UNIVERSITY OF CALIFORNIA AT LOS ANGELES UNIVERSITY OF CALIFORNIA, LIBRARY, A-OS ANGELES, CALIF. SUPPLEMENTARY EDUCATIONAL MONOGRAPHS PublitheJ in conjunction with THE SCHOOL REVIEW and THE ELEMENTARY SCHOOL JOURNAL Vol.1 August 27, 1917 No. 4 Whole No. 4 ARITHMETIC TESTS AND STUDIES IN THE PSYCHOLOGY OF ARITHMETIC By GEORGE S. COUNTS Professor of Education in Delaware College THE UNIVERSITY OF CHICAGO PRESS CHICAGO, ILLINOIS 278 4 5 COPYSIGHT 1917 BY THZ UNIVERSITY OF CHICAGO All Rights Reserved Published August 27, 1917 Composed sad Printed By The University oi Chicago Press Chicago, Illinois, U.S.A. TABLE OF CONTENTS AFTER I. INTRODUCTORY STATEMENT II. THE NATURE OF THE TEST AND COLLECTION OF DATA .... 4 The Test Series A and B of the Courtis Tests The Test Described Addition Subtraction Multiplication Division Fractions Time Allowance Collection of Data The Cleveland Test The Grand Rapids Test Summary Statement III. GENERAL RESULTS 21 Determination of Standards Derivation of a System of Weights The Use of the Test Distributions Accuracy Summary IV. TYPES OF ERRORS S3 Addition Subtraction Multiplication Division Fractions Fractions of Like Denominators Fractions of Unlike Denominators Summary iii iv CONTENTS CHAPTER PAGE V. A COMPARISON OF THE ARITHMETICAL ABILITIES OF CERTAIN AGE AND PROMOTION GROUPS 78 Age Groups Method Results Promotion Groups Methods Results Summary VI. A COMPARISON OF THE ARITHMETICAL ABILITIES OF CERTAIN RACE GROUPS 112 Method Results Conclusions VII. SUMMARY AND CONCLUSIONS 121 The Nature of the Test and Collection of Data General Results Types of Errors A Comparison of the Arithmetical Abilities of Certain Age and Promotion Groups A Comparison of the Arithmetical Abilities of Certain Race Groups CHAPTER I INTRODUCTORY STATEMENT This investigation is a study of the arithmetical abilities or attainments of school children as measured by an arithmetic test. The study naturally falls into two divisions, the first including chapters ii, iii, and iv, the second, chapters v and vi. In the former, the test used in the investigation is described, and results are discussed which throw light on its use. In the latter, two special studies are made in which the test is used as a measuring instrument. These five chapters will now be described in greater detail. In chapter ii it is shown that there is a need for a spiral test in the "fundamentals" of arithmetic to be used in diagnosing city, school, class, and individual weaknesses in the various operations included in the term "fundamentals." It is further pointed out that Series A and B of the Courtis standard tests are inadequate to meet this need. The test then, as developed, composed of 15 sets of different types of examples, is described and analyzed. This is followed by a statement concerning the collection of the data upon which the remainder of the study is based. The purpose of chapter iii is fivefold: (i) In order that the test may be of the greatest value educationally it is necessary that standard attainments for children in the various grades in each of the 15 sets be determined. This is done on the basis of results from Cleveland and Grand Rapids. The validity of these results is discussed from the standpoint of the Courtis standard scores. (2) A system of weights is derived by which it is made possible to convert the scores made by a particular group or individual in the 15 different types of arithmetical operations into a single score to represent general arithmetical attainments of the individual or group. (3) The use of the test is discussed in detail, the method by which it may be employed to diagnose city, school, class, and individual weaknesses being shown. (4) Distributions of the scores 2 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC made by groups of children in the typical operations are discussed for the purpose of indicating the different types of individual reaction to examples of varying degrees of complexity and for the purpose of pointing out certain differences in the responses made to the "fundamentals" and to fractions. (5) The degree of accu- racy with which the various types of examples are worked is shown, accompanied by a comparison of the curve of accuracy and the curve of "rights" for one of the sets. Chapter iv is a study of errors, in which the types of errors made by children in working the different kinds of examples are analyzed. It is of value to the teacher to know what sorts of errors she may expect from the pupil when the latter encounters the different arithmetical operations. The frequency of these errors is also determined hi order that the teacher may be able to apply the proper amount of emphasis at the various points of difficulty. Because of inability to isolate kinds of errors made in connection with some types of examples, since the study was confined to an examination of records made by pupils, this study is incomplete. It is necessary that it be supplemented by experimental data. The problem presented by the study in chapter v is, in the first plate, the problem of measuring the attainments of various groups of children for the purpose of discovering differences in four age groups throughout Grades 3-8 inclusive. In the second place, a study is made of certain promotion groups for the purpose of dis- covering differences. This division of the study has three parts: the first relates to the fast and slow pupils and is confined to the records of pupils in Grade 8-2; the second is concerned with a group of pupils repeating because of failure to do the work of the grade, a group repeating because of sickness, transfer of school, or similar cause, and a group of pupils making normal progress, the data for this study being secured from pupils in Grade 7-2 only; the third has to do with a group of pupils in Grade 8-2 who had failed below the sixth and another group who had failed above the fifth grade. The differences found are analyzed and inter- preted. In chapter vi a problem of the same general type as that of the previous chapter is encountered. The problem here is to deter- INTRODUCTORY STATEMENT 3 mine whether or not there are differences in arithmetical attain- ments which follow racial lines. Owing to the meagerness of the data, this study is confined to five races, or nationalities, Americans, Hollanders, Germans, Swedes, and Slavs. Owing to the fact that this entire study has been made on the basis of records made by pupils, it is in many particulars incomplete and tentative, for there are many matters that cannot be deter- mined by an examination of records. Furthermore, the conditions under which the records were made were not sufficiently under control. It is therefore evident that it is necessary to supplement this study by experimentation. CHAPTER II THE NATURE OF THE TEST AND COLLECTION OF DATA THE TEST In connection with the Cleveland Survey the demand arose for an arithmetic test to measure the presence and absence of arith- metical attainments in the school children of that city. The sort of test desired was one that would, on the one hand, show the general standing of the city as a whole in the "fundamentals" of arithmetic and would, on the other hand, be diagnostic in its char- acter, indicating school, class, and individual weaknesses in each of the different types of operations which enter into the solving of the more complex examples in each of the four fundamental operations. SERIES A AND B OF THE COURTIS TESTS It was felt by those in charge of the survey that no test had as yet'been devised which would exactly fit their needs. Series A of the Courtis tests was unsatisfactory because, as Mr. Courtis himself has said, "the standards derived from the use of Series A .... are either complex or of questionable value, owing to the uncer- tainty of their meaning." 1 Tests Nos. i, 2, 3, and 4 of this series are merely tests of knowledge of the tables in the four fundamental operations, and, since a pupil may know his tables perfectly and yet be quite unable to solve any of the more complex examples, and vice versa, these tests by themselves are of little value. Test 8, the only other test of the fundamentals in this series, is of doubtful value. In the first place, the form in which the examples appear is not the form to which the child is accustomed. For example, when called upon to add two or more numbers, the pupil does not ordinarily have them presented to him in this form, 304-735+ 123= . In order to work the example he must copy the three 1 S. A. Courtis, Manual of Instruction for Giving and Scoring the Courtis Standard Tuts, p. 7. 4 THE NATURE OF THE TEST AND COLLECTION OF DATA 5 numbers in column form. This consequently makes necessary the copying of figures in the test, or else the performing of the operation in a wholly unaccustomed manner. In the second place, the use of the symbols introduces another factor. A pupil might be able to perform the required mathematical operation perfectly, yet fail on an example in this test because of unfamiliarity with the symbols. If it is desired to test the knowledge of symbols, a separate test should be devised for that purpose. In the third place, a particular score in this test may mean almost anything because of the com- plex nature of the test. For example, what may a score of "four" mean? It may mean either strength or weakness in any one of the four operations, or it may mean anything between these extremes. Thus, since Series A is found to be quite unsatisfactory, let us turn to Series B of the Courtis tests. The latter, when used as a supplement to the former, or rather when substituted for Test 8 of that series, represents a distinct improvement over the earlier tests. The four tests in Series B are composed of four sets of com- plex examples in the four operations. Test i involves the addition of columns of 9 three-place numbers, Test 2 the subtraction of eight-place numbers from eight- and nine-place numbers, Test 3 the multiplication of four- by two-place numbers, and Test 4 the division of four- and five- by two-place numbers. Series B supplemented by Series A is very good so far as it goes, but it does not go far enough. It makes possible a measure of the general attainment in each of the fundamental operations, but does nothing more. In a word, it is not diagnostic. For instance, sup- pose we have a pupil who knows his addition tables perfectly, as indicated by a record made in Test i of Series A, but fails miser- ably on Test i of Series B. These two facts about the pupil are worth knowing, but are of comparatively little value unless supple- mented by other facts. Why he fails on the second test is not known. It may be because of failure to bridge the attention spans, or of inability to "carry," but the test throws no light on the question. It is just at this point that Series A and B of the Courtis tests break down. It is necessary to introduce, between the very simple type of example in the first series and the highly complex 6 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC type in the second series, tests representing types of intermediate complexity. This is, in fact, the logical evolution of Mr. Courtis' own system and is actually embodied in principle in his Standard Practice Tests. THE TEST DESCRIBED Since no existing test quite met the needs of the members of the survey staff, they took upon themselves the task of devising one. In this work the co-operation of Mr. Courtis was secured, with the result that to him is due whatever merit the test, as it now stands, may possess. In order that the reader may get a clear impression as to the nature of the test, and that the discussion may be the more easily followed, the test is here reproduced in full. Passing over for the moment the first page of the test folder, since it does not constitute a part of the test, the test is seen to be composed of 15 sets, desig- nated as Sets A, B, . . . . O. An examination of the test shows it to be composed of four sets in addition (A, E, J, M), two in subtraction (B, F), three in multi- plication (C, G, L), four in division (D, I, K, N), and two in frac- tions (H, O). Since the pupil begins with Set A and takes each set In its proper order, the spiral character of the test is apparent, a feature which deserves some further comment. The several sets in each operation are arranged in the test in the order of their com- plexity, but with them are interwoven the sets of the other opera- tions. Thus a pupil first works on a set of examples in addition, then passes successively to sets in subtraction, multiplication, and division before encountering addition again. This changing from one type of operation to another lessens the strain on the pupil which is involved in a prolonged test of this sort. ADDITION As indicated above, there are 4 sets in addition. Set A involves the addition of the simple combinations, Set E the addition of columns of 5 one-place numbers, Set J the addition of columns of 13 one-place numbers, and Set M the addition of columns of 5 four-place numbers. The 65 examples of Set A were taken from Test i, Series A, of the Courtis tests; the 16 examples of Set E, THE NATURE OF THE TEST AND COLLECTION OF DATA 7 ARITHMETIC EXERCISES Name___ Age today. Vein Months Grade. School. Teacher. Date today. Have you ever repeated the arithmetic of a grade because of non- promotion or transfer from other school. If so, name grade Explain cause Inside this folder are examples which you are to work out when the teacher tells you to begin. "Work rapidly and accurately. There are more problems in each set than you can work out in the time that will be allowed. Answers do not count if they are wrong. Begin and stop promptly at signals from the teacher. A B C D B F G H A R I J K L M N O A R STUDIES IN THE PSYCHOLOGY OF ARITHMETIC SET A Addition 1 6 904 1 7932 136 2 6 5 1 2 3 7604 589 3 897 8 2148 023 7 2 196 5679 5 7 1 4 7 031 2 5675 869 6 9 854 9 8021 350 4 2 974 5 7480 392 3 2 380 2 1960 418 5 624 5 1637 9 04 7 4 318 9 0234 865 SET B Subtraction 9 7 11 8 12 1913 4 12 -i -1 6 _1 3 J) jr 8 3 6 8 11 12 5 10 6 11 15 10 12 JL 9 7 1 2 J) 7 8 _9 4 2 7 13 3 10 1615 4 8 J. * 7 2 5 1 3 9 2 3 4 10 13 10 9 5817 6 11 J. I 5 1 4 569 5 12 15 5 16 7816 9 11 .2. 