March 12, 2024
Room to Grow Math
Season 4
Episode 3

Room to Grow - a Math Podcast

Balancing Instructional Modalities

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Room to Grow - a Math Podcast

Balancing Instructional Modalities

Mar 12, 2024
Season 4
Episode 3

Room to Grow Math

In this episode of *Room to Grow*, our hosts look for the balance between instruction that is teacher-driven, traditional lecture-style, and inquiry-based, discovery-style lessons. They recognize the value of both types of teaching, understanding that there is a time in learning for both exploration and for direct and explicit teaching.

The conversation offers explanation of what conditions may require different teaching strategies, based on the goals and content of the lesson as well as how students are responding to and progressing (or not) toward intended learning.

The common theme between these approaches is student sense-making, and our hosts each share a personal example of taking opportunities to encourage sense-making in students.

We encourage you to explore the resources below, referenced in this episode:

- NCTM President Kevin Dykema’s President’s message that sparked this episode: https://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Kevin-Dykema/Balancing-Instructional-Strategies-in-the-Math-Classroom/
- TI’s Building Concepts lesson on structure in solving equations: https://education.ti.com/en/t3-professional-development/for-teachers-and-teams/online-learning/on-demand-webinars/2016/building-concepts-foundations-for-success-in-expressions-and-equations
- A sample problem-based curriculum for middle school (NOT the one Curtis’ son uses!): https://curriculum.illustrativemathematics.org/MS/teachers/what_is_pbc.html

Did you enjoy this episode of *Room to *Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com.

Be sure to connect with your hosts on Twitter and Instagram: @JoanieFun and @cbmathguy.

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In this episode of *Room to Grow*, our hosts look for the balance between instruction that is teacher-driven, traditional lecture-style, and inquiry-based, discovery-style lessons. They recognize the value of both types of teaching, understanding that there is a time in learning for both exploration and for direct and explicit teaching.

The conversation offers explanation of what conditions may require different teaching strategies, based on the goals and content of the lesson as well as how students are responding to and progressing (or not) toward intended learning.

The common theme between these approaches is student sense-making, and our hosts each share a personal example of taking opportunities to encourage sense-making in students.

We encourage you to explore the resources below, referenced in this episode:

- NCTM President Kevin Dykema’s President’s message that sparked this episode: https://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Kevin-Dykema/Balancing-Instructional-Strategies-in-the-Math-Classroom/
- TI’s Building Concepts lesson on structure in solving equations: https://education.ti.com/en/t3-professional-development/for-teachers-and-teams/online-learning/on-demand-webinars/2016/building-concepts-foundations-for-success-in-expressions-and-equations
- A sample problem-based curriculum for middle school (NOT the one Curtis’ son uses!): https://curriculum.illustrativemathematics.org/MS/teachers/what_is_pbc.html

Did you enjoy this episode of *Room to *Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com.

Be sure to connect with your hosts on Twitter and Instagram: @JoanieFun and @cbmathguy.

00:00:00:00 - 00:00:32:04

Joanie

Is the best instruction for students on their math learning a inquiry based approach or is the best kind of instruction where it's direct, explicit, sort of teacher led, teacher driven. You know, when I think of that, I think of like a lecture style learning experience. Today's episode of Room to Grow was inspired by a recent NCT president's message on balancing instructional strategies.

00:00:32:06 - 00:00:57:03

Joanie

Curtis and I unpack the common and sometimes polarizing approaches of inquiry based instruction and direct explicit instruction. Spoiler alert We argue for a balance of the two, but we also present some thoughts about how this connects to student learning, conceptual understanding and procedural fluency, and to mathematical flexibility. We hope you enjoy the conversation as much as we did.

00:00:57:03 - 00:00:59:01

Joanie

So let's get growing.

00:01:00:03 - 00:00:59:01

Curtis

Well, Joanie I am really excited to be recording The Room to Grow podcast again with you today. We've got a great conversation and I think prepared for our listeners. And yeah, it's going. It's going to be good. This topic is one that it's become kind of a really good conversation for me with some of my coworkers and some of some of my my friends even.

00:01:27:02 - 00:01:50:12

Curtis

This becomes a conversation that you end up having in the in the public space quite a bit. And so I'm really excited to be chatting with you today, thinking a little bit about this idea of instructional strategies and and the different approaches that we hear kind of talked about in in social media and in the news and all of those kinds of things.

