LA 333 ■If ' : ''W %WM MONOGRAPH No. 3 Arithmetic Survey Newark, New Jersey Harvard University, Library of the Graduate School of Education Arithmetic Survey in the Public Schools of Newark, N. J. BOARD OF EDUCATION NEWARK, NEW JERSEY DECEMBER 1919 i A 333 -fa As i?if LIBRARY OF CONGRESS s Table of Contents PAGE Foreword 5 Introduction 7 Chart I — Accuracy Tests, Newark Compared with Woody Standards 8 Woody Tests, Series B 9 Chart II — Newark, Four Fundamentals Combined, Compared with Woody Standards Combined 10 Stone Reasoning Test 11 Chart III — Reasoning Tests, Newark Compared with Stone Test and with Three Cities 12 Table I — Scores Made by Each School in Newark in the Woody Tests, Series B, and in the Stone Reasoning Test. . .opp. p. 13 Analysis of Conditions as Shown by Table 1 13 Table II — Comparison of Groups of Schools 15 Distribution of Pupils' Scores 17 Chart IV — Distribution of Scores Attained in Addition, Multiplica- tion, and Reasoning °PP- P- 17 Chart V — Per Cent, of Accuracy in Each of Four Fundamentals. . . 18 Accuracy 19 Table III — Per Cent, of Accuracy in the Four Fundamentals 19 Discussion of Individual Problems 19 Addition Scale 20 Table IV — Addition Problems Incorrect or Not Attempted 21 Subtraction Scale 22 Table V — Subtraction Problems Incorrect or Not Attempted 23 Multiplication Scale 24 Table VI — Multiplication Problems Incorrect or Not Attempted.. . 25 Division Scale 26 Table VII — Division Problems Incorrect or Not Attempted 27 Reasoning Test 28 Conclusions 30 Foreword The survey in penmanship in the schools of Newark proved to be so suggestive and valuable that, upon the recommendation of the Super- intendent of Schools, a survey of arithmetic was authorized by the Com- mittee on Instruction and Educational Supplies. This survey was made under the direction of Mr. Elmer K. Sexton, Assistant Superintendent in charge of the Department of Reference and Research. Mr. Sexton's report of the survey, which follows, is very suggestive and shows clearly wherein classroom work in the subject is weak or strong and where the emphasis should be placed to improve weak work. It also indicates needed adjustments in the course of study. Among the important facts revealed by the survey is the satisfactory condition in grades one to four inclusive, where excellent results are obtained in the fundamentals. It is fair to infer that, in the main, the course of study for these grades is well adapted to the ability of the children, and that the methods of teaching are effective. The record of the fifth grade is also gratifying. The great contrast between the results in the fifth and the sixth grades is striking. This is probably due to the fact that the work of the sixth grade is less interesting, there is little that is strong in its appeal, and the operations in denominate numbers constitute the new features of the work. The survey reveals the need of continued drill in the fundamentals in the upper grades. The weakness shown in the formal work is due to the fact that in these grades the problem work, or so-called "thought- work," requires a very large part of the time. Neither phase of the work can safely be neglected, because the children are at that period of life when habits are in the forming. The habit of doing accurate work in the fundamentals is of such great value that it must be fixed. There must be a better balance between the formal and the thought work in the upper grades. DAVID B. CORSON, Superintendent of Schools. Report of Arithmetic Survey On April 21, 1919, the Superintendent recommended to the Com- mitee on Instruction and Educational Supplies that a survey be made of the Newark schools in the subject of arithmetic. The Board approved the recommendation, and included in the resolution authority for making a survey in spelling. Tests in arithmetic were given in all schools of the city May 29 to June 2, 1919, inclusive. The tests given were the Woody Scales, Series B, and the Stone Reasoning Test. In order that they should be given as uniformly as possible and to the greatest advantage to the pupils, teachers were selected to conduct the tests who were well qualified to conduct them in an efficient manner and to secure from the pupils their best work. A teacher was sent to each school (in no case the school where she regularly taught), where she conducted all of the tests for that school. Uniform directions were given to these teachers covering the procedure for giving the tests. The test papers were then sent to the Research Department, where they were scored and the results tab- ulated. For the purpose of comparing the work done in the various schools, all pupils in the 4A, 6A, and 8A -grades were tested, except in those schools having four classes of one grade, in which case the pupils in only two of these classes were tested. A sufficient number of pupils of the other grades between 4A and 8A were tested to secure an accurate median in these grades to complete the line of the graphs for all grades above the 4B, in order that the work of this city might be compared with the work of other cities. From the 60,000 papers obtained as a result of these tests a sufficient number (31,405) were scored to give an accurate measure of the schools of the city for purposes of comparison. The principal aims in mind in making the tests were : 1 . To compare the work of the Newark schools in arithmetic with the work in other cities. 2. To compare the work in the various schools of Newark. 3. To observe, if possible, the efficiency of groups of schools having peculiar types of children or peculiar organization. 