May 14, 2024
Room to Grow Math
Season 4
Episode 4

Making Sense of Mathematics

Room to Grow - a Math Podcast

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Room to Grow - a Math Podcast

Making Sense of Mathematics

May 14, 2024
Season 4
Episode 4

Room to Grow Math

In this episode of *Room to Grow*, our hosts discuss ways to support student sense-making in teaching and learning mathematics. Building out on the ideas shared in Episode 3 of *Room to Grow*, Curtis and Joanie dive more deeply into what it means for students to “make sense of mathematics.”

They discuss what it looks and sounds like when students are making sense, as opposed to just repeating back learned ideas, and consider which classroom structures and teacher moves might best support students’ sense-making.

They acknowledge that sense-making is not more or less important than learning mathematical skills and fluency, but that it is a part of deep learning and of a student’s ability to generalize their understanding.

As Peter Liljedahl says in *Building Thinking Classrooms, *“The goal of building thinking classrooms is not to find engaging tasks for students to think about. The goal of thinking classrooms is to build engaged students that are willing to think about any task.”

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

- The Standards for Mathematical Practice, now referenced in most states’ math standards and originally published by the Council of Chief State School Officers and the Governor’s Association;
- Jo Boaler’s
*Math-ish*book and website; - Peter Liljedahl’s book and website
*Building Thinking Classrooms*, and Robert Kaplinsky’s blog post about why you should read it.

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 discuss ways to support student sense-making in teaching and learning mathematics. Building out on the ideas shared in Episode 3 of *Room to Grow*, Curtis and Joanie dive more deeply into what it means for students to “make sense of mathematics.”

They discuss what it looks and sounds like when students are making sense, as opposed to just repeating back learned ideas, and consider which classroom structures and teacher moves might best support students’ sense-making.

They acknowledge that sense-making is not more or less important than learning mathematical skills and fluency, but that it is a part of deep learning and of a student’s ability to generalize their understanding.

As Peter Liljedahl says in *Building Thinking Classrooms, *“The goal of building thinking classrooms is not to find engaging tasks for students to think about. The goal of thinking classrooms is to build engaged students that are willing to think about any task.”

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

- The Standards for Mathematical Practice, now referenced in most states’ math standards and originally published by the Council of Chief State School Officers and the Governor’s Association;
- Jo Boaler’s
*Math-ish*book and website; - Peter Liljedahl’s book and website
*Building Thinking Classrooms*, and Robert Kaplinsky’s blog post about why you should read it.

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;26;15

Joanie

How can I help kids use representations to make sense? And I think for me that that kind of the most logical thing as well, let them come up with their own representations first, because that will build on their own understanding. And then how do how do we help them see connections between their representation and another representation. Like that's true sense making, not here's how you do tape diagrams.

00;00;26;15 - 00;00;56;11

Joanie

Step one draw this. Step two divide that, like procedural sizing representations, does not lead to sense making. In today's episode, we focus on how we can support students in making sense of mathematics. We believe this is a topic near and dear to the hearts of our listeners, as well as to Curtis. In me, we wonder about ways of knowing what it looks like and sounds like in the classroom when students are making sense, and all the ways.

00;00;56;11 - 00;01;11;15

Joanie

Since making is baked into the standards for mathematical practice, we talk about the importance of intentional planning that allows for students sense making through tasks, questioning, and time. There's a lot to unpack here, so let's get growing.

00;01;14;02 - 00;01;43;25

Curtis

Well, Joanie, here we are again. It is mid-April and we are recording the Room to Grow podcast. I'm very excited to be, doing this again with you today. Today we have a really cool topic that we're going to be talking about, this idea of making sense of mathematics, a topic that is near and dear to our hearts. And I think probably just about every math teachers heart on the planet. So, really excited to be chatting with you about this again today. Yeah. Me too. Curtis.

00;01;43;25 - 00;02;07;23

Joanie

Yeah. Me too. Curtis and I think, you know, as we've gone through our planning conversations for this particular episode, I think we have touched on so many of these ideas and several other of the topics that we've discussed, and I'm excited to kind of pull it all together under this frame of making sense of mathematics. And when we're talking about making sense of mathematics, we're talking about students making sense of mathematics.

