# Numberless Word Problems

“They just add all the numbers. It doesn’t matter what the problem says.”

This is what a third grade teacher told my co-worker Regina Payne while she was visiting her classroom as an instructional coach. She didn’t really believe that the kids would do that, so she had the class come sit on the carpet and gave them a word problem. Sure enough, kids immediately pulled numbers out of the problem and started adding.

She thought to herself, “Oh no. I have to do something to get these kids to think about the situation.”

She brainstormed for a few moments, opened up Powerpoint, and typed the following:

Some girls entered a school art competition. Fewer boys than girls entered the competition.

She projected her screen and asked, “What math do you see in this problem?”

Pregnant pause.

“There isn’t any math. There aren’t any numbers.”

She smiles. “Sure there’s math here. Read it again and think about it.”

Finally a kid exclaims, “Oh! There are some girls. That means it’s an amount!”

“And there were some boys, too. Fewer boys than girls,” another child adds.

“What do you think fewer boys than girls means?” she asks.

“There were less boys than girls,” one of the students responds.

“Ok, so what do we know already?”

“There were some girls and boys, and the number of boys is less than the number of girls.”

“Look at that,” she points out, “All that math reasoning and there aren’t even any numbers in the problem. How many boys and girls could have entered into the competition?”

At this point the students start tossing out estimates, but the best part is that their estimates are based on the mathematical relationship in the problem. If a student suggested 50 girls, then the class knew the number of boys had to be an amount less than 50. If a student suggested 25 girls, then the number of boys drops to an amount less than 25.

When it seems like the students are ready, she makes a new slide that says:

135 girls entered a school art competition. Fewer boys than girls entered the competition.

Acting very curious, she asks, “Hmm, does this change what we know at all?”

A student points out, “We know how many girls there are now. 135 girls were in the competition.”

“So what does that tell us?”

Another student responds, “Now that we know how many girls there are, we know that the number of boys is less than 135.”

This is where the class starts a lively debate about how many boys there could be. At first the class thinks it could be any number from 0 up to 134. But then some students start saying that it can’t be 0 because that would mean no boys entered the competition. Since it says fewer boys than girls, they take that to mean that at least 1 boy entered the competition. This is when another student points out that actually the number needs to be at least 2 because it says boys and that is a plural noun.

Stop for a moment. Look at all this great conversation and math reasoning from a class that moments before was mindlessly adding all the numbers they could find in a word problem?

Once the class finishes their debate about the possible range for the number of boys, my co-worker shows them a slide that says:

135 girls entered a school art competition. Fifteen fewer boys than girls entered the competition.

“What new information do you see? How does it change your understanding of the situation?”

“Now we know something about the boys,” one of the students replies.

“Yeah, we know there are 15 boys,” says another.

“No, there are 15 fewer, not 15.”

Another debate begins. Some students see 15 and immediately go blind regarding the word fewer. It takes some back and forth for the students to convince each other that 15 fewer means that the number of boys is not actually 15 but a number that is 15 less than the number of girls, 135.

To give you a heads up, after presenting to this one class she ended up repeating this experience in numerous classrooms across our district. After sharing it with hundreds of students, only one student out of all of them ever guessed the question she actually asked.

Do you think you know what it is? Can you guess what the students thought it would be?

I’ll give you a moment, just in case.

So all but one student across the district guessed, “How many boys entered the art competition?”

That of course is the obvious question, so instead she asked, “How many children entered the art competition?”

Young minds, completely blown.

At first there were cries of her being unfair, but then they quickly got back on track figuring out the answer using their thorough understanding of the situation.

And that is how my co-worker got our district to start using what she dubbed Numberless Word Problems – a scaffolded approach to presenting word problems that gets kids thinking before they ever have numbers or a question to act on.

Recently we shared this strategy with our district interventionists and several of them went off and tried it that week. They wrote back sharing stories of how excited and engaged their students were in solving problems that would have seemed too difficult otherwise. This seems like a great activity structure for struggling students because it starts off in a nonthreatening way – no numbers, how ’bout that? – and lets them build confidence before they ever have to solve anything.

Do I think that every word problem should be presented this way? No. But I do think this is a great way to prompt rich discussion and get students to notice and grapple with the relationships in problem situations and to observe how the language helps us understand those relationships. To me this is a scaffold that can help get students to attend to information and language. As many teachers like to say, standardized tests are as much reading tests as they are math tests.

