Anderson Norton, Ph.D., Making the Shift: Moving From Additive to Multiplicative Thinking

Mike Wallus, Vice President for Educator Support


One of the most important shifts in students’ thinking during their elementary years is also one of the least talked about. I’m talking about the shift from additive to multiplicative thinking. If you’re not sure what I’m talking about, I suspect you’re not alone. Today, we talk with Dr. Anderson Norton about this important but underappreciated shift.

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Anderson Norton’s research is driven by a desire to understand how humans have access to knowledge as powerful and reliable as mathematics. Throughout his career, building upon Jean Piaget's genetic epistemology, he has learned that many philosophical questions about the nature of mathematics have psychological answers. He grounds his research in psychological models of students' mathematics, and collaborates with psychologists and neuroscientists to find these answers.

In 2022, he authored a related book, published by Routledge: The Psychology of Mathematics: A Journey of Personal Mathematical Power for Educators and Curious Minds.


Developing Fractions Knowledge

Teaching Children Mathematics


Mike Wallus: One of the most important shifts in students' thinking during their elementary years is also one of the least talked about. I'm talking about the shift from additive to multiplicative thinking. If you're not sure what I'm talking about, I suspect you're not alone. Today we talk with Dr. Anderson Norton about this important but underappreciated shift. 

Welcome to the podcast, Andy. I'm excited to talk with you about additive and multiplicative thinking.

Andy Norton: Oh, thank you. Thanks for inviting me. I love talking about that.

Mike: So, I want to start with a basic question. When we're talking about additive and multiplicative thinking, are we just talking about strategies or operations that students would carry out to find a sum or a product of a problem? Or are we talking about something larger?

Andy: Yeah, definitely something larger, and it doesn't come down to strategies. Students can solve multiplication tasks—what, to us, look like multiplication tasks—using additive reasoning. And they often do. I think they get through a lot of elementary school using, for example, repeated addition. If I gave a task like, “What is 4 times 5?”, then they might just say, “That's 5 and 5 and 5 and 5,” which is fine. They're solving a multiplication problem, but their method for solving it is repeated addition, so it's basically additive reasoning. But it starts to catch up to them in later grades where that kind of additive reasoning requires them to do more and more sophisticated or complicated strategies that maybe their teachers can teach them, but it starts to add up, especially when they get to fractions or algebra.

Mike: So, let's dig into this a little bit deeper. How would you describe the difference between additive and multiplicative thinking? And I'm wondering if there's an example of the differences in how a student might approach a task or a problem that could maybe highlight that distinction.

Andy: The main distinction is with additive reasoning, you're working within one level of unit. So, for example, if I want to know—going back to that 4 times 5 example—really what I'm doing is I'm working with ones. So, I say I have 5 ones and 5 ones and 5 ones and 5 ones, and that's 20 ones. But in a multiplication problem, you're really transforming across units. If I want to understand 4 times 5 as a multiplication problem, what I'm saying is, “If I measure a quantity with a unit of 5, the measure is 4.”

Just to make it a little more concrete, suppose my unit of measure is like a stick that's 5 feet long, and then I say, “OK, I measured this length, and it was four of these sticks. So, it's four of these 5-foot sticks. But I want to know what it is in just feet.” So, I've changed my unit. I'm saying, “I measured this thing in one unit, this stick length, but I want to understand its measure in a different unit, a unit of ones.” So, you're transforming between this one kind of unit into another kind of unit, and it's a 5-to-1 transformation. So, I'm not just doing 5 plus 5 plus 5 plus 5, I'm saying every one of that stick length contains 5 feet, five of these 1-foot measures. And so, it's a transformation from one unit into another, one unit for measuring into a different unit for measuring.

Mike: I mean, that's a really big shift, and I'm glad that you were able to describe that with a practical example that someone could listen to this and visualize. I think understanding that for me clarifies the importance of not thinking about this in terms of just procedural steps that kids would take to either add or multiply; that really there's a transformation in how kids are thinking about what's happening rather than just the steps that they're following.

Andy: Yeah, that's right. And a lot of times as teachers, or even as researchers studying children, we're frustrated like the kids are when they're solving tasks, when they're struggling. And so we try to give them those procedures. We might give them a visual model. We might give them an array model for multiplication, which can solve a lot of problems. You just sort of think about things going vertically and things going horizontally, and then you're looking at an area or a number of intersections. So, that makes it possible for them to solve these individual tasks. And there's a lot of pressure on teachers to cover curriculum, so we feel like we have to support them by giving them these strategies. But in the end, it just becomes more and more of these complicated strategies without really necessitating the need for something we might call a “productive struggle”; that is, where students can actually start to go through developmental changes by allowing them to struggle so that they actually develop these kinds of multiplicative structures instead of just giving them a bunch of strategies for dealing with that one task at a time.

Mike: I'm wondering if you might share some examples of what multiplicative thinking might look like or sound like in different scenarios. For example, with whole numbers, with fractions or decimals … 

Andy: Uh-huh.

Mike: … and perhaps even in a context like measurement. What might an educator who was listening or observing students' work, what might they see that would indicate to them that multiplicative reasoning or multiplicative thinking was something that was happening for the student?

