Lb C COURTIS STANDARD PRACTICE TESTS IN ARITHMETIC TEACHER'S MANUAL For use with the Card-Cabinet Edition BY S. A. COURTIS Supervisor of Educational Research, Detroit Public Schools. Contains full instructions for the proper use of the Standard Practice Tests in Arithmetic, together with sample graphs and records and suggestions for the diag- nosis and remedy of the difficulties of in- dividual children, by means of which alone it is possible to effect any radical improvement in the efficiency of teaching. WORLD BOOK COMPANY YONKERS-ON-HUDSON, NEW YORK STANDARD PRACTICE TESTS Personal Note to Teachers: The Courtis Standard Practice Tests in Arithmetic were , designed to decrease the amount of a teacher's routine drudgery j and to improve the efficiency of his teaching. Measurement | proves that when the tests are rightly used both objects are ac- comphshed. Measurement also proves that the degree of suc- cess attained depends upon how far the teacher comprehends the purpose of this new tool, and uses the opportunities it pro- vides for adjusting the drill work to the needs of the individual children. Will you not, therefore, study the instructions care- fully until you understand the essential features of the system, then make such adjustment of the general method to your local conditions as will enable you to a. Measure your class to determine the initial ability of its members. (See pages 6, 7-11) b. Eliminate from the drill class those who have (or reach) standard ability. (See pages 6, 11-13) c. Give to each of the other members drill upon those les- sons where drill is needed. (See pages 6, 13) d. Permit each individual to practice in his own way and to grow at his own rate. (See pages 6, 20-27) e. Give exactly the assistance needed to each child that fails. (See pages 27-44) f. Measure the efficiency of your teaching. (See page 22) If you will do this faithfully and intelligently, the efficiency of your teaching of the four operations with whole numbers will rise from the conventional 5% to 10% to from 60% to 75%. Copyright, 1914, 191 5, 1916, by World Book Company. Yonkers-on-Hudson, New York. '• . CSPTA^TM:Ca Bd-2 TEACHER'S MANUAL Ten Essential Points 1. The purpose of the Practice Tests is to develop ability that will transfer to regular arithmetic work, and to all situa- tions in which computations are used ; not to give a few children a chance to show how rapidly they can finish the series of lessons. 2. Lack of ability to "cipher" will prove a handicap in the life of any man or woman. On the other hand, once standard degrees of ability have been attained, further drill is a waste of time and effort. Children of standard ability should be set at more profitable work. 3. Ten minutes a day, day after day, spent in intense, purposeful, snappy practice, has proved adequate to develop proper habits of speed and accuracy. Why use more? 4. Skill cannot be developed in growing children by in- struction, or by any other form of personal exertion on the part of the teacher. Consequently the teachers who best inspire their classes to voluntary effort will secure the best returns. 5. Mere repetition does not develop skill; it is repetition accompanied by the conscious desire to improve that brings results. Mark for growth^ not for the number of lessons completed. 6. Remember always that "nothing succeeds like success." It is easy to get a child to try once, but he will not keep on Q*yA i ^ej 4 STANDARD PRACTICE TESTS trying unless his efforts bring success. The Practice Tests will automatically set for each child each day a task within his reach. If you reward growth in proportion to effort, the children will do the rest. 7. There has always been some cheating in school work, and some children will always cheat. The best safeguard against cheating is not the repressive vigilance of authority, but the development of ideals of honesty and self-mastery. 8. The surest sign of faulty use of the Practice Tests is the speed that is due to excessive effort and nervous strain. The speed that is desired is the perfection of execution pro- duced by thoroughness of preparation. The speed that is merely hurry makes for exhaustion. Do not force speed. Inspire conscientious practice and the question of speed will take care of itself. 9. Under no circumstances forget that when a child fails, there is a reason, and that only as you discover the true reason will you be able to render real assistance. Neither age, experi- ence, training, nor "pull" will make a wrong diagnosis effec- tive. It pays to take the time necessary to determine the causes of the difficulties of individuals, because then, and then only, can you apply the proper remedy. 10. Through appeals to basic instincts, drill work with the Practice Tests becomes so interesting and enjoyable that it takes on the nature of play. This in no way decreases the value of the results secured, however. See that the children look forward to the drill period as to play time, and have a little human interest yourself in the records made. TEACHER'S MANUAL Section I General Description Each child should be supplied with a Student's Record and Practice Pad and the teacher with a cabinet of lesson cards and a Teacher's Manual. Each lesson card in the cabinet consists of a number of examples of one type, the types being so chosen that the range is from the simplest examples to the most difficult a child in the grades is called upon to solve. Further, the mastery of the examples on each card insures the mastery of some one of the many component elements that enter into skill in the four operations. In addition, for instance, one of the early lessons is designed to [teach the combinations, another the bridging of the tens, another carrying, another control of the attention span. Four of the cards (Lessons 45, 46, 47, and 48) are study cards for the use of children who have trouble in learning the combinations. Four other cards (Lessons 13, 30, 31, and 44) are test cards to be used in determining what practice a child needs, and in measuring the efficiency of the child's efforts. These eight cards, strictly speaking, are not practice lessons, although their use is essen- tial to the success of the plan. The lessons are issued in two editions. Form A and Form B. The examples in these two forms differ, but the two forms are equal in the number and difficulty of the examples in each lesson. In filling an order, equal quantities of Form A and Form B are supplied; the two should be equally distributed in each class. It makes no difference whether a child begins to work on Form A or Form B. The corresponding lessons of either form serve as a test for the study and practice put upon the other. No one lesson in the same grade should take any longer to complete than any other lesson. The remaining forty cards are practice lessons and are used as follows: A lesson card, practice side up, is placed under the topmost tissue-paper sheet of the Student's Practice Pad. The child sees the examples through the paper but does his work directly upon the tissue-paper sheet. The num- ber of examples in each lesson has been so chosen with respect to the diffi- culty of the examples that all the lessons require the same working time. That is, an eighth-grade child of standard ability, for instance, will require three minutes to finish each and every lesson.^ At the end of the practice period, the card is taken out, turned over that the answer side is uppermost, and again placed under the tissue- paper sheet. It is then easy for the child to compare his answers with the true answers and find his own mistakes. The perforated tissue sheets are now torn out, but only the papers the children judge to he perfect are handed in^, so that the teacher is relieved of much drudgery of correction of papers. The lesson cards are printed on strong cardboard and may be used again and again, and year after year. It is the supply of tissue-paper sheets that is renewed each year. ^Except those marked "Double Time," which require six minutes. 6 STANDARD PRACTICE TESTS The scores made each day are recorded by the child in the Student's Record, and graphs drawn that the child may see the effect of his own practice. In other words, the children take care of their own progress in every possible way, that the teacher may . be relieved of the mere mechanical details of preparation and marking, in order that he may give his time to the actual teaching of children. Program At the beginning of the term, the first activity is measurement. The teacher gives Lesson 13 (Test A) to determine which children need drill on Lessons 1-12. Those who have perfect scores are excused from drill work until the rest of the class has completed these lessons. Their drill time is spent upon other work. The next day the remaining children take Lesson 1. The third day those who are successful take Lesson 2, and sc on. But those who failed on Lesson 1 spend the third drill period in practicing the examples in the lesson, and the following day try the lesson again, to test the effectiveness of their practice. Only when Lesson 1 has been completed successfully do they go on to Lesson 2. In this way each child masters the simple work before attempting the more complex; each child practices only on the type of example in which he falls below the standard, and each child progresses' as fast or as slowly as his native powers and personal efforts allow. At the end of two months each child in a class of 40 may be working on a different lesson. To a teacher reading the foregoing description for the first time, it may seem, that it would be very confusing to have children working on different lessons and progressing at different rates. It must be remembered, however, that as far as the teacher is concerned his work is the same whether the child is on Lesson 1 or on Lesson 40, and that at the start (the most confusing time) all the children work on the same lesson. By the time the number of lessons in use reaches four or five, the system will be running smoothly and will give no trouble. There are two important reasons' why eve.ry teacher should be willing to make whatever effort may be necessary to adopt this new form of class- room procedure. The first one is that experimental psychology has proved that children differ enormously in their natural powers and rates of growth, so that only that teacher can be efficient who adapts school work to the needs of each individual child. No teacher, by mere intuition alone, can make such adjustments for fifty children, while the system described above will do it automatically. Once you have made the system your own, you will get better results than ever before and with less effort. The second reason is that the system itself has been a gradual evolu- tion. It is based primarily upon a careful, scientific measurement of the relative effect of the various factors which condition teaching, and the original plan has been modified by the contributions of many successful TEACHER'S MANUAL teachers. Each device has been adopted in response to a real need. The system as a whole has proved practical in the hands of a large number of teachers and is rapidly being extended to other subjects. You cannot afford, professionally, not to master this new and efficient tool. Section II Detailed Instructions for Each Day of the First Two Weeks Do not begin the Practice Test work until the second week of school, or until the organization of your class is completed, so that you have some knowledge of the personal characteristics of the children in it. Monday Give to each child a card of Lesson 1 and a Student's Record and Practice Pad.^ Show them how to put it in their practice pads under the topmost tissue sheet, practice side up, so that the ex- amples may be seen through the paper. Explain the necessity for lifting the tissue-paper sheet well up from the rest of the pad, and pushing the card as far up into the stub as possible so that it will be held firmly in place and not slip. Have the children read aloud the instructions for starting and stopping as given in the Student's Record, page 3. Go through the motions once or twice until you are sure every child understands. Then let them w^ork on Lesson 1 for a half-minute interval, and score their papers, following the instructions. This, of course, is mere practice, to make them familiar with the general procedure. Explain that the first real work will be to find out which children need the drill. Collect all the cards of Lesson 1. Distribute the cards of Lesson 13 and have the children put them in position in the practice pads ready for work. Warn the children that to study a card before the proper time is cheating. Then put the Lesson 1 cards back into the cabinet in their proper place. A little care in returning cards to the cabinet will enable you to keep your material in good order. Always put a card after the numbered separator, with the top of the card to the left, the face of the card towards you. You will then be able to select Form A or B as need arises. Assign the task of filing the returned cards each day to the children who are excused from drill. It is ex- cellent practice in an important form of office work. Have all the pads and cards put away ready for the work of the next day. Tuesday For the testing work, the teacher needs a timepiece showing seconds. A dollar watch having a second hand does very nicely. A football timer (price $2.50 at any store that sells sporting goods) is very much better, as 1 If the children buy this material themselves, be sure that all are supplied before you begin explanations. Low sixth grade, 4:}i minutes High fifth grade, 4^ minutes Low fifth grade. 