SELECTIONS ILLUSTRATING EXCELLENCE IN DRILL AND IN CONCRETENESS
From the system ranking next to the best in drill.
Grade III B
I. Objective.
1. Work.
a. Fractions. Review previous work. Teach new fractions; 7ths, 10ths, and 12ths.
b. Notation, numeration, addition and subtraction of numbers to 1000.
c. Liquid and dry measures.
d. United States money.
e. Weights.
2. Objects and Devices.
a. Counting frame.
b. Splints, disks for fractions, etc.
c. Shelves.
d. Liquid and dry measure.
e. United States money.
f. Scales.
II. Abstract.
1. Work.
a. Counting to 100 by 2’s, 10’s, 3’s, 4’s, 9’s, 11’s, 5’s, beginning with any number under 10; counting backwards by same numbers, beginning with any number under 100.
b. Multiplication tables. Review tables already studied. Teach 7 and 9.
c. Drill in recognizing sum of three numbers at a glance; review combinations already learned; 20 new ones.
2. Devices.
a. Combination cards, large and small.
b. Wheels.
c. Chart for addition and subtraction.
d. Fraction chart.
e. Miscellaneous drill cards.
f. Pack of “three” combination cards.
Prince’s Arithmetic, Book III, Sects. I and II.
Speer’s Elementary Arithmetic, pp. 1-55.
Shelves: See II A.
Combination Cards: large and small. These cards should contain all the facts of multiplication tables 3, 6, 8, 7, and 9. As:—
| 7 × 1 | 2 × 7 | 7 ÷ 1 | 21 ÷ 3 | |
| 1 × 7 | 7 × 3 | 14 ÷ 2 | 21 ÷ 7, | etc. |
| 7 × 2 | 3 × 7 | 14 ÷ 7 |
For use of these cards, see directions in I B.
Wheels for Multiplication and Division:
See directions under II A.
Chart for Adding and Subtracting:
For directions, see II B and II A.
Add and subtract 2’s, 3’s, 4’s, 5’s, 9’s, 10’s, 11’s, 12’s, 15’s, and 20’s.
Fraction Chart shows, ½, ¼, ⅛, ⅓, 1/6, 1/9, 1/12.
Miscellaneous Drill Cards:
For directions, see I A.
“Three” Combination Cards:
For use, see I A.
Grade III A
I. Objective.
1. Work.
a. Fractions previously assigned.
b. Notation, numeration, addition, subtraction, multiplication, and division of numbers to 1000.
c. Long and square measures.
d. Weights.
2. Objects and Devices:
a. Counting frame.
b. Splints, disks for fractions, etc.
c. Shelves.
d. Scales.
II. Abstract.
1. Work.
a. Counting to 100 by any number from 2 to 12, inclusive, beginning with any number under 10; counting by same numbers backward, beginning with any number under 100.
b. Multiplication tables—all tables.
c. Drill in recognizing sum of three numbers at a glance; review combinations already learned; 20 new ones.
2. Devices.
a. Combination cards—large and small.
b. Wheels.
c. Chart for adding and subtracting.
d. Chart for fractions.
e. Miscellaneous drill cards.
f. Pack of “three” combination cards.
Prince’s Arithmetic, Book III, Sects. III to VI, inclusive.
Speer’s Elementary Arithmetic, pp. 56-104.
Shelves: See II a.
Combination Cards: large and small. The cards should contain all the facts of the multiplication tables 11 and 12, also the most difficult combinations from the other multiplication tables. As:—
| 12 × 1 | 12 ÷ 1 | 24 ÷ 2 |
| 1 × 12 | 12 ÷ 12 | 24 ÷ 12, etc. |
| 12 × 2 | 12 ÷ 2 | |
| 2 × 12 | 12 ÷ 3 |
For use of cards, see directions in I B.
Wheels for Multiplication and Division:
See directions under II A.
Chart for Adding and Subtracting:
For directions, see II B and II A.
Add and subtract 6’s, 7’s, 8’s, 13’s, 14’s, 16’s, 17’s, 18’s, and 19’s.
Review other numbers under 20.
Chart for Fractions shows all fractions already assigned.
Miscellaneous Drill Cards:
For directions, see I A.
From the system ranking best in concreteness.