4. To ascertain as far as possible where the weaknesses lie, both as to school, grade, or phase of subject, and as to methods of instruction. The professional spirit in which this survey has been received is highly commendable and indicates that all teachers and principals are very willing to have the work of their pupils measured and to learn wherein their school or class is weak or strong. It means that improve- ment must inevitably follow. Conclusions concerning a certain school should not be drawn too quickly, as this is a survey in one subject only, but in a subject which 19 PUBLIC SCHOOLS OF NEWARK, N. J. Chart I Woody Accuracy Tests * A/ewarA jr June, ,9/9 Compared tv/ffy Woody Standards IB n /6 IS I* 13 IX II /o oft _ Subtraction m Mu/f/p//C<7f7'on . Divis/on Newark resu/fs shown by ba/f year grades- Woody resu/fs shown Ay -fo // year ■ y redes H-B V-A SB SA 65 <*A IB ja 8B &a ARITHMETIC SURVEY lends itself easily to measurement. The facts in the case should be followed with an open mind in order that the school system may reap the maximum benefit from this survey. The results are partly exhibited by means of graphs and tables. WOODY TESTS, SERIES B Chart I represents the ability of Newark pupils in abstract work in the four fundamental operations in arithmetic, together with the Woody standards as ascertained by a very careful examination of nearly 5,000 pupils conducted in the main by Mr. Woody himself. It may be said, however, that in giving this test and securing these stan- dards, Mr. Woody placed no time limit upon the pupils while they were working, whereas ten minutes was given to the Newark pupils. This, of course, would cause a lowering of the Newark standards, as compared with the Woody standards. The question might arise whether or not ten minutes is not ample time in which to do this work, if the pupils have been well trained in the four fundamental operations, except, per- haps, in the lower grades where pupils are given some examples covering subjects with which they are not yet familiar. Good work in the funda- mentals is almost always done rapidly. The Woody tests were made in the early part of the year, the Newark tests in the latter part, so the Woody standard is shown on a line with the B grades rather than the A. The chart shows that addition in the Newark schools is the poorest of the four fundamental operations when compared with the Woody standard and that multiplication is the best. Addition is a very difficult process to teach unless the work is organized, and systematically devel- oped and drilled upon. Much more drill is of necessity given the multi- plication combinations in all subsequent mathematical work than is given the addition combinations. It is easier to put the multiplication tables into; practice through figures than the addition combinations, and consequently pupils can read through a multiplication example much more rapidly than down a column of figures. The line in addition begins slightly above the Woody standard in the 4B, crosses it between the 4B and 4A, and remains below with an ever-widening gap through the upper grades. The line of subtraction has a varying course and finally falls below the Woody standard in the 7A and continues below through the eighth grade. The multiplication line crosses the Woody standard between the 5A and 6B, and remains below until it reaches the 8A, where it surpasses the Woody standard. The result in division is shown to surpass that found by the Woody test in the lower grades tested to a greater degree than any of the other operations. The line crosses the Woody standard between the 6B and 6A grades and finishes somewhat below. It will be observed that, in the lower grades, the work in the Newark schools surpasses the Woody standards in all of the four fundamental operations. This would tend to show that the primary grades of the city have been doing their work well, exceptionally well, while the upper grades have not kept up the good standing. 10 PUBLIC SCHOOLS OF NEWARK, N. J. N is S3 I Vi u 3 ARITHMETIC SURVEY 11 The information found on Chart I has been combined for Chart II, which shows the work of the Newark schools in the four fundamental operations combined, as compared with the combined Woody standards in the four fundamental operations, together with the percentage of accuracy of Newark pupils in fundamentals. This brings out more graphically the facts which a careful study of Chart I shows, viz. : that the work of the primary grades in the Newark schools is well done in the four fundamentals and that the work continues better than the Woody standard until between the 5A and 6B, where the line falls below the Woody standard and continues below through the eighth grade. (See also Chart III, where the reasoning line falls below in nearly the same place.) It is also quite evident from both charts that all "A" classes are stronger than the "B" classes. There seems to be no explanation for this unless it is psychological. It may take less mental effort for a teacher to promote from 5B to 5A than from 5A to 6B, where there is a change of grade. The 5A grade appears to be the strongest grade of all those tested. This is rather surprising, because the 5B and 5A teachers have long thought that the work laid down for them in the course of study was rather heavy. The line drops slightly between the 4A and 5A, but the line to the 5B is very nearly straight with the 6B and subsequent grades. The decided drop may be partly due to the very high standing of the 5A in all tests. It is very noticeable that the 5A grade in every opera- tion stands higher in proportion than any other grade examined. This, of course, makes the contrast with the 6B and subsequent grades very much greater than it otherwise would be. The percentage of accuracy, as shown on Chart II, follows very closely the results in the fundamentals in the various grades. The ques- tion may arise whether accuracy leads to good results or good results lead to accuracy. At any rate, it is evident that they are closely corre- lated. It may be observed, also, that the 5A grade represents the highest relative point in accuracy of all the grades tested, while the falling off of accuracy in the 6B, 6A, and 7B is not so marked as in the results in the four fundamentals. STONE