5}i minutes High fourth grade, 5}i minutes Low fourth grade, 634 minutes 8 STANDARD PRACTICE TESTS it starts from zero, and shows minutes and seconds of elapsed time more clearly than a regular watch. It is of the utmost importance that the teacher use scientific care in keeping exact time. H the time interval varies from day to day, the chil- dren's scores will vary for no apparent reason, and the whole force of the timing be lost. Give the signal to start just as the second hand reaches the sixty mark, and give the signal to stop sharply at the end of the interval. The time to be allowed for the various grades is as follows: High eighth grade, 3 minutes Low eighth grade, Z]/^ minutes High seventh grade, 3^/2 minutes Low seventli grade, 3^ minutes High sixth grade, 4 minutes All lessons in any one grade are to have the same time allowance. Those to whom these standards seem high are, of course, at liberty to change these time allowances to suit their own ideas. The standards above, however, have been set after a careful investigation of the actual speed of work of children in the grades, and are believed to represent the speeds at which children can work without strain. Speed is apparently fixed by the maturity of the individual, so that there is an op- timum speed for each age. H a teacher has several grades in one room, he may adopt one of several methods. The best way is probably to start all the children together, and give each grade its own signal to stop. For instance, in a room having both A 6th and B 7th children, the teacher should start all together, and at the end of 3^ minutes should say "B 7th, stop^. Hands up. Score your papers," and at the end of four minutes, "A 6th, stop. Hands up. Score your papers." Other methods are to use the time of the youngest grade, or of the largest grade, but either of these methods means a lack of adjustment for some children. If this is done, however, it is better to have the time too long than too short. Still anothet method is to use the time of the oldest class for all, but require all other classes to do but a part of the test. The following table shows the per cent of each lesson that should be completed by each grade for each time allowance: TEACHER'S MANUAL Time Grades Allowanc e, 8 7 6 5 4 Minutes H L H L H L H L H L 3 100% 92% 85% 80% 75% 70% 63% 57% 42% 38% 354 100 92 87 81 76 68 62 56 52 3>^ 100 93 87 82 73 66 61 5Q 3^ 100 94 88 79 71 65 60 4 100 94 84 76 69 64 414 100 89 81 74 68 m 100 90 82 76 534 100 91 84 5>4 100 92 Q% 100 For instance, if a teacher used a time allowance of four minutes with his oldest class (High 6th), for the low sixth he would set 94% of the examples in Lesson 1, or 68 (72 x 94%) examples as the number to be completed. This is a good method, but a little difficult to handle. The teacher should work out and post on the board the number of examples to be completed by eaeh grade for each lesson. Work may be done in either pencil or ink. Probably ink is to be preferred, particularly in the upper grades, as children should learn to write rapidly and neatly with ink. Most commercial work requires a permanent record, and is done in ink. Having decided the plan to be followed, give Lesson 13 to all, following the instructions in the Student's Record. The answers' to the test cards are not given on the back of the cards as in the regular lessons, but are found in the Teacher's Manual on pages 54-57. Note that there are two forms of Lesson 13, A and B. Both sets of answers will, of course, have to be read. Have the children exchange papers, and mark each wrong answer with a cross as you read the correct answers from the manual. In checking similar work in business, it is customary to read the figures in order without giving them their place values. The figures should be read in groups of three, with scarcely perceptible pauses between tlie periods. Thus 3,456,789 would not be read three million, four hundred fifty-six thou- sand, seven hundred eighty-nine, but three (pause), four five six (pause), seven eight nine. If the answers are read slowly, the scoring of the children may be depended upon in all but the lowest grades. An answer Is to be counted wrong if it is illegible because of poor figures, or if it has been written over, or erased, or corrected in any way after the first writing, even if the error was discovered before the answers were read. The reason for this drastic rule is twofold: (1) it prevents cheating, and (2) it emphasizes the need of close attention and absolute accuracy. 10 STANDARD PRACTICE TESTS The purpose of the Practice Tests is to develop habits of accuracy and this cannot be done if any form of correction is tolerated. Experience has shown that absolute accuracy in first-draft, straight-ahead work is easily attainable by from 30% to 50% of the children, and that the accuracy of the remaining children can be brought to very high levels if the ideal is constantly kept in mind> Have the children count the number of examples tried, and the number right. These results will be called their scores. The small figures written to the right and above certain examples in the lessons save time in counting. Teach the children to go back to the small figure nearest the point where they stopped, and count only from that point on.^ In this way they will never have to count riiore than four examples. In getting the number right, it is usually better to count the number wrong and subtract from the number tried. The scores from Test A are to be written in the summary on page 5 of the Student's Record. Have the children point to their open records on their desks, while you go rapidly up and down the aisles making sure each record is in the right place. Take pains with the records for the first week, and you will have no trouble later. The child should understand that the keeping of a neat record book is part of the work, that in business and in life they will often have occasion to make such records, and that the two things demanded are accuracy and neatness. Exhibit sample record books from time to time, of the wor.st in the room as well as the best, that the children may have some idea of what is wanted. The record book provides an opportunity for a kind of valuable training which is too often neglected by teachers of arithmetic. Ask the children who completed all the examples in the test to stand, then ask of those standing, that all who had one or more examples wrong sit down. Collect the perfect papers. Then ask those who came within one example, in either speed or accuracy, of having perfect papers, to stand. Collect these papers also. Finally, collect the remaining papers. Then ask how many think they could do better if they took the same test again the following day. Nearly all will respond. Promise to give them another trial and have the pads put away. Reliability of Results Every teacher should understand that measurement of human ability is the measurement of a variable quantity. The amount and character of the work a person can do in a given time varies from hour to hour and from . * Those who do not accept the opinions given, should make up regulations to fit their own convictions. ^That is, a child who had tried 37 examples, would go back to 35, then count, 36, 37, instead of counting the whole 37 examples. TEACHER'S MANUAL 11 situation to situation. Fortunately, however, if the conditions of testing are kept constant, about half the children will have almost identically the same scores on the second day, and of the remaining children, all but about 10% will make the same score within two examples more or less. There are usually, however, about 10% of the average class whose scores will vary widely from their true abilities. Two tests are, therefore, better than one, particularly at the beginning of the testing work, and it will pay to repeat the test the next day. Teacher's Scoring After school or at some convenient time go over the papers handed in. Place them over the proper page in your Manual and make sure the children have made no mistakes in scoring. Enter the scores in your record on page 45. Then fill out a report similar to that on page 51. Record the number of children who had perfect papers, and the number who missed but one example; find what per cent each of these numbers is of the total membership of the class. Be sure to date the report and to mark it "First Trial." Such report should be carefully preserved, as comparisons with later tests will show the efficiency of the teaching. Wednesday Begin by having the children who on Tuesday used Form A exchange cards with those who used Form B. This will tend to eliminate the effect of any special "cramming" that may have been done. When all have a different form from the one used Tuesday, give the test as before, scoring the papers and entering the records, col- lecting the papers, and scoring and recording them as before. Comparison of the two results will show how reliable the first test was. Selection of Those Who Do Not Need Drill Test A covers simple work in the four processes. Children who had perfect papers do not need the drill contained in the first twelve lessons. Moreover, experiments have proved that such children not only do not need drill, but are likely to he injured by it, so that they would have lower scores after taking the drill lessons. Therefore, put on the board the names of all who had perfect scores in both tests and make it plain that they are excused from the drill work in Lessons 1-12, and are to spend the drill time in study upon such assignments as the teacher may make. Try to make the other children understand that the practice lessons are to develop in them the abilities which the perfect children already have. It is of vital importance that the children understand that their daily practice and success in the various drills are simply ^a means to an end, and that the real measure of their success will be their scores when Test A is reached ^ts regular position as Lesson 13. 12 STANDARD PRACTICE TESTS Warning. Lessons 13, 31, 32, and 44 are not practice lessons but tests. They should not be studied or practiced except under test conditions. With children who cannot be trusted these lessons should be collected at once and given out only as tests. The teacher who is lax in his care of this point will be deceived by his results and dishonest in any comparisons he may make with the results of other teachers. Alternative Plans The instructions above call for the elimination of those who had perfect papers both days. Some teachers extend this to include those who were perfect either day, while still others include those who missed but a single example in either speed or accuracy. The author advises the latter for Grades 4, 5, and 6, but absolute perfection both days for Grades 7 and 8. He favors giving those who missed but a single example on one of the days a third chance, counting the best two out of the three scores. The course to be followed from this point on depends somewhat upon the conditions within the class. If only a few qualify for release from Lessons 1 to 12, it is best to begin the next day with Lesson 1 and follow the series through in regular order according to the instructions given be- low. If, however, half of the class or more than half the class are to be excused, give Tests B (Lessons 30 and 31) to these children on Thursday. If, again, half the class' have perfect papers, give Test C (Lesson 44) ^ In other words, children who fail on Test A should start on Lesson 1, those who succeed on Lesson 13 but fail on Test B should start on Lesson 14, while those who have perfect papers v/ith the first two tests but fail on Test C should start on Lesson 31. That is, the tests divide the series of lessons into three groups, Lessons' 1-12, 14-29, 32-43, each rnore difficult than the one, before it, so that by means of the tests the teacher is able to start each child at the exact point in the series where he needs drill. Children who complete all the tests successfully do not need the slightest drill work in the four operations, as they already have more than average adult ability in these skills.^ The author and the publishers hereby give emphatic warning that the drill lessons are designed only for children who need them, and that they should not be held responsible for the bad effects and loss of efficiency sure to follow the use of the drills with children who have already attained the desired goal. Failure to determine the needs of children and to adjust individual work accordingly is one of the greatest factors operating to decrease the effectiveness of almost all the drill work found in common practice. The method of handling the class work also depends somewhat on the conditions within the class, and somewhat upon the individual preference of the teacher, ^o^rie like to have the class complete each group of lessons * Note that the time allowance for Test C is six minutes. ''Children who complete all the tests successfully but who are careless and inaccurate in their regular arithmetic work, should be put back into the drill class until they prove that they can transfer their skill to the regular work. TEACHER^S MANUAL 13 as a unit. That is, no child works on any lesson beyond 13 until 90% of the class or more have completed Lesson 13. Then Test B is given and the second group of lessons is carried through. This method divides into small groups the free time of the children who are excused from drill because of their ability. One after another escapes from the drill class until all have finished the first group of lessons, then all begin on the second group, and so on. Other teachers give Test A, then Test B to those eliminated by Test A, and Test C to those eliminated by Test B. Each child then starts to work directly upon the lesson corresponding to his needs. Under this method, however, when a child finishes the series he has no more work to do, so that if already of standard ability he may have one long unbroken period without drill or test. Under the first method, he would at least be tested once every two months and his free time broken into short periods. The author advises the first plan, and the instructions which follow are based upon it. When, however, most of the class would be excused from drill to wait for one or two children to complete the first group of lessons, it is better at once to adjust the work of the class to the group of lessons needed by the largest number of children. Each teacher must decide for himself the plan that best suits the needs of the class. Thursday Collect and return to the cabinet the cards for Lesson 13, then supply all the children not excused from drill with the card for Lesson 1. Give the test under standard conditions, have the children score their own papers and record their scores in their daily records. ^ Note that if a child is absent, when he returns he is to record ''ab" (absent) in the place of his scores. Some score is to be recorded by each child every day. This means that the record book of every child in the room should in- te at any time during the year just how many days the practice tests have en in use. If a child enters late, his score for his first lesson should be writ- «i opposite the day indicated by the books of the rest of the class. Children o are excused from drill should write *'ex" for a score. If the drill period omitted for any reason, have all the children write '' om " for a score, ese records are necessary to compare the efficiency of teaching under differ- ent conditions. It is evident that it is not right to compare results in a class having practice every day with one having practice but three times a week. Such records will make possible comparisons on the basis of loo days' prac- tice, or any other equal interval. They are of the utmost importance in de- termining the character of the changes that need to be made to render the system still more efficient. When the work of scoring and recording has been completed, ask those who had perfect papers to stand. Collect their cards and papers, giving them in exchange cards for Lesson 2, w^hich ^ On page 6. 14 STANDARD PRACTICE TESTS they should put into their pads in position ready for work the next day. Have them write *' 2 " in the column headed "Lesson" in their Student's Record opposite Day 2, then put everything away ready for the drill the next day. Next have the children who missed on Lesson 1 stand. Tell them that their scores show that they need practice, and that on Friday they are to spend the drill period in study and practice on Lesson 1, and that the test on Monday will show how well they study. Have these children write " 1 " opposite Day 2 in the col- umn headed "Lesson," and "pr" ("practice) for their score, so that they will remember what they are to do. They, of course, retain the cards for Lesson 1. Teacher's Scoring Go over the perfect papers carefully. At first the children will score carelessly, leaving it to you to detect their mistakes. For the first week~or so, therefore, rescore the papers carefully, but keep a record in your manual of each child who turns in a paper in which you find mistakes. Such children, of course, should study Lesson 1 instead of trying Lesson 2 on Friday. Put the names of such children on the board and for a day or two refuse to accept perfect papers from them until their scoring has been checked by other children. If you make it a disgrace to hand in an imperfectly scored paper, and if you keep a record of the children from whom such papers come, it will not be long before you will need to examine only the papers from certain children. After two weeks, the daily scoring and recording should not take more than ten minutes of your time. You should always remember that mistakes are likely to happen with any one. Even your own scoring will not be perfect. It is important only that you detect any cheating that may be attempted. It is very foolish painfully and tediously to rescore each day all the papers handed in. Such over-con- scientiousness is just as much to be criticized as a failure to detect dishonesty m the few children who are sure to try to deceive you. Therefore use judg- ment in scoring and do not wa^te time on such unprofitable work when it is not needed. In the same way, use judgment in the degree of perfection required for perfect papers. Do not have the same standard for all. If an indifferent child is finally induced to make a little effort, and turns in a paper poor in figures and neatness but perfect in results, accept it and give it the praise it deserves, waiting until the habit of effort has been well established before requiring perfection in the minor elements. On the other hand, set before the able children who lead the class the highest standards possible. In the same way, the standards at the end of a term should be higher than those at the beginning. TEACHER'S MANUAL 15 Teacher's Record Record each day in the record provided in this Manual the papers accepted as perfect, by putting the number of the trial in the proper space. This will keep the progress of the children in such form that you can tell instantly the condition of the class and of individual children. A glance at the illustration on this page will show the children who are progress- ing rapidly and those making little progress. A similar glance at the record of any one child will tell the character of his work. For instance, a record of 15, 8, and 2 trials for Lessons 1, 5, and 8, respectively would mean that a child was making very satisfactory progress as the number of trials needed for each additional lesson was steadily decreasing, but a child whose record was five trials each for these three lessons would be doing little more than memorizing the answers. TEACHER'S RECORD SHEET Name ^ . .^.-^^C?:?^. 5cAoo/J=l5^rr»?tru:t;^ 2^ 3 Z 2 3 3/^Z / 2 a / / 29 a<^7?'^^^xy /3y..,rur^ — 10 IS S S 4- 7 S a 6\4- 2 S i 3 -2 24- 3 J^n^ j^JUy<:A. S 5 — 3\5\S 4 S ^ 3 /2 The record shows that John Smith had a score of 24 examples right (average of two initial tests) at the beginning of the work, that the number of trials required to complete a lesson steadily decreased, and that his score in the final test was perfect. May Brown has a similar record although her ability was less at the start, so that her gain in the test (from 10 to 24 examples right) was greater although she has not yet overcome all her difficulties (24 examples instead of 29). Both these records indicate excellent work. The record of Tom Black, however, is not satisfactory. His tests show no gain, and the number of trials for each lesson is not changing much. He is probably merely learning the answers, and needs special study and assistance. Look out for and avoid records of this type. The Teacher's Record is meant for your own use. It is not essential to the work, except that by means of it you can prevent children saying that they handed in a lesson when they did not. It is also of value in erfabling you to systematize the assistance you give to individuals. Some teachers have used with good results a large piece of cardboard ruled to correspond to the record in the Manual, but with squares large enough to permit colored or gilt stars to be stuck on for each perfect paper. Such a graphic record of the work of the class is a great spur. The only objection to it is the emphasis v^hich it puts on the number of lessons com- 16 -STANDARD PRACTICE TESTS pleted. This can be offset somewhat by giving to the child who has made the largest gain for the day the privilege of putting on the stars for that day. To find the child making the largest gain, ask all those gaining ten examples or more to raise their hands. Then go up the scale, 11, 12, 13, etc., or down, 9, 8, 7, etc., until but a single hand is raised. Friday Lesson 1 Ask those who are to study Lesson 1 to raise their hands. Have them take out their pads and begin work.i If they prefer to practice by writing the answers, let them, but get them started before giving Lesson 2 to the other children, so that there can be no ques- tion of their practice work being counted as a test. The study should continue during the whole time the other children are taking the test, scoring the papers, etc. At the end of the period, comment on the probable gains on Monday of one or two individuals who have studied well, or who have wasted their time. This study period, properly handled, can be used to prove to the children the value of ten minutes of concentrated effort. Teaching how to study is one of the most important features of a teacher's work. From this time on some children should be studying and some taking tests each day. Out of a class of fifty children, therefore, not more than from ten to twenty perfect papers are likely to be handed in on any one day. Some of the children, particularly after a few days when the children begin to see the possibilities of study, will want to practice at home. In general this should be allowed, but only when it results in gain. There is such a thing as over-training, and it is well to keep track of the effects of home study. The interest, encouragement, and pride of parents in the progress of the child are some of the best spurs to effort, but the inordinate ambitions and lack of appreciation of the efforts the child may be making are also some of the most harmful influences. Lesson 2 Give and score Lesson 2, following the procedure for Lesson 1. Those who have perfect papers should be given the card for Les- son 3. Those who fail should prepare for a study period on Lesson 2. Be sure each day to have ALL record their scores for the day, and the work for the following day. Monday Upt.to this time the drill period will have taken 25 or 30 minutes each day, but beginning with the second week, keep a score in your record book of the total interval from the time you first say ''Prepare for Arithmetic Drill " to the time you are ready to take up other work. ^ If there is time in the daily program, much more effective work can be done by having this practice and study done during a regular study period, so that every child is ready for a test every day. ^ TEACHER^S MANUAL 17 Make this period a drill also in close attention, quick actions, and efficient work. At the end of two weeks the drill period should take not more than ten minutes per day in every grade except the 4th. Here fifteen minutes may be required. To spend more time is conclusive proof that your control of the class or your power of organization is less than that of hundreds of other teachers who have succeeded in keeping within the time limits given. On this day there may be four groups of workers in your class: (1) children excused from drill; (2) children trying Lesson 3 for the first time; (3) children studying on Lesson 2; (4) children trying Lesson 1 for the second time. From this time on, however, the children themselves will know what to do so well that they will take charge of their own work. Your work is merely to time the test, whether the children are on Lesson 1 or 10, and to record the results. Be sure this day publicly to commend the children who studied Lesson 1 well and show a large gain in today's scores. Emphasize GAIN THROUGH STUDY, and not completion of the lessons only. Tuesday This day will be a study period for those who missed Lessons 1, 2, and 3, and a test period for those who succeeded on Lessons 1, 2, and 3. Follow the same procedure as before. You will quickly discover that one or two children need to be watched in regard to the way they make their records, one or two in planning for the work they are to take next. As rapidly as possible get the whole pro- gram down to a military routine in charge of the children themselves. Omit unnecessary details, such as standing, raising of hands, etc., and have the work carried through with the least possible help from you. In three weeks your time, from the end of the test interval to the end of the ten- minute period, should be free for helping some child who has repeatedly failed. In a month or six weeks, the whole drill period may be put in charge of individuals who are excused from drill, thus giving them valuable training in leadership and setting your time free from individual work. A teacher can reach a final efficiency of 75-85% by systematically helping two children a day. Wednesday, Thursday, Friday No further instructions are necessary. On Friday collect all the Stu- dent's Records! and make sure the records are being properly kept. If the above program has been followed, each book should contain two records for Test A in the summary on page 5, and eight records opposite the first eight days. Two sample records are given in the figures on the following page: 18 STANDARD PRACTICE TESTS Sample Records Boy A Boy B c Scores G Scores Q ^3 ■♦-* o TJ 4^ & s x>D I 4> J3 Q J H 2 Q hJ H C^ 1 1 72 72 1 32 25 2 2 C5 CO 2 Pr Pr 3 Pr Pr 3 32 • 30 4 2 70 69 4 Pr Pr 5 2 Pr Pr 5 43 43 6 2 70 70 6 Pr Pr 7 3 07 67 7 51 50 8 4 70 70 8 Pr Pr Monday Omit the drill on this day and spend the time teaching the children to graph their scores. Prepare on the board a copy of the first graph page (page 10 in the Student's Record). Select some child who had a low score in Lesson 1 at the start, and who has made a good gain, but has not completed the lesson. If there are none in the class, use the scores of Boy B in the illustration above. Copy the record on the board. Tell the children that you are going to show them how to draw a picture of these scores, that they may all see the gain that is being made. Have them all turn to page 10 and find the column marked "1st Trial." Point to the score for the number tried (32 in the illus- tration, on the following page) and have the children move their pencils up the column until they find this number, then draw a short horizontal line through it. Illustrate your instructions in the diagram on the board. In sim- ilar fashion have the score for each trial found and marked. The children should then use their rulers and draw a line from one number to the next. The result of the reqord above will look like the figure on the following page. TEACHER'S MANUAL 19 Graph Sheet FOR Lesson No. 1. . .72 examples Lesson No. 2... .70 examples Lesson No. 3... 