Room to Grow Podcast

Season 5 Episode 1:1 Centering Student Reasoning in Conceptual Understanding and Procedural Fluency

00;00;02;00 - 00;00;30;20

Joanie Intro

In today's episode of Room to Grow, Curtis and I revisit the many conversations in the math education space around balancing conceptual, understanding and procedural fluency. We consider how these two ideas are not discrete, but that they work together to build students reasoning and understanding of mathematics. We find connections to standards, policy, and student motivation in today's conversation, and we hope you'll have some takeaways to use in your classroom tomorrow. So let's get growing.

00;00;33;02 - 00;01;19;22

Curtis

Well, Joni, I am very excited. For today's conversation, we are, recording the Room to Grow podcast yet again. And I am really excited because we've just had the opportunity to spend a couple of weeks together over the last month at a couple of different conferences, attending sessions and bumping into each other as the random meetings would happen. And so we've got different experiences and things, but really, you know, today we've got a cool conversation, a topic that is near and dear to my heart and it's, it's a soap box for me. And it's actually a soap box for one of our coworkers, too. Because, I think he and I have many, many conversations at length about the topic we're talking about today.

00;01;19;28 - 00;01;53;18

Joanie

I love it, I think I think it's actually a soapbox for a lot of people. Curtis. And really, when we were talking about this being our topic for today's conversation, I was like, well, we've already talked about this, but I think despite all the conversation that we've had, there continues to be nuance and there continues to be learning and understanding about how, you know, we think about instruction and how we think about deep student learning.

So, I would love to kind of hear you talk about a framework or a like, what's the topic for today? What is it that we're going…

00;01;53;20 - 00;01;54;12

Curtis

Yeah.

00;01;54;14 - 00;02;00;04

Joanie

And how is it different than, the, the previous conversations that we've had around similar ideas?

00;02;00;11 - 00;04;38;20

Curtis

Well, I, you know, I think so here's, here's what we're wanting to talk about today.

I think that, you know, we see, at math conferences and really in the math community, I mean, if you go and you pick up a copy of the math teacher leader, articles or, you go out and you look at any teacher blogs, and things where, folks are just talking about math and sort of our needs around teaching mathematics. One of the things that, everyone is talking a lot about is this idea of cultivating, understanding, cultivating deep conceptual understanding of math topics. And,

we see that many of those conversations drive towards cultivating conceptual understanding, coming through this idea of inquiry or inquiry style learning where the students kind of have a discovery opportunity put in front of them. The maybe the there aren't a whole lot of guardrails,

put on these things that sometimes maybe there are, but but really, they're, they're, they're experiences that students have  and they, they seem to be a little more wide open and not quite so much, of the, the, procedural pieces, put in place. Right. We're allowing students to kind of explore and try and all of these, these ideas and, and those are wonderful ways I want to start off by making sure that we, we identify that, the focus on that is not bad.

And, and really, it's a good, good thing that we're, we're trying to really help math teachers, be exposed to more than the drill and kill, piece of the puzzle. But I think today, one of the things that we want to make sure that we don't lose is that, procedural understanding is important one, but also procedural understanding and procedures and, and looking at, algorithms and the way that we do execute problems are good opportunities if taken correctly or if taken correctly.

Sounds like I know what I'm talking about and I don't. So, if taken it taken in a, in a, in a, in a slow and considerate way those provide opportunities for cultivating that same kind of deep understanding. If we slow down and we ask questions and we try to explore why the procedures work the way that they do.

00;04;38;23 - 00;06;42;13

Joanie

Right. Exactly. And I'm really thinking, you know, as we've, as we've sort of developed this idea and talked about what we want our conversation to be grounded in today. I think that a lot of these, the conversations around this balance, right, like,

00;04;54;18 - 00;04;56;24

Curtis

Yes. Yes.

00;04;56;27 - 00;05;05;04

Joanie

you know, presidents messages, there's been all sorts conference sessions, all sorts of things around finding a balance between conceptual understanding and procedural fluency and I think where our conversation has gone and where both of us in our independent ways, but also coming together, it's understanding that it's not one or the other. Right.

