ilii?*^ ■e 5JcuJ lotk Hnte OJalbsB of Agricttlture At ffiornell Mmueraitg atljaca. 3S. ?. UNIVERSITYOFPITTSBURGH BUm^EJIN UNIVERSITY EXTENSION 1919-mo «:; A STUDY OF ARITHMETIC IN: > WESTERN PENNSYLVANIA ' , VOL. 15- rtLT 1, 1819 NO. 20 '.Th« TJiiiversity 6(f Pittsburgh Bulletin is published by the -Dpi-' versity af Pittsburgh and is Issued thirty times a year. Entered March 13, 1914, at the Post Office at Pittsburgh, Pa., ag second-, -.class- matter , under an Act of Congress of August 24, 1912 DlViSION OF UNIVERSITY EXTENSION Orgainization and Activities I. Gekbrai,. Edugation Section . FuSlicatibns , - Appointment Bureau / '\ Student Employment , ' ' ' Traiiher Appointment General Alunini Appointment General InfoMiiation Sefvicp Educational Meetings and Cohvehtions. ,' / . --^ ■ i ":"-' -'-"-■: _ -.. ^ ' ' ^' ' II. ExTR^-MURAt iNSTRyCTION Dbpartment ' ' ''!'' Formal Instruction . „ Class Instruction - ■ Correspondence- Sti;dy Lectures - , ~ , - Conimunity Center ^> ' Americanization :■ _> School and: Social Sur^i^eysy Investigation, Research III. PUBUC SSRVICS PBPAK*MEN*r t' . -- ^■ informal Instruction :- ' -/ '_ Package Library Bureau > Visual Bureau School Relations High School Visitation Interscholastrc Contests /Student Welfare, IV. BuSIKeSS AND COMMBRCIAI, DfiVElOPMENJf business Stiryeys Business Short Courses Cooperative Work with Commercial Clubs I^or information, address ' JamSs HbrbBrt Kauey, Director University Extension, JJniversity of Pittsburgh, Pittshurgh, Pa. UNIVERSITYOFPITTSBURGH BULLETIN UNIVERSITY EXTENSION 1919-1920 A STUDY OF ARITHMETIC IN WESTERN PENNSYLVANIA V-'' * ■ ' ^ / , CONTENTS FAoe A Study of Arithmetic in Western Pennsylvania. _ 4 Districts Cooperating _ _ 4 First Conierence - 5 Uniformity of Procedure. - ~ - 5 Supplementary Instructions for Giving the Tests <5 Time for Giving the Tests 7 Who Should Give the Tests _ 8 Second ConeerEnce 8 Checking the Tests.... _. 8 Computing the Medians ~ 8 Calculation of Grade and District Scores 9 A Superintendent's Use of the Graphs 9 Third and Fourth Conference _ 10 Analysis of District Graphs _ _ _ 10 Crises ia Teaching the Fundamentals and How to Meet Them 11 Addition 11 Subtraction _ _ _ _ 15 Multiplication 16 Division . _ 18 Individualized Teaching _ 19 ■Laws Relating to Practice and Practice Material 21 Motivation of Drills - 22 Administrative Changes 23 The Results of the Tests _ _ 24 Tabulation of Data 24 Mediaii Rate and Accuracy _ 24 Efficiency _ 24 Striking Differences in the Achievement of Districts 25 Significance of These Differences _ _ 26 Relation of Rate and Accuracy. _ 27 Comparison of Western Pennsylvania Medians with the General Medians - 42 State Medians - - _ 43 Comparison of Western Pennsylvania Medians with State Medians SO Comparison of Western Pennsylvania in Number of Examples Attempted with General Medians and State Medians 50 Comparison of Western Pennsylvania in Accuracy with General Median and State Medians - 53 Appendix _ 54 PREFATORY NOTE The general plans for this cooperative study were made by Dr. C. B. Robertson, Director of the Extension Division, recently deceased. Co- operative relations were established with the Bureau of Measurements of the University of Pennsylvania, by whom the plan agreed upon for this year's work had been evolved during the previous year. Dr. J. H. Kelley, recently appointed Director of the Extension Division, cooperated heartily in the project. Mr. Homer E. Cooper, Head of the Extra Mural Department of the Extension Division, arranged for the conferences and all supplies. The writer, associated with the Extension Division for the latter half of the year 1918-19 to conduct this study, directed the con- ferences, tabulated the data, and edited the bulletin. Acknowledgment is due the many superintendents, supervising prin- cipals and teachers for their ready cooperation throughout the entire project, and to the county superintendents and assistant county super- intendents through whose encouragement many districts participated. The Extension Division is planning to continue this cooperative work next year and will later submit plans for the year's work. Thomas J. Kjrby, Professor of Secondary Education, University of Pittsburgh. Copies of this bulletin may be had free by addressing the above at the University of Pittsburgh. A STUDY OF ARITHMETIC IN WESTERN PENNSYLVANIA In order to coordinate the work in measurement already in progress in Western Pennsylvania, and to further stimulate it, the University of Pittsburgh through the Extension Division offered the services of the writer to conduct conferences with groups of school men interested in a more scientific method of giving tests and utilizing the findings of them. A very hearty response followed this offer, and conferences were ar- ranged at once with groups of superintendents and supervisory officers, the first test taken up being the Courtis Arithmetic Test, Series B. ♦Districts Cooperating. The following are the names of the districts with the superintendents and supervising principals who cooperted in the study and returned the required data: Ambridge Supt Charles S. McVay Avalon _ Supv. Prin. S. Todd Pearley Aspinwall Supv. Prin. F. D. Keboch Boswell Supv. Prin. A. L. Jeffrey Bellevue Supv. Prin. T. E. Garber Ben Avon _ Supv. Prin. H. M. Merritt Berlin Supv. Prin. J. C. Speicher Braddock Supt F. C. Steltz Bradford Supt J. E. Butterworth Carrick _ _ Supv. Prin. W. H. Sprenkle tChartiers Township (Allegheny Co.) Supv. Prin. H. E. Patton Conemaugh Supt. J. N. Uhler Coraopolis _ Supt C. E. Hilborn JCarnegie Supt T. J. George tConnellsville Supt. S- P. Ashe East Pittsburgh Supv. Prin. H. L. Koons Edgewood _ Supv. Prin. F. H. Remaley Edgeworth Supv. Prin. H. E. Vannatta Elizabeth _ _ Supv. Prin. T. M. Buck *Note — In the State of Pennsylvania there are four classes of dis- tricts. First Class Districts are those having over 500,000 population; Second Class, having a population of 30,000 to 500,000; Third Class, having a population of 5,000 to 30,000; Fourth Class, having a population less than 5,000. Those which co-operated are Third and Fourth Class Districts. tData received too late to be included. iData included only part of the classes of fourth and fifth grades, hence not included. Elwood City Supt. C. F. Becker Emsworth Supv. Prin. Grace A. Courtney Etna - _ Supv. Prin. E. S. Rice Glenfield _ Supv. Prir R. E. Hager Glenshaw Supv. Prin. Mary R. Jeffery Glassport _ Supv. Prin. J. S. Hart Knoxville _Supv. Prin. Milo H. Miller Leetsdale Supv. Prin. H, H. Poole Lucerne Township (Fayette Co.) Supv. Prin. R. E. Jensen Meyersdale Supv. Prin. W. H. Kretchmann Mifflin Township (Allegheny Co.) Supv. Prin. C. A. Edmundson Millvale Supv. Prin. C. C. Williamson Mt. Lebanon Supv. Prin. Elizabeth E. Emerick Natrona. _ _ _ Supv. Prin. M. C. Harner Oakmont _ Supv. Prin. W. Lee Gilmore Perryopolis _ Supv. Prin. W. L. Fife Pitcaim _ _ _ Supv. Prin. C. G. Pierce Port Vue Supv. Prin. A. E. Leffler tRankin _ Supv. Prin. C. L. Wilson Reserve Township (Allegheny Co.). ..Supv. Prin. Martha Manning Sharpsburg Supv. Prin. J. J. Donovan South Fork. Supv. Prin. W. C. Crawford South Brownsville Supv. Prin. D. D. Watkins St. Clair Borough Supv. Prin. Alice Milligan Springdale Supv. Prin. P. R. Anderson West Homestead Supv. Prin. F. L. Rose West View Supv. Prin. H. W. Hennon Wilkinsburg _ _...Supt. J. L. Allison Wilmerding. Supv. Prin. Chas. W. Shaffer