Promotion, Retardation, and Elimination
- Author:
Edward L. Tiiorndike, Ph.D.,
Teachers College, Columbia University. II.
Promotion and the Course of Study.
It is desirable that the course of study should be stated in terms of objective achievement grade by grade, so that teachers may know what their pupils are supposed to accomplish.1 It would be desirable also to have this series of stages of achievement correspond to equal time-units for the average, or better, the modal child, i. e. to have each grade in succession represent what would be a year’s or half year’s work for him if all the children stayed to complete the course, or if elimination were random with respect to school ability.2 As things are, it is desirable to have each grade represent a year’s or a half year’s work for the modal child who enters that grade. There is no demonstrable tendency in the city schools as a group to depart from the second standard, except in the first grade. Individual cities, of course, may seem to be acting unwisely in making ostensibly equal grades really very unequal. Before passing judgment on any city, however, its practice over several years must be studied and all the circumstances determining its policy must be considered. The apparent departure of making grade eight too short as compared with grades three to seven, may be entirely due to the greater elimination during Ihe year in grade eight of those who, if they stayed, would fail of promotion. The same fact must be considered in connection with the apparent shortness of the third and fourth years of the high school, although in this case there is perhaps a real error in making the first year of the high school too hard in comparison with the last two.
The first grade is probably longer for those who enter it ‘With allowances, of course, first for individual differences among pupils, the work of a grade not being required to he done by all with exactly the same degree of excellence; and secondly for different courses of study for different types of pupil.
2The reasons for this are economy and convenience. More rapid courses for more gifted pupils might well crowd the work in some semesters and relax it in others with still greater economy. On the other hand, slower courses might at times be more economical.
than later grades are for those who enter them, although of course not nearly so much longer as it seems. The best practical solution may be, not to lessen its work, but to add an easier preparatory grade and admit to the first grade only the pupils who are ready for it?those who can reach the standard of the modal pupil in the normal time.
Promotion and Retardation. Retardation is commonly taken to mem the fact that a pupil is in a lower grade than he would be if he had begun school at the usual age and had progressed one grade each year. This raises certain difficulties in making allowance for systems which give seven or nine grades to the work usually done in eight, and lacks the objectivity and uniformity which would be gained if we could establish certain definite amounts of achievement in terms of knowledge, power, skill, etc., to be expected at each age, and could use “retardation” for the degree of inferiority of a child to the amount of achievement to be expected at his age. But until such standards of school progress are available, and until children are measured by them, wTe may profitably use the customary definition of retardation. Accepting this definition of retardation, our figures suggest two facts not hitherto sufficiently emphasized. There is no support whatever in fact for the doctrine that the retarding force is greater in the early than in the later grades (grade one being left out of the question). Indeed, the same pupil will commonly spend a considerably longer time in grades six, seven, and eight than in grades two, three, and four. Certain pupils are not retarded in grades six, seven, and eight for the sole reason that they are not there to be retarded,?they have been eliminated. If all pupils stayed in school until twenty, and the present standards of promotion were maintained, retardation would be measurably greater in grades six, seven, and eight, than in grades two, three, and four.
In these facts of promotion and failure there is no support whatever for the doctrine that retardation by non-promotion at the end of the year is an injustice to the pupil retarded. As a matter of fact, there is probably far more injustice done to the gifted one-seventh who are not promoted “doubly,” i. e. allowed to complete a grade in less than a year, than is done to the one-seventh who fail of promotion in one year. Systems of promotion need to be fitted to individual differences in capacity, to be made more flexible, rather than to be made easier for those who now fail. It is of course true that teachers may exaggerate the importance of satisfactory achievement in one grade as a prerequisite for success in the following grade, that they may exaggerate the bad effects upon the zeal of a school from treating competent and incompetent pupils alike in promotion, and that they may even be stupidly unjust in a few cases. But with rare exceptions, teachers refuse promotion to a pupil only because they honestly think he is not fit to do the work of the next grade, and that it is for the common good not to let him attempt it; and in a majority of cases they are right. Special industrial and trade schools in which pupils who make slow progress in the typical elementary schools could be given a trial at another sort of education, would be more to the advantage of the eleven year old pupils now found in the third grade, the twelve year olds in the fourth, and the thirteen year olds in the fifth grade, than such a relaxation of standards in the typical school as would allow the less scholarly children to progress in it at the speed now expected of the modal child.
