June 30, 2009
1
Sharon A. Edwards, Robert W. Maloy,
& Gordon Anderson
University of Massachusetts Amherst
“Math is language, too,” Phyllis and David Whitin (2000)
remind us in their informative book about reading and writ-
ing in the mathematics classroom. This means that students in
elementary school math classes are learning two distinct, yet
related languages—one of numbers, the other of words. These
languages of numbers and words are combined in math word
problems, a standard feature of the academic curriculum and a
key part of high-stakes math achievement tests for all students.
Math word problems are intricate language constructions
—they contain unfamiliar words, complex combinations of
text and numbers, and considerable amounts of information
to decode and organize. Young readers who are confused and
distracted by everyday language, math words, or combinations
of both may know how to do the necessary math operations,
yet answer incorrectly because they do not clearly compre-
hend what the question is asking them to do. Others may be
confused about the math operations needed for the problem.
Those who struggle with both the reading and the math face
the biggest challenges. These youngsters are the least success-
ful with math word problems, performing worse than students
with only math difculties or students experiencing neither
reading nor math difculties (Fuchs & Fuchs, 2002).
Literacy coaches charged with helping teachers support
students as readers, often focus on literary texts. Less atten-
tion is given to helping teachers support students to read math.
Drawing on our experiences as elementary school teachers,
a college faculty member who coordinates a literacy tutoring
program, and a computer scientist who works on the design
of intelligent tutoring systems, we focus this article on strate-
gies for addressing the reading challenges found in math word
problems. Using fourth grade questions from the Massachu-
setts Comprehensive Assessment System (MCAS) test, we
identify seven specic word and math language comprehen-
sion challenges. For each challenge, we propose strategies that
literacy coaches and teachers can use to support students in
understanding and solving math word problems.
We focus on fourth grade test examples because that grade
is a crucial mid-point for elementary school students. Math
learning at fourth grade establishes a foundation for success
in upper elementary school and beyond. In our state, ex-
amples from the fourth grade test apply to third grade as well
since word problems begin to have more complex formula-
tions at that level. Fifth and sixth grade questions follow the
same formats as those at fourth grade. With these ideas as a
starting point, we invite you to examine math word problems
from your state and school district exams (or school math
textbook series), identify word and math language compre-
hension challenges, and propose other strategies that teachers
can use to support elementary school students.
Unfamiliar Vocabulary
Challenge. It is hard for students to solve math word prob-
lems containing terms and phrases unfamiliar or unknown
to them. For example, questions from recent Massachusetts
MCAS tests, an exam given to all public school fourth
graders in the state, included words not usually heard in ev-
eryday speech such as item, stadium, depositing, bow, and
repair. Reading and understanding these kinds of words
pose challenges for some students. They may be more
familiar with shopping for milk, bread, or candy rather than
for these items. They may not attend events at “stadiums”
or have heard adults talk about making a “deposit” on an
apartment or into a bank account.
In other questions, unfamiliar phrases confuse young
readers. Consider the following example:
“Haley swam 22 laps each day for 18 days. Then
she swam 25 laps each day for 10 days What was
the total number of laps she swam over the 28
days?” (Massachusetts Department of Education,
2006)
“Swam laps” is a difcult term for young readers, particu-
larly those who do not swim back and forth in a pool. It is
possible not to understand what Haley was doing.
Strategy. One strategy is to ignore unfamiliar or confus-
ing words and try solving the problem with the words the
child knows. In the Haley question, a youngster can read
the problem while ignoring the words swam and laps. The
reader can learn to recognize that Haley did something 22
times each of 18 days and then 25 times each of 10 days.
Multiplying 22 x 18 and 25 x 10 and then adding those two
totals will produce a correct solution without the student
needing to understand what it was that Haley did.
Reading Coaching
for
Math Word Problems
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Proper Names
Challenge. Word problems sometimes include proper names
that are unfamiliar or unknown to young readers. The 2007
MCAS for fourth graders included 29 proper names in 39
questions, including Mr. Gomez, Ms. Rodriguez, Angelina,
Rhonda, Elin, Ms. Lin, Kiki, Pedro, Kyle, Ryan, Rosetta, and
Shannon. Unfamiliar names may distract young readers from
the essential mathematical information in a problem.
