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 SCHOOL DOCUMENT NO. ^-1918 
 
 BOSTON PUBLIC SCHOOLS 
 
 ARITHMETIC 
 
 DETERMINING THE ACHIEVEMENT OF PUPILS IN 
 COMMON FRACTIONS 
 
 Bulletin No. XV. of the Department of 
 Educational Investigation and Measurement 
 
 
 jiJlsiR>, ;i©lB';\;' ',;;',,' >',^:; ; ;;'. 
 
 BOSTON 
 
 PRINTING DEPARTMENT 
 
 IQ18 
 

 In School Committee, Boston, June 26, 1918. 
 Ordered, That four thousand (4,000) copies of a 
 bulletin on ''Determining the Achievement of Pupils 
 in Common Fractions," prepared by the Department 
 of Educational Investigation and Measurement and 
 approved for publication by the Board of Superin- 
 tendents at its meeting on June 19, 1918, be printed 
 as a school document. 
 
 Attest : 
 
 THORNTON D. APOLLONIO, 
 
 Secretary. 
 
 I 
 
 ,' -ij. 
 
/ 
 
 CONTENTS, 
 
 Introduction . . . . • • • • 
 I. Previous Work in Fractions in Boston 
 
 Summary of Results in Addition of Fractions, 
 Summary of Results in Subtraction of Frac- 
 tions 
 
 II. Multiplication and Division of Fractions 
 
 Extent of Tests 
 
 Types of Fractions .... 
 Construction of Tests 
 Giving Tests and Correction of Results 
 III. Analysis of Results .... 
 
 Achievement 
 
 Analysis of Results in Grades VII and VIII 
 Diagnosis of Results in Each Test 
 Test 1 
 Test 2 
 Test 3 
 Test 4 
 Tests 
 Teste 
 Testy 
 Analysis of Results in Grade VI 
 IV. Plan of Diagnosis for Teacher 
 V. Summary and Conclusions . 
 
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 387243 
 
Digitized by the Internet Archive 
 
 in 2007 with funding from 
 
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 http://www.archive.org/details/arithmeticdetermOObostrich 
 
INTRODUCTION. 
 
 As fast as practicable the Department of Educational 
 Investigation and Measurement has undertaken to 
 extend educational measurement in arithmetic beyond 
 the four fundamental operations covered by the Courtis 
 Tests. To that end from time to time the department 
 has given tests in the four processes of common fractions, 
 and in a more limited way in problem work. This 
 bulletin, prepared by the Assistant Director, Mr. Arthur 
 W. Kallom, covers the work done thus far with the 
 addition, subtraction, multiplication, and division of 
 common fractions. Particular attention is directed to 
 Mr. Kallom's discussion of the various types of problems 
 involved in computation with fractions, and also to the 
 suggestions offered of ways and means of improving 
 unsatisfactory achievements of pupils. 
 
 It is frequently argued in educational discussions that 
 command over the tools of an education should be 
 acquired by the end of the sixth grade. Obviously, 
 ability to compute with common fractions is merely a 
 means to a desired end and not an education in itself. 
 This study shows that either (a) not as much ability 
 to use common fractions is being developed before 
 the end of the sixth grade as should be, or (b) the period 
 of the first six years in the elementary school is too 
 short to furnish pupils with all the tools of an education. 
 
 The manuscript for this bulletin was approved for 
 publication by the Board of Superintendents at its 
 meeting on June 19, 1918. 
 
 FRANK W. BALLOU, 
 
 Assistant Superintendent in Charge. 
 
.^..rrSCHOOL DOCUMENT NO. 5. 
 
 DETERMINING THE ACHIEVEMENT OP PUPILS IN 
 COMMON FRACTIONS. 
 
 I. Results of Previous Tests in Fractions. 
 
 Tests in the four fundamental operations have been 
 given in the City of Boston during the past five years. 
 During this time the gain in the amount of work done has 
 been from 12 per cent to 17 per cent. This gain in 
 amount of work done has been accompanied by an actual 
 increase in the accuracy with which the work was com- 
 pleted. This increase by which pupils are being graduated 
 or are being promoted into the next grade with varying 
 degrees of superiority up to 17.7 per cent over the 
 results which were being obtained previous to the giving 
 of the Courtis standard tests, is due directly or indirectly 
 to the system of educational measurement as established 
 in Boston. 
 
 The ability to handle integers does not constitute, 
 however, the sum total of the tools necessary for the 
 child in order that he may do arithmetic. Fractions, in 
 one form or another, play a large part in the arithmetical 
 work of the pupil. That we might know how well the 
 pupils are doing their work in common fractions, a plan 
 was organized in 1915 to give tests in addition, subtrac- 
 tion, multiphcation, and division of fractions in successive 
 years to a group of approximately 1,000 children in 
 Grades VI, VII, and VIII in an experimental way. 
 The tests were organized by the department in such a 
 way as to determine not only what the ability was to do 
 the various operations, but also if the pupils failed, in 
 what type of examples in any given operation the pupil 
 failed. Of the two phases of the work, the latter is of 
 the greater importance. It is not enough to say pupils 
 fail to do addition of fractions with a speed or an 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 7 
 
 accuracy which seems desirable. One must go further 
 and show the types of examples in which pupils fail. 
 
 Further, if a single test be given in a certain operation 
 and a pupil fails, it becomes the work of the teacher to 
 determine in what particular the pupil fails. This 
 enables the teacher to place the emphasis upon the work 
 where it belongs and not waste her efforts and those of 
 the pupils in drilling on matter which needs no drilling. 
 
 Addition of fractions may be divided into fourteen 
 types; * subtraction of fractions may be divided into 
 nine types similar to those in addition, the difference 
 in number being due to the fact that it is impossible to 
 reduce any answer in subtraction to a mixed number. 
 The types in multiplication and division will be analyzed 
 in the succeeding pages of this bulletin. 
 
 Summary of Results in Addition of Fractions. 
 The results of the tests in addition of fractions were 
 published in School Document No. 3, 1916. The data 
 upon which the conclusions were drawn are shown in the 
 following tables. Table I shows the type of examples 
 used in the six tests together with the time allowance. 
 
 TABLE L 
 Showing Examples Used in Tests in Addition of Fractions, 
 
 
 
 December, 1915. 
 
 
 Addition 
 
 of 
 
 Fractions.— 
 
 -Test 1.— Time, 
 
 2 Minutes. 
 
 m \ 
 
 
 (2) f^ 
 
 (3) fe 
 
 (^) To 
 
 1 
 
 _4 
 
 
 1 
 
 14 
 
 7 
 
 11 
 
 7 
 i2 
 
 Addition 
 
 of 
 
 Fractions.— 
 
 - Test 2.— Time, 
 
 2 Minutes. 
 
 ™i 
 
 
 <., 1 
 
 (« i 
 
 « 1 
 
 1 
 _6 
 
 
 3 
 
 1 
 
 \2 
 
 7 
 
 * See School Document No. 3, 1916. "Determining the Achievement of Pupils in Addi- 
 tion of Fractions." 
 
8 
 
 SCHOOL DOCUMENT NO. 5. 
 
 Addition of Fractions. — Test 3. — Time, 2 Minutes, 
 
 I (2) I (3) I (4) g 
 
 II 1 U 2 
 15 2 14 3 
 
 (1) 
 
 Addition of Fractions- 
 
 - Test 4. 
 
 (1) \ (2) 1 
 
 (3) 
 
 9 1 
 
 10 4 
 
 
 Time, 
 
 Minutes. 
 
 4 
 
 9 
 
 5 
 
 8 
 
 (4) 
 
 Addition of Fractions. — Test 6. — Time^ 
 
 1 5^ 3 
 
 6 12 8 
 
 (^) ii) 
 
 (2) 
 
 Minutes. 
 12 
 10 
 
 (4) 
 
 Addition of Fractions. — Test 6.- — Time, 2 Minutes. 
 
 9^ 
 10 
 
 (2) 
 
 (3) 
 
 1 
 
 8 
 9^ 
 10 
 
 (4) 
 
 12 
 ]_ 
 10 
 
 Table II shows the medians obtained as a result of 
 
 the tests. 
 
 TABLE IL 
 Summary Sheet — City Medians. 
 