REASONING TEST Chart III presents the results of the reasoning test compared with the average of three cities where the test has been given by outside and disinterested experts — Salt Lake City, San Francisco, and Butte— and with Salt Lake City and San Francisco the two cities nearest the size of Newark. Salt Lake City presents a better showing than Newark and is highly conmmended in the report of the committee, of which Mr. Elwood B. Cubberley, of Leland Stanford, Jr., University, was chair- man. He says, "From these results it is clear that the schools of this city rank high in the ability of their children to reason." The reasoning ability of the Newark pupils is well up to the Salt Lake City standard and somewhat above the average of the three cities up to the 6B grade. It then falls rapidly in 6A and 7B below the three- 12 PUBLIC SCHOOLS OF NEWARK, N. J. % k * £ 1 S> V .Vi J? ^ 1 S^ ^ \ ^3 s $ -k t (§ W & 1 S Q *> N § $ 1 ^ >: 1 ^ ■* ^o S *s * <3 ••ft *> 2 "t ^ ^ § '?P QO ^ ARITHMETIC SURVEY 13 city average, but returns in 7 A, 8B and 8A, finishing much above the three-city average, but still below the Salt Lake City standard, and makes a greater per cent, of increase than any of the three cities. The Newark schools have a high record in the reasoning test when compared with the three cities, although it is not as good as that of Salt Lake City. The per cent, of gain in reasoning power from the 5B to 8A is as follows: Salt Lake City 184%, Newark 153%, San Francisco 143%, average of the three cities, 186%. The lower per cent, advancement in Newark is largely due to the excellent start in the lower grades and the subsequent loss in the 6A and 7B grades. By comparing Chart II and Chart III we find that the difficulty, both in the fundamental operations and in the reasoning, occurs in nearly the same grades, that the pupils come up from the primary school through the fifth grade in excellent condition, but that during the next few half-year grades there is a decided falling off, strengthening again as the line approaches the eighth grade. The decided bend in the lines of the two graphs should be carefully considered. The cause of this loss of efficiency may be found in the course of study or in the topics treated in these grades which perhaps have not been so thoroughly organized as to subject matter, nor as systematically drilled upon as those of other grades. When this deficiency is removed Newark will stand very high in both formal work and in reasoning. ANALYSIS OF CONDITIONS AS SHOWN BY TABLE I Table I gives the results in the 4A, 6A, and 8A grades of the four fundamental operations and the reasoning test in the 6A and 8A grades. The figures representing these results have also been totaled for pur- poses of general comparison. The Woody and the Stone standards are given at the top of the table, together with the Newark standard. From this table each principal can compare the work of his school with that of the other schools in the four processes for each grade given, and in reasoning for the 6A and 8A grades. He can find which grade is weakest or which process is weakest. If a grade is weak, he can learn which process of that grade is weak ; or, if a process is weak, he can learn in which grade the weakness lies. In short, he can locate strong or weak points and give less or more time to them, improve the methods of instruction, or more carefully organize his drill lessons, as seems neces- sary. It will be seen that school number 19 is below the Newark median in all four operations in the 4A grade, while it is well above in the 6A and 8A grades. The 4A grade in this school is taught by an inexpe- rienced teacher who has not yet been able to control the class, and the pupils have not only not improved in their work, but have actually learned very bad habits. This emphasizes the importance of good teachers. Some schools show a decided improvement in the four fundamentals from the 4A to the 8A grade, as found in schools number 42 and 36. School number 42 ranks 19th in 4A, 16th in 6 A, and 1st in 8A. School TABLE I Table of Scores Made by Each School in Newark in the Woody Tests Series B and in the Stone Reasoning Test Tune. 1919 Woody Median. 11.0 8.0 7.0 5.0 16.0 12.0 15.0 10.0 18.5 14.5 18.0 14.0 STONE REASONS Q TEST 12.06 9.46 11.16 7.54 40.18 15.13 12.63 15.49 11.37 54.45 16.73 14.36 18.19 13.33 62.31 3.27 6. SO Average Oitlei 4 A 6 A 8 A 5.27 8.60 Newark Aver- age Grand A S M D Total A S M D Total A S M D Total Total 6A 8A Total School No. 1 12.67 1.80 11.56 8.39 34.42 12.33 11.79 13.56 10.50 4S.1S 15.50 14.56 17.33 13.72 61.11 143.71 4.84 9.29 14.13 2 11.20 9.12 9.17 5.00 34.49 14.80 10.92 14.71 9.38 49.S1 15.35 13.18 16.*0 11.79 56.82 141.12 4.22 6.81 11.03 3 11.92 9.47 11.50 7.89 40.78 12.22 10.11 12.50 8.28 43.11 16.14 12.50 17.20 11.23 57.07 140.96 2.31 7.15 9.46 4 11.77 9.83 10.75 8.43 40.78 14.33 11.96 15.00 10.86 52.15 15.25 13.70 16.93 12.55 58.43 151.36 6.45 S.85 15.30 5 12.13 9.69 10.63 5.65 38.10 15.65 12.42 14.75 12.25 55.07 16.40 13.92 1S.38 13.07 61.77 154.94 4.90 6.36 11.26 6 11.50 9.47 11.17 8.59 40.73 14.44 11.71 14.00 10.83 50.98 16.00 14.15 15.90 12.88 58.93 150.64 4.74 8.35 13.09 8 12.50 9.58 10.78 8.18 41.04 14.50 11.71 15.44 11.08 52.73 16.S6 14.45 17.17 12.69 61.17 154.94 4.85 8.79 13.64 9 12.97 10.80 12.12 9.88 45.77 15.23 13.17 15.40 12.40 56.20 17.43 14.55 18.00 13.S8 63.86 165.83 4.80 10.28 15.08 10 11.38 9.45 11.36 7.17 39.36 15.77 11.38 16.80 12.36 56.31 16.69 15.17 18.17 14.06 64.09 159.76 6.18 11.96 18.14 11 12.11 9.11 11.22 7.79 40.23 15.97 13.18 16.33 12.00 57.48 16.59 14.83 18.71 13.29 63.42 161.13 5.86 11.18 17.04 12 13.31 9.74 12.09 8.33 43.47 16.00 13.00 16.35 11.88 57.23 17.10 14.69 18.17 14.05 64.01 164.71 6.41 10.07 17.38 13 11.65 9.61 10.88 8.06 40.20 14.70 13.21 15.23 11.56 54.70 17.08 14.73 17.96 13.55 63.32 158.22 5.82 10.59 16.41 15 11.81 9.64 11.19 8.52 41.16 15.3S 12.25 15.56 11.50 54.69 17.25 14.41 