00;05;16;20 - 00;05;17;13

Curtis

Right.

00;05;17;16 - 00;05;23;13

Joanie

And, and it's almost like they're not discrete from one another.

00;05;23;16 - 00;05;26;01

Curtis

I think that's the biggest thing.

00;05;26;04 - 00;05;42;273

Unknown

conceptual understanding of procedures. Right. So rather than saying it's conceptual understanding and it's procedural fluency and find the balance between those two, I'm sort of the language. It's sort of popping out in my head is mathematical reasoning

00;05;43;00 - 00;05;43;29

Curtis

Yes.

00;05;44;02 - 00;05;56;22

Joanie

reasoning through understanding concepts and it's reasoning through procedures and it's reasoning through how all of that fits together to reason with mathematics. So for me,

00;05;56;29 - 00;05;59;02

Curtis

Yes, yes.

00;05;59;05 - 00;06;42;13

Joanie

reasoning while we're working through developing conceptual understanding. And as you said, sometimes that's with a much more open, you know, loose task where students have opportunities to explore and go in their own direction and come up with ideas that are not as structured.

And there are other, maybe there's a rule or a procedure that isn't easily explored, isn't easily discovered. It's something that students just need to be told and, practice and understand. Right. And, and neither of those types of extremes in our teaching practice should be without reasoning. Reasoning needs to be the underlying thread through all of it.

00;06;42;13 - 00;00;00;20

Music break

End of Segment 1

Start of Segment 2

00;06;50;02 - 00;07;06;28

Curtis

You know, I love that you bring up this idea of reasoning because I think that's I really do, I agree, I think that's where we're really trying to drive toward is in all of the things that we do in our math classrooms, we want to cultivate deep mathematical reasoning.

And I think that's I think that's really what we're actually maybe driving toward is, is that idea of reasoning, you know, a couple of weeks ago, we had Pam Harris on the podcast and this concept of, you know, mathematics and figuring out, what's going on. And she told a story about her son, but I'm not going to steal her her story. You guys can go, back and listen to the October podcast, but I,  I wanted to kind of jump in here and thinking about. I always tell stories about my son through Truitt, and I have, I have one that is, sort of, you know, related to this idea of he's got a broad space, right?So we just, we were. I don't even remember where we were. We were out on a walk, I think, walking along the lake. And my son says to me, says, hey, dad,

00;08;01;00 - 00;08;11;06

Unknown

did you know the gym at school is about twice as big as our has? About twice the area as our house. Maybe including the attic.

00;08;11;09 - 00;08;57;00

Curtis / Joanie / Curtis

Like, he just says this like we're walking along. There's birds, there's fish, there's, you know, whatever. And my son is thinking, this is where my son's brain is, right? He's just thinking about, you know, there's surface area of the gym floor is probably about twice as big as our house. And so I started doing a little bit of, a quick arithmetic. I'd never thought about this before. In my head, I'm like, well, I know how big a basketball court is, and I know they probably got a little edges around the area and thought, yeah, actually, the size of your gym, it's it probably is about twice as big as our house is. I said, how did you even think about. Well, dad, half of it is about the same as our house. So twice as much. And so he's just thinking about these things. 

00;08;53;06 - 00;08;54;12

Joanie

Of course. Dad, come on.

00;08;57;03 - 00;09;13;19

Curtis

Of course. Dad, come on.

And so he's kind of got these, like, wide open spaces just thinking conceptually about surface area and being able to break down the rooms in our house. And first of all, I was just fascinated by the fact that he could look at individual rooms, sum them all up, and then because he has no idea how big or small our house is.

00;09;13;19 - 00;09;24;19

Joanie

No, of course not. A 10 year old doesn’t go we live in, ya know, an 1800 square foot house or my friend has a 3000 square foot house. That's not their language. 

00;09;29;24 - 00;11;24;12

Curtis

That's not their language. It's not the way he thinks. But just visually he can like, put all those together. And somehow at some point in his life, he did. And then he, he's doing this. So this is, this is that like wide open spaces right there, there, there, wasn't a task. There was, maybe maybe he had a task at school where they were talking about surface area. And so then he started. But it it wasn't so much about that.

