Coliittitiiii llnxtitr^itp 
 
Class _kj3 
 Book 
 
 
 66 
 
 Gop3Ti^litN". 
 
 COPYRIGHT DEPOSnv 
 
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A Comparative Study of Achievement 
 in Country and Town Schools 
 
 By Norman Frost, Ph.D. 
 
 TEACHERS COLLEGE, COLUMBIA UNIVERSITY 
 CONTRIBUTIONS TO EDUCATION, NO. Ill 
 
 Published by 
 
 "Vmthtrs (Snllsg?, dnlutnbta Ulmtisrattg 
 
 New York City 
 
 1921 
 
 "f-'onoi 
 
 ■'•aofr 
 
Copyright, 1921, by Norman Frost 
 
 i^^y;;; 
 
 (^ilG i5 1921 m'quiddy printing CO. 
 
 NASHVILLE 
 
 g)ClA6305<J7 
 
ACKNOWLEDGMENTS 
 
 THIS study has been made possible by the help of a number of peo- 
 ple. Thanks are extended to the school officials who have fur- 
 nished data or permitted tests to be given in the schools in their charge. 
 Indebtedness is acknowledged to the many members of the faculty of 
 Teachers College, Columbia University, who have assisted by con- 
 structive criticism. Special thanks are due Dr. N. L. Engelhardt, Dr. 
 G. D. Strayer and Dr. E. L. Thorndike for advice, suggestion and 
 encouragement. 
 
 N. F. 
 
CONTENTS 
 
 I. Previous Objective Studies of Achievement in Country 
 
 Schools 7 
 
 II, Purpose and Scope of the Present Study 15 
 
 III. Achievement Measured by the Trabue Language Scales . 18 
 
 IV. Achievement Measured by the Courtis Standard Research 
 
 Tests in Arithmetic Series B 33 
 
 V. Achievement Measured by the Thomdike Silent Reading 
 
 Scale Alpha 2 56 
 
 VI. Summary and Conclusions 61 
 
 Appendix 68 
 
A COMPARATIVE STUDY OF ACHIEVEMENT 
 IN COUNTRY AND TOWN SCHOOLS 
 
 CHAPTER I 
 
 PREVIOUS OBJECTIVE STUDIES OF ACHIEVEMENT IN 
 COUNTRY SCHOOLS 
 
 An accurate comparison of school systems in objective terms is 
 necessary for the proper solution of problems in regard to the organi- 
 zation and administration of schools. Such a comparison is made 
 possible by the use of standardized tests in the systems to be compared, 
 and has been made with great profit in many city school systems. To 
 a much more limited extent the same kind of comparisons have been 
 made among rural schools. 
 
 The Ohio Survey Commission Report, 1914, on pages 133 and 134, 
 reports the median score on the Hillegas scale of 97 eighth grade 
 composition papers from Delaware (city) as 521.6, and the sim- 
 ilar median of 118 eighth grade composition papers from Dela- 
 ware (county) is reported as 387.6. On pages 135j^. of the same 
 report results of handwriting scores by the Ayres Scale are given. 
 These results are from 1,397 pupils in 176 rural schools in 21 coun- 
 ties in Ohio ; from 214 pupils in the three upper grades of the rural 
 schools of Delaware County ; from 312 pupils in the three upper grades 
 in the city schools of Delaware ; and from a small city in Iowa, based on 
 results obtained by I. King and H. Johnson, whose study was originally 
 published in the Journal of Educational Psychology (III: 514-520). 
 Tables are given which show, among other things, that in the small 
 Iowa city 98 per cent of the eighth grade papers were scored over 50 
 on the Ayres Scale ; that 77.8 per cent of the like papers from the Ohio 
 rural schools made like scores ; 73.5 per cent from the Delaware County 
 (rural) ; and 91.3 per cent from the city of Delaware. In conclusion 
 it is stated that "the handwriting in both Delaware city and Iowa city 
 is on the whole better than in the rural schools . . ." and that 
 "there is greater uniformity in the city schools." 
 
 Results of work with the Courtis Standard Research Tests in 
 Arithmetic Series B and Starch's Arithmetical Scale A are reported 
 on pages 33j^. of A Study of Rural Schools in Travis County, Bul- 
 letin of the University of Texas, December, 1916. Tables are given 
 showing grade results of the Courtis tests, and the discussion is sum- 
 
 7 
 
8 Study of Achievement in Country and Town Schools 
 
 marized in this statement : "In every instance the number of examples 
 attempted and the number of examples worked correctly were lower 
 in this group of country schools than for the corresponding grades in 
 Boston and Detroit, and among the 11,800 pupils tested in Iowa. 
 Sometimes this difference was as great as 50 per cent, the average dif- 
 ference being approximately 25.5 per cent." Concerning the results of 
 the Starch test the following statement is made: "As a test for the 
 reasoning ability of these pupils on arithmetical problems, Starch's 
 Arithmetical Scale A was used. The results from the fourth to the 
 eighth grade were duly tabulated. Each grade scored below the scale 
 standard established for it. The fourth grade was 11 per cent below 
 standard; the fifth grade 19 per cent; the sixth grade 5 per cent; the 
 seventh grade 6 per cent ; and the eighth grade 8 per cent." 
 
 In 1916 the State Board of Education in Connecticut made a num- 
 ber of studies of the schools in certain of the towns (townships) of 
 that state. The tests used were not standardized, but they were uni- 
 form for all of these investigations. The subjects covered were arith- 
 metic, reading, language, history, geography and spelling. Compara- 
 tive tables for different towns in the state are given in these reports, 
 which were published separately as bulletins of the Connecticut State 
 Board of Education. The following quotations give an idea of the 
 character of the findings : "The accomplishment of the pupils as shown 
 by the tests given is only from fair to poor. Especially is this true 
 in view of the retardation in the schools." {Educational Enquiry, 
 Trumbull, p. 29.) "While the results in this fundamental test are not 
 satisfactory, they nevertheless compare favorably with the work of 
 pupils in the towns of Seymour, Westbrook and Bloomfield." {Edu- 
 cational Enquiry, New Hartford, p. 24.) "It would appear from the 
 above that the children cannot add with facility and accuracy, although 
 some progress is being made from grade to grade." {Educational 
 Enquiry, Seymour, p. 23.) "It is obvious that most pupils in these 
 towns are unable to interpret and solve correctly simple problems." 
 {Educational Enquiry, Westbrook, p. 25.) 
 
 In an unpublished survey of the Haldane school. Cold Springs, N. Y., 
 made in 1917, a table on page 30 gives a comparison of median achieve- 
 ments in English composition by grades (Hillegas Scale, Nassau 
 County Extension), which includes grade medians for the schools of 
 Mobile County, Ala., and for the city of Mobile. In every grade re- 
 ported the median for the city schools is higher than that for the 
 country schools. 
 
Previous Objective Studies 9 
 
 The Survey and Report of The Virginia Public Schools Education 
 Commission, published in 1919, gives comparative results for country 
 and city schools for the following tests : Thorndike Reading Scale A2, 
 Virginia Reading Test Sigma 8, Virginia General Examination Exer- 
 cise 1, Ay res Spelling Scale, Starch Scale for Measuring Handwriting, 
 Woody Arithmetic Scale Series B, Courtis Standard Research Test 
 in Arithmetic Series B, and the Nassau County Supplement to the 
 Hillegas Scale for Measuring English Composition. About sixteen 
 thousand different children were examined with from six to forty dif- 
 ferent tests. Of these sixteen thousand children about five thousand 
 were in grades three to seven of rural white schools. The general 
 character of the results is indicated by the following quotations from 
 the report : "In summary it may be said that the condition of reading 
 in the Virginia city schools is fairly satisfactory. Such deficiencies 
 as do exist will be easily adjusted when once attention is called to them. 
 In the rural schools, however, there is a deficiency that should be met 
 by vigorous remedial measures." (p. 121.) In regard to handwriting 
 the report has this to say: "The scores for rural schools having four 
 or more rooms compare favorably with those for city schools, and the 
 progress is regular from grade to grade. The average progress is 
 four-tenths of a scale step per year, which is less than it should be by 
 Starch Standards. The one-room country school shows the poorest 
 record." (p. 122.) In regard to spelling it says: "The most notable 
 deficiencies are in the one-room school (all grades) and in the seventh 
 grade for every group tested. In three cases the deficiency is a year 
 or more of progress as measured on the Ayres Scale." (p. 123.) 
 Concerning arithmetic the report states that "The average seventh 
 grade score for Virginia city children at the end of the year is 15.9, 
 which is the sixth grade Woody Standard for the beginning of the 
 year, and it is 15.1, or less than the sixth grade Woody Standard for 
 the seventh grade in the best graded rural schools. For the one-room 
 rural schools it is only 13.5." (p. 124.) "What is true of addition 
 is generally true of the results in subtraction, multiplication and divi- 
 sion." (p. 125.) In general conclusion in regard to the results in 
 rural schools the report says : "The inferiority of the small rural school 
 is apparent from the figures given. In reality that inferiority is even 
 greater since in all non-city schools children are on the average about 
 a year older than city children and in the one-room schools children 
 are on the average about a year and a half older than city children in 
 each of the upper grades of the elementary schools. In arithmetic the 
 
10 Study of Achievement in Country and Town Schools 
 
 inferiority of the one-room school is perhaps most marked, children 
 in such schools being on the average one grade behind children in the 
 larger non-city schools." (p. 129.) 
 
 E. H. Taylor, in an article, "A Comparison of the Arithmetical 
 Abilities of Rural and City School Children," published in the Journal 
 of Educational Psychology (V: 461-466, October, 1914), reports the 
 results of Courtis arithmetic tests given by himself and the county 
 superintendent to most of the children in a county in Illinois. Tests 
 were given to 309 children in 28 schools. Results are tabulated and 
 compared by grades with scores of 7,008 children, published by Courtis 
 in Elementary School Teacher (XII: 133/). The comparison shows 
 that the rural schools are consistently below the others ; grades 3 and 
 4 of the rural schools are approximately a grade below, grades 5 and 6 
 are almost two grades below, grade 7 is more than one grade but less 
 than two grades below, and grade 8 is at least two grades below. 
 
 Richard Zeidler reports in Elementary School Journal (XVI : 542- 
 555) tests of efficiency in the rural and village schools of Santa 
 Clara County, CaHfornia. Courtis arithmetic tests were given by the 
 writer to 587 children in grades 5 to 8 inclusive. Results are compared 
 with a group of 20 small Western cities, with Salt Lake City, and with 
 Butte. The writer concludes that "From these tables it will be noted 
 that rural and village schools in Santa Clara County, in every subject 
 and in every grade, fall below the median scores for the cities. This 
 inferiority ranges as high as 600 per cent, as is shown in division in 
 the fifth grade of Santa Clara County compared with the fifth grade 
 division in Salt Lake City, Utah. It must be admitted, however, that 
 Salt Lake City has an enviable record." ". . . It will be seen that 
 rural and village schools of Santa Clara County fall, on the average, 
 fully two grades below Salt Lake City, and one grade below the twenty 
 small cities." 
 
 The same author reports tests in silent reading in the rural 
 schools of Santa Clara County, California, in Elementary School 
 Journal (XVII: 55-62). Starch Reading Tests were given by the 
 writer in 26 rural schools. Eight of these schools were one-teacher 
 schools, 7 were two-teacher, 7 were three-teacher, and 4 had more 
 than three teachers. The results are tabulated in detail, and compared 
 with results from San Jose. Mr. Zeidler says : "In comparing the 
 median number of words read per second in the rural schools with 
 the medians for the San Jose schools, it will be seen that the rural 
 schools compare very favorably with the city schools. The fifth grade 
 
Previous Objective Studies 11 
 
 falls 2/10 below, while the seventh and eighth grades score higher 
 than the city schools of that community. But if we make a compari- 
 son of both the rural schools of Santa Clara County and the city 
 schools of San Jose with the standardized scores, we find that both fall 
 below, with the single exception of the seventh grade in the rural 
 schools, which surpasses the standards by 1/10." "In the reproduction 
 of the thought, the rural school children fall below the children of the 
 city of San Jose. There is a marked difference in the third grade, 
 where the median for the rural schools is 7.5 as against 12.5 in the 
 city." 
 
 Charles L. Harlan is the author of "A Comparison of the Writing, 
 Spelling and Arithmetic Abilities of Country and City Children," pub- 
 lished in Educational Administration and Supervision (II: 560-573). 
 Tests were given to 359 pupils in 21 rural schools in a "typical" county 
 in Pennsylvania. Handwriting was scored on the Ayres Scale and 
 compared with Freeman's average based on a study of handwriting 
 in 56 cities. No material difference appears between the results in 
 Mr. Harlan's study and the averages obtained by Freeman, either in 
 the quality or the speed of writing. Three hundred and thirty-seven 
 pupils were tested in spelling, words being used from the Ayres list. 
 Results are tabulated, and Mr. Harlan concludes : "A comparison of 
 the above averages shows an advantage of from seven to fourteen 
 words per one hundred in favor of the city pupils. The country 
 averages are uniformly lower than the city averages. This is no mere 
 statistical accident. The pupils of the rural schools do not spell as 
 well on written tests as do the children of the city schools." Courtis 
 Arithmetic Tests Series B were given to 255 pupils. The comparison 
 is with the Courtis Standards. "In summarizing, one may say that in 
 the fundamental operations of arithmetic the pupils of the rural 
 schools rank approximately two grades below the pupils of the city 
 schools." "The improvement in arithmetic from grade to grade is 
 less for country children than for city children." 
 
 M. E. Haggerty, in Studies in Arithmetic, Indiana University 
 Studies, No. 32 (September, 1916), reports results from 5 counties 
 in Indiana. Results are tabulated, and it is concluded that "These 
 data point to the conclusion that the district schools are more efficient 
 in procuring achievement in the function measured than the graded." 
 (p. 68.) This is the only study found that reports an actual advan- 
 tage in favor of the rural school. Mr. Paul Mort is directly responsi- 
 ble for the data collected. It is not definitely stated how the material 
 
12 Study of Achievement in Country and Tozvn Schools 
 
 given in this study was gathered, but it is to be inferred from the ex- 
 tent of the study that the testing was done by the teachers. 
 
 In A Report on the Use of Some Standard Tests, Wisconsin State 
 Department of Public Instruction Studies in Educational Measure- 
 ments in Wisconsin, Bulletin No. 1, W. W. Theisen reports results 
 for 36,564 children in Wisconsin, of whom 15,825 were in rural schools. 
 He summarizes on page 22 as follows : ''Judged by the Ayres stand- 
 ards, rural children in Wisconsin do not spell well. They average not 
 less than 10 points below the standard in every grade. However, 
 when compared with other classes of Wisconsin schools, as will be 
 seen from Table 1, they are not conspicuously poorer nor better spell- 
 ers than the pupils in the cities and villages." 
 
 The same author in the same report gives results of penmanship 
 scored according to the Thorndike Handwriting Scale from 141 rural 
 schools in 28 counties. He summarizes (p. 85) : "Rural schools on the 
 whole make the best showing of Wisconsin schools in writing. Their 
 quality is not inferior to that of other classes of schools, and their 
 speed is more nearly in accord with desirable standards in most grades." 
 
 The Sixty-Ninth Report of the Public Schools of the State of Mis- 
 souri (June 30, 1918) reports a series of investigations as to rural 
 school conditions in that state. Courtis Standard Reading Test No. 2 
 was given to 6,056 pupils in 547 schools in 82 counties. Among the 
 findings it is stated (p. 74) that "A second observation based on the 
 table is that in all three phases of reading ability the rural school chil- 
 dren in these grades (2-8) are below the standard set by Courtis." 
 
 The Kansas City Scale for Measuring Handwriting was used for 
 scoring papers of about 8,000 pupils in 86 counties in the same state. 
 The results are reported in detail by grades, and it is concluded (pp. 
 80-81) that "The quality or form of handwriting of the rural school 
 pupils is far below the accepted standards, grade for grade. The 
 speed is above the accepted standards, except in the seventh and eighth 
 grades." 
 
 Results are also reported from 6,445 pupils in grades 4 to 8 inclu- 
 sive and from 87 counties for the use of Courtis Standard Research 
 Tests in Arithmetic, Series B. (pp. 81-85.) It is concluded that 
 "On the whole the rural school children in Missouri are not up to the 
 ability of the city pupils as shown by the Courtis general tabulations. 
 This is particularly true of the sixth, seventh and eighth grades. In 
 the eighth grade it was pointed out that the Missouri rural school pu- 
 pils were below in every particular." 
 
Previous Objective Studies 13 
 
 In the same report (pp. 85-87) results are given for a spelling test 
 based on the Ayres' Spelling Scale given to 8,326 pupils in 611 schools 
 in 86 counties. In conclusion it is stated (p. 87) that "The rural 
 school pupils in Missouri are below the standard attainment for the 
 city school pupils in 84 of the larger cities of the United States, and 
 the writer of the report of which this section is a summary says also 
 that the rural school pupils are below the attainment of the ten town 
 schools in the state studied by him." 
 
 In 1918, in Report on Spelling and Penmanship in Country Schools, 
 Louisiana State Department of Education, Bulletin No. 1, June, 
 1918, John M. Foote reported the results of spelling test from the Ayres 
 list given to 4,584 pupils in rural schools in Louisiana. He concludes 
 (p. 11) that : "While it may be that the superiority of the large type of 
 school as compared with the small type is decisively reflected in the 
 instruction in some of the subjects taught, it must be concluded that 
 in so far as spelling ability in the rural schools is concerned no such 
 decisive difference exists. The results show clearly that there is a 
 difference in favor of the large school, but the assumption that the 
 classroom results are very much better is not warranted." 
 
 The same author in the same bulletin reports handwriting results 
 according to the Ayres Scale for 4,073 pupils in 114 rural schools, 
 and concludes (p. 24) that "It will be seen by reference to the table 
 and diagram that there is comparatively little difference in both qual- 
 ity and speed between the five groups. It will be noted that the dif- 
 ference in quality is less than in speed. The situation is similar to that 
 found in spelling. The evidence therefore points to the conclusion 
 that the pupils in the large schools are slightly superior, but that no 
 great difference prevails." 
 
 The same author, in Silent Reading in Rural and Village Schools, 
 Louisiana State Department of Education, Bulletin No. 9, July, 1919, 
 reports results from work with Monroe's Standardized Silent Reading 
 Tests with 4,315 pupils in 96 schools located in 15 parishes. He con- 
 cludes (p. 8) that "A comparison of the three groups A, B, and C 
 reveals a distinct difference in favor of the larger school. With the 
 exception of the fourth grade the scores of group B are better than 
 those of group C. Group A scores are higher in every instance than 
 groups B and C. This superiority of the large type of school is in 
 agreement with results reported for Santa Clara County, California." 
 
 Still later, in A Study of Arithmetic in Rapides Parish, Louisiana 
 State Department of Education, Bulletin No. 11, September, 1919, Mr. 
 
14 Study of Achievement in Country and Tozun Schools 
 
 Foote reports results for the use of the Courtis Arithmetic Test Series 
 B in the schools of Rapides Parish, Louisiana. In this study schools 
 are grouped in four classes, those in the city of Alexandria in group 
 A, eight high schools in group B, eight graded schools in group C, and 
 twenty-eight one- and two-teacher schools in group D. Mr. Foote 
 concludes (p. 12) that "Superior results are obtained in the large type 
 of schools." (p. 13.) "Group A and D vary in speed by 1.0 year 
 and in accuracy by 2.0 years. It is therefore evident that the large 
 schools as represented by groups A and B obtain results that are dis- 
 tinctly superior to those obtained by groups C and D." 
 
 George A. Grim, in "Spelling in Northampton County," University 
 of Pennsylvania Bulletin, Vol. XX, No. 1, October, 1919, Sixth An- 
 nual Schoolmen's Week Proceedings, reports spelling results for the 
 rural and urban sections of Northampton County. The results are 
 reported for grades 2 to 8 inclusive, and in five of the seven grades 
 reported the urban schools surpass the rural. The differences re- 
 ported are not great in any case. 
 
 Cyrus D. Mead, in The Spelling Ability of the Plumas County Chil- 
 dren, California State Board of Education, Supplement of California 
 Blue Bulletin, December, 1919, reports on the spelling ability of the 
 pupils of Plumas County, California, as shown by a test from the 
 Ayres list. He concludes (p. 5) that "The pupils of this county are 
 . . . just about of the average ability of city children over the 
 country." 
 
 The above references include all the data easily available dealing in 
 an objective way with the comparative results of instruction in rural 
 and in urban schools. The difficulty of collecting data in rural schools, 
 where distance is great and classes small, makes the time element in 
 collecting any large number of cases very great and seems to have pre- 
 vented the collection of extensive material. The studies reported in- 
 clude 2 studies of composition work ; both of these favor the urban 
 school. There are 6 studies of handwriting reported, and of these 3 
 favor the urban school, 2 show negHgible differences, and 1 favors the 
 rural school. Nine studies of arithmetic are reported, and 8 of these 
 favor the urban school, while 1 favors the rural school. Four studies 
 of reading are reported, and all of them favor the urban school. 
 Seven studies of spelling are reported, 3 of which favor the urban 
 school and 4 of which find negligible differences. In all 28 studies are 
 reported, of which 20 find the results better in urban schools, 6 find 
 negligible differences, and 2 find results superior in the rural schools. 
 
CHAPTER II 
 PURPOSE AND SCOPE OF THE PRESENT STUDY 
 
 The studies of results of instruction in rural schools reported in 
 Chapter I are in objective terms, and seem to indicate a situation that 
 is quite generally inferior to that in cities and towns. All of the com- 
 parisons are made, however, upon a grade basis, which assumes that 
 the third grade, or the eighth grade, or any other grade should be the 
 same in one system as in another. That grades are not in fact equal 
 is shown by results. Under the present diversity of conditions it is 
 entirely possible that grades should not be equal. 
 
 Particularly is this true in country schools as contrasted with city 
 schools. The shorter terms, poorer attendance, and inherent difficul- 
 ties involved in the small school make it probable that grading in these 
 schools should be different from that in city schools. For example, 
 in Madison County, Kentucky, the school year for the smaller schools 
 is only six months. This means that a boy or girl entering school at the 
 age of six and progressing normally would at the age of fourteen com- 
 plete the eighth grade, having attended school a total of 48 months. A 
 child entering one of the larger schools of the country, such as that in 
 Richmond, would have nine months' schooling each year, and would, 
 if he progressed normally, complete the eighth grade with a total of 72 
 months in school ; that is, with 24 months' more schooling than the 
 country child received. 
 
 Again, in some of the states the school system is organized on the 
 basis of seven grades in the elementary school, so that a direct compar- 
 ison of a fifth grade in a seven grade system with a fifth grade in an 
 eight grade system is impossible. 
 
 Further, the use of grade standards must assume that tests are given 
 at the same time in the school year. This would be necessary either 
 to compare the system tested with another system or with the work 
 in the same system in other years, since the standard for any grade 
 must be different at the beginning and at the close of the school year. 
 It is not always convenient, or even possible, to give tests in any partic- 
 ular part of the school term. There will be material gain in the ease 
 of using standard tests if it can be made possible to give them at any 
 convenient time and to obtain results comparable with those from other 
 places and given at other times. 
 
 15 
 
16 Study of Achievement in Country and Tozvn Schools 
 
 Then, too, in any system of schools some pupils are likely to fail of 
 promotion each year. This causes retardation, and is much more se- 
 rious in some systems than in others. Any consideration of the effi- 
 ciency of a school system must account for this factor. Grade stand- 
 ards do not do so. 
 
 It is for such reasons as these that the Nassau County Survey, pub- 
 lished as a bulletin of the University of the State of New York (De- 
 cember, 1917), in reporting the measurements of the achievements of 
 pupils, carefully refrains from making any comparison of the urban 
 and rural schools, and makes the following statement (p. 146) : "The 
 classification of school districts by the approximate size of the schools 
 has not been continued in this chapter, because of the possible injus- 
 tice which might thereby be done to the small schools, in which the 
 exact classification of pupils into eight grades is not always possible. 
 There might have been possible a more accurate comparison of the 
 results from large and from small districts, if the results had been 
 tabulated by age rather than by school grade. A comparison of the 
 ability children have in a rural school by the time they are 'twelve 
 years old' with the ability of children of the same age in a large village 
 school might possibly be more significant than a comparison of the 
 ability between children of the 'sixth grade' in the two types of schools. 
 For several reasons, however, we have not been able to make such 
 comparisons between the achievements of pupils of the same ages in 
 districts of various sizes." 
 