Promotion and Elimination.
The estimates of elimination by grades, reported by the author in “The Elimination of Pupils from School,” have been attacked on the ground that the percentages of “hold-over” or repeating pupils are far larger in the earlier grades than in the later, and that adequate correction was not made for this fact. I have elsewhere shown that I did make this correction, and now shall show that my correction was adequate.
We can estimate the number of pupils who continue to any given grade in two ways. What these are will be clearer, if we take first an arbitrarily simple case and analyze it. Suppose first that for thirty years or so the population of a community is stationary, that no ones dies before twenty-five, that there is no immigration or emigration, that one hundred pupils begin school each year, that every one stays in school until the end of the high school, and that every one begins at the beginning and spends just one year in each grade. Then the number of pupils in each grade “will be one hundred. Suppose that in each grade all of those entering it spend just two years; the number in each grade will be two hundred, or twice as l’irge as the number beginning school in one year. Suppose that in each grade 84 per cent of those entering it stay just one year, and 1G per cent just two years.
After sucli action has- been under way long enough, the numbers in all the grades will still be alike, but each will be one hundred and sixteen, or 1G per cent larger than the number beginning school in any one year.
Suppose now that of those entering each grade 84 per cent stay just one year and IB per cent just two years, and also that in every year of the thirty years one-half of the children in the sixth grade in June leave school. Then we should have as the relative sizes of our grades in the middle of thirty year period 11G, 11G, 116, 116, 116, 116, 58, 58, 58, 58, etc. The proportion which grade seven was of any early grade would represent the proportion of pupils beginning school who continued to the seventh grade, which is, of course, one-half. The proportion which the seventh grade was of the number beginning school in one year (58 per cent) would be an overestimate of the proportion continuing in school to the seventh grade, for the same reason that the process in the sixth grade would give 116 per cent continuing in school. What is required is the proportion which the number beginning the seventh grade in one year is of the number beginning school in one year. This case may be generalized in the form of two laws:?1. Disregarding growth of population, immigration, emigration, and death, if the rate of progress in a grade of those entering it, i. e. the frequency and degree of their retardation or acceleration, is equal for all grades, any decrease of a Inter as compared with earlier grades is due to elimination; and 2. Disregarding as before all factors save retardation and elimination, if in any grade there is an excess of retardation over acceleration, the ratio of those found in that grade at the beginning of one year, if there has been zero elimination, will be over 100 per cent of those beginning school in one year, and by an excess proportionate to the exccss of retardation in that grade.
It is therefore obvious that the percentage of pupils beginning school who are retained to any grade, can not be measured by the percentage which the pupils in that grade are of the number beginning school in one year. If the latter figure is taken as a base, the other figure must be ihose beginning, that grade in one year.
Mr. Ayres is right in supposing that when age enrolments are given, the number of pupils beginning school in one year can be inferred with substantial accuracy from them, but he is obviously wrong in supposing that the ratios of the enrolment of the later grades to the number beginning school in one year measure the retention to those grades.
If retardation is equal in all grades, then, as we have seen, the numbers in the grades at the beginning of the year give us by their differences the elimination (disregarding growth of population, etc.). If it is unequal, we must correct for it. We have shown that it is approximately equal from the second grade to the third year of the high school inclusive, in the sense that in any June the proportion of pupils destined, if they stay in school, to repeat the grade, is for these grades in order .1225, .14, .1475, .16, .1425, .15, .125, .21, .20, .10, .05. But it is likely that those so destined will leave school before the next years enrolment record is taken, more often than will those who did not fail; and it is likely that this excess elimination of those who fail will be greater in the higher grades than in the lower.
This implies the possible need of a second correction, for the excess elimination of non-promoted over promoted pupils, and for the increase in this excess as we pass to later and later years. The data of table V, taken in conjunction with the percentages failing of promotion, allow an impartial though not precise estimate to be made.