67 examples Lesson No. 4... .70 examples LESSO N.NO. 73 7a 7a 73 73 73 73 73 73 73 73 73 73 73 73 71 71 71 71 71 71 71 71 71 71 71 71 71 71 70 70 70 70 70 70 70 70 70 70 70 69 S9 69 69 89 69 69 69 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 67 «7 67 67 67 67 67 67 67 67 66 66 66 66 66 66 66 66 66 66 66 66 63 6S 65 6S 6S 65 65 e* 64 64 64 64 64 64 6t 64 64 64 64 64 64 64 «3 63 63 63 63 63 63 «i 63 63 63 63 63 63 63 63 63 63 63 6a 63 63 61 61 61 61 61 61 61 61 61 61 61 61 61 88 68 58 58 58 SS 58 58 58 SS 58 S8 58 58 67 57 57 57 57 56' 56 56 56 56 56 56 56 58 SM SS SS 65 5S 55 SS 55 55 SS SS 53 53 53 53 S3 S3 S3 S3 S3 53 53 SI 51 51 /«- 51 51 51 51 r* SO SO SO SO SO SO SO SO 48 49 49 49 49 49 49 48 48 48 48 48 48 4B 48 48 48 47 47 47 »47 47 47 47 47 47 46 46 1 46 46 46 46 46 46 46 46 46 45 45 45 45 45 4S 45 45 45 45 44 44 ^ 44 44 44 44 44 44 44 37 I 37 37 37 31 31/ 31 31 31 31 31 31 31 31 31 31 31 31 INSTRUCTIONS: After each trial, in the column corresponding to the num- ber of the trial, draw a short horizontal line through your score in examples trud. Usmg a ruler, draw a heavy line from this point to the score marked in the previous column. In hke rranner draw a curve for Rights, using a heavy broken line. More than one graph can^be drawn on this page; see Model, page 7. When you have completed the lesson successfully, hand in this record book with your paper. The dotted line is drawn in precisely similar fashion to represent the scores for rights. Discuss the meanings of the ups and downs of the figures until the children are able to "read" the curves easily. Then put a second series of scores on the board and have the graphs drawn, or let the children who have tried any test three times or more draw the graph of their own scores. From this time on, at the end of the drill period each day, ask **How many have tried the lesson they are on three times? Start your graphs." Then, "How many have tried the lesson they are on five times or more? Bring me your graphs." Ordinarily but one or two children will come forward. Have them bring their record books to you open in such a position that the graph is right side up to you.^ You can then see at a glance whether their scores are rising or falling. Praise those whose scores have gone up. Retain the books of those whose scores for several days have shown no gain, or a loss. A single low score may have no meaning. Low scores are often made just before a big gain. But three days of low scores or three days without any gain means almost without exception that the child is merely wasting his time. It is ^The graph for the score for the day should of course be drawn first. 20 STANDARD PRACTICE TESTS better for him to have no drill at all than merely to go through the motions without benefit day after day. A child should be permitted to try a lesson fifteen or twenty times, or more, as long as he is steadily gaining, even though the gain is small, but it is criminal to let a child keep on wasting his time or )ven injuring himself by improper study and effort. Therefore, stop the drill of such children until you have time to work with them as individuals, and either diagnose the trouble (see pages 27-44) or make some radical change in the method of study being followed. When no diagnosis can be made, either from lack of time or because the cause of the difficulty is obscure, it is a good plan to let the child go on to the next lesson for a time. For some children, intermittent practice on three or four lessons seems to be better than continuous practice on one lesson. The graphs tell the whole story of a child's progress at a glance, and no teacher can afford to neglect this feature of the work. If done systematically, it will take less than half a minute each day to select the children in need of assistance. The children, after once starting a graph, add to it each day immediately after recording their scores. No further instructions are necessary. The mechanical features of tbe work should become more and more routine, and the teacher should be able to give his time and attention more and more to the direct personal assistance needed by the children who fail. The teacher's true function is this giving of assistance and encouragement, and his success will depend directly upon how thoroughly he discharges it. Therefore, the remainder of the manual will be devoted to giving such directions as are possible for dealing with the common difficulties which children are likely to have in working on these lessons. Section III Diagnosis and Remedy of Individual Defects In the following pages will be found a discussion of some of the more common difficulties which occur in working with children, together with suggestions for dealing with the same. The teacher should read these, through rapidly so that he may know the general content, but they will require careful reading only as the need arises. In the same way, when a child has difficulty with a particular lesson, read the comments on that lesson; for each is designed to bring to light and to remedy a particular difficulty. General Difficulties Cheating. Some children always have cheated, and some children always will. However, the teacher should do his utmost to develop ideals TEACHER^S MANUAL 21 of honesty and self-mastery, and should value the opportunities for moral training which the Practice Tests provide. One form of cheating consists in learning the answers by heart, and writing them in place without doing the work. Even in honest study on the same test day after day, there is danger that the answers will be impressed upon the mind. The remedy is to work the examples in a different way each day; one day from the left of the sheet to the right, another from right to left, the next from top to bottom, and so on. If in spite of changing the order of work, a child complains of knowing the answers, have him change from Form A to Form B, and vice versa. This is one of the reasons for supplying the tests in two different forms. If children understand that they must really gain in ability from their practice so much that after they have completed Form A of any lesson, they will be able to do Form B successfully also, it will go a long way towards eliminating the cheating. In other words, the remedy for cheating is to detect it, and to substitute the right ideal for the poor one. In similar fashion, when sudden and remarkable gains are made so that cheating is suspected, give the child the other form of the same lesson, and let the test be conducted under your immediate supervision. If the child's scores fall to the old level, it is almost a certain sign of cheating. The penalty should be to begin again at the beginning. Another form of cheating is the writing of the answers on the tissue- paper sheet. Some children will even try putting the lesson card in place answer side up. To detect this, walk through the aisles at the time of the test and make sure the cards are put in position correctly. Where this form of cheating is suspected, once in a while have the children get all ready for a test, then instead of giving the signal "Start," collect the pads and spend the ten minutes looking through the papers. A more difficult form of cheating to detect is the careful preparation beforehand of perfect papers, and the substitution of such papers for the paper actually written during the class time. A test under close supervision ^wi ll be effective here also. I^K^ Teachers should exercise great vigilance at all times, as children love to " boast of fooling the teacher, and are sure to. defend themselves by the statement that "Every one in the room does it." Unfortunately too few adults know how to interpret such statements correctly, and the effect on the teacher's reputation is disastrous. Take it for granted that every child is honest and if you speak of the subject at all, discuss it from the side of ideals of honesty, growth, and self-mastery. At the same time, be wise and vigilant. Suspect especially those children who do well in the daily drills, but fail on the tests or in their regular arithmetic work. 22 STANDARD PRACTICE TESTS Transfer. Another serious problem is* the question of transfer. It is well-nigh inconceivable that a child can do perfect work in the drills day- after day, yet fail in the tests' or in regular arithmetic work, but laboratory experiments i^rove that for some children the slightest change in the most trivial conditions under which work is done is enough to throw well- fixed habits out of gear. Apparently the remedy is to bring the possibility of transfer to the conscious attention of the child. If he is careless in regular work, have him prepare the examples of the lesson for any day in the form of one of the lesson cards, and have him practice it until he can do it perfectly. The teacher must keep consistently to the position that imless the speed and accuracy of the drill work transfer to other situations, they are valueless. In this connection, the teacher will do well to call the attention of the pupils to the difference between their habits of study and work on their ordinary lessons and on the drill lessons. The wise teacher will, from time to time, give set tasks to be completed, and set lessons to be studied, in a given time. Both children and adults should constantly strive for maximum concentration of attention and effort during their working periods. Efficiency. The term "efficiency" has been used so loosely in educational discussions as to be almost meaningless, but the adoption of definite standards of achievement makes possible scientific definition. By the efficiency of any test will be meant the per cent the perfect papers are of the total number; by the efficiency of the teaching during any period, the per cent the children who attain standard ability are of the total number below standard at the start. For instance, if in a class of fifty, five children are perfect in Test A at the beginning of the term, the efficiency of the test is 5/50 or 10%. If, after seven weeks' work. Test A is given again and thirty children have perfect papers, the efficiency of the second test is 30/50 or 60%. The efficiency of the teaching is 55%. That is, there were 45 children below standard at the beginning and 25 children have been brought up to standard by the teaching. The efficiency is 25/45 or 55%.* Such a measurement of efficiency is open to the objection of unequal units, since to raise one child to the standard is not the same as to raise to the standard another child of very different initial ability. Practically, however, for all unselected classes of twenty children or more, the objection of unequal units is not valid. Teachers interested in their own professional ability should measure carefully the efficiency of their teaching year after year. To be sure, it sometimes takes courage to face the disagreeable truth, for present-day efficiencies are low ; but without such measures it is impossible to tell whether one is improving or not. Particularly valuable would be comparisons of the * For practical purposes it is sufficient to subtract initial from final efficiency ,- 60%— 10%=50%. , TEACHER^S MANUAL 23 efficiency of different methods used by the same teacher with different groups of children of the same grade, and of approximately equal initial ability. Types. The diagnosis of educational ills and the prescription of appro- priate remedies is foreign to the average teacher's thinking and practice. To aid in this work each lesson of the Practice Tests has been made to reveal and remedy a different difficulty. Unfortunately, however, there are certain general defects which operate to obscure the smaller difficulties, and many of these are due to the fact that individual children exhibit extreme mental and physical peculiarijties. In the discussions which follow, however, nothing should be construed as meaning that a class can be divided into groups of different types which can be handled as groups, each in its own way. For nothing is farther from the truth. Each individual is a law unto himself and is better able to detect the methods of study and work best suited to his needs than any teacher. Just ask yourself how far you would be willing to have some one else determine what you are to eat, or read, or how you should spend your time. So in school the demand of teachers for the use of certain methods, or certain orders of growth, is just as irksome to the children. The proper thing to do is to set the desired goal before the child, and let him choose his own method of reaching that goal. The teacher, however, needs to have a sympathetic understanding of what is going on in the mind of the child, that he may make the adjustments proper to the method chosen. Therefore, the effect of various factors sometimes operating will be discussed. Age. In nearly every class will be found young, precocious children and over-age, retarded children. Unfortunately, physiological age and psycho- logical age do not always agree. A child may have a mind so keen and alert that he can understand perfectly sixth-grade work at the age of ten, yet his bones and muscles may be no more mature than those of other children of ten. For such children the standards of speed, if not easily attained, should be lowered to correspond to their muscular age, whatever that may be; that is, lowered to a standard that can be easily reached with ordinary effort. In other words, when a child has difficulty in reaching the required speed, if he is also under age, or gives evidence of immaturity for his years, lower the standard so that the task set is within his reach. On the other hand, the over-age child is usually slow in developing. Be content with slow, steady gains, but push the practice as far as you can. Physically he will be capable of standing a large amount of drill, but