00:01:50:12 - 00:02:19:06

Curtis

And even our publications thinking about instructional approaches and the balance maybe that needs to be had between them because it seems like and it's happened over and over, that we have this pendulum in mathematics that swings from from one side to the other of instructional strategies. And so just having a conversation around that I think is going to be a really good one today.

00:02:19:07 - 00:02:41:13

Joanie

Yeah, I'm excited for this conversation and partly because this is a this is a topic that can be polarizing and not that that's our intent to go to like some extreme argument kind of space. But I think these are the interesting conversations where, you know, Curtis, we prepare, we we can kind of go, Yeah, well, I see the argument for that and I also see the argument for this.

00:02:41:13 - 00:03:01:20

Unknown

And I think you sort of you gave the punch line a way in a really subtle way. There when you talked about balance. And I think that's, you know, that's where we maybe hope that we're going to land at the end of this conversation is on balance. So let me just kind of stop in back up just a second or pause for just a second and really name out what we're talking about.

00:03:01:20 - 00:03:43:20

Joanie

One of the, one of the several catalysts to us having this conversation today was a recent president's message that Kevin Diekema, the president of NCT, released, and it was titled Balancing Instructional Strategies. I think it was December of 2023. It was Kevin wrote that up and it was, it was, you know, his thinking on kind of this battle between is the best instruction for students on their math learning a inquiry based approach or is the best kind of instruction where it's direct, explicit sort of teacher led, teacher driven.

00:03:43:22 - 00:04:05:10

Joanie

You know, when I think of that, I think of like a lecture style learning experience or even the well, I don't want to jump into that quite yet. So so as we sort of said, like we of course believe that there is balance to that, but I think there's a lot to unpack here before we kind of go to our our end point, which is balance is what's right, which is also said Kevin Diekema says.

00:04:05:10 - 00:04:31:10

Joanie

So I want you to talk a little bit about what what your thought is in terms of let's talk about extremes. So what is your thinking about why an inquiry based approach? And I want you to give me a chance to talk about this, too, because I have some stories here. What is your argument that an inquiry based approach 100% of the time with every mat topic is not effective?

00:04:31:12 - 00:05:01:10

Curtis

So I think this is an interesting thing and how we even get here is a question that I think is is worth exploring, because I don't think that anyone who is advocating for an inquiry based approach to teaching topics would stand by a statement that you should only ever do inquiry based approach. I don't think anybody would ever stand by that statement.

00:05:01:11 - 00:05:37:09

Curtis

And and so how do we even get to folks thinking that that is what is being spoken or told is I think the pervasiveness of those kinds of approaches being talked about in articles and research and on the Internet and in social media and, you know, in your name, your social media platform video that you might be doing, folks are talking primarily about that type of instruction.

00:05:37:09 - 00:06:30:13

Curtis

The sessions that you go to at a conference often end up looking that way and promoting that kind of thinking. And so the idea then becomes if you're just a casual observer of each of these things, you begin to think, my gosh, is that what I need to be doing in my classroom? Or as a parent, you know, looking at what is being talked talked about in those educational spaces, my gosh, is that what my students are being exposed to only?

Right. This is the only kind of homework I ever see my students looking at. This is the only kind of thing that I ever see my students dealing with in in instruction. So that's how we end up in that space of thinking that, my gosh, 100% inquiry is exactly what we should be doing. Yeah, So why it might not be a good idea.

00:06:30:15 - 00:07:07:13

Curtis

Is there just certain times when that happens and when when students may get lost or frustrated in a space that, you know, struggle goes from productive to unproductive? There you go. And and if we lean towards only doing inquiry based things, we can often lose sight of the scaffolding that needs to happen or the the questioning that needs to happen to help students get freed from that unproductive, unproductive struggle.

00:07:07:14 - 00:07:48:11

Curtis

And the demonstration or the little example that gets you over the hump, right? Yeah, right. And I see that all the time when I'm dealing when I'm talking with my and my son, talking about his mathematics homework. Right. We we, we test the waters. And if he's finding a bit of frustration or he's starting, I can sense right. The spiral starting to happen. And so it's up to me as his as his tutor at home to start to see that and and say, I'm not just going to leave him out to dry, to get frustrated, frustrating struggle. I'm going to I'm going to step in and get him over the hump. What is it about the thing that he's getting frustrated about?

