SPEAKER_01 0:00

Welcome to Room to Grow. I'm Curtis Brown. And I'm Joni Fenderberg. We work together at Texas Instruments, and we're glad you're here. We're looking to continually improve our practice, and we understand that you are too. We hope that you'll find this podcast as a room for you to grow along with us as we wrestle with and explore ideas about teaching math even better. In our final episode of season one, Curtis and I discuss the importance of balancing conceptual understanding and procedural fluency in math. We start by sharing our individual definitions of conceptual understanding and why we think it is sometimes perceived as more important than procedural fluency. We consider what happens to students' understandings of mathematics when these are out of balance, and we really wrestle with the question of which order they should be taught. This conversation only begins to touch on these important ideas, and I know we'll explore them further in season two. But in the meantime, we hope you enjoy considering these ideas alongside us. So let's get growing.

SPEAKER_00 1:01

Well, Joni, thank you again for joining me again today. It's so fun to just get to uh record another session. This is actually going to be our last session for 2021, which um it's amazing this year has gone by so fast. Um, and we've gotten to do a whole bunch of really fun stuff together in recording these sessions. So um I'm happy to have one more uh chance to record a new session with you today.

SPEAKER_01 1:26

Yeah, me too, Kurt. And it's, I mean, it's it seems weird to call episode 11 a milestone, but it does kind of feel like one for us, like this being our last episode of 2021, which we're gonna call season one. And, you know, I know you and I have had some conversations already to start talking about and thinking about season two, also. So uh super excited to be wrapping up our first season and uh hope that our listeners enjoy this last conversation that we have planned today. I know I am super excited for this conversation. I think this is a topic that you and I uh are both super passionate about, and uh really there's a lot of interesting things to unpack on this one.

SPEAKER_00 2:04

So I think so. I think so. This is gonna be a really good one, uh, chatting about um really this idea of um sort of the balance or in some cases the imbalance uh of conceptual understanding and procedural fluency. And I feel like this is a pendulum that's been swinging in mathematics for probably hundreds of years, at least the last hundred years for certain. Yeah. Um, but maybe even longer than that, is this idea of should we be spending our time looking at conceptual understanding and really that's our main focus? Or should we be spending our time uh focusing on procedural fluency and that's like our main focus? And and and you'll find folks who are passionate on both sides of this fact. And and really this is one of those places where you and I we really uh have some interesting thoughts, I think, but uh we want to strike a good balance between the conversation, right? Like that's our ultimate goal. And we want to provide a place for everyone to to have uh some thoughts um and to wrestle through some things. And I think that's what's gonna happen today. So let's uh let's dive in.

SPEAKER_01 3:19

Okay. Well, here's where I think we should start. I I know that when you say conceptual understanding and procedural fluency, specific to mathematics education, those those terms mean different things to different people or they conjure up different ideas for different people. So why don't we start since we're gonna be talking today about effectively balancing conceptual understanding and procedural fluency? Let's start by just calling out right here at the beginning what we mean by each of those terms. What when we say conceptual understanding, what is it that we're talking about? We say procedural fluency, what is it that we're talking about? And certainly acknowledging that, you know, we're not we're not the end all be all of what those definitions are. There's certainly lots of room for uh interpretation, but just to frame the conversation for our listeners um and anchor our listeners in what we mean. So, Kurt, when I say conceptual understanding to you, what is that, what do you think of? What does that mean to you?

