degree of efficiency required for the successful pursuit of the vocations into which children will later go. (6) The degree of achievement required at each grade in order to insure such efficiency in the end. (c) The time allotment necessary to secure such efficiency at each step. (d) The proper relation of efficiency in the fundamentals to efficiency in other arithmetical subjects. (e) The proper relation of efficiency in the fundamentals of arithmetic to the necessary requirement's in every other subject which schools teach and to the other important school interests. At the present time only the barest beginning has been made in determining the adequate answer to any one of these demands, and until such an answer is actually determined we must content ourselves with the highly arbitrary standards which we have. In this connection it is interesting to compare the median scores in Indiana schools with similar scores from lowa^ and Kansas*. These medians are given in Table I, and shown graphic- ally in Figures 7 and 8. The Kansas medians are derived from 19 cities; those from Iowa are from 9 cities. As in Indiana, the total number^ of children is somewhat less than 10,000 in each State. Graphed on the Indiana Standard both States appear higher, Iowa strikingly so. Searching the Indiana table of medians it is difficult to find any city as high as the Iowa median in all points. For the sake of comparison with the best of the Indiana cities, Table V is constructed showing the 5 cities containing the highest scores and the median scores of the 5. If the Indiana study had been based upon these 5 cities the Indiana scores would have been clearly above those of Kansas but not equal to those of Iowa. 3An unpublished study by Ernest J. Ashbaugh. 'Walter S. Monroe. A Report of the Use of the Courtis Standard Research Tests in Arithmetic in Twenty-four Cities. Emporia, Kan. 12 Indiana University Studies The best Indiana scores, however, are above the Iowa median. If all the Indiana cities could have done as well as the best they could have equalled Iowa. This fact gives plausibility to the Iowa scores and renders them a more practical ideal than they would otherwise seem to be. It should be noted, however, that no State score equals Mr. Courtis' ideal of 100 per cent accuracy. Some cities, however, do occasionally reach or approach that level and more often in division than in any other process. Comparison of 1915 Scores and 1914 Scores from Nine Cities Opportunity offers for comparison of the 1915 scores in nine cities with the 1914 scores in the same cities. These- are cities 3, 5, 6, 7, 8, 9, 12, 15; and 18. The medians for the two years are shown in Tables III and IV. Table III shows the 1914 fifth, sixth, and seventh grade medians with the medians of the sixth, seventh, and eighth grades of 1915. Here you pre- sumably have a year's growth in the same children. (This is, of course, not exactly true, for the composition of the grades has changed somewhat 'within that time.) With but rare exceptions the children of each grade in each city have made a substantial gain in achievement, thus showing that with the schools working as they do, the pupils have the opportunity to grow. Did the schools actually improve their teaching methods during the year? The answer to this question is found in Table IV where the several grades of 1915 are compared with the corresponding grades of 1914. There seems little doubt that improvement oc- curred in the eighth grade but not so markedly in the other grades. Some cities improved more than others; some did not do so well as in the previous year. As a whole, the improvement is not marked. The teachers seem either to have approached the limit of their power to teach, or they do not know how to make improvement in their methods. TABLES AND CHARTS :i4 Indiana University Studies 1—1 OS H 1— ( <) Q :? 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'^ "5 3 J? jE 3! 1? "i-r- 3 tn N ■« -S -2 S5 a s g ? ^ ■? ;g r~r- — • • • • • • -• — • — • — • — 4t — • — •- ^ ^ N ^ *l O- ■» _J5 ;^_J3. «<©ir.T3i vs M e ^ ^ s ":E — Sr- 5^ 5E ! — • • •- A ?. 1, tr- N -^ 2? ^ w 2 ill HI 3 31 58 Indiana University Studies ^0 80 70 60 EXA MPLES 15 10 5 6A 7B 7A 8B SA ^^^ """ «9/?~' — __ ^55^ I)EP. ^_„,s-- • ^-^ :=:=Hir • 1)ER ATS ATS. RTS HTS. Fig. 9. Change in Attempts, Rights, and Dependability in Addition FKOM PeBRUABT TO JUNR 11. ARITHMETIC TESTS IN THE RURAL SCHOOLS OF FIVE INDIANA COUNTIES By Paul R. Moet. Introduction / This study is based upon Courtis Arithmetic Test reports of the tests made in the spring of 1915 in the counties of Wabash, Huntington, Lake, Warren, and Randolph. Wabash county was reported by Superintendent Robert K. Devricks; Huntington county by Superintendent Clifford Funderburg; Lake county by Supervisor E. A. Whitney; Warren county by Superintendent Harry Evans. Superintendents Funderburg and Devricks made much of this study possible by cooperating in giving a question- naire in their schools. The first part sums up the general conditions in these counties and points out a few comparisons. The second is a discussion of some of the factors influencing achievement in the fundamental processes of arithmetic. The third is a discussion of the results obtained from retarded pupils (i.e. those who have failed one or more times) in eight townships. Two plans were followed to insure dependability in the con- clusions of the second part. In the first place, no conclusions were drawn from any set of comparisons unless the induction from those comparisons agreed with that from the comparison of the "regulars" (i.e. those who are in the classes with which they entered school) in those eight townships for which the distributions of the scores were made on the basis of retardation. This eliminat- ed the influence of the retarded pupils. In the second place, no conclusions were drawn that were not borne out by the results in at least two counties, and thus by two widely different sets of conditions. The limitation arising from the small number of cases con- sidered is offset to some degree by the large representation of schools (67 schools in 8 townships, alone). This tends to eliminate the influence of abnormal local conditions. 60 Indiana University Studies Results prom the District and Graded Schools of Five County Systems In this division I attempt but little more than to give the descriptions of the systems concerned as they were reported. All of Wabash county with the exception of one township has had the eight months' term for a number of years. The township excepted has had it for but three years; There are seven townships in the county each having from one to five graded (consolidated or combined)^ schools of from two to four teachers each, and each, with one exception, having a number of district schools. The ratio of children in district schools (upper four grades considered) to those in graded schools is 376 to 757. The system of township supervision has been practiced thruout the county for a number of years. In this system the principal of one of the township high schools has all the township schools under his direct supervision. The teaching of arithmetic is begun in the first grade. In Huntington county the length of term varies from six and one-half to seven months. Each township has one graded school with two teachers. The ratio of district school children to graded school children is 315 to 71. The township supervision system was in use in one township only, and then for but two years, preceding the giving of the tests.^ ■ No special supervision is given as to time of beginning of the teaching of arithmetic. Lake county had had the nine months' term for six years preceding the giving of the tests. There are ten townships under the supervision of the county superintendent, which are, with one exception, for the most part consolidated. There are also five villages under the county superintendent, but these have separate school boards. The four Warren county towns are under the supervision of the county superintendent. Only one Randolph county school was reported. This is a consolidated school having both local and county supervision. The length of recitation periods for the same type of schools varies very little. 'Combined schools are those supplanting two or three district schools and generally have no more than two teachers. Consolidated schools are those supplanting several district schools. 'In the fall of 1915 a supervision system similar to that in use in Wabash county was adopted. Haggerty: Studies in Arithmstic l{ /^ — S — I S — 3 — S — 5 — = — I \p, £ — tr/S — R — % — S — S — S-^ RS -2 E;"S SSaSS2°^^ t^-o o^t nw- I* ft • 9 4 * 9 — ^— • 1— • ' • « • • • ■ ■ — S • 1 — :* fir' — • g. o- N i -J — • — a ^ » o • — • — • — • — • • £• J5 o » (^ -o m T tn N — S- ?, a -• • ■ * »ir • J » • • • • • • I n cj ^ o *l\ X /F" o ifi T n w -« uh I g . ■ ■ — 9. g. • A* *^ t ' ^l — 9:—? • fc » * — » — S — ' ^^ SSS'^ txs QQ tsi -o in VI 2^32 • I — • • » > » ■ m T (^ w O ■ ■ • — ■ • > ■ / S 2 3 2 "^ * ^1 \ * -o <« T "n n -• I J • » ft •: ♦ • *-* — V~* — f * ft •: • •: ^ o .s <2 ^ ^ <>« O On flO t- O ^ rt M a - ■• • • ■ c-r-^ • — ^ • m • • 1 C 2 o^ 41 (> tP] *^X^ t n n - me • • • ■• • • • J^ 1^* » ^ « • • • • m • ' "3 — T -r^ 3E ■«) OOvMf^^Ooi^nN'^ ft • • • • ^j » n • ■ — • • • 1 S 0> 60 N ■* ^/ I ^ rt N " rf m m « * * fi It • '•*' f 1 ■ • • ■ ■ 1 n t r, 3— — ^ \ /■ ^ *^ — * *■ t. "%' — sr D K f— • • • • • ■• \3 ft ^J > • > o ov Qo r- -o i""- yt ^ t^ n M - i • • ■■-• • • ■ • — ^- ' 9 ^ 90^ » « > ■ ' • « pj O Ov tc o m ly m ^ .8 -a A — / >» J^ , • ■ ■< • 1 — f* ^j ' ■ a 1 00 (^ JS tf) X^"^ ^-"^ w - ^ ^J » • > • - j; • • 1*— • — '" • ^ • a > • • 1 g o^ cr e^ -o o^ I ■ » • ■ • ^ ' • '-*'^ • ■ m • • — ♦ •-k:7 *»■ * • t« so oi ^It^* -^^ — • • • • • • ■ m ^^ — • ■ 9 m ■ • • ci — o o^ cc t- 'O in-Tfn n — -K S Vt« 7-S < • • — ^' ■m ' — • •— «> I pR S S * ^-J,^ ■ ' = <1 ^-1 w • • • • • ' V, — • • • • • '• 1 ■t S S — rZ -^ k.