 To really measure school efficiency it is necessary to show the prog- 
 ress, not only of those who are promoted, but of all the children ; and 
 to do this in terms of some unit common to all schools, city or country, 
 seven grade or eight grade system, and long or short school term. 
 There will be added convenience if such a unit can be so devised as 
 to be usable at any time during the school year. 
 
 The chronological age of school children is such a unit. If the prog- 
 ress made by children can be shown, say between the ages of 7 and 12, 
 a measure that is definite, universal, and approximately equal at any 
 time during the school year is obtained, provided only that children of 
 both ages are tested at the same time. The question is whether such 
 progress can be shown. 
 
 This study undertakes: 
 
 First, to show that the difference in performance in school subjects 
 of children of different ages can be obtained ; 
 
Purpose and Scope of Present Study 17 
 
 Second, to show that this difference is a measure of school efficiency 
 which may be used to measure schools or school systems ; 
 
 Third, to apply this measure to a system of country schools ; 
 
 Fourth, to compare the results in this country system of schools with 
 the results in certain city school systems. 
 
 The country school system to which this measure is applied is that 
 of Madison County, Kentucky. This county lies a little east of the 
 central portion of the state, and includes both blue-grass and mountain 
 land. In the 1910 census it ranks 20th in the land area among the 
 counties of the state, 5th in the value of all farm property, and 12th 
 in the value of all farm crops. Corn and tobacco are easily the lead- 
 ing crops. The county is one of the wealthier and larger counties of 
 the state. There are two normal schools, the Eastern Kentucky State 
 Normal at Richmond and a normal department in connection with 
 Berea College at Berea. The number of public white schools is 68, 
 six of which have three or more teachers and a term of nine months. 
 The rest of the schools have only one or two teachers and a six-month 
 school year. A county supervisor of schools is employed to assist the 
 county superintendent. There is no reason to suppose that the schools 
 of this county are particularly worse than those of surrounding counties. 
 
 For the purposes of this study three tests were selected : Trabue 
 Completion-Test Language Scale B, Courtis Standard Research Tests 
 in Arithmetic Series B, and Thorndike Silent Reading Scale Alpha 2. 
 These tests were given in grades 3 to 8 inclusive of all the public 
 schools for white children in the county during November and Decem- 
 ber, 1919.1 The results are tabulated separately for the schools hav- 
 ing a six-month term and for those having a nine-month term. 
 
 In order that the results in Madison County may be compared with 
 results in other places, data are included as follows : From Louisville, 
 Kentucky, data for each of the three tests used ; from Paterson, New 
 Jersey, and St. Paul, Minnesota, data for the Trabue Language Scale ; 
 from Louisiana, Arkansas City and Salina, Kansas, and Hibbing, 
 Minnesota, data for the Courtis Standard Research Tests in Arithme- 
 tic Series B ; from Hamilton Township, New Jersey, and Amsterdam, 
 New York, data for Thorndike Silent Reading Scale Alpha 2. 
 
 In order to show clearly the method followed, the results of the 
 Trabue Completion-Test Language Scale B are worked through in de- 
 tail in the next chapter. 
 
 ^ Five of the one-teacher schools were not in session when visited, and no re- 
 sults from them are given. 
 
CHAPTER in 
 ACHIEVEMENT MEASURED BY THE TRABUE LANGUAGE SCALE 
 
 Table I shows the age-grade distribution of scores made in the 
 Trabue Language Scale B by pupils in the Madison County schools 
 having a term of 6 months. The table reads as follows : There are 
 8 pupils 7 years of age in the third grade, 1 of whom made a score of 
 3, 2 a score of 4, 2 a score of 5, 3 a score of 6, etc. In this table and 
 throughout the study the age used is that of the last birthday at the 
 time the test was given. A child is considered as 10 years old who is 
 as much as 10 years old and less than 11 years old. 
 
 It is evident from the total number of cases occurring for the re- 
 spective ages that the distributions are incomplete for most ages ; prob- 
 ably for all. There should be at least as many 8-year-old children, for 
 example, as there are 9-year-olds. This incompleteness for the younger 
 ages represents children in the first and second grades, which were not 
 tested, or children who have not yet entered school. Since these chil- 
 dren have not yet been promoted to the third or higher grades, it is 
 evidently the opinion of the teachers that they could not do so well as 
 the children who are in the third or higher grades. It follows that the 
 distributions for children of the younger ages are lacking in the num- 
 ber of cases for the lower scores. The distributions for the older 
 children, on the other hand, may be considered as lacking in the num- 
 ber of cases for the higher scores, since the older children not appear- 
 ing in the table are those who have either completed the eighth grade 
 before reaching the age indicated, or who have for some reason stopped 
 attending school. 
 
 If the difference in performance in school subjects of groups of 
 children of different ages is to be a valid measure of the efficiency of 
 a school system, it is evident that the age-groups in the respective sys- 
 tems jnust be comparable. That is to say, either these groups must 
 include all children of the age indicated, or the selection must be shown 
 to be by random sampling only, or the selection must be shown to be 
 influenced only by known causes acting equally and in the same direc- 
 tion, or it must be possible to so correct the age-groups given as to 
 meet substantially the above conditions. 
 
 The age-groups presented in Table I do not meet the first of these 
 
 18 
 
TABLE I : Age-Grade Distribution of Trabue Language Scale B Scores in 
 THE 6-Month White Schools of Madison County, Ky. (November and 
 December, 1919) 
 
 Age 
 
 Grade 
 
 Scores 
 
 Total 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 
 7 
 
 3 
 4 
 5 
 
 
 
 1 
 
 2 
 2 
 
 1 
 
 2 
 
 1 
 
 3 
 
 1 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Total 
 
 
 1 
 
 
 
 1 
 
 5 
 
 3 
 
 5 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 16 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 4 
 5 
 
 6 
 1 
 
 
 2 
 1 
 
 1 
 
 1 
 
 3 
 3 
 
 2 
 
 1 
 
 10 
 
 7 
 1 
 
 '"4 
 
 2 
 1 
 4 
 
 1 
 
 
 
 
 
 
 
 
 
 
 27 
 
 1 
 
 1 
 
 
 
 1 
 
 
 
 
 
 22 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Total 
 
 
 7 
 
 
 3 
 
 2 
 
 6 
 
 3 
 
 18 
 
 4 
 
 7 
 
 1 
 
 1 
 
 1 
 
 
 
 1 
 
 
 
 
 
 54 
 
 
 
 
 
 
 
 g 
 
 3 
 4 
 5 
 
 6 
 6 
 
 
 6 
 2 
 1 
 
 "i 
 
 12 
 
 1 
 
 2 
 3 
 
 9 
 
 7 
 4 
 
 3 
 2 
 
 2 
 3 
 
 7 
 
 
 
 
 
 
 
 
 
 
 
 40 
 
 
 3 
 4 
 
 2 
 1 
 
 3 
 
 .... 
 
 1 
 1 
 
 
 
 
 
 
 33 
 
 
 
 
 
 
 
 20 
 
 
 
 
 
 
 
 
 Total 
 
 
 
 12 
 
 
 9 
 
 1 
 
 13 
 
 51 20 
 
 5 
 
 12 
 
 7 
 
 3 
 
 3 
 
 1 
 
 2 
 
 
 
 
 
 
 93 
 
 
 
 
 
 
 
 10 
 
 3 
 4 
 5 
 6 
 
 7 
 8 
 
 3 
 1 
 
 1 
 
 1 
 
 1 
 
 5 
 3 
 
 2 
 
 1 
 3 
 
 9 
 6 
 4 
 1 
 1 
 1 
 
 1 
 2 
 4 
 2 
 
 i 
 8 
 3 
 2 
 
 '"3 
 5 
 1 
 
 1 
 2 
 3 
 1 
 
 2 
 3 
 4 
 2 
 
 1 
 .... 
 
 
 
 
 
 
 
 30 
 
 
 1 
 2 
 
 
 
 
 
 
 31 
 
 
 
 2 
 
 
 
 
 
 
 
 32 
 
 
 1 
 
 
 
 
 
 
 11 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Total 
 
 
 4 
 
 1 
 
 ■? 
 
 1 
 
 10 
 
 6 
 
 22 
 
 9 
 
 17 
 
 9 
 
 7 
 
 11 
 
 9 
 
 4 
 
 1 
 
 
 
 
 107 
 
 
 
 
 
 
 
 
 
 11 
 
 3 
 
 4 
 5 
 6 
 
 7 
 8 
 
 3 
 1 
 
 
 3 
 1 
 
 1 
 
 
 3 
 3 
 
 2 
 
 3 
 2 
 
 5 
 10 
 
 3 
 
 1 
 
 2 
 1 
 3 
 
 1 
 2 
 4 
 1 
 2 
 
 1 
 4 
 3 
 
 3 
 4 
 5 
 1 
 
 .... 
 
 5 
 2 
 
 1 
 1 
 
 4 
 1 
 1 
 
 
 
 
 
 
 
 19 
 
 
 
 
 
 
 
 28 
 
 
 2 
 
 
 
 
 
 
 34 
 
 
 
 2 
 
 1 
 
 2 
 
 
 .... 
 
 16 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 1 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Total 
 
 
 4 
 
 
 5 
 
 
 8 
 
 6 
 
 19 
 
 6 
 
 10 
 
 8 
 
 13 
 
 8 
 
 8 
 
 2 
 
 1 
 
 3 
 
 2 
 
 1 
 
 
 104 
 
 12 
 
 3 
 4 
 5 
 6 
 
 7 
 8 
 
 1 
 1 
 
 1 
 
 1 
 
 
 1 
 1 
 2 
 
 1 
 
 1 
 
 3 
 9 
 
 8 
 
 1 
 
 2 
 
 4 
 
 3 
 
 .... 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 
 3 
 5 
 3 
 
 6 
 
 2 
 
 6 
 
 2 
 
 1 
 
 .... 
 
 5 
 1 
 
 "2 
 4 
 2 
 
 1 
 4 
 4 
 7 
 2 
 
 
 
 
 
 
 
 25 
 
 
 
 
 
 2 
 1 
 
 2 
 
 
 
 
 
 
 31 
 
 
 
 - 
 
 1 
 
 ___ 
 
 1 
 1 
 1 
 
 
 
 
 
 27 
 
 
 
 
 
 
 16 
 
 
 
 
 
 
 
 
 
 
 3 
 
 .... 
 
 
 7 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 Total 
 
 
 2 
 
 
 
 
 A 
 
 4 
 
 91 
 
 in 
 
 11 
 
 17 
 
 7 
 
 8 
 
 18 
 
 6 
 
 3 
 
 
 ■^ 
 
 
 
 116 
 
 
 
 
 
 
 
 
 13 
 
 3 
 4 
 5 
 6 
 
 7 
 8 
 
 1 
 
 
 3 
 
 1 
 
 
 1 
 1 
 
 "i 
 1 
 
 1 
 
 
 
 
 1 
 
 2 
 
 4 
 5 
 
 
 
 
 
 
 
 
 
 
 6 
 
 
 2 
 6 
 
 1 
 
 4 
 2 
 2 
 1 
 
 2 
 2 
 6 
 2 
 
 i 
 
 2 
 2 
 1 
 
 '"2 
 
 .... 
 
 1 
 
 --- 
 
 .... 
 
 ... 
 
 
 
 16 
 27 
 
 
 
 
 
 
 2 
 3 
 1 
 
 2 
 
 '"4 
 
 1 
 2 
 5 
 
 
 
 
 
 23 
 
 
 
 
 
 
 
 2 
 
 
 
 
 12 
 
 
 
 
 
 
 
 
 
 1 
 
 
 16 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 Total 
 
 
 
 4 
 
 
 2 
 
 3 
 
 14 
 
 9 
 
 10 
 
 12 
 
 13 
 
 7 
 
 6 
 
 7 
 
 8 
 
 2 
 
 
 1 
 
 
 100 
 
 14 
 
 3 
 4 
 
 5 
 6 
 
 7 
 8 
 
 "2 
 
 
 "\ 
 
 
 1 
 1 
 
 
 
 .... 
 
 2 
 2 
 
 '""5 
 2 
 
 1 
 2 
 2 
 
 1 
 
 '"2 
 3 
 2 
 2 
 
 
 
 
 
 
 
 
 
 
 3 
 
 
 1 
 
 2 
 
 1 
 
 .... 
 
 
 
 
 
 
 
 
 
 11 
 
 
 3 
 3 
 
 2 
 2 
 
 1 
 .... 
 
 1 
 
 3 
 
 2 
 6 
 
 1 
 
 I 
 5 
 
 
 
 
 
 18 
 
 
 
 
 
 
 
 
 
 15 
 
 
 1 
 
 
 
 
 
 
 1 
 2 
 
 
 
 
 13 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 22 
 
 
 
 
 
 
 
 
 
 
 
 
 Total 
 
 
 3 
 
 1 
 
 1 
 
 
 2 
 
 
 5 
 
 5 
 
 12 
 
 10 
 
 5 
 
 10 
 
 3 
 
 13 
 
 9 
 
 3 
 
 
 
 
 82 
 
 
 
 
 
 
 15 
 
 3 
 4 
 5 
 6 
 
 7 
 8 
 
 
 
 
 
 
 1 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 1 
 
 .... 
 
 — - 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 1 
 1 
 4 
 
 1 
 1 
 1 
 
 
 
 1 
 1 
 
 2 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 2 
 
 ""3 
 
 
 
 
 
 6 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 3 
 
 1 
 
 .... 
 
 
 17 
 
 
 
 
 
 
 
 
 
 
 
 
 Total 
 
 
 
 
 
 __ 
 
 
 2 
 
 
 1 
 
 2 
 
 2 
 
 7 
 
 3 
 
 5 
 
 3 
 
 4 
 
 3 
 
 1 
 
 
 
 33 
 
 16 
 
 4 
 5 
 6 
 7 
 8 
 
 
 
 
 
 ■ 
 
 
 ■ 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 1 
 
 3 
 1 
 
 3 
 4 
 
 1 
 
 
 
 
 
 9 
 
 
 
 
 
 
 1 
 
 
 
 
 
 1 
 
 4 
 
 4 
 
 
 
 
 16 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Total 
 
 
 
 
 
 
 1 
 
 
 
 1 
 
 2 
 
 1 
 
 4 
 
 3 
 
 4 
 
 7 
 
 1 
 
 4 
 
 
 
 
 28 
 
 19 
 
20 Study of Achievement in Country and Tozvn Schools 
 
 conditions, in that they do not include all the children of the ages 
 indicated. Take, for example, the group of 12-year-old children, the 
 largest group presented. This shows 10 children in the third grade, 
 25 in the fourth, 31 in the fifth, 27 in the sixth, 16 in the seventh, and 
 7 in the eighth. Since the distribution of 12-year-old children through- 
 out the grades is not rectilinear, as is shown by the part of the distri- 
 bution given, it seems unlikely that if a full distribution were available 
 it would stop at the lower end with 10 children in the third grade or 
 at the upper end with 7 children in the eighth grade. It seems likely 
 that there would be some 12-year-old children in grades below the 
 third and in grades above the eighth. This opinion in the latter in- 
 stance is substantiated by the fact that some 10- and 11-year-old chil- 
 dren are in the eighth grade this year. If they progress normally, 
 they will complete the work of the elementary school this year, and 
 will not be enrolled in the school next year. The same thing is likely 
 to have happened last year, so that there are now some children who 
 are 12 years old and have completed the work that the school offered 
 and are therefore not in school. Also, there are 13-, 14-, and even 15- 
 year-old children in the third grade. There is every reason to sup- 
 pose that there are 12-year-old children in grades lower than the third. 
 In addition to the above considerations, it is known that there were 
 some children absent from school on the day the tests were given. 
 
 As regards the second possibility mentioned for comparable groups : 
 Is the selection by random sampling only? In so far as the selection 
 is due to absence from school on the day the test was given, the se- 
 lection may be considered random as to ability in the test concerned. 
 The chief causes of absence mentioned by the teachers were work, bad 
 weather, illness, and indifference on the part of the parents. The 
 work for which the children were kept at home during the time these 
 tests were being given (November and December) was usually work- 
 ing tobacco or killing hogs. While it is likely that more of the older 
 children than of the younger ones were kept at home for work, still 
 some of the younger ones were kept at home for this purpose. The 
 discrepancy in absences between the older and younger children in re- 
 gard to absence for work is reversed and probably neutralized by the 
 discrepancy in absences due to bad weather and roads. The absences 
 due to illness of the children and to indifference on the part of the 
 parents may be assumed to be about the same for children of all ages. 
 
 On the other hand, the selection due to the inclusion of only those 
 children in grade 3 to 8 inclusive is not random. The older children 
 
Achievement Measured by Trabue Language Scale 21 
 
 in grades 1 and 2, that is, those who are not included in the distribu- 
 tion given, are in those grades because, in the opinions of the teachers, 
 they are not so able as the children of the higher grades, or because 
 they have for some reason become so retarded in their school work as 
 to make them distinct from other children. In other words, these 
 children are, in the opinions of the teachers, of less ability in school 
 work than those of corresponding age who have been promoted to the 
 higher grades. This means that the distributions given in Table I 
 do not include the children who have the lower degrees of ability in 
 school work. In the same way the younger children who have com- 
 pleted the eighth grade and are therefore not included in the table are, 
 in the opinions of the teachers, the children having the higher degrees 
 of ability in school work. All of which goes to show that the selec- 
 tion of cases for the age-groups as they are presented is not random. 
 
 The next consideration is whether the selection of age-groups, in 
 such respects as it is not random, is due to the same causes acting 
 equally and in the same direction, for all the schools to be measured, 
 and in each of the age-groups. As is shown above, there is decided 
 selection due to the failure to include children in grades 1 and 2, and 
 those who have completed the eighth grade. This number of children 
 is determined by the rate of progress of children through the grades. 
 This rate of progress is unequal even for different schools within the 
 same system, as has been shown repeatedly by studies widely sepa- 
 rated both in time and locality. A good and recent example is fur- 
 nished by the age-grade tables on page 90 of An Educational Study of 
 Alabama, United States Bureau of Education Bulletin, 1919, No. 41. 
 
 The possibility still remains of correcting the tables presented. In 
 order to correct an incomplete distribution such as that presented it 
 is necessary to have a reasonably large part of the complete distribu- 
 tion. In the data presented the most nearly complete distributions are 
 those for the ages 10, 11, 12 and 13. The distribution of 12-year- 
 olds is the largest given, which indicates that it is most nearly complete. 
 Taking this distribution as a sample, how may it be corrected ? 
 
 Unfortunately the form of this distribution is not exactly that of 
 the normal frequency curve. This is shown by that portion of the 
 distribution which is given. It might also be inferred from the fact 
 that in most school systems it is easier for a child to become retarded 
 than it is for a child to make two grades in one year.^ This natu- 
 
 ^ Strayer and Thorndike, Educational Administration, p. 38. 
 
22 Study of Achievement in Country and Tozvn Schools 
 
 rally causes a skewed distribution. To correct a skewed distribution for 
 truncation by the development of mathematical formulae is a lengthy 
 and complicated process, and one that is liable to serious error. Fur- 
 ther, since the degree of skewedness and of truncation may both vary 
 for every distribution to be studied, such correction is obviously too 
 tedious and cumbersome a method to be used in a measure of school 
 efficiency that is to be easily available for busy school men. 
 
 If an age-grade table of all the children in all the grades were avail- 
 able, it might serve as a basis for calculating the number of children in 
 grades 1 and 2. It does not furnish a ready basis for finding the num- 
 ber who have completed the eighth grade. Also, for Madison County 
 schools no such table is available.' 
 
 The problem resolves itself into an attempt to find the number of 12- 
 year-old children who are in either the first or second grade, or who 
 have completed the eighth grade. The 13-year-old children in the 
 third grade this year probably were in the second grade last year when 
 they were 12 years old. The probability is that last year the number 
 of 12-year-old children in the second grade who were not promoted 
 to the third grade was about equal to the number of 13-year-old chil- 
 dren in the third grade who were not promoted to the fourth grade. 
 Inasmuch as there has been no particular change in the school popu- 
 lation of Madison County during the last few years, and in view of 
 the above probabilities, the number of 13-year-old children in the 
 third grade may be considered a fairly reliable indication of the num- 
 ber of 12-year-old children in the second grade.- Reference to Table 
 I shows this number to be 6. Similarly, the 14-year-old children in 
 the third grade may be considered a measure of the 12-year-olds in 
 the first grade. This number is 3. 
 
 In much the same way it may be supposed that the number of 11- 
 
 ^To get a census showing the ages of children in the first and second grades 
 of the Madison County public schools it would be necessary to visit each school, 
 and in the case of children who do not know their age, to visit the home. 
 
 " In order to test the validity of this assumption a study was made of the 
 age-grade tables for rural children and for children in places having a popula- 
 tion of less than 2,000 given on page 90 of An Educational Study of Alabama, 
 United States Bureau of Education Bulletin, 1919, No. 41, and the age-grade 
 table for pupils in non-city schools given on page 309 of the Survey and Report 
 of the Virginia Public Schools Education Commission. The actual number of 
 10-, 11-, 12- and 13-year-old children in grades 1 and 2 was compared with the 
 number of such children estimated to be in these grades on the basis of the 
 above assumption. The difference between the actual number of such children 
 and the estimated number was reduced to a percentage basis showing the per 
 cent of error in each estimate. Twenty-four estimates were made, and the 
 median per cent of error, direction of error being taken into account, was — 4^. 
 
Achievement Measured by Trabue Language Scale 23 
 
 year-old children in the eighth grade this year is the same as the num- 
 ber of 11-year-old children in the eighth grade last year. These should 
 be added to the 12-year-old distribution this year. Reference to Table 
 I shows this number to be 2. In the same way the number of children 
 who are 12 years old this year and who completed the eighth grade 
 two years ago is estimated as 2. 
 
 Using these estimates, the following distribution of 12-year-old chil- 
 dren is obtained: 3 in grade 1, Language Scale scores unknown; 116 
 in grades 3 to 8 inclusive, distributed according to Language Scale 
 scores ; 2 who completed grade 8 last year, Language Scale scores un- 
 known; and 2 who completed grade 8 two years ago. Language Scale 
 scores unknown. 
 
 Estimating corrections for the 10-, 11- and 13-year-old distributions 
 on the same basis as for the 12-year-olds, gives Table IL In this 
 table the figures in the columns under the captions XI, X2, X9, XIO, 
 and XI 1 are the estimated number of children of the respective ages 
 in grades 1, 2, 9, 10 and 11 respectively.^ 
 
 These corrections make the distributions for the ages 10, 11, 12 and 
 13 reasonably complete. There is still some elimination of children in 
 the grades to be accounted for, and there is the possibility that some 
 of the children of these ages may not have entered school. These 
 considerations are probably not so serious, however, as the fact that 
 the scores of the children represented by estimated additions to the 
 distributions are unknown. It is known, however, that the children 
 in the lower grades are likely to have made very low scores, while, 
 on the contrary, the children who have completed the eighth grade 
 would probably have made relatively high scores. Allowing these 
 cases to remain undistributed at the respective ends of the distribution, 
 the medians, Q's and P.E.'s of Table II are obtained. 
 
 The distributions of scores in this table show a great amount of 
 overlapping for the different ages. Twenty per cent of the children 
 in the 10-year-old group have scores superior to those of 50 per cent 
 of the 13-year-old children. This overlapping is in accord with the 
 results obtained with grade distributions. 
 
 ^ An attempt was made to estimate the necessary additions for these distribu- 
 tions on the basis of an age-grade table showing a form of distribution similar 
 to that in Madison County. The age-grade table for the Alabama schools in 
 places of less than 2,000 population was used. No estimate of children who 
 had completed the eighth grade seemed feasible on this basis, and the medians 
 calculated on the basis of the distributions corrected in this way showed co- 
 efficients of unreliability greater in every instance than those of the medians cal- 
 culated on the basis of the above corrections. 
 
24 Study of Achievement in Country and Town Schools 
 
 As in the overlapping of grade distributions, so in this overlap- 
 ping with age distributions, the unreliability of the individual measures 
 gives an indication of greater overlapping than really exists.^ In this 
 case there is an added element tending to exaggerate the amount of 
 overlapping in the undistributed scores at both ends of the age distri- 
 butions. These scores tend to increase the inter-quartile range, and 
 consequently the Q. 
 
