word problems

Last week, between work with grade-level bands, I had a complete K-6 staff for about an hour. I wanted to tackle “story problems,” so I started by showing them “How Old Is the Shepherd,“ by Robert Kaplinsky:

Seriously, take the 3:07 to watch it. You’ll see that 75% of 8th graders presented with the nonsensical problem, “There are 125 sheep and 5 dogs in the flock. How old is the shepherd?” answered with a number.

I had given my close friend and colleague, Debbie Nichols, a sneak peek at the video. She often helps me think things through, and is one of the teachers I am profiling heavily in my book. She decided to give the same problem to her 1st and 2nd graders to see what would happen. We both expected the younger kids to do better on the nonsense problem, mostly because they haven’t been trained to pluck-numbers-and-do-something-with-them the way the older kids have. In our experience, younger kids are great sense-makers.

Debbie changed the problem to read, “There are 25 sheep and 5 dogs in the flock. How old is the shepherd?” Partway through, she wondered about vocabulary, and added a second question, “There are 25 kids and 5 dogs in the classroom. How old is the painter?” Debbie interviewed her students individually, recorded it on video, and sent me her notes with the subject header WOW:

After individual questioning, Debbie brought the kids to the rug and asked the whole class, “There are 25 sheep and 5 dogs in the classroom. How old is the teacher?” Students quickly fell in line behind some opinionated students, and came to consensus that the teacher was 30 because 25 + 5 = 30.

Wow indeed. What’s especially striking to me is how many students admitted they didn’t understand the problem, and still gave a numeric answer anyway. Some of the nonsense answers even sound kind of right–like they are imitating the way a math answer should sound. “The answer is 8. I counted by 9s.” (As much as I love the brain made of pink worms, the counting by 9s answer is the most fascinating one to me.)

The following day, Debbie posted, “There are 4 kids and 3 chickens in the room. How old is Mrs. McCabe?” (Mrs. McCabe is another teacher in the building.) Students wrote their answers on the chart paper. Take a peek:

43, 7, 7, 40, 64, 70, 2, 2, 10, 9, 30, 36, 100, 34, 7, 44.

Nobody wrote, “I can’t tell because this problem makes no sense.”

43 and 34 come from students taking the 4 children and 3 chickens and using them as digits in a new number.

7 came from students adding the 4 children and the 3 chickens.

I suspect 70 comes from a student adding the 4 and 3, then deciding 7 was too young, so they made it 70.

40, 64, 30, 36, and 44 may come from students disregarding the information in the scenario, and just making a reasonable guess about Mrs. McCabe’s age (56). The 40 and the 30 might be similar to the 70, in that the numbers 4 and 3 were in the problem, and they were clearly too young, so 40 and 30 sounded reasonable. (In this group of answers, if students were thinking about reasonable ages for teachers, that’s something to build on. My next step would be to ask them for evidence in the question.)

2, 10, 9, and 100 remain mysteries to me.

I can see in the anchor chart that Debbie took out her purple marker and tried to help students recognize the lack of relationship between the number of chickens, the number of children, and Mrs. McCabe’s age. She had them try to come up with a question that would make sense from the chickens and children, like how many legs are there?

After reading Debbie’s notes, I wondered what my own children would do with this problem. I asked my 5-year old daughter, (who asked to be called D), “There were 4 children and 3 chickens in a room. How old is the teacher?”

“9.”

“Where’d the 9 come from?”

“No, wait, that’s not enough. 90.”

“Where’d the 9 come from and where’d the 90 come from?”

“Well, I added the 3 and the 4, and that made 9. But that’s not enough. Like, Maya is 7, and that’s 2 years less than 9, and she’s not old enough to be a teacher! So I made the 9 a 90.”

“Can you show me how you added 3 and 4?”

She counted on her fingers.

“Oops. It’s not 9. It’s 7. So she’s 70.”

I was quiet for a while, thinking about a question.

“We have 2 dogs and 1 fish, right?”

“Yeah.”

“So, we have 2 dogs and 1 fish in our house. How old is D?”

She laughed uproariously. “Mommy, that doesn’t make sense! The dogs don’t have anything to do with how old I am! 2 and 1 is 3, but I’m 5!”

“Really? OK. Let me try again, then. We have 2 dogs and 1 fish in our house. You are 5 years old. Let’s say we bought another fish. How old is D now?”

She was hysterical at this point. When she calmed down, I said, “OK, let’s go back to this question again. There were 4 children and 3 chickens in a room. How old is the teacher?”

“OH!”

I recognize that these nonsense word problems are contrived. I think they’re revealing, though. In particular, I think they show some problematic beliefs our students have about doing math:

1. All math problems have to be answered with numbers.
2. All math problems can be answered.
3. It’s normal for math not to make sense.