18.45 14.16 64.27 160.12 4.60 10.31 14.91 16 12.08 9.81 11.30 7.50 40.69 16.50 12.42 16.15 11.50 56.57 18.13 14.20 17.50 12.67 62.50 159.76 5.14 8.49 13.63 17 11.9a 9.48 11.32 8.63 41.36 15.50 13.11 15.93 11.50 56.04 17.14 14.81 18.21 13.36 63.52 160.92 5.78 9.96 15.74 18 12.46 9.96 11.42 8.50 42.34 16.07 13.32 17.14 13.00 59.53 17.60 15.00 19.29 14.11 66.00 167.87 6.90 12.78 19.68 19 11.10 8.75 10.S3 5.59 36.27 16.50 14.22 15.83 12.17 58.72 17.18 14.47 1S.80 13.44 63.89 158.88 4.86 9.59 14.45 20 10.33 9.25 10.75 7.75 38.0S 14.83 11.69 13.00 10.25 49.77 16.67 14.07 17.17 13.25 61.16 149.01 4.02 9.20 13.22 22 12.10 9.65 11.93 9.57 43.25 15.63 12.95 15.30 11.60 55.48 16.43 14.61 18.68 13.89 63.61 162.34 5.52 11.56 17.08 23 12.53 9.53 11.08 8.50 41.64 16.50 12.92 16.30 11.88 57.60 16.89 14.85 19.15 13.75 64.64 163.88 7.21 11.41 18.62 24 12.07 9.54 10.75 7.50 39.86 15.27 13.42 15.41 11.30 55.40 15.64 13.44 18.07 12.75 59.90 155.16 4.71 8.37 13.08 25 12.15 9.40 11.94 6.44 39.93 15.47 13.00 16.00 11.67 56.14 16.75 14.75 18.67 13.71 63.88 159.95 5.78 8.64 14.42 27 10.96 9.48 10.89 7.00 38.33 15.29 13.40 16.63 11.50 56.82 17.06 14.13 18.21 12.81 62.21 157.36 5.11 8.01 13.42 28 11.57 9.63 11.43 6.41 39.04 15.33 13.50 15.83 11.55 56.21 17.50 14.75 18.39 13.72 64.36 159.61 5.86 10.95 16.81 30 10.53 9.52 10.80 6.93 37.78 14.29 12.06 14.92 10.46 51.73 16.25 14.29 18.17 13.13 61.84 151.35 4.09 9.02 13.11 31 12.67 8.67 10.73 7.00 39.07 14.00 11.60 15.90 11.33 52.83 16.3S 13.75 18.30 13.25 61.68 153.58 5.27 9.90 15.17 33 13.03 9.69 11.68 8.00 42.40 16.15 12.94 17.19 11.94 58.22 17.55 14.70 18.60 12.91 63.76 164.38 5.79 10.56 16.35 85 13.22 9.63 10.85 8.43 42.13 15.64 13.85 15.35 10.86 55.70 16.64 14.14 17.07 12.94 60.79 158.62 5.11 12.06 17.17 36 11.73 9.15 10.93 5.92 37.73 14.59 13.22 16.50 11.44 55.75 17.60 14.90 19.23 14.77 66.50 159.9S 5.29 11.10 16.39 37 12.00 8.86 10.00 7.50 38.36 15.27 13.18 15.17 11.17 54.79 16.83 14.05 16.90 12.08 59.86 153.01 4.40 8.21 12.61 38 12.87 9.83 11.40 7.36 41.16 14.93 12.83 15.25 11.90 54.91 16.70 14.64 18.29 13.05 62.68 159.05 4.89 10.81 15.70 39 11.43 9.17 11.56 7.38 39.54 15.50 13.29 16.20 12.22 57.21 17.70 15.00 18.55 14.35 65.60 162.35 6.79 10.30 17.09 40 11.75 9.65 11.21 7.00 39.61 15.03 12.33 16.00 11.25 54.61 16.39 14.50 18.35 12.70 61.94 156.16 4.94 8.31 13.25 41 12.61 9.85 11.34 9.24 43.04 15.44 12.56 15.10 12.29 55.39 16.67 14.55 18.21 13.28 62.71 161.14 6.32 9.57 15.89 42 12.62 9.50 10.96 7.S3 40.91 15.00 12.94 15.88 12.50 56.32 17.38 15.16 19.81 15.40 67.75 164.98 5.34 9.85 15.19 43 11.89 9.57 11.44 S.13 41.03 15.00 12.70 15.67 10.93 54.30 15.50 13.00 17.83 13.00 59.33 154.66 5.34 8.50 13.84 45 12.65 9.66 11.72 7.33 41.36 14.81 11.88 15.31 11.20 53.20 16.70 14.40 17.75 13.30 62.15 156.71 5.64 9.65 15.29 47 13.28 9.86 11.67 9.39 44.20 15.80 13.69 17.21 11.83 58.53 17.31 13.6S 18.11 12.92 62.02 164.75 4.93 8.54 13.47 48 11.41 8.88 10.39 6.27 36.95 13.40 10.00 11.89 7.70 42.99 15.25 13.25 17.30 11.80 57.60 137.54 2.96 6.95 9.91 14 11.38 10.00 12.14 10.94 11.42 13.40 10.00 11.50 12.50 12.14 9.36 8.77 9.04 9.13 9.06 10.19 9.27 9.44 9.4S 9.39 11.83 10.14 11.39 10.60 10.80 11.33 10.S3 9.43 11.45 11.29 6.64 5.80 6.10 5.83 8.10 S.00 7.38 6.38 8.50 son 39.21 34.71 38.67 36.50 39.38 42.92 37.4S 36.75 41.93 40.8S 16.25 15.00 13.88 14.17 13.35 12.75 10.94 11.13 16.5S 16.50 13.92 14.60 12.20 11.50 U.OO 10.64 58.3S 55.75 49.74 50.54 97:59 90.46 88.41 87.04 39.38 42.92 37.48 36.75 41.93 40.88 4.84 6.18 5.76 4.78 4.84 21 6.18 26 5.76 34 4.78 7 29 44 46 50. 51 A V.ldi ion. S— ! Subtract on. M— I lultiplic ation. D- -Din'sioi . 14 PUBLIC SCHOOLS OF NEWARK, N. J. numbe- 36 is 40th in 4A, 20th in 6A and 2nd in 8A. These schools show a commendable improvement, but they also show that the work of the primary grades is not up to the average. On the other hand, the comparative standing of the grades of other schools, presents a reverse condition. This occurs less frequently. For example, school number 9 begins as 1st in 4A, is 15th in 6A, and 16th in 8A. School number 47 is 2nd in 4A, 3rd in 6A, and 23rd in 8A. While there is a decrease in efficiency from the 4A to the 8A grade in these schools, the work of the primary grades was exceptionally well done. The results in the reason- ing test follow in a large measure, the results in the four fundamentals. A school which loses position in the four fundamentals as it moves toward the higher grades is not able to show results in reasoning. Some schools vary decidedly with reference to. processes. Schools number 22 and 36 are weak in addition; numbers 1, 15, and 31, in sub- traction; numbers 5, 8, 9, 13, 28, 35, and 41, in multiplication; and num- bers 16, 27, 33, and 47, in division. Many of these schools show great variance also by grades. The analytic processes are naturally the reverse of the synthetic processes and follow them. There are, however, several schools where the results in the analytic processes are much better than those in the synthetic, as in schools numbers 9, 13, 22, and 28. This condition is illustrated by giving the rank of these schools in each of the four funda- mentals and grouping them under synthetic and analytic processes, as follows : Synthetic Analytic School Addition Multiplication Subtraction Division 9 7 13 3 1 13 27 30 5 13 22 21 7 11 4 28 18 25 2 19 Totals 73 75 21 More attention to the synthetic processes might have materially raised the general standing of these schools. On the other hand, in schools numbers 16, 19, 25, 31, 33, and 47, the results in the synthetic processes are far better than in the analytic. The rank of these schools in the two processes is as follows : Sj nthetic Analytic School Addition Multiplication Subtraction Division 16 5 22 18 29 19 14 17 25 23 25 10 3 17 18 31 26 9 37 28 33 1 1 6 15 47 2 4 7 12 Totals 58 56 110 125 These schools could readily profit by giving more attention to the analysis of the synthesis, although from two investigations subtractive