^Q.n in T» m r* _• S n to » CD i5 P O D H 13 !S o « < S o O O D^ 62 Indiana University Studies 5 a /1SS;| ^1 I 5 5N|'fc>4R/S • • • R "a s 1 S'l ?;l SIS'S-SSo-"?^' m' ■ "' |S 1 — s— 8 -a I -^ \% ^ -y- ^ ? s. ?. *^ £ 2 — t — ?~"S — 3 — § g 5 s '^ ' * S Sv • "■ S *• fT* ■s--| — -s — ? — R — s — & "M jj — sr- ^-SSS'!-'^ ^ i< 1 5 ?! ? § -gr * S.|\\S V • Ji f- ~ a 1 = S ?! 5 = OO^X ^-OOl T O M 1 fT* Is O O' — ^2^*f''^">^'^ W " 1 * £ ?■ ■^ 1 •■ ■ ■ • • ■ 5 "■■ttV ?■ 5 a ^ — . /t;*w .... . .. . . .. - < « in • — • — I— U- — • — «) i: CO ^ 1 o « o rd % 0. o ^ 55 C s K .s g w o gj U '.s a rv| ii n N d o fl bH Haggerty: Studies in Arithmetic 63 Table I groups the results from the different systems so that they are comparable. Figures 1 and 2 ^ show the standing of the county systems compared with the average of the twenty Indiana cities tested in 1914, which average is represented by the lines marked "Indiana Standard". I wish to call attention to but three points here: the improvement shown by the second testing of Warren county, the uniformity of the results from the different systems, and what it means for a system to be up to standard. Superintendent Harry Evans of "Warren county, in discussing the improvement shown by the second testing, states that after giving the first tests he gave the results to the teachers and pointed out a few weaknesses, but did not introduce any con- structive work. He thinks, however, that the pointing out of the weaknesses probably served as a stimulus to both teacher and pupil. He also mentions the fright of the children at the first testing. As for the second point, an examination of Table I shows but a slight degree of variation in results from different systems as compared with the twenty Indiana cities* tested in 1914. There are of course many factors entering into this, some of which are discussed later in this Study, but supervision probably has as much to do with the variations in the case of the city scores as any other one factor. Might, then, the uniformity in the systems here given be due to their comparative lack of supervision? The point as to the meaning of the standard remains. It will be noticed that in a great many places the scores of the county systems exceed the Indiana Standard. If we are to take this standard in any case as the minimum score which a child should have, and the maximum necessary for him to have, we might con- clude that no further work is needed in these classes in funda- mentals of arithmetic. This conclusion would be erroneous for when the median score of a group is equal to the standard only 50 per cent of the group have a score equal to or exceeding the standard. There is still work to be done with the low 50 per cent. These individual cases should be examined carefully to discover whether the low scores are the results of wrong grading or of some obstacle. Attention can be given to these while the standard sin flgvires 1 and 2 tlie greater the height of the left end of the graph, the greater the "speed". The greater the slope upward toward the right the greater the per cent of accuracy or "dependability"- Equal heights represent equivalent speeds thruout. Equal slopes represent equivalent dependabilities. *See Second Annual Conference on Educational Measurements, (Indiana University Bulletin, XIII, No. 11), p. 26. 64 Indiana University Studies xnnwB ■p i -puadaa Q " paadg . (3 -i^IIiq^ !§ -pnadaa 9 « K S| paads fl ^IIpq'B .0 •Q 3 -pnadSG +3 paadS d mn<¥e •2 -pnadaa g paadg siidnj: JO jaqratiN AIIM'E i? -pnadaa s-i paads . s Amq'B i-B -puadaa 3 J3 §t paads ^ a AlTW a -pnadaa CS 03 eS paads P ^ailM'E -pnadaa -0 paads STtdnj JO jaquinjsL % § °S .9 fi l^V d o a ^a Oa o CO »o 1 i> U3 l> rH -* t^ CO ■* ic ■<* CO w SO lO t* t> CD CD ^ I> CD I> ■* O 1-1 a t> 00 l> CD lO Tji lO tP ■^ •* b- CD r- co CO TP o OS O CO M ,_t m ■* lO i> 00 O) tH CO CO o> 00 w >o •^ CD CQ < CQ 03 CD CD lO lO H u CD CD t> En >^ lO CO iH H o CO CO O »o lO CD r> 00 CO -^ CO CO CO t^ ■* ^ -*< i-H iH CQ ^ (N t>. >» - crt 3 -Q XJ CD fe fe CM o O ^ += >> d J CO '. >> +3 ■^i s ■§ c p( s s o o " *= 03 "S u h Hi o a 1 Hfe W 1 1 J1 bD c3 ^ 00 O lO CO OS o »o ■* iO t* lO CD iC CO CO CD »o S CO U3 I> I> 00 O CO ■^ CO OS CD CO CD N N CO m Tt* ^ 00 00 t> lO 00 00 CD ■ ,H o CO CO -* CO 00 b- t> CS 00 00 00 o CD iO « CD CD CO c^ 10 a »o CO CS b- CO CO l> l> CO n OS cO Tt< o (N lO Tt< (M (N r-'rft n on tH lO CD »o CD lO t> lO lO lO OS 00 O W ■^ CO CO •* lO ■«*< o CO rH (N o o lO ■* CO I> t> iH eo OS W t> »o CO o to to CO lO o CO CO l> X lO 00 ■* lO co CO to a t> lO CO 00 1-1 lO 10 co »o to CO CD »o o Tjf o ■* CO CO CO CO CO CD ■* ■ t- IC CO i.0 lO rt «3 Ttl lO M « (N 1 CD 10 N O rH ■* CO CD CD rt lO 00 I> CD b- rH O O OS J> 00 •^ X^ ^ lO rfi CO CD lO '# t^ CO l> iM M I-- O I> Tt* S§ ; Oi b- ■ o o b- 00 ■ CD iC o Eh O 03 CD ,-1 CO CO I> CD CD (N lO lO l> CO M CO (N 00 3 % I- ^ 11' 1 Entire county... . Four townships. . Three townships. ; 66 Indiana University Studies jC(H(iqB ■PS -puadaa 0" paadg < s AlIRB s-l -dnapa a B i3 a ^t paadS f^ s ^^HTq^ i-l -puadaa ^ paadS d rnvn^e •43 -puadaa 13 t3 t fl Ua t> at 1 X 00 u i> OS GO N (D 00 « CO CO rH 00 i-l iH ^ t^ (N M I-- l> GO I> CO 01 .H OJ iH ^ CO CO 50 to (0 !0 ■<:J< 1> (N . Oi l> OS ini M »o (N OS -* ^ iO 00 00 ]> CO »o »o M CO t> ^ CO I> CO t> CO < OS CO CO H CO l> CO Iv 01 "* b- «3 00 >^ H l> (N 12; 00 OS U n rH l> Q CO CD i-I N 00 00 l> I> la t> CO CO (N rH rH >> >. !i ;3 -O ^ I^ &^ 4:1 •»:> >) • 0} • ^S^ rt ■§ d pas 8g8 .fe t. .J) p a Hft. H ■ : fj ^§ rf '^3 ^ ^ll 1 ^&S 1 ■^ »o M (N ■* 00 W ^ 00 ffl g CO 00 CO X CO OS CO '^ ;:! OS I> « 6 00 00 01 b- fe 00 (N l> ■^ (^ 01 l> co 00 ^- 00 CO r-l 00 5 co r-l CO 10 g s? i 00 00 8 I-l CO CO CO Tt< •* 00 CO CD OS t^ s CO 1 i § 00 CO 10 ^ ■<3< « I> CO 00 OJ l> t 00 =0 00 05 CD 00 rH rH lO 05 CD 00 00 0> N OS CD 00 10 10 CO « S IV CO to ^» 00 b- CO 00 t^ 00 2 O) rH CO so CO 00 CD CO a a ^ g (N 1-1 tH ^ Oi ^ t> tD t> CO to ifS (N CO rH CO CJ CT b- CO 00 10 © iCi CS] OJ N rH iH ■* rH O CD CO to l> 1> lO oi N C3 ,-( i-\ ^ ^n N 1> M T-t N CO « I> CO CD CO CO b- CO t> M 00 O I> »C b- CQ h3 ^ CD I> O o 0) N 00 n CO r-t Ti* o t> I> 00 03 H 00 1> CD CJ 00 00 HH (4 o »c o Eh lO CO o m S IC l> lO 00 IN t> O M Ol 00 « CO >> >> Fh h ce & S 3 p' t^ J3 -Q (U o fe fe eiH CM o o +3 +^ t»i II '. '. m ■ ca a ill sil oi -^ « •s S £ fi o a Hh&H : ci ■S B 111 1 ^w>: 1 68 Indiana University Studies 50 per cent are working on something else. Theoretically, a system might be made perfect in this manner without changing the median of the group. The progress would show in the form of decreased variabihties. (Variability shows how far the individuals in a distribution are distributed from the median; a decreased variabihty shows that the individuals are clustering more closely to the median.) Such analytical work is now being attempted in Wabash county. The conditions of the entire township are first studied in a township institute; then the teachers with the help of the supervisor take up their individual problems, using the individual score sheets as guides. Care is taken to keep the teachers from overrating the importance of this work. Factors Affecting Achievement in the Fundamentals of Arithmetic The Graded vs. the District School. — The district scores exceed the graded as follows: (1) Wabash county regulars (Table II)' in 8 out of 11 determining cases ;^ (2) Huntington county regulars (Table II), 9 out of 15; (3) Wabash county totals (Table I) in 10 out of 12; (4) those from the sixth and the seventh grade in three townships in Lake county (Table I) in 4 out of 6. The graded exceed the district in the Huntington county totals (Table I) in 8 out of 15 determining cases. These data point to the conclusion that the district schools are more efficient in procuring achievement in the functions measured than the graded. This is probably due to the emphasis placed on the newer lines of work, such as drawing and manual arts, in the graded schools. The fact that Huntington county, having but two teachers in each graded school, has not emphasized the newer hnes so much probably accounts for the exception to the conclusion given in the last statement of the summary of the data. ^Explanation to Table II: The Variabilities are from the totals. The Regulars are those who have made a grade each year. The Retarded are those who have repeated one or more grades, one or more times. The Accelerated are those who have skipped one or more grades so that they are in advance of the classes with which they entered school. The Graded groups are those who have spent more than one-half of their school life in schools with more than one teacher. The District groups are those who have spent more than one-half of their school life in one-teacher schools. "A "case" is the comparison of scores for one function, such as addition in any grade: a marked difference in scores is considered a "determining case" Haggerty; Studies in Arithmetic 69 The Length of Term. — Table II shows that the shorter term regulars exceed those of the longer term in 8 out of 14 determining nases in Wabash county, and in 8 out of 12 in Huntington county. The same explanation applies here as in the previous paragraph. This condition also is probably due to the teachers being "crowded down" to the older subjects in the curriculum. County Superintendents Devricks and Funderburg have both offered further explanations which have had important local application in our inter-township studies.