the returns from the time expended will be small. Be sure to reward such gains in proportion to effort and not to amount. Temperament. There are three extremes of temperament to be noted. One of these is the hasty, energetic child, careless and inaccurate in all school work, and often difficult to manage. Such children will respond to the element of "record breaking" in drills, and do unusually well. It is 24 STANDARD PRACTICE TESTS common experience to find a child, classed as dull and difficult in ordinary school work, leading the class in drills. This merely indicates that the child has natural ability and can do well when he chooses to exert himself, but that the ordinary work of the school has made no appeal to him. Children of this type will develop in speed first, then slowly bring up their accuracy. The other extreme is the phlegmatic child who insists upon having everything he does exactly right, but is correspondingly slow. Such chil- dren grow slowly in speed, and need the ideal of rapid, efficient action as much as the other type need ideals of conscientious accuracy. The teacher is called upon to appreciate the struggles of both and should let each grow in his own way, commending the gain of one in speed as much as the gain of the other in accuracy. It is the appreciation of effort, and encouragement at points of difficulty, that make a teacher's work successful. The third type Is the nervous, self-conscious child who "goes to pieces" on all examinations, and consequently dreads and hates the drill work. The difficulty to be overcome here is not mathematical at all, and the remedy is less difficult than might be imagined. Give such children permission not to take the tests on any day they feel themselves getting nervous. Give them trials alone before or after school, if necessary, but always keep to the front the ideal of the development of such self-mastery that they will be able to take their tests with the class. The method here is suggestion and physical training. Call the attention of the child to the fact that when he is "nervous" his muscles are always tense and his breathing shallow. Show him how to relax, and how to breathe deeply. Interrupt his test at the least sign of nervousness, with the suggestion "Relax," "Take a deep breath." Make the point that mastery of himself is more important than the mastery of the drill lesson, and the child will soon respond. No more valuable assistance can be rendered by a teacher than the delivery of such a child from the habit of worry, and the development of poise and control. Physical Condition. The effect of this factor is too often disregarded by teachers. On the one hand, they are likely to take to themselves credit for the rapid mental progress shown by certain children in their classes, when the real reason for growth is merely that the children have reached a growth stage in their development and could not possibly be kept from learning; and on the other hand, they are likely to be unduly discouraged by the failure of certain children to respond to their efforts. The truth of the matter is that there seem to be in some children well- marked periods of rapid mental growth, and conversely there may be long periods where normal practice seems to produce no results. The teacher's task is at best a difficult one, and when one has done everything one knows* how to do, and the scores of what is apparently a normal child remain stubbornly constant, it is sometimes comforting to know that at least the practice may be preparing for a period of rapid development severa!! months later. Just as a physician has many cases which he is unable to cure. TEACHER^S MANUAL 25 so the teacher will have many children where the difficulty will resist every effort to remove it. In cases where a child's memory is variable, where on some days the child does well, and seems to make good progress, and on other days" to lose the gain and do poorly, suspect some physical disturbance. Inquire as to home conditions, food, sleep, recent disease, etc. In poor districts, children are often required to labor long hours after school, and with loss of sleep, insufficient food, and physical exhaustion there is little energy left for school work. Until we have state control of the conditions of child life out of school hours, little can be done in such cases unless teacher or prin- cipal cares to give personal time and attention in trying to remedy the social conditions in the home. Severe 4isease may operate to weaken memory and Intellectual powers for several years. It rnay also serve as a stimulus to growth. In cases of impaired memory, the teacher should not be discouraged by frequent slumps and lapses. If the practice is continued through t\yo or three years, the time will finally come when normal conditions will be attained. The period from 11 to 13 (eighth grade) is a difficult period, particularly for girls. Great physical changes are taking place in the body, and new im- pulses and ideas are making themselves felt in the mind. Unfortunately, also, school work at this age is often a deadly, mechanical grind. The develop' ment curves of many school abilities frequently show marked drops at the sixth grade. Experiments seem to prove, however, that the cause is wholly mental. If the work makes sufficient appeal, the sixth grade may be a place of unusual growth. Keep the play element uppermost, and make the drills "fun." Mental Traits. Children differ markedly In their natural aptitudes. The two extremes most often noted by teachers are the mechanical and the rea- soning types. In nearly every class there will be some one child who leads the class in the drill work, but who falls away behind on the problem work. There is often also a child of marked intellectual understanding who is slow and inaccurate in the drills. The Practice Tests are admirably suited to handling such children. Praise the mechanically minded child and help him in every way you can to finish the drill work and to feel that he is succeeding. Then give him special assignments in the problem work, to be studied in the drill period. Take these assignments from the second-grade work if necessary, but have them within the reach of the child's understanding, and carry him along slowly and successfully for a time in the problem work. The time will soon come when he will make rapid growth. On the other hand, It Is of the utmost Importance that a child of great reasoning powers should not be handicapped by poor habits of work in the Work he is capable of doing. If such a child, through the persistent efforts of a teacher, finally attains to efficient control over the mechanical 26 STANDARD PRACTICE TESTS skills, his productivity as a mature individual will be greatly increased. Be very strict, but patient, with such children. Accuracy will be hard for them to acquire, but no other class of children will ultimately give so great a return for efforts expended. Speed. Every individual is capable of working at very different speeds, yet every individual probably has his own normal speed. In demanding a certain rate of work from the children, the idea is not at all to "speed them up"; that is, make them work at high tension. The speed that is demanded is the average or natural speed for a given grade as determined by the measurement of many thousands of children of that grade. A child who cannot make this speed without strain will ordinarily be found to be inaccu- rate in his work, or to have poor habits of work. It takes time to make mistakes. Speed in addition, for instance, means the existence of a com- plex habit, means that there has been sufficient repetition to render various component actions which enter into the habit automatic and perfect. Too many teachers do not know the value of the repeated working of a single example, until the habit of automatic action has been established. When signs of a strain to make a given speed are evident, break the test up into small sections — a single example, then two, four, etc. Have these practiced until the child knows them by heart, until they can be worked through smoothly at high speed without strain. Keep up the practice until all parts of a test have been so handled, and finally the speed, not only for that test, but for other and more difficult tests, will be found to have reached the proper level. One exception to the above is to be noted. Some children are born with nerves and muscles predestined to slower action than normal. Very little can be done in such cases, and the standard should be reduced to the level of such children's ability. The one thing the writer can suggest is to set the new standard slightly above their present achievements, and little by little raise the speed. Such children, of course, have to work at higher nerve tension than normal children, if they are to overcome their natural handicap of slowness, but one should be careful to demand but a very gradual increase in speed. Mental Deficiency. Much attention is now being paid to the organization of special classes for backward and defective children, but teachers should be careful not to be hasty in assuming that a child's poor work is due to this cause. The perfectly balanced child, either mentally or physically, is very rare. All have mental tendencies and bias which in degree are worthy of the name mental defects. Measurement has shown that a child may be able to learn addition easily, yet have great ditficulty with subtrac- tion, while for the next child it may be subtraction that is easy and addition that is hard. Such tendencies are passed along from parents to children in precisely the same fashion as red hair or blue eyes. The teacher should note particularly that where such defects operate to hinder the development of the child in the elementary grades, a condition of mental retardation TEACH ER^S MANUAL 27 often arises which makes the child seem wholly defective. If you have such children in your class, watch them closely to discover in what they are interested, what kind of activities they can do well. Give them as much as possible of such work. Once a door has been found into the mind, very rapid development may take place along many other lines, and the child may finally approximate normal conditions. Miscellaneous. It would be impossible to list all the special and peculiar difficulties which fall within the experience of any grade teacher, and each worker must depend upon his own powers of observation to discover, and his own ingenuity to invent, methods of treatment for the same. Certain difficulties, however, are inherent in the four processes themselves, and these the Practice Tests are designed to remedy. Each lesson will now be discussed in detail. Practice Lessons Lesson 1. Subject, Addition Special Phase, Knowledge of the Combinations The examples in this lesson represent the next step Typical examples beyond the addition combinations, and it is the belief of the author that the combinations should be learned by practicing such examples. There is evidence that it takes longer to study the separate combinations, and then learn to add, than it does simply to learn to "~ — " "" "^ add. For the use of those teachers who take the op~ posite view, the hundred fundamental combinations are given in Lesson 45, together with tests for the same. The answers to these should not be written, but recited orally. To prove to a child that he does not know his combinations, have him put the test card in his pad and write all the answers. Then, with the answers in position, find the time required for him simply to read the answers. Remove the card and find the time required to give the answers from memory. Unless the two times are closely the same, the child's speed and accuracy may be benefited by a study of the combinations'. Note that on the test card the fundamental facts are divided into three groups, so that a child need waste no time on groups that he already knows. Do not let him say, "Five and four are nine." Teach him to read the answers only. Thus, for the combinations shown in the margin the child should neither think nor say more than 4 9 2 "9, 17, 8." He must read the answers in exactly the way that he 5 8 6 reads words without spelling the letters. "" As soon as a child by study has decrease3 the time required to recite from the test card the answers to the combinations, try him again on Lesson jl. If his scores show no improvement, it is proof positive that for him the study of the separate combinations has not been of value. Only about 6 8 9 6 4 5 3 8 7 2 1 4 28 STANDARD PRACTICE TESTS three or four children out of ten will profit much by such a direct study of the combinations. A better plan is to have the child practice directly on the examples in the lesson. Have him add aloud one example until he knows it by heart. Then take two, three, one row, two rows, etc., until the child can do a limited portion of the test satisfactorily. He will then know what is expected of him and how to study, so that he may be left to his own study, the teacher following the effect of his efforts from the graph of his daily scores. In practicing on Lesson 1, have the child add from the bottom to the top of each example. When these additions become too familiar, reverse the order and let him add from top to bottom. Be sure to emphasize the need of adding, not merely remembering the answers. Many teachers object to giving the child the same examples to work over and over again. They feel that mere memory of the results in a particular example is of no value. This is wrong. In each form of Lesson 1 all the important combinations occur at least once, and many twice. If a child practices on Lesson 1, Form A, until he knows it by heart, and is then tested with Form B, his scores will show a surprising gain although the examples are not the same in the two tests. In similar fashion, by the time he has learned by heart all the addition lessons in the series, he will have covered the combinations in so many different arrangements that no new material can be given him that he will not be able to add without study. However, each teacher will have to experiment and prove these statements for himself. For- very many, conviction is born of experience only. The surest symptom of lack of knowledge of the combinations is count- ing. Look out for counting with fingers, toes, tongue, nodding the head, etc. By practice on one example until it is learned by heart, convince the child that it is both quicker and easier "to remember" than "to count." Even the most persistent counters will have a few simple combinations which they do not count, and these may be used as a basis of explanation. Children, often have difficulty with particular combinations, as 7-|-8 or 9-}-4. If repeated practice fails to break down such difficulty, use one of the roundabout methods, as breaking the 4 into 3+1, adding the 9 to the 1, and then adding the 3 to the sum. Or try to establish some associa- tion, as the sum of any number and nine is always one less unit in the next tens, — 4, 13. But such methods should never he used except as a last resort. Every thought takes time and energy, and it is wrong to make a child think two thoughts to get a given sum when one will do. Children occasionally fail in this test for other reasons. One of the brightest children in a certain class made almost no progress in Lesson 1, although his scores were very low. Close observation of the boy at work showed that he was counting. A test of the combinations proved that he knew them verv well. A little experimental testing soon discovered the fact TEACHER^S MANUAL 39 that the child's trouble came wholly on the second addition. In the example in the margin the 11 was given instantly, but the child was unable 5 to add the 5 to the 11 held in mind. Nor could he count very rapidly. 