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The Bureau of Educational Research was established by 
 act of the Board of Trustees June 1, 1918. It is the purpose 
 of the Bureau to conduct original investigations in the field 
 of education, to summarize and bring to the attention of 
 school people the results of research elsewhere, and to be of 
 service to the schools of the state in other ways. 
 
 The results of original investigations carried on by the 
 Bureau of Educational Research are published in the form of 
 bulletins. A list of available publications is given on the 
 back cover of this bulletin. At the present time five or six 
 original investigations are reported each year. The accounts 
 of research conducted elsewhere and other communications 
 to the school men of the state are published in the form of 
 educational research circulars. From ten to fifteen of these 
 are issued each year. 
 
 The Bureau is a department of the College of Education. 
 Its immediate direction is vested in a Director, who is also 
 an instructor in the College of Education. Under his super- 
 vision research is carried on by other members of the Bureau 
 staff and also by graduates who are working on theses. From 
 this point of view the Bureau of Educational Research is a 
 research laboratory for the College of Education. 
 
 Bureau of Educational Research 
 
 College of Education 
 
 University of Illinois, Urbana 
 
BULLETIN NO. 44 
 
 BUREAU OF EDUCATIONAL RESEARCH 
 COLLEGE OF EDUCATION 
 
 HOW PUPILS SOLVE PROBLEMS 
 IN ARITHMETIC 
 
 By 
 
 Walter S. Monroe 
 
 Director, Bureau of Educational Research 
 
 PUBLISHED BY THE UNIVERSITY OF ILLINOIS, URBANA 
 
 1929 
 
UNIVERSITY 
 5100 12 28 6412 ™ nm?. 
 
PREFACE 
 
 The problem studied in the investigation reported in this bulle- 
 tin is an important one. Arithmetic is generally thought of as afford- 
 ing a large portion of the opportunities for reflective thinking in the 
 elementary school, and it has been assumed that much of the train- 
 ing which pupils receive in this process is secured by solving arith- 
 metical problems. If it is true, as the present investigation indicates, 
 that pupils do not think reflectively in solving problems, it is obvious 
 that their learning in the field of arithmetic is not what it is assumed 
 to be. The reader of this bulletin, however, should bear in mind 
 that the investigation deals only with the way in which pupils now 
 solve problems. No attempt was made to ascertain if pupils could 
 be taught to solve problems by thinking reflectively. 
 
 The data for the investigation were collected during the school 
 year of 1926-27. Mr. John A. Clark, Assistant in the Bureau of 
 Educational Research, supervised the scoring of the test papers. 
 The tabulation of the data, however, was not completed until after 
 Mr. Clark left the Bureau of Educational Research. Work on other 
 projects during the school year of 1927-28 delayed the completion 
 of the report. 
 
 Walter S. Monroe, Director 
 
Digitized by the Internet Archive 
 
 in 2012 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/howpupilssolvepr44monr 
 
LIST OF TABLES 
 
 Table I. Provisions for Comparisons of Pupil Responses to Various 
 
 Types of Problem Statements 10 
 
 Table II. Variations in Pupil Responses to Problems When Technic- 
 al Terminology (T) was Substituted for Simplified Terminology 
 (S)— Grades VI, VII, and VIII 13 
 
 Table III. Variations in Pupil Responses to Problems When Irrele- 
 vant Data (I) were Substituted for Relevant Data (R) — Grades VI, 
 VII, and VIII 14 
 
 Table IV. Variations in Pupil Responses When Language Which is 
 Abstract (A) in Nature was Substituted for Language Which Pre- 
 sents a Real or Concrete (C) Setting. Grades VI, VII, and VIII. . 16 
 
 Table V. Number and Frequency of Errors in One Thousand Test 
 
 Papers 18 
 
 Table VI. Variations in Pupil Responses to Problems When Technical 
 Terminology (T) was Substituted for Simplified Terminology 
 (S)— Grade VI ' 27 
 
 Table VII. Variations in Pupil Responses to Problems When Technical 
 Terminology (T) was Substituted for Simplified Terminology 
 (S)— Grade VII 28 
 
 Table VIII. Variations in Pupil Responses to Problems When Technical 
 Terminology (T) was Substituted for Simplified Terminology 
 (S)— Grade VIII 28 
 
 Table IX. Variations in Pupil Responses to Problems When Irrelevant 
 
 Data (I) were Substituted for Relevant Data (R) — Grade VI. .29 
 
 Table X. Variations in Pupil Responses to Problems When Irrele- 
 vant Data (I) were Substituted for Relevant Data (R) — Grade VII . 29 
 
 Table XI. Variations in Pupil Responses to Problems When Irrelevant 
 Data (I) were Substituted for Relevant Data (R) — Grade VIII. . 30 
 