-' The two most im- portant of these are the influences of close supervision and local school interest. Influence of the Quality of Gradation on the Class Score. — The good gradation that results from efficient supervision is effective not only in obtaining better results from the individuals, but may affect the score of a class without influencing the scores of the individuals in the class. Suppose, for instance, that several children whose scores were equal to the fifth grade standard were wrongly placed in the sixth grade. Here these scores would fall below standard and would lower the sixth grade score. Were this incorrect gradation a common thing thruout a school the grade scores would be low. Were the gradation corrected the grade scores would rise. An investigation showed this to account for more than one-half the difference between two sets of scores, in one system tested. This principle probably helps to account for high scores in the eighth grade in some instances, since the passing of the children from that grade is the only place the superintendent gets a hand in gradation in many county systems. Supervision. — Applying the principles developed in the pre- ceding paragraphs to the comparison of the results from Wabash and Huntington counties we find two conditions in each county that tend to make it stand the higher. For Huntington county these are the shorter term and the greater per cent of district schools. For Wabash county they are the better gradation re- sulting from the township supervision system, and the definite standards set for attainment in arithmetic from the first grade up. Reference to Table II shows that the Wabash county regulars exceed in 14 out of 16 determining cases, while Figure I shows that the totals exceed in all 15 determining cases and that it has less variability in 19 out of 29. We may conclude, then, that the 'For the Inter-townshlp study of Wabash county see Wabash County Teachers' Manual, 1915-16, pp. 47-62. 70 Indiana University Studies H o o ffl - lO I.* N t^ t^ ■^ t- (N >0 U5 M « 00 b- CC 05 X W p t^ (D t- £^ « Oi CO th o ■ (N ■ ■ ■ T}1 .^ lO T}1 CD tH (M ^ O ■ CD lO o to I^ o ■ (N ■ ■ ■ T}l 1> lO O PO lO rW lO lO CD Ol (N CD CO O ■ N ■ • ■ CO CO l> I> O ■ lO O 00 CD ■ l> N C Variability. . Regulars. . . . Retarded. . . Accelerated. (N c ■* 1 t^ l> I> o ■* cc CO CO u? o m t> [> O CO t> f- tO O CO l> 00 t^ OS 00 rH b- t* « M H =D lO CO lO CD iH CO CO CO l> O l> l> I> ^ ^ -CO t> I> l> ' Tji CO O »0 »0 U3 1> CO 00 to CO CD w -^ -* CO « N «i -^ ti V d -a ri ? e8 E- mp3 I CD N 1 t- 00 CD CD lO t^ CO CD CO ^ lO O (O CO CO r-! (M O 00 00 w •H CO o b- i> b- OS 00 o X 00 OS lO o o lO CD lO OS O 00 I> 00 b- b- lO N . lO (N CO PM P O CO (N CO K CO CO CD H U »0 OS .-H »0 ■* CD H m HH 00 (N lO Q lO CD lO rH b- ■<* t- CD t^ OS ■<4< -"t CD b- CO - rH O CO 00 00 00 lO U3 "5 M 00 CO t> CD I^ OS T-l W t> CO ■* ed -d d -a tal gul tar 1 E- Wffi 1 Haggerty: Studies in Arithmetic 71 10 m X N c» CD (M CD 00 eo CD in 00 M CO dJ l> CD Tjf CO O 1> N N N O CO I> 00 b- ■* CD co 00 00 00 o N O lO 1> CO I> l> M ^ «■ 00 CD CO m CO 00 CD CO ■<*< ■* lO CD CO lo O CO CD CO CD 1> O CO X CD f-H t> I> 00 N 00 •^ lO o lO lO lO c CD 00 I> CD I> (N CO co OC ■^ CO -d nJ Tl "5 d 0) OD h KM 1 g o 12 ■ lO CO CO (d CD O 10 CD s 00 o o 00 ^^ o In (N O 00 00 10 00 lO CO lO 10 o CD 00 o o 00 Oh != o M w §s lO ■<# b- CO lO o o Ol lO lO lO CO lO -1« lO CD lO O CO CO CO CO N 00 lO 01 CO CD 00 00 CD (N CO lO lO o CD Oi 00 CD CO o 00 00 CO 10 1) c a> ID P3K 72 Indiana University Studies A^inq'B I -pnedaa >■ Q paads i jt^niq'E -pnadaa .9 B .^ paaflS s fl ^TOiqB < ^ -pnadaa pi paads DQ d X'jjiTq's s -puadaa ■tf 1-1 00 l> 00 lO w ■^ ta o ■ CJ n I-t iH 1-1 r-t w4 lO iH OS Tfl 00 iH « t^ 00 (N ^ ■^ »0 CO (N iH <^ 1-1 rH iH CC ^ Ol (M CO CO (N CD CO J^ o o CO o o - C( o OJ O CO tH r-f r-f lO OJ t> cn o ■^ tH I-( CO w Tt< iH O 03 M e; 00 « o J ""I ^ l> 00 OJ (M O- « o o o o O (M .H .-( l-t 7-t t> CT h- CO O lO r- lO lO r- CO OC iH lO o M 00 M 05 lO CO t- to ■ -d r ^1^ (■ a B-E ''i 1 s e t> 'm^< 1 t- O r OJ OJ OJ O O 00 rH CO OS i-l iH W W I> N O t> £N CO ^ rH T-l tH O 00 o 00 t^ 00 CO CO o iH rH iH rH iH iH P3 O ^ CD CD CO CO OJ CO O 00 iH CO fl4 u O 00 M 00 P Q CO 00 O X OJ X b- CD OJ lO t^ 01 O OJ 1-1 (N l> ■* 00 t-- 00 i lO 00 iH o o o I-t 1-1 1-1 CO (M CO CD CO CO CO CO »o 00 K 00 ' CO (N CO 1^ tj 3 ■o %u ! E- MM 1 § gs^ (N (N CO « CO ^ i-( T-t rH 1> i-c CO I> W I> 00 lO CO cq ec rH iH T-l rH O lO 00 H OJ M O l> w H 1> CD b- •* »o o 00 00 OJ 1> 00 1> w o t^ OJ OJ OJ 10 O CO lO CO »o ^ -^ Tj< W 00 w O CO rH IC rH CO si TJ t< V c9 « 311 b PIK 1 Haggerty: Studies in Arithmetic 73 ^ 00 •* © 03 11.9 12.7 10.5 ^g^ 12.9 14.0 11.8 (N CO rH 00 00 00 12. S 14.0 12.0 ■■^ ^ n , © CD to 10.8 11.2 10.0 O tD I^ CD (M (N ■^ CO (D 00 00 00 . IM O O CO C! CO CO M 00 to to CO 00 CO t- Gi O 05 m o 00 i> t> r- 10.4 10.6 10.0 ■* o »c lO lO lO ira ^ o 00 00 Cl ■# lo r^ lO (M (M c E- SB S3K U2 W N 11 SJ OJ 00 CO 1> o M M (N iH I-i rH O O O W (N iN J-i T^ T^ iH CO O 00 00 r- 1 o o o (M O O N o cn o 00 a> 00 tS O 10 O rH O lO 00 CO to to to '^ to o CO 00 X t> Its o CO ^ (M w' -d p3 tI _; -13 t- cfl P d e5 M II 74 Indiana University Studies ID a a g •ffl (3 -pnadea pagdg 1 •3 1 -pnadaa paadS I s 1 -pnedaa psads i -pnaflaa paadg JO jsquin^ a a o 1 ■ s -pnadaa paadg § 1 1 xiniq'B -puadea paads o CO -pnadaa paads o i < -pnadao: paads siidiid: JO jsqinti^ to r-l h- w o »o « »0 lO 1> ■* OJ CO (N O • « -* CO M lO lO N '■* lO CD O IM 00 r-l O CD lO CD 00 ■* ■* lO CO (N CD CD !> O ffl a W O ■ iH I> CO CO Ol r* ^ lO 1> Ol ^ rt lO T*i CO ■<* o ■Til CO CO • IM CO CO • lo CO 00 0> ■ t<5 »o o o ^ r^ CO m ^ rH ^ ■* CO J rH t> o t^ »0 CD CO 00 M - iH CO CO CO CD n OS CO (N U5 lO U3 CD O t^ l> O CO CO CO CD ^ • o (N (N 1-4 • 1-1 ■ ^ ■■^ m ■s^ a I"? S 1 •3 cS (S 'S e >»»< 1 m t-- a> lO lO CO o o o -* -^ ^ CO CO r- CO o o w o CO CD CO t^ O (N 00 iH tn Pm P O c> o o n IT o o p" >o CO < w o Cf o CO ■^ CO 0- o o ir CO o l> r> o cc IT o »o IT t^ lO C^ o o ir »o If o o cc l> l> ■* lO b- -d ■a 3 J3 c cd ^g O lO CD CO IN CO CO (N ^ O OS ■* CO 5 CO CO OJ CO ^g 3 « 00 U3 lO (N CD 3 CO 00 lO CD CD CO CO to CD CD CD' 00 cq 01 si ■^ "H Haggerty: Studies in Arithmetic 75 ss lO M •* (N Tfl O IC -* CD O <0 "O (0 lO CO 10 O CO (O N <£) OJ lO O CD to l> w to lO to lO o CO to 1> CO P O « 00 CO lO CO r-« ^ sss E-i O iC i> »o « CO « ^ (NOW -^ CO CO CO O M lO CO CD §S3 ■* O CO (0 CO CD ^ O O lO lO to CO O CO CO 00 CO r- OS 00 Tt* (N 1 c CO "D «P3 i-H lo in I (O CD lO lO 00 r-( ■* ■* -^ CO o o lO lO "O CO W M CO CD to CO CD CD (M « 00 I> l> CD lO o to ■(}< lO T)< -* CO o CO CD CD CO Ph r^ 00 O ■* CO ^ O P3 00 ■* lO M H ^ 01 lO lO O CO ■* CO S 4. CD O IC ■* CD ■* ciL OS O O IC CO CO CD I> lO 00 O 00 CO N CO (N ■* tH lO ■* lO ■^ N lO CO CD CD CO i-l W oi -d C<3 T) ■jsl O a> a> F P3P5 76 Indiana University Studies 1 1 1 00 s ■ a < a o '> '■3 5 -puadaa GO < O : O W (N (M in » IM GO X CO o rH »o o w 00 00 OS OT «■ ■* I> 00 1> O '43 o S -pnadaa X «D O « O (O (M I> CO Ol -4 =^ 9 ^ I> CO « ■i w Ph O P5 o o M ai t-t Q OS O ■* CO I> CD paads CD rt m CO o • (N ■ ■ ■ M Oi 00 O (rt lo »o OS OS GS CD ■* 00 00 oi 1> 1 o S CO -puadQQ W M -"^f 01 O |> ,-H 1> CO 00 00 m i> I> 00 t^ OS CO CD ! O l> CO ' paads M l> CM « lO Oi O 00 ^ 01 lO 00 O OS o tH 1— I OS CO I> CS O 00 o ■3 ■a < -pnadaa CO ■^ in OS o CD CM CD lO t> e in 00 i-H CO in CD CO "O paads l> 1>' OS lO o N "^ l> t^ r-( I> l> CO t> !> I> O OS -^ t^ t-, l> sndnj JO jaqninN; 00 ■ Oi ■* »o OS ■ -^ -rtJ 00 00 OS 1-4 CO 05 CO IN CO CO H Q S o f s -puadaa CO OS M ^ O b- IM 00 CO O CO CO o 00 00 i> co o o (D CD lO (N CO ■* l> 00 CO paadg lO iH CM 01 »ra . CO ■ ■ ■ a CD I> lO CO CO 00 en CD i> in ■B 1 Pi "43 ■3 -puadaa ^ 00 O lO ITS CO (N l> lO 00 ■^ (TO rH CD l> -^ 00 o t^ in CO in paads CO CC IM ^ lO t- 00 t^ CO o o o t^ t> l> CD CO o t- 00 I> 1 1 -puadaa OS PO m i-( O CO CM CD 1> t'- ■# l> o CO CO CO rH O N 1> f- N paads t> 00 CO »o O ■ .H ■ ■ ■ DO OS 00 ifl o o o 00 00 00 t> ^ rt< 00 OS X O jtdliqu -puadaa ,-( ira ^ rH lO CO (N CD CD CD 00 O IN lO CO lO §g§ paads I> OS iH P3 lO 1> 00 I> CD t> in o I> E-^ 00 l> CM O I> X N sijdnd: JO jaquin>j r^ ' iO a m 0)0 CO CO l> X CO ■* Total Variability. . Regulars. . . . Retarded. . . Accelerated. Total Regulars.. . . Retarded. . . c CO "a IB l> 00 io CO a oi d OS i-i CO 00 00 1> CO ■<*< 00 o O) Oi 00 (M 00 1> CD iCi CO CO N o r^ N i> to CO O rH r-1 00 l> 00 to 00 CO iC to i> to 00 O 00 ira to ic 00 CO I> I> 00 IN i MOO I> l> I> 05 O 00 00 O 00 lO t> lO CD to to CD 00 l> r- N o ; ■* CM tM c E- Regulars.. . . Retarded. . . GO 00 CO (M M O CO CO 00 CO j> to lO o o 00 05 CO to C Tj* CD [> to (N lO U3 O O 05 CD r> 113 00 o o r* CO N E2 o M i> 00 CO W H ^ lO lO 00 O CD t^ lO w Hln CO l> lO ti 1 (M O 00 I> 00 to ^ c^ o 00 0= 00 CO lO (N CO CD CO CO CO 05 t^ CO CO ^0 05 iC (M C^ m" "^ 1 "rt !^ te ^ ^pi; 11 78 Indiana University Studies lessening of emphasis on fundamentals attendant on lengthening the term and increasing the teaching staff is more than offset by the improved gradation and specific standards that accompany supervision. The Achievement op the Retarded A comparison of the scores of the regulars and retarded in Wabash and Huntington counties shows that in Wabash county the regulars exceed in 7 out of 13 determining cases, and in Huntington county in 14 out of 15. Following up this point, TABLE III— COMPARISON OP PROFICIENCY IN FUNDAMENTAL PROCESSES OF ARITHMETIC WITH RETARDATION A = Addition; S = Subtraction; M = Multiplication; D = Division. WABASH COUNTY Grade 5 Grade 6 Grade 7 Grade S Average Retardation in Years Leading in Courtis ' Shorter term group.. . . Longer term group. . . . Shorter term group.. . . Neither Longer term group. . . . 1.9 1.3 D A S M 1.3 1.6 A S M D 1.4 1.5 S M D A 1.2 1.2 A D SM HUNTINGTON DOUNTY Average Retardation / Shorter term group.. . . in Years \ Longer term group. . . . Shorter terTn ffrmin. 