SPEAKER_00 0:00

Welcome to Room to Grow. I'm Curtis Brown. 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. So let's get growing. Today, Curtis and I discuss mathematical exploration and its role in effective instruction. We consider how the opportunity to explore builds motivation and interest and contributes to a growth mindset for students. We tackle the ongoing challenge of finding time and give some suggestions for fitting exploration into your classroom routines. Let's go. So, Kurt, I'm really excited to have some conversation with you today about this idea of mathematical exploration. You and I've had lots of conversation, and I know this is super a big passion of yours. So why don't you start us off by telling us why you believe that giving students the opportunity to explore with mathematics is important.

SPEAKER_02 1:02

Well, Joni, thank you. Yeah, I'm really excited to talk about this. It is a big passion of mine. I think partly because I'm a math nerd, and so it comes naturally for me to just want to explore and be curious. But um I think that that the purposes behind playing are are really twofold. And I call it playing, mathematical exploration. Um, so I'll use play quite a bit in this conversation for sure. But um I think that it's twofold. There are two pieces. One, I think that that students they um they get a chance to be curious. Uh we are innately curious creatures, um, and it starts early on, um, especially in in elementary school. And I I love having conversations with my nine-year-old who's in third grade right now. Um, I I absolutely love having math conversations, and he brings them up. I I don't go to him and have, you know, hey, Tegan, let's talk about this. No, we don't have that conversation right away on my time. It's him coming to me, and so it's his curiosity, which is so great that he comes and does that. And it's natural for him to be curious about these different things. And so um, I think, first of all, it gives the kids an opportunity to explore and play and be curious, and it's a no-pressure environment. And that's one of my one of my passions on this, is that um it the pressure, the environment that we create um for the students needs to be um sort of a no-pressure environment. There is a sense of those questions and and and it's a hot topic right now, or a hot way to say things. What do you notice and what do you wonder? Um, and and I think the premise behind those things is really, really good. Um, but the intent needs to be if a student comes back and doesn't have this giant um aha moment or this great big thing, we don't we don't need to come away with that nugget. Like they don't have to come home with that nugget, they shouldn't feel that pressure. Um, because if they don't feel that pressure, it becomes fun, it becomes uh play, if you will. And all those discoveries and all of those pieces come from their curiosity, right? So they also then in the second part of it, they get to own it, right? Everything that they come up with is something that they discovered uh on their own. And so I think, you know, there's there's twofold, right? There's this, we have this no-pressure environment to go play and explore, and it and it plays to our um innate curiosities. And so I think that's hugely, hugely important, right? Um, and and students kind of get to go play and discover what they want to discover. But then the second piece of that comes right after it, and that is the fact that they own it. Anything that they come back with, anything that they um come up with from their exploration, from their playing, is something that they own. And it actually makes me think of um when I was in college um and I guess high school maybe, um, there were computer games um that uh when you kind of went on the screen, uh the bulk of the screen was was black or or you know, hidden, grayed out, where you couldn't actually see it. And your character um would be in one corner of the screen or whatever. And as you navigated around through the screen, well, then that those portions of the screen would become visible and stay visible to you. And I kind of think a little bit about that um in my in my own experience, right? So every time that I've gotten to go and play with mathematics, there are other people who have discovered those things before. I'm not talking about folks who are writing PhDs and um you know doing their dissertations on different things. I'm not talking about that. I'm talking about, you know, there's there's this whole world of mathematics that has been discovered. Um but it hasn't, it isn't my world yet. I don't, I don't have visibility into it. And so it's kind of this same thing. Like if I'm going through and being able to discover and play and explore and find this, these areas out on my own, then I then I get to own them a little bit, and and it's my discovery. Um, and so that's I think those two things, right? It plays to my curiosity, it's fun, which opens up the frontal cortex, and there's all kinds of brain science about that stuff too. But it's fun. And then second, so we have the engagement, but now we get the ownership because I'm exploring it, I'm discovering it, so it's now my mathematics. And I think that's kind of the twofold piece. Um, that's my big benefit there.

SPEAKER_00 5:55

I think as you were talking, especially about that ownership piece, but also about the play piece. I think one of the things that, you know, when the Common Core state standards came out or college and career ready standards, you know, a lot of states are not necessarily following Common Core, but the whole idea there was to circle back to this idea that actually math makes sense. And I think for for kids, too often they get caught in this, you know, a direct instruction rather than an exploratory mode in the classroom. And they they're under this conception that there's like a way to do each kind of math problem. And the purpose of math class is to learn the way to do it, right? And and then it becomes this big memorizing game where, you know, the things that you were talking about in terms of exploring and taking away that pressure that just allows kids to have the freedom to, you know, think a little more loosely, I guess, without feeling like there's a right answer they're supposed to get at. Uh, I think it really ties kinetically, really tightly back to that idea of math makes sense and we should have all kids. And you're right, I think it's more natural that curiosity for the younger kids, but I think we need to figure out a way to get that curiosity back for middle and high schoolers too, and even as adults, right? Like to to try and break through this myth that math is about rules, really. Math is about um, you know, understanding. Math is about math makes sense. So uh ways to explore that.

