Teacher's Manual For First Year Algebra Scales Hotz Published by QTeacfjerg College, Columbia 3infter*ftp New York City 1922 Teacher's Manual For First Year Algebra Scales By HENRY G. HOTZ, Ph.D. Professor of Secondary Education University of Arkansas Published by Eeadjer* College, Columbia 33ntoer*tt|> New York City 1922 Copyright, 1922, by Teachers College, Columbia University PREFACE This manual is compiled for the purpose of assisting teachers of mathematics in the administration and practical use of my First Year Algebra Scales. There is a feeling that the original mono- graph, which appeared in the Teachers College Contributions to Education series, is too technical and, consequently, too difficult for most teachers to read intelligently and to determine from it with ease how to apply the scales most profitably. Suggestions concerning the purpose of these scales are incorporated in this manual in as simple and direct form as possible. Special training in statistical methods is not necessary for their comprehension. Besides the Tentative Standards of Achievement proposed in the original monograph, scores more recently obtained in various cities and through school surveys have been included. Suggestions on presentation and diagnosis of results have also been added. In order that the progress of a class may be more scientifically determined, a revision of the tests, with the exercises arranged in duplicate or alternate scales of equal difficulty, will be published in the near future. HENRY G. HOTZ University of Arkansas Fayetteville, Arkansas 438402 CONTENTS PAGB Description of the Scales 7 Selection of Scales to be Used 8 When to Give the Tests 9 Directions for Administering the Tests 10 Directions for Scoring the Papers 12 Directions for Determining the Median or Class^Score \ . . 21 Standards of Achievement Tentative Standard Scores 26 Scores Attained in Various Other Cities 27 Graphical Representation and Statistical Interpretation of Results 34 Analysis of Errors 40 Value of the Scales 44 Bibliography 46 Teacher's Manual Fpr First Year Algebra Scales The First Year Algebra Scales were first published in Since then they have been extensively used by teachers, school administrators, directors of educational research, and in various school surveys. I. DESCRIPTION OF SCALES The scales consist of five different sheets of algebraic exercises designed to measure the ability of pupils in elementary algebra. They are : 1. Addition and Subtraction 2. Multiplication and Division 3. Equation and Formula 4. Graphs 5. Problems The first two scales, it will be seen, are designed to test the achievement of students in the fundamental operations, involving integral, fractional, and radical expressions; the second two, to test the ability of students in handling the instruments of quanti- tative thinking; while the last is composed of verbal problems of the type usually stressed in the first year of algebra. The exercises in each scale are arranged in order of difficulty; that is, each scale begins with exercises so easy that they can be solved by practically every member of a class. Each succeeding exercise, however, becomes increasingly more difficult so that the last ones in each scale can be solved by only a relatively small number of students who try them. Two series of each scale are offered Series A and Series B. Series B is the longer and contains from eleven to twenty-five exercises in each scale. Series A is only about one half as long and contains from eight to twelve exercises in each scale. It covers just as wide a range of difficulty and has the added advantage of having the intervals between successive exercises and problems approximately equal: that is, Ex. 3 of a given scale is as much more difficult than Ex. 2, as Ex. 2 is more difficult than Ex. I. 1 Hotz, Henry G.: First Year Algebra Scales. Teachers College, Columbia Univer- sity, Contributions to Education, No. 90. 8 FIRST YEAR ALGEBRA SCALES In determining individual and class scores, the factor of primary importance is not so much how many exercises an individual can solve correctly in a given time, but rather how far along on the scale of exercises, arranged in order of increasing difficulty, he can perform satisfactorily. In other words, the pupil is measured almost entirely by the point which he reaches on the scale. For this reason the tests may very properly be technically called "scales," and are characterized as "difficulty tests" or "power tests" by specialists in educational measurements. The scales were derived from data obtained from tests given to over 16,000 high-school students. The schools which cooperated in standardizing the scales varied all the way from the small rural high school to the large cosmopolitan high school. Classes were tested in eighty- four high schools located in the states of Massa- chusetts, Connecticut, Rhode Island, New York, New Jersey, Ohio, Wisconsin, Missouri, Oklahoma, Colorado, and Washington, and the results subjected to intricate statistical treatment. The difficulty of each exercise, or its position on the scale, was determined by the percentage of pupils solving each exercise correctly. 2 2. SELECTION OF SCALES TO BE USED Series A will be found on the whole to be more satisfactory than Series B, especially where the time available for testing purposes is limited. This is particularly true if the purpose of the test is primarily to determine degrees of attainment. If, however,- the purpose of the test is mainly diagnostic, that is, to discover difficulties which the students are encountering, Series B should be used. It contains a richer variety of exercises and, consequently, a greater number of type processes. This makes it possible for teachers to make a more complete analysis of the mistakes made by pupils. If only one scale can be used, it should be the Equation and Formula Scale, because it is more comprehensive and so tests a much wider range of functions. At least two scales should be used, however, and the scale which undoubtedly comes second in importance is the Problem Scale. If Series A is used there will be ample time to give both during a single class period of forty minutes. 