 TABLE II 
 
 Age Distribution, 10 to 13 Inclusive, of Trabue Language Scale B Scores 
 
 IN THE 6-MoNTH WhITE ScHOOLS OF MaDISON CoUNTY, Ky. 
 
 
 
 Scores 
 
 
 
 
 
 Age 
 
 
 
 
 
 
 
 Total 
 
 
 XI 
 
 in 
 
 X2 
 10 
 
 
 
 1 
 1 
 
 2 
 
 3 
 1 
 
 4 
 
 in 
 
 5 
 
 6 
 
 99 
 
 7 
 q 
 
 8 
 17 
 
 9 
 
 q 
 
 10 
 
 7 
 
 11 
 
 11 
 
 12 
 
 ? 
 
 13 
 4 
 
 14 
 
 1 
 
 15 
 
 16 
 
 17 
 
 18 
 
 X9 
 
 XIO 
 
 Xll 
 
 
 10 
 
 
 
 
 136 
 
 11 
 
 fi 
 
 in 
 
 4 
 
 
 fi 
 
 
 8 
 
 6 
 
 11 
 
 6 
 
 in 
 
 8 
 
 13 
 
 8 
 
 8 
 
 2 
 
 1 
 
 3 
 
 2 
 
 1 
 
 
 2 
 
 
 
 122 
 
 n 
 
 3 
 
 fi 
 
 2 
 
 1 
 
 2 
 
 
 4 
 
 4 
 
 21 
 
 10 
 
 11 
 
 17 
 
 7 
 
 8 
 
 18 
 
 6 
 
 2 
 
 
 3 
 
 
 
 2 
 
 2 
 
 
 129 
 
 13 
 
 2 
 
 3 
 
 2 
 
 
 4 
 
 
 2 
 
 3 
 
 14 
 
 9 
 
 10 
 
 12 
 
 13 
 
 7 
 
 6 
 
 7 
 
 8 
 
 2 
 
 
 1 
 
 
 7 
 
 2 
 
 2 
 
 116 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 median 
 
 75 per- 
 centile 
 
 
 25 pe 
 centil 
 
 r- 
 e 
 
 Q 
 
 
 P. E. 
 
 in 
 
 6.63 
 
 .37 
 
 9.00 
 
 
 2.00 
 
 
 3.50 
 
 Mil— M10= .87 
 
 .311 
 
 11 
 
 7.50 
 
 .34 
 
 10.73 
 
 
 4.68 
 
 
 3.02 
 
 M12— Mll = 1.52 
 
 .283 
 
 ^?. 
 
 9.02 
 
 .30 
 
 12.04 
 
 
 6.48 
 
 
 2.78 
 
 M13— M12= .73 
 
 .305 
 
 13 
 
 9.75 
 
 .35 
 
 13.00 
 
 
 6.92 
 
 
 3.03 
 
 M13— M10 = 3.12 
 
 .316 
 
 Another notable feature of this table is the great discrepancy be- 
 tween progress at different ages. The improvement the 11 -year-old 
 children show over the 10-year-olds is .87 of a step ; that of the 12- 
 year-olds over the 11-year-olds is 1.52; and that of the 13-year-olds 
 over the 12-year-olds is .73 of a step. The progress in language ability 
 as measured by this test is almost as great from ages 11 to 12 as the 
 combined progress from ages 10 to 11 and from ages 12 to 13. 
 
 The total progress for the three-year period from 10 to 13 inclusive 
 is 3.12 with a P.E. of .316.- Since these schools were in session for 
 6 months each year, the total school time for the three years is 18 
 months. This makes the average progress per month for the period 
 .173 of a step. 
 
 The results of the Trabue Language Scale B test in the white schools 
 of Madison County having a 9-month school term are handled in the 
 
 ^ Thorndike, E. L. : Mental and Social Measurements, p. 93. Kelley, T. L. : 
 "The Measurement of Overlapping," Journal of Educational Psychology, X : 
 458#. (December, 1919). 
 
 ^ Thorndike : Mental and Social Measurements, p. 194. 
 
Achievement Measured by Trabue Language Scale 
 
 25 
 
 same way as are those from the schools having a 6-month school term. 
 Table III gives the facts for these schools. 
 
 From this table it seems that the overlapping of language perform- 
 ance of children of different ages is about as great in the 9-month 
 schools as in the 6-month schools. Some of the 10-year-old children 
 have scores above the median score of the 13-year-olds. The same 
 causes are operating to make this overlapping appear greater than it 
 really is. 
 
 TABLE III 
 
 Age Distribution, 10 to 13 Inclusive, of Trabue Language Scale B Scores 
 IN the 9-Month White Schools of Madison County, Ky. 
 
 
 
 
 Scores 
 
 
 
 
 
 Age 
 
 
 
 
 
 
 
 Totdl 
 
 
 XI 
 
 9 
 2 
 
 X2 
 17 
 
 
 
 
 
 1 
 
 2 
 
 
 
 3 
 
 4 
 
 8 
 
 5 
 
 7 
 
 6 
 18 
 
 7 
 7 
 
 8 
 17 
 
 9 
 
 10 
 1'' 
 
 11 
 
 8 
 
 12 
 
 8 
 
 13 
 
 3 
 
 14 
 I 
 
 15 
 ? 
 
 16 
 1 
 
 17 
 1 
 
 18 
 
 X9 
 
 XIO 
 
 XI 1 
 
 
 10 
 
 
 
 
 132 
 
 g 
 
 7 
 
 
 9 
 
 1 
 
 ■^ 
 
 ^ 
 
 H 
 
 (^ 
 
 16 
 
 5 
 
 8 
 
 1 
 
 7 
 
 4 
 
 S 
 
 ? 
 
 1 
 
 1 
 
 
 1 
 
 
 
 101 
 
 n 
 
 3 
 
 ? 
 
 S 
 
 
 2 
 
 1 
 
 fi 
 
 3 
 
 8 
 
 4 
 
 8 
 
 14 
 
 12 
 
 12 
 
 12 
 
 15 
 
 10 
 
 11 
 
 7 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 139 
 
 n 
 
 4 
 
 3 
 
 7 
 
 
 
 
 
 1 
 
 2 
 
 4 
 
 4 
 
 10 
 
 9 
 
 5 
 
 1 
 
 9 
 
 11 
 
 11 
 
 2 
 
 -- 
 
 
 10 
 
 1 
 
 i 
 
 101 
 
 
 
 
 
 
 
 
 
 p. E. trii 
 
 e 
 
 
 
 
 
 
 
 
 Median 
 
 — obtaine 
 median 
 
 d 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 
 Q 
 
 
 P.E. 
 
 10 
 
 7.42 
 
 .32 
 
 
 10.33 
 
 4.37 
 
 
 2.98 
 
 Mil— M10= .73 
 
 .290 
 
 11 
 
 8.15 
 
 .31 
 
 
 10.84 
 
 5.31 
 
 
 2.76 
 
 M12-M11=2.97 
 
 .284 
 
 n 
 
 11.12 
 
 .30 
 
 
 13.81 
 
 8.09 
 
 
 2.86 
 
 M13— M12 = 1.09 
 
 .295 
 
 13 
 
 12.21 
 
 .36 
 
 
 14.97 
 
 9.02 
 
 
 2.97 
 
 M13— M10=4.79 
 
 .304 
 
 Note. — In these and succeeding tables XI is the estimated number in grade 1, 
 scores unknown. X2, X9, XIO and XI 1 represent the corresponding facts for 
 grades 2, 9, 10 and 11 respectively. MIO is the median for 10-year-old children, 
 age last birthday at the time the test was given. Mil, M12 and M13 are the 
 medians for the ages 11, 12 and 13 respectively. 
 
 Also, the progress between the different ages is unequal, as in the 
 case of the children in the 6-month schools. The improvement between 
 ages 10 and 11 is .73 of a step ; between ages 11 and 12 it is 2.97 steps ; 
 and between ages 12 and 13 it is 1.09 steps. The total improvement for 
 the three-year period from ages 10 to 13 inclusive is 4.79 steps, with a 
 P.E. of .304. Since there are 9 months of school each year, the total 
 school time represented is 27 months. The average monthly improve- 
 ment, therefore, is .177 of a step. The corresponding improvement 
 for the 6-month schools is .173 of a step. 
 
 This comparison would be valid and significant if it could be as- 
 sumed that the quality measured, ability to complete correctly sen- 
 tences in which certain words are missing, was improved only by school 
 
26 Study of Achievement in Country and Town Schools 
 
 work. Such, however, is not known to be the case. It is entirely 
 possible that this quality improves by a natural development as the age 
 and experience of the subject increases. It is possible, and even prob- 
 able, that this ability to complete sentences is improved both by natural 
 development and by school work. The conclusion from the above 
 comparison, therefore, is that schooling and natural development ac- 
 complish an improvement of this ability of 4.79 steps between the 
 ages of 10 and 13 inclusive for the children in places in Madison 
 County having a 9-month school term, and of 3.12 steps in places in 
 the same county having a 6-month school term. 
 
 TABLE IV 
 
 Age Distribution, 10 to 13 Inclusive, of Trabue Language Scale B Scores 
 IN Certain White Schools of Louisville, Ky. 
 
 Age 
 
 
 
 Scores 
 
 
 
 Total 
 
 
 XI 
 4 
 
 1 
 
 X2 
 
 12 
 4 
 
 X3 
 
 36 
 12 
 
 1 
 
 2 
 
 3 
 
 4 
 
 1 
 4 
 
 5 
 
 "2 
 
 6 
 
 8 
 5 
 
 7 
 
 2 
 1 
 
 8 
 
 18 
 13 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 X9 
 
 
 10 
 
 4 
 6 
 
 32 
 39 
 
 17 
 9 
 
 43 
 55 
 
 17 
 11 
 
 29 
 67 
 
 6 
 13 
 
 14 
 26 
 
 1 
 1 
 
 3 
 17 
 
 
 
 
 247 
 
 11 
 
 2 
 
 
 
 
 6 
 
 
 292 
 
 P 
 
 
 1 
 
 4 . 
 1 - 
 
 
 
 
 2 
 2 
 
 1 
 
 2 
 3 
 
 3 
 
 1 
 
 21 
 10 
 
 3 
 5 
 
 19 
 22 
 
 12 
 
 8 
 
 43 
 53 
 
 19 
 13 
 
 77 
 65 
 
 lb 
 
 27 
 
 70 
 46 
 
 4 
 6 
 
 27 
 20 
 
 4 
 1 
 
 4 
 
 7 
 
 "26 
 
 332 
 
 1"? 
 
 
 
 
 316 
 
 
 
 
 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 
 75 per 
 
 
 25 per- 
 
 
 Q 
 
 
 P.E. 
 
 
 
 
 
 mediad 
 
 centile 
 
 
 centile 
 
 
 
 
 
 in 
 
 11.32 
 
 .24 
 
 13.48 
 
 
 7.37 
 
 
 3.11 
 
 Mil— M10= 1.58 
 
 .174 
 
 11 
 
 12.90 
 
 .14 
 
 14.37 
 
 
 10.36 
 
 
 2.00 
 
 M12— Mll = 1.56 
 
 .120 
 
 1? 
 
 14.46 
 
 .13 
 
 16.37 
 
 
 12.34 
 
 
 2.01 
 
 M13-M12= .15 
 
 .120 
 
 13 
 
 14.61 
 
 .14 
 
 16.58 
 
 
 12.50 
 
 
 2.04 
 
 M13— M10=3.29 
 
 .174 
 
 Table IV gives the facts in regard to the Trabue Language Scale B 
 test for certain of the white schools of Louisville. The papers on 
 which this table is based were given, collected and scored by the 
 Psychological Laboratory of the Louisville Board of Education. 
 
 The total improvement in ability to complete sentences in these 
 schools for the three-year period from ages 10 to 13 inclusive is 3.29. 
 The P.E. of this gain is .174, a little over 5 per cent of the gain. The 
 probable deviation of the true measure of this gain from the measure 
 obtained is only about half as great as in the Madison County schools, 
 either those having a 9-month school term or those having a 6-month 
 term. The total gain made is a little greater than in the case of the 
 Madison County 6-month schools and is not quite so great as in the 
 case of the Madison County 9-month schools. The median ability of 
 the respective age-groups, on the other hand, is materially higher than 
 
Achievement Measured by Trabue Language Scale 
 
 27 
 
 is the case in any of the Madison County schools. The median for the 
 Louisville 10-year-old children is higher than for any of the age-groups 
 of the 6-month schools, and is higher than all except the 13-year-old 
 group for the 9-month schools. The median ability of the Louisville 
 13-year-old group is 14.61 ; that for the Madison County 9-month 
 schools of -corresponding age is 12.21, and for the 6-month schools is 
 9.75. The 13-year-old children in Louisville have an ability to com- 
 plete sentences that is 50 per cent greater than the like ability of chil- 
 dren of the same age in the 6-month schools of Madison County. 
 
 The progress made each year in the Louisville schools is unequal, 
 as is the case in the Madison County schools. In Louisville the prog- 
 ress from 10 to 11 is 1.58; from 11 to 12 it is 1.56; and from 12 to 13 
 it is .15. 
 
 TABLE V 
 
 Age Distribution, 10 to 13 Inclusive, of Trabue Language Scale B Scores 
 
 IN Paterson, N. J. 
 
 
 
 Scores 
 
 
 
 
 
 
 Age 
 
 
 
 
 
 
 
 Total 
 
 
 XI 
 
 
 5 
 
 1 
 2 
 
 2 
 
 7 
 1 
 
 3 
 
 2 
 
 4 
 
 5 
 6 
 5 
 
 5 
 
 4 
 4 
 
 2 
 
 6 
 
 23 
 11 
 
 8 
 
 7 
 
 18 
 10 
 
 8 
 
 8 
 
 31 
 17 
 15 
 
 9 
 
 20 
 16 
 16 
 
 10 
 
 34 
 34 
 39 
 
 11 
 
 23 
 27 
 19 
 
 12 
 
 15 
 29 
 32 
 
 1 
 
 1 
 
 1 
 2 
 
 3 14 
 
 3 4 
 8 11 
 5 18 
 
 15 
 
 2 
 4 
 9 
 
 16 
 
 17 
 
 18 
 
 19 
 
 X9 
 
 XIO 
 
 
 10 
 
 1 
 7 
 9 
 
 
 
 
 
 
 209 
 
 n 
 
 2 
 3 
 
 2 
 3 
 
 1 
 
 1 
 
 
 
 200 
 
 12 
 
 
 
 1 
 
 
 8 
 
 
 224 
 
 ii-t 
 
 
 
 
 
 1 
 
 1 
 
 2 
 
 6 
 
 2 
 
 14 
 
 13 
 
 25 
 
 20 
 
 29 
 
 2 
 
 6 17 
 
 16 
 
 12 
 
 9 
 
 1 
 
 
 42 
 
 8 
 
 246 
 
 
 
 
 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 
 
 75 per- 
 
 25 per- 
 
 
 Q 
 
 
 
 P.E. 
 
 
 
 median 
 
 
 centile 
 
 centile 
 
 
 
 
 
 
 10 
 
 9.03 
 
 .12 
 
 
 10.01 
 
 7.02 
 
 
 1.49 
 
 
 Mll-M10=2.27 
 
 .128 
 
 11 
 
 11.30 
 
 .16 
 
 
 12.82 
 
 9.06 
 
 
 1.88 
 
 
 M12— Mll= .48 
 
 .141 
 
 12 
 
 11.78 
 
 .16 
 
 
 13.80 
 
 9.87 
 
 
 1.96 
 
 
 M13— M12 = 1.52 
 
 .180 
 
 13 
 
 13.30 
 
 .24 
 
 
 16.87 
 
 10.82 
 
 
 3.02 
 
 
 M13-M10=4.27 
 
 .170 
 
 Table V gives the facts in regard to the Trabue Language Scale B 
 for schools in Paterson, New Jersey. The data from which this table 
 is compiled were collected by the department of school administration 
 of Teachers College, Columbia University. 
 
 The total gain in ability to complete sentences in the Paterson schools 
 is 4.27. This gain is greater than in the case of any other schools pre- 
 sented in this study. The P.E. of this gain is .170, which is about the 
 same as that for the like measure in Louisville. Since the gain in this 
 case is greater, the proportion of the probable error to the gain is a 
 little less; in Paterson 4 per cent as against a little more than 5 per 
 cent in Louisville. The median ability of the 13-year-old children in 
 the Paterson schools is 13.30 as against 14.61 in the Louisville schools. 
 
28 
 
 Study of Achievement in Country and Town Schools 
 
 CHART I 
 Progress in Trabue Language Completion Test Scores Between Age 
 Groups in the Public Schools of Madison County, Ky. ; Louis- 
 ville, Ky. ; Paterson, N. J.; and St. Paul, Minn. 
 
 SCORES 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 II 
 
 (0 
 
 9 
 
 8 
 
 T 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 I 
 
 
 AGES 10 
 
 MADISOR COUNTY 
 9-MONmSCMOOLS 
 
 MADISON^tlf 
 e-NONTfiSCHOOLS 
 
 12 
 
 13 
 
Achievement Measured by Trabue Language Scale 
 
 29 
 
 This median ability of the 13-year-old children in Paterson, however, 
 is greater than is the like measure for the 6-month or the 9-month 
 schools of Madison County. 
 
 The yearly gains are again unequal. From 10 to 11 the gain is 
 2.27; from 11 to 12 it is .48; and from 12 to 13 it is 1.52. 
 
 Table VI gives the facts in regard to the Trabue Language Scale B 
 in St. Paul, Minnesota. This table is based upon data printed in the 
 St. Paul Survey. 
 
 TABLE VI 
 
 Age Distribution, 10 to 13 Inclusive, of Trabue Language Scale B Scores in 
 St. Paul, Minn. Based on Tables Published in the St. Paul Survey 
 
 Age 
 
 Scores 
 
 Total 
 
 10 
 
 
 
 4 
 
 1 
 
 1 
 
 2 
 
 15 
 2 
 
 1 
 2 
 
 3 
 
 4 
 1 
 1 
 1 
 
 4 
 
 19 
 8 
 3 
 5 
 
 5 
 
 9 
 6 
 4 
 
 1 
 
 6 
 
 30 
 24 
 13 
 10 
 
 7 
 
 24 
 8 
 13 
 11 
 
 8 
 
 56 
 39 
 42 
 21 
 
 9 
 
 33 
 25 
 21 
 16 
 
 10 
 
 63 
 53 
 55 
 41 
 
 11 
 
 41 
 30 
 39 
 39 
 
 12 
 
 52 
 53 
 77 
 61 
 
 13 
 
 " 26 
 35 
 45 
 54 
 
 14 
 
 24 
 50 
 50 
 66 
 
 15 
 
 12 
 8 
 30 
 40 
 
 16 
 
 4 
 11 
 20 
 29 
 
 17 
 
 1 
 3 
 4 
 14 
 
 18 
 
 19 
 
 X9 
 
 417 
 
 11 
 12 
 13 
 
 '"■4 
 
 7 
 
 2 
 .... 
 
 "38 
 
 359 
 422 
 453 
 
 
 Median 
 
 P. E. true 
 
 —obtained 
 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 11 
 12 
 13 
 
 10.32 
 11.41 
 12.24 
 13.34 
 
 .13 
 .14 
 .11 
 .12 
 
 12.28 
 13.55 
 14.05 
 15.29 
 
 7.96 
 9.03 
 10.13 
 11.13 
 
 2.25 
 2.26 
 1.96 
 2.08 
 
 Mil— M10=1.18 
 M12— Mll= .83 
 M13— M12 = 1.10 
 M13-M10=3.11 
 
 .120 
 .106 
 .098 
 .106 
 
 The total gain in ability to complete sentences in this case is 3.11. 
 This gain is less than in any of the other systems studied. The median 
 ability of the 13-year-old children, on the other hand, is 13.34, which 
 is higher than the like ability of the same age-group in any of the other 
 systems studied except Louisville. The P.E. of the gain in these 
 schools is .106, which is less than 4 per cent of the gain made. The 
 reliability of the measure in St. Paul seems a little greater than in any 
 of the other systems considered. The progress from age to age is un- 
 equal. From ages 10 to 11 it is 1.18; from ages 11 to 12 it is .83 ; and 
 from ages 12 to 13 it is 1.10. 
 
 Chart I shows the progress for the different systems both relatively 
 and absolutely. It is significant that while the progress in the Madison 
 County schools having a 9-month school term is relatively a little 
 greater than the progress in the city schools presented, the total result 
 is less. In other words, although the children gain as much or a little 
 more in ability to complete sentences between the ages of 10 and 13, 
 they have less of this ability when they began and less when they get 
 
30 
 
 Study of Achievement in Country and Town Schools 
 
 through. This would seem to indicate either later entrance in school 
 or poorer work with the lower grades, possibly both. 
 
 The following table summarizes the total gain in ability to complete 
 sentences, as measured by the Trabue Language Scale B, between the 
 ages of 10 and 13, for the respective school systems considered. It 
 also shows the P.E. of this total gain, and the number of cases on 
 which this measure of gain is based. 
 
 School Systems 
 
 Total 
 
 GAIN 
 
 P.E. OF 
 
 TOTAL 
 
 GAIN 
 
 Total 
 frequencies 
 
 
 10 to 13 
 
 10 and 13 
 
 Madison County, 6-month schools 
 
 Madison County, 9- month schools 
 
 Louisville 
 
 3.12 
 4.79 
 3.29 
 4.27 
 3.11 
 
 .316 
 .304 
 .174 
 .170 
 .106 
 
 252 
 233 
 563 
 
 Paterson . .. _ 
 
 455 
 
 St. Paul 
 
 870 
 
 
 
 It should be noted that the reliability of the gains given, as indicated 
 by a relatively small P.E., is much greater in the city schools than in 
 the Madison County schools. This is due in part to the greater num- 
 ber of cases. That it is not wholly due to this cause is indicated by 
 the fact that the reliability is greater for the Madison County 9-month 
 schools than for the 6-month schools, though the number of cases is 
 slightly less. It is possible that this greater reliability is due in part 
 to a more uniform type of work in the city and town school systems 
 than in the country schools. 
 
 It is also to be noted that the combination of elimination, natural 
 development and schooling over a three-year period increased the ability 
 to complete sentences 4.79 steps in places in Madison County having 
 a 9-month school term,. 4.27 steps in Paterson, 3.29 steps in Louisville, 
 3.12 steps in places in Madison County having a 6-month school term, 
 and 3.11 steps in St. Paul. If it may be assumed that the amount of 
 this increase due to natural development is the same in all these places, 
 the efficiency of the schools in training and eliminating children be- 
 tween the ages of 10 and 13 in such ability in language as is measured 
 by the Trabue Language Scales would rank in the above order. 
 
 This ranking does not agree with that obtained by ranking in order 
 of the median ability of the 13-year-old children. Ranking in this 
 way, Louisville comes first with a median of 14.61, St. Paul second 
 with 13.34, Paterson third with 13.30, Madison County 9-nionth schools 
 fourth with 12.21, and Madison County 6-month schools last with 9.75. 
 
Achievement Measured by Trabue Language Scale 
 
 31 
 
 This discrepancy may be due to a difference in the work with chil- 
 dren under 10, to a greater irregularity of attendance of the younger 
 children in some systems than in others, to a difference in the amount 
 of gain due to natural development because of different environment, 
 to a difference in care in scoring papers, to a difference in intellectual 
 level of the communities, or to a combination of these causes. 
 
 The following table shows the yearly gains in each of the systems 
 of schools studied : 
 
 School Systems 
 
 
 Yearly Gains 
 
 
 
 10 to 11 
 
 11 to 12 
 
 12 to 13 
 
 Madison County, 6-month schools _ . _ 
 
 .87 
 
 .73 
 
 1.58 
 
 2.27 
 
 1.18 
 
 1.52 
 2.97 
 1.56 
 
 .48 
 .83 
 
 .73 
 
 Madison County, 9-month schools.. 
 
 1.09 
 
 Louisville 
 
 .15 
 
 Paterson . 
 
 1.52 
 
 St. Paul 
 
 1 10 
 
 
 
 Total .... . ... ... 
 
 6.63 
 
 7.36 
 
 4.59 
 
 
 
 This table seems to indicate that the greatest gain is made from 
 ages 11 to 12, but that material gain may be made during any of the 
 yearly periods from ages 10 to 13 inclusive. 
 
