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The Bureau
act of the
of the
Bureau
was
of Educational Research
Board
of Trustees
June
1,
1918.
established by
the purpose
It is
to conduct original investigations in the field
summarize and bring
of education, to
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
Bureau of Educational Research are published
bulletins.
A
of available publications
list
back cover of
this bulletin.
is
of research conducted elsewhere
to the school
men
form of
given on the
At the present time
original investigations are reported each year.
on by the
in the
or six
five
The accounts
and other communications
of the state are published in the form of
educational research circulars.
From
ten to fifteen of these
are issued each year.
The Bureau
Its
is
a department of the College of Education.
immediate direction
an instructor
vision research
staff
is
vested in a Director,
in the College of Education.
is
carried on by other
members
and also by graduates who are working on
this point of
who
Under
is
also
his super-
of the
Bureau
theses.
From
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
Director,
Bureau
S.
Monroe
of Educational
Research
PUBLISHED BY THE UNIVERSITY OF ILLINOIS, URBANA
1929
UNIVERSITY
5100 12 28 6412
™ nm?.
PREFACE
The problem
tin
is
studied in the investigation reported in this bulle-
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 training which pupils receive in this process is secured by solving arithmetical 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
The reader of this bulletin, however, should bear in mind
to be.
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
Table II. Variations in Pupil Responses to Problems When Technical Terminology (T) was Substituted for Simplified Terminology
(S)— Grades VI, VII, and VIII
Table III. Variations in Pupil Responses to Problems When Irrelevant Data (I) were Substituted for Relevant Data (R) Grades VI,
VII, and VIII
Table IV. Variations in Pupil Responses When Language Which is
Abstract (A) in Nature was Substituted for Language Which Presents a Real or Concrete (C) Setting. Grades VI, VII, and VIII.
Table V. Number and Frequency of Errors in One Thousand Test
.
Papers
10
13
14
16
18
Table VI. Variations in Pupil Responses to Problems When Technical
Terminology (T) was Substituted for Simplified Terminology
27
(S)— Grade VI
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 Irrelevant 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 Presents 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 Presents 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 Presents 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 concernis identified and answered; denominate number facts
are recalled; numbers are read and copied. However, the response cannot be adequately described by enumerating the responses to such elements. Reflective thinking is involved. It should be noted that the total response to a verbal problem includes 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
ing a functional relationship
total response. 1
Elementary-school teachers and other observers of the performances
of pupils in the field of arithmetic probably would criticise this statement by pointing out that it describes how pupils should solve problems, rather than how they do solve them.
It is the purpose of
this study to answer certain questions relating to the actual procedure 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
the statement,
it
was
it
occurred.
If this
called concrete.
activity
was indicated in
of a problem
The terminology
was considered technical when the terms used were those that are
commonly employed by
specialists in a particular field.
The terminology 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.
Bulletin No. 44
8
The
which information was sought
specific questions for
may
be
stated as follows:
1.
What
is
the difference between the responses of pupils to problems stated
same problems stated in concrete terms?
abstractly and the
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:
solving a problem incorrectly in principle,
4. In
what procedure do pupils
follow?
In these questions, the phrase "responses of pupils to problems"
refers only to the principle or plan of solution
No
throughout the study.
attention was given to errors of calculation.
The
Four separate tests were conTest B, Test C, and Test D. With one exception, 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
Problems 1 and 7 were presented in the same form on all four
I.
In Test A, Problem 2 is stated in simple terminology (S), all
tests.
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
construction of the tests.
structed
— Test A,
relevant (R), and the setting
statement of the problem
is
is
concrete (C).
The
difference in the
the change from simple terminology to
In Test C, the problem is stated in simple
although
the data given are relevant (R), and the
terminology (S),
In
Test D, technical terminology (T) is
setting is abstract (A).
used, irrelevant data (I) are included, and the setting is abstract
(A). The four forms of Problem 2 are as follows:
technical terminology.
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?
B — TRC.
Mr. Jones was allowed successive discounts of 20% and
the list price of a new furnace which he bought from Smith and Company.
Jones paid $551, what was the list price of the furnace?