 Table XII. Variations in Pupil Responses When Language Which is 
 Abstract (A) in Nature was Substituted for Language Which Pre- 
 sents a Real or Concrete (C) Setting — Grade VI 30 
 
 Table XIII. Variations in Pupil Responses When Language Which is 
 Abstract (A) in Nature was Substituted for Language Which Pre- 
 sents a Real or Concrete (C) Setting — Grade VII 31 
 
 Table XIV. Variations in Pupil Responses When Language Which is 
 Abstract (A) in Nature was Substituted for Language Which Pre- 
 sents a Real or Concrete (C) Setting — Grade VIII 31 
 
HOW PUPILS SOLVE PROBLEMS IN ARITHMETIC 
 
 The problem of this study. In another place, the author has 
 described the general process of solving verbal problems in arith- 
 metic as follows: 
 
 Meanings are connected with words and symbols; the implied question concern- 
 ing a functional relationship is identified and answered; denominate number facts 
 are recalled; numbers are read and copied. However, the response cannot be ade- 
 quately described by enumerating the responses to such elements. Reflective think- 
 ing is involved. It should be noted that the total response to a verbal problem in- 
 cludes the determination of the calculations to be performed plus the response to the 
 example formulated. Reflective thinking is involved in only the first phase of the 
 total response. 1 
 
 Elementary-school teachers and other observers of the performances 
 of pupils in the field of arithmetic probably would criticise this state- 
 ment by pointing out that it describes how pupils should solve prob- 
 lems, rather than how they do solve them. It is the purpose of 
 this study to answer certain questions relating to the actual pro- 
 cedure that pupils follow in solving problems in arithmetic. 
 
 It seemed to the writer that the nature of pupils' responses to 
 changes in the statement of a problem would furnish an indication 
 of how they proceed in solving it. 2 Consequently, a series of tests 
 was devised which made possible a comparison of responses of pupils 
 to certain types of statements of the same problem. A statement 
 of a problem was considered abstract if there was no reference to 
 the activity in which it occurred. If this activity was indicated in 
 the statement, it was called concrete. The terminology of a problem 
 was considered technical when the terms used were those that are 
 commonly employed by specialists in a particular field. The term- 
 inology was considered to be simple when it consisted of words and 
 phrases in general use. A third modification in the statement of a 
 problem was secured by introducing irrelevant data. The way in 
 which these variations were introduced in the statement of certain 
 problems is described later under the heading of the construction of 
 the tests. 
 
 x Monroe, W. S. and Clark, J. A. "The Teacher's Responsibility for Devising Learning Exercises 
 in Arithmetic," University of Illinois Bulletin, Vol. 23, No. 41, Bureau of Educational Research Bulletin 
 No. 31. Urbana: University of Illinois, 1926, p. 22. 
 
 2 The word "problem" is used throughout this bulletin to designate what is sometimes called a 
 verbal problem. Arithmetical exercises frequently designated as examples are not included. 
 
8 Bulletin No. 44 
 
 The specific questions for which information was sought may be 
 stated as follows: 
 
 1. What is the difference between the responses of pupils to problems stated 
 abstractly and the same problems stated in concrete terms? 
 
 2. What is the difference between the responses of pupils to problems stated in 
 simple terms and the same problems stated in technical terminology? 
 
 3. What is the difference between responses of pupils to problems in which all 
 the data given are needed for solving the problem and the same problems stated so 
 as to include some irrelevant data? 
 
 After the data had been collected, a fourth question was added: 
 
 4. In solving a problem incorrectly in principle, what procedure do pupils 
 follow? 
 
 In these questions, the phrase "responses of pupils to problems" 
 refers only to the principle or plan of solution throughout the study. 
 No attention was given to errors of calculation. 
 
 The construction of the tests. Four separate tests were con- 
 structed — Test A, Test B, Test C, and Test D. With one excep- 
 tion, 3 the test problems were taken from texts designed for the 
 seventh grade. The wording of these problems was varied, and the 
 various forms of statements arranged as shown in Section I of Table 
 I. Problems 1 and 7 were presented in the same form on all four 
 tests. In Test A, Problem 2 is stated in simple terminology (S), all 
 of the data given are relevant (R), and the setting is concrete (C). 
 In Test B, technical terminology (T) is used, all the data given are 
 relevant (R), and the setting is concrete (C). The difference in the 
 statement of the problem is the change from simple terminology to 
 technical terminology. In Test C, the problem is stated in simple 
 terminology (S), although the data given are relevant (R), and the 
 setting is abstract (A). In Test D, technical terminology (T) is 
 used, irrelevant data (I) are included, and the setting is abstract 
 (A). The four forms of Problem 2 are as follows: 
 
 Problem 2 
 
 Test A — SRC. During a sale, Smith and Company reduced the price of furnaces 
 20%. When the purchaser paid cash, they gave an additional 5% off the sale price. 
 Mr. Jones bought a furnace at the sale and paid $551 cash for it. What price did 
 Smith and Company originally ask for the furnace? 
 
