Season 2 | Episode 6 – Making the Shift: Moving From Additive to Multiplicative Thinking - Guest: Anderson Norton, Ph.D.

ROUNDING UP: SEASON 2 | EPISODE 6 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. GUEST BIOGRAPHY 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 . RESOURCES Developing Fractions Knowledge Teaching Children Mathematics TRANSCRIPT 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 ...