Understanding
Word Problems
By LEE V. STIFF, North Carolina State University, Raleigh, NC 27695
F
or several years now, I have been
asked to share with junior and senior
high school mathematics teachers in North
Carolina ways to improve students' reading
comprehension of word problems. My work
with teachers and students has given me
the opportunity to field-test several strategies for improving reading skills. One such
strategy uses comprehension guides (Earle
1976; Herber 1978).
In general, comprehension guides help
students understand written prose at three
levels. The reader who correctly answers
the questions What does the author say?
What does the author mean by what is
said? and Which ideas based on your previous experiences relate to the ideas expressed by the author? reads at the literal ,
interpretive, or applied levels of comprehension, respectively. Two of these levels
are important to an understanding of word
problems: the literal and the applied, or operational, levels. Accordingly, a comprehension guide for a word problem consists
of literal statements, the word problem, and
operational statements.
Literal statements express factual information found in the word problem. Users of
a guide should decide what the word problem actually says and then identify the important facts therein. In addition, users
should determine the specific question that
is to be answered. Often, students read
poorly at the literal level of comprehension
because of a poor vocabulary. Sometimes
students read too well , adding information
that seems to follow but is not given. The
use of comprehension guides gives students
the opportunity to address both types of
comprehension difficulties.
Operational statements express mathematical computations or procedures
needed to solve the problem. Users of a
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guide should decide which mathematical
concepts and operations are required to
obtain a solution. Word problems are often
difficult because students do not try to combine known information with information
found in the statement of the problem. Students must learn to construct solutions on
the basis of all the available information.
Guides contain solutions and partial solutions, so students have the opportunity to
identify possible solutions.
Consider this typical mathematics problem for junior high school students:
A local university hired 15 students to
clean the soccer stadium after a match.
They earned $4.35 an hour. They worked
8 hours. How much did the university pay
to have the stadium cleaned?
Students frequently struggle with this type
of problem because it is difficult to keep the
information in it straight. (Did each student
work 8 hours or did all 15 students work a
total of 8 hours?) Some information is factual: A local college employed fifteen students. Other information must be constructed: The cost to the university to clean the
stadium was (8 x 15 x 4.35) dollars. Comprehension guides help students sort out information found in word problems.
M ak ing a Comprehension Guide
A teacher should follow three steps when
preparing a comprehension guide for students' use. (Students can prepare them on
their own with some practice.)
1. The first step is to identify a word
problem. The most convenient and obvious
source of problems is the mathematics textbook.
2. Next, teachers should construct declarative statements to express the literal
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and operational content of the word problem. Declarative statements are assertions
that, when read, become the assertions of
the reader. This gives the student the sensation of understanding or comprehending
the text of the problem. Literal and operational declarative statements are created
by asking, respectively, each of two
questions: What is the essential information given or asked for? and What computations or procedures must one use to
solve this problem?
3. Not all declarative statements should
express accurate information. An important
step in creating a comprehension guide is
to construct both true and false statements
about the problem at the literal and operational levels.
Literal statements should precede the
statement of the problem! At first, teachers
may find this arrangement awkward, but I
have found that it enables students to read
the problem with a specific purpose in
mind. As students improve in their abilities
to extract information from word problems,
the positions of the literal statements and
the word problem can be switched. The
statement of the problem can then precede
the literal statements. Operational statements are always placed last in comprehension guides. The number of declarative
statements in a guide is arbitrary; however,
a one-page guide seems to work best.
In the development of a comprehension
guide, it is important to construct powerful
distractors. My experiences with comprehension guides indicate that well-chosen
distractors stimulate mathematical thought
and dialogue among students. Good distractors force students to think about information and how it should be used. Declarative statements that express typical errors
in comprehension made by students are the
best distractors. In table 1, statements l-4
and ll-3 are examples of good distractors
because they represent common misconceptions of students.
Statements I-A and ll-1, in table 1, are
examples of another powerful construct,
the incomplete statement. These statements
TABLE 1
Comprehension Guide- Stadium -cleaning
Problem
I. Check all items that correctly identify information contained in the problem and what is to
be found .
__1. The university hired 15 people to clean
the stadium.
__ 2. Each student earned S4.35 for each
hour worked .
__ 3. The soccer match lasted 8 hours.
__ 4. The students worked a combined
number of 8 hours.
__ 5. Each student worked 8 hours.
What is to be found?
