Pam Harris  00:01

Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam. 

Kim Montague  00:08

And I'm Kim. 

Pam Harris  00:09

And we're here to suggest that mathematizing is about thinking and reasoning; about creating and using mental relationships. We answer the question, if not algorithms, then what?

Kim Montague  00:22

Okay, Pam. So I'm really excited about today's podcast, because today we're going to talk about the quintessential Pam Harris thing, the thing that will rock your world, the thing you're known for, this is your stick, right? Friends, this is the podcast that you're going to want to go back and listen to when you have a piece of paper and a pen. This is the development of mathematical reasoning.

Pam Harris  00:46

Today, we're going to talk about the development of mathematical reasoning, which is my way of encapsulating several important ideas together. We're gonna talk about some of them today, that we're all on a journey mathematically and if we know where we fall, if we know where our students are, if we know sort of we are on this landscape, how they're reasoning, then can inform everything we do in our teaching. It's for students, for teachers, for adults, it's for all of us. It's a framework for viewing mathematics teaching and for evaluating how students are reasoning, and therefore helping them move forward from that place.

Kim Montague  01:26

So Pam, before you talk to us about some details, let's describe the graphic for listeners who haven't seen it yet. Picture a large oval with a series of smaller embedded ovals, each larger oval, including the ones inside of it kind of like a hierarchy. One leads to the next but continuous rather than linear, and each one of those ovals represents a different type of reasoning.

Pam Harris  01:51

Yeah, so the upshot about how and why I created this is, if you picture that outer oval and that outer oval, at least for our purposes in K-12 education, is Functional Reasoning. So there's math sort of beyond that, but we're kind of in high school sort of trying to get to this idea of reasoning about functions. And if we want students to be able to reason about topics in high school like functions, relations, x's and y's, graphs, and tables, and equations, they need to be able to what we build on that, they need to be able to reason proportionally. So Proportional Reasoning is in that next inside oval, because we have to own that in order to get to the outside oval of Functional Reasoning. But in order for students to think and reason proportionally they need to be able to reason multiplicatively, not get answers to multiplication problems, but actually change the way they think. So Multiplicative Reasoning is in the next inner oval. And in order for them to reason multiplicatively they need to reason additively. And so Additive Reasoning is in the next inner oval. And in order to reason additively what do we build that on, they must be able to solve problems using counting strategies. So Counting Strategies is the most inner oval.

Kim Montague  03:03

So little bitty students start learning how to solve problems with Counting Strategies. And then they develop Additive Reasoning. And then that builds to Multiplicative Reasoning and then Proportional Reasoning. And finally, in high school, you get to that Functional Reasoning. But also on the graphic, you have some Spatial Reasoning and something else?

Pam Harris  03:21

Yeah, so there's, there's two types of reasoning that should be developed all the way along. So if we really are helping kids develop these kinds of reasonings, we're really changing their brain to be able to think more sophisticatedly. We do that using spatial models. We use models of thinking that are graphic and spatial in nature. And so we build their Spatial Reasoning along that whole time. And also we represent the relationships they're using in a broad general way that we can use variables to sort of talk about things in generalities. That is building their Algebraic Reasoning. So each of those, as we're building each of those different sort of encapsulated within each other reasonings, we're also building Spatial Reasoning and Algebraic Reasoning the whole time.

Kim Montague  04:14

So let's talk about the development in general, why did you come up with it? And why does it even matter?