9 6 3 8 J) 57 1 4 Au. Ru. THE NATURE OF THE TEST AND COLLECTION OF DATA g SET C Multiplication 349054 2 7 8 2 6 1 2 9 7 4 6 9 5 All. ^^Ml Ru. 9 5 4 8 762 051 3 3 9 6 5 7 4 1 2 _6 8 7 7 087 631 3 8 9 2 9 4 3 1 4 -i 4 8 9 041 3 5 4 6 2 8 8 7 9 3 1 3 7 4 6 8 032 092 6 3 7 5 9 5 4 6 SET D Division ^^__ __^ 4)32 6)36 2 7)28 9)9_ 3)21 6^48 21 5)10 22 4)24 7)63 6)0 8)32 128 5)30 8)72 IJO 9)36 12Z 2)10 7)42 I)l_ 6)18 3)6 4)20 7)49 123 2 )JL 6)6_ 3)27 8)64 1)2 4)16 5)0_ 3)24 9)63 2>J 8)24 7)7 2)18 6)42 3)0 7)21 4)4 3)15 9)8_1 7^0 10 STUDIES IN THE PSYCHOLOGY OP ARITHMETIC SET E Addition At,. Ru. 5 2 9 288 2 >8 057 Jt JL JL 2 6 8 3 5 4 8 6_ _8 1 4 2 5 J^ 4 9 6 7 5 1 3 5 -i . 626 772 833 549 JL JL JL 8 5 5 9 1 6 3 3 i JL 4 8 5 JL 1 3 4 7 1 2 8 9 =i JL SET F Subtraction . ^- 616 1248 456 709 1365 618 i^MMM^MM 1092 472 716 344 1267 1335 509 419 707 277 .^ ___ 816 335 1157 908 1355 908 616 258 519 324 MW_W 1236 908 1344 818 1009 768 >269 295 SET G Multiplication 1269 772 ^~MW 615 527 6789 2 854 286 2345 6 2345 9735 2 5 8642 9 ^ 9735 2468 9 3 6789 6 ^-i 3579 3 2468 7 5432 9876 4 8 8642 5 ^- 3579 7 9876 4 5432 3689 8 5 2457 6 ^^^M 9863 4_ 7542 7 THE NATURE OF THE TEST AND COLLECTION OF DATA u SET H Fraction* 3.1 64 4,1 l_ 5 m . 99 24 9 7 5 5 99 5 1 JL-.L 9 9 B m _2 m 7 7 6_ 7 9 M JL-. 2 - 7 7 JL-.L 7 9 2_ 3_ _JL = ^.+J_= A_= 88 88 77 99 99 .1 Jl 8 8 JL 9 9 SET I Division 4)55424 7)65982 2)58748 5^41780 9)98604 6)57432 3)82689 6)83184 8)51496 9)75933 8)87856 4)38968 At*. Rtt. 12 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC SET J Addition 79472967789432 5251969180531 1 44894265573776 28148471414766 62435 07821 860 852 6 8 55585335213936 13152973139549 86324213372657 31973367942345 24676806898422 98317561445892 98596567546894 SET K Division 21~2T3 52)1768 417779 22JT62 31~8""T 427966 235731 72)1656 81)972 73)1679 21)294 62)1984 31)527 52)2184 41)984 3273*84' 51)2397 82)1968 71^3692 227484 41)1681 337~f9~3 61)1586 53)1166 3l7~"96 Au. Ru. THE NATURE OF THE TEST AND COLLECTION OF DATA 13 SET L Multiplication Ate. fto. 8246 3597 29 73 5739 8JJ 2648 46 mm^HHmmm^m 4268 7593 37 64 SET M Addition 6428 58 2123 5142 1679 0376 5555 4955 6331 9314 6808 5507 8563 207 PMMMMHMM 3691 4526 7479 2087 8165 7493 8937 8625 9016 6345 4091 6487 2783 3844 7591 4883 8697 6166 1341 7314 5226 9149 6268 2883 8467 7725 2584 0251 8331 0058 7535 5493 2398 5223 3918 9397 7337 6158 2674 3732 9669 4641 5114 7919 8154 8243 6429 9298 7404 2575 SET N Division 67)32763 48)28464 97)36084 59] 129382 78)69888 88^344^6 6 9)40296 3 8)26562 14 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC SET O Fractions At.. RU. 11 1 _ 9 2_ 3 5 15 + 6 ~ 14 4 ~ 5 X T = JL L- 19 11 6 21 ~ 6 X 20~ 12 8 ~ 1 3 5 11 5 2_ _M> ' ^ 1^" ^^" 7 x io~ 6 15 12 8 20 1 3 3 3 3 ^^M ' " 21 6 4 18 8 10 ^-^ _ Instructions for Examiners Have the children fill out the blanks at the top of the first page. Have them start and stop work together. Let there be an interval of half a minute between each set of examples. Take two days for the test; down through I the first day, and complete the test on the next day. The time allowances given below must be followed exactly. Set A.. 30 seconds Set B 30 seconds Set C 30 seconds Set D 30 seconds Set E 30 seconds Set F....._ 1 minute Set G 1 minute Set H 30 seconds Set I J minute Set J 2 minutes Set K 2 minutes Set L .....3 minutes Set M 3 minutes Set N 3 minutes Set O 3 minutes Have the children exchange papers. Read the answers aloud and let the children mark each example that is correct, "C." For each set let them count the number of problems attempted and the number of C's and write the numbers in the appropriate columns at the right of the page. The records should then be transcribed to the first page, verify the results set down by the pupils. Please THE NATURE OF THE TEST AND COLLECTION OF DATA 15 the 14 examples of Set J, and the 12 examples of Set M were taken wholly or in part from Lessons 4, 23, and 27, respectively, of the Courtis Standard Practice Tests. These 4 types of examples in addition were chosen because the solution of the examples in each succeeding set involves a mental process not present in the immediately preceding set, which marks it off as a type. Thus Set A represents the very simplest sort of addition, the combining of 2 one-place numbers. In Set E the pupil must not only combine two numbers, but must hold this sum in his mind and combine it in turn with a third number, and so on through four combinations. At first glance Set J seems to be of the same type as Set E, the difference being merely one of quan- tity, but such is not the case. Twelve combinations must be made instead of four. Now the span of attention has limits. Anyone who has ever attempted to add a long column of figures knows what this means. The addition of one figure after another from the first figure in the column to the last is not one continuous process, but is broken up into segments. That is, the individual adds up to a certain point, holds the sum in his mind as the atten- tion wavers, and then continues the addition of the column as the attention returns. This is called "bridging the attention spans" and is a mental process called forth in the addition of the long columns in Set J. There is one other operation that the pupil must learn to perform successfully before he can become a competent adder, and that is "carrying." For testing ability to perform this operation Set M appears in the test. In the addition of these columns the pupil must "carry" a result forward from the addition of one column to the next. Thus the 4 sets in addition indicate ability or lack of ability (i) in performing the simple addition com- binations, (2) in adding a third number to a sum secured by the addition of two numbers,. (3) in bridging the attention spans, and (4) in "carrying." SUBTRACTION There are but 2 sets in subtraction in the test. The first, Set B, is made up of the simple combinations; and the second, Set F, involves the subtraction of three-place numbers from three- and 16 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC four-place numbers. The examples in the former set were taken from Test 2, Series A, of the Courtis tests, and those in the latter from Lesson 20 of the Courtis Standard Practice Tests. Subtraction is confined to 2 sets because, for diagnostic pur- poses, they are sufficient. The only operation that is added in the more complex forms of subtraction, not found in the simple com- binations, is that of borrowing. This is demanded in the examples of Set F just as much as in the larger examples. 1CULTIPLICATION Multiplication appears in 3 sets. Set C involves the simple combinations, Set G the multiplication of four-place by one-place numbers, and Set L the multiplication of four-place by two-place numbers. The 50 examples in Set C were taken from Test 3, Series A, of the Courtis tests; the 20 examples in Set G were specially devised under the supervision of Mr. Courtis for this test; and the 8 examples in Set L were taken from Test 3, Series B, of the Courtis tests. The first set tests knowledge of the tables. In the second the pupil must "carry" results forward. And in the third, Set L, the operation is further complicated by the demand for knowledge of the mechanics of handling the product of the multiplication and the second term of the multiplier. The addition of the partial products is also introduced. DIVISION Four sets are given over to division, D, I, K, and N. The simple combinations appear in Set D, the division of five-place by one-place numbers in Set I, the division of three- and four-place numbers by two-place numbers in Set K, and the division of five- place numbers by two-place numbers in Set N. The 49 examples in the first set were taken from Test 4, Series A, of the Courtis tests; the 12 examples in Set I were taken from Lesson 31 of the Courtis Standard Practice Tests; and the other two sets, K and N, made up of 25 and 8 examples, respectively, were specially devised for this test. As in the sets for the other three operations, the attempt was here made to introduce into the test examples embodying the THE NATURE OF THE TEST AND COLLECTION OF DATA 17 different types of difficulty that are encountered in division. Set D tests knowledge of the tables. Set I is made up of more complex examples in short division which differ from the examples in Set D by the introduction of the operation of carrying. Sets K and N are sets in long division. The former represents the very simplest type of this operation, since there is no carrying required in the multiplication and no borrowing in the subtraction. The latter, on the other hand, is much more complex, involving both carrying and borrowing. FRACTIONS For the purpose of testing the ability of pupils to apply the four fundamental operations to the working of fractions 2 sets of fractions were placed in the test, Set H and Set O. Both sets, the one made up of 24 examples and the other of 12, were specially devised for this test. The examples in Set H are very simple, involving the addition and subtraction of fractions of like denominators. In Set O frac- tions of unlike denominators are to be added, subtracted, multiplied, and divided. These sets of fractions, it will be noted, differ from the other sets in that they are not homogeneous. In the first there are two different types of operations to be performed, and in the second there are four. This is freely acknowledged as a defect. But, since the test was to be used in the survey, it was necessary that its scope be limited; and, since the testing of attainments in fractions was felt to be more or less experimental, it was thought that the fractions should be sacrificed rather than the fundamental operations. TIME ALLOWANCE A word should be said about the time allowances given to the several sets. The child is not allowed to begin with the first set and to work an indefinite time on it or any following set. On the contrary, as indicated by the time allowances given on the last page of the test, the pupil is allowed to work a specified time on each set. This time ranges from 30 seconds for the easier sets to 3 minutes for the more difficult sets. In each case the attempt was made to make the tune allowance large enough to enable even l8 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC the slowest pupil to work at least one of the examples, and yet small enough to prevent even the most rapid pupil from exhausting the possibilities of the set. Thus the test is a speed test with a definite time allowance given to each of the 15 sets. COLLECTION OF DATA The data on which the present study is based were secured from two sources, viz., Cleveland and Grand Rapids. Since there were slight differences in the tests themselves and in the giving of the tests in the two cities, each will be treated separately. THE CLEVELAND TEST The test as given to the children of the Cleveland schools was slightly different from that just described. In the Cleveland test the result "21" was repeated so frequently in Set K that some of the pupils taking the test, after working several examples of the set and finding the answers to be "21" in almost every instance, wrote down " 21 " as the answer to the remaining examples without actually working them. In the light of this experience Set K was modified so as to avoid the repetition of this result. Set L was modified by giving more space for working the examples, because the Cleveland results showed that insufficient space had been given. Set O was also modified. In its earlier form the examples in the addition of fractions constituted one column, those in sub- traction another, those in multiplication another, and those in division another. When they appeared in this form it was found that, quite frequently a pupil would select the examples in multi- plication and avoid the more difficult examples of the other opera- tions. To place a check on this tendency the four types of examples were intermingled, as seen in the test in its present form. The tests were given on June 4, 7, and 8, 1915, to the B sections of Grades 38 inclusive. The teachers gave the tests, following the instructions given on the test sheet and certain other instructions sent out to the principals of the schools from the office of the super- intendent. 