Season 5 Episode 7:

Procedural Fluency from Conceptual Understanding

 00:00:00:00 - 00:00:02:00

Opening music

 00;00;02;00 - 00;00;34;00

Joanie

In this episode of Room to Grow, Curtis and I share our thoughts about students building procedural fluency from conceptual understanding. We start by trying to define these ideas and share how they often get conflated with the instructional approaches of inquiry based instruction versus direct instruction. We talk about the right order for learning procedural fluency and conceptual understanding, and what it all means for classroom teachers at the end of the day. We hope you enjoy listening as much as we enjoyed having the conversation. So let's get growing.

00;00;36;12 - 00;01;43;09

Curtis 

Well, Joanie, I am super excited to be recording again with you today. We are recording. I think it's now our 52nd episode intro podcast. So we've, been doing this a little while, and I'm, I'm really excited to have another opportunity to chat with you today and today, we're continuing the series that we've had going on all of 2025. This series on the Principles to Actions book celebrating the 10th anniversary of the book. And, really just I'm excited for today because I think this is a topic of conversation that, we'll get lots of people thinking, potentially lots of people talking, amongst themselves and at, schools and at district levels and all around. So, thinking about this, we've been doing this for a long time, this this idea of balancing, procedural fluency and conceptual understanding. So today we're looking at the principle, the, the, the teaching practice of building procedural fluency from conceptual understanding. 

00;01;43;20 - 00;03;20;09

Joanie

Yeah. I really am excited for this conversation. I think this is one of the, the meatier of the mathematics teaching practices. And I also think this is one that was a little, I don't know if it if it's innovative or something that was, you know, new to a lot of people when this book was published first ten years ago, I think the, the release of the Common Core State Standards, where there was actually mathematical language in the standards around conceptual understanding and differential that from fluency with procedures, which is I think, you know, how most people in the general population think about math, they think math is fluency with procedures.

And oh, if if you're good at math, that means you can get the correct answer really quickly to computation kinds of problems. So yeah, so I'm really excited for this conversation, Kurt, because I think there's, there's a lot to unpack here. So why don't we start by just starting with some clarity about what these things even mean? Because, you know, we've had a lot of conversation in the past about some tangential topics, including teaching, instructional approaches. And I think these, these ideas get confused with those instructional approaches. So what do we actually mean by procedural fluency. And what do  we actually mean by conceptual understanding. So if we can get clear on that then we can kind of unpack What does it mean to build procedural fluency from conceptual understanding. 

00;03;20;15 - 00;06;17;03

Curtis

 Yeah I think that's a I think that's a good place to start thinking about. What do we mean with understanding a concept versus being able to put together and execute and be fluent with a procedure. And I was I was talking just a minute ago with you about this, you know, thinking about multiplication with, array models and trying to get the grasp of a sense, really, of what's actually happening there.

It's a little bit difficult to get that sense when I'm using sort of maybe a traditional algorithm where I'm lining up, things by place value and carrying out these multiplications, line by line, right. Going, going through sort of a set of steps. It's a little bit difficult to get a sense, really, of what multiplication is doing, right. What that procedure is actually executing where I if I'm looking at this from, say an array model, if I'm, if I'm modeling this procedure, this concept of, of multiplication with an array, I get a visual picture, I get an actual sense of what do I mean when I'm going to take 57 and multiply it by 25.

Right.

And, and it's, it's cool to be able to say and students can say, hey, I've got now 57 lines of 25 or 27 lines that are 57 long, right? But to actually see that picture happen is, is an important piece of the puzzle, right. And I think, I think maybe in my description of that, there's some pieces and parts that we can talk about. What are the what do we mean when we say procedural fluency? Procedural fluency being I can execute that multiplication that relatively quickly, I can flow through whatever algorithm, whether it's a model or an array, or if it's actually sort of the traditional algorithm. I'm I'm following that procedure quickly, fluently. I'm not getting mixed up. I'm doing right. So there's that's the procedural fluency part of this thing where the conceptual understanding is being able to then interpret that response, whatever 57 times 25 is do got do that one quickly for me so I have the value. But now that I have those 57 lines or 57 rows of 25 long, or versus vice versa, 27 lines of 50, to 57 long, whatever that, that statement is, I there's a conceptual understanding of being able to now tie that response, whatever that numeric value is to the picture. Yeah. Or the, the result that happened there.