 Test B — TRC. Mr. Jones was allowed successive discounts of 20% and 5% on 
 the list price of a new furnace which he bought from Smith and Company. If Mr. 
 Jones paid $551, what was the list price of the furnace? 
 
 3 Problem 3 was not taken directly or indirectly from any text. However, it represents a very 
 simple type of problem that has a high frequency of occurrence in all texts. 
 
How Pupils Solve Problems in Arithmetic 9 
 
 Test C — SRA. An amount was reduced 20%. After an additional reduction of 
 5% of the remainder was made, the final remainder was $551. What was the original 
 amount? 
 
 Test D — TIA. A man borrowed $551 for 60 days at 7% to pay a bill on which 
 successive discounts of 20% and 5% were allowed. What was the original amount 
 of the bill? 
 
 The variations of the statements of the other problems may be 
 ascertained by referring to Appendix A. The arrangement of the 
 various forms of statement in the four tests is indicated in the first 
 section of Table I. 
 
 Plan of securing equivalent groups. It is obvious that it would 
 not be satisfactory to have the same pupils respond to two or more 
 statements of the same problem. Hence, it was necessary to secure 
 four groups of pupils that were equivalent in ability to solve the prob- 
 lems selected. The method of random sampling was employed as 
 a means of securing these equivalent groups. In order to have each 
 of the tests given to a random sample of pupils, the four tests were 
 arranged in alternate order so that when distributed to the pupils 
 in the class, the first, fifth, ninth, thirteenth, and so forth, would 
 receive Test A; the second, sixth, tenth, fourteenth, and so forth, 
 would receive Test B; the third, seventh, eleventh, fifteenth, and so 
 forth, would receive Test C; the fourth, eighth, twelfth, sixteenth, 
 and so forth, would receive Test D. Since the tests were to be given 
 in a large number of classes, it seemed that this plan of sampling 
 would provide equivalent groups. 3a 
 
 The equivalence of the groups on which the comparisons were 
 based was still further insured by means of a carefully planned 
 arrangement of the different forms of statements in the several 
 tests. The six types of problem statements considered are repre- 
 sented by the letters S, T; R, I; C, and A. 4 Each statement of a 
 problem necessarily possesses three of these characteristics, one of 
 each pair. In this investigation, each statement was formulated so 
 that its characteristics w T ere present in one of the following eight 
 combinations: SRC, TRC, SIC, SRA, TRA, TIC, SIA, TIA. These 
 various forms of statement were distributed among the different 
 tests for each problem as indicated in Section I of Table I. This 
 distribution was planned so as to provide the maximum data for 
 answering the questions given on the preceding page. A study of 
 Table I will reveal that no two comparisons of the same factors were 
 made between the same groups of pupils. 
 
 3a These groups were equivalent not only in arithmetical ability but also in respect to teachers, 
 textbooks, and other factors. 
 *See p. 8 and 20. 
 
 0. OF ILL LIB. 
 
10 
 
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How Pupils Solve Problems in Arithmetic 11 
 
 Provisions for making comparisons. Table I presents in sum- 
 mary fashion the provisions for comparing pupil answers to the 
 various problem statements. Section II shows that provision for 
 the comparison of the responses to simple terminology (S) with 
 those made to technical terminology (T) was made in the case of 
 each of ten problems. 5 In the case of Problems 2, 5, and 11, all of 
 the data were relevant (R) and the setting concrete (C). In the case 
 of Problem 2, the comparison is between Groups A and B; for Prob- 
 lem 5, it is between Groups C and D; and for Problem 11, it is 
 between Groups B and C. In the case of Problems 4 and 8, the state- 
 ment contained irrelevant data (I) and provided a concrete setting 
 (C) ; the comparison in the former being made between Groups B 
 and D, and in the latter between Groups A and C. The rest of the 
 table should be interpreted in the same manner. 
 
 Pupils tested. The problems for the tests were taken from those 
 
 that seventh-grade pupils are asked to solve, but it was thought wise 
 
 to have the tests given also in a few sixth and eighth-grade classes. 
 
 -.Usable data w r ere secured from the different groups as follows: 
 
 ^ Sixth grade, 775; seventh grade, 5902; eighth grade, 2579; total 9256. 
 