Windber _ _„Supt. W. W. Lantz tData received too late to be included. In the tables each district is given a number which it retains through- out the study. However, in the tables the districts are not arranged in alphabetical order. Slight corrections made in grade medians after data were submitted may make it difficult in some cases for a superintendent to recognize his district. To any such, the number assigned his district will be sent on request. First Conference Unifdrmiiy of Procedure. Four conferences were planned for each group, though in some instances these were combined to expedite the procedure. In the first conference, the problem of scientifically giving the tests was handled. The contents of Folders A and BA were thor- oughly discussed and supplemented with a view to obtaining the highest degree of uniformity possible in giving the tests. Following implicitly the instructions worked out by Courtis and the supplementary ones agreed upon in the conference was held as the first criterion of scientific pro- cedure. The following sheet of supplementary instructions was prepared to bring more forcibly to the group the fundamentals in the Courtis in- structions, and a copy given to each examiner to further effect uniformity of procedure: SUPPLEMENTARY INSTRUCTIONS FOR GIVING THE COURTIS ARITHMETIC TESTS Tou are supposed to be thoroughly conversant with the content of Folder A and Polder BA. Before entering a room have clearly in mind your part in the procedure as follows: 1. Preparation 1. Attitude of examiner with the children. See Folder A, page 3. 2. What preliminary remarks are you to make? Folder A, page 4. Omit all that is in parenthesis. 3. How are the papers to be passed? Folder A, page 4. Pass them. 4. Turn papers and read instructions aloud. Emphasize in the instructions : Folder A, page 3. a. Writing of answers on the test. Write one to show them. b. They are not expected to be able to do all. c. Marked for speed and accuracy. More important to have answers right than to try a great many. d. Take problems in order to the right. 5. Then immediately proceed to fill in blanks. Have papers turned face down when finished. 6. Just what do you tell the children as to how you will prepare them and start them off together? Folder A, page S. 7. Give them practice in turning the paper for a uniform start. Folder A, pages 4-5. II. Giving ths Tests 1. Timing — ^At SS seconds say At 60 seconds say Folder BA, page 1. a. Then the examiner should write down the time of begin- ning. 2. How much time do you allow for the test? Folder BA. 3. How and when do you end the test? Folder BA, page 1. 4. Immediately have children count number of problems finished. Folder BA, page 1. Have this number entered. Folder BA, page 1. 5. Read answers and let each child check his own answers, placing at the left of a problem C for correct, X for wrong. Pass among them as you read. Have number correct entered. Folder BA, page 1. 6. Have paper folded ready for subtraction, then turn face down. 7. Proceed as before, beginning at "Giving the Tests" above. 8. Then proceed to multiplication and division. 9. Collect papers. Fasten them together. III. Checking Rh;sui,ts and Entbeing Scores 1. Have teachers or examiners check results as indicated on Folder A, page 6, and Folder BA, item 6. Note I. N. F. papers. 2. Have scores entered for each class in duplicate on Class Rec- ord Sheet No. 1 (Red). Folder BA, item 7. Also fill in blanks on record and duplicate from, "City .• to Date: 1st Trial " inclusive. Also "Total Scores" and number of "I. N. F." 3. Do not do any computing of medians until next conference. 4. Bring with you to second conference tests for each class and a Class Record Sheet No. 1 in duplicate for each class. The writer's experience in giving tests had revealed that it is quite necessary to tell children, unaccustomed to taking tests, to write their answers on the sheet as indicated under I-4-a. of this sheet. The need of repeating verbally the instructions regarding speed and accuracy was emphasized. Many mistakes in timing tests occur even when Courtis' regulations are followed, due to the fact that the examiner does not write down the time of beginning the test, hence the supplementary in- structions under Il-l-a. No examiner should entrust so important a feature as the time element in tests to his memory or unconfirmed ob- servation of the time. Time for Giving the Tests. In this conference the time for giving the tests was agreed upon. As nearly as was possible all the tests were given within the same week and even on the same day, but in some cases the tests could not be given at the time agreed upon. In some schools, due to the unusual loss of time in the first half of the year, classes were not so far advanced as usual. This resulted in a few instances in not giving the Division test to the fourth grade. The same fact may also have affected the scores to some extent. In this conference, the proper form of distributing scores on Class Record Sheet No. 1 was taken up and illustrated by the use of an actual set of papers from a class pre- viously examined. To further enforce uniformity, the members of the conference were given at least one test in which the procedure previously discussed and agreed upon was demonstrated by the writer. The members took the test, scored the papers and entered records as the pupils would be ex- pected to do. Who Shoui,d Give the Tests. It was agreed that those present in the conference should conduct the tests in their schools to insure con- formity with the procedure decided upon. In one instance the writer met the entire teaching force in conferences. They in turn gave the tests. In one other instance, the superintendent and his supervisors who had been present in the conferences held similar conferences with their teachers who then conducted the tests. While this introduced an addi- tional variable, it has the advantage of inducting the teaching force into testing, and bringing a fuller appreciation and sympathy for the move- ment Second Conference Checking the Tests. In the second conference, reactions of the examiners upon their experiences in giving the tests were first heard. There was general agreement that uniformity had prevailed. Then the tests were checked for any inaccuracies. In as much as these had been previously checked by the teachers almost no errors were found. Class Record Sheets No. 1 were also checked. In a few instances the scores were found improperly entered and had to be corrected. In general they showed a thorough understanding of the method and a high sense of the value of absolute accuracy in handling the original data. Computing the Medians. Most of the time of this conference was spent in deriving the method of computing the median rate, median ac- curacy and efficiency. Folder D was used as the source of information with the understanding that Courtis' instructions should be followed even though other methods might appeal to individual members. Some dis- cussion was given to the merits of the median, and especially to Courtis' meaning of efficiency, but the fundamental point was to insure ability on the part of each member to compute accurately these central tendencies on his own papers. Special cases that often lead to mistakes were handled to forestall errors. Then the group computed enough of the medians on their