c'r/.rfe© V^iQ-O Colorado State Teachers College BULLETIN SERIES XX JULY, 1920 NUMBER 4 A, Comparative Study of Three Diagnostic Arithmetic Tests GEORGE WILLIAM FINLEY Published Montlily by State Teachers College, Greeley, Colorado Entered as Second-Class Matter at the Posto-lfice at Greeley, Colorado, under the Act of August 24, 1912 A COMPARATIVE STUDY . of Three DIAGNOSTIC ARITHMETIC TESTS by George William Finley, B. S., M. S., Professor of Mathematics in Teachers College Greeley, Colorado FOREWORD This study was undertaken with a view to making a comparison of the results obtained by the use of different arithmetic tests. Those chosen for comparison were the Cleveland Survey Tests, the Woody Scale, and the Monroe Diagnostic Tests. All three of these purport to be diagnostic in their nature, and if this be true they should lead to approximately the same conclusions concerning the arithmetical abilities of the children tested. It was with a desire to determine whether they do this or not that this study was made. The study is divided into two parts. Part I gives a discussion of the value of arithmetic tests in general and a description of the tests used. Part II gives the results obtained by giving the three different tests to a group of children and the conclusion reached from these results. A Comparative Study of Three Diagnostic Arithmetic Tests PART I In recent years there has been a most remarkable development of all kinds of educational tests and measurements. Of course it has always been necessary for teachers to measure their pupils’ attainments in some fashion or other. Some children were promoted at the end of the year while others were retained in the same grade. This was done because the teacher judged that in the one case sufficient progress had been made to enable the children to do the work of the next grade, while in the other such progress had not been made. In order to arrive at these conclusions the teacher had to measure the achievements of the various children in the grade. Again at the end of each month teachers were called upon to “grade” the pupils in the various subjects that they happened to be studying. This again called for the measuring process. But the sort of measuring done was of a very indefinite kind. It was made up very largely of the teacher’s estimates of the child, and into it entered a great many things besides the ability to do certain specific things. Then, too, the teacher’s knowledge of the specific abilities of the children was exceedingly limited. It is true so-called tests and examina¬ tions were given but they were of such a nature as to test the abilities of the children only in a very general way. In fact they were often said to test the children’s general ability in this, that, or the other subject, whereas, as Ave now know, there is no such thing as general ability in a subject. There are, in fact, as many separate abilities in even a single subject as there are different types of mental activities involved. Another difficulty with these tests was that they lacked uniformity. If a child did not do as well in a test in arithmetic this week as he did last week it was taken to mean that he was losing ground. This might not be at all true. The tests were different and therefore there was really no basis for comparison. Again, if a child in the sixth grade got a grade of 90% in arithmetic while one in the eighth grade got a grade of 70'% this fact did not give any basis for comparing the abilities of these two children. Their grades were obtained upon entirely different tests. This then was the state of things up to within the last twenty years. At the present time, however, quite a different state of affairs obtains. Tests and scales have been developed and standardized so that a teacher need no longer be in doubt about how her pupils compare with other pupils in the same grade, with pupils in other grades of the same school, with pupils in other school systems, or, best of all, with their own jirevious records in any specific ability. The Courtis Standard Research tests were not given in this experiment, but as all of tlie scales have been built, to a greater or less extent, upon them they will,be discussed here. Inspired by the work of Rice and Stone, the pioneers in the field of tests and measurements in arithmetic, Mr. C. A. Courtis took up the task of developing a set of standard tests. He worked out a set, now known as series A, which he gave to thousands of children in different parts of the country. Five thousand children were tested in Detroit; 33,000 in New York; 20,000 in Boston, and many others in smaller systems. In scoring these ])apers perhaps the most remarkable fact brought out was the wide range of variability shown by the children in any given grade. Some children in the sixth grade, for instance, made scores lower than the average of the third grade while others exceeded the average of the eighth grade. In spite of this fact, however, Mr. Courtis found that the scores for the children of the sixth grade tended to be grouped about a certain standard of excellence which was a little lower than that about which the scores of the seventh grade children tended to be grouped and higher than that of the fifth grade. This lead to the establish¬ ment of certain standards of excellence for the different grades in the par¬ ticular abilities tested by these examples. Series A of the Courtis tests includes eight separate tests, each one con¬ taining more examples than the swiftest child could complete in the time allotted. The tests are thus a measure of speed as well as of accuracy. These eight tests take up the combinations in addition, subtraction, multiplication and division, speed copying of figures, one-step reasoning problems, abstract examples in the four fundamentals and two-step reasoning problems. After using this series for several years Mr. Courtis, and others as well, found that it was not satisfactory in several respects. In the first place it was too expensive in both thne and money. Then again it did not given an adequate test of the abilities most needed by the pupil. It tested the pupil’s knowledge of the addition combinations but did not give much information concerning his ability to apply this knowledge to the addition of columns of numbers. The same is true of the other operations. He found also that there was practically no relation between a child’s ability to give the addition combinations and his ability to