Make Sense of Problems…
This issue we look at the first Common Core Mathematical Practice, “Make sense of problems and persevere in solving them.” What are some ways that teachers support their students as they make sense of a Problem of the Week?
Noticing and Wondering/Forget the Question
When teachers do PoWs in class, a very popular activity for getting students started at making sense of the problem is to display, hand out, or read out loud the problem scenario, without a specific math problem to be solved. Then we ask students to share what they Notice about the scenario, and what they Wonder.
With younger students, we often read the problem aloud and ask, “What did you hear?” first. Older students might read the problem as they listen, and then write down or tell a partner what they noticed and wondered.
We also sometimes do a whole group brainstorm or a class go-round so everyone can hear the noticings and wonderings of their classmates. Making a big public list of noticings and wonderings also helps students have something to come back to when they get stuck or want to make sure their answer is right.
You can find a print-friendly Scenario Only version of each Current PoW in the blue box with all the teacher resources, to make it easy to share the PoW without telling students the specific problem they’ll be solving. We like the Scenario Only version because not having a specific problem:
- helps keep some kids from blurting out an answer right away, and helps them stay focused on understanding
- helps students who are scared of getting wrong answers… anything they can notice or wonder is valued, and there’s no question to get wrong
- gives the students a chance to come up with a problem to solve, via their wonderings, which helps them be motivated to actually solve it
- shifts the focus from getting the answer and being done to coming up with as many math ideas as possible
To learn more about I Notice, I Wonder brainstorming, you can visit the Understand the Problem strategy page, read this story by Annie Fetter [PDF] about the first time she did I Notice, I Wonder with students, and check out these articles about noticing and wondering in the ComMuniCator: http://mathforum.org/pow/teacher/articles.html.
Change the Representation
Annie tells a story about doing some math with her niece, Olivia, who was in second grade at the time. Olivia was a very good reader, and Annie figured it was important for her to know that being good at reading can really help you do math. (Olivia’s parents are an anthropologist and a fine silver jewelry maker, so Annie feels like she always needs to make her nieces and nephew think about math!)
Annie asked her to think about this story: “A man has three bunches of roses. Each bunch has six roses.”
Olivia immediately said, “He’s got 18 roses.” Annie told Olivia that a lot of students think the man in the story has nine roses — to which Olivia exclaimed, “WHAT?!? Why would they think that?” Annie and Olivia talked about how she was good at understanding stories, and how that helped her also be good at math. Olivia thought that was pretty cool — she wouldn’t be tricked into adding the three and the six if she stopped to think about what was happening in the story. The conversation helped Olivia think about how she could use her strengths to connect to math, and it helped Annie think about the importance of supporting students to understand the story behind a math problem.
When we work with students who are stuck, our first question is often, “What’s going on in the story?” or “What is this story about?” That gives us a window into what the students do and don’t understand about the context.
Annie also wondered if Olivia, who likes to draw (though not as much as she likes to read), was able to visualize stories in her head as she read them. We’ve started to help students make visualizing and illustrating a part of their sense-making routine too. Especially for those discouraged students who leave their sense-making ability at the door of math class, drawing a picture can be a great way into a math story. Many of the same students who would answer “9″ for the bunches of roses problem would be able to draw an accurate picture of 18 roses if they are just asked to “draw a picture illustrate this story.”
If your students are having trouble connecting to the action in story problems, seeing what the problem is about, or connecting the problem to their own experience, you might try:
- Asking stuck students, “How would you illustrate this problem?”
- Dividing into groups of no more than 4 and each group coming up with a skit to act out the action in the problem.
- Inviting small groups to draw a picture to help them solve the problem, and then sharing out each picture before going on to solving the problem. Talk about what makes the pictures similar and different, and which would be most useful, and why.
Share and Compare
Lots of PoW submitters write and tell us, “When I first solved the problem I was thinking this, but then when we talked about it as a class, I realized I was wrong.” Sometimes it takes outside input to realize hidden assumptions!
For example, I just wrote the commentary to accompany student solutions to the PreAlgPoW “Totolospi.” It was a tricky probability problem and students had a range of answers. Several submitters wrote to say, “At first I thought there were only four possible outcomes, but after we talked as a class I realized there were eight.”
A great moment to have those class discussions is after students have had a chance to read the problem for themselves and start doing some initial work. Questions come up like, “What do they mean by ‘all the possible tosses?’” or “Wait, how many possibilities are there for each die?” or even “What’s a cane die?” Pausing the group early on for a “distributed summary” — a chance for students to quickly report on their current thoughts and questions — can help problem-solvers get on the same page before they commit to a lengthy process based on a wrong assumption.
Good questions to ask for a distributed summary early in problem solving are:
- Are there any questions that have come up about the story?
- Is there anything in your thinking that you’re not too sure about?
- How are you understanding the story?
- What’s one idea that you’re trying right now?
- Is there anything you’ve ruled out? An answer that’s definitely too high or too low?
- What’s your best estimate/hypothesis about the answer right now? How could you test that?
All of these strategies help students engage in sense-making, and avoid doing random calculations without thinking.
… And Persevere in Solving Them
The other key component of Mathematical Practice #1 is perseverance. How can we support students to persevere when they see math as too hard, too frustrating, or even too boring?
At the Math Forum, our focus in perseverance is on supporting students to revise their submissions to the PoWs — we’ve found over the years that that’s when most of the learning happens, and since our only contact with most PoW members is through mentoring student submissions and reading their revisions, well, that’s what we’ve gotten good at.
Here are some principles that work well at encouraging students to persevere and revise:
- Value what the student has done so far — there is nothing more encouraging than feeling heard! We always find at least one thing to say to show that we recognize the thinking the student has done and appreciate it.
- Remember to praise the things the student did well — there was a study once with two teams of bowlers who watched video of their bowling and got feedback. One team got all feedback about what they could do better (i.e. what they’d messed up and needed to change). The other team got all feedback about what they were already doing well. Guess which team improved most? The team that got only specific positive feedback. It’s counter-intuitive but true!
- Stick to one or two steps you want the student to take next. We’ve found that students only respond to at most two requests, and so if you don’t choose your top two, your students will choose for you… Sticking to one or two requests means you can prioritize!
- Baby steps! Trying to make every student an expert in one revision isn’t going to work and is frustrating. What’s one thing each student could do to make a little bit more progress?
- Consider what this student would find interesting? For example,
- instead of “How did you know the answer was eight?” (kind of a boring question) we might ask “Some students said the answer was four because RFF was the same as FRF (order doesn’t matter). How would you explain why you’re right and they’re wrong?” (fun because you get to argue!).
- instead of “Can you write a rule or equation?” we might ask “What would happen if there were 100 of them? 1000? 1,000,000? How could you figure it out?” (more fun because big numbers are fun and because the student has a reason to write a rule, they aren’t just doing it ’cause we told them too).
If you’d like to read more about helping students persevere, you might enjoy this article by Suzanne entitled “Supporting Sense-Making and Perseverance,” or the article by Math Forum Teacher Associate Marie Hogan entitled “Pairing Like Students for Persevering in Problem Solving.”
We hope that some of the strategies for sense-making help students get started, and that through feedback and multiple revisions we can help them persevere on the PoWs as well!