SPEAKER_00 4:16

Well, probably the first thing uh that comes to mind for me is uh a little bit about what was in adding it up. Um, if you found and read that that article, um, that book rather. And it was one of the best things that have ever happened uh to me was somebody handed that to me uh as a new teacher when it came out. I I was uh I needed that. Um and it and it really was beneficial to me. But here's the definition that that um is given there, and that is it come it's comprehension uh of mathematical concepts, operations, and relations. Okay. And and that sounds, that's like a lot of fancy work and and and idea, but it's it's we're stepping away from the uh the steps that go into every little thing, right? And instead it's the ideas and the um the processes that go on behind the scenes of each of those things. And I I have a hard time explaining this without getting into um maybe concrete examples of this kind of thing. So when I think about this, um, I spend a lot of time working with my son um in his arithmetic, that he's still working on learning uh the the skills of of working with numbers. And right now he's in fourth grade and we're multiplying two digit uh values by two digit values. And and so this this the concepts behind that are the things like um multiplying eight tens by five ones, right? And then eight tens by four tens, right? And talking about it in that fashion and in that way that we're visualizing, hey, we've got eight, 10 blocks and we're gonna make an array that's gonna be four, 10, 10 rods wide or whatever, and trying to grab those two things side by side. So that's really the conceptual understanding, at least in my mind. And again, I don't wanna be come out there and say, hey, I've got all the answers or I'm the expert on this. Like this is just what conjures up in my mind, and I think probably a lot of math teachers' minds.

SPEAKER_01 6:27

Yeah, I think so too. And I think this can get uh even hold a little bit of controversy, right, in terms of what this means. But I'm gonna I'm gonna share how I think about conceptual understanding. And I I totally agree with what you said. I would just extend it a little bit to say, I think it's easier to describe what happens when you have conceptual understanding than to say what conceptual understanding is. So for me, it's it's understanding not just what to do, but why, and why mathematical relationships or even procedures or or mathematical facts why they're true. And to be able to, you know, justify is maybe a little overused, but like really support. And then there's a flexibility that comes with conceptual understanding, where then you can say, okay, when I'm presented with a brand new situation that's not like anything I've ever studied in mathematics before, I can bring ideas together in a new and unique way. That's to me, you have conceptual understanding if you can do that, where you can bring ideas together that you haven't been taught how to bring together before. So it's this flexibility and this it's this ability to apply your understanding to a situation that's completely unique to you. So that that's one of the pieces that I would add to what you said. But I think part of what makes this complex is that conceptual understanding, the way that we're we've talked about it and the way that you know we're gonna engage in conversation today, it doesn't live in isolation. It's not this like thing that can just exist by itself, right? It's it's part of this bigger picture of mathematical understanding of mathematics learning. So I think it's hard to nail down specifically what it is because it actually is um it plays with procedural fluency. And even, you know, you referenced adding it up. Like adding it up just defines um mathematical proficiency with five strands. So it is conceptual understanding, it is procedural fluency, but it's also strategic competence and adaptive reasoning and uh and a productive disposition. And again, all of those things sort of thread together in such a way that um, you know, pulling one of them out certainly unravels the rope. It's that it's that rope image in adding it up, right? Like the five little strands that weave together to make a strong rope. And it's definitely a less strong, um, a less robust understanding of mathematics if you're missing one of those. But you really can't talk about just one all by itself.

SPEAKER_00 9:09

No, and I think that's I think that ultimately that's what you and I are trying to talk about today in this idea of balancing, striking the balance between uh the this this pendulum that we've seen fly back and forth and the discussion is that we really can't just say, hey, we're really gonna develop um conceptual understanding, like that's gonna be our focus, and procedural fluency will just happen as a result of that. I I mean, I I I'll admit I've probably been guilty of even saying something very similar uh to that thought process because it sort of made sense to me to say, hey, like if I if I can kind of manage uh the concepts that are coming up, or if I can, if I can use what I know about mathematics to kind of explore and develop um this new idea, like then the procedural piece will come. But there the people that that talk about practice and those kinds of things, I mean, it's true. Like if I sit down, I mean, there's a reason why people invented flashcards, is because it did work in some form or fashion. Like this is it's right, right? It does what it's supposed to do, um, which is help me memorize facts. Like I can memorize facts from that, but do I understand them? Well, then that discussion becomes a little bit different, right? And so we need both. We need to be able to talk about both sides of that of that coin. And and and really, in in the way adding it up talks about it, really, you you are putting all of those five pieces together to create this one big overarching idea.