2 For a complete account of the method employed in locating each exercise on a linear scale, consult Hotz, Henry G.: First Year Algebra Scales. WHEN TO GIVE THE TESTS 9 Whenever it is possible to do so, all five of the tests of a given series should be used, since the achievement on all of the tests gives a much more reliable indication of a pupil's ability than the results from one or two tests would give. Teachers have found it most practicable to use the tests in rotation somewhat as follows : 1. At the end of three months Addition and Subtraction Scale Equation and Formula Scale 2. At the end of six months Multiplication and Division Scale Problem Scale 3. At the end of nine months Equation and Formula Scale (repeated) Graph Scale Whenever it is desired to use the same scale a second time, it is advisable to select it from a different series. It is feared that unless at least six months' time has elapsed since a given test was used some of the practice effect may survive. 3. WHEN TO GIVE THE TESTS The scales may be used very profitably as early as the end of the third month of the school year. Tentative Standards of Achievement, 3 based upon the 16,000 papers of the original study, were compiled for three-, six-, and nine-month intervals. It is, therefore, much more satisfactory for comparative purposes to submit these scales to algebra classes immediately after they have studied algebra for three, for six, or for nine months. However, data on the achievement at other intervals are being collected constantly, much of which is included in this pamphlet, 4 and as time passes more and more information with regard to the achievement that may reasonably be expected at other intervals will become available. The scales are not intended to be used beyond the first year. For this reason very few results from classes having had algebra more than ten months have been reported. 8 See p. 26. 4 See pp. 27-34- 10 FIRST YEAR ALGEBRA SCALES 4. DIRECTIONS FOR ADMINISTERING THE TESTS 1 . Preliminary. Before passing the papers see that the desks are cleared and pupils are provided with pencils. For the graph test rul- ers should also be provided. Then make the following statement: "I am going to give you a test to see how well you can solve exercises in algebra. Papers will be passed to you with the printed side down. Please leave them so until I tell you to turn them over." 2. Pass the papers, or have them distributed by the pupils in the front seats, with the printed side down (Series B, first page down) and the top end away from the pupil. 3. When all are ready say to the class: "Turn your papers. Write your name in the first blank space," etc. (The number of blank spaces to be filled out may be determined by the one giving the test. It is not necessary to have all filled in. Some teachers simply have pupils write their names in the upper right hand corner of the blank page so as to prevent the pupils from seeing any of the exercises before all of the directions have been given.) 4. Then repeat one of the following series of directions, depend- ing upon the tests to be given: Addition and Subtraction Scale "Attention! The exercises on these sheets are in addi- tion and subtraction collection of terms. Take the ex- ercises in the order in which they are given. Work as many as you can and be sure you get them right. Work directly on these sheets and do not ask anybody any questions. When you have worked all the exercises you can, lay aside your pencils and remain quiet so as not to disturb those who are still working. You will have twenty minutes in which to work (Series B, forty minutes], Start." Multiplication and Division Scale "Attention! The exercises on these sheets are in multi- plication and division. Take the exercises in the order in which they are given. Work as many as you can and be DIRECTIONS FOR ADMINISTERING THE TESTS 1 1 sure you get them right. All answers must be reduced to their simplest forms. Work directly on these sheets and do not ask anybody any questions. When you have worked all the exercises you can, lay aside your pencils and re- main quiet so as not to disturb those who are still work- ing. You will have twenty minutes in which to work (Series B, forty minutes). Start." Eqiiation and Formula Scale "Attention! On these sheets you are given a number of equations and formulae to solve. Take the exercises in the order in which they are given. Solve as many as you can and be sure you get them right. Work directly on these sheets and do not ask anybody any questions. When you have worked all the exercises you can, lay aside your pencils and remain quiet so as to not disturb those who are still working. You will have twenty minutes in which to work (Series B, forty minutes). Start." Graph Scale "Attention! On these sheets you are given a number of graphs. Read each question carefully and then do as you are told to do. Take the exercises in the order in which they are given. Answer as many questions as you can and be sure you get them right. Work directly on these sheets and do not ask anybody any questions. When you have answered all the questions you can, lay aside your pencils and remain quiet so as not to disturb those who are still working. You will have twenty-five minutes in which to work. Start." Problem Scale "Attention! On these sheets you are given a number of questions to answer. Take the exercises in the order in which they are given. Answer as many questions as you can and be sure you get them right. In all the prob- lems which call for the equation, for example No. 4, simply state the equation which will solve the problem. Take for example, this problem: A coat and hat cost $30. The coat cost 5 times as much as the hat. Find the cost of each. The equation would be x + $x $30. 12 FIRST YEAR ALGEBRA SCALES (Write the equation on the board.) Work directly on these sheets and do not ask anybody any questions. When you have answered all the questions you can, lay aside your pencils and remain quiet so as not to dis- turb those who are still working. You will have twenty-five minutes in which to work (Series B, forty minutes). Start." 