QJorncU UntuEraUg library 3ltl;aca, Ntxa ^otk BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE 1891 The date' shows w?hen this volume wasUaken. iiik coipy the ca the Ufararian. To renew this bock coipy the call No. an'dgive to the librarian. _ '^^ home">i|sb^ules 16 . ■ s ^ ' '" ' -4' '' 1 •>-.• ■\T-irr - All Books- siibject to Recall AU borrowers must regis- ter in the library to borrow books for hon)e use. All books must be re- turned at eafl of college year for TOpection and repairs. J Limited books i^st be re- turned within the four week limit and not renewed. Students must return all books before leaving town. Officers should arrange for the return of books wanted during thair absence from town. Volumes of periodicals and of pamphlets are held in the library as much as possible. For special pur- pos,es they are given out for a limited time. , Borrowers should not use their library privileges for the benefit of other persons. Books of special " value and gift books, when the giver wishes it, are not allowed to circulate. Readers are asked to re- port all cases of . books marked or mutilated. Do not deface books by marks and writing. Cornell University Library BF456.N7 W91 Measurements of some achievements in ari 3 1924 031 015 146 oljn 4 MEASUREMENTS OF SOME ACHIEVEMENTS IN ARITHMETIC BY CLIFFORD WOODY, Ph.D. TEACHERS COLLEGE. COLUMBIA ONIVERSITY CONTRIBUTIONS TO EDUCATION, NO. 80 PUBLISHED BY ^eutl^a QIalbgr, (Eolnmbta HInitterattg NEW YORK CITY 1916 A Copyright, 1916, by Clifford Woody Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031015146 CONTENTS PART I I. Introduction i II. The Arithmetic Scales and Their Uses., 3 1. Directions for Administering the Tests 2. Directions for Scoring the Tests 3. Directions for Determining the Class Score 4. Tentative Standards of Achievements III. The Value and Uses of the Scales 23 IV. The Limitations of the Scales 24 PART II I. The Derivation of the Scales 25 1. History of the Scales 2. Probable Error (P.E.) Taken as the Unit of Measure 3. Scaling the Problems in Addition for Each Grade 4. Measuring the Distance Between the Grades 5. Location of the Zero Point 6. Referring All the Problems to Zero II. Tables of Crude Data From Which Scales Were Developed 55 ACKNOWLEDGMENTS The author wishes to acknowledge his indebtedness to those whose aid has made this study possible. He wishes to thank the superintendents, principals, and teachers who so willingly cooperated in the collection of the data. He feels especially indebted to Professor George D. Strayer, Professor Edward L. Thorndike, and Dr. Marion Rex Trabue for their valuable suggestions and criticisms, which have given to the study what- ever merit it possesses. C. W. MEASUREMENTS OF SOME ACHIEVEMENTS IN ARITHMETIC PART I Section i. INTRODUCTION The purpose of this monograph is to set forth the results of an attempt to derive a series of scales in the fundamental opera- tions of arithmetic. Thus the problem is closely related to the general movement for the measurement of educational products by means of objective scales. The method followed in the development of these scales is most clearly related to the methods used by Dr. Buckingham^ in the development of his Spelling Scale and by Dr. Trabue in the Completion-Test Language Scales.'' In the development of these scales the fundamental idea was to derive a series of scales which would indicate the type of problems and the difficulty of the problems that a class can solve correctly. Accordingly, each of the scales is composed of as great a variety of problems as the fundamental operations can well permit. These problems, beginning with the easiest that can be found, gradually increase in difficulty until the last ones in each series are so difficult that only a relatively small percentage of the pupils in the eighth grade are able to solve them correctly. In the determination of the relative difficulty of these problems, the relative per cents of correct answers obtained by submitting them to large numbers of school children were taken as a basis. Two distinct series of scales in each of the fundamental opera- tions have been derived. Series B contains only about half as many problems as Series A. Series A thus has a greater power of diagnosing the weaknesses of a class and is recommended where there is ample time for testing. Series B was derived ^ Buckingham, B. R., Spelling Ability, Its Measurement and Dis- tribution, 1913. 2 Trabue, Marion Rex, Completion-Test Language Scales, 1916. 2 Measurements of Some Achievements in Arithmetic especially for use where the amount of time that can be devoted to measuring is very limited. Part I of this monograph is devoted especially to the scales and their uses. Specific directions for administering the tests and scoring the results are given in detail. A statement of the values and limitations of the scales is also given in this part. Part II deals with the history and the method of the deriva- tion of the scales. It also includes many tables of crude data from which the scales were developed. Section II. THE ARITHMETIC SCALES AND THEIR USES I. Directions for Administering the Tests These scales are useful as measures of achievement in the fundamentals of arithmetic either of a class or of a whole school system. Series A is more valuable when the amount of time for testing is plentiful. Series B was especially constructed for use in measuring school systems where the amount of time for testing purposes is limited. Both series of tests are adminis- tered in the same way. The Addition and Subtraction Scales can be used in grades two to eight inclusive; the Multiplication and Division Scales, in grades three to eight inclusive. These scales may be sub- mitted in any order to the pupils. They may be given in imme- diate succession or with