subtraction was shown to be better than additive subtraction. ARITHMETIC SURVEY 15 • It will be seen from Table I that school number 18 stands first, without a close competitor. A careful analysis of this school's work places it as 14th in addition, 3rd in subtraction, 14th in multiplication, and 8th in division in the 4A grade ; 4th in addition, 8th in subtraction, 3rd in multiplication, and 1st in division, in the 6A grade; 3rd in addi- tion, 3d in subtraction, 2nd in multiplication, and 5th in division, in the 8A grade; making it stand 7th in 4A, 1st in 6A, and 3rd in the 8A grades; also 5th in addition, 2nd in subtraction, 1st in multiplication, and 3rd in division, in all three grades. This gives it first place in the four fundamental operations in the whole city. In the reasoning test the. school stands 2nd in 6A and 1st in 8A, making it first in reasoning. There are a number of schools that stand very high in the list. Attention is called to numbers 18, 23, 12, 33, 39, 22, 11, 28, 10, and 41, as the ten that stand highest in the combination of results in funda- mentals and reasoning. In only one of these schools does the compar- ative standing in reasoning fall below that in the fundamentals. On the other hand, numbers 48, 2, 3, 37, 6, 20, 30, 24, 5, and 8; represent the ten lowest. In five of these ten .schools the comparative standing in reasoning is lower than in the fundamentals. The correlation between the comparative standing in fundamentals and the comparative standing in reasoning is very close. While there is naturally correlation between them in the work of individuals due to mental ability, one fails to see how this natural correlation can exist when schools are compared, where all kinds of minds are supposed to be found. This same condition exists when we take any group of schools presenting the highest stand- ard and compare it with a similar group with a correspondingly low standard. The conclusion then can fairly be drawn that pupils do not learn to reason well until they have mastered the machinery by which they work out their steps in the reasoning process. For the purpose of comparing types of schools, the schools have been grouped as follows : Those with pupils of Italian parentage, those of Hebrew parentage, those of prosperous Americans, those of less prosperous Americans, alternating schools, and all year schools, with the results as indicated in Table II. This table contains the sum of the scores in the four fundamentals in the 4A and 8A grades, and the per cent, of improvement from 4A to 8A ; the score in reasoning in the 6A and 8A grades and the improvement found between these grades. To this is added, for purposes of comparison, the average age of graduation in the various groups for the school year 1918-19 and the same records for the whole city. TABLE II Comparison of Groups of Schools Average age Accuracy Reasoning of graduates No. of Sum of Medians Per Cent. Averages Per Cent. 1918-19 Schools 4A 8A Imp. 6A 8A Imp. Yrs. Mos. Italian 6 38.57 59.13 S3 3.91 7.80 99 13 11 Hebrew 7 41.42 63.35 53 5.01 8.91 78 14 3 Prosperous American 9 41.03 62.21 52 5.70 10.33 81 14 4 Less Prosoerous American 6 39.49 62.04 57 5.28 9.42 78 14 4 Alternating 8 38.69 60.44 56 4.21 8.57 104 14 1 All Year - 5 38.80 58.32 50 3.82 7.61 99 *13 10 Whole City 40.18 62.31 55 5.27 9.60 82 14 2 * On account of the epidemic the first class graduated from the all year schools during the school year 1918-19 was in January, 1919, making the average nearly one month older than the two» following classes. 16 PUBLIC SCHOOLS OF NEWARK, N. J. It will be seen that the Hebrew schools stand in the 4A in the highest position and finish in the 8 A in the highest position, making 53% increase in fundamentals, while in reasoning they stand third highest in the 6A and finish third highest in the 8A, making a moderate per cent, of increase. The average age of graduates of this group is one month higher than for the whole city. They appear to be especially strong in fundamentals but this strength is not maintained in reasoning, where they are surpassed by both groups of Americans. They appear to give closer attention to formal work (as shown also by the writing survey) than the Americans, but cannot reason as well. This group contains one alternating and one all year school, and is to that extent affected by them. The Italian schools start lowest and finish next to the lowest with a small per cent, of increase in the fundamentals. The results in rea- soning are also very poor, although the per cent, of increase in reasoning is large partly due to the fact that their score in the 6A is very low. The 8A reasoning is next to the lowest. The average age of the graduates of this group is three months below the average for the city and the lowest of all except all year schools. This result may be affected by the fact that this group contains three alternating and four all year schools. The contrast between the schools of the prosperous American and those of the less prosperous American children is somewhat surprising. The schools of the less prosperous American children begin below those of the prosperous American children and finish below, but make a greater degree of progress in fundamentals, while in reasoning the same thing is repeated except the degree of progress is not so great in the schools having the less prosperous American children as in the schools having the prosperous American children. We cannot escape the con- clusion that there are a sufficient number of less prosperous Americans residing in these districts, who are less prosperous because of inferior mentality, to influence the results of these tests. Pupils of this type are frequently difficult to control. Many of them come from homes, where from various causes the conditions are such as not to contribute to a high standard of efficiency and the whole school is influenced by them. Because of the conditions enumerated