Helping Students Build Mathematical Relationships: An Interview With Pam Harris

In many issues of M Magazine, we share a feature that explores a guest math educator’s viewpoint on important issues in math instruction. In this issue, Marian has a conversation with Pam Harris about how to help students deepen their math understanding and build mathematical relationships and connections mentally. This interview has been edited for length and clarity.

The views and opinions expressed in this interview are those of Pam Harris and do not necessarily reflect the views or opinions of Rubicon, a Savvas Company or Marian Small.

Pam Harris

Pam Harris is a mom, a former high school math teacher, a university lecturer, and an author who shares her expertise all over the world through in-person and virtual workshops. Pam also hosts the Math is Figure-out-able podcast.

MARIAN: Can you tell us a little about yourself?

PAM: Although I’d already been teaching for several years, my frame of reference for the nature of mathematics started to shift when I had my son. We counted a lot as he grew. I couldn’t not bring math into it, even though at that point I knew very little about how young children develop math.

Despite that, my son thought and reasoned about math in ways I had never conceived of. One day in first grade, he came home with a paper on subtraction with regrouping. He had the correct answers, but there were no marks on his paper other than the answers. I said, “Are you cheating?” He said, “No, I didn’t understand what my teacher was doing, so I made up my own way.” From my perspective at that time, that wasn’t something you do. In math, you wait until the teacher tells you what to do, and then you mimic what the teacher did. I wasn’t sure what to make of his strategy, but since it worked, I let it slide.

In second grade, he came home with the same paper. I asked, “Did you tell your teacher how you subtract?” He said, “I couldn’t remember what I did last year, so I made up another way.”

Faced with a kid who had invented not one, but two algorithms of their own, I couldn’t help but be curious. “Fascinated” might be a better word. He was so intrigued and playful with relationships in a way I had never seen before. My high school students were compliant; they did what I told them to do, but they weren’t as interested or engaged. I wanted to help my high school students reason the way my son did. So, I read journal articles and research studies and dove into some of the curricula at that time. I went into my kids’ classrooms and said, “May I please experiment?”

I tried a lot of things to see if they would help students understand better. I had only ever mimicked teachers because that was what I had been told to do. That was clearly the rule of the game, and I followed that. In fact, I followed it so well that I aced math. I was fast, and that meant I was good at math. It was quite the midlife crisis when, as a high school math teacher, I realized that all that stuff I’d been doing was what I now call “fake math.”

To be precise, I’m not calling what I was doing “fake.” I’m not calling people fake. I’m not calling effort fake. But it wasn’t real mathematics. It wasn’t what mathematicians do. And I now argue that we can teach students to do what mathematicians do at a grade-appropriate level. I started to see that we can figure out math based on what we know and then build more relationships and connections mentally to figure more things out. Then we know more, and it doesn’t have to be handed to us. That process of figuring out what we don’t know from what we do builds the power of reasoning that math class should be focusing on in the first place.

It’s not some fuzzy, undefined reasoning. It’s content and reasoning together, but it’s not rote memorization. So, I would make that juxtaposition: for many years, I thought that math was rote “memorizable,” that math was something you learned the same way you learned the names of capital cities. Now I firmly proclaim math is “figure-out-able.”

MARIAN: If somebody said to me, “We’ve heard about Pam Harris, and we’re thinking about inviting her to a conference,” what would I say is unique about you?

PAM: I think most of my colleagues at conferences would advocate for conceptual understanding. But most people’s desired outcome is still that students get good at following the steps of algorithms. The problem is that because an algorithm can be used without any reasoning, they are terrible tools for teaching reasoning. I say it’s not about getting kids good at algorithms; it’s about developing the way they think so that they’re reasoning the way mathematicians do—they’re building content. But my definition of content is the standards, the curriculum, and not that the end goal is an algorithm. I help teachers realize their own paradigm, because until we realize our own paradigm, we don’t realize that we have the option to shift it.

I think many teachers think that education is a bandwagon of trying new things. But I’m not offering a bandwagon pendulum shift; I’m offering a paradigm shift. And once we get that paradigm shift, the whole landscape of how we teach changes to “Oh, I can use almost any resource.” But it’s not about resources necessarily; it’s about how to use those resources.

I think some teachers might believe that math is about rote memorizing and mimicking procedures because it “worked for them.” But I put that in quotes because it looks like it worked because they got good grades. But there are other teachers who actually developed relationships despite the algorithms, and they were able to successfully do real math. I submit that if someone had been actively helping them create those mental relationships, they could have gone much further, faster.