 Addition of Fractions, December, 1915. 
 
 
 3% 
 1^ 
 
 Test 1. 
 
 Test 2. 
 
 Test 3. 
 
 Test 4. 
 
 Test 6. 
 
 Test 6. 
 
 Grade. 
 
 .2 
 
 1^ 
 
 
 
 < 
 
 i 
 
 1^ 
 
 < 
 
 11 
 
 < 
 
 J 
 
 -el's 
 
 is 
 
 d 
 
 
 VIII 
 
 VII 
 
 VI 
 
 1,130 
 1,243 
 1,265 
 
 20.7 
 16.6 
 10.7 
 
 88.0 
 87.0 
 80.0 
 
 11.6 
 10.1 
 
 7.7 
 
 74.0 
 73.0 
 66.0 
 
 8.4 
 7.3 
 6.5 
 
 47.0 
 46.0 
 42.0 
 
 6.0 
 6.3 
 4.0 
 
 68.0 
 69.0 
 70.0 
 
 6,9 
 6.3 
 4.6 
 
 62.0 
 65.0 
 61.0 
 
 6.4 
 5.7 
 4.4 
 
 47.0 
 48.0 
 49.0 
 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 9 
 
 The following conclusions were drawn as a result of the 
 analysis of the tests.* 
 
 1. The factors that enter into the problem of adding 
 fractions are much more complex than those that enter 
 into the problem of adding integers. 
 
 2. The errors were largely due to the failure of pupils 
 to reduce consistently either to lowest terms or to mixed 
 numbers. This faihng on the part of many children 
 to use the principle of reduction would seem to indicate 
 that the method, now largely in use, of teaching such 
 reductions by themselves, has failed to produce satis- 
 factory results. In view of this fact, would it not be 
 well to teach reductions as such, in connection with the 
 subject of addition of fractions? This would at least 
 make a closer connection between the two operations, 
 and thereby tend to form the habit of writing the answer 
 in its best form. 
 
 3. Eight per cent of the pupils in Grade VI, 11 per 
 cent in Grade VII, and 5 per cent in Grade VIII were 
 unable to do the simplest problems in the addition of * 
 fractions. 
 
 4. Drill and individual work given the children in 
 Grade V of selected schools in the spring at the sug- 
 gestion of the department showed its effect in the work 
 of Grade VI in the late fall. This was evidenced by an 
 increase in both speed and accuracy over that obtained 
 in the entire city and in two cases over that shown by 
 the whole number of pupils in the grade in which the 
 selected groups were enrolled. 
 
 Summary of Results in Suhtr action of Fractions. 
 The results of the tests in subtraction of fractions 
 were not published because they were not materially 
 different from the results, of the tests in addition. The 
 following table shows the types of examples used in the 
 five tests together with the time allowance. 
 
 * School Document No. 3. 
 
10 SCHOOL DOCUMENT NO. 5. 
 
 TABLE III. 
 
 Showing Examples Used in Tests in Subtraction of Fractions, 
 
 December, 1916. 
 
 Subtraction of Fractions. — Test 1. — Time, 2 Minutes. 
 (1) \ (2) I (3) I (4) f^ 
 
 1 1 1 A 
 
 _4 _4 JS 16 
 
 Subtraction of Fractions. — Test 2. — Time 
 (1) \ (2) I (3) 
 
 1 3 
 
 Subtraction of Fractions. — Test 3. 
 (1) I (2) I (3) 
 
 1 _3 
 
 i2 i2 — _ 
 
 Subtraction of Fractions. — Test 4- — Time, 2 Minutes. 
 (1) 4. (2) 6 (3) 6 (4) 6 
 
 2i 5| 2f ^ 
 
 Subtraction of Fractions. — Test 5. — Time, 2 Minutes. 
 (1) 9i (2) 7A (3) 7,V (4) 74 
 
 iJ 6f 4f 2fv 
 
 In the tests in addition the addition of mixed numbers 
 was not included although it is very probable that there 
 would have been some difficulty in disposing of the sum 
 of the fractions, especially if the sum were more than 
 an integer. This same phase occurs in the multiplication 
 of mixed numbers by an integer and will be pointed out 
 in its proper place. The subtraction of mixed numbers, 
 however, is a vital problem especially when the fraction 
 in the subtrahend is larger than the fraction in the 
 minuend. Because of this. Tests 4 and 5 were given 
 upon this type of example. Table IV shows the medians 
 in speed and accuracy in the subtraction of fractions. 
 
 - Time, 
 
 2 Minutes. 
 
 2 
 3 
 
 (4) 
 
 3 
 
 4 
 
 3 
 11 
 
 
 5 
 
 9 
 
 - Time, 
 
 2 Minutes. 
 
 7 
 9 
 
 (4) 
 
 7 
 10 
 
 1 
 12 
 
 
 8 
 15 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. U 
 
 TABLE IV. 
 Summary Sheet — City Medians. 
 
 Subtraction of Fractions, Deoember, 1916. 
 
 
 Pupils. 
 
 Test 1. 
 
 Test 2. 
 
 Test 3. 
 
 Test 4. 
 
 Test 5. 
 
 Grade. 
 
 
 1^ 
 
 f 
 
 II 
 1^ 
 
 
 II 
 
 d 
 
 |l 
 
 d 
 .2 
 
 
 VIII 
 
 1,239 
 1,283 
 1,499 
 
 22.5 
 19.7 
 15.1 
 
 91.0 
 84.0 
 73.0 
 
 7.3 
 6.0 
 4.9 
 
 86.0 
 85.0 
 76.0 
 
 6.1 
 5.6 
 4.6 
 
 65.0 
 61.0 
 51.0 
 
 18.0 
 14.2 
 11.9 
 
 99.0 
 97.0 
 85.0 
 
 6.4 
 5.2 
 4.6 
 
 81.0 
 
 VII 
 
 66.0 
 
 VI 
 
 64.0 
 
 
 
 The following summary shows the number of districts, 
 the number of grade classes, the grades tested, and the 
 total number of pupils included in the test in addition 
 and subtraction of fractions. 
 
 Addition of Fractions. 
 December, 1915. 
 
 Number of elementary districts 12 
 
 Number of grade classes 91 
 
 Given in grades VIII, VII, VI 
 
 Number of pupils 3,638 
 
 Subtraction of Fractions. 
 
 December, 1916. 
 
 Number of elementary districts ... . . . 10 
 
 Number of grade classes 102 
 
 Given in grades VIII, VII, VI 
 
 Number of pupils , 4,021 
 
 II. Multiplication and Division of Fractions. 
 
 Extent of Tests. 
 The following summary shows the number of districts, 
 the number of grade classes, the grades tested, and the 
 number of pupils included in the test in multiplication 
 and division of fractions, given in December, 1917. 
 
 Number of elementary districts ..... 10 
 
 Number of grade classes 95 
 
 Given in grades VIII, VII, VI 
 
 Number of pupils 3,513 
 
12 SCHOOL DOCUMENT NO. 5. 
 
 Types of Fractions. 
 
 An extended analysis was made in addition and sub- 
 traction of fractions to determine the various types 
 with which the pupil came in contact. An analysis on 
 a similar basis of the processes of multiplication and 
 division of fractions was not believed necessary. In 
 these two processes the separation into types depended 
 upon the character of the multiplier and multiplicand 
 or the dividend and divisor. 
 
 The process of division in fractions is either one of two 
 procedures. In one case one proceeds to find how many 
 times one number of a certain denomination is contained 
 in a larger number of the same denomination. This is to 
 determine how many measures of a certain length there 
 are in a measure of a different length. This type of 
 procedure has beea called division hy measuring. In the 
 other case one proceeds to separate a number into a 
 certain number of parts. This type of division is called 
 division hy parting. These two types, measuring and 
 parting, became the basis upon which the three tests in 
 division of fractions were formulated. In view of these 
 conditions the following types were selected: 
 
 Multiplication. 
 Integer multiplied by fraction. 
 Fraction multiplied by integer. 
 Mixed number multiplied by integer. 
 Integer multiplied by mixed number. 
 Mixed number multiplied by fraction. 
 Fraction multiplied by mixed number. 
 Mixed number multiplied by mixed number. 
 Fraction multiplied by fraction. 
 
