SPEAKER_00 0:00

Welcome to Room to Grow. I'm Curtis Brown. And I'm Joni Funderberg. 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.

SPEAKER_01 0:19

So let's get growing. Welcome to the first episode of Room to Grow. We're very excited to kick off this project and so happy you decided to join us. In this episode, Joni and I will tackle our first and what will be likely many visits to the ideas in mathematical modeling. Today we will talk about what is modeling really and what do our students think when they hear the word modeling. Also, we'll begin to unravel the problem of finding time to do it in our classes and our opinions on how to get started. So here we go.

SPEAKER_00 0:49

Okay, so in this conversation about modeling, Kurt, let's let's first talk about what we really mean by modeling. There's this word modeling floats around all the time, and I just want to be sure that we establish from the get-go of this conversation what are we talking about?

SPEAKER_01 1:04

Yeah, I think this is a huge conversation and a huge topic. It's very hot uh right now. It's something that that a lot of people want to talk about. And I know for me in the classroom when I was teaching first, I know my first thing that I went to, my thought process was um problems that were contextual or you know, sort of our typical um kind of uh word problem and that sort of thing. That was where my mind kind of went. Um, and it still does initially, probably the first time that I think about it, that sometimes to seems to be the first thing that comes to my mind. Um, but then when I started thinking about um and looking at the common core standards and particular the math practice for um reading through that, um I kind of picked up a a number of things. Um, probably the biggest one for me though, is that what we're looking to do in these math modeling tasks and thinking about mathematical modeling and doing it with our students, one of the very first things that I kind of picked up when I started reading that was that it's it's focused on the mathematics that they know. And it's really getting the the focus is on what students are doing and what students come up with. Um, and I believe that that's that's maybe at the core of all of this conversation that we're gonna have here today, just talking about mathematics, is that that we really want to be thinking about what students know and what students are thinking and what they're coming to the table with. And that might even be one of the things we want to talk about here, thinking about, you know, when you said modeling to me, I had this series of things that kind of came to my mind, one of them being uh word problem. And the next maybe thing that came to mind was maybe a larger sort of data collection and coming up with some sort of equation, like that, you know, that conversation started to go through my mind, and then maybe the next thing was some representation of mathematics. All of those things are things that came through my mind, but I think uh you've mentioned this before. What happens when our students hear that word modeling? That's a huge question. Do you have like that's a that's a big deal?

SPEAKER_00 3:19

It's a really big deal, and I love that you brought that up because I think too often we're focused. I know when I was in the classroom, when my focus when I was planning a lesson was more about what am I gonna do? Like how what are my moves? What are the questions I'm gonna ask? What are the images I'm gonna show to my students as I think through my instruction? And it really wasn't until maybe the last half of my teaching career. I taught for almost 20 years, and it was only the last 10, I would say, that I actually started to plan from the student perspective, right? What is the student gonna do? So a lot of time I spent as a classroom teacher, I wasn't thinking about when I say modeling, hey, we're gonna do a modeling task today, what are my students thinking that means? And I really wish I could go back and first of all apologize to some of my early career students, like, oh, sorry, I could have been so much better for you. But I would love to go back and just take a moment to like call that out with students and say, like, what do you what do you think about when I say this and really help get everyone directed so that they understand what the expectation is. And I'm I'm thinking how that connects back to what you said around the language for math practice four, where modeling in math practice four is is pulling out this the math that students know. And I think for teachers who haven't done this regularly in their practice, they're probably gonna find that their students like struggle with that, right? The kids don't necessarily know how to do that automatically if it's not something that's been part of our regular classroom practice. So taking some things that help establish a culture where students are, oh, wait, this isn't about trying to come up with the answer the teacher wants or the that the answer that's in the back of the book. This is about how do I make sense of this of this contextual problem and what math do I know that I can bring to help solve it? So intentionality, just taking the time to discuss that and be sure that kids are on board with what it is you're trying to do, I think goes a long way.

SPEAKER_01 5:20

Yeah, I think that's I think that's huge, getting the students to understand that what we really want them to do is take some time and and look at the situation and think about the mathematics that they've come to the table with. What are some creative ways that I can apply the math that I know to this situation and and start to make sense of this in a way that that I can can grasp a hold of? And are there ways that I can put together pieces of my math experience that that maybe I haven't put them together in that way before? Or you know, try something new? I think that was one of my biggest things when I started teaching statistics was just getting my students to get out of of the box of here's this prescribed um sort of method you approach every one of these problems with. And instead to look at the scenario, think about what's available to me, what are the tools I have in my toolbox, if you will, and and kind of figure out how to put them together and and make sense of the situation or the problem and and think about it as more than just um teaching a math concept in context, or in their case, solving a math problem in context, and instead like figuring out how to represent and make sense of a math problem or a math situation, um, in a in a real world scenario that that has a real meaning and a real outcome. Um I think that's I think that's probably the the biggest things that that as I started teaching statistics, as I started getting into that, that's the world that that I was forced into. And that was really a good thing for me in my math teaching to kind of be able to start thinking uh in that way. I think that's huge. Yeah. So getting students to do that, you know, when I mention modeling to my students, you know, they could think of things from a a model of a clothing, a fashion model or something, um, to the little plastic toys that they build up um or even plastic blocks that they link together to make some sort of a figure. So they might think of all kinds of different things when they think of this idea of modeling. And we want to be really specific that we're talking about more than that.