add a long column of figures. He therefore devised a second group of tests known as series B. This group consists of four tests, one for each of the fundamental operations. Test 1 involves the addition of columns of 0 three-place numbers; Test 2 the subtraction of eight-place from eight- and nine-place numbers; Test 3 the multiplication of four- by tAvo-place numbers, and Test 4 the division of four- and five-place numbers by two-place numbers. These tests have also been thoroughly standardized. These Courtis tests are of great value to the teacher or supervisor of arithmetic. They furnish an instrument by means of which he may deter¬ mine the degree of excellence reached by a grade or an individual in any one of the four fundamental operations. But they are not primarily diagnostic in their nature. Whatever diagnosis is made by their use is general and not specific in its nature. They do show, for instance, that a certain grade is low in addition, but they give no suggestion as to just which one of the several abilities required in addition is at fault. Then, too, they are limited to the field of the four fundamental operations with integers. Realizing these facts a number of-investigators have been at work devis¬ ing tests that would be primarily diagnostic in their aim. Three such tests or scales have been devised and used to a considerable extent, viz., the Cleveland Survej'^ tests, the Woody scale, and the Monroe tests. We shall consider them in the order given. THE CLEVELAND SURVEY TESTS When Dr. Judd and his co-laborers started the Cleveland Survey they looked over the field of existing tests and scales in arithmetic and decided that none of those that had been developed up to that time would meet the needs of the situation. The Courtis tests seemed to be the most 'promising but they were open to serious objections. Series A they felt to be unsatisfactory for the same reasons as those already given in this discussion. Series B used as a supplement to series A would constitute a decided improvement. But even this combination did not go far enough to suit them. By using the combination they saw that they could measure general attainment in each of the four fundamental operations but nothing more. In other words the test would not be diagnostic. For 6 instance, a pupil might shoM’ by his ^A'Ol•k on Test 1, Series A, that he knew his addition tables perfectly, and yet he might fail utterly on Test 1 of Series B. These facts, they argued, would be Avorth knoAAung, but they would be of comparatively little value unless supplemented by other facts. The question of why he failed on the second test would remain unanswered. It might be because he failed “to bridge the attention spans,” or because of his inability to “carry,” but the tests would give no indication, as to which it was. In order to throAV light upon this question it was necessary to introduce betAveen the simple types of the first series and the more complex types of the second some intermediate forms. These investigators accordingly secured the co-operation of Mr. Courtis and worked out Avhat are knoAvn as the Cleveland Survey Tests in Arithmetic. These tests are here reproduced in full. .They consist of 15 sets, designated A, B,-0. There are four sets in addition (A, E, J, M), tAvo in subtraction (B, F), three in multiplication (C, G, L), four in division (D, I, K, N), and tAvo in fractions (H, O). This gives a spiral arrangement, as the pupil begins Avith Set A and takes each set in its proper order. In the sets involving addition. Set A, which is simply Test 1 of Series A in the Courtis Standard tests, requires simply a knowledge of the combina¬ tions. Set E requires the addition of columns of five one-place numbers. This, then, is a new type. The pupil must combine the first two numbers and must then hold this sum in mind while he combines it in turn with the next number. Set' J requires the addition of 13 one-place numbers. This again introduces a new element, “bridging the attention span.” It is a Avell known fact that the addition of a long column of numbers is not one continuous process. The individual rather adds up several numbers, pauses for a moment Avhile the attention Avavers, then continues the addition. The fourth set, M, requires the addition of columns of five four-place numbers. This brings in another mental process, that of “carrying.” The four sets then indicate ability or lack of ability (1) in addition combinations, (2) in adding several numbers in a column, (3) in “bridging the span of attention,” and (4) in “carrying.” The tests contain but two sets in subtraction. Set B tests the knowledge of the subtraction combinations, while set F, the subtraction of three- from three- and four-place iuimbers, tests a knowledge of borrowing. This covers the field of subtraction. In multiplication there are three sets. Set C gives the simple combina¬ tions, Set G, the multiplication of four-place by one-place numbers, tests a knowledge of “carrying,” while set L, ^le multiplication of four- by two- place numbers, requires a knowledge of the mechanics of handling the multi¬ plication by a second number in the multiplier and of the addition of the partial products. In division there are again four tests. Set D tests a knowledge of the simple combinations. Set I, the division of five- by one-place numbers, intro¬ duces “carrying.” Set K, the division of three- and four- by tAvo-place num¬ bers, brings in the simplest type of long division, involving no carrying in the multiplication, and no borrowing in the subtraction. Set N is the more complex type of division requiring both carrying and borrowing. These tests attempt also to diagnose the ])upirs ability in fractions in addition to his ability in the fundamentals with integers. For this purpose Sets H and O were introduced. Set H requires addition and subtraction of fractions having a common denominator, Avdiile in Set 0 fractions of unlike denominators are added, subtracted, multiplied and divided. The Cleveland Survey tests carry out the plan of the Courtis Standard tests as to time allowance. The time limit ranges from 30 seconds to 3 minutes, d'he plan Avas to give sulficient tinie for even the sloAvest pupil to Avork out at least one example but not enough to allow the swiftest to finish them all. 7 Aritl metic Exercises Cleveland Survey Tests Name.Age today. Years Months Grade. School.. Room. Teacher..