And then it brings me back to thinking about whenever he was asking me. So that's one end of the spectrum. But then on the other end of the spectrum, when he's doing his homework and he's got lined up, the, the, you know, a three digit number times a one digit number. And he is multiplying. And we have a conversation about the length, the language of three times, you know, the first, the one digit, maybe it's a seven, three times seven is 21.

And and he's putting the 20 portion up above the next, the ten digit there. And we talk about why is that two. What is that two representing is it's two tens. You're representing, you're putting two tens up there so you can add them later. And so we have this conversation. That's the opportunity that I'm talking about. So we have approximating the area of our house comparing it to a gym and having an estimation conversation and there's conceptual things happening there. And then there's conceptual things happening whenever we talk about doing this procedural understanding this, like knowing how to multiply this three digit number times three and putting the this 20 representing this 20 by using a two in the tens place and saying, I have two tens here and now I'm going to multiply. And then I'm going to add that there's so many things that go on there. And I think that's what I'm trying to maybe drive at.

00;11;24;14 - 00;15;41;25

Joanie

Yeah I, I, I always love how you use your kids and the way your kids think and explore the world to help make our points, but I, I really appreciate this so much and the idea of the connection between the understanding of a concept and the operations of procedures,

feel like procedures have gotten a bad rap and I just want to be cautious because there's a lot of, I don't know, there's a lot of noise out in the field right now, I think.

And in particular, you know, Kurt, my role at, in our work together is focused on policy. So I have exposure to a lot of different policy conversations and things that, you know, I didn't have access to as a classroom teacher. I wasn't aware of, you know, what's happening legislatively around math instruction. And there is this temptation to look at standardized test scores and say, you know, something is wrong with mathematics education because students are not achieving and scores are not getting and they're not improving despite the changes that we're trying to make in the classroom. And I think there is this temptation, you know, when you've been in the field for a lot of years, like I have, you see this like cycling back, let's start talking about this again. So I think there's a lot of, there's a lot of noise external to math education around a refocus on procedures and skills and ensuring that students can accurately and effectively do mathematical computation.

And I think there is a danger in contradicting that there's a danger in saying no inquiry is more important and conceptual understanding is more important and we should be focused on that at the sacrifice of being able to do procedures and get answers. So I love the, the bringing together of those two ideas, because I think it actually is the right solution. And another lens that I'm bringing is, is standards. So in I'm in Colorado as you well know, and I'm on the committee that is revising our Colorado high school math standards. We're just doing high school. But some really great conversations have happened among our group as we've looked at the previous standards that have been in Colorado's standards since 2011, I think 2012, they've been around for a long, long time. And as we think about the opportunity to revise and make these better in a way that actually changes classroom practice and creates different experiences for students, we're wrestling with this same thing of, you know, there, there are procedures, there are sort of skills that students need to be able to do with high school mathematics. But when those skills live outside of reasoning and outside of conceptual understanding, we would argue they're not actually doing math, they're just doing computing. And in our current age of technology and AI, you know, we gotta pay attention to those things. I was just at the California math Council conference and although I didn't attend this session, I heard somebody talk about Robert Koplinski session, and he said, if we say we're preparing students for college and career and we're not teaching them how to use AI effectively, then we're just lying to ourselves into them. Like what? What does mathematics look like in high school? In an age where a computer can do any calculation much faster than a human can and much more accurately than humans can? So what what AI can't do, in my opinion, is the reasoning piece. And, Right. How you just described your conversation on the walk with your son and how do we, how do we support students in thinking through those patterns and recognition and making those connections to their lives, and then being able to see those concepts through the development of the procedures that they need to learn in order to do the computation and the skill required of the mathematics they're learning.