 Division. 
 Integer divided by fraction (measuring) . 
 Fraction divided by integer (parting) . 
 Mixed number divided by integer (parting) . 
 Integer divided by mixed number (measuring). 
 Fraction divided by fraction (measuring). 
 Mixed number divided by fraction (measuring). 
 
 n 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 13 
 
 The two types, fraction divided by mixed number and 
 mixed number divided by mixed number, are not 
 included because they do not conform to either the 
 parting or measuring criterion. In practical work we 
 neither have to perform such examples as dividing a 
 fraction into 3| parts nor finding how many 3| inches 
 there are in \ of an inch. Neither are we required to 
 perform such examples as dividing a mixed number into 
 3 1 parts nor finding how many 3| inches there are in 
 4i inches. 
 
 Of course, it is recognized that common fractions are 
 taking less and less place in our practical life, the proc- 
 ess giving way more and more to the use of the decimal 
 fraction. However, there is still use for the common 
 fraction having a small denominator, and it is still a 
 part of the required work in our courses of study. This 
 being true, it is pertinent to ascertain what results we 
 are achieving. 
 
 Construction of the Tests. 
 In constructing the tests the department decided 
 in the light of previous experience with addition and 
 subtraction of fractions that multiplication and division 
 might be given at one time. In order to decrease the 
 number of tests, two types were placed in a test. For 
 example, multiplication of an integer by a fraction and 
 multiplication of a fraction by an integer comprised 
 Test 1. As will be seen in Table V, the other tests 
 were made in a similar way. In the analysis of the 
 results the two types will be discussed separately. 
 There was an effort to keep the tests within the realm 
 of the practical. In all cases the terms of the fractions 
 involved were kept small. A fraction multiplied by a 
 fraction is not in any test but is included in the process 
 of multiplication of mixed number by a fraction. The 
 latter type was used because it was considered more 
 difficult. If this be true, a pupil might be able to do 
 the former type but unable to do the latter. However, 
 ability to do the latter would include abihty to do the 
 
14 SCHOOL DOCUMENT NO. 5. 
 
 former. Table V shows the types of examples and the 
 time allowance for each test. 
 
 TABLE V. 
 
 Showing Examples Used in Tests in Multiplication and Division of 
 Fractions, December, 1917. 
 
 Multiplication of Fractions. — Test 1. — Time, 2 Minutes. 
 (1) I X 6 (2) i X 8 (3) f X 12 (4) 12 X A 
 
 Multiplication of Fractions. — Test 2. — Time, 4 Minutes, 
 
 (1) 2461 (2) 5731- (3) 275 (4) 456| (5) 189 
 5 5 8| 2 5i 
 
 Multiplication of Fractions. — Test 3. — Time, 2 Minutes. 
 (1) 4| X i (2) 7i X I (3) 5^ X I (4) |- X 2f 
 
 Multiplication of Fractions. — Test 4- — Time^ 5 Minutes. 
 
 (1) 321 (2) 84J (3) 29i (4) 25i (5) 191 
 69 J 791 28i 17f m 
 
 Division of Fractions. — Test 5. — Time, 2 Minutes. 
 (1) i -^ 8 (2) 9 -^ f (3) 6 + i (4) 8 -^ t 
 
 Division of Fractions. — Test 6. — Time, 4 Minutes. 
 (1) 5678J ^ 5 (2) 27891 -^ 4 (3) 2467 ^ 8i 
 
 (4) 6752 ^ 12| 
 
 Division of Fractions. — Test 7. — ■ Time, 3 Minutes. 
 (1) f-4 (2) 3f ^i (3) 5t-f (4) 6f-^f 
 
 Giving of the Tests and Correction of Results. 
 
 Following the plan developed in 1912 and continued 
 
 since the department was organized,* twenty-five 
 
 Normal School seniors were trained to give the tests 
 
 in a uniform manner. The tests were given to 1,290 
 
 * Ballou, F. W., "Training Normal School Seniors in Educational Measurement," School 
 and Society, Volume V., No. 108, pages 61-70, January 20, 1917. 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 15 
 
 pupils in Grade VI, 1,196 pupils in Grade VII, and to 
 . 1,027 pupils in Grade VIII in December, 1917. 
 
 The old course of study for Grade V requires : 
 
 Multiplication of fractions and mixed numbers and 
 integers; finding fractional parts of integers includ- 
 ing the cases where the parts so obtained are mixed 
 numbers. 
 
 Thus the sixth grade may begin their work without 
 a knowledge of division of fractions, and it is possible 
 that division of fractions may not have been taught 
 during the first three months of the school year. In 
 spite of this knowledge, it was decided to test in Grade 
 VI for two reasons. First, that it might be known just 
 what the status of Grade VI actually is on a city- 
 wide basis in multiplication and division of fractions; 
 and second, to find out what is done by those schools 
 which did more work than was actually required by 
 the course of study. 
 
 After completing the work, the examiners brought the 
 tests to the office of the department and all the work of 
 correction and tabulation was done by members of the 
 department. Certain rules were formulated for the 
 correction of results. 
 
 (a) All results which were not reduced to lowest terms or to 
 
 mixed numbers were called wrong. 
 (5) The papers on which children added or subtracted the 
 
 fractions were counted as I. N. F. papers. (Instructions 
 
 Not Followed.) 
 
 (c) All other papers, regardless of how the child did the 
 
 examples, were scored as right or wrong. 
 
 (d) The form of doing the work did not count against the child 
 
 if his answer was correct. 
 
 (e) Some children did not do Test 1, but started upon Test 2, 
 
 owing to confusion in understanding the directions. 
 Any paper showing no work at all in Test 1 was marked 
 I. N. F. in all tests. (Instructions Not Followed.) 
 These were very few. 
 
16 
 
 SCHOOL DOCUMENT NO. 5. 
 
 (/) If in Tests 5, 6, and 7 (division of fractions) pupils 
 multiplied, the papers were not niarked I. N. F. 
 (Instructions Not Followed.) This was because there' 
 is much confusion between the two processes and many 
 pupils really multiply when they think they are divid- 
 ing. If any other process was used the test was marked 
 I. N. F. (Instructions Not Followed.) 
 
 III. Analysis of Results. 
 Achievement. 
 Table VI shows the results for the entire number 
 of pupils tested. In the first column is shown the grade, 
 followed by a column showing the number of pupils 
 tested in each grade. Under each test is given the speed 
 median and the accuracy median for each test and 
 grade. The table is to be interpreted as follows: In 
 Grade VIII, 1,027 pupils were tested. These pupils 
 attained a speed median of 11.1 examples with an 
 accuracy median of 93 per cent in Test 1. In Test 2 
 the speed median was 8.8 and the accuracy median was 
 63 per cent. Thus, reading across the page on the first 
 line one will find the medians in speed and accuracy for 
 each test for Grade VIII. The table shows the same 
 facts for Grades VII and VI. 
 
 TABLE VL 
 Summary Sheet — City Medians. 
 
 Multiplication and Division of Fractions. 
 
 
 
 Multiplication. 
 
 Division. 
 
 
 TEST 1. . 
 
 TEST 2. 
 
 TEST 3. 
 
 TEST 4. 
 
 TEST 5. 
 
 TEST 6. 
 
 TEST 7. 
 
 Grade. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 CO 
 
 
 i 
 
 02 
 
 < 
 
 i 
 
 11 
 
 P 
 
 .i 
 
 p 
 
 < 
 
 
 ^1 
 
 13 
 03 
 
 1^ 
 
 < 
 
 d 
 .2 
 
 03'^ 
 < 
 
 vm 
 
 1,027 
 
 11.1 
 
 93 
 
 8.8 
 
 63 
 
 7.6 
 
 85 
 
 4.7 
 
 
 
 10.1 
 
 75 
 
 3.3 
 
 29 
 
 10.3 
 
 79 
 
 VII 
 
 1.196 
 
 8.4 
 
 88 
 
 7.7 
 
 38 
 
 6.4 
 
 81 
 
 4.2 
 
 
 
 8.2 
 
 59 
 
 2.9 
 
 
 
 8.5 
 
 68 
 
 VI 
 
 1,290 
 
 6.2 
 
 13 
 
 8.2 
 
 
 
 4.7 
 
 
 
 5.6 
 
 
 
 5.4 
 
 1 
 
 3.2 
 
 
 
 4.9 
 
 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 17 
 
 It will be noticed that the accuracy medians for 
 Test 2, multiplication of mixed number by integer or 
 integer by mixed number, Test 4, multiplication of 
 mixed number by a mixed number, and Test 6, division 
 of mixed number by an integer, are especially low. 
 