above, many of the best teachers seek transfers to other schools. The group of schools of the prosperous American children finishes slightly below the city average in 8A in fundamentals and the highest of any group in the 8A reasoning. This group contains one alternating school but no all year school, while the average age of graduates is two months higher than the city average. The group of schools of the less prosperous American children contains neither alternating school nor all year school. The group of alternating schools begins lowest in the fourth grade and finishes third from the lowest in fundamentals, while in reasoning it begins third from the lowest and finishes third from the lowest. In this group of eight we have three Italian schools, two of the five all year schools and one school of the prosperous Americans. The average age of graduates is two months lower than the city average influenced largely by the Italian and all year schools. ARITHMETIC SURVEY 1/ All year schools begin low in the 4A in fundamentals, and finish lowest in the 8A, making only 47% of progress, while in reasoning they begin lowest, finish lowest and make 110% of progress, due, largely, to the low start. This group of schools contains three Italian and one Hebrew school while the average age of graduates is four months lower than the city average. In general, the high record in 8A in fundamentals is perhaps due largely to a desire on the part of the pupil to do careful work and on the part of the teacher to require efficient results. The schools composed largely of Hebrew children and the schools composed largely of Ameri- can children show the best results. Mentality, age or foreign parentage affects the work in the four fundamentals much less than in reasoning. Reasoning, since it involves the interpretation of printed matter, must be influenced by any lack of ability to> understand English or by mentality. The American pupils, therefore, should excel, but there appears to be no good reason other than the intelligence of the pupils for the standing in both the fundamentals and reasoning of the pupils of the all year and Italian schools compared with those of the Hebrew schools. The results in the all year and alternating schools may be influenced by the predominance of pupils of Italian parentage and because the thorough organization of these schools is not yet completed. DISTRIBUTION OF PUPILS' SCORES Chart IV shows the distribution by grades of pupils' scores in addi- tion and multiplication compared with the distribution by grades of pupils' scores in the reasoning test. The results of the work in the four fundamentals are not as diversified as are the results in the reasoning test. This is natural and leads to the conclusion that mental ability is not as important in securing results in the four fundamentals as in reasoning, and that we have in each grade pupils who may do quite uniform work in the fundamentals but who cannot do uniform work in reasoning. The improvement from grade to grade is not uniform in any case. In addition the greatest improvement occurs in 5A, in multiplication 5A, and in reasoning, perhaps 8A. The distance between these lines may be due to grading, or to the standard set by the course of study. The chart also shows that a great degree of proficiency in funda- mentals may be attained early in the grades and that the subsequent improvement is slight for each grade. The reasoning ability or interpre- tation of written problems appears to develop later and as the pupils become older and reach the higher grades the improvement by grades is very marked, as shown by the lines indicating the grade averages. It is interesting to note that many pupils of the lower grades can do better work than others in the upper grades. Two-tenths of one per cent, of the pupils in the 4A do as good work in addition as the average 8A pupil and 17% in the 6A do as good work as the average 8A pupil ; Chart IZ Distribution ot ■Scores Attained in Addition, /lu/tip/icat/on and Reasoning 18 PUBLIC SCHOOLS OF NEWARK, N. J. <.S ^ N 1 I "5 ■5 fcX'> §^ N. 1 4°- ~* ARITHMETIC SURVEY 19 while 3.6% of the 8A pupils do no better work than the average 4A pupil and 36.2% do no better than the average 6A pupil. In reasoning 5.5% of the 6A pupils could do as well as the 8A pupils while 9% of the 8A pupils can reason no better than the average 6A pupil. ACCURACY The accuracy of the work in the four fundamentals by grades is shown in the following table : TABLE III Per Cent, of Accuracy in the Four Fundamentals Per Cent. Correct on Total Attempts Grade Addition Subtraction Multiplication Division Average 4A 68.9. 61.3 60.3 56.1 62.2 5B 71.1 72.2 63.2 64. 67.6 SA 75. 5 76.4 74.8 69.8 74.1 6B 77.7 79. 74.2 71.1 75.5 6A 79.9 79.6 78. 72.9 77.6 7B 80.2 82.6 78.6 74.4 78.9 7A 82.6 85.7 84.3 78.6 82.3 8B 84. 87.4 84. 82.8 84.5 8A 85.7 90.6 88.1 84.9 87.3 Total City 76.8 76.1 73.8 70.2 74.2 (See Charts II and V for graphs showing the average per cent, of accuracy by grades.) Chart V shows the per cent, of accuracy graphically. It will be noticed that accuracy in addition improves the least of any of the four fundamentals. This is probably due to the fact that addition examples of medium length do not often occur in our books in arithmetic, nor is there a strong motive created to secure accurate results in this one of the four fundamentals, while the other processes more frequently occur in sub- sequent work. The per cent, of accuracy in the reasoning test based on the number of problems reasoned correctly is 92.9% in the 6A, and 94.3% in the 8A ; while the per cent, of accuracy based on the total number of examples attempted is 57.9% in 6A, and 74.8% in 8A. The manipulation of the figures involved in these problems is very simple indeed, so that if pupils understand what to do they can very easily do it. The test is based almost entirely on the interpretation of the written problem and not the