Season 5 Episode 1: 

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Opening music

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Joanie

Happy new year.

In this episode of Room to Grow, Curtis and I begin to unpack the mathematical teaching practices from Nick TMS principles to actions. Celebrating its ten year anniversary this year. Today we talk about facilitating meaningful mathematical discourse, and we connect these ideas to those from math practice. Six attend to precision. 

We think you'll enjoy this conversation, so let's get going.

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Curtis  

Well, Joanie, I am so excited to have you in the office again. We are recording the December no January podcast. It is December. We are recording the January podcast for 2025 that year. Five season five. Yeah. We were just pontificating on the fact that we are five seasons in and having fun out, and we still have room to grow.

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Joanie

Yeah. Curtis, great to be here in Dallas with you. I think there's, something special that happens when we're sitting across the table from each other instead of across the screen from each other.

But I'm also really excited as we think about 2025 and kind of the approach we want to take with the podcast. So I'm going to kind of start with this, but then I know we're going to backtrack to kind of set up

today's episode. We thought because this year is the 10th anniversary of

 Actions, which has been just such an increase table resource for educators over the last ten years.

We thought it would be cool to dedicate eight of this year's episodes to the eight mathematical teaching practices. So today we're going to focus on facilitating meaningful mathematical discourse as our practice. But we didn't start with that as the topic. So I want to back up a little bit and let you set us up for the start of the conversation.

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Curtis

For sure. So thinking about that, the fact that we were really wanting to highlight and maybe celebrate the principles to actions, the mathematics teaching practices there, I was thinking about some things and looked at the NCTM my NCTM blog, the digests that we get, on a daily basis.

And I saw a question. Yes, yes. There always there's so many fruitful things that that happen in that. So if our listeners are not a part of that, one, should probably become members at NCTM, and then two start receiving that, digest. It's fantastic. It is often the beginning of much fruitful conversation between you and I and this, this time I saw someone post about how they were planning to or needing to meet a student, with their IEP and in particular, the student had audible or, or, auditory accommodation for having a test or assessment read to them. And so there was a discussion about how would you read

a mathematical expression that included parentheses with order of operations, kind of as maybe the core of what they were trying to get to.

And there was discussion and questions about how would you read this and not give away or given as an advantage to that student? By the way that you read it to a student that would be taking the test on a piece of paper that way. So that was an interesting discussion, and it led me to be thinking about the precision with which we use, or the precision we, we use to speak about mathematics in our classrooms.

And the expectations that we have on students and all of those things, which then led me to the

standards for mathematical practice in the Common Core attend to precision. And so thus our conversation now.

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Joanie

Yeah. And then we kind of built it out to say, okay. So yeah. So just connecting a couple more dots. But for sure attending to precision is part of that. What was posted within that for sure. But I just want to pause too, because, you know, much like the principles to Actions is ten years old, the Common Core or even older than that. So, you know, it could be we, we sort of talk about the eight mathematical teaching practices, like everybody knows that for sure. And I'm thinking back ten, 12 years ago where we really spent a lot of time unpacking those. And what do those actually mean? And I think that one in particular, SMP six had created a lot of confusion for people because when you think about that from a mathematical lens, you're thinking accuracy and computation. And how many decimal places should I round to? And you know that it sort of conjures up, meaning that is important, for sure. Those are important things of that SMP, it's really talking about precision of communication. And that's how we kind of took the next step over to math discourse. Maybe we can sit for a minute here and kind of unpack these, because there's a lot of terms we're going to be throwing around that are not necessarily interchangeable. So how are we thinking about precision of language discourse and conversation?

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Curtis

Well, you said I you said something when we were preparing for this conversation about students discourse and really the idea of students being able to own their own language, right, in discourse allowing them to kind of communicate what they're thinking, to highlight what they're thinking. Or at least that sparked in my mind this idea that discourse, when we say the word discourse, there becomes sort of this, I'm, I'm attempting to communicate to you what I am thinking. My, my communication to you is about my thinking. I'm not regurgitating to you something that I've learned from you or I've said from you. It's, it's me trying to communicate what's going on inside of my inside of my head, between my ears and trying to get that out to in a way that you can understand what I'm thinking. 

Right. 

And that it's mine.

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Joanie

yes, I love that. The way that you spelled that out, because I think conversation might be a part of discourse, but conversation is like, not necessarily what we're talking about.