 CONCLUSIONS 
 
 1. The difference in performance as measured by the Trabue Lan- 
 guage Scale B of school children of different ages can be obtained. 
 
 2. Since this test measures a quality not proved to be solely, or even 
 largely, determined by school training, this difference is not a reliable 
 measure of school efficiency. 
 
 3. This difference does measure the improvement of children in the 
 respective systems as to the function measured, such improvement be- 
 ing due to school training and other influences. 
 
 4. The children in the Madison County 6-month schools show less 
 improvement during the three-year period from 10 to 13 inclusive than 
 do the children in any of the other systems except St. Paul. 
 
 5. The 13-year-old children in the Madison County 6-month schools 
 show less ability in the function measured than do the children of like 
 age in any of the other systems studied. 
 
 6. The children in the Madison County 9-month schools show rea- 
 sonable improvement during the three-year period from 10 to 13 in- 
 
32 Study of Achievement in Country and Tozvn Schools 
 
 elusive, but the ability of the 13-year-old children to complete sen- 
 tences is less than the like ability of 13-year-old children in any of the 
 systems studied except the 6-month schools in the same county. 
 
CHAPTER IV 
 
 ACHIEVEMENT AS MEASURED BY THE COURTIS STANDARD 
 RESEARCH TESTS IN ARITHMETIC SERIES B 
 
 At the same time that the Trabue Language Scale B was given in 
 the schools of Madison County, the Courtis Standard Research Test 
 in Arithmetic Series B was used. 
 
 The problems involved in finding the difference in performance of 
 different age-groups are practically the same as in the work with the 
 Trabue Language Scale B. Distributions for the respective age- 
 groups are completed on the same assumptions, and the same age- 
 groups are used throughout. The number of examples correctly com- 
 pleted has been used as the score. 
 
 The work in addition, subtraction, multiplication and division is 
 presented in sections 1, 2, 3 and 4 respectively of this chapter. 
 
 Section L Addition 
 Table VII gives the data in addition for the children 10 to 13 years 
 of age inclusive in the Madison County schools having a 6-month term. 
 A new problem is presented in this table in that the medians of the 
 
 TABLE VII 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Addition Rights in the 6-Month White Schools of Madison County, Ky. 
 
 
 Scores 
 
 
 Age 
 
 
 Total 
 
 
 XI 
 
 X2 
 
 
 
 65 
 54 
 62 
 45 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 X9 
 
 XIO 
 
 XU 
 
 
 10 
 
 11 
 5 
 3 
 
 2 
 
 19 
 11 
 5 
 3 
 
 25 
 20 
 19 
 16 
 
 9 
 12 
 19 
 13 
 
 7 
 10 
 5 
 8 
 
 6 
 
 6 
 
 7 
 
 11 
 
 
 1 
 2 
 4 
 
 1 
 
 
 
 
 
 
 144 
 
 11 
 
 4 
 8 
 5 
 
 
 
 2 
 4 
 13 
 
 
 
 126 
 
 12 
 
 2 
 
 
 
 2 
 4 
 
 
 140 
 
 13 
 
 
 1 
 
 2 
 
 123 
 
 
 
 
 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P. E. 
 
 10 
 
 .64 
 
 .07 
 
 1.52 
 
 .09 
 
 .71 
 
 Mil— M10= .23 
 
 .072 
 
 11 
 
 .87 
 
 .11 
 
 2.37 
 
 .28 
 
 1.04 
 
 M12— Mll= .13 
 
 .092 
 
 12 
 
 1.00 
 
 .12 
 
 2.48 
 
 .43 
 
 1.20 
 
 M13— M12= .71 
 
 .147 
 
 13 
 
 1.71 
 
 .21 
 
 4.47 
 
 .57 
 
 1.90 
 
 M13— M10=1.07 
 
 .136 
 
 10- and 11 -year-old groups fall in the to 1 interval, and that the 25 
 percentiles of all age-groups fall in the same interval. It may be 
 
 33 
 
34 Study of Achievement in Country and Town Schools 
 
 questioned whether the children in grades 1 and 2, grouped under 
 the captions XI and X2, should be considered as having a score of 0, 
 or whether an attempt should be made to distribute their scores. Tab- 
 ulation of the pupils' test papers shows that there are 147 children in 
 grade 3, and that 127 of these grade 3 children made a score of 0. 
 Since 127 out of 147 children made a score of and since it may be 
 safely assumed that children in grades 1 and 2 would not do so well 
 as children in grade 3, the score of these first and second grade chil- 
 dren has been considered as 0. Scores given under the caption have 
 been considered as distributed evenly between and 1. 
 
 The large number of scores in the to 1 interval as well as the 
 scores considered as indicate that the ability in addition required to 
 do any of the examples in the Courtis Standard Research Tests in 
 Arithmetic Series B is so great as to preclude measurement of many 
 of the individuals tested further than to show that their ability is less 
 than the lowest ability measured by this test. The great number of 
 cases also reduces Q. 
 
 TABLE VIII 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Addition Rights in the 9-Month White Schools of Madison County, Ky. 
 
 
 
 Scores 
 
 
 
 
 Age 
 
 
 
 
 
 Total 
 
 
 XI 
 
 X2 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 X9 
 
 XIO 
 
 Xll 
 
 
 10 
 
 9 
 1 
 3 
 
 16 
 9 
 
 1 
 
 23 
 21 
 21 
 
 14 
 13 
 19 
 
 19 
 12 
 15 
 
 11 
 
 7 
 24 
 
 16 
 10 
 17 
 
 6 
 
 7 
 
 5 
 9 
 13 
 
 5 
 2 
 6 
 
 
 1 
 
 2 
 
 1 
 
 1 
 3 
 1 
 
 
 
 
 127 
 
 11 
 
 1 
 1 
 
 
 
 94 
 
 12 
 
 2 1 
 
 
 1 
 
 
 131 
 
 13 
 
 4 
 
 3 
 
 7 
 
 15 
 
 11 
 
 11 
 
 17 
 
 8' 
 
 5 
 
 3 
 
 
 2 1 
 
 
 1 
 
 8 
 
 1 
 
 1 
 
 98 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 
 2.07 
 
 .21 
 
 4.20 
 
 .26 
 
 1.97 
 
 Mil— M10= .18 
 
 .206 
 
 11 
 
 2.25 
 
 .26 
 
 4.75 
 
 .64 
 
 2.05 
 
 M12— Mll = 1.02 
 
 .194 
 
 12 
 
 3.27 
 
 .19 
 
 4.89 
 
 1.40 
 
 1.74 
 
 M13— M12= .54 
 
 .186 
 
 13 
 
 3.81 
 
 .25 
 
 5.68 
 
 1.70 
 
 1.99 
 
 M13— M10 = 1.74 
 
 .198 
 
 The total gain in ability to add for the three-year period from ages 
 
 10 to 13, as measured by the difference in the median scores of the 
 10-year-old and the 13-year-old children who took the test, is 1.07, 
 with a P.E. of .136. This P.E. is slightly over 12 per cent of the total 
 gain indicated. 
 
 The improvement in ability to add from year to year, as measured 
 by the difference between the median scores of the respective age- 
 groups, is uneven, as it is in the case of language. From ages 10 to 
 
 11 the improvement is .23; from ages 11 to 12 it is .13; and from ages 
 
 \ 
 
Achievement Measured by Courtis Arithmetic Tests 
 
 35 
 
 12 to 13 it is .71. The improvement from ages 12 to 13 is practically 
 twice as great as the combined improvement from ages 10 to 11 and 
 from ages 11 to 12. Apparently the first real teaching of such ability 
 to add as is measured by the Courtis Standard Research Tests in Arith- 
 metic Series B in these schools is to children between the ages of 12 
 and 13. 
 
 Table VIII presents the facts in regard to addition in the 9-month 
 schools of Madison County. The same statistical method is used as 
 that used with the data from the 6-month schools. 
 
 The total improvement in ability to add in these schools for the 
 three-year period from ages 10 to 13 inclusive is 1.74. The P.E. of this 
 improvement is .198, or slightly under 12 per cent of the total gain. 
 The accuracy of the measurement for the 6-month schools is indi- 
 cated by a P.E. showing slightly over 12 per cent of the total gain ; 
 just about the same degree of reliability. 
 
 The improvement by years varies for these schools as it does for 
 
 TABLE IX 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 Addition Rights in Louisville, Ky. (White Schools) 
 
 Age 
 
 Scores 
 
 Total 
 
 in 
 
 XI 
 
 8 
 5 
 
 X2 
 
 16 
 8 
 5 
 
 X3 
 
 43 
 16 
 8 
 5 
 
 
 
 12 
 10 
 6 
 
 8 
 
 1 
 
 16 
 11 
 10 
 9 
 
 2 
 
 19 
 21 
 22 
 19 
 
 3 
 
 38 
 28 
 23 
 18 
 
 4 
 
 33 
 31 
 25 
 25 
 
 5 
 
 27 
 35 
 34 
 29 
 
 6 
 
 28 
 29 
 26 
 26 
 
 7 
 
 14 
 
 28 
 28 
 30 
 
 8 
 
 18 
 27 
 37 
 24 
 
 9 
 
 10 
 21 
 
 28 
 25 
 
 10 
 
 5 
 10 
 13 
 
 18 
 
 11 
 
 6 
 \0 
 13 
 13 
 
 12 
 
 1 
 4 
 6 
 13 
 
 13 
 
 1 
 
 2 
 9 
 9 
 
 14 
 
 3 
 4 
 1 
 
 15 
 
 1 
 
 "2 
 4 
 
 16 
 
 "3 
 2 
 
 2 
 
 17 
 
 1 
 1 
 1 
 1 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 X9 
 
 289 
 
 11 
 
 
 
 1 
 
 
 
 
 
 304 
 
 1? 
 
 1 
 
 1 
 1 
 
 "25 
 
 304 
 
 13 
 
 
 1 
 
 
 
 306 
 
 
 
 
 
 
 
 iledian 
 
 P. E. true 
 — obtained 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 
 11 
 
 12 
 13 
 
 3.77 
 5.62 
 6.73 
 7.46 
 
 .21 
 
 .18 
 .18 
 .27 
 
 6.26 
 8.22 
 9.14 
 12.80 
 
 .43 
 3.17 
 4.08 
 5.08 
 
 2.91 
 2.52 
 2.53 
 3.86 
 
 Mll-M10=1.85 
 M12— Mll = l.ll 
 M13— M12= .73 
 M13— M10=3.69 
 
 .170 
 .155 
 .202 
 .213 
 
 the 6-month schools. From ages 10 to 11 it is .18; from ages 11 to 12 
 it is 1.02; and from ages 12 to 13 it is .54. The greatest gain occurs 
 between the ages of 11 and 12. This occurs a year earlier than the 
 greatest gain in the case of the 6-month schools. 
 
 The median achievement for children of each age is materially higher 
 than is the case in the 6-month schools. For the 13-year-old children 
 this median achievement is 3.81 in the 9-month schools as against 
 1.71 in the 6-month schools ; a difference of 2.10 in favor of the schools 
 having the longer term. 
 
36 Study of Achievement in Country and Town Schools 
 
 For the purposes of comparison, data are presented in Tables IX 
 to XIII inclusive from certain other school systems. 
 
 Table IX, for Louisville, Kentucky, is based on data in the Psycho- 
 logical Laboratory of the Louisville Board of Education, and is for cer- 
 tain of the white schools of the city. The tests were given under the 
 direction of the Psychological Laboratory during the school year prior 
 to June, 1918. 
 
 Table X, for Louisiana, is based on tests given by the parish super- 
 visors in thirteen Louisiana parishes. The schools tested were white 
 schools selected by the supervisors as neither the best nor the worst 
 in their respective parishes. They range in size from one teacher to 
 
 TABLE X 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Addition Rights in Certain White Schools in Louisiana 
 
 Age 
 
 Scores 
 
 Total 
 
 
 XI 
 
 24 
 13 
 
 7 
 
 X2 
 
 43 
 24 
 13 
 
 7 
 
 X3 
 
 58 
 43 
 24 
 13 
 
 
 
 48 
 47 
 45 
 50 
 
 1 
 
 31 
 38 
 37 
 35 
 
 2 
 
 24 
 27 
 51 
 42 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 X9 
 
 XIO 
 
 Xll 
 
 
 10 
 11 
 12 
 
 23 
 28 
 41 
 31 
 
 12 
 20 
 30 
 98 
 
 10 
 16 
 18 
 16 
 
 6 
 10 
 11 
 16 
 
 4 
 8 
 11 
 
 8 
 
 1 
 2 
 6 
 6 
 
 -.1 
 
 1 
 
 "5 
 
 1 
 1 
 1 
 
 1 
 
 
 
 
 
 
 
 
 287 
 
 1 
 1 
 
 
 
 
 
 1 
 
 1 
 
 17 
 
 
 
 280 
 
 
 
 
 1 
 
 1 
 1 
 
 .... 
 
 299 
 
 13 
 
 
 
 
 277 
 
 
 
 
 
 
 
 
 
 
 
 Median 
 
 P. E. true 
 
 — obtained 
 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 11 
 12 
 13 
 
 .38 
 1.34 
 2.46 
 2.79 
 
 .09 
 .13 
 .12 
 
 .15 
 
 2.46 
 3.64 
 4.20 
 5.10 
 
 0.00 
 0.00 
 
 .68 
 .98 
 
 1.23 
 1.83 
 1.76 
 2.06 
 
 Mil— M10= .96 
 M12— Mll = 1.12 
 M13— M12= .33 
 M13— M10=2.41 
 
 .094 
 .106 
 .114 
 .102 
 
 TABLE XI 
 
 Age DiSTRreuTiON, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Addition Rights in Arkansas City 
 
 
 Scores 
 
 Total 
 
 
 XI 
 
 X2 
 
 X3 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 X9 
 
 
 10 
 
 15 
 8 
 3 
 1 
 
 54 
 
 15 
 
 8 
 
 3 
 
 65 
 54 
 15 
 
 8 
 
 54 
 49 
 63 
 34 
 
 46 
 33 
 54 
 32 
 
 19 
 35 
 37 
 36 
 
 29 
 17 
 33 
 39 
 
 14 
 
 29 
 34 
 37 
 
 8 
 10 
 
 22 
 32 
 
 6 
 4 
 
 8 
 12 
 
 2 
 3 
 6 
 4 
 
 
 
 
 
 312 
 
 11 
 
 2 
 8 
 6 
 
 
 
 
 259 
 
 12 
 13 
 
 4 
 
 3 
 5 
 
 ..... 
 
 294 
 254 
 
 
 Median 
 
 P. E. true 
 
 — obtained 
 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 11 
 12 
 13 
 
 .40 
 1.10 
 2.10 
 3.33 
 
 .03 
 .11 
 .12 
 .13 
 
 1.00 
 3.01 
 4.22 
 5.01 
 
 0.00 
 
 0.00 
 
 .75 
 
 1.64 
 
 .50 
 1.50 
 1.73 
 1.68 
 
 Mil— M10= .70 
 M12— Mll = 1.00 
 M13-M12 = 1.23 
 M13— M10 = 2.93 
 
 .072 
 .098 
 .106 
 .082 
 
 sixteen teachers. In some cases only a limited number of the pupils 
 were tested in each grade. When this was done, the teacher was asked 
 
Achievement Measured by Courtis Arithmetic Tests 
 
 37 
 
 to select pupils neither the brightest nor the dullest. This manifestly 
 gives a selection that is not accurately representative of the Louisiana 
 schools ; it seems likely that the selection tends to be above rather than 
 below the median situation, and the range is less in proportion as the 
 teachers and supervisors have succeeded in avoiding the extreme cases. 
 Tables XI and XII, for Arkansas City and for Salina, are based on 
 tabulations made from papers collected by the Bureau of Educational 
 Measurements and Standards of the State Normal School at Emporia, 
 
 TABLE XII 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 Addition Rights in Salina, Kan. 
 
 Age 
 
 Scores 
 
 Total 
 
 
 XI 
 
 16 
 4 
 2 
 
 X2 
 
 20 
 16 
 4 
 
 2 
 
 X3 
 
 43 
 20 
 16 
 
 4 
 
 
 
 46 
 15 
 14 
 13 
 
 1 
 
 43 
 49 
 24 
 13 
 
 2 
 
 44 
 42 
 38 
 30 
 
 3 
 
 44 
 36 
 48 
 27 
 
 4 
 
 26 
 36 
 57 
 40 
 
 5 
 
 20 
 37 
 43 
 
 32 
 
 6 
 15 
 
 n 
 
 29 
 40 
 
 7 
 
 14 
 8 
 6 
 
 16 
 
 8 
 
 12 
 8 
 17 
 
 9 1 
 
 3 
 
 8 . 
 6 
 11 1 
 
 Oil 
 1 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 X9 XIO 
 
 
 10 
 
 
 
 
 
 
 
 
 
 
 
 
 335 
 
 n 
 
 9 "i 
 
 5 
 
 1 
 4 
 4 
 
 
 
 
 
 
 
 
 
 2 
 
 ""2 
 5 
 
 "2 
 
 296 
 
 1? 
 
 
 
 
 
 
 
 
 
 311 
 
 n 
 
 1 
 
 
 
 
 
 
 2 
 
 
 
 274 
 
 
 
 
 
 
 
 
 
 Median 
 
 P. E. true 
 
 — obtained 
 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 
 11 
 
 12 
 13 
 
 1.98 
 3.06 
 4.16 
 5.25 
 
 .12 
 .13 
 .12 
 .15 
 
 3.89 
 5.10 
 5.70 
 7.25 
 
 .10 
 1.34 
 2.46 
 3.24 
 
 1.89 
 1.34 
 1.62 
 2.00 
 
 Mil— M10=1.08 
 M12— Mll = 1.10 
 M13— M12 = 1.09 
 M13— M10=3.27 
 
 .120 
 .114 
 .114 
 .120 
 
 TABLE XIII 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 Addition Rights in Hibbing, Minn. 
 
 
 
 Scores 
 
 
 
 
 
 Age 
 
 
 
 
 
 
 Total 
 
 
 X2 
 16 
 
 X3 
 
 34 
 16 
 
 
 
 34 
 23 
 11 
 3 
 
 1 
 
 27 
 30 
 16 
 4 
 
 2 
 
 31 
 20 
 25 
 
 8 
 
 3 
 
 32 
 32 
 16 
 15 
 
 4 
 
 24 
 
 11 
 10 
 
 5 
 
 19 
 20 
 19 
 13 
 
 6 
 
 12 
 
 12 
 24 
 9 
 
 7 
 
 12 
 17 
 17 
 16 
 
 8 
 
 7 
 
 12 
 19 
 18 
 
 9 
 
 3 
 9 
 16 
 11 
 
 10 
 
 2 
 9 
 13 
 11 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 X9 
 
 XIO 
 
 
 10 
 
 1 
 8 
 4 
 10 
 
 1 
 3 
 
 7 
 2 
 
 1 
 4 
 3 
 
 1 
 
 1 
 3 
 3 
 
 1 
 
 1 
 5 
 2 
 
 
 
 
 
 
 
 
 
 
 257 
 
 11 
 
 
 
 
 
 
 
 
 
 
 244 
 
 I*? 
 
 2 
 
 "2 
 
 2 
 2 
 
 
 
 
 2 
 
 4 
 33 
 
 ■""4 
 
 222 
 
 13 
 
 
 
 1 
 
 1 
 
 
 179 
 
 
 
 
 
 
 
 P.E. true 
 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 
 P.E. 
 
 10 
 
 2.56 
 
 .16 
 
 4.78 
 
 .41 
 
 2.18 
 
 
 Mil— M10=1.44 
 
 .164 
 
 11 
 
 4.00 
 
 .21 
 
 7.22 
 
 1.73 
 
 2.74 
 
 
 M12— Mll = 2.41 
 
 .206 
 
 12 
 
 6.41 
 
 .25 
 
 9.34 
 
 3.21 
 
 3.06 
 
 
 M13— M12 = 2.22 
 
 .322 
 
 13 
 
 8.63 
 
 .45 
 
 15.12 
 
 5.36 
 
 4.88 
 
 
 M13— M10=6.07 
 
 .297 
 
38 Study of Achievement in Country and Tozvn Schools 
 
 Kansas. The tests were given and scored by the teachers during the 
 two years prior to June, 1917. 
 
 Table XIII, for Hibbing, Minnesota, is based on papers collected by 
 the Department of Research of the Hibbing schools. The tests were 
 given by the teachers and scored by clerks in the fall of 1919. 
 
 In the following summary the total improvement in ability to add 
 for the three-year period from ages 10 to 13 inclusive is given for each 
 system of schools reported, and also the median ability of each 13-year- 
 old group. This median for the 13-year-old group may be considered 
 in a way as a measure of the result obtained in any system with chil- 
 dren up to the age of 13. 
 
 School Systems 
 
 Hibbing 
 
 Louisville 
 
 Salina 
 
 Arkansas City 
 
 Louisiana 
 
 Madison County, 9-month schools 
 Madison County. 6-month schools 
 
 Total gain 
 
 10 to 13 
 
 inclusive 
 
 Td.O? 
 3.69 
 3.27 
 2.93 
 2.41 
 1.74 
 1.07 
 
 Medians 
 
 13-year-old 
 children 
 
 8.63 
 7.46 
 5.25 
 3.33 
 2.79 
 3.81 
 1.71 
 
 It is significant that the ranking in order of median ability of 13- 
 year-old children so nearly coincides with ranking in order of total 
 improvement for the three-year period from 10 to 13 inclusive. The 
 only difference is that the order of Louisiana schools and the Madison 
 County 9-month schools is inverted in the two rankings. The ability 
 to add is a mechanical ability, taught definitely in the schools, and very 
 little affected by the process of natural development which was so 
 potent a factor in the case of language ability as measured by the Tra- 
 bue Language Scale. 
 
 Differences in method of giving the tests and in detail of procedure 
 in different places affect the medians of the 13-year-old groups of chil- 
 dren. These elements of error may be of determining size in cases 
 where the tests are given by interested parties, such as teachers. When 
 the difference between medians for groups of children of different 
 ages is taken as a measure, the sources of error referred to above 
 neutralize each other, since it may be assumed that the errors will be 
 as great for one group of children as for another group, the tests be- 
 ing given at the same time in the same system and with the same 
 
Achievement Measured by Courtis Arithmetic Tests 
 
 39 
 
 general directions or lack of directions. The ranking in order of the 
 total gain in ability to add over the three-year period from ages 10 to 
 13 inclusive is therefore the more significant ranking. 
 
 The following summary shows the yearly gain in each of the sys- 
 tems of schools studied : 
 
 School Systems 
 
 Hibbing 
 
 Louisville 
 
 Salina 
 
 Arkansas City 
 
 Louisiana 
 
 Madison County, 9-month schools 
 Madison County, 6-month schools 
 
 Total '. 
 
 Yearly Gains 
 
 10 to 11 
 
 1.44 
 1.85 
 1.08 
 .70 
 .96 
 .18 
 .23 
 
 6.44 
 
 11 to 12 
 
 2.41 
 1.11 
 1.10 
 1.00 
 1.12 
 1.02 
 .13 
 
 7.89 
 
 12 to 13 
 
 2.22 
 .73 
 1.19 
 1.23 
 .33 
 .54 
 .71 
 
 6.95 
 
 The extreme variation in the amount of yearly gain is significant 
 in that it shows so great a lack of uniformity in the time of teaching 
 so fundamental a process as that of column addition. The totals in- 
 dicate that as a rule greater progress is made in teaching this process 
 between the ages of 11 and 12; that the period from ages 12 to 13 is 
 next; and that the least gain is made from ages 10 to 11. 
 
 CONCLUSIONS 
 
 1. Such difference in performance in addition as is measured by 
 the Courtis Standard Research Tests can be obtained for school chil- 
 dren of different ages. 
 
 2. Since this test measures an ability trained in the schools and not 
 greatly dependent on natural development, this difference in perform- 
 ance is a valid measure of school efficiency in teaching this process, 
 
 3. Material gain in ability to add may be made with children dur- 
 ing any of the age periods, 10 to 11, 11 to 12, or 12 to 13. 
 
 4. The difference in ability of children of different age-groups is a 
 valid measure, even though there may have been some irregularity in 
 the giving of the tests. 
 