 These pupils represented forty-one cities in Illinois. 6 
 
 The validity of data. The principal questions relative to the valid- 
 ity of the data are: (1) Are the problems representative of those that 
 seventh-grade pupils are asked to solve? (2) Were equivalent groups 
 secured? The judgment of the author determined the problems 
 selected. As stated on page 8, they w T ere taken from texts designed 
 for the seventh grade. No analysis of these texts was made to deter- 
 mine the frequency of occurrence of various types of problems, but 
 a previous study 7 had made the author familiar with the various 
 types of problems in arithmetic texts. The selection was limited 
 somewhat by the requirement that the problem be one whose state- 
 ment could be varied according to the plans of the experiment. 
 Although it was evident from the test papers that the pupils in several 
 of the classes tested had not studied certain of the types of problems 
 represented at the time the tests were given, the author believes 
 that the problems are sufficiently representative of those assigned in 
 the seventh grade for the purpose of this experiment. 
 
 5 As has been stated previously, Problems 1 and 7 were presented in the same form on all tests. 
 
 These cities are: Beardstown, Belvidere, Benton, Blue Island, Canton, Centralia, Champaign, 
 Chicago, Chicago Heights, Cicero, Clinton, Danville, DuQuoin, Edwardsville, Elmhurst, Evanston, 
 Forest Park, Granite City, Harvey, Herrin, Jacksonville, Joliet, Kewanee, LaGrange, Lyons, Madison, 
 Marion, Moline, Mt. Carmel, Oak Park, Oglesby, Ottawa, Pekin, Quincy, Rockford, Rock Island, 
 Streator, Taylorville, West Frankfort, Westville, Zeigler. 
 
 7 Monroe, W. S. and Clark, J. A. "The Teacher's Responsibility for Devising Learning Exercises 
 in Arithmetic," University of Illinois Bulletin, Vol. 23, No. 41, Bureau of Educational Research Bul- 
 letin No. 31. Urbana: University of Illinois, 1926. 92 p. 
 
12 Bulletin No. 44 
 
 The second question, regarding the equivalence of the groups 
 tested, was discussed on page 9. 
 
 Simple versus technical terminology. The data relative to the 
 differences in responses, due to changing the terminology from simple 
 to technical, are summarized in Table II. 8 The data for the differ- 
 ent problems are not consistent. A few problems were attempted by 
 slightly more pupils and larger per cents of the solutions were 
 correct in principle when they were stated in technical terminology, 
 but the opposite condition prevails in the majority of the problems. 
 Considering the differences in the per cent of problems attempted 
 and the differences in the per cent of problems correct in principle, 
 only seven of the twenty are negative. Approximately the same 
 conditions prevail in Tables VI, VII, and VIII, only twenty-one of 
 the sixty differences being negative. Hence, in general, the data 
 support the thesis that the use of what is called technical terminology 
 increases the difficulty of arithmetical problems, but the exceptions 
 indicate that such terminology does not necessarily make a problem 
 more difficult. The most marked exception is Problem 3. Exam- 
 ination of the two statements of this problem shows that the essential 
 difference between the simple and the technical statement is the use 
 of "amount of a bill" in the latter in the place of "total cost" in the 
 simple statement. Apparently, "amount of bill" was more easily 
 understood by these pupils than "total cost." This probably indi- 
 cates that "amount of bill" was more frequently used in the problems 
 they had solved and hence a technical terminology becomes "simple" 
 when it is used frequently. In other words, pupils as now taught 
 learn to respond to certain words and phrases and when unfamiliar 
 terms are substituted a larger proportion of the pupils make incor- 
 rect responses. 
 
 The effect of introducing irrelevant data. The facts relating to 
 the effect of introducing irrelevant data in the statement of a prob- 
 lem are summarized in Table III. 9 Except in the case of Prob- 
 lem 4, this table indicates that the presence of irrelevant data 
 makes the problem more difficult, although several of the differ- 
 ences are small. The largest difference in per cent of solutions 
 correct in principle is for Problem 6. In this problem, the 
 lengths of the three sides of a triangle were given and the ir- 
 relevant fact was the altitude. Apparently, many of the pupils, 
 especially those in Grades VI and VII, had become so accustomed 
 
 8 The data are given separately for the three grades in Tables VI, VII, and VIII in Appendix B. 
 9 The facts are given separately for the three grades in Tables IX, X, and XI in Appendix B. 
 
How Pupils Solve Problems in Arithmetic 
 
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How Pupils Solve Problems in Arithmetic 15 
 
 to making an addition response to a request to find the perimeter 
 that they added the quantities given without noting what they were. 
 