4, The suggestion was made that he picture the 11, as written on the 7 paper underneath the 5 in position for adding. This he was able " to do readily, and after a very little practice showed a good gain in his scores. The child probably had a strong tendency to depend upon visual imagery. The practice tests will readily select the children in need of such special assistance, but this in itself is of little value. The critical factor is the skill of the teacher to diagnose conditions and to prescribe remedies. How- ever, do not be discouraged if you have had no experience in such work. Very few teachers have. The important thing is a willingness to try and the power to profit by your experiences and by your mistakes. Lesson 2. Subject, Subtraction Borrowing Subtraction is the easiest of the four processes. There is very little more to it than knowledge of the combinations, and the process of borrowing. The correlation between the combinations and the . 21 operation itself is much higher than in addition. The combinations 9 will be found in Lesson 46. When a child has low scores in Lesson 2, test him on the combinations. Make very sure he knows those with a minuend larger than ten. Teaching borrowing is merely teaching the habit of seeing a num- ber as 1 and the next lower digit. Thus, in the example shown in the 63, margin, show the child that it might be written as in the second 7 form. The 1 is used with the three and the five is brought down. ' 4 7 Practice the child in seeing 5 as ^^ 8 as -j^* etc., then teach him to link the 1 to- the figure at the right. The one remaining point is dis- ^3 crimination between the situation where borrowing is needed, and where it is not. Do the first work with the single combinations, and then with Lesson 2. 50 1. 7 Do not trouble the child with explanations about the place value of the 1 that is borrowed. Do not expect him to realize that it is really a "ten." Teach children to borrow by imitation, just as you would teach them to use a spoon or to shoot a bow and arrow. A few children will refuse to imitate until they understand the reason for the actions, and in such cases the reasons should be given. But for most children the time for such explanations is when skill has been fully developed. In particular avoid the y. "seven from three I cannot take. Therefore. I borrow-" etc. Do not I 30 STANDARD PRACTICE TESTS allow the children to say "seven from three." As with the addition com- binations, they should read the remainders without saying or thinking the subtrahend or minuend. Many children in the upper grades are unable to subtract without repeating the whole formula learned in the primary grade. For the child who does not readily get the "remainder" idea, try the "completion" idea (Austrian subtraction). For instance, instead of saying "seven from thirteen," one can just as well think "seven and how many are thirteen." Both of these are to give place immediately to the mere response to the subtraction situation, "6." In Austrian sub- §3 traction, in the example in the margin, said out in full, seven and 7 how many are 13, the child thinks next, one and how many are "7^ six. That is, the one from the thirteen is carried as in addition. That is one of the great merits of the Austrian subtraction. It has, however, nothing like the superiority which its advocates claim for it, but is of value as an alternative method for certain children. Some teachers may object that the discussions above deal wholly with the mechanical phases of the development of skill. This is true. Under- standing and skill are two separate phases of training which have no necessary connection with each other. The motive for subtraction must come from real situations within the child's experience, and the discussion above takes it for granted that the child in whom the skill is to be developed has already had the necessary experiments with concrete situations demanding subtraction to develop understanding of what subtraction is and why the process is necessary. The one mechanical element possibly connecting the two is the appreciation of the remainder. The child who subtracts 7 from 63 should realize that the 5 and 6 make fifty-six. If he is unable to perceive them as a single number, have him write the numbers from fifty to seventy in a single column. Have him point to sixty-three, then to the seventh number below it. Work in this way until the child develops a proper number sense. Lessons 9, 17, 24, 33, 37, and 41 also deal with subtraction but present no new difficulties and will not be referred to again. They simply give practice in subtraction with larger and larger numbers. Lesson 3. Subject, Short Multiplication Without Carrying In multiplication, as in addition, there is little correlation be- tween knowledge of the combinations (or tables) and the ability to 31 multiply. The combinations given in Lesson 47 should not be __2 studied as such, although the tests' are convenient in locating difficulty with particular combinations. As in addition, so in multiplication, do not let the child say "two times one, two times three." He knows he wants to multiply, and the 2 and the 1 are both on the TEACHER^S MANUAL 31 paper and present to his mind through visual stimuli. The re- sponse is all that should be given. Thus, in drill on combina- tions the child should say or think only 28, 24, 45. So in mul- T c k tiplying 31 by 2 the child should think and write only 2, 6 and — — — read 62, omitting all other responses. As in addition, the most efficient way to learn the combinations is repeated practice on the examples in Lesson 3. If a child does not know the result of a given combination, tell him, then h^ve him practice the example over and over until it is remembered readily, even if the same example has to be tried once or twice a day for a week or more.. Lesson 4. Subject, Short Division Without Carrying In division, as in subtraction, the correlation between the fundamental facts and the resulting operation is much higher than for addition and multiplication. The combinations and 4)284 tests will be found in Lesson 48. Be sure that the child who has had trouble with Lesson 4 has at least a fair command of the combinations, then let him practice on the examples of Lesson 4. In these simple divisions there is no need of "four into 28," etc. It is possible to give the response only as in the previous lessons. Lesson 5. Subject, Addition Bridging the Tens Many a child who knows his combinations perfectly can- not succeed in column addition, because for all sums above 18 3g +7 = he must count. In this test make plain to the children that the lesson is preparation for column addition (Lesson 8), and if the children add the 7 to the 8, and then carry one to the three, they are miss- ing the benefit of the lesson. The child should see the 38 and the 7 and think of 45 only. If he must say anything, it should be 38, 7, 45. Many children have difficulty in bridging the tens. The fault is usually with the concept of the number of sequence. Counting by tens is sometimes 32 STANDARD PRACTICE TESTS a help. Writing the numbers down in order and moving the hand as in 47 46 45 44 43 42 41 40 39 38 37 is worth trying. Still an- other device in common use is the adding of a given digit to a column of figures each of which ends in a given digit. For instance, for practice on the 8-|-7 combination three columns would be used as follows: A B C 98 8 27 88 78 58 78 58 72 68 28 63 58 68 88 48 88 45 38 98 26 28 18 18 18 38 37 8 48 91 7 would be added to each num- ber in turn. Children with poor memories should emphasize the tens, as thirty-eight instead of the conventional thirty-eight. After the idea of bridging the tens is once established, practice soon does the rest. Lesson 6. Subject, Multiplication Carrying To establish the idea of carrying, let the child write at first both partial products, as shown in the margin, so that he can see the way the number car- ried is added to the next partial product. Then have him practice the same example over and over, do- ing the carrying mentally until the w^ork goes smoothly. Then have him try another example and a third, until sure the idea is established. Practice will do the rest. 32 5 32 5 10 15 160 TEACHER^S MANUAL 33 Be sure the children have the right habits of work. In multiplying 32 by 5 the child should say or think only 10, writing the zero, then 15-16, writing the 16. If the child has difficulty in omitting the unnecessary words, let him practice on very easy examples, as 32, 23, 42, 24, etc., making use of the same combinations over and over again. Difficulty in omitting un- necessary words usually means lack of practice on Lesson 3. Lesson 7. Subject, Division Carrying The first point of difficulty in examples of this type is the recognition of the quotient when the dividend is not an even multiple of the divisor. Let the child write 3x4=12) 3)138 C 3 X 4=12 I I 3 X 5=15 j then fill in the 13 and 14 between 12 and 15. Under ques- tioning he will be able to work out 3)13=^4 and 1 over, 3)14= 4 and 2 over. If this does not help, have him make out for different divisors a. list of dividends like the following: 3-f-3=l 4r-^-3=l and 1 over 5-f-3=l and 2 over 6-f-3=2 7 etc. The second difficulty is the holding in mind of the product while the subtraction is made. Help the child to work the example out in full as in long division, so that he may see each step of the process, then have him perfect the short division habit by much practice on a few examples before attempting to use it generally. In division, as in multiplication, omit unnecessary words. Most chil- dren, however, will require an extra step as follows: In working 3)138 they will say or think 4-12-1-18-6, although many are able to reduce this to simply 4-18-6. Such efficient habits increase speed, decrease mental strain, and are much to be desired. 34 STANDARD PRACTICE TE S T S Lesson 8. Subject, Addition Process, Column Addition Column addition is a complex habit in which a number of elements enter. This lesson combines and extends the abilities developed by Lessons 1 and 5. The child should say 9, 10, 15, 3 19, 25, 28 in one breath smoothly, without break. The problem 6 here is, not addition, but the reading of the partial sums. Just 4 as pronouncing words is not reading, so merely announcing the 5 various partial sums one after another is not column addition. 1 In working with individuals, do not be content with correct an- 7 swers. Put before the child smooth, continuous adding as the 2 goal. Here again, practice on first one, then two, then small ' groups of examples will enable a child finally to finish the whole lesson easily within his time allowance. Speed and accuracy without strain are therefore the test as to whether the child is adding in the proper manner. This is one of the important lessons. The child will soon begin to see the sums in groups as 9, 19, 28 or 10, 19, 28. Do not force this, but praise it highly when it occurs, providing only it does not involve skipping about in the columns. The grouping must not interfere with the onward progress of the addition. Any habit that does is vicious. A child may fail in this lesson because he has a very short attention span. The symptoms and remedy for this trouble are given in Lesson 20. Lesson 9. Subject, Subtraction See Lesson 2. Lesson 10. Subject, Long Multiplication Process Without Carrying The examples in this lesson call for no carrying, so that a child's whole attention may be given to the process. The new points in the lesson are the placing of the partial products and the 21 addition to obtain the final product. Teach these by the imita'lion 13 method, showing the child what to do, but not by telling him why 63 it is done. It is in the upper grades that the explanation should 21 be discussed. The child who had trouble with Lesson 1 is quite 273 likely to fail here. Remedy: Review Lesson 1 just before trying Lesson 10. TEACHER'S MANUAL 35 Lesson 11. Subject, Long Division The examples in this lesson call for no carrying. See that the child gets the steps of the process well established: divide, 14 multiply, subtract, bring down. In every example, if the first 21)294 figure of the divisor is used as a trial divisor the quotient will be 21 the true quotient. Therefore, teach here the use of the first "m figure of the divisor as a trial divisor. (See Lesson 19.) g^ Teach also two checks which should always be applied before — and after subtraction: Is the product larger than the partial dividend? Is the remainder larger than the divisor? To aid the child in discriminating between the dififerent cases, prepare a number of examples incorrectly worked, and have him tell what is the matter with each. Illustrations 4 Product larger than divi- 2 Remainder larger than di- 62)1984 dend, indicating that cor- 62)1984 visor, indicating that corre- 248 responding digit in quo- 124 sponding digit in quotient tient is too large. 