SPEAKER_02 7:26

Yeah, that's such a big thing. Um, and I I'm sorry, I didn't mean to interrupt you there, but I I'm not sure while you were talking, I was thinking about um something that I I heard from uh a math hero of mine, James Tanton. And then I know we've talked about him before. Um, and I absolutely love he has this whole deal about um learning math uh historically, and I'm probably misquoting that the way that he says it. But the idea being that um if we go about learning math and discovering math the way it was discovered the first time, it it makes sense, right? Like so if we go back and we think about how math was thought about or a particular topic was thought about um in the in the origin, you know, in the history, as far back as we can kind of go, and then we follow the pattern of discovery through these things, math makes sense, right? And it's logical, it's the you know, it's very reasonable to come up to the come to the conclusions um that were come to via the the vehicles that other people have gone. And so um, but you get to do that from the perspective of, hey, I didn't know this, and now I'm playing with it, and ooh, yeah, I see this connection, and ooh, yeah, now I've got it. So it does make sense um to go and play and discover.

SPEAKER_00 9:03

So just to play devil's advocate a little bit here, I know you and I have had lots of conversation about one of the biggest uh, you know, factors in terms of teachers changing their teaching is time. So when a teacher's feeling this pressure of, I've got so much curriculum that I have to get through and so much math that, you know, I'm responsible for ensuring my students know before they move on to the next grader course, this feels like one of those like, oh, mathematical exploration or mathematical play, like I don't have to do that. Like let's get, you know, let's get serious and just do mathematical learning. How, you know, what would be your response to that? How how can teachers think about the time issue and sort of reconcile, you know, that struggle that's that's constant and continuous for them and this idea of the importance of exploration?

SPEAKER_02 9:56

So um I think there's two, again, there's probably a twofold response to this. Um, the first one is by doing this play, I'm investing in the future. And I I don't mean that to sound um, you know, sort of I I don't know. I what I mean is that you're investing in the future of your school year, not necessarily just the fact that you're investing in students and their lives and the future of the planet and whatever. That's also good, and that's why we do what we do. But um, what I really mean is that we're investing in the future of my school year, right? So if I do this, say in the fall, and we we pick an important topic um in the fall that we're gonna go and we're gonna play with because I know in the springtime we're gonna come back, and this topic is gonna be a big uh player later in the school year because of something else we're gonna be needing to do. If I have done a good job of allowing students to play and explore and really own their own learning in the fall, I've invested and I save the time later in the school year. And so, yeah, it's gonna cost me some time in the fall. It might cost me some significant amount of time to allow my students the freedom to play for a little bit. And it might feel like I'm behind. But when I get to that time in the spring, I'm gonna make up for it. I'm gonna catch back up, I'm gonna be able to go faster. My students are gonna be able to pick things up because they've already got a really strong uh foundation. The second response to that, and I know you and I have had big conversations about this, is that um this is one of those things that you hear people talk about as keynote speakers in conferences or in sessions or in professional development webinars, whatever it might be, you go to one of those things and you hear people talk about this and you go away and you think, oh my gosh, I'm not doing that enough. I've got to do that 183 days out of my school year. And the the answer is no, you can't and you shouldn't be trying to do this. Instead, we should be thinking about just the big topics, right? Just the really important topics, and I I'll follow that up with the ones that lend themselves to this. Um, I've got a nice high school example. I'm still thinking about an elementary example for this, but a high school example of this is thinking about um how the parameters, say, of a quadratic um in standard form affect the graph. Right? If I've if I kind of have the time to play with this concept um and and invest in playing with what does A do uh to the shape of that quadratic, to the position of the vertex, whatever, um, what does B do? What does C do. I can then uh make connections later in the school year really much faster because I've got a good feel for what do these values accomplish, what do these values um feel like? And then I'm also investing in their later um school, whenever they maybe get into calculus and we're looking at derivatives and things. I mean, there's some, there's so many great things that can come from that playing and that exploration.