00;02;08;10 - 00;02;33;10

Joanie

and thinking about, you know, what does that what does that look like? What does that mean? And how do we how do we ensure that that's happening, as educators? Because like you said, this is something that surely every math teacher wants their students to make sense and understand. And, I'll just kind of start off with, you know, a quote that takes me back to my district math coordinator days when, when the new Common Core State standards came out.

00;02;33;12 - 00;02;56;18

Joanie

I remember watching these videos that were published by the the lead writers of the Common Core standards that were intended to help people make sense of them a little bit. Right, to understand what the intent behind the writing of the standards were. And I will never forget, I can even just picture his face right now. One of the videos, Bill McCollum, says math makes sense.

00;02;56;20 - 00;03;18;13

Joanie

And then he went on to talk about how even in his position at the college level, he so often see students that come in thinking that math is just a set of rules and procedures and, and never understand that there are bigger concepts at work there, but that math makes sense, which is just such a powerful, you know, three line quote that that has stuck with me for a long time.

00;03;18;13 - 00;03;33;20

Joanie

So, you know, to frame our conversation today, I'm gonna throw a question at you to respond to how do we know if students are actually making sense of the mathematics that we're learning, or if they're just thinking about math as a set of rules and procedures to apply?

00;03;33;29 - 00;03;57;19

Curtis

So I think that's a that's a really good, question and I wondered if you were going to throw that at me or if I was going to get a chance to throw that question to you on this. And, and I don't know that I have a good, perfect answer for that. I don't even know if there is a good, perfect answer to that. I know in a lot of our conversations pre podcast, we were talking about this idea of how do you even measure understanding?

00;03;57;19 - 00;04;25;02

Curtis

Because in most of the measurement ways that we do measurement, there's some level of procedure that that has to happen. And I think that's where we struggle. or at least I did, in terms of trying to measure my students understanding, because there is some level of, you know, structure, and procedure and getting to an answer, that that happens.

00;04;26;00 - 00;04;57;11

Curtis

so, so thinking about what does, understanding look like, for, for a kid and for my students in my classroom, I think about, my son Teagan, and we were looking at, a math test that he was taking, getting ready for the state, exam, recently. And I was asking him some questions around, you know, statistics and the bar graphs and, the, the statistical plots that he was looking at.

00;04;57;14 - 00;05;37;15

Curtis

And it seemed to me, that that was a place where I could get a level of understanding, for him because there wasn't, because we were having a real conversation. And maybe that's, maybe that's the way that we can get at it in our classrooms. As, as, as much as we possibly can. Is, is around conversation or, or even, these essay type responses, if we're actually trying to use this in some sort of, an assessment space.

00;05;37;22 - 00;06;45;04

Curtis

But I think it, it, it does boil down to the conversations and, and being able to have students explain, the way and the thought process they, that they use to execute whatever the task is, that we ask them, had them doing so, you know, in the case of Teagan, when we were looking at, and I think it was a histogram and I don't, I don't remember the content or the context, rather, of the, of the histogram itself. But thinking about when I was prompting him about that, that histogram and sort of his interpretations of it, and we were I was asking him, you know, some typical questions about shape and, about mean and about median and, and where did those kind of lie in the distribution and getting some of his feedback on how he was thinking about those questions? I asked him this is really the only way I had any level of of understanding his understanding.

00;06;45;04 - 00;07;35;16

Joanie

I like it, I like it well, as you were talking, I was thinking one of the things that makes, sense, making hard to assess is it's not. It's not necessarily like the end goal. Right? Like making sense is something that happens all the way through the learning. You've got to do some making sense at the beginning of learning a new concept, because you've got to be able to anchor it to previous information and already know knowledge and, you know, put it within the greater landscape of your knowledge. And then as you develop, understanding or learn a procedure or, you know, start to get a better grasp on the mathematics, then there's room for another layer and depth of sense making.