00:07:48:11 - 00:08:16:09

Curtis

Okay, so how can I get him over that hump? How can I give him? And sometimes it literally has to be a direct instruction sort of statement. I have to tell him no, this actually means execute this. Yeah. Here's why. Yeah. okay, Dad, Thanks. And then he's able to get back into and explore or continue down the path he needed to be on.

00:08:16:09 - 00:08:44:07

Joanie

I, I really like this example that you just gave with Teagan. Don't you love how we use our the children learners in our lives all the time? All the podcasts we're going to, we're going to be out of luck once they all grow up and move on right. Anyway, know what? One of the things I just want to kind of call out that I just heard you say within that that I think is one of the maybe myths or extreme thinkings that we're hoping this conversation will bust a little bit.

00:08:44:07 - 00:09:12:09

Joanie

Is it It's fluid, right? Like what you described is you could sit down with Teagan and start you know, here's here's a task I we a mathematical task we want to work on that's going to help you to develop understanding of an important mathematical idea. I you're going to ask some general questions. You're going to, you know, give him the like get him started on the task, but try to pull out his own thinking.

00:09:12:09 - 00:09:46:17

Joanie

So under maybe a formal definition, that would be inquiry based, where he's exploring his own understanding and he's making sense of the task without, you know, a direct path to follow that you just modeled for him and now he's just going to repeat the path that you just took. But then as he's in that exploration phase, as he is doing inquiry based learning with that task, you can pivot in the moment when you see there's a need right here.

00:09:46:17 - 00:10:19:03

Joanie

Okay, this this exploratory approach he's taking right now is is going to lead him to a point of frustration or a point of shutdown that I don't want to have happen. Now, notice I'm not saying it's going to lead him to wrong ideas. And we've had this conversation many times before. There's a lot of value in letting kids thinking go to something that is a misconception or an early conception of an important idea, because that's how we correct those conceptions.

00:10:19:03 - 00:10:57:14

Joanie

That's how we can address them head on. So but I love the point that you made of like product of struggle shifting to unproductive struggle. And that is not you know, there's not like a checklist for that necessarily that can be a very kid specific moment, specific situation. So being able as a teacher to have the flexibility to pivot in the moment and say, okay, with this topic, with this student in this moment, inquiry is not is not effective.

00:10:57:14 - 00:11:18:23

Joanie

In fact, I can see that this might start to work against this students learning. So I'm going to jump in with a moment of direct instruction. That doesn't mean you totally like, never mind. Teagan ripped the paper out of his hands and, you know, start writing on a whiteboard and asking him to copy things down. Right? Like, it doesn't have to be the extreme.

00:11:18:23 - 00:11:31:00

Joanie

There can be a moment of some direct information that you put into that in a very strategic way that helps you and him get to the desired outcome of that task or that lesson.

00:11:31:00 - 00:11:38:00

End of segment - Music break

00:11:38:15 - 00:12:31:12

Curtis

Now, thinking about the the opposing piece of this, you know, for a long time, the idea of let's sit in in rows, let's look at a chalkboard or a dry erase board or, you know, some sort of electronic responsive board has been, you know, a method of instruction, right.

Thinking about here's an example that has these values, that has this function. And we are going to look at how we and name your name, your mathematical process that we're going to do to this particular thing. And I'm going to just here's the first step. Here's the second step. Here's the third step. Now, you guys all mimic exactly what I do right?

00:12:31:14 - 00:12:58:12

Curtis / Joanie

That's a great way to teach someone how to fly fish. That's a great that it is. I mean, if if my if my idea is to get this the my kid to have a real understanding of the execution of of a great cast of a skill. Yeah. That is, that is a, that is a great way to, to do that.

00:12:58:14 - 00:13:39:15

Curtis

And so if the execution that I'm looking to, to get is some sort of procedure of doing these things. Yeah sure. But if I need to have the student understand what they're doing, why when that sort of procedure applies. Right. We need some exploration, we need some level of ability to to go forth and try stuff and discover, hey, this didn't work then or so.

Right. Does that work that way? Needs to be a good question to be able to ask or have the students be able to ask.

00:13:39:15 - 00:14:21:13

Joanie

Yeah, I love that. And I think what what popped into my head as you were describing that difference is sense making, right? Like showing a skill and giving that direct instruction to getting an answer like that might be the most efficient way to get students to get to an answer.