Sure

Getting students talking is a thing that's got its own challenges. And I'm sure we'll tap into some of those issues today, too. But this idea of discourse is really about intentionally, like getting a student to communicate how they think and reason in a way that others can actually understand.

for sure, then having another student respond to that, either with their own way of thinking something it made them think of. But the idea around mathematical discourse is not just about, oh, I want them talking, and I want words coming out of their mouth, and I want them using precise math language. No, it's about really communicating, like you said, what's in their brain? How do I get somebody else to understand in such a way that they can respond and that back and forth response and, and collective sense making is focused. It's focused on the mathematics. The sense of the mathematics. So discourse as a strategy for learning is the key idea here. Right. Like this is a math teaching practice. It is an effective teaching practice. There's research to back this. But I don't want to lose focus of the of the whole point, which is to facilitate better and deeper student understanding of the mathematics.

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Curtis

Right. And I mean if you go back and you look at the mathematics teaching practice in the Principles to Actions book, it spells out pretty much exactly what you just said, which is effective teaching of mathematics. I'm just going to read from, from the book Effective Teaching of Mathematics facilitates discourse among students with this purpose to build shared understanding of mathematical ideas. And this is how by analyzing and comparing student approaches and arguments. This idea of discourse has a purpose, has a direction, has a goal. Right. We intend to build understanding by having this discourse, me communicating what's going on between my ears, you being able to understand me well enough to know what I'm thinking, and then to be able to analyze and, and communicate what's happening between your ears and come back and forth and back to forth. And by the two of us working together or multiple if it's a group of students. Right, we build that mathematical understanding.

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Beginning of Segment 2

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Joanie

So, okay, with that kind of established, I think it might be helpful to kind of get into some of the, maybe the actual practical application of this.

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Curtis

for sure.

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Joanie

So for instance, I said earlier, like students talking, that's not necessarily discourse, but it is a necessary but not sufficient condition.

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Curtis

For sure. Hard to discuss.

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Joanie

You know, all the classrooms that I have been in, both as the teacher and as, you know, observer or coach or support person in, in a classroom. There are a lot of kids who aren't comfortable speaking in the classroom.

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Curtis

for sure.

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Joanie

So maybe we could start with just some ideas about how can I even get to meaningful mathematical discourse that, you know, helps build shared understanding of mathematical ideas like that could feel daunting if I'm in a position where my kids don't even want to talk.

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Curtis

Well, I think you said it just now that students talking, while it is not the end, is a required part of this. And so how do we build? And I think this, you know, we've had a lot of conversations around culture and classroom culture and safe spaces and, and really kind of building up our classroom norms in such a way that that students are willing to talk, that they are willing to, to share and to communicate and ideas are respected back and forth. And I, you know, I think it goes to, first of all, that classroom culture has to be established that this is a place where all the values on the table are considered, are valuable, are, you know, a part of the conversation, right? And whether they are mathematically, directly, correctly, related to what we're talking about or not is not the problem, not the point.

The point is getting students to communicate what's going on in between their brains and have or in between their brains and in between their ears, in their brains, and having them feel safe enough, having them have a space where they are willing to participate is the first step.

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Joanie

I totally agree. And then, and then building on that, once you have that classroom culture, then I think they're, you know, not for us to enumerate all the ideas and resources that are out there, but there are a lot of resources out there for providing some structures and scaffolds that help students build up that confidence in being able to, share verbally what their thinking is.

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Curtis

For sure.

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Joanie

I, I wanted to share one idea, that when I, when I was in my district math coordinator role, I had an elementary counterpart, and she brought this up to me in terms of this idea of like getting encouraging students to talk and share their thinking.

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Unknown

And I was kind of debriefing and observation with her and like, this teacher knew I was coming.

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Unknown

This was something we were working on together and just sort of like, okay, talk about your thinking. Like there was no build up. It was just like, I've never asked you to talk about your thinking in class before. And now the district math coordinator is here in the room, so I want you to talk. Now do it. Do it well.

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Unknown

And it's like you can't just, like, turn on a dime and expect kids to come along with you. But her, her advice was, you know, think about language development. And when you think about babies,

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Unknown

they can understand words before they can speak them.

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Curtis

for sure.

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Unknown

So giving students plenty of time to hear the mathematical ideas before we're asking them to voice their own.

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Unknown

And that might mean like the old think pair share kind of strategy, right? Where like I might practice with somebody else or listen to somebody else's thinking, so they, they understand what they hear before they can speak and they can speak before they can write.