__A. The total number of hours worked by
the 1 5 students
__ B. The expense of the university for one
worker
__ C. A student"s hourly wage
__ D. The cost of cleaning the stadium
__ E. A student·s weekly wages
A local un iversity hired 15 students to clean the
soccer stadium after a soccer match. They earned
$4.35 an hour. They worked 8 hours. How much
did the university pay to have the stadium
cleaned?
II. Correctly identify the operations or procedures
needed to solve the problem.
__ 1. 8
X
15
__ 2. Time= number
worked .
of
students x hours
_ 3 . (8 + 15) X $4.35
_4. 15 X ($4.35 + 8)
__ 5. 8 X 15 X $4.35
__6. Cost = wages x time.
__7. $4 .35
X
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provide correct but incomplete information,
computations, or procedures. Incomplete
statements are used to help students see
that solutions are often obtained by considering pieces of the problem.
A third type of statement is the procedural statement. Statement ll-6, in table
1, is an example. These statements let the
teacher determine whether students have a
more general understanding of the underlying relationships in a problem.
Table 2 provides a second example of a
comprehension guide. It is instructive to
note the types of statements contained in
the guide. Statements I-3 and ll-2 are distractors; 1-B and II-4 are incomplete state-
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TABLE 2
------------------------
Comprehension Guide- Carpeting Problem
I. Che.ck all items t~at correctly identify information contamed m the problem and what is
to be found.
_ _ 1. Dee Morgan bought a new home
__2. The livi ng room is rectangular.
__ 3. The carpet costs S12 .25 a square yard .
_ _ 4. T he measurements of the living room
are 3.4 m by 5.6 m.
__ 5. Dee enjoys decorating her living room .
What is to be found?
__A . How much the carpet will cost
__ B.
__ C.
_ _ D.
_ _ E.
The area of the living room
The dimensions of the living room
Time needed to lay the carpet
The number of square meters in the
living room
Dee Morgan's rectangular living room measures
3.4 m by 5.6 m. What is the area of the room?
How much wi ll wa ll -to-wall carpet for the living
room cost at S12.25 a square meter?
II Correctly identify operations or procedures
needed to solve the problem.
_ _ 1. Length x wid th = cost
_ 2 . 12.25 m x 5.6 m
_ 3. Area x cost
_ _4 . 3.4 m x 5.6 m
_ 5 . Length x w idth x cost
_
6. 3.4 m x 5.6 m x S12.25
ments; and statement II-3 expresses a computational procedure related to the solution
of the problem.
U s ing Guides
Although compreh ension guides can appropriately be used as an occasional substitute
for primary mathematics instruction, they
are most effective when used once or twice
a week to review mathematical topics that
have been formulated into word problems.
Guides offer students an opportunity to
sharpen newly learned mathematical skills
and concepts, perhaps in preparation for a
unit test. If used r egularly, guides are a
good way to stimulate the kind of mathematical reflection and discussion that
characterize good problem solvers.
Heterogeneous ability groups of three or
four students each should be created. Each
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group considers the same problem(s). Each
student reads the guide for a given word
problem and makes a decis ion about which
statements ought to be identified as correct.
Once group members have had an opport unity to decide individually about
s tatements, they then discuss the merits of
any selection. Students must be given an
opportunity to defend or reject any selection. At the end of the discussion, each
group identifies what it considers to be the
correct selections. The give-and-take of this
activity generates not only lively debates
but also mathematics learning!
Time limits should be placed on discussions before a session begins. (Sometimes students don't want to stop the
debate!) It is a good idea to monitor the
progress of each group while their decisions
are being made. In so doing, teachers can
u sually prevent serious errors in reasoning
and evalua te the inter actions of each
group. This monitoring, of cours e, requires
the teacher to move from group to group
during the discussions. At the conclusion of
the group discussions, t he teach er should
indicate to the class which statements are
"judged correct." Undoubtedly, more class
discussion will be needed to r econcile differen ces in opm10n about st atements
judged to be correct.
Comprehension is a key factor in improving r eading in mathematics (Henrichs
and Sisson 1980). Comprehension guides
give teachers the means to attack this reading problem as students improve th eir ability to solve word problems. However, mathematics teachers should provi de r eading instruction in mathematics to students who
do not have deficient r eading skills (Krulik
1980). I agree with Herber (1978) who
stressed that mathematics teach ers should
t each reading function ally, t h at is, teach
the reading skills needed to learn mathematics content and teach thes e skills concurrently wit h mat hematics. M athematics
teachers do n ot have to be reading t eachers
to remedy s tudents' difficulties in understanding word problems.