such intervals of time intervening as is most convenient. In the development of the scales subtrac- tion and multiplication were given in succession on one day and addition and division on the next day. The writer recommends that for Series B all tests be given in succession. If the measurements by these scales are to be valid and com- parable, it is necessary that the same standard of procedure be followed in giving the tests and in scoring the results as was followed in the original development of the scales. The same individual should give all of the different tests. He should give the same instructions to every class. He should have the same manner in each class room. In giving the " specific directions " to the class he should use as nearly as possible the same emphasis and intonation. He should not stress one part of the directions more than another part. It is highly important that the teacher or the one in charge of the room remain silent (saying nothing to the children indi- vidually or collectively during the time of giving the tests). When ready to distribute the tests, place one face downward on each desk. Insist that the pupils do not turn the papers 3 4 Measurements of Some Achievements in Arithmetic Series A' ADDITION SCALE Name Are you a boy or girl? In what grade are you?... (1) (2) (3) (4) (5) (6) (7) (8) (9 2 2 17 53 72 60 3+1= 2+5+1= 20 3 4 2 3 — 45 26 37 10 2 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 46 15 156 19 2065 S^^} »(20) (21) (22) ( •23) (24) (25) $ .75 $12.50 $8.00 547 i+4 = 4.0125 1+1+1+1 = 1.25 16.75 5.75 197 1.5907 .49 15.75 2.33 4.16 685 678 4.10 8.673 94 456 6.32 393 525 240 • 152 (26) (27) (28) 1 ~~(29) (30) (31) (32) 12i i+l+i = f+i^ = 41 2J , 113.46 i+i+i = 62i 2i 6i 49.6097 l?f 5i 3J 19.9 371 — 9.87 (34) (35) .0086 18.253 6.04 - (33) (36) (37) .49 J+l = 32= 2 ft. 6 in. 2 3rr. 5 mo. 16i .28 3 ft. 5 in. 3 yr. 6 mo. 12 .63 .95 4 ft. 9 in. 4 yr. 9 mo. 5 yr. 2 mo. 21 32;: l!69 22 6 yr. 7 mo. .33 .36 1.01 .56 .88 (38) .75 .56 1.10 25.091 + 100.4+25+98.28+19.3614 = .18 .56 iThe scales are printed in large type, on separate sheets, 8|' x 11", with ample space for the insertion of answers. The Arithmetic Scales and Their Uses Series A SUBTRACTION SCALE Name When is your next birthday? How old will you be?.. Are you a boy or girl? In what grade are you? (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 8 6 2 9 4 11 13 59 78 7 — 4= 76 S 1 3 4 7 8 12 37 60 (12) (13) (14) (IS) (16) (17) (18) (19) (20) 27 16 SO 21 270 393 1000 S67482 2i — 1 = 3 9 2S 9 190 178 S37 106493 (21) (22) (23) (24) (2S) (26) 10.00 3i — 1= 8083646S 8| 27 4 yds. 1 ft. 6 in. 3.49 49178036 S| 12i 2 yds. 2 ft. 3 in. (27) (28) (29) (30) S yds. 1 ft. 4 in. 10 — 6.25= 7Si 9.8063 — 9.019 = 2 yds. 2 ft. 8 in. S2i (31) (32) (33) (34) (35) 7.3—3.00081= 1912 6 mo. 8 da. S 2 6| 3J — lf= 1910 7 mo. IS da. = 2| 12 10 6 Measurements of Some Achievements in Arithmetic Series A MULTIPLICATION SCALE Nafne When is your next birthday? How old will you be? Are you a boy or girl? In what grade are you? , (1) (2) (3) (4) (S) (6) (7) 3X7= 5X1= 2X3= 4X8= 23 310 7X9 = 3 4 (8) W (10) (11) (12) (13) (14) (15) 50 254 623 1036 5096 8754 165 235 367 8 6 840 23 (16) 17) (18) (19) (20) (21) (22) 7898 145 24 9.6 287 24 8X51 = 9 206 234 4 ,05 2J (23) (24) (25) (26) (27) (28) (29) liX8= 16 JX{= 9742 6.25 .0123 iX2 = 21 59 3.2 9.8 i^?^ .. (3^) (32) (33) (34) 2.49 12 15 6 dollars 49 cents 2i X 3i = i X * = 36— X— = 8 25 32 (35) (36) (37) (38) (39) 987J 3 ft. 5 in. 2i X 4J X li = .0963* 8 ft. 9J in. 25 5 .084 9 The Arithmetic Scales and Their Uses 7 Series A DIVISION SCALE Name When is your next birthday? How old will you be? Are you a boy or girl? In what grade are you? (1)_ (2) (3) (4) (5) ( 6) 316 91 27 41 28 115 91 36 3139 (7) (8)_ (9)_ (10) (U) (12) 4t-2= 910 111 6X = 30 2113 2 -=- 2 : (13) (14) (15) (16) (17) 4 1 24 lbs. 8 oz. 8 ] 5856 J of 128 = 68 1 2108 SO + 7 : (18) (19) (20) (21) (22) 13 1 65065 248 H- 7 = 2.1 ] 25.2 25 19750 21 13.50 (23) (24) (25) (26) 23 1 469 75 ] 2250300 24001 504000 12 1 2.76 (27) (28) (29) (30) i of 624 = .003 1 .0936 3i ^ 9 = f-i-5 = (31) (32) (33_) 5 3 9| -I- 3f = 52 1 3756 4*5 (34) (35) (36) 62.50 4- IJ = 531 1 37722 9 ] 69 lbs. 9 oz. 8 Measurements of Some Achievements in Arithmetic Series B ADDITION SCALE Name When is your next birthday? How old will you be? Are you a boy or girl? In what grade are you? (1) (2) (3) (S) (7j (10) 2 2 17 72 3 + 1 = 21 3 4 2 26 33 — 3 — — 35 (13) (14) (16) (19) (20) 23 25+42= 9 $ .75 $12.50 25 24 1.25 16.75 16 12 .49 15.75 — IS 19 $8.00 547 5.75 197 2.33 685 4.16 678 .94 456 6.32 393 525 240 152 (33) (36) .49 2 yr. 5 mo. .28 3 yr. 6 mo. .63 4 yr. 9 mo. .95 5 yr. 2 mo. 1.69 6 yr. 7 mo. 22 !33 .36 1.01 .56 .88 .75 .56 1.10 .18 .56 (23) (24) (30) + i= 4.0125 2i 1.5907 6f 4.10 3f 8.673 — (38 ) 25.091 + 100.4 + 25 + 98.28 + 19.3614 = The Arithmetic Scales and Their Uses Series B SUBTRACTION SCALE Name When is your next birthday? How old will you be?.. Are you a boy or girl? In what grade are you? (1) (3) (6) (7) 8 2 11 13 5 17 8 (9) (13) (14) (17) 78 16 SO 393 37 9 25 178 (19) 567482 106493 (20) 21-1 = (31) 7.3 — 3.00081 = (24) 81 SJ 31 (25) 27 121 (27) 5 yds. 1 ft. 4 in. 2 yds. 2 ft. 8 in. inu lo Measurements of Some Achievements in Arithmetic Series B MULTIPLICATION SCALE Name When is yoiir next birthday? How old will you be? Are you a boy or girl? In what grade are you? , (1) (3) (4) (5) 3X7= 2X3= 4X8= 23 3 (8) (9) (11) (12) 50 254 1036 5096 3 6 8 6 (13) (16) (18) (20) 8754 7898 24 287 8 9 234 .05 (27) (29) 6. 