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Unknown

So like this idea of like thinking about the progression of language and thinking about the progression of mathematical language the same way, right?

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Curtis

for sure. Yeah,

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Unknown

So that’s why I brought up think, pair, share a half a beat earlier than…it's that idea of like, first they have to be able to have time and space to even know what their own concept is, and then they need an opportunity to sort of practice it. So maybe it's internally saying, you know, this is what I'm going to share. And then I'm going to go to my partner and be able to share out loud.

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End of segment 2

Beginning of Segment 3

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Curtis

that brings up an idea for me. Okay, so in this example of your teacher saying, okay, go, go do this, go talk. It makes me wonder, you know. Doctor Knighten put together a great article this month, publication.

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Joanie

Yes, she did.

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Curtis

And the, the title of it was give your students the Gift of Time. Now, what's fantastic is when I read that, I read something into it that maybe she didn't have or she didn't put in the words in the document. Maybe she intended it also. But the idea that,

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Unknown

you know, students can take get the time to develop the language, right, and how they develop that language is the same that our our babies learn to understand language.

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Joaine

 Right

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Curtis

They hear it. They hear it from Mom and Dad speaking. 

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Joanie

Right. 

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Curtis

And so hearing others speak that mathematical language means that me as the teacher, I have to hold myself to a level of precision and precise mathematical language.

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Joanie

Absolutely.

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Curtis

That I, that I, you know, there may be things that I would say in regular conversation, that are not

precise mathematical language, that sometimes can help a student understand a concept totally.

But if I don't maintain high precision in my mathematical language, my students are not going to hear that and then be able to repeat it, or think it, or have it in their brain that they understand it, that they can begin to speak it and then write it and developing and understand it. So I do have to hold myself to a pretty high level of mathematical precision as a teacher, right?

Sometimes I can I can step down if I'm trying to meet a student where they're at, I can use language that I need to, to get to them. But if I don't start with the with the precisely language I'm going to be in trouble.

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Joanie

yeah. And I'm thinking this. I wasn't anticipating this. Babies learning to speak would be such a powerful analogy, but I think it kind of is.

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Curtis

for sure it is

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Joanie

And I’m thinking a lot about what you just said, that there there's a there is a point in the learning progression where we don't have to insist that students are precise,

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Curtis

for sure.

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Joanie

like I believe as a, as a student is developing their understanding of a mathematical concept, it's, it's helpful to let them use informal language, inaccurate language.

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Curtis

For sure. 

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Joanie

Just like, you know, babies when we do, you know, a nursery rhyme or pat a cake like everything doesn't have to be precise, right?

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Curtis

for sure.

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Joanie 

it's like, what's the intent? So I think there is a time in teaching in the time a time in the progression of an exploration of a concept where the informal sort of casual language and, and again, it's back to that, letting students practice like if my, if my goal

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Unknown

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Unknown

just want to understand your thinking, then use all the words.

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Unknown

And as we get better at being able to describe and when and when we need that nuance of no, I didn't mean that, I meant this, then we bring in that precision of language,

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Curtis

For sure. For sure

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Joanie

something cute about a baby saying psghetti. You don’t want to correct it right away.

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Curtis

My, my my youngest son.The speed lemon is we have the speed lemon. And recently it's now become the speed limit, and recently it’s become the speed lemont., we've get the T on the end, but, no, that's you brought up. And so we've been talking kind of in this, in this higher space in sort of general terms. But to make it very practical, I want to think I actually want to take this right down because I'm thinking very much about this development and the timing in which I might have, certain conversations about precision versus more casual and and accepting informal sort of, language about certain things. For example my, younger son is multiplying right now multiplying two digit numbers, by one digit numbers. Right. So I don't remember what the problem was. I think it was, nine times 23 or something like that. Right. So we're multiplying nine times the quantity 23 and the model or method that they are currently working on is, is sort of breaking that.

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Joanie

Right.

00;18;48;15 – 00;19;35;00

Curtis

And adding those value components 20 and three, multiplying nine times 20 and then multiplying nine times three. And then adding those two values together is the process that they're kind of going through. And they have tese two boxes sort of array on a, on a paper and off to the side. They've got nine times 23. And then they're supposed to write the 20 above one side and then the three up on the other side. And I was talking to him about a couple of things. One, I wanted him to said there wasn't there and there wasn't even a blank for it. But I said, I want you to write instead of just 20 and three.