5. When the time is up, say "Stop" and collect the papers. Most of the pupils will have finished before this time. Those who have not, in all probability have done all they can. A warning, stating the amount of time left, should be given three minutes before time is called for the tests of Series A and five minutes in advance for those of Series B. With many classes which have had less than nine months of algebra, and especially those which have had only three months, it is perfectly safe to call time before the full time allowed for that particular test has elapsed. Students may be provided with scratch paper for their own use. It has been found to be most satisfactory to pass quietly down the aisles a few minutes after the test is started and give each pupil a sheet of scratch paper. Pupils will find it more convenient, how- ever, to work directly on the question sheets. For all but the problem test it is desirable to have as much of the work as possible on these sheets. 5. DIRECTIONS FOR SCORING THE PAPERS In scoring the papers, answers are to be marked either right or wrong. All answers which may be accepted as correct are given on pages 13 to 19 of this manual. A few incorrect answers are also listed, to indicate more definitely the types of answers which must not be accepted. No credit is given for answers that are partially right. This procedure, though somewhat arbitrary, greatly simplifies the task of scoring the papers. It saves time and offers less chance for variation in scoring the results. This last factor, uniformity in scoring, is an absolute essential in order that valid comparisons between different school systems may be made. After each paper is scored, it has been found most convenient to record the total number of exercises solved correctly in the upper right-hand corner of the test sheet. DIRECTIONS FOR SCORING THE PAPERS ANSWER KEY TO ADDITION AND SUBTRACTION SCALE Problem No. Answers Problem No. Answers I 9? 2r j 2 ^2 2 ? 5* 156 r2 ^ 2 . ( ,_ 2)( , +2) 4 9-6* 9 6* 5 6x +2 4 ' 4 4 6 7 1 5-3<* 8 5* 20 (a + i) (a*-a+i)' 9 10 5< + 6 40-13* ii a 2 -66 +4 21 40-13* 12 4*- 1 3c 6c * 3 -i9*+30 I i 15 8' 16 6 a 2* 22 23 24 5iVs; n t 5 Vs + s Vs 6g-7. 7^6a a 2 4 ' 4 a 2 14 FIRST YEAR ALGEBRA SCALES ANSWER KEY TO MULTIPLICATION AND DIVISION SCALE Problem No. Answers Problem No. Answers I 2iy 14 4*- 8; 40 -2) 2 3" r ^ & 6 3 8a 2 6 2 J 5 a(m n) ' am an 4 30 16 si*y 5 6m c 2 2 p 2 5 y(P~9Y 3pr-27 ii ioa3 + 3 3 a 2 -52a + 9 /^ T */2 -? V _1_ Q -v-^ 'IX -\- Q ^ o*^ I V . * O**' 1 V 12 w 2 10 21 3 \^ 3/ 3*^ 9 22 4 8 13 _4. 3a 23 3 DIRECTIONS FOR SCORING THE PAPERS ANSWER KEY TO EQUATION AND FORMULA SCALE Problem No. Answers Problem No. Answers I 2 2 3 15 I 3 3 2 16 154 4 5 7 2 17 El R 6 2; not 2= 2 18 ~;not^; 7 9 i nor x=- 2 8 2 19 10, 5 (both roots) 9 4 20 -5, -i*; not -* = 5 10 15 16 21 ,-,,,-4- . ii 12 40 2 .30 4 7'7 22 23 ;? i 13 15 24 is. rr.i \j Vir r^ 14 m=2,n = 4 * 25 1 * Where two results are ordinarily required in an answer, the exercise is marked correct if the work is done correctly up to the point where only one value is obtained and stopped at that. If, however, the student makes an error in solving for the second value, the problem is scored incorrect. 16 FIRST YEAR ALGEBRA SCALES ANSWER KEY TO PROBLEM SCALE Problem No. Answers I 3* 2 m r 3 a + b 4 X + IOX = 132. ($12 and $120) 5 980 V 6 4* + 40 = 240; 2x -|- 2(x 20) = 240; 4^ - 40 = 240. (50 ft. x 70 ft.) 7 3* - 136 = 836; 2x 136 = 836 - #; a; + y = 836 and# = = 2y 136. (324 children and 512 adults) 8 -T-; not dw dwx = r 9 (X + 12) (x - 4) = x 2 . (6 ft. X 6 ft. and 2 ft. X 18 ft.) 10 2i:5f = 20 : x ; x : 20 5 ft. 9 in. : 2 ft. 6 in. ; x : 20 = 69 :30 ; eight times as high as the man; not 2.6 : 5.9 = 20 : x. (46 ft.) ii x + y = 5000 and + -42 = ^2; 100 100 3 * + 4.(=;ooo x} T*? 7 ?* ri'iv 1- onn - r 1 v fTy 100 IOO : 7 2 > -OS* I 20( -4^ - J 7 2 - ($2800 and $2200) 12 40* = 5^ 5(* - 2); 55* = 40(* + 2); - - 2 = (293 1 A miles) 40 55 13 x + y = 20 and 50^ + 65^ = 1200; 50^ + 65(20 x) = 1200; not SQX + 65(20 x) = 12. (6% Ibs. and 13^3 Ibs.) 14 5 * 2 = i* D;5(# io) 2 = 180. (i6in. X i6in.) The equations given above are those which are usually found. Modific?tions, which in the end equal the same, may be accepted. For example, 4* = 240 + 40 is the same as 4* 40 = 240, and = is the same as 69 : 30 = x : 20. Where the prob- 30 20 lems have been worked out and the correct answers are given, they are to be scored as correct, though such a procedure on the part of students is to be discouraged. DIRECTIONS FOR SCORING THE PAPERS ANSWER KEY TO GRAPH SCALE Problem No. 1 . 2000 2. 24 3- 3330 to 3350 (inclusive) 5- 6. and 9- - ,. 11 -3 to -9 (inclusive) TEARS * 6 a 10 la 14 ia ia ao 18 FIRST YEAR ALGEBRA SCALES 7- \ 1 Spaos l- unit 8. 72. Sb 'to 32 3 FEET IO. DIRECTIONS FOR SCORING THE PAPERS x = 3 y = 2. / r 1 Space 1 Unit ii. In twelve weeks; the thirteenth week. 3 s f fa 8 10 12. m it WEEKS 20 FIRST YEAR ALGEBRA SCALES The score assigned to each pupil for each test is the total number of "rights," that is, the total number of exercises solved correctly in the test. Individual scores made on the various tests by the different pupils of a class may be recorded on a sheet similar to Fig. I. City School.. Teacher. Date Grade. Score s on Each Test I II III IV V Median Score Standard Score FIG. i Individual Class Record Sheet Used in Recording the Results for Each Class DETERMINING THE MEDIAN OR CLASS SCORE 21 6. DIRECTIONS FOR DETERMINING THE MEDIAN OR CLASS SCORE The median number of exercises correctly solved is used in connection with these scales as the class score. Though not entirely accurate scientifically, it is the most readily computed and is, therefore, for all practical purposes the most satisfactory measure of the achievement of a class. The median score represents the number of exercises solved correctly by just fifty per cent of a class. That is, there are just as many students in a class who solve a larger number as there are students who solve a smaller number of exercises. In order to determine the median point of the achievement of a class, it is necessary to make a distribution table of the results of a test. Such a table shows the number of pupils who were unable to solve a single exercise correctly, the number who solved one exercise correctly, two exercises, three exercises, etc. Sample dis- tributions for four of the tests, Addition and Subtraction, Multi- plication and Division, Equation and Formula, and Problems, are shown in Table I, (page 22). Another distribution is given in Table II, (page 23). 