own papers to fix the proper method. It was agreed to have all 8 these computations completed for the third conference to which these Class Record Sheets should be brought for further use. Calculation of Grade and District Scoees. The proper entry of class grades in Record Sheet No. 3 and Supervisory Graph was discussed and demonstrated also. One of these sheets was to be made for each building and one for the school district. It was agreed, where there were two or more classes in one grade to enter the score for each class and then average these scores for the final score for that grade in a building. Likewise, when these grade scores were assembled in a Record Sheet No. 3 and Supervisory Graph for the district, they were averaged for the final grade score for the district. The graphs were then to be made up from these data. A SupEkintendent's Use of the Graphs. Since these Record Sheets and Graphs would constitute the factual presentation of the functions tested, the members were asked to transfer their graphs to large sheets which could be used to greater advantage in the next conference and also in meetings to be conducted later by them with their teachers. The making of a graph for each building on which every class record would be shown was urged in order that the individual class and teacher might not be lost sight of. A similar graph for the district showing the scores for the grades of each building was also urged in order to compare schools. This of course was to contain a single graph made quite promi- nent representing the entire district. A Teacher's Use op the Tests and Class Record Sheets. In ad- dition to this discussion of the best presentation of facts for the super- intendents and principals, the question of the best use of the tests and class record sheets to be made by the teachers was discussed. It was agreed that devices to get the teacher to keep the entire distribution of her class in mind were necessary. While the median is the best single measure of the performance of a class the individual scores must not be lost sight of. One device presented by a member of the conference, in addition to that of keeping a distribution of the scores, was to draw a vertical line through Class Record Sheet No. 1, as nearly as possible at the median point for rate for this class, and likewise a horizontal line through the median point for accuracy. This divides the class into four groups each of which presents a diflferent teaching problem, those in the upper right quadrant being rapid and accurate in comparison with the median of this class, those in the lower right quadrant being rapid and inaccurate, those in the upper left quadrant being slow and accurate, and those in the lower left quadrant being slow and inaccurate. Like- wise lines of different color can be drawn at the points of the Courtis medians and similar comparisons made. These devices prepare the teacher for the need of emphasizing the individual in the group and his special needs. It was also pointed out that by study of the individual test papers the teacher could discover many facts pertinent for diagnosing the formance of individuals and devising remedial measures. The final problem left for the group for the next conference was the further use superintendents, supervisors, principals and teachers could make of the data contained in the tests, the record sheets and the graphs, in determining the quantity and quality of performance of schools, classes and individuals, and in diagnosing difficulties and deciding upon remedial measures. Third and Fourth Conferences Analysis of District Graphs. In the third conference, it was first ascertained that the medians and efficiency had been properly computed and that the results of these had been correctly entered in Record Sheet No. 3 and recorded in the Supervisory Graph. Then each member pre- sented the graphs of his school showing where there was close agree- ment with the general median and marked variation from it. Attention was directed to steps of advancement between grades which gave evidence of good classification and likewise to lack of such steps of progress and to reversions by any particular grade or grades, which were evidences of questionable classification for the ability in question. Not only was the standing of a grade with reference to the curve of median development noted, but likewise its position in the curve for the particular district reporting. For many of the marked variations, those reporting offered explanations in the light of facts which they regarded as peculiar to this situation. Where a grade or a district was found to be superior in any process, inquiry was made as to the method of teaching used and the time devoted to this particular kind of work with the hope of being able to discover any excellence of method contributing to the superiority. In general the results were sufficiently short of the general medians, to stim- ulate further search for the shortcomings or "crises" involved and to highly motivate a consideration of remedial measures with which to attempt improvement. When the districts reporting were found to be so universally below the general median, the question was raised as to what data were used in the computation of the general median in as much as most of the state medians so far reported have also been much below the general median. Three conditions were proposed in explanation. The efficiency in the four fundamentals has declined since Mr. Courtis made his tabulations in 10 1916, or the cities and towns from which the medians were then obtained are no longer reporting, or many other cities and towns are now report- ing whose scores are lowering the high medians of those cities compris- ing the 1916 tabulations. The conference then turned attention to a discussion of the "Crises in Teaching the Fundamentals and How to Meet Them.'' The following is an outline of points with a summary of the discussion : CRISES IN TEACHING THE FUNDAMENTALS AND HOW TO MEET THEM I. ADDITION 1. Counting. This was very frequently reported as a shortcoming for a high percentage of pupils. a. Causes. The following were agreed upon as contributory causes : (1) Not knowing the forty-five fundamental combinations. (2) Not knowing the decade combinations. (3) Knowing the above combinations in isolation, but not using them in the addition of a column. Some children can add the fundamental combinations when placed as such on the board with celerity and likewise the decade combinations, but in adding columns of some length they count much of the time. b. Remedial Measures. The following remedial measures were pro- posed. (1) Learning the forty- five fundamental combinations moti- vated by speed contests. Children should become able to give these in 30 to 60 seconds. A timfe record should be kept. In this way a child has opportunity to compete against his own previous record, thus having a motive for improvement whether he is the slowest or the fastest in his class. (2) Practicing on Decade Combinations. The fact that a child knows 7 and 6 is no guarantee that he knows 17 and 6, 27 and 6, etc. The element of contest with time record should be provided for here, too. It was also pointed out 11 that it is better to practice these decade combinations witli 6 6 the larger number below, as 17_ 27^ etc., rather than in 17 27 the form' 6 6 usually used. No data