 It is probably true that there is no great use for the 
 type of work shown in these three tests in practical life, 
 but the business world does require it to some extent; 
 business courses in our high schools require the processes, 
 and the new course of study requires this work. In 
 view of these three conditions, it was thought best to 
 include these three tests in order that we might have 
 some facts on which to base the development of our 
 work in multiplication and division of fractions. 
 
 Analysis of Results in Grades VII and VIII. 
 
 The analysis of results which is given in this bulletin 
 is based wholly upon a study of the wrong examples in 
 the work performed by pupils in the test given in Decem- 
 ber. It is perfectly possible that a pupil who does the 
 work and reaches the right result may be doing it in an 
 inefficient and round-about manner. When correcting 
 large numbers of papers, his work does not attract the 
 attention that is attracted by a pupil who does many 
 examples and gets none or only a few right. 
 
 This study is based, then, upon those papers which 
 showed low scores in accuracy. Furthermore, owing to 
 the low degree of accuracy in Grade VI, due largely to 
 lack of knowledge, the analysis is based upon work in 
 Grades VII and VIII. In a study like the present one 
 a piece of work done by a person ignorant of the process 
 has little or no value. The value of a study of this kind 
 comes from studying results of pupils who are supposed 
 to have been taught the process. An analysis of Grade 
 VI will be made in a later part of the bulletin. 
 
 It is impossible for the report of a study to be as helpful 
 to a teacher as if the individual teacher had made the 
 study for herself. It is only when the teacher will take 
 the work of her class room and make some similar analy- 
 
18 SCHOOL DOCUMENT NO. 5. 
 
 sis, seeking to find out why the pupil makes the failure 
 and just what the pupil does in making the failure, that 
 we are going to make great gains in the treatment of 
 individual pupils. The analysis is given here rather in 
 detail in the hopes that it may act as a guide and stimu- 
 late some teachers to undertake this rather laborious 
 but extremely interesting work for the good of the indi- 
 vidual who is having trouble with his fractions and school 
 work in general. 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 19 
 
 
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20 SCHOOL DOCUMENT NO. 5. 
 
 Table VII shows the general situation in regard to 
 failures in Grades VII and VIII. In the compiling of 
 this table, it was considered (1) that a pupil had failed 
 to do a certain type if he did not get at least one example 
 right among those he attempted, (2) that a pupil did 
 not fail if he used the correct method even though he 
 did not get at least one right answer. The table is to be 
 interpreted as follows. In Test 1, 3.1 per cent of Grade 
 VIII failed in the multiplication of an integer by a fraction 
 and 2.6 per cent failed in the multiplication of a fraction 
 by an integer. In Test 2, 17.9 per cent failed in the 
 multiplication of a mixed number by an integer and 
 30.6 per cent failed in the multiplication of an integer 
 by a mixed number and so on. 
 
 In multiplying a mixed number by a mixed number, 
 Test 4, two forms were used as illustrated below. 
 
 (a) Vertical method : (6) Horizontal method : 
 32i 32i X 69| = 
 69i 97 139 _ 13483 _ 
 3 ^ 2 " 6 ~ ^"^^^^ 
 
 XU6 
 
 23 
 
 288 
 192 
 
 2247J 
 
 In this paper whenever the example was done similarly 
 to illustration (a), it has been termed the vertical method, 
 and when (h) the mixed numbers were reduced to 
 improper fractions, it has been termed the horizontal 
 method. When the example was done by the vertical 
 method, the eighth grade failed in 90.4 per cent of the 
 cases; when done by the horizontal method the same 
 grade failed in 4.6 per cent of the cases. 
 
 In Test 6 the pupils were required to divide a mixed 
 number by an integer (Examples 1 and 2) or an integer 
 by a mixed number (Examples 3 and 4). Two possi- 
 bilities of doing the work present themselves. First, 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 21 
 
 pupils may reduce the mixed numbers to improper 
 fractions and follow the general rule for division of 
 fractions. 
 
 For example (a) : ^^-o^ r 17035 _ 
 
 ODYO^ -^ O = — -r- 5 = 
 
 Second, they may, when the fraction is in the dividend, 
 do the example by either short or long division as it 
 stands. 
 
 For example (h) : 5678 J -^ 5 
 5 )56781 
 
 1135 Rem. = 3 J 
 
 ^3 •- " 3 ^ 5 " 3 
 11351 Ans. 
 
 When the mixed number is in the divisor, they may 
 place the example on the paper as though they were doing 
 an example in long division, multiply both dividend and 
 divisor by the denominator of the fraction and proceed 
 as in long division. 
 
 For example (c): 2467 -- 8i 33)9868^ ^^^' 
 
 66 
 gl)2467 ^ 
 
 4 4 297 
 
 33) 9868 298 
 
 297 
 
 1 
 
 These two processes will be termed (a) process of 
 inversion and (b and c) process of long division. 
 
 Diagnosis of Results in Each Test 
 
 TEST 1. 
 Type of Examples Used in Test 1. 
 
 (1) i X 6 (2) * X 8 (3) f X 12 (4) 12 X A 
 
22 SCHOOL DOCUMENT NO. 5. 
 
 In this test the pupils were required to multiply a 
 fraction by an integer (Examples 1, 2 and 3) or an 
 integer by a fraction (Example 4). About 13 per cent 
 failed in the seventh grade and about 3 per cent failed 
 in the eighth grade in each type. In such a simple test 
 the chances of making errors are limited; so they fall 
 very largely into two groups. In one group the pupils 
 find the answer by multiplying the integer by one of the 
 terms of the fraction and adding the other. 
 
 For example: i X 6 = 14 (6 X 1 + 8) or 
 i X 6 = 49 (6X8 + 1) 
 
 The pupils in the second group multiply both numer- 
 ator and denominator by the integer. 
 
 ^ , 7 ^ c 56 (8 X 7) 
 
 Forexample: -X8 = -^g^gj 
 
 Cancellation gives little trouble because compara- 
 tively few pupils use this method of shortening the 
 procedure. In some cases there was evidence of can- 
 celling by dividing the integer by the numerator, but 
 these cases were few. There seemed to be a mixture of 
 processes in the minds of some pupils because a few 
 inverted one or the other of the factors. . 
 
 TEST 2. 
 Type of Examples Used in Test 2. 
 
 (1) 246i (2) 2731 (3) 275 (4) 456^ (5) 189 
 
 5 5 8f 2 5i 
 
 In this test the pupils were required to multiply a 
 mixed number by an integer (Examples 1, 2 and 4) or 
 an integer by a mixed number (Examples 3 and 5). 
 The percentage of failure for the first type for Grade VII 
 was 34.3 per cent and for Grade VIII, 17.9 per cent. 
 For the second type the percentage was nearly twice as 
 much, being 53.4 per cent and 34.3 per cent respectively. 
 This great difference was due very largely to the con- 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 23 
 
 struction of two examples, the fourth, 456yX 2, and the 
 thirteenth, 379i X 3. In many cases pupils had the 
 fourth example right and also the thirteenth, if they 
 reached this example, and no others. In these two 
 examples the multiplication of the fraction by the 
 integer gives the fractional part of the product without 
 further reduction. 
 
 In this test there are three chief sources of error. 
 
 (a) If the fraction be in the multiplicand, the multi- 
 plication of the fraction by the integer in the multiplicand 
 and if the fraction be in the multiplier, the multiplication 
 of the fraction by the integer in the multipher. 
 
 For example: 246^ 
 5 
 
 
 275 
 
 81 
 
 49i 
 1230 
 
 = i X 246 
 = 5 X 246 
 
 6 = 1X8 
 2200 = 8 X 275 
 
 1279i 
 
 Ans. 
 