SPEAKER_00 13:17

I really appreciate this answer. So I'm thinking about, you know, really when we started the conversation, it's mathematical exploration and mathematical play, which maybe sounds like it's not the same thing as learning, right? It sounds like a different, like, oh, this is something other than learning. Like, be sure you're taking the time to play. Like that sounds like this off thing. But really, what I'm hearing you say is this is it's actually part, an important part of the learning process, right? So the idea of investing in the later learning, it's really about that the quote exploration part that you do early is actually just the students' initial thinking around this big mathematical idea. And it allows them to formulate and, like you said, own their the initial ideas of mathematics that then when more formal instruction comes later on in the year, that's maybe the more formal instruction where you have a specific learning target in mind, and there's you know, a specific mathematics you want them to engage with during a lesson, it's actually their third or fourth time working with that mathematics rather than their first time working with that mathematics. So that that idea of like, and and you said investing in the future, and it made me think of uh a catchphrase I used to use when I was providing professional learning for teachers, and that's go slow to go fast, right? Like you take the time to do this now and that it saves you time in the long run. Right, right. Like direct instruction might be faster on the front end, but when students don't remember it and they don't own it and they haven't made their own sense of the mathematics, then really you've actually just slowed yourself down long term. So what feels like slow in the beginning is actually buying you the opportunity to go a little bit faster. Yeah, absolutely. You know, as we engage down the road. So I love this idea, I love that shift in thinking, right? It's not about, it's not about either or. It's really about, okay, how does this fit into the the bigger, the bigger picture of what I'm learning? So let's I I want to push a little bit deeper on this idea of it's not every math topic. So how would you how would you recommend that a teacher think about, okay, which are the ones, what are the topics that lend themselves? And I'm not asking to like come up with a list, but what kind of thinking would you do to decide? Like, is this an important enough concept that I should give my students the opportunity to explore and play here? Or is this something where, you know, just the whatever open-ended problem I want to give them for this lesson is enough? What are some of the guidelines you would use to decide, you know, what to what to take the time to do this for and what not to?

SPEAKER_02 16:20

So I would probably think about um topics that I see connections um for the future. So this is a big planning thing for me as a teacher. I've got to be thinking about um and looking at the curriculum and looking at the the topics that I'm going to be connecting for my students um throughout my school year and then in the classes that come after me in the future, um, what are the things that maybe what I do here connects directly to further on out, right? So I think those two those two things are big, looking for those large veins of connection uh among uh topics. That's probably the first, the first one. Um the second, I feel like are are the ones that come up naturally for our students, the ones that that seem to be somewhat um interesting to our students and and on their own, right? I know my my nine-year-old um who is in third grade, um last year in second grade, they had just introduced this idea of uh fractions to him. Um and so he came home and we were talking about uh fractions. As a matter of fact, this was right at the very beginning of the of the pandemic. Um, and we were talking about um hotels. And I don't even know how we came up with this and why this even made sense to us. But um we started looking at hotels and uh windows being open and closed, and we were shading um you know what amounted to be an array, right, uh, of this hotel. And I had just a single-story hotel that I was working with with him. So we were talking about ribbons and breaking it into pieces, but I didn't call it that. We were talking about the hotel. And we were shading the picture and talking about how many of the of the piece, and we were talking about fractions and comparing them. And I even had a little number line underneath of it, and we were doing this stuff, and we were just playing, asking what if questions, and he was he was doing most of the guiding um in terms, oh no, daddy, we need it. What if we shade these ones over here? Let's talk about that. So it was all his own just kind of fun exploration. So it was the thing that came up naturally. So I think of those two things, uh, of course, the the ones that are connected to the future, and fractions happens to be very connected to much of mathematics, right? We we do a lot of things that deal with that, right? So, you know, looking into middle grades and you think about proportional relationships, you look up into high school and you think about um things like rate of change, and uh, you know, this fraction just kind of runs right through that whole, yeah, that whole piece. Absolutely hugely connected topic. Um, but it also was one that was just fun and interesting. I mean, how many kids don't like talking about? And I I know this is one that kind of gets some some resistance in the math world, but like how many kids don't like talking about how much of the pizza have I got, right? So we've got this thing and we can talk about that, and we can connect it to a number line, and we've got this visual representation, and we get to talk about how many slices does Joey and Bobby get and whatever. Like it can become kind of a fun exploration and playtime with some mathematics, um, but it's such a foundational piece um for the rest of of what they're going to be doing in their in their world. And so I think that's I think that's super huge and important.