Mathematics: If the children are actually doing work which has social value, they must gain accurate knowledge of the activities in which they are engaged. They will keep a record of all expenses for materials used in the school, and will do simple bookkeeping in connection with the store which has charge of this material. In cooking, weights and measures will be learned. The children will also keep accounts of the cost of ingredients. Proportions will be worked out in the cooking recipes. When the children dramatize the life of the trader, in connection with history, they have opportunity to use all standards of measurements. Number is demanded in almost all experimental science work; for instance, the amount of water contained in the different kinds of fruit, or the amount of water evaporated from fruits under different conditions (in drying fruits). All plans for wood work will be worked to a scale and demand use of fractions. When the children have encountered many problems which they must solve in order to proceed with their work, they are ready to be drilled on the processes involved until they gain facility in the use of these. The children should be able to think through the problems which arise in their daily work, and have automatic use of easy numbers, addition, subtraction, multiplication, short division, and easy fractions.
As one reads these two samples of excellence he must find that each is so excellent in its one strong feature that it is not good; that work according to either must suffer; that what each needs is what the other has. Such a synthesis is represented in the next illustration.
A Combination of Excellences
September. 1. Measure height, determine weight. From records determine growth since September, 1905. 2. Learn to read thermometer. Make accurately, scale one fourth inch representing two degrees on paper one inch broad. Find average temperature of different days of month. Practice making figures from 1 to 100 for the thermometer scale. Count 100 by 2’s. 3. Make temperature chart. 4. Measure and space calendar, making figures of size appropriate to inch squares. Learn names of numbers to 30. 5. Make inch-wide tape measure for use in nature study, number book and cubic-inch seed boxes. 6. Review telling time. A. In addition to above; analyze numbers from 11 to 40 into tens and ones. Walsh’s Primary Arithmetic to top of page 10.
October. Problems on calendar,—number of clear, of cloudy, and of rainy days in September. Compare with September, 1905, 1904, 1903, 1902; temperature chart and thermometer; height and weight. Lay off beds for tree seeds; plant the same. Make envelopes for report cards. Drill on combinations in the above. Make rod strings and hundred-foot strings for determining distance wing seeds are carried from plants. Practice making figures from 1 to 100 for thermometer scale. Develop table of tens. A. In addition to the above analyze numbers from 40 to 50 into tens and ones. Primary Arithmetic, pp. 10-22. Teach pupils to add at sight.
November. From wall calendar count number of clear days, of cloudy days, and rainy days in October. Compare with September; with October of 1905, of 1906. Find average daily temperature; 8.30 A.M., 1 P.M. What kind of trees grow fastest? Measure growth of twigs of different kinds of trees. Compare this year’s growth with that of last year and of year before last. Compare rate of growth of different kinds of trees, as oak, willow, Carolina poplar, and elm. Develop table of 5’s from lesson with clock dial; review 2’s and 10’s. Practice making figures from 1 to 100 for the thermometer scale. Learn words representing numbers as well as figures. Make seed envelope. A. Analyze numbers from 60 to 65 into tens and ones. Primary Arithmetic. B, pp. 17-26; A, pp. 39-49.
Last six weeks of first term.—Continue finding average daily temperature. From wall calendar count number of clear, of cloudy, and of rainy days in November. Compare with November, 1906, 1905. Continue measurements on growth of trees. Drill on telling time from clock dial. Practice making figures from 1 to 100 for thermometer scale. Continue learning words representing numbers. Review tables of 2’s, 5’s, 10’s; learn table of 3’s. Primary Arithmetic. B, pp. 27-40. Analyze numbers from 11 to 30 into tens and ones. Primary Arithmetic. A, pp. 49-61. Analyze numbers from 66 to 100 into tens and ones. In January review all facts in number book. Drill on tables.
(Only the first one half of the third year’s course shown.)
The system from which this last selection is taken had the following remarkable rankings: 3d best in general excellence, 2d best in concreteness, and 5th best in drill. And as measured by the tests of this study, this system stood 4th from the best in abilities, and spent a little less than the medium amount of time.
CHAPTER XIX
MEASURING RESULTS IN EDUCATION
Efficiency in any line of human endeavor depends upon our ability to evaluate the results which are secured. No one would question the progress which has been made in education during the past hundred years; but one may very justly inquire concerning the efficiency of the work that has been done from the standpoint of the money which has been spent, and the effort and devotion of those who have engaged in teaching. In the mercantile pursuits it has been noted that seven out of every ten failures can be charged directly to a lack of knowledge of facts. Such investigations as we have had in education tend to prove that a like situation is to be found in this field. The failures in education, whether due to a lack of economical use of the funds available, to an inefficient system of organization, or to unintelligent practices in method, are, on the whole, not to be charged to a lack of devotion on the part of those who have given their lives to the schools. Until it is possible to measure the results achieved, the facts of success or failure cannot be established.