1.4 1.4 A S M D 1.5 1.3 1.7 1.4 1.5 1.1 Leading in Courtis Tests Neither , Longer term group. . . . S AMD ASM D A S M D Table III shows that in 2 out of 3 determining cases in Wabash county and in all 3 in Huntington county there is positive corre- lation between amount of retardation and lack of proficiency in the fundamental processes of arithmetic. Considering the fact that 69 per cent of those who failed in Wabash county in 1915 eighth-grade diploma examinations, and 87 per cent of those in Huntington county fell below 60 per cent in arithmetic, is there not some evidence towards a con- clusion that lack df proficiency in the fundamentals has been a cause of poor arithmetic work and thus of retardation? At least there is a relation between these in the cases studied. This may, however, be the result of factors affecting similarly the work in both general arithmetic and fundamentals. III. THE EFFECTS OF SIX WEEKS' DAILY DRILL IN ADDITION By Maky a. Kerr, Principal of Department School, Bloomington. Near the last of February, 1915, in the third week of the new semester, the Courtis Test, Series B, was given thruout the Bloomington schools. The three arithmetic teachers in the Department school (which comprises all the children in grades six, seven, and eight in the city), two of whom had had no previous experience with these tests, scored all the sixth, seventh, and eighth grade papers and tabulated the results. The papers of Teacher A were scored by Teacher B, those of Teacher B by Teacher C, and those of Teacher C by Teacher A. The results were tabulated by sections and then by grades. Table I gives the median scores by half-years. TABLE I— MEDIAN SCORES (February, 1915) Num-^ ber ' in Grade Addition Subtraction Multiplication Division Grade 2 a S si bo ii 1^ a < S Is " IS 1 Rigllts Depend- abiUty p. 1 (6 " 6B 6A 7B 7A SB 8A 83 83 63 67 as 62 8.7 9 9.7 9.8 11.4 11.5 5.5 5.3 6.6 6 6.9 6.3 64 59 60 62 61 55 8 9.1 9.7 10.4 11 13 5.5 7.1 7.6 7.6 8.8 10.3 69 79 79 73 80 79 6.7 7 8 8.5 9 10.1 4.3 4.6 6 5.5 6 6.2 66 66 63 65 67 62 6 6.6 7.4 8.8 9.4 10.4 4.6 6.3 6 7.1 8.2 8.9 76 81 81 81 87 86 The results revealed several things that needed to be remedied, but after considerable study and comparison it was felt that addition was the fundamental operation in which our children showed the least growth and dependabihty from grade to grade. Note that the gain in examples right from grade 6B to grade 8A is only .8 of an example. The 6A median, the lowest, is 5.3; the SB, the highest, is 6.9; a difference of only 1.6 examples. The range in dependability is from 55 per cent to 64 per cent and lowest in grade 8A. 80 Indiana University Studies It was decided to try a systematic drill in addition. Each pupil was given his scores in all of the fundamental operations and under his teacher's direction compared his individual scores with those of his section, his grade, and the Courtis Standard for his grade. All this was done to reveal to him his own condition and to get him into the proper attitude to help himself. The plan of drill decided on was as follows: 1. Time. — The first 5 minutes of each recitation period. The drill period covered the last 6 weeks of the semester. 2. Stimulus. — Accuracy. Each day the children were started off on the drill with the instruction, "Get as many right answers as possible today." Teachers and pupils kept daily records of scores made. The teachers frequently advised those who seemed not to be making proper progress. They suggested to different pupils what they felt those pupils needed in order to improve. These suggestions were based on observation of the child's procedure during the drill and on the daily records kept. The children themselves were often able to locate their own peculiar difficulties and were anxious to remedy them. 3. Examples. — Mimeographed examples in addition beginning with 5-figure columns were used at first. The number of columns remained 3 thruout the drill, but the number of figures per column was increased one per week up to 9 figures. The number of ex- amples for each drill exercise was 5. As a further incentive pupils who 3 days out of 5 had 100 per cent accuracy were to come to the assembly room on Friday for a contest with all others who had attained similar scores. This requirement was raised to 4 out of 5 and of the last two weeks 5 out of 5 correctly solved. This drill continued for 6 weeks or until the close of the year, when the Courtis Test, Series B, was again given. Table II shows the results. The 423 children in Table I who took the February test and the 6 weeks' drill are the 423 in Table II. Only those who were in for both tests and the drill are included in any of the tables. Haggerty: Studies in Arithmetic TABLE 11.— MEDIAN SCORES (June, 1915) 81 Grade 6B.... 6A. . . . 7B.... 7A. . . . 8B.... 8A. . . . Total.. Addition Num- Grade « IQ^S Subtraction M Ifl'g Multiplication ■a a ^ a3 Division 83 83 63 67 65 62 423 9.7 6.9 72 8.1 6 74 6.6 4.4 67 10.5 7.5 71 9.1 7.2 79 7.4 4.7 64 10.8 8.1 75 9.5 7.6 80 7.8 4.7 61 11.8 8.4 71 10.1 7.2 71 8.7 5.4 62 12. 9.3 78 10.8 8.3 77 9.4 6.2 66 13.7 10.4 76 11.9 9.7 82 10.2 6.7 66 5.8 6.7 7.3 8.8 9.2 11.1 M 4.3 5.4 6.4 7.5 8. 9.8 &3 • 75 81 88 85 87 89 The change from February to June in median scores and dependability is shown in Table III and gra.phically in Figure I. Note the growth from grade to grade in examples right from 6.9 in 6B grade to 10.4 in grade 8A, a gain of 3.5 examples. Note also that 6.9 was the highest score made in February and that by the 8B grade. In June 6.9 was the lowest score and was made by 6B. In accuracy all classes were above 70 per cent, and the gain from February to June ranged from 8 per cent in the 6B grade to 21 per cent in the 8A grade. TABLE III.— MEDIAN SCORES: ADDITION (Febkuart and Jtinb) Grade 6B. Grade 6A. Grade 7B. Grade 7A. Grade 8B. Grade 8A. J February ■ ' ' [June fPebruary ' ■ \jiine /February ■ ■ * \june /February ' ' ■ \ June ^February ■ ■ ■ \June /February ! 11.5 • ■ ■ \June 1 13 . 7 6—4908 82 Indiana University Studies _JUNC OCPUtDAtMUTY _^ June ATJ — rtB Arj- r ;: K 3 a 7 ___- JuaeKtj- Teb Etj- 1 6B SB SA 7B 7/\ Fig. 1. Scores in Attempts, Rights, and Dependability in Addition FOR Both Febhijary and June. The score on the ordinate indicates the number of problems; the symbols along the abscissa indicate the successive grades from 6B to 8A. Together with the gain in median scores from February to June, we must consider the change in variability. If this gain has been accompanied by an increased per cent of variabihty its value is questionable. Table IV shows what happened. In attempts the variability in grades 6B and 7A shows no change. In grades 6A, 7B, and 8A there is less variabihty. In 8B there is a slightly greater variability. But note what happened in dependability. Haggerty: Studies in Arithmetic 83 In every grade the variability was lessened, showing that the children composing the group were much better graded in June than in February so far as this single ability is concerned. TABLE IV.— VARIABILITY IN ADDITION, BY PER CENTS (Febbuaky and June, 1915) Pbbeuaby June Attempts Rights Attempts Eights Grade 6B 22 21 26 18 21 21 27 21 23 30 28 20 22 IS 17 18 22 18 20 20 Grade 6A Grade 7B Grade 7A ''O Grade SB 17 Grade 8A The range of variability in attempts and rights in the 20 Indiana cities in 1914 ^ is presented in Table V. Bloomington's variability is included for comparison. TABLE v.— VARIABILITY IN ADDITION: BLOOMINGTON COM- PARED WITH TWENTY INDIANA CITIES (May, 1914) Variability IN Attempts Vabiabilitt in Rights 20 Indiana Cities Bloomington 20 Indiana Cities Bloomington Lowest Highest February June Lowest Highest February June Grade 6 22 31 6B 22 6A21 22 15 42 68 • 6B27 6A21 20 20 Grade 7 22 30 7B 26 7A 18 17 18 35 J f 60 -1 7B 23 7A30 16 20 Grade 8 2Q 32 8B21 8A21 22 18 32 c 63 8B 28 8A20 17 15 ' M. B. Haggerty — Arithmetic: A Co-operative Study in Educational Measurements (.Indiana University Studies, No. ^7). 84 Indiana University Studies TABLE VI.— MEDIAN SCORES IN ADDITION: BLOOMINGTON COMPARED WITH TWENTY INDIANA CITIES (May, 1914i Accuracy Sixth Grade — Highest Median of Twenty Indiana Cities Bloomlngton Median, 6B Bloomington Median, 6A Seventh Grade — Highest Median of Twenty Indiana Cities Bloomlngton Median, 7B Bloomington Median, 7A Eighth Grade — Highest Median of Twenty Indiana Cities. Bloomington Median, 8B Bloomington Median, 8A Attempts Bights 8.9 5.6 9.7 6.9 10.5 7.5 9.4 6.4 10.8 8.1 11.8 8.4 10.3 7.2 12 9.3 13.7 10.4 65 72 71 68 75 71 69 78 76 A comparison of Table V with Table VI reveals the following facts : 1. Bloomington's medians in speed, rights, and dependability are higher than the highest median of the twenty Indiana cities. 2. Bloomington's variability in speed is less in most grades and in no grade greater, while in rights in every grade the varia- bility is lower, than any of the twenty Indiana cities. Figure II, which is a copy of the Indiana Standard (1914), presents graphically Bloomington's position on the scale in both February and June. Now from all of these facts and comparisons are we not justified in stating as our first conclusion that the drill had the desired effect? You must so conclude if desired effect is to be determined by higher medians and less variability. Just here, however, we find ourselves raising several questions which seem worth answering so far at least as any of the data we have can answer them. First, was there any transfer of increased abihty in addition to the other fundamental operations? Figure III presents the conditions "before and after taking" in all of the fundamentals. At a glance, it can be seen that the drill in addition produced in- creased skill in addition and in addition only. Note that in the other operations the lines are almost coincident. We conclude, therefore, that the drill in addition did not affect skill in the other fundamental operations. Haggerty: Studies in Arithmetic 85 y it?.", " /IS ?: S 2 ' i a s 2^sf* f f- * * * * ?i S_ ?;^^4s4*S2?! = S»-i'-5.RtAiS b: ^1 !. i 1 ij,8>| ^>^ J, 4 s ^ s-^- l< Sjl^l is !ii = iS.'t'!it*?. S o: 1 C 1 * • 2 5 = ^i \^\/' f- J ' ? f. fi S(^ ?, ?l '; ?, is "' ^ ?! 