Good Problems Based On Important Math: An Interview With Patrick Vennebush

Patrick Vennebush

My other book, One Hundred Problems Involving the Number 100, was published by NCTM in 2020. The idea behind that book was to provide interesting problems for students. I felt like there was a dearth of these problems in classrooms, so I wanted to provide ones that teachers could use that would be accessible and intriguing for students. For each problem, there is information about what a teacher might do ahead of time to set it up or how a teacher might use the problem at the beginning of a lesson and then build from it.

MARIAN: In the [One Hundred Problems] book, you talk about the characteristics of good problems. You’ve included things that would not surprise people, such as they’re intriguing, they’re low floor/high ceiling, and they invite collaboration. But one thing you say that is particularly interesting to me is that the problems should be “based on important math.” What do you mean by that?

PATRICK: There are a lot of important math topics we want students to learn as they move through their education, and there are a number that are just sort of “there.”

I remember in fifth grade, I did a problem that would not be considered appropriate now. It was about a person who found butts of cigars, and he could take 5 butts and put them together to make a new cigar and then smoke that one. He had 25 cigar butts to start. How many new cigars could he make? The idea was that those 25 butts would make 5 cigars, but then once he smoked those, the 5 remaining butts would make 1 more, so he could actually get 6 cigars from the 25 butts. There was something in there about exponential growth, but that wasn’t the way we perceived it. It was just a puzzle that we were given.

I like to find things that involve important math, bring out that important math for students, and really push those kinds of ideas. Sudoku puzzles are big right now, and I get frustrated when people tell me, “I love doing sudoku. It’s my math fix.” And I’m like, “Well, it’s not really a math problem; it’s a logic problem.” You could do it with letters instead of numbers, and it wouldn’t be any different, whereas a KenKen^® problem actually involves operations and thinking about number combinations. Some might question whether KenKen problems represent important mathematics. I think they do, unless you’ve done 200 of them and understand the idea behind them, and then you can move on and do something else.

PATRICK: I think a problem has to be something where you don’t immediately know how to do it. Otherwise, it’s just an exercise. That’s pretty consistent with Common Core literature as well.

A problem can allow students to think about some important mathematics on a deep level, and it can set the stage for a lesson. What I hear a lot is, “I couldn’t give my kids that problem because we haven’t learned about _____.” Even with the cigar problem I mentioned, teachers would say, “Well, I’d have to teach them about exponential growth or exponential functions before they could attempt to do that.” And I’m like, “Well, I did it in fifth grade, and I didn’t know anything about that.”

So, those are the kinds of problems that I like to do with teachers. I’ll give them something to think about, and hopefully it’s meaty enough and interesting enough that they sink their teeth into it. How would they then use that to set up a lesson? My hope is that it would work in the same way in the classroom—we’ve got this big, meaty problem, and we’re going to pull the important mathematical ideas out of that.

PATRICK: We have a curriculum that covers all the Common Core, and teachers tell me on a regular basis that they can’t possibly get through all of it. And I think it’s true. My argument is that it’s better for kids if they have a chance to think deeply about particular topics or math problems rather than racing through. If you get through 50 percent of the curriculum and kids retain 80 percent because it’s done well, you’re ahead compared to if you get through the whole curriculum and kids retain only 20 percent of it. I think spending that extra time allows kids to think about it more deeply, with a higher level of retention, and there will potentially be less review the next year, or less reteaching, which saves time in the long run.

MARIAN: One of the things that’s been interesting for me is that there’s a lot more collaboration allowed in math classrooms than there used to be—even at the high school level. But, for the most part, we don’t believe in collaboration on assessment, only on learning.

In Quebec, even in the elementary grades, they work on what they call situational problems, which are elaborate scenarios where there’s a lot of information, and a lot of math can be pulled out of these scenarios. And because of the nature of situational problems, students doing provincial tests are allowed to talk to one another for a while before they start their work. Once the discussion part is over, students finish the work individually. But they work together at the beginning.

I think it’s a smart practice because that’s how adults work. If you ask me to do something and I’m not sure what to do, I’d chat with you for a while and think about it, and then I’d go away and do the work. Except for this one case in Quebec, we haven’t accepted that in assessment. Do you think that collaboration in assessment could be something that could change things for people?