00;07;35;18 - 00;08;08;25

Joanie

And then if you talk about like getting to the end of quote unquote understanding, and this is where assessment comes in, right? Where we're we're kind of when we assess, we're trying to determine the end product of student learning. What, what do they know at this moment in time. Right. And that sense, that sense making around a particular math topic might not end at a unit test or an exit ticket from a daily lesson, or even a state standardized assessment where, you know, this particular math topic is not going to be assessed formally.

00;08;08;25 - 00;08;30;23

Joanie

Again, the sense making doesn't end there. So that that idea of can we continue to make sense. And I think for me that's one of the things that I love about doing teacher professional learning is I, I continue to make sense of mathematics myself, even, you know, as an adult with lots of years of studying these topics behind me and lots of exposure to them.

00;08;30;23 - 00;08;55;05

Joanie

So, I also want to tie back to the, the standards for mathematical practice, because I think those are ways to help us frame like what specifically does that look like when a student is making sense? And, you know, of course, in the first standard for mathematical practice, we see makes make sense of problems and persevere in solving them so that that word is like coming in there, like make sense?

00;08;55;05 - 00;09;22;23

Joanie

It's right there. The very first two words of that malpractice. But we see sense making through. I mean, I would argue we could probably go into every single one of those eight math practices and find sense making in there somewhere. But I really like from this first one some of the language in there to think about whether students are making sense, explaining to themselves the meaning of a problem.

00;09;22;23 - 00;09;50;20

Joanie

So I really love that idea of having students be metacognitive and self-reflect on their own understanding. Like, if I can't explain to myself, then there's no way I could be making sense. I also really like to change course if necessary, evaluate their progress, and change course if necessary. There's some sense making around wow, what I'm doing is not productive, is not getting me closer to what I'm trying to understand.

00;09;50;20 - 00;10;20;27

Joanie

So that ability to recognize when I don't understand or when something isn't making sense. And I can, and I can change that. A nd then just they're continually ask themselves, does this make sense? And how does this make sense? That kind of reminded me, too, of the, the ask yourself questions that, Grace and Amy talked about when we had them on our podcast of a few months ago. So those are some of the things that I'm thinking about.

00;10;20;27 - 00;10;28;28

Music break - End of segment 1

00;10;28;28 - 00;11;22;13

Curtis

You know, it's really interesting. And there's there's been kind of a cool conversation that's happening. I don't know if you've seen it, in the my NCTM blog this last week or I guess it's a digest, that, is posted out there and there's been a conversation around hundreds of charts. And I know neither of us were elementary, teachers, but, just thinking about that and watching the conversation, and I don't pretend to be an expert on numeracy or even how students establish or understand number. but there is a conversation about reflecting, the 100th chart. You know, typically, if you think about it like, I know when I was in school and even, you know, watching my, my own kids, in school, and even this really, you know, the numbers charts that I've seen that are, area based.

00;11;22;16 - 00;12;01;24

Curtis

Right. So you have a multiplication chart rather that that goes, you know, 1 to 100, but it's top down. It's literacy based. It's the way we read articles, but there's been a conversation about reflecting that and doing it from the bottom up, these, these hundreds of charts will be interesting from the bottom up. And in doing, you know, counting the counting numbers 1 to 100, bottom up and the conceptual or the making sense, connections that can happen if we do it that way.

00;12;01;28 - 00;12;27;13

Curtis

So I there's a lot to this idea of, looking at and making sense of, mathematics. And even from the number perspective, the way that we, present things to students. And that actually kind of leads me to thinking a little bit, about another question, which is, you know, we talk a lot about teaching in the podcast.

00;12;27;13 - 00;12;58;08

Unknown

We talk about a lot about, shifting our perspectives and shifting the, the maybe the methodologies or, even growing in some of the ways we approach, and plan. But if we're thinking about sense making, how does that how does that look different in my classroom than maybe, maybe it did before or what are some of the things that are inherent in a classroom where making sense is, sort of the primary focus?