Of course, no one would deny that real progress is made by the process of trial and success, both in the art of teaching and in the practice of administration. It is true, too, that we shall have to depend in considerable measure upon demonstration as a means of bringing about improvement in current educational practice. It is none the less true, however, that scientific work in education will furnish the basis for the more rapid elimination of the mistakes in current practice, as well as point the way for improved organization of teaching. The science of education will, in its development, occupy relatively the same position with reference to the art of teaching that the science of medicine occupies with respect to the art of healing. The progress which has been made during the past twenty-five years in the art of farming would never have been possible without the scientific work that has been done in agriculture.
Aside from the fact that we are only beginning to have a profession of education, many other factors have entered to delay the progress in the direction of standardizing our work by means of accurate measurement of the results achieved. One of the most comforting of the fallacies which are at times urged against the attempt to measure results is found in the popular statement that the only criterion by which the success of school work can be measured is found in the ultimate success of the individuals who are subjected to the process. The most inefficient teacher in the most poorly equipped school, if his period of service has been long enough, will point to the success of a few of the boys who once attended that particular school, as proof of the adequacy of the work which is now being done. The failures are never brought to mind. The fallacious reasoning found in such an appeal is all too common in our educational discussion. To take a selected group of individuals, who have, because of native ability, and possibly because of favorable environment, achieved distinction; and to claim that this success is due to our system of education, may be satisfying to our pride, but cannot appeal to our good judgment. The only available measure of the success of the work done in any particular school is to be found in the changes which are brought about in boys and girls, young men and young women, during the period of their school life.
It has been argued, too, that that which is most worth while in education cannot be measured. Those who advance this argument speak continually in terms of “atmosphere,” “spirit,” and the like. There are two replies to be made to this contention. The one is that any power which the teacher has, whether it is called influence, or ability to teach arithmetic, must result in some change in the children who are taught. Another equally valid answer is to be found in the fact that the best teachers of arithmetic, of literature, of geography, of history, and the other studies are, at the same time, the teachers whose influence we value most in the school.
We have been hopeful that the sciences of biology, psychology, sociology, and economics would, in their development, solve the problems of education. No one would deny the significance of the work done in these fields as fundamental to the development of scientific work in education. No one is fully equipped to undertake investigation in the field of education without preliminary training in these fundamental sciences. Progress in the science of education has come, however, through the efforts of those men of sound fundamental training who have attacked the problems of education as such, rather than through the work of the biologist, psychologist, sociologist, or economist. If we should wait for the sciences mentioned to solve our problems, progress would indeed be slow.
Those who are unacquainted with modern statistical methods as applied in the social sciences have at times felt that it was impossible to measure large groups of individuals who differ in ability, in interest, and in environment. It is impossible within the limits of a brief chapter to make clear the validity of such measurement. It may be confidently asserted, however, that the measurement of a large group of individuals is, on the whole, more satisfactory than the attempt to measure a single individual. We can be more sure of the accuracy of our results in comparing two groups of children of a thousand each, than we could in the attempt to measure accurately a single individual with regard to ability in school subjects.
A most persistent objection to the measuring of results comes from those who feel that it is not fair to compare individuals or groups who are not alike in all particulars. They would claim, for example, that we cannot compare children in spelling ability when one group comes from homes in which the English language is spoken, while the other comes from the homes of those who speak a foreign language. It is probable that this objection is due to a belief that measurement will result in a comparison of the present situation without any regard to the growth or development which has characterized the group. If we derive units of measurement in spelling, manifestly the attempt would be to measure the changes which have been brought about in any group in terms of units which are comparable. If group one shows ability ten, having advanced during the year from ability seven, it will be considered just as satisfactory as the advance which has been made in group two, which has moved from ability eight to ability eleven. In other words, the purpose of measurement is never to attempt to make all alike. It is rather to discover differences and the reason for their existence; but most of all to give us some adequate means of determining progress or change.
Let us suppose again, in a matter of business administration, that one school shows a much higher per capita cost than another. This does not prove that one school is more efficiently managed than another. What it does do is to suggest that some adequate reason is to be found for the difference which exists. In like manner, one city may show a much higher cost for janitors’ salaries than does another. This may suggest investigation, but it does not prove that the city with the higher cost for janitors’ service is inefficiently managed or extravagant in its expenditures. It may be that the city that spends a relatively large amount for janitorial service actually gets more per dollar for the money which it spends than does the city with the smaller cost. It is always a purpose of measurement to discover discrepancies and to raise problems.