 2206 Ans. 
 
 (b) Placing of the second partial product one place 
 3 the left of the first partial product. 
 
 For example: 275 
 
 81 
 
 i X 275 
 8 X 275 
 
 
 206i = 
 2200 = 
 
 » 
 
 22206i Ans. 
 
 (c) Multiplication of the denominator of the fraction 
 by the integer and adding the numerator. 
 
 For example: 246^ 
 5 
 
 26 = 5 X 5 + 1 
 1330 
 
 1356 
 
 The kind of error in (a) develops because it is not clear 
 in the minds of many pupils which integer is to be mul- 
 tiplied by the fraction. If the fraction be in the mul- 
 tiplier, the integer of the multiplier is multiplied. by the 
 
24 SCHOOL DOCUMENT NO. 5. 
 
 fraction and this product given as one of the partial 
 products. It is not strange that this should be done on 
 account of the drill which has been given in reduction 
 of mixed numbers to improper fractions. In other words 
 an old habit is at work, for the pupil has not yet appre- 
 ciated that it is not the same thing but something entirely 
 different. For those pupils who persist in doing this 
 work, individual attention is probably the only method 
 of eradicating the error. 
 
 The kind of error noted in (b) is due probably to the 
 same cause, viz., the following of an old habit. In 
 multiplication of integers the pupil was taught that he 
 must place the second partial product one step to the left 
 of the first partial product. When the process in frac- 
 tions is performed, the pupil follows the same habit 
 unless he is led to see the difference and a large amount 
 of practice in the correct method of doing the work is given. 
 Some pupils will need more of this practice than others 
 before the new habit is fixed. 
 
 The third source of error noted in (c) develops through 
 the multiplication of the fraction by the integer. For 
 example, it was a fairly common error in such an example 
 as 2461 X 5 to call 5 X | == 26. That is, apparently the 
 example was done exactly as though it was reduced to 
 an improper fraction and then the denominator, 5, 
 thrown away. 
 
 Another habit is probably at work in this case which 
 is not generally taken into consideration. In teaching 
 multiplication of integers emphasis is placed upon the 
 fact that the product of one number by another is larger 
 than either of the factors. To have a pupil realize 
 that a number may be multiplied by another such that 
 the product is smaller than one of the factors and that 
 when both factors are fractions the product is smaller 
 than either of the fractions, means that the pupil must 
 break old habits and form new ones. The ability to meet 
 this new experience and use it means a large amount of 
 drill before the old habit can be modified to meet the 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 25 
 
 new conditions. Unless this drill is adequate such 
 errors as those just pointed out are likely to occur. 
 
 Emphasis is being placed upon estimating the answer 
 in many schools, but even though a pupil is able to make 
 an approximate estimate of the product in these 
 examples, he will not be able to trace the error until he 
 appreciates the possibilities of these three types of 
 error. 
 
 Many strange methods were used which do not fall 
 into the foregoing groups. The following examples are 
 illustrations of these: 
 
 341 
 
 71 
 
 = 341 -T- 3 
 
 7061 
 
 7 
 
 1131 
 
 5651 = 8 X 706 + 3 
 
 56i 
 
 = 113 ^ 2 
 
 4942 = 7 X 706 
 
 2387 
 
 = 7 X 341 
 
 10593 Ans. 
 
 2556^ Ans. 
 
 If teachers will give a test similar to this one to their 
 classes, it is more than likely that some of these strange 
 ways of doing the work will manifest themselves. It 
 should be clear that class work does not reach these 
 individuals and if the pupil is to learn the correct method 
 it is only through individual help. 
 
 TEST 3. 
 Type of Examples Used in Test 3. 
 
 (1) ^ Xi (2) 7i X f (3) 5i X f (4) f X 2f 
 
 In this test the pupils were required to multiply a 
 mixed number by a fraction (Examples 1, 2 and 3), or 
 a fraction by a mixed number (Example 4) . The per- 
 centage of failure was about 17 per cent for Grade VII 
 and 8.7 per cent for Grade VIII in each test. 
 
 The greatest difficulties in this test are shown in (1) 
 the reduction of the mixed number to an improper 
 fraction and (2) in the process of cancellation. The 
 
26 SCHOOL DOCUMENT NO. 5. 
 
 first type of failure shows itself in many ways. For 
 example, some pupils multiply the two fractions and 
 add the integer. 
 
 For example: 4| X | = 4/:^. 
 
 Some pupils consider the integer as another factor 
 instead of a part of one of the factors. 
 
 17 1 5 K^ o2 20 ( 5X2X2 20). 
 
 For example: I X 2f = - J-^^^^ = -j^ 
 
 Another common method was to multiply the integer 
 by the numerator of the fraction and add the numerator 
 of the fraction which is a part of the mixed number. 
 
 For example- ^ X 3^ - ^-^ ( ^ X ^ + ^ - 15). 
 J^or example. , x 6^ - ^^ ^ ^^^ - ^^^^ 
 
 There were many other improper methods of finding 
 the product of a mixed number and a fraction. 
 
 In cancellation the difficulty came in canceling before 
 the reduction of the mixed number to the improper 
 fraction and also in canceling the integer of the mixed 
 number. Examples of this type of error seem unneces- 
 sary. 
 
 TEST 4. 
 Type of Examples Used in Test 4. 
 (1) 32i (2) S^ (3) 29f (4) 25f (5) 
 
 79i 
 
 8 
 3 »' 2 
 
 In this test the pupils were required to multiply a 
 mixed number by a mixed number. 
 
 There were 78 per cent of the pupils who attempted 
 to do the work vertically and 22 per cent who did it 
 horizontally. Of those who did the work vertically 92 
 per cent failed to do the work correctly, and 1 per cent 
 had the method correct but made errors in the work, 
 while the remaining 7 per cent had the correct answer. 
 Of those who did the work horizontally 7 per cent failed 
 to do the work correctly, 45 per cent had the method 
 correct but made errors in the work, while 48 per cent had 
 the correct answer. The two methods will be con- 
 sidered separately. 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 27 
 
 (a) In a study of the vertical method it is noticeable 
 that only a small percentage of the pupils (8 per cent) 
 even get the right method. There were three common 
 erroneous ways of doing the work. 
 
 1. The product or sum of the fractions added to 
 product of the integers. 
 
 For example : 
 o2-o o2-o 
 
 288 =9X32 
 
 or 
 
 288 =9X32 
 
 192 = 6 X 32 
 
 
 192 = 6 X 32 
 
 i = 4Xi 
 
 
 1 = 4 + 1 
 
 2208J 2208f 
 
 By far the largest proportion of the pupils who did 
 the work vertically found their answers in this manner. 
 2. A disregard of one or both of the fractions. 
 
 For example : 32 J 
 
 321 
 
 69i 
 
 69J 
 
 16 = i X 32 
 
 or 288 = 9 X 32 
 
 288 = 9 X 32 
 
 192 = 6 X 32 
 
 192 
 
 
 2224 2208 
 
 3. Process correct except that the product of the fractions 
 is omitted. 
 
 For example : 32 J 
 
 691 
 
 
 23 = 
 
 JX69 
 
 16 = 
 
 iX32 
 
 288 = 
 
 9 X32 
 
 192 = 
 
 6X32 
 
 2247 
 
 (b ) The chief error when the work is done horizon- 
 tally consists of inverting one or both of the fractions 
 after the mixed numbers have been reduced to improper 
 fractions. 
 
28 SCHOOL DOCUMENT NO. 5. 
 
 It is not the intention of this bulletin to point out the 
 method to be followed in multiplication of mixed num- 
 bers. There are three facts, however, that are signifi- 
 cant. 
 
 1. The larger proportion, 78 per cent, of the pupils 
 did their work vertically either through habit, or because 
 the examples were placed in a vertical position in the 
 test, or because of ignorance, and therefore followed the 
 line of least resistance. 
 
 2. Of those who did the examples vertically and used 
 the right method, only a small percentage made errors. 
 
 3. There was a large percentage, 45 per cent, who used 
 the correct method when doing the work horizontally 
 but made errors in the work. It is not fair, however, 
 to draw the conclusion that the better method is the 
 vertical method, because of the high percentage of 
 accuracy. The number of pupils concerned is not large 
 enough to make this conclusive. 
 