00;12;58;10 - 00;13;25;07

Joanie

Oh, I love this question. I don't know that when I reflect on my own days in the classroom, I don't know that sense making was really a part of something I did every day. I think that it requires a teacher's intentional planning. Oh yeah, to give the opportunities for that to happen. Right. So, some of the things I'm thinking about first and foremost is time.

00;13;25;10 - 00;14;23;00

Joanie

We have to give kids time in order to make sense. It's not something that, you know, they're going to hear a new idea or after the introduction of a concept is going to immediately click for them, like giving them the time and space, without direct instruction, without even necessarily, you know, the teacher's voice to just think and, and wrestle with and engage with a mathematical idea with, with some maybe guidance and structure around how they can make sense of it. So I think time is the most important thing. And being able to I know it's also one of the biggest things that teachers struggle with. There's always this pressure for time. And I know I don't think I taught a single day where I didn't feel like, oh my gosh, I have so much to cover. so relieving ourselves from that burden so that we can create this space.

00;14;23;00 - 00;14;51;01

Joanie

And then the other component that I think is really important from my perspective is planning our questioning, thinking about how we anticipate students are going to engage and ensuring that we have open ended questions that allow us to guide students towards the sense making. and I think that comes from like really intentional planning ahead of time, thinking about, okay, here's the concept.

00;14;51;01 - 00;15;23;17

Joanie

I'm trying to get the mat, here's where I think they are. And then here are some ways I can guide them to to connect that gap. Or maybe here's a question I could ask to get a better sense of where they are. And then based on what they say, here's what I might ask next to help guide them to that sense making. And, and Curtis, I would love. I know when we are planning for this conversation. You talked a little bit about the sage on the stage guide on the side. Would you, would you kind of jump in and elaborate on that a little bit? Because I thought that was a really powerful way to think about this.

00;15;23;17 - 00;15;50;16

Curtis

Yeah, I think I mean, what I, what I remember talking about on this in and, you know, it's a, I mean, we hear that conversation a lot.

That's, that's been a, you know, been a statement that's put out there many times over. Right? This idea of we need to not be the sage on the stage, and we need to be, more the guide on the side. But I one of the things that that I'm passionate about, that does not mean that all of a sudden the guide on the side is, I mean, you're still guiding, right?

00;15;50;16 - 00;17;02;26

Curtis / Joanie / Curtis

Like you're not just completely hands off. You're not. You're not just stepping away and just standing in the corner because my grin on your face let the kids just run rampant with, a particular thing. It, it is more about. And I think there's a huge, skill. and, and even a practice. right. So this is, this is a developed thing, over time in the questioning and the, the, the thinking about how and how can I help guide this conversation so that the students have opportunity plenty of it to make and reason, and make sense and reason, about this task or whatever. And, and you know, I was thinking about this. Does it require, like, a real task? Does it require a super heavy. You know, we hear a lot about implementing great tasks and certainly there's a lot of value in having tasks that are, you know, low floor, high ceiling, open, you know, types of questions with lots of pathways in the middle of them.

00;17;03;01 - 00;17;31;17

Curtis

And certainly those things are lhugely important. But does it require that. Well, no, I think I can I think I can, allow my students good reason inning even with tasks that maybe don't meet all of those, all of those requirements or have all of those things out there. I think the bigger the bigger thing that I was trying to get at with this was, you know, there's a balance to it, right?

00;17;31;17 - 00;17;58;10

Curtis

Sage on the stage says, all the information comes from me, the order of my desk. Right. The way that my desks are oriented, everything is pointed at me. You know, I remember early in my teaching career, I even had, in my room, not only did I have rows, and in my room, but I actually had rows that were in sort of a U-shape.

00;17;58;12 - 00;18;27;28

Curtis

I mean, do you want to talk about. Wow, you want to talk about, you know, putting my students in the position of, you know, sit and get, you know, early on in my teaching career, this is how I feel about you, right? Like, I yeah, I and that was not a perspective that I understood that that that it was, you know, if you would have asked me was I was I, you know, trying to be the sage on the stage, I would have said, no, I'm trying to, you know, I want my students.