SPEAKER_01 10:50

You know, NCTM put a lot of attention into this in all of the the standards work that they've done really since 1989. Um, but I think it's gotten, especially this term conceptual understanding, has gotten so much attention in the last 10 years. I know that uh when I was, I, you know, my background working in a district and I was supporting middle and high school math teachers when our state, Colorado, adopted the Common Core State Standards. And this idea of conceptual understanding was a really big, like new idea. And it's actually not new. I mean, adding it up predates Common Core by almost 10 years, and and CTMs work with standards, you know, predates by another 10 or 12, even before that. So it's it's not new, but I think it conceptual understanding got so much attention in Common Core and then, you know, whatever new adapted versions of Common Core that states have adopted in terms of college and career readiness. And I don't think it's because it's new, I think it's because it's hard. And I think it's because it's it's lacking. You know, I know we've we've shared often, Kurt, that as math educators, and I would bet this is true for a lot of our listeners too, right? When you tell people at a dinner party or somebody that you meet on an airplane or whatever, like, oh, I'm a math teacher. Oh, I I didn't like math. I was never good at math. Like there's this sort of common thing. And I think a lot of that is tied back to because a focus on conceptual understanding wasn't a part of our math education for a lot of us when we were children and we were growing up. Right. It was a focus on procedures and answer getting. And and people, I think a lot of people still, even adults now, think that that's what it is to be good at math. That means you can get the right answers really quickly. Um and they don't think about how important it is to know that there's something behind those right answers. Like I was thinking when you were talking about flashcards, there are people who would be like, what do you mean why five times eight is 40? It just is. Like that's just a fact. There is no why behind that. Like even the fact that there is something to understand deeper than what's the answer to five times eight is evidence that that conceptual understanding has not been an appropriate part of math education for a long time.

SPEAKER_00 13:06

That's exactly yeah, that's exactly right. And and I mean, we can get lost in that um discussion about um, you know, which one should we have and which one should we. I I don't think that we want to get too buried in that conversation, but I do think we should get into the conversation um a little bit about what is the balance or or maybe even why the balance between these two things. Because I think, you know, just like you said, we we can err on one side, we can err on the other side, we we can uh argue make really good arguments for why both things are good. Right. And I think that's why we want to maybe start this conversation or or or continue this conversation about why the balance now is important. And I don't think that you and I having this conversation about uh balance is uh is a new one, right? I think we're echoing the work of many, many people out there um who are talking about this and saying how these two things work together. Um, and how do we how we actually get towards uh a really good foundational understanding of the mathematics that's um at work in in all of these things that we're talking about? And so um I think maybe let's take our conversation in that direction. So why is it that we should have uh balance between these two things?

SPEAKER_01 14:30

I like that. Yeah, I like that a lot because I think there is there is maybe even a little bit of culture wars out there around which one is more important. So I I think I think you could comment on it. I would imagine our listeners. Yeah.

SPEAKER_00 14:44

There's a culture war in my household about which one is more important.

SPEAKER_01 14:49

That's hilarious, Kurt. I'm gonna I want to hear more about that. So cue up a story for me.

SPEAKER_00 14:54

Um that might be an offline one. I'll get in trouble. My wife might listen to the podcast. I don't know. We'll see.