00;06;17;03 - 00;09;38;19

Joanie

Well, and not even the result. I love that you gave this example because I'm like constructing that array in my head. And I'm thinking about, you know, base ten to. Yeah. Or even extending that up to polynomial multiplication and thinking. Right. Algebra piles. Right. Like and and it's not only seeing the array of 57 times 25 where I can point to here are the 25 groups of 57 or here are the 57 groups of 25. Like I can make that really visual, but not only that, having that visual connection to understand what multiplication is actually doing, right. Like there's multiplications actually a pretty complicated one because there's more to multiplication than just this repeated addition, right? So it's more than just thinking about 25 groups of 57 or 57 groups of 25. Like that's a that's an initial way to think about it. But then you get into fraction multiplication and you've got to think about scaling instead. So anyway yeah but, but I don't think that's it. Yeah. Yeah. And I was thinking as you were talking like not only can I see the product in that array, but I can also see all the pieces of the procedure, right. So like I could point out where is the seven times five. That would be the first step of the standard algorithm for multiplying those two two digit numbers. And where where do I actually see 20 times 57. And where do I see five times 57? So, it really provides that opportunity to unthread the standard algorithm and really point to what's going on here. So that we're getting at the procedural fluency. Procedural fluency for me is about being able to get to computation answers quickly and accurately and efficiently. Right. And that doesn't always mean the standard algorithm. Oftentimes it does. But I'm thinking just quick, I want to throw this in here. Since you brought up the multiplication example, I was listening earlier this week I was listening to, a Different Math podcast, and Pam Harris was the guest, and she was talking about when her son was in school and learning multiplication. And she said, do you know your fives? And he's like, no. And she was like, what do you mean you don't know your fives multiplication tables? And he's like, I don't need to know my fives, mom. I know my tens. And once I know my tens, I just take off and then I know my fives. And she talked about how mind blowing that was for her talking to this fourth or fifth grade kid. And I'm like, that's, that's what we're going for, right? Like, her son had this understanding that there's a relationship, there's a multiplicative relationship.

And I don't have to know my fives if I know my tens. So being able to really quickly know what 57 times five is. Well, because I know what 57 times ten is, and then I can just take half of it, right? So for me, that's the difference. Like, do I have to actually line the numbers up and do the computation and carry the one and, you know, put the magic zero in like that? That may be a procedural understanding, but being able to quickly say, I know 57 times five because I know 57 times ten, that to me is where you get that conceptual understanding.

00;09;24;26 - 00;09;52;22

Curtis

Yeah. Yeah. Not I mean wow. Yeah. That's a, that's a, that's a great example of what we really mean when we start, pulling together conceptual understanding.

00;09;52;22 - 00;12;32;00

Joanie

Yeah. Nine is what I think. That's what I think gets conflated though is you know, we have there's a lot of talk out there right now around what, what I guess we're calling the math wars, which is about instructional approaches. Right. So I think that again, going back to like what society thinks math is, society thinks math is, oh, I know, 57 times 25.

I, I can I can do that quickly in my head. And, and I would imagine I mean at least the people I encounter, you know, they think what I'm doing in my head is multiplying is the standard algorithm, right? Like, oh, I can do the standard algorithm and keep track of all those numbers quickly in my head.

Whereas actually I might think about 57 times 25 a lot differently than that using using some number sense and reasoning. But my point is, when we're teaching something like the standard algorithm, it's very it, it's step by step, right? It's direct instruction. We don't we don't typically ask kids to explore and discover the standard algorithm for themselves. What we what we do is we say, okay, here's how you here's how you do two digit multiplication. First you multiply this number and that number, and you write this here, and you write that there. And right. You go through this step by step. And I think that is traditional. How many of us I certainly learned math that way. Here are the steps to take to solve this kind of problem, where you're told what to do. And then your job as the student learning the mathematics is to repeat those steps as they were taught to you. And then and then there's this other idea of inquiry based, where, you know, that gets a little bit conflated with conceptual understanding, because it's often how conceptual understanding is approached instructional, where we give students the opportunity to explore and make sense and not direct their thinking, not direct their approach, but let them access it themselves.

