SPEAKER_01 0:00

Welcome to Room to Grow. I'm Curtis Brown.

SPEAKER_00 0:02

And I'm Joni Fenderberg. We work together at Texas Instruments, and we're glad you're here. We're looking to continually improve our practice, and we understand that you are too. We hope that you'll find this podcast as a room for you to grow along with us as we wrestle with and explore ideas about teaching math even better. We think you're gonna love this episode on helping students struggle productively. Curtis and I believe that supporting productive struggle is one of the most challenging things a teacher faces in their classroom. We talk about the nuance and individualism inherent in ensuring that student struggle is productive and doesn't cross over to unproductive frustration. We explore the idea of the learning pit and how to use this concept as a guide during interaction with a student who is in the midst of productive struggle. Finally, we walk listeners through a situation where I was struggling, and Curtis helped move me through the struggle to find a solution to an interesting geometric puzzle that was originally unrelated to the podcast. We really hope you enjoy this conversation. Let's grow.

SPEAKER_01 1:06

Well, Joni, I'm really excited again today to just get a chance to visit with you and talk to you about mathematics. And in this case, we're going to be talking about students and students struggling productively. This is going to be a big topic and it has been a big topic as of late in the math world, thinking and talking about productive struggle and really this whole idea of what does that mean and where does that fit in my classroom and in my students' experience. So I think let's start there in just kind of getting this idea going about what is productive struggle, maybe maybe an example of not productive struggle or unproductive struggle and kind of going from there.

SPEAKER_00 1:50

Great. Yeah, I'm excited for this conversation too, Curtis. And I think this is a this is such a big topic and such a nuanced topic because productive struggle is not something concrete that you know we can define that says if a student is doing this, they are struggling productively, or if they are not doing this, they are not struggling productively. And the approach towards encouraging productive struggle is really individualized, even more so than other things that we attend to in math education. So I think it might make sense to talk a little bit about productive versus unproductive, because as a classroom teacher, I want to be able to recognize when a student's struggle is unproductive so that I can, you know, back up and sort of reassess the situation and provide support that shifts away from unproductive struggle. So I really believe that a lot of kids go to unproductive struggle sort of naturally. I think there's this concept of learned helplessness that is really common in math classes where students have had experiences with teachers where when they don't understand, they raise their hand and they sort of like get this attitude of, uh, I don't have to do anything until the teacher comes over and explains it to me. So this idea of struggling through is not something that necessarily comes naturally, particularly for a student who's really been exposed to very traditional, you know, math sort of instruction over a long period of time. And I think I've shared on previous episodes of the podcast that I'm I'm working one-on-one with my fourth grade nephew, Andrew. And I've I just noticed a couple of weeks ago in working with him that, and something became challenging. And I could see his level of frustration increasing. Like I could see in his face that he was struggling to understand something, and he was even saying, I don't understand this. And and it, I pushed him too far. Like it got to the point where his struggle became unproductive. And he literally just like had a little mini meltdown and then shut down. And it was like, okay, we're done for today. There's no going back from this. Um, but being able to sort of say, I can reflect back on that and say, oh, this was the moment where he went from productive struggle where he was like, oh, this is hard. And I don't quite understand. I don't know what the answer is immediately, to I'm so frustrated, I just don't even want to engage in the conversation anymore.

SPEAKER_01 4:20

Yeah, I think that's an experience I've I've kind of got a similar thing. And I've I know I've talked about my son uh on the podcast as well. So just thinking about um my conversations with him, and it's it's always interesting. So around my house, it's it's pretty common for him to come up to me and ask me some mathematical question, some arithmetic question. So, daddy, what's 99 plus 77? Or, you know, and and come to me with these uh kind of continuing, you know, always these questions. And it's it's so fun to have him doing that. And it's interesting that when he comes up with the problem or when he comes up with uh the situation, um often he is much more willing to spend his time and his effort on the challenging issue that he's brought up than if I prompt him from it. And so I think it it really is an example of him being bought in and that perseverance kind of rises uh from his kind of interest or uh, you know, desire to know, or at least his desire to challenge me uh about a particular problem, right? I mean, he's always trying to come in and well, daddy, what's that? And and he's always wanting to, he's wanting to stump dad. Um, and then I said, well, I don't know. So let's figure out how we do this. And and he'll kind of help me work out how to do a particular problem. But it's because I think it's it's mainly because it's his curiosity, his little passion, his little desire in trying to do uh that particular problem. And so his perseverance is much better uh when it's that way rather than uh a problem that, say, I come up with, or, or maybe he's he's doing working on homework and it's it's challenging or whatever. Um, then his perseverance has dropped a little bit because maybe his motivation is down a little bit, his interest is down a little bit. And so trying to find good ways uh to bring that back up is always uh an interesting challenge for me as a dad mentor uh in that space.