00;15;42;03 - 00;19;15;01

Curtis

Yeah. I think that's exactly what we're driving at, is this idea that, you know, we have, I was I was thinking about Tiguan. You know, I, I do, I enjoy, using my son's, experiences with mathematics, on this podcast because, you know, it's like we're walking through, and seeing this happen and he's been recently…You and I did a session actually at CSM, on this of solving systems of linear equations and really kind of walking through, what exactly we're talking about, even right here, this, this idea of there's a procedure, there's some procedures that that happen in solving systems of equations. And there are reasons why some of those procedures are acceptable procedures.

 And I've, I'm slowing Teagin down. He's good at carrying out the procedures. He knows he's like me x y, you know, do the thing right. Show me the rule. I can figure out. I can see the pattern. I can just do the thing and we just go. But I'm focusing on slowing him down enough to say, look,

do you do you know why you can do this? Do you know why this is okay? Let's have a conversation about why this is okay. Why can we take this equation and use this this method called elimination. What are we eliminating and, and why? Why is it an acceptable algebraic move to put these two equations together? Add them or subtract them and, and then eliminate one of these variables, isolate one of these other variables and say now we have an equation that is in terms of one variable only that we can actually solve. And having the conversation about what does equals mean, what does it mean that these two equations are, you know, that these two expressions or sorry, equations have equivalents. What is equivalence? What are we talking about in terms of this x in this y. When you say solution what does that mean. And he said it to me last night. I almost hugged him. Knowing my son hugging isn't always a thing for him, but I wanted to just reach over. I just wanted to reach over and hug him because we've now, we've now transitioned to solving systems of linear inequalities. And he, he just kind of he laid into me dad a solution is talking about this. It makes both of these statements true. And, I'm like, wow, that's awesome. You know, that he's got this idea in his head now. He's kind of understanding that that intersection isn't just a visual thing, but it's it's the solution set. It's it's right. And so he is now kind of associated not just the numbers but the numbers as a part of a solution set. And if I can get him talking about, lines as the solution set, like the visual representation of the solution set to an equation for a line, now I'll know we're getting somewhere, but so far he's got the point. He's got the point. He's got the coordinate point. And so I'm just I'm super excited about that kind of thing.

00;19;16;26 - 00;19;21;29

Music break

End of Segment 2 

Start of Segment 3

00;19;21;29 - 00;23;12;17

Joanie

Yeah. And again, that's the very similar ideas we've been talking about with our standards or vision. So one of the things that we did under the, the big conceptual category of algebra in Colorado is like we really stopped and pulled out some things that we felt like were assumptions or maybe sort of skimmed over in the previous standards.

And this is all related to the specific example. You were just talking through thinking through systems of equations and thinking through, solutions and even all the way back to the idea of what's the difference between an expression and an equation and being really intentional about that because, I don't, I don't think student I think a lot of students conflate the two.

They'll see, you know, two x plus five and think that that. Okay, I just I have to do something. Right. I also was reading an article. I was digging through some old, mathematics teacher learning and teaching from NTM, the journals. I actually get the paper copies, which I love, and they stack up my only argument as they stack up in my office and I'm like, okay, I got it. I got to get through some of these. But, I was reading through one from back in April, and I'll link this article in our show notes, but it was written by Karen Karp, Sarah Bush, and Barb Dougherty that I'm sure I'm saying her last name wrong. But they're the folks that wrote, the Rules that expire article that was really big.

And they’re writing this article, the title of the article was From “Rules That Expire to Deeper Mathematical Thinking.” And I just thought there was so much great language in here that talks about the ways that, the focus on rules for students without reasoning, without meaning making leads to trouble. Right? So, they can misapply what they're learning. They can over generalize what they're learning and they just talked about that in the rules that expire concept.

How sometimes the way we're giving instruction like leads them right down the path of, of overgeneralization, of misunderstanding or of misapplication of the, the actual rules of the mathematical concept. So they argue for this idea of including reasoning within and building up one of the lines that I really liked In the article is building students thinking so that they persevere in thinking critically, become attuned to looking for pattern, using multiple representations and making conjectures, and develop a curiosity about problems that are out of the norm from what they most often experience in a more arithmetic focused approach. And I was like, yeah, that it's not just can they do and understand the math and get the right answers in the math, but does it create this curiosity and exploration that…talk about a soapbox moment Curt. I wish everybody could see the smile that's growing on your face right now because that is that is such your passion. But when you have that, when you have support for that ability to reason and make sense of what you're doing, whether it's procedural or whether it's, you know, a contextual problem and an inquiry based lesson, that curiosity and exploration is just unmatched in terms of student engagement and motivation for learning. So I just think that's such a key component to helping kids learn and persevere and engage with mathematics that we can't diminish. And we can't, we can't decrease that.