TABLE V. DATA SHOWING INVERSE CORRELATION BETWEEN FAILURES AND RAPID PROMOTIONS. Galesburg, III. Per cent of pupils completing grade, who require more than one year to complete it. 9 1 H 2 H 3 Ii 4 H 1898 26.3 13.3 23. (j 17.5 17.4 31.8 26.3 45.0 Kansas City, Mo. Per cent of total enrolment remaining more than 200 days in the grade. 1905 15.9 12.5 16.3 19.8 16.5 17.8 10.0 1907 15.8 13.8 19.8 26.4 28.3 24.7 15.3 Springfield, III. Per cent of June enrolment remaining more than one year in the grade. 1908 24.8 21.1 18.7 27.6 16.5 15.9 15.9 | 14.7 1*2.6 ! 14.2 12.6 11.1 2.9 ! 5.7 2.6 j 1.3 4.8 13.9 6.8 i 12.5 1.1 6.1 0 I 0 WlLLIAMSPORT, Pa. Per cent of Juue enrolment remaining more than 180 days in the grade. 1907 27.1 17.1 1S.1 20.4 12.3 8.9 11.7 9.8 8.9 1908 20.5 11.5 13.6 14.3 14.2 11.4 8.6 5.6 6.1
The figures for Kansas City, Mo., 1007, Springfield, 111., 1907, and Williamsport, Pa., 1908, are taken from Ayres, L. P., Laggards in our Schools, p. 75. In reasoning from these facts we must consider first that Springfield and Williamsport are cities which from grade seven to the last grammar grade have very much lower percentages of non-promotions than do cities in general, and that consequently these bare percentages of repeaters make the retention of pupils who fail in any given year seem lower than it would be in cities in general. What we should use is not the relative frequency in the different grades of pupils repeating the grade, but the relative value in the different grades of the ratio of pupils repeating, to the pupils failing of promotion. These facts are given in table VI.
TxVBLE VI. PER CENT OF PUPILS FAILING OF PROMOTION WHN REMAIN OVER A YEAR IN THE SAME GRADE. Grade Springfield Williamsport 1907 and 1908 1907 and 1908 1 138 73 2 114 73 3 9G 74 4 102 79 5 89 GO 6 G8 63 7 39 70 8 45 77 9 79 1 H 60 2 H 47 3 H 18 4 H 0 These statistics are somewhat mysterious if taken at their face value. In Springfield, the pupils in the early grades seem to repeat a grade although promoted from it; in Williamsport, even in grades one, two, and three, over one-fourth of the pupils failing of promotion seem to be eliminated from school; and in Kansas City, by any rational estimate whatever of June enrolments, there still results the appearance of a large number of pupils in grades two to six (next to the last grammar) inclusive, who spend over a year in a grade without being recorded as having failed of promotion in it!3 Obviously we need to know just how the figures were computed in the superintendents’ offices before we can interpret them.
One fact is certain,?pupils who fail of promotion even as late as the third year of the high school do remain in school and make the grade’s enrolment larger than the number beginning it in one year. The following statements are probably also true. In WilaT!iis is fn lio pxnlflP’ort l>v the system of promotion in vo<rne i” Kansas City. A class may lie advanced from one grade to the next at any time of the year, when the work of the previous grade has been covered.
liamsport the likelihood of a “failure” becoming a “repeater” the following year is as great in the last three grades as in the first three (see table VI). In Springfield one-tenth of the failures in grade five seem to be eliminated before the next year’s enrolment is counted, one-third of those in grade six, nearly three-fifths of those in grades seven and eight, and two-fifths, one-half, and fourfifths, for the first, second, and third years respectively of the high school. Jn Galesburg those who fail in grades six, seven, and eight, must continue in school nearly if not quite as often and as long as those who do not fail; otherwise there could not be so many pupils repeating these grades. Kansas City occupies a position with respect to the elimination of failures in late grades between Springfield and Williamsport.
On the whole I estimate that of pupils failing of promotion in the last grammar grade about one-third are eliminated before the next year’s enrolment is counted; of pupils failing in the next to the last grammar grade, about one-fourth; of pupils in the sixth grade, about one-fifth; and of pupils in the fifth grade about onesixth. If these estimates are fair, the failures in grades six, seven, and eight continue to the following year at least eight-tenths as often as those promoted. At all events, Mr. Ayres is certainly wrong in supposing that only “a few?a very few?pupils get to the seventh or eighth grade, fail of promotion, and repeat the work of the grade.”4 In Galesburg about half of the last grammar grade is made up of such repeaters, and in Kansas City about oneeighth. In Springfield about half of those failing repeat the grade, and in Williamsport four-fifths.