 The introduction of irrelevant data in Problem 4 10 resulted in a 
 larger per cent of solutions correct in principle. Tables IX, X, and 
 XI reveal additional cases of this phenomenon. In Problem 4 the 
 irrelevant data are furnished by enumerating the articles purchased. 
 The presence of these items, together with the other changes in the 
 phraseology, appears to have made the problem easier. The expla- 
 nation of this phenomenon is, probably, to be found in the use of 
 "total amount of a bill was $1500," in the place of "merchandise worth 
 $1500." On page 12 it was pointed out that "amount of a bill" 
 appeared to be a very familiar expression. If this is true, it is not 
 difficult to understand why the pupils responded more satisfactorily 
 when irrelevant data were included in the problem. Hence, the 
 problem became easier, not because the irrelevant data were intro- 
 duced, but because another change was made which more than over- 
 balanced this change. 
 
 Abstract versus concrete terminology. Table IV 11 shows that, 
 in general, a slightly larger per cent of pupils attempted the problems 
 when they were given a real or concrete rather than an abstract 
 setting, Problems 2 and 8 showing the greatest differences. The 
 per cent of solutions correct in principle is approximately the same 
 for the two types of statements, except in the cases of Problems 6 
 and 9. The differences for these problems are —8.0 and 8.2, respec- 
 tively. Examination of these two problems does not suggest a prob- 
 able explanation of this variation. It appears that the introduction 
 of phrases descriptive of a concrete setting for a problem makes 
 little or no difference in the responses of pupils. They respond to 
 abstract problems about as satisfactorily as they do to those having 
 a concrete setting. 
 
 Easiest form of problem the one pupils have learned to solve. 
 The data of Tables II, III, and IV, and more especially those of the 
 detailed tables in Appendix B, suggest to the writer that the ter- 
 minology with which pupils have become familiar in their study of 
 arithmetic is the easiest for them. Although this hypothesis could 
 not be proved without information regarding the problem statements 
 in the texts used by the pupils tested, it appears consistent with 
 what we know about learning. If it is valid, this hypothesis implies 
 that, in general, pupils do very little or no reflective thinking in 
 solving arithmetic problems. Instead, they learn to make rather 
 
 10 See p. 21 and 22 for the four statements of this problem. 
 
 n The data are given separately for the three grades in Tables XII, XIII, and XIV in Appendix B. 
 
16 
 
 Bulletin No. 44 
 
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How Pupils Solve Problems in Arithmetic 17 
 
 fixed responses to certain types of statement. Hence, when a differ- 
 ent form of statement is used, they make no response or an inaccur- 
 ate one. There are exceptions, of course; some pupils do reason in 
 such situations. But the data collected in this investigation suggest 
 that a considerable number do not think reflectively when they 
 respond to arithmetic problems. This conclusion is reinforced by 
 the data described in the following paragraphs. 
 
 Procedure in incorrect solutions. For the purpose of studying 
 the incorrect solutions, 250 seventh-grade papers of each test were 
 selected at random. The errors in principle were tabulated separ- 
 ately for each test. Table V gives the number of erroneous solu- 
 tions, the number of different kinds of erroneous solutions, and the 
 number of unique solutions; i.e., those wmich occurred only once. 
 Considering the nature of the problems, the number of solutions 
 incorrect in principle is surprisingly large, but the number of differ- 
 ent kinds of erroneous solutions is probably more significant. 
 Examination of the test papers revealed many solutions that seemed 
 to have little or no rational basis. An attempt was made to identify 
 the solutions in which there was evidence of a partial comprehen- 
 sion of the problem. Only 28 of the 253 erroneous solutions of Prob- 
 lem 2 were so classified. The remaining 225, or 89 per cent, w r ere 
 such that there seemed to be no evidence that the problem was 
 even partially understood. 
 
 It may be argued that Problem 2, which involved successive 
 discounts, was so difficult that it did not reveal the nature of pupil 
 responses to problems. Problem 3, which seems to be relatively 
 simple, was solved incorrectly in 80 different ways, but in only 18 
 of these was there evidence of partial comprehension. In view of 
 the simple nature of this problem, it is astonishing that seventh- 
 grade pupils should give 62 different solutions that contained no 
 suggestion of even a partial comprehension of the problem. There 
 were 115 different erroneous solutions for Problem 4, which was 
 the easiest in the list, but in only 18 of these was there evidence of 
 partial comprehension of the problem. It therefore appears that a 
 very large proportion of the pupils tested made a response to the 
 problem without understanding it. Since the comprehension of a 
 problem is the first step in a rational solution of it, we have here 
 additional evidence in support of the hypothesis that, in general, 
 pupils do little or no reflective thinking in solving arithmetic prob- 
 lems. 
 