PATRICK: I do. But I worry about how teachers and administrators would feel about kids working together and how they would deal with individualized assessment scores. But it sounds like, in this particular situation, they’ve figured out a way to overcome that. I think it’s a great idea.

One of the things that we promote with Bridges in Mathematics is this idea of observational assessment. When kids are solving problems together and playing games together, we say to the teacher, “Go talk to them while they’re doing it, and use that as part of the assessment information about what students know.” When they do that, they’re seeing them in the moment; it’s not just something pulled out of mid-air.

The other way that I’ve seen assessment done, which is still individualized and is the same idea, is you give students a situation without a problem attached to it and tell them to write down everything they know about the situation. Then have them turn over the sheet, and on the back, it might say, for example, “If you haven’t yet found the area of the rectangle, go ahead and do it now.” You get kids to show you everything they know before asking them to show you one particular thing.

MARIAN: When you were creating the problems for your book about 100, or when you’re creating tasks in your other work, what is top of mind? Is it that it’s a real-life problem? Is it that it’s a thinking problem? What are those top-of-mind things?

PATRICK: The first thing is always if kids will find it interesting enough to want to engage in it. Another thing is whether a kid can read it. I’m also interested in how it is presented as a real-life context, because I find that real-life problems can sometimes be harder for kids than if the problems don’t have a real-life context.

I love context, but I want to make sure it’s a kid context. I love mortgage problems, but that’s because I’m 51. So, 16-year-olds might care about car insurance and car payments, but that’s about as far as it goes in terms of adult things. If I teach a kid to think, I think that’s far more important than them knowing about mortgages. I consider if the context is interesting. Is there a little twist to it? Does it bring out important mathematics that I want kids to be able to see in a positive way? I also like to make sure it’s not full of a lot of jargon.

There’s a problem that I like to do with kids where I ask them to find the product value of their name. So, a = 1, b = 2, and so on, and they multiply their letters. Then I ask if they can find another word that has the same product value as their name. It’s all about the factors of the numbers and things like that. I like when kids are not thinking about factors at first, and then it slowly appears to them—there’s something exciting about that.

The product value of “Marian” is 29 484. In this problem, Patrick asks students to find a word with the same product value as their own name.

MARIAN: It’s interesting to me that you mentioned jargon, because vocabulary is also something about which people have different opinions—whether it’s important that kids use precise words or whether you should be using precise words in the problems. Are you saying that to make a problem inviting, you almost have to stay away from those uncomfortable words?

PATRICK: A problem is not just a problem; it’s a launch to something. With the earlier problem about factors, I’m hopeful that after kids have a chance to think about it, there would be some discussion about it, and the word “factor” would come out naturally. When it comes out, we’d say, “We call this a factor, but what does that feel like to you?” And students would come up with their own definitions.

MARIAN: One thing that teachers worry about is that some students are way behind other students in the same classroom. Do you feel it’s important that teachers find ways to ensure that everyone’s involved?

PATRICK: I have a lot of empathy for teachers. Dealing with that is really hard. I think there are a number of things teachers can do: sometimes it is different tasks and potentially different groups, but there’s also collaboration—having kids work together, like you talked about with assessment. Students can all get into the problem together, and then there could be some individual time for work.

At the end of the day, we’re responsible for all kids learning a whole lot of math, so I don’t want to continually differentiate to the point that some kids never get the important math that they need. That’s a big fear of mine, especially when I think about equity issues and some of the research that has come out about teachers’ beliefs about different demographic groups and what kids are able to do and what they’re not.

My first job was in a rural area in Pennsylvania, and the demographics from a race standpoint were pretty homogeneous, but there were differences in terms of socioeconomic status and parental involvement and things like that, so there were often biases in that way. Some teachers were like, well, that kid’s not going to be able to handle that, so I’ll give them a different, easier problem. But that’s going the wrong way. So, I think we do need to help teachers. I don’t have all the answers. It’s a tough one, for sure. But my hope is that maybe there’s a rich problem that kids talk about for a little bit, as you were saying, and they have enough of the background about it that they can then start to think about the ideas within it. Even if they don’t get all the way to closure in a full answer, that’s okay.

MARIAN: Thank you very much for your time, Patrick. You’ve given teachers a lot to think about in terms of good problems and launching students to a place of deeper thinking.

PATRICK: Thanks, Marian.