And I think that it's often, a misconception that conceptual understanding has to be taught through inquiry, and that procedural fluency has to be taught by direct instruction or explicit instruction. So I think those two things get, mixed up. And I would like for us to try to tease them apart. So what's your thought about how what is meant by conceptual understanding and procedural fluency is different than how we would teach them what our instructional practices might be?

00;12;32;00 - 00;12;42;10

Music break

End of Segment 1

Start of Segment 2

00;12;42;08 - 00;15;30;15

Curtis

I think you lined it out beautifully there. The idea of, inquiry is exactly what you said. This idea of allowing the students to discover or explore or, or get into a space, a problem, a task, and begin trying to do their own pattern recognition, their own sort of. Well, what if I try this thing? But the tools in their toolbox are the procedures that they, that they have and I think, you know, when I think about, even this problem that we were just talking about with, you know, some large two digit times two digit number, 57 and 25 and, and thinking about how do I, how do I explore that problem, how do I turn my kids loose with that problem in some sort of inquiry based thing?

Yeah. It may lend itself better maybe to you to developing that concept, but that wouldn't be that. That is different. So the the instructional approach of inquiry, like inquiry, is an instructional approach. Conceptual understanding is sort of a result if that like that. Yeah I do and, and in the same way direct instruction is is an instructional method. Right. And, and procedural fluency is, is a result. Right. And and I think we can do both. We can mix them together. Right. Like we can do these things. We can weave them together. And I know I'm kind of leading the conversation to something we're going to talk about in just a minute. But we don't have that. One does not always result in, so direct instruction could if I, if I do it well, we can actually talk about a concept. Yeah I can, I can directly tell you some things. I can show you some things that. Yeah maybe it might be, it might happen better if you had inquired and discovered it on your own. But sometimes I have to give you enough for you to get over the hump, to be able to find out or figure out that thing.

Right. And so there is some there is some requirement of direct, there's some requirement of inquiry and discovery. We don't want to just create a whole bunch of little robots doing calculator, calculator math. Right. We don't want a whole bunch of little calculators. We have calculators for that. What we want is we want to, like you said, build sense makers. We want to have students make sense of things.

00;15;30;13 - 00;18;02;18

Joanie

Yeah. For sure. And I want to come back to what you were saying, because I think even just saying, like the way to teach conceptual understanding is through an inquiry based approach and the way to teach procedural fluency is through a direct instruction approach. I don't think that's even true either.

So I think that, you know, I'm thinking back to, Pam, the example of Pam Harris's son. Like, how did he get that? I don't have to know my fives if I know my tens. Like there's some procedural thinking still in there, right? First. Like, he's combining this like, why? Why would why did your face even light up when I was saying that? Why did my face light up when I was listening to that in the car earlier this week? Well, it's because we know the the conceptual understanding of multiplication, and we know that, multiplying by ten is the easiest multiplication. Like, you know, very early on, students can learn their tens because it's really easy to do. It's easy to do mentally.

So this idea of like what cues us all in is that I'm going to do the easy multiplication rather than the harder multiplication. I'm going to use the easy computation and the relationship. I understand. But when I think about how did her son figure that out, it's likely through pattern recognition and the teacher drawing his attention, or his classmates and his attention to these relationships.

Right. So like at some point he had to understand that five is half of ten, and that relationship holds through multiplication. Like, that's a huge concept. And I doubt that it was as simple as his teacher saying, anytime you're encountering a multiplication by five, you should just multiply by ten and then divide the product in half. Like I doubt that that was taught not right, but it could be that the teacher drew the attention through, you know, maybe a multiplication chart.

Like, what do you notice about the row of multiplying by ten compared to the row of multiplying by five? So, so that idea of of guided instruction I think is may be better than talking about inquiry versus explicit and direct. I think there's a middle ground. Right. It's not black or white. There's a way to kind of do some direct guidance to make the inquiry effective, if that makes sense.

00;18;02;20 - 00;21;39;15

Curtis

Well, that's what I it's, it's semantics at some point where we start throwing these words around. But I, I was just thinking about, you know, you, you use the word drew, drew their attention. You could just as well say, directed their attention and, and, you know, hey, look over there. Because if somebody asks for directions, I'm pointing them in a in a direction, right, right, right.