are at hand to prove this statement, but appealing to a psychological law, "Have a habit formed in the way it is to be used," the validity of the suggestion is apparent because in adding either up or down a column the large number is first in order as anyone will realize by testing his procedure in adding a column. (3) Forming the habit of using these combinations when add- ing a column. This can be done by having a pupil add aloud. If he pauses to count ask him what combination he needs, thus assisting him to recognize these combina- tions as a part of his process and to incorporate them into his habit of column addition. The writer has found that a sheet containing the combinations with answers to them in the hands of a pupil who knows them in isolation but does not use them, will greatly assist this pupil to realize the value of using the combinations and to form the habit of using these in his column addition. (4) Practice in which the new habit is used. It is not prac- tice that makes perfect but the right kind of practice. Any situation in which the pupil reverts to his old adding habit should be avoided. 2. Needless Language. Some reported pupils saying: "Four and six are ten and eight are eighteen, etc." a. Cure. Practice on fundamental and decade combinations, omitting this excessive language, giving only answers. Then encourage and re- ward the use of the new habit in adding columns. 3. Persistent Error in One or More Combinations. These are not easy to diagnose, but are more frequent than commonly thought. They can be discovered when the pupil gives the combination aloud or better still through the use of the Studebaker or other similar device where written answers to the combinations are demanded. When found the right response should be impressed, and occasion given for its com- plete fixation. 12 4. Pupils Fairly Rapid but Inaccuratb. This was cited as one of the most common causes of low efficiency. How can the difficulty be diagnosed? Pupils should first be diagnosed as suggested above for counting; then for persistent errors. If the difficulty is discovered here it should be treated as indicated above. However, there are many pupils who reveal none of these shortcomings, but are inaccurate. Some teachers try to remedy this by admonishing children to add more slowly. This in most cases augments errors because of changing the pupil's nat- ural pace. What is the difficulty of these children? No doubt they are like the majority of adults, who, despite the fact of long practice in addi- tion, joined with the fact that they have no persistent errors in com- binations, and know all the decade combinations, still make errors in addition. What safeguard is there for this condition? a. Checking. The only cure for this situation that the writer has found is checking results. Pupils should be encouraged to add as rapidly as possible up, then down the column. If the answers agree the pupil should proceed to the next column. In general, tests have not encouraged checking, but rather relying upon a single performance, a practice at variance with the best social custom. Someone asks should all pupils be taught' to check their results. No; any pupil who can qualify under Courtis' Standards of "Efficiency" without checking should be encouraged to d^end upon one addition. The writer found in his testing only two adults who added with absolute accuracy. The efficiency score for all districts tested shows a median of about six per cent. In other words, a very small percentage of children in grades four to eight add with absolute accuracy and at the rate prescribed by Courtis. Some object to checking because it cuts down the rate of accomplishment. This is true, but it does not slow down the real working rate. It also provides for a degree of accuracy which social life demands, and it effects a speed suffi- cient for ordinary computation. 5. Pupils Slow and Inaccurate. Many children come under this caption. They should be diagnosed for counting, and if found wanting, tested as indicated above. Their inaccuracy would probably come under conditions mentioned above. 6. Pupils Slow but Accueatb. Most of these children count and should be taught accordingly. 7. Carrying. It is difficult to tell how much inaccuracy is due to rriistakes in carrying. The chief point of discussion was whether the carried figure should be written on the paper. On this there was sharp 13 division. The arguments advanced for writing the figures were first as an aid in checking, second that most persons engaged in computation write down this figure, having found it an aid to efficiency. In a cursory investigation of this point the writer found that about seventy-five per cent of bookkeepers, auditors and accountants write this figure, while about twenty-five per cent do not. The arguments against it were the time consumed, the marring of the paper and the fact that the memory was trained when required to retain this figure. The first two arguments are not valid, the third is not psychological. Until there are some experi- mental data to determine the proper procedure, the weight of evidence seems to be with writing the carried figure, especially if the results are to be checked. 8. Grouping. The question was raised as to whether children should be taught to group digits to make ten. The experimental data* on this are unfavorable to such teaching. The writer while in New York City tested one school in which much stress had been placed on grouping digits into tens. Apparently problems had been prepared in this school which were abnormally adapted to such a procedure. When these chil- dren were given problems not especially arranged for grouping in tens, they were bewildered and made a very unfavorable showing because they spent their time trying to group by tens when there were few such oppor- tunities. Such a method is too highly specialized to teach generally to elementary school children. Individual children may develop such short cuts, but it is better to spend the time with the majority of the children at least in teaching them to add rapidly successive combinations as they occur in chance arrangement. 9. Practice. Practice in which each pupil has the greatest incentive to use the new habits and to abandon his old ineffective ones is necessary. Practice material should afford a pupil opportunity to measure his prog- ress and to compete with his own past performance and that of other pupils. The writerf in an investigation involving more than seven hun- dred fourth and fifth grade children practicing for about 60 minutes using sheets, a sample of which is found in thel appendix of this study, found an improvement of about fifty per cent in ra.te, the accuracy being almost unaffected. Examples used for practice should closely ap- proximate those on which pupils will be tested in school and in life. •Eighteenth year book. National Society for the Study of Education, Part II, p. 82. tKlrby: Practice in the Case of School Children, pp. 21-23. 