SPEAKER_00 19:49

Yeah. I'm thinking too about your your fractions example, because yeah, you as you so eloquently said, that's a thread that goes all the way up, you know, calculus and beyond, right? That concept of of a fraction. Um, yeah, and it's relevant. And I think uh I think well, I thought it was eloquent. Um, but I was just thinking, like uh it it also made me think, and I know we'll we'll do a future podcast on this idea of um you know one of the Nctm mathematics uh teaching practices, which is uh illicit and use evidence of student thinking. Boy, I'm the opposite of eloquent right now. Um, but this idea of being able to give students the opportunity to explore around fractions, share out their thinking. And then again, that's laying the groundwork for me as a teacher down the road to say, oh, remember how, you know, Andrew thought about this idea with fractions, or, you know, when when Tegan shared his example, you know, that really is a great way for the teacher to be able to connect back to the student thinking as they formalize the mathematics later on in the in the lesson. So I think that's so important. And, you know, again, I'm I'm getting more and more convinced throughout this conversation of how important this idea of of exploration really is and how it's going to contribute to deepening the learning of every single other student. The other, the other thing that I think is really cool here that again can connect back to the motivation for how do I sort of justify taking the time to do this as a teacher, this is a great opportunity to tap into the language development as well. Because if if students are in an exploration kind of phase and a you know, a low pressure kind of phase, and I would imagine you you maybe did some of this with your son, or at least saw the opportunity to do this with your son. You know, when you're just having these sort of informal curiosity conversations, you know, that's a great opportunity to throw in the formal mathematical language and help your student, you know, help your student or your child or you know, whoever you're talking to understand like. You know, oh, that's called this in math. And, you know, I'm thinking I was, I was listening to one of the podcasts that I enjoy listening to, and they were talking about how do we learn. And the guy was arguing that like the very first step of learning any new thing is understanding the language of it. So he was suggesting like doing story, like read a story about the topic. So, you know, there you it's just an opportunity to grasp the ideas and learn the language because you have to have the language to get further in the study. You know, how often have you picked up a book about a topic you don't know that much about? And you can literally read a paragraph over and over again and be like, yeah, I have no idea what that what that says because I don't know the word. So I think this is a great way too to like it's the ownership piece, but it's then taking it that step further and saying, okay, here's the language that is associated with the ideas that you just came across. And um, you know, got again gives gives that power and ownership to the kids too.

SPEAKER_02 23:02

That's right. It gives opportunity for students to develop that language and you, as the teacher, an opportunity to be able to say, Hey, when you noticed XYZ pattern, that's actually called whatever. And so you get it, you get that opportunity to give the students um words for what they're seeing and what they're experiencing and what they're discovering. And so, yeah, I I totally agree that not only do we now have ownership of the concept, and the students have gotten to, you know, kind of do their curiosity thing, and it really feeds that curiosity. Um, you know, once you've had an opportunity to discover something in mathematics, you want to do it again. Like that's a good feeling, that's a positive feeling towards mathematics. So you want to do it again. So now there's a desire to learn this. Now there's a desire to kind of come back to math and do something more. And then they've now had an opportunity to learn some actual language, right? They've opened the door when they've described it in their own words. And I think that's important, right? You describe it in your own words, but then you have the opportunity as teacher to say, well, that's a great way of saying it. But here's the math term that we use um in other math courses to talk about this. Like you really get the opportunity to connect what they think of as the concept to the actual math term. And so um, yeah, I think there's huge, huge amounts of value in mathematical exploration um for your classes.

SPEAKER_00 24:36

Yeah. I think this ties back to um, you know, the the last episode we recorded where we were talking about ensuring that every student learns. And this idea of exploration is it's an it's an open door for every student, right? Every student's going to be able to engage in the way that makes sense to them and to go as far with the exploration as, again, makes sense to them. So it's an opportunity for the educator to give power and value to the student thinking because whatever it is they explore, their own sense making is what's important. Like that's the goal of spending the class time to do the exploration is to allow the student to make their own sense of the mathematics. So it's not only freeing for the student who isn't trying to, you know, master the concept or match the specific response or or mathematical process that the teacher is looking for, but it's really about um whatever you come up with is exactly what you were supposed to come up with. And then the the teacher can then value that from all students. So um, yeah, it's great. I love how we're making connections back to our previous podcasts. And um, I think this is a great topic. And I'm I can see why this is something I know you've been enthusiastic about for a long time. And I've really enjoyed, you know, you pushing my thinking and really helping to convince me that this would be something I think teachers should try and find the time for. Pick one.

SPEAKER_02 26:06

That's all. Just pick one. Don't don't try to do it for everything for this school year. Pick one thing and um give your students one day. I wouldn't even say let's do this for for multiple days or or you know, give them one day, or if if one day sounds like too much, give them a half an hour to go and and just play with. And, you know, if you if you're in high school, put them on a graphing uh device and and let them uh explore dynamically something. If you're you know in elementary, maybe you've got some uh manipulatives and some handheld things that you can do there, or maybe there's some some great um, you know, technology pieces out there that that work well for that. So um, but pick one and and give them a no-pressure situation to go go and play. Um and just see what kinds of things your kids come up with. I think you'll be surprised.

SPEAKER_00 26:58

Yep. Awesome. Well, thanks for the great conversation. I'm I'm really excited. I'm actually gonna go try some of this with, you know, my nieces and nephews. I don't have kids at home anymore, but certainly the kids that I encounter in my life and you know, thinking about mathematical exploration for my own self too. So thanks for great conversation, Kurt. I'm looking forward to the next one. 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.