 In multiplying mixed numbers horizontally the pupil 
 must take the following steps to do the work: 
 
 Reduction of one mixed number to improper fraction. 
 
 1. Integer multiplied by the denominator. 
 
 2. Add the product to the numerator. 
 
 3. Write the improper fraction. 
 
 Reduction of second mixed number to improper 
 fraction. 
 
 4. Same as 1 (above) for second fraction. 
 
 5. Same as 2 (above) for second fraction. 
 
 6. Same as 3 (above) fo^ second fraction. 
 Multiplication of improper fractions. 
 
 7. Numerator of first fraction multiplied by numer- 
 
 ator of second fraction. 
 
 8. Denominator of first fraction multiplied by 
 
 denominator of second fraction. 
 
 9. Reduction of improper fraction to a mixed num- 
 
 ber to find the answer (7 -J- 8). 
 
 Thus in doing this test multiplication and division of 
 integers play an important part. When one considers 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 29 
 
 that the median accuracy in multiplication of integers 
 is 80 per cent and the median accuracy in division of 
 integers is 90 per cent, such a large percentage of error 
 as shown in doing this type of examples by the horizon- 
 tal method leads one to suspect that there must be some 
 factor present which is not being considered but has an 
 important influence upon the result. 
 
 The error may occur in any one of the nine steps noted 
 in the foregoing. An error in the early steps may result 
 in an increased error in the answer. Which method 
 should be used in multiplication of mixed numbers 
 depends on two important questions to neither of which 
 do we have an answer. 
 
 1. By which method is the pupil most likely to obtain 
 the correct answer? 
 
 2. How long does it take pupils to learn the method? 
 With sixth grade pupils it may be possible that the 
 
 time used to teach the vertical method would be out of 
 proportion to the results achieved through a greater 
 percentage of accuracy providing this method is more 
 likely to produce accurate results. It may be that pupils 
 of sixth grade ability are not mature enough to learn 
 the vertical method without an unreasonable expendi- 
 ture of time. If this be true, the teaching of this process 
 according to this method should be left to the high school 
 classes. Whatever may be the truth in the case, if we are 
 going to teach multiplication of mixed numbers, 50 per 
 cent of the eighth grade class should obtain a higher 
 accuracy than 0. 
 
 
 
 TEST 
 
 5. 
 
 
 
 
 Type of Examples 
 
 Used 
 
 in 
 
 Tests 
 
 3 
 
 4 
 
 -8 (2) 
 
 9 -V- f 
 
 (3) 
 
 6 
 
 -1 
 
 (1) i - 8 (2) 9 -V- f (3) 6 - I (4) 8 ^ f 
 
 Tests 5, 6, and 7 consisted of examples in division of 
 fractions. In Test 5 the pupils were required to divide 
 an integer by a fraction (Examples 2, 3, and 4) or a 
 fraction by an integer (Example 1). The percentage 
 of error was very large, being 35.4 per cent for the first 
 
30 SCHOOL DOCUMENT NO. 5 
 
 type and 49.5 per cent for the second type in Grade 
 VII and 25.3 per cent and 42.9 per cent for Grade VIII 
 in the respective types. 
 
 The chief cause for failure to get at least one example 
 right in such a large percentage of cases is due to failure 
 to invert the divisor. Either the pupil does not invert 
 the divisor or, not knowing which is the dividend and 
 which the divisor, inverts the dividend. A notable fact 
 is that when the divisor is an integer the chance of 
 failure is nearly doubled. Evidently the pupil does not 
 know the possibilities in this case. 
 
 As pointed out under the discussion of the fourth test, 
 habit plays a very important part. The particular habit 
 which probably influences the results in this instance is 
 one formed in work with integers. Here the pupil 
 learned that the answer in division must be smaller 
 than the dividend. There certainly comes a new experi- 
 ence into the hfe of the pupil when he sees for the first 
 time a division example in which the answer is larger 
 than the dividend. Unless it is made very clear it must 
 be a difficult thing for pupils to understand how one can 
 divide 4 by J and get an answer of 16. To get such a 
 large answer seems to violate all their previous concep- 
 tion of the meaning of division. It may be possible 
 that the teaching of the idea of partition and measuring 
 as pointed out in the earlier part of the bulletin would 
 be a material help in conquering this difficulty. 
 
 The working of old habits may then be a partial 
 explanation of the cause of such a large percentage of 
 error in such simple examples as those given in Test 5. 
 
 The method of doing the examples was very largely 
 the method of inversion. There were a few cases of 
 the longer and more laborious method of reduction to 
 a common denominator and then dividing one numera- 
 tor by the other. These, however, were isolated cases 
 and need only be mentioned in passing. 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 31 
 
 (1) 56784 
 
 TEST 6. 
 Type of Examples Used in Test 6. 
 
 -- 5 (2) 27891 ^ 4 (3) 2467 ^ 8^ 
 
 (4) 6752 4- 12i 
 
 Table VIII shows the results in Test 6. The table is 
 to be read as follows: in doing the type mixed number 
 divided by an integer, 18.3 per cent of the eighth grade 
 failed to do correctly, even in method, any of the 
 examples, 60.7 per cent did the work correctly in at 
 least one example, and 21 per cent used the correct 
 method but did not have a single example correct. 
 The rest of the table is read in a similar way. 
 
 TABLE VI I L 
 Showing Results Attained in Test 6. 
 
 Division of Fractions. 
 
 
 Mixed Number Divided by 
 
 Whole Number Divided by 
 
 
 Whole Number. 
 
 Mixed Number. 
 
 
 INVERSION. 
 
 LONG DIVISION. 
 
 inversion. 
 
 LONG DIVISION. 
 
 Grade, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 § 
 
 
 ^ 
 
 
 
 u 
 
 J 
 
 
 
 (0 
 
 
 ij 
 
 
 
 
 +3 O 
 
 .jj . 
 
 
 
 -»^ 2 
 
 
 
 
 
 
 f^-e 
 
 ii 
 
 "S-3 
 e^^ 
 
 ^■s 
 
 Pi S 
 
 Jl 
 
 el's 
 
 cJ t 
 
 6& 
 
 flif 
 
 d U 
 
 ' 
 
 n^ 
 
 6-^ 
 
 6^ 
 
 6-^ 
 
 6^ 
 
 c")!^ 
 
 a^ 
 
 6-^ 
 
 6^ 
 
 
 
 uf4 
 
 
 
 ^rt 
 
 
 ^•s 
 
 Mrt 
 
 u^i 
 
 
 uK 
 
 v" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ph 
 
 (u 
 
 PLH 
 
 fin 
 
 P4 
 
 Pt 
 
 pL. 
 
 ^1 
 
 PLH 
 
 Pk' 
 
 Ph 
 
 (U 
 
 vni 
 
 18.3 
 30.7 
 
 60.7 
 40.3 
 
 21 
 29 
 
 83.4 
 87.2 
 
 15.5 
 10.3 
 
 1.1 
 
 2.5 
 
 20.0 
 40.5 
 
 31.4 
 15.4 
 
 48.6 
 44.1 
 
 97.3 
 99.2 
 
 2.7 
 .5 
 
 
 
 vn 
 
 .3 
 
 The failure, when the work was done by inversion, 
 was in inverting the wrong fraction as pointed out in 
 analysis of Test 5. When the work was done by long 
 division, the failure was the inability to dispose of the 
 remainder. The large percentage of failures in both 
 types when done by long division does not necessarily 
 
32 SCHOOL DOCUMENT NO. 5. 
 
 show anything because there was considerable evidence 
 that this type of division of fractions had not been 
 taught by many teachers. It is indeed possible to come 
 to the same conclusion as reached in the analysis of 
 Test 4, viz., if this type is going to be taught (and it 
 is required by the new course of study) it should be 
 taught effectively enough so that the median accuracy 
 should be more than 39 per cent in the eighth grade. 
 
 TEST 7. 
 Type of Examples Used in Test 7. 
 