3 3 2 * 1 S- J, f. S !-. !^ Si f. 1 of ^' ' .^~T-^>-. -^". '1. .' . '. .' . ' . ' •* bI 6|?iaag3~if."!>!-nf. i I ,- .\ -.' t'i^ ^^, \ :'.'•, ^'^ 1 i 1 4 2 = li s 1 i'- s !i !r ^ ;> !■ < a: X 1, : .- . ', ' ', ,^'^^ ! /. ': /,s i< &^2S121! 32 e-« f--a tr.rrrtrj- 1 5 1 ^ 4 i 1 s. ii.'Y" 1 !. J ^ ' 'vi lAifjIJ lf,RS:|{,l|!.Sf,S.J,!, I 1^|| J -'V^r^J- V 1 • S S ' < a * * * >■ "!■ 5 S, 5 f- S !, i J, ^ i ^ J ■ , ■ "■ >^l l\ ■ f ." , \ o 'Rsssif-iAin;, 1 || ^ 4 ^ i |iw^, * ^. ^ ^ < 5 i< s — 1 55 = I5-1S--S i -7 >' '■ - i5 t h i ^ . ;^j V " ,', : .'■ -.«> ErS 2S - SS2**^* ■" -rr. tl- fT- ^ li * ^*r^^-^ y ', 1 ,' ' ' '' < ' — 1 s- ,j 5 s J i 1!, i J, !, '. " - a r\ - ' ^ , « - o ". 1 '-K £ S. i « J ?. * - '^ i 1 K 5 i i| S >• '- ^ i< ' R ;- 1 JSf-'l !.•?.!.■ ^ k i i. i \^ f' '^ •- •' U) — S — S~ S5|*sj.i !.?■?■- {\ T. 5 A * 1 ?> - - < ii I— I . CL 86 Indiana University Studies 65 Fig. 3. Scores in Attempts and Rights for February and June in all Half-grades from 6B to 8A. The lowest quadrangle is addition; the second, subtraction; the third, multiplication; and the fourth, division. Haggerty: Studies in Arithmetic 87 A second question, and a very important one is, who were benefited by the drill? Was everyone? Courtis tells us that "Under existing methods of uniform training for all members of a class, the children whose natural aptitudes are in accord with the particular form of practice used, profit quickly and largely by the practice. Appropriate tests show that the number of such children does not exceed one-third of the membership of the average class and is usually much less. All the other children, however, either fail to profit by the practice or are positively injured by it. And again, "Recent studies yield some idea of the degree of success possible under average classroom conditions. One-third of children will respond readily to any systematic training: a second third may be easily reached by the teacher able to diagnose individual needs and ingenious enough to adjust the work accord- ingly. The remaining third represent those heavily handicapped by nature."^ Let us see how far our results bear out the conclusions of Mr. Courtis (Table VII) . If one-third of the pupils will respond readily to any systematic training, and a second third may be reached by the expert teacher, then we should have effected a desirable change in somewhat less than two-thirds of our pupils. Altho no one of the arithmetic teachers claimed to be an expert, two of them did try to discover causes of individual failures in their classes and to suggest to pupils devices for improvement. Always, the teacher's effort was to convince a pupil that he could remedy his own defects if he really wished to do so. Table VII shows that 66 per cent of the 423 gained in dependability with an average gain of 27 per cent. Thirty per cent lost in dependability with an average loss of 19 per cent, and 4 per cent made no change. Of the 15 pupils who made up 'this 4 per cent, 5 pupils had 100 per cent on both tests. Three were 80 per cent or above, 1 was per cent, 1 was 14 per cent, 1 was 33 per cent, 2 were 50 per cent, 1 was 67 per cent, and 1 was 70 per cent. In number of examples right 70 per cent made a gain with an average of 3.6 examples, 22J per cent lost with an average loss of 2 examples, and 7i per cent made no change. Are the results from such a drill as this a fair means of judg- ing teachers? The conditions of the drill were uniform in all grades and with all the teachers. Teacher A had 132 pupils in 2S. A. Courtis. — Teachers' Manual for Standard Practice Tests, p. 4. 88 Indiana University Studies 8 sections from 6B to 8B inclusive. Teacher B had 122 pupils in 7 sections from 6B to 8A inclusive. Teacher C had 169 pupils in 8 sections from 6B to 8A inclusive. Each teacher had about the same number of weak and strong pupils. Are we justified in rating these teachers according to the results they succeeded in, attaining with their pupils? I believe we are. Per cent of Pupils Making Gain in Dependability and Rights Accuracy. Rights. Teacher A 60 63 Teacher B 67 75 Teacher C 69 75 Teacher B and Teacher C, it is clear, were able to arouse about the same per cent of their pupils to make better scores and a larger per cent than Teacher A. This finding is in perfect accord with the rating of these three teachers by Professor H. G. Childs, based on the records of three observers who, independent of each other, and at different times, visited the regular classroom work. Mr. Childs also correlated these ratings with the gross change in results obtained by the pupils of each of the teachers, from February to June. Teacher C ranked first and Teacher B a close second, but there was a wide gap between these two and Teacher A. Haggerty: Studies in Arithmetic 89 H pq m O D pq < H Q pq g M A 3 !^ f* " o C o" 5§ n fl nco be O - t-< cd 60 !^ 'S ® " M T3 B " O B o Si t> (0 n .3 ^1 (D Per cent los- ing OS W OS ■* O 05 N N N rH (N (N r-l n Num- ber los- ing ■* OS N CD CO (N CD N l-l iH rH T-t -H 01 2 1 Total exam- ples gained 136 194 159 178 206 207 Av.3.6 Per cent gain- ing OS !> CO t> O C U3 CD IN CD « 3 O t- 1 Num- ber gain- ing OS CD CO »0 < B O o Per cent mak- ing no change in ^ CO CO p 5 1< Num- ber mak- ing no change •^ m c^ -^ £> 1 »o to 3 Total points lost CD (N CO >H U3 t^ »o CO CO n CO lO CO «3 M s OS . iH i-H (N o -^ e> cc § t— 1 fl «3 P9 S o o O r-l -* §1 o CO >> o n (N o o o .H H ^ 8 o o c N V CN rH P Q s 5;i t, OS CO 03 ■* M O He IS s H 'a S (M >. M !0 aa 00 Oi U3 ffl ■^ N ■7H »o H m c5 :? ^ o 02 1 a < )—\ 1— 1 t— I > c -t^ fC <{ ff < n < re t "5 s CC CD l> r- 00 a E- ^ i ■? (D a ■a -z t- ^ E- (If 1 £ g £ H ■ C e c c C C Haggerty: Studies in Arithmetic 91 SUMMARY OF TABLE VIII Pabt I Fifty-six per cent of the boys gained in dependability and in number of examples right. Sixty-five per cent of the girls gained in dependability and in number of examples right. Sixty per cent of the 423 boys and girls gained in dependability and in number of examples right. One and one-half per cent of the boys gained in dependabiltiy and lost in number of examples right. Two and one-half per cent of the girls gained in dependability and lost in number of examples right. Two per cent of the 423 boys and girls gained in dependability and lost in number of examples right. Two and one-half per cent of the boys gained in dependability and had same numbers of examples right. Two per cent of the girls gained in dependability and had same number of examples right. Two per cent of the 423 boys and girls gained in dependability and had same number of examples right. Part II Five and one-half per cent of the boys lost in dependabihty and gained in number of examples right. Bight per cent of the girls lost in dependability and gained in number of examples right. Seven per cent of the 423 boys and girls lost in dependability and gained in number of examples right. Twenty-five per cent of boys lost in dependability and lost in number of examples right. Seventeen and one-half per cent of girls lost in dependability and lost in examples right. Twenty-one per cent of the 423 boys and girls lost in dependabihty and lost in number of examples right. Four and one-half per cent of boys lost in dependability and had same number of examples right. Two and one-half per cent of girls lost in dependability and had same number of examples right. Four per cent of the 423 boys and girls lost in dependabihty and had same number of examples right. Part III Two per cent of boys had same dependability but gained in number of examples ri^ht. One per cent of girls had same dependability but gained in number of ex- amples right. One and one-half per cent of the 423 boys and girls had same dependabihty but gained in number of examples right. 