00;23;12;24 - 00;23;28;23

Curtis

My head is just spinning right now. There's so many little things that you just said that I just, I want to talk about that. And I'll, I'll focus in on, on just the, the importance of the curiosity piece. I just, I can't overemphasize that the little fire that gets, ignited inside of students, that engagement,

The, the, the wonder, you know, we, we see that as a, as a conference theme, right?

Curiosity, wonder. Ignite all of these things are our words and terms that we use and conference themes and it's because we as math teachers, somewhere along the line, some one, some experience ignited in us a passion for understanding of mathematics. Somewhere along the line, we got this curiosity moment. I mean, Gail Burrell will, will tell you she I was at Northwest Math Conference with her at the beginning of the month. Well, the beginning of October. And, she came up to me all excited about this problem that she had just asked AI to generate, which, by the way, when you said when Robert Kaplinsky said what he did about teaching students to utilize AI that's I agree with that statement so much because you need to know what did it generate what I wanted to generate. Can I filter this? Gail came out to me so excited about this problem and she said, here's this thing. And, and it was, a problem. I don't even know if it was exactly what she was trying to generate, so much as what it did generate gave her an opportunity for exploration. And it was, you know, solving a problem where we had, a square root.

00;25;03;05 - 00;28;14;18

Unknown

So square root of five minus x is equal to, linear expression, x minus one or x plus one or something like that. And the process by which you go about solving that, introducing extraneous solutions or solutions that don't make the original statement true. But because of the transformation of functions that you have to do, you make an assumption there, introduces this.

And so I'm, I'm thinking about, oh, this is so great. And I'm so excited about that. And I played with this the rest of the day. I played with representations of this. And one of the things that comes to mind because when we've talked about this on the podcast before, the importance of multiple representations and being able to make connections among representations of mathematics, right? And I went and I was playing with this graphically and trying to represent, okay, so algebraically, this is what I do. What does that look like when I do this graphically?  How does that kind of apply right over here? And then looking at the, the transformation of this problem set that goes along the side, and that's the kind of curiosity that we're trying to generate in students, that's the kind of excitement. Now, they're not all going to connect with this this way. I'm a math nerd. I wear the shirt I've got, like, there's no, there's no doubt that I am a math nerd. And not every student is always going to be that. But what we can generate in them is a desire to kind of understand this procedure, a desire to understand why is it we humans, we want to know why. So that's just the that's part of the human condition. We want to know why. And so can we kind of link up that want to know why with the procedures that we're doing we can do that. It's teachers us. We can do that.

And sometimes it's just the way we present it. Sometimes it's just the energy with which we come to the classroom and we say, wow, check out this thing. That fire alone can be enough.

But sometimes it's making this unexpected connection, this unexpected thing that goes out into a real world application. Maybe it's the, you know, distance cars take to stop because of the different pressures that they can put on the brakes. And you get these great relationships that happen there. Or maybe it's, maybe it's something as simple as, as, you know, addition sort of representation in elementary spaces like kindergarten and, and in first grade and in doing some of these things tangibly and talking about sharing and opening things up. Right. Like there's all kinds of ways to, to get students hands on these things and create that experiential sort of curiosity. And, and that doesn't only happen when the students have a wide open task or an inquiry based thing. It can be happening when we're working on procedural tasks.