1 The scoring was done by the pupils under the super- vision of the teacher. 1 Charles H. Judd, Measuring the Work of the Public Schools, p. 245. THE NATURE OF THE TEST AND COLLECTION OF DATA 19 THE GRAND RAPIDS TEST The test as described in this chapter was given to both sections of Grades 3-8 inclusive on February 28 and 29 and March i, 2, and 3, 1916. A great deal more care was taken here than in Cleve- land to insure the results against error. In the first place, the writer was present at a meeting of the principals from all of the schools, where the test was carefully gone over and the method of giving the test explained. In the second place, the request was made that one person, preferably the principal, do all the timing in each school, and that the testing be begun in the lower grades and proceed upward, so that the examiners might be somewhat experienced in the giving of the test when the more important grades were tested more important because it is only in the upper three grades that the children are able to work examples in all the sets. In the third place, the teachers and the pupils in the Grand Rapids schools were familiar with the Courtis practice tests. The teachers were consequently to some degree experienced examiners, and the children were acquainted with the signals for beginning and stopping work. In the fourth place, the writer personally con- ducted the tests in 50 classes in 8 schools. TABLE I NUMBER OF CLASSES TESTED Grade Cleveland Grand Rapids Total z. . 85 64 149 4 87 62 149 c. . 9 58 148 6 87 C7 140 7. . 86 46 132 8 85 zi 116 Total 520 314 834 From these two sources, as shown in Table I, results were secured from 834 classes, 520 in the Cleveland schools and 314 in the schools of Grand Rapids. The number of classes is given rather than the number of children tested because the medians in the general tables to be discussed in the following chapter are medians of class standings and not of individual standings. 20 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC SUMMARY STATEMENT In summary, the present test is a speed test which measures attainments and indicates weaknesses in the four fundamental operations and fractions. In addition it tests knowledge of tables, the ability to add short columns, to bridge the -attention spans, and to "carry"; in subtraction it tests knowledge of the tables and the ability to "borrow"; hi multiplication it tests knowledge of the tables, ability to "carry," and ability to add in connection with multiplication; in division it tests knowledge of the tables, ability to "carry" in short division, and ability to solve two types of examples in long division, the one involving neither "carrying" nor "borrowing" and the other involving both; and it tests the ability to apply these four fundamental operations to the working of examples in fractions. The test was given to, and results secured from, 834 classes in the schools of Cleveland and Grand Rapids. In both cities the test was given almost entirely by the teachers. In Cleveland the teachers were inexperienced in giving tests, while in Grand Rapids they were all more or less familiar with the Courtis tests. CHAPTER III GENERAL RESULTS DETERMINATION OF STANDARDS In order that the test may be of the greatest educational value it is necessary that standard scores be determined for the several grades in each of the sets. These scores must of course be deter- mined empirically, that is, on the basis of what children actually do. As indicated in the previous chapter, more or less valid results were secured from two large school systems, Cleveland and Grand Rapids. These results were tabulated, and measures of central tendency computed. The median was chosen for this measure for two reasons: (i) since it is not disproportionately affected by an extreme case, it in large measure eliminates errors due to over- timing or undertiming; (2) the median is easily computed. These two facts make the median a highly desirable average, especially when such an enormous body of material must be handled as is necessarily the case in the survey of a large school system. The method used to secure a final average score for each of the sets of the tests was as follows: First, the median of each of the 90 Cleveland schools (more or less depending on the grade) in a par- ticular grade was found; secondly, the median of these medians was computed to get an average for Cleveland as a whole; thirdly, the same thing was done for Grand Rapids; fourthly, the medians of the two cities were averaged to get tentative standard scores for the different sets of the test. The test was given to the B sections only in Cleveland and to both sections in Grand Rapids, but since it was given in Cleveland at the close of the term (June) and in Grand Rapids at the beginning of the term (February, March), in order to get the standard score the results from the lower sections in Cleveland were averaged with results from upper sections in Grand Rapids. These standard scores found by averaging the Cleveland and Grand Rapids medians appear in Table II. An examination of the 22 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC table shows that the average scores made in Set A, simple addition, by third-grade pupils in the time allowance (30 seconds) was 13.4 examples; by fourth-grade pupils, 17.1 examples, etc. The absence of a score, as in the earlier grades for Sets H, K, L, N, and O, indicates that the pupils in that grade were unfamiliar with the type of operation demanded. TABLE II AVERAGES OF MEDIAN SCORES IN EACH ARITHMETIC TEST FOR GRADES 3-8. CLEVELAND AND GRAND RAPIDS Set Grade 3 4 5 6 7 8 A 13-4 8.9 6-5 6-3 4-3 2.O 2.O I7.I 12.8 II.7 IX .4 5-o 4-5 3-6 21. 9 16.6 14.8 15-0 5-9 6.6 5-i 5-6 i-7 3-9 5-6 2-7 3-4 i.i 24-9 19-5 16.8 17.7 6.7 7-7 5-5 6.0 3-i 4-4 7-0 3-2 4-i 1.6 3-3 27.0 21. I 18.2 20.3 7-4 9.1 6.0 7-7 4-0 5-i 9-4 3-8 4-7 1.9 4-3 28.9 25-8 19.9 22.8 8.0 10.6 6.7 8.6 4-7 6.1 11.4 4-4 5-4 2-4 S-2 B c D E F... G H I 0.6 1.9 I.O 3-o 4-o i-7 2.4 0.8 J-- K...*.. L M i-4 N o It is freely conceded by the writer that, because of the com- plexity of the test and the difficulties encountered in following the time allowances, and because of the fact that the test was quite largely given by persons with little or no training in testing, it is very likely that many errors were made in the giving of the test. Now the important question that arises is the nature of the errors made. If they were of a compensating sort that is, if it were purely a matter of chance whether the examiner overtimed or undertimed the errors made in one direction were offset by those made in the other. If, on the other hand, the errors were of the cumulative type that is, if for any reason the examiners tended to overtime more than undertime, or vice versa the errors would GENERAL RESULTS not offset one another, and an error would enter into the final results. On first thought it would seem that, since the tests were being given in connection with a survey to determine the standing of a city in arithmetical attainments, as well as the relative stand- ings of the individual schools within the city, there would be a tendency for the teachers to overtime rather than to undertime. Some light may be thrown on our problem if we turn to Table III. In the description of the test it was said that five of the sets A, B, C, D, and L were taken over from Series A and B of the Courtis standard tests. It is therefore possible to make a comparison between the Cleveland-Grand Rapids average for each of these sets and the Courtis standard scores. This comparison is made in Table III and Diagram i. TABLE III RESULTS OF CLEVELAND AND GRAND RAPIDS TESTS COMPARED WITH COURTIS STANDARDS GRADE SCORE SET A B C D L 3 /Cleveland and Grand Rapids. . 13-4 13-0 17.1 17.0 21.9 21.0 24.9 25.0 27.0 2Q.O 28.9 31-5 8.9 9-5 12.8 12.5 16.6 15-5 19-5 19.0 21. 1 22. 25-8 24-5 6.5 8.0 ". 7 n-5 14.8 15.0 16.8 18.5 18.2 20.5 19.9 22.5 6-3 8.0 IX. 4 n-5 iS-o 15-0 17.7 18.5 20.3 22. 22.8 24-5 4 /Cleveland and Grand Rapids. . 1-7 0.8 2.7 2.O 3-2 2.8 3-8 3-3 4-4 4.0 5 /Cleveland and Grand Rapids. . 6 /Cleveland and Grand Rapids. . 7 /Cleveland and Grand Rapids. . 8 /Cleveland and Grand Rapids. . The Courtis standards are supposed to represent June attain- ments, while the Cleveland-Grand Rapids average represents February or March attainments, almost a half-year behind the Courtis standard. A word should be said, however, about Set L. Very little reliance can be placed upon this comparison because the 24 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC Courtis standards in this case are purely tentative. For that reason no graphical representation is made of the comparison. DIAGRAM i. Results secured from Cleveland and Grand Rapids in Sets A, B, C, and D compared with the Courtis standards. From the table and the diagram it is seen that, so far as the seventh- and eighth-grade attainments in the four sets of simple combinations are concerned, the Courtis scores are higher than the Cleveland-Grand Rapids scores. In the third grade the same GENERAL RESULTS 25 difference is noted, while in the intermediate grades the scores quite closely agree. Two facts deserve comment. In the first place, the differences on the whole favor the Courtis scores, and the differences are about as great as they should be in view of the fact that the Courtis scores represent an advantage of almost half a grade. The presumption is strong, therefore, that the errors which are very likely to have accompanied the giving of the test were of the compensating type in these four sets at least. In the sec- ond place, there seems to be a characteristic difference in the forms of the two curves of progress from grade to grade. The Courtis scores indicate uniform progress from grade to grade, while the Cleveland-Grand Rapids scores show, though not emphatically, to be sure, the progress to be less rapid with each successive grade. In other words, the latter tends to resemble in certain respects the typical learning curve. Thus it would seem that the question has not yet been answered whether, during progress through the grades, the limits which are ordinarily set to improvement through practice are completely offset by the maturing of the pupil. The Courtis results seem to indicate that these limits are offset, while the results from Cleveland and Grand Rapids seem to point to the contrary. To return to the question of error in the final result, what may be said concerning the reliability of the scores for Sets E to O inclusive? Arguing from the Courtis standard scores, the scores for A, B, C, and D seem to be free from any considerable error. Through experience in giving the test the writer has come to the conclusion that accurate timing is more difficult in connection with these first sets than with the later sets, because of the short time allowances in the former. The same absolute error in two given cases is relatively a greater error where the time allowance is small than where it is large. It would at least seem that there is no reason for thinking that the later sets were not given as accurately as the first four. To this last statement an exception should possibly be made in the case of Set H. To this set an allowance of but 30 seconds is given, while a minute is given to each of the two preced- ing sets. There is undoubtedly a tendency for the examiner to allow a minute for this set also, so that there is probably a cause of error operating in Set H that is not present in the other sets. 26 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC From the foregoing it would seem that, arguing from the Courtis standards to Sets A, B, C, and D, and from these four sets to the remaining sets, with the possible exception of H, the average scores for the several sets made by the pupils in the lower sections in the Cleveland grades and the upper sections in the Grand Rapids grades constitute reliable standards for midyear attainments. These standards may therefore be tentatively accepted, subject of course to revision as returns are secured from other cities. DERIVATION OF A SYSTEM OF WEIGHTS It is desirable for certain purposes that some method be found of equating the scores made in the different sets by a particular system, school, class, or individual so tnat a single score, the sum- mation of the scores made in the several sets, may be obtained to indicate general attainment in the "fundamentals." In order to do this, a unit must first be found in terms of which the score made in each of the sets may be stated. In essence the equating of the sets resolves itself into a state- ment of their relative difficulties. There are two factors that con- stitute the criteria of difficulty: the first is speed, the second is