 5. That all the school systems investigated are doing better work in 
 the teaching of addition than are the schools of Madison County. 
 
40 Study of Achievement in Country and Town Schools 
 
 6. That the schools in Madison County having a 6-month school 
 term are doing poorer work than the schools having a 9-month term. 
 That is to say, the one- and two-teacher schools are doing poorer work 
 than the larger schools. 
 
 Section 2. Subtraction 
 
 The data presented in subtraction are tabulated from the same sets 
 of papers used for addition. The subtraction tests were given under 
 the same conditions as those in addition, by the same people, and were 
 corrected and collected in the same way. The statistical procedure 
 has been the same in tabulating and calculating. 
 
 Table XIV gives the facts in regard to subtraction in the Madison 
 County 6-month schools. 
 
 TABLE XIV 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Subtraction Rights in the 6-Month White Schools 
 
 OF Madison County, Ky. 
 
 
 Scores 
 
 _ 
 
 
 Age 
 
 XI 
 
 X2 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 X9 
 
 XIO 
 
 XU 
 
 Tota 
 
 10 
 11 
 12 
 13 
 
 10 
 6 
 3 
 
 2 
 
 19 
 10 
 6 
 3 
 
 76 
 69 
 61 
 
 48 
 
 13 
 11 
 13 
 12 
 
 7 
 6 
 13 
 11 
 
 5 
 6 
 10 
 
 8 
 
 8 
 8 
 8 
 8 
 
 2 
 3 
 
 7 
 5 
 
 1 
 
 " "3 
 
 4 
 
 
 
 
 
 
 
 
 
 
 141 
 
 3 
 
 4 
 2 
 
 
 
 
 
 1 
 
 2 
 4 
 11 
 
 ..... 
 4 
 
 ..... 
 
 125 
 
 2 
 
 1 
 2 
 
 
 
 137 
 
 
 1 
 
 
 123 
 
 
 Median 
 
 P. E. true 
 
 —obtained 
 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 11 
 12 
 13 
 
 .54 
 
 .67 
 
 .97 
 
 1.70 
 
 .05 
 .08 
 .17 
 
 .25 
 
 1.05 
 1.79 
 3.67 
 5.05 
 
 .08 
 .22 
 .41 
 .53 
 
 .48 
 
 .78 
 
 1.63 
 
 2.26 
 
 Mil— M10= .13 
 
 M12— Mll= .30 
 M13— M12= .73 
 M1.3— M10 = 1.16 
 
 .058 
 .111 
 .188 
 .162 
 
 In these schools the median ability in subtraction for children 10, 
 11 and 12 years of age is between and 1, and the 25 percentile for 
 each of the age-groups presented falls within the same interval. Ap- 
 parently most of the children in these schools under 14 years of age 
 possess an ability in subtraction that is so slight that it cannot be 
 measured by this test. 
 
 The total gain in ability to subtract for the three-year period from ages 
 10 to 13 inclusive, as measured by the difference between the median 
 scores of the 10-year-old children and the 13-year-old children, is 1.16. 
 The P.E. of this measure of gain from the true measure is .162, almost 
 14 per cent of the total gain. The probable actual deviation from the 
 difference obtained is reasonably small, but the very small difference 
 
Achievement Measured by Courtis Arithmetic Tests 
 
 41 
 
 obtained makes this probable deviation quite large proportionately. 
 The same thing is true in the case of addition. 
 
 The improvement from year to year increases consistently. From 
 ages 10 to 11 it is .13 ; from ages 11 to 12 it is .20 ; and from ages 12 to 
 13 it is .73. 
 
 Table XV gives the facts in regard to subtraction in the Madison 
 County 9-month schools. 
 
 The median ability in subtraction of each of the age-groups is above 
 
 1. The 25 percentiles of the 10- and 11-year-old groups fall in the 
 
 to 1 interval. 
 
 TABLE XV 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Subtraction Rights in the 9-Month White Schools 
 
 OF Madison County, Ky. 
 
 
 
 
 Scores 
 
 
 
 Age 
 
 
 
 
 
 Total 
 
 
 XI 
 
 X2 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 X9 
 
 XIO 
 
 Xll 
 
 
 10 
 
 9 
 2 
 3 
 
 17 
 
 9 
 2 
 
 38 
 26 
 25 
 
 12 
 12 
 8 
 
 6 
 5 
 6 
 
 6 
 6 
 16 
 
 17 
 6 
 10 
 
 11 
 9 
 13 
 
 7 
 5 
 14 
 
 5 
 6 
 10 
 
 3 
 6 
 
 7 
 
 ] 
 
 1 
 
 '"I 
 3 
 
 2 
 
 
 
 
 
 
 
 124 
 
 11 
 
 
 
 
 1 
 1 
 
 '"\ 
 
 — - 
 
 98 
 
 12 
 
 
 1 
 
 
 1 
 
 130 
 
 13 
 
 4 
 
 3 
 
 3 
 
 8 
 
 13 
 
 11 
 
 7 
 
 9 
 
 .9 
 
 6 
 
 10 
 
 4 
 
 4 
 
 i 
 
 ---- 
 
 1 
 
 
 8 
 
 1 
 
 1 
 
 103 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 Median 
 
 —obtained 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 
 1.66 
 
 .26 
 
 4.88 
 
 .17 
 
 2.35 
 
 Mil— M10= .34 
 
 .270 
 
 11 
 
 2.00 
 
 .33 
 
 5.83 
 
 .51 
 
 2.66 
 
 M12— Mll=2.50 
 
 .280 
 
 12 
 
 4.50 
 
 .31 
 
 7.05 
 
 1.31 
 
 2.87 
 
 M13— M12= .77 
 
 .289 
 
 13 
 
 5.27 
 
 .35 
 
 8.42 
 
 2.59 
 
 2.91 
 
 M13— M10=3.61 
 
 .279 
 
 The total gain for the three-year period from ages 10 to 13 inclusive 
 is 3.61, as against a gain for a similar period of years in the 6-month 
 schools of 1.16. The P.E. of the gain in the 9-month schools (3.61) 
 is .279; a little less than 8 per cent of the total gain. The probable 
 actual deviation of the true from the obtained difference in the case 
 of the 9-month schools is greater than in the case of the 6-month 
 schools, .279 as against .162, but in proportion to the respective gains 
 it is less. 
 
 The progress from year to year is uneven. From ages 10 to 11 it is 
 .34; from ages 11 to 12 it is 2.50; and from ages 12 to 13 it is .77. 
 By far the largest gain occurs between the ages 11 and 12. 
 
 Tables XVI to XX inclusive present comparative data from other 
 school systems, as was done for addition. In the following summary 
 the total improvement in ability to subtract for the three-year period 
 from ages 10 to 13 inclusive is given for each of the systems of schools 
 
42 Study of Achievement in Country and Toivn Schools 
 
 TABLE XVI 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Subtraction Rights in Certain White Schools of Louisville, Ky. 
 
 
 Scores 
 
 
 
 Airp 
 
 
 
 Total 
 
 
 XI 
 
 X2 
 
 X3 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 10 11 
 
 12 
 
 13 
 
 14 15 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 X9 
 
 
 10 
 
 10 
 
 18 
 
 45 
 
 t; 
 
 15 
 
 99 
 
 9q 
 
 M 
 
 '7 
 
 97 
 
 14 
 
 IS 
 
 11 
 
 4 4 
 
 ? 
 
 1 
 
 
 1 1 
 
 
 
 
 
 
 
 
 
 
 286 
 
 Ij 
 
 5 
 
 10 
 
 18 
 
 
 
 n 
 
 P 
 
 IS 
 
 99 
 
 '7 
 
 •^9 
 
 •'7 
 
 11 
 
 31 2inn 
 
 1' 
 
 3 
 
 ■i 
 
 1 1 
 
 1 
 
 1 
 
 
 1 
 
 
 1 
 
 
 
 
 313 
 
 12 
 
 
 5 
 
 10 
 
 
 
 (^ 
 
 n 
 
 1'' 
 
 '0 
 
 '8 
 
 V 
 
 "^7 
 
 34 
 
 23 - 
 
 R iq 
 
 11 
 
 11 
 
 in 
 
 Q 3 
 
 ? 
 
 5 
 
 ? 
 
 ■> 
 
 
 1 
 
 
 
 
 324 
 
 13 
 
 — - 
 
 
 5 
 
 3 
 
 1 
 
 1 
 
 2 
 
 15 
 
 26 
 
 20 
 
 36 
 
 29 
 
 31 21 30 
 
 16 
 
 11 
 
 13 
 
 5 7 
 
 2 
 
 3 
 
 2 
 
 2 
 
 2 
 
 
 
 1 
 
 26 
 
 310 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 in 
 
 4.02 
 
 .23 
 
 6.42 
 
 .10 
 
 3.16 
 
 Mil— M10=2.74 
 
 .184 
 
 11 
 
 6.76 
 
 .19 
 
 9.31 
 
 3.73 
 
 2.79 
 
 M12— Mll = 1.13 
 
 .162 
 
 n 
 
 7.89 
 
 .18 
 
 10.76 
 
 5.39 
 
 2.68 
 
 M13— M12 = 1.65 
 
 .162 
 
 13 
 
 9.54 
 
 .20 
 
 12.78 
 
 7.12 
 
 2.83 
 
 M13— M10=5.52 
 
 .184 
 
 TABLE XVII 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 Subtraction Rights in Certain Schools in Louisiana 
 
 
 
 Scores 
 
 
 
 Age 
 
 
 
 
 Total 
 
 
 XI 
 
 X2 
 
 X3 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 X9 
 
 XIO 
 
 xu 
 
 
 10 
 11 
 
 n 
 
 24 
 
 46 
 
 52 
 
 5' 
 
 '>^ 
 
 38 
 
 13 
 
 I"! 
 
 in 
 
 W 
 
 1 
 
 3 
 
 1 
 
 
 
 
 
 
 
 
 
 
 297 
 
 14 
 
 24 
 
 46 
 
 46 
 
 35 
 
 30 
 
 99 
 
 ''I 
 
 IP 
 
 P 
 
 f 
 
 4 
 
 5 
 
 1 
 
 1 
 
 ? 
 
 
 
 
 1 
 
 
 
 289 
 
 7 
 
 14 
 
 24 
 
 42 
 
 23 
 
 28 
 
 36 
 
 35 
 
 29 
 
 20 
 
 S 
 
 11 
 
 9 
 
 8 
 
 6 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 304 
 
 13 
 
 
 7 
 
 14 
 
 36 
 
 18 
 
 29 
 
 29 
 
 34 
 
 21 
 
 25 
 
 11 
 
 12 
 
 12 
 
 5 
 
 3 
 
 i 
 
 
 
 
 16 
 
 1 
 
 i 
 
 277 
 
 
 
 
 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 median 
 
 75 per- 
 centile 
 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 
 .31 
 
 .13 
 
 3.75 
 
 
 0.00 
 
 1.87 
 
 Mil— M10=1.10 
 
 .120 
 
 11 
 
 1.41 
 
 .14 
 
 3.98 
 
 
 0.00 
 
 1.99 
 
 M12— Mll = 1.97 
 
 .134 
 
 u 
 
 3.38 
 
 .17 
 
 5.65 
 
 
 .73 
 
 2.41 
 
 M13— M12= .78 
 
 .156 
 
 13 
 
 4.16 
 
 .19 
 
 6.79 
 
 
 1.68 
 
 2.55 
 
 M13— M12 = 3.85 
 
 .144 
 
 TABLE XVIII 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Subtraction Rights in Arkansas City 
 
 Age 
 
 
 
 Scores 
 
 
 Total 
 
 
 XI 
 
 X2 
 
 X3 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 X9 
 
 
 10 
 
 18 
 
 5'' 
 
 60 
 
 37 
 
 31 
 
 25 
 
 '>5 
 
 ?3 
 
 16 
 
 7 
 
 8 
 
 
 1 
 
 
 2 
 
 
 
 
 
 305 
 
 11 
 
 8 
 
 18 
 
 52 
 
 33 
 
 31 
 
 22 
 
 ?4 
 
 16 
 
 18 
 
 7 
 
 17 
 
 3 
 
 1 
 
 2 
 
 5 
 
 
 
 
 
 257 
 
 12 
 
 4 
 
 8 
 
 18 
 
 29 
 
 22 
 
 31 
 
 26 
 
 45 
 
 25 
 
 25 
 
 19 
 
 16 
 
 7 
 
 2 
 
 3 
 
 2 
 
 
 1 
 
 
 283 
 
 13 
 
 1 
 
 4 
 
 8 
 
 21 
 
 10 
 
 18 
 
 25 
 
 31 
 
 20 
 
 37 
 
 26 
 
 25 
 
 11 
 
 7 
 
 a 
 
 4 
 
 2 
 
 ---- 
 
 1 
 
 254 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 median 
 
 75 per- 
 centile 
 
 25 per 
 centile 
 
 
 Q 
 
 
 P.E. 
 
 10 
 
 .60 
 
 .11 
 
 3.23 
 
 0.00 
 
 
 1.61 
 
 Mil— M10= .96 
 
 .122 
 
 11 
 
 1.56 
 
 .16 
 
 4.29 
 
 0.00 
 
 
 2.14 
 
 M12— Mll = 2.51 
 
 .141 
 
 17. 
 
 4.07 
 
 .17 
 
 6.17 
 
 1.53 
 
 
 2.32 
 
 M13— M12 = 1.38 
 
 .148 
 
 13 
 
 5.45 
 
 .17 
 
 7.59 
 
 3.06 
 
 
 2.26 
 
 M13— M10=4.85 
 
 .130 
 
Achievement Measured by Courtis Arithmetic Tests 
 
 43 
 
 TABLE XIX 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Subtraction Rights in Salina, Kan. 
 
 Age 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Scores 
 
 
 
 
 
 
 
 
 
 
 
 
 Total 
 
 
 XI 
 
 15 
 4 
 3 
 
 X2 
 
 19 
 15 
 4 
 3 
 
 X3 
 
 46 
 19 
 15 
 4 
 
 
 
 17 
 11 
 3 
 5 
 
 1 
 
 21 
 19 
 5 
 3 
 
 2 
 
 33 
 13 
 10 
 4 
 
 3 
 
 30 
 14 
 
 27 
 8 
 
 4 
 
 41 
 40 
 36 
 17 
 
 5 
 
 35 
 
 32 
 27 
 31 
 
 6 
 
 31 
 3fi 
 45 
 »5 
 
 7 
 
 23 
 23 
 28 
 37 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 X9 
 
 XIO 
 
 
 in 
 
 20 
 30 
 33 
 
 28 
 
 6 
 18 
 27 
 20 
 
 4 
 8 
 19 
 24 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 341 
 
 11 
 
 7 
 11 
 13 
 
 4 
 
 8 
 6 
 
 9 
 
 3 
 
 7 
 
 
 
 
 
 
 
 
 
 
 
 
 295 
 
 1^ 
 
 I 
 
 3 
 3 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 
 2 
 1 
 
 "~2 
 
 309 
 
 13 
 
 
 
 
 2 
 
 1 
 
 266 
 
 
 
 
 
 
 
 
 Median 
 
 P. E. true 
 
 —obtained 
 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 
 n 
 
 12 
 13 
 
 3.65 
 5.39 
 6.54 
 7.35 
 
 .19 
 .19 
 .16 
 .15 
 
 5.96 
 7.79 
 8.87 
 9.72 
 
 .30 
 2.44 
 
 4.28 
 5.72 
 
 2.83 
 2.62 
 2.29 
 2.00 
 
 Mil— M10=1.74 
 M12— Mll = 1.15 
 M13— M12= .81 
 M13— M10=3.70 
 
 .162 
 .148 
 .134 
 .150 
 
 TABLE XX 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Subtraction Rights in Hibbing, Minn. 
 
 
 
 Scores 
 
 
 
 
 Age 
 
 
 
 
 
 Total 
 
 
 X2 
 16 
 
 X3 
 
 35 
 16 
 
 
 
 33 
 20 
 3 
 
 1 
 
 9 
 6 
 
 6 
 
 2 
 
 10 
 11 
 
 3 
 
 12 
 9 
 13 
 
 4 
 
 14 
 25 
 11 
 
 5 
 
 22 
 15 
 12 
 
 6 
 
 20 1 
 20 1 
 
 n 1 
 
 7 8 
 
 6 16 
 
 6 22 
 
 q IS 
 
 9 
 
 15 
 18 
 IS 
 
 10 
 
 4 
 
 19 
 19 
 
 11 
 
 9 
 12 
 
 25 
 
 12 
 
 6 
 6 
 11 
 
 13 
 
 5 
 
 7 
 
 7 
 
 14 
 
 3 
 4 
 13 
 
 1 
 
 5 16 
 
 . 3 
 2 4 
 fi 7 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 X9 
 
 XI 
 
 
 10 
 
 
 248 
 
 11 
 
 2 
 
 ? 
 
 1 
 3 
 
 2 
 
 1 
 
 
 
 
 
 238 
 
 12 
 
 1 
 
 1 
 
 4 
 
 
 227 
 
 13 
 
 — - 
 
 
 2 
 
 2 
 
 4 
 
 4 
 
 10 
 
 9 
 
 3 
 
 9 16 
 
 13 
 
 10 
 
 19 
 
 15 
 
 5 
 
 7 
 
 
 6 1 
 
 4 
 
 2 
 
 
 — 
 
 1 
 
 
 32 
 
 4 
 
 179 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 
 75 per- 
 
 25 per- 
 
 Q 
 
 
 
 P.E. 
 
 
 
 median 
 
 centile 
 
 centile 
 
 
 
 
 
 10 
 
 4.64 
 
 .30 
 
 7.93 
 
 .33 
 
 3.80 
 
 
 Mil— M10=2.16 
 
 .240 
 
 11 
 
 6.80 
 
 .25 
 
 10.02 
 
 3.72 
 
 3.15 
 
 
 M12— Mll=2.39 
 
 .212 
 
 12 
 
 9.19 
 
 .23 
 
 11.93 
 
 6.26 
 
 2.83 
 
 
 M13— M12=2.15 
 
 .260 
 
 13 
 
 11.34 
 
 .36 
 
 15.87 
 
 8.10 
 
 3.88 
 
 
 M13— M10=6.70 
 
 .284 
 
 reported, and also the median ability of each 13-year-old group. As 
 in addition, so in subtraction, this median for the 13-year-old group 
 may be considered in a v^ay as a measure of the result obtained in 
 any system with children up to the age of 13. 
 
 School Systems 
 
 Total gain 
 
 10 to 13 
 
 inclusive 
 
 Medians 
 
 13-year-old 
 children 
 
 Hibbing 
 
 Louisville 
 
 Arkansas City 
 
 Louisiana 
 
 Salina 
 
 Madison County, 9-month schools 
 Madison County, 6-month schools 
 
 6.70 
 5.52 
 4.85 
 3.85 
 3.70 
 3.61 
 1.16 
 
 11.34 
 9.54 
 5.45 
 4.16 
 7.35 
 5.27 
 1.70 
 
44 
 
 Study of Achievement in Country and Tozvn Schools 
 
 The ranking in order of total gain made for the three-year period 
 from ages 10 to 13 inclusive corresponds quite well with the ranking 
 in order of medians for the 13-year-old children. This correspond- 
 ence is not so close as was the case with addition ; there are four dis- 
 placements in the ranking. As was the case with addition, the rank- 
 ing in order of total gain for the three-year period is the more sig- 
 nificant, since the use of a diflference as a measure tends to neutralize 
 errors in the giving and scoring of the test. 
 
 It is interesting to note that the rank in gain in ability to subtract is 
 not the same as that in ability to add. Arkansas City and Louisiana 
 both come ahead of Salina in subtraction, and not in addition. This 
 difference between the rank of these systems for the two processes 
 would seem to indicate that subtraction needs to be taught ; that im- 
 provement in addition does not necessarily carry with it exactly pro- 
 portionate ability to subtract. 
 
 The following summary shows the gain for each yearly age-period 
 in the systems studied : 
 
 School Systems 
 
 Yearly Gains 
 
 
 10 to 11 
 
 11 to 12 
 
 12 to 13 
 
 Hibbing- ._ .-.. 
 
 2.21 
 
 2.74 
 
 .96 
 
 1.10 
 
 1.74 
 
 .34 
 
 .13 
 
 2.34 
 1.13 
 2.51 
 1.97 
 1.15 
 2.50 
 .30 
 
 2.15 
 
 Louisville __ _- 
 
 1.65 
 
 Arkansas City ._ . 
 
 1.38 
 
 Louisiana ..- 
 
 .78 
 
 Salina . . . 
 
 .81 
 
 Madison County, 9-month schools . 
 
 .77 
 
 Madison County, 6-month schools . - _ - 
 
 .73 
 
 
 
 Total 
 
 9.22 
 
 11.90 
 
 8.27 
 
 
 
 The variations are as great in the amount of yearly gain as they 
 were shown to be in addition. As in addition, so in subtraction, the 
 greatest yearly gain seems to be between the ages of 11 and 12. In 
 subtraction the next greatest gain seems to be between the ages of 10 
 and 11, while in the case of addition the second greatest gain was be- 
 tween the ages of 12 and 13. The smallest gain is made between the 
 ages of 12 and 13. 
 
 CONCLUSIONS 
 
 1. The conclusions reached in regard to addition (p. 39) hold in the 
 case of subtraction. 
 
Achievement Measured by Courtis Arithmetic Tests 
 
 45 
 
 2. Improvement in addition does not necessarily carry with it pro- 
 portionate ability to subtract. 
 
 Section 3. Multiplication 
 
 The data presented in multiplication are from the same sources as 
 those for addition and subtraction, and the method of treatment has 
 been the same throughout. 
 
 Table XXI gives the facts in regard to multiplication in the Madison 
 County 6-month schools. 
 
 TABLE XXI 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Multiplication Rights in the 6-Month White Schools 
 
 OF Madison County, Ky. 
 
 
 Scores 
 
 
 Age 
 
 
 Total 
 
 
 XI 
 
 X2 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 X9 
 
 XIO 
 
 Xll 
 
 
 in 
 
 9 
 
 8 
 4 
 2 
 
 19 
 9 
 6 
 
 4 
 
 82 
 75 
 62 
 57 
 
 16 
 11 
 12 
 15 
 
 3 
 11 
 
 16 
 
 8 
 
 5 
 8 
 12 
 9 
 
 4 
 2 
 
 9 
 5 
 
 1 
 2 
 3 
 3 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 140 
 
 11 
 
 
 
 
 
 
 
 
 
 
 2 
 4 
 10 
 
 """2 
 4 
 
 .... 
 
 124 
 
 1? 
 
 4 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 1 
 
 136 
 
 18 
 
 4 
 
 
 
 
 
 
 
 124 
 
 
 
 
 
 
 
 
 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 Median 
 
 —obtained 
 
 75 per- 
 
 25 per- 
 
 Q 
 
 
 P.E. 
 
 
 
 median 
 
 centile 
 
 centile 
 
 
 
 
 10 
 
 .51 
 
 .04 
 
 .93 
 
 .08 
 
 .42 
 
 Mil— M10=.ll 
 
 .036 
 
 11 
 
 .62 
 
 .05 
 
 1.27 
 
 .21 
 
 .53 
 
 M12— Mll = .31 
 
 .094 
 
 12 
 
 .93 
 
 .14 
 
 3.16 
 
 .38 
 
 1.39 
 
 M13— M12 = .05 
 
 .142 
 
 13 
 
 .98 
 
 .18 
 
 3.77 
 
 .43 
 
 1.67 
 
 M13— M10 = .47 
 
 .111 
 
 The median of each of the age-groups given falls below 1, and even 
 the 75 percentile of the 10-year-old group falls between and 1. It 
 seems that most of the children in these schools below the age of 14 
 have an ability in multiplication that is less than can be measured by 
 the Courtis Standard Research Tests in Arithmetic Series B. 
 
 The total gain in ability to multiply during the three-year period, ages 
 10 to 13 inclusive, is only .47. The P.E. of this statement of gain is 
 .111, something over 25 per cent of the gain itself. The gain noted is 
 very slight. Apparently these 6-month schools in Madison County are 
 failing to teach most of the pupils under 14 how to multiply. 
 
 The improvement from year to year necessarily is very slight, since 
 the total improvement is so small. From ages 10 to 11 the improve- 
 ment is .11 ; from ages 11 to 12 it is .31 ; and from ages 12 to 13 it is 
 .05. The greatest gain in any one-year period is .31 from ages 11 to 12. 
 