25 iX2 = 3.2 ,, (33) (35) (37) (38) 2i X 3J = 987i 2i X 4i X U = .0963i 25 .084 (24) (26) 16 9742 21 59 The Arithmetic Scales and Their Uses ii Series B DIVISION SCALE Name When is your next birthday? How old will you be?.. Are you a boy or girl? In what grade are you? m_ (2) (7) (8)_ 316 9^27 4-5-2= 910 ( 11) (14) (15) (17) 2 1 13 81 5856 i of 128 = SO -i- 7 = (19) (23) (27) (28) 248 -5- 7 = 23 1 469 | of 624 = .003 1 .0936 (30) (34) (36) i -i- S = 62.50 -T- li = 9 1 69 lbs. 9 oz. 12 Measurements of 'Some Achievements in Arithmetic over until they are told to do so. When all have their pencils in hand, say, " Turn your papers over and answer the questions at the top of the page." (The number of questions to be an- swered can be determined by the one giving the tests. It will take less time and cause less confusion if the one giving the tests will repeat the question and tell the children what to write. For example say, " The first line asks, ' What is your name?' Write your name," etc.) When all the questions have been answered repeat the follow- ing formula of specific directions. If you should happen to be giving the Addition test say, " Every problem on the sheet which I have given you is an addition problem, an " and problem." Work as many of these problems as you can and be sure that you get them right. Do all of your work on this sheet of paper and don't ask anybody any questions. Begin." For each scale in Series A, allow twenty minutes; for each in Series B, allow ten minutes. It is important that the time be kept accurately and that all of the children quit work when the signal " Stop " is given. Most of the children will have finished before that time. Those who do not have done, in all proba- bility, all they can; at least they have taken as much time as it takes the average class to complete the test. The only variation in procedure in giving any of the other tests is the substitution in the formula of specific directions of the expressions subtraction or " take away problems," multipli- cation or " times problems," and division or " into problems," for the expression addition or " and problems." The expres- sions " and," " take away," " times," and " into " problems are used so as to make clear to the children what process is to be involved. It is possible that teachers use these expressions in the lower grades instead of " addition, subtraction, multiplica- tion and division problems." There is a great variation in the names applied to the subtraction process. In giving the original tests it was necessary to find out how the teacher designated the process and then use her terminology. 2. Directions for Scoring the Tests In scoring the tests the standard for marking a problem cor- rect is absolute accuracy, and, wherever .possible, reduction to The Arithmetic Scales and Their Uses 13 TABLE I: Answers to Problems PROBLEM ADDITION SUBTRACTION MULTIPLICATION DIVISION 15 3 21 2 2 9 6 S 3 4 98 6 32 5 5 98 69 4 6 97 4 1,240 13 7 4 5 63 2 8 8 47 150 9 87 41 1,524 1 in 80 .3 4,361 5 il 168 16 8,288 6-1/2 not 6+1 12 59 24 30,576 J3 64 7 70,032 61bs. 2oz. not 6 2 14 67 25 6,600 732 15 425 12 5,405 32 if, 70 80 71,082 31 It 844 215 29.870 7-1/7 not 7+1 is 10,966 463 5 616 5,005 19 $2.49 460,989 38.4 35-3/7 not 20 $45.00 1-3/4 14.35 12 21 $27.50 6.51 60 390 10 ^ 87^ 3 46 0.13 11 2/3 31.658,429 10 2°;S+9"^' 24 18.3762 3-1/8 42 30004 25 2, not 16/8 nor 14-3/8 21/32 210 2/1 26 125. not 1 yd. 2 ft. 3 in. 574,778 .23 123-4/2 = 2 not 63 in. 27 7/8 2 yds. 1ft. 8 in. 20^00 546 not 81 in. 28 1 not 4/4 nor 3-3/4 or 3.75 .12,054 31.2 29 12-1/4 not 23-1/2 not 1/4 not 2/8 7/18 11.3/4 = 23-2/4 = 30 12-5"/8*not .78V3^ 89.64 3/20 or .15 11-13/8 = 15/8 31 217^1413 4.29919 9/40 2-1/12 32 1-1/2 not 6/4 1 yr. 10 mo. $51.92 or nor 1-2/4 = - 23 da. 51 dol. 92 cts. 2-17/30 33 10.V5^ 13/60 8-3/4 "7^2(23°' 34 13/24 3-1/4 not 1/4 50. 3-2/8 = 1/4 35 10 ft. 8 in. or 2-1/4 not 24693-3/4 71-7/177 or 10-2/3 ft. 2-2/8 = 1/4 71.04 36 22yrs.Smo.or 17 ft. 1 in. 7 lbs. 11-2/3 22-5/12 yrs. '^■- ^ 1^^- ' 9 oz. ^« 968^1^24 .0080902-1/2 or 38 268.1324 .00809025 39 79 ft. 1-1/2 in. 14 Measurements of Some Achievements in Arithmetic its lowest terms. If the results are to be comparable with the results and values established in these scales, only those answers should be accepted as correct which are found in Table I. These are the answers which were accepted in the original development of the scales. A few incorrect answers are also listed in order to offer less chance for variation in the scoring of the results. 3. Directions for Determining the Class Score For the determination of the class score, two different methods have been derived. The first method was derived especially for use in Series A, where there is no definite attempt to place the problems on a linear scale with equal steps between them. By this method, after the problems have been marked as right or wrong, enter the results on a score sheet similar to the one given in Table II. Thus a complete record of the particular problems solved by each child is obtained. To complete the class score, find the number of pupils in the class that solved each problem correctly. Divide the number by the total number in the class so as to get for each problem the per cent of the class that solved it correctly. Since, in the development of these scales, that problem which can be solved correctly by just 50 per cent of the class is taken as the best measure of the achievement of the class, select those five prob- lems which come nearest to being solved by just 50 per cent of the class.^ Table III gives the established value for each prob- lem in the different processes. From Table IV find the amount that must be added or subtracted to the values given in Table III for each of these selected problems to find just what difficulty a problem would need be in order that just 50 per cent of the class could solve it. Take the average of these five determina- tions and let it represent the class score. This means that a problem of that difficulty can be solved by just 50 per cent of the class in question. 