accuracy. Since accuracy by itself means almost nothing, since the number of examples attempted likewise means but little, and since the number of examples worked correctly in a given time includes a measure of both speed and accuracy, it seems to the writer that the latter is as valid a gauge of difficulty as any that might be chosen. Having accepted this criterion, the equating of the sets is a very simple matter. A second's work might be taken as the unit. Then, if it required on the average three seconds to work one example and two seconds to work another, their relative difficulties would be as three is to two. For the sake of conven- ience, however, we may take the average time required to work an example in one of the sets as a unit. A value of i . o is then given to each of the examples worked in that set, with values for the examples of the other sets varying inversely as the speed with which they can be worked. In the present study it is suggested that the average time required to work an example in Set A be accepted as the unit, and GENERAL RESULTS 27 that each example correctly worked in this set be therefore given a value of i . o. An example of this set is chosen because of its size and stability. It is a smaller quantity than any other unit would be, because on the average a pupil works more examples of this type than of any other. It is more stable than any other, there being least variation from individual to individual in the records made in this set. A second suggestion is that the system of weights be derived from the records made by eighth-grade children. It is true that the relative difficulties of the sets are not the same from grade to grade. Set N is, for example, not only absolutely much more difficult for the fourth grade than for the eighth, but relatively much more difficult. Of course it is possible to make a system of weights for each grade, but that has its disadvantages. The thing desired is a system of weights that will show progress from grade to grade. If the system is changed with every grade, there is no intelligible relation between the score made by one grade and that made by the grade above or the grade below. Again, it would seem that the relative difficulties of two sets should be determined on the scores made by individuals who have attained some degree of mastery over both sets rather than over but one. Finally, the system of weights should be derived from the eighth-grade scores because those scores represent the final achievement, under the present school organization, resulting from formal training in arithmetic. In Table IV the system of weights is shown and the method of deriving them is indicated. In the first horizontal column of the TABLE IV DERIVATION OF SYSTEM OF WEIGHTS Set A B c D F G H I J K L M N O Eighth-grade score Time allowance in seconds 28.9 30 as- 8 30 30 22.8 8.0 30 10.6 60 6.7 60 8.6 30 4-7 60 6.1 1 20 11.4 1 20 4-4 i So 5-4 180 ?8o i So Score per 30 seconds 28.9 19.9 22.8 8.0 5-3 3-35 8.6 a. 35 i-53 2.8s 73 .90 .40 .87 Weight (relative difficulty) I.O ,1.12 1.45 1.27 3.6i S.4S 8.63 3.36 12.3 18.9 IO.I 39-5 32.1 72.2 33 3 table are the average eighth-grade scores for the 15 sets; in the second are the time allowances in seconds; in the third is the aver- age score per 30 seconds for each set; and in the fourth are the 28 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC weights, or measures of relative difficulty. However, since the time allowance varies from set to set, the system of weights as pre- sented in this table requires some revision. For, as the weights now stand, each of the sets with time allowances of 3 minutes has just six times as much influence in determining the total score as has any one of the sets with time allowance of 30 seconds. It is there- fore necessary, in order that each set may have the same influence on the total score as any other set, to modify the weights to that end. This revised system of weights appears in Table V. The TABLE V EQUATION OF TIME ALLOWANCES Set A B c D E F G H I J E L M N O Weight (relative difficulty) Time allowance in seconds I .0 30 1. 12 30 I -45 30 1.27 30 3-61 30 S-.45 60 8.63 60 3.36 30 12.3 60 18.9 1 20 IO. I 1 20 39-S 1 80 32.1 1 80 72.2 180 33-3 1 80 Weight after equating time allowances I.O 1. 12 1-45 1.37 3.61 a. 73 4.31 3-36 6. is 4-73 2-S3 6.58 5-35 12. 5-56 same result would have been secured if in Table IV the time allow- ance had been neglected entirely and the weights computed on the average scores as they stood. Such a method, however, would have been misleading. As shown by the two steps taken in evolving the systems of weights, it now does two things: (i) it equates the examples on the basis of difficulty, and (2) it equates the time allowances of the several sets. THE USE OF THE TEST In a previous chapter it was pointed out that the test was evolved for the purpose of diagnosing weaknesses of one sort and another in school systems, schools, classes, and individuals. The method by which the test may be used for doing this thing will here be demonstrated. We shall first make some comparisons between two large city systems Cleveland and Grand Rapids. By using the system of weights just described, it is possible to get a single score to represent the arithmetical attainments of each grade for each of these two cities. These scores are given in Table VI and graphically repre- sented in Diagram 2. GENERAL RESULTS 29 Turning to the diagram, we see that there are considerable differences between the two curves, especially in the lower grades. TABLE VI COMPARISON OP TOTAL SCORES MADE BY GRADES 3-8 m CLEVELAND AND GRAND RAPIDS City Grade 3 4 5 6 7 8 Cleveland 90 22 181 132 260 243 3i8 325 370 383 43 439 Grand Rapids The superior attainment of the Cleveland children in the lower grades indicates relatively greater stress on arithmetic in this period. Work in arithmetic is begun earlier in Cleveland than in Grand Rapids, but this initial advantage is not maintained. From the showing that the latter city has made, the conclusion would seem to be justi- fied that a large expenditure of time on arithmetic in the lower grades is of comparatively little importance in securing high attainment in the eighth grade. However, this is a very general result and by itself is of little value because its meaning is vague and uncertain. The important thing for either city to know is its weak points. The detailed records by which this is possible are found in Tables VII and VIII and Diagrams 3, 4, 5. In Diagram 3 there appears a graphic com- parison of the records made by the pupils of the two systems in each of the four sets in addition, A, E, J, M. The first glance at these curves shows that the relations between the two cities are about the same here as indicated by the general results, viz., superiority of Cleveland in the lower, and superiority of Grand DIAGRAM 2. Comparison of total scores made by Grades 3-8 in Cleveland and Grand Rapids. STUDIES IN THE PSYCHOLOGY OF ARITHMETIC Rapids in the upper, grades. In the simple addition combinations, Set A, there is little difference between the two systems, except in TABLE VII MEDIAN SCORES IN EACH ARITHMETIC TEST FOR "B GRADES -3-8, CLEVELAND SECTIONS OF Test Grade 3 4 5 6 7 8 A. . 13-4 9-3 6-5 6-3 4-3 2.O 2.0 17-8 13-4 12.0 12.4 5-3 4-9 3-9 22.2 17.2 15-5 15-7 6-3 6.7 5-2 5-o 2.O 4.0 6.8 2-5 3-2 i-3 24.8 19.8 16.6 18.5 6.8 7-5 5-5 5-5 3-i 4-4 8-5 2.8 3-8 i-7 3-i 26.7 21-5 17.7 20.8 7-5 8.6 5-9 7-7 4.0 4-9 IO.I 3-2 4-4 2.O 4-i 27-5 26.0 19.0 22.5 7-8 IO. I 6.6 8-5 4-7 5-7 12.5 3-9 5-i 2.6 5-5 B c D E F... G H I 0.6 1.9 i.i 3-2 4.0 i-7 2-5 0.8 T K. . L M i-4 N O. TABLE VIII MEDIAN SCORES IN EACH ARITHMETIC TEST FOR GRADES 3-1-8-2. GRAND RAPIDS Grade lest 3-1 3-a 4-1 4-a 5-1 S- 6-1 6-2 7-i 7-2 8-1 8-a A ii. 8 13.4 13.6 16.4 2O. 3 21. ? 22.8 2$ .O 26 *i W | M rt (V) r< k, ct H GENERAL RESULTS would be difficult to say. However, in the other cases the statement is certainly true. This would indicate less individual TABLE XII DISTRIBUTION OF 100 PUPILS IN EACH GRADE SET J 1 Score I a 3 4 5 6 7 8 9 IO ii 12 13 14 4-2 5-i 5-2 6-1 6-2 7-i 7-2 8-1 8-2 Attempts. Rights.... Attempts . Rights Attempts . Rights.... Attempts. Rights.... Attempts. Rights.... Attempts . Rights.... Attempts . Rights. . . . Attempts. Rights.... Attempts. Rights.... 21 2 18 18 21 IO 17 4 18 2 IO 4 13 I 12 30 23 2O 27 17 20 10 38 IO 19 9 13 3 8 2 14 5 16 30 8 *r 14 3i 21 22 21 17 19 'P 22 J/ 24 JO 15 JO 18 14 8 '7 ,1 9 28 ii 24 18 24 15 -T7 21 JP 19 17 8 5 J I 7 2 JO" 8 JO" 8 18 7 JO" 8 23 13 *P IO JS 16 2 I 9 i jj 6 15 8 ^ 9 2? 14 18 IS 22 II 3 9 8 17 I 12 2 3 7 5 5 3 5 3 8 6 14 8 Jj 3 J J J 2 3 2 5 2 7 3 8 4 P 3 j 2 7 3 i 2 2 3 i 5 i 3 / 3 7 2 I J I J I 2 I 2 I I 2 J 4 8 J J I i 6 I 7 12 J 2 2 4 I 10 2 TABLE XIII DISTRIBUTION OF 100 PUPILS IN EACH GRADE SET M Score o I 2 3 4 s 6 7 8 9 IO II 13 4-2 5-i 5-2 6-1 6-2 7-i 7-2 8-1 8-2 Attempts Rights j 20 5 22 4 14 19 21 17 33 6 16 4 ii 2 14 36 2O 16 14 22 26 JO 24 JO 21 8 ii 4 19 J 12 4 19 2O IO 33 14 27 i9 22 21 16 21 *5 19 15 18 P 15 7 16 15 7 2J 7 22 9 JO 16 30 19 25 IS 22 16 18 20 2^ 18 3 J Attempts 2 2 16 7 21 II 23 10 18 IS 25 18 2p 21 23 18 2 J Rights 16 Attempts 4 2 d 2 JJ 6 16 ii jo" IO 19 7 20 8 3 Rights 6 IS Attempts. . . . 4 2 6 II 4 7 2 J2 8 8 3 2 I J I J 2 <5 a 4 3 JO a J J I iRights 3 8 Attempts Rights i 7 Attempts. . . . 2 I 2 2 3 I 3 2 2 .... Rights 5 5 12 Attempts. . . . 2 J J I I Rights i 2 9 J 9 Attempts. . . Rights i 3 j Attempts . . . Richts 2 3 9 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC variation on the simple addition examples than on the more complex. The character of the relation between the curves for the "rights " and for the "attempts" is a third matter deserving attention. On the average the curve for the "rights" is flatter than that for the "attempts." This is emphatically true in Sets J and M, the more complex types. Thus there is less tendency among the pupils to vary in the number of examples attempted than in the number solved correctly. This is probably explained by the fact that the number of examples attempted is controlled quite largely by the physical limitations on speed, since the character of the operation in each of these types of examples is familiar to all the pupils. In working the examples correctly, on the other hand, another factor is involved, and that is the factor of right and wrong associations. A more strictly mental limitation is here added to the physical limitation just mentioned. TABLE XIV DISTRIBUTION OF 100 PUPILS IN EACH GRADE SET L * Score o i a 3 4 5 6 7 & /Attempts 8 20 27 26 14 28 8 19 3 17 2 17 2 10 2 7 I ii 37 21 32 25 22 26 14 24 II 20 9 18 3 17 6 20 26 6 38 12 31 19 34 22 29 25 I? 22 13 24 15 23 2 I II 4 20 12 29 14 21 10 22 19 21 21 23 14 s ' x lRights 26 J Attempts . . . 2 2 II 4 12 7 19 ii 23 15 27 16 24 17 3 i 5 3 5 i 6 3 17 8 22 7 15 6 2 1 Rights 12 16 2 13 I 12 I 10 , /Attempts I "^Rights 4 f 1 Attempts . . . 2 6 - 2 lRights 3 /Attempts. . . II 7 \Rights 4 /Attempts . . . 10 I II 2 16 4 7 iRights I 6 i 5 Q /Attempts '"^Rights.. i o (Attempts . . , 8 - 2 lRights I 4 Since the foregoing characteristics are not peculiar to the dis- tributions in addition, but are common to the distributions in each of the other three fundamentals as well, it is not necessary to present tables and graphs setting forth the distributions in these three GENERAL RESULTS 43 operations. We have an entirely different proposition in the case of fractions. For this reason, therefore, Tables XIV and XV are accompanied by Diagram 9, which graphically portrays the facts found in the tables. In the first table appear the distributions for the several grades in Set L, multiplication; in the second the dis- tributions in Set O, fractions. TABLE XV DISTRIBUTION OF 100 PUPILS IN EACH GRADE SET O Score i a 3 4 5 6 7 8 9 10 II 12 /Attempts .... 5 ~ 2 1 Rights I II 3 27 2 13 7 3i 3 16 3 22 2 13 I 10 I IO 2 9 20 28 8 18 6 18 7 23 3 22 3 19 4 19 13 2 IO 7 7 16 7 13 3 12 6 16 ii 12 II I 16 13 15 8 7 5 6 8 7 10 7 ii 7 5 3 JO 7 3 7 f 1 Attempts. . JO 6 14 9 S 5 P IO 13 ii '4 IS -rj 8 '4 5 5 i 10 ii 12 6 12 7 II 4 IX I 9 9 ii 5 P 8 9 5 7 i 7 i jj 2 J2 3 JO 4 8 4 d 4 14 4 jj 3 12 3 P 2 3 i 4 i 4 i d i P 3 8 i jj 15 22 27 "18 I 14 I J 1 Rights 13 , f Attempts. . 6 - 2 \Rights ..... IO 9 I 9 I 9 f Attempts. . 1 Rights IS f Attempts. . 2 1 Rights.., 6 o (Attempts. . '"^Rights 2 7 2 IO n (Attempts. . 8 - 2 \Rights P . .... 4 The diagram is of the same order as the previous one and there- fore requires no explanation. The similarity between the curves for Set N and those for the sets in addition just discussed is appar- ent. Let us therefore turn at once to a comparison of the curves of this set and of Set O. Perhaps the most obvious feature in the comparison is the relation between the curves for the "rights" and the curves for the "attempts" in Set 0. Here, in direct contrast to the sets in the fundamentals, the curves for the "attempts" present a much more flattened appearance than the curves for the "rights." This would seem to indicate that, whereas in the fundamentals the knowledge of the character of the operation to be performed was common property for practically all the pupils, in fractions the character of the operation is not known by all. To those familiar with the method of handling fractions, or to those who think themselves familiar with it, it is a simple matter to attempt a 44 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC Grad GraJe. 