(Continued on page 215)
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7. Let N 0 be an y positive integer.
NH
1
=
3Nk + 1 if Nk is odd
(h alt if NH 1 = 1)
{
Nk /2 if Nk is even
Thus, if N 0 = 11, the correspondin g chain is
11, 34, 17' 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2,
1. This chain has fourteen links. Write a
program that will calculate the length of
each chain for t h e nu mbers 1 t hrough 100.
The program sh ould a lso r eport th e mean,
median, or moda l lengths.
8. Write a pr ogram t h at will tak e a
paragraph and gen erate a frequency distr ibution of the lett ers a, e, i, o, a nd u.
Summary
Students need to learn basic st atist ical concepts and s kills. Beyond this, however, they
need to h ave practice in a pplying t h ese fundamental ideas. The micr ocomputer can
help teachers offer suppor t ive learning experiences, as well as in teresting and relevant problems. We n eed to con t inue to develop a ctivities t hat will use th e computer
to enhance learning, specifically in this
most important conte nt area.
REFERENCES
Carlson, Ronald J . "Buffon's Needle Problem on a
Microcomputer." Mathematics Teacher 74 (November 1981) :638-40.
RESOURCES
Classroom Computer News. Box 266, Cambridge, MA
02138
The Computing Teacher. University of Oregon,
Eugene, OR 97403
Conduit, Box 388, Iowa City, lA 52244
Educational Comput~r. Box 535, Cupertino, CA 95015
Minnesota Educational Computing Consortium, 2520
Broadway Drive, St. Paul, MN 55113 •
Unde rstanding
Word Proble ms
(Continued from p age 165)
My observations of students using comprehension guides in the classroom have
convinced me that students can benefit
from sh aring learning experiences and that
the use of an ambiguity or a d istractor is
sometimes the clearest way to make a point.
Classes are frequently noisy, ideas are invariably dispu ted, and students ar e naturally animated , but students are always involved and th inking. Compreh ension guides
work. I encourage you to try t hem.
BIBLIOGRAPHY
Cheek, Earl H., J r., and Martha C. Cheek. " Organizational Patterns: Untapped Resources for Better
Reading." Reading World 22 (May 1983) :278-83.
Denmark, Tom. "Improving Students' Comprehension
of Word Problems." Mathematics Teacher 76 (January 1983):31 34.
Conference Board of the Mathematical Sciences, National Advisory Committee on Mathematics Education. Overview and Analysis of School Mathematics .
Grades K - 12. Reston, Va.: National Council of
Teachers of Mathematics, 1979.
Earle, Richard A. Teaching Reading and Mathematics.
Newark, Del. : International Reading Association,
1976.
Culp, G., and H. Nickles. An Apple for the Teacher.
Monterey, Calif.: Brooks/Cole Publishing Co., 1983.
Francis, Richard L. "Word Problems : Abundant and
Deficient Data." Mathematics Teacher 71 (January
1978):6- 11.
Goodman , Terry. "Statistics for the Secondary Math·
ematics Student." School Science and Mathematics
81 (May 1981):423.
National Counci l of Teachers of Mathematics. An
Agenda for Action: Recommendations for School
Mathematics of the 1980s. Reston, Va.: The Council,
1980.
Swift, Jim. "Challenges for Enriching the Curriculum :
Statistics and Probability." Mathematics Teacher 76
(April 1983) :268- 69.
Panur, Judith, et al., eds. Statistics: A Guide to the
Unknowtl. 2d ed. San Francisco: Holden-Day, 1978.
Henrichs, Margaret, and Tom Sisson. " Mathematics
and the Reading Process: A Practical Application to
Theory." Mathematics Teacher 73 (April 1980):25357.
Herber, Harold L. Teaching Reading in Content Areas.
Englewood Cliffs, N.J.: Prentice-Hall, 1978.
Krulik, Stephen. "To Read or Not to Read, That Is the
Question!"
Mathematics
Teacher 73 (April
1980) :248- 52.
Perlman, Gary. "Making Mathematics Notation More
Meaningful." Mathematics Teacher 75 (September
1982) :462- 66. •
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