00;28;14;20 - 00;29;07;26

Joanie

Procedural skill. Yeah. Yes. I 100% agree. So as, as we shift towards, you know, wrapping up this conversation, I want to go to the place that we always go to which is what does this mean for teachers in the classroom? And why do you think, like, why are we continuing to have this conversation and why are we continuing to find additional nuance and additional ways to talk about the importance of how reasoning influences conceptual understanding and procedural fluency? And, you know, kind of challenging maybe some of the previous language around those being an either or or something that we need to balance instead of something that works together, like how do we, how do we think about actually making these kinds of shifts in the classroom? What are your thoughts about what a teacher can do differently tomorrow as a result of having listened to our conversation today?

00;29;07;29 - 00;31;58;19

Curtis

I, I love that question. So thinking about how do I do this differently tomorrow? I think the, I think the first thing I want to ask is why I may have a standard tomorrow that I'm focused on. I may have a lesson, that I've got some standards that, you know, maybe, maybe I'm in a classroom or in a school where, you know, administration expects me to have the standards written on the wall.

Maybe they're just the numbers. Maybe it's the name. Maybe it's the whole standard completely written out. I don't know. Maybe that's it's something that's written in my lesson notes. Maybe it's it's, you know, the kids don't see that, but it's something. So I have something tomorrow that I'm driving toward. I want my students to be able to accomplish at the end of the day tomorrow.

And I think the question, the first question that I'm going to ask is why?  Not only why am I doing it? Why is this important? Because that's a good question. Like, if it's not important, why am I doing it? That's a good question. But I, I think that most of the mathematics, if not all of the mathematics that we actually are trying to get in front of students, all of it is important.

So I think I think that's a pretty easy question. Hopefully. But maybe a different question is why does why does this work the way that it works? Like is there something in here where I'm going to tell my students, talk to my students about how to factor things and, and you know what? Today we are we are just going straight at this procedurally, we are diving into look, this leading coefficient has this value so we're going to work with that. These two you know the the the B term. The linear term has this this number. It has these factors. Let's work on can we establish some sort of pattern here that's going to help us be able to break this thing down into factoring a quadratic. Right. And, and instead of just line up the table, here's the thing. This is what we do. Why does it work that way? Is there some way that I can represent this? Maybe it's a graphical representation. I can mix together a graph and a table that kind of gets me a little bit more understanding. Maybe there's some cool graphic organizer way of breaking down the relationship that helps me visualize,

the relationship that we're talking about. But I think the question if I had, you know, had to sum this all up into one word, it would be to ask my my ask myself the question, why? Why, why is it important and why two why does it work this way? And is there some way that I can help my students see why it works this way? I think that's maybe the the one thing I think of when I think about this, this, this conversation we've had today.

00;31;58;24 - 00;32;30;03

Joanie

And I just want to take what you're saying and extend it even a little bit further and a little bit into more specificity, because I think what you were saying that I totally agree with what you were saying. But as we're exploring why is this important and why does it work? I think that a couple of key ideas that are like, what should I do differently tomorrow is really providing opportunities for students to explore patterns and to look for patterns and to describe patterns and to recognize patterns

00;32;30;06 - 00;32;31;22

Curtis

100%.

00;32;31;25 - 00;32;34;20

Joanie

and also doing that through multiple representations.  I think that's another really key factor, in helping get to reasoning. And it's not just multiple representations for multiple representations sake, but for the purpose of making those connections, recognizing the patterns and seeing the underlying mathematics that we can see through multiple representations that we don't see when we're focused on a single representation.

00;33;00;18 - 00;33;01;22

Curtis

Yes.

00;33;01;25 - 00;33;02;13

Joanie

I love that. I think it's doable. And I really want to wrap us up with, I'm going gonna, I'm going to own the last Word in today's podcast, Curtis, because I saw a great quote, in, another article written by these same three authors in this same issue that I just think is the perfect place for us to end.

And the quote is to change your practice, you have to practice change.

00;33;26;06 - 00;33;29;14

Curtis

That's awesome.

00;33;29;17 - 0;33;51;10

Joanie Outro

Well, that's it for this time.

Be sure to check the show notes for the resources we mentioned and others you might want to explore. We would love to hear your feedback and your suggestions for future topics. And if you're enjoying learning with us, consider leaving a review to help others find us and share the podcast with a fellow math educator.

See you next time!