00;18;28;00 - 00;18;50;25

Curtis

I never would have understood even the orientation of my classroom putting me as the sage. Right. on the stage. So thinking about there are lots of ways that we can get at this idea of putting ourselves on the side, allowing students, the opportunity to, in the space, to think you brought up time. I think that's huge.

00;18;50;28 - 00;19;25;02

Curtis

I think there's also the idea of giving the student, you know, removing the pressure, giving them, setting up the task or setting up the activity in such a way that that they have space to, do multiple drafts of their thinking. Right, right. So, you know, the, the, the, the folks in the focus on, you know, I know, Peter Little at all book and thinking building thing in classrooms and the idea of vertical learning surfaces with, you know, a nonpermanent writing surface.

00;19;25;02 - 00;20;06;20

Curtis

Right. So dry erase board or, you know, sort of some sort of, sticky note page that's nonpermanent, whatever it is, giving the students multiple opportunities to rehash their thinking, be on their feet and, and do the that lends itself to the sense making about a scenario or about a math problem or a math task or, something of that nature. So those, those kinds of things come, come to mind whenever we're, we're talking about, you know, this idea of what what does my teaching look like when my emphasis is on the making sense of mathematics?

00;20;06;20 - 00;20;06;20

Music break – End of segment 2

00;20;16;03 - 00;20;41;13

Curtis

I think there's also plenty of opportunity to think about, you know, we're both high school background and think, you know, so our focus tends to often be on, on the things that we taught or whatever. But I think there's opportunity, you know, Pam Harris talks, and then does a lot with numeracy and has the focus or the quote math is figure out a bowl.

00;20;41;16 - 00;21;13;07

Curtis

Right. So that's, that's her thing. Right. Is is sense making. And I see her, posts and I think she's still doing it. I haven't looked on Twitter or X, I guess in a long time to know if she's still doing this, but she used to lead these, these math talks on, on there, where she would post a problem that had some sort of a unique number thing in it and ask people to post how they worked it and the way that they thought about doing the problem.

00;21;13;13 - 00;21;49;29

Curtis

And there was always these fascinating ways that people, ridiculously creative ways that people made sense of, the tasks that she put out there. And I just, I wonder if there's opportunity for us to do those kinds of things that maybe aren't in, in high school, that aren't curriculum based activities necessarily. Right? They're, they're older. Right. So our kids are not necessarily doing a lot of the number, numeracy things now, in high school.

00;21;49;29 - 00;22;13;21

Curtis

But do we provide opportunities just once in a while to allow the students and I an opportunity to make sense? It isn't so much about the numbers in numeracy, but it's the process of making sense. How do I break this thing down? How do I how do I put it into a bite sized chunk? How do I get close enough, that I can that I can think about this thing?

00;22;13;21 - 00;22;30;08

Curtis

And what was it that made sense about my close enough answer? not because that particular problem was, the thing that we're focused on in high school, but it's the process of getting students to make sense and think and do that.

00;22;30;08 - 00;22;58;04

Joanie

I like that. I, I think bringing up, you know, this idea of non-curricular tasks. I think about that too, as a way to break the cycle of and I know we talked about this before of, you know, in math class, sometimes students can get the impression that what I'm supposed to do is figure out what answer the teacher wants me to give here. Or, gosh, we've just spent the last week solving, solving equations in one variable.

00;22;58;04 - 00;23;21;07

Joanie

So I must if the teacher's giving me this, I must apply that skill. Even if the context of the problem doesn't. That doesn't create any sense making. Right? So I love that idea of going to a non-curricular task to both get students sort of break the thinking about here's the math we've been working on. So, every problem I give you is going to be an application of this math topic we've been working on.

00;23;21;09 - 00;23;49;23

Joanie

And also like kind of resetting them to just make sense. Just use your brain. Just use what you know, your interaction with the world. It's back to Bill McCollum. Map make sense. So helping kids break free from what are the rules and procedures to just how do I reason about this and how do I tap into what I know and understand, about this particular mathematics to make sense of it?

00;23;49;23 - 00;24;14;08

Joanie

And I want to tie back a little bit to you were talking about tasks. And, you know, one of the things I really wrestled with in preparation for this is, is it about the task? Like, there's always been so much emphasis on finding these rich tasks that allow students to reason, but I think kids can reason even about, you know, a worksheet of practicing a procedure.

00;24;14;10 - 00;24;42;01

Joanie

You know, you could you could look at a series of back to your point about the hundreds chart, right? It's very procedural. Memorization of facts is important. Or, you know, knowing how those numbers are all relate to each other and what the patterns are. But their sense making involved in that too. So, I just think it's important to think about all of the opportunities we have as educators to present students with questions that allow them to make sense.

00;24;43;02 - 00;25;14;21

Joanie

And for sure, some of those resources, are hugely important in doing that. The other thing that I was thinking about in preparation was, the use of representations. So yeah, we use models, right? Like we use drawing things or even like, a table to draw a graph or all these different ways that we represent the mathematics and I think representations can be such an important component of sense making for students.

00;25;14;23 - 00;25;46;15

Joanie

And at the same time, I think there can be danger in teaching a representation for the sake of learning the representation and without like it's possible to focus on a representation without bringing sense making alongside of it. So oh wow, students. Yeah. Student like how do you how do you avoid that? How do you not just make it about, you know, oh, we're supposed to solve this problem by, you know, drawing a double number line?

00;25;46;15 - 00;26;05;07

Joanie

Mom, I can't I can't do it the way you're helping me understand it, because I have to do it with a double number line, and it's like, oh, the representation is getting in the way of this sense. Yeah, right. So, like, really thinking about as an educator, like how, how can I help kids use representations to make sense?

00;26;05;10 - 00;26;36;01

Joanie

And I think for me that that kind of the most logical thing as well, let them come up with their own representations first, because that will build on their own understanding. And then how do how do we help them see connections between their representation and another representation. Like that's true sense making, not. Here's how you do tape diagrams. Step one draw this. Step two divide that, like procedural erasing representations, does not lead to sense making. Oh wow. It's my mic drop moment.

00;26;36;01 - 00;26;50;10

Curtis

Yes, hey that is a that is a mic drop moment I, I, I just about jumped out of my seat, when you were saying that because, in my head and in a, your mic drop moment is exactly it.

00;26;51;04 - 00;27;18;02

Curtis

We, we often and I'm guilty of it too. Proceduralise those representations that are intended to help students make sense. and yeah, I was watching Poor Teagen, he gets to he gets picked on a lot. I was I'm I'm going to bring up Truett here in just a second too. But and my, our podcast listeners are going to feel like they know my kids one of these days.

00;27;18;04 - 00;27;46;10

Curtis

I'll bring them on the podcast someday. Now. yeah. So poor Teagan, we were we're working on, number lines, adding, adding integers the other day. And Teagan can see the numbers in the expression and knows how to evaluate that. Yep. He can look at the expression, you know, ten plus negative three equals.

00;27;46;10 - 00;28;18;28

Curtis

And he can give me seven pretty fluently. we were working on, we were working on a, a, his Chromebook. He was working on an online assessment of this, whatever homework problem or whatever. And the there was a math represented. And I remember very vividly, it was a multiple choice problem. it was the, the initial expression was nine plus negative four.

00;28;19;00 - 00;28;43;03

Curtis

And, and actually, I'm not even sure that there was an initial expression. It was just the, it was just the arrow diagram. So on the picture got it. There was a, an, an arrow slightly above the number line pointing to the right, starting at zero, going out to positive and going to nine, and then one above that.

00;28;43;05 - 00;29;25;20

Curtis

That was going from positive nine left four units. He was supposed to pick the two possible expressions that were that were there, that aligned with that diagram, that aligned with that diagram, and he, I think was very much caught up in a literacy sort of format. He was reading top down, and he continued over and over and over to be looking for expressions that started with a negative value, because he was seeing the first that was the top arrow and it was going left.