And again, reflecting back on my own classroom days, and especially in my early career and especially even later in my career, when I was under pressure to get through content, when I felt like, they need to at least see it, you know, those kinds of when I was saying those kinds of things to myself, going to here, let me show you now you replicate and look at that.

00:14:21:13 - 00:14:47:06

Joanie

You can spit out the right answer. But there wasn't sense making involved in that. And there's a place for right answers And and I'm I'm all about that. Yeah. But right answers without sense making is not math in my opinion. So when you need to shift to why does it work that way or when doesn't it work or, you know, what are we even doing here is an algorithm.

00:14:47:06 - 00:15:20:17

Joanie

But what's the mathematical idea behind that algorithm that allows us to follow these steps and get that outcome like that sense making is is really challenging. If not, I might even go so far right now as to say impossible to do because we're all human, we're all individuals. And how we make that, how we make sense, how we learn is completely dependent on our own personal background and experiences and understandings.

00:15:20:17 - 00:15:50:16

Joanie

And I've said for a long time, like, I can't really learn unless I anchor it to something I already know. I have to fit new learning into the broader landscape of my knowledge as a whole. And that can't if if that could be proceduralized and generalized that every person could learn by just doing this or asking that we would have figured that out by now. That's just not how not how learning happens.

00:15:51:00 - 00:16:34:19

Curtis

Now, I think you So a couple of things happened in my brain just now, which could be dangerous and who knows where this conversation may end up going now because I've got an interest in thought it, as you were talking about that and thinking about sense making. Right. This idea of sense making and when does mathematics so mathematics itself kind of brush up against sense making completely become all about sense making or and when is it just this, this procedural thing?

00:16:34:19 - 00:17:06:13

Curtis

And I was thinking about, you know, these these things that traditionally get brought up in arguments about this topic we can predict and Amanda and I'm the end you know it's not it's not hard, I don't think, to convince someone that there is great value in the sense making around completing the square or the sense making around learning to factor quadratics and in the things that that are needed right in the patterns that are developed.

00:17:06:13 - 00:17:34:21

Curtis

And and believe me, there's lots of ways to develop those patterns, right? There's tons of ways. And some would air towards what some might think of as some sort of some sort of lecture style thing. You can explore those things with. Q and A and having questions. Lots of big classroom conversations. You can you can explore those things as individuals or in small groups and, and do different kind of learning exploration.

00:17:34:21 - 00:18:13:16

Curtis

So lots of those. I feel like we don't get as much pushback. I feel like sometimes we get a lot of pushback in the discussion around the development of numeracy. And you mentioned multiplying a two digit number by a two digit number earlier in our pre conversation when we were talking about this and I was thinking about if what I'm after is the execution of, Hey, I need to know what 15 times 17 is, I need to know that number outside of any context.

00:18:13:18 - 00:18:40:06

Curtis

Well, okay, some sort of direct instruction, memory route memorization of you take this one and you do this multiplication and you carry out these set of okay, sure, we can make that a set of routine procedures. And the next time you see 17 times 15 or even you might be able to extrapolate that out to, you know, multiply 63 times 74.

00:18:40:12 - 00:19:18:06

Curtis

Okay, great. You can probably routinely memorize some set of instructions and you know that the result of this multiplication action of this two digit number of times that two digit number results in this number, but being able to approximate responses, being able to apply that to some kind of approach where, hey, it's no longer a two digit number of times, a two digit number, it's now a decimal that happens to have a decimal in the middle of this.

00:19:18:06 - 00:19:55:14

Curtis

So now I'm multiplying 3.2 times 2.7. Just the addition of that decimal makes that procedure difficult. Now, I don't understand what's going on. Right. And the reason I'm bringing this up is this is exactly what Teagan and I were talking about the other night that made that made me think of this and realizing that if I know how to multiply a two digit number of times the two digit number, I can carry that out if I ignore the decimal places.

00:19:55:14 - 00:20:32:16

Curtis / Joanie

But realizing and understanding what those decimal values and place value and how place value impacts the result of this multiplication, those are the things we talk about when we're talking about deep sense making. Right? Exactly. And those require that little bit of extra conceptual development in order to apply the same algorithm. Right. I'm going to apply the same algorithms to both of them have two digits right in both of and we're executing the same steps, but it doesn't look the same.