• •.Date today. Have yon ever repeated the arittnnetic of a grade because of non-promo¬ tion 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. Woik rapidly and accurately. There are more problems in each set than you can work out in the time that Avill be allowed. Answers do not count if they are Avrong. Begin and stop promptly at signals,from the teacher. 1 1 1 A 1 1 B 1 C 1 1 1 1 1 D 1 E 1 1 1 1 F 1 1 G H A 1 1 1 1 1 1 1 1 'l 1 1 1 i 1 1 1 1 1 1 1 R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Rank 1 1 1 1 1 i 1 1 1 1 -V- 1 1 1 1 1 1 1 1 1 1 I 1 J 1 K 1 L 1 :\I 1 N 1 1 1 1 1 1 1 0 1 1 1 1 1 1 A 1 I I 1 1 1 1 1 1 1 1 1 K 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Rank III II 1 1 i 1 1 1 1 8 9 SET D—Division— 3)9 4)32 6)36 2)0 6)48 1)1 5)10 2)6 8)32 1)8 5)30 8)72 2)10 7)42 1)1 6)18 1)3 2)8 6)6 3)27 5)0 3)24 9)63 2)4 6)42 3)0 7)21 4)4 SET E- —Addition— 5 2 9 2 6 2 8 8 8 3 2 8 0 5 4 0 5 7 0 8 4 1 6 6 8 6 2 6 8 5 7 7 ■ 2 5 9 8 3 3 1 6 5 4 9 3 3 5 1 3 8 . 8 SET F- —Subtraction 616 1248 1365 456 .709 618 1267 1335 707 509 419 277 1355 908 519 616 258 324 1009 768 1269 269 295 772 SET G- —Multiplication— 2345 9735 8642 2 5 9 9735 2468 6789 9 3 6 5432 9876 8642 4 8 5 5432 3689 2457 8 5 6 Ats. Rts. 7)28 9)9 3)21 4)24 7)63 6)0 1)0 9)36 1)7 3)6 4)20 7)49 8)64 1)2 4)16 8)24 7)7 -2) 18- 3)15 9)81 7)0 1 4 9 4 6 7 2 5 1 '5 3 5 4 4 3 4 1 3 0 4 7 8 1 2 5 8 9 5 4 6 1092 716 472 344 816 1157 335 908 1236 1344 908 818 615 854 527 286 6789 2345 2 6 3579 2468 3 7 • 3579 9876 7 4 9863 7542 4 . 7 10 SET H—Fractions- SET H—Fractions— 1 Ats. Rts. 3 1 6 4 4 1 8 7 -i-= -- = — H-= — -= 5 5 9 9 9 9 • 9 9 1 5 3 1 1 4 6 2 _1-= -= — -1-= — -= 9 9 7 7 7 7 7 7 2 4 4 1 5 1 6 5 _1 _|_= --= — + —= —• - -= 9 ■ 9 5 5 8 8 9 9 7 1 5 2 5 2 8 1 -1-= -= H-= — -= . 9 9 7 7 9 9 9 9 1 3 6 1 2 1 5 4 __ _j_= - —. = ■ + —= — — ■—■ = 8 - 8 8 8 7 7 9 9 2 6 4 3 4 2 7 5 —+ —== --= - + —= — -= 9 9 8 8 7 7 9 9 SET I—Division— 4)55424 7)65982 2)58748 5)41780 9)98604 6)57432 3)82689 6)83194 8)51496 9)75933 8)87856 4)38968 SP]T J—iVddition— 7 9 4 7 2 9 6 7 7 8 9 4 3 2 5 2 5 1 9 6 9 1 8 0 5 3 1 1 4 4 8- 9 4 2 6 5 5 7 3 7 7 6 2 8 1 4 8 4 7 1 4 1 4 7 6 6 0 7 8 2 1 1 4 6 8 5 2 2 6 8 6 2 4 3 5 7 0 4 1 8 6 0 9 1 5 5 5 8 53 3 5 2 1 3 9 3 6 1 3 1 5 2 9 7 3 1 3 9 5 4 9 8 6 3 2 4 2 1 3 3 7 2 6 5 7 3 1 9 7 3 3 6 7 9 4 2 3 -4 5 2 4 6 7 6 8 0 6 8 9 8 4 2 2 9 8 3 1 7 5 6 1 4 4 5 8 9 2 9 8 5 9 6 5 6 7 5 4 6 8 9 4 SET K—Division— 21)273 i 52)1768 41)779 22)462 31)837 42)966 : 23)483 72)1656 81)972 ■ 73)1679 21)294 62)1984 31)527 52)2184 41)984 32)384 i 51)2397 82)1968 71)3692 22)484 41)1681 33)693 61)1586 53)1166 31)496 SET L—Multiplication— 8246 3597 5739 2648 29 73 85 46 4268 7593 6428 8563 37 64 58 207 — -- — — 11 SET M—Addition- 7493 8937 8625 2123 5142 3691 9016 6345 4091 1679 0376 4526 6487 2783 3844 5555 4955 7479 7591 4883 8697 6331 9314 2087 6166 1341 7314 6808 5507 8165 5226 9149 6268 9397 7337 8243 2883 8467 7725 6158 2674 6429 2584 0251 8331 3732 9669 9298 0058 7535 5493 4641 5114 7404 2398 5223 3918 7919 8154 2575 SET N—Division— 67)32763 48)28464 97)36084 59)29382 '78)69888 88)34496 69)40296 38)26562 SET 0—Fractions- 11 1 9 1 ■ 3 5 -1-= —:-= — X — = 15 6 14 4 5 6 5 2 5 19 11 5 6 21 __ _ _ 6 20 12 8 1 3 5 11 5 2 — X — = -:-= -h — = 6 10 6 15 12 8 20 1 3 3 1 _ 3 3 21 6 8 10 Ats. Rts, 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 the sets of examples. Take two days for the test; give down through I the first day, and complete the test on the next day. The time allowances below must be followed exactly. Set A. . . . Set F.... . . 1 minute Set K. . Set B... . ... 30 seconds Set G.... . . 1 minute Set L.. Set C.. . . . . .30 seconds Set H. .. ... 30 seconds Set M. . Set D. . . ... 30 seconds Set I.... . . 1 minute Set N.. Set E.. . ... 30 seconds Set J.... ... 2 minutes Set 0.. 2 minutes 3 minutes 3 minutes 3 minutes 3 minutes Have the children exchange papers. Read the ansAvers 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. Please verify the results set down by the pupils. 12 THE WOODY SCALES The Woody scales are the results of another attempt to devise a series of tests for measuring achievements in the four fundamental operations of aritlimetic. The author of the scales makes the statement that the funda¬ mental aim was to devise a series which would indicate the type of prob¬ lems and the difficulty of the problems that a class could solve correctly. Each test is, therefore, composed of as great a variety of problems as possible. They are arranged in the order of increasing difficulty, beginning with the easiest that can be found and gradually increasing in difficulty until the last can be solved by only a small per cent of the pupil^ in the eighth.grade. The degree of difficulty of each problem was determined, not by analysis, but by sub¬ mitting the tests to a large number of children and computing the difficulty of each problem from the number of children that were able to solve it. In building the scales under the above outlined plan -the author made up tests containing as great a variety of problems as possible and submitted them to a large number of children. The results of these tests showed that the preliminary tests did not conform to the plan adopted. They did not show an arrangement of problems such that they were solved by a gradually increasing per cent of the pupils from one grade to the next higher. There were large gaps between certain problems. These defects were remedied by introducing extra problems to fill up these gaps and by dropping out such problems as were solved by a higher percentage of pupils in the lower grades than in the higher grades. This methbd of construction has been severely criticised. It is main¬ tained that if we are to measure arithmetical abilities with any degree of certainty we must include in our tests problems that exercise all the im¬ portant types of arithmetical abilities, whether or not this gives us a list of problems gradually increasing, in difficulty. This criticism is undoubtedly just to a certain extent. At least it is safe to say that if we are to use the Woody scales intelligently we must know their limitations. These scales are published in two series, A and B. Series A is the more complete, while series B is made from series A by leaving out part of the problems, and is intended to be used by those who can devote but a limited time to giving the tests. Series A was used in this study and is given here in full. 13 Series A Addition Scale By Clifford Woody City. County. School. Date. . . Name..When is your next birthday? How old will you be?.Are you a boy or a girl?.. . . In what grade are you?.Teacher’s name. ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ( 6 ) ( 7 ) ( 8 ) ( 9 ) 2 2 17 53 72 90 3 + 1 =• 2 + 5 + 1 = 20 3 4 2 45 26 37 10 0 ' Z 30 25 ( 10 ) ( 11 ) ( 12 ) ( 13 ) ( 14 ) ( 15 ) ( 16 ) ( 17 ) ( 18 ) 21 32 43 23 25 +,42 = 100 9 199 2563 33 59 1 25 33 24 194 1387 35 17 2 16 45 12 295 4954 — — 13 — 201 15 156 2065 — 46 19 - — ( 19 ) ( 20 ) ( 21 ) ( 22 ) ( 23 ) ( 24 ) ( 25 ) $ .75 $12.50 $8.00 547 ]^ + >^ = 4.0125 % + % + 78 + Vs = 1.25 16.75 5.75 197 1.5907 .49 15.75 2.33 685 4.10 — - 4.16 678 8.673 .94 456 6.32 393 — 525 240 152 ( 26 ) ( 27 ) ( 28 ) ( 29 ) ( 30 ) ( 31 ) ( 32 ) i2y. Vs +1/4 + % + 1/4 = 43/4 2 % 113.46 3/^ + % + 1/4 = 62% 21/4 63/8 49.6097 12% 5% 33/4 19.9 37% - - 9.87 — .0086 18.253 6.04 ( 3 . 3 ) .49 ( 34 ) ( 35 ) ( 36 ) ( 37 ) .28 /6 + % = 2 ft. 6 in. 2 yr, 5 mo. .63 3 ft. 5 in. 3 yr. 6 mo. 121/8 .95 4 ft. 9 in. 4 yr. 9 mo. 21% 1.69 5 yr. 2 mo. 323/4 .22 6 yr. 7 mo. — .33 .36 1.01 .56 •88 ( 38 ) .75 25.091 + 100.4 + 25 + 98.28 + 19.3614 = .56 1.10 .18 .56 14 Series A Subtraction Scale By Clifford Woody City. County. School.. Date. Name.When is yoiir next birthday? How old will you be ?.Are you a boy or girl ?. In what grade are you ?.• • ..Teacher’s name. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 8 6 2 9 4 11 13 59 78 7 — 4= • 76 5 0 1 3 4 7 8 12 37 60 (12) (13) (14) (15) (16) (17) (18) (19) (20) 27 16 50 21 270 393 1000 567482 23/4 — 1 = 3 9 25 9 190 178 537 106493 (21) (22) (23) (24) (25) (26) 10.00 372 — y2 = 80836465 27 4 yds. 1 ft. 6 in. 3.49 49178036 574 1278 2 yds. 2 ft. 3 in. (27) (28) (29) (30) 5 yds. 1 ft. 4 in. 2 yds. 2 ft. 8 in. 10 — 6.25= ‘753/4 5274 9.8063 — 9.019 = (31) (32) (33) (34) (35) 7.3 — 3.00081 = 1912 6 mo. 8 da. 5 2 678 378 — 178 = 1910 7 mo. 15 da.-= 278 12 10 15 Series A Multiplication Scale By Clifford Woody City. County. School. Date.. Xame..*.When is your next birthday?. How old will you be?.Are you a boy or girl?. (1) (2) (.3) (4) 0 llCVlllt (5) (6) (7) 3X7 = 5X1 = 2X3 = 4X8 = 23 310 7X9 = 3 4 (8) (9) (10) (11) (12) (13) (14) (15) 50 254 623 1036 5096 8754 165 235 3 6 " 7 8 6 8 40 23 (16) (17) (IS) (19) (20) (21) (22) 7898 145 24 9.6 287 24 8X5%= _ 9 206 234 4 .05 21/0 (23) (24) (25) (26) (27) (28) (29) 11/4 X 8: = 16 % X % = 9742 6.25 .0123 1/8X2 = 2% 59 3.2 9.8 (30) (31) (32) (33) (34) 2.49 12 15 6 dollars 49 cents 21/2 X 3y2 = 1/2 X 1/2 = .36 25 32 8 (35) (36) (37: ) (38) (39) 987% 3 ft. 5 ill. 2% X 41/2 X 11/2 = .09631/8 8 ft. 91/2 in. 25 5 .084 9 16 Series A Division Scale By Clifford Woody City. County.. Name. . .'. How old will you be ?. In what grade are you ?. . School. Date. .When is your next birthday ?, (1) (2) (3) (4) (5) (6) 3)6 9)27 4 )28 1)5 9)"^ 3)39 (7) (8) (9) (10) (11) (1 2) 4^2 = 9)0 ryr" 6 X . . . . = 30 2)13 2 - - 2 = (13) (14) (15) (16) (!' 7) 4) 24 lbs.. 8 oz. 8)5856 14 of 128 = 68)2108 50 ^ - 7 = (18) (19) (20) (21) (22) 13)65065 248 -- 7 = 2,1)25.2 25)9750 2)13.50 (23) (24) (25) (26) -23)469 75)2250300 2400)504000 12)2.76 (27) (28) (29) (30) Ys of 624 = .003).0936 31/2 = 9 = 3 / 4 - - 5 = (31) 5 3 4 5 (32) 9% -- 33/4 = (33) 52)3756 (34) (35) (36) 62.50 ^ 11/4 = 531)37722 9)69 lbs. 9 oz. The addition scale begins with 2-1-3 and includes the addition of in¬ creasingly difficult exercises. It brings in fractions, both with common denomi¬ nators and with different denominators, mixed numbers, decimals, and com¬ pound numbers of two denominations. The subtraction scale is made up of problems involving numbers of the same kind as those in the addition scale. The multiplication scale includes the simple combinations, multiplica¬ tion of integers by integers up to four figures in the multiplicand and two in the multiplier, a fraction by a fraction, a decimal by a decimal, and a compound number by an integer. The division scale includes the simple combinations, short division, long division up to the division of a number of from five to seven digits by one of two or three digits, division of mixed numbers, fractions, decimals and compound numbers. fn giving these tests the time allowed is practically unlimited; twenty minutes being allowed for each test. In this length of time nearly all the pupils will have completed all the problems that they can solve. These tests 17 are then “power” tests rather than “speed” tests such as those devised by Courtis. Another way in which the Woody scales differ from the Courtis tests is that in the latter the problems in a given test are of equal difficulty, while in the former they are of varying degrees of difficulty. This being the case it became necessary for Mr. Woody to adopt some unit by means of which the degree of difficulty of each problem could be stated. The unit' adopted was the Probable Error (P. E.) of the school grade distribution. The median achievement of a grade distribution, i. e., a problem that is solved by exactly 50% of the grade, is taken as the measure of the achievement of the grade. The P. E. of a grade distribution is that distance along the base line of a surface distribution from the median point to the perpendicular on either side of the median which cuts off twenty-five per cent of the cases. The P. E. of the grade’s distribution is the limits of the middle 50% of the grade. In other words if exactly 50% of a class are able to solve a problem correctly, then 25% of that class should be -able to solve a problem that is at least one unit (P. E.) more difficult, and 75% of that class should be able to solve a problem one unit less difficult. THE MONROE TESTS The third series of tests included in this study is the one devised by Walter S. Monroe. This author starts out deliberately to construct a series of tests of the operations in arithmetic that will include all or nearly all of the types of examples encountered in arithmetical work. He points out the fact that existing studies show that there are as many arithmetical abilities as there are types of examples and argues that any test that is to be really diagnostic must include all the important types within its scope. According to Mr. Courtis there are six types of operations in the addition of integers, four in subtraction, nine in multiplication, and ten in division. Kallom has analyzed the addition of two fractions and reached the con¬ clusion that there are fourteen types of examples. Mr. Monroe, without mak¬ ing a very careful analysis, carries the discussion of types on through frac¬ tions