It has been contended that it is not important to derive scales or units of measurement on the ground that the scientific study of education is significant only in so far as it has to do with a careful investigation of the processes involved in growth. Those who make this contention seem to feel that a careful study of the way in which children come to form habits, to grow in power of reasoning, or in ability to appreciate, will give us the final answer concerning the methods to be employed in teaching. The difficulty with this point of view is that human beings, even though they be trained in investigation, are fallible. The only final test of the success of any method, however carefully derived, and however much of it may depend upon a knowledge of the processes involved in growth in the particular aspect of mental life which is involved, is to be found in the result achieved. Theoretically a method may seem to be perfect, and yet in terms of the results which are secured it may prove to be a failure. If the results are not accurately measured, if we do not derive scales of measurement, we can never be certain of our conclusions with regard to the method to be employed in bringing about any particular type of mental growth or development.
Possibly the one element in the situation which has operated to retard development in the direction of accurate measurement of results, more than any other, is the tendency in education to appeal to authority, and the corresponding lack of devotion to scientific investigation. It is, of course, much easier to solve the problems which one meets by taking the opinion of those who have had experience in the field. No one would deny the value of the judgment of our great educational leaders. The fact remains, however, that these same leaders would be the last ones to place their own opinion in opposition to the results obtained from a careful scientific investigation. Indeed, it is in no small measure due to the insistence of these leaders that we are coming to have adequate investigations with regard to our educational practice.
It has seemed necessary to discuss at some length the objections which have been made against the attempt to measure results in education, rather than to devote more space to a discussion of the work which has already been done. All students of education are familiar with the early work of Rice, and with the later contributions of Thorndike, Ayres, Cornman, and many others who have contributed to the literature of educational investigation. Possibly the most significant piece of work that has been done is Thorndike’s scale for measuring handwriting.[32] We may reasonably hope to have scales derived for the measurement of abilities in other subjects.
In administration, considerable work has already been done with reference to the cost of education, both as regards the relationship of expenditure for education to other expenditures, the question of a proper distribution of money within the educational budget, and of the proper distribution of state school funds.[33] We can, of course, hope for much more significant work in this field as more adequate systems of accounting are introduced and more satisfactory reports are issued. It is noteworthy that in those school systems in which an attempt has been made to check up expenditures carefully, remarkable savings have been made. We have not yet reached the limit of possible reduction of expenditure without the sacrifice of our present efficiency. Much work has been done on problems of school organization, yet the problems of retardation and elimination will be satisfactorily treated only as we secure more accurate records concerning attendance, scholarship, health, promotions, and demotions, such as are provided for by the genetic records now kept in some of our more progressive school systems. The problems of departmental work and individual instruction can never be satisfactorily solved until we measure accurately the results secured under different systems of organization.
Implicit in all of the argument which has been advanced in favor of measuring results is the contention that education means change. If changes are brought about in the children who are being educated, then there must be the possibility of measurement. These changes may take place in habit, in knowledge, in methods of work, in interests and ideals, and in power of appreciation. Probably no one would question the possibility of measuring the change which takes place in the formation of habits. We have quite commonly been willing to measure growth in knowledge by tests which demand that students not only remember facts, but that they show some ability to apply them. Whether or not a student commands a particular method of work can be determined by observation of his method of procedure as well as by the results that he secures. If interests or ideals are changed, there necessarily follows a change in activity. Any real power of appreciation will be accompanied by some change in expression.
The fact that we do not yet have scales or units of measurement which will enable us to evaluate accurately the results obtained in all of the different forms of school activity is not an argument against the possibility of measurement. In any field the development of units of measurement is dependent upon careful investigation, and upon a realization of the imperfection of the units already used. It is only as we insist upon measurement that we can hope to have our units refined. Take for example the problem of grades or marks which are commonly assigned to students as a measure of their efficiency in doing school work. Any investigation of these units will show that there is very great variation in their application by different members of the teaching corps.
The way to bring about a remedy is not to abolish all marks or grades, but rather to study the problem of the proper distribution of marks, and, if necessary, to weigh differently the marks of different instructors. The more imperfect the unit of measure which we now apply, the greater the need for insisting upon accurate measurement.
The first step in the development of scientific inquiry in any field is found in accurate description of the phenomena involved. The demand that we measure results in education is simply a demand that the basis for scientific investigation be made available by means of this accurate description of the situation as it exists. Some investigators in education have already been able to take the further steps in scientific inquiry which have enabled them to foretell with considerable accuracy the results which might be expected in education under given conditions. Further progress is, however, dependent upon that sort of measurement which will discover problems which are not now clearly defined or which have not yet been thought of. Of course, as inferences are made in the light of the problems suggested, there will be still further necessity for accurate measurement. When those who are charged with the responsibility of determining educational policy appeal to fact rather than to opinion, when we are able to evaluate accurately that which we achieve, educational progress will be assured and a profession of education will have been established.