Where do these beliefs come from?

In the last week, as I’ve been mulling all this over, I’ve been revisiting some of the books I have about the intersection of reading comprehension and math, like Comprehending Math by Arthur Hyde, Mathwise by Arthur and Pamela Hyde, and From Reading to Math by Maggie Siena. This quote of Siena’s about the foundations of reading jumped out at me:

“Children must…expect the things they read to mean something and expect to be satisfied by that meaning” (17).

Do we teach the same expectation in math? Or do we teach students to answer every problem with a number, guessing if they must, and it’s OK if it doesn’t make sense?

I’m still scratching my head over where this message comes from with my kids and Debbie’s students. D has grown up during the writing of my book, when I am hyper-aware of the math messages I am sending and hearing. Debbie’s students had a great year of kindergarten with a teacher who emphasizes making sense, and now they’re with Debbie, who teaches math for understanding. And yet, 100% of them answered the nonsense question with a number.

As a fan of CGI, I know children are naturally sense-makers. But I also know that reading mathematical problems is a special kind of reading, and students need instruction in it. Historically, teachers have used two different types of instruction for reading word problems:

1. Teach students to “decode” math problems with “keywords,” like “in all means add.”
2. Teach students to recognize unnecessary information, red herrings, and traps that “they put in the problems to trick you.”

On keywords, some of us have been having fun over on Twitter, creating a list of problems that show why it’s a doomed strategy. For example, Tommy buys 3 bags of avocados. There are 4 avocados in each bag. How many avocados did Tommy buy in all? Hmm. I thought in all meant add?

As for focusing on the traps “they” are putting in the problem, I am no fan of this strategy either. What sort of message are we sending kids with this teaching? That there are rooms full of nasty adults, rubbing their hands together, trying to set traps that catch nice little children taking math tests? Ahem. Though there may be some truth to that image, I refuse to cede mathematics to the standardized-test and curriculum writers who write crappy, trappy math problems. I want to snatch math back, and teach students to see the beauty and usefulness of math around them, and to enjoy the journey through a perplexing, puzzling problem. So red herrings and tricks be damned!

What do we do instead? How do we teach students to read math problems for understanding in a way that will yield empowered students who expect to make sense? I’m looking for resources on this question, so please pass them along in the comments. In the meantime, let me share three of my favorite approaches. They all have something in common, which is that they are all strategies to make it impossible for students to leap right to answering the question. All three approaches force students to slow down and make sense, first.

1. The Math Forum at Drexel University is a fantastic group of people who are all about teaching students to make sense of math. One of the strategies they have been promoting is Notice and Wonder, where teachers share a scenario without a question, and ask students what they notice and wonder. You can read about it in Max Ray‘s Powerful Problem Solving, by following Annie Fetter, or at a whole bunch of sites here, here, here, here, here, here, and here.

2. Brian Stockus wrote a great blog called “Numberless Word Problems,” in which he described a co-worker removing the numbers from a word problem. Again, this strategy eliminates the option of racing to an answer, and introduces students to the idea that we can do quite a bit of mathematical thinking about quantities without knowing what they are, which Kate Nowak framed as the the rich idea at the heart of algebra:

3. In Mathwise, Art and Pamela Hyde wrote, “Getting students to slow down and think about a problem is not always easy, especially if they are used to calculating answers quickly to one-step translation problems. We have found that students can be encouraged to think through their assumptions with an intriguing type of problem called “Fermi questions'” (66). Fermi Questions are mathematical questions where answers seem impossible, but we can get close by making some assumptions and then approximating:

• How many piano tuners live in Chicago?
• How many kids could fit in the gym with no furniture inside?
• How many hairs are on your head?

From a teaching point of view, Fermi Questions can be fantastic for helping kids realize they are making assumptions and connections and using their prior knowledge in mathematics.

All three of these strategies–Notice and Wonder, Numberless Word Problems, and Fermi Questions–force students to slow down and make sense of the situation before worrying about the answer.

I’m hoping to learn more about high quality instructional strategies for math teachers that are rooted in what we know about teaching reading comprehension. Annie Fetter presented on this idea at NCTM in New Orleans, and I think it’s an idea with long, strong legs. The connection between making sense in literacy and math is something I talked about in the workshop last week, and it seemed to resonate with Shawna Coppola, a wonderful literacy specialist. I loved her notes:

Making sense is the thread that ties everything together, in every content area. Right? If our students arrive having already internalized the message that making sense isn’t part of math, or that math doesn’t make sense, or that word problems are just a bunch of numbers hidden in words and traps, we have our work cut out for us. Time for some intentional, creative, inquiry-based teaching that empowers students to make sense.