Help students make sense of a problem: See it in action

Making sense of a problem helps students develop a growth mindset by focusing them on their understanding rather than on just getting the correct answer. Often students learn a math rule without understanding why the rule makes sense. Making sense of a problem deepens students' understanding of the material, helps other students in the class see the problem from a different perspective, and encourages students to work through their misunderstandings and gaps of knowledge by making and learning from their mistakes.

Next, watch Cathy Humphries, a math teacher, help her students make sense of a problem and hear from Professor Jo Boaler.

Cathy Humphries: “Before class today, we're going to have a rule, and some people already know the rule, but I don't care about the rule right now. What I want to know is, can anybody make sense of this? ... So right now, we have two different answers. We have Answer 6, and Leslie's saying it's because you divide—there's three one-thirds in one, and then two times three is six. And then we have the other theory, and put your hands down. The other theory is the answer is 1-1/2, and we don't know why except that the rule works. So the rule working is not good enough today. So why does it make sense, why does Leslie's answer make sense? What does 1-1/2 make sense? I want to know why it makes sense. Excuse me. This is really interesting. The side of the room that I talked to, about half of the people thought it was six, and about half of the people thought it was 1-1/2. And then I found Sam willing to make an argument for 1-1/2. So Sam would you go to the board and draw your picture. He's going to erase the other pictures so far.”

Sam: “Well, if you have this, and that's, it's not even, but if that's one whole, and if these, pretend these two parts are shaded, and that's two-thirds, right? And this one doesn't count now. So that's two-thirds, and this is the other one-third. And so if you draw—this is a whole. See, that's the whole right? And if you take these two and put them here, like it was, then you still have these, then you still have the one-third left, and one-third is half of two-thirds, and so you just add one-half.

Cathy Humphries: Now remember the day we talked about—convince yourself, convince a friend, and convince a skeptic. Anybody want to—let's see if anybody can challenge Sam? Push on his thinking a little bit?

Student 1: I'll challenge Sam.

Sam: I thought you were going to call on me.

Student 1: Okay, that's exactly what I was thinking, and then Michael convinced our table, we had a couple at our table here, and we thought it was 6. But then I'm pretty much convince that...

Cathy Humphries: Zach or Ben, are you willing to talk about what [Student 1] said that convinced you?

Student 2: Everything.

Student 3: I thought it was 1-1/2 in the beginning. He was the skeptic.

Cathy Humphries: You were the skeptic. This will be really great. What made you change your mind?

Student 2: It made more sense what he wrote on the board.

Cathy Humphries: Can you explain it now?

Student 2: I think so.

Cathy Humphries: Okay. Would you try please? If you can explain what somebody else said, then you're on notch further up in your ability to understand it.

Professor Jo Boaler, Mathematics Education Expert, Stanford University: So in the Cathy video we watched, we saw a task that could definitely be all about performance—one divided by two-thirds. If that was given to students as a question, definitely they could see that as one quick answer that they're meant to show. But Cathy transformed that task by asking students about the different ways they could make sense of one divided by two-thirds, and that opened it up, along with the culture of her class that's all about learning and exploring. That opening encouraged opportunities for a growth mindset.

So Cathy has a really strong focus in her classroom on a particular norm that's about encouraging students to be sense makers. Nothing goes in her classroom unless the students can make sense of it. So there are different things within the norm of sense making. She encourages different ways to make sense with different representations, encouraging students to draw and to visualize and to show things in different ways. She also develops a culture of collaboration and shared work. Students really make sense of mathematics when they talk about it and when they reason mathematically with each other.

Something else she does is she gives public space to the sense-making norms. So she holds whole class discussions. Now when teachers have those whole class discussions, it's a chance for them to really instantiate norms of what they think is important with all the students, what math is, and what's important in the class.