We shall always have most excellent work in teaching done by those for whom scientific investigation, as such, means little. The investigations made will, however, modify the practice of these same teachers through changed demands and because of the demonstration of the validity of the new method of work by those who can appreciate the significance of results achieved by investigators. It is certainly to be expected that open-minded teachers will experiment for themselves and will aid in the work of the expert who must use the schools as his laboratory. Much depends upon the coöperation and good will of all who are engaged in teaching. It is not too much to expect that the spirit of investigation will be found in large measure to characterize those whose privilege it is to provide the situation in which intellectual development is stimulated.
An example of a study involving the coöperation of the pupils, teachers, and supervisory officers of twenty-six separate schools or school systems is Dr. C. W. Stone’s study on Arithmetical Abilities and Some of the Factors Determining them. The following quotations from Dr. Stone’s study indicate the type of result which we may hope to get from such investigations.
In Reasoning
The scores for the reasoning problems were determined from the results of two preliminary tests—one, giving one hundred 6th grade pupils all the time they needed to do the problems as well as they could in the order as printed (see page 11); and another, giving one hundred 6th grade pupils all the time they needed to do the problems as well as they could in the reverse order from that as printed. The results as tabulated below in table II show that scores for reasoning problems of Grade VI pupils can be very definitely arranged in a scale on the basis of relative difficulty. Just what the scale should be can only be determined by determining the form of distribution and the location of the zero point. From what is known of these the scale of weighting shown in the last column of table II is believed to be the best, and this is the one employed in the computations of this study. However, in order to enable the reader to satisfy himself as to which is the best method, the scores of the twenty-six systems were calculated on each of three other bases—(1) counting each problem reasoned correctly a score of 1; (2) counting each problem reasoned correctly a score based on the ratio of its difficulty as shown in the next to the last column of table II; and (3) counting the scores made on only the first six problems for which presumably all pupils of all systems had ample time. See Appendix, p. 98.
In both reasoning and fundamentals the scores used as a measure of the achievement of a system were computed by combining the scores of one hundred pupils. Where more than one hundred pupils were tested, the papers used were drawn at random, the number drawn from each class being determined by the ratio of its number to the total number tested in the system. Where less than one hundred pupils were tested, the combined scores made were raised to the basis of one hundred pupils.
TABLE II
PRELIMINARY TESTS
Reasoning—Unlimited Time
100 Different Pupils Tested Each Time
| Number of Problems | % Reasoned Correctly as Printed | % Reasoned Correctly as Reversed | Average % Reasoned Correctly | Weight According to Average % Correct | Weight Used as Probably the Best |
| 1 | 95 | 92.6 | 93.8 | 1 | 1 |
| 2 | 86 | 82.2 | 84.1 | 1.1 | 1 |
| 3 | 94 | 89 | 91.5 | 1 | 1 |
| 4 | 80 | 83 | 81.5 | 1.5 | 1 |
| 5 | 88 | 86 | 87 | 1.1 | 1 |
| 6 | 69 | 57.4 | 63.2 | 1.5 | 1.4 |
| 7 | 70 | 80 | 75 | 1.25 | 1.2 |
| 8 | 29 | 44 | 36.5 | 2.6 | 1.6 |
| 9 | 19 | 15.5 | 17.2 | 5.45 | 2 |
| 10 | 24 | 27.4 | 25.7 | 3.6 | 2 |
| 11 | 17 | 7.5 | 12.3 | 7.6 | 2 |
| 12 | 7 | 16.4 | 11.7 | 8 | 2 |
Precautions observed to make the Scoring Accurate
The simplicity of the tests made the scoring comparatively easy; and with the observance of the following precautions it is believed that a high degree of accuracy was attained. (1) In so far as practicable, all the papers were scored by a single judge—only two persons being employed on any phase of the work for the entire twenty-six systems; (2) each problem was scored through one hundred or more papers, then the next followed through, etc.; (3) the score for each part of each problem, the errors, etc., were entered on a blank provided with a separate column for each item; (4) where there was doubt as to how the score should be made, the scorer made a written memorandum of how the case was finally decided and this memorandum served as the guide for all future similar cases.
What the Scores Measure
As used in this study the words achievements, products, abilities, except where otherwise qualified, must necessarily refer to the results of the particular tests employed in this investigation. That some systems may achieve other and possibly quite as worth-while results from their arithmetic work is not denied; but what is denied is that any system can safely fail to attain good results in the work covered by these particular tests. Whatever else the arithmetic work may produce, it seems safe to say that by the end of the sixth school year, it should result in at least good ability in the four fundamental operations and the simple, everyday kind of reasoning called for in these problems. It does not then seem unreasonable, in view of the precautions previously enumerated, to claim that the scores made by the respective systems afford a reliable measure of the products of their respective procedures in arithmetic.