I want you to write 20 plus three. I want the plus sign in the middle, because what you really have is the quantity 20 plus three. So we're multiplying nine times the quantity 20 plus three. I wanted him to start thinking about that. So I was asking him to put the plus sign in there, even though his teacher didn't have it.

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Joanie / Curtis

When you’re coming back to those partial products, the plus sign there

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Curtis

The plus sign matters to do with right. Right. Well what do we do? Yeah. Right. And so then he we were we were multiplying. And then he said, okay, so I'm going to take nine times he said do you see. He said nine times 20. And he said, well nine times two is 18.

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Curtis / Joanie

And then I stick on a zero, I add as I add a zero is what he said, I add a zero said, well, you don't add zero. I mean, what would I adding zero do? And he started to kind of mess with that. And I said, I said, I said, well, adding you don't, you don't add zero. I said, you're multiplying by ten. I'm okay if you want to say it that way. If you want to say, I'm going to multiply nine times two and then I'm going to, I'm going to multiply by by ten. Fabulous. Like that's good. That's great. Because he knew almost immediately that it was 180. But we talked about nine times, two tens. I use that language a lot with him. Nine times, two times, nine times, two oh 180 180 instead of nine times two and add a zero, right. So this, this idea of being very precise, both about the addition right, the partial product and then summing those two together at the end. But then also this, this idea of what is that quantity actually what does that 20 really. Not that that two. Right. It's not a process. Right. And so, and this idea of out of zero, I think well wait, wait a minute. Adding a zero means something different. And so it was interesting to kind of watch him and try to hold him to that higher precision language. Now, if I was in my high school classroom, oh, I can't tell you how many times I pulled out a zero or something equivalent. I said, just stick a zero on the end. Just stick a zero on the end.

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Joanie

We try to, you know, in the name of I want to make it more understandable. Right. And, and again, from a really good place of I want to speak to them using informal language. I want it to make sense to them. We could be actually really confusing them and detracting from the mathematics.

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Curtis

for sure.

00;21;47;13 - 00;21;49;07

Joanie

you know, I love that you share that story. And again, it just makes Oh, how many things I did I did wrong in the classroom. And again, it's not about always being precise every single time.

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Curtis

for sure.

00;22;01;04 - 00;22;34;09

Joannie

But you know, when we think about that consolidation of a lesson where we're really trying to, you know, like, hey, hey, this is the meaningful mathematics today. This is the thing we really want to drive home to really call out the that precise language and name it. We might informally say we're adding a zero right here. But actually, you know, to be able to call that out and make it at the forefront of students minds…well, let's just say it the right way all the time and assume that they'll catch on and understand the mathematics.

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Curtis

Right. Right now I think there's a, I think there's a real, important piece, that you just, you just hit, which is doing the precise language and speaking with that precise language is super important.

But then also being able to kind of show what it actually what actually happens as a result. Right.

It looks like I wrote 18 and put a zero on the end. It isn't what I did. It isn't how this worked. We really did multiply nine times two tens, right? We multiplied nine times that quantity 20 right, which is 180.

But I don't in my head think of 920 stacked on top of each other. In my head I think of, well, I know nine times two is is 18 and I know ten times 18 is 180. That's what's going on in my head. But it's important to be able to kind of show those things and call that out. Right? Right.

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Joanie

And allow the students the opportunity to talk about their own sense making of that. Yeah. And back to, you know, the early Common Core days. I might be going off on a tangent here, so I won't I won't go too long here. But the, the standard algorithm is not designed to teach understanding. The algorithm is designed to be efficient. So doing things like, you know, nine times two, right. Eight and carry the one like we do that for efficiency because that's the mathematics.

00;24;07;21 - 00;24;23;03

Curtis

Right. Absolutely. That's, that's 100% it. These algorithms are built for efficiency and it's wonderful to have them. We should be efficient there. There's, there's an importance to the efficiency. But not at the expense.

00;24;23;03 - 00;24;44;08

Joanie

gives us the opportunity to again make that explicit with, when, when we because I'm going to do that all the time as a teacher. Right. I’m gonna say the things that are not mathematically precise. So I think, I think again calling out this is about when we're getting at big mathematical ideas. That's when we want to focus on discourse. That’s when our precision of language. It's that understanding of what's happening mathematically. And then as we get fluent, you know, as we're more proficient, like, I'm not going to spend time with 11th graders talking about the standard algorithm.

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Curtis

for sure.

00;25;00;14 - 00;25;01;12

Joanie

Why I carry the one.