92 Indiana University Studies One per cent of boys had same dependability but lost in number of examples right. One-half per cent of girls had same dependability but lost in number of examples right. One per cent of the 423 boys and girls had same dependability but lost in number of examples right. Two per cent of boys had same dependability and same number of examples right. One per cent of girls had same dependability and same number of examples right. One and one-half per cent of the 423 boys and girls had same dependability and same number of examples right. Next we were interested in seeing what had been the effect of placing the stress on accuracy. In the summary of Table VIII we find in Part I that 60 per cent of the 423 pupils gained in de- pendability and in number of examples right at the same time. This is a typical case: February score: 8 attempts; 5 rights; 62 per cent accuracy. June score: 9 attempts; 7 rights; 77 per cent accuracy. Two per cent gained in dependability and lost in number of examples right. For example: February score: 8 attempts; 5 rights; 62 per cent accuracy. June score: 6 attempts; 4 rights; 67 per cent accuracy. Two per cent gained in dependability and had the same number of examples right. For example: February score: 8 attempts; 5 rights; 62 per cent accuracy. June score: 7 attempts; 5 rights; 71 per cent accuracy. This 64 per cent of the 423 pupils, we feel safe in saying, were benefited by the drill. In Part II of the summary we note that: (1) Seven per cent of the 423 pupils lost in dependability but gained in number of examples right. Here is a typical score of this sort: February score: 8 attempts; 6 rights; 75 per cent dependability. June score: 10 attempts; 7 rights; 70 per cent dependabiUty. (2) Twenty-one per cent lost in dependability and in number of examples right. For example: February score: 8 attempts; 6 rights; 75 per cent dependability. June score: 8 attempts; 5 rights; 62 per cent dependability. (3) Four per cent lost in dependability and had s^ame number of examples right. For example: February score: 8 attempts; 6 rights; 75 per cent dependabihty. June score: 9 attempts; 6 rights; 67 per cent dependability. Haggerty: Studies in Arithmetic 93 This 32 per cent evidently were not benefited by the drill. We have not found any satisfactory explanation for all of this but we have had raised for us some interesting problems, at which we hope to work. In this group we have such scores as this: February: 11 attempts; 11 rights; 100 per cent dependability. June: 13 attempts; 10 rights; 77 per cent dependability. Should a pupil making such score on the February test have been subjected to the drill? We think not. This score was made by a 6B girl, and showed standard abiUty in speed for grade seven (Courtis Standard) and 100 per cent accuracy. Here are the scores of another 6B pupil: February: 5 attempts; 5 rights; 100 per cent dependability. June: 5 attempts; 1 right; 20 per cent depeudabiUty. Would this lad have been benefited more by a speed stimulus ? We had scores like this: February: 3 attempts; 2 rights; 67 per cent dependability. June: 3 attempts; rights; per cent dependability. The above is the score of a 6B boy who belongs in the class "heavily handicapped by nature." We find many scores like this: February: 8 attempts; 7 rights; 87 per cent dependabiUty. June: 9 attempts; 7 rights; 78 per cent dependabihty. Here, while the figures show a drop in dependabiUty, we feel that the girl has not been harmed by the drill. The fact that 60 per cent gained in dependabihty as they gained in speed and that 21 per cent lost in dependabihty as they lost in speed leads us to feel that these two stimuli are inseparably related, but in different ways for different children. We should like to know what would have been the effect on the 21 per cent if we had used the speed stimulus instead of the accuracy stimulus. 94 Indiana University Studies TABLE IX.— DEPENDABILITY BY QUARTILES, GRADE 6B A DlSTRIBtTTION OP THE 83 6B PuPILS INTO QUABTILES, BaSED ON DE- PENDABILITY AS Shown in the February Test, and the Change in Dependability in These Same Groups as Shown by the June Test. Lower Quartile Lower Half Upper Quartile Upper Half ^ , . fFebruarv 19.5 43 ■ 35.5 54.8 89.5 77.5 80 Average Dependability J ^ «"i ud,i y 73.6 Gain Percent Making •! Loss 72 19 9 74 21 5 28.5 62.5 9 44 51 No change 5 For those who made /(I) Total gains made. . gains \(2) Average gain 571 38 960 30 75 12.5 273 15.1 For those who made fCD Total loss made.. . . losses \(2) Average loss 70 17.5 149 21.3 367 28 547 26 For those who made no change, scores were . . < 33 33 100 100 100 100 Did THE Pupils Who Needed to Gain Profit by This Drill? Table IX is for the 6B grade and for dependability only. What is true of this grade is true of all grades. The 83 pupils were arrayed according to their dependability scores made in February test, beginning with 100 per cent and ranging down to 0. The median was determined, as well as the upper and lower quartiles. Then the score made by each pupil in June was placed just opposite his February score and the change computed. Notice what happened to the lower quartile. The average dependability of the lower quartile in February was 19.5 per cent, in June it was 43 per cent. Seventy-two per cent of the lower quartile gained in dependability with an average gain of 38 per cent each. Nineteen per cent lost in dependability with an average loss of 17.5 per cent. Nine per cent or 2 pupils made no change from February to June. The score of one was 0, of the other 33 per cent. The pupils whose scores fell into this lower quartile were undoubtedly in greatest need of improvement. Note that 72 per cent did make an average gain of 38 per cent. The 19 per cent who lost in dependability and the 9 per cent who made no change we found were 6 boys, 2 of whom at least were "border- line" cases. The average age of the 6 boys was 13- years and 3 Haggerty: Studies in Arithmetic 95 months, this being more than one year older than the average for the grade. The lower half shows that 74 per cent made a gain, that the average gain was 30 per cent, and that the average of the group was raised from 35.5 per cent to 54.8 per cent. The average dependability of the upper quartile in February was 89.5 per cent and in June was 77.5 per cent. Eight and one- half per cent made a gain, 62.5 per cent lost with an average loss of 28 per cent. To the 285 per cent who made a gain we may add the 9 per cent who made no change, since their scores were 100 per cent on both tests. This 37J per cent were benefited by the drill, but what about the 62^ per cent? To many of these the loss was due to overspeeding. For example, such a score as this: February: 7 attempts; 7 rights; dependability 100 per cent. June: 8 attempts; 7 rights; dependability 87 per cent. In some cases there was no change in number of attempts but a loss of one in rights. As for example: February: 11 attempts; 10 rights; 90 per cent dependability. June: 11 attempts; 9 rights; 82 per cent dependability. These cases do not represent serious losses, but this can hardly be said of such a case as this: February: 5 attempts; 5 rights; 100 per cent dependability. June: 5 attempts; 1 right; 20 per cent dependability. Here is a clear case of over-speeding: February: 11 attempts; 10 rights; 91 per cent dependability. June: 15 attempts; 10 rights; 67 per cent dependability. We admit frankly that we do not know how to account for the losses, but have merely attempted to indicate some probable reasons for them. IV. AN EXPERIMENTAL STUDY OF THE EFFECTS OF DRILL IN ARITHMETICAL PROCESSES UNDER VARYING CONDITIONS By Herman Wimmeb, SupBrintendent of Schools, Roohelle, 111. Introduction. — This study of the conditions and effects of drill was made with the pupils of the fifth, sixth, seventh, and eighth grades at Bremen, Ind. These grades averaged about thirty-five pupils each, except the sixth grade, which had forty-four pupils. The sixth grade was divided for this work into two groups equal in number and apparently equal in abiUty. The two sections of .this grade were designated as 6E and 6W. As a measure of efficiency the Courtis Standard Test, Series A, was used. Each grade was given the first test at the beginning of the twelve weeks of work. At the end of six weeks the second test was given to all, and at the end of twelve weeks all were given the third test. The papers of all pupils who missed any one of the three tests were discarded. All of the tests were given by the writer and all of the papers were scored by him or under his direction. The arrangement of the drill for the several grades is shown in Table I. The time for drill was always subtracted from the regular class time devoted to arithmetic. TABLE I.— SCHEME OF DRILL First six weeks Second. six weeks Grade 5 5 minutes daily 5 minutes daily. Grade 6E 5 minutes daily; J reasoning, f fundamentals. Same as first six weeks. Speed emphasized. Grade 6W 15 once per week, f to reasoning, 1 to fundamentals. Same as 6E. Accuracy empha- sized. Grade 7 No drill 5 minutes daily on reasoning. No drill 5 minutes dally on fundamentals. The chief results are shown in Tables II to IV and in the more analytic tables later on. In these tables the number of problems solved is not given in any case. All figures used are per cents. These per cents are the gains made by the pupils over the grades made on the similar tests taken six weeks earlier. Haggerty: Studies in Arithmetic 97 All per cents are based on the average of the group considered. In part of the tables both "attempts" and "rights" are con- sidered and tabulated separately for test six which is made up of one-step reasoning problems, test seven which is made up of longer examples in the fundamentals, and test eight in which the work is two-step reasoning. Courtis states that it is not necessary in tests one to five, inclusive, which deal with very small numbers used in the fundamental operations and the copying of figures, to compute the number of examples that are right, inasmuch as the child's ability is shown just as adequately by his speed. TABLE II.— PER CENTS GAINED IN THE SECOND TEST BY EACH CLASS OVER THE SCORES MADE ON FIRST TEST Items of Test Grade Grade Grade Grade 5 6E 6W 7 4.7 10.4 11.6 11.1 6.6 25.0 22.8 11.4 3.1 11.9 24.5 12.1 63.4 25.0 54.9 11.4 36.9 20.4 43.3 11.1 -1.0 43.3 48.2 12.1 48.6 82.2 72.0 16.3 12.9 11.9 5.0 11.2 3.9 3.8 13.0 11.3 42.2 74.6 81.6 17.8 3.8 68.6 132.1 16.6 Grade 8 1. Addition 2. Subtraction 3. Multiplication. . 4. Division 5. Copying figures ^ /Attempts JEights _ fAttempts (Eights „ /Attempts \Eiglits 11.2 11.1 10.9 11.1 10.8 12.1 13.5 10.6 12.0 16-. 8 95.0 TABLE III.— GAm IN PER CENTS SHOWN BY AVERAGES OP SCORES MADE IN THIRD SERIES OP TESTS OVER AVERAGES MADE IN SECOND SERIES Items of Test Grade 6E Grade 6W Grade 7 Grade 8 1. Addition 2. Subtraction. . . . 3. Multiplication. . 4. Division 5. Copying figures, _ fAttempts \Eiglits. J fAttempts ■ iRights „ /Attempts lEights 9.3 14.9 8.4 18.4 — .8 20.7 18.8 6.6 19.6 1.4 5.1 3.7 6.2 18.2 7.4 .0 22.6 16.0 2.7 3.3 11.7 5.1 1.7 3.2 —2.6 7.4 —3.9 18.3 25.3 2.7 15.4 20.0 12.8 11.5 11.7 14.4 15.5 17.2 6.0 3.5 8.5 18.4 —5.6 —5.1 98 Indiana University Studies TABLE IV.