1 The work of scoring may be greatly economized by omitting the scoring and entering on the score sheet of the problems which will not figure in the determination of the 50 per cent right point. Thus in an eighth grade class the first twenty or more problems can most certainly be neglected. A little experience will teach the scorer what problems he needs to score for a given class. The Arithmetic Scales and Their Uses 15 TABLE II Sample Score Sheet pupils' names X i 34567 NO. OF PROBLEM 910)1 12 1314 1516 17 ISIS eoeue £3 ETC. "1 " A ^ ^ <=d -^ ^ J. = J u „ ^ ^ "?: ^ ^ = r- '> ?= . ' "^ >- ^ l>- [-^ [^ -N M -^ M p — ^ r- p-. ^.^^ Ko. Wttlug Esoh Problaii J( Betting aioh Picblm _ To illustrate the determination of the class score, the five problems in addition which came nearest to being solved by- just 50 per cent of the pupils in a certain third grade class of 61 pupils were problems Nos. 14, 17, 16, 18 and 15. These i6 Measurements of Some Achievemen'ts in Arithmetic TABLE III Established Value of Each Problem in Each Scale NO. OF PROBLEM ADDITION subtraction multiplication division 1 1.23 1.06 .87 1.57 2 1.40 1.48 1.05 2.08 3 2.50 l.SO 1.11 2.18 4 2.61 1.50 1.58 2.31 5 2.83 1.70 2.38 2.40 6 3.21 1.75 2.62 2.46 7 3.26 2.18 2.68 2.56 8 3.35 2. 51 2.71 3.05 9 3.63 2.57 3.78 3.16 10 3.78 2.65 3.79 3.20 11 3.92 2.88 4.09 3.49 12 4.18 2.90 4.26 3.59 13 4.19 2.96 4.71 3.96 14 4.85 3.64 4.72 4.06 15 4.97 3.70 4.73 4.60 16 5.52 4.35 5.05 4.67 17 5.59 4.41 5.20 4.98 18 5.73 4.42 5.24 5.16 19 S.7S 5.18 5.38 5.26 20 6.10 5.52 5.63 5.31 21 6.44 5.70 5.72 5.36 22 6.79 5.75 5.83 5.48 23 7.11 5.76 5.83 5.56 24 7.43 5.91 5.89 5.58 25 7.47 6.77 6.29 5.78 26 7.61 7.07 6.30 5.91 27 7.62 7.21 6.58 6.04 28 7.67 7.38 6.85 6.43 29 7.71 7.41 6.97 6.76 30 7.71 7.41 7.00 6.83 31 7.97 7.49 7.07 6.87 32 8.04 7.52 7.07 6.88 33 8.18 7.69 7.29 7.22 34 8.22 7.72 7.50 7.24 35 8.58 7.84 7.65 8.17 36 8.67 7.66 8.23 37 8.67 8.02 38 9.19 8. S3 39 8.61 problems were thus solved correctly by 54, 54, 48, 48 and 68 per cent of the class, respectively. Table IV tells how much to add to or subtract from the established value given in Table III The Aritlimetic Scales and Their Uses 17 \i ^ =t p: o o •5 e 3 «^ CO i8 Measurements of Some Achievements in Arithmetic COMPOSITION OP SCALES IN "SERIES B" Addition Subtraction Multiplication Division NO. OF NO. OF NO. OF NO. OF PROBLEM VALUE PROBLEM VALUE PROBLEM VALUE PROBLEM VALUE 1 1.23 1 1.06 1 .87 1 1.57 2 1.40 3 1.50 3 1.11 2 2.08 3 2.50 6 1.75 4 1.58 7 2.56 S 2.83 7 2.18 5 2.38 8 3.05 7 3.26 9 2.57 8 2.71 11 3.49 10 3.78 13 2.96 9 3.78 14 4.06 13 4.19 14 3.64 11 4.09 15 4.60 14 4.85 17 4.41 12 4.26 17 4.98 16 5.52 19 5.18 13 4.7r 19 5.26 19 S.7S 20 5.52 16 5. OS 23 S.57 20 6.10 24 5.91 18 5,24 27 6.04 21 6.44 25 6.77 20 5.63 28 6.43 22 6.79 27 7.21 24 5.89 30 6.83 23 7.11 31 7.49 26 6.30 34 7.24 24 7.43 35 7.84 27 6.58 36 8.23 30 7.71 29 6.97 33 8.18 33 7.29 36 8.67 35 7.65 38 9.19 37 38 8.02 8.53 for each of the problems in order to estimate the value of a prob- lem that w^ould be solved correctly by just 50 per cent of the class. Thus For 54 per cent add .15 to 4.85 54 per cent " .15 68 per cent " .70 48 per cent subtract 48 per cent " = 5.00 " 5-59 =574 " 4-97 =5-67 .07 from 5.52 = 5.45 ■07 " 5-73 = 5-66 Average 5.50 The average of these 5 determinations (5.50) represents better than either single measurement the degree of difficulty that a problem must have in order that just 50 per cent of this class can solve it correctly. The class score for any other class can be computed in a similar manner. The second method for the determination of the class score was derived especially for Series iB where there was a definite attempt to place the problems on a linear scale with equal steps between them. This method introduces a certain amount of error, but for all practical purposes it is a satisfactory measure. By this method the median number of problems solved cor- The Arithmetic Scales and Their Uses 19 TABLE IV For Use in Estimating the Degree of Difficulty Required IN A Problem so That Just 50 Per Cent of the Class CAN Solve it Correctly subtract ADD 10% 1.90 50% 0.00 11 1.82 51 .03 12 1.74 52 .07 13 1.67 53 .11 14 1.60 54 .15 IS 1.54 55 .19 16 1.48 56 .22 17 1.42 57 .26 18 1.36 58 .30 19 1.30 59 .34 20 1.25 60 .38 21 1.20 61 .41 22 1.15 62 .45 23 1.10 63 .49 24 1.05 64 .53 25 1.00 65 .57 26 .95 66 .61 27 .91 67 .65 28 .86 68 .70 29 .82 69 .74 30 .78 70 .78 31 .74 71 .82 32 .70 72 .86 33 .65 73 .91 34 .61 74 .95 35 .57 75 1.00 36 .53 76 1.05 37 .49 77 1.10 38 .45 78 1.15 39 .41 79 1.20 40 .38 80 1.25 41 .34 81 1.30 42 .30 82 1.36 43 .26 83 1.42 44 .22 84 1.48 45 .19 85 1.54 46 .15 86 1.60 47 .11 87 1.67 48 .07 88 1.74 49 .03 89 1.82 90 1.90 rectly is taken as the imeasure of the achievement of any class. By the median number of problems solved is meant such a number of problems ,that there are just as many pupils who solve a greater number as there are those who solve a less number. In order to determine the median point of the achievement of the class, it is necessary to make a distribution table, show- 20 Measurements of Some Achievements in Arithmetic ing the number of pupils who were unable to solve a single problem correctly, the number who solved one problem, two problems, three problems, etc. As examples of this sort of distribution we may take the following: TABLE V Number of Times Each Addition Problem was Solved Correctly 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Class I . . . 1 2 3 4 1 2 3 5 6 11 1 5 2 4 1 1 Class II . . . 3 3 4 4 7 5 3 4 3 1 1 Class III . . . 