6-2 ISP Grad i/o /Vo, o/ DIAGRAM 9. A comparison of the distribution of "attempts" and "rights" in Set L (multiplication) and Set O (fractions) for Grades 5-8. GENERAL RESULTS 45 large number of the examples. This is not true of the fundamentals. For instance, take an example in long division. Even though the method of working such an example is perfectly familiar, it requires considerable time to work it because it is a long process. Speed can be developed only through much practice by making a large number of reactions quite automatic. The actual process involved in working an example in fractions, such as is found in Set O, is, on the other hand, a relatively short one. Now, since so far as attempt- ing the examples is concerned it is just about as easy to attempt one of the examples as another, those pupils who are familiar with the method of solving fractions attempt a large number, or all of them. Those, on the other hand, who are unfamiliar with the method are able to attempt but a few. In this way the curve for " attempts" becomes flattened. The curve for "rights" is less flattened because of the composition of the test set. As will be pointed out later, the examples in the multiplication and division of fractions are easier than the other two types. This causes the distribution of "rights" to be largely confined to six examples. Since there is no such factor operating to narrow down the distri- bution of "attempts," the curve for the "rights" is elevated in comparison. Another fact indicated by the diagram which bears somewhat on this same matter is the increase of the percentage of pupils attempting all the examples in Set O up to Grade 7-2 and then a decrease in the percentage to Grade 8-2. An examination of Table XV gives further evidence on this same point. It is seen that there is a constant increase in this percentage from Grade 5-2 through Grades 6-1, 6-2, and 7-1 to Grade 7-2, where the maximum of 27 per cent is reached. Then there is a decrease to 18 per cent in Grade 8-1, and a further decrease to 14 per cent in Grade 8-2. This is a very significant fact. An inspection of the work actually done by the pupils on this set indicates a tendency among them to substitute various "easy" methods for the correct methods in work- ing the examples. For example, a pupil may add two fractions by adding their numerators and their denominators. This takes less time than the right method. Thus, by substituting invalid for valid methods the pupil is enabled to complete the set in a relatively 4 6 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC short time. As the pupil matures he gradually develops greater speed, and this probably accounts for the increase in the number of pupils attempting all the examples of the set up to Grade 7-2. The decrease from this point on is probably due to the weeding out of these invalid and short methods through increasing familiarity with fractions. Set O- Sa* M-AJj.tie* / Gra. = JL ft ' io 10 2 k DIAGRAM 23. Typical errors made in multiplying fractions of unlike denominators In this same section (h) we find another peculiar type of error. The work on all these examples was done by the same pupil. The pupil has learned to cancel, but not to use the results of cancella- tion, except in the first of the examples, where it seems that he has used the " 2 " because of inability to find anything else to put down as a result. In comparing Cleveland and Grand Rapids certain differences are noted. The children of the former seem to be more inclined to find a least common denominator than do the children of the latter city. This is partially accounted for by the difference in the organization of the test when given to the two groups, already referred to. In the inversions of terms and of results Grand Rapids monopolizes all the errors found. In other respects the records from the two cities are not greatly different. TYPES OF ERRORS 75 Turning now to Table XXXIV and Diagram 24, we find the facts presented for the last type of examples found in Set O, divi- sion. Failure to invert the divisor (section a, Diagram 24) seems to be the most frequent error. Another interesting and logical error is the dividing of the numerator of one of the fractions by the numerator of the other and the denominator of one by the denomi- nator of the other (section b, Diagram 24). The rule is that the larger quantity is divided by the smaller rather than the term of the dividend by the term of the divisor. A very interesting example TABLE XXXIV FREQUENCY OF TYPES OF ERROR IN DIVISION OF FRACTIONS OF UNLIKE DENOMINATORS EIGHTH GRADE FREQUENCY TYPE or ERROR Cleveland Grand Rapids Total Failure to invert 32 26 en Mistakes in fundamentals 27 2O 47 Numerators divided, denominators divided 2 2O 22 L. C. D., numerators added 14 6 2O Numerators added, denominators added z 7 Inversion of dividend 2 e. 7 L. C. D., numerators subtracted a 6 Miscellaneous 2 4 6 Method obscure 14 12 26 Total IOO IOO 2OO of this error is found in section g of the diagram. Here the pupil has introduced decimals into the operation. Another error deserv- ing mention is that reproduced in section e. The dividend is inverted instead of the divisor. The other types of errors have all appeared in connection with the other types of examples and have been discussed; hence nothing further need be said regard- ing them. There also seem to be some differences between Cleveland and Grand Rapids in the division of fractions. Children of the former city fail to invert; that is, they employ the method of multiplica- tion more frequently than do children of the latter city. This is probably due in a measure to the difference in the organization of 76 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC the test as given in the two cities. Cleveland children show greater weakness in fundamentals here as in the other operations. The addition suggestion already discussed is found to operate on division for the Cleveland children. The Grand Rapids children, on the other hand, seem prone to divide the one numerator by the other and the one denominator by the other. These statements cover the chief differences. 20 . i 20 ??_._. ??j_JL il 21 ' 6 126 21 * 6 3! 21 ' 6 42 a be ?<>j.JL 21 ?2_.__1 _L J.2. 33 21 ' 6 27 21 ' 6 120 21 ' 6 42 d t f 11 . 5 _2.6 12 ' 8 1.4 5 . n = a.i 6 ' 15 2.3 20 _. I _ 20 21 ' 6 "sTs DIAGRAM 24. Typical errors made in dividing fractions of unlike denominators SUMMARY 1. In the addition of the simple combinations the general propo- sition seems to be established that on the average those combina- tions whose sums exceed ten are more difficult than those whose sums are less than ten. To ,this general statement there are indi- vidual exceptions which indicate the formation of peculiarly strong associations, some being right and others wrong. These peculiar associations vary among different groups. This would indicate that the formation of the association is to be accounted for in terms of the experience of the group rather than in the character of the combination itself. 2. In the simple subtraction combinations "bridging the tens" is found to be a relatively much more difficult operation than in the addition combinations. Freakish errors, on the other hand, are found to be less frequent in the former than in the latter. The TYPES OF ERRORS 77 understanding of the meaning of zero seems to accompany the maturing of the pupil. This is indicated by a relatively large per- centage of errors made on the combination i o by fifth-grade pupils, whereas this combination presented but little difficulty to pupils in the eighth grade. 3. Practically all the errors made in the simple multiplication combinations are made in those combinations in which zero enters as one of the terms. Furthermore, it is a more difficult mental operation to multiply a quantity by zero than to perform the reverse operation, to multiply zero by the quantity. And a pupil may have difficulty with the zero in the simple combinations, yet be quite able to handle it in the more complex examples, and vice versa. In the complex multiplication examples the most frequent error is made in multiplying. 4. In the simple division combinations the most frequent error is made in dividing a quantity by itself. The result given is zero, showing a confusion between the division and subtraction processes. In long division the demand for multiplication accounts for most of the errors. 5. The typical errors made in working fractions indicate, as a general rule, a slavish adherence to the mechanics of fractions and show emphasis upon method rather than upon an understanding of the process. There consequently follows a great deal of confusion of methods on the part of the pupil. 6. In the addition and subtraction of fractions of like denomi- nator there is a tendency to add both numerators and denominators in the one case and subtract them in the other. 7. In the working of fractions of unlike denominator those involving subtraction are found to be the most difficult, followed in order of decreasing difficulty by those involving addition, divi- sion, and multiplication. Multiplication of such fractions is shown to be especially easy. 8. In the application of each of the fundamental operations to fractions there seem to be certain types of errors which recur again and again. Careful attention on the part of the teacher to these typical errors would be worth while. CHAPTER V A COMPARISON OF THE ARITHMETICAL ABILITIES OF CERTAIN AGE AND PROMOTION GROUPS One of the great values of a standard test is to throw light on our methods of instruction, the general organization of our courses of study, our system of promotion, and so on, through an analysis of what children do under these different influences. The present study represents an attempt to compare the arithmetical abilities or attainments of certain age and promotion groups. It therefore falls into two divisions, closely related and dealing largely with the same problem, the one concerning itself with differences in groups of children classified according to age, and the other with differences in groups classified according to rates and causes of promotion or non-promotion. Although these two divisions of the study are very closely related, they will be treated separately for the sake of convenience. AGE GROUPS The purpose of this division of the investigation is to find out whether or not there are any differences in the arithmetical abilities which accompany differences in the age of pupils in the same grade, that is, whether the under-age group is at all different from the over-age group, or whether the intermediate or normal group differs from either of these. Of course the test employed is quite inade- quate to indicate all differences in arithmetical abilities, but in so far as the test is adequate the nature of the differences, if any exist, will be analyzed. METHOD The data upon which this part of the study is based were secured from the results of the arithmetic test given to the children in the B sections of Grades 3-8 inclusive of the Cleveland schools. The giving of the test has already been discussed and therefore need not be taken up here. 78 ABILITIES OF CERTAIN AGE AND PROMOTION GROUPS 79 On the first page of each folder the age of the pupil taking the test was called for, as indicated in the reproduction of the test in chapter ii, except that it called for age in years only, not in years and months as in the revised test. Thus we had recorded the age of each of the children taking the test. It was therefore possible to group the pupils in each grade according to age. Now it will be remembered that, while the median results for Cleveland as a whole seemed to be devoid of any considerable inaccuracy, there was some doubt as to the accuracy of any par- ticular record, owing to the fact that the test is a complicated one involving time allowances difficult to administer exactly, and to the fact that it was given by teachers with little or no training in giving tests of this sort. Thus it is evident that, if our comparisons are to be valid, some method must be adopted which will eliminate those errors which may have been made in timing. Furthermore, an examination of the results of any standard test secured for the various schools and classes of a large city system shows that there are large differences from school to school and from class to class that are to be accounted for by differences in the training which the pupils in the different schools and classes have received. This must also be taken care of by our method; other- wise differences between two age groups might be due to differences in training rather than to differences in age. Thus it is seen that there are two factors which might