Problem Solving: Choose the Operation
What Is It?
The process of "choosing the operation" involves deciding which mathematical operation (addition, subtraction, multiplication, or division) or combination of operations will be useful in solving a word problem. For example, one way to solve the following problem is to think of it as a problem of subtraction, e.g.:
If there are eighteen students, and six students are not here today, how many are present?
18 - 6 = ?
In comparison, the following problem can be thought of as a problem solved by addition.
If there are twelve students in class today and six students are absent, how many are there in all?
12 + 6 = ?
Why Is It Important?
Choosing mathematical operations is an important part of the larger process of translating English sentences into mathematical expressions. Success depends upon two things:
(a) the ability to understand the literal meaning of the sentence
(b) the ability to express this meaning mathematically
Students who cannot understand the literal meaning of the sentence will not be able to express it mathematically, even if they have the necessary mathematical skills. (Imagine trying to solve a word problem in a language you don't know, such as Arabic.)
Even if students can understand the literal meaning of the sentence, they will not be able to solve the problem unless they can also express this meaning mathematically. In other words, successful solutions to word problems involve both reading skills and mathematical skills. In particular, choosing an operation involves, in part, identifying language clues that suggest mathematical interpretations. Consider the following examples.
If there are eighteen students, and six students are not here today, how many are present?
If there are twelve students in class today and six students are absent, how many are there in all?
The phrase "not here" conveys the concept of taking away—or subtraction. Alternatively, the phrase "in all" may signal a problem solved by addition.
Instead of teaching how to solve word problems as a separate concept, teachers should embed problems in the mathematics-content curriculum. When teachers integrate problem solving into the context of mathematical situations, students recognize the usefulness of strategies (NCTM, 2000).
Teachers must make certain that problem solving is not reserved for older students or those who have "got the basics." Young students can engage in substantive problem solving and in doing so develop basic skills, higher-order-thinking skills, and problem-solving strategies (Trafton and Hartman 1997).
How Can You Make It Happen?
Choosing the operation is a difficult skill for some students, especially those struggling with reading. There is no single solution. A combination of strategies will work best.
Identify Key Words
It may help to work with students to identify certain words that are commonly associated with mathematical operations. For example, the following phrases or words often suggest which operations to use. Consider displaying a table such as this in your classroom and add words and phrases as you find them in word problems.
Addition Subtraction Multiplication Division in all
total
sum
both
combined
altogether
how many
perimeterfewer
left
how much change
how many more
how much more
less
difference
minus
remainstotal
in all groups
area
times
rate
twicehow many each
how many groups divided equallyIt may also help to have students take turns thinking out loud as they work through word problems. For example, consider the following problem.
Juanita took twenty dollars to the mall. She bought a headband for three dollars and a bracelet for seven dollars. How much did she have left?
A student might think aloud (or write) something like this:
First I added three plus seven dollars because it said "three dollars and seven dollars" so I knew that meant to add. So, that was ten dollars. Then I subtracted ten dollars from twenty dollars because it said "How much did she have LEFT" so I knew that meant to subtract.
Get to the Bottom of the Problem
While the "key word" approach may provide hints, many problems do not provide overt clues. For example, to understand the following problem, one must understand the meaning of the words absent and present. There is no substitute for understanding the vocabulary of a word problem and what it means. This involves finding the important pieces of information, and may require students reading the problem several times, and/or students putting the problem into their own words.
Draw a Picture
Drawing a picture or diagram is often a good intermediate step in translating a word problem into a mathematical expression. For example, consider the following word problem.
If there are eighteen students, and six students are not here today, how many are present?
This problem may be represented graphically using a picture. You could draw eighteen children in a row, then cross out six of them.
Or a table such as this:
present present present present present present present present present absent absent absent present present present absent absent absent Once presented in this way, the problem may be more easily seen as a problem of subtraction, because we are clearly "taking away" some parts from the whole. Consider having student create their own standard visual representations for problems involving addition, subtraction, multiplication, and division—then have them practice choosing from among their representations given particular word problems.
Unnecessary Information
It is important to encourage students to read an entire problem before starting to solve it-deciding which information is important and which information is not needed. One method is to have them practice with problems that have too much information, such as:
Emma rode her bike the same distance as Michael. It is 12 miles from Emma's house to school, 4 miles to the library, and 1 mile to the playground. If Michael and Emma rode a total of 26 miles, how many miles did Emma ride?
Can students find 13 miles as an answer? Discuss the incorrect answer they might have found if they didn't focus on the important information. Have students create their own word problems that contain too much information, and challenge each other to solve them.