Perhaps you can use this activity structure when students are seeing a new problem type for the first time and then fade away from using it over time. Or maybe you have students who have been doing great understanding word problems, but lately they’re rushing through them and making careless errors. This might be an opportunity to use this structure to slow them down and get them thinking again.

Either way, if you do try this out, I’d love to hear how it went.

[UPDATE 2: I’ve created a page on my blog devoted to numberless word problems. Check it out for more resources.]

## 43 thoughts on “Numberless Word Problems”

1. howardat58

Brilliant !
I think that the same general problem exists with the introduction of algebra and “variables”. How many kids would agree, or even recognize, that
is “algebra” ?

1. bstockus Post author

Well obviously if you learned the area formula in elementary school, that couldn’t possibly be algebra. 😉

I do like how the conversations this elicits get kids talking about ideas they may not have considered before.

1. bstockus Post author

Thanks! Glad you like it. It’s funny that you mention wordless word problems because my co-worker and I have a habit of calling them wordless problems instead of numberless word problems. I’m curious, what is an example of an actual wordless word problem?

2. Ashli

Really like the conversations that can get started here with the open, numberless situation. I used something similar in my high school classes and it often led to enlightening conversations with regards to how the students thought about math. Happy to see the strategy scales well!

3. Laura

Loved this post!! I hear so many times as a math coach that the kids “just don’t get it”. This is a great way to scaffold THINKING about a problem before actually solving it!!

4. Kari

I have been using this strategy with my intervention kiddos, and it sure has worked wonders! My second graders are notorious for adding the numbers together, no matter what the words say, so getting them to focus on the ‘story’ of the problem has helped reset their thinking.

1. bstockus Post author

Yay! Glad to hear it’s beneficial for your students. Is this a strategy used by general classroom teachers on your campus as well, or is it just you using them in intervention?

5. Tina C.

Hello! This post was recommended for MTBoS 2015: a collection of people’s favorite blog posts of the year. We would like to publish an edited volume of the posts and use the money raised toward a scholarship for TMC. Please let us know by responding via email to tina.cardone1@gmail.com whether or not you grant us permission to include your post. Thank you, Tina and Lani.

6. Pelayom6

This is a great post. I really like how one numberless word question got the students to really think on what the question was aking. From there, students started plugging in numbers to see if it could be solved and if it would make sense!

1. bstockus Post author

Yeah, the fascinating thing to me is how much reasoning and thinking the students did about one problem. It’s a great example of breadth vs depth. The teacher could have given a worksheet full of these types of problems, but my bet is the students would got much more out of this one problem.

1. Pelayom6

Probably so….im student in college and a future teacher. This definitely is something to keep in mind. Although, I plan on teaching the lower grades, I was wondering, can you still use this kind of numberless word problems with them? Or would it be too hard for them to process it?

7. Dee

Reblogged this on Life in a Post Harlem Renaissance and commented:
A brilliant way of checking for understanding. Often times the children are eager to jump right in the algorithm for find the “right” answer when all I want them to do is take the time to think about the problem and how to solve it.

8. Cay

I love the idea of taking the numbers out! I’ve always had my students think of the word problems as stories first and identify the character(s), setting, and problem – then focus on the numbers. With numberless problems I can scaffold this process and develop more open-ended and student-generated questions – leading to more engagement and deep thinking. Thanks!

1. bstockus Post author

This makes my mathematical heart happy. I love that you have already been connecting these types of problems to stories. That’s a great step to ensure students are making sense of the problem rather than just looking to compute whatever they can find. Best of luck as you try them out! You’re welcome to report back here how it goes. I’d love to hear about your experiences.

9. Tim Erickson

Lovely! I have also seen a terrific teacher, years ago, do this, kind of inside out: “Okay, we’re writing an article. The headline is, ‘6 x 8 = 48.’ What’s the story?”

10. lovemathweb

Amazing!! I think I will introduce each new concept this way, I love the idea. Thank you for sharing. I will share it with my school as well.

11. Josh Rosen

Thanks for this post-
I like the idea of giving students some information and then asking them to come up with the possible questions that could follow from that information.

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