working: of it. DISCUSSION OF INDIVIDUAL PROBLEMS A brief review of the errors made in the individual problems of these tests may serve to emphasize some of the weak points they have brought to light. The Woody Scale for each of the four fundamental operations is printed, followed in each case by a table showing the number of times each problem was worked incorrectly or not attempted. (See Tables 20 PUBLIC SCHOOLS OF NEWARK, N. J. IV, V, VI, and VII.) The number of times each problem was worked correctly can easily be found. The Stone Reasoning Test is also printed, followed by a discussion of the results obtained on each individual problem. Addition Scale (1) (2) (3) (5) (7) (ID 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 15 19 .49 15.75 (21) (22) (23) (24) (30) $8.00 547 Vz + y 3 = 4.0125 2J4 5.75 197 1.5907 6H 2.33 685 4.10 3 34 4.16 678 8.673 .94 456 6.32 393 525 240 152 (33) (36) (38) .49 2yr. 5 mo. 25.091 + 100.4 + 25 + 98.28 + 19.3614 = .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 In examining the results of the examples set in the addition scale, it was found that the answer to the first example in addition was frequently given as 6, which might have resulted either by multiplying or by adding the number of the example with the other figures. In example 7, where the example was stated horizontally, the sign was very frequently dis- regarded. Thirty-seven per cent, of the pupils in the 4A, 14% of those in the 6A, and; 21% of all pupils failed to work it correctly. Failure to observe the signs when the example was stated horizontally was also shown in example 14, where 19.3% of all grades failed. Table IV shows that many pupils did. not attempt No. 14. The percentage of failures in numbers 7, 14, and 16 seem to indicate that these examples presented exceptional difficulties to our pupils. (The examples in the Series B tests are supposed to present equal steps of difficulty.) In example 16 many pupils obtained the answer 70, having omitted the figure 9 at the top of the column. The answer to example 23 was frequently given as I. In the 8A, 51% failed in example 36, 37% in example 33, and 25% in example 30. 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I o 28 PUBLIC SCHOOLS OF NEWARK, N. J. second figure of the quotient. The seventh and eighth grade pupils failed more frequently in this example than in the 27th. The errors in the 28th example seemed to be largely by schools. In a 6A class of one school, 24 out of 44 pupils worked it incorrectly. By multiplying in example 30 many gave 3fy as the answer. In some cases the errors were caused by their not knowing how to handle the 5, and in other cases by the inversion of the %. This example gave 72% of failures, 63% in grades above the 5B and 46% in the 8A. In the 36 example 80% of the 8A failed. Division of denominate numbers, however, is not to be found in our course of study, having been purposely omitted. The 4A contains eight zero papers, with five of the highest papers having a score of 12 out of a possible 15. The 6A contains three having a score of 2, and five perfect papers. The 8A has two with a score of 6, and 98 perfect papers, or 9% perfect. Reasoning Test Solve as many of the following problems as you have time for; work them in order as numbered: 1. If you buy 2 tablets at 7 cents each and a book for 65 cents, how much change should you receive from a two-dollar bill ? 2. John sold 4 Saturday Evening Posts at 5 cents each. He kept y 2 the money and with the other Yz he bought Sunday papers at 2 cents each. How many did he buy? 3. If James had 4 times as much money as George, he would have $16. How much money has George? 4. How many pencils can you buy for SO cents at the rate of 2 for 5 cents? 5. The uniforms for a baseball nine cost $2.50 each. The shoes cost $2 a pair. What was the total cost of uniforms and shoes for the nine ? 6. In the schools of a certain city there are 2,200 pupils; y 2 are in the primary grades, % in the grammar grades, % in the High School and the rest in the night school. How many pupils are there in the night school? 7. If 3y 2 tons of coal cost $21, what will S l / 2 tons cost? 8. A newsdealer bought some magazines for $1. He sold them for $1.20, gaining 5 cents on each magazine. How many magazines were there? 9. A girl spent Y% of her money for car fare, and three times as much for clothes. Half of what she had left was 80 cents. How much money did she have at first? 10. Two girls receive $2.10 for making button-holes. One makes 42, the other 28. How shall they divide the money? 11. Mr. Brown paid one-third of the cost of a building; Mr. Johnson paid Yi the cost. Mr. Johnson received $500 more annual rent than Mr. Brown. How much did each receive? 12. A freight train left Albany for New York at 6 o'clock. An express left on the same track at 8 o'clock. It went at the rate of 40 miles an hour. At what time of day will it overtake the freight train if the freight train stops after it has gone 56 miles? The Stone Reasoning Test presents an entirely different phase from that of the four fundamentals. The formal arithmetic required is very simple, most of the difficulty being due to the proper interpretation of the written problem. This test is constructed so that each problem offers increasing difficulties, and very few pupils are expected to finish in the time allowed — fifteen minutes. Every step in the problems was marked correct if reasoned correctly, whether the correct results were obtained or not. Most of the failures appeared to be due, first, to inability to reason, and second, to inability to interpret the printed statement, and to fully recognize the value of each word. It is quite evident from these test papers that teachers should train pupils to interpret oral and written statements more carefully. This can be done very easily in almost all subjects, or if necessary by specially prepared written or printed direc- tions, requiring on