I am not, putting them on the train tracks and saying, figure it out. Yeah. And saying, go, because the train tracks will keep you there. I, I'm and this is a semantics thing that I, I'm getting caught up in. So just bear with me for a second. No, no, you're good, because I know you're. I don't want to distract here.

I don't want to distract from where we're headed in this conversation, but I, I am caught up on this idea of, direct, instruction. I think often what goes through our minds at least goes through my mind but it's changing in these, in these last 30 seconds, What goes through my mind when somebody says direct instruction is telling them, I am telling you this is the thing, right?

I'm telling you, this is how you do this, right? This number here, put this number one do this do. Yes. We're telling. That's what goes through my mind when I hear direct instruction. I think that's what most of us do. I would agree, but there is a I'm I am going to occasionally need to direct guide. This is the word you used. But it's still I'm still directing you, to a thing. And so it is a fine nuance. It is a, it is a, a semantic, conversation. They're like thinking about, well, what is the word we really mean? But I, I think you you laid it out nicely for, hey, we're drawing some a students’ attention. He didn't probably just all of a sudden say that occasionally my son Truett does that. And it's just bizarre to me when I'm sitting here chatting with him about mathematics and all of a sudden he says, well, I see blah, blah, blah. And I'm like, how did how did you even know to think that? And, and I think we are surprised often we we are if we give students the opportunity, we can be surprised often at their creativity.

However, those moments don't always happen by themselves. Agree. There's some prompting that you can give. There's some guidance that you can give if there's you know, we had a conversation, I think, last month about questioning. Right. And, and the idea of of how we develop understanding through the questions we ask. And, and it is this same kind of thing.

We can use questions. We can ask questions in order to guide them or direct them to a particular, outcome or to a noticing, you know, his teacher may or may not have pointed out the having concept. Right. But in the example that I was just we were just working on 25 times 57, I've been trying in my head like, how am I going to do that? I can't like, do this. As we've had this conversation, I finally came up with one I want if I multiply the same one I came up with, if I multiply 57 by 100 and divide by four and divide by four.

00;21;39;15 - 00;26;50;29

Joanie

Or divide by two and divide by two again, that's what I was thinking. Which is why I brought up Pam's son. Because, yeah, that's in my head now and again. Okay, so a couple things I want to respond to. First of all, yeah, I agree with the like direct sort of creates this conception that we mean, I'm going to tell you exactly what to do. I'm going to give you one process. You need to do it exactly like I did it.

That's how I also think of direct instruction. And I remember back, when I was supporting teachers in a district math coordinator role and working with middle and high school teachers about trying to bring more conceptual understanding into their into their lessons and trying to, you know, really think about mathematical knowledge and mathematical, skill as has how also being able to understand concepts and not just being able to produce procedures.

And the word that I use because people do conflate this conceptual understanding with inquiry based instruction, the word that I used that maybe is a little softer than direct instruction is explicit. Right? So helping students explicitly understand the mathematics that's going on. So, I just wanted to kind of throw that word out. There is a word that we could use that isn't we don't mean to tell them exactly how to do it and how to think, but we do mean make it clear and make it obvious to them what's happening mathematically. So I really I really like that word. And then I also want to talk about the fact that when you have both conceptual understanding and procedural fluency, then and procedural fluency is about this being able to choose the right strategy. So I'm going to tell a story I know I've told before on the podcast, but hopefully I didn't tell it in our last episode.

So, it's not too recent. But I have twin niece and nephews. Andrew is the nephew I'm always talking about on the podcast, because I've had the opportunity to work with Andrew on his math for a lot of years. And good news he doesn't call me for math help very often anymore, which I take that as a win. Although I kind of am sad. I'm kind of like miss doing math with Andrew, but I remember back when they when they were maybe in third grade and, my brother and sister in law were first saying, hey, Andrew's really struggling and, you know, can we talk? I had a colleague of mine do a little sort of interview assessment on both of them, actually. because Emma. Anyway, side note, there, my friend did this interview assessment and to talk about like what? To understand what they understood about math and also how they thought about math and I kind of sat and just even eavesdropped on the interviews. And I remember Emma is a very procedural thinker. Emma is very much like I was when I was learning math. Like, tell me the pattern, tell me the steps, and I will repeat it and I will get it right every time. Andrew was struggling because he wasn't coming up with his math facts quickly. He wasn't, you know, following computation the same way that his teacher was wanting him to in third and fourth grade, which is why he thought he was bad at math. What we learned is Andrew has incredible number sense, and he builds all of his computational thinking from his number sense. Emma is very procedural and doesn't go to her number sense. So, the example where this became obvious was when my friend said in this interview and the kids didn't have paper and pencil, they had to do all their computation mentally and the mental computation was, what's 204 -199? And Emma had to like, I could see her looking up and she was kind of nervous, and it took her a long time to get there. And she's like, five so she finally got there. Andrew, when he got the same question, was, gave us this look like, are you kidding me?