SPEAKER_01 15:02

You don't want you don't want yeah, you don't want to do that. She might listen to the podcast indeed. Okay. Well, I'm thinking about some specific examples, and I would bet our listeners have these examples too. So I remember being in a training um when I was a classroom teacher, and the trainer was helping us try to understand this idea around, you know, teaching kids more than just answer getting ideas and answer getting techniques, but really focusing on, you know, building a robust understanding of actual mathematics. So they were were contrasting two separate videos of um young girls doing mathematics. So the first one was this uh maybe a first or second grader. Um, and she was doing a double-digit subtraction problem. So a two-digit number minus a two-digit number. And it was something like 56 minus 28 or something like that, where um there was gonna have to be regrouping or borrowing happening um within the um within the procedure. So she they're videotaping this girl and she sets up uh the standard algorithm for subtraction, and she does the subtraction incorrectly. So 56 minus 28, instead of doing, you know, looking at the ones digits of each of those numbers uh and reading it as six minus eight, she what often I think kids do at that age is eight minus six. She just subtracted the smaller number from the larger number. So she ended up with an incorrect answer. And the interviewer didn't correct her, just handed her a hundreds chart and said, Okay, can you show me that same subtraction using a hundreds chart? And this was a tool that the student had familiarity with. So she used the hundreds chart and she got a different answer than she had gotten when she applied the standard algorithm because there was an error in her standard algorithm. And she was really confused. And you see her go back and try to like redo it. At first, she tries to like subtract by tens on the hundreds chart, where she just you know goes a row down and a row down and then and then do the eight. Um, and she gets a different answer than she had gotten with the standard algorithm. So then she goes back and actually counts back 28 spaces one at a time from 56 and gets the same answer, which is actually the correct answer, but it differs from what she got of the standard algorithm. And she's so cute, she gives us like, huh, this little confused face. And the and the interviewer just says, So what do you think is going on here? And she's like, I don't get it. And the interviewer asks her, Well, which one do you think is right? You know, she explains that I get this answer with this with this way, and I get this answer with the hundreds chart. And the interviewer asks, which do you think is right? And she points to the standard algorithm. And she says, This is the way my teacher does it. So this has to be right. And and she can't get past it. And it's it was just so painful to watch. Like when she used the hundreds charts and she used reasoning about the numbers, she was able to get to the correct answer. But when she applied the algorithm, there was a misconception in the application of the algorithm, but she was convinced that that was right because that's the way my teacher does it. So it was super interesting. And then and then we were contrasted with a video of like a sixth grader who's doing um, who's representing mixed numbers as improper fractions. And again, she makes a mistake, but she catches her mistake and fixes her mistake in the explanation with the interviewer. And um so so it's a similar thing where when she does the procedure, she makes an error and then she reasons through it and catches her error and goes back and fixes it. And the interviewer is like, Well, how do you know which one is right? And she's like, Well, this is this one makes sense because I know this, right? So going when she when when the student is anchored in conceptual understanding, she can reject that, oh, I made a mistake in my computation. Whereas the other student is focused on the procedure, and this is I'm telling a very long story here to make a point. But it's the idea of like when conceptual understanding is solid, then errors in procedure can be found and made sense of more easily. So I think there is an argument for saying you've got to have conceptual understanding to help correct procedural fluency errors. But I think there's also uh argument the other way. So what do you think would be what what do you think would happen when there's more procedural fluency? Actually, I'm not even sure which way I just argued, Kurt. What would you add to what I the story I just told you?