and decimals and reaches the conclusion that there are at least 86 significant types of examples in the fundamental operations of arithmetic (integers, 30; common fractions, 36; decimal fractions, 20 to 40). This is exclusive of those involved in the writing and reading of numbers, in the tables of denominate numbers, and in the solution of problems. The 21 tests devised by Mr. Monroe contain 61 of these types. These tests are given in limited lengths of time so that they measure both speed and accuracy. In this respect they differ from the Woody tests and agree with the Cleveland tests. In fact Mr. Monroe argues that arithmetical abilities are “two dimensional,” and that any attempt to measure them must take this fact into consideration. He admits, however, that the usual class-room pro¬ cedure is to measure power only Avithout much regard to speed. The 21 tests are given here in full. They are printed in four different folders. The first two, containing tests 1-11, deal Avith the four fundamental operations Avith integers; the third, tests 12-16, deals Avith common frac¬ tions; the fourth, tests 17-21, Avith decimal fractions. The fourth folder Avas not used in this study. 18 Part I—Tests 1-6. Bureau of Educational Measurements and Standards Kansas State Normal School, Emporia, Kansas DIAGNOSTIC TESTS IN ARITHMETIC Operations With Integers Devised by Walter S. Monroe Name..Age today. Years Months City.••. Grade. Room. School. Teacher.Date today. Instructions to Examiners Have the pupils fill out the blanks at the top of this page. Have them start and stop work together. Use a stop watch if one is available; if not, use an ordinary watch with a second hand and exercise care to allow just the exact time for each test. Allow an interval of half a minute or more between tests. Require the pupils to close the folder as soon as the signal to stop is given, in order to make certain that they do not spend this rest period working on the next test. If the pupils need to sharpen pencils before going on, allow this to be done. The following time allowances must be followed exactly: Test 1—30 seconds. Test 4—1 minute. Test 2—30 seconds. Test 5—3 minutes. Test 3—1 minute. " Test 6—2 minutes. Have the children read the following directions: “Inside this folder are examples which you are to work out when the teacher tells you to begin. Do not open this folder before the teacher gives the signal. Work rapidly and accurately. There are more examples in each test 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. Place the test in position on your desk so that you can open it quickly when the signal is given to begin, but do not open it until the signal is given.” After all of the tests have been completed have the pupils exchange papers. Read the answers aloud and have the children mark each example that is correct “C.” Count the number of examples attempted and the number of “C’s” and write the numbers in the proper spaces at the top of the tests. Examples partially completed or partially right are not counted. Before collecting the papers have the records transcribed to the first page. The teacher should verify a sufficient number of records to make certain that the pupils hav« marked the papers and transcribed the results correctly. 1 1 1 Test .1 I 1 2 1 3 1 1 1 1 4 5 6 1 1 1 Number of examples attempted].j.j. '. 1 1 1 1 1 1' 1 Number of examples right.|.j.j. 19 Test 1—ADDITION. At Rt 4 5 7 0 2 9 2 0 6 3 7 8 1 7 6 7 3 2 1 2 8 7 8 4 4 3 4 0 9 0 3 9 3 • 4 G 5 8 8 5 4 4 1 0 0 7 6 6 3 0996 5 5211877 521 1 8 77,43309 At Test 2 —SUBTRACTION. Rt. . . . 37 94 60 27 39 41 77 53 5 8 3 6 7 8 3 — • 9 65 80 • 92 70 68 58 26 43 2 4 5 3 2 9 9 8 95 50 36 34 44 25 63 57 4 7 1 8 6 • — 7 9 , At.... Test 3- -MULTIPLICATION. Rt... . 6572 6750 5863 3754 2845 6 9 2 5 8 4936 9327 8274 8409 6391 4 7 3 6 9 5482 8609 3679 2758 4658 2 5 8 4 7 9653 3174 2874 7901 2179 3 6 9 2 At.... 5 Test 4- -DIVISION. Rt:.... 8)3840 4)7432 7)2534 3)8430 6)4680 9)8577 2)6370 5)9310 8)7512 4)3820 7)9653 3)5781 6)6720 9)5373 2)5130 20 Test 5—ADDITION. 7862 6809 8941 5917 5013 7623 7910 4814 1761 5299 9845 9007 5872 6601 8522 6975 3739 3496 1046 1227 — — — — 8758 2462 1247 4319 2350 9869 3573 2358 3197 4572 1081 5795 2338 6420 7805 4314 5917 6772 9864 1249 — — --- --- Test 6 —DIVISION. 82)3854 43)1591 63)3591 94)4042 83)5312 42)672 62)2108 93)5022 84)7140 41)3567 64)5312 92)6624 6772 At. . . . Rt. . . . 7864 1249 6028 7883 8975 6535 8240 9005 2340 9869 1573 2319 6794 3203 — — — 6794 3283 7917 5420 7805 4304 4570 7642 9027 8028 7803 9975 ' 8758 2462 1247 At.... Rt 74)2664 31)1953 21)1344 53)4452 71)5183 32)2304 23)782 51)2703 73)6278 33)1386 24)984 52)3484 21 Part II—Tests 7-11. Bureau of Educational Measurements and Standards Kansas State Normal School, Emporia, Kansas DIAGNOSTIC TESTS IN AKITHMETIC Operations With Integers Devised by Walter S. Monroe Name.Age today. Years Months City. Grade. Room. School. Teacher.Date today. Instructions to Examiners Have the pupils fill out the blanks at the top of this page. Have them start and stop work together. Use a stop watch if one is available; if not, use an ordinary watch with a second hand and exercise care to allow just the exact time for each test. Allow an interval of half a minute or more between tests. Require the pupils to close the folder as soon as the signal to stop is given, in order to make certain that they do not spend this rest period working on the next test. If the pupils need to sharpen pencils before going on, allow this to be done. The following time allowances must be followed exactly: Test 7—2 minutes. Test 10—2 minutes. Test 8—3 minutes. Test 11—4 minutes. Test 9—1 minute. Have the children read the following directions: “Inside this folder are examples" which you are to work out when the teacher tells you to begin. Do not open this folder before the teacher gives the signal. Work rapidly and accurately. There are more examples in each test 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. Place the test in position on your desk so that you can open it quickly when the signal is given to begin, but do not open it until the signal is given."’ After all of the tests have been completed have the pupils exchange papers. Read the answers aloud and have the children mark each example that js correct “C.” Count the number of examples attempted and the number of “C’s” and write the numbers in the proper spaces at the top of the tests. Examples partially completed or partially right are not counted. Before collecting the papers have the records transcribed to the first page. The teacher should verify a sufficient number of records to make certain that the pupils have marked the papers and transcribed the results correctl 3 ^ 1 1 1 Test . 