1 2 4 6 7 4 4 5 5 4 3 2 1 According to these distributions 52 pupils are in Class I, 37 pupils in Class II, and 48 pupils in Class III. Now let us proceed to find the median achievement for each of these class distributions. Since there are 52 individuals in Class I the median point evidently falls between the achievements of the 26th and 27th pupils. Let us begin yvith the individual who was unable to solve a single problem correctly and count the two individuals who solved two problems, the three who solved three problems, and so on till we come to the steps that includes the 26th indi- vidual. Now if we are to indicate the exact point in the achieve- ment of the pupils where there are just as many pupils who solve a greater number of problems as where there are those who solve a less number, it is .necessary to count 5 of the 6 individuals who solved 10 problems correctly. Thus on the assumption that the individuals are distributed over any step at equal distances from one another, the median point is 5/6 of the distance through this step. Hence: the median achieve- ment of this class, i.e., the median number of problems solved, is 10.8 problems correctly solved. Similarly there are 37 pupils in Class II. The middle case is the 19th pupil, who is the fifth pupil in step 4. There are 18 pupils who solve a greater number of problems and 18 who solve a less pumber of problems. Thus the exact median point in the achievement of the class lies in the middle of that frac- tion of a steip assigned to the 19th pupil. Thus the median The Arithmetic Scales and Their Uses 21 point is *TJ^ of the distance through the 4th step. Hence the median achievement for this class is 4.6 problems solved correctly. The distribution for Class III represents a peculiar difficulty in the calculation of its median. There are 48 pupils m this class and evidently the median point falls between the 24th and the 2Sth individual. However, it happens that 24 of the pupils solve more than seven problems and 24 of them solve less than seven. Probably the wisest assumption to make is that the 4 pupils on step 6 take up all of that step and the 4 pupils on step 8 take up all of that. If this is assumed, then the median falls on step 7, probably at 7.5 since any given dis- tance on a scale is best represented by its middle point. Thus the median achievement for the Class III is 7.5 problems solved correctly. By similar computations the medians of any dis- tribution can be obtained. By the comparison of the medians thus determined, we get a very satisfactory measure of the achievement of any class on the basis of the total number of problems correctly solved. 4. Tentative Standards of Achievement While these new scales have not been used in measuring suf- ficient numbers of children to warrant the establishment of definite standards of achievement, it was thought well to indicate some tentative standards. These tentative standards have been derived from the actual achievements of the children tested with the preliminary tests. The fact that these tests were given during the first part of the school year should be kept in mind when comparison is made with tentative standards shown in Tables VI and VII. Table VI contains the tentative standards for Series A. TABLE VI Tentative Standards of Achievement for Series A SUBTRACTION MULTIPLICATION DIVISION 1.44 2.96 1.89 2.54 4.22 4.05 3.21 5.47 5.53 4.94 6.46 6.72 5.87 7.31 7.26 6.59 7.64 7.93 7.16 grade ADDITION II 3.12 III 4.99 IV 6.11 V 6.99 VI 7.95 VII 8.65 VIII 9.01 22 Measurements of Some Achievements in Arithmetic These standards were derived according to the first method given for the determination of the class score. They are based upon the degree of difficulty which the problems must possess in order that just 50 per cent of the class can solve them. Thus, if a problem in addition has 3.12 units of difficulty it will be solved by 50 per cent of the second grade; if it has 4.99 ^units of difficulty it will be solved by 50 per cent of the third grade, etc. Table VII contains the tentative standards for Series B. TABLE VII Tentative Standards of Achievement for Series B subtraction multiplication division 3 6 3.5 3 8 7 5 10 11 7 12 15 10 13 17 13 14.5 18 14 These standards have been derived according to the second method for determining the class score. They are based upon the total number of problems that were correctly solved in each grade. Thus in the second grade in addition, the median achieve- ment was 4.5 problems, in the third grade, 9 problems correctly solved, etc. ^ ' ' 3RADE ADDITION II 4.5 III 9 IV 11 V 14 VI 16 VII 18 VIII 18.5 Section III. THE VALUE AND USES OF THE SCALES 1. The scales themselves contain 148 problems which in- volve many of the fundamental principles of arithmetic. A child who understands and can solve all of these problems correctly probably knows more arithmetic than the average eighth grade child. 2. These scales are useful in that the value of each problem is known, and from these values the value of other problems can easily be determined. 3. The scales are useful in measuring the achievements of any class or of a whole school system. Since all the pupils in all the grades are measured by the same scales, the amount of progress from grade to grade can be definitely determined. Comparisons ,can be made with similar grades in other build- ings or school systems. If the measurements show, for instance, that a certain sixth grade class is unable to solve a greater num- ber of problems correctly than a fifth grade class in the same school system, the cause of this condition should be investigated. In such ways the tests should prove useful to those in charge of school systems. 