It's five. Duh. Like it was so easy for him. And in both cases, the interview follows up with how did you think about that? And Emma's like, well, I, I imagined 199 under 204 in my head. And I had to because four minus nine doesn't work, I had to borrow. But then it was 20. So then I had to. And she's explaining how it took her so long to get there, because she's trying to mentally do the standard algorithm for subtraction, whereas Andrew said, I just counted backwards from 204 to 199. It's only five, so that that to me is procedural fluency, right? The flexibility to choose what's the right approach here. And you have that flexibility when you can make sense of what's happening and you can think about, like Andrew wasn't stuck on the way to do subtraction is the standard algorithm he was making sense of what do these numbers mean and what does subtraction mean? And how can I use what I understand about those things to get to an answer?

00;26;51;29 - 00;28;44;11

Curtis / Joanie / Curtis

That's beautiful. In my head, I was, adding one. There you go. A same idea, right? Like you're using the fact that 199 is one away. I'm sorry. I'm putting words in your mouth. Maybe that's not. I mean, that's oh, that's exactly what I was doing. But, you know, what's funny is, my first thought was actually, if you had asked me to add those two, I went to sort of this memorized thing like I have unfortunately, as a, however, rolled up person, I am, I, I have now formalized what we hope students, to do sort of as a sense making exercise. I have now said, oh, anytime I'm working with these large things, and I see that one of them ends in a nine, I'm gonna add one and make it easier. Yeah, well, I'm going to figure out how to make that the next larger one. Yeah. But the operation that you're doing, whether it's addition or subtraction, is a opposing.

Right? So if I, if I add if I add one to both then I can figure out the subtraction easily. If I take one from this one and add it to that, that's how I figure out how you use an easily and and I've, I've, I've gotten it to the where it's now a procedure and that's I don't think a good thing in my head honestly because then it's easy to apply the wrong procedure. Like if I, if, if I'm not thinking conceptually about what's going on here, it's not just applying a root procedure. Yeah, yeah, yeah, it's easier to do the wrong one when it's this. I always do that. Exactly. And that's what happened.

00;28;44;11 - 00;28;55;10

Music break

End of Segment 2

Start of Segment 3

00;28;55;10 - 00;31;49;15

Joanie

I'm going to tell another story. I'm full of stories. Today we had our Colorado Math Teachers conference a couple weeks ago, and we had a presenter who was wonderful, by the way, and I would love to have her on the podcast. So I'm going to withhold her name, just in case we don't get her. But anyway, she had the like, we split the room so elementary folks went into one half of the room with another presenter, and she was working just with the secondary, middle and high school math teachers, and she gave us a problem that was a ratio problem, right? Was something about like two cups of pancake mix make enough pancakes for three people. And she said, I want you to use that relationship to find out how many cups of pancake mix would you need to feed 35 people? And she's like, I don't want you to do it procedurally. I don't want you to do it algebraically.

And I was sitting at this table with five high school math teachers and one sixth grade teacher, and the five high school math teachers couldn't do anything except set up a proportional relationship and cross, multiply and then divide to find the answer. Like that was literally the only way they could think about the problem. And their sixth grade teacher started drawing double number lines.

And then she did a little, you know, strip diagram. And then she's just making a graph, like she had 100 different ways to solve it. And I just thought, how powerful is that? That as high school teachers were so stuck on procedures and like, here's how you solve this kind of problem. Now, I'll tell you, the high school teachers had the answer. And you can anybody listening is thinking in their head right now like, oh, it's not going to come out an even number, right? It's going to come out with a fractional number of cups of pancake mix, which is the beautiful part of the problem. But to understand what a proportional relationship is and what does a ratio represent, and how can we use this multiplicative thinking to to understand what's underlying was so much more obvious from the sixth grade teachers thinking than it was. And as she was explaining to the rest of the table how she used these different models that weren't computational, that weren't just an algebraic process to think through this problem, that the high school math teachers minds were blown. And it was it was so cool. And it made I mean, I don't know that there's one moment, but it made me think about how much of what I taught in high school.