14 II. SUBTRACTION 1. Not Knowing the Combinations. This was evidenced by count- ing and unusual hesitation. a. Remedial Measures. Learning the hundred fundamental combina- tions in the manner suggested on fundamental combinations in addition. With these thoroughly automatized, the difficulties of subtraction are materially lessened. In as much as there are no decade combinations to consider the problem is simpler. The writer has found very few cases of children who knew the subtraction combinations but who did not use them in working problems. The probable reason that this situation arises less often in subtraction than in addition is that in subtraction the com- binations recur in a form almost identical with those learned in isolation. Giving the results of these combinations comprises almost the entire operation in subtraction. Each of these is less true in addition where many more factors are involved. 2. Austrian Method. Some members of the conferences were dis- posed to favor the Austrian Method. It was pointed out that the ex- perimental data* are not favorable to the use of this method. No school system included in this study had taught this method exclusively enough or for a long enough period to afford evidence of its efficiency. In answer to the objection that the Austrian Method is used in giving and receiving change, it was suggested that somewhere in the school course sufficient training and practice might be given under conditions as nearly approximating social practice as possible in order to take care of this highly specialized social need. 3. Checking. Here as in addition the need of establishing a habit of checking results was agreed upon. It seems that this is almost a universal custom in business practice, which has not been reduced to habit in school children. There is need of investigation and experi- mentation to see whether the addition of the remainder and subtrahend is a better method of checking than going over the subtraction a second time. •Eighteenth year book: National Society for Study of Education, Part >age 83. II, page 83 15 4. Borrowing. The question was raised as to the comparative effi- cacy of the two methods, borrowing from the minuend and adding to the subtrahend. It was pointed out that sufficient experimental data were not available to answer this question. However, if from the vast number of school systems tested in subtraction over the United States, the results of those using each system could be compiled and compared, some data with which to answer this question might be obtained. A further ques- tion was raised as to the advisability of writing the borrowed unit. No experimental data are at hand with which to answer this, but on the grounds of the psychological law, "Do not form a habit which later is to be broken," this practice was discouraged. 5. Practice. After diagnosing individual errors and deficiencies and ministering to them as indicated above it was pointed out that prac- tice must follow in which the newly acquired knowledge is incorporated into the large procedure, to aid which it was primarily learned. Too often after an error m method is pointed out to a child and a correction made he is permitted to practice under conditions which not only allow the use of his old habit but encourage it This practice, of course, does not get the results sought for. Again it is not mere practice, but the right kind of practice that brings improvement. Practice may even result in retrogression. Problems used for practice should approximate quite closely those on which pupils will be tested whether in a formal school test or in life's test. III. MULTIPLICATION 1. Multiplication Combinations. Here again the most commonly reported cause of inefficiency was not knowing the combinations. a. Remedial Measures. The writer has found sheets like the sample given in the appendix to this study of great value both for diagnosing difficulties and errors, and for practice in multiplication. First any errors that are habitual must be discovered and corrected. If to a child nine sevens are sixty-four, this needs correction not practice. After errors are diagnosed and corrected, right practice must follow on the combina- tions. The use of some device such as that mentioned above makes pos- sible completion both with the pupil's own previous score and tliat of other pupils. Much of the time ordinarily consumed in teaching the multiplication tables can be saved by means of such a device. 16 2. Zero in the MuWip^iER. Zeroes in the multiplier were reported as a source of common error. a. Remedial Measures. One reason for this is the common failure to teach this situation. It is the writer's opinion that such matters should be taught with as little explanation as possible, the fundamental test of the teaching being, can the pupil handle the situation. Hence after pupils are shown how to handle the zero, proper practice in this should be provided. Here again special sheets should be prepared in which this situation occurs frequently with other situations as simple as possible. Such a practice sheet should contain problems like the following, ar- ranged according to difficulty: 3726 3726 3726 3726 3726 40 404 4004 4040 4000 Such problems allow the maximum effort on the function need- ing improvement and comply with a law of drill which needs more careful attention if economy of learning is to be present, viz. : "Arrange drill to give the maximum practice to the function to be improved." By having such a sheet on which pupils can work, the individual who needs assistance on this point can be cared for without taking the time of the entire class. This matter of preparing material to handle special difficulties, very effectively makes possible individualized teaching, much of which is necessary in economical drill. 3. Checking. Pupils do not check their results in multiplication. A very large percentage of children will never perform the fundamental operation accurately without checking. Is it not time to devise our prac- tice and tests with a view to encouraging this practice of checking which generally has to be learned and applied after leaving school in situations demanding accuracy in computation. The common complaint charged up against school children taking positions where computation is neces- sary is not a lack of speed, but inability to reach a dependable result. This habit of checking can be automatized only through practice where exceptions are not allowed to occur. Nothing affords clearer evidence of ineptitude and indifference in this matter, than the pupil who finishes a problem either in arithmetic or algebra and turns to the teacher for her nod of approval or disapproval of its accuracy. 4. Practice. Practice in multiplication as elsewhere must be so ar- ranged so as to give the maximum of time and effort to the improve- 17 ment of multiplication. Many devices offered in text books and practice sheets necessitate copying problems, thus using valuable time in laboriously copying figures, which is checked up against learning multiplication. Then, too, this practice must involve difficulties that have been discov- ered. Too often, in the review, the problems are^ chosen haphazardly, and peculiarly difficult situations are omitted; hence ability to handle them fades out before the test situation arises. Practice should approximate as nearly as possible life situation and those in which pupils willl be tested. The time element with record should enter into practice. IV. DIVISION 1. Combinations. Pupils were quite frequently reported slow and inefficient in the use of division combinations. One cause of this is the failure to give drill on the combinations where a remainder is involved. Most of the practice given in school is on the combinations involving the converse of the multiplication combinations. Three sheets have been devised by the writer involving the combinations needed for division, a sample of which may be found in the appendix of this report. By the use of such sheets the contest element can be introduced and standards of accomplishment for the various grades determined. Fourth and fifth grade classes with about an hour of drill with these sheets increased their speed in writing the answers about seventy-five per cent.