00;25;01;12 - 00;25;06;25

Curtis

Why is there a one or a zero on the end of this thing? For sure. 

00;25;06;25 - 00;25;14;29

Joanie

I mean not that some of them could probably use that, but yeah, I just kind of want to frame because

I don't, I don't want our listeners to think we're suggesting that you always have to be precise with your language.

00;25;14;29 - 00;27;42;01

Curtis / Joanie

No, I think I and I think that goes back to even the baby analogy that you had.

Right. Is it's it's great,

Early on hearing the cutesy things and, and us using real language to try to give them accurate understanding, but then casual things come up, right as they as they fully understand the meaning of words, they can then be used in different ways, right? They can we can we can reapply them, or we can use a different word that's related to that word.

Right.

And they can assume these things. It's part of developing language. And I think the same thing is true in what we're talking about in these mathematical ideas. 

Right. 

And we we know enough about this particular concept that we can we can handle that. We can handle the, the casualness, we can handle sort of the, the confusion, imprecise language, because we are we already know the precise part of it, and we're just utilizing something that maybe is more convenient or quicker way of communicating the idea. Sometimes my my kids get my kids get on to me occasionally about my, necessity or my, obsession, even sometimes, with precision. And, you know, when we, when we say something, we have a joke at my house at some point in my life my parents were discussing the, circumference of the earth at the equator.

You know, typical dinner. This is this is the kind of thing that happened in my house. Okay? And my mother, I think my mom, said something that was somewhat imprecise, and my dad says 25,000.

And it's that it's that, obsession with. You have to say it right. You have to say it exactly the right way. That's not what we're intending in our math classrooms. And we're not saying that every single time you talk about just the distributive property that you're supposed to be using the distributive property of multiplication over division or, sorry, over additions, like this whole, like, this whole mouthful.

We're not saying that, I hope not. We're not.

00;27;42;06 - 00;27;46;18

Joanie

Well, I think the bigger idea of what we're talking about here is one of the key messages,

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Unknown

that is in, well, not principles to actions, but, taking action. The additional series that they made. They were grade band focused. And as I was reviewing this chapter in preparation for the podcast

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Unknown

they suggested that meaningful mathematics discourse is not improvizational.

00;28;03;25 - 00;28;07;07

Unknown

Rather, it is something for which teachers can plan.

00;28;07;07 - 00;28;09;27

Curtis

Oh man. Yes.

00;28;09;27 - 00;29;00;15

Joanie

I think that’s what we’re trying to get at. When we're thinking about the trajectory of our lesson and we're thinking about what do we want, how are we going to get students to talk about their thinking and reasoning? How are we going to get discourse where they're responding to one another?

You know, how are we going to foster that culture where we we encourage disagreement and encourage students to defend their, their thinking to one another. And, all of that, like you, we have to plan for that. Like expecting to be able to do it in the moment is really challenging. And that allows us to differentiate like, okay, where am I going to be soft and where am I going to pull in that precision of mathematical language? And where am I going to ensure that the, the mathematical, the mathematics is understood because that precision of language has been brought in.

00;29;00;15 - 00;29;54;22

Curtis / Joanie

for sure. And I think that, and that also goes back to Doctor Knighten’s post about giving our students the gift of time because planning for those precise conversations and planning for that student discourse and knowing and anticipating what students are going to be saying and anticipating and ordering and thinking about now, what will, what, what, is the order in which I want to, reveal these things and kind of investigate these things that students, I think will say, right.

Having all of that, that takes time. That takes an incredible amount of time to plan and think about and to be ready for and then to execute. We can't rush through those things. And so, you know, being ready for those things and planning for them is, is so important.

00;29;54;22 - 00;30;05;24

Joanie

Right. And I think too, by doing that, you know, I'm thinking also about the equitable teaching practices and thinking about discourse as a strategy for an equitable classroom.

00;30;05;24 - 00;30;06;11

Curtis

for sure.

00;30;06;11 - 00;31;00;00

Joanie

And because, you know, as we as we started today, saying discourse is really about students owning their own thinking and explaining their own thinking and communicating what's between their ears to other students like that is building their mathematical identity, their mathematical agency.

You know, to have that the the confidence that comes along with I have something valuable to contribute and, and I can communicate in such a way that others understand me, just such a powerful force in, in building student identity. So I think of that is such a, it's such a great avenue for, inclusion in your classroom and, you know, tapping into those, the potential of every student and thinking of every student in the valuing of every student.

00;31;02;25 - 00;31;20;28

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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