I didn't ever come around and help kids understand the the concepts that would have helped them get better at the procedures. So I, I want to let you respond, but I am also recognizing time, and I want to kind of circle us back to one component of this that we haven't talked about yet. So I'll shut up for a minute, let you respond.

00;31;49;15 - 00;34;07;29

Curtis

That's all right. No, I, I've, I've, I've sat at that table. Yeah, I've, I've sat at that table and I have stumbled through trying to come up with something other than your proportional reasoning. Set up the, the really set up two ratios, set them equal to each other across multiple, you know. Right. I and I am 100% like I love the fact that you had that opportunity to kind of observe that and and experience that and that your teachers did too. And that's that's so cool that I what I, I want to I want to be careful. I love that there's so much conceptual understanding happening there. Right? Yeah. And and so much more experience that can happen there too. And you're the person who led the conversation was very intentional, right. In in saying we're not we're not interested in efficiency right now.

We're not we're not trying to get to the answer as quickly as possible, as quickly as possible. I want to I want to know that you have, grasp hold of what is going on. Yeah. When I ask you to figure out how to serve 35 people. Yeah, with this pancake mix and and and what is that relationship between these two things? Is what she's really asking there. Right. And how do we kind of come up with what that relationship looks like, how these things vary together, how we actually can can okay. Now that we know how they vary together, let's see how we get to that. 35 people servings. Yeah, serving for 35 people. And what we need to do that. So I, I love that that's that that's the experience there. But just calling out that it's not it's not bad that the high school teachers went straight to the efficient. Sure. Absolutely. Because that's what I mean. Eventually we are wanting to be efficient with things, but the fact that they had an opportunity to go and try to really concentrate on what's going on here, I think that's I think that's the beautiful part.

00;34;08;06 - 00;35;32;09

Joanie

Yeah, I do too, and I think, I think it's a really nice connection of this mathematics teaching practice. Right. Like the procedure of fluency is important. Let's let's be real. If I'm going to host 35 people for pancakes, I don't I don't I don't want to draw a tape diagram and spend 15 minutes figuring out how many cups of pancake mix. I'm going to set up a proportion cross multiply, and I'm going to get the answer fast. But, but the idea of building that from conceptual understanding, is, is what we want to have happening in classrooms. We want students to have the experience of both because both are how they build true mathematical knowledge that that they can then apply to new situations.

Okay. So the last thing I want to I want us to explore, Kurt, is one of the criticisms that I often hear about this mathematics teaching practice or or concerns about this is it people think it's implying that you have to do conceptual understanding first before you go to procedures. So they read this literally as an order explanation that in order to build procedural fluency, you have to first have conceptual understanding. So I would love to you just kind of respond to that. What do you what do you think about that?

00;35;32;09 - 00;39;31;28

Curtis

Well, I think that that's I think there's some things that are implied here. So I was reading back through the sentence just now, and, and I'm going to, I'm going to, I'm going to read it because I'm not sure we did at the beginning, but if we did, I apologize.

I don't think we did. So the, the, the description of this practice says this effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual mathematical problems. So I do think and I want to be I want to be very careful here because I do think that, you know, the wording was chosen purposefully there.

Yeah, that there is a there is a foundation of conceptual understanding. I've got to be able to sort of build procedural fluency based upon understanding what's happening in this concept. However, I think that there are times and often many of them where skills are important to be able to build this, this set of, of conceptual understanding. And I think also there are procedures and procedural flexibility, if you will, that comes from, you know, exploring these ideas in, in different ways.