Andy: So, it really is that sort of transformation of units. Like imagine, I know something is nine-fifths, and nine-fifths doesn't make a whole lot of sense unless I can think about it as nine units of one-fifth. We have to think about it as a measure like it's nine of one-fifth. And then I have to somehow compare that to, “OK, it's nine of this one unit, this one-fifth unit, but what is it of a whole unit? A unit of one?” So, having an estimate for how big nine-fifths is, yes, it's nine units of one-fifth. But at the same time, I want to know how big that is relative to a one. So, there's this multiplicative nature kind of built into tasks like that, and it's one explanation for why students struggle so much with improper fractions.

Mike: So, I'm going to put my teacher hat on for a second because what you've got me thinking is, what are the types of tasks or experiences or even questions that an educator could put in front of students that would nudge them to make this shift without potentially pushing them to a place where they're not quite ready to go yet? 

Andy: Hmm.

Mike: Could you talk a little bit about what types of tasks or experiences or questions might help provide a little bit of that nudge?

Andy: Yeah, that's a really good question, because it goes back to this idea that students are already solving the kinds of tasks that should involve multiplicative reasoning, but they might be using additive strategies to do it. Those strategies get more and more complicated, and we as teachers facilitate students just, sort of, doing something more procedural instead of really struggling with the issue. And what the issue should be is opportunities to work with multiple levels of units and then to reflect on their activity and working with them. 

So, for example, one task I like to give students is, I'll cut out a piece of construction paper and I'll hand it to the student. And I'll have hidden what I'm going to label a whole, and I'll have hidden what I'm going to label to be the unit fraction that might be appropriate for measuring this thing I gave them. So, I'll give them this piece of construction paper and I'll say, “Hey, this is five-sevenths of my whole.” Now what I've given them as a rectangular strip of paper …

Mike: Mm-hmm

Andy: …without any partitions in it. I've hidden the whole from which I created this five-sevenths. I've hidden one-seventh, and I've put them away, maybe inside of envelopes. So, it becomes like a game: “Can you guess what I have in this envelope? I just gave you five-sevenths. Can you guess, what is this five of? What is the unit that this is five of?” and “What is the whole this five-sevenths fraction is?” So, it's getting them thinking about two different levels of units at once. They've been given this one measurement, but they don't know the unit in which it's measured, and they don't even have visually present for them what the whole unit would be.

So, what they might do, is they might engage in partitioning activity. Sometimes they might partition what I give them into seven equal parts instead of five because I told them “five-sevenths” and five-sevenths to them, that means, “partition it a seventh.” Well, that could lead to problems, and if they see that their unit is smaller than the one I have hidden, they might have to reason through what went wrong, “Why might have you gotten a different answer than I did?” So, it's those kinds of activities—of partitioning or iterating a unit, measuring out with a unit, and then reflecting on that activity—that give them a basis for starting to coordinate these units at higher and higher levels and, therefore, in line with Amy Hackenberg’s framing, develop multiplicative concepts.

Mike: I think that example is really helpful. I was picturing it in my head, and I could see the opportunities that that affords for, kind of, pressing on some of those big ideas. 

One of the things that you made me think about is the idea of manipulatives, or even if we broaden it out a little bit, visual models. Because the question I was going to ask is, “What role might a visual model or a manipulative play in supporting a shift from additive to multiplicative thinking?” I'm curious about how you would respond to that initially. And then I think I have a follow-up question for you as well.

Andy: OK. I can think of two important roles for visual models—or at least two for manipulatives—and at least one works with visual models as well. 

But before answering that, the bigger answer is, no one manipulative is going to be the silver bullet. It's how we use them. We can use manipulatives in ways where students are just following our procedures. We can use visual models where students are just doing what we tell them to do and reading off the answer on paper. That really isn't qualitatively any different than when we just teach them an algorithm. They don't know what they're doing. They get the answer, they read it off the paper. You could consider that to be a visual model, what they're doing on their paper or even a manipulative; they're just following a procedure. 

What manipulatives should afford is opportunities for students to manipulate. They should be able to carry out their mental actions. So, maybe when they're trying to partition something and then iterate it, or they're thinking about different units. That's too much for them to keep in mind in their visual imagination. So, a visual model or a manipulative gives them a way to carry those actions out to see how they work with each other, to notice the effects of those actions. 

So, if the manipulative is used truly as a manipulative, then it's an opportunity for them to carry out their mental actions to coordinate them with a physical material and to see what happens. And visual models could be similar, gives them a way to sort of carry out their mental actions, maybe a little more abstractly, because they're just using representations rather than the actual manipulative, but maybe gives them a way to keep track of what would happen if I partitioned this into three parts and then took one of those parts and partitioned into five. How would that compare to the whole? So, it's their actions that have to be afforded by the manipulative or the visual model. And to decide what is an appropriate manipulative or an appropriate task, we need to think about, “OK, what can they already do without it?” And I'm trying to push them to do the next thing where it helps them coordinate at a level they can't just do in their imagination, and then to reflect on that activity by looking at what they wrote or looking at what they did. 