account for differences between two groups that must be eliminated. The first of these is differences in giving the test. Overtiming or under- timing would favor or prejudice one group with reference to another. The second of these is differences in training. One group may show superiority over another because its members have had a more effective course of training. In order, therefore, that any compari- sons which are made may be valid, it is necessary that the groups compared be homogeneous as to the conditions under which the test was given and as to training. Of course there are other minor factors which may have influence, but it is believed by the writer that if these two are taken care of the comparisons will be valid. After an examination of the records had showed that it would be possible to secure data on four age groups, the records made by 8o STUDIES IN THE PSYCHOLOGY OF ARITHMETIC the pupils of a grade in a particular school were divided into two groups on the basis of sex; then each of these was thrown into four groups on the basis of age. Since, however, the age of each pupil was given in years only, it was impossible to divide the boys and girls of an entire class into four equal groups. For instance, sup- pose we have a third-grade class of 20 children. It is probable that 10 of these will be boys and 10 girls. Of the 10 boys it is prob- able that i will be seven years old, 4 eight years old, 4 nine years old, and i ten years old or more. Now in order that each of our four age groups may be equally influenced by the giving of the test to this class and by the training which the class has received, it is necessary that this class be equally represented in each group. Since there is but one pupil in the lower age group and but one in the upper, one must be taken at random from each of the inter- mediate groups. The same method is followed for the girls. Thus from this class of 20 pupils but 4 boys and 4 girls have been taken, because the ages were given in years. This method of selecting pupils for each of the age groups was continued until there were secured records from 50 boys and 50 girls for each of the age groups in each grade from the third to the eighth inclusive. This made a total of 100 records for each group in each grade, or 400 records for each grade, making a grand total of 2,400 records, upon which this study is based. In order to secure this number, the records made by 40-50 schools were ana- lyzed for each grade. And it should be reasserted that the four age groups in each of the grades (100 pupils to the group) represent experience in taking the test as nearly identical as it is possible to make it, and also, after allowing for differences due to transfer from one school to another, identical training so far as training in the school is concerned. The facts concerning the ages of these groups in the several grades are given in Table XXXV. Here the average age of each group is given. And it should be added that the range of the ages in any one group is practically confined to two years except in the case of Group IV, in which the range is about three years. This means that while the ages of the pupils of one group are not identi- cal, because of differences in the same grade from school to school ABILITIES OF CERTAIN AGE AND PROMOTION GROUPS 81 in this respect, the groups are quite homogeneous as to age. The table shows that the difference between the average age of each group and the average of the next older group in each grade is at least a year in every instance, and in some cases it is considerably more. Thus the groups are seen to represent real age differences. TABLE XXXV AVERAGE AGE OF EACH OF FOUR AGE GROUPS IN GRADES 3-8. 50 BOYS AND 50 GIRLS PER GROUP IN EACH GRADE Gr jup I II III IV 2 . 7 7 8 8 O ii 6 4 8.7 9.8 IO.Q 12.7 e. . 0.7 10.7 12. 13.6 6 10. <; n. 6 12. 14.4. 7. . II .4 12. "J I 2 ..? I^.O 8 12 . 1 12 . I 14. 2 1C. f After these records had been secured they were all carefully regraded, lest any error due to the scoring of the pupils should prejudice the results. They were then tabulated, both the number of examples attempted and the number of examples worked cor- rectly in each of the sets of the test. And, finally, average "rights," average "attempts," and accuracy were determined for each age group in each grade for each of the sets. RESULTS The detailed facts concerning the number of examples worked correctly by the four groups in the six grades appear in Table XXXVI. In order that the table may be made perfectly clear to the reader an explanation is necessary. The Roman numerals, I, II, III, and IV, represent the four age groups in each grade. Group I is the under-age group, Groups II and III are the interme- diate or normal groups, and Group IV is the over-age group. Keeping this explanation in mind, we read that in the third grade Group I, the under-age group, worked correctly on the average 16. 6 examples in Set A, 10. 7 in Set B, and so on. Group II, the younger 82 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC of the two intermediate groups, averaged 15.9 examples correctly worked in Set A, 9.1 in Set B, and so on. In this same way the TABLE XXXVI AVERAGE "RIGHTS" IN EACH SET FOE EACH OF FOUR AGE GROUPS IN GRADES 3-8. DATA FROM 2,400 PUPILS Set Third Grade Fourth Grade Fifth Grade I II in IV I n III IV I n III rv A.., 16.6 10.7 7-1 7.8 4.2 2.6 15-9 9.1 6.0 5-8 4.2 1.8 16.5 10.4 6.3 6.9 4-7 2.3 16.1 9-3 7.6 6-5 4-7 I.O 19.9 15-0 14.0 13-9 5-6 4. ^ 20.4 13-1 13-8 13-5 5-7 4-7 19.7 13-5 13-3 12.3 5-0 4.6 21.2 14-3 13-0 12-5 6-3 4.8 23.6 19.1 16.3 18.2 6-3 7.6 22.8 18.5 14.9 16.6 6.1 6.9 22.7 17.9 iS-5 16.0 6.1 6-7 24.2 17.7 16.0 15-6 6.4 6.3 B c D E F... G 1.8 0.8 o-5 1-7 O.2 i-4 0.7 0.4 1.6 O.I 1-4 0.8 0.4 i-7 1.6 I.O 0.4 1.6 3-7 2.8 1.2 3-i 3-9 i-7 2.6 o-S 3-6 2.O 0.8 3-o 3-8 i-5 2-3 0.4 3-4 2-5 I.O 2-9 3-4 1.2 2.2 0-3 3-4 3-i 0.9 3-4 3-6 i-4 2-5 0-3 4-8 4-5 2-3 3-7 6-7 2-3 3-2 i.i 0.2 4-9 4-7 1.9 3-7 6.1 2.4 3-0 I.O O.I 4-8 3-7 1.8 3-7 5-8 2.O 2.8 0.8 0.2 4-3 39 1.8 3-6 59 i-7 2-5 0.8 0.2 H I T K.. L M O.Q 0.8 0.9 i.i N o A Sixth Grade Seventh Grade Eighth Grade I II III IV I II III IV I n III IV 25.0 21. I6. 3 19.6 6-3 7-8 5-3 6.6 3-o 4-2 8.6 2.4 3-8 1-3 4-6 25.2 2O. 2 17.7 19-5 6.6 7-8 5-3 6-5 3-i 4-2 8.4 2.6 3-6 1-4 4-3 25.2 19-3 17-4 18.8 6-3 6.8 5-i 5-8 2-5 4-0 8.2 2.2 3-i I.O 3-4 25-3 20. o 18.1 19.2 7.0 7-5 5-2 6-5 2-5 4-7 7.6 2.4 3-7 I.O 3-5 26.2 22.4 17-4 21.6 7-7 9.0 S '* 7-8 4-3 5-0 10.7 3-1 4.8 1.9 5-9 28.3 21. 1 I8. 3 22.3 7-2 8.5 5-6 7-9 3-6 4-7 10.2 2.8 4-i 1.6 4-8 29-3 23-8 19.4 21.8 7.6 8.6 6.1 8-5 3-5 4-7 9-9 3-i 4-2 1.6 4-4 28.7 22.3 19.7 21.4 7-4 8.0 5-6 8.2 3-2 4-6 9.6 2-3 3-8 1.2 3-9 30.1 27.0 18.5 23-3 8.2 10.7 6.8 9-5 5-o 6.1 13-2 4.0 5-4 2.6 7-5 27-3 25-5 18.6 22. S 78.5 9-4 6.2 8.7 4-4 5-4 12. 1 3-3 4.6 2-3 6.0 30.1 26.2 19.9 23-1 7.6 9.6 6.0 8-4 4-i 5-4 ii. 6 3-i 4-5 2.0 5-5 27-7 24.4 18.9 22.5 7-5 8.7 5-9 8.6 3-7 5-2 ii. i 3-3 4-3 1.8 4.8 B C D E F... G H I T K.. L M N O scores for the other two groups hi the third grade may be read, as well as the scores for all four groups in each of the remaining grades. 83 A glance at the table is sufficient to show that there is a tendency for the average score to diminish in passing from Group I to Group II, from Group II to Group III, and from Group III to Group IV in each of the sets in each of the grades. There are exceptions, of course, and the differences encountered in passing from a younger group to an older one vary in degree with the sets and with the grades. Since the more important facts presented in Table XXXVI are presented graphically in the diagrams which follow, we shall now pass to them. In these diagrams comparisons are made between two groups only Group I, the under-age group, and Group IV, the over-age group because these two groups represent the extremes. In the four sections of Diagram 25 the comparisons are made between the two groups throughout the six grades in the four sets in addition, A, E, J, and M. A few very interesting differences are to be found in comparing the two curves. In the simpler sets, A and E, the over-age group seems on the whole to be superior to the under-age group, while in the more complex sets, J and M, and especially in the latter, the superiority of the under-age group is quite marked. There also seems to be a difference in the relations of the two curves in the lower and the upper grades. In the former the differences between the attainments of the groups are less in evidence than in the latter. That is, the diagram would indicate that the differences between the under-age pupils and the over-age pupils become more noticeable as we proceed upward through the grades; and this is especially true of the records made in the more complex examples. Passing now to Diagram 26 we come to a similar comparison of records made in subtraction. The records made by the two groups in the two sets of examples in subtraction are here graphically pre- sented. The same conclusions may be drawn from this comparison as were drawn from the comparisons of the records of the two groups in the four sets of addition, viz., that the differences are less marked in the simpler than in the more complex set and that the superiority of the under-age group increases with the progress through the grades. 84 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC The records made in the three sets in multiplication, C, G, and L, by the two groups of pupils under comparison are shown in Dia- CrfHJp X (Under-flfc) Group /F '(0i/d& DIAGRAM 29. A comparison of records made by two age groups through Grades 3-8 in fractions (Set O). 88 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC more clearly than any of the others thus far examined the increas- ing superiority of the under-age group as we proceed from the less to the more complex types of operation. This is indicated, with exceptions of course, by the increasing divergence of the two curves as we pass from left to right in the diagram. Cro SJ^ ?20 Cr-ottjalff (.Over-Age) -- H 7 IT A S DIAGRAM 30. Average "units" made in each set by two age groups in eighth grade. This same matter is approached from a slightly different angle in Table XXXVII and in Diagram 31. By use of the system of weights just referred to, the average "rights" made by each of the groups in each of the sets throughout the six grades has been con- verted into the terms of the "unit." The 15 sets were then classi- fied into six groups on the basis of complexity. Into the first group were put Sets A, B, C, and D, for the examples of these sets are clearly the most simple examples in the test. Into the second group were put Sets E, F, and G, representing the examples of the ABILITIES OF CERTAIN AGE AND PROMOTION GROUPS 89 next degree of complexity. Set I was not included in this group because the returns seemed to indicate that it is of much greater difficulty than any of the other three sets. Set H was put in a class by itself because of the peculiar type of reactions made to it by the pupils. Sets I, J, and K were then put into the fourth TABLE XXXVII AVERAGE NUMBER OF "UNITS" MADE IN CERTAIN GROUPS OF SETS BY EACH OF FOUR AGE GROUPS IN GRADES 3-8. DATA FROM 2,400 PUPILS Set Third Grade Fourth Grade Fifth Grade I II III IV I n III IV I II in IV A, B, C, D. . 49 30 3 ii 5 42 26 2 10 4 47 29 3 10 5 45 29 3 10 6 75 48 9 32 3i 72 5 7 29 27 70 46 8 29 24 72 51 IO 31 26 92 65 IS 49 45 ii 87 62 16 45 44 6 85 61 12 44 38 n 87 59 13 43 34 ii 247 E, F, G. . H I, J, K.. L, M, N O Total 98 84 94 93 195 185 177 190 277 260 251 Sixth Grade Seventh Grade Eighth Grade I II m IV I II in IV I II m IV A, B, C, D. . 97 67 22 60 52 26 99 68 22 60 53 24 96 64 19 55 43 19 97 68 22 56 4 8 19 103 78 26 77 69 33 107 73 27 70 59 27 112 76 29 69 61 24 no 73 28 66 49 22 117 88 3 2 94 86 42 112 80 29 84 75 33 117 79 28 80 68 3i 112 7 6 29 76 67 27 386 E, F, G H I, J, K.. L, M, N O Total 324 326 296 310 386 363 37i 348 459 413 403 group because they were considered to be less complex than the last three sets in the fundamentals, L, M, and N, which were put into the fifth group. Set O was, like Set H, kept by itself because it is different from the other sets, the examples being more complex and having been worked with a larger percentage of error than those of the other sets. Although there may be serious question con- cerning some of these groups, the writer is of the opinion that the four groups composed of Sets A, B, C, and D, Sets E, F, and G, Sets L, M, and N, and Set O, respectively, do represent groups of examples of increasing complexity. 28 20 ./* 4 S 6 7 8 3 4- !