* 2. Finding the Triai, Quotient. The condition reported most fre- quently as a cause of low efficiency in long division was the undue time consumed in finding the trial quotient. If this element can be isolated and practice given therein, material improvement in the entire process results. This can be done by an extension of the plan of the sheets mentioned above to involve such situations as the following: 45^ — 21s and remainder 6©:= — 21s and remainder 90= — 21s and remainder One member of a conference reported that a plan similar to this had been used in his schools. The scores in this school for long division were very high. Dr. Thorndiket says that such a scheme forms one of the best introductions to long division. •Lioc. clt., p. 59. fThorndlke: Education, p. 133. 18 3. Zero in the Quotient. Another factor reported frequently which impaired the efficiency in long division was failure to place a zero in the quotient when the divisor was not contained after a figure of the dividend was brought down. The proper handling of this situation should be demonstrated to pupils failing therein ; then they should be given a list of problems to work, prepared to enforce the correct solufion. Again these problems should contain this element often repeated, with other ele- ments simple. The following is suggestive of such a list as could be made: 35(3698 3S|3868 12SI38622 125|37622 Such a sheet used by a pupil under proper direction would serve to de- velop efficiency in this particular situation. 4. Checking. The same failure to check results was in evidence here as noted in the other operations, and the need was quite as apparent Some belief was expressed that going over the division a second time would be a more effective method than the one commonly advocated. Most adults whom the author has questioned go over the division the second time. This is a problem for experimentation. The method of checking by casting out the nines is simple enough to be taught and perhaps is suflfi- ciently different from the routine itself, that children would use it if taught to do so. 5. Practice. Here again practice sheets should be provided which would afford the maximum amount of effort on division. These should afford opportunity for brief intensive practice with the opportunity of contest and measuring improvement. In this practice, situations found to be pittfalls should occur frequently enough to insure facility in their handling. The Courtis Practice pads and the Studebaker Drills are well adapted to this use. V. INDIVIDUALIZED TEACHING 1. Diagnosis. Much that has been said in the preceding pages re- garding diagnosis of peculiar cases demands individual teaching and treat- ment. After tests have been given it is necessary to know the exact situation that is retarding any particular pupil. These are so varied, as has been indicated, that group treatment is often very uneconomical. It is not the same teaching problem to assist a pupil who adds very rapidly, but inaccurately, that it is to aid a pupil who adds very slowly but ac- 19 curately. The former most likely needs to be taught to check his results, the latter needs assistance in the mastery and application of the com- binations. A thorough diagnosis should always be made of extreme cases preceding practice. This is as necessary to success in treatment of cases in arithmetic as in medicine. 2. Individuai, Differences. Nowhere are individual differences more forcibly brought to attention than in a class record of a Courtis Test. Here is a record of a seventh grade class in which one child added 24 problems with 24 correct, while another child added 9 with only 5 correct. Between these are all varieties of performance. It is quite instructive to the teacher to make a complete distribution of the records of her class showing their scores in the four tests. She will not only find wide varia- bility among pupils, but she will find that the same individual may be strong in multiplication but weak in division. Stone found that pupils had "Arithmetical Abilities" in contrast to the older conception that ability to do arithmetical work was an entity. The teacher need not feel that she can overcome these individual differences. The facts are that uniformity of effort on the part of the children of a class will tend to accentuate them. In an investigation made by the author* in which about one hour of drill was given to a large number of children, those of highest initial ability were the ones that made the greatest gains, the result being greater individual differences after the practice than before. The teacher can, however, obtain greater uniformity in accomplishment, by excusing from practice those pupils who reach and maintain satis- factory standards. 3. Materials. Individualized teaching demands materials that will allow one pupil to do one thing while others are doing something else. The Studebaker material is well suited to individual work. If all but two members of a class have attained a standard in subtraction, it is necessary to have material with which these two pupils can practice at given times when the remainder of the class has some other assignment. If one pupil is unable to handle zero in the multiplier, it is necessary to have a special sheet of material on which he may practice after he has acquired the proper method. If a pupil does not know the fundamental combinations in addition a sheet containing these, on which he may prac- tice, is invaluable. In any of these cases, there would be a waste of time if there were not at hand material that could be assigned according to needs. Much of the time given to practice, after the majority in the class has attained a standard, needs be utilized on such individual ma- terial. Differences in teaching results are often due to differences in materials. •Log. cit., pp. 64-68. 20 VI. LAWS RELATING TO PRACTICE AND PRACTICE MATERIAL The following laws were used in the evaluation of methods and ma- terials for practice. 1. GrvB Drui as Nearly as Possible in the Way Liee Defers the Situation. On -this basis the form for decade combinations in addition in which the larger number comes first was advocated. On the same grounds it was pointed out that written practice in multiplication com- 8 binations in the form 4 was better than 4x8 or 4 8s, because the first is the form that life offers. All drill exercises and material should be examined in terms of this law. 2. Do Not Form a Habit Which Later Must Be Broken. Many teachers in teaching subtraction encourage marking out of the digit, writ- ing a digit one less and writing the borrowed one by the digit of next lower order. This habit must be broken later if speed is attained. Hence it should not be formed. In one school where unusual results were found in long division the pupils from the beginning of learning the process were not permitted to use an eraser at the blackboard, to prevent the careless guesses usually made as to a trial divisor and the written multiplications often used to verify or falsify these guesses. Likewise the teacher who encourages placing a row of zeros in the partial product is not in accord with this law. Counting in addition is doubtless a by- product of counting in the early stages of number work, which of course is quite legitimate. However, in the early stages of abstract addition care should be taken to form the habit of adding by combinations. 