So, it's always that: Carrying out actions in slightly more powerful ways than they could do in their mind. That's sort of the sense in which mathematics builds on itself. After they've reflected on what they've done and they've seen the results, now maybe that's something that they can take as an object, as something that's just there for them in [their] imagination so they can do the next thing: adding complexity.

Mike: OK. So, I take it back. I don't think I have a follow-up question because you answered it in that one. What I was kind of going to dig into is the thing that you said, which is, there's a larger question about the role that a manipulative plays, and I think that your description of a manipulative should be there to manipulate to help kids carry out … 

Andy: Mm-hmm.

Mike: … the mental action and make meaning of that. I think that piece to me is one that I really needed clarified, just to think about my own teaching and the role the manipulatives are going to play when I'm using them to support student thinking.

Andy: And I'll just add one thing, not to use too many fractions examples, but that is where most of my empirical research has been, was working with elementary and middle-school children with fractions. But I have to make these decisions based on the child. So, sometimes I'll use these Cuisenaire Rods, the old fraction rods, the colored fraction rods. Sometimes I'll use those with students because then it sort of simplifies the idea. They don't have to wonder whether a piece fits in exactly a certain number of times. The rods are made to fit exactly. And maybe I'm not as concerned about them cutting a construction paper into equal parts or whatever. So, the rods are already formed. But other times I feel like they might be relying too much on the rods, where they start to see the brown rod as a 4. They're not even really comparing the red rod, which fits into it twice. They're just, “Oh, the red is a 2; the brown is a 4. I know it's in there twice because two and two is four.” So, you start to think about them, whole numbers. And so sometimes I'll use the rods because I want them to manipulate them in certain ways, and then other times I'll switch to the construction paper to sort of productively frustrate this idea that they're just going to work with whole numbers. I actually want them to create parts and to see the measurements and actually measure things out. 

So, it all depends on what kind of mental action I want them to carry out that would determine [which] manipulative as well. Because manipulatives have certain affordances and certain constraints. So, sometimes Cuisenaire Rods have the affordances I want, and other times they have constraints that I want to go beyond with, say, construction paper.

Mike: Absolutely. 

So, there's kind of a running theme that started to develop on the podcast. And one of the themes that comes to mind is this idea that it's important for us to think about what's happening with our students’ thinking as a progression rather than a checklist. What strikes me about this conversation is this shift from additive to multiplicative thinking has really major implications for our students beyond simple calculation. And I'm wondering if you could just afford us a view of, why does this shift in thinking matter for our students both in elementary school, and then also when they move beyond elementary school into middle and high school? Could you just talk about the ramifications of that shift and why it matters so much that we're not just building a set of procedures, we're building growth in the way that kids are thinking?

Andy: Yeah. So, one big idea that comes up starting in middle school—but becomes more and more important as they move into algebra and calculus, any kind of engineering problem—is a rate of change. So, a rate of change is describing a relationship between units. It's like, take a simple example of speed. It's taking units of distance and units of time and transforming them into a third level of unit that is speed. So, it's that intensive relationship that's defining a new unit. When I talk about units coordination, I'm not usually talking about physical units like distance, time, and speed. I'm just talking about different numerical units that students might have to coordinate. 

But to get really practical, when we talk about the sciences, units coordinations have to happen all the time. So, students are able to be successful with their additive reasoning up to a point, and I would argue that point is probably around where they first see improper fractions. (chuckles) They're able to work with them up to a point, and then after that, things [are] going to be less and less sensible if they're just relying on these additive sort of strategies that each have a separate rule for a different task instead of being able to think more generally in terms of multiplicative relationships.

Mike: Well, I will say from a former K–12 math curriculum director, thank you for making a very persuasive case for why it's important to …

Andy: (chuckles)

Mike: … help kids build multiplicative thinking. You certainly hit on some of the things that can be pitfalls for kids who are still thinking in an additive way when they start to move into upper elementary, middle school and beyond. Before we go, Andy, I suspect that this idea of shifting from additive to multiplicative thinking, that it's probably a new idea for our listeners. And you've hinted a bit about some of the folks who have been powerful in the field in terms of articulating some of these ideas. I'm wondering if there are any particular resources that you'd recommend for someone who wants to keep learning about this topic?

Andy: Yeah. So, there are a bunch of us developing ideas and trying to even create resources that teachers can pick up and use. Selfishly, I'll mention one called “Developing Fractions Knowledge,” used by the U.S. Math Recovery Council in their professional development programs for teacher-leaders across the country. That book is probably, at least as far as fractions, that book is maybe the most comprehensive. But then, beyond that, there are some research articles that people can access, even going in Google Scholar and looking up units coordination and multiplicative reasoning. Maybe put in Steffe's name for good measure, S-T-E-F-F-E. (chuckles) You'll find a lot of papers there. Some of them have been written in teacher journals as well, like journals published by the National Council of Teachers of Mathematics, like Teaching Children Mathematics materials that are specifically designed for teachers.

Mike: Andy, thank you so much for joining us. It's really been a pleasure talking with you.

Andy: OK. Yeah, thank you. This was fun.Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.