> 6 7 ^ M. / J8 36 34 J2 30 26 2<5 24 Gat O S 6 7 8 2 J * 5 * 7 8 DIAGRAM 31. A comparison of average numbers of "units" made in certain groups of sets by two age groups through Grades 3-8. ABILITIES OF CERTAIN AGE AND PROMOTION GROUPS 91 JX Thinking this last statement to be a safe proposition, Diagram 3 1 has been constructed, in which the average records made by the under-age and the over-age groups in these four groups of sets are compared. Thus, as we pass from one section of the diagram to the next, we proceed from records made in sets of simple examples to records made in more complex examples. Here again the fact is very clearly brought out that the superiority of the under-age group is not at all conspicuous in the simpler operations, but becomes more and more marked as the more complex operations are encountered. One more diagram should be presented before we leave this phase of the problem for the purpose of bringing out more clearly the differ- ences met with as prog- ress is made through the grades. In order to bring the cumulative force of records made in the entire test to bear on this DlAGRAM 3*--A comparison of the average V/J.J. C4.1 V/ UV--O L. lf\J k/v/tl.L \Jll CliAO - *t , * 11. ., 1 1 numbers of units made in all sets through problem, the average Grades 3-8 by two age groups, numbers of "units" made by each of the age-groups in the 15 sets are added to get a single score to represent each group in each grade. A graphic representation of this comparison is shown in Diagram 32, from which it is plain that the difference between the total scores made by the under-age and the over-age groups becomes consistently greater (except at the sixth grade) from grade to grade. Before closing the comparison between the age groups on the basis of the number of examples worked correctly, two summary 2 8 9 2 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC tables should be presented, Tables XXXVIII and XXXIX. In the first is presented the average number of "rights" made in each set by the four age groups in Grades 3-8 combined. By applying our own system of weights to this table we obtain the second, in TABLE XXXVIH AVERAGE "RIGHTS" IN EACH SET FOR EACH OF FOUR AGE GROUPS IN GRADES 3-8 COMBINED Set Average Rights I II III IV A.. 23.6 19.2 14.9 17-4 6.4 7-0 4-7 5-3 2-7 4-0 7-2 2-3 3-S 1.2 3-0 23-3 17.9 14.9 16.7 6.2 6-5 4-5 S-i 2-3 3-8 6.8 2.1 3-i i.i 2-5 23-9 18.5 *5-3 16.5 6.2 6-4 4-4 S-o 2.2 3-7 6-5 1.9 3-o I.O 2-3 23-9 18.0 iS-5 16.3 6-5 6.2 4-3 5-2 2.1 3-8 6-3 1.8 3-0 0.9 2.1 B C D E F... G H I J K L M N o which is found a statement of the average number of "units" made in the entire test of 15 sets by each of these age groups in the com- bined six grades. These tables bring out nothing that has not TABLE XXXIX AVERAGE "UNITS" MADE IN EACH OF FOUR AGE GROUPS IN ALL SETS BY GRADES 3-8 Group I II III IV Average "units" made. 290 272 265 262 already been discussed, but merely present in summary form what has already been included in the other tables. They may therefore be passed by without further comment. ABILITIES OF CERTAIN AGE AND PROMOTION GROUPS 93 Now, to sum up briefly the facts that have been discovered with reference to the number of examples worked correctly by the pupils TABLE XL AVERAGE "ATTEMPTS" IN EACH SET FOR EACH OF FOUR AGE GROUPS IN GRADES 3-8. DATA FROM 2,400 PUPILS Set Third Grade Fourth Grade Fifth Grade I II III IV I II III IV I II III IV A 17-3 "5 8.6 9.0 4-9 4-o 2.6 1.8 1-4 2.9 16.6 9.9 7-2 4-7 3-5 2-4 i-5 2.8 I7.I II. I 7-5 8.2 5-3 3-7 2-3 1.8 3-2 17.1 10.6 9.0 7-9 5-4 3-8 2.8 1.8 1.6 3-2 2O. I 15-4 15.0 14-5 6.1 5-7 4-3 3-7 2.2 4-3 4.8 3-0 3-7 20. 6 13-4 15-3 14-5 6.2 6.0 4-3 3-0 1.9 4-5 4-6 2-9 3-5 19.9 14.1 14-5 5-5 6.0 4-2 3-4 2.1 4-4 4-4 2-7 3-6 21.7 14-7 13-7 6.6 4-4 4.0 2.2 5-0 4-9 3-3 4-2 24.0 19-3 I 7 .6 18.6 6.8 8-3 5-5 6-3 5-0 7-2 3-7 4-5 22.9 19.0 16.7 17.2 6.6 7-7 5-6 6-5 2-7 4-8 6.8 3-6 4-2 22.9 18.3 17.2 17-0 6-7 7.6 5-6 6.0 2.8 6-7 3-6 4-3 24.5 18.2 18.0 16.7 6.9 7-7 5-4 5-8 2-7 5-o 3-4 4-2 B c D E F G H I J.. K. . L M 2.1 1.9 2.0 2-3 N O o-5 0-5 0-3 O. 2 I.O 0.8 x.3 1-7 A.. Sixth Grade Seventh Grade Eighth Grade I II III IV I II III IV I II III IV 25.2 21.3 18.4 20. o 6-7 8-5 5-9 9-2 3-8 5-5 9-3 3-6 4-9 1.9 7-2 25-4 20.5 20. o 20. o 7-o 8.6 6.2 9-3 3-8 5-6 9.0 3-9 5-2 2.2 7-5 25-7 19-7 19.6 19.4 6.8 8.0 5-8 8.9 3-4 5-3 9.0 3-7 4-6 2.0 7-3 25-7 20.5 20.7 20. o 7-5 8.7 6.2 9.8 3-6 6.1 9.1 5-4 2.0 7.8 26.4 22.7 19-3 21 .9 8.2 9-9 6.6 10. 1 4-9 6-5 11.4 4-4 6.1 2-4 8.4 28.6 21.4 21. 22-9 7.8 9.8 6-5 10.9 4-3 6.1 II. O 4.6 5-5 2.4 7-9 29.7 24-3 22.5 22.5 8.! 9-9 6.9 11.7 4-4 6-3 10.7 4-6 5-8 2-5 8-4 29.0 22.9 22.8 22.2 8.1 9-5 6.9 11.4 4-4 6-3 10.8 4-4 5-7 2-3 8.0 30-4 27.2 21. 23-8 8-5 11. 6 7-5 11.9 5-6 7-4 13-5 5-3 6.8 3-o 9-3 27.7 25-8 21.2 23.1 8.0 7-o 10.8 7.0 12.6 S-o 6.1 2.8 8.9 3-4 26.5 22.3 23-9 8.2 10.6 6.9 ii. 8 5-0 12.4 4-7 5-9 2.6 8.7 28.1 24.7 21.4 23-1 8.0 9-9 6.8 n. i 4-5 6.8 11. 8 4-8 6.0 2-5 8.4 B c D E F G H I J K L M... . N O in these four age groups, it may be said: (i) that there are differ- ences, (2) that on the average the younger groups are superior to 94 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC the older, (3) that this superiority is more marked in the later than in the earlier grades, and (4) that it is also more marked in the handling of the more complex than in the handling of the simpler types of examples. We pass now to an examination of the records made by the pupils in these four age groups for the purpose of discovering differ- ences in the number of examples attempted in the various sets. These facts are presented in detail in Table XL. Since this table is identical in form with tables already explained, our attention may be directed at once to the facts themselves. A glance is sufficient to show that no such differences are to be found in the comparison of the "attempts" made by the four groups as were found in the com- parison of the "rights." Especially is it true that in the simpler sets there is no tendency for the under-age pupils to attempt more examples than do the over-age pupils. In the more complex sets, however, there is such a tendency, especially in the upper grades, but to a much less marked degree than in the case of the "rights." Let us therefore turn to Table XLI and Diagram 33, in which are presented the total scores attempted in the entire test by each of the groups in each of the grades. This total score has been TABLE XLI AVERAGE "UNITS" ATTEMPTED IN ALL SETS BY EACH OF FOUR AGE GROUPS IN GRADES 3-8 Grade Group I II III IV 133 250 327 390 454 517 121 243 37 403 447 486 130 239 3" 386 466 489 136 267 312 412 454 469 4 e. . 6 7 8 determined by using the system of weights already employed in obtaining total scores of "rights." The diagram shows very clearly that there is no clear difference between the two extreme groups. In three of the six grades the over-age group attempts more ABILITIES OF CERTAIN AGE AND PROMOTION GROUPS 95 "units" than does the under-age group, while in only two grades is the reverse the case. In Tables XLII and XLIII the facts concerning attempts are put in summary form. In the first table we have the average number of examples attempted in each set by each of the four age groups in Grades 3-8 combined, while in the second table these set averages are reduced to a single score by the use of the system of weights. Table XLII shows no clear tendency of any one group to take the lead; one group forges ahead in one set only to fall back in another. Table XLIII likewise shows no differences worth considering. In summary it may therefore be said that the study reveals no clear differences between any two of the four age groups in numbers of examples attempted in the various sets. A third phase of the problem now presents itself, although it is implied in what has gone before, and that is the question of accuracy. Of course, since it has been found that the under-age pupils excel in "rights" and on the average attempt no more than do the over-age pupils, it necessarily follows that the former have attained a greater degree of accuracy. How- ever, since this is only a general impression, it will be worth while to make a special study of accuracy. 510 480 42.0 390 360 330 300 210 V" IIH 98.^ 98.1 B 08. i 07.7 O7.4 96.6 c 89.6 88 4 88.6 87.4 D 96.6 95.6 94.9 94.2 E Q7 . 7 O2 .4. 01 .6 OI . 2 F... 87.5 84.7 84.2 80.6 G 86.4 85.0 84.3 80.3 H 74-3 72.8 68.3 7I.I I 76.8 72.6 69.6 64.9 T 75. 5 73.2 71.7 71.2 K. . 93- 3 02.^ 89.9 86.3 L 67.3 62.4 59 -6 54-8 M 74.0 68.9 67.6 64.1 N 71. 1 65.5 57 .2 Si-6 O 60. 3 S9.2 52.2 47.3 ioo STUDIES IN THE PSYCHOLOGY OF ARITHMETIC All Sets Diagram 35. In a previous paragraph the comment was made that differences in accuracy between the two extreme age groups did not increase in the same degree with progress through the grades as was found to be true in the case of differences in "rights." This point is brought out very clearly in Diagram 35, which shows that, while there is some slight increase in the superiority in accuracy of the under- age group over the over- age group, as we pass up through the grades the in- crease is of small signifi- cance. Table XL VI is a sum- mary in which there is presented the average accuracy made in each set by the four groups in the six grades combined. An examination of this table shows it to be quite re- markable, for there are only two cases in the entire table where passing from the percentage of accuracy made in a particular set by one of the age groups to the percentage of accu- racy made by the next older group is not accom- panied by a decrease in (i) in Set C the accuracy loo 9f 90 SS 80 75 70 55 2$ 20 IS 10 S DIAGRAM 35. A comparison of percentages of accuracy made in all sets by two age groups through Grades 3-8. accuracy. The two exceptions are: of Group II is 88.4 per cent and that of Group III is 88 . 6 per cent; (2) in Set H the accuracy of Group III is 68 . 3 per cent and that of Group IV is 71.1 per cent. From this it is certainly evident that on the average the younger pupils of a grade are more accurate than the older pupils. ABILITIES OF CERTAIN AGE AND PROMOTION GROUPS 101 The facts presented in Table XL VI for Groups I and IV are presented in Diagram 36 in graphic form. The one thing empha- sized by this diagram, in addition to the mere fact that the under- age group is more accurate than the over-age group, is that the difference becomes more marked as we pass from the simpler to the more complex examples, until the greatest difference is found in Set O, fractions. Group 2T (.1/ae/e.r-Aji) Group jy (.Over- H J J ABC 27 T DIAGRAM 36. Average accuracy made in each set by each of two age groups in Grades 3-8 combined. Table XL VII presents a final statement of the relative accuracy of the four groups in all the sets in Grades 3-8 combined. These facts merely emphasize what has already been said, and they need be discussed no further. As a brief summary of the facts relating to the accuracy of the four age groups in the tests it may be said: (i) that there are differ- ences; (2) that these differences show the younger pupils to be on the average more accurate in their work than the older pupils; 102 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC (3) that these differences are on the whole quite uniform from grade to grade; and (4) that the differences are more evident in the more complex than in the simpler examples. TABLE XLVII AVERAGE ACCURACY IN EACH OF FOUR AGE GROUPS IN ALL SETS IN GRADES 3-8 Group I n m IV Accuracy 84.0 81.3 78.8 76.8 PROMOTION GROUPS The purpose of this second division of the investigation is to throw some light on promotion practices, as found in Grand Rapids, and their relations to arithmetical attainments. Although the paucity of the data has made it quite impossible to make the case conclusive in any instance, the attempt is made in this study to do four things: (i) to determine differences in arithmetical attainments of three promotion groups in the eighth grade the fast, the regular, and the slow group; (2) to determine differences among these same pupils when regrouped on the basis of age, for the purpose of finding out whether or not the age or the promotion factor is the more important; (3) to determine differences in arithmetical attainments of three other promotion groups in the seventh grade, "regular" pupils (those making normal progress), "irregular" pupils (those repeating because of transfer of schools, sickness, etc.), and failures (those repeating because of failure to do the work of the grade); and (4) to determine differences in arithmetical attainments of two more promotion groups in the eighth grade, the one composed of pupils failing below the sixth grade, the other of those failing above the fifth grade. METHOD This study is based entirely on records made by children in the Grand Rapids schools. An examination of the arithmetic folder used hi the survey of that city shows that on the front page of the ABILITIES OF CERTAIN AGE AND PROMOTION GROUPS 103 folder the following question was asked of the pupil taking the test: "Have you ever repeated the arithmetic of a grade because of non- promotion or transfer from other school ? If so, name grade Explain cause." In addition to answers to this question, through special request the children from a number of the schools indicated the fact whether they had ever skipped one or more grades. Thus records were secured showing normal progress, progress below nor- mal, and progress above normal, and the cause of slow progress repeating was ascertained in the cases where it occurred. In comparing the fast, regular, and slow