 The results of instruction in multiplication in these schools are very 
 meager. 
 
46 
 
 Study of Achievement in Country and Town Schools 
 
 TABLE XXII 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Multiplication Rights in the 9-Month White Schools 
 
 OF Madison County, Ky. 
 
 
 
 Scores 
 
 
 
 Asje 
 
 
 
 
 Total 
 
 
 XI 
 
 X2 
 
 
 
 1 
 
 2 
 
 3 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 X9 
 
 XIO 
 
 Xll 
 
 
 10 
 
 9 
 
 1 
 3 
 
 16 
 9 
 
 1 
 
 37 
 31 
 
 32 
 
 4 
 
 10 
 14 
 
 12 
 7 
 14 
 
 14 10 
 
 7 12 
 
 23 16 
 
 12 
 11 
 10 
 
 6 
 4 
 
 
 
 
 
 
 
 
 
 
 120 
 
 11 
 
 
 
 3 
 
 '"2 
 
 — - 
 
 1 
 1 
 
 
 
 96 
 
 12 
 
 3 
 
 2 
 
 i 
 
 
 131 
 
 18 
 
 4 
 
 3 
 
 13 
 
 10 
 
 11 
 
 
 1 9 
 
 12 
 
 11 
 
 5 
 
 
 
 
 2 
 
 8 
 
 1 
 
 1 
 
 101 
 
 
 
 
 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 
 P.E. 
 
 10 
 
 .94 
 
 .21 
 
 3.85 
 
 .13 
 
 1.86 
 
 Mil— M10= 
 
 .76 
 
 .206 
 
 11 
 
 1.70 
 
 .26 
 
 4.58 
 
 .45 
 
 2.06 
 
 M12-M11 = 
 
 1.36 
 
 .200 
 
 12 
 
 3.06 
 
 .20 
 
 4.56 
 
 .87 
 
 1.84 
 
 M13— M12 = 
 
 .80 
 
 .216 
 
 13 
 
 3.86 
 
 .29 
 
 6.25 
 
 1.52 
 
 2.36 
 
 M13— M10= 
 
 2.92 
 
 .222 
 
 Table XXII gives the facts in regard to multiplication in the 9- 
 month schools of Madison County. 
 
 The median ability in multiplication in these schools for 10-year-old 
 children is .94; a Httle less than 1. The 25 percentile of the 10-, 11- 
 and 12-year-old groups is less than 1 in each case. The situation 
 here presented is materially better than that in the 6-month schools. 
 
 The total gain in ability to multiply for the three-year period from ages 
 10 to 13 inclusive is 2.92. The contrast between this gain and that in 
 the 6-month schools (.47) is marked. The P.E. of this gain of 2.92 
 is .222, a little over 7 per cent of the gain. The probable actual de- 
 viation of the true from the obtained measure is greater in the case of 
 the 9-month schools than in the case of the 6-montli schools, but in 
 proportion to the gain made it is much less. 
 
 The improvement in ability to multiply from ages 10 to 11 is .76; 
 from ages 11 to 12 it is 1.36; and from ages 12 to 13 it is .80. The 
 gain during any one of the year periods is much greater than the gain 
 for the entire three-year period in the 6-month schools. The great- 
 est gain is from ages 11 to 12, as is the case in the 6-month schools. 
 
 Tables XXIII to XXVII inclusive give comparative data in regard 
 to multiplication from other school systems. In the following sum- 
 mary the total improvement in ability to multiply during the three-year 
 period from ages 10 to 13 inclusive is given for each of the school sys- 
 tems studied, and also the median ability for the respective groups of 
 13-year-old children. The schools are listed in the order of amount of 
 improvement made. 
 
Achievement Measured by Courtis Arithmetic Tests 
 
 47 
 
 TABLE XXIII 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Multiplication Rights in Certain White Schools in Louisville, Ky. 
 
 Age 
 
 Scores 
 
 Total 
 
 XI 
 
 10 
 5 
 
 X2 
 
 10 
 5 
 
 X3 
 
 43 
 16 
 10 
 
 5 
 
 
 
 7 
 
 10 
 2 
 3 
 
 1 
 
 15 
 
 8 
 9 
 3 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 
 10 
 11 
 12 
 13 
 
 23 
 11 
 U 
 9 
 
 34 
 17 
 18 
 18 
 
 32 
 37 
 31 
 22 
 
 30 
 41 
 34 
 32 
 
 26 
 36 
 38 
 43 
 
 21 
 35 
 35 
 40 
 
 12 
 32 
 41 
 38 
 
 5 
 21 
 34 
 34 
 
 7 
 14 
 15 
 18 
 
 3 
 3 
 17 
 13 
 
 
 
 
 
 
 
 284 
 
 7 
 9 
 13 
 
 5 
 6 
 
 8 
 
 1 
 
 2 
 2 
 
 
 
 
 309 
 
 ""■4 
 
 2 
 
 1 
 
 320 
 305 
 
 
 
 
 
 Median 
 
 P. E. true 
 
 — obtained 
 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 11 
 
 12 
 13 
 
 3.82 
 5.98 
 7.05 
 7.43 
 
 .21 
 .14 
 .15 
 .14 
 
 6.11 
 
 8.17 
 9.17 
 9.46 
 
 .28 
 4.00 
 4.80 
 5.50 
 
 2.91 
 2.08 
 2.18 
 1.98 
 
 Mil— M10 = 2.16 
 M12— Mll = 1.07 
 M13— M12= .38 
 M13— M10=3.61 
 
 .158 
 .127 
 .120 
 .152 
 
 TABLE XXIV 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Multiplication Rights in Certain White Schools in Louisiana 
 
 Age 
 
 Scores 
 
 Total 
 
 XI 
 
 24 
 13 
 
 7 
 
 X2 
 
 42 
 24 
 13 
 
 7 
 
 X3 
 
 61 
 43 
 24 
 13 
 
 
 
 69 
 63 
 
 74 
 50 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 X9 
 
 XIO 
 
 Xll 
 
 
 10 
 11 
 12 
 13 
 
 24 
 31 
 36 
 33 
 
 21 
 30 
 
 28 
 35 
 
 21 
 26 
 34 
 25 
 
 12 
 16 
 34 
 
 27 
 
 8 
 11 
 21 
 23 
 
 4 
 9 
 12 
 17 
 
 2 
 7 
 5 
 16 
 
 
 
 
 
 
 
 
 
 289 
 
 4 
 6 
 3 
 
 1 
 3 
 2 
 
 '"3 
 
 
 1 
 
 1 
 
 1 1 
 
 1 
 
 1 
 
 17 
 
 
 
 280 
 
 1 
 
 1 
 
 
 300 
 
 1 
 
 275 
 
 
 Median 
 
 P. E. true 
 
 — obtained 
 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 11 
 12 
 13 
 
 .23 
 
 .95 
 
 1.88 
 
 2.98 
 
 .06 
 .12 
 .13 
 .17 
 
 1.85 
 3.23 
 4.26 
 5.70 
 
 0.00 
 
 0.00 
 
 .41 
 
 .97 
 
 .92 
 1.61 
 1.92 
 2.86 
 
 Mil— M10= .72 
 M12-M11= .93 
 M13— M12 = 1.10 
 M13— M10=2.75 
 
 .080 
 .106 
 .136 
 .117 
 
 TABLE XXV 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Multiplication Rights in Arkansas City 
 
 Age 
 
 Scores 
 
 Total 
 
 XI 
 
 X2 
 
 X3 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 X9 
 
 
 10 
 11 
 12 
 13 
 
 14 
 
 8 
 4 
 1 
 
 51 
 14 
 
 8 
 4 
 
 61 
 51 
 14 
 
 8 
 
 63 
 41 
 49 
 30 
 
 44 
 21 
 25 
 13 
 
 24 
 40 
 38 
 22 
 
 24 
 22 
 50 
 24 
 
 11 
 21 
 31 
 3C 
 
 6 
 15 
 
 22 
 22 
 
 
 
 
 
 
 
 
 298 
 
 9 
 28 
 28 
 
 6 
 16 
 16 
 
 2 
 
 2 
 
 24 
 
 8 
 4 
 4 
 
 
 
 
 
 258 
 
 2 
 3 
 
 1 
 6 
 
 2 
 4 
 
 . 
 
 296 
 245 
 
 
 Median 
 
 P. E. true 
 
 — obtained 
 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 11 
 12 
 13 
 
 .36 
 1.71 
 3.20 
 4.51 
 
 .06 
 .14 
 .15 
 .19 
 
 1.78 
 3.84 
 5.13 
 6.88 
 
 0.00 
 
 0.00 
 
 .97 
 
 2.01 
 
 .89 
 1.92 
 2.08 
 2.43 
 
 Mil— M10=1.35 
 M12— Mll = 1.49 
 M13— M12 = 1.31 
 M13— M10=4.15 
 
 .098 
 .127 
 .150 
 .126 
 
48 Study of Achievement in Country and Town Schools 
 
 TABLE XXVI 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Multiplication Rights in Salina, Kan. 
 
 
 Scores 
 
 
 Age 
 
 
 Total 
 
 
 XI 
 
 X2 
 
 X3 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 10 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 X9 
 
 XIO. 
 
 
 10 
 
 16 
 3 
 3 
 
 22 
 
 16 
 
 3 
 
 1 
 
 48 
 22 
 16 
 3 
 
 31 
 19 
 
 8 
 6 
 
 27 
 17 
 8 
 8 
 
 20 
 22 
 11 
 13 
 
 23 
 23 
 29 
 11 
 
 18 
 21 
 29 
 31 
 
 48 
 52 
 45 
 
 27 
 
 24 
 22 
 .51 
 46 
 
 10 
 
 8 
 19 
 29 
 
 9 
 10 
 33 
 
 28 
 
 4 
 17 
 27 1 
 19 1 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 334 
 
 11 
 
 i 5 
 9 19 
 15 
 
 2 
 3 
 
 7 
 
 "7 
 4 
 
 2 
 1 
 
 8 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 267 
 
 1? 
 
 
 
 
 
 
 
 2 
 4 
 
 "i 
 
 333 
 
 13 
 
 - 
 
 — 
 
 2 
 
 
 
 
 
 2 
 
 275 
 
 
 
 
 
 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 
 3.13 
 
 .18 
 
 5.32 
 
 0.00 
 
 2.66 
 
 Mil— M10=1.41 
 
 .155 
 
 11 
 
 4.54 
 
 .18 
 
 6.23 
 
 1.39 
 
 2.44 
 
 M12-M11 = 1.64 
 
 .148 
 
 12 
 
 6.28 
 
 .16 
 
 8.84 
 
 4.18 
 
 2.33 
 
 M13— M12= .53 
 
 .156 
 
 13 
 
 6.81 
 
 .20 
 
 9.18 
 
 4.86 
 
 2.66 
 
 M13— M10 = 3.68 
 
 .162 
 
 TABLE XXVII 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Multiplication Rights in Hibbing, Minn. 
 
 
 
 Scores 
 
 
 Age 
 
 
 
 Total 
 
 
 X2 
 18 
 
 X3 
 
 35 
 16 
 
 
 
 34 
 19 
 
 1 
 
 23 
 11 
 
 in 
 
 2 
 
 29 
 
 24 
 1? 
 
 3 
 
 35 
 25 
 17 
 
 4 
 
 20 
 23 
 27 
 
 5 
 
 21 
 
 42 
 29 
 
 6 
 
 16 
 
 25 
 
 27 
 
 7 
 
 12 
 19 
 
 26 
 
 8 
 
 9 
 10 
 18 
 
 9 
 
 5 
 15 
 18 
 
 10 
 
 4 
 8 
 10 
 
 11 
 
 1 
 1 
 3 
 
 12 
 
 1 
 2 
 9 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 X9 
 
 no 
 
 
 in 
 
 261 
 
 11 
 
 2 
 6 
 
 1 
 1 
 
 "4 
 
 1 
 1 
 
 
 
 
 244 
 
 12 
 
 1 
 
 4 
 
 
 230 
 
 13 
 
 
 
 3 
 
 6 
 
 4 
 
 12 
 
 12 
 
 13 
 
 13 
 
 17 
 
 16 
 
 11 
 
 12 
 
 12 
 
 8 
 
 3 
 
 
 2 
 
 
 
 33 
 
 4 
 
 181 
 
 
 
 
 
 
 
 
 
 p. E. true 
 
 
 
 
 
 
 
 
 Median 
 
 —obtained 
 median 
 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 
 2.77 
 
 .18 
 
 
 5.17 
 
 .41 
 
 2.38 
 
 Mil— M10=2.32 
 
 .140 
 
 11 
 
 5.09 
 
 .17 
 
 
 6.92 
 
 2.62 
 
 2.15 
 
 M12— Mll = 1.39 
 
 .140 
 
 12 
 
 6.48 
 
 .18 
 
 
 8.97 
 
 4.42 
 
 2.27 
 
 M13— M12 = 2.17 
 
 .220 
 
 13 
 
 8.65 
 
 .32 
 
 
 12.59 
 
 5.63 
 
 3.48 
 
 M13— M10=5.88 
 
 .220 
 
 As is the case in addition and subtraction, the ranking in order of 
 gain made in abiHty for the three-year period from ages 10 to 13 inclu- 
 sive does not correspond exactly with the ranking in order of ability of 
 
 School Systems 
 
 Hibbing 
 
 Arkansas City 
 
 Salina 
 
 Louisville 
 
 Madison County, 9-month schools. 
 Louisiana 
 
 Madison County, 6-month schools. 
 
 Total gain 
 
 Medians 
 
 10 to 13 
 
 13-year-old 
 
 inclusive 
 
 children 
 
 5.88 
 
 8.65 
 
 4.15 
 
 4.51 
 
 3.68 
 
 6.81 
 
 3.61 
 
 7.43 
 
 2.92 
 
 3.86 
 
 2.75 
 
 2.98 
 
 .47 
 
 .98 
 
Achievement Measured by Courtis Arithmetic Tests 
 
 49 
 
 13-year-old children. The measure of gain made is probably the 
 more significant measure, as for addition and subtraction. 
 
 It must be borne in mind that the steps of improvement are not 
 necessarily of equal value. It may be more difficult to teach a child 
 who cannot multiply to multiply well enough to do one example cor- 
 rectly than it is to teach a child who can do one example correctly to do 
 two examples correctly. In other words, because Hibbing makes twice 
 as much improvement as Louisiana does not necessarily mean that the 
 teaching in Hibbing is twice as good as that in Louisiana. This dif- 
 ficulty in evaluating the results of this test is not due to the fact that 
 a difference between the median abilities of age-groups is used as a 
 measure, but rather to the manner in which the test is constructed. 
 The unit in which the result is stated is not one of the relative diffi- 
 culty with which those results are obtained. 
 
 The following summary shows the gain for each yearly age period 
 in the systems studied: 
 
 School Systems 
 
 
 Yearly Gains 
 
 
 10 to 11 
 
 11 to 12 
 
 12 to 13 
 
 Hibbing. . _.._. . 
 
 2.32 
 
 1.35 
 
 1.41 
 
 2.16 
 
 .76 
 
 .72 
 
 .11 
 
 1.39 
 1.49 
 1.64 
 1.07 
 1.36 
 .93 
 .31 
 
 2.17 
 
 Arkansas City _. . _ . .. 
 
 1.31 
 
 Salina 
 
 .53 
 
 Louisville- ._.--_ 
 
 .38 
 
 Madison County, 9-month schools 
 
 .80 
 
 Louisiana .. - . _ .. 
 
 1.10 
 
 Madison County, 6-month schools. . 
 
 .05 
 
 
 
 Total 
 
 8.83 
 
 8.19 
 
 6.34 
 
 The variations in amount of yearly gain are as great as they are in 
 addition and subtraction. The greatest gain seems to be between 
 ages 10 and 11 in this case, while in both addition and subtraction the 
 greatest gain seems to come between ages 11 and 12. The smallest 
 gain appears between ages 12 and 13. 
 
 The conclusions in regard to multiplication are the same as those for 
 addition (p. 38) and subtraction (p. 44). 
 
 Section 4. Division 
 The data presented in division are from the same sources as those 
 for addition, subtraction and multiplication, and the same method of 
 treatment has been followed. 
 
50 Study of Achievement in Country and Tozvn Schools 
 
 Table XXVIII gives the facts in regard to division in the 6-month 
 schools of Madison County. 
 
 TABLE XXVIII 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 Division Rights in the 6-Month White Schools of Madison County, Ky. 
 
 
 
 
 
 
 Scores 
 
 
 
 
 Age 
 
 
 
 
 
 
 
 
 Total 
 
 
 XI 
 
 X2 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 X9 
 
 XIO 
 
 Xll 
 
 
 in 
 
 10 
 
 6 
 
 17 
 10 
 
 98 
 83 
 
 7 
 15 
 
 
 4 
 
 1 
 
 1 
 
 5 
 
 
 2 
 2 
 
 
 
 
 
 
 
 
 139 
 
 n 
 
 1 
 
 
 1 
 
 
 2 
 
 
 
 126 
 
 12 
 
 3 
 
 6 
 
 78 
 
 16 
 
 
 9 
 
 7 
 
 
 6 
 
 4 
 
 1 
 
 
 
 4 
 
 2 
 
 
 138 
 
 1'^ 
 
 2 
 
 3 
 
 70 
 
 16 
 
 
 8 
 
 5 
 
 
 1 
 
 
 1 
 
 
 
 11 
 
 4 
 
 2 
 
 123 
 
 
 
 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 median 
 
 
 75 per- 
 centile 
 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 
 .43 
 
 .03 
 
 
 .78 
 
 
 0.07 
 
 .35 
 
 Mil— M10= .13 
 
 .028 
 
 11 
 
 .56 
 
 .04 
 
 
 .94 
 
 
 .18 
 
 .38 
 
 M12— Mll= .20 
 
 .053 
 
 12 
 
 .76 
 
 .09 
 
 
 2.05 
 
 
 .32 
 
 .86 
 
 M13-M12= .04 
 
 .078 
 
 13 
 
 .80 
 
 .10 
 
 
 2.15 
 
 
 .36 
 
 .89 
 
 M13-M10= .37 
 
 .072 
 
 In these schools the median ability in division of each of the age- 
 groups from ages 10 to 13 inclusive is between and 1. For the ages 10 
 and 11 the 75 percentiles are in the same interval. It seems that in 
 division, as in multiplication, the majority of the children under 14 
 years of age in the Madison County 6-month schools are unable to 
 work correctly any of the examples given in the Courtis Standard 
 Research Tests in Arithmetic Series B for division. 
 
 The total gain in ability to divide during the three-year period from 
 ages 10 to 13 inclusive is from .43, the median for the 10-year-old chil- 
 dren, to .80, the median for the 13-year-old children, a difference of .37. 
 The P.E. of this difference is .072 — about 20 per cent of the gain. 
 This gain is very small for so long a period of time at the age when 
 material progress is being made in this ability in other school systems. 
 
 As would be expected where the total improvement is so small, the 
 yearly gains are very small and uneven. From ages 10 to 11 the gain 
 is .13; from ages 11 to 12 it is .20; and from ages 12 to 13 it is .04. 
 The P.E. of the gain from ages 12 to 13 is .072, which is almost twice 
 as much as the gain. This means that the gain in this year is almost 
 negligible. 
 
 Table XXIX gives the facts in regard to division in the Madison 
 County 9-month schools. 
 
Achievement Measured by Courtis Arithmetic Tests 
 
 51 
 
 TABLE XXIX 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 Division Rights in the 9-Month White Schools of Madison County, Ky. 
 
 Age 
 
 Scores 
 
 Total 
 
 
 XI 
 
 X2 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 X9 
 
 XIO 
 
 Xll 
 
 
 10 
 
 9 
 
 1 
 3 
 5 
 
 16 
 9 
 1 
 3 
 
 60 
 
 46 
 45 
 17 
 
 5 
 6 
 13 
 15 
 
 8 
 11 
 9 
 
 8 
 
 5 
 3 
 10 
 11 
 
 9 
 
 7 
 13 
 
 7 
 
 4 
 2 
 11 
 6 
 
 3 
 
 2 
 8 
 6 
 
 1 
 
 2 
 8 
 3 
 
 1 
 4 
 
 2 
 3 
 
 1 
 1 
 1 
 3 
 
 1 
 1 
 1 
 
 1 
 
 
 1 
 
 
 
 
 
 124 
 
 11 
 
 
 1 
 1 
 
 8 
 
 
 
 96 
 
 12 
 
 2 
 
 — 
 
 .... 
 
 1 
 
 1 
 
 
 129 
 
 13 
 
 1 
 
 99 
 
 
 
 
 
 
 Media 
 
 n 
 
 P. E. true 
 
 — obtained 
 
 median 
 
 75 per- 
 centile 
 
 1 
 
 j 25 per 
 
 ! centile 
 
 
 Q 
 
 
 P.E. 
 
 10 
 11 
 12 
 13 
 
 .61 
 
 .82 
 
 2.27 
 
 3.13 
 
 .12 
 .16 
 .21 
 .33 
 
 2.37 
 2.90 
 5.31 
 6.37 
 
 .11 
 
 .30 
 .63 
 .98 
 
 1.13 
 1.30 
 2.34 
 2.69 
 
 Mll-M10= .21 
 M12— Mll = 1.45 
 M13— M12= .86 
 M13— M10=2.52 
 
 .122 
 .190 
 .264 
 .210 
 
 In these schools the 25 percentile of each of the age-groups from ages 
 10 to 13 inclusive falls between and 1, and the medians for the 10- 
 and 11-year-old groups each fall in the same interval. A much larger 
 proportion of the children reported are able to work some of the ex- 
 amples correctly than is the case in the 6-month schools. 
 
 The total gain from ages 10 to 13 inclusive in this case is 2.52 as 
 against .37 in the case of the 6-month schools in the same county. The 
 P.E. of the gain for the 9-month schools is .21, about 8.5 per cent of the 
 gain. The probable actual deviation of the true measure of gain from 
 the measure of gain given is almost three times as great as the like 
 probable deviation in the case of the 6-month schools, but the gain in 
 the case of the 9-month schools is so much greater that proportion- 
 ately the probable deviation is smaller. 
 
 As in the 6-month schools, so in these, the yearly gains are unequal. 
 From ages 10 to 11 the gain is .21 ; from ages 11 to 12 it is 1.45 ; and 
 from ages 12 to 13 it is .86. The gain for each year after the first is 
 much greater than the total gain in the 6-month schools for the entire 
 three-year period. 
 
 Tables XXX to XXXIV inclusive give comparative data from other 
 school systems in regard to division. In the following summary the 
 total gain in ability to divide during the three-year period from ages 10 
 to 13 inclusive is given for each of the systems studied, and also the 
 median abihty of the 13-year-old children in the respective systems. 
 The schools are listed in the order of the amount of improvement 
 made. 
 
 In this table it is interesting to note that the ranking in order of 
 amount of total gain corresponds exactly to ranking in order of the 
 
52 Study of Achievement in Country and Town Schools 
 
 TABLE XXX 
 
 Age Distribution, 10 to 13 Inclusi\'e, of Courtis Series B Arithmetic Tests 
 Division Rights in Certain White Schools of Louisville, Ky. 
 