Test
3
However,
Problem 3 was not taken directly or indirectly from any text.
simple type of problem that has a high frequency of occurrence in all texts.
it
5% on
If
Mr.
represents a very
How
C
Test
— SRA.
Pupils Solve Problems in Arithmetic
An amount was
5% of the remainder was made,
reduced 20%.
9
After an additional reduction of
the final remainder was $551.
What was
the original
amount?
D— TIA. A man
Test
successive discounts of
borrowed $551 for 60 days at 7% to pay a bill on which
were allowed. What was the original amount
20% and 5%
of the bill?
The
variations of the statements of the other problems
ascertained
by
referring to
various forms of statement in the four tests
section of Table
may
The arrangement
Appendix A.
is
be
of the
indicated in the
first
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 problems selected. The method of random sampling was employed as
a means of securing these equivalent groups. In order to have each
random sample of pupils, the four tests were
arranged in alternate order so that when distributed to the pupils
of the tests given to a
in the class, the first, fifth, ninth, thirteenth,
and so
receive Test A; the second, sixth, tenth, fourteenth,
forth,
would
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 represented 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
w 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
its
characteristics
T
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.
distribution
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.
Bulletin No. 44
10
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How
Pupils Solve Problems in Arithmetic
Provisions for making comparisons.
mary
Table
I
11
presents in sum-
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
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 Probthose
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 statement 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.
;
The problems
were taken from those
it was thought wise
to have the tests given also in a few sixth and eighth-grade classes.
-.Usable data w 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
Pupils tested.
for the tests
that seventh-grade pupils are asked to solve, but
r
^
The
validity of data.
The
principal questions relative to the valid-
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 ere taken from texts designed
for the seventh grade. No analysis of these texts was made to determine 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 statement 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.
ity of the
T
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 Bulletin No. 31. Urbana: University of Illinois, 1926. 92 p.
Bulletin No. 44
12
The second
tested,
question, regarding the equivalence of the groups
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 different 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. Examination 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
Apparently, "amount of bill" was more easily
understood by these pupils than "total cost." This probably indicates 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-
simple statement.
rect responses.
The
The
effect of introducing irrelevant data.
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 Problem 4, this table indicates that the presence of irrelevant data
makes the problem more difficult, although several of the differ-
The
ences are small.
correct
in
lengths of
principle
the
three
for
sides
Problem
of
relevant fact was the altitude.
especially those in
8
9
per cent of solutions
largest difference in
is
6.
In
this
problem, the
the irof the pupils,
a triangle were given and
Apparently,
many
Grades VI and VII, had become so accustomed
The data are given separately
The facts are given separately
for the three grades in
for the three grades in
Tables VI, VII, and VIII in Appendix B.
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
Tables IX, X, and
phenomenon. In Problem 4 the
irrelevant data are furnished by enumerating the articles purchased.
larger per cent of solutions correct in principle.
XI
reveal additional cases of this
The presence
of these items, together with the other
changes in the
The explaphenomenon 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"
phraseology, appears to have
made
the problem easier.
nation of this
appeared to be a very familiar expression. If this is true, it is not
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 introduced, but because another change was made which more than overbalanced this change.
difficult to
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, respectively. Examination of these two problems does not suggest a probAbstract versus concrete terminology.
able explanation of this variation.
It
appears that the introduction
of phrases descriptive of a concrete setting for a
little
or no difference in the responses of pupils.
problem makes
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
II, III, and IV, and more especially those of the
Appendix B, suggest to the writer that the terminology 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
of
Tables
detailed tables in
solving arithmetic problems.
10
See
p.
21
n The data
Instead, they learn to
and 22 for the four statements of this problem.
are given separately for the three grades in Tables XII, XIII,
and
make
rather
XIV in Appendix
B.
Bulletin No. 44
16
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How
Pupils Solve Problems in Arithmetic
17
Hence, when a differor an inaccursome
pupils
course;
do reason in
exceptions,
of
There
are
ate one.
investigation
suggest
collected
in
this
But
data
situations.
the
such
think
reflectively
they
number
not
when
do
that a considerable
is
reinforced
This
conclusion
by
respond to arithmetic problems.
following
paragraphs.
the data described in the
fixed responses to certain types of statement.
ent form
of
statement
is
used, they
make no response
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 separately for each test.
tions, the
number
number
Table
V
gives the
number
of erroneous solu-
of different kinds of erroneous solutions,
and the
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 different 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 comprehension of the problem. Only 28 of the 253 erroneous solutions of Problem 2 were so classified. The remaining 225, or 89 per cent, w 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 seventhgrade 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 probof
r
lems.