Strategy. Even though students may not be able to read
one or more of the names they encounter, most of them
recognize that the unknown word is a name. When this
happens, teachers can encourage young readers to substi-
tute the rst letter of the proper name, their own names,
or the names of friends or family members. When names
are made familiar or abbreviated to a letter, it is easier to
concentrate on the mathematics of the problem.
Sentence Structure and Syntax
Challenge. Word problems on state and national tests and
in commercially published math textbooks are written in
compositional, not conversational English. As such, they
are not easily recognizable for some young readers. Con-
sider the following problem:
“Mr. Thomas walks every day. The distance that
he walks each day is between 4 miles and 8 miles.
Which of the following could be the total number of
miles Mr. Thomas will walk in 30 days?” (Massa-
chusetts Department of Education, 2006)
Strategy. Mathematician George Polya’s (1973) classic
problem solving framework offers a model for teaching
students how to understand word problems. To solve any
problem, Polya proposed the following series of steps
(Figure 1).
Figure 1: Problem Solving Steps
Step 1 What kind of ques-
tion is this?
Identify the question type•
Connect to already learned •
approaches
Step 2 What is the question
asking for?
Find the keywords•
Step 3 What information
am I given to solve
the problem?
Focus on relevant information•
Organize information in a •
table or drawing
Step 4 How can I solve this
problem?
Use the information already •
given
See if the problem can be bro-•
ken down into smaller steps
Eliminate obvious wrong •
answers
Step 5 Did I solve the
problem?
Decide if you solved what is •
being asked
In Polya’s framework, a problem solver rst understands
what type of problem is being posed, then claries what
is being asked for, investigates the problem to see what
information is already given, formulates a plan for solving
the problem, and checks the computational work for any
missteps or errors before nalizing an answer.
While Polya’s problem solving steps delineate what to do
mathematically, his ideas apply to comprehending prob-
lems in order to solve them. When addressing the rst
three questions in this problem solving sequence (“What
type of problem is this?” “What is the problem asking?”
and “What do I know to solve the problem?”) focus on
reading words and numbers. Teachers can use Polya’s
questions to guide students toward understanding.
Math Terminology
Challenge. Math terminology in word problems is a
comprehension as well as a decoding challenge. Stu-
dents need to understand and recall mathematics terms
and concepts expressed in words, such as total; number
sentence; even and odd numbers; equations; expressions;
greatest and least; equal; and probability.
Many math language terms have a counterintuitive, con-
versational meaning. Consider the following problem:
“Last month, 3801 people ate at Tony’s Pizza.
This month, 2765 people ate at Tony’s Pizza.
How many more people ate at Tony’s Pizza last
month than this month?” (Massachusetts Depart-
ment of Education, 2007)
“How many more” suggests adding to nd a total, but
“more” in the question requires subtracting the smaller
number from the larger to nd the correct answer.
Further issues arise in the case of words such as differ-
ence, product, or represents whose meanings are differ-
ent in math language than in everyday language. Product
is something one buys at the store as well as the answer
to a multiplication question. Difference is assessed visu-
ally or vocally as well as by subtracting. Represents is
a political term that is not automatically translated as
“stands for.”
Strategy. Teaching math vocabulary has been shown to
improve student performance on math tests (Gifford &
Gore, 2008). Teachers can purposefully teach math vo-
cabulary. For example, students can be taught that “Total
Number of Different Combinations” problems require
multiplication, as in the following question:
“Mr. Mitchell is ordering special sweatshirts for
his students. The chart below shows his choices
for size, color and pattern.
Choices for Sweatshirts
Size Color Pattern
Small
Medium
Large
White
Yellow
Flowers
Plants
June 30, 2009
3
What is the total number of different combinations of
1 size, 1 color, and 1 pattern that Mr. Mitchell can or-
der? (Massachusetts Department of Education, 2005)
Mr. Mitchell’s combination problem is solved by multi-
plying 3 sizes x 2 colors x 2 patterns = 12 different com-
binations, a strategy that students can use whenever they
encounter a combinations problem.
