MATH WORD PROBLEMS PART 1: WHY THE “KEY WORDS” APPROACH DOESN'T WORK!

If you were to ask a large group of math students at any level what the hardest problems in math are, the majority would most likely say, “Word problems!” Word problems are often tough for students because they add an extra level of complexity to math. They require students not only to perform mathematical operations, but also to do these things:

• Read and comprehend the text of the problem
• Understand what is being asked (the question to answer)
• Create a mathematical equation to answer the question
• Determine which parts of the equation are provided and which are still needed
• Determine how to solve the equation if parts of the equation are missing

Students who are English language learners or who have challenges with reading comprehension are likely to just find the numbers in the problem and guess at how to use those numbers to find the solution. (“Maybe if I just multiply the two numbers I see…?”) To help solve this problem, teachers often teach students to watch for key words in word problems—words that will indicate whether to add, subtract, multiply, or divide. 

When using the “key words” approach, we generally tell our students that “altogether” means “add,” “fewer” means subtract, “double” means multiply by 2, and “share” means divide, for example. Would it surprise you to know that there is no research to support this practice of using key words to help solve word problems? And that it’s even true that saying the word “divide” means that we need to divide isn’t always accurate?

Let’s find out why this is the case by looking at key word “fails.”

Altogether

Let’s start with “altogether.” We tell students that when the problems ask how many objects there are altogether, we add. Will that work for this problem?

Sara bought 4 boxes of crayons with 24 crayons in each box. How many crayons did she buy altogether?

Oops! That didn’t work! Maybe we’ll do better with the word “fewer.” 

Fewer

When we talk about how many “fewer” one person has than another person, that means subtract, right? Let’s check it out with this problem:
 

Zach won a bunch of prizes at the fair. Ben won 6 fewer prizes. If Ben won 2 prizes at the fair, how many prizes did Zach win?

Subtracting 6 minus 2 will not get us the correct answer of 8 prizes! 

Each

Let’s try something with multiplication. Often we teach students that the word “each” means multiply, as in this problem:

Three bags were sitting on the counter. In each bag were 20 apples. How many apples were on the counter?

So, “each” works as a signal to multiply, right? Let’s try it again:

Three bags of apples were sitting on the counter. Each bag held the same number of apples. If the bags held 60 apples altogether, how many apples were in each bag?

Oops! It looks like we can’t rely on the word “each” to tell us which operation to use, either! 

Double

Ok, at least the word “double” must mean multiply by 2. Let’s try that:

Three months ago, Jason decided to start saving his allowance money. In just three months, he has been able to double the amount in his bank account. If he has $100 now, how much money did he have when he started saving his allowance money three months ago?

Multiplying 100—or 3—by 2 wouldn’t solve that problem correctly, would it? 

Divide

Finally, let’s check out the one that may seem the most strange: that “divide” doesn’t mean divide!

If Katie divides 12 sheets of paper into fourths, how many pieces of paper will she have?

Key words like more, less, and share don’t work much better. If children rely on these key words to pick the correct operations to solve word problems, they will often obtain incorrect answers and become frustrated that key words didn’t actually tell them the correct operation! 

In my next post, I’ll explain another word problem approach that is much more reliable than using key words—and approach that will help even students with math learning disabilities conquer word problems!