SPEAKER_00 6:26

Yeah. I I like that you brought up that curiosity and, you know, the the sort of self-directed approach because I that ties so nicely back to our previous episode where we talked about that, right? With mathematical play and mathematical exploration. So I'm sure somebody listening could could go back and make some connections between what we talked about in that episode. So I think we've really hit the nail on the head around what makes this productive struggle so tricky, in that, you know, you were able to really describe it with your son, and I can describe it with my nephew because we know both of them really well. And we know, you know, their response and the and and how to redirect and get back on track. So I want to just sort of say those things plainly as we think about uh teachers and students in the classroom. Like what are the the different factors that we know can contribute to a student's ability to struggle productively and things that a teacher needs to know or bring to the forefront of their mind as they think about supporting students through productive struggle? So I'm thinking about, you know, personality is definitely one of them and the role of curiosity. Have I, you know, is this something that the student is in, you know, genuinely curious about, or is this something that they're view viewing as an external expectation that they might not necessarily feel, yeah, you know, the willingness to engage. What are some of the other factors that you think teachers need to consider as they gauge levels of productive struggle for their individual students?

SPEAKER_01 7:56

So I think um when I when I think about it, I I think about uh classroom environment uh a little bit. So our environment, this the the way students maybe feel about our particular classroom or the particular experience, right, when they come into math class. Am I am I feeling like it's a safe space for me to open my mouth? Am I feeling like uh I I've got a voice here that matters and and that I'm heard. So I think that's I think that's one piece of the puzzle that we certainly as teachers have uh significant control over. Um, but I think there are other external factors, things of of, you know, just what's my experience outside of school right now? Right. I mean, I I'm carrying in, I may be carrying in all kinds of things uh into school that that really, I mean, they're uh they're unrelated to my schooling, but they certainly impact my schooling uh tremendously. So, you know, you you know the hungry, angry, lonely, tired uh sort of space, right? Um, and I think all of those uh play into that. So just knowing our students and their their experience and where they're coming from, uh, I think is also uh a big, big deal. I think the pressures that um teachers um feel um that they need to put on to students or that they need to kind of ask. Uh, I know because I I felt like I needed to say things to my students and challenge them all the time and kind of really kind of help them along. Um, and I know my students felt that pressure. Um, you know, and so I think there was this, there was a space, um, there's a space there where the students go, oh my gosh, I gotta come up with something, I gotta think of something. And that can really lock it down, right? I think we talked about that that in the in um other other at other times, like this idea of curiosity really coming from maybe taking away some of that pressure. And so there's another space where um teacher has a pretty good amount of control, right? Of just um allowing the student to maybe have uh relax with with uh uh the learning there, right? Does that make sense?