74 is too small. Be sure to teach that in such cases the work must be begun over again. Two very common mistakes by children are subtraction when the product is too large, and continued division without bringing dov/n a new figure when the product is too small. 74, the remainder, is larger than 62, so the child has divided again. Proof by multiplication and a dis- cussion of place value will remedy the trouble. For very dull pupils, teach long division by using single digits for divisors. Work for the omission of all unnecessary words'. For those who have learned the Austrian subtraction, the process of division can be greatly sim- plified. This is the real advantage of the method, and the writer believes that all should learn Austrian division, whether they learn the addition or not. The actual work on the division example shown in the margin woul'd Le Illustrations 48 212 62) 1984 248 is subtracted 62)1984 248 from 198 124 504 74 496 62 8 124 124 36 S T A NDARD PRACTTCE TESTS as follows: the child would see 6)19 and write the 3, would see 3 X 2 and think 6 under the 8, then two more would make the 6 into 8, so that he would write the remainder 2. 32 The rest of the work is 18-1-124-2-0-0. That is, only the 62)1984 remainders are written, the multiplication and subtraction 124 being carried on mentally. The Austrian method of bor- rowing (carrying) makes this possible. The advantages are that it saves much work and time; the disadvantages, from the point of view of the teacher, are that mistakes are harder to locate since more of the work is done mentally. Lesson 12. Subject, Addition Carrying In the first examples in this lesson the sum of each col- umn falls in the tens, so that the number to be carried is 1. 9 5 6 6 8 In the second part the sums are all in the twenties, and in 3 4 2 3 2 the last part the sums are sometimes one and sometimes the 6 4 7 5 6 other. In adding the example in the illustration, a child should think 8, 16, writing the 6, then 6 (5+1), 9, 15, or better 6, 15, or 10, 15. That is, the number to bQ carried should be added to the first addend. A surprising number of children wait until the addition of the next column is completed before adding the number carried. Children^ with weak memories may have to write the number carried at the top of the next column, but this should not be" permitted except as a last resort. All such devices take time and alter the original example so that checking is difficult. Lesson 13. Subject, Test A The child has now completed the main elements of the four processes', and his scores in Lesson 13 should be compared with his previous scores to see his gain. Some of the children, in spite of apparent good work, will show no gain or even loss. In such cases suspect either cheating (see page 20) or nervousness that may arise from knowing that the past work is being tested. In the latter case repeat the test, using first Form A and then Form B, until the nervousness wears ofif. There may also be one or two cases of legitimate lack of transfer. These will repay careful study. Very little is known about the conditions which prevent transfer and the writer will welcome reports of the experiments of others, particularly if causes and remedies are discovered. The writer advises that each child be allowed to take Lesson 13 as he reaches it in regular order. He also advises that Lesson 13 be given to the entire class (including those excused and those who have not yet reached it) thirty-six days after the first test. Six days should be taken for the work, TEACHER\S MANUAL 37 giving first Form A and then Form B, Lesson 13, then Lessons 30, 31, both Form A and Form B, in order to determine those who did not need the next group of drill lessons. (See page 6.) At this time 75% or more of the class should complete Test A successfully. Be sure to record the results from these general tests in the summary on page 47. Lesson 14. Subject, Multiplication The Zero Difficulties Many children develop difficul- ties in handling zeros in multiplica- cor\ n i e tion and division examples. Four ^^^ *^3 ^^^ 231 cases for multiplication are given in ^^ ^^ ^^ ^^^ this lesson. Say nothing about such 12600 21090 630 693 difficulties unless the child makes such 420 231 mistakes. Then deal with each child 4830" 2379*^ individually and show the analogy be- tween the handling of zeros and the handling of the other figures. When ciphers occur at the end of numbers, have the children simply annex the proper number of ciphers to the product of the significant figures. Note that in some of the examples where a zero occurs in the multiplicand, carrying is called for. This is a new point, but should make little trouble after Lesson 12. When the zero occurs in the multiplier, there should not be three partial products, but two only. The zero means that there is no mul- tiplier in the place where it occurs. The child may write one zero in the product to hold the place, if he must, but the form given in the illustration is better. Lesson 15. Subject, Division Zero Difficulties, Without Carrying This lesson is the converse of Lesson 14. The zero difficulties are a frequent source of inac- 690 510 302 curacy, particularly the third form in the illustration. When it is needed, proof by multipli- 71)48990 3)1530 31)9362 426 93 cation is often a good way of 639 \ 62 showing a child the reasons for ^39 o2 the methods used. 31 x 32 will not yield 9362, while 31 x 302 will. 38 STANDARD PRACTICE TESTS Lesson 16. Subject, Addition Review and Combination. Column Addition and Carrying See Lessons 8 and 12. Lesson 17* Subject, Subtraction See Lesson 2. Lesson 18. Subject, Long Multiplication With Carrying 582 37 Omitting unnecessary words will make for ease and 4074 accuracy of carrying, as the number to be carried does not 1746 need to bfe held in mind for so long a time. 21*5^4 Lesson 19. Subject, Long Division With Carrying The examples in this lesson are all the simplest case 72 in long division with carrying. The first figure of the 63)4536 divisor is the trial divisor, and the trial quotient is the 44^ true quotient. Make the point that the child does not know the "63" combinations; that it would be easier to divide by 60, the nearest round number, and that by dividing by 60 with the zero canceled is the same as dividing by 6. All of this is direct preparation for later lessons dealing with the more difficult cases. Lesson 20. Subject, Addition Attention Span Many children who have no trouble with Lesson 8 will fail on this lesson and have many answers wrong. The reason is a 9 psychological and not an arithmetical one. The human mind 2 is so constructed that it is difficult for it to give continuous 4 attention to any stimulus for a long interval. There are waves 8 or pulses in one's attention, and when the attention is at a 2 low phase, the mind is very likely to be diverted by sights, 7 sounds, or thoughts of a different character. Experiment 5 shows that most children can add steadily for six additions. 3 In the illustration this would mean that a well-trained child 6 would be able to say 16, 20, 21, 27 easily and smoothly, that 1 30 and 35 would come a little less readily, and that the child 4 would apparently stick on 35-|-7, going over and over it with- 8 out being able to name the sum. If the column had been 8 added 16, 21, 30, 42, 52, 56, the apparent difficulty would be — to add 56 and 2. 126 126 TEACHER^S MANUAL 39 The difficulty is in the mind, however, and not in the combinations. The adding activity seems to give rise to nerve currents v^hich have no habitual path of discharge, and so are dammed up, as it were, by the adding activity until the time comes when they are strong enough to throw the entire adding mechanism out of gear. They are then released and dis- charged along some nerve path in the body, causing involuntary movements, as sighs, frowns, etc., or altering the tension of certain muscles or the activity of certain internal organs, etc., etc. As a person has more and more practice in addition he learns to do two things: (1) to hold firmly in mind the last partial sum during the period of disturbance; and (2) con- sciously to interrupt the adding activity and give the confusion currents a chance to discharge. The writer, for instance, repeats the sum to be re- membered, as "35, 35, 35," and at the same time consciously moves his eyes away from the column being added, simultaneously taking a long breath. He is then ready to continue the adding actively for another interval. The degree of difficulty caused by such confusion currents varies enormously in different people. Some show no signs of it whatever. Some have an attention span of but three or four additions. There is evidence tending to show that the average span is about six or eight additions. These are the cases that usually make difficulty. Lesson 20, therefore, with columns twelve additions long, should bring all such trouble to light. It will also serve to detect children having a span several examples long. For instance, if a child misses approximately every fourth or fifth example, suspect the same difficulty, but a long span. The difficulty is easily recognized by its symptoms. If the child hesi- tates at regular intervals in a column, if he goes over and over a given addition and is apparently unable to think at all, if he gives up in the middle of a column and begins again, the difficulty is almost sure to be one of this nature. The remedy is obvious. Teach the child to recognize the difficulty when it occurs, to avert his attention momentarily by lifting his eyes and taking a deep breath, to keep his place in the column by pointing with his pencil, and to remember correctly the partial sum. Children are apt to get entangled in the situation, and to go over helplessly the same addition so "long, that when the crisis occurs and the mind clears, the additions which had been made previously are totally forgotten. Lesson 21. Subject, Multiplication Merely Longer Examples for Further Practice 40 STANDARD PRACTICE TESTS Lesson 22. Subject, Division Second Case This lesson covers the second case, where the trial divisor is one larger than the first figure of the di- visor, but the trial quotient is the true quotient. 50 A child without much imagination or number 49 ^^ sense will have difficulty in recognizing 50 as 48 the nearest round number to 49. Have such 47 49)3087 294 ^ child actually write out on paper the whole 46 of the tens in which the divisor falls, as shown 45 in the illustration. He can then see that 49 is 44 nearer to fifty than forty and appreciate that 43 when the second figure of the divisor is larger 42 than five, he must use, not the first figure of 41 the divisor, but the next larger digit as the 40 trial divisor. If much practice in finding the nearest number does not help, have the child multiply 63 by each of the numbers from 40 to 60 and divide each product by the multiplier from which it came. If the child uses the first figure of the divisor as the trial divisor in all these cases, he will make mistakes in four or five cases. That is, his first trial quotient will not be the true quotient. The teacher must remember that the use of a trial divisor was originally the result of much experience, and the child without either experience or imagination will not appreciate readily the value of a device which eliminates many mistakes. The result will be a slow speed and the irritation which comes from making errors. Successful use of the proper trial divisor can easily be shown to "pay." Lesson 23. Subject, Addition Combination of Attention Span and Carrying See Lessons 19 and 20. Lesson 24. Subject, Subtraction See Lesson 2. Lesson 25. Subject, Multiplication Practice with Longer Examples TEACHER^S MANUAL. 