00:20:32:16 - 00:20:39:17

Curtis

And the answer doesn't come out to be the same because we have to figure out how to deal with those decimal approximation of those decimals.

00:20:40:00 - 00:21:24:06

Joanie

However, I want to jump on this because I agree and applying the same algorithm for a two digit times a two digit will get you most of the way there. And you can finish just by sense making, right? Like maybe I don't remember how the decimal thing works, but if I understand, you know what 2.3 and what 3.7 are numerically like, if I can make sense of them as quantities then after I've done the the you know, manipulation of the numbers in the two digit by two digit algorithm, I can sort of figure out what makes sense for where the decimal place might go.

00:21:24:12 - 00:21:43:17

Curtis / Joanie

I my goodness, I'm so excited to hear you say that. I'm like jumping out of my seat because this was the thing that was happening in our conversation with Tegan and I. So he was getting so frustrated because he was getting these and he was and he wants to just remember the rule. I can't remember the rule.

00:21:43:17 - 00:22:04:01

Joanie / Curtis

What's the rule is that this rule is at that rule. Well, he was getting super frustrated because he kept getting answers wrong in this digital curriculum that he was playing with, which, by the way, I'm not a fan, but I'll go I'll I'll get there later. We can have that conversation another day and I'll try not to get on that soapbox right now.

00:22:04:06 - 00:22:39:15

Curtis

His he was getting frustrated because he was typing in the results of his of his multiplication and getting things wrong. And it was not because he didn't know how to do carry out the execution. It was this typing in thing that was happening. As soon as he realized that what he could do was apply some some sense to the outcomes, he could he could carry out the algorithm just fine and get the numbers down.

00:22:39:20 - 00:23:03:12

Curtis / Joanie

And now he has this long string of digits that he's got to find out what its approximate value should be. And then he goes, yeah, well, because this one, this one rounds this way and this one round that's about that one result should be about this. And so he could just pick the value, he could just stick this in there or choose the right multiple choice answer because all the distractions were ridiculous.

00:23:03:12 - 00:23:23:04

Curtis / Joanie

Yeah. Yeah. And so he could choose the right answer. Not because he multiplied and got every single digit, but because he realized the sense he realized what was what should be about the answer. Yes. Yes. Awesome. So the sense piece of this, proud Dad moment. So powerful…yes.

00:23:23:06 - 00:23:30:00

End of segment - Music break

00:23:30:00 - 00:24:04:01

Joanie

I want to tell a counter story because I worked with my nephew Andrew this weekend, too, and he he's going through in his math class right now. There starting to solve algebraic equations. And this homework was, you know, multi-step equations. So they've learned the distributive property and they've learned about combining like terms. And so it was like, I don't know, there are seven or eight problems or whatever in the homework. And, you know, the first one is a number times a quantity equals another number of times a quantity, and you've got to distribute.

00:24:04:01 - 00:24:25:14

Joanie

And then, you know, maybe on one side there's like terms to combine and then you've got to, you know, get all the variable terms together on one side and get all the constant terms together on another side. Well, we're working through well, like problem five, like we're well into this like pattern of, you know, execute this algorithm. And Andrew's brain thinks that way.

00:24:25:14 - 00:24:52:06

Joanie

So he's very successful, just like plowing through these. Like, these are easy. These are fine. He's doing great. We get to an example that it kind of broke the pattern of difficulty, right? This was maybe the easiest problem on the entire assignment, and it was five or six close to the end. And the problem was two times the quantity x minus three equals 20.

00:24:52:08 - 00:25:11:12

Joanie / Curtis

And he starts like, okay, I need to distribute the two. And I was like, Wait, wait, wait, wait, wait, wait, wait. This was like sort of me taking this moment to say, I know he's got the algorithm down. He can crank through. He might be he knows the distributive property he makes that he knows the distributor property. He knows how to combine like terms he knows how to solve.

00:25:11:12 - 00:25:29:18

Joanie

When there are variables on both sides. There is no doubt in my mind that he is fine with that. I wanted to stop him and go like, let's let's talk about what this is we're actually doing. And I couldn't have this conversation when there's distributing on both sides and different like terms to be combined. But I could have it for this example.