 (1) f-i (2) 3f-^i (3) 5|-| (4) 6f -^ f 
 
 In this test the pupils were required to divide a frac- 
 tion by a fraction (Example 1) or a mixed number by a 
 fraction (Examples 2, 3 and 4). In the eighth grade 
 about 20 per cent and 24 per cent failed respectively in 
 each type, and in the seventh grade 39 per cent and 33 
 per cent failed respectively. The general cause of failure 
 was the same as in Test 5, viz., difficulty with inversion 
 in one form or another. 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 33 
 
 > 
 
 5 
 
 •a 
 
 c 
 
 C 
 
 '■3 
 a 
 
 w .S 
 < 1 
 
 t/3 
 
 C 
 (]> 
 
 1 
 
 •;^u^^ J8J 001 
 
 fH 
 
 iH O 
 
 o 
 
 o 
 
 O t>. 1-1 iH iH 
 
 •ainuaojQj gi 
 
 O 
 
 § g 
 
 o 
 
 o 
 
 o o o o o 
 
 •umysf^ 
 
 O 
 
 o o 
 
 o 
 
 o 
 
 o o o o o 
 
 1 
 
 •^uao aaj qOI 
 auiiiOBay; laquini^i 
 
 o 
 
 O l^ 
 
 o 
 
 o 
 
 <N -* O O -* 
 
 * 
 
 •an^naojaa gi 
 
 o 
 
 o o 
 
 o 
 
 o 
 
 o o o o o 
 
 •uBipaK 
 
 o 
 
 o o 
 
 o 
 
 o 
 
 o o o o o 
 
 1 
 
 •^nao aaj OOX 
 3utqoB8H aaquinj^ 
 
 o 
 
 O JO 
 
 o 
 
 o 
 
 (N O O O t^ 
 
 •8Xi:^tiaoj8d: 52, 
 
 o 
 
 ^ s 
 
 o 
 
 o 
 
 => g ° "= § 
 
 •UBTP9K 
 
 o 
 
 o o 
 
 o 
 
 o 
 
 o o o o o 
 
 
 •^uao J8J OOT 
 
 o 
 
 O <N 
 
 o 
 
 o 
 
 05000 
 
 •^|I:^u^0J^^I gi 
 
 ° 
 
 o o 
 
 o 
 
 o 
 
 o o o o o 
 
 •uBipan 
 
 o 
 
 o o 
 
 o 
 
 o 
 
 o o o o o 
 
 CO 
 
 1 
 
 •^uao J8J OOT 
 SuiqoBay; jaquin^ 
 
 o 
 
 ^ ?3 
 
 o 
 
 ;^ 
 
 - s ° «=• ^ 
 
 •an^uaojaj gi 
 
 o 
 
 § s 
 
 o 
 
 8 
 
 o 1 o o o 
 
 •UBipaj^ 
 
 o 
 
 ° § 
 
 o 
 
 o 
 
 ° g * ° § 
 
 
 •^uao jaj GOT 
 Suiqo^ay^ jaquini^ 
 
 o 
 
 00 »o 
 
 o 
 
 ^ 
 
 (N Tjf O --I ■* 
 
 •an^naojad gi 
 
 o 
 
 g 2 
 
 o 
 
 s 
 
 8 8 = °^ 
 
 •oBrpaj^ 
 
 o 
 
 g § 
 
 o 
 
 o 
 
 g o o o o 
 
 1 
 
 •^uao jaj 001 
 autqa^ay; aaquin^ 
 
 t^ 
 
 «> ^ 
 
 <N 
 
 00 
 
 S J: ° ^ S 
 
 •an^^uaojaj gi 
 
 § 
 
 8 i 
 
 o 
 
 § 
 
 8 I ° 8 g 
 
 •n^ipajxr 
 
 o 
 
 2 8 
 
 o 
 
 8 
 
 8 8 ° ° g 
 
 
 ' 
 
 (N 
 
 ^ i 
 
 1^ 
 
 8 
 
 § s §■ ^ s 
 
34 
 
 SCHOOL DOCUMENT NO. 5. 
 
 Analysis of Results in Gxade VI. 
 
 Attention has already been called to the fact that 
 
 the pupils of the sixth grade were apparently unable to 
 
 do the work in multiplication and division of fractions. 
 
 It ^ems worth while, however, to call attention to the 
 
 FIGURE 1. 
 Median Scores for Grade VI in Ten Schools in Multiplication of Fractions. 
 
 Median Scores for Tests 5, 6, and 7 are all 0. 
 
 /O ZO JO 40 JO GO 70 ^0 'fo /Oi 
 
 1%. 
 
 11 
 
 30 \ 
 
 37 
 
 bO 
 
 Sb 
 
 b% 
 
 ^0 
 
 4h 
 
 bl 
 
 Iv- 
 
 --^ 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 — 
 
 / 
 
 " — .^_^ 
 
 
 
 
 
 
 
 L,-^**^ 
 
 T^ 
 
 ^:^^^ 
 
 "-^^Z^ 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 i 
 
 S, 
 
 
 
 
 
 
 
 
 
 ^ 
 
 >^ 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 :z^^=^^^ 
 
 zz:^^^ 
 
 --^^S^^ 
 
 
 , — ■ — ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '"■^ 
 
 \ 
 
 
 
 ' 
 
 
 
 
 
 
 
 Test I Test Z 7cst3 flsr ■^ 
 
 great variation in the accuracy of the various schools 
 in the work of multiph cation and division of fractions. 
 
 Table IX shows the results in accuracy in the ten 
 schools which were tested. In the first column is shown 
 the school designated by a number. Under each test 
 is shown the median or the 50 percentile score, the 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 35 
 
 75 percentile score, which is the score above which are 
 25 per cent of the cases and below which are 75 per cent 
 of the cases, and also the number of pupils who obtain 
 a rank of 100 per cent accuracy. That is, the first line 
 is read as follows: in school No. 12 the pupils in Test 1 
 
 FIGURE 2. 
 
 Seventy=five Percentile Scores for Grade VI in Ten Schools in Multiplication 
 
 and Division of Fractions. 
 
 Seventy-five Percentile Score for Test 6 is 0. 
 
 /Z 
 
 X7 
 
 30 
 
 S7 
 
 bo 
 
 SU 
 
 (y^ 
 
 40- 
 
 ^b 
 
 hi 
 
 10 
 
 ZO 
 
 50 
 
 r^ 
 
 ^ 
 
 T^^^ 
 
 ^^^ 
 
 ^:^ 
 
 
 
 
 
 
 
 
 
 *-v^ 
 
 "^ 
 
 ^ 
 
 / 
 
 ^ 
 
 ^^ -~ 
 
 
 H^ 
 
 '^^^ 
 
 &^^ 
 
 — 
 
 -~J 
 
 _ 
 
 
 
 
 
 
 
 
 >. 
 
 ^<. 
 
 ___"ir»7 
 
 
 
 
 
 
 
 — ^^ "^ 
 
 ^ 
 ^ 
 
 
 
 — 
 
 — 
 
 ' 
 
 ^ 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 ! 
 
 ^ 
 
 
 
 ■"'^'^~v>^ 
 
 
 
 ^^ 
 
 ^^ 
 
 
 
 T^sT / Test a, Tcard - TesT 4 TcstS — - TesT 7 ^ 
 
 have a median of 0. The 75 percentile point is at 20 
 and 7 pupils reach 100 per cent. In Test 2 the median 
 and 75 percentile is and there are none who reach 
 100 per cent. The . results in the other tests are in 
 succeeding columns. The record of the other schools 
 is read in a similar way. Attention is called to the 
 
36 SCHOOL DOCUMENT NO. 5. 
 
 scores attained by Schools 30, 60, 62, 61, in con- 
 trast with Schools 12, 57, 40. 
 
 Figure 1 and Figure 2 show the data of Table IX in 
 graphic form. Figure 1 shows the variation in the 
 medians. The records for Tests 5, 6, and 7 are not 
 given because they are all and would be drawn over 
 the line showing the results for Test 4. Figure 2 shows 
 the variation in the 75 percentile scores for the ten 
 schools. The record for Test 6 is not drawn because 
 all the schools had a record of the same as in Test 4. 
 These two graphs emphasize the great variation in the 
 results much more strongly than Table IX. 
 
 As before pointed out, it was not expected that the 
 results of Grade VI would be high because of lack of 
 preparation in Grade V, but neither was so great a 
 variation expected among the different schools. This 
 difference in variation is probably due to a difference in 
 procedure in the different schools. 
 