The Data
The source of the data used to help answer the above questions is some six thousand test papers gathered from twenty-six representative school systems. Copies of the tests may be found in Part I, pages 10 and 11; as may also a statement of conditions under which the tests were personally given by the author, page 13; and the method of scoring, pages 15 to 18.
The achievements are considered from two standpoints—(1) the scores and mistakes of the systems as systems, (2) the scores of individual pupils as individuals.
Table III gives the scores made in reasoning by each of the twenty-six systems, counting all the problems that were solved, and weighting them according to the last column of table II. The Roman numerals used in the left-hand column to designate the systems are those that fell to each system by lot. As seen by the column headed scores made, the systems are arranged according to number of scores, i.e. system XXIII made three hundred fifty-six points, the lowest score, and is placed first in the table; system XXIV made four hundred twenty-nine points, and is placed second, etc. System V, having made the highest score, is placed last in the table.
ACHIEVEMENTS OF THE SYSTEMS AS SYSTEMS
Measured by Scores Made
TABLE III[34] TABLE IV
| Scores of the Twenty-six Systems in Reasoning with Deviations from the Median. Scores from all Problems | Scores of the Twenty-six Systems in Fundamentals with Deviations from the Median. Scores from all Problems | ||||||
| M[35] = 551 | M = 3111 | ||||||
| Systems in Order of Achieve- ment | Scores Made[36] | Devia- tions from the Median | Devia- tions in Per Cent of the Median | Systems in Order of Achieve- ment | Scores Made | Devia- tions from the Median | Devia- tions in Per Cent of the Median |
| XXIII | 356 | -195 | -35 | XXIII | 1841 | -1270 | -41 |
| XXIV | 429 | -122 | -22 | XXV | 2167 | -944 | -30 |
| XVII | 444 | -107 | -19 | XX | 2168 | -943 | -30 |
| IV | 464 | -87 | -16 | XXII | 2311 | -800 | -26 |
| XXV | 464 | -87 | -16 | VIII | 2747 | -364 | -12 |
| XXII | 468 | -83 | -15 | X | 2749 | -362 | -12 |
| XVI | 469 | -82 | -15 | XV | 2779 | -332 | -11 |
| XX | 491 | -60 | -11 | III | 2845 | -266 | -8 |
| XVIII | 509 | -42 | -8 | I | 2935 | -176 | -6 |
| XV | 532 | -19 | -3 | XXI | 2951 | -160 | -5 |
| III | 533 | -18 | -3 | II | 2958 | -153 | -5 |
| VIII | 538 | -13 | -2 | XVII | 3042 | -69 | -2 |
| VI | 550 | -1 | -2 | XIII | 3049 | -62 | -2 |
| I | 552 | 1 | 2 | VI | 3173 | 62 | 2 |
| X | 601 | 50 | 9 | XI | 3261 | 150 | 5 |
| II | 615 | 64 | 12 | IX | 3404 | 293 | 9 |
| XXI | 627 | 76 | 14 | XII | 3410 | 299 | 10 |
| XIII | 636 | 85 | 15 | XXIV | 3513 | 402 | 13 |
| XIV | 661 | 110 | 19 | XIV | 3561 | 450 | 14 |
| IX | 691 | 140 | 20 | IV | 3563 | 452 | 14 |
| VII | 734 | 183 | 33 | V | 3569 | 458 | 15 |
| XII | 736 | 185 | 34 | XXVI | 3682 | 571 | 18 |
| XI | 759 | 208 | 38 | XVI | 3707 | 596 | 19 |
| XXVI | 791 | 240 | 44 | XVIII | 3758 | 647 | 21 |
| XIX | 848 | 297 | 54 | VII | 3782 | 671 | 22 |
| V | 914 | 363 | 66 | XIX | 4099 | 988 | 31 |
The middle column gives the deviations from the median, which is that measure above and below which one half the cases lie. In this table the median is five hundred fifty-one. These deviations serve to show the differences in scores made; and they are also employed in computing the measures of variability and relationship. The third column is the deviations in per cent of the median. It affords another expression of the difference in size of scores made by the systems.
Table IV reads exactly as III, the scores[37] being those made on all problems of the test in fundamentals. These two tables give some general help on the nature of the product of the first six years of arithmetic work. One very evident fact is the lack of uniformity among systems; another is the lack of correspondence of relative position among the systems in the two tables. With the exception of systems XXIII and XIV, no system occupies the same relative position in the two tables, e.g. system XXIV stands second from the lowest in reasoning and eighteenth from the lowest in fundamentals. This fact is more accurately summarized in the coefficients of correlation, table XV, p. 37.