00:25:29:20 - 00:25:46:08

Joanie

I was like, Let's stop and talk about this for a second. Put your pencil down. Let's just talk about this problem. This is saying two and then there's something in parentheses equals 20. Well, what does that mean mathematically? And he's like, Well, two times something equals 20. And I was like two times something. Don't even look at the problem.

00:25:46:08 - 00:26:06:08

Joanie

Just think in your head two times something equals 20. And he's like ten. And I was like, Yeah. And I said, okay, let's go back to the problem. So where is ten? And he's like, the thing in parentheses has to be ten because it has to be two times ten equals 20. So seven like that fast. He had it.

00:26:06:08 - 00:26:34:03

Joanie

He's like the thing. And I was like, Wait, wait, wait, let's slow down and say what we just did in your head. Like, I want you to, like, own this new way of thinking about this problem. And to me, it's that's what I thought of when you were describing Teagan, like having this moment of Wait, I understand decimal numbers, and I understood and that I can get an approximation using rounding that will allow me to identify the right answer in a set of multiple choice responses.

00:26:34:03 - 00:26:56:08

Joanie

Like for sure, I'm going to do the more efficient thing right? Which the kids think it's a shortcut or an easier way when really like, no, that's just like actually brilliant mathematics when you can stop and say, I know if two times the number is 20, then the number has to be ten, and if X minus three has to be ten, then X has to be.

00:26:56:08 - 00:27:13:01

Joanie / Curtis

I totally screwed that up. What I want anyway, you know what I like about what you're saying? I totally know what I mean. I think it was X plus three because there was a seven. there was a seven. Okay, totally. You totally get X minus three earlier. So I, I but I blew the math. I apologize, but you get the point, right?

00:27:13:03 - 00:27:46:06

Joanie

Like so. And, and for, for both instances, we did not diminish the student's ability to understand the computation by slowing them down and asking them to make sense. Right. Right. So the I, I feel like we're maybe shifting over into a different conversation than inquiry versus direct instruction, but I just think that it's such a great example of sense making and skill have to go together.

00:27:46:08 - 00:28:06:21

Joanie / Curits

And I don't want to simplify it so much to say that a skill becomes direct instruction and sense making becomes inquiry based. Because I think it's a little more fluid than that. But I guess I'm just trying to like circle back around to where we started with fly fishing and yeah, yeah. So I kind of want to just bring us back to the lake now.

00:28:06:22 - 00:28:40:06

Joanie / Curtis

It makes sense. That balance as you just brought up though, is is at maybe the heart of a lot of this is we, we often we also often label inquiry based instruction as development of conceptual understanding. Yes. And we label direct instruction as building procedural fluency. Right. We tend to kind of trouble those that way. And I don't think there should be an equal sign between each of those in each of those cases.

00:28:40:08 - 00:29:01:11

Curtis

Right. I think you can do and I'm not going to try to give tons of examples of these kinds of things, but I think you can do because I want to go back to something you just said in your story, but I think you can do, you know, the development of of a concept and the development of a conceptual understanding through some level of direct instruction.

00:29:01:13 - 00:29:26:17

Curtis

And I think you can develop procedural fluency through some level of inquiry, right? Like, okay, so, so equating those, I think is sometimes sometimes happens in people's minds, certainly often does in my own, like I get, I get, I catch myself having that thought and I go, wait a minute, no, that's not exactly what, what these mean. And so I have to be careful.

00:29:26:19 - 00:29:58:09

Curtis

But I want to I think that's where you brought that. And I and I love that that you brought that up in this conversation. Right. To bring us back to the table. Yeah. So real quick and and I've never done this on the podcast before, but this is I can't help it. So, you brought up this development of understanding structure of that equation and looking at the development of structure in that equation that you were describing there.

00:29:58:09 - 00:30:23:02

Curtis / Joanie

And yeah, okay. In all of the other ones there is distributive property be done on both sides. The quantity inside of of the parentheses on both sides of the equation were not equivalent, right? So it wasn't, you know, five times this quantity, multiple occurrences of the variable. Yeah. It was way too much to just Yeah. Way too much reason to enhance opportunity to have that this development of structure.

00:30:23:04 - 00:30:54:06

Curtis

But I love the fact that you slowed him down to answer that to to say, hey, do you recognize what you're doing here. Hold on. There is actually an activity that was developed for this purpose on the T.I website was a part of the Building Concepts series of activities where we look at developing the procedure, developing a structure, understanding of a structure in solving an equation.

