# 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.]

# Talking Up Talking Points

I have been talking up Talking Points ever since I got home from Twitter Math Camp in July. Don’t know what Talking Points are? No problem. You can learn more about them on the Twitter Math Camp wiki. We had a group led by @cheesemonkeysf who dove deeply into this topic and shared their work with the rest of us. When you have a chance, I suggest reading the document titled About Talking Points. In the meantime, here’s a brief summary:

Talking Points are simply a list of thoughts. They are statements which can be factually accurate, contentious, or downright wrong. They can be thought-provoking, interesting, irritating, amusing, smart, simple, brief or wordy.

That definition aside, it’s not so important to know what Talking Points are as it is to know what to do with them and why.

Often children (and even adults) are asked to discuss their ideas or work together in groups, but it quickly becomes apparent that those involved don’t really know what is being asked of them.

You have those participants who love to talk, but they don’t really know how to consider other people’s views. On the other hand, you have participants who are quiet. They find it hard to join in the conversation.

The role of the teacher is to make explicit the kind of talk that is useful, and that is where the Talking Points activity comes in.

The activity stimulates speaking, listening, thinking, and learning. It offers ways in to thinking more deeply about the subject under discussion. It gives everyone a chance to say what is on their mind, so that others can decide whether they agree or disagree.

If you want to try out the activity, you need to create small groups of about 4 people per group. Give each group a list of Talking Points. (You can write your own, or you can start by using some of the ones on the Twitter Math Camp wiki.) The group will engage in 3 rounds per Talking Point.

Round 1

• One person reads a Talking Point
• Go around the group. Each person says whether they AGREE, DISAGREE, or are UNSURE about the statement AND WHY.
• The most important part of this is that there is NO COMMENT by anyone else in the group. Their job is to listen.
• After everyone has had their turn, proceed to round 2.

Round 2

• Go around the group again. Each person says whether they AGREE, DISAGREE, or are UNSURE about their own original statement OR about someone else’s statement they just heard AND SAY WHY.
• Again, there should be NO COMMENT from anyone else in the group while someone is speaking.
• After everyone has had their turn, proceed to round 3.

Round 3

• Go around the group one final time. Each person simply states whether they AGREE, DISAGREE, or are UNSURE about the original statement. The group takes a tally and moves on to the next Talking Point.

When time is up, the facilitator can choose whether to have the participants complete a group self-assessment. (Check out the wiki for an example.) This gives them a chance to reflect on the discussion and how their group worked together.

The facilitator can also do a whole group debrief. I highly recommend doing this because it helps the participants reflect together, and it also gives the facilitator a chance to point out and emphasize to everyone behaviors and ways of talking that were effective. Here are sample questions that could be discussed:

• Who in your group asked a helpful questions and what was it?
• Who in your group changed their mind about a Talking Point? How did that occur?
• Who in your group encouraged someone else? How did that benefit the conversation?
• Who in your group provided an interesting additional idea and what was it?
• What did your group disagree about and why?

You might be thinking to yourself, “Oh this activity is just like ___.” I’ve had several people tell me that, but once they have participated in their first Talking Points activity, they see how much deeper the conversation is than in the other activities they were thinking of.

The first time I tried this activity was with our district interventionists. This school year we are leading PD sessions with them about 10 times spread across the year, and one of the recurring themes is knowing their learners in general and using strategies to foster a growth mindset in particular. Before we dove into this topic, we created some Talking Points to get them talking with one another about their beliefs about intelligence and learning. Here is the set we used:

Talking Points About Intelligence and Personal Qualities

1. Your intelligence is something very basic about you that you can’t change very much.
2. You can learn new things, but you can’t really change how intelligent you are.
3. No matter how much intelligence you have, you can always change it quite a bit.
4. You can substantially change how intelligent you are.
5. You are a certain kind of person, and there is not much that can be done to really change it.
6. No matter what kind of person you are, you can always change it substantially.
7. You can do things differently, but the important parts of who you are can’t really be changed.
8. You can always change basic things about the kind of person you are.

The Talking Points were successful in getting the interventionists to start talking about ideas of growth mindset without us have to tell them anything. Many of them already felt strongly that intelligence can change (which made me happy considering the population they serve!) but I appreciated the discussion in a few of the groups where one person was able to point out some nuance in the language that made the others in the group consider the statement more carefully.

For example, in one group someone raised the point that she has an uncle with Down syndrome, and over the course of his life she doesn’t feel that his intelligence has really changed all that much. This personal experience made it difficult for her to fully agree with the Talking Points statements.

When the groups were finished discussing their Talking Points, we debriefed as a whole group. One of the recurring comments they made was how nice it was to have a chance to give their opinion and feel like it was heard by the rest of the group. Because of the “No Comment” rule, they knew that no one was going to interrupt them, and if someone tried, the other group members quickly reminded them of the rule.

A few weeks later, I was asked to share Talking Points with our district leaders at a Vertical Leadership Team meeting. This meeting consisted of all of the principals from all of our elementary, middle, and high schools as well as many other district leaders, about 120 folks in total. No pressure! I was nervous about what they’d think, but happy to have the forum to share the activity.

As with the interventionists, I was using the activity to lead in to a discussion of growth mindset research, but this time I revised the statements because I wanted them to each feel unique. In the first set I used, I felt like several statements said the same thing but in different ways. Here are the statements I finally settled on:

Talking Points About Learning and Intelligence

1. You can learn new things but you can’t really change how intelligent you are.
2. You are a certain kind of person, and there is not much that can be done to really change it.
3. When you are learning something new, you should avoid making mistakes at all costs.
4. You are smart when you can complete tasks quickly and accurately.
5. The people who are the best in their field tend to be naturally good at what they do.

I only had 20 minutes to introduce Talking Points, share how to do them, have everyone in the room try them, and then debrief. It was a rush to get it all done, but it went smoothly enough. I definitely think 30-35 minutes is probably better for a first introduction.

I love walking around and listening in on different discussions while the groups are going through the Talking Points. While listening to the district leaders, I loved hearing one woman say, “Well, I did agree with the statement, but now I disagree because of what you two just told me. I just didn’t know that before.” It’s not required that anyone change anyone else’s mind during Talking Points, but it sure is powerful that it has that ability to happen based on just three short rounds of sharing opinions.

Someone in my department used Talking Points just over a week ago with a team of instructional coaches. While each grade level has some common ELA vocabulary that is used, the way it is used has not always been consistent across grades. Before the coaches started preparing for upcoming PD sessions, she had them go through some Talking Points about the vocabulary. She said it was such a quick and powerful way to gain clarity as a group before they started their planning.

Ever since I’ve shared Talking Points, it has started to slowly spread in my district, and I’m excited to see the creative ways it is used to deepen conversations among both students and staff.