As seen from its heading, table XXII gives the systems in order of achievements. These serial standings are derived from tables III and IV. Reading from the top, system XXIII has an average serial standing of one, being lowest in both reasoning and fundamentals; system XXV ranks three in average serial standing, being fourth from lowest in reasoning and second from lowest in fundamentals; the readings for the other systems are similar.
Column Heading Keys:
Sys = Systems
A = Average serial standing
B = Serial standing in reasoning
C = Serial standing in fundamentals
D = Serial standing in time expenditure
E = Week minutes devoted to arithmetic
F = Week minutes devoted to all subjects
G = % of time to arithmetic
TABLE XXII TABLE XXIII TABLE XXIV
.. = no time assigned.
Tables XXIII and XXIV keep the same order of systems and show the time expenditure. The first line of table XXIII reads,—system XXIII ranks fourteenth from the lowest in time expenditure, with 1150 week minutes devoted to arithmetic, 9675 week minutes devoted to all subjects, the 1150 week minutes devoted to arithmetic being 12 per cent of the 9675 week minutes devoted to all subjects. Similarly for the other systems, e.g. system XXV with a serial standing in abilities of three, and a serial standing in time expenditure of two, spends 722 week minutes on arithmetic, and 8700 week minutes on all subjects, arithmetic costing 8 per cent of all the school time. The reader will recognize that the third column, which gives the time devoted to all subjects for one week of each of the first six years, gives the only new data of this table, column two being the same as given in table XXI and the first and fourth columns being derived from the others.
Probably the first essential shown by this table is the lack of correspondence between the serial standing in time cost and the serial standing in abilities; e.g. the system with the lowest time cost is found by referring to table XIII to be system XXII, which is seen in table XXII to rank four and one-half in average abilities. Similarly, the system that ranks fifteenth in time cost, ranks fifth in abilities, etc. Another noticeable showing is the wide variability in the school time of the systems. It will be seen to vary from 7200 to 9900 week minutes. This time includes recesses, and it means that lengths of school days vary from an average of four hours to five and one-half hours. And if the names of the systems were given, it would be recognized that almost invariably the longer school hours are accompanied by the least amount of variation in program, such as physical education, field trips, assemblies, etc. Perhaps the other most striking fact of this table is the wide variation in the per cent of time devoted to arithmetic. It varies from 22 per cent for system IV to 7 per cent for system XXII, a difference of more than three to one.
As table XXIV is part of the discussion of factors in time expenditure, its sample readings are given under that heading, page 62.
The Relation of Time Expenditure to Abilities Produced
The reader found one indication of the relation, or lack of relation, between time cost and products in tables XXII and XXIII. Each of the three following tables expresses these same facts.
TABLE XXV
Comparison of the Achievements of the Systems having Less than
Median Time Cost with those having More
| Combined Scores of the Thirteen Systems | ||||
| With less than median time cost | With more than median time cost | With less than median time cost | With more than median time cost | |
| Including home study | Without home study | |||
| Reasoning | 7,519 | 7,893 | 7,277 | 8,135 |
| Fundamentals | 40,751 | 40,273 | 37,165 | 43,859 |
TABLE XXVI
Ratio of Time Expenditures to Abilities[38]
| Systems | Average Ratios | Reasoning Ratios | Fundamental Ratios | |||
| Serial Standing of Systems | Time Cost to Reasoning and to Fundamentals | Serial Standing of Systems | Time Cost to Reasoning | Serial Standing of Systems | Time Cost to Fundamentals | |
| IV | 1 | 2.26 | 1 | 3.99 | 4 | .520 |
| XXIII | 2 | 1.92 | 2 | 3.22 | 1 | .624 |
| XVII | 3 | 1.65 | 3 | 2.88 | 7 | .421 |
| XIII | 4 | 1.54 | 4 | 2.55 | 3 | .533 |
| XX | 5 | 1.45 | 7 | 2.36 | 2 | .535 |
| XVIII | 6 | 1.41 | 5 | 2.48 | 13 | .336 |
| XIV | 7 | 1.40 | 7 | 2.36 | 6 | .438 |
| VIII | 8 | 1.39 | 8 | 2.33 | 5 | .457 |
| IX | 9 | 1.353 | 9 | 2.25 | 5 | .457 |
| XVI | 10 | 1.352 | 6 | 2.40 | 18 | .304 |
| XV | 11 | 1.31 | 11 | 2.20 | 8 | .422 |
| VII | 12 | 1.28 | 12 | 2.14 | 9 | .415 |
| XXIV | 13 | 1.24 | 10 | 2.21 | 21 | .270 |
| II | 14 | 1.22 | 14 | 2.02 | 7 | .421 |
| VI | 15 | 1.20 | 13 | 2.04 | 11 | .354 |
| I | 16 | 1.15 | 15 | 1.93 | 10 | .363 |
| III | 17 | 1.05 | 16 | 1.77 | 16 | .331 |
| XII | 18 | 0.943 | 17 | 1.55 | 13 | .336 |
| XXV | 19 | 0.941 | 17 | 1.55 | 15 | .333 |
| X | 20 | 0.93 | 18 | 1.53 | 14 | .335 |