22 FIRST YEAR ALGEBRA SCALES TABLE I DISTRIBUTION TABLE SHOWING SCORES ATTAINED FOR FOUR OF THE TESTS, OKMULGEE, OKLAHOMA 6 City Okmulgee School High Remarks 9-month group State Oklahoma Teacher Date June 1921 . Grade pth Number of Pupils Making Score Indicated No of IMvy. Ul Examples Addition Multiplica- Equation Correct and tion and and Problems Graphs Subtraction Division Formula 25 24 23 22 21 2O 19 18 17 16 15 14 13 12 6 6 12 . II 5 ' 8 12 ; 10 17 9 12 2 9 7<5 9 12 6 ^ 3 8 -r/ 7 21 19 ii 41 7 ii 14 9 10 44 6 7 12 9 16 U^ 5 7 9 7 19 4* 4 5 6 2 18 3 i 3 4 4 2 12 ; '* 2 2 4 t, I O Total 97 98 96 98 Median 8.74 8.19 9.0 5-79 Standard 7-9 7-9 7.8 5-6 Form used by Bureau of Educational Research, University of Illinois, Urbana, 111. DETERMINING THE MEDIAN OR CLASS SCORE 23 Table II also indicates in a clear and concise way the method of computing the median class score. 6 Since a thorough knowledge of the method of calculating the median is essential to the proper use of these scales, it is urged that teachers who are not familiar with educational statistics make a careful study of this table. TABLE II SAMPLE DISTRIBUTION OF SCORES MADE ON EQUATION AND FORMULA TEST, SERIES B Score Number of Pupils Making Each Score Computation of Median 20 i. After checking the problems correctly 19 solved, count the check marks in each 18 paper, and indicate the number in the 17 II 2 proper place in column 2. 16 1111 4 2. Find the total number of scores (N). 15 1 1 i 3 . 14 Mill 5 3. Median equals middle score = 13 Illll II 7 2 12 III 3 II IO Illl Ill 4 3 N is Thus, = ^ = i7# 9 II 2 ' 2 2 . 8 Beginning at i in the third column 6 5 1 1 I I and counting up, it is necessary to count 3>^ of the 7 to make 17^, thus: 4 * 17 1 A = 1 + 1+2 + 3+4+3 and 3 2 3^ of the 7. I Put 3^ as a numerator over 7 and O add to 13, the step on which the 7 occurs, thus: Total Scores (N) 35 Median = 13 + ^ = 13.5 Median Score 13-5 Errors in computing the median or class score are very common ; and, when comparison is to be made with standard scores, an error of one-half point in its computation may do serious injustice to a class. As a further safeguard against error in this important de- tail, four additional class distributions are submitted in Table III, and a complete discussion of the method of determining the median or class score of these distributions follows. 6 The method suggested here is an adaptation of the plan used by Clifford Woody in his The Woody Arithmetic Scales and How To Use Them, p. 19. 24 FIRST YEAR ALGEBRA SCALES TABLE III SAMPLE DISTRIBUTION OF SCORES MADE BY FOUR DIFFERENT CLASSES ON EQUATION AND FORMULA TEST Number of Problems Solved Class I Class II Class III Class IV o i I 2 2 2 i I i 3 2 2 3 4 I I 2 5 4 I 2 3 6 6 5 3 2 7 3 4 I 8 3 3 I 3 9 2 2 2 4 10 I I I 2 ii I I I 12 13 I 14 I 15 Total 26 19 16 22 Median Score 6.5 7 .6 6.0 7-5 According to this table, there were twenty-six students in Class I, nineteen students in Class II, sixteen students in Class III, and twenty-two students in Class IV. In Class I one student solved one problem correctly, two solved two problems correctly, two solved three correctly, etc. To find the median score of this class, it is necessary to find the point in the distribution of the class where there are just as many students who solved a greater number of problems as there are students who solved a smaller number. Since there are twenty-six students in the class, this point is obvi- ously midway between the scores made by the thirteenth and fourteenth students, counting down in the distribution. That is, to include the thirteenth individual with the poorer group, it is necessary to count three of the six students who solved six prob- lems; and, since it is assumed that the individuals are distributed STANDARDS OF ACHIEVEMENT 25 evenly through a step at equal distances from one another, the median point is just one half of the distance through this step, from six to seven. Therefore, the median score of this class is 6.5 problems solved correctly. In Class II there are nineteen students. Thus, there are 9.5 individuals both above and below the exact median point in the distribution of this class. To include 9.5 individuals with the poorer group, it is necessary to count 2.5 of the 4 students who solved 7 problems. Hence, the median point is just - - of the 4 distance through the 7th step, which makes the median score of this class 7.6 problems solved correctly. Class III illustrates another situation still. There are 16 students in the class, and in counting out the 8 individuals for the poorer group, we exactly take up all the cases in the 5th step. The fact to be noticed here is that the median point is raised clear through the 5th step. The median score for this class, therefore, is 6.0 problems correctly solved. Class IV is here included to assist in the solution of another dif- ficulty which is often encountered. There are 22 individuals in the class. Counting from the top of the distribution, the 1 1 cases for the poorer group take us, as seen above, entirely through step 6. Likewise, counting from below, to include n cases, we have to go clear through step 8. From this it appears that the median point could be located all the way from 7 to 8 in the class distribution. Since, however, any given distance on a scale is best represented by its middle point, the median score of this class should be 7.5 problems solved correctly. 7. STANDARDS OF ACHIEVEMENT Tentative standards of achievement were proposed in the orig- inal monograph. 7 When these were published, the scales had not been used very extensively, and some doubt was expressed with regard to the reliability of the tentative standards. There is as yet, however, no conclusive evidence that any of these standards should be materially revised. There is some evidence that the tentative standards are on the whole a little too high. On the other hand, whenever the tests 7 Hotz, Henry G: First Year Algebra Scales, p. 41. 