General conclusion. Although the data of this investigation are
not entirely consistent, they appear to substantiate the conclusion
Bulletin No. 44
18
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How
Pupils Solve Problems in Arithmetic
19
that a large per cent of seventh-grade pupils do not reason in attempting to solve arithmetic problems.
Relatively few pupils follow the
many of them appear to perform almost random calculations upon the numbers given. When
they do solve a problem correctly, the response seems to be determined largely by habit. If the problem is stated in the terminology
with which they are familiar and if there are no irrelevant data,
plan described on page
their response
lem
is
is
7.
likely to
Instead,
be correct.
On
the other hand,
expressed in unfamiliar terminology, or
relatively few pupils appear to
not attempt to solve
it
attempt to reason.
if
the prob-
"new" one,
They either do
if it is
a
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 purpose 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 characteristics
problem statements are represented by symbols as
of the
follows:
S
Simple terminology
Technical terminology
Abstract
Concrete
All data relevant to problem
Irrelevant data included
T
A
C
R
I
Problem
Mr. Duncan drove
1
6375 miles one year and found
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.
Test A.
his car
at the end of the year that 340 gallons of gasoline
Test
Problem
A — SRC.
During a
sale,
2
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?
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
Test
price of the furnace?
Test
— SRA.
C
tional reduction of
An amount was
5%
of the
reduced 20%. After an addiremainder was made, the final remain-
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?
der was $551.
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 technical 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
were formulated.
Without describing in detail the variations in the remaining corresponding problems, each of the statements of corresponding problems will be listed under the problem, test, and combination the
particular statement represents.
different tests
Test
Problem
A — TRA.
and
at 40c a yd.,
B— SIA.
3
Find the amount of a
for 10 yd. at
bill for
3} S yds. of material
85c a yd.
x
Find the total cost of Z /i
yards of material, 3
inches wide, at 40c a yard, and 10 yards of material, 40 inches wide,
at 85c a yard.
Test
—
l
Test C
SRA. Find the total cost of Z /i
yards of material at
40c a yard, and for 10 yards at 85c a yard.
D— TIC.
Mrs. Henderson made the following purchases:
wide at 40c per yd.; 10 yds. of curtain
material, 40 inches wide, at 85c per yd.
Find the amount of the
Test
x
Z /2 yds. of ribbon, 3 inches
bill.
Test A — TIA.
Problem 4
The
following articles were purchased: 5
cleaners, 12 parlor lamps, 7 rugs,
amount
30 days,
of the bill
5%
i
cash,
vacuum
and a suite of furniture. The total
Terms: 90 days, net; 60 days, 2%;
was $1500.
10%. Find the net amount
if
cash
is
paid.
Bulletin No. 44
22
B — TIC. The
Test
Company buys
Hall
vacuum cleaners, 12
Brown and Com-
5
parlor lamps, 7 rugs, and a suite of furniture from
pany, wholesalers, for a total of $1500. Terms: 90 days, net; 60
2%; 30 days, 5%; cash, 10%. Find the net amount paid to
Brown and Company if the Hall Company pays cash.
days,
TestC
— TRA.
A dealer buys merchandise worth
2%;
90 days, net; 60 days,
amount if cash is paid.
D— SIC.
Test
30 days,
5%;
The Hall Company buys
Terms:
Find the net
$1500.
10%.
cash,
vacuum cleaners, 12
Brown and Com-
5
parlor lamps, 7 rugs, and a suite of furniture from
pany, wholesalers, for a total of $1500. The bill must be paid not
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
later
,
Problem
Test A — SIA.
them
sold
A man
purchased 50 articles at $1.50 each and
Allowing 20% of the selling price for exwas made on each article? This is what per cent
at $2.25 each.
how much
penses,
5
of the selling price?