1 7 1 8 1 1 9 10 11 1 1 1 Number of examples attempted..|.|. 1 1 i 1 1 ! Number of examples right.|.|. 1 1. 22 Test 7—ADDITION. 7 6 6 8 2 1 2 6 8 7 7 9 3 2 6 8 0 9 9 8 5 5 9 1 3 2 3 1 0 9 3 5 6 6 7 5 5 4 8 0 1 1 1 1 0 0 4 6 7 8 8 ’ 7 7 7 1 4 7 7 5 3 5 5 0 3 7 5 4 2 4 5 3 4 6 6 4 2 4 1 5 4 5 7 5 3 2 4 6 9 7 9 7 Test 8 - -MULTIPLICATION. 4857 5718 36 92 9625 6123 23 64 1253 5376 38 76 Test 9—SUBTRACTION. 739 1852 975 367 948 906 508 1371 1284 447 843 966 1910 / 3o 1056 361 478 591 831 954 1077 360 483 704 At. Rt. 8 3 2 6 9 5 7 9*9 4 3 7 8 8 5 1 1 6 4 9 0 7 8 4 4 9 7 2 2 1 2 8.8 3 1 0 7 6 9 3 3 8 8 6 3 2 3 9 9 4 5 3 0 9 0 6 2 2 3 1 8 7 8 1 1 1 2 0 7 '6 9 1112 0 7 6 0 2 2 2 3 1 7 8 3 4 4 4 5 At Rt 6942 4065 58 47 7486 9027 75 89 3786 5492 49 At 53 Rt 1087 516 962 821 239 • 325 730 1853 897 508 162 258 877 1190 619 618 739 257 1328 939 1316 872 654 827 8 3 4 8 5 2 7 4 2 9 5 5 5 23 At. . Test 10—MULTIPLICATION. Rt. . 560 807 617 . 840 730 609 37 59 508 80 96 70 435 790 940 307 682 870 308 60 38 42 409 40 780 502 ■ 386 150 850 401 56 68 207 90 72 80 817 460 730 605 392 590 109 30 52 84 306 30 At. . Test 11—DIVISION. Rt. . 47)27589 79)36893 36)28296 68)31824 96)56064 28)21980 57)22572 89)25365 48)32304 76)36708 67)39932 98)46844 24 Part III. Tests 12-16. Bureau of Educational Measurements and Standards Kansas State Normal School, Emporia, Kansas DIAGNOSTIC TESTS IN ARITHMETIC Operations With Common Fractions Devised by Walter S. Monroe Name.Age today. Years Months City. Grade. Room. School. Teacher.Date today. Instructions to Examiners Have the pupils fill out the blanks at the top of this page. Have them start and stop work together. Use a stop watch if one is available; if not, use an ordinary watch with a second hand and exercise care to allow just the exact time for each test. Allow an interval of half a minute or more between tests. Require the pupils to close the folder as soon as the signal to stop is given, in order to make certain that they do not spend this rest period working on the next test. If the pupils need to sharpen pencils before going on, allow this to be done. The following time allowances must be followed exactly: Test 12—1% minutes Test 13—2 minutes. Test 14—1 minute. Have the children read the follo\fing directions: “Inside this folder are examples which you are to work out when the teacher tells you to begin. Do not open this folder before the teacher gives the signal. Work rapidly and accurately. There are more examples in each test 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. Place the test in position on your desk so that you can open it quickly when the signal is given to begin, but do not open it until the signal is given." After all of the tests have been completed have the pupils exchange papers. Read the answers aloud and have the children mark each example that is correct “C.” Count the number of examples attempted and the number of “C’s” and, write the numbers in the proper spaces at the top of the tests. Examples partially completed or partially right are not counted. Before collecting the papers have the records transcribed to the first page. The teacher should verify a sufficient’ number of records to make certain that the pupils have marked the papers and transcribed the results correctly. Test 15^—2 minutes. Test 16—2 minutes. 1 1 1 Test . 1 12 13 1 1 1 14 15 16 1 1 Number of examples attempted..|.|. 1 1. f ( 1 1 1 Number of examples right.|.1. 1 1. 25 Test 12.—ADDITION. At Reduce your answers to lowest terms. Rt . I 1 3 2 5 2 — 4 -= - 1 -= _ 1 -= 6 3 10 5 9 3 5 1 1 1 5 7' — H-= — 4-- = -1 - = 6 2 8 2 ■ 6 12 3 1 1 1 1 7 — H-= -1-= - 1 -= 4 2 3 12 2 10 3 5 5 1 1 5 — + —= - 1 -= -1-= 4 12 - 8 4 2 12 I 2 4 7 5 3 — “1-= - 1 -= - 1 -= 6 3 5 10 8 4 Test 13—SUBTRACTION. Reduce your answers to lowest terms. At . Rt . 3 2 5 3 1 2 — -= —-- - 4 5 6-4 2 7 7 1 2 1 5 2 — -=— --—- -- 10 6 3 2- 6 15 3 1 7 1 2 3 — - -= 1-^ 4 3 9 6 3 5 5 3 3 2 7 3 — -= - -= 6 8 4 7 12 8 5 3 8 4 4 1 — -= --- ---—- - =: 6 5 15 9 5 3 Test 14—MULTIPLICATION. At . : . Reduce your ansAvers to lowest terms. Rt . 2 3 - 2 3 5 3 — X — = — X — = — X — = 3 4 5 7. 12 5 4 2 1 3 1 1 — X — = — X — = — X — = 9 5 3 8 2 3 2 3 4 1 7 4 — X — = — X — = — X — = 5 4 5 3 12 7 3 1 2 1 1 1 — X — = — X — = — X — = 8 4 7 6 3 2 4 5 4 7 1 • 3 — X — = — X — = — X — = 15 8 5 9 6 10 26 Test 15—ADDITION Reduce your answers to lowest terms. 1 3 3 -+-= —+ 6 5 12 4 1 1 - 1 -= —" + 9 6 3 1 2 7 2 3' 10 3 5 1 -1-= - -h 8 6 7 2 2 3 - \ -= -h 5 3 10 Test 16—DIVISION. Reduce your answers to lowest terms. 2 1 4 5 3 7 5 5 3 6 8 7 1 1 2 2 3 3 4 8 3 7 11 5 4 1 2 5 2 5 Rt 3 1 5 2 4 5 15 9 1 3 3 4 1 1 10 15 4 3 —+ —= 7 5 At Rt 3 2 8 3 7 4 12 9 2 3 3 4 1 1 4 6 5 4 12 9 5 8 4 7 3 8 2 5 1 4 2 3 4 5 8 9 3 4 3 7 27 Part IV, Tests 17-21. Bureau of Educational Measurements and Standards Kansas State Xormal School, Emporia, Kansas DIAGNOSTIC TESTS IN ARITHMETIC ^Multiplication and Division of Decimal Fractions Devised by Walter S. Monroe Name. ^...Age today. Years Months City. Grade. Room. School. Teacher.Date today. Instructions to Examiners Have the pupils fill out the blanks at the top of this page. Have them start and stop work together. Use a stop watch if one is available; if not, use an ordinary watch with a second hand and exercise care to allow just the exact time for each test. Allow an interval of half a minute or more between tests. Require the pupils to close the folder as soon as the signal to stop is given, in order to make certain that they do not spend this rest period working on the next test. If the pupils need to sharpen pencils before going on, allow this to be done. The following time allowances must be followed exactly: Test 17—30 seconds. Test 20—30 seconds. Test 18—30 seconds. Test 21—30 seconds. Test 19—30 seconds. Have the children read the following directions: “Inside this folder are examples which you are to work out when the teacher tells you to begin. Do not open this folder before the teacher gives the signal. Work rapidly and accurately. There are more examples in each test 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. Place the test in position on your desk so that you can open it quickly when the signal is given to begin, but do not open it until the signal is given.” After all of the tests have been completed have the pupils exchange papers. Read the answers aloud and have the children mark each example that is correct “C.” Count the number of examples attempted and the number of “C’s” and write the numbers in the proper spaces at the top of the tests. Examples partially completed or partially right are not counted. Before collecting the jDapers have the records transcribed to the first page. The teacher should verify a sufficient number of records to make certain that the pupils have marked the papers and transcribed the results correctly. Test . 