 
 Scores 
 
 
 Age 
 
 
 Total 
 
 
 XI 
 
 10 
 5 
 
 X2 
 
 16 
 10 
 5 
 
 X3 
 
 43 3 
 16 2 
 10 1 
 5 1 
 
 1 
 
 24 
 3 12 
 6 12 
 
 1 9 
 
 2 
 
 31 
 16 
 15 
 13 
 
 3 
 
 24 
 23 
 18 
 16 
 
 4 
 
 32 
 28 
 22 
 13 
 
 5 
 
 21 
 25 
 
 22 
 21 
 
 6 
 
 20 
 
 22 
 23 
 34 
 
 7 
 
 13 
 20 
 27 
 23 
 
 8 
 
 9 
 18 
 27 
 30 
 
 9 
 
 11 
 19 
 
 23 
 25 
 
 10 
 
 3 
 21 
 
 23 
 26 
 
 U 
 
 3 
 
 12 
 10 
 
 25 
 
 12 
 
 4 
 12 
 17 
 18 
 
 13 
 
 2 
 9 
 11 
 13 
 
 14 1 
 
 7 
 12 
 6 1 
 
 5 16 
 
 17 
 
 18 
 
 19 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 X9 
 
 
 10 
 
 
 
 
 
 
 
 
 
 
 
 297 
 
 11 
 
 3 3 
 
 i 3 
 2 6 
 
 2 
 6 
 2 
 
 1 
 4 
 
 
 
 
 
 
 1 
 1 
 
 ""26 
 
 308 
 
 12 
 
 ] 
 
 
 
 
 319 
 
 13 
 
 ) 
 
 
 
 
 336 
 
 
 
 
 
 
 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 
 2.82 
 
 .19 
 
 5.60 
 
 .17 
 
 2.71 
 
 Mil— M10 = 3.02 
 
 .192 
 
 11 
 
 5.84 
 
 .24 
 
 9.68 
 
 2.68 
 
 3.50 
 
 M12— Mn = 1.77 
 
 .204 
 
 12 
 
 7.61 
 
 .23 
 
 10.83 
 
 4.17 
 
 3.33 
 
 M13— M12 = 1.15 
 
 .191 
 
 13 
 
 8.76 
 
 .21 
 
 12.05 
 
 5.80 
 
 3.12 
 
 M13— M10=5.94 
 
 .176 
 
 TABLE XXXI 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 
 
 Division Rights 
 
 IN 
 
 Certain White Schools 
 
 IN 
 
 Louisiana 
 
 
 
 
 Scores 
 
 
 
 Age 
 
 
 
 Total 
 
 
 XI 
 
 X2 
 
 X3 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 X9 
 
 XIO 
 
 XI 1 
 
 
 in 
 
 24 
 14 
 
 7 
 
 43 
 24 
 14 
 
 59 
 
 43 
 24 
 
 106 
 90 
 103 
 
 16 
 
 27 
 32 
 
 17 
 19 
 31 
 
 9 
 16 
 
 22 
 
 4 
 16 
 23 
 
 4 
 9 
 13 
 
 2 
 7 
 8 
 
 '"h 
 
 7 
 
 1 
 3 
 
 8 
 
 1 
 
 3 
 
 7 
 
 
 
 
 
 
 
 
 
 286 
 
 11 
 
 1 
 
 1 
 
 2 
 
 
 
 2 
 1 
 
 1 
 1 
 
 
 
 282 
 
 12 
 
 1 
 
 1 
 
 1 
 
 
 305 
 
 13 
 
 .... 
 
 7 
 
 14 
 
 75 
 
 37 
 
 18 
 
 15 
 
 26 
 
 22 
 
 13 
 
 10 
 
 12 
 
 6 
 
 4 
 
 3 
 
 
 1 
 
 
 16 
 
 1 
 
 1 
 
 281 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 
 .16 
 
 .02 
 
 .83 
 
 0.00 
 
 .33 
 
 Mil— M10= .50 
 
 .060 
 
 11 
 
 .66 
 
 .09 
 
 2.70 
 
 0.00 
 
 1.35 
 
 M12— Mll= .48 
 
 .092 
 
 12 
 
 1.14 
 
 .12 
 
 3.80 
 
 .30 
 
 1.75 
 
 M13— M12=1.27 
 
 .138 
 
 13 
 
 2.41 
 
 .19 
 
 5.85 
 
 .65 
 
 2.60 
 
 M13— M10=2.25 
 
 .120 
 
 TABLE XXXII 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Division Rights in Arkansas City 
 
 
 
 Scores 
 
 
 
 
 Age 
 
 
 
 
 
 Total 
 
 
 XI 
 
 X2 
 
 X3 
 
 
 
 1 
 
 3 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 X9 
 
 
 in 
 
 16 
 
 8 
 
 49 
 16 
 
 60 
 49 
 
 86 
 81 
 
 43 
 38 
 
 22 
 10 
 
 1 
 
 9 7 
 6 9 
 
 2 
 9 
 
 "'6 
 
 "5 
 
 - 
 
 7 
 
 1 
 1 
 
 
 
 
 
 
 
 
 295 
 
 11 
 
 2 
 
 
 
 
 
 
 
 257 
 
 n 
 
 4 
 
 8 
 
 16 
 
 82 
 
 45 
 
 37 
 
 1 
 
 9 19 
 
 11 
 
 15 
 
 1? 
 
 
 6 
 
 in 
 
 q 
 
 
 
 1 
 
 
 
 
 294 
 
 13 
 
 1 
 
 4 
 
 8 
 
 42 
 
 29 
 
 25 
 
 2 
 
 3 19 
 
 25 
 
 23 
 
 15 
 
 
 5 
 
 8 
 
 10 
 
 5 
 
 1 
 
 3 
 
 — - 
 
 1 
 
 1 
 
 248 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 
 Median 
 
 — obtained 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 
 Q 
 
 
 P.E. 
 
 10 
 
 .26 
 
 .04 
 
 1.23 
 
 0.00 
 
 
 .61 
 
 Mil— M10= .42 
 
 .053 
 
 11 
 
 .68 
 
 .08 
 
 2.07 
 
 0.00 
 
 
 1.03 
 
 M12— Mll = 1.14 
 
 .102 
 
 12 
 
 1.82 
 
 .14 
 
 4.50 
 
 .55 
 
 
 1.97 
 
 M13— M12=1.83 
 
 .150 
 
 13 
 
 3.85 
 
 .20 
 
 6.43 
 
 1.24 
 
 
 2.59 
 
 M13— M10=3.39 
 
 .121 
 
Achievement Measured by Courtis Arithmetic Tests 53 
 
 TABLE XXXIII 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Division Rights in Salina, Kan. 
 
 ARe 
 
 Scores 
 
 Total 
 
 335 
 297 
 
 10 
 11 
 
 XI 
 
 16 
 3 
 3 
 
 X2 
 
 21 
 16 
 3 
 3 
 
 X3 
 
 47 
 21 
 16 
 3 
 
 
 
 48 
 36 
 22 
 11 
 
 1 
 
 41 
 21 
 18 
 17 
 
 2 
 
 39 
 29 
 28 
 20 
 
 3 
 
 27 
 37 
 25 
 
 27 
 
 4 
 
 28 
 17 
 32 
 25 
 
 5 
 
 21 
 29 
 33 
 29 
 
 6 
 
 21 
 16 
 29 
 31 
 
 7 
 
 9 
 16 
 25 
 14 
 
 8 
 
 12 
 25 
 
 28 
 20 
 
 9 10 11 
 
 2 .. 2 
 7 5 4 
 9 21 8 
 11 18 14 
 
 12 
 
 1 
 4 
 10 
 
 8 
 
 13 
 
 5 
 5 
 10 
 
 14 15 16 
 
 2 V. "2 
 2 1 2 
 2 ._ 2 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 "2 
 
 23 
 
 24 
 "2 
 
 X9 
 
 XIO 
 
 12 
 13 
 
 2 
 5 
 
 ""2 
 
 322 
 275 
 
 
 Median 
 
 P. E. true 
 
 — obtained 
 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 11 
 12 
 13 
 
 1.86 
 3.60 
 5.45 
 6.80 
 
 .15 
 .21 
 .19 
 
 .22 
 
 4.43 
 6.85 
 8.26 
 9.56 
 
 0.00 
 
 .95 
 
 2.86 
 
 3.54 
 
 2.21 
 2.95 
 
 2.80 
 3.01 
 
 Mil— M10= 1.74 
 M12— Mll = 1.85 
 M13— M12 = 1.35 
 M13— M10=1.96 
 
 .158 
 .176 
 .176 
 .166 
 
 TABLE XXXIV 
 
 Age Distribution, 10 to 13 Inclusive, of Courtis Series B Arithmetic Tests 
 
 Division Rights in Hibbing, Minn. 
 
 Age 
 
 Scores 
 
 Total 
 
 
 X2 
 
 X3 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 8 
 
 9 
 
 10 
 
 11 I 
 
 2 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 X9 
 
 XIO 
 
 
 10 
 
 16 
 
 35 
 16 
 
 98 
 60 
 28 
 14 
 
 11 
 22 
 10 
 10 
 
 14 
 21 
 20 
 4 
 
 11 
 16 
 12 
 4 
 
 17 
 13 
 26 
 11 
 
 11 
 15 
 19 
 
 8 
 
 12 1 
 
 13 1 
 18 1 
 
 9 1 
 
 1 4 
 3 11 
 3 14 
 3 12 
 
 4 
 10 
 16 
 
 9 
 
 5 
 12 
 13 
 10 
 
 1 
 7 
 
 4 1 
 
 5 1 
 
 5 2 
 9 1 
 3 3 
 
 8 
 
 i 
 
 3 
 6 
 
 1 
 
 '5 
 
 2 
 
 — 
 
 ~" 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 
 
 
 258 
 
 11 
 
 1 
 3 
 
 1 
 
 "2 
 2 
 
 1 
 
 "i 
 
 
 
 
 
 
 
 
 243 
 
 1' 
 
 2 
 
 1 
 
 1 
 
 1 
 
 
 1 
 
 4 
 32 
 
 "1 
 
 229 
 178 
 
 13 
 
 
 
 
 
 
 
 Median 
 
 P. E. true 
 
 —obtained 
 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 P.E. 
 
 10 
 11 
 12 
 13 
 
 .79 
 3.15 
 5.97 
 9.44 
 
 .17 
 .27 
 .27 
 .45 
 
 4.50 
 
 7.48 
 
 9.73 
 
 15.12 
 
 .13 
 
 .74 
 
 2.96 
 
 5.18 
 
 2.18 
 3.37 
 3.38 
 4.92 
 
 Mil— M10=2.36 
 
 M12— Mll=2.82 
 M13— M12 = 3.47 
 M13— M10=8.65 
 
 .172 
 .220 
 .327 
 .172 
 
 size of the median for the 13-year-old children. This is not true in 
 the case of addition, subtraction, or multiplication, though in each case 
 there is a marked similarity in the ranking. 
 
 School Systems 
 
 Total gain 
 
 10 to 13 
 inclusive 
 
 Medians 
 
 13-year-old 
 children 
 
 Hibbing 
 
 Louisville 
 
 Salina 
 
 Arkansas City 
 
 Madison County, 9-month schools 
 Louisiana 
 
 Madison County, 6-month schools 
 
 8.65 
 5.94 
 4.96 
 3.39 
 2.52 
 2.25 
 .37 
 
 9.44 
 8.76 
 6.80 
 3.65 
 3.13 
 2.41 
 .80 
 
CHAPTER V 
 
 ACHIEVEMENT AS MEASURED BY THE THORNDIKE SILENT 
 READING SCALE ALPHA 2 
 
 The Thomdike Silent Reading Scale Alpha 2 was used to measure 
 reading in the schools of Madison County at the same time that the 
 tests were given in Language and Arithmetic. This scale is designed 
 to be used as a group test, and the directions for scoring are given on 
 the basis of groups of pupils such as would be found in one room of a 
 city school system. In country schools such groups do not exist, and 
 therefore individual scores were assigned each pupil, on the basis of 
 tables prepared by Dr. Truman Lee Kelley.^ 
 
 Using the individual scores assigned on the basis of these tables, the 
 remaining problems involved in finding the differences in performance 
 in reading of different age-groups are practically the same as in the 
 work with the Trabue Language Scale and the Courtis Standard Re- 
 search Tests in Arithmetic. Distributions for the respective age- 
 groups are completed on the same assumptions, and the same age- 
 groups are used throughout. Scores used are in terms of difficulty as 
 assigned by Thorndike. 
 
 Table XXXV gives the facts in regard to reading in the Madison 
 County 6-month schools. The table reads, there are estimated to be 
 8 children 10 years old in the first grade (XI) whose scores in read- 
 ing are unknown, 10 children 10 years of age with scores unknown 
 are estimated to be in the second grade (X2), 1 child 10 years old 
 made a score of 2.0, 5 made a score of 2.2, 3 a score of 2.6, 3 a score 
 of 2.9, 2 a score of 3.2, etc. 
 
 The total gain in ability to read for the three-year period from ages 
 10 to 13 inclusive is from 4.50 to 5.26, a difference of .76. The P.E. of 
 this difference is .09, or about 12 per cent of the total. This gain is 
 very small for a three-year period, and the P.E. of the gain is relatively 
 quite large. 
 
 The improvement from year to year is very uneven. From ages 10 
 to 11 it is .03 ; from ages 11 to 12 it is .59; and from ages 12 to 13 it is 
 .14. The results indicate rather indefinite work in reading. 
 
 ^ Kelley, T. L. : "Thorndike's Reading Scale Alpha 2 Adapted to Individual 
 Testing," Teachers College Record, XVIII: 253#. (May, 1917). 
 
 56 
 
TABLE XXXV 
 Age Distribution, 10 to 13 Inclusive, of Thorndike Silent Reading Scale Alpha 2 Scores in the 6-Month White Schools of Madison County, Ky. 
 
 
 
 Scores 
 
 
 Age 
 
 
 
 
 
 XI 
 
 X2 
 
 2.0 
 
 2.1 
 
 2.2 
 
 2.3 
 
 2.4 
 
 2.5 
 
 2.6 
 
 2.7 
 
 2.8 
 
 2.9 
 
 3.0 
 
 3.1 
 
 3.2 
 
 3.3 
 
 3.4 
 
 3.5 
 
 3.6 
 
 3.7 
 
 3.8 
 
 3.9 
 
 4.0 
 
 4.1 
 
 4.2 
 
 4.3 
 
 4.4 
 
 4.5 
 
 4.6 
 
 4.7 
 
 4.8 
 
 4.9 
 
 5.0 
 
 5.1 
 
 5.2 
 
 5.3 
 
 5.4 
 
 5.5 
 
 5.6 
 
 5.7 
 
 5.8 
 
 5.9 
 
 6.0 
 
 6.1 0.2 
 
 6.3 
 
 6.4 
 
 6.5 
 
 6.6 
 
 6.7 
 
 6.8 
 
 6.9 
 
 7.0 
 
 7.1 
 
 7.2 
 
 7.3 
 
 7.4 
 
 7,, 5 
 
 7.6 
 
 7.7 
 
 X9 
 
 XIO 
 
 XII 
 
 10 
 
 8 
 
 10 
 
 1 
 
 _._ 
 
 5 
 
 -.- 
 
 .._ 
 
 
 3 
 
 
 ... 
 
 3 
 
 
 
 2 
 
 6 
 
 
 
 4 
 
 1 
 
 
 
 1 
 
 12 
 
 1 
 
 2 
 
 2 
 
 11 
 
 
 
 2 
 
 18 
 
 3 
 
 5 
 
 4 
 
 1 
 
 
 ? 
 
 ? 
 
 ? 
 
 4 
 
 
 •-> 
 
 " 
 
 
 
 — 
 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 11 
 12 
 
 4 
 3 
 
 8 
 4 
 
 
 
 4 
 
 2 
 
 
 
 
 2 
 
 i 
 
 ... 
 
 4 
 
 1 
 
 
 2 
 
 4 
 
 1 
 
 5 
 6 
 
 
 
 6 
 4 
 
 1 
 4 
 
 
 
 1 
 1 
 
 5 
 6 
 
 ... 
 
 5 
 
 1 
 
 ... 
 
 9 
 
 in 
 
 
 1 
 
 1 
 
 2 
 3 
 
 9 
 
 in 
 
 1 
 
 5 
 
 5 
 10 
 
 7 
 8 
 
 1 
 
 1 
 
 9 
 
 ] 
 
 2 
 
 9 
 
 '3 
 
 ■^ 
 
 1 
 
 3 
 4 
 2 
 
 
 3 
 4 
 
 7 
 
 
 2 
 2 
 2 
 
 4 
 
 2 
 
 1 
 
 7 
 7 
 
 ""2 
 
 1 
 
 "'3 
 
 "I 
 
 ' i 
 
 
 
 
 
 
 
 — 
 
 "'2 
 
 — 
 
 — 
 
 13 
 
 1 
 
 3 
 
 z. 
 
 z. 
 
 1 
 
 
 
 z. 
 
 i 
 
 i: 
 
 
 2 
 
 i: 
 
 2 
 
 :i 
 
 1 
 
 
 Z 
 
 2 
 
 
 
 
 
 5 
 
 
 2 
 
 1 
 
 9 
 
 ... 
 
 
 7 
 
 10 
 
 1 
 
 8 
 
 4 
 
 2 
 
 3 
 
 1 
 
 2 
 
 
 4 
 
 
 2 
 
 1 
 
 
 "i 
 
 1 
 
 ::: 
 
 
 
 
 '" 
 
 "i 
 
 1 
 
 3" 
 
 "1 
 
 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 
 
 Age 
 
 Total 
 
 Median 
 
 — obtained 
 
 75 per- 
 
 25 per- 
 
 Q 
 
 
 
 P E 
 
 
 
 
 
 median 
 
 centile 
 
 centile 
 
 
 
 
 
 
 
 10 
 
 122 
 
 4.50 
 
 .10 
 
 5.01 
 
 3.22 
 
 .89 
 
 
 Mil— M10= .03 
 
 .09 
 
 
 11 
 
 107 
 
 4.53 
 
 .12 
 
 5.23 
 
 3.24 
 
 .99 
 
 
 M12— Mll= .59 
 
 .09 
 
 
 12 
 
 131 
 
 5.12 
 
 .09 
 
 6.06 
 
 4.37 
 
 .84 
 
 
 M13— M12= .14 
 
 .09 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 13 
 
 117 
 
 5.26 
 
 .11 
 
 6.59 
 
 4.59 
 
 1.00 
 
 
 M13— M10= .76 
 
 .09 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 TABLE XXXVI 
 Age Distribution, 10 to 13 Inclusive, of Thorndike Silent Reading Scale Alpha 2 Scores in the 9-Month White Schools of Madison County, Ky. 
 
 Afp 
 
 
 
 Scores 
 
 
 10 
 
 XI 
 
 6 
 
 '"'3 
 3 
 
 X2 
 
 13 
 6 
 
 
 
 2.2 
 
 2 
 
 1 
 
 2.3 
 
 2.4 
 
 2.5 
 
 2.6 
 
 1 
 1 
 
 2.7 
 
 1 
 
 2.8 
 
 2.9 
 
 5 
 1 
 3 
 
 3.0 
 
 "i 
 
 3.1 
 
 3.2 
 
 2 
 2 
 
 3.3 
 
 5 
 
 2 
 3 
 
 3.4 
 "i 
 
 3.5 
 
 1 
 1 
 2 
 
 3.6 
 
 5 
 2 
 2 
 
 3.7 
 
 "2 
 
 3.8 
 
 3.9 
 
 4.0 
 
 2 
 
 1 
 
 4.1 
 
 4 
 5 
 2 
 1 
 
 4.2 
 
 4.3 
 
 4 
 1 
 2 
 
 4.4 
 "I 
 
 1.5 
 
 7 
 5 
 3 
 
 2 
 
 4.6 
 
 1.7 
 2 
 
 "i 
 
 4.8 
 
 3 
 
 1 
 2 
 3 
 
 4.9 
 
 2 
 3 
 5 
 
 2 
 
 5.0 
 5 
 
 "3 
 
 2 
 
 5.1 
 
 ' 8 
 
 "i 
 
 2 
 
 5.2 
 
 3 
 3 
 4 
 
 2 
 
 5.3 
 "2 
 
 5.4 
 
 "2 
 
 1 
 
 5.5 
 
 5 
 5 
 3 
 
 4 
 
 5.6 
 
 2 
 2 
 2 
 
 5.7 
 
 1 
 
 2 
 4 
 
 2 
 
 5.8 
 
 1 
 1 
 
 4 
 3 
 
 5.9 
 
 1 
 2 
 1 
 1 
 
 6.0 
 
 3 
 4 
 6 
 3 
 
 6.1 
 
 "i 
 
 1 
 
 6.2 
 
 1 
 4 
 6 
 3 
 
 6.3 
 
 "2 
 3 
 
 6.4 
 
 1 
 
 ""i 
 
 1 
 
 6.5 
 
 1 
 2 
 3 
 
 8 
 
 6.6 
 
 3 
 4 
 9 
 3 
 
 6.7 
 
 3 
 
 1 
 5 
 
 7 
 
 6.8 
 2 
 
 "3 
 
 2 
 
 6.9 
 
 "i 
 
 "3 
 
 7.0 
 
 1 
 2 
 1 
 3 
 
 7.1 
 
 1 
 2 
 1 
 1 
 
 7.2 
 
 i 
 3 
 1 
 
 7.3 
 
 1 
 2 
 2 
 
 7.4 
 
 7.5 
 
 7.6 
 
 7.7 
 
 7.8 
 
 7.9 
 
 8.0 
 
 8.1 
 
 8.2 
 
 8.3 
 
 8.4 
 
 8.5 
 
 8.6 
 
 8.7 
 
 1 
 
 X9 
 
 XlO 
 
 xu 
 
 11 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 1 
 7 
 
 ""2 
 I 
 
 
 1? 
 
 1 
 
 1 
 
 
 1 
 
 
 1 
 
 
 1 
 
 
 
 
 
 
 
 
 
 13 
 
 3 
 
 
 
 
 
 1 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 P. E. true 
 
 
 
 
 
 
 
 Age 
 
 Total 
 
 Median 
 
 —obtained 
 median 
 
 75 per- 
 centile 
 
 25 per- 
 centile 
 
 Q 
 
 
 
 P. E. 
 
 10 
 
 109 
 
 4.55 
 
 .15 
 
 5.51 
 
 2.98 
 
 1.26 
 
 Mil— M10= 
 
 .75 
 
 .14 
 
 11 
 
 84 
 
 5.30 
 
 .17 
 
 6.25 
 
 3.75 
 
 1.25 
 
 M12— Mll = 
 
 .66 
 
 .13 
 
 12 
 
 107 
 
 5.86 
 
 .10 
 
 6.65 
 
 4.95 
 
 .85 
 
 M13— M12 = 
 
 .59 
 
 .10 
 
 13 
 
 83 
 
 6.45 
 
 .13 
 
 7.01 
 
 5.14 
 
 .93 
 
 M13— M10=l 
 
 .90 
 
 .12 
 
Achievement Measured by Thorndike Reading Scale 57 
 
 Table XXXVI gives the facts in regard to reading in the 9-month 
 schools of Madison County. 
 
 The total gain for the three-year period in this case is from 4.55, 
 the median ability of the 10-year-old children, to 6.45, the median 
 ability of the 13-year-old children, a difference of 1.90. This gain is 
 just 2.5 times as great as that made in the 6-month schools of the 
 same county. The median abilities of the 10-year-old children are 
 about alike. The comparison of amount of gain in the two cases is 
 valid, since the scores given are in terms of difficulty for the children. 
 It must be remembered, however, that there is no proof that the gain 
 in each case can be attributed solely to school work. It is possible 
 that influences outside of the school contribute materially to the re- 
 sult. The conclusion can only be that school influence on reading and 
 other possible influence on reading accomplish between them 2.5 times 
 as much in places having a 9-month school term as do the same in- 
 fluences in places having a 6-month school term. 
 
 The P.E. of the gain in the 9-month schools (1.90) is .12, about 
 7 per cent of the gain. The probable actual deviation of the true dif- 
 ference from the obtained difference in these schools is a little larger 
 than the like deviation in the 6-month schools, but in proportion to the 
 gains made in the respective schools the deviation is less in the 9-month 
 schools. 
 
 The yearly gains are much more nearly uniform than in the 6-month 
 schools. From ages 10 to 11 the gain is .75 ; from ages 11 to 12 it is 
 .56; and from ages 12 to 13 it is .59. 
 
 Tables XXXVII to XXXIX inclusive give comparative data in re- 
 gard to reading from certain other school systems. The data from 
 Louisville are based upon tests given by the Psychological Laboratory 
 of the Louisville Board of Education. Data for Hamilton Town- 
 ship, New Jersey, were collected by the Psychology Department of 
 the Trenton State Normal School. The data from Amsterdam, New 
 York, were collected by the Department of Educational Administra- 
 tion of Teachers College, Columbia University. 
 
 In the following summary the total improvement in silent reading dur- 
 ing the three-year period from ages 10 to 13 inclusive is given for each 
 of the school systems studied, and also the median ability for the re- 
 spective groups of 13-year-old children. The schools are listed in the 
 order of the amount of improvement made. 
 
58 Study of Achievement in Country and Town Schools 
 
 
 Total gain 
 
 Medians 
 
 School Systems 
 
 10 to 13 
 inclusive 
 
 13-year-old 
 children 
 
 Amsterdam. _ -- -- 
 
 2.85 
 1.90 
 1.81 
 1.58 
 .76 
 
 6 55 
 
 Madison County, 9-month schools -. 
 