4. Perhaps the most valuable use of the scales lies in the diagnosing power of the class mistakes. The writer was con- vinced during |the process of scoring these test papers, nearly 20,000 in all, that the mistakes of a class tend to be grouped around some central tendency. The great variety of the prob- lems in these scales and the fact that the problems in each of the various operations proceed from the simplest to the more difiScult problems aid greatly in locating the weaknesses of the class. If a large number in a class fail to invert the divisor in the problems in division of fractions, or if a large number in a class fail to locate the decimal point properly in the problems in multiplication of decimal fractions, a teacher should know immediately that these classes need more practice in these par- ticular processes. In a like manner, by locating the particular types of problems missed, one should be able to direct the work of a class more intelligently. 23 Section IV. LIMITATIONS OF THE SCALES 1. It is possible that with a greater number and variety of pupils the value of some of the problems might be somewhat changed. However, the children tested were from widely sepa- rated districts in Indiana, New Jersey, Connecticut, and New York. They represent children from many classes of society and from many nationalities. Moreover, much variation existed in the methods of teaching and in the school room practices. Thus the writer believes the values established are well founded. 2. On the scales as now presented the value of some of the problems may be slightly altered due to the fact that they are located in different positions from those in which they were located on the preliminary lists of problems. The exact amount of , this alteration can be determined only by further testing with the scales. 3. The scales as now presented might be slightly bettered if two or three more diificult problems were added to each of them. The scales probably would be bettered if problems could be found of such difficulty as to make the steps between them of exactly equal distance. However, for practical purposes, the effects of these two defects can be disregarded. 4. The value of these scales may be somewhat affected by their more extended use. As teachers become more acquainted with them, they may drill especially upon them. Therefore, it would be much better if several series of such scales of the same difficulty as these should be developed. 1 5. These scales are not intended to give a definite measure of an individual child. But, if we can measure approximately how difficult a problem a child can solve and then supplement this problem with a large list of problems similar in nature and in difficulty, we can get a fairly accurate measure of the achievement of the child. 6. The relative difficulty of these problems was determined from the achievements of school children in grades two to eight inclusive. It is probable that for adults and teachers the rank- ing would be in a different order. Only further testing can substantiate this point. 24 PART II Section I. DERIVATION OF THE SCALE I. History of the Scale The completed scales as shown in Part I of this monograph have been developed from about 20,000 test sheets. The first preliminary .series of tests were given to a number of pupils in the public schools of Indiana and New Jersey. The pre- liminary series of tests consisted of a sheet of problems in addition and likewise one in subtraction, multiplication, and division. In constructing these preliminary lists there was a definite attempt to select problems of as great a variety as the fundamental processes would permit. .There was also an at- tempt to begin the series in each process with the easiest problem that could be found and then gradually to increase the difficulty of each succeeding problem until the last ones in the series would be correctly solved by only a small percentage of the pupils in the eighth grade. By the selection of problems of such varied types and by giving the same lists of problems to pupils in all grades, it was thought that the diagnosing power of the lists would be greater and that the amount and the nature of the progress of one grade over another could best be determined. The preliminary lists of problems in addition were given to 908 pupils, in subtraction to 916 pupils, in multiplication to 868 pupils, and in division to 696 pupils. |The results of these preliminary lists showed that some of the problems were poorly selected and that they should be discarded. When the prob- lems were ranked according to the total percentage of pupils solving them correctly, the results showed large gaps existing between the problems in particular portions of the series. Guided by the results of these preliminary lists new lists were constructed. Only those problems of the original lists were chosen which were solved by a gradually increasing per- centage of the pupils as one proceeded from the lower to the higher grades. If a problem were solved by a higher per- centage of the pupils in the lower grades than in the higher 25 26 Measurements of Some Achievements in Arithmetic grades it was rejected. Wherever there tended to be too large a step between two consecutive problems in the original series an attempt was made to interpose two or three problems of intermediate difficulty. ( From the last week in October till the end of the second week in December, 191 5, pupils were tested with these new lists of problems. These pupils were from seven different school systems located in Indiana, New Jersey, Connecticut, and New York. The addition problems were given to 4,489 pupils, the subtraction to 4,423 pupils, the multiplication to 3,922 pupils, and the division to 3,660 pupils. These pupils were distributed fairly equally from the second to the eighth grades inclusive. All of the tests were given by the writer himself with the exception of those given to the pupils in two small school sys- tems in Indiana.