26 FIRST YEAR ALGEBRA SCALES have been submitted to classes in large high schools, where the teaching is generally more efficient and where there is more careful selection of the subject matter taught in elementary algebra, the results invariably surpass these standards. TABLE IV TENTATIVE MEDIAN STANDARDS OF ACHIEVEMENT, SERIES A Three-Month Group Six- Month Group Nine-Month Group Addition and Subtraction 5-o 6.8 7-9 Multiplication and Division 5-3 6-3 7-9 Equation and Formula 4-9 7-i 7-8 Problem Test 4-3 4.9 5-6 Graph Test 2.8 (four and one-half 5-6 months) TABLE V TENTATIVE MEDIAN STANDARDS OF ACHIEVEMENT, SERIES B Three-Month Group Six-Month Group Nine-Month Group Addition and Subtraction 9-7 12.9 14.4 Multiplication and Division 9.6 14.0 16.3 Equation and Formula 7.8 14-3 16.0 Problem Test 54 6-5 7-5 Graph Test 3.7 (four and one-half 7.2 months) STANDARDS OF ACHIEVEMENT TABLE VI SUMMARY OF MEDIAN SCORES ATTAINED IN VARIOUS CITIES WITH FIRST YEAR ALGEBRA SCALES, SERIES A ADDITION AND SUBTRACTION SCALE City No. of Months of Algebra Studied 3 6 8 9 16 14 Illinois Cities 6-9 7-3 Rural Schools 4-3 4-9 Virginia Small City Schools 5-2 5-9 Large City Schools 3-6 5-3 5-o North 1 Rural Sch ls 3-3 3-7 4-5 4.9 ' , \ Small City Schools Carolina T ~. c , , J Large City Schools 2.9 3-6 5-4 3-9 3.9 4-3 54 Atlantic City, N. J. 7-4 Reading, Pa. 3-9 Providence, R. I. (Moses Brown School) 6-7 10.5 Fayetteville, Ark. (University H. S.) 5-i 5-3 7-5 Fayetteville, (City H. S.) 5-3 Little Rock, Ark. 4-9 6.0 Potsdam, N. Y. 6-3 Saratoga, N. Y. 54 Elmira, N. Y. 6.1 Whitehall, N. Y. 6.9 Cities of Original Study 5-0 6.8 7-9 MULTIPLICATION AND DIVISION SCALE No. of Months of Algebra Studied City 3 6 8 9 IO 14 Illinois Cities 7.2 74 Athens, Ohio 3-8 4-8 5-2 Reading, Pa. 3-9 Providence, R. I. (Moses Brown School) 7-7 10.4 Fayetteville, Ark. (University H. S.) 5-6 5-9 6.8 Lockport, N. Y. 6.7 Potsdam, N. Y. 5.8 Saratoga, N. Y. 6.2 Elmira, N. Y. 5.9 Amsterdam, N. Y. M. Cities of Original Study 5-3 6-3 7-9 28 FTRST YEAR ALGEBRA SCALES TABLE VI SUMMARY OF M EDIAN SCORES ATTAINED IN VARIOUS CITIES WITH FIRST YEAR ALGEBRA SCALES, SERIES A EQUATION AND FORMULA SCALE No. of Months of Algebra Studied ity 3 6 8 9 10 14 Illinois Cities 77 7-9 1 Rural Schools 1.6 3-9 Virginia \ Small City Schools 4.6 5-8 J Large City Schools 4.0 54 5-1 ) Rural Schools 3-2 4.2 47 4.9 North Carolina > Small City Schools 3.1 2.8 1.5 5.1 J Large City Schools 3-3 4-5 5-5 Reading, Pa. 1.8 Atlantic City, N. J. 6.2 Chicago, 111. (Research Study by Eleanora Harris) 6-3 6-9 Providence, R. I. (Moses Brown School) 9.0 9.8 Fayetteville, Ark. (University H. S.) 5-9 6-3 8.6 Philadelphia, Pa. 6.8 Little Rock, Ark. 6.0 6.8 Lockport, N. Y. 7-3 Potsdam, N. Y. 6.4 Saratoga, N. Y. 54 Whitehall, N. Y. 7.8 Amsterdam, N. Y. 8.1 Cities of Original Study 4.9 7-1 7-8 STANDARDS OF ACHIEVEMENT 2 9 TABLE VI SUMMARY OF MEDIAN SCORES ATTAINED IN VARIOUS CITIES WITH FIRST YEAR ALGEBRA SCALES, SERIES A GRAPH SCALE No. of Months of Algebra Studied City 4X 6 8 9 10 12 14 Chicago, 111. (Research Study by Eleanora Harris) 4-7 Fayetteville, Ark. (University H. S.) 4-5 6.0 Providence, R. I. (Moses Brown School) 6.5 6.8 Reading, Pa. 1.8 Illinois Cities 6.2 5-0 Wellington, Kans. (1919) 6.6 Wellington, Kans. (1920) 6.4 Wellington, Kans. (1921) Rapid Group 7.8 Average Group 4.8 Slow Group 4.6 Mississippi (Two Schools) 4.2 South West City, Mo. Hackensack, N. J. 2-3 5-6 5-2 7-3 74 Amsterdam, N. Y. 2-5 Cities of Original Study 2.8 5-6 FIRST YEAR ALGEBRA SCALES TABLE VI SUMMARY OF MEDIAN SCORES ATTAINED IN VARIOUS CITIES WITH FIRST YEAR ALGEBRA SCALES, SERIES A PROBLEM SCALE No. of Months of Algebra Studied City 3 6 8 9 10 H Illinois Cities 6.4 5-0 Athens, Ohio 1.9 2-5 3-8 Reading, Pa. 2.O Chicago, 111. (Research Study by Eleanora Harris) 4-3 Providence, R. I. (Moses Brown School) 5-8 7.2 Mount Holly, N. J. 3-6 Fayetteville, Ark. (University H. S.) 4-3 4-3 6.2 Saratoga, N. Y. 4.2 Elmira, N. Y. 4-9 Amsterdam, N. Y. 5.5 Cities of Original Study 4-3 4-9 5-6 STANDARDS OF ACHIEVEMENT TABLE VII SUMMARY OF MEDIAN SCORES ATTAINED IN VARIOUS CITIES WITH FIRST YEAR ALGEBRA SCALES, SERIES B ADDITION AND SUBTRACTION SCALE No. of Months of Algebra Studied City 3 6 8 9 10 12 14 Cleveland, Ohio 14.7 Fayetteville, Ark. (University H. S.) 94 New Jersey Cities I2.O Racine, Wis. (1919) 13-0 17-3 Racine, Wis. (1920) 13-6 18.8 Wisconsin Cities (1918) II. 2 14.4 Wisconsin, 22 Cities (1921) 14.8 Wellington, Kans. (1919) 18.1 Wellington, Kans. (1920) 15-9 Wellington, Kans. (1921) Rapid Group 19-5 Average Group 14.1 Slow Group 10.8 Elizabeth City, N. J. 12.8 Andover, Mass. 10.0 South West City, Mo. 17.0 Hackensack, N. J. 12.0 14-5 18.0 2O.O Paragould, Ark. 10.8 12.6 12. 1 Cities of Original Study 9-7 12.9 14.4 FIRST YEAR ALGEBRA SCALES TABLE VII SUMMARY OF MEDIAN SCORES ATTAINED IN VARIOUS CITIES WITH FIRST YEAR ALGEBRA SCALES, SERIES B MULTIPLICATION AND DIVISION SCALE No. of Months of Algebra Studied City 3 6 8 9 10 12 H Cleveland, Ohio 14.8 New Jersey Cities 154 Racine, Wis. (1919) 13.6 18.1 Racine, Wis. (1920) 14.7 18.6 Waukesha, Wis. 14.0 Wisconsin Cities (1918) 11.4 17.4 Wisconsin, 23 Cities (1921) 16.2 Wellington, Kans. (1919) 18.3 Wellington, Kans. (1920) 17.4 Wellington, Kans. (1921) Rapid Group 17.0 Average Group 14.0 Slow Group I3-I Andover, Mass. 14.2 Ironwood, Mich. II. Hackensack, N. J. 10.4 15-9 16-5 19-5 Paragould, Ark. H-3 I3-I 14.9 Cities of Original Study 9.6 14.0 16-3 STANDARDS OF ACHIEVEMENT 33 TABLE VII SUMMARY OF MEDIAN SCORES ATTAINED IN VARIOUS CITIES WITH FIRST YEAR ALGEBRA SCALES, SERIES B EQUATION AND FORMULA SCALE No. of Months of Algebra Studied City 3 . 6 8 9 10 12 14 Cleveland, Ohio 14.9 Fort Smith, Ark. 7.2 9-9 New Jersey Cities 16.3 Racine, Wis. (1919) 17-5 Racine, Wis. (1920) 15-9 Wisconsin Cities (1918) 17.2 Wisconsin, 28 Cities (1921) 14.2 Wellington, Kans. (1919) 18.1 Wellington, Kans. (1920) 16.7 Wellington, Kans. (1921) Rapid Group 20. 