I shall now show that Ayres’ method of estimating elimination by comparing the number beginning school in one year with the number continuing to the later grades, if properly applied, gives substantially the same estimates of elimination as those in “The Elimination of Pupils from School”.
Consider first the following facts:?of the pupils enrolled in June in grades two, three, four, and five, 12.3 per cent, 14 per cent, 14.8 per cent, and 16 per cent respectively fail of promotion; 87.7 per cent, 86 per cent, 85.2 per cent, and 84 per cent respectively are promoted. Call the number of pupils at the beginning of one year in grades two, three, four, and five, Pop. 2, Pop. 3, Pop. 4, and Pop. 5, respectively.
‘Ayres, Leonard P. Laggards in our Schools, p. 03. 262 TI1E PSYCHOLOGICAL CLINIC. Call the number failing in one year, f2, f3, f4, and f5, respectively. Call the number promoted in one year, p2, p3, p4, and pf>,. respectively. Call the number skipping one of these grades in one year, s2, s3, s4, and s5, respectively. Call the number eliminated from school in one year, otherwise than by death, before reaching (he grade, e2, e3, e4, and e.r), respectively. Then, disregarding increase of population, death, and migration into and out of the school system, (1) Pop. 3 = p2 + f3 ?{- s2 ? so ? en (2) rop. 4 = p3 + 1”4 + S3 ? s4 ? e4 (3) Pop. 5 = ] >4 -j- f 5 + s4 ? s5 ? e5 Since s2 = s3 = s4 = s5 approximately, the equations (1), (2), and (3) become (4) Pop. 3 = 112 -)~ fo ? e3 (5) Pop. 4 = p.3 + f4 ? e4 ((>) Pop. 5 = p4 f5 ? e5 Suppose now that Ayres were correct in his supposition there is no elimination in grades two, three, and four (that is, e3, e4, and e5 all equal 0), and in his estimate that Pop. 2 ? 1.29 x the number beginning school in one year Pop. 3 = 1.28 X ” Pop. 4 = 1.20 X ” ” ” ” ” ” Pop. 5 = 1.06 X ” ” ” ” ” ” ” ” Call the number beginning school in one year, A, then 1.28 A should = (p2 + f3?) or F (87.7 X 1.20 A) -f (14 X 1.28 A) l 1.20 A ” = (p3-f f4) or | (SC X 1.28 A) + (14.8 X 1-20 A) | 1.0G A ” = (p4 + fa) or [ (85.2 X 1.20 A) -j- (14.3 X 1.00 A) 1 But they do not. The numbers required to satisfy the equations are 1.31, 1.28, and 1.17. In other words, as will be made clear later, Ayres’ 1.28, 1.20, and 1.06 are too high. Call the numbers beginning these grades in one year, 1>3, b4, and b5. Since s2 = s3 ?- s4 = s5, if there is no elimination (death, increase of population and migration being disregarded) 1)3 = p2, b4 = p3, and b5 = p4. Then if Ayres’ suppositions were correct, b3 should = (87.7 X 1.29 A) or 1.10 A b4 ” =(80 X 1.28 A) or 1.03 A b5 ” = (85.2 X 1.20 A) or .02 A But these results (*1.10 A, 1.03 A, and .92 A) are absurd, for the PROMOTION, RETARD A TI ON, ELIMINATION. 263 numbers of pupils beginning these grades in one year should, if there is no elimination, be each 1.00 A. Ayres’ ratios for Pop. 3, Pop. 4, and Pop. 5 to A of 1.28, 1.20, and 1.06, are therefore too high, and unless e3, e4, and e5 are positive quantities, the ratios are also in wrong proportions to one another. To be correct they must give results for b3/Pop. 3, b4/Pop. 4, and b5/Pop. 5, as .064, .038, and .020, if account is kept of increase of population and death; or if account is kept of these two factors otherwise, as Ayres-’ says is done by liini, they must give results summing to approximately 3.00 and in a progression paralleling .064, .038, and .020. Ilis 1.28, 1.20, and 1.06 are incorrect in all being too large and in decreasing relatively too rapidly. Consider now what results if, first, we divide Ayres’ 1.20, 1.28, 1.20, and 1.06 all by 1.13; that is, assume that lie overestimates the ratios of grade populations to number of pupils beginning school in one year, bv 13 per cent, so that the true ratios are 1.14, 1.13, 1.06, and .04; and if, secondly, we assume also that the figures for e3, e4, and e5 which 1 gave in ‘’The Elimination of Pupils from School” are correct (e3 = .03 A, e4 = .07 A, and d5 = .00 A). We then have Pop. 3 = (87.7 X 1.14 A) + (14 X !-13 A) ?