SPEAKER_00 10:04

It does. And it's another one of those tricky balance things, right? I can I could relate to you saying, you know, when you're in the classroom putting pressure on your students, I know I did. I wanted my students to take their learning seriously and I wanted them to, you know, give effort to my class and, you know, pay attention and, you know, try and learn this thing and work through challenges. But I think we also need to sometimes remember to step back and and and remind kids to just sort of relax. Like it's okay. This is, you know, this is algebra two. This is not, you know, neuroscience. This is a brain surgery. This is like you're just learning, uh, you know, kind of uh get a get a more solid perspective. And I think that balance as a teacher is tricky. Like you don't want to constantly send the message of pressure, pressure, and this is important and you know, we're serious and pushing to excel all the time. And you don't also don't want to send a message of, yeah, it's all right, doesn't matter. You don't want that sort of laissez faire message all the time, too. But you know, and in reflecting for this episode, I try to draw on some of my own personal experiences around productive struggle. And one of the ones that popped into my mind is the work that I'm doing in the gym with a personal trainer right now and trying to build up some of my strength. And the the best thing about having a personal trainer is he'll challenge me. He'll, he'll put weights on that. I'm like, I never would put that weight on the barbell for myself. I would think that is way too hard for me. Um, and at the same time, he's great at reading my face. And he'll, you know, when he sees me getting frustrated when a weight is too heavy and maybe I can't complete all the reps he asked me to do, and he'll say, Joni, just have fun with it. And I just think that that's such a great reminder. And I think, you know, I I want to be able to go back and say that to my students in the classroom. Hey, just have fun with it. It's okay.

SPEAKER_01 11:50

Yeah, yeah. I think that's so that's a huge what a great, what a great takeaway from from your trainer and a good example too, because I I say the same thing thinking, man, I should really, I really owe my students a a bit of an apology for the amount of pressure that I put on them, right? Um, you must take this seriously all the time. And this is mathematics. We don't have fun in here. No, it it we really need to be thinking about this as you know, we we absolutely want them to both remember that it is uh you know, serious business. It is their education, but at the same time, you know, we're not in life and death situations here in our math in our classrooms, right? We're not uh dealing with those things, you know, beyond. And so trying to set that expectation helps helps them maybe be able to relax and and look at things um and be able to enter that idea of productive struggle maybe a little bit faster. We're talking about this idea of it being productive struggle. And so when we use the word struggle, there's uh, you know, there's sort of a connotation that comes with that. So it's gotta be something, it's gotta be hard. Um, you know, there's something there's something to that uh piece of this. But I think that it's it's probably a little bit more than that. Um, that it's not just, hey, we need to present them with something that's a challenge or something that's just not simple. Um, but what else is there maybe to producing um tasks or um engaging experiences for our students that are in fact uh able to produce productive struggle with?

SPEAKER_00 13:35

Yeah, I think it there is some, there are some characteristics I should say around a lesson or a task or an activity that we want to uh invoke productive struggle. So I think understanding that it needs to be, it needs to be not simple, because if it's simple, then there isn't any struggle at all. But it also needs to not be, you know, impossibly difficult. It needs to be something that is a challenge, but also that I believe it's uh a challenge I can reach. It's something that feels attainable to me, not like, oh my gosh, like I can't even imagine myself starting on this math task. Uh, I think there needs to be this view of like, oh, I know a little something about this, but the answer or the pathway to the answer is not immediately obvious to me. Um, and then I think so. I think there's some characteristics around the task itself, but I also think there's some characteristics around the instructional strategies that teachers use and the thing that the teachers are going to pay attention to as they're going through uh instruction around that task. So I think it's about, first of all, let me back up too and say it could there could be a task that has the right level of difficulty to allow for productive struggle of students. And a well-meaning teacher could inadvertently frame the task or give some, you know, background or a launch to the task that actually takes away some of the struggle. So, in in one of my first thoughts about in approaching a productive struggle lesson or in uh, you know, wanting to engage students in that is try to give as little information as possible, especially in the beginning. Like hold back on any hints you might be thinking about or any framing. You know, the whole idea is just to ground students in their own ability to think and reason and uh, you know, resist that temptation to sort of simplify it down or um proceduralize something that that potentially has, you know, the possibility for productive struggle.

SPEAKER_01 15:42

Yeah, I think there's something, there's definitely something to that, right? Uh making sure that first hurdle is something you know jumpable, but then I also feel like I accomplished something uh when I jump it, right? Um and I I I liken this to um when when I'm trying to help someone learn to to code uh something, uh this is something that I I I love to see and I and I want to make sure that that whoever it is has small successes um to be able to kind of continue going, right? Because that builds the engagement, right? If I if I see that I've had a little success and I've made a good made good progress, then that encourages me to kind of keep going in this, in this, uh, in this challenge. And so what we want to make sure is that inside of the task and the way we implement it in our classroom, that that what we're doing is is giving small success uh a chance every time, right? And that we don't take away those by giving too much scaffolding or too much uh helpful hints or or whatever it is that we that we do, right? Joni actually in preparation for this, um you and I looked at this image. And I think we're gonna put this in the show notes or at least make a link to it uh out there. And it would be helpful actually uh to have this pulled up here. It's a learning pit image um that you showed me. And as we were talking about uh a task that we're gonna we're gonna talk about here in just a few minutes, um, we were talking about how do we frame this and how do we frame the the language around the kinds of help that we uh have for students. So Joni, can you talk us through that image? I'm gonna try to pull it up here uh and look at it here in our notes.

SPEAKER_00 17:35

Definitely. Yeah, and we will include this in the slow in the show notes so people can pull this up and and take a look at it and even dig into some of the other resources that were created around this image. So this is part part of some work out of an organization called the Challenge Learning Group. And this idea of the learning pit and the learning challenge were created by a person named James Nottingham. These images are pretty easy to find on the internet. But this idea of the learning pit is that when when learning starts, we can sort of imagine a student, you know, standing on the ground thinking, oh, okay, you know, I have this task and I think I know how to do this. So I'm gonna progress into it. And as they progress in, again, this is a task that's well aligned with that level of difficulty that we were describing just a few moments ago, uh, they they are gonna reach a point where, huh, this maybe isn't what I thought it was gonna be. Like that first initial challenge comes and there's kind of this eye-opening, like, oh, wait, I'm I I actually don't know exactly what to do here. And at that point, the student might be at the place where they're confused. And in the image that I hope folks can pull up and look at, this is when the student is sort of standing at the bottom of the pit, right? They're they're in that mode of, wow, I'm stuck and I'm confused and I don't really know what to do. Then the image progresses by showing then the student next pulling on a rope. So there's a rope that is uh hanging from the pit that they can hang on to to then pull themselves up and out of the pit. And that's when the the attitude shifts from I don't know what to do, I'm confused, to wow, I'm gonna actually have to work at this and putting in the effort to overcome that challenge. And then the next image, like the students almost pulled themselves out where they're starting to feel the results of that effort of pulling themselves up. And then finally they're up and out of the learning pit and feeling the success of having accomplished the problem. So uh I think it's a really nice visual. And I think there's actually some nuance to this visual too, that that you and I talked about. And we'll unpack that a little bit deeper in just a minute. But this idea of the rope, I think is really important. And I wanna, I I would love for us to explore. I would love you to share some of your thinking around. I mean, it doesn't need to be a rope, it could be, you know, steps in a wall or a ladder. You know, there's there's lots of ways to visualize that pulling out. But I I think what's important is that the student is pulling themselves up by the rope. It's not that the teacher comes down and boosts the student or that the teacher even throws the rope necessarily. So I I I'm curious to know your thoughts about that sort of transition from I'm confused to I need to work hard at this and I want to work hard at this.

SPEAKER_01 20:33

So yeah, Joni, there were a couple of things that kind of jumped out at me. I hadn't seen this before. Um, you showed it with me a couple of shared it with me a couple of days ago when we were um prepping for this. And and I um I I was really quick. I I thought, man, this is such a cool, cool image because there's two things happening here. And you you pointed uh pointed one of them out, and that is that the that the student has a rope that they are actually the ones pulling up on, right? That there isn't another person called out in this in this image. It's the student actually pulling themselves and and putting the effort in, and I need to work hard at this. And so that part of it I think is really, really cool. But I I found it interesting, and I don't know if this was intentional or not um by the artist who drew it, but the the child in the in the middle where they're saying, I'm confused, they're kind of at the bottom of the pit and they're in that that space where I'm stuck. Um, their back is actually to the rope. Um, the and so they're facing away from that wall. And I thought, you know, what what it is that we as teachers are trying to do, because it at this moment uh struggle could pre could quickly become unproductive. Right. This is the time when I as a teacher step in, right? And and the student is no longer um being able to make progress. Or realize that they, you know, that that this is still doable. They're kind of in that space of, man, I I'm really just stuck. And the the image of the student being uh facing away from that rope was powerful to me because what I was thinking then as teacher, I'm thinking, oh my gosh, what I need to do is help this student turn around. That's all I need to do. I don't need to hand him the rope. I don't need to say, hey, there's a rope over there. All I've got to do, all I have to do as a teacher, because it, I mean, and whatever image, it doesn't have to be a rope. It could be the stairs, like you mentioned, or whatever. But it all I have to do is get the student to turn around and and allow them then to see that that the foresight or the the sight, really, to be able to see, oh, hey, there's a pathway here. Um, hey, here's an here's something that I can that I can use. And so I uh I guess what I was trying to make the analogy there is that that that when we provide that help, when we ask that probing question, or um maybe we make this statement about um here's something you do know, um, or tell me something that you do know. Um that statement or that little probing question should do one thing. It should help the student turn around. It should help them go from I don't see the rope to, oh, there is actually a rope over there. But it should not hand them the rope and say, hey, climb this rope. Right. Like that there's there's a careful um amount of questioning, I think, that that um we as teachers and it takes practice and and whatever it definitely does um learning how to turn them around, but not give them the rope.

SPEAKER_00 23:54

Yes. I I love how you noticed that. And I know we're gonna shift into our conversation about the specific example that you and I engaged in a week or so ago. That was so so cool to illustrate that. But I just want to reiterate too, uh well-meaning teachers will hop down there into the pit with their students and say, Hey, don't get frustrated. There's a rope right here. Let me show you. You just climb the rope like this, and then you get out, and then you have, you know what I mean? You feel that success. And and I think I again reflecting back, I it probably sounds from our podcast like I was a terrible teacher and I did a lot of things right, but I do reflect back on the things I could have done better. And that's definitely one of them. That was definitely one of them. And shifting our thinking, and I actually did do this before I left the classroom, shifting our thinking from our goal as teachers is to make math easy to our goal as teachers is to help students learn. And learning isn't always easy. Um, I think that's so important. So it and I also just want to talk about the cultural piece, and this will be a topic for an upcoming episode where we're gonna talk about how to create that kind of a classroom culture, but there has to be a classroom culture around the attitudes and the approaches that we want students to engage in during productive struggle. And the work for creating that classroom culture actually probably happens long before they're given a task that would put them in the position for productive struggle. So I just want to acknowledge that it's not as simple as an in-the-moment thing that you know, this isn't one of those things that a teacher can turn around and try tomorrow necessarily. They need to be sure they have that cultural piece already set up and that they've got some relationships and an understanding of their students because of the all of the nuance that's involved in in this kind of work in the classroom. So we had such a great experience trying to uh engage in a little productive struggle experience ourselves. So let me start by uh kind of setting up, and again, we'll put a link in the snow in the show notes. I keep wanting snow notes, like what is that about? We'll put a link in the show notes. Um and maybe even some of the images that were part of this conversation that we had around here. But this came from um a friend of mine shared a puzzle that was in, I think it was Scientific American or something, years and years ago. And here's the frame. So we just ask our listeners to visualize if they can picture a square. So a quadrilateral with four equal sides. And then picture just dividing, not don't physically divide this on paper. If you want to draw your square, it's probably helpful. But um, picture the upper right 25% of the square. So a smaller square in the upper right of the original square removed. And that ends up creating like an L-shaped that is from a square with 25% removed. So the goal, once you have that L shape, is to divide that shape into four congruent parts. So take that L shape and divide it so that you have four pieces that have equal size and shape. So that was the task that that you and I engaged in.

SPEAKER_01 27:26

That's exactly the task. And I just want to um give you everybody a spoiler alert. We are gonna go through this. And so if you want to solve that problem, if you want to take your time and actually um mess with that, uh pause the episode um, or you know, come back and listen uh in a little bit because we are actually gonna go through uh some of the solution for this problem. So just spoiler alert. Uh we are gonna we are actually going to to go through this. So just uh just a heads up. And so um that's such a great problem. It's such an interesting task because um, you know, I think it meets several of the things that we were talking about. I I know what a square is, I can remove a corner of a square, and I understand, I I have a grasp on what a congruent shape might be. Um, and so there I can immediately begin going um at uh you know ideas for what would that congruent shape maybe look like? What are maybe some strategies I can start uh thinking as a student? Like I know I need, you know, the these same size, same shape uh you know, figures, and I need four of them. So I got to break this thing into four congruent uh pieces. So um that was that is that's a really cool task. What a cool way to kind of get get us started here. So Joni, you want to talk us through maybe a little bit about what what you were thinking um as you got started?

SPEAKER_00 29:01

Yeah. So I actually had spent quite, well, some time, maybe 20 minutes, 30 minutes with this problem before I shared it with you, Curtis. And the first approach that I had was, okay, I can I can sort of see three equal parts pretty easily, right? Like that quarter that I removed leaves three quarters of the square that I know are congruent. So I I kind of had to wrestle a little bit with my temptation to just go to three and work past that for four. So my first thought was well, if I can divide the shape in half and get two congruent parts, then I can just, then I can think about dividing one of those in half again, and I'll have my four congruent parts. So the first thing that I did was draw a diagonal line basically from the lower left corner of the L shape up to the corner of where the 25% had been removed. So I had a diagonal line. And it didn't take me more than a couple of minutes to realize after I drew that line that I wasn't gonna be able to divide that. It was kind of a trapezoidal shape now. Uh, I wasn't gonna be able to divide that trapezoid into two equal parts with uh with that line there. So then I shifted my thinking a little bit and was thinking about um just starting at, I started at the top part of the L and trying to like create sort of a and when I showed you my thinking, you described it as like an in and out, right? Like I had drawn this on grid paper. So I kind of traced around some of the squares. I because I was thinking about the area. I I was able to count like how many squares were in the L shape and realize how many I wanted in each of the four congruent parts. And then I was just sort of messing around with how to create like an odd shape that had that area that then maybe I could duplicate if I did some shifting and rotating or something. So that was my initial approach. And I and I shared with you, I don't think I had shared my approach with you though. I just shared the problem with you. Right. And you got to a solution much more quickly than I had.

SPEAKER_01 31:09

So I yeah, I jumped in and I had drawn myself uh an image uh there and I went through the steps and um pretty quickly I did some of the similar thinking, you know, thinking about the triangle. Can I can I break this, uh, can I connect those two corners and then make uh you know, split this this two shapes each into two congruent shapes and and run with that. Um I too came pretty quickly uh to the observation that that wasn't going to get me where I wanted to go. Um and so I went back to this and began looking at um, I drew a much smaller square initially than you did, um, which I think lent itself to me coming up with a pretty fast solution because as soon as I was able to kind of step back and realize that we were looking for four uh pieces, I started drawing uh or looking for ways to break those uh four pieces up. And I knew that bend was gonna make a difference. And so my focus, just similar to yours, was on that that uh sort of that that corner piece, if you will. And what am I gonna do down in that corner uh that would make sense to kind of help me? And then realizing the next thing that I had to realize was um that uh orientation doesn't affect congruence. And as soon as I picked up those pieces, I was then able to see, oh my gosh, well, I've removed this one. Um, and if I I had a uh four by four square. So if you imagine um each side was four by four, which meant that each of my remaining little squares, the little three that were left over, were each two by two. Um and so that gave me 12 total units. So I knew I had to have only three units in each one. And it was pretty easy to orient those three little square units um to come up with the shape that I needed that would fit uh in each of those and and be congruent and would fit all the way around. So then I so once I had a solution, it was kind of interesting because that I quickly then shifted into okay, how do I help Joni get to this solution without like giving it away? I didn't want to be the spoiler alert. And so this was the space where you actually showed me the pit, the learning pit uh drawing, which was great.

SPEAKER_00 33:35

Yeah.

unknown 33:36

Yeah.

SPEAKER_01 33:37

And so we had this whole conversation about the student needing to turn around and how do I ask the right questions and what what is the right question so that I'm not giving you the rope. But um, and so I had to really begin to try to investigate the task a lot more and try to understand what was, in fact, the pattern here that got me to the space where I was. Um, and so I think, you know, one of the things, just side note as we're as we're talking about this, teacher content knowledge um of the area in which we are trying to uh encourage our students in in productive struggle is so important. Us knowing all of the connections between the different uh you know representations of the mathematics or perhaps the different uh pieces of a particular puzzle or whatever it is we're trying to work on with our students is so important. Not just what is the solution or what's the procedure to get from A to B, but um all of the different variances uh uh that get us from A to B. Having an understanding of all of the connections between all of those and among all of those is super important to my ability then to be able to not hand you the rope, but just help you to turn around so you can see the rope on your own. Um, and so let's jump back to where you were. So you had a larger, you had a larger square, you had an eight by eight square. And so you had lots of little pieces.

SPEAKER_00 35:09

I did. And I and I was really intentional about that when I was initially trying to solve the problem. I was like, oh, I want a bigger square because I felt like it would be easier to see. I felt like if I had a small square that I would, you know, I wouldn't be able to engage with the parts the way that I wanted to. I know that sounds strange, but I I appreciate you bringing up the the perspective that you had, that you had to know and understand not just what the solution was, but why it was the solution. And, you know, I know in our, you know, in the moment you actually took some time to figure that out. But from my perspective, the other thing that you did was then really become an investigator, like we talked about in a previous episode of this podcast, where you were like I was showing you the approaches that I had taken. I showed you the first thing I tried. I told you what my thought process was, then I showed you the next thing I tried. And I I could I you were able to ask me questions that allowed you to understand my approaches, what I was thinking about, and the things that I knew didn't work, like where I had hit roadblocks. Um, and I just want to say how important that time of you questioning was because it allowed you to know what did I already think about, sort of back to that rope analogy, right? It allowed you to know, okay, she's already looked up and to the left and she's already looked down and to the right, and she started to turn her body but only 45 degrees instead of, you know, all the way around 180 degrees to see the rope. Like you, you had a really good understanding of what my thought process had been. And what was so great about that for me as a learner is the questions that you ask me built on how I was already thinking about the problem. You didn't ask me a question that caused me to just ditch everything I'd tried and start over. I think that would have taken me to unproductive struggle in that moment. Like, oh, I I wasn't even, I wasn't even close, like is how that might have played out. But instead, you were able to say, hey, when you're doing these ins and outs, like you got me to think about a tessellation and you got me to think about the importance of that, like, okay, it's it's not necessarily gonna be a a reflection, or it's not gonna be the exact same image, exactly in the same orientation. Like, I need to be thinking about rotating. And I even remember saying to you at one point, like, I want to be able to like cut it out and move it around. And you really jumped on that, like, yes, think about that. And so your ability to give me a guiding question that actually built on how I was already thinking about the problem. For me, that was what turned me around to see the rope. That was the moment like that you helped me build on what I had already been thinking, but also pulled me away from what was maybe getting me off track. So it was both you're talking about the tessellation, bringing up the tessellation, and you're talking about the size of my square. The fact that I had was working with such a large square with such a large area might be making it more complicated. And that's all literally, Kurt, that's all you said. Um, and then I could go down like, okay, maybe I'm gonna try a smaller square and wait, let me be strategic about what's I don't want a two by two, like that's too small. Um, and and to get strategic about, oh, if I try a, you know, a six by six, then I don't get an easily divisible number and I'm gonna have to be breaking up these little area squares into smaller parts. So I just want to acknowledge, like, it was about you understanding the problem, but it was also really about you understanding my thinking so that you could draw on my thinking and and and just give me that push in the right direction.