SPEAKER_00 19:48

That's okay. That's okay. I I I'm so I was listening to your story and and thinking about these two students and thinking about this quote that you shared with me from an article um, or rather a blog post by Grant Wiggins. And I I was looking at that and thinking about what you were talking about there, uh, how these two students are are um using procedures. And I made this point earlier when we were talking about uh this stuff that that procedures and um procedural fluency and facts and applying those things, those standard algorithm kinds of things. And I and I I want to be really careful here. Um because I I feel like I I feel like we could um take this to, hey, we're we're bashing on standard algorithms, or we're we're saying that uh teaching uh methodologies for accomplishing things like completing the square or factoring uh a polynomial, or for heaven's sake, we should ever mention multiplying two binomials together and this idea of foil, right? I I don't wanna I don't want to come out here and just say, hey, we need to bash on all of those kinds of things and this idea of Nix the tricks and all of that stuff, which I mean, there's certainly good pedagogy behind all of those statements. But I also don't want to say like, hey, we need to just be on this uh bandwagon of of always doing conceptual understanding. And I I'm I'm gonna maybe make this might be the theme of the podcast here, is me coming back and making sure that we established we're not bashing on procedures here. So I I again I don't want to pick on the the students' use of um these algorithms, but those algorithms feel like to me a set of train tracks. And I get on these train tracks and I can basically just push the the forward button. And as long as I'm as long as I am okay with following the rules of those train tracks, I don't go too fast, I make sure my work is neat. I kind of I go through the steps one at a time and I just follow the the cookbook, if you will, then I'm gonna end up at at the appropriate response at the end, right? I'm gonna be able to get the arithmetic done. I'm gonna be able to compute um the the multiplication of those two binomials together, and I'm gonna come out with this polynomial that's gonna be great, and we're gonna have this that that's gonna be my answer, and that's awesome. Um but the idea of that conceptual understanding um allows me the ability to realize when I've gotten off the tracks, to realize my mistake when I, when I've when I've gone too fast for the tracks and I've derailed, or or you know, I've I've gotten off track and I took I took a wrong turn, the switch wasn't in the right spot, or whatever. Like however you want to take that train analogy. I I just I feel like that conceptual understanding is are the guardrails that kind of keep me from falling off of the train tracks uh too much, right? And and as long as I've got things that I can tie to, things that I know previously, things that I have held on to, I can get I I mean, I have this conversation with my student, with my uh student, with my student, my son. He's my student. I mean, he lives with me, but he's my student. He's a life student. Um, so I had this conversation with my son yesterday. We were multiplying uh two-digit numbers, and he had uh multiplied, uh it was um 70 something. Uh I don't remember, but it was 70 and 80. Okay, so it was 70 something times 80 something. And he had multiplied, um, one of them was like 80. Okay, so maybe it was 76 times 80. I'm just making this up. All right. He he multiplied the seven, the 70 and the 80. I'm having to concentrate to make sure that I say that right. He multiplied the 70 and the 80, and he he only saw the one zero, right, from the 80. Uh-huh. He multiplied the 76, and he he said eight times 70. 8 times 70 is is 560. And so he had this, he had this 560 uh written down. And he just, I mean, he was dead upset that he was getting the right right answer because he kept saying eight times 70, 8 times 70. And I was like, You are right. 8 times 70 is 560. But what do you have right here? Is what you have eight times 70? And he argued with me for a minute, but ultimately I I asked him to draw himself a picture of this, you know, make make this little array and and fill out your 80 or 80 and your 76. And then then finally we were able to get to the point of oh I'm multiplying 80, 80, 810s times seven tens. And so I should have 5600. I should have five 5600s. And I'm also being really careful with him. And I don't know, some somebody can come back and give me feedback and say, you're doing it wrong, Kurt. Um, I was a high school teacher, so I'm totally willing to admit this. All right. But I I really am trying to concentrate on him talking about in terms of the units. I I really want us to get down to can we be talking about in terms of the unit eight tens times seven tens, we get 56 tens, right? 50 are sorry, so 56 hundreds, right? So would just this idea of of pulling those together and trying to really okay, 10 times 10's 100, 8 times 6 is 50, 8 times 7 is 56. So I've got 5600s and and really thinking this through in terms of those concepts. That does that make sense about why I I where those two things kind of come together is the concept, the conceptual understanding helps me realize when I've gotten off the tracks.