pupils the method of selecting records of pupils employed in connection with the study of the age groups was adopted for the purpose of eliminating the factors of differences in giving the test, and differences in training. The same method was used in selecting records for the other pro- motion groups. In this way 150 records were secured for the first division of the study, representing 50 "fast" pupils, 50 "regular" pupils, and 50 "slow" pupils. For the second division of the study these same 150 records were used, merely being grouped in a differ- ent way. For the third division of the study, 162 records were secured, 54 for each group. The data for the last part of the study were the most meager, for it was possible to obtain only 32 records for each of the groups. The records thus selected were carefully scored; the attempts and rights were tabulated; and average attempts, average rights, and percentages of accuracy were computed. RESULTS As already stated, the first division of this study relates to three promotion groups, the first designated as the "fast" group, com- posed of pupils who have skipped one or more grades; the second designated as the "regular" group, made up of pupils who have neither skipped nor repeated; and the third designated as the "slow" group, composed of pupils who have repeated one or more grades. Each group is represented by 50 pupils in Grade 8-2. The average number of examples worked correctly in each set by the pupils in each of these groups is shown in Table XL VIII. 104 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC The total number of "units" made by each of the groups, as deter- mined by the system of weights, is also presented in this table. In general, the differences found here are of the same order as those discovered in the study of the age groups. The "fast" pupils are decidedly superior to the "slow" pupils, while the "regular" pupils occupy an intermediate position. These differences are more marked in the more complex than in the simpler examples. TABLE XLVIII AVERAGE "RIGHTS" MADE IN EACH SET BY THREE PROMOTION GROUPS. GRADE 8-2. DATA FROM 150 PUPILS SET ^r A a C D E F G H I J K L M N O $* Fast 30.6 25-1 21. 1 22.7 7-7 10.9 6.8 8-4 4-i 5-9 10.7 4-9 5-8 2-3 6.6 442 Regular 29-9 24.1 2O.7 22.8 7.8 IO.I 6-7 8. 5 4-i 5-3 10.7 4-4 5-i 1.8 5-3 414 Slow 30.2 26.1 20.6 22.4 7-2 10.3 5-9 8.1 3-7 5-4 10.2 4-1 5-2 i-7 4-7 399 The facts for the "attempts," made by the same groups, appear in Table XLIX. Although the differences between the groups are not quite so marked here as in the case of the "rights," they are very substantial, being larger than those found in the study of the Cleveland age group. TABLE XLIX AVERAGE "ATTEMPTS" MADE IN EACH SET BY THREE PROMOTION GROUPS. GRADE 8-2. DATA FROM 150 PUPILS SET ! GROUP H| A B C D E F G H I J K L M N O H Fast 30.7 25-4 22.2 23-4 8.2 11.7 7-6 11.4 4-9 7-5 10.9 6.6 7-2 2-9 9-1 5l6 Regular. . . 3 -2 24.4 21.7 23-3 7-6 10.8 7-5 10.9 4-9 7.0 II. 6.2 6-7 2-9 8.2 492 Slow 3-5 26.5 22.1 23-1 7-7 ii. 4 6.9 ii. 4 4-9 6.9 10.4 5-7 6-7 2.7 7-9 487 The average accuracy achieved by the three groups in all the sets is shown in Table L. As would be suspected from the facts presented concerning "rights" and "attempts," the "fast" group is the most accurate. ABILITIES OF CERTAIN AGE AND PROMOTION GROUPS 105 In order to discover the relative importance of the factors of promotion and of age, the 150 pupils used in the study just dis- cussed were regrouped on the basis of age, the 50 youngest being placed in one group, the 50 oldest in a second group, and the remain- ing 50 in a third group. TABLE L AVERAGE ACCURACY IN ALL SETS. THREE PRO- MOTION GROUPS. GRADE 8-2 Accuracy Fast 8x o o to ^ O M -^ 9 M saaqiO r, 2 * * iS o> 10 oo SABIS M : : to : :^. 00 sapaMg n M to M : O M M siapuBjiOH o o 10 ^00 O * M M . M n pnox M'^SS^ M - O o waqJO n M M -00 M tO M oo SABjg -*:::::: : : :* 00 O sapaAvg SUBULiaf) M Mnn M -00 >o M- SjapuBnoH tOM 1^00 M 00 : O o 10 suBouauiy ^ O fO cOO M : to !o l^jox ?J?M t?M M C* t^ 1 swqiQ Ocor-^OD MOO t~ N 0^ 7 6AB IS I/) H M M . MOO J N sapaMg i ' M . M CO M CO M O ETTBUUaf) M :, MO -O to siapuBnoH *o^ : to - 10 <* 1O 10 snBDuaray C0to^ *0>0 8 l*V>x tO O *O O M O" M OQ M N i swqiO co^oo^tor. W M tOOO SABIS ::: |u, J 9 sapa^s : : : MM : M o SUBUIWO MM :oto >. to uapmnoH M * M MOOO i suBDuauiy COO M OO ^ to M^ o to *t M MO O tO O Ol MO V 10 1*4 X M O O* to r- O v TtOOO 00 q?o M w 6ABJS *::: M :, M sapaMg H M 10 M -to o M O sutmijsf) : :, COMOV, IO SJ9pUBIIOJT M " M 5 suBOuamy 1O M M tO CO M M 2 SCHOOL a 8 ' 2 t* v c ' a ) - ? " ;3 iijijii 'me South Division. Furner Union i ABILITIES OF CERTAIN RACE GROUPS 115 races had equal representation in each of the classes from which records have been taken, or if we had sufficient data so that equal numbers of the races might be taken from each class, as was done in the study of age groups, the question would be a very simple one. But, since the races are not equally represented in the classes and the data are strictly limited, some method must be found of elimi- nating differences of school training and methods of giving the test. Referring to Table LXI again, let us consider the problem as related to the records of the pupils in Grade 8-2, and for the moment let us confine ourselves to the records made by the Americans and the Hollanders. Five of the American pupils in this grade are taken from the Coldbrook School, while but 2 Hollanders are taken from the same school. Now if the training received in the Coldbrook is superior to that received in the other schools, and if the records made by the 50 American pupils and the records made by the 50 Hollanders in the grade be averaged without eliminating this factor of difference of training, the American group would be given an advantage because of its greater representation in a superior school. The method adopted for eliminating differences in school train- ing is as follows: The records made by the entire grade in one school are taken as a base, and a system of coefficients is computed by the use of which the records made in the other schools may be converted into the records of the school taken as a base. Thus the thing that is actually done is to determine what the records of the different pupils and groups of pupils would have been if they had all been in the same school, subjected to the same school influences. To be more concrete, let us again turn to Table LXI and indicate specifically how the method is applied in dealing with the race groups in Grade 8-2. The table shows that the data have been taken from Grade 8-2 of six schools. The next thing that must be done is to find the average records made by these six grades in each of the 15 sets of the test. These facts are presented in Table LXII. Now, taking the average score made in the Union School as a base, a coefficient is determined for each set in each school by dividing the record made in the Union School by the record made in each of the other schools. Thus a set of coefficients is found by which the records of the five schools may be converted into the records n6 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC made by the Union School, which means that differences in school training and differences in giving the test are overcome. By using the set of coefficients for any one of the schools, it is possible to TABLE LXII AVERAGE "RIGHTS" MADE IN EACH SET BY GRADE 8-2 IN Six SCHOOLS A B C D F G H I J K L M N O Union 28.3 23-5 16.8 19.0 7-5 9-3 6.7 5 j 4-7 5-4 9-4 4-9 4-9 1.9 4-8 Coldbrook 28.8 28.8 22.3 27.2 8-3 13-5 7-9 8.8 5-5 6.8 12.2 6-5 6.8 3-i 5-3 Diamond 3i-7 25-5 20.7 22.7 8.0 10.3 6.4 12.0 4-7 5-8 10.3 4-9 5-0 2-3 4-3 East Leonard 34-8 30.8 21-5 24.8 8.9 12.9 9-3 14.2 6-7 6.9 I3-I 6.1 6.6 4-i 6.1 Lexington 32.0 30.8 23-5 24.6 9.0 12.3 7-7 7.8 4-7 6.7 II. O 5-8 6-5 2.7 6.0 South Division. . 32-5 26.5 20.5 23.0 7.6 IO.O 6-3 8-5 4.0 5-0 10-3 4-3 5-3 1.8 4-0 determine what sort of a record a group of pupils in that school would have made if they had been trained and tested in the Union School. These coefficients are presented in Table LXIII. TABLE LXIII VALUE OF EACH EXAMPLE IN EACH SET FOR GRADE 8-2 IN EACH OF Six SCHOOLS IN TERMS OF RECORD MADE BY GRADE 8-2 IN UNION A B C D E F G H I J K L M N Union I.OO I.OO I.OO I.OO I.OO i .00 1 .00 I.OO I.OO I.OO I.OO I.OO I.OO i .00 I.OO Coldbrook. . Diamond. . . East 0.98 0.89 0.82 0.92 0.75 0.81 0.70 0.84 0.90 0.94 0.69 0.90 0.85 1.05 0.58 0-43 0.85 I.OO 0.79 0-93 o-77 0.91 o-75 I.OO 0.72 0.98 0.63 0.83 0.91 1. 12 Leonard. . Lexington. . South 0.81 0.88 0.76 0.76 0.78 0.72 0-77 0.77 0.84 0.83 0.72 0.76 0.72 0.87 0.36 0.65 0.70 1 .00 0.78 0.81 0.72 0.85 0.80 0.84 0.74 0-75 0.46 0.70 0-79 0.8o Division. . 0.88 0.89 0.82 0.83 0-99 0.93 i. 06 0.60 1.18 i. 08 0.91 1. 14 0.92 i. 06 I. 2O All that remains now is merely to point out the way in which these coefficients are applied to the race groups. Referring again to the section of Table LXI which presents the facts for Grade 8-2, we find 5 American pupils in the Coldbrook School. The total number, of examples worked correctly in each of the sets by these 5 pupils is determined. Turning to Table LXIII we find the Coldbrook coefficient for Set A to be 0.98. The total number of "rights" made by the 5 pupils is multiplied by this quantity. The same thing is done for each of the other sets. Then we pass to the 4 pupils in the Diamond School and repeat the process, and like- ABILITIES OF CERTAIN RACE GROUPS 117 wise with the Americans in each of the other schools. Then the total scores thus made in each of the sets by the American pupils in the six schools are added. This grand total is then divided by 50, since that is the entire number of American pupils in the grade, and an average score is secured for each of the sets. Like procedure is followed for each of the other four race groups. An objection which may be raised to this method is that in eliminating differences due to training and to the giving of the test race differences are also eliminated. The answer to this objection is that it would be valid if mixed schools that is, schools in which several races are represented were not used for the study. To the extent that a difference between the average records made by two schools is due to the presence of an inferior or a superior race group in one of the schools, the method does eliminate race differ- ences as well as differences due to school training and methods of giving the test. But, since the schools are mixed, no one race can greatly affect the average score of a school. The objection thus resolves itself into the question of the racial composition of the classes from which the records used were taken. We must there- fore refer again to Table LXI, in which the composition of each of these classes is given in detail. If it be remembered that the cate- gory of "others" includes representatives of races other than the five used in this study, pupils of mixed parentage, and pupils con- cerning whom there was some doubt as to race, the table shows that in no single instance does one race represent a majority of the class, and in only two cases, Diamond, Grades 6-2 and 7-2, does one approach representing a majority. Thus it is quite evident that the differences between any two schools cannot be attributed in any appreciable degree to the predominance of any race in one or the other of the schools. It may be that the method used in this study does minimize to some small extent racial differences, but to a very small extent if at all. RESULTS The average scores made in each set as computed by the method just outlined, by each of the five race groups in Grades 6-2 to 8-2, are found in Table LXIV. An examination of the table seems to Il8 STUDIES IN THE PSYCHOLOGY OF ARITHMETIC I " $ I O to ^ O HI vo to OO OO O ^f co O IH SABIS ON t^ O ** O to ^ CO ^"OO ^" to HI to M M M IH OO ^ O PI COOO VO CO t^ HI TfM PI OO O 7 O Pi O OO tOO O co^O O^OIH ^ CO PI IH M M Tf 1>.OO t^- IH IH t- O O O COOO OO to Tf PI d HI HI O CO OCOIH r>. t^. HI OO P> OIH c< to^"CO O ^O I s * ^ to M to to PI to CO P* PI PI HI M O COTJ-IH t^oOOcoO>or^iOM t^ M O *^OHI t^Otot^^"toO ^*toc4 to H O O tooo PI IH M r^o IH co PI co t*~ CN 3 M M IH O OOO OO to COO O to to HI to CO PI M M O ^- ^- O PI i^ t^ O COOO to to to O O PI PI PI M PI PIOO'*IOIOO^-HIIOHIOOPIO'* PI PI PI PI COOOIHOOHIHIIOHIMOCOMOOPI SA13IS Pi Pi OO HI vo OO tocoiOOtoioiH co CO PI HI N t^PIHIPIlOPICOHICOOl^Ot^^HI t^oo o r- tooo to Tt co tooo to << M co PI HI HI HI H HI Otow QtoOcoO 'tO'tPi OO M PI OO O t^OO to IO CO IOOO 'I- to HI CO CO PI HI HI O t^w wt^d MvOOOOO Ococo COO P O HI vo OO O OO to to PI ^00 ^" ^* HI co ^" PI co O HI SABJS l>- HI O to l^OO O l>. lOt^lOlOWO PI PI PI PI OO too O to -t Pi tooo co HI i^. OOO l^ 7 ptSSp1 lN-0>oooc.O HI Tj-MOOOO 'i'OO t^ O 't 1 Tj- O O HI l-00 IOO PI ^O CO Tf HI P PI HI Ol PI to t^ TJ- co Pi to OO co PI OO OO OOO O SABjg CO PI HI HI HIHIOOPI^"OHIPI OO O O OO ^"O OO 7 PI PI HI HI l^-Ol^Pl O PIO t^O OO tooo COOO 3 PI PI HI HI piHipiOPii^Ot^Pir^i^Tt-coPiO PI PI HI HI HI >O cotOPI Tf^rtO O COO Pi P< co HI PI PI M HI "