1 1 17 1 18 1 1 1 1 i 19 1 1 I 1 20 1 21 Number of examples attempted. . 1 1 . 1 . 1 1 1 . 1 1 . 1 1 1 ' I i Number of examples I'ight 1 . 1 . -W 28 Test 17—DIVISION. At Rt The correct answer for each example with the exception of the decimal point is given at the side immediately after the letters “Ans.” Write the answer in its proper position and place the decimal point in its proper place. Place ciphers before or after the answer when they are necessary. .03)16.2 Ans.: 54 .07)1.82 .06)7.44 Ans.: 124 .08).952 .02).144 Ans.: 72 .08)40.8 .03)47.4 Ans.: 158 .07)8.61 .09)5.76 Ans.: 64 .04).348 .02).748 Ans.: 374 .03)89.1 .09)94.5 Ans.: 105 .01)5.48 .04)9.84 Ans.: 246 .07)'.238 Test 18—MULTIPLICATION. Place the decimal point correctly 657.2 67.50 5.863 .7 .03 .6 46004 20250 35178 932.7 82.74 8.409 .08 .4 .07 74616 33096 58863 367.9 27.58 4.658 .2 .05 .8 7358 13790 37264 574.6 82.47 7.462 .06 .9 .02 34476 74223 14924 Test 19—DIVISION. Ans.: 26 .05).415 Ans.: 83 Ans.: 119 .04)87.6 Ans.: 219 Ans.: 51 .09)3.42 Ans.: 38 Ans.: 123 .05).965 Ans.: 193 Ans.: 87 .06)51.0 Ans.: 85 Ans.: 297 ^ .05)6.85 Ans.: 137 Ans.: 548 .06).288 Ans.: 48 Ans.: 34 .08)44.8 At. Rt. in the folloAving products: Ans.: 56 375.4 28.45 4.936 .09 .2 .05 33786 5690 24680 ‘ 639.7 54.82 8.609 .3 .06 .9 i9191 32892 77481 965.3 31.74 2.874 .04 .7 .03 38612 22218 8622 834.7 54.32 7.842 .5 .08 .4 41735 43456 31368 At Rt The correct answer for each example with the exception of the decimal point is given at the side immediately after the letters “Ans.” W.rite the answer in its })roper position and place the decimal point in its proper place. Place ciphers before or after the answer when they are necessary. .4)148. Ans.: 37 .9)65.7 .7).301 Ans.: 43 .3)47.7 .2).548- Ans.: 274 .4)744. .9).756 Ans.: 74 .8)672. 5) 865 Ans.: 173 .3)684. .2)7.92 Ans.: 396 .4)352. .7)3.22 Ans.: 46 .5) .710 .1)9.42 Ans.: 942 .6).852 Ans.: 73 .6)1.68 Ans.: 28 Ans.: 159 .6)8.34 Ans.: 139 Ans.: 186 .3)117. Ans.: 39 Ans.: 84 .7)59.5 Ans.: 85 Ans.: 228 .6)93.6 Ans.: 156 Ans.: 88 .3)16.2 Ans.: 54 Ans.: 142 .8)376. Ans.: 47 Ans.: 142 .2)74.2 Ans.: 371 29 Test 20—MULTIPLICATIOX. At Et. . Place the decimal point correctly in the following products: 487.5 57.28 6.294 4065. 967.5 .62 9.5 .28 5.1 8.4 302250 544160 176232 207315 712700 61.32 7.465 7486. 907.2 14.53 .17 4.3 .76 .39 6.2 104244 320995 558936 353808 90086 5.376 8637. 549.3 84.74 8.637 .91 2.4 5.7 .83 1.6 489216 207588 313101 703342 138192 5194. 784.1 36.74 2.893 4936. .49 .72 3.5 .68 9.4 254506 564552 128590 196724 463984 Test 21—DIVISION. At Rt The correct answer for each example, with the exception of the decimal point, is given below the quotient, after the letters, “Ans.” Write the answer in its proper position and place the decimal point in its proper place. Place ciphers before or after the answer when necessary. .47)2758.9 8.2)38.54 79.) 36.893 • .43)1591 Ans.: 587 Alls.*:* 47 Ans.: 467 Ans.: 37 3.6)2829.6 74.) 26.64 .68)31.824 3.1)1953. Ans.: 786 Ans.: 36 Ans.: 468 Ans.: 63 96.) 5606.4 .63)35.91 2.8)21.980 94.) 4.042 Ans.: 584 Ans.: 57 Ans.: 785 Ans.: 43 .57)22572. 2.1)140.7 89.) 253.65 .53)4.452 Ans.: 396 Ans.: 67 Ans.: 285 Ans.: 84 4'.8) 32304. 83.)531.2 .76)367.08 4.2).672 Ans.: 673 Ans.: 64 Ans.: 483 Ans.: 16 30 PART II ' Having in mind the purpose and character of the'tests to be used we may now turn to the main question at issue in tlie study, viz.,, do the different tests agree as to results? If they do the fact may be taken as a strong indi¬ cation that they are all well suited to their purpose. If they disagree then certainly one or more of the tests is faulty in some respect or else they do not measure the same abilities. The tests were given on six, successive school days, beginning October 23, to a group of about 60 eighth grade pupils in Manhattan, Kansas. The order followed was Cleveland tests, Monroe tests, and Woody scales. The tests were all given and the scores checked by the author. Care was exercised to see that conditions were as nearly identical in the different tests as it was possible to make them. The results of the tests are shown in Tables 1 to 6, and diagrams 1 to 6. Table I shows a comparison of the standard scores and the class scores for the number of problems solved correctly and the per cent of accuracy in each of the Cleveland tests. The standards shown here are the averages of the Cleveland, Grand Rapids and St. Louis median scores in the 8B sections. Table 2 gives the standard scores and class scores in attempts and in per cent of accuracy for the Monroe tests. In both of these tables the tests are arranged in such order as to bring together all the tests in each of the four fundamental operations. Tables 3, 4, 5 and 6 show the results of the Woody tests. These results are shown in graphic form in diagrams 1, 2, 3 and 4. In these diagrams the horizontal lines represent the grades, the vertical lines the tests and the figures at the points of intersection the standard scores of the different grades in the indicated tests. The broken line represents the class scores as determined by this series of tests. Comparison of Standard and Class Scores Table 1 — Cleveland Survey Tests Table 2 — Monroe Tests. Standard Class Standard Class Scores Scores Scores Scores Test Rts. Ac. Rts. Ac Test Ats. Ac. Ats. Ac. A . . .29.8 99 24.5 99 1 . . 12.7 100 12.5 100 E ..... . 7.8 94 5.2 93 7 . . 5.4 79 4.9 81 ,T . . 5.6 78 3.7 70 5 . . 6.1 66 • 5.4 62 M . . 5.3 76 4.6 87 2 . . 8.9 100 7.9 100 B . . 25.2 99 18.2 95 9 . . 8.5 97 8.1 100 F . . 10.2 90 7.1 83 3 . . 6.2 84 5.6 86 C . . 19.7 89 16.3 87 8 . . 6.5 73 6.1 81 G . . . 6 9 88 5.5 90 10 . . 6.6 82 4.9 90 L 4 7 69 3.6 69 4 . . 4.6 88 4.9 100 D . . 22.3 97 18.7 98 6 . . 4.5 100 3.4 100 I . . 4.7 84 2.8 70 11 . . 3.4 68 3.0 100 K . . 10.8 95 7.6 94 12 . . 9.8 73 7.8 76 N . . 2.4 81 1.5 68 15 . . 8.5 59 6.5 76 II . . 