3. In Practice the Maximum Eeeort Should Be Given to the Function to Be Improved. The Courtis Standard Practice Tests are said to be superior in their results to the Maxon Drill Cards. In the latter much of the effort goes into copying, while in the former the max- imum effort is on the function in question. Drills with flash cards where one pupil responds are not productive of high efficiency because but one pupil is sufficiently active at a time. Drill so arranged that every pupil 21 gives an active individual response is more efficient. If a pupil needs drill in handling a zero in the quotient, a card of problems should be at hand which would give practice to this particular need. 4. Children Should Know the Results of Their Previous Per- formances. Children have little pleasure in trying to surpass a per- formance which is only vaguely in the mind of the teacher and "subject to change without notice." All experimentation has revealed the validity of this principle. Hence material should be chosen which provides readily for the exercise of this law. 5. Drills Should Be Short, Spirited and Motivated. VII. MOTIVATION OF DRILLS 1. Let Pupils Know the Results of Their Previous Performance. This is one of the most effective means of motivation. 2. Let Pupils Know the Standard of Performance That Is Ex- pected OF Them in Each Grade. Too often the teacher is unable to answer such a question as "How well should a fourth year pupil add?" Supervisors in the past were no more definitely informed. It used to be embarrassing to the writer when advocating standards of performance to be compelled to answer such a question as the above with a discursive evasion. Are we not ready to let pupils know what is expected of them in the four fundamentals at least for each grade? In each room the standards of accomplishment should be conspicuously displayed with a list of those who are efficient. At present the best standards are those from the Courtis Tests. These could be placed on the blackboard as the requirements for belonging to the "Efficiency Roll," which might well take the place of the old "Honor Roll." Then efficiency can be defined as ability to do this number of problems of the difficulty and length given by Courtis in the time specified with absolute accuracy. When pupils know definitely what is expected of them, and when they are supplied with the practice material with which they can develop and test themselves, they may be expected to present themselves to the teacher for a final test for entrance among the efficient. Having attained this coveted goal, a pupil should then be held only for a demonstration of this ability at rather regular intervals. Such a scheme as the above works when there are grade standards for the broad jump, the high 22 jump, the sprint and the base ball throw. And much of the practice re- quired in these is done by the individual on his own time. How much will knowing the standards required of them motivate children's work in arithmetic? VIII. ADMINISTRATIVE CHANGES Most of the points discussed above have been problems of teaching. The following administrative changes were suggested and discussed : 1. Change in Time Schedule. If there was a suspicion that lack of time was contributory to the lack of efficiency in arithmetic in a school system, more time might be given to it. This discussion led to a decision on the part of some members of a conference to make a study of time schedules in order to have further facts at hand. 2. Chance oe Emphasis on Phases of Arithmetic. Some super- intendents and principals expressed the belief that too much time was being given to reasoning problems in their schools in the fourth grade and too little emphasis to such ability as the Courtis Tests demanded. They felt that a change in emphasis was necessary in their course of study. 3. Dividing a Class Into Sections According to Ability. These sections would necessarily be shifting sections, as the same pupils do not display the same degree of efficiency in each of the four fundamentals. 4. Arranging Contests Between Classes, Any Class Having the Right to Ch.u,lenge. 5. Change in Methods of Teaching. These have been discussed above. 6. Change in Materials. Superintendents and principals expressed the view that materials for drill complying with the principles set down above were essential to high efficiency and also to economy of time. There was a feeling that the Courtis tests should be used to measure the efficacy of various practice materials and that money expended for such materials should be invested in the light of the effect it would pro- 23 duce. The materials most often recommended as known' to be effective were: a. Studebaker Economy Practice Exercise in Arithmetic. b. Courtis Standard Practice Tests in Arithmetic. c. Laurel Fundamentals of Arithmetic. d. Thorhdike Addition Sheets. e. Kirby Multiplication and Division Sheets. f. Self devised sheets. Such materials should be devised by teachers to take care of difficult situations, such as zero in the multiplier and zero in the quotient. Most school systems have means of duplicating such sheets. THE RESULTS OF THE TESTS Tabulation op Data. In Tables I to V are given the median rate, accuracy and efficiency for each district, each grade in a separate table. There is also presented the median rate, accuracy and efficiency for the forty-five districts, also the general median and the Courtis standard. In the tabulation the districts are arranged in third class districts — cities, towns or boroughs having a population of 5,000 to 30,000, and fourth class districts — shaving a population of less than 5,000. Separate medians are given for the districts included in these two classes. The score in efficiency has been included in order to provide opportunity for com- parison, as well as a consideration of its merits. The data from which these tables were prepared were duplicates of Class Record Sheet No. 1, and Record Sheet No. 3 and Supervisory Graph which were sent to the writer for tabulation. Though these had been checked in the confer- ences they were carefully checked again. Some corrections were neces- sary though in general the work had been accurately and faithfully done. Median Rate and Accuracy. No discussion of the meaning of these terms seems necessary. The writer has felt that the term "Examples Completed" would convey more accurately Mr. Courtis' meaning than "Example? Attempted," in as much as an example does not count in the score unless entirely finished even though it has been begun. However, the terminology of Courtis has been followed in this study. Efficiency. Mr. Courtis defines efficiency in terms of rate and ac- curacy. In deciding upon a rate for each grade, he has set standards which his long experience has shown him are valid as objectives. Still 24 it is possible that the rate is somewhat arbitrarily determined. In