I help teachers realize their own paradigm, because until we realize our own paradigm, we don’t realize that we have the option to shift it.

MARIAN: You talk a lot about routines. Tell us about your favourite routines.

PAM: My favourite routine is using problem strings. When I was researching different ways of thinking, I tried many things, but I found that I grew most when I would do a well-constructed string of problems that was intended to help me notice and use relationships.

Problems strings allow us to mentally build those patterns and relationships and refine them to use them with bigger or gnarlier numbers. I found them so useful that I began to use them with my kids, with my students, and with teachers in professional learning. I found I could get my message across in a quicker, more concise way using a problem string.

Often, in a problem string, I’ll present an easy first problem that everybody can access. We don’t spend a lot of time on it. And then I’ll present the next problem. I might say, “Could you use the first problem to help you with the second one?” So, for example, if I had 70 – 40 as the first problem, the next problem might be 70 – 39. And I might let students solve it however they want. But I might ask, “Did anybody use the first problem to solve the second?” If someone did, then I’d share that strategy and model it and make it visible—it’s so important to make it visible. In fact, I’ll say one of the biggest reasons to use problem strings is if you do them well, you are modelling student thinking so that students can latch on to those relationships.

If no one used the first problem, then I might say, “Could you?” So, I use some subtle pedagogical strategies to get kids to consider different strategies. It’s not me telling them “All right, everybody, today we’re going to subtract too much. The steps are first you do this and then this.” It’s not that at all, because that would become another thing for students to rote memorize. And if students haven’t already built enough to be able to hang that on, then it’s going to be another series of steps they don’t understand.

Sample Problem Strings:

Developing a Subtraction Over Strategy

33 − 10

33 − 9

56 − 20

56 − 29

72 − 40

72 − 39

42 − 18

MARIAN: Do you try to make sure that there are alternative strategies to get to the same place? Because you often can get to the same place thinking in different patterns.

PAM: That’s an excellent question. I think there’s only a small subset of main strategies, and I’m going to focus on one of those strategies with each problem string. With subtraction, for example, I think there are four main strategies. So, a particular problem string is going to focus on one of them. On a different day, I will do a series of problem strings to focus on a different strategy. My goal is that students own all four strategies—with the important endpoint that once they’re familiar with all of them, I don’t tell them which one to use. Once they own them, they have a choice.

Often, people think there are all these strategies out there, so we should let students find the one that works best for them and leave them alone. But that’s not choice. If they only have one way to look at subtraction, they’re stuck there. They need to own the major relationships so that then, given the numbers, they can let the numbers influence how they solve the problem. That’s choice.

So, for example, for 72 – 39, a student could choose to do what we call the “over strategy.” That’s when they think, “I’m going to think about 72 – 40. That’s too much, and I’m going to adjust.” So, that’s one of the major strategies: I’m going to do something a bit too big, and then I’m going to compensate.

That over strategy shows up in addition. If I think about 48 + 29, I can think about 48 + 30 to help me. But I wouldn’t want to do that if the problem was 48 + 33. I don’t want to do 48 + 40 and then back up a lot. Now I could flip that around to 33 + 48 to get 33 + 50 and back up. But I would probably want to use one of the other three major strategies given different numbers. So, numeracy means that you are allowing the numbers and the structures to influence the relationships that you use to solve the problem.

MARIAN: Do you encourage teachers to have kids name these strategies, or do you name them for the teacher?

PAM: I encourage teachers to pull language out of their students as part of the process of learning. I think naming/describing strategies is part of the learning process because it’s part of students saying to themselves, “How would I describe the relationships I’m using? What words would I put to that?” And that helps students themselves gain clarity as they try words and then say to their peer, “Does that say what we’re using?” And then their peer says, “Well, I think I would use these words.” And as they negotiate those words to describe the relationships they’ve been using, students gain clarity and get more secure: “Oh, those are the relationships we’re talking about.” So, the name itself is not very important, but the process of describing is.

The one caveat I would put about the name is it needs to be mathematical. When we first started doing the work, I would work with teachers and we would say, “Oh, it’s the Marian strategy.” But we quickly found that when students moved up to the next grade, the teacher would say, “Tell me what you’re doing.” They would say, “I’m using the Marian strategy,” which of course did not help the new teacher understand what they were doing. So, we want to have mathematical terms so that when a student says, “Oh, you know, I’m adding a bit too much and then I have to adjust back,” the teacher can say, “Oh, I know what you’re doing.” Or, if a student says, “I’m doubling one factor and halving the other to get an equivalent problem that’s easier to solve,” the teacher can say, “I know the relationships you’re using,” and now we’re communicating mathematically.