9.3 77 5.6 89 13 . . 7.8 71 6.8 81 0 . . 5.7 68 3.5 47 14 . . 13.5 75 9.6 93 16 . . 8.5 59 9.7 82 31 Table 3—Woody Addition Scale Table 4 —Woody Subtraction Scale Xo. of Xo. Getting % Getting Xo. of Xo. Getting % Getting Problem EachProb. EachProb. Problem Each Prob. Each Prob. 1. 98 1. .58 98 o .■ . 59 100 o .59 100 3. .59 100 3.. .'. .58 98 4. .59 100 4. .59 ' 100 5. 98 5. .59 100 6.;. 98 6. .59 100 .58 98 - 100 8. .58 98 8. .58 98 0. ..58 98 9 100 10. 95 10. 100 11. .56 95 11. .57 '96 12. .55 93 12. .59 100 13. 100 13. .58 98 14. . 55 93 14. .57 96 15. 92 15. ..'. 55 93 16. .55 93 16. .57 96 17. .52 88 17. 90 IS. .55 93 18. .54 92 10. 92 10 50 85 20. .53 90 20. .53 90 21. .47 SO 21. .44 75 oo .41 70 oo .54 92 23. 92 >3 49 84 24. .49 84 24. .51 87 25. 90 •^5 43 73 26. .47 SO 26. .45 76 27. 87 27. .37 , 63 28. .53 90 28. .45 76 20. .44 75 29. .51 87 30. .43 73 30. .45 . 76 31. .37 63 31. .39 66 32. .45 76 32. .31 52 33. 61 33. .42 71 34. .48 81 34. .36 61 35. ...36 61 35. .40 68 36. .36 61 37. .30 51 38. .24 34 Standard Score. 9.01; Class Score, S.76 Standard Score, 7.G4; Class Score, 7.00 32 Table 5— -Woody Multiplication Scale Table 6—Woody Division Scale No. of No. Getting % Getting No. of No. Getting % Gettin Problem EachProb. EachProb. Problem EachProb. Each Prol 1. . 58 98 1. . . . . 55 96 2. .59 100 2. . . . . 57 100 3. .59 100 3. .. . . 57 100 4. . 59 100- 4. . . . . 57 100 5. .59 100 5. .. . . 57 100 6. . 59 100 6. .... 57 100 7. .^58 98 7. .. .. 57 100 8 . .59 100 8. . . . . 55 96 9 55 93 9 55 96 10. . 56 95 10. .. . . 57 100 11. .58 98 11. . . . . 55 96 12. .58 98 12. .... 55 96 13. . 51 86 13. . . . . 51 89 14. .58 98 14. . . . . 57 96 15. ..56 95 15. . . . . 52 91 16. .44 75 16. . . .. 51 89 17....... .53 90 17. . . . . 52 91 18. .54 92 18. . . . . 41 68 19. .55 93 19. .. . . 51 89 20. .55 93 20. . . . . 48 84 21. .56 95 21. . . . . 53 93 22. .56 95 22. . . . . 49 86 23. .54 92 ‘23. . . . . 36 63 24. . 53 90 24. . . . . 43 75 25. .48 81 25. . . . . 42 74 26. .45 76 26. . . . . 44 77 27. . 53 90 27. .... 49 86 28. .50 85 28. . . .. 45 79 20. .54 52 29.. . 60 30. .51 87 30. . . . . 36 63 31. .52 88 31. .. . . 39 68 32. .43 73 32. . .. . 43 75 33. .46 78 33. . . . . 36 63 34. . 42 71 34. . . . . 33 58 35. .34 57 35. . . . . 22 39 36. .34 57 36. .. . . 8 14 37. .36 61 38. .25 42 39. .27 46 Standard Score, 7.93; Class Score, 8.19 Standard Score ,7.16; Class Score, 7.1 33 As a whole the group made the poorest showing in the Cleveland tests and the best in the Woody tests. This is undoubtedly due in part to the fact that the Cleveland tests were given first. It also indicates that the Cleveland standards are higher than either of the others. In the Cleveland tests the scores are all below standard; only one of them reached seventh grade standard, five are between seventh and sixth, seven between sixth and fifth, and two below fifth grade. In the Monroe tests the score in one test is above standard, those in six tests are between seventh and eighth grade standards, and those in the three remaining tests are below sixth grade standards. On the Woody scale three are above standard and one between seventh and eighth grades. There is then, even in this general statement, a serious discrepancy be¬ tween the results obtained from the Woody scales and those obtained from the other two tests. Using the first named the teacher or supervisor would be led to the conclusion that these pupils did not need much more drill on the fundamentals. Using either of the others he would come to exactly the opposite conclusion. But leaving the standards out of consideration let us see. how the results agree as to the strength or weakness of the group tested in the different operations. Both the Cleveland and the Monroe tests show weakness in addition, the former to a greater extent than the latter, a lesser degree of weakness in subtraction and multiplication and irregularity in division and in fractions. The Woody tests agree with this showing in a general way, but they put subtraction considerably above any of the other operations. Turning now to a study of the particular abilities in the various opera¬ tions let us see what the different tests show. Test A, Cleveland, shows the group to be below sixth grade attainment in knowledge of addition combina¬ tions. The Monroe tests do not include problems of this character, but the Woody addition scale has two problems. Nos. 1 and 7. Neither of these shows any weakness here as both were solved correctly by all but one member of the group. Test E, Cleveland, addition of o figure columns of single digits, indicates slightly better than fourth' grade attainment, the weakest point in addition. Test 1, Monroe, 3 figure columns of single digits, shows between seventh and eighth grade attainment, the highest point in addition. Of course these tests are not identical in character and these results seem to indicate that they are not even of the same type. Problem 2, Woody addition scale, a column of three figures, was solved correctly by everj’^ member of the group, showing no weakness in this character of work. Test J, Cleveland, addition of long columns requiring the bridging of the memory span, shows a score below fifth grade attainment, but slightly better than test E. Test 7, Monroe, gives a score below seventh grade standard, the weakest point in addition. The Woody scale does not give a problem of this character. Test M, Cleveland, column addition four numbers wide and five deep, gives a score equal to seventh grade standard, the highest point in addition. Test 5, Monroe, of exactly the same character, also gives a score equal to seventh grade standard. Problem 18, Woody addition scale, was solved correctly by 93% of the class, a showing which agrees fairly well with the other two results. Test B, Cleveland, subtraction combinations, shows a score a little below sixth grade. The Monroe tests do not include'this type, but problems 1 to 7 and problem '10, Woody subtraction scale, show no weakness at all, being solved correctly by practically every member of the class. Test F, Cleveland, subtraction involving borrowing, gives a score between fifth and sixth grade standards. Test 9, Monroe, gives a score between seventh and eighth grade standards. Problems 16, 17, 18, 19 and 23, Woody sub- 34 DIAGRAM 1 Median Scores in Rights, Cleveland Test 0rades 62 7^ 62 5^ 4 f 3/ Ad Sub. Mu/. D/v. Fr. 7 3 4 .3 98 l s A 6 5 . 5 Z /. 02 / 97 6: 9 4 .7 2 23 4 7 /. 08 2 9. 3 3 70 7 . f s 2 4. •