This post was originally published on mathteacherbarbie.com. If you are reading this elsewhere, you are reading a stolen version.
The problem with word problems
Word problems in math class tend to be highly contrived. They feel different than the same scenarios in the real world. Whether they are actually higher pressure than the same situation in real life or not (I mean really, what’s more important: solving a real-world problem or solving a math-class problem), they also tend to release those negative brain chemicals that get in the way of our thinking processes.
Word problems in math class
- Require reading skills that some students may not have. Some real-world scenarios also require this, but many or most communicate the information verbally or in other ways.
- Give all of the relevant information all at once. Real world problems tend to be revealed over time, giving us a moment to integrate each new piece of knowledge before being bombarded with more.
- Often give only the relevant information and no extra. In the real world, we generally understand we may have extra information. It’s rare to have too much info in a math class problem, leading us to believe that every bit of info must be important, even if it’s not.
- Give the numbers and the context at the same time. Though this is similar to the second bullet, it more specifically leads us to focus on what we think is important in math class — the numbers — and ignore the context. If we were to face the problem in the real world, we would naturally focus first on the context. It’s in the context that what we’re supposed to do with the numbers becomes clear. The context provides the structure and operations. Until we know what we’ll do with them, the numbers only muddy the waters.
How to read a word problem
Read for the context, the structure, the story. Ignore the numbers until later. The numbers don’t matter until we understand what the question is.
There are a lot of problem-solving models out there. Most of them start with “read the problem twice” or “read the problem three times.” If they’re being particularly thoughtful, they’ll tell you what to read for each time. If these truly worked, we would have already settled on the best one instead of having so many different approaches! Instead, I propose something a little less restrictive than counting your read-throughs.
Read for context, not quantities.
Acknowledge that the numbers really don’t matter. If you needed to know how many total socks you have and some of them are in the laundry and some are in the drawer, you first strategize before you go count them. With that context, you know that you can count the clean socks in the drawer and count the stinky socks in the laundry and then add them together. Then, after realizing that structure, is when you would go count and do.
When I first read a word problem, in fact, I have trained my eyes to basically skim over the numbers. When I read them aloud to students, I mumble my way over the numbers or substitute the phrase “some number of” instead of saying the actual numbers. Downplaying them. Remember above how we said that putting the numbers and the context both up front on the same level was unique to word problems and part of why they feel contrived and unnatural? Culturally, though, we see math as being about the numbers (even though math is arguably more about patterns, but that’s a topic for another post) that we as a culture gravitate first toward the numbers in math class, even if we’d be able to set those same numbers aside outside of math class.
The numbers will still be there to look back at when you’re ready. But you don’t really need them yet. Make sure everyone understands the context, or what’s going on in the problem. That’s the important part and is what gives structure to the upcoming math problem.
How to think about a word problem
Picture or model the situation. If you can visualize, can you see what’s going on in your mind’s eye? If you can’t visualize images out of words, then can you draw or build a model of what’s happening to help you see or “see” it?
Especially with younger children, put the problem in the context of a larger story. Children explore the world through story. It’s a context they’re used to. It also makes the whole process more fun.
Identify what the question is. Often there will be an actual question mark, but not always. What is it that the problem is actually asking you to find?
Write your answer sentence early. You don’t have to know the number solution to format your answer. Write out a complete sentence answering the question you identified, just leave a blank (or a smiley face or a flower or a lightning bolt or the phrase “some number of”) instead of the actual number. If this is harder than usual sentence writing for you, you may not have identified the actual question.
Make a “reasonable guess” at the answer. What would be realistic? What is your estimate? This will be very useful later in the process.
Make a humorously high and a humorously low guess at the answer. Then plug them into your answer sentence and laugh about them.
Identify the mathematics you will need. Is this combining different-sized parts into a whole (addition), trying to find out about one of the parts of a whole (subtraction), combining same-sized groups into a larger group (multiplication), dividing a group into same-sized parts (division), something else? I talk about some ways to tackle this step in the article Add, Subtract, Multiply, Divide in Word Problems? Beyond Key Words.