00;29;25;20 - 00;29;46;17

Joanie / Curtis

He's reading from he's reading from top to bottom. So he's not starting at the number line. And looking above it, he's starting at the top arrow exactly going oh that's an arrow pointing to the left four unit. So I must start with negative. I must be starting with negative four. And and but it starts with the nine. So it doesn't make sense.

00;29;46;17 - 00;30;08;21

Joanie / Curtis

Why it starts at the nine and goes to the. Yeah. Oh and so so yeah that would be so messing him up. Him seeing the not seeing the bottom arrow starting at zero like he was so caught up in the top arrow being an arrow to the left. He kept choosing the negative nine even though he knew it wasn't negative or not mean.

00;30;08;22 - 00;30;34;17

Curtis

He could have told me the result was 510 times over. Nope. All we had was the expressions, but we had this visual representation. We were trying to make sense, and I'm not faulting the visual expression. I'm just I'm just putting it out there that sometimes we were proceduralize these things and students get in this routine of trying to make sense of the representation and procedural eyes.

00;30;34;17 - 00;30;59;23

Curtis

This representation, instead of thinking about the mathematics. And we get lost in it. so often. On the flip side of that, I want to celebrate the visualizations that that happen in my younger son's mind, he was solving a problem. it was some, ratio. He was trying to figure out how many cookies were in each group kind of problem.

00;30;59;26 - 00;31;31;00

Curtis

It was a nice division problem. and he, he had it was, it was, 56 cookies and they were eight students or whatever. And he was needing to come up with, with or. No, it was seven students, I apologize. It was seven students. And in his mind, he saw seven groups of seven is 49. and then he said, and seven more is 56.

00;31;31;02 - 00;31;53;00

Curtis

And so, yes, he was supposed to do this division thing in his head, and he even had that written out on the paper, but he could see the square number and he talks. I think I've talked about that before. He sees those squares and he sees the squares. And then, yeah, he was able to tack on seven more to get the 50 and then builds from there, which is great.

00;31;53;00 - 00;32;13;28

Curtis

That's great sense making. So I mean I don't I don't want to jump. You know it says celebrating one representation and then having the proceduralized version in the other. child. It was just when you were talking about that, both of those two stories were in my mind going, my gosh, you're saying exactly what I think of.

00;32;14;01 - 00;32;46;22

Curtis

And really, when we think about, you know, how folks have, you know, over, over time, rejected even what was introduced with the Common Core and long before the common Core, other methods of these representations of shared representations and people trying to do conceptual understanding, that we have sometimes demonized those representations because not because the representations were bad, but because the representations were proceduralized and the sensory was stripped out.

00;32;46;24 - 00;33;10;11

Curtis / Joanie / Cutris

And it's like, well, yeah, of course the, the standard algorithm would be simpler or faster or whatever, if what we're doing is just a procedure trying to get the answer. Yes. But if what we're trying to do is make sense, then maybe we can, then maybe we can use this representation to help us make sense of the mathematics that we're trying to accomplish.

00;33;10;11 - 00;33;18;19

Music break – End of segment 3

00;33;18;19 - 00;33;42;11

Joanie

I just want to chime in here to Curtis, a comment that, a comment that we shared with one another when we were planning for this. Like, we're not suggesting that this idea of sense making and engaging with conceptual understanding is the end all be all. In mathematics, there is a role for procedures. There's a role for getting answers quickly, efficiently and accurately.

00;33;42;11 - 00;34;12;19

Joanie

Like, we're not suggesting that you should throw all of your lessons that are procedure lessons out the window and always focus on sense making. I think what we're trying to get at is, is what an integral part of learning that is, and how, again, just thinking of weaving it throughout the entire learning process and not thinking about it as, replacing other parts of learning or as it not just being an end goal like an end of unit, you know?

00;34;12;19 - 00;34;37;21

Joanie

Oh, I gotta put, I've gotta put a sense making question on my final unit assessment. No, this should be I loved what you said earlier about, like, having real conversations throughout the teaching process that just allow students to express themselves. Not in a like you have to say this. You know, these words have to be included in your written explanation or whatever, but just to let them talk about how they make sense.