Our job as teachers is to build the major relationships. So, I feel like what I bring to the work is to say, “Here are those major relationships. Let’s build them in kids.”

MARIAN: Is most of the work you do focused on calculations, or do you also address things that are not calculations but still mathematical? Do you spend time on that kind of stuff too?

PAM: Absolutely. You might not see it in a first presentation with me because in an introductory session I would do with teachers, I have only a short amount of time. So, my biggest goal is to help them shift their paradigm. So, I will typically use a calculation problem to do that. Though with high school teachers, I’ll often use a function problem. One of my favourite high school ones has to do with function notation and graphing lines and the connections between graphs and tables, and all that happens in one twenty-minute problem string.

I have problem strings that are about geometry, and I have problem strings that are about data and statistics. Most of my problem strings are in context, so if I’m doing a multiplication problem string, it is often something like “If there is 1 pack of gum that has 27 sticks, how many sticks would be in 10 packs? How many sticks would be in 5 packs? How many sticks would be in 9 packs?” So, that context pulls out what it means to use the distributive property in multiplication. So, one of the things I emphasize is the meaning of operations, the properties, and how they come out in operations, but that all comes as an outcome of kids diving in and solving problems. And then we make sense of and generalize what they were doing. So, it’s not an upfront “Today we’re going to talk about the commutative property”; it’s “Let’s do these tasks. What were you noticing?” Students might say, “We were rotating these arrays, and the areas were always the same.” If we write that as a sentence, we call that the commutative property. So, it’s a just-in-time, not a just-in-case, kind of approach.

MARIAN: Is there anything else you want to tell our readers?

PAM: Can I ask you a question?

MARIAN: Of course.

PAM: I’ve seen some of your work, and you have some really nice thinking tasks, or reasoning tasks. I think that’s a unique contribution that you’ve given to the work. You’ve helped all of us go, “Oh, that’s a nice one. Kids will have to play with some relationships to think about that.” So, I would guess that you were more like my son, the kind of kid who was playing with numbers.

MARIAN: But I was also very obedient and did whatever my teacher wanted. That’s all they ever heard. They didn’t know what was going on in my head ever. So, I knew how to do both things.

PAM: That’s exactly like Kim, my co-host on the Math is Figure-out-able podcast. Talk about a teacher who influenced me. When I was reading all that research and diving into my kids’ classrooms, hers was one of the classrooms I dove into. And what was interesting to me is that if I dove into a classroom of a teacher who was only compliant, who was like me and had just memorized, I didn’t learn a lot. I would watch them teach, we would try new things, and we’d both kind of stumble along.

But when I watched things happen in Kim’s class, she would ask students questions. I’d be like, “Wait, how did you know to ask that question?” And it was because she had mathematical relationships in her head that she could nudge toward.

She might say, “Were you thinking about quarters?” and the kid would go, “I was thinking about quarters.” And she was like, “Tell me more about quarters.” I would ask, “How did you know the kid was thinking about quarters?” She’d say, “Because I was thinking about quarters.” I thought, “Well, I want to think about quarters.”

Part of what I learned from her was that there are major things that mathy people think about and play with. I’ll give you some examples. Mathy people know their partners of 10 and partners of 100. That’s a thing they’ve played with so that when they’re subtracting 100 – 88, they don’t really think too much about it because they’ve thought a lot already in the past—about 88 and 12 making 100—so it becomes almost a non-problem.

MARIAN: It’s interesting because I think of the same thing you said in a very different way. I think compliant kids learn partners for 10 and 100 because their teachers tell them to, but they’re doing it only because their teachers tell them to. They aren’t doing it because they think it’s useful. I think what’s different about the mathy people is that their whole mindset is “How can I make this simpler?”

And I don’t think that’s the mindset of most kids. The mindset of most kids is “How do I do it the way my teacher wants?” So, I think it’s about helping kids see that when you’re doing what you call “Figure-out-able” stuff, what you’re really saying is “Let’s take a hard question and make it easy.” And I don’t think most teachers bring that to kids’ attention—that that is the game. The game is making something ugly pretty.

PAM: Well said. And our job as teachers is to build the major relationships. So, I feel like what I bring to the work is to say, “Here are those major relationships. Let’s build them in kids.” I think our work dovetails nicely.

MARIAN: Yeah, that’s lovely. Thank you for your time.

PAM: Thanks, Marian.