Right. So, so which one comes first, the chicken or the egg. And and which one is priority for me. Right. I it's you mix them well. You integrate them well. Right. We want scrambled eggs. Love it. I mean, this is, this is, this is exactly the the thing I, I think this whole idea of which one is better and which one do we have to do first and which one? It changes depending on the day and on the content and on the set and where my kid is and how they think and, and and what we're trying to get to if efficiency and making sure that I've got the ability to multiply my tens or my fives is, is what we're being measured on, and we want you to be a little calculator. Well, then memorization is a thing. Like I'm doing it. Yeah, but if what I want you to be able to understand is what's happening whenever I ask you to multiply, I serve 35 people. Yeah. With, you know, this, this pancake mix ratio. And this is what, this is how much you can serve with this many, this many cups.

So how much does it take to serve 35 people? I think there's a little bit more manipulation and understanding of of how things vary together. That has to happen. Yeah. In order for me to get there. And so the right order. Sure. There's times when introducing a concept. I might want to do some procedures with you first and have you ask, why does that work?

Right. Because now you're vested and I can help you. We can start exploring why. Right. And that's where the concept kind of comes in. Why did you manipulate this this way versus sometimes I might want to lay out this tape diagram and have you start building things on this and have you recognize, oh, so when I do this, that's what's happened. Here's what's happening. And so I think I think there's this interweaving that has to happen. And it isn't necessarily a right order quotations around. Right. I don't know that it's always one or the other. And sometimes and maybe most of the time it's both. 

00;39;31;28 - 00;42;10;05

Joanie

It's both. I totally agree with that. And I think even the I didn't bring that pancake mix up example to be able to tie back to it here, but I think it's a great way to point out like it's never we never stop learning.

You know, this is why we called this podcast Room to Grow. Like it doesn't matter how much math. You know, there's always another way you can understand it. And that's what I actually saw happen with these high school teachers. Right? They definitely had their procedural fluency. They definitely knew how to figure out how many cups of pancake mix do I need for 35 people. But by watching their sixth grade teacher come up with different diagrams and different, representations, that explained why the answer was what the answer was, I think really opened their eyes to what? Why is it that I'm multiplying by two and dividing by three like they knew that that's the computation I need to do multiply by two divided by three, but having a having a deeper understanding of why that is, and what does it actually mean, you know, that that gave them a deeper level of understanding about proportional relationships and, and about that process. So I agree with you. I think, I think they they have to go together and the other thing I want to say is like the folks that that read this very literal as saying, you have to do conceptual understanding first and then build procedural fluency like how learning doesn't work that way, like we don't master something and then move on to the next thing, right?

Like we're constantly coming back and that was certainly my moment when I became a math teacher. That's when I really learned math, because it was this coming back and revisiting ideas from a different perspective and a different layer. And I think I could I could do procedural fluency first. I could be really good with the standard algorithm like my niece was.

And then as soon as she hears her brother say, oh, I just thought about this. And now she's stronger at her procedural fluency, right? Because she has that a better understanding of a different approach. So yeah. So all of that to say, I totally agree, I don't think it's I don't think it's one before the other. And ultimately you have, you know, practically you're going to choose to do one approach before the other, but it should be a constant weaving and a constant balance of both. I think where we get in trouble is when we do one without the other.

00;42;10;07 - 00;43;05;20

Curtis

I think that's exactly it. I think the, the danger is when we start to abandon one or prioritize one, right over the other, and we aren't allowing the flow of the learning process to kind of guide where what do I need to do right now in order to further develop the sense making?

And I think I, you know, as we move maybe into the last part of this podcast, thinking about, whoa, what does this mean, then for me and my classroom and, and for me, maybe as a coach as I'm, you know, talking with my teachers that I'm working with or, you know, me as a parent, me as I'm helping my whatever role home or whatever my role is. What, what does this mean for me? What are we hoping that that students have as a result of this conversation?

00;43;05;27 - 00;43;58;21

Joanie

Well, I think it's easy. I think we've been talking about it from the very first comments of this episode, and that's around making sense. And, and if we think about instruction, should prioritize student sense making. It's, it's back to what you were just saying about the order.

Sometimes sense making is going to come from seeing the procedure first and then backing into what's the conceptual understanding that led to that procedure. And other times, sense making is going to be about exploring the concept first, so that as I learn the procedure, I have understanding to anchor the procedure in. So again, I think I think the bottom line is if we prioritize sense making, that balance will come and the order will be what it needs to be.

Show Close

00;44;00;23 - 00;44;15;20

Joanie 

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!

00;44;15;21 - 00;44;23;00

Music out