00;34;37;21 - 00;35;01;02

Joanie

And I'm thinking about all the resources that we have as educators to help us understand and to give us strategies for how we might do this. So, certainly when I was talking about the representations and making connections between how individual students might have thought about representing a mathematical concept, I was thinking about the five practices for orchestrating productive mathematical discussions.

00;35;01;02 - 00;35;44;21

Joanie

Right. Like that's a great structure for eliciting different ways of thinking about mathematics. And then orchestrating. It's not there's not a better word orchestrating a, procedure, a way for students to share their thinking and to help students see the connections between different representational representations and different ways of making sense about a problem. And I also think there's a lot of power in, that effective mathematics teaching practices are NTM published in Principles to Actions years ago, and the Taking Action series do such a nice job of helping unpack those by the different grade bands, but I'm zeroing in on the eighth one.

00;35;44;21 - 00;36;09;06

Joanie

Elicit and use evidence of student thinking like if I, really trying to find out how students are thinking mathematically and bringing evidence of their thinking into my instruction. I think I'm going to be tapping into their sense making and giving them, again the space and the voice to be able to talk about how they make sense.

00;36;09;06 - 00;36;36;04

Joanie / Curtis

And then, again, I think one of the things I saw often in all the classrooms I was in was, failing to summarize the lesson. And this is a great way to. Yeah. You know, connecting how students are thinking differently is a great way to summarize. Now, I think you I think you hit on it and that collecting and that constructing and orchestrating that conversation.

00;36;36;07 - 00;37;06;10

Curtis

You know, we've been saying this over and over again. And I just as you were talking about this, I even just realized, like, what the phrase making sense, the first word in that making is, is it is a construction word like we are putting together. We are doing since we are doing this, this, construction of, of the sense.

00;37;06;10 - 00;37;37;12

Curtis / Joanie / Curtis

Right? We are making sense. Making things come together like it's a construction sort of thing. And I that was it. That just hit me as we were talking about this. And you were you were talking. It's not a passive thing. It's an act of passing. Oh, I like that. It is active. So the study is active in and we are active as teachers in that, in that same, in that same space, all of this, this making sense of mathematics, you know, you brought it up and we've iterated it.

00;37;37;12 - 00;38;07;12

Curtis

And I won't reiterate hopefully too much the, the idea that, you know, we do need procedural things. We you need to be able to get to answers. and we do need to be able to have evaluations and assessments and know what students know and can carry out. Right, in order to be able to, to fit into the system, that we've got of education and, and classes and, and moving students along that system.

00;38;07;14 - 00;38;36;27

Curtis

But you know, in reading and preparing for this, this podcast and thinking about this, I looked at, some summaries of Peter Little adults, book building, thinking classrooms. And one of the quotes that, was brought out and I read it off of, Robert Kaplinsky s blog, about this, about Peter's, book. And he shared the quote, from the book that says this says the goal of and it's building thinking classrooms.

00;38;36;27 - 00;40;13;10

Curtis

But I would argue that even just the goal of, of our mathematics teaching, is not to find engaging tasks for students to think about the goal of thinking classrooms. And again, I would say just mathematics teaching is to build engage students that are willing to think about any task. I mean, what we're doing and why we care so much about making sense of mathematics is not necessarily that every student who's graduated high school and into college or whatever or off into their career, is all of a sudden capable at, you know, 20 years down the road of being able to factor quadratic or, you know, perform some task that we had them do in their algebra one or algebra two class. It's that that person, 15, 20 years post-high school, has retained the sense making process that we taught them in our in our classrooms. I mean, that's ultimately the goal of what we're trying to do in our math classes. When we talk about making sense, yes, we want to make sense about the math for the time being because we have a, you know, we have a course that we're teaching and we have standards meet, we have assessments and all those things, but ultimately the things that we want them to be, to be lifelong is the ability to construct sense, to be able to make sense of things.

00;40;13;12 - 00;40;18;16

Joanie

And there’s your mike drop moment.

00;40;18;19 - 00;40;35;10

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.

00;40;35;12 - 00;40;42;13

Joanie

See you next time!

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