^ The tests were given and the results scored according to the instructions given for administering the tests in Part I of this monograph with the one exception that no time limit for the solution of the problems ,was used. It was felt to be highly important, if the difficulty of each problem was to be firmly established, that each child should have a chance to solve each problem. All of the tests were scored by the writer himself and thus the personal element in scoring was reduced to a minimum. The standard for marking a problem right or wrong as pre- sented in Part I of this monograph was arbitrarily adopted. It was decided that a problem to be marked correct must be absolutely accurate and, wherever possible, reduced to its lowest terms. Otherwise, the problem was marked wrong. How- ever, before adopting this arbitrary standard an effort was made to gain from teachers and supervisors of arithmetic the stand- ards by which they marked a problem right or wrong. It was almost unanimously agreed that a problem must be absolutely accurate and reduced to its lowest terms. Thus the arbitrary standard adopted by the writer is in accordance with the best practice exercised in the teaching of arithmetic. The results of these tests were recorded in two ways: I. The pupils were distributed according to the number of 1 Those giving the tests in these two systems were men who have had experience in giving tests and who could be trusted to carry out the writer's directions. Derivation of the Scale 27 TABLE VIII Distribution According to the Number of Addition Problems Solved GRADE grade GRADE GRADE GRADE GRADE GRADE II III IV V VI VII VIII 38 3 21 41 37 15 37 a 36 30 82 55 35 2 37 96 72 34 4 45 91 70 33 1 51 90 76 32 8 34 75 46 31 13 45 83 45 30 1 13 45 49 27 29 26 51 57 20 28 2 35 33 53 18 27 1 32 36 48 19 26 1 2 46 37 34 10 25 11 40 34 34 4 24 5 54 37 16 4 23 3 33 75 29 27 1 22 6 47 64 25 8 2 21 1 11 42 77 15 7 1 20 10 54 54 15 4 19 26 65 49 6 3 18 43 56 43 5 2 17 4 47 75 28 3 16 3 64 72 10 2 15 7 70 42 7 14 7 54 24 3 13 13 44 14 12 10 40 18 1 11 43 39 16 1 10 31 33 7 9 46 35 5 8 38 23 2 7 35 16 1 6 36 10 2 5 69 14 1 1 4 48 8 2 3 43 4 1 2 17 4 1 13 6 1 25 4 rested. . 489 615 602 687 633 917 544 an . . . .. 6.819 14.509 18.321 23.073 29.774 32.446 33. 25percent. . 4.505 10.902 16.201 20.532 25.625 28.872 31.667 75 " " . . 9.929 16.894 20.694 26.206 33.446 35.070 35.903 Quartile 2.712 2.996 2.247 2.837 3.910 3.099 2.118 28 Measurements of Some Achievements in Arithmetic TABLE- IX Number in Each Grade that Solved Each Problem in Addition Correctly problem grade grade grade grade grade grade grade no. II III IV V VI VII VIII 1 388 456 499 654 622 896 541 2 433 582 595 681 630 913 542 3 392 593 595 680 626 911 539 4 326 468 521 659 614 901 540 5 323 501 554 673 629 914 544 6 279 530 , 565 668 628 911 542 7 259 538 565 679 631 915 542 8 220 474 542 665 624 911 544 9 165 530 568 667 613 880 522 10 152 531 570 663 623 895 539 11 190 543 577 663 608 886 535 12 52 399 541 657 620 896 537 13 32 229 373 627 622 901 537 14 37 405 541 664 627 900 534 15 23 328 499 627 602 876 530 16 6 238 387 567 533 806 500 17 22 288 431 551 500 787 475 18 8 208 386 539 511 801 SOS 19 1 92 246 399 436 662 457 20 1 87 307 555 586 883 528 21 1 71 276 498 564 839 498 22 49 204 441 528 814 489 23 4 34 308 490 771 500 24 4 99 296 521 385 25 3 14 213 397 651 457 26 2 11 192 423 682 483 27 10 178 369 678 470 28 14 166 414 693 448 29 8 131 300 591 409 30 2 33 157 403 684 462 31 3 34 164 317 490 344 32 4 157 373 674 421 33 3 57 235 461 290 34 15 128 271 338 684 432 35 4 57 169 318 558 392 36 2 40 179 537 354 37 1 20 176 529 359 38 1 9 155 24tf 274 No. Tested. 489 615 602 687 633 917 544 Derivation of the Scale 29 problems solved correctly. Table VIII represents the distribu- tion for the problems in addition. Beginning at the lower left- hand corner, Table VIII shows that 25 out of 489 pupils in the second grade, and 4 out of 615 pupils in the third grade were unable to solve a single problem, etc. This table also shows the median achievement of each grade distribution. The median achievement of a class is such a number of problems correctly solved that there are just as many pupils who solve a greater number of problems as there are those who solve a less number. This table shows the range in the numbeu of problems correctly solved that will include the middle 50 per cent of the pupils. It also shows the variability in terms of the quartile, or, as it is sometimes designated, the " semi-inter- quartile range." 2. The results were tabulated in another method so as to record the number of pupils who solved each individual problem correctly. Thus Table IX shows that 388 out of 489 pupils in the second grade solved problem No. i ; 433 pupils solved problem No. 2, etc. From these two crude summaries given in Tables VIII and IX the addition scales have been developed.^ 2. P.E. AS A Unit of Measure It may be said that we have always measured pupils in the fundarnental operations of arithmetic. It may be said that schools and school systems have likewise been measured. No doubt this is true. Whenever a teacher says that one boy is better in addition than another boy, in a certain sense, she measures him. Whenever we compare one individual with another individual, one quality with another quality, or one class with another class, we are measuring. Such standards of measurements as these are no doubt inaccurate and changeable. Whenever a teacher measures a class by means of an examina- tion she tends to have a more constant and more objective meas- urement. The relation of the different questions of the exam- ^ Similar tables for the problems in subtraction, multiplication, and division will be found at the end of Part 11. In the discussion of the derivation of the scales I shall show in detail the method by which the scale in addition was developed, and shall not discuss the other processes. However, I shall include the final values of each problem in each of the other processes and the most important tables of crude data from which the established values were deter- mined. 