6 Average Group 14.4 Slow Group II.2 Elizabeth City, N. C. 12.8 Andover, Mass. 12.8 Ironwood, Mich. 6-4 Cold Springs, N. Y. (Holdane School) 13-5 Hackensack, N. J. 9.0 14.5 2O.O 20.3 Paragould, Ark, II.O 12.5 15-8 Cities of Original Study 7.8 14-3 16.0 34 FIRST YEAR ALGEBRA SCALES TABLE VII SUMMARY OF MEDIAN SCORES ATTAINED IN VARIOUS CITIES WITH FIRST YEAR ALGEBRA SCALES, SERIES B PROBLEM SCALE No. of Months of Algebra Studied City 3 6 8 9 10 12 14 Cleveland, Ohio 6-3 Fort Smith, Ark. 4-7 5-0 New Jersey Cities 7.0 Waukesha, Wis. 7-9 Wisconsin Cities (1918) 8.2 Wisconsin, 31 Cities (1921) 7-i Wellington, Kans. (1919) 9.8 Wellington, Kans. (1920) 94 Wellington, Kans. (1921) Rapid Group 11.4 Average Group 7.8 Slow Group 5-8 Elizabeth City, N. C. 8.1 South West City, Mo. 8-3 Mississippi (Two Schools) 4-7 6-3 Cold Springs, N. Y. (Holdane School) 5-8 Hackensack, N. J. 5-5 5-3 5-9 54 Paragould, Ark. 7.0 6.1 74 Cities of Original Study 54 6-5 7-5 8. GRAPHICAL REPRESENTATION AND STATISTICAL INTERPRETATION OF RESULTS After the papers for one or more of the tests of a class have been scored, and the median has been computed, the results should be entered in tabular form on an Individual Class Record Sheet similar to the one shown in Fig. I (p. 20). In large school systems it is usually necessary to combine the individual scores of various classes. For this purpose the tabula- tion shown in Table I (p. 22) has been devised. In order to save time, teachers very frequently use this form of tabulation for individual classes as well and compute the medians directly from these dis- tributions. REPRESENTATION AND INTERPRETATION OF RESULTS 35 TABLE VIII SUMMARY OF RESULTS IN EQUATION AND FORMULA TEST, SERIES B, FORT SMITH, ARKANSAS Score Three-Month Group Six-Month Group 20 i 19 18 I 17 2 16 2 15 6 14 i 4 13 2 6 12 3 7 II 9 5 IO 10 9 9 26 7 8 22 ii 7 35 6 6 28 9 5 26 3 4 12 4 3 7 4 2 7 I 9 O 4 Number of Pupils 2OI 87 Median Score 7.2 9-9 Table VIII represents, in somewhat greater detail, the results of one of the tests in a city high school having several algebra classes. It shows the distribution of the group that has studied algebra for three months, and the distribution of the group that has studied algebra for six months. Such a table indicates the extent of variation within a group and also the excessive amount of overlapping that exists between the two group distributions. The facts in this table may be more strikingly portrayed if presented graphically as in Fig. 2. In this graph the results of two groups are drawn upon the same scale, one placed above the FIRST YEAR ALGEBRA SCALES other. In constructing these graphs, the -X"-axis represents the scores attained by the various pupils, and the F-axis the per- centage of pupils making each score. Graphs of this type furnish a most efficient means for showing : (l) The wide range of abilities within a group. (2) The scores Per Cent of Pupils 16 10 _r Three -Month Group i (201 Pupils) Per Cent of Pupils 2 4 6 Score 14 16 18 20 Six-Month Group (87 Pupils) _. _- 18 20 Score FIG. 2. Distribution and Median Scores, Equation and Formula Test, Series B, Fort Smith High School, Fort Smith, Ark. most frequently made by pupils of a group. (3) The extent to which a group falls short or surpasses the median standard achievement, as indicated by the distance between the median for the group and the standard median. (4) The amount of progress made from group to group, as indicated by the distance between the group medians. (5) The amount of overlapping between the groups. In Fig. 2, REPRESENTATION AND INTERPRETATION OF RESULTS 37 this last factor, the excessive amount of overlapping, is undoubt- edly the most significant. It will be seen that 29 per cent of the pupils of the three-month group do as well as or better than, the median pupil in the six-month group, and that about 24 (24.4) per cent of the pupils of the six-month group fall below the achieve- ment of the median pupil in the three-month group. It is sometimes desirable to represent graphically the scores attained by each individual pupil in a given test. This is done in Fig. 3> where the results obtained from an entire class in the Addi- tion and Subtraction Scale are exhibited. Here each pupil's score Score Standard Class, lie di_sn_ 1 10 123 6 7 6 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 (Each line represents a pupil's ecor) FIG. 3. Individual Scores of a Six-Month Group, Addition and Subtraction Test, Series A, Moses Brown School, Providence, R. I. is represented by a vertical line, the height of the line indicating the size of the score. Thus the relative standing of each and every pupil is vividly portrayed. It will be seen that the best student in this class does more than five times as well on this test as the poorest student. When a composite picture of the achievement of a pupil on all five tests is desired, a graph similar to that in Fig. 4 is suggested. The scores of five different pupils on each of the five different tests are here exhibited. The short horizontal lines indicate the median score of the class. Pupil A, it is evident, surpassed the median scores of the class in all of the tests; pupils B, C, and D fell below these medians in one or more tests; and pupil E did very poorly in all but the Multiplication and Division Test, in which he did exceptionally well. The relative standing of the members of a class is more readily determined from this graph if pupils are arranged roughly in order of excellence of achievement. 3o FIRST YEAR ALGEBRA SCALES In Fig. 5 the median scores made in a city high school on all of the tests in 1920 are compared with the scores made in the same school in 1919 and with the standard scores. It is at once evident that the median scores made in 1920 are all well above the stand- ard but not quite as high as the 1919 median scores. A most interesting graph of results from the same school is re- produced in Fig. 6. These data were obtained during the school year of 1920-21, and clearly illustrate the advantage of homo- geneous grouping of pupils. At the beginning of the year all first year algebra pupils were divided into three sections on the basis 5s rr ; - =-. r -j .s r. " JS :r_ - -- JS -_ - - | I Z ', 4 i ] 1 2 4 E ! J 7 i I ] 1 3 < 5 1 2 345 Letters'represent different pupils; perpendicular lines, pupil's individual scores in (i) Addition and Subtraction, (2) Multiplication and Division, (3) Equation and Formula, (4) Graph, (s) Problem, tests respectively; horizontal cross bars: class median; - - - standard median. FIG. 4. Individual Scores Attained by Pupils from a Six-Month Group on All Tests, Series A, Moses Brown School, Providence, R. I. of results obtained with the Otis Intelligence Tests. The pupils making the highest grades in the intelligence test, 29 in all, were placed in one group and covered a year and a half of work in algebra in nine months. Twenty-two pupils were placed in the slow section and about 70 in the normal group. The bright group was tested at the end of six months and the other two groups at the end of nine months. The graph indicates that the normal group, covering the usual amount of ground, made a satisfactory showing on all the tests; and, moreover, that the bright group, at the end of six months, as shown by the results, possessed the ability to solve the algebra exercises that was far superior even to that represented by the nine-month standard. REPRESENTATION AND INTERPRETATION OF RESULTS 39 20. 