Pam Harris  04:20

So there's a few reasons. Let's get into a couple of them. We don't need calculators, we need reasoners. Now, maybe it used to be true years ago that we did need people to be a shopkeepers and bookkeepers. We needed them to be able to calculate, but we don't anymore. We have technology to do that. And so we don't need kids that just perform steps to get answers. We need kids who know what to put in technology or how to even program the technology. We need kids that can think about the problems at hand and know how to pull the numbers out, know how to know the relationships between them. It's not enough for students to get answers. And so a huge thing, if you can picture these ovals that are within each other, if you picture any one of those levels or domains, it's not enough for kids to get answers in those domains. Kids actually have to develop that kind of reasoning. We need to change their brain structure, more complex connections and relationships. So for example, if I'm in Multiplicative Reasoning, I might have kids getting answers to multiplication problems, but they might be thinking in additive ways or using counting strategies. Well, then I need to work with them to get them thinking more sophisticatedly, to get them thinking more multiplicatively. The more they're not thinking in one level, the harder it will be for us to move on to the next level. So then we end up with kids at Proportional Reasoning, where kids are supposed to be solving Proportional Reasoning problems, and they're trying to use Counting Strategies. That's untenable. Any middle school teacher can tell you that unfortunately, that's where we are too often. And so we really need to know where students are and then how to help them develop through these levels to help them increase their sophisticated thinking. So that then they can move on to the next level they can continue to increase their sophistication of thinking. So just the other day, a teacher said to me on social media, that she thanked me for changing her whole frame of reference when she talks to kids. And I said, "Tell me more about that." She said, "So I used to talk to kids just to sort of get answers from them, see how they're doing. Maybe I did some social emotional stuff, or I would sort of expect correct answers." She said, "Now. Now I ask kids to determine where they are. And then my goal is to work with them to help them think about becoming more sophisticated thinkers. What kinds of questions could I ask them, to nudge them to, to consider the problem in a more sophisticated way to think in terms of bigger chunks of numbers or think more multiplicatively." Because she knows the Development of Mathematical Reasoning, that gives her a sense of where kids are and then how you can help them continue to develop.

Kim Montague  07:00

That's so good. Knowing these types of reasonings is so freeing to me, because that means that we aren't stuck where we are. We can develop the next level reasoning at any time.

Pam Harris  07:12

Yeah, so I'd love to give you an example of a personal friend of mine. So there's a now personal friend. So there's a gal Holly, on our team, who several years ago, I met her in actually a graduate class. So, she was in a graduate math class, and I did a short stint. And I think, I don't think it was at that moment, I think it was the next time she and I worked together, she came to a workshop that I did. And I was talking about moving kids, helping kids develop Additive Reasoning if they were using counting strategies. And so I brought up a thing that sometimes teachers use to help students that get stuck, kids that can't find answers to problems using counting strategies. And so sometimes we teach kids what they called touch math or dot math. And if I can just describe it really quickly, by the way, I'm not advocating this. If I could describe it really quickly, when kids are adding, say five plus six, they would have the kids write down the numeral five, write down the number six. And then on that numeral five, there's some places where they've told kids to write five dots. So kids sort of memorize that there are five dots that go on the numeral five, and then they write down the number six, and then they write six dots. They draw six dots and the numeral six. And then to add five, plus six, they literally start with the dots. So they would go 1, 2, 3, 4, 5 on the five, and then they would keep going on the six,  6, 7, 8, 9, 10, 11. And that sort of can help kids kind of touch things. And the kids can just use the counting that they can do in order to solve a problem like five plus six. Well, then when kids get to adding two digit numbers, if say they're adding 52 and 64, then they line up the five and the six and the two and the four and they draw the dots. And then they literally are thinking, counting because they're counting all of those dots. And so I had just taught this group of teachers about how to help kids move from Counting to Additive Thinking, and I said, "So what's the problem with this dot math or touch math approach?" And Holly, her eyes got really big. And she looked at me. She said, "Wait a minute, wait a minute. That's how I learned how to add." And I said to her, "What are you thinking? You're a little hot there. You're a lot of energy coming out of you right now. Like, what's going on?" She goes, "That's, that's, that's why I can't learn higher math, because I'm stuck in Counting Strategies. Are you saying I could learn to think additively?" And I was like, "Well, are you? In the string that we just did the problem string we just did, were you thinking in terms of bigger chunks of numbers.?"And she's like, "I was. But honestly, Pam, if you give me a problem, right now, I would probably draw the dots." And I said, "Do you have to?" And I'll never forget the look on her face. She just looked like I'd set her free. She goes, "No! I can absolutely think in terms of big chunks of numbers." And so then I continued to work with Holly she came to every workshop, I did everything she'd get her hands on. And she built her Additive Reasoning and then she built her Multiplicative Reasoning and now she's on her team. And she presents for us, she's a great mathematician. But we had to get her out of, we had to, like free her up from saying, don't get stuck in that Counting Strategy. You can think additively. Once she could think additively, she could think multiplicatively, begin to build that Multiplicative Reasoning. And now that she could think multiplicatively, she's building her Proportional Reasoning, we can all sort of progress. And that's the, that's the brilliant message of the Development of Mathematical Reasoning is, no matter where you are, okay, that's where you are. Now, let's continue to build your brain structure. Let's make you a more dense thinker.

