2. How do they work?

Numberless Word Problems fundamentally change the way students have to engage with word problems. My favorite story to demonstrate this is the story of how they originated in my previous school district:

“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. Regina didn’t really believe that the kids would just add all the numbers, so she had the class come sit on the carpet and gave them a word problem. Sure enough, without considering the situation at all, 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.

Finally, one of the students shared, “There isn’t any math. There aren’t any numbers.”

Regina 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 dropped 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 which 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. How impressive is that?

Once the class finishes their debate about the possible range for the number of boys, Regina 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.

A child challenges this statement with, “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 135, number of girls.

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 Regina 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 planned to ask.

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.

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

That of course is the obvious question, so instead Regina 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 Regina Payne got our district to start using what she later dubbed Numberless Word Problems.

Did you notice?

  • Regina not only took the numbers out of the problem, but she also removed whole sentences so students only had to make sense of a part of the story at a time.
  • Regina asked purposeful questions such as, “What math do you see in this problem?”, “So what does that tell us?”, and “What new information do you see? How does it change your understanding of the situation?”
  • Students were invested in making sense of the situation as they reasoned about the quantities and relationships in the problem.
  • Students made thoughtful estimates and defended the reasonableness of their estimates.
  • Regina invited the students to predict what question would be asked given the situation. Saving the question for last meant that the students couldn’t give in to their “compulsion to calculate.” Until she posed a question, there was nothing to solve.

Taking out the numbers made the problem make more sense. Now it’s time to try out a Numberless Word Problem in your classroom!

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