 6 45 
 
 Louisville . - -- 
 
 6.90 
 
 Hamilton Township.. .. . 
 
 6.77 
 
 Madison County, 6-month schools . . 
 
 5.25 
 
 
 
 The ranking in order of amount of improvement for the three-year 
 period from ages 10 to 13 inclusive does not correspond exactly with 
 the ranking in order of abihty of 13-year-old children. As has been 
 pointed out in the case of language and arithmetic, the medians for 
 13-year-old children may be affected by constant errors in any of the 
 respective systems, such as a tendency to overrun the time allowance, 
 a little greater leniency in scoring papers in one place than in another 
 either by the teachers or by the examiner, in giving directions, etc. 
 Such errors are compensating when the difference between the medians 
 for two ages is used as a measure, since these errors would be constant 
 for any given place and time for pupils of the diiTerent ages. In view 
 of this, it is probable that the ranking in order of total gain in the above 
 table is the more significant ranking. 
 
 The amount of gain in the respective systems is directly comparable, 
 since the scale used is expressed in terms of units of difficulty for the 
 children. We can say that the results in Amsterdam for the three- 
 year period from 10 to 13 inclusive are about four times as good as 
 are the results in the Madison County 6-month schools, and about 
 twice as good as the results in Hamilton Township. However, not 
 all of the improvement noted is known to be the result of school pro- 
 cedure. A good share of the gain in ability to read silently may be 
 due to forces outside of the school, such as library facilities, home in- 
 fluence, newspapers, etc. Indeed, it is sometimes charged that school 
 work in reading stresses oral work too much.^ The conclusion from 
 the above table is therefore limited to the statement that the sum of 
 school influence in reading and outside influences in reading is about 
 four times as great in Amsterdam as in the Madison County 6-month 
 schools for the equal period of time from ages 10 to 13. If equality 
 of outside influences making for improvement in silent reading might 
 be assumed for the different systems studied, there would be left the 
 single variable element in the school work on silent reading. It does 
 
 ^ For an example of this see Klapper, Teaching Children to Read. 
 
Achievement Measured by Thorndike Reading Scale 
 
 59 
 
 not seem probable, however, that influences other than those of the 
 school which make for improvement in silent reading are equal in 
 cases such as Amsterdam, New York, and the smaller schools of Mad- 
 ison County, Kentucky. 
 
 The following summary shows the gain for each age period of a 
 year in the systems studied : 
 
 School Systems 
 
 Yearly Gains 
 
 
 10 to 11 
 
 11 to 12 
 
 12 to 13 
 
 Amsterdam.. . . __. 
 
 1.67 
 .75 
 .94 
 
 .72 
 .03 
 
 .62 
 .56 
 .42 
 .34 
 .59 
 
 .56 
 
 Madison County, 9- month schools 
 
 .59 
 
 Louisville.. . ._ ._ . _ . 
 
 .45 
 
 Hamilton Township . 
 
 .52 
 
 Madison County, 6-month schools . . 
 
 .14 
 
 
 
 Total 
 
 4.11 
 
 2.53 
 
 2.26 
 
 
 
 The variations in the amount of yearly gain are large, as is the case 
 in language and arithmetic. The greatest gain seems to be from ages 
 10 to 11, and each year after this shows less gain than the preceding 
 year. It appears that greater gain is made by younger children than 
 by older ones within the limits of the age-groups studied. 
 
 CONCLUSIONS 
 
 1. The difference in ability in silent reading, as measured by the 
 Thorndike Silent Reading Scale Alpha 2, can be obtained for children 
 of different ages. 
 
 2. Since the function measured by this scale may be improved by 
 influences outside of the school, as well as by school training, this dif- 
 ference cannot be considered as a wholly reliable measure of school 
 efficiency in teaching silent reading. 
 
 3. This difference does give a reliable measure of the improvement 
 in silent reading due to all causes during the three-year period from 
 ages 10 to 13 inclusive. 
 
 4. Since the improvement measured by this scale is stated in terms 
 of difficulty for the children, the differences obtained in the respective 
 systems studied may be compared directly. 
 
 5. The differences obtained may be compared for two systems of 
 schools, even though the testing was done at different times in the re- 
 spective systems. 
 
60 Study of Achievement in Country and Town Schools 
 
 6. The diflferences obtained are valid, even though there may be 
 constant errors in the giving of the tests, such as might be caused by 
 having the tests given by the teachers. 
 
 7. Greater gain in silent reading seems to be made by the younger 
 children within the age limits studied (ages 10 to 13 inclusive). 
 
 8. Material gain in silent reading may be made during any part or all 
 of the period from ages 10 to 13 inclusive. 
 
 9. The results in the Madison County 6-month schools are much 
 poorer than in any other system studied. 
 
 10. The results in the Madison County 9-month schools compare 
 favorably with those in the other systems studied. 
 
CHAPTER VI 
 SUMMARY AND CONCLUSIONS 
 
 This study was undertaken with four specific objects in mind. 
 These objects are set forth in Chapter II, and are stated as follows : 
 
 First : To show that the diflFerence in performance in school sub- 
 jects of children of different ages can be obtained. 
 
 Second: To show that this difference is a measure of school effi- 
 ciency which may be used to measure schools or school systems. 
 
 Third : To apply this measure to a system of country schools. 
 
 Fourth : To compare the results in this country school system with 
 the results in certain city school systems. 
 
 The results of the study are considered in the light of these aims. 
 Sections 1, 2, 3 and 4 of this chapter deal respectively with the pur- 
 poses stated. 
 
 Section 1 
 
 The purpose as stated is "to show that the difference in perform- 
 ance in school subjects of children of different ages can be obtained." 
 
 With the tests used such differences have been obtained for chil- 
 dren from 10 to 13 years of age inclusive. As the work progressed, 
 certain difficulties became apparent which seem to justify the following 
 statements : 
 
 The tests to be used in obtaining age differences in performance 
 must be of such a nature that identical tests can be given in identical 
 manner to all of the grades tested. Such a spelling scale as that de- 
 vised by Ayres could not give age differences because the 12-year-old 
 child in the third grade would have a score on a test different from 
 the test used for the 12-year-old child in the sixth grade. The results 
 for children of like age, but in different grades, would be in terms that 
 could not be directly compared. 
 
 The tests should be given in as many as possible of the grades of 
 the elementary school. The tabulations in this study include results 
 for grades 3 to 8 inclusive and in some cases for grades 4 to 8. With 
 results from children in these grades it was found necessary to esti- 
 mate the number of children in the lower grades of the respective 
 ages studied, and it seemed unwise to attempt to find medians for 
 
 61 
 
62 Study of Achievement in Country and Tozvn Schools 
 
 groups of children under 10 years of age because of the limited num- 
 ber of children of the lower ages tested. If the tests could have been 
 given in grade 2, it would have been possible to calculate medians for 
 children under 10, and the necessity of calculating the probable num- 
 ber of 10-to- 11 -year-old children in grade 2 would have been avoided, 
 and scores could have been assigned them. 
 
 Full age-grade tables of all the children in all the grades of the 
 systems studied should be prepared. Such tables give a very desirable 
 check on the number of children tested, and furnish a basis for esti- 
 mation of the number of children of any given age in grades not tested. 
 
 Tests used should be such as will test the ability of the children in 
 the lower grades as well as the ability of those in the higher grades. 
 The work with Courtis Standard Research Tests in Arithmetic showed 
 that these tests do not measure the ability of lower grade children, 
 since a great many children were unable to work correctly even one 
 of the examples given. A test that includes problems less difficult than 
 those of this series would be desirable. 
 
 Tests used must be individual tests rather than group tests ; at least 
 it must be possible to assign individual scores on the basis of the tests 
 given. This is necessary in order that the individual scores of chil- 
 dren of any given age may be taken from the different grades in which 
 these children are tested and thrown into a single distribution on an 
 age basis. 
 
 For work in country schools there is an added need for individual 
 tests rather than group tests, since groups of children of like age or 
 grade in any given country school are likely to be small. 
 
 The above statements may be summarized as a set of directions for 
 the guidance of those giving tests which are to be tabulated on an age 
 basis. The concise directions are as follows : 
 
 1. Select tests that are identical for all the grades to be tested. 
 
 2. Test as many of the grades as possible. 
 
 3. Make full age-grade tables of all the children in all the grades. 
 
 4. Select tests that measure the ability of all the children. 
 
 5. Use individual rather than group tests. 
 
 Section 2 
 
 The second purpose of this study as stated is "to show that this dif- 
 ference is a measure of school efficiency which may be used to meas- 
 ure schools or school systems." 
 
 In so far as the tests used measure functions affected solely, or even 
 
Summary and Conclusions 63 
 
 largely, by school training, the difference in performance of children 
 of different ages seems to furnish a measure of school efficiency. In 
 the case of the Trabue Language Scale B the function measured has 
 not been shown to be fundamentally one dependent on school training. 
 In the case of the Courtis Standard Research Tests in Arithmetic 
 Series B it seems that the function measured is to a very large degree 
 at least one dependent on school training ; therefore in this case the 
 difference obtained is one that is chiefly the direct result of the school 
 training received by the children and of elimination, and for that rea- 
 son may be considered a measure of the effectiveness of that training 
 and of elimination. In the case of the Thorndike Silent Reading Scale 
 Alpha 2 the function measured is one which is greatly affected by 
 school training, but which may also be greatly affected by influences 
 outside of the school. 
 
 Even though the function measured be one which is affected by in- 
 fluences other than school training, it may be argued that the work 
 of the school should so supplement other influences brought to bear 
 upon the children as to secure for them reasonable development of 
 the function considered. If this conception of the work of the school 
 is accepted, the difference between performance of children of differ- 
 ent ages seems a sound measure of the degree to which the school has 
 succeeded in its effort to secure the desired development. 
 
 In any case in which the difference in performance of children of 
 different ages is considered as measuring some function other than 
 one dependent on school training for its improvement, the difficulty 
 lies in the nature of the test rather than in the fact that the age-differ- 
 ences are used. The same criticism would apply to the same meas- 
 urement if grade standards were used. 
 
 There are a few advantages that the use of age-differences as a 
 measure of school efficiency seems to have over the use of grade 
 standards. One of these is that the testing may be done at any con- 
 venient time in the school year, with the possible exception of the first 
 three or four weeks of the school term. This is due to the fact that 
 the birthdays of any age-group of children, say the 10-year-olds, are 
 distributed throughout the year in the same way as are the birthdays 
 of any other age-group, say the 13-year-olds. This means that the loss 
 of children from the group because they reach a higher age will be 
 compensated by the accretion of other children from the next lower 
 age-group. Moreover, this loss and accretion continues throughout 
 the school year and for each age-group, so that the difference between 
 
64 Study of Achievement in Country and Town Schools 
 
 any two age-groups is a constant quantity. During the first few weeks 
 of school there may be a shghtly uneven loss in school ability due to 
 the vacation period, since it has never been shown that the retaining 
 power of children of different ages is the same. After school is once 
 well under way this uneven loss is overcome, and the testing may be 
 done at any convenient time. 
 
 Another advantage in the use of age-diflferences as a measure 
 of school efficiency lies in the fact that errors such as those caused by 
 having the tests given by the teachers are compensated, since such er- 
 rors are identical for children of all ages. 
 
 A further advantage in the use of age-differences lies in the fact 
 that the effects of poor grading or great retardation are considered in 
 the measurement of improvement. The necessity of correcting grade 
 results for retardation is avoided, since the score of the retarded child 
 of any given age has entered into the measurement of that age-group. 
 
 The advantages mentioned above are of particular significance for 
 work in country schools. The difficulties involved in giving tests 
 where the schools are scattered, the grading poor, and the groups of 
 children frequently very small, make it especially desirable to give 
 tests in these schools at the most convenient time of the year and to 
 have the tests given by the teacher. 
 
 Section 3 
 
 The third purpose of this study as stated is "to apply this measure 
 to a system of country schools." 
 
 This purpose has been accomplished and the results given in the 
 treatment of the respective tests. 
 
 Section 4 
 
 The fourth purpose of this study as stated is "to compare the re- 
 sults in this country school system with the results in certain city 
 school systems." 
 
 The Madison County 6-month schools have been compared through- 
 out this study with the schools in the same county having a 9-month 
 school term. In addition to this comparison, the results of the Trabue 
 Language Scale B have been compared with results from Louisville, 
 Kentucky ; Paterson, New Jersey ; and St. Paul, Minnesota. As has 
 already been pointed out, this comparison concerns the results of 
 school training and also other influences. The 9-month schools in 
 Madison County made a gain in the ability to complete sentences 53 
 
Summary and Conclusions 65 
 
 per cent greater than the gain made over the same three-year period 
 in the 6-month schools. None of the systems studied made a gain so 
 great as that of the Madison County 9-month schools, and all of them 
 except St. Paul made gains greater than did the Madison County 
 6-month schools. St. Paul made practically the same gain as did these 
 schools. 
 
 If the comparison is made on the basis of the final ability attained 
 by the children (medians for 13-year-olds), the Madison County 6- 
 month schools are poorer than any of the others studied. Louisville 
 13-year-old children possess 50 per cent greater ability to complete 
 sentences than do those in the smaller schools of Madison County. 
 The children of the same age in the 9-month schools possess 25 per 
 cent greater ability than do the children of the 6-month schools. The 
 10-year-old children in Louisville possess greater ability to complete 
 sentences than do the 13-year-old children in the 6-month Madison 
 County schools, and so do the 12-year-old children in the 9-month 
 schools, the 11-year-olds in Paterson and the 10-year-olds in St. Paul. 
 
 Whichever basis of comparison is used, the Madison County 6- 
 month schools are at the bottom. If the age achievement is used 
 as the basis of comparison, the Madison County 9-month schools are 
 only kept from the bottom by the presence of the 6-month schools 
 among the systems compared. 
 
 In comparing the work in arithmetic it must be remembered that 
 the results are in terms of examples worked correctly in a given length 
 of time, and that it cannot be assumed that it is twice as difficult to 
 add 8 examples as it is to add 4 in the same time. Bearing in mind 
 this caution, it seems, on the basis of the results obtained, that while 
 the pupil in the Madison County 6-month schools is increasing his 
 ability to add to such an extent that he can add one more example in 
 the time allowed, the pupil in the Madison County 9-month schools is 
 increasing his ability to add by 1.6 examples, the pupil in Louisiana by 
 2.2, the one in Arkansas City by 2.7, the one in Salina by 3.0, the one in 
 Louisville by 3.4, and the one in Hibbing by 5.7 examples. 
 
 In subtraction the situation is much the same. While the pupil in 
 the Madison County 6-month schools is improving his ability to sub- 
 tract by 1 example, the one in the 9-month schools of the same county 
 is improving his ability by 3.1, the one in Salina by 3.2, the one in 
 Louisiana by 3.3, the one in Arkansas City by 4.2, the one in Louis- 
 ville by 4.8, and the one in Hibbing by 5.8 examples. 
 
66 Study of Achievement in Country and Town Schools 
 
 While, the pupil in the 6-month schools of Madison County is im- 
 proving his ability to multiply to such an extent that he can multiply 
 one more example in the time allowed, the pupil in Louisiana improves 
 his ability so that he can multiply 5.8 more examples in the time al- 
 lowed, the pupil in the Madison County 9-month schools improves by 
 6.2 examples, the one in Louisville by 7.7, the one in Salina by 7.8, 
 the one in Arkansas City by 8.8, and the one in Hibbing by 12.5 ex- 
 amples. 
 
 In division the situation is even worse. While an improvement of 
 one example is being made by the pupil in the Madison County 6- 
 month schools, there is an improvement of 6.1 by the pupil in the 
 Louisiana schools, of 6.8 in the Madison County 9-month schools, of 
 9.1 in Arkansas City, of 13.4 in Salina, of 16.1 in Louisville, and of 
 23.4 in Hibbing. 
 
 If the comparison be put on the basis of achievement of 13-year-old 
 children, the situation in the Madison County 6-month schools does 
 not seem to be much better. The median 13-year-old child in these 
 schools can neither add, subtract, nor multiply as well as the median 
 10-year-old in Hibbing, and he can barely divide as well. The Madi- 
 son County 9-month schools do better on this basis, but the median for 
 the 13-year-old children is below the median for the 11-year-old chil- 
 dren in Hibbing in addition, subtraction, multiplication and division. 
 
 The comparison of the results of the Thorndike Silent Reading 
 Tests shows conditions which are not quite so disastrous for the 
 Madison County 6-month schools in that the differences are not so 
 great. On the basis of improvement, while the pupil in the Madison 
 County 6-month schools is improving his ability to read by one step 
 on the Thorndike scale, the pupil in Hamilton Township improves 2.1 
 steps, the one in Louisville 2.4, in the Madison County 9-month schools 
 2.5, and in Amsterdam Z.7 steps. On the basis of age achievement 
 the median 13-year-old child in the Madison County 6-month schools 
 is poorer than the median 11 -year-old child in Hamilton Township, 
 Louisville, and Amsterdam, and almost as poor as the median 11 -year- 
 old child in the Madison County 9-month schools. 
 
 The comparison of the Madison County 6-month schools with the 
 other schools tested may be summed up in the statement that the chil- 
 dren in these schools have less ability along every line tested than have 
 the children of the same ages in other schools, and that they are learn- 
 ing more slowly, except that the children in St. Paul are improving as 
 slowly as they do in language. 
 
Summary and Conclusions 67 
 
 Final Summary 
 
 The following four statements seem justified concerning the four 
 purposes for which this study was undertaken : 
 
 First: The difference in performance in school subjects can be ob- 
 tained for children of different ages. 
 
 Second: This difference is a measure which may be used to meas- 
 ure schools or school systems, and has certain advantages for this pur- 
 pose, especially in rural schools. 
 
 Third : This measure has been applied to a system of country 
 schools. 
 
 Fourth : The Madison County 6-month schools compare unfavor- 
 ably with all of the other schools measured. The Madison County 
 9-month schools compare more favorably. 
 
APPENDIX 
 
 For the convenience of any who may wish to supplement the re- 
 port made in Chapter I of scientific studies on the results of instruc- 
 tion in rural schools, or who may later wish to bring this report up to 
 date, this statement of the educational literature investigated is made. 
 
 As a first step in searching for such data the following surveys were 
 studied. In each case the tables showing the results were investigated 
 to find what scores might be included from rural schools in the com- 
 parisons made. 
 
 SURVEYS 
 
 Akron, Ohio. 
 
 Alabama (three counties). 
 
 Alabama.* 
 
 Alton, 111. 
 
 Amsterdam, N. Y. 
 
 Anderson, Ind. 
 
 Arizona.* 
 
 Ashland, Ore. 
 
 Atlanta, Ga. 
 
 Baltimore, Md. 
 
 Binghamton, N. Y. 
 
 Blaine, Wash. 
 
 Bloommgton, Ind. 
 
 Boise, Idaho. 
 
 Boston, Mass. 
 
 Bridgeport, Conn. 
 
 Buffalo, N. Y. 
 
 Brookline, Mass. 
 
 Butte, Mont. 
 
 Chicago, 111. (Educational Commission). 
 
 Cincinnati, Ohio. 
 
 Cleveland, Ohio. 
 
 Cleveland, Ohio.f 
 
 Colorado. 
 
 Cold Springs, N. Y. (Haldane School). 
 
 Colorado (rural schools). 
 
 Columbia, S. C. 
 
 Connecticut (Educational Commission). 
 
 Connecticut (counties). 
 
 Dansville, N. Y. (High School). 
 
 Dayton, Ohio. 
 
 Delaware.J 
 
 Delaware (School Buildings). 
 
 Denver, Col. 
 
 Detroit, Mich. (Recreational). 
 
 Elyria, Ohio.* 
 
 Evansville, Ind. (Vocational). 
 
 Framingham, Mass. 
 
 Gary, Ind.* 
 
 Gary, Ind. J 
 
 Georgia (Counties). 
 
 Grafton, W. Va. 
 
 Grand Junction, Col. 
 
 Grand Rapids, Mich. 
 
 Great Neck, L. I. 
 
 Greene County, Ind. 
 
 Greenwich, Conn. 
 
 Hammond, Ind. (Industrial). 
 
 Harrisburg, Pa. 
 
 Herkimer, N. Y. 
 
 Huron County. 
 
 Illinois (State). 
 
 Indiana (Vocational). 
 
 Janesville, Wis. 
 
 Jefferson County, Ind. (Vocational). 
 
 Kansas (high schools). 
 
 *United States Bureau of Education. 
 tRussell Sage Foundation. 
 JGeneral Education Board. 
 
 68 
 
Appendix 
 
 69 
 
 Lane County, Ore. 
 
 Los Angeles, Cal. 
 
 Leavenworth, Kan. 
 
 Louisville, Ky. 
 
 Maricopa County, Ga. 
 
 Maryland (State). J 
 
 Memphis, Tenn.* 
 
 Minneapolis, Minn. (Vocational). 
 
 Missouri (Saline County). 
 
 Montclair, N. J. 
 
 Nassau County, N. Y. 
 
 Newburgh, N. Y. 
 
 New Orleans, La. (Vocational). 
 
 Newton, Mass. 
 
 New York City. 
 
 North Dakota.* 
 
 Oakland, Cal. 
 
 Ogden, Utah. 
 
 Ohio. 
 
 Oklahoma. 
 
 Oklahoma (county). 
 
 Oswego County, N. Y. 
 
 Paducah, Ky. 
 
 Paterson, N. J. 
 
 Port Townsend, Wash. 
 
 Portland, Ore. 
 
 Reading, Pa. 
 
 Rochester, Minn. 
 
 Rockford, 111. 
 
 St. Louis, Mo. 
 
 St. Paul, Minn. 
 
 Salt Lake City, Utah. 
 
 San Antonio, Texas. 
 
 San Francisco, Cal. 
 
 San Mateo, Cal. 
 
 Saskatchewan, Canada. 
 
 Solvay, N. Y. 
 
 South Bend, Ind. 
 
 South Dakota.* 
 
 Stamford, Conn. 
 
 Syracuse, N. Y. 
 
 Travis County, Texas. 
 
 Utica, N. Y. 
 
 Van Wert, Ohio. 
 
 Vermont.f 
 
 Virginia (Educational Commission). 
 
 Washington.* 
 
 Waterbury, Conn. 
 
 Westchester, N. Y. 
 
 Wilmington, Del.* 
 
 Windsor County, Vt. 
 
 Wisconsin (State). 
 
 Winston-Salem, N. C. 
 
 Wyoming.* 
 
 The next step in the search for data bearing on the objective results 
 of instruction in rural schools was a study of A Selected Bibliography 
 of Certain Phases of Educational Measurements, prepared by Edna 
 Bryner and published in the Seventeenth Year Book of the National 
 Society for the Study of Education, 1918. The titles listed were 
 exainined, and whenever there seemed any possibility that the article 
 might contain any data concerning country schools the article was stud- 
 ied. One bulletin by E. J. Ashbaugh, entitled The Arithmetical Skill 
 of Iowa School Children, University of Iowa Extension Bulletin, No. 
 24, November, 1916, was not found. In all, twenty-eight of the ref- 
 erences were examined, and of these five were found to contain data. 
 
 As a final step in the search for material bearing on the comparison 
 of country and city schools by the use of objective standards, a study 
 was made of the monthly lists of current educational publications pre- 
 
 *United States Bureau of Education. 
 fRussell Sage Foundation. 
 JGeneral Education Board. 
 
70 Study of Achievement in Country and Town Schools 
 
 pared by the United States Bureau of Education, beginning with the 
 list for June, 1917, and extending through the list for October, 1920. 
 In this study every title and subtitle was considered, and in case 
 there seemed a possibility that objective data might be found, the 
 reference was investigated. In this study only two articles which 
 seemed to bear in any way on this topic were not found. These refer- 
 ences are: (1) Johnson, W. E., "Reading, Writing, Arithmetic and 
 Spelling in City and Town Schools of South Dakota in 1917-1918," 
 Bulletin of the Northern Normal and Industrial School at Aberdeen, 
 S. D., Vol. XII, No. 2 (October, 1918) ; and (2) Heckert, J. W., 
 "The Cleveland Survey Tests in Arithmetic in the Miami Valley," 
 Elementary School Journal, XVIII : 447-457. 
 

 
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