the part of the pupil silent interpretation followed by ARITHMETIC SURVEY 29 action. It is very frequently the habit of teachers not to allow pupils to make mistakes, whereas it would be much better if the pupil actually made the mistake under the supervision of the teacher and then was brought face to face with the results of his own incorrect interpretation. In the Newark tests many pupils copied the questions before answering them ; others had an elaborate system of working out the formal arithmetic of the problem, and still others analyzed on scratch paper and then copied the work on the answer paper. Many pupils went to the other extreme and merely copied the answer on the final paper, doing the work either without figures or on scratch paper. In this way many failed to receive any credit for a partially correct example, as no work was shown on which any credit could be based. A very noticeable improvement in reasoning and in neatness was shown between the 6A and 8A grades. There were classes in which all pupils showed very commendable work as to reasoning and appearance. As a general rule, the papers that were neatly arranged were much more accurate than those that were carelessly arranged. This was also true in the work in fundamentals. The papers from some schools seemed to indicate that the dollar sign was considered a very unimportant factor. In fact, it seemed in some cases as though the children were not required to use it at all. In many cases the children could not spell the name of their school, and it was very common to find the name of the teacher misspelled. One teacher's name was spelled in eighteen different ways. In example 1 the word "each" was the cause of most of the failures, resulting in the pupils using 7^ as the cost of two tablets. In the third step of the second example many multiplied instead of divided. The fourth example was very poorly done; in many cases 50$ was divided by 2 and again by 5^. The fifth example presented very little difficulty ; when errors occurred they were generally due to placing the $18 in the cents column when adding it to $22.50. Many pupils worked the sixth example as though it read "J4 of the remainder and z /g of those then remaining." There were many errors, however, in getting % of 2,200. Very few errors occurred when the pupils added the fractions and sub- tracted from %. In the seventh example, many who failed expressed the first step 3^-^-21, and many divided in the second step instead of multiplied. In the eighth example, many expressed the first step in this form $1 — $1.20 = $.20 gain. The complications increased toward the end of the set and were mastered by only the best minds. In the ninth example the stumbling block was "half of what she had left." The reasoning here began to be wide of the mark. The errors in the tenth were generally due to the decimal point. Most pupils failing on this example had $3.00 or $.30 for making one buttonhole. Some of the pupils were able to solve social questions better than mathematical questions, and suggested that each be given one-half of the money. One pupil suggested that each girl be given half of the buttonholes to work. The eleventh example was worked incorrectly in most cases — many completed the first step and stopped. In the twelfth, many pupils reasoned out \ 2 /<-> hours later, which was partially correct and for which full credit was given. 30 PUBLIC SCHOOLS OF NEWARK, N. J. Thirty-two of the 2,352 6A pupils received a score of zero on this test and 29 a score of less than one point out of a possible 17.2 points. Ninety-one of the 1,511 8A pupils received a score of less than 5 points. This discussion of individual problems seems to present a dis- couraging picture, but it is an array of errors only for the purpose of profiting by the knowledge of those errors. There were many excellent papers, many classes did highly commendable work, and many schools stood high. This must not be lost sight of. CONCLUSIONS From the foregoing the most important conclusions are : That of the four fundamentals the results in addition are poorest when compared with the Woody standard, caused probably by the lack of organization of this subject in the lower grades where the combina- tions are supposed to be learned, and by lack of subsequent rational and systematic drill in the higher grades. That the results in multiplication are the best of the four funda- mentals. That in some schools-the results in the analytic processes are much better than those in the synthetic processes. That the synthetic and analytic processes seem to be mutually beneficial. That the results in arithmetic in the primary grades are well up to the standard but that the improvement is not maintained in the middle grades. That the weak grades seem to be 6B, 6 A, and 7B, both as to funda- mentals and as to reasoning. The higher grades reclaim some of the lost ground. The cause of the loss should be a matter of study and investigation. That the results in reasoning as a whole are good and except for the loss in the 6A and 7B grades the work of the Newark schools deserves commendation. That schools which stand high in fundamentals also stand high in reasoning and vice versa. The reasoning seems to be very much impaired when the machinery with which the problems are worked is weak. That the "A" classes are stronger than the "B" classes, and that the 5A grade is the strongest of all tested. That accuracy necessarily accompanies any satisfactory amount of good work accomplished. That the improvement from grade to grade is not uniform. That there are pupils in the lower grades who can greatly surpass some in the upper grades in work which both have been taught. That the age of the pupils affects considerably the results of their work. That the work of the Italian, alternating, and all year schools is poor but that the age is lower than the average. That a good teacher affects very materially the results.