SPEAKER_01 26:18

I think that's really interesting because I think the you know, the two-digit multiplication, two-digit by two-digit is a great example. I think of how many adults I know who can go through that algorithm, you know, stay on the train tracks as you described, but not really make sense of the place value work that they're doing there. Um, and and I think that's often because of how they were instructed. Like there's this perception that we're we're taught procedures as rules, not as the result of the structure and the mathematical relationships, right? Those things are often taught separated. So I love how you described the ability to combine procedural fluency and conceptual understanding in the sound standard algorithm. Like the standard algorithm is not one or the other, it's both. It's but are we thinking about it in such a way that brings that out? So I think this is a perfect transition into talking about, you know, how do we think about what teachers can do to ensure that kids are learning both procedurally and conceptually? And I think, you know, back to when Common Core came out, one of the pushbacks or that we saw even like the new math movement of, you know, a few decades ago is this idea of students learning different ways, different algorithms. Multiplication is another great, it is a good example here. I had never heard of the lattice method of multiplication until my own children were in like third and fourth grade, and they come home and they're like drawing these boxes and these diagonal lines across and this really, really wacky thing for multiplication. Um, and and how many times I and the other high school teachers that I taught with were like, what are these crazy elementary teachers doing teaching them to multiply this lattice multiplication? It's really similar, Kurt, to what you described as your box method for your son, where it's it takes it take it it breaks basically it down into single digit by single digit. And then there's these diagonal lines that help keep track of place value. But when that's taught as a trick, um it's it's not an efficient strategy, right? So it could be just taught as like, here's here's a way to multiply numbers, which is how it was taught to my students, like or to my children. It's like how you, you know, you put this number here and you put that number there and you draw this line and you add this way. I have a 29-year-old daughter-in-law who I adore, and she still does multiplication using lattice multiplication. It is like a super inefficient strategy. But that wasn't meant to be a procedure. That wasn't meant to be the way to get the most efficient way to get the correct answer. It's meant to be a way to understand what's happening with place value. Yeah. So I think it can be conflated. I think it's there's so much responsibility on the teacher to understand, like when we're exploring something like lattice multiplication or the box method, or uh, you know, multiplying polynomials, like FOIL is uh FOIL is a mnemonic device for remembering a series of steps. Yes. It doesn't help us uh it doesn't help us understand like why do we do first times first and first times outer and outer times inner and you know, last times like why do we do that? So it's it's not about it's it's so much pressure, is the only way I can say it, on the teacher to be able to frame the experiences that students have to know like what is my purpose right here? Am I trying to help them understand a bigger idea about this mathematical concept? Or are we trying to get to a quick and efficient answer? If we're trying to get to a quick and efficient answer, we should be talking about procedures. We should be talking about standard algorithm. But if we're trying to build the other side of the coin, which is this robust understanding of what is the mathematics here, then the instruction needs to be framed differently.

SPEAKER_00 30:40

So I'm gonna go um I'm gonna go off script here a little bit and ask you a different question. Um, because I I'm as you're thinking, as you're describing that and as you're thinking about it, and I think about my son, it it's tempting to me. And I want to just know your thoughts on this. Um it seems to me that we can go uh down the road of hey, our goal is procedural fluency, our goal is efficiency and time and making sure that kids can do the computation or perform whatever the the mathematical uh uh gymnastics that they need to be able to perform. Um, you know, we want them completing squares, we want them being able to find uh and determine zeros of polynomials. We want them to be able to compute derivatives, we want them to be able to multiply two-digit uh values times two digit values efficiency and accurately, right? Like those are things that we ultimately want to get to because those are important. But we don't want uh students to get there without an ability to understand or be able to correct themselves when things go wrong. And so it seems to me that beginning with conceptual understanding, beginning and building the concept seems to me to be the place to start. And then we'll we'll we'll wrap up with or we'll finish with procedural fluency once we've kind of got the concept. And I think about things in calculus that, you know, I I I never taught calculus, but um I I'll stand right, I'll stand here and tell you that I I believe pretty strongly that I understand the concepts um that are that are there in an introductory calculus course. And I can talk to you about the rate of change of a rate of change and related rates and and how things move and link together and how integrals and derivatives are are related, right? I can I can talk to you about those concepts and the way that they look graphically and analytically. So I can do those things procedurally. I'm very weak, and so I'm just not there. But I I think what I'm going for here is uh do you think that maybe the right place to start? And man, that's not even a good way to phrase that because I don't like taking a stand. Um is there a is there a a correct way to do this? And if there is, where should we begin? Should we be beginning with conceptual understanding and then pushing towards procedural fluency? Or should we be intermixing them somehow? Like what is your response to that?