 > 
 IV. Plan of Diagnosis for Teacher. 
 
 As a result of this study of common fractions, how 
 may a teacher effectively check the work in her grade? 
 How may a teacher determine and keep in some perma- 
 nent form a record of the pupils' ability in common 
 fractions? The following form of record (page 37) has 
 been used and proved to be an effective method of 
 doing this work. This sheet may be duplicated so that 
 each pupil may have a copy. It was planned to be used 
 as follows : 
 
 The teacher may give examples in addition similar to 
 the types illustrated in the sixth grade course of study, 
 School Document No. 19, 1917, and if the pupils get the 
 answers right the type may be checked in column marked 
 ''R'^ if wrong it may be checked in column marked 
 ^'W^\ These checks in the '^ wrong'' column should 
 be changed as fast as the pupil has mastered the 
 type. After giving a series of lessons covering the 
 various types and recording results, the teacher has a 
 record of the ability of each individual in the room 
 showing his strength and weakness. If a pupil fails in 
 
ACHIEVEMENT OF PUPILS IN COMMON FRACTIONS. 37 
 
 a problem, it can be determined immediately by refer- 
 ring to the record and by asking a few judicious questions, 
 whether the difficulty is in the mechanics of the prob- 
 lem or in the problem itself, or both. In any case the 
 teacher can easily tell to what extent she is required to 
 give help to the pupil. 
 
 Record in Common Fractions of 
 
 Name School 
 
 Age Room 
 
 Grade 
 
 
 R. 
 
 W. 
 
 
 R. 
 
 W. 
 
 Addition. 
 Type 1 
 
 
 
 Multiplication. 
 
 Fraction by an integer 
 
 Integer by a fraction 
 
 Integer by a mixed number, 
 
 Mixed number by an integer, 
 
 Fraction by a mixed number. 
 
 Mixed number by a fraction, 
 
 Fraction by a fraction 
 
 Mixed number by a mixed 
 number 
 
 
 
 2 
 
 
 3 
 
 
 4 
 
 
 5 
 
 
 6 
 
 
 7 
 
 
 8 . 
 
 
 9 
 
 
 10 
 
 
 11 
 
 12 . . 
 
 
 13 
 
 
 14 
 
 
 Mixed numbers 
 
 
 
 
 Subtraction. 
 
 ("Without "Borrowing.") 
 
 Type 1 
 
 
 • 
 
 Division. 
 
 1 
 j 
 
 Fraction by an integer 
 
 Integer by a fraction 
 
 Integer by a mixed number. 
 
 Mixed number by an integer. 
 
 Mixed number by a fraction. 
 
 Fraction by a fraction 
 
 
 
 2 
 
 
 3 
 
 
 4 
 
 
 5 . . 
 
 
 6 
 
 
 7 
 
 
 8 
 
 
 9. . . . 
 
 
 Fraction from mixed number, 
 
 Mixed number from mixed 
 number (with "borrowing"), 
 
 Fraction from integer 
 
 Fraction from mixed number. 
 
 Mixed number from integer . . 
 
 Mixed number from mixed 
 number 
 
 
 
 
38 SCHOOL DOCUMENT NO. 5. 
 
 Summary and Conclusions. 
 
 1. The median accuracy in all but the simplest 
 tests in multiplication is strikingly low in some schools 
 and high in others. The range of variation in the 
 medians of the ten school districts tested extends from 
 to 92 per cent. 
 
 2. Analysis of results in multiplication of mixed 
 numbers and division of a mixed number by an integer 
 and of an integer by a mixed number seems to indicate 
 a lack of drill in these types commensurate with their 
 difficulty. A large percentage of the pupils show an 
 utter lack of knowledge of the process. 
 
 3. In the tests in division of fractions, the chief 
 source of error is in the apparent inability of the indi- 
 vidual pupil to distinguish between the dividend and the 
 divisor. This results in an inversion of either dividend 
 or divisor and sometimes both. 
 
 4. The low percentage of accuracy in Tests 4 and 6 
 where the process consists of a number of steps leads 
 one to think that some factors are influencing the results 
 which are not usually considered as important. 
 
 5. The ineffectiveness of the instruction as indicated 
 by the large variation within the class is again shown in 
 these tests in multiplication and division of fractions. 
 Class room drills tend to increase the difference between 
 the individuals of the class by increasing the abihty 
 of the bright pupil and not reaching the slow pupil. The 
 difficulty of the individual can only be reached by indi- 
 vidual instruction whether that pupil be advanced or 
 retarded. The waste through nonpromotion, poor 
 attendance, and other causes may be eliminated. It is 
 highly important that we find out the reasons for failure 
 through the analysis of results and apply the remedy 
 needed in each individual case. 
 
ANNOUNCEMENT. 
 
 Bulletins published bx.the Department are distributed by the Secretary 
 of the School CommittfeB, who will, so far as the supply on hand permits, 
 fill mail applications; f(|jtf copies when such requests are accompanied by 
 the price indicated. 
 
 No. I. 
 
 No. 
 
 II. 
 
 No. 
 
 III. 
 
 No. 
 
 IV. 
 
 No. 
 
 V. 
 
 No. 
 
 VI, 
 
 No. 
 
 VII 
 
 No. 
 
 VIII. 
 
 No. 
 
 IX. 
 
 No. X. 
 
 No. 
 
 XI. 
 
 No. 
 
 XIL 
 
 No. XIII. 
 
 No. 
 
 XIV. 
 
 No. 
 
 XV. 
 
 Provisional Minimum and Supplementary Lists of Spelling. 
 
 Words for Pupils in Grades I to VIII. 
 School Document No. 8. 1914. Out of Print. 
 Provisional Miminum Standards in Addition, Subtraction 
 
 Multiplication and Division for Pupils in Grades IV tr 
 
 VIII. 
 School Document No. 9. 1914. Out oj Print. 
 Educational Standards and Educational Measurement. 
 School Document No. 10. 1914. Out of Print. 
 Spelling, Determining the Degree of Difficult}^ of Spelling 
 
 Words. 
 School Document No. 10. 1915. Out of Print. 
 Geography. A Report on a Preliminary Attempt to Measure 
 
 Some Educational Results. 
 School Document No. 14. 1915. Out of Print. 
 English. Determining a Standard in Accurate Copying. 
 School Document No. 2. 1916. Price, 7. cents. 
 Arithmetic. Determining the Achievement of Pupils in the 
 
 Addition of Fractions. 
 School Document No. 3. 1916. Price, 7 cents. 
 Report on High School Organization and Expenditures, 1916. 
 Printed for local distribution only. 
 Penmanship. Determining the Achievement of Elementary 
 
 School Graduates in Handwriting. 
 School Document No. 6. 1916. Price, 7 cents. 
 Arithmetic. The Courtis Standard Tests in Boston, 1912-1915. 
 
 An Appraisal. 
 School Document No. 15. 1916. Price, 7 cents. 
 Spelling. The Teaching of Spelling. 
 School Document No. 17. 1916. Price, 7 cents. 
 Standards in Silent Reading, with Suggestions on How 
 
 Teachers May Test Their Pupils in Silent Reading. 
 School Document No. 18. 1916. Price, 7 cents. 
 Arithmetic. The Value to the Teacher, to the Principal and 
 
 to the Superintendent of Individual and Class Records 
 
 from Standard Tests. 
 School Document No. 22. 1917. Price, 7 cents. 
 A Plan for the Promotion of Teachers from Merit Lists. 
 School Document No. 2. 1918. Price, 7 cents. 
 Arithmetic. Determining the Achievement of Pupils in 
 
 Common Fractions. 
 School Document No. 5. 1918. Price, 7 cents. 
 
THIS BOOK IS DUE ON THE LAST DATE 
 STAMPED BELOW 
 
 AN INITIAL FINE OF 25 CENTS 
 
 WILL BE ASSESSED FOR FAILURE TO RETURN 
 THIS BOOK ON THE DATE DUE. THE PENALTY 
 WILL INCREASE TO SO CENTS ON THE FOURTH 
 DAY AND TO $1.00 ON THE SEVENTH DAY 
 OVERDUE. 
 
L