30 Measurements of Some Achievements in Arithmetic ination to one another, however, is unknown. All the questions may be of equal difficulty, or one may be several times as diffi- cult as another. The chief value of a scale as a means of meas- urement is ^that it is made up of a number of distinct units whose value is known and remains constant. Such a scale can be used by different people in making similar measurement and the results will be comparable. On the linear rule the unit of measurement is the inch or centimeter; on the thermometer, the degree. Everyone knows what is meant when we speak of in inch, a degree, or any fractional part thereof. These amounts are very definite and always have the same meaning. Moreover almost any one can make reliable measurements with a rule or with a thermometer. In the building of these arithmetic scales there has been a definite attempt to approximate as closely as possible the accur- acy and the constancy of the ruler or the thermometer. The difficulty of each problem has been established and its position above a selected zero point determined. The problems have all been placed in their relative positions on a projected linear scale. In the scales of Series B a definite attempt has been made to select problems with equal amounts ,of difficulty between them. The unit of measure of difficulty on these arithmetic scales, which corresponds to the inch on the ruler or to the degree on the .thermometer, is what is called in statistical terms the Median Deviation or Probable Error. (P.E.) Before taking up the significance of the median deviation let us discuss the normal surface of frequency. In the construc- tion of these scales, it has been assumed that achievement in the solution of problems in the fundamental processes is dis- tributed according to the normal surface of frequency. Fur- thermore it has been assumed that the variability of any grade from the second to the eighth is equal to that of any other. These assumptions are based upon the well-established principle that intellectual abilities are distributed in the same way as are physical traits. If we should arrange one thousand men, selected at random, in ,a row according to their height, we should find a very large group of men in the center who are about medium height. On one end of the row would be a few very short men and on the other end would be a few very tall men. Likewise Derivation of the Scale 31 if we assume that achievement in the solution of problems in the fundamental processes in any grade is distributed normally, then we should expect to find a large number of the class solv- ing about the same number of problems; furthermore we should expect to find a few dull pupils who can solve but just a few problems and a few bright pupils who can solve more than the average number of problems. The so-called normal curve illus- trating such a distribution is reproduced in Fig. i. The properties of the normal curve have been most accurately determined. Let us assume that Fig. i represents the achievement in the solution of problems among a large number of third grade pupils. very few few average many very many Fig. I. Normal surface of frequency showing the distribution of achievement in the solution of problems. The space enclosed between the curve and the base-line repre- sents all of the pupils arranged according to the number of problenn solved. The height of the curve above the base-line indicates the number of pupils in the class solving the relative number of problems shown on the base-line. Each pupil is represented by an equal amount of the enclosed area. Thus, at the extreme left the curve is very near the base, which indi- cates the small number of pupils who were able to solve only a very few problems. In the middle the curve is distant from the base-line representing the large number of pupils who solved an average number of problems; at the extreme right the curve is yery near the base, which indicates the small number of pupils who are able to solve many more than the average num- ber of problems. If our assumption with regard to the achievement in the solu- tion of problems is true, then the graphic representations of the 32 Measurements of Some Achievements in Arithmetic tables of distribution according to the number of problems solved must be similar to Fig. i. Figs. 2 to 8 inclusive represent graphically the distribution of the achievement in the solution of the addition problems throughout the various grades. These figures on the whole correspond fairly well to the normal curve of distribution. It will be seen that in the second grade distribution the curve is somewhat skewed to the left. This is probably due to the fact that a great number of the teachers were just beginning to teach the fundamental operations to their classes. It will also be seen that the distributions for grades seven and eight are skewed somewhat to the right. This indicates the need for one or two more difificult problems at the end of the addition series. It will be noted from the distribution tables in the back of this monograph that the distributions for the other processes con- form to the normal curve better than the foregoing figures. 1^ o=Median Score. 1 i 3 ^ S (, 1 i *i 10 ft It 15 It IS n, IT II i* s k 7 t 9 It 11 It fi t It /T IW If ia V ttiiV^ Kf K< 27 Kff tf M X »2 J3 Vt 3f »k *? 32 Fig. 6. Distribution according to the number of Addition Problems solved in Sixth Grade. 34 Measurements of Some Achievements in Arithmetic nJ yu nJ I ! 1 13£= 1 ! -S. ! I / t S f 5- 4 T t q 10 II H 13 /i^ IS ti J7 II 11 M il 11^X3 t-f is%m a li 30 St JK SJ af iS»i7^t Fig. 7. Distribution according to the number of Addition Problems solved in Seventh Grade. lZiHSI.7it LJ1 H i