00:30:54:08 - 00:31:22:12

Curtis / Joanie

I'll leave it at that. I'm not going to go advertise this whole activity, but I will put a link in the show notes favorite activities because it showed me something that I never really thought about. I would have been just like your nephew carrying through this this distributive property blanketly because that's just the procedure. I knew how to solve exactly set of problems.

00:31:22:12 - 00:32:06:05

Curtis

Here's my question that was being asked through the lens of I got this toolbox of things that I can use. Hey, wouldn't it be really efficient if I recognized this kind of structure in the places where that happens now? How do you get that kind of recognize, how do you get that recognition practice? There's practice of realizing, so when this quantity that's inside the parentheses can can be strategically isolated and say, two goes really nicely into 20, I can, I can say I mean, if it had been 30 or even 37.

00:32:06:07 - 00:32:27:21

Curtis

Yeah. The distributive property, it would have been a different conversation. It would have been a different conversation. 30. Okay, great. Two goes into 30 nice and evenly and you get 15, but it might not have come quite as quickly. The 20 was the thing that really to me drew the structure. Like, wow, that's right, too. That's right. Ten is 20.

00:32:28:01 - 00:32:40:09

Curtis

So this thing, this quantity has to be ten. That is the light bulb moment for your nephew. Yeah. Happen there. So yeah, I love it. I get super excited about that kind of thing.

00:32:40:09 - 00:33:16:18

Joanie

No. And I love that you brought that up because I think this is a great place for us to start. Go to our wrap up because what you just described is what I believe every teacher is trying to create in their students. Right? We want them to be able to get correct answers. We want them to be able to apply algorithms appropriately if that's the most efficient way. But we also want them to know that they can use sense making and their understanding of the concepts as part of those two things. Like that's part of how you get two answers.

00:33:16:18 - 00:33:53:06

Joanie

That's part of how you how you reason through, you know, a set of mathematical tasks. And it, it, it reminds me and this will be another podcast, I don't want to go too far down this road, but when I was a department chair at high school and having conversations with parents or kids who had been over accelerated, who had been pushed to early into higher level math classes, where that instruction became about like that rote application of here's the procedure, now crank out 50 of them and these kids could get it right every single time.

00:33:53:08 - 00:34:41:17

Joanie

And I would I can't tell you how many times I said, Boy honors Algebra two honors pre calc calculus AP, maybe these are the places where kids hit the wall where their ability to just throw a strategy and repeat it over and over again, you know, to mimic and and look for pattern recognition like they hit a wall. They, they, there reaches a point where that's not enough where you have to the sense making isn't there. You really, really struggle and I could speak to that as the department chair because that's what I experienced myself as a mathematics learner. For me, my wall was calc two. When I took calc two, I was like, oh my gosh, I have been faking my way through. I don't really understand what I'm doing. I'm just good at cranking out answers.

00:34:41:17 - 00:34:50:01

Joanie

So I just think that's such a great way for us to wrap. Curtis to say this is this is what we mean by learning mathematics well.

00:34:50:01 - 00:35:41:15

Curtis

yeah, I think that's a great place for us to wrap up in this conversation, to bring it full circle and thinking about the the article and really the conversation that that started this podcast for us, the advocacy for balance is huge. And I want to just say this, that not just the advocacy for balance, but the freedom as a teacher to acknowledge, Hey, I'm the one in here in my classroom with my students at this moment. Yes, I make the choice about what is the best way to help my student make sense of this problem and execute for correct answers.

00:35:41:20 - 00:36:12:11

Curtis

Right. I think both of those things need to be applied and understood. And I just I'm advocating for you, teacher. You know, we can talk about this all day, but I'm not the one in in front of your class teaching, right? Joani You're not the one in front of the class teaching. There are millions of teachers out there doing this every day and advocating for their students and doing doing the right thing.

00:36:12:11 - 00:36:25:21

Curtis

And I'm just trying to to remind us that, hey, we, we support you. We know it's hard. We know the balance is difficult. yes, but you have the freedom to choose.

00:36:25:21 - 00:36:35:03

Joanie

And thank you, thank you to all those teachers out there and all those who are supporting those teachers out there for helping kids get to that place.

00:36:35:03 - 00:37:00:21

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.

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