 General conclusion. Although the data of this investigation are 
 not entirely consistent, they appear to substantiate the conclusion 
 
18 
 
 Bulletin No. 44 
 
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How Pupils Solve Problems in Arithmetic 19 
 
 that a large per cent of seventh-grade pupils do not reason in attempt- 
 ing to solve arithmetic problems. Relatively few pupils follow the 
 plan described on page 7. Instead, many of them appear to per- 
 form almost random calculations upon the numbers given. When 
 they do solve a problem correctly, the response seems to be deter- 
 mined largely by habit. If the problem is stated in the terminology 
 with which they are familiar and if there are no irrelevant data, 
 their response is likely to be correct. On the other hand, if the prob- 
 lem is expressed in unfamiliar terminology, or if it is a "new" one, 
 relatively few pupils appear to attempt to reason. They either do 
 not attempt to solve it or else give an incorrect solution. 
 
APPENDIX A 
 
 PROBLEMS USED IN TESTS 
 
 In the following pages are reproduced the various forms of the 
 twelve arithmetic problems presented to more than nine thousand 
 pupils of the sixth, seventh, and eighth grades in Illinois for the pur- 
 pose of obtaining data in regard to the actual procedure that pupils 
 follow in solving problems in arithmetic. Four different statements 
 are given for each problem, with the exception of Problems 1 and 7, 
 which appeared in the same form on all four tests. The character- 
 istics of the problem statements are represented by symbols as 
 follows: 
 
 S Simple terminology 
 
 T Technical terminology 
 
 A Abstract 
 
 C Concrete 
 
 R All data relevant to problem 
 I Irrelevant data included 
 
 Problem 1 
 
 Test A. Mr. Duncan drove his car 6375 miles one year and found 
 at the end of the year that 340 gallons of gasoline had been consumed. 
 At 21c a gallon, what was the cost of gasoline for one mile? 
 
 This same statement was used for Tests B, C, and D. 
 
 Problem 2 
 
 Test A — SRC. During a sale, Smith and Company reduced 
 the price of furnaces 20%. When the purchaser paid cash, they 
 gave an additional 5% off the sale price. Mr. Jones bought a fur- 
 nace at the sale and paid $551 cash for it. What price did Smith 
 and Company originally ask for the furnace? 
 
 Test B — TRC. Mr. Jones was allowed successive discounts of 20% 
 and 5% on the list price of a new furnace which he bought from 
 Smith and Company. If Mr. Jones paid $551, what was the list 
 price of the furnace? 
 
 Test C — SRA. An amount was reduced 20%. After an addi- 
 tional reduction of 5% of the remainder was made, the final remain- 
 der was $551. What was the original amount? 
 
 Test D—TIA. A man borrowed $551 for 60 days at 7% to pay 
 a bill on which successive discounts of 20% and 5% were allowed. 
 What was the original amount of the bill? 
 
 20 
 
How Pupils Solve Problems in Arithmetic 21 
 
 In Test A, the technical term, successive discounts, was restated 
 in simplified language. A real setting was added by using words 
 such as sale, Smith and Company, and Mr. Jones. Only relevant 
 data were used. Thus Problem 2, Test A satisfied the theoretical 
 requirement of having simplified (S) terminology, relevant (R) data 
 only, stated concretely. 
 
 In Test B, the technical terms, successive discounts, and list price, 
 were retained. Relevant data only were used. The words, Mr. 
 Jones, Smith and Company, and sale gave a real setting. Thus the 
 combination of factors represented by the symbol TRC was satisfied. 
 
 In Test C, simplified terminology was substituted for the tech- 
 nical terms, successive discounts. Relevant data only were used. 
 No attempt was made to give a real setting to the problem, but 
 rather, the problem was stated abstractly. In the latter respect, 
 Test C differs from Test A. 
 
 In Test D, the technical terms, successive discounts, were used. 
 Irrelevant data, 60 days at 7%, were introduced and the problem 
 did not present a real setting. 
 
 This procedure describes how the corresponding problems of the 
 different tests were formulated. 
 
 Without describing in detail the variations in the remaining cor- 
 responding problems, each of the statements of corresponding prob- 
 lems will be listed under the problem, test, and combination the 
 particular statement represents. 
 
 Problem 3 
 
 Test A — TRA. Find the amount of a bill for 3} S yds. of material 
 at 40c a yd., and for 10 yd. at 85c a yd. 
 
 Test B — SIA. Find the total cost of Z x /i yards of material, 3 
 inches wide, at 40c a yard, and 10 yards of material, 40 inches wide, 
 at 85c a yard. 
 
 Test C — SRA. Find the total cost of Z l /i yards of material at 
 40c a yard, and for 10 yards at 85c a yard. 
 
 Test D — TIC. Mrs. Henderson made the following purchases: 
 
 Z x /2 yds. of ribbon, 3 inches wide at 40c per yd.; 10 yds. of curtain 
 
 material, 40 inches wide, at 85c per yd. Find the amount of the 
 
 bill. 
 