— AVERAGE GAINS MADE IN SECOND TESTS OVER FIRST (UPPER LINES OF FIGURES). GAINS MADE IN THIRD SERIES OVER SCORES MADE IN SECOND SERIES (LOWER LINES) Grade 5 Grade 6B Grade 6W I Grade 7 Grade 8 All tests, 1-8 inclusive. 19.6 Fundamentals: 1, 2, 3, 4, 7 rights, and 7 attempts Reasoning: 6 attempts, 6 rights attempts, 7 rights , . . . .'{ Beasonings: 6 rights and 8 rights/ only 26.2 34.3 11.1 14.7 12.9 67.2 11.5 75.9 11.9 45.1 8.8 18.1 6.9 83.5 13.8 102.1 10.6 12.9 9.1 11.4 4.7 15.7 18.6 16.5 19.1 19.5 8.6 11.1 13.3 34.6 — .4 54.2 — .8 TABLE v.— NORMAL GROWTH OP SIX WEEKS WITHOUT DRILL, GRADES 7, 8 Grade 8 Ail tests Fundamentals Reasoning, attempts and rights Reasoning, rights only ; 19.5 11.1 34.6 54.2 Normal Growth without Drill during Six Weeks. — Table V shows the results in grades 7 and 8 where during the first six weeks there was no drill. These pupils generally did not know that they were to be given a second and third series of tests and so were not preparing to make better grades. The table shows that the average of the two gains made by these grades on the whole series of eight kinds of problems was 16.233 per cent. In funda- mentals both grades made approximately the same gain, but in reasoning grade 8 far surpassed the grade 7. The average for both grades in reasoning, taking into consideration correct answers only, was 35.36 per cent. The normal growth for the six weeks is probably not nearly so large as the figures would indicate. The reason why the pupils were able to increase their grades so much was that at the second series of tests they knew just what kind of work to expect in each Haggerty: Studies in Arithmetic 99 test and were better able to use all of their ability in making a good showing. By examining Table IV it is seen that the pupils of all grades and in all kinds of work almost without exception made a greater gain in the second series of tests' over their previous work than they were able to make on the third series over the second. This was on account of the condition just mentioned. The proper way to arrive at a proper answer to this problem would be to give a class three sets of tests without drill at any time and see what growth was shown between the second and third tests. A httle light, perhaps, is thrown on this by Table IV, where we find that during the second six weeks while the grade 7 pupils were practicing on reasoning exclusively they made a gain of 4.66 per cent on fundamentals, while during the same period when grade 8 pupils were practicing on fundamentals they seemed to have made no growth at all in reasoning. Comparative Progress with Drill and without. — Table VI gives an unequivocal answer to the question as to whether drill is worth while. The smallest difference in averages was shown in funda- mentals where the fifth and sixth grades, who had drill during the first six weeks, were able to excel themselves by only 4.356 per cent more than the seventh and eighth grade pupils who were being given no drill. In all other cases the gain is large. In reason- ing where correct answers only are considered,, the difference in gain in favor of those groups that took regular- drill is 32.7 per cent. Evidently it pays to give regular drill work in arithmetic. TABLE VI.— COMPARATIVE PROGRESS WITH AND WITHOUT DRILL. FIGURES BASED ON GAINS MADE IN SECOND TEST OVER FIRST Grades 5, 6 Grades 7, 8 Difference - Drill No drill "Drill All tests, attempts and rights (see Table I) 33.3 15.6 68.0 68.1 16.2 11.3 25.1 35.4 17.1 4.4 32.9 32.7 Comparison of Progress with Drill on Fundamentals and Pro- gress with Drill on Reasoning. — From Table VII it seems that about the same progress was made by the two groups, the group that had a drill daily in fundamentals and the group that had the 100 Indiana University Studies same amount of dfill in reasoning. However, it is very interesting to see along what lines each gained. Those who had drilled in reasoning made a large gain in reasoning, while the group that had practiced fundamentals exclusively made a slight loss in reasoning. The group that had prac- ticed fundamentals made a gain of 13.34 per cent over their former record in fundamentals, while the group which had drilled on reasoning made a gain of only 4.66 per cent in fun- damentals. Practices in reasoning gives somewhat larger gains than practice in fundamentals only. These figures seem to show clearly that we get in arithmetic what we drill for. The most profitable form of drill seems to be one in which part of the time is devoted to reasoning and part to fundamentals. TABLE VII.— PROGRESS WITH DRILL ON REASONING COM- PARED WITH DRILL ON FUNDAMENTALS IN SECOND SIX WEEKS No drill in first 6 weeks. During the second 6 -weeks, grade 7 was given 5 min- utes daily drill in reasoning ; grade 8 had 5 minutes daily drill in fundamentals. Grade 7 Grade 8 All tests 9.1 4.7 18.6 19.1 8.6 13.3 .4 .8 Drill for Speed and Drill for Accuracy. — The difference in progress made by the two groups, one being drilled for accuracy and the other for speed, is not very large. As the results are group^ in Table VIII, the speed group is shown to have made the greater gain in fundamentals, in reasoning when correct answers only are considered, and in the average of all tests. The only group of tests on which the accuracy pupils excelled was in reasoning problems where both attempts and rights are considered. By referring to Table III, it is seen that a large gain was made in Test No. 7 by the pupils who were practicing for speed (Grade 6E). Test No. 7 is made up of problems in addition, subtraction, multiplication, and division in which large numbers are used. As the greater gain in nearly all cases is with the pupils Haggerty: Studies in Arithmetic 101 who practiced for speed, it would seem that this form of practice is preferable. I believe that speed generally takes accuracy along with it and that the pupil who is speediest in the manipulation of figures makes the fewest mistakes. TABLE VIII.— COMPARISON WITH DRILL FOR SPEED AND DRILL FOR ACCURACY (SECOND 6 WEEKS) Grade 6E (Drilled for speed) Grade 6W (DrUled for accuracy) Difference in favor of speed section All tests Fundamentals Eeasoning, rights only 11.1 12.9 11.9 6.9 10.6 2.3 6.0 1.3 The Best Distribution of Time for Brills. — Tables IX shows that the better progress in every group of tests was in favor of the pupils who were given one fifteen-minute drill once per week over the pupils who were given a five-minute drill five times per week. Fifteen minutes once per week seems much better than five minutes per daj' for drill work. Much better results were attained; there was a saving of time, not only the ten minutes difference per week, but a saving of time that was lost in taking up and laying aside a special drill each day; there was also a saving of energy on the part of. the teacher in preparing for and administering one drill rather than five per week. The biggest difference was in reasoning. This is perhaps ex- plained when we consider that not much could be done in training for reasoning in drills only five minutes in length, especially when each was taken partially for reasoning and partially for funda- mentals. TABLE IX.— DISTRIBUTION OF TIME OF DRILLS IN GRADE 6 (FIRST 6 WEEKS) Grade 6E (5 minutes daily) Grade 6W (15 .minutes once per week) Difference in favor of IS minutes once per week All tests Fundamentals Reasoning, attempts and rights Beasoning, rights only 34.3 14.7 67.2 75.9 45.1 18.1 83.5 102.1 10.8 3.4 16.3 26.2 102 Indiana University Studies Comparison of the First Six Weeks' Gain under Drill in Each Grade. — Grade 6W pupils who had fifteen minutes of drill once per week made the greatest gains in all tests. The figures of Table X do not show the real gain so far as a comparison of the seventh and eighth with the fifth and sixth grades goes. This tabulation is unfair to the two upper gradfes. The reason is this: In all grades, regardless of the kind of drill or whether they had drill, greater gains were shown on the second test than on the third. The reason for this is that the pupils knew so much better how to go about the work in the second test than in the first that they were able to make great gains. The third test found them not much better acquainted with the nature of the tests than they were the second time, and they were thus not able to make such great gains. In Table X, the gains shown for the fifth and sixth grades are for the second test, while the gains of the seventh and eighth grades are for the third test. The fifth grade pupils were at a great disadvantage in each series of tests in those parts in which reasoning was required, not so much on account of the reasoning required but on account of the difficulty which they encountered in reading the problems, or at least in reading them rapidly. Too large a proportion of their time was consumed in the work of reading the problems. The child's ability to read is put to the test in sets six and eight of each series quite as well as his ability to reason. It seems that our sixth grade pupils made mgre progress on account of drill work than the other grades. Whether this would be true with other groups of pupils and other teachers in charge, I am unable to say. There was a smaller proportion of pupils in the sixth grade who were not doing passing work than in any other grade tested. This would have a tendency to skew the results. I believe that no adequate conclusion can be formed from these tests as to which grades will profit most from drill work. TABLE X.