41 Lesson 26. Subject, Division Third Case, Where the First Figure of the Divisor Is the Trial (Divisor, but the True Quotient Is One Smaller than the Trial Quotient The teacher's task here is to develop caution and judgment n the child. In this lesson, the method previously followed 89 )roduces incorrect results. The beginner is apt to feel that 63)5607 the selection of the true quotient is only guesswork, but the 504 teacher must lead him to see that the mere determination of "567 a trial quotient is not enough, that he must not proceed with 567 the work of division until he has estimated at least the probable ■"-" effect of the second figure in the divisor upon the product. Gradually he will come to see that when the dividend is al- most an exact multiple of the first figure of the divisor, the true quotient will probably be one less than the trial quotient. In a large number of such cases he will get the true quotient on the first trial, and the greater the number of successes the better his judgment. He will also be learning another lesson of great value, that there are situations in life in which it pays not to go ahead until the consequences of an action have been carefully determined. Lesson 27. Subject, Addition Practice with Larger Examples Unfortunately, when the number of examples is so few, the answers are soon learned. Have the child make up his own examples for practice, and alternate Form A and Form B for tests. Lesson 28. Subject, Multiplication Practice with Larger Examples Lesson 29. Subject, Division The Fourth Case, Where the First Figure of the Divisor Must Be Increased by One to Obtain a Trial Divisor, and the Second Trial Quotient Must Be Increased by One to Get the True Quotient 79 This case is, of course, very much like the last (Lesson 3g\ 2844 26), except that the difficulty comes in the second figure of the 252 quotient instead of the first. Judgment must be built up slowly "ooj. in the same way as before. qoa 42 STANDARD PRACTICE TESTS Lessons 30, 31. Subject, Test B These two lessons form a single test, and only the children who are perfect in both tests, or have but a single example wrong in the two days' work, should be excused from drill on Lessons 14-29. The score in Test B is the sum of the scores for Lessons 30 and 31, not either score alone. The reason for the two tests is that the examples are so long that only a few can be done in three minutes. In a single test covering the four opera- tions, the result would be based upon so few examples of each operation that its reliability would be low. Even under the present arrangement, any child excused from drill because of success in this test, but inaccurate in daily work in arithmetic, should be put back into the drill class. Test B should be given again. as a general test 48 days after the first trial. Be sure to compare the results of the second trial of this test with the first, and to record the same in the summary, on page 47 of the Teacher's Manual. Read again page 22. Following the second trial of Test B, give Test C, Lesson 44, Forms A and B, to determine the children who should be excused from drill on Lessons 32 to 43. Note that Test C measures endurance and calls for double the usual time allowance. In Grades 4, 5, and 6 only the exceptional child will reach the last group of lessons. It is expected that about 75% of the children will finish Lesson 31 in a half year. In the sixth, seventh, and eighth grades, therefore, more and more children will be excused from the earlier lessons, so that eventually 75 or 80% will finish all the lessons. However, do not expect this the first year the tests are used. If 30% finish all the lessons in the eighth grade the first year, the class is better than the average. Lesson 32. Subject, Addition Addition of Numbers of Different Lengths 14896 635 Difficulties in this lesson are likely to be wholly difficulties 74 of attention. Make sure the child understands that he is to add 380 merely the figures that appear in any one column, and show him 7492 how to follow up a column. Practice will do the rest. • *6 ^ 85 294 54957 I TEACHER^S MANUAL 43 Lesson 33. Subject, Subtraction Practice with Long Examples Lesson 34. Subject, Multiplication Practice with Long Examples Lesson 35. Subject, Division Practice with Long Examples Lessons 36, 37, 38, 39. Subject, Endurance These examples are taken from the writer's Research Tests, Series B, Forms 1 and 2, and are the smallest examples that will measure all the com- ponent abilities that enter into each operation. They also measure endur- ance. It is not enough that a child be able to figure correctly for a short interval. He must be able to keep at it for some time. The time interval is accordingly doubled, and the confusion of scoring papers, etc., of the children who have not reached these tests is an added factor of difficulty. In business offices computations must be carried on in spite of noise, con- fusion, and interruption. Help the child to get the necessary poise and concentration in the busy classroom. Practice is the only remedy needed for failure. Forms 3 and 4 of the Research Tests Series B^ are composed of entirely different examples of equal value, and are available to measure the results of the use of the Practical Tests, or for other research work. Lessons 40, 41, 42, 43. Subject, Copying The examples in these lessons are based upon the writer's Research Tests, Series A, Test 7. Measurement has proved the very great need for training in copying. The difficulties that arise, so far as is known, are due wholly to lack of concentration or attention. Lesson 44. Subject, Test C Allow six minutes for this test. This test measures endurance and copying in the four operations. In almost all commercial work records must be copied from slips to books, and from one book to another. Such copying calls for a peculiar kind of atten- tion, which may or may not be generated by the previous work. Every answer must not only be correct, but written in the correct space in the answer column. ' * These may be obtained from S. A. Courtis, 82 Eliot St., Detroit, Mich. 44 STANDARD PRACTICE TESTS The child who completes Test C successfully has no further need of drill work, except the incidental drill of daily use of computation in his arithmetic work. As with previous tests, be sure to keep on page 49 a record of your results on both the first and second trial 6f this test. Lessons 45, 46, 47, 48. Subject, Combinations in the Four Operations For use, see Lessons 1 to 4. Conclusion The foregoing lessons have been designed to cover every known diffi- culty in the development of ability in the four operations with whole num- bers. Unfortunately, the collection of such difficulties has been a recent activity, and the author will, therefore, welcome letters from teachers giving symptoms and remedies for difficulties that have not fallen within his teach- ing experience, that the series of lessons may be made more complete. For the same reason he will welcome results of tests, and summaries of the time required to complete the different groups of lessons, etc., that the same may be completely standardized. TEACHER'S RECORD SHEET Name. .School. Grade.. Name of Children Score Number of Trials to complete successfully Lesson Score ! Test A 1 2 3 4 5 6 7 8 9 10 11 12 te.t A 1 • I 1 2 1 3 -- 4 5 1 \ O 1 7 1 1 8 1 1 1 1 1 i 1 10 1 1 1 1 11 1 1 1 1 1 12 1 1 1 1 13 1 14 ! ir> ■ 1 , 1 1 t ir, i ! 1 1 17 f 1 — 1 • 18 lO 20 21 22 23 24 1 1 1 1 1 — 1 1 1 1 1 1 1 ! 1 1 1 1 1 t 1 1 1 25 1 1 1 1 1 1 1 1 1 2(5 1 1 1 1 III 27 1 1 1 i III 28 1 1 1 i 1 1 1 1 29 1 1 1 -1 1 1 1 30 1 1 Forward 45 TEACHER'S RECORD SHEET (Continued) Name. Schools Grade. Name of Children Score Number of Trials to complete successfully Lesson Score Te.t A 1 2 3 4 5 6 ^ 8 9 10 11 12 te.t A 31 32 1 33 34 35 36 37 38 39 40 41 42 43 1 44 45 ! 46 j 47 1 48 1 j 49 50 51 52 53 1 I 54 ■ 1 r 55 ! 1 1 56 1 1 1 57 1 ! 1 i 58 — 59 — 60 Forward 46 I TEACHER'S RECORD SHEET (Continued) Name School. Grade. Name of Children "el Number of Trials to complete snccetsfnlly Lesson i feit B 14 16 16 17 18 19|20 21 22 23 24 26 26 27 28 29 tlTt B 1 1 »> 1 ;{ 1 4 1 5 1 6 i . 1 7 1 8 1 9 j lO 1 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ' 2.-» 26 27 28 29 \ 30 Forward 47 TEACHER'S RECORD SHEET (Continued) Name. .School Grade. Name of Children i Number of Trials to complete snccessfnlly Lesson 1 Test B 14 16 16 17 18 19|20|3122 2324 25 2627 2829 Test B 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 — 48 49 50 51 52 53 54 55 i 1 56 57 58 59 (JO Total 48 TEACHER'S RECORD SHEET (Continued) SchooL .... Grade.. Name of Children Score Number of Trials to complete successfully Lesson Score Test C 32 33 34 35 36 37 383940 41 42 43 test C 1 2 - 3 1 4 5 6 7 8 9 lo ! 11 12 . 13 14 15 - 16 17 18 1 19 1 20 1 1 21 1 1 22 1 23 1 1 24 ! 1 25 I 1 20 27 28 29 30 ! _. Forward 49 TEACHER'S RECORD SHEET (Continued) Name.. School. Grade.. Name of Children Score Number of Trials to complete snccessfnlly Lesson Score Test C 32 3334363637 38394041 42 43 test C 31 32 1 33 1 34 35 36 1 37 1 1 38 1 39 1 1 40 1 1 1 i 41 ! 1 1 1 42 1 - - 1 43 1 1 44 ] 45 46 1 ! 47 1 1 1 48 1 1 1 1 49 j i 1 50 1 j ] 51 1 1 1 52 1 1 1 53 1 1 1 1 1 64 1 1 1 1 • 1 i 1 i BS 1 1 ! 1 1 i 1 56 1 ! 1 1 1 1 1 1 57 { j 1 1 1 1 58 Mini 1 1 59 1 1 1 1 1 1 1 60. 1 M 1 1 ! 1 Total 1 Report of Test Teacher. SchooL Grade. Room. Date- Form. 1st Trial 2d Trial Total number of children in class Number having perfect papers. Per cent having perfect papers Number missing but one example. Per cent missing but one example Number excused from Lessons Nos. Per cent excused from Lessons Nos.. Efficiency. Efficiency Previous Trial. Gai am. 51 SUMMARY-TIME COST Teacher To be filled out at end of term School Grade Number of Days No. Lessons Finished Names 1 Tests .essons Practice Excused Omitted or Absent Total 1 2 8 4 5 O 1 7 8 9 10 11 12 13 14 15 16 - __- 17 18 19 20 21 22 23 24 25 26 1 27 28 29 30 Forward 52 SUMMARY-TIME COST (Continued) Teacher. To be filled out at end of term School Grade.. Number of Days No. Lessons Finished Names Tests Lessons Practice Excused Omitted or Absent Total 31 32 33 ~ 34 35 36 37 • 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 63 54 55 56 57 58 59 69 1 Totals • cent of time sav ^d (Divide the total of the excused column by the sum of the totals of tests, lessons, prac- tice, and excused columns.) 53 54 STANDARD PRACTICE TESTS ANSWERS Lesson No. 13 Test A Lessons 1-12 Form A Add ' Subtraction 96 50 68 71 23 74 19 28 41 36 37 42 25 35 34 75 68 85 Multiplication 744 966 299 2982 1488 2397 19 21 13 Division 37 82 TEACHER'S MANUAL 55 ANSWERS Lesson No. 13 Test A Lessons 1-12 Form B Add Subtraction 58 61 78 49 16 94 33 84 37 24 40 35 29 37 47 96 50 / 47 Multiplication 399 672 559 1887 2788 2993 24 23 23 Division 42 62 56 STANDARD PRACTICE TESTS ANSWERS Lesson No. 30 Test B Part 1 Lessons 14-29 Addition Form A 4816 5767 6199 Form B 6119 4866 5797 Subtraction Form A 64879321 88225099 18115955 FormB 76884659 82657718 96538845 TEACHER^S MANUA L 57 ANSWERS Lesson No. 31 Test B Part 11 Lessons 14-29 Multiply Form A 579014 585354 FormB 741228 416698 Divide Fonn A 861 973 Form B 971 862 58 STANDARD PRACTICE TESTS ANSWERS Lesson No. 44 Test C Lessons 32-43 Form B Double Time Form A 531813 9776756 5353488 6987 Form B 510157 7676759 75844608 7897 Contents Page Personal Note to Teachers 2 Ten Essential Points 3, 4 Section I General Description 5 Program 6 Section II Detailed Instructions — Monday 7 Tuesday 7-11 Wednesday 11-13 Thursday 13-16 Friday 16 Monday 16, 17 Tuesday 17 Wednesday, Thursday, Friday 17, 18 Monday 18-20 Section III Diagnosis and Remedy of Individual Defects — Cheating 20, 21 Transfer 22 Efficiency 22 Types 23 Age 23 Temperament 23, 24 Physical Condition 24, 25 Mental Traits 25 Speed 26 Mental Deficiency 26, 27 Miscellaneous 27 Practice Lesson 1 27-29 2 29,30 3 30,31 4 31 5 31,32 6 32,33 7 33 8 34 9 34 59 60 STANDARD PRACTICE TESTS Practice Lesson 10 Page 34 11 35,36 12, 36 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39: 40. 4.1. 42. 43. 44. 45. 46. 47. 48. 43, 39 40 40 40 40 41 41 41 41 42 42 42 43! 43 43 43 43 43 43 43 4^ 43 44 44' 44 44 4^ Teacher's Records — Daily Record 45-50 Report of Tests 51 Summary 52, 5i I Index Addition Lessons — Lessons 1, 5, 8,12,16,20,23,27,32,36,40 Page 27, 31, 34, 36, 38, 38, 40, 41, 42, 43, 43 Alternative Methods 12 Answers — Lessons 9 Tests , 54-58 Reading of 9 When correct 9 Attention Span 38 (Austrian — Division 35 Subtraction 30 Borrov^ing — subtraction 29 Bridging Tens 31 Carrying — Addition 36 Subtraction 29 ^ Multiplication 32 X Division 33 Cases in Division — I 38 II 40 III 41 IV 41 Combinations — Addition 27 Subtraction 29, 30 Multiplication 30, 31 Division 31 Copying 43 Correction of Lessons — Children 9 Teacher 11 Detailed Instructions 7 Diagnosis of Defects 20 Difficulties 20 61 62 Division — Lesso Pages Efficiency .^ Elimination THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS ^M'^i'f^ ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE F^UR^ SOeRDUE. ^"^ **°° ^^ ^"^ SEVENTH SIy 9, 35, 39, 43 1, 43, 43, 43 23 Forms A ar General De Graphs — Drav Meat Sam] Use Habits of ^^ Add Subl Mul Div; Home Stu Illustratio Dia Div Grs Re( Immaturi Instructic Judgmen In In Measure Memory Mental Mental Multipli -1^ 9 iVJAY 10 g^ ^ Lessons 11 5 5 LD 21-95m-7.'37 ; 28, 34, 38, 42 Pa^e 30, 32, 34, 37, 38, 39, 40, 41, 43, 43, 43 24 Nervousness 24 Periods of Grov^th ^ TEACHER^S MANUAL 63 Physical Condition 24 Practice : 16, 38 Precocious Children 23 Process. — Addition 27 Subtraction 29 Multiplication 30 Division 31 Program 6 5 Records — f Children 18 Teachers 15, 45-50 Eeliability of Results 10 Reports of Tests • 51 Elesearch Tests 43 Retarded Children 23 Sample Records — Graphs 19 wScoring — ' I Children , 10 Speed , 26 vSubtraction — Lesson 2, 9, 17, 24, 33, 37, 41 Page 29, 34, 38, 40, 43, 43, 43 Summary 52, 53 Teacher's Record — Scoring , 14, 15, 45-50 liTeaching — Graphs 19 How to Study 16 ^Temperament 23 Tests A : 6, 7-10, 36 Tests B 6, 12, 42 Tests C ....6, 12, 43 Time Allowances 9 Transfer 22 Types .;.,. ,,... 23 Warnings :.*.;*. . • . V : .• 12 Coi Shi PRACTIC Two series lesson sheet= taining 150 covers the provide grr aSorded by FORM A- packages, i of one less(- set of 48 pi FORM E- packages, < of one lessi_ set of 48 P STUDE^_ A pamph] and graph_ gether wit package o TEACHF A 64-pagf essary ins- material. TEACHh Pages 45- Price, net SPECIE I Studen- sheet in Price, n€_ cabin; cards. L CABIN cards. !_ STUDE Edition., TEACI the prop_ TEACI SPECI- THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. 270ct'55PW Atfii r; l3iDec*58fc . l REC^D LD DEC15t95l> Test IBERS e Edition id Form B - ; containing o. jrm covers th.| to provide 5 afforded by i bf 48 cards. of 48 cards. JD AND daily record ording the 4^ )ns for using pon which thel net 10 cents. lAL itaining all thej le proper use cents. )f Teacher's M I Teacher's id Practice p£ rms A and B. 18 separators, 48 separators, for the Card-Ej {cessary instruct LD 21-100m-7,'39(402s) iUSHEl YONKERS-ON-HUDSUN, iNt-w xr^K CHICAGO ATLANTA DALLAS MANILA Pamphlet Binder Gaylord Bros- Inc. Stockton, Calif. T.M.Reg.U. S.Pat. Off. THE UNIVERSITY OF CAUFORNIA UBRARY I m