Tag Archives: problem solving

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 throw a final wrench in to the discussion, she asks, “So what question could I ask you about this situation?”

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 1: I wrote a follow up post about writing numberless word problems if you’d like to learn more.]

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

Kickoff! #ElemMathChat

Tonight we kicked off a new weekly Twitter chat, #ElemMathChat. Hooray! As the name implies, the chat is designed for elementary school folks to talk about math.


I’ve been so excited to get this chat started! For the past two years, I’ve been a member of the MathTwitterBlogoSphere whose membership is primarily composed of middle and high school teachers. There are a few of us elementary-minded folks. We have appreciated all of the interactions we’ve had with the MTBoS. However, after meeting up at this year’s Twitter Math Camp, we decided our mission this year is to grow the elementary-side of the MTBoS.

And so it begins.

Tonight’s chat was a huge success! We had a great turnout with educators from around the US and Canada. (Thanks for catching my mistake @ChrisHunter36!) Everyone seemed excited about having a forum to discuss elementary math specifically. One person even commented that she was happy to have a place where she could be taken seriously. She said she’s tired of being considered “cute” for teaching first grade.

Our topic for the first chat was balancing problem solving with teaching/covering math skills. If you want to catch up on the conversation, you can check out the Storify put together by @davidwees. While the overall conversation was energetic and interesting, I was left a tiny bit disappointed.

I think it’s because I was the one who suggested this topic. Balancing problem solving and covering math skills is something I have struggled with myself as a teacher, and now as a district curriculum specialist, I am hearing from numerous teachers who are struggling to find the same balance themselves. So going in, I had some clear ideas of what I wanted to talk about and get out of the discussion.

The first question was “How do you define problem solving in the elementary math class?” This generated some interesting discussion. Some key points that rose to the surface for me were that problem solving involves thinking critically, collaborating, and using math as a tool. I especially like the “math as a tool” metaphor because it gives meaning to why we’re learning it in the first place. I think it’s often an implied message, but one educators need to try to make more explicit. I also liked how people described problem solving as a time to make kids get out of their comfort zones and make their brains sweat. I love the image that conjures in my mind.

The interesting thing that came out of this first question is that everyone seems to have different ideas about what problem solving is. Some people talked about it in a way that sounded like solving word problems, whereas others referred to rich and engaging tasks that focus more on the process than the endpoint. This is one area where Twitter chats can frustrate me. The conversation is happening so fast with so many people talking simultaneously that it can be challenging to pull the threads together into a coherent whole.

Maybe that’s what I need to learn how to do as a moderator. Instead of following my script of questions, I could have stopped and made question 2 be “So I’ve heard problem solving described as ___, ___, ___, and ___. What is one definition we can all agree on?” The conversation over the rest of the hour felt weaker because we didn’t necessarily have an agreed-upon definition to base our discussion on.

Question 2 also had some problems: “How do you define math skills?” This is where I had a clear idea of what I meant, but the majority of the group was on a different wavelength. Since we had just talked about problem solving, everyone seemed to think that I meant the Standards for Mathematical Practice or general thinking skills that are needed to solve problems. What I really meant, and I did try to clarify, are the nuts and bolts skills that teachers need to teach their kids: adding and subtracting whole numbers within 1,000, multiplying fractions with whole numbers, interpreting dot plots, and measuring angles, to name a few.

Here’s an example to illustrate the tension I was thinking about when suggesting this week’s topic. Learning a skill like long division takes time and effort. It is a very structured thing to do, but until students understand it, they are prone to making many errors. Can I do a few problem solving activities and have my kids somehow come away from the experiences as masters of long division?

You may be thinking right now, “But kids don’t actually have to know long division in order to solve problems. They just need a strategy that makes sense to them.”

I agree with you. However, in Texas and in Common Core, the standards do explicitly state that students learn to divide using the standard algorithm. So like it or not, it’s a skill that students are expected to learn.

Here’s where the tension comes in. Long division is just one skill. There are numerous other skills students are also expected to master in any given grade. How do you ensure the nuts and bolts mastery while at the same time providing ample opportunity for the types of activities that require critical thinking, collaboration, and brain sweating?

And please don’t take any of this the wrong way; I don’t fault anyone in the chat for not providing me a satisfying answer. To be honest, I don’t think a one hour Twitter chat is going to be the place to find concrete answers to big questions like this. It doesn’t mean I don’t want answers (hence the tiny bit of disappointment I felt), but I have learned over the past two years what Twitter can and can’t do.

What it can do is bring together like-minded people to fuel conversations and build relationships. The more I connect with people on Twitter, the more I get to know them. I can start chatting with them outside of our weekly chats. Perhaps I ask for help with a problem I’m having, or perhaps we set up a Google Hangout to have an actual conversation about a particular issue (good-bye 140 character limit!), or maybe we even collaborate on a proposal for a national conference.

Valuable professional relationships can grow from short, weekly conversations. It’s why I’m still here two years later, and it’s why I’m excited to get this specific chat launched. I’m eager to meet like-minded elementary folks and start forging some new professional relationships.