 Problem 4 
 
 Test A — TIA. The following articles were purchased: 5 vacuum 
 cleaners, 12 parlor lamps, 7 rugs, and a suite of furniture. The total 
 amount of the bill was $1500. Terms: 90 days, net; 60 days, 2%; 
 30 days, 5% i cash, 10%. Find the net amount if cash is paid. 
 
22 Bulletin No. 44 
 
 Test B — TIC. The Hall Company buys 5 vacuum cleaners, 12 
 parlor lamps, 7 rugs, and a suite of furniture from Brown and Com- 
 pany, wholesalers, for a total of $1500. Terms: 90 days, net; 60 
 days, 2%; 30 days, 5%; cash, 10%. Find the net amount paid to 
 Brown and Company if the Hall Company pays cash. 
 
 TestC — TRA. A dealer buys merchandise worth $1500. Terms: 
 90 days, net; 60 days, 2%; 30 days, 5%; cash, 10%. Find the net 
 amount if cash is paid. 
 
 Test D — SIC. The Hall Company buys 5 vacuum cleaners, 12 
 parlor lamps, 7 rugs, and a suite of furniture from Brown and Com- 
 pany, wholesalers, for a total of $1500. The bill must be paid not 
 later than 90 days from the day of purchase. 2% will be taken off 
 if paid in 60 days; 5% off if paid in 30 days; or 10% off if cash is paid. 
 Find the amount Brown and Company receives if the Hall Company 
 
 pays cash. n , , - 
 
 ^ J Problem 5 
 
 Test A — SI A. A man purchased 50 articles at $1.50 each and 
 sold them at $2.25 each. Allowing 20% of the selling price for ex- 
 penses, how much was made on each article? This is what per cent 
 of the selling price? 
 
 Test B — SIC. The Student Supply Store in a university dis- 
 trict bought 50 fountain pens at $1.50 each and sold them at $2.25 
 each. Allowing 20% of the selling price for the running expenses 
 of the store, how much was made on one fountain pen? This is 
 what per cent of the selling price? 
 
 Test C — SRC. The Student Supply Store bought fountain pens 
 at $1.50 each and sold them at $2.25 each. Allowing 20% of the 
 selling price for the running expenses of the store, how much was 
 made on one fountain pen? This is what per cent of the selling 
 price? 
 
 Test D — TRC. The Student Supply Store purchased fountain 
 pens at $1.50 each and sold them at $2.25. Allowing 20% of the 
 selling price for the running expenses of the store, the profit is how 
 much per fountain pen? The profit is what per cent of the selling 
 
 Problem 6 
 
 Test A — SI A. One side of a triangle is Z x /i ft., another side is 
 8J4 ft., a third side is 8% ft., and the distance from the top point 
 to the base is S^g ft- What is the sum of the sides? 
 
 Test B — TRC. A pennant was cut so that its base was 3}/% ft., 
 the top side was 8J4 ft., and the lower side was 8% ft- What was the 
 perimeter of the pennant? 
 
How Pupils Solve Problems in Arithmetic 23 
 
 Test C — TIA. The base of a triangle is 3% ft. One side is 834 
 ft., the other 8; 3 s ft., and the altitude, 8}£ ft. What is the perimeter 
 of the triangle? 
 
 Test D — TIC. A pennant was cut so that its base was Z x /i ft., 
 the top side was 8J4 ft., the lower side was 8?£ ft., and its altitude 
 S}^ ft. What is the perimeter of the pennant? 
 
 Problem 7 
 
 Test A. Walter sold his bicycle to Henry for $36 and gained 
 25% of the selling price. If Walter had wished to make 40% of the 
 cost, what would have been the selling price? 
 
 This same statement was used in Tests B, C, and D. 
 
 Problem 8 
 
 Test A — SIC. A crew of 4 men working for 8 hours with a steam 
 shovel, dug a basement 40 feet long, 24 feet wide, and 8 feet deep. 
 Mr. Thomas paid them 40 cents a cubic yard for this work. How 
 many cubic yards were taken out in digging the hole? 
 
 Test B — SRA. How many cubic yards are taken out in digging 
 a hole 40 feet long, 24 feet wide, and 8 feet deep? 
 
 Test C — TIC. A crew of 4 men working for 8 hours with a shovel 
 dug a basement 40x24x8 for which Mr. Thomas paid them 40 
 cents a yard. How many yards were excavated? 
 
 Test D — SRC. Mr. Thomas hired men to dig a basement 40 
 feet long, 24 feet wide, and 8 feet deep. How many cubic yards of 
 dirt were taken out in digging the hole? 
 