SPEAKER_01 33:26

I love that you asked me this question because I the honest answer is I don't know. And maybe even a better answer is I think it depends. And here's why. Um, you know, I want to reference to that Grant Wiggins blog blog post that you talked about. We're gonna definitely link this one in the show notes because there's a lot of really interesting ideas in here. But one of the most key ideas in here for me that really resonates with me and my teaching experience and my parenting experience is that students need to anchor new learning to previous learning. They have to build on something that they understand. So I can see arguments both ways that there are times where really focusing on the concept first is going to help a student understand that concept better. They're not gonna get lost in the procedure and not care about the concept. And I also think there are times where or students for which starting with the concept doesn't allow them to connect to anything. So I I really think it depends on the kid and it depends on the topic. So I'm thinking about, you know, my own son, I've said this before on previous podcasts. If he didn't understand why we were doing something, he he couldn't like his brain wouldn't open enough to be able to synthesize the procedure until he had the why, the concept. Like he always needed the concept first to anchor to. I've taught students where we going through the procedure of double-digit, you know, two digit times two-digit multiplication, we have to have the procedure and their ability to like put numbers in the right place of the standard algorithm for that, before we can open up the door and say, you know, actually this eight is 80 and this seven is 70. Because starting with that wouldn't have worked for those students. So it's I it's part of why teaching is such an incredibly complex craft. I just don't know that there is a right answer to that. For me, the right answer is start with the one that allows you to anchor to something students already know so that they can build off of something they already understand rather than try to generate new knowledge from nothing. Wow. I want to turn that, I want to turn that question back at you though. What is what are your thoughts? Is it which which one should you start with?

SPEAKER_00 36:01

You mean I don't get to, you don't, I did I don't get to escape by just saying, boy, what a great answer there, Joni. And that was such a great place for us to end this podcast. I um so I'll be honest with you, I really feel pretty strongly um about the the idea. And maybe it's it's um, maybe it's just me. Maybe it's just my own person, personality kind of getting in the way here. And I love the way that you responded there about talking about considering the the student. Um, because um you know we've had this conversation a few times in our podcast, right? About that, that we really are investigators and learning our own students and being able to figure out who they are and and what they need best, right? And what's gonna serve them best. And so ultimately, yeah, you're right. That's probably the best answer is to know my students, know the situation, and uh work with what makes it best for them. But I'll be honest, I I really um I really like the concept of getting at the concept first. Um, getting at the building blocks of um each piece first. And I think there may be two reasons why I think this. I think first of all, um, and it probably it goes to some things that are that are in that Grant Wiggins post, um that that concepts are the building blocks by which we make new information. So it's really, really difficult for me to start from complete scratch. Everything has a frame of reference. Uh, all of my experiences, both in mathematics and really just in life, have a frame of reference. They they have something that's previous knowledge. I mean, apart from being an infant, um, and even maybe then, right? Because consciousness before the before we are born. So, so thinking about uh all of that. So I I I really think that getting at the building blocks, the concepts that are the building blocks that allow me to be able to find some new procedure that build allow me to be able to uh experience some new procedure, put them together in some new way, that is sort of the way that I can get at something, right? I I just blindly going through a set of numbers and putting numbers where you tell me they belong in order to come up with a response feels um frustratingly scary and maybe even meaningless to me. I'm gonna probably make people mad by saying all of that. But I just I I feel like I I feel like if I'm just going through the motions, I'm not connecting it to anything. And yeah, I could memorize it, but it's not gonna get me anywhere further. And I can't certainly can't build on it, right? Because I can't tell you why I did that in order to maybe build it into how I'm gonna multiply two uh polynomials later on, right? Or two binomials later on. I I can't I can't like build those two things together and see that they're connected. Um, and so I'm not gonna be able to build on that. Um so for me, I I really do feel somewhat passionate about that that I think we should be starting with concepts uh and then working towards procedures, but not, and let me stress this um not ignoring those procedures. Procedures end up being a huge goal. We've got to be able to do them efficiently, but not at the sacrifice of doing concepts first, in my opinion.

SPEAKER_01 39:54

Yeah, yeah, I like that. Well, Curtis, there's so much more to talk about. And so much. I know our time is short. Um, I I think we're definitely gonna have additional conversations on this topic. And um, we have actually have a significant number of resources that we've talked about and mentioned in this episode too. So uh I want to encourage all of our listeners to check those out in the show notes and send us some feedback, and uh, we'll definitely line up another conversation on this topic for season two. But uh great way to wrap up season one. As always, I've enjoyed the conversation with you. I think I'm walking away with uh more questions and more ideas buzzing around my head than when we started. So to me, that's a success.

SPEAKER_00 40:35

For sure. Same for me. Same for me. Well, Joni, let's sign off and uh we will uh see everyone in 2022. Well, that's it for this time.

SPEAKER_01 40:45

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.