# Ep 5: The Development of Mathematical Reasoning

July 21, 2020 Pam Harris Episode 5
Math is Figure-Out-Able with Pam Harris
Ep 5: The Development of Mathematical Reasoning

This episode is so important! Pam and Kim describe the Development of Mathematical Reasoning, and how it transforms our understanding of what it means to learn and teach mathematics. They discuss how freeing it feels to know that everyone can develop their reasoning, and how the Development of Mathematical Reasoning empowers teachers to know how best to help their students.
To help you determine where you and your students are in the Development of Mathematical Reasoning, here's a free download! Use the How Do You Reason guide to know how you, your friends, and students are reasoning.
They also announce the new Development of Mathematical Reasoning Workshop! A new free online workshop. So exciting! Be sure to tell all your colleagues about this amazing opportunity to learn how to help anyone develop reasoning.

Talking Points

• What the Development of Mathematical Reasoning looks like
• We don't need calculators, we need reasoners
• One way people get stuck in counting strategies
• The freeing power of progressing in reasoning
• New free online workshop!

Pam Harris:

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

Kim Montague:

And I'm Kim.

Pam Harris:

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:

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:

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:

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:

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:

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:

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:

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

Pam Harris:

Kim Montague:

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:

Kim Montague:

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."

Pam Harris:

Kim Montague:

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 And we'll also put it -- Sorry, 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. I was gonna say that one more time so they could catch it. It's at mathisfigureoutable.com/reason.

Pam Harris:

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:

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: