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Opening music

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Unknown

In this episode of Room to Grow, Curtis and I wrap up our series on the mathematics teaching practices from Principles to actions. We discuss the importance of supporting productive struggle in learning mathematics by first exploring what it means to struggle productively, and we settle on a term we actually like better than struggle. We go on to share stories and examples of how classroom culture and teacher content knowledge are the core components of effectively engaging, productive struggle in the math classroom. Want to grapple these ideas with us?

Let's get growing.

00:00:37:03 - 00:01:01:14

Curtis

Well, hey, Joni, I am so excited to be recording this podcast with you the Room to Grow podcast yet again. Today we are midsummer, in Texas. So today the temperature, I think, going to be 101 degrees. So I'm very glad to be inside in my air conditioned office here to be able to be recording this with you.

00:01:01:14 - 00:01:32:16

Curtis / Joanie / Curtis

And today we've got a really great topic. We are continuing our, series. Celebrating. In fact, we're wrapping up. That's exactly right. We are wrapping up our series today on the Principles to Actions publication from NC, TM, celebrating the 10th anniversary, of that publication and the great work that was done there, kind of outlining the teaching practices, the mathematics teaching practices, and giving some, some real, definition, around those.

00:01:32:16 - 00:02:01:20

Curtis

So really, that has set the foundation for our entire, year so far this year. And, we are excited now to to have our final conversation in that series. Today, we are talking about the mathematics teaching practice. Supporting productive struggle in learning mathematics. So that's, that's our topic today. We're going to be looking at and I know we did a previous podcast conversation on this and folks can go look for that.

00:02:01:20 - 00:02:23:13

Curtis / Joanie

I think it was in our season one. Yeah, it was July of 2021 that we had a great conversation about productive struggle. And for anyone who's listening today that hasn't already listened to that, that one, I would encourage them to go back and listen. You know, we're going to be intentional about just not repeating that conversation.

00:02:23:15 - 00:02:46:16

Joanie

So kind of going a little bit deeper. But there were some great nuggets in there. We talked about, the idea of the Learning Pit, which I still think about all the time as a great metaphor for, you know, what we mean by students experiencing productive struggle and what the role of the teacher is in, in that and ensuring that the struggle stays productive and not destructive or, you know, turning into frustration.

00:02:46:16 - 00:03:08:12

Joanie

So yeah, that was a great episode. But, I'm excited to continue to build out our thinking around this today. And if it's okay with you, I think I'll start with reading like we've kind of started the each of yeah, I think that's a great idea. Each of the episodes in our series on the principles to actions, mathematics teaching practices with reading like how it's how that practice is defined by enactment in principles to action.

00:03:08:12 - 00:03:40:08

Joanie / Curtis

So for this one, it's described this way. Effective teaching of mathematics consistently provides students individually and collectively with opportunities and supports to engage in productive struggle as and they grapple. I think it's much to say as they grapple, I think it does say as they grapple. So sorry, as they grapple with mathematical ideas and relationships. I love that term grapple with, and I think we should, we spend some time here today unpacking that idea of grapple with.

00:03:40:08 - 00:04:35:07

Curtis

But, really, this this is a lot. Whenever you, read this in as the whole, you know, we're looking at a consistent thing that's happening in my classroom. So I'm providing opportunity. Yeah. All of this sounds an awful lot. And I think we had this conversation before we started recording that. This sounds an awful lot like classroom culture like that. That really as much as any of the teaching practices, this one in particular is really pretty dependent upon implementation and thinking about, you know, me as a teacher, what are my routines? What are the things that I've established both in my classroom and in my preparation that kind of facilitate me being able to provide opportunities and support for students to to struggle or grapple, grapple with mathematics in a productive manner, because you can certainly struggle unproductive.

00:04:35:11 - 00:04:58:16

Curtis / Joanie

But I think, you know, we really want to focus on what are the things that maybe I couldn't do, that helps me be able to provide this opportunity for my students. Yeah, yeah. I want to go there to what the teacher does and and to the discussion around, classroom culture, because I agree. I think that is I didn't think of it as, as plainly as you said it.

00:04:58:16 - 00:05:22:02

Joanie

That productive struggle actually is about classroom culture. I was more thinking, oh, it's really highly dependent on a healthy classroom culture. But, but before we go there, I do want to slow down for a second and talk about this word grapple, because when we bring that word into our minds and think about, you know, what does that look like, sound like, feel like to grapple with something?

00:05:22:04 - 00:06:29:13

Joanie

It's a different image I get than if I think about what does it mean to struggle? Even if I'm saying productively struggle, right. Like struggle has this real negative connotation of of, you know, an uncomfortable challenge. And kids struggle with mathematics. We use that as a way to describe kids that aren't being successful. Right. So thinking about grapple, though, grapple is more I'm wrestling with this idea. I'm, trying to make sense of something. Yes. And that just does such a better job of painting the visual image, or the sort of circumstantial way of describing what productive struggle in learning mathematics should look like. So thinking about we want students to be engaging in that wrestling with idea, but we don't want them to be in the like painful frustration of that struggle and, and, and lack of success that struggle can construe.

00:06:29:18 - 00:07:38:11

Curtis

Yeah, I think I grapple also to me. So along the same lines, but maybe even another dimension to it, grapple gives me that that visual image or that that there's an intention, there's an intentionality behind it. There's a purpose in in my grappling, I am I'm, I'm intentionally grappling with this thing because I want to figure it out or because I, I need to learn or I need to make sense of this thing or this topic. You know, when I think of, I'm grappling with a lot of times we use that terminology with like, I'm, I've got a decision I need to make or I'm, you know, there's a, there's some life choice that's going on or someone has given me some feedback and I'm trying to figure, you know, trying to grapple with that feedback to figure out how I should process it.

What should I do with it? There's there's just another level of intentionality where struggle. Yeah, there is a lot of potentially negative connotation, even when you use that, that qualifier productive in front of it. 

00:07:38:11 - 00:09:23:14

Joanie

Right. And I think to the struggle everybody will go too hard. Right. So if you were, playing devil's advocate against the mathematics teaching practices or this one in particular, it could be perceived as like, oh, I, I give kids a really hard challenge problem every day.

Therefore I'm creating productive struggle in my classroom. Well, it's not always about the thing being hard, right, right. It's it's about and and and that's what I appreciate about even how you were just describing grapple. Right. Like when we think about grappling with a life decision, it's not because there's one right and one wrong answer, or one right and multiple wrong answers to the decision you might be making. Sure. It's it's the it's the weighing of different things, and it's the bringing ideas together in a way that you're not used to. Right. So grappling is more and I think grappling as a, as a synonym for word active struggle in mathematics is more about thinking about the math in a way that you haven't before. So you're working to make connections that aren't obvious or aren't obvious to you in this moment. And, and I think that's why the beginning part of this statement that consistently providing students individually and collectively with these opportunities, right, that individual part, I think, is another component of what makes this a challenging practice to implement, because we now we've said over and over again on our podcast, Curtis, that students learn at different rates and students bring very different backgrounds to class.

00:09:23:14 - 00:09:40:08

Joanie / Curtis

And that's just normal. That's just the state of of learning mathematics. So addressing collective and individual productive struggle is is really important and not a trivial skill. I yes, absolutely.

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Music break

End of Segment 1

Start of Segment 2

00:09:50:10 - 00:11:16:04

Curtis

I think even, you know, to try to set us up for a conversation we were just having actually I sent you, an image. So my family was just on vacation. And, in the restaurant, one of the restaurants we were eating, and there was a, American flag made out of baseballs. It was an inaccurate American flag made out of baseballs. It was personal scale, which for some reason didn't bother my my son. Truitt. Who is, you know, he's always asking really fun math questions. And we were trying to look at the, the in the upper left hand corner and I'll, we'll figure out how to include the image. I'll try and do that in the show notes. Yeah. So people can follow along. But we were trying to figure out how many stars, quote unquote, there were in the blue area of the, of the representation of the American flag here. And I might by by talking about it, I almost steal people's thinking, a little bit. So spoiler alert, I don't want to do that. So maybe you pause it and go look at the picture and figure out how many stars and think about how you, would do it. But I'm going to just describe it for folks that maybe don't have the show notes in front of them. There's rows of baseballs, and in the blue section of the flag in the upper left hand corner, there were alternating blue and white baseballs in, what are their seven?

00:11:16:06 - 00:11:57:19

Curtis / Joanie / Curtis

Seven rows? 7 or 7? There's seven rows and then there's, ten baseballs. So alternating blue and white baseballs back and forth. And so there were 70 total, baseballs in that rectangle, half of the blue. And like, the blue corner part of the American flag image. And so we were having a discussion about how many blue breaks or how many white baseballs there were there.

And my son Truitt was, was trying to explain to me. And Joanie, I just I think I just figured out what he was talking about looking at. That's like literally just I just heard something different. I in my head, I just did it. I just saw it based on that description. Yes, yes. So I just saw it, differently.

00:11:57:20 - 00:12:47:02

Curtis

And I saw now what he was talking about that he had to do to minus one anyway. So, I'll let you guys kind of just think about this, but really what what was interesting to me, I'm not going to tell you how I thought about it. We can have a discussion about that at a later time if you run into me somewhere. But I, I what I, what I thought was cool and why I'm even bringing this up now is that we spent more of our time talking about the procedure and the process by which my son, through it, and myself and others at the table were coming up with the number of baseballs. Yeah, much more time, in fact, so much time that I didn't even it didn't even dawn on me to question that there weren't the right number of baseballs in the in the blue section to represent the states.

00:12:47:04 - 00:13:06:12

Joanie / Curtis

So spoiler alert there were not 50 white baseballs, 50 white. There were not 50 so-called stars on this image. And it just totally didn't even bother me that the there weren't or that or the kids or the kids, because we were so caught up in trying to we were we were all about the process and the grappling.

00:13:06:14 - 00:14:15:18

Curtis

That's why I brought this up. We were in the grappling. There were some intentionality. We were trying to think about how we would do it. There was, there was not necessarily a right way and a wrong way. There were lots of different ways of of being able to, to communicate this, this number of stars and the thing and really it was more about how do you communicate how you figured it out.

Yes. And I think that was just, you know, there was some level of of productive struggle and grappling that was going on there. That was just fun. And I think it's, you know, we were talking about culture. We were sitting around the table having a just a discussion. Yeah, it was fun. There was no dinner with your family. Yeah. There was no big. You know, it wasn't like we had some big exam had let's do a math problem now. And you know, for perspective and Truitt, he just turned, It was the day before or two days before his birthday, you know. So he just turned ten. So he's you know, that's kind of the age level we're talking about with this kid asking these questions and trying to grapple with how many stars there were on the flag.

00:14:15:21 - 00:14:52:10

Curtis / Joanie

Yeah. So the culture, though, I think is such a big deal, when we do these things, this, this conversational this, let's work on it. Let's think about it kind of culture. Yeah. And and the other thing I'm thinking about that you just really nicely exemplified with this personal story of a family conversation about, you know, this image of baseballs or this, I guess it was actual baseballs you were looking at, not just an image, but, you know, this is the kind of thing that we can do to bring in effective, productive, productive struggle in mathematics.

00:14:52:10 - 00:17:37:05

Joanie

And I think about, you know, years ago when I was supporting teachers and they would say, you know, why do I have to have kids show five different ways to solve a problem, like once if they if if we can just teach the standard algorithm, they'll get the right answer every time. Why isn't that good enough? And I think what you just described is answers that for me in a couple of ways.

One is the opportunity to struggle productively is in understanding how somebody else thought about it. Right. So it's not just what's the right answer. Right. If if your conversation at the dinner table had been how many white baseballs are there in the star, you know, blue star section of that portrayal of the American flag? You know, you could have count. There weren't enough that you couldn't count them, right. You couldn't just counted them and said, well, the answer is 35. But that's not an interesting conversation, right? Like, my answer is one thing, but the the thing that made the conversation interesting and the thing that created grappling within your family discussion was, was thinking about different ways to get to that number. Yeah. And how did you think about it? And can you help me understand how you thought about it? And I love that. Five minutes ago you had an moment, because when we were talking about this 20 minutes ago, you were like, oh, I don't think I even really understood what Truitt was trying to do. And now you do.

Right? So like having these moments of understanding different ways of thinking. And it's that idea, like the reason we show multiple ways to get solutions is not because we want to have as many, you know, solution paths in our toolkit. It's not about like, let's showcase every single possible way, that could just turn into tediousness, but how can we think about different ways, and how do the different ways help us understand the math differently?

How do they give us a more robust understanding of what's going on? So getting much more rich, deep thinking, mathematical thinking than just obtaining an answer is a huge, important part of why we support productive struggle. And then the other thing I was thinking about that I want to throw in here because, you know, I've been this was a summer of conferences for me and several of the conference that I went, conferences that I went to were policy focused. And I kept hearing, I've been hearing a lot, but especially in the policy space, a lot of conversation about workforce readiness and durable skills. Right? So, yes, students need to be able to read and write and know their history and all of that when they graduate from high school. But they also need to be critical thinkers, and they also need to be able to communicate.

00:17:37:05 - 00:18:02:15

Joanie / Curtis

And they also need to be able to be creative and have flexibility and persevere. And these are all things that come from having the opportunities and supports to productively struggle and learning mathematics as well. I think they they do they come from developing that and but they develop in, in a sandbox space. And let me say that let me say that, maybe a little more explicitly.

00:18:02:21 - 00:19:59:12

Curtis

I think they come from low stakes, spaces. So, so another thing about the fact that this was a family conversation, and I know we can't just drag our families around and not every kid, you know, we can't just visit people at their homes and and have dinner with them, have family conversations. Well, let's learn math while we eat dinner together. Although we should do, we should, we should, we should. That should be a part of our homes. But I'm not sure that that is necessarily in every student's home. A way that that can, can happen. And so how do we do that in our classroom? So I think, I think this idea of sandboxing and low stakes environments, which, which means, you know, we can't spend all of our time in low stakes environments, I get that. I understand that we have we have testing and state standards, and we have other things we have to be able to do, which I think also speaks to something in, in the description that I hadn't noticed before, or at least that it, it's it kind of popped out to me. So it says effective teaching of mathematics consistently provides students individually and collectively with opportunities. But what it doesn't say is that it always or that it every single time we're in the math classroom. So I'm trying to relieve maybe a little bit of the pressure also to say that that yes, you should do this is very, extremely important. But it it can't always be every day. Right. It can't always be every conversation. Is this so. But it has to be. It has to be there. But I don't feel the pressure that it has to always be there, I guess is what I'm trying to say. So the the low stakes thing is, is really important. And finding time for my students to engage in low stakes things that that matter to them that we can that we can give them an opportunity to, to productively grapple.

00:19:59:13 - 00:20:20:00

Curtis / Joanie

I'm going to start using that productively. I think we should I think we should just say grapple for the rest of the episode. Yeah, I, I totally agree with that. I love the idea of a low stakes sandbox and thinking about that, because that helps. That helps open the door for our discussion to go to, how do I how do I create the classroom culture where students can struggle productively?

00:20:20:00 - 00:22:35:19

Joanie

Because let me tell you, I have been in classrooms. I have been the teacher in classrooms where there wasn't that feeling of safety, that feeling of low stakes. I can remember my very first teaching job. I took over at, the beginning of the second semester. So mid-year I came in me to, Yeah, that's. Those are fun situation. You've already got a different teacher, has established the culture, and the kids all know each other, and then you're new. And that's fun. But my first teaching job, one of my classes was honors precalculus. And these were kids who were either juniors who were hoping to take AP calculus, either AB or BC in their senior year, or these were seniors that were, you know, filling out their college applications and or waiting to hear where they got admitted.

And it was very much a high stakes environment and very much a competitive feel within that classroom. And I had a handful, 3 or 4 students that would come to my office, I would say every day after school. I don't think that's an exaggeration. There were a handful of times that they didn't, but they I basically learned fairly quickly, like they did not feel comfortable asking questions in class, so they would come after school and ask those questions one on one.

And I, I it took me most of the semester and some great mentoring from the teachers that I shared my office with to understand that there's an unhealthy competitive culture in my class where where kids didn't feel safe asking questions and putting productive struggle into my repertoire at that time would have been a major disaster. It just would have been like, I knew.

I knew there were these four kids because they came to see me. But how many others were in there feeling like, I have to know the right answer? Because if I don't, I look dumb in front of my peers and that's very high stakes. Especially, I don't know if it's more so in middle and high school, but, you know, at all ages that ability to be vulnerable and say, I don't know or get something wrong and not feel like my classmates are going to judge me.

00:22:36:01 - 00:23:02:07

Joanie / Curtis

So how can we think like, what are some of the components of the teacher's role in creating that classroom culture that can be that low stakes sandbox that you were describing? What would that look like for a teacher to be able to do that? Well, I'm I'm looking and I'm thinking about what it my person, my personal experience to was was I did not step into an honors pre-calculus course.

00:23:02:07 - 00:25:32:09

Curtis

I stepped into, really a, what we call double blocked algebra. Course, we had the other end of the spectrum. We had students taking, algebra for the second or third time, and we actually had them twice, twice in the day. So we didn't have them two hours in a row, but we had them, one class period, two different class periods. And so that was kind of a different sort of experience. And I really, truly, I think it it has a lot to do with our responses and our the way we as teachers respond in the classroom and the teaching moments in the times when we are, you know, instructing. I, I hesitate to use the word direct instruction, but, you know, when we are, are talking and showing and, and demonstrating, the mathematics, and, and asking questions and get, you know, prompting students, you know, establishing one, telling them that it's a safe environment, but then to the demonstration that it's a safe environment, that that when a student responds in a manner that is, maybe got a misconception or, or, conception, that is as pre conception, you know, thinking, you know, pre knowledge sort of thing. Our response to that is one way. Right. The encouraging and and tell me more what you're thinking and and what that means though is me as a teacher when I'm asking tell me more about what you're thinking. I've got to have an I've got to have enough, teacher content knowledge to be comfortable in asking the tell me more question, because what I'm really searching for is where can I help them make a connection? Where can I help them make a connection? That's what's going on in my head, is I want to know what's you're thinking, because somewhere in what you're thinking, there's a connection to what we're doing.

And if I can find what it is that you're thinking that has the connection, then I can start to lead you down the path for making that connection. Exactly. And and it has to do with my, my level of response. And if I don't have the content knowledge to be comfortable to ask that prompting question to say, tell me more, I, I can't be comfortable very easily.

00:25:32:10 - 00:25:57:18

Curtis / Joanie

And I also can't establish for my students a level of comfort in my classroom to be able to ask that question. Right, exactly. I think, oh my gosh, you said so many golden nuggets in the middle section right there. I'm going to go back and replay that in my mind multiple times. But yes, I would agree. And I want to kind of stop in and re-emphasize a couple of things that you said that I think are so, so crucial.

00:25:57:18 - 00:29:12:02

Joanie

One is that teacher content knowledge, because I really believe that the the resistance to doing these kinds of rich mathematical tasks that allow students to engage in discussion. And, you know, a lot of the things that we've been talking about when there's resistance and on the part of a teacher to do this, I think it's often based in their their own lack of depth of understanding, because to guide students, to understand the mathematics really deeply, I have to understand the mathematics really deeply and maybe even a lot more than I'm trying to get my students to, and for sure. Right. Like, I think it is about that being able to make connections, to find out where kids are in their understanding and to move them along toward where they need to be. I've got to have a lot of knowledge to, to be able to do that. So I think that's super, super important. And I want to put a pin in that and, and talk through a little bit more in a minute about how, how do teachers build up their content knowledge if they are aware, like, gosh, I don't understand this really deeply and I need to and what do I do about that? But the other thing is just how you talked about the response to to students. And I, I love that and agree that that's so important and wanted to kind of talk that through, even with some practical examples. I know years ago I saw a protocol and I'll put this I'll, I'll dig it out, see if I can find it and put it in our show notes for this episode. But it was about, the teacher using a protocol during the warm up portion of the class period that she called my favorite node. So she would, you know, give students a, math problem to do. And then they would submit answers and she would talk about several of the answers that were incorrect, like, oh, this is wrong, but this is my favorite wrong answer because this allows me to help you see that, blah blah blah, blah, blah.

So for instance, if you if I think I'm just trying to think of a simple like algebra, one example, if students were solving an equation like, you know, negative three x equals 15 and they got an answer of x equals five, the teacher might go, oh, this is my favorite. No. Because this is a really common mistake to make. Right. Like yeah we're dividing the person. Understood that division is what's needed to solve for X. But if we're dividing by a negative three, you know, or you could say if I put five in to in place of the X, would I get a true statement? Like there are so many ways to talk about that. Right. So we're not just accepting when students give wrong answers are making it safe for them to give wrong answers. But celebrating wrong answers as a, as a, as an opportunity for us to more deeply understand what's going on. And I could even think about I never tried this, but if I get to go back to the classroom, one of the things I might do is challenge my students to come up with a wrong answer that and and then, you know, swap with a classmate and see if they can figure out what was the common mistake that was made.

00:29:12:02 - 00:29:32:15

Joanie / Curtis

Because that will get you at, you know, a deeper level of understanding if you can think about I know the right way to think about this problem. So I also know some of the wrong ways or the traps or the misconceptions that would lead me to to have, incorrect mathematics happening, I could tell. So here's an exercise you could have your students do.

00:29:32:17 - 00:29:54:14

Curtis

Okay. And, and this actually comes from so, my time working at, at laying the foundation. And when we were writing our diagnostic sort of, exams for each of the units, sort of at the, at the beginning of these units, we had these diagnostic or even at the end of these units, we had these diagnostic exam questions and we wrote them.

00:29:54:14 - 00:30:20:12

Unknown

They were multiple choice questions. And we we made absolutely sure that every distractor had a real like it needed to be diagnostic. Right. It needed to tell me what the student knew about the thing that we were asking the question about. And so every single one of those multiple choice answers was, in fact, a distractor. That was possible because I made a misconception or I made some stuff up or a common error.

00:30:20:12 - 00:30:46:13

Unknown

Oh, I love it. And so having your students write their own multiple choice questions for topics to to give them an opportunity to say, here's a common mistake that I make, right? Yeah. So that was always the I used to think, yeah, yeah. It was I could, I could say, well, here's the one I always do. And then I tried to be creative about some of the other kinds of things that would happen, with those questions.

00:30:46:13 - 00:30:56:04

Unknown

But that's a cool way to kind of get students thinking about, you know, hey, write a multiple choice question on this topic. Yeah, I love that.

00:30:56:04 - 00:31:06:04

Music Break

End of Segment 2

Start of Segment 3

00:31:06:12 - 00:31:27:06

Joanie

The other thing that I want to tap into, and in terms of that, how the teacher responds to students. And I know I've talked about this in other episodes, but I just can't I can't pass up an opportunity to talk about it again. Let's go. And that is really thinking about how what role the teacher plays in how the students in the class perceive one another.

00:31:27:07 - 00:31:58:13

Joanie / Curtis / Joanie

Right? So we all know that in a typical classroom, there are this idea of a teacher's pet, right? That that wouldn't be a thing if that didn't happen in classroom. Right? If that weren't perception of students that, oh, the teacher likes that kid because that kid always knows the right answers or something along those lines. And, and, and then also the perception that happens when a student might feel like they don't understand to the same level as their classmates do.

00:31:58:15 - 00:34:22:04

Joanie

You know, I can think about pairing kids up for a partner quiz and some somebody getting paired with a student that doesn't ever, you know, complete their homework or, you know, does poorly on exams and feeling like, oh, I don't want to be with that kid because they don't know what they're doing. But what I'm trying to get at is the way the teacher interacts with students can help create those perceptions. And there's this idea of capital in the classroom, both social capital and academic capital. And it's there's a huge role that teachers can play in contributing to the social and academic capital of of kids in the room. And I think it's particularly important for students who may not have social and academic capital outside of my math class. I do things to make sure that they do so, you know, ensuring that I value all students perspectives and that I and that I express that value and in a very authentic way. Not like, oh, thank you for trying, but that's the wrong answer. But even even something like the favorite no protocol like, oh, I'm so glad you brought that up. Because here's here's another aspect of this mathematics that I want us to explore together. So responding to a student's incorrect answer in that way is completely different than just saying, no, that's not it. Who else has an answer? You know what I mean? Or even putting the student on the spot and like, well, what did you do? And let's diagnose your error right here in this moment. Like, that's not what I'm talking about. I'm talking about really expressing that social and academic value that all students bring to the class. And, and building on that, like there is value in the, the most, basic foundational understanding, even if you can't get all the way to the answer, the very first thing that you thought about is something that everyone in the class can learn from. So as a teacher, taking the approach of of building that layer of understanding, using actual student thinking and actual student work is such an important way to build that social and academic capital. And and that to me creates that safety and value. Because if every student feels like I have something to contribute, even if I'm don't always have the right answers, I don't always understand.

00:34:22:05 - 00:34:47:09

Joanie / Curtis

I'm not always the fastest. I still bring value to our classroom learning community. Yeah, I think that's a that is. So that is super important. That idea that, you know, each of those answers provides an opportunity. And each of those, of those students, misconceptions provides learning potential for everyone else, in the classroom as well. Right?

00:34:47:09 - 00:35:59:19

Curtis

I think all of those are highly, highly valuable things. And I'm going to give you another opportunity for another of your soap boxes here in just a second, because what I'm about to say, that can't happen. Yes, teacher content knowledge is huge, but teacher content knowledge alone then relies on me to be quick on my feet. Yeah, and I just, you know, that is also not something that we necessarily want to be dependent upon. We've got to be able to be prepared and teacher preparation. You know, me as a teacher taking the time and thinking about and and again, this goes back to my discussion earlier about, you know, it it it isn't necessarily going to be every single day that every single lesson and every single one of these things is this huge grappling whatever.

But if if I am taking appropriate time to be appropriately prepared, yet knowing all the things that I know about the topic, and I can kind of think about, okay, here's the things that I think students are probably going to bring up. Like it's that whole anticipation, piece of the puzzle of knowing pretty well what my kids are probably going to say.

00:35:59:21 - 00:36:27:13

Curtis / Joanie

That gives me a better chance of having the right reaction. Oh, I'm so glad you brought that up. Because and here's the connection. And we've just got to be yeah. Prepared I, I agree and I appreciate the open soapbox opportunity. I will gladly step up on it. You know, you know, teacher planning is one of my soapbox topics.

00:36:27:13 - 00:37:48:08

Joanie

That's 100% sure. And I love how you framed that to Curtis, because it it takes me back to the well, what if I don't have the content knowledge because I think those things go hand in hand. So if I'm thinking I'm, you know, there's I'm, I'm working with my students on a big mathematical idea and this is a prime opportunity to grapple with something to ensure that there's deeper understanding and not just an answer getting approach to this mathematical topic. I want to pick a good task, and then I want to have a really solid plan around implementing that task. And that's going to involve building out my own knowledge about it and what you just said, anticipating what students are going to do with it, anticipating what they're going to do correctly, anticipating where they're going to get stuck, anticipating possible things that are going to distract them or take them off in an unproductive direction. And I just believe the best way to do that is to do it with other teachers. And yes, for me, that is the perfect way to spend PLCs. Let's say were we're an algebra one plc, we're going to have kids engage and grapple with this task. Let's talk about how we think they're going to solve it, what we're thinking about, the different types of students in your room and the different levels of backgrounds and understanding they're bringing.

00:37:48:10 - 00:38:16:07

Joanie / Curtis / Joanie

How is this how is, you know, Curtis going to approach it? You know, he has he always gets the right answer faster than anybody else. So what is he going to do. And and you know, what about Andrew. Not just not fast but I'm trying to come up with sample student names here. You know, if you think about like a typical kid, think about five sort of typical kids in your class and plan around, what do I think that kid is going to do?

00:38:16:13 - 00:40:07:01

Joanie

And and when you're a new teacher or when you're inexperienced with the content you're teaching, talking to other teachers is the best way to both build out your content knowledge and build out your anticipation skills. Right? When you're a first year teacher, it's very hard to know what kids are going to do because you haven't had that experience before. So I just think there's such power in PLCs and planning with other educators around a specific task, both to help build my own understanding of the content and to help, you know, plan for how how students are going to approach it. Because you're right. If if I have already thought through, you know, hey, here are four possible, paths that students might go down that are going to actually take them in a different direction than, the math I'm trying to get them to, I can plan for what's the question?

I can ask them to redirect or refocus or connect them back. Right. Like you were saying before, like I have to be able to ask the probing question to understand what their thinking is, but then I have to be able to think, what am what am I get? What seed am I going to plant with them to make the connection to what I want them to think about? You know, in our previous episode on this topic, we talked about The Learning Pit and this image of of students being stuck in the bottom of a pit with not knowing what to do and and not being able to get out. And the image had a little rope and it's like we just want them to see the rope. We don't need to go climb it for them. We don't even need to hand it to them. We just need to help them see it. And you, again, encourage folks to go back and listen to that. You had such great language around. Knowing what they're thinking helps me know what to say, so they can see the rope so they can see that connection and then start to pull themselves out of that.

00:40:07:03 - 00:40:38:08

Joanie / Curtis

Of the grappling moment and into the successful I understand moment. Yeah. I mean, it's, it's it's just that that idea of being able to establish opportunities and supports. Right. So we want to be able to do these in a and I'm, I'm quoting directly off of the art out of the, the practice itself that, you know, we provide, students with opportunities and supports both individually and collectively for being able to grapple with mathematics.

00:40:38:10 - 00:41:39:22

Curtis

And and the opportunities are the tasks and the and the times in our classroom and the supports that we're talking about are really the ways that we as teacher react, the ways that we as teacher anticipate the way as we as teacher get our teacher content knowledge up to a place where, where I can, support my students as they grapple and as they kind of take the the off the beaten path, methods in my son included, when he's trying to figure out how many stars there are on a, on a wall, representing the US flag. So I just, I think that, you know, this idea, this, this whole, teaching practice being written around students grappling with mathematical ideas, it does depend an awful lot on it, and I, I trying to hesitate. I'm trying to not put all the pressure on us as teachers. And you know, we know this is a learning this is a growing experience.

00:41:39:22 - 00:42:11:01

Curtis / Joanie

We know all of us are developing, these skills over time and over practice, over repeated attempts to do this in our classrooms to make make sure that it's productive. But it does it does depend a lot on on me as the teacher setting it up. Right. It does. And and I love that that I think, I think the place where we end is talking about the idea of grappling as a teacher, like grappling with, how do I do this?

00:42:11:01 - 00:42:37:16

Joanie

Well, how do I get my home? How do I help students grapple? Teachers need to grapple with how do I help students grapple? And I think for for both levels, a teacher trying to effectively implement productive struggle with students and for students effectively engaging in productive struggle. The outcomes of that done well are so fabulous. Everybody loves the feeling of being able to do something hard.

00:42:37:18 - 00:42:42:17

Unknown

Yes. So just let's just remember that it's worth all the effort, both for us and for them. 

00:42:46:00 - 00:43:07:09

Well, that's it for this time. Be sure to check the show notes for the resources we mentioned and others you might want to explore. We would love to hear your feedback and your suggestions for future topics. And if you're enjoying learning with us, consider leaving a review to help others find us and share the podcast with a fellow math educator.

See you next time!

00:43:07:11 - 00:43:16:12

Music  

Hey!