SPEAKER_01 38:56

So I think there were a couple of things that you called out there that are important to note. And and one of them, um, for for me as teacher, I I didn't know going into that what you um what you were gonna know or what you were not gonna know. I I didn't. Um, and so I had to take some time as teacher to ask you questions, right? I had to investigate, like you said, I had to become an investigator. Um, and and then I had to take some time to formulate. And I I want to just um give encouragement to folks that like it it it really is something that that um we as teachers have to slow down and and be able to um say, okay, I asked you this question, this was your response. You know, in my mind I've got to, I don't have to do this on the like immediately. I can take the time to say, hmm, let me think about that. And and and just allow that question to kind of sit for a second before I ask the student another question. Um, so that so that between the two of us, it really feels like we're getting someplace. And now the student is still working on the problem and maybe waiting for my next, maybe waiting for my next question, but I and and I'm really thinking in my mind, okay, so they know this and they know this. Okay, what's the way that I can formulate that? And you I verbalize this a lot in our conversation. I was thinking to my, I was saying out loud, okay, so how do I get you to this next little step? And and um ultimately my questions, um, I was trying to get you to see this pattern of threes, that each of those remaining pieces were divisible by three. And so the numbers of remaining uh squares were divisible by three. And so maybe there's something to do with this idea of the threes uh fitting in there. That's why that four by four square worked out so nicely for me because of my resulting piece had three pieces in it. Um and so I was intentionally trying to get you there.

SPEAKER_00 41:02

That number three, like that, that really was helpful. And again, you didn't you didn't tell me try a four by four square, but you did say maybe try a different size square. And that got me into thinking, okay, what's like what's a smaller number that I can engage with? And that really that that's where the click happened that allowed me to find the solution, which was to get down to like each of those four congruent parts is gonna only have three squares in it. And at that point, Kurt, I was way beyond productive struggle. I was like, oh, I got this because there aren't that many ways to orient three little blocks. So I'm gonna, I'm gonna, you know, be able to try all of the options and one of them is gonna work. And sure enough, the here's here's the get here's the giveaway. If anybody still hasn't worked out this puzzle and wants to try it, it's it's there are a little L shelf, uh, an L shape. The four congruent parts are all L-shaped as well. So one sort of sits in the crook of the L and then the others are are reflected upside down on the other side. So um getting it down to the three and knowing like, oh, there are there are not that many permutations of three. And that's why when I was working with the much larger square, oh my gosh, my my area of each piece was 12. And there were just so many ways to get uh a shape with area 12 that I was gonna overwhelm myself with possibility before I came to the solution.

SPEAKER_01 42:25

Right, right. So let's go back and just kind of think now, uh, just to summarize um our conversation here and and maybe bring us back to to maybe the the end here. And that is so these questions that we ask as teachers, these guiding or or or or really I feel like they're more probing questions than they are guided questions. Um the the ideal uh scenario there is to help the student connect to the student's thinking. We're not helicopters. And uh I I I was thinking about this, you know, we're not actually dropping down some little thing here, lifting the student out of the pit and putting them someplace else, right? So we're we're not like taking them completely out of their thinking from where they were before and putting them in a whole new space of thought. We're making a connection, even if what we want to do is miles away. Like, because it's possible and even likely that we're gonna be there. But there's got to be a path that connects all the way through, right? Because the student does have some valid thinking. And so what we're gonna try to do is make a connection between each little step and and and probe just enough to get them to turn and then turn again and then turn again and make the next, you know, to make the progress over there to to that rope, um, so that it that it's on their space. And so we're really intentionally validating, validating the student uh approach and the student thinking, um, and then trying to help them redirect their thinking, right, so that they can actually see uh that space because the idea is to get them to turn, not to hand them the rope. Right.

SPEAKER_00 44:06

Exactly. It's reminding me of the uh the principles to actions, the eight mathematical teaching practices, and one is about questioning. And and they talk about probing questions that allow the teacher to better understand what students are thinking and how students are approaching the work. And then another set of questions that push the students thinking, you know, from where they are toward the in the right direction towards the intended learning. So uh that could be another source of support for our listeners to really narrow down on what the right questions are because again, Kurt, we're in this. This is such a nuanced topic, you know, knowing what makes productive struggle and being able to be adaptive for different students. This is another place where we're gonna need to be adaptive. We're gonna need to really be able to do the investigator role and dig down into what is a student thinking, and then connect that student thinking back to the broader landscape of learning around this topic and be able to guide the student in a way that doesn't, you know, hand them the rope and remove the learning for them. 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.