-03 A Pop. 4 = (8G X 1.13 A) + (14.7 X 1.00 A) ?.07 A Pop. 5 = (S5.2 X 1.00 A) -+- (l(i X .04 A) ?.00 A These equations result in 1.13, 1.06, and .06, in other words they are closely approximated by the 1.13, 1.06, and .04 of our hypothesis. If we assume that practically all of e3 and e4 (say ninetenths) failed the previous year, and that of e5 two-thirds failed the previous year, the above equations give for 1)3 -(- those who would count in 1 >3 had they not been eliminated, for 4b -f~ those who would count in b4 had they not been eliminated, and for b5 + those who would count in b5 had they not been eliminated, 1.00 A, .97 A, and .03 A respectively; that is, there is a close approximation to the proper relations of the numbers beginning grades three, four, and five in one year, one to another. The approximation will be still closer if we suppose that smaller percentages of those eliminated had failed the previous year. The fact is that Ayres’ figures for grades three, four, and five can be derived from the known facts of retardation only by supposing the elimination in grades two, three, and four to be almost exactly as I estimated it in “The Elimination of Pupils from Schools”, and by decreasing by 13 per cent his estimate of the ratios sLaggards in our Schools, p. 54. of grade populations to the number of pupils beginning school in one year.
The same principle holds good for the sixth, seventh, and last grammar grades. If we assume the ratios of grade populations to the number of pupils beginning school annually as 13 per cent smaller than his (that is, .80, .63, and .45), assume the elimination grade by grade as stated in my “Elimination of Pupils from School”, and then for any grade add those promoted from a previous grade to those held back in the same grade, and subtract those eliminated, we get .76, .64, and .45, a very close approximation to our .80, .63, and .45. The equations are Pop. G = [(84 x .94 A) -f (14.3 X -SO A) ? -13 A] Pop. 7 = [(85.7 X .SO A) -f (15 X .G3 A) ? .14 A] Pop. last grammar = [ (85 X .63 A) + (12.5 X .45A) ? .14A] It appears therefore that Ay res’ estimate of the number of pupils beginning school in one year is relatively too small (or his estimate of grade populations relatively too large), and that instead of 1.29, 1.28, 1.20, 1.06, .90, .71, and .51, he should use 1.14, 1.13, 1.06, .94, .80, .63, and .45. If now, instead of assuming, as Mr. Ayres does, that all the non-promoted in grades six, seven, and eight are surely eliminated before the next year’s enrolment is counted, we estimate how many of them are on the basis of the facts reported for Galesburg, Kansas City, Mo.; Springfield, and Williamsport, and replace Ayres’ ^ |>n” or per cent which each grade’s enrolment is of the … . b number of pupils beginning school in one year, by ^ or per cent which the number of pupils beginning the grade in one year is of the number of pupils beginning school in one year, we have Pop. 4 = b4 + ys f4 Pop. 5 = b5 + % f5 Pop. G = bG + % f0 Pop. 7 = b7 + % f7 Top. last grammar = b last grammar + % f last grammar which give b4 = (1.06 A ? .14 A) or .91 A b5 = ( .94 A ? .125 A) or .815 A bG = ( .SO A ? .091 A) or .709 A b7 = ( .63 A ? .07 A) or .56 A b last grammar = (.45 ? .038 A) or .412 A
The corresponding figures in “The Elimination of Pupils from School” were .90, .81, .68, .54, and .40. We know that elimination has grown a little less severe in the last seven or eight years. It appears, therefore, that Ay res’ method for estimating elimination, if used without error and as for the year 1900, would give results in full agreement with those stated in “The Elimination of Pupils from School”. It is certain that Ayres’ estimates of elimination are too low, because of his astonishing error in assuming that all of the non-promoted children in the sixth, seventh, and eighth grades leave school before the next year’s enrolment is counted,?unless, of course, he has made some error of equal amount in the opposite direction.
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