A second strategy is to invite students to create their own
informational placemats and posters as memory guides to
math terminology and strategies. Children can diagram
phrases to remind them about key terms—the answer to 4 x
4 is a product while the answer to 6 + 6 is a sum. A product
and a sum are both a total, and a total means all. Songs and
skits, particularly ones students compose themselves, also
help students learn math terms and their denitions.
Multiple Math Operations
Challenge. Word problems may involve multiple math-
ematical operations and when they do, some students
may not understand that more than one step is needed.
The problem of Haley swimming laps each day requires
multiplying twice (22 x 18 and 25 x 10) before adding
the subtotals. Students who do realize the need for mul-
tiple math operations might make the mistake of adding
22 + 18 + 25 + 10 to get 75, a wrong answer that may be
one of the possible answer choices.
Strategy. Invite students to change the text of the prob-
lem, but not the numbers. Instead of Haley swimming
laps, the text could be Haley shooting basketballs. The
basketball fans in the class will relate to the idea of mak-
ing 22 baskets for 18 days followed by making 25 baskets
for 10 days more easily than swimming laps. In this way,
those children will recognize that more than one math
operation is needed to answer the problem.
Alternatively, as students identify the multiple math opera-
tions needed, they can sketch a picture, make a chart, or
draw a map displaying what they must do step-by-step.
Visuals representations help students understand the need
for multiple mathematical operations.
Words and Numbers
Challenge. Math word problems blend words and numbers
in ways that can create confusion for young readers.
Consider the following problem:
“Mr. Jordon is buying 3 CDs. Each CD costs
$18.99 including tax. Which is the best estimate
of the cost of the 3 CDs?” (Massachusetts Depart-
ment of Education, 2001)
The numbers in the problem (3 and $18.99) are embedded
within sentences that appear straightforward, yet the two
words including tax are easily missed, creating an oppor-
tunity for students’ calculation to be correct while their
answer choice is incorrect.
Strategy. Invite students to compose their own math word
problems, math comics, and math stories as another way to
understand how writers blend words and numbers together
to pose questions (Edwards, Maloy & Verock-O’Loughlin,
2002). They can author informative problems using child-
engaging language and incorporating child-familiar topics
such as shopping, food, music, sports, and pets, as in fourth
grader Kelsea’s math word problem using fractions.
Kelsea’s Fraction Problem
“There were 50 Labrador puppies—16 were choco-
late, 12 were yellow, and 22 were black. Then I got
16 golden retriever puppies. There were 66 pup-
pies. What fraction of the total number of puppies
were chocolate, yellow and golden?”
As students compose their own problems, teachers or coach-
es can point out the importance of the information written in
the question. Kelsea’s word problem results, 16, 12, and 16
equals 44/66, a fraction that can be reduced to two-thirds.
Visual Displays of Information
Challenges. Word problems may require students to read
and interpret charts, graphs, pictures, and other visual dis-
plays of information. These visual displays can be confus-
ing even to adult readers (Tufte, 2001). Reading visuals
involves interpreting both words and numbers presented
not in sentences or paragraphs, but in rows, lines, circles
and other congurations.
Strategy. Design and construct charts, graphs, and visuals
about topics and questions that students want to ask friends
and family. To glean ideas about how to visually display in-
formation after it is collected, or to nd interesting questions
to pose, go online to USA TODAY at http://www.usatoday.
com/snapshot/news/snapndex.htm. News-related graphs
and questions are categorized and displayed. These visual
displays of information are easy to read and understand.
Conclusion
Math word problems have been a relatively understudied
component of math and literacy learning (Powell, Fuchs,
Fuchs, Cirino, & Fletcher, 2009). They present complex
and multifaceted issues, including the seven challenges de-
scribed in this article. While we discussed these challenges
one by one, some problems present multiple challenges.
Literacy coaches and teachers need wide-ranging strate-
gies in order to support children as they improve their skills
in reading and mathematics. By using novelty, exibility,
and creativity of response, together they can help students
deepen and broaden an understanding of the languages of
words and numbers found in math word problems.
June 30, 2009
4
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