ac- curacy he has taken absolute accuracy, the standard demanded in busi- ness computation. One may differ with Courtis as to the rate-factor in his definition of efficiency, but there is no doubt that social demands are for an efficiency combining a reasonable speed with absolute accuracy, such as Mr. Courtis has proposed. According to his standards a class is efficient in the ratio that the number in the class, who can complete the required number of examples in a given time with absolute accuracy, is to the entire number in the class. It should be noted that a class may attain the Courtis Standard in median rate and accuracy if the class is only SO per cent efficient. For example in fourth grade addition where the Courtis standard is a. median rate of six examples attempted and 100 per cent accuracy, these scores can be attained and yet a class have only SO per cent efficiency. So while Mr. Courtis in his folders speaks of a class 100 per cent efficient, which or course is a desirable goal, SO per cent of efficiency is consistent with his standard median rate for each grade and his standard median accuracy of 100 per cent. Few studies have reported efficiency, but it is included in this one with the hope that greater emphasis may be given to establishing in chil- dren the habit of arriving at results for whose accuracy they are willing to vouch and a dependability in computation socially valuable. Striking DiFFERfiNcfis in the Achievement of Districts. An ex- amination of Tables I to V reveals a wide variation in the achievement of these cities, towns and boroughs. In fourth grade addition, Table I, district 23, has a median rate of 9.3 examples attempted and an accuracy of 70, while district 42 has a median rate of 4.8 examples and an ac- curacy of SO. In subtraction the range is from a median rate of 10.4 examples attempted and an accuracy of 93 for district 34, to a median rate of 3.7 examples attempted with an accuracy of 33 for district 42. In multiplication district 25 has a median rate of 7.4 examples and an accuracy of 60, while district 42 has a median rate of 2.4 examples and an accuracy of 34. In division the range is from district 41 having a median rate of 2.8 examples, accuracy SO, to district 34, having a median rate of 7.2 examples, accuracy 86. The variation in efficiency is striking. In this grade the range is from a zero median per cent to 29 per cent of children who can add at the standard rate prescribed by Mr. Courtis with absolute accuracy. In subtraction one district has a median effi- ciency of 39 per cent. In fifth grade subtraction, Table II, the range of the median scores in attempts is from 11.2 to S.4, and in accuracy 90 to 33 per cent. In 2S division the range is from 7.5 to 3.6 examples attempted, while the former district has an accuracy of 70 per cent and the latter 32. In the sixth grade addition, Table III, the highest median score in attempts is 12.6, the lowest 5.2, yet the former has an accuracy of 74 per cent, the latter 53. In subtraction, district 30, with a median of 14.2 examples attempted, more than doubles the score of district 42 with a median score of ii.l examples attempted, yet the one has a median ac- curacy of 90, the other 66 per cent. In the seventh grade division. Table IV, the range of the median scores in attempts is from district 32, with a score of 12.8 examples to district 44, with a score of 5.3 examples, while the former has a median accuracy of 95 and the latter 58 per cent. In the eighth grade addition. Table V, district 4 has a median num- ber of problems attempted almost 100 per cent greater than district 45, yet the median accuracy for the former is 84 and the latter 54 per cent. In subtraction the range is from district 41, with a median rate of 8.2, accuracy 70, to district 4, having a median rate of 15.5 examples, ac- curacy 94, or district 24 with a median rate of 14.5 examples, accuracy 100 per cent. In multiplication, district 5 has a median rate in examples attempted 183 per cent higher than district 37, while the median per cent of accuracy of the one is 99 and the other 79. In division, district 2 has a median of 6 examples attempted, accuracy 80 per cent, while district 30 has a median of 15 examples attempted and a median accuracy, of 100 per cent. Here the range in efficiency for addition is from a zero median to 15 per cent. One district attained a median efficiency of 64 per cent in division. The Significance of These Differences. Such differences need careful consideration. In general if a district has a low median in one operation it is likewise low in the others. Are these differences due to a difference in the amount of time scheduled for arithmetic? Are they due to a difference in the value placed in different districts upon skill in such work as these tests measure? Are they due to differences in social and economic conditions of districts? Or are they due primarily to differences in effectiveness of teaching and supervision of arithmetic? Would tests in other phases of arithmetic or in other studies reveal the same general differencs in favor of the same districts? Certainly where such differences exist, it is safe to say that in some districts the funda- mentals are being taught too well or in others they are being taught with such ineffective results that children leave school with an insufficient ability in these fundamental operations. Those districts whose medians are at the extremes, and far out of accord with the general tendency of performance, should study their 26 situation, not only those having the extremely low scores, but any whose scores may be abnormally high. If a satisfactory explanation can not be found, then some adjustment should be attempted to bring the results into accord with accepted standards. Many such adjustments, both teach- ing and administrative, have been discussed in previous sections of this bulletin. Many superintendents have already made adjustments and are expecting to measure their results and report on them. Relation op Rate and Accuracy. In general the teaching which has been getting the best results in speed has also gotten the best results in accuracy. Following is the score in rate and accuracy for the five districts having the highest rates, and for the five having the lowest rates, together with their scores in accuracy in fourth grade addition. Table I. Similar illustrations can be found in the other operations. — Highest Five — — Lowest Five — Rate Accuracy Rate Accuracy 9.3 70 4.8 50 9.2 70 4.9 59 8.6 72 4.9 47 8.0 78 S.2 50 7.5 62 5.4 54 It is apparent that teaching which has resulted in low speed has not brought a high degree of accuracy but the reverse. It is encouraging too that teaching -which has resulted in high speed has combined with it a factor that has produced comparatively high accuracy. In fourth grade addition. Table I, 13 out of the 22 districts that stood above the median in speed for their own group, also stood above the median in accuracy; in subtraction, 17 out of 22; in multiplication 15 out of 21 ; in division 13 out of 18. This relation persists in the eighth grade. Table V. In addition 13 of the 22 districts surpassing the median rate for their own group, also surpassed the median accuracy ; in subtraction 15 out of 22 ; in multiplica- tion 18 out of 22 ; in division 15 out of 22. 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