 Problem 9 
 
 Test A — TRC. The list price of caps is $3.50. During a sale, 
 Walter bought a cap on which a discount of 15% was given. What 
 is the net price Walter paid? 
 
 Test B—TRA. The list price is $3.50; discount is 15%. What 
 is the net price? 
 
 Test C — SIC. The Johnson Clothing Company gave a reduc- 
 tion of 15%, on all goods bought on June 12. At the sale, Walter 
 bought a cap which usually sold for $3.50; a shirt, for $4.50; and a 
 tie for $1.50. He gave the clerk a ten-dollar bill. What did Walter 
 pay for the cap bought on June 12? 
 
 Test D — SRA. One article usually sells for $3.50. It was sold 
 for 15% less than the usual price. What price was paid for the 
 article? 
 
24 Bulletin No. 44 
 
 Problem 10 
 
 Test A — SRC. Mr. Jones shipped to his agent in Chicago, a 
 carload of peaches, 410 bushels, which he sold at $1.25 a bushel. 
 The agent received 8% of the selling price for selling the peaches. 
 The other expenses were: freight, $75.65; drayage at Chicago, 
 $8.50; baskets, $49.20; picking and packing, $50.20; carting to the 
 station, $18; refrigeration, $55. How much a bushel did Mr. Jones 
 receive after expenses were taken from the amount the agent received 
 for the peaches? 
 
 Test B — TRA. A commission of 8% was paid to an agent who 
 sold 410 bu. of fruit at $1.25 per bu. Other expenses were $75.65, 
 $8.50, $49.20, $50.20, $18, and $55. How much per bu. net was 
 received? 
 
 Test C — SI A. The sale of 410 bushels at $1.25 a bushel was 
 made by an agent who usually sold a total of $200,000 worth each 
 year. He received as his share 8% of this amount collected. Other 
 expenses were $75.65, $8.50, $49.20, $50.20, $18 and $55. How 
 much did the owner receive for one bushel after all expenses were 
 taken from the amount collected? 
 
 Test D—TIA. The sale of 410 bushels at $1.25 a bushel was 
 made by an agent who usually sold a total of $200,000 worth each 
 year. A commission of 8% was paid. Other expenses were $75.65, 
 $8.50, $49.20, $50.20, $18 and $55. How much per bushel net was 
 received ? 
 
 Problem 11 
 
 Test A — TIC. Mr. Thomas deposited $800 in a savings account 
 and $325 in a checking account with the First National Bank, 
 which pays interest at the rate of 4% compounded semiannually. 
 How much will Mr. Thomas have in his savings account at the end 
 of two years? 
 
 Test B — SRC. Mr. Thomas deposited $800 in a savings account 
 with The First National Bank, which pays 4 per cent interest every 
 six months. The interest, when due, is added to the amount depos- 
 ited. How much will he have at the end of two years? 
 
 Test C — TRC. Mr. Thomas deposited $800 in a savings account 
 with the First National Bank which pays 4% interest compounded 
 semiannually. How much will he have at the end of two years? 
 
 Test D — TRA. A man deposited $800 in a savings account of 
 a bank which pays 4% interest compounded semiannually. How 
 much will be due at the end of two years? 
 
How Pupils Solve Problems in Arithmetic 25 
 
 Problem 12 
 
 Test A — SRA. A man had an average income last year of 
 $79.17 a month. His average monthly expenses were $60.50. If he 
 had $632.80 at the beginning of the year, how much did he have 
 at the end of the year? 
 
 Test B — TIA. A man had an average income last year of $79.17 
 a month. His average monthly expenses were $60.50 of which $350 
 was spent each year for food and clothing. If he started the year 
 with a balance on hand of $632.80, wmat was the balance at the end 
 of the year? 
 
 Test C — SIC. Mr. Williams had an average income of $79.17 
 a month. His average monthly expenses were $60.50 of which $350 
 was spent for food and clothing. At the beginning of the year Mr. 
 Williams had $632.80 which he had saved previously. How much 
 did he have at the end of the year? 
 
 Test D — SI A. A man had an average income of $79.17 a month. 
 His average monthly expenses were $60.50 of which $350 was spent 
 for food and clothing. At the beginning of the year he had $632.80 
 which he had saved previously. How much did he have at the end 
 of the year? 
 
APPENDIX B 
 
 TABLES SHOWING DETAILS OF VARIATIONS IN PUPIL 
 
 RESPONSES TO DIFFERENT TYPES OF PROBLEM 
 
 STATEMENTS 
 
 The following tables give for each of three grades — sixth, seventh, 
 and eighth — the data relative to the differences in pupil responses to 
 six types of problem statements. For the meanings of the symbols 
 used, see page 20. 
 
 26 
 
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UNIVERSITY OF ILLINOIS-URBANA 
 
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