Tag Archives: reasoning

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.]

Exploring MTBoS: Mission #1

Starting this week I’m taking off on an 8-week adventure Exploring the MathTwitterBlogosphere (Explore MTBoS for short). I’ve been loosely connected to the MTBoS since last August when Dan Meyer encouraged educators to start blogging. Like many people, I went all in for a while, but then life got in the way, and I haven’t really maintained my blog so much lately. Thanks to the Explore MTBoS program, I will at least be blogging and making connections for the next eight weeks, and perhaps it will give me the motivation to keep it going after the eight weeks are up.

Mission #1

We had to choose from two prompts. I chose:

What is one thing that happens in your classroom that makes it distinctly yours? It can be something you do that is unique in your school…It can be something more amorphous…However you want to interpret the question! Whatever!

For whoever happens to read my blog for the next part of this mission, I’m actually out of the classroom currently. I was an elementary school teacher for 8 years, and for the past four years I’ve been a math curriculum developer. However, just because I’m out of the classroom doesn’t mean my memory has gone foggy or anything.

With regards to math education in particular, what made my classroom distinctly mine, even though I got the idea from a co-teacher, was Problem of the Day (or P.O.D. as my kids liked to call it). As the name implies, the students were presented a new problem at the beginning of every math class.

At the time, I had a specific goal for doing Problem of the Day. The high stakes test in Texas, the TAKS test (which is now the STAAR), had six objectives and the sixth objective was called “Mathematical Processes and Tools”. It was a doozy of an objective because it wasn’t really about any particular math concepts. Rather it was about asking students a variety of questions that required problem solving and reasoning. Supposedly having good teaching methods while teaching the core content was enough to prepare students for Objective 6, but after many years in the classroom I knew that my students could easily be thrown for a loop by those questions. So during Problem of the Day I often used Objective 6 questions from released TAKS tests.

(As an aside: Looking back, I’m not proud that I focused on doing this for test prep. I am not a fan of high stakes tests, but the reality at the time is that it was my responsibility to prepare my students and this is the method I chose to try. As it turns out, it worked out amazingly well, and I see now that I could use Problem of the Day, or a related structure, to actually enhance my general math teaching.)

So as I said, I presented a new problem every day. Our school used a problem solving structure called FQSR (Facts, Question, Solve, Reflect). My students would divide their paper into a grid and label each section F, Q, S, or R to represent their work in that section. The first thing they had to do after they read the problem was to write down whatever facts they felt would help them solve the problem. Then they had to write the question they were being asked. (This actually made for some great conversation and also gave me some wonderful insights into how students comprehended what they were reading.) Next, they had to solve the problem in whatever way made sense to them. Finally, they had to write a response (reflection) that explained why they did what they did and what their answer to the question was.

When they were done, they would bring it up to me to read over their work. I wouldn’t tell them if they were correct or incorrect. Rather, I would ask them questions or point out where I was confused while looking at their work. The student would go sit down and use my questioning to continue working on their solution. Sometimes they would start over, sometimes they would elaborate more in their reflection, whatever they felt they needed to do. If I got a line of students waiting to see me, it was their job to share their work with each other in line while I continued reviewing work. Sometimes students would come up and see me 3, 4, or even 5 times to continue getting feedback on their solution. All the while, I never verified whether their answer was correct.

After it seemed like most of the class was ready to continue, we moved to the presentation phase where students got up and shared their solution with the class. They stood up at the front and shared their work using our document camera. I stood in the back to make it clear that I wasn’t running the show. I let students ask the presenter questions to clarify. I would also ask questions to clarify. Usually we made it through 2-3 students before having a discussion about whether we could all agree on an answer. By this point students were generally in agreement (for good or ill), and I would finally give the answer.

When first starting P.O.D., I knew my students were going to be weak at showing their work and even weaker at writing their reflections. For the first few weeks, I would choose one of the students and I would model the solution and reflection sections based on their work. They would tell me what they did and I would talk about how I would show/write that on my paper. I did this for much longer than a teacher would normally feel comfortable, but I can tell you that it paid off big time. My students’ responses got better and better because they had worked with me to model what it means to write about math thinking. They understood the value of telling what nouns actually go with the quantities they were computing with, for example.

You’d think this would be a boring activity because I forced a structure on them day in and day out, but my kids loved it. Maybe it’s because of the classroom culture I fostered, maybe I had weird kids, or maybe it’s because I wasn’t the voice of authority. Sure, I would give feedback as they worked, but so did other students. Sure, I asked questions during someone’s presentation, but I was always in the back of the room, not in a place of control. Also, I didn’t ask as many questions as my students did. I was “with” them, not “above” them.

While my students learned a lot from doing P.O.D., it was a valuable experience for me as well. I learned that word problems can be much trickier than you’d think. Here are two examples. (I’m making up the wording, but the essence of the problems is the same.)

1. Matt baked 24 cookies. He ate 5 and his sister ate 6. How many cookies did they eat?

I kid you not, every year I’ve presented a problem with similar wording, my students invariably subtract to find the answer. Generally they do 24 – 5 – 6 to get 13. I’m sure you can guess why: Because cookies were eaten, and that just means the amount is going to go down. It just has to.

I LOVE talking about this problem with students during P.O.D.. (This actually isn’t an objective 6 TAKS question. I just snuck it in every year because I knew it would trip them up and lead to great discussion.) Even after talking about the problem with students, and finally getting a few of them to recognize their error in comprehending the question, I still have students after a good 15-20 minute discussion still unclear why the answer is 11. And I’m okay that not all of them get it by the end. Doing P.O.D. is about the process of learning to comprehend, reason through, and solve problems. I can take a loss here and there for the greater victory of developing strong problem solvers over time.

2. Jamal is going to the movies. He buys popcorn for $2.65 and a soda for $3.25. What information is needed to determine how much change Jamal received?

This is another problem that I love because it shows me very clearly that students can read words and completely ignore them. It also shows me that they make a lot of assumptions. Finally, it makes it clear why there is a step in FQSR where you identify the question – because it’s not always what you think it’s going to be! I was floored at how many of my students had temporary blindness when they got to “What information is needed to determine…” Once they got to “…how much change Jamal received?”, all of a sudden their sight returned and they started doing some computations with numbers. If you’re like me, you’re probably wondering how it didn’t occur to them that they had absolutely no idea how much money Jamal handed the cashier, but that did not phase a class of 22 fourth graders one bit. They happily presented me their solutions to the problem. It wasn’t until the class discussion that finally the idea was raised that a student wasn’t actually sure how much money Jamal had. I said that’s an interesting point and decided we should reread the problem together to see if we missed something. As we read “What information is needed to determine…” I stopped and asked my students what those words meant. Finally it dawned on them what they were being asked to do. It was a wonderful a-ha moment for them.

If you’re with me until now, thanks for taking the time to read all of this. While blog posts are encouraged to be on the concise side, I have lots to say, and saying it gets me excited and reinvigorates me.

Sure, in the end I did P.O.D. for test prep, and sure it turned out to be super effective with regards to my students’ scores on the objective 6 questions that year, but it turned out to be about so much more than that. It was about empowering students and helping them become the mathematical thinkers I wanted them to be all along. It gave me practice serving more as a coach and resource rather than as the voice of authority in my classroom, and it taught me a lot about how my students reasoned about solving problems. Now, if only I could have been on a TEAM of teachers that did roughly the same thing I wouldn’t have to be sharing it now as something I’m proud of that made my class distinctly mine.