—
B SIC. The Student Supply Store in a university disbought 50 fountain pens at $1.50 each and sold them at $2.25
Test
trict
each.
Allowing
of the store,
20%
of the selling price for the
how much was made on one
what per cent
running expenses
This is
fountain pen?
of the selling price?
C— SRC.
The Student Supply Store bought fountain pens
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
Test
at $1.50 each
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
Test
A — SIA. One
a third side
8J4
to the base
ft.,
is
S^g
Test B — TRC.
is
ft-
A
Problem
6
side of a triangle
8%
ft.,
What
is
is
x
Z /i
ft.,
another side
is
and the distance from the top point
the
sum
of the sides?
pennant was cut so that
the top side was 8J4 ft., and the lower side was
perimeter of the pennant?
its
base was
3}/% ft.,
8% ft- What was the
How
— TIA.
The base
C
Test
the other 8;3 s
of the triangle?
ft.,
Pupils Solve Problems in Arithmetic
ft.,
of a triangle
and the
is
altitude, 8}£
3%
ft.
ft.
23
One
side
is
834
What is the perimeter
D — TIC. A
x
ft.,
pennant was cut so that its base was Z /i
the top side was 8J4 ft., the lower side was 8?£ ft., and its altitude
S}^ ft. What is the perimeter of the pennant?
Test
Problem
Test A.
25%
Walter sold
of the selling price.
7
Henry for $36 and gained
Walter had wished to make 40% of the
his bicycle to
If
what would have been the selling price?
This same statement was used in Tests B, C, and D.
cost,
Problem 8
Test
A — SIC. A
Test
B — SRA. How many
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?
cubic yards are taken out in digging
a hole 40 feet long, 24 feet wide, and 8 feet deep?
— TIC.
C
Test
A crew of
dug a basement 40x24x8
cents a yard.
Test
How
D — SRC.
4
men working
for 8 hours with a shovel
which Mr. Thomas paid them 40
many yards were excavated?
for
Mr. Thomas hired men to dig a basement 40
and 8 feet deep. How many cubic yards of
feet long, 24 feet wide,
dirt
were taken out
Test
A — TRC.
in digging the hole?
Problem 9
The
list
price of caps
is
Walter bought a cap on which a discount of
is the net price Walter paid?
Test
is
B—TRA. The
list
price
is
During a
$3.50.
15% was
$3.50; discount
is
sale,
given.
What
15%.
What
the net price?
— SIC.
The Johnson Clothing Company gave a reducon 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
C
tion of 15%,
Test
for
15%
article?
D— SRA. One
less
It was sold
was paid for the
article usually sells for $3.50.
than the usual
price.
What
price
Bulletin No. 44
24
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
receive after expenses were taken from the
a bushel did Mr. Jones
amount the agent received
for the peaches?
Test
B— TRA. A
commission
8%
of
was paid to an agent who
Other expenses were $75.65,
sold 410 bu. of fruit at $1.25 per bu.
$8.50, $49.20, $50.20, $18,
received?
How much
and $55.
per bu. net was
— SIA.
C
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
Test
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
made by an agent who usually sold
year.
A
commission
of
bushels at $1.25 a bushel was
a total of $200,000 worth each
8%
$8.50, $49.20, $50.20, $18
was paid. Other expenses were $75.65,
and $55. How much per bushel net was
received ?
A — TIC.
Problem
11
Test
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 deposited.
How much will he have at the end of two years?
Test
C— TRC.
Test
D — TRA. A man deposited $800 in
4% interest compounded
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?
a bank which pays
much
will
be due at the end of two years?
a savings account of
semiannually.
How
How
Pupils Solve Problems in Arithmetic
Problem
25
12
A — SRA.
A man had an average income last year of
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
$79.17 a month.
B — TIA. A man
had an average income last year of $79.17
average
monthly
expenses were $60.50 of which $350
a month. His
If he started the year
for
and
clothing.
food
was spent each 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
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
a month.
D — SIA. 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?
Test
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
and eighth
six
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—
For the meanings
types of problem statements.
used, see page 20.
26
sixth, seventh,
in pupil responses to
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