16. 10 A d. and Mult, and Equat Sub. Div. and Form. lij Problem Graph Scores. 1919^3 Scores, 190 Standard Scores FIG. 5. Comparison of Median Scores Attained in 1919 and 1920, Series B, Wellington High School, Wellington, Kans. 20- 16- | - 1 ^ | J5 \ ml In Add. and Mult, and Equat. and Graph Problem Sub. Div. Form. Standard Scores Scores of Bright Group at nd of Six MonthSBza Scores of Normal Groups Scores of Slow Grouper^ FIG. 6. Comparison of Median Scores Attained by Bright Group, Median Group, and Slow Group in All of the Tests, Series B, Wellington High School, Wellington, Kans. 40 FIRST YEAR ALGEBRA SCALES 9. ANALYSIS OF ERRORS There is a tendency on the part of teachers and administrators to use standard tests merely for the purpose of determining de- grees of attainment. Large numbers, the majority of the teachers perhaps, consider the net results, as shown by individual and class scores, as the all-important objective in the use of standardized tests. Such knowledge of total scores achieved is valuable, but it is merely the first step in the process of securing greater efficiency in instruction through the use of standard testing devices. Tests have other values which are much more significant to the class- room teacher. They should be used much more extensively to reveal weaknesses in teaching and to aid in the diagnosis of dif- ficulties encountered by pupils. In the analysis of the results of a test, the types of problems causing special difficulty are usually quite readily disclosed, but it is a much more intricate matter to determine with any degree of precision the mental processes that may be responsible for the various errors. Three studies on the analysis of errors most frequently made in algebra are here reported in Tables IX, X, and XL TABLE IX CLASSIFICATION OF 443 ERRORSS MOST FREQUENTLY MADE BY PUPILS OF THE FORT SMITH (ARK.) HIGH SCHOOL IN THE EQUATION AND FORMULA TEST, KjL*Ju% ^^ *4 L SERIES B 1. Performing wrong operation in solving for unknown: Ex. 7. z = 6 or 2 = 6 3 3 12 2 2 = 2=6 3 3 2. Error in sign in transposition: Ex. 4. 5^ + 5 = 61 30 5a - 30 = 61 + 5- 3. Simple arithmetical errors: Ex. 3. 3* = 9 - 3 3* = 9- s Fort Smith Survey: Classification of Errors in Algebra, made under the direction of A. M. Jordan, of the University of Arkansas. ANALYSIS OF ERRORS 4 1 4. Error in using the four fundamental operations of algebra : Ex. 5. 7 3 = 12 4 low = 8. Ex. 8. c 2 (3 4c) = 12 c 2 6 - 8c = 12. 5. Adding denominators in addition of fractions: 3 4 2 JL- JL 7 2* 6. Incomplete solution: Ex. 12. 4;y + 3? = 30 7 y = 30. Ex. 18. 4* = 2 - x = \. 7. Error in sign in division: Ex. 6. - 32 = - 6 2 = 2. 8. Error in copying: Ex. 14. 7m $n = 12 -jn 3 = 12. 9. Using exponent for coefficient: Ex. 19. p2 5p - 50 - 3P = 50. 10. Error in substituting the value of the unknown in a formula Ex. 1 6. Area of a triangle = \ bh. Find area when b = 10 ft. and h = 8 ft. Area of triangle = 5 X 4 = 20. 11. Solving for wrong unknown in a formula: Ex. 17. RM = EL, solve for M. EL FIRST YEAR ALGEBRA SCALES 4. Performing the wrong operation in solving for unknown Error in sign in transposition Simple arithmetical errora Error in using the four fundamental operations in algebra Adding denominators in addition of fractions 6. Incomplete solution 7. Error in sign in division 8. Error in copying 9. Using exponent for* coefficient 10. Error in substituting the value of the unknown in a formula 11. Solving for the wrong unknown in a formula 12. Unclassified 443 100* FIG. 7. Distribution of 443 Errors made by Three and Six-Month Groups on Equation and Formula Test, Series B ANALYSIS OF ERRORS 43 TABLE X SUMMARY OF ERRORS MADE BY PUPILS IN 36 WISCONSIN HIGH SCHOOLS ON FOUR TESTS, SERIES B, MARCH AND APRIL, 1921 Per Cent of Total No. Errors ADDITION AND SUBTRACTION TEST Failure to deal with parentheses correctly 3 8 Failure to write the denominator of a fractional answer 24 Wrpng process (adding instead of subtracting, for example) 18 Writing the sum of the numerators for a new numerator and the sum of the denominators for a new denominator 4 Errors whose cause could not be discovered Miscellaneous errors . 8 Total ioo MULTIPLICATION AND DIVISION TEST Mistakes in dealing with exponents 37 <~ Using the wrong process 14 <- Mistakes in the use of signs 13 ^ Mistakes in factoring 11^ Failure to deal with parentheses correctly 6 ** Errors whose cause could not be discovered 1 1 Miscellaneous errors 8 Total ioo EQUATION AND FORMULA TEST Failure to change signs when transposing 28 >- Multiplying by the coefficient of the unknown in order to solve .... 18 i~ Use of wrong process 10 *" Mistakes in solving literal formulae Errors in dealing with parentheses 7 u Mistakes in substitution in formulae 7*" Finding one root only in solving a quadratic 7 ** Dividing by the numerator of the coefficient of the unknown in order to solve 6 ^ Errors which could not be explained 5 Miscellaneous errors . 4 Total ioo j PROBLEM TEST Errors due to ignorance of fundamental relationships (Those in length, breadth, thickness, and volume for example) 52 ** Errors due to misreading the problem 4 1 u * Errors which could not be explained 7 Total ioo Osburn, W. J.: Survey of Algebra Instruction in Wisconsin High Schools. These pupils had studied algebra for six months. The total number of pupils tested varied from 1055 to 1635. Some of the schools did not find it possible to give all four tests. 44 FIRST YEAR ALGEBRA SCALES TABLE XI CLASSIFICATION OF ERRORS 1 " MOST FREQUENTLY MADE BY PUPILS OF THE FORT SMITH (ARK.) HIGH SCHOOL IN THE^PROBLEM TEST, SERIES B (w*^Lt*JA 1. Incorrect operation indicated, usually due to failure to comprehend the problem. Caused 55 per cent of the errors. Prob. I : If a coat costs x dollars, how much will 3 coats cost? Answer: x 3. 