--COMPARISON OF THE FIRST SIX WEEKS' GAIN UNDER DRILL, GRADES 5 TO 8 Figures show the gains of grades 5-6 during first six weeks and grades 7-8 during second six weeks. The seventh and eighth grades had no drill work between the first and second tests. Grade 5 Grade 6£ Grade 6W Grade 7 Grade 8 All tests 19.5 14.1 23.4 26.2 34.3 14.7 67.2 75.9 45.1 18.1 83.5 102.1 9.1 4.7 18.6 19.1 8 6 Fundamentals 13.3 Reasoning, attempts and rights Reasoning, rights only — .4 — .8 V. EXPERIMENTS WITH COURTIS PRACTICE PADS By Floba Wilber, Principal of Fort Wayne Training School. Nature of the Study. — This paper reports an investigation of the value of such individual daily practice as is afforded by the Courtis Standard Practice Pads, where the drill to be given is determined by the individual needs of the members of the class. In order to test the results of the practice, Series B of the Courtis Tests was given in September, 1914, to the fifth and sixth grades in the training department of the Fort Wayne City Normal School. On the basis of these tests the beginning section of each grade was divided into two equal groups. As no two pupils equal in each test in both "attempts" and "rights" could be found, the standing of each pupil was determined by finding the average of his attempts and rights in the four fundamental operations. Of two pupils having such equal scores, one was put into each group. Thus the individual scores were as nearly equal as possible, and the combined scores of the groups were not more than one example apart. TABLE I.— AVERAGE OF ATTEMPTS AND RIGHTS FOR POUR FUNDAMENTAL OPERATIONS FOR 28 PUPILS Grade 5 Grade 6 Group I Group II Group I Group II 3.5 3.5 6.5 6.875 2.75 2.75 6.0 5.75 2.625 2.625 5.5 5.75 1.875 1.875 5.375 5.375 2.5 2.25 4.25 4.25 2.25 1.625 4.125 4- 3.625 4.5 2.5 3.125 Totals 19.125 19.125 34.25 35.125 The fifth grade group gave 4f minutes of the daily period devoted to arithmetic to this practice and the sixth grade group gave 4 minutes. The other group in each grade used the full period for regular work. Except for the time given to the practice, both groups were taught together. 104 Indiana University Studies 5 2 p m n m g O ^; B " ^ =° t= S O 5 < P o OQ O Q o I— I Qj P4 I" n n ^ oa CO Ui CO 1 1^ ". " 1 -g. o on o ^ O o ^ ui '.JH s m •Vr o CO (N CN ^^ 'It* CO CO 5 ■a s M -e o o o O O i-t CO o •* 1 P> ra Oi 1 |1 S 1 *"* (N 1 O >o ■* iQ eo il ^ s esi U3 ■* (N h3 eg M 1 «> CO CO o lO p o CO 00 CO i:^ o s 11 s '^ g ^ rg CO h 1 to CO "* iO -* 01 U3 CX9 . s CDJ3 CO ■* §-s s o •S, (5 -J CM ^^ o s g." 1 US (^■^ & CO o tti O (M ^ T»1 US s ^ §■ m CO o l> t^ QO a oo g 8-< iJ s g 1 1 CO (M lO >n co -^ , t^ <>i oS-B II s -a t- eo iO CO «o to lO 00 r- t> tg t- il s o - "^ (M « ■* o ■* o to (Nl Tf '"' " LO Tj- m (M «o '^ 3 o •^ "^ t- •o "* « '* 00 US s '"' '"' '"' 3 □0 00 U3 to Tft as CO CO CO CO '^ CO 00 cs « oo l^ -# o 3 "^ c» " CO CO 2 ^ CS in s o '"' ^^ cs eo 2 ea 2 - c» OO ■^ s cs o o o o oo CO *< 00 ro in o Tj* ,-1 to (N o •-4 >o g (N r- ■«< ,-1 CO O IM •-4 U2 lO r^ lO '"' «= fJO co at CO y* *"* 00 to to CD -* «, ■° 1- ' : C3 •^, D s3 @ J 1 .E IS 1 106 Indiana University Studies In May Series B was given a second time. The results have been computed from the differences between the scores of the two tests. They can be used as suggestive only, inasmuch as before school closed the number of pupils remaining in each group (7) was much too small for positive conclusions. Speed. — In Table II are found the scores in attempts and rights for each pupil in each operation Jiogether with the group scores, which are the totals of the individual scores, and the group medians. Using the group scores as the basis of computa- tion, Table III gives the per cent of gain made. during the year by each group and the difference in gains by practiced and un- practiced groups. TABLE III.— PER CENT OF GAIN (IN MAY OVER SEPTEMBER) IN SPEED OF PRACTICED AND UNPRACTICED GROUPS Unpracticed Practiced Gain of Prac- ticed over Unpracticed Gbadb 5 Addition 30 27 12 43 28 51 76 43 54 148 144 60 49 88 129 13 27 136 Division 144 Ghape 6 Addition 17 Subtraction, . . 21 37 Division 53 Thus the speed gain made in addition during the year by the fifth grade unpracticed group is 30 per cent, while the practiced group gained 43 per cent. In the sixth grade the unpracticed group made a gain of 43 per cent and the practiced group 60 per cent. Comparing the gain made by the practiced group over the unpracticed, it is found in grade 5 to be 13 per cent in addition, 27 per cent in subtraction, 136 per cent in multiplication, and 144 per cent in division. In the sixth grade it is 17 per cent in addi- tion, 21 per cent in subtraction, 37 per cent in multiplication, and 53 per cent in division. The gain in speed in the fifth grade is particularly noticeable, and varies greatly. The sixth grade gains do not vary so much. The greatest gain in both grades is in multiplication and division. Haggerty: Studies in Arithmetic 107 Accuracy. — Table IV shows in a similar way the gain made by each group in May in accuracy. Thus the unpracticed group in the fifth grade gained 30 per cent in addition and the practiced group 37 per cent, making a gain for the practiced over the un- practiced of 7 per cent. In subtraction this gain was 20 per cent, in multiplication 5 per cent, while in division the practiced group fell 1 per cent below the unpracticed group. In the sixth grade the gain of the practiced over the unpracticed is in addition 8 per cent, in subtraction 19 per cent, in multiplication 2 per cent, and in division 6 per cent. The greatest gain in accuracy is in subtraction. The gains in both grades are much the same in each operation except in division, where the practiced group's gain over the unpracticed is 7 per cent greater than the corres- ponding gain in the fifth grade. TABLE IV.— ACCURACY PER CENT OF GAIN (IN MAY OVER SEPTEMBER) IN ACCURACY OF PRACTICED AND UNPRACTICED GROUPS Unpracticed Bracticed Of Practiced over Unpracticed Grade 5 30 29 26 43 12 2 18 28 37 49 31 42 20 21 20 34 7 Subtraction Multiplication 20 5 — 1 Gbade 6 Addition Subtraction Multiplication 8 19 2 6 Tables V and VI give the medians for each group in attempts and rights together with the gain of each and the gain of the practiced over the unpracticed. 108 Indiana University Studies TABLE v.— MEDIAN ATTEMPTS (IN MAY AND SEPTEMBER) OF PRACTICED AND UNPRACTICBD GROUPS Unpracticed Practiced Gain Made Bt Gain of Septem- ber May Septem- ber May Un- prac- ticed Prac- ticed Prac- ticed over Un- prac- ticed Gbade 5 Addition 5.5 5.8 4.3 3.3 6.7 9.5 5.8 4.8 6.5 7.4 4.8 3.5 8.8 11.5 9.3 7.8 6.3 5.8 3.2 2.5 8.5 9.5 6.3 4.8 8.6 8.9 7.5 5.7 12.5 14.5 10.8 8.8 1 1.6 .5 .2 1.1 2. 3.5 3. 2.3 3. 4.3 3.2 4. 5. 4.5 4. 1.3 Subtraction Mullipiication. . . . 1.4 3.8 3. Grade 6 Addition 2.9 Subtraction Mujtiplication. . . . Division 3. 1. 1. TABLE VI.— MEDIAN RIGHTS (IN MAY AND SEPTEMBER) OF PRACTICED AND UNPRACTICED GROUPS Unpracticed Practiced Gain Made Bt Gain of Prac- Septem- ber May Septem- ber May Un- prac- ticed Prac- ticed ticed over Un- prac- ticed Grade 5 Addition . ... 1.5 2.5 .5 2.5 7.5 3.5 2.8 4.3 '6.5 2.3 2.3 5.8 8.5 6.3 5.5 1.8 2.3 1.5 .5 4.5 6.3 2.3 2.5 6.5 6.8 5.8 3.5 6.5 11.2 7.3 7.5 2.8 3. 1.8 2.3 3.3 1. 2.8 2.7 4.7 4.5 4.3 3. 2. 4.9 5.0 5.0 1 9 Subtraction Multiplication.... 2.5 2.5 7 Grade 6 Addition Subtraction Multiplication Division 3.9 2.2 2 3 Haggerty: Studies in Arithmetic 109 Comparison of Speed and Accuracy. — That the practice did not train in speed at the expense of accuracy is shown by a comparison of the May medians of the practiced group with the Indiana medians in attempts and rights {Indiana University Bulletin, Vol. XII, No. 18). Table VII shows the per cent of superiority of the practiced groups over the Indiana median in attempts and rights. This superiority is greater in rights than it is in attempts, being especially marked in the fifth grade. Only at one place — • sixth grade addition — is the superiority in attempts higher. A study of Table VIII showing the Indiana medians of de- pendability and that of the practiced groups in May gives similar results. TABLE VII.— COMPARISON OF INDIANA AND PRACTICED GROUP MEDIANS Attempts Bights Excess of su- Indiana Median Prac- ticed Median Per Cent Su- periority Indiana Median Prac- ticed Median Per Cent , Su- periority periority in riglits over at- tempts Gbade 5 Addition 6.6 7.3 6.3 4.5 8.6 8.8 7.5 5.7 30 21 19 27 3.6 5. 3.9. 2.6 6.5 6.8 5.8 3.5 81 36 49 35 51 Subtraction MultipUcation... . Division 15 30 > 8 Gbadb 6 Addition 7.4 8.9 5.7 12.5 14.5 8.8 69 63 54 4.4 6.5 4.8 6.5 11.2 7.5 48 72 56 —21 Subtraction Division 9 2 110 Indiana University Studies TABLE VIII.— COMPARISON OF DEPENDABILITY BETWEEN INDIANA AND PRACTICED GROUPS Grade 5 Gkade 6 Indiana Median Practiced Group Median Indiana Median Practiced Group Median 55 68 61 57 72 77 69 67 59 73 67 74 67 Subtraction 76 ]\Xu]tipllcation 55 Division. 79 V Summary. — On the basis of group scores and medians each group made a considerable gain in each operation. The gain of the practiced group was the greater in each opera- tion except in fifth grade division. This lack of gain may bear a relation to the general lack of dependability in division in the fifth grade as shown in the report of the twenty cities of Indiana to which reference has already been made. In speed the gain is considerable, and there is a clear tho not large gain in accuracy. The gains made by practice vary greatly in the four operations both by individuals and by grades. The experiment gives encouragement to the idea that such practice as can be given to individual pupils upon the fundamental arithmetical operations in accordance with individual needs is justified. However, this experiment cannot be conclusive on account of the small number of pupils engaged.. Cornell university Library QA 135.H22 Studies in af;*";,?.™!