How to start when you don’t know how
An example might make the next points clearer, so let’s start with a word problem. Ry ate $\frac{1}{2}$ of a pizza. Zoey ate $\frac{1}{3}$ of a pizza. How much more pizza did Ry eat than Zoey?
- Make sure you can write that answer sentence, even if you don’t know what number goes in the blank yet. For our example: Ry ate _____ more pizza than Zoey.
- Pay attention to units of measurement. For our example, here we’re talking about portions of a pizza. We could even fill in more of our answer sentence to reflect this: Ry ate _____ of a pizza more than Zoey.
- Use simpler numbers.
- Sometimes our brains just won’t cooperate and just can’t ignore the numbers. But sometimes those numbers (and our prior bad experiences with them) throw us off. (This happens a lot when the numbers are fractions, for example.) Instead of allowing that fear response to shut down all thinking, temporarily substitute in “easier” numbers to read through the problem and figure out how to solve it. For our example: “Ry ate 3 pizzas. Zoey at 2 pizzas. How much more pizza did Ry eat than Zoey?” Or think about a pizza cut into 12 slices. Then Ry ate 6 slices and Zoey ate 4. Is the subtraction more obvious now? Ry ate $6-4$, or 2 more slices of pizza than Zoey.
- But don’t forget to redo that with the original numbers! Now we know that the relationship is a subtraction one, so we can set up $\frac{1}{2}-\frac{1}{3}$ and tackle just the fraction arithmetic when we need to instead of letting it hijack the whole process.
- Try a similar problem with a more familiar context.
- Like changing the numbers, this can help us understand the structure of the problem, just by taking out the piece that’s getting our brain stuck. Just don’t forget to go back and do the original problem too once you figure out the math!
- Try something you know is wrong.
- You might laugh together.
- You might gain insights.
- You might see structures you hadn’t noticed.
- For our example, we’re pretty sure that Ry didn’t eat 15 more pizzas than Zoey, but can you a) imagine it, and b) prove it?
- Draw pictures or build models that represent the situation. Then see if you can act it out, or put it in the context of a larger story or situation. For our example, cut out circles to represent pizzas. Then remove half of one for the amount that Ryan ate. Remove a third of the other for the amount Zoey ate.
Many of these may be difficult to convince a child to try when they just want to get through the homework as fast as they can. However, if you can establish the pattern of trying them, a) they’ll get faster, b) your child (and you!) will understand more, and c) homework will start to feel more like play. And we know that children learn best through play.
How to gather the information from a word problem
Once you do the prepwork to understand the problem in a meaningful way (all the tips above and anything else you can think of), getting the numbers and other information you need from the word problem becomes a lot easier! The most common way of “gathering” your information is just to make a list of the numbers along with the units they count at the top of the part of the paper where the work’s going to be done. Don’t just write $\frac{1}{2}$ and $\fract{1}{3}$, though. Label them. Write Ry and Zoey beside the fractions that go with them. Write “of a pizza” after them. Think of it in the same way as jotting down notes as someone yammers at you, dropping the important bits of information in the middle of the conversation and expecting you to remember them later. What will you need to write beside those bits of information to remember what they represent and how they fit into the story?
How to check the answer
Sometimes we can do a reverse operation to check our answer. In our example, we could add Zoey’s $\frac{1}{3}$ of a pizza plus the difference we find to make sure that they total to Ry’s $\frac{1}{2}$. Check it with modeling and cutting wedges (pieces) from the circles if that helps. Most often, though a quick check for “is this reasonable” will work. If you did the estimations when you were figuring out the problem, is the answer close to the one you estimated? Is it far away from the wild guesses you knew were wrong? Is it a number that makes sense and doesn’t feel ridiculous in the context? Does it fit in your answer sentence without seeming completely out of place?
If you want more ways to check the reasonableness of your answer, see my post 10 Ways to Check for Reasonableness
How to put it all together
If you drafted your answer sentence at the beginning, all you have to do now is to fill in that blank!
You’ve Got This!