2. Conditions of the problem apparently understood, but the work left in incomplete form. Caused 15 per cent of the errors. Prob. 7. The total number of circus tickets sold was 836. The number of tickets sold to adults was 136 less than twice the number sold to children. How many were sold of each? Answer: x = No. of children's tickets sold 2x 136 = No. of adults' tickets sold. L-, 3. Failure to comprehend the problem, perhaps due to confusion and to use of technical terms. Caused 10 per cent of the errors. Prob. 6. The width of a basket ball court is 20 feet less than its length. What is the length and width of the court if the perimeter (distance around) is 240 feet? Equation: x x 20 = 240. L, 4. Inverting the order of terms in subtraction and in division. Caused 5 per cent of the errors. Prob. 2: A man is m years old: how old was he r years ago? Answer: r m. Prob. 5. The distance from Chicago to New York is 980 miles. If a train runs v miles an hour, what is the time required for the run? Answer: -^- 980 <-- 5. Simple arithmetical errors. Caused 5 per cent of the errors. 6. Attempt to solve problems containing two unknowns with only one equation containing the two unknowns. Caused 2 per cent of the errors. 10. VALUE OF FIRST YEAR ALGEBRA SCALES These scales may be used by teachers for three distinct and very useful purposes. They may be used (a) to indicate attainment, (b) to measure progress, and (c) to diagnose difficulties. Scales which increase in difficulty by approximately equal steps furnish a most reliable objective means for determining the actual 10 Fort Smith Survey: Classification of Errors in Algebra, made under the direction of A. M. Jordan, of the University of Arkansas. VALUE OF FIRST YEAR ALGEBRA SCALES 45 achievement of a student or a group of students. Any one of the scales may be used for this purpose, though the Equation and Formula Scale is perhaps to be preferred, since, as previously stated, it is a more comprehensive test. It is well to keep in mind also, in this connection, that a low median class score is not always, nor even quite generally, due to poor instruction. Any one or a com- bination of several causes may be operating to keep a class score down. It is the duty of the teacher, however, to study these causes and to learn which ones are affecting the efficiency of the instruc- tion, in order that proper remedial measures may be applied. This it is possible to accomplish with a much greater degree of certainty when the teacher knows the actual standard of achievement a class has attained. Such knowledge furnishes the teacher with a fact basis upon which to proceed and a motive with which to operate. The extent of progress made by a class can be quite scientifically measured by submitting the same scale to a class at intervals of about three months. Teachers should be cautioned very specifi- cally, however, not to do any drill work upon the exercises or problems appearing in the scales. If it is feared that some of the practice effect may survive, it is suggested that another scale in the same series, or the same scale in a different series, be used for the second test. 11 The most desirable method of measuring progress, very naturally, would be to have another parallel series of scales similar and equal in difficulty to those of Series A, and it is to be hoped that such a series will soon be constructed. For diagnostic purposes the scales of Series B have been found to be more serviceable. They offer a richer variety of exercises and, therefore, a greater number of type processes. Hence, a more com- plete analysis of the mistakes made by students, and the difficulties they encounter, is made possible. Finally, it must be stated emphatically that these are primarily power tests and as such should never be used for purposes of drill. Furthermore, with the time limits as now fixed, they are speed tests to a limited extent only. If a pure speed test is desired, the Stand- ard Tests 12 devised by Dr. H. O. Rugg could be used to advantage. These would be particularly useful in determining whether a class has had sufficient drill upon the fundamentals. 11 See suggestions for using the scales in rotation, p. 9. 12 See School Review, October, 1917, 25:546. 46 FIRST YEAR ALGEBRA SCALES II. BIBLIOGRAPHY I. Material in Periodicals and Books: Cawl, Franklin R.: Practical Uses of an Algebra (Hotz) Standard Scale. School and Society, July 19, 1919. Harris, Eleanora: Study of the Hotz First Year Algebra Scales and the Rugg- Clark Standard Algebra Tests. Master's Dissertation, University of Chicago, 1919. Hobbs, James B.: Results from Giving the Hotz First Year Algebra Scale Tests to a Six-Eight Month Group. School and Society, October 16, 1920. Hotz, Henry G.: First Year Algebra Scales. Contributions to Education, No. 90, Teachers College, Columbia University, New York City, 1918. (This is a tech- nical monograph giving a detailed account of the statistical methods employed in the derivation of the scales.) Ramsey, J. W. : A Study of the Intelligence of Paragould (A rk.} High School Students. Master's Dissertation, Peabody College for Teachers, 1921. II. Material in School Surveys and Educational Reports: Survey of the Virginia Public Schools. Part II. Educational Tests. Survey of the Reading (Pa.) High School. Survey of the School System of Mount Holly, N. J. Survey of the Public Schools of Hackensack, N. J. Survey of the Philadelphia Schools. Survey of Public Education in North Carolina. Biennial Report, 1916-1918, State Department of Public Instruction, Madison, Wisconsin. Report of Division of Educational Tests, 1919-20, Bureau of Educational Re- search, University of Illinois, Urbana, 111. The scales may be secured from the publishers: Bureau of Publications, Teachers College, Columbia University, New York City, or from the following distributing centers: Public School Publishing Co., Bloomington, Illinois; Bureau of Educational Measurements and Standards, State Normal School, Emporia, Kansas. T GENERAL LIBRARY UNIVERSITY OF CALIFORNIA-BERKELEY RETURN TO DESK FROM WHICH BORROWED This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. J2/V55U MAR 2 9 1955 iff OCT26'67-10 LD 21-100m-l,'54(1887sl6)476 Gaylord Bros. Makers Syracuse, N. "V. ' PAT. 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