Kim Montague  10:32

Oh, I'm smiling, so big. I love Holly. So, Pam, let me ask you another question. You used to have a different graphic name. And on your team, we used to call this the progression, right? We just said, "Hey, the progression." Why the change?

Pam Harris  10:48

That's an excellent question. And there are a few major reasons. So one of the reasons is there's a lot of people out there that started to develop things called progressions. And I didn't want my thing to get kind of lost in this sea of things called progressions. So that's one reason. Another is the graphic used to be, it was far more linear. It was do this then do that. And people got the wrong idea. I actually had somebody on social media, there was a high stakes test question with data on population. And a correct answer could be that you could model it linearly, like you could model it with a successive additive pattern happening, or you could model it exponentially with successive multiplicative pattern happening. And the idea of the question was to compare those two models and see, which looked like it would bear out over time. And a teacher shot back to me and said, "Oh, but you shouldn't model it linearly with a linear equation, because that's Additive Reasoning. And once we move to Multiplicative Reasoning, like the exponential functions, then we leave Additive Reasoning behind. We shouldn't ever use that again." And I was like, "No, that's not the idea." Like we don't stop counting. We just don't want to count when we should be thinking additively and we don't stop adding, but we don't want to add when we should be thinking multiplicatively. And so there's this idea of this more continual embeddedness happening. And so the graphic now shows more of an embedded landscape sort of a perspective. It also has this connotation of more of an asset perspective, that what can you do? What do you own? And then we want to nudge you on from there. It's less of a deficit perspective, where it's more like looking at what you can't do or what you don't own. It's like, where are you? What can you do? It's a positive asset perspective. And then the last reason might be because the word development, and I'm going to give Cathy Fosnot credit for helping me think about this word. She talked about that mathematicians can develop and that it's about development. It's not about rote, memorizing things. It's not about the number of rules and procedures that you could spit out. But it's more like we expect kids to develop as they grow up that we can develop mathematicians. And that word development sort of connotates that we can all do it. Everybody develops, everybody grows up. And so that's more inclusive. It's not about mathy people, math brains, math genes. No, we can all develop mathematically just like Holly, who was convinced she was sort of couldn't do higher math, it's because she was stuck in Counting Strategies. And once we sort of opened her mind, like there's this other thing, and you could do it, then she absolutely kept developing. And we can all develop to the extent that we want to in math. So those are the major reasons that we sort of gave it a look change to a different kind of graphic and a name change to call it the Development of Mathematical Reasoning. We need teachers to know this progression. We need them to better understand where their students are, and then how to move them forward, how to help them develop and become more and more sophisticated in the levels of reasoning.

Kim Montague  13:51

And now that we know that these types of reasoning exist, you might be thinking, "How do I determine where I am and where my students are?" We have a free download today that we're really excited to share and it's called, "How Do You Reason?", and you can download that and use it to help you determine where you are, where your students are. You can find that at mathisfigureoutable.com/reason. And we'll also put it -- Sorry, I was gonna say that one more time so they could catch it. It's at mathisfigureoutable.com/reason.

Pam Harris  14:25

And we'll also put it in the show notes. So you could go to the show notes and download it from there as well.

Kim Montague  14:29

Also, if you are interested in specific examples and more details about each of the type of reasoning, we are beyond excited to announce that Pam is putting out a free online workshop called "The Development of Mathematical Reasoning". It's been in the works for a while now. We can finally announce it. It is a free four week online course. And it really is designed to help everyone understand more deeply the domains of each reasoning. You can register now at mathisfigureoutable.com/freeworkshop.

Pam Harris  15:05

And if you happen to be listening to this podcast some other time go ahead and check it out because you get on the waitlist for the next time that we offer it at mathisfigureoutable.com/freeworkshop I am super excited about this free workshop. Kim, you know that we've worked on this for a while. I've been wanting to put this out to give teachers a chance to learn this important information. We're so excited to offer it free again go to mathisfigureoutable.com/freeworkshop to sign up. You are going to love it! Alright, so if you wouldn't mind, like the podcast give us a review at your favorite podcast hosting site so more people can find it. Check us out on the website mathisfigureoutable.com We'd love to have you join us at Math Strat Chat if you are interested to learn more math and you want to help students become mathematicians, then the Math is Figure-Out-Able Podcast is for you. Because Math is Figure-Out-Able!