| XI | 21 | 0.913 | 20 | 1.48 | 12 | .346 |
| XIX | 22 | 0.91 | 19 | 1.50 | 17 | .311 |
| XXI | 23 | 0.83 | 21 | 1.37 | 19 | .293 |
| V | 24 | 0.67 | 23 | 1.06 | 20 | .272 |
| XXII | 25 | 0.65 | 22 | 1.08 | 23 | .219 |
| XXVI | 26 | 0.64 | 24 | 1.05 | 22 | .227 |
The above details are compiled from the scores of individual systems as given in tables III and IV, the median cost being that given in table XXI. As measured by the time used in school, the thirteen systems with less than the median cost stand slightly the better; and as measured by the time including home study, the thirteen systems with more than the median time cost stand somewhat the better. The time used in school is doubtless the more exact measure, but, as shown in table XXI, some systems depend on home study to a considerable extent. Hence both measures are used. The lack of relation indicated in this general way is shown more accurately in the table, page 263, in terms of coefficients of correlation.
The order of systems in this table is determined by the first column, which gives the average serial standing as determined by the ratios of time to products. The right-hand column under each heading gives the ratio of time expenditure to abilities produced, and the left-hand column gives the serial order of that system as measured by the highness of the ratio, i.e. highness of cost per unit of product; e.g. in system IV the ratio of time to reasoning is 3.99 (see fourth column), the highest ratio in reasoning (determined by dividing the time cost, 1854 week minutes, by 464, the points made in reasoning). The ratio of time to fundamentals in this system is .52; giving an average ratio of 2.26. That is to say, the ratio of time to abilities in system IV is as 2.26 to 1, the highest among the twenty-six systems.
That there is no direct ratio between time expenditure and abilities is again shown by this table. For example, system XXII, which spends the least amount of time (see table XXI), ranks fourth from the lowest in abilities (see table XXII), ranks 25th, that is, next to the highest, in ratio of time cost to abilities produced; and, what is even more striking, system XXVI, which spends third from the least amount of time, ranks third from the highest in abilities and 26th or highest in the ratio of time cost to abilities produced.
That a large amount of time expended is no guarantee of a high standard of abilities may again be convincingly seen by comparing the ratios of the five systems spending the smallest amount of time with the five spending the largest. Of the five spending the least time, the average ratio is .80, which corresponds with the 23d or the 3d from the best in ratio; and of the five spending the greatest amount of time, the average ratio is 1.57, which corresponds with the 4th poorest in ratio.
The last three tables have each shown the decided lack of relationship between time cost and abilities produced, and hence for these systems it is evident that there is practically no relation between time expenditure and arithmetical abilities; and, in view of the representative nature of these twenty-six systems, it is probable that this lack of relationship is the rule the country over.
This is not to say that a certain amount of time is not essential to the production of arithmetical abilities; nor that, given the same other factors, operating equally well, the product will not increase somewhat with an increased time expenditure. What is claimed is that, as present practice goes, a large amount of time spent on arithmetic is no guarantee of a high degree of efficiency. If one were to choose at random among the schools with more than the median time given to arithmetic, the chances are about equal that he would get a school with an inferior product; and conversely, if one were to choose among the schools with less than the median time cost, the chances are about equal that he would get a school with a superior product in arithmetic.[39]
So far, then, as ability in arithmetic means ability to handle such foundation work as is measured by the tests in this study, this “essential” has not necessarily suffered by the introduction of other subjects and the consequent reduction of its time allotment.
One would need to read the whole study to appreciate fully the nature of the investigation. From the pages quoted, however, it must be apparent that: (1) schools and school systems vary greatly in the results which they secure in arithmetic; (2) the excellence of the work done is not directly proportional to the time expended. We will find it necessary to revise our opinions with regard to the organization of school subjects, the allotment of time, the methods of teaching, and the like, in proportion as we have careful investigation in these several fields.
For Collateral Reading
E. L. Thorndike, The Principles of Teaching, Chapter XVI.