SPEAKER_00 7:37

Right. So for for me, one of the non-negotiables around a modeling task, and again, I I I want to emphasize too, we've said we've we've made some reference to math practice for, in the standards of uh, or in the common core state standards rather, in the high school version, there's a whole modeling conceptual category. And that's that involves the modeling cycle, and that's something a little bit different than what we're talking about, too. We're talking about a modeling task or a modeling experience that students can engage in all throughout. And it it's it's less formalized than what's in that um in the modeling cycle, which we should talk about in a future podcast. And I think we have that in our plans. But for me, one of the non-negotiables around a modeling task to bring out the ideas of math practice for is that it needs to be in context. It needs to be something that is connected to something in the real world. It's not just an abstract math experience, it's not just manipulating, you know, mathematical symbols or working within a single representation, but it's something that students could have potentially have some experience with or that could be described so that they could picture it in their minds. So anchored in context is one of the non-negotiable qualities for me of a good modeling task.

SPEAKER_01 9:00

Yeah, I agree with that uh non-negotiable that's anchored in context. I also think that um when when I'm choosing a modeling task, when I'm thinking about a modeling task that I'm going to um going to choose, I I think it needs to be rich enough that there's got to be a multiple approaches. Like that I need to be able to have this open enough sense that you know perhaps we all start off with the same space, but each of my students can approach it from a slightly different perspective, and there's not this obvious uh sort of uh prescribed set of steps that my students go through to come up with their own solution. And this kind of brings me to think about some of the projects that we've worked on at TI. I've had an opportunity to do some really fun things with um Gail Burrell um from Michigan State University and Tom Dick from Oregon State University. We've worked on a project with mathematical modeling, and one of the key things that we wanted to do with that set of projects was make it absolutely non-prescriptive, um, as far from prescriptive as possible. We present a scenario to the students and to the teacher, but then just let them go and come up with your own sort of graphical model, your own sort of simulation, if you will, or perhaps a a way of rating things. Like all of those pieces are things that the students or the teachers come up with on their own rather than us prescribing, okay, you should do a graph that looks like this. Right. Or you should come up with an equation that looks like this. So I I agree with you. I think you know, making sure that it's in real-world context or at least um applicable context, and then I think also a level of openness that allows for multiple approaches, I think is another piece to that puzzle.

SPEAKER_00 11:01

That's so important. And yeah, just a just an extra shout out to you and Gail and Tom for those uh those activities are awesome. In fact, the uh greatest of all time activity with the quarterback rankings, we had some great conversation with my family while we were watching the Super Bowl on that one, so that was a really fun one. I appreciated that.

SPEAKER_01 11:21

Fantastic. So now we're talking about adding something that really is a large time investment. Because if I'm gonna allow my students to take time to consider the math that they know and come up with a creative solution and think about that, like what do I do to find time to do modeling well with my students? How do I do that?

SPEAKER_00 11:51

Yeah, it's a it's a great question, and I think even more relevant now, you know, we're we're recording recording this as we're maybe hopefully getting close to the end of the COVID era, but you know, time has always been and will always be the most precious commodity in a classroom for sure. And really, I think that it's a little bit of a mind shift, but it's also a recognition that that's a very real constraint that we have to operate under as teachers. So yeah, we make decisions every day as teachers about what we're gonna take the time to, you know, cover, what we're gonna take the time to have students engage with, and what we're simply not. And so I think it's partly it's just that, right? And I think part two of the mind shift, I know when I was working with teachers and the and the new standards came out and this became a thing that we tried to pay attention to, it was first of all, like don't try to do this every day. Don't even try to do this every week. You know, pick like three big mathematical ideas from your curriculum, whether it's a course or um a grade level, and then find a good modeling task that aligns with that really big idea. So the idea is that you're you're taking the time when it's the most valuable. And it's not just like, oh, this is something I have to do, you know, oh, we're gonna have modeling Mondays in my class. It's not about that. It's about how can we use these modeling experiences for students to actually develop a deeper level of understanding of that important mathematics. So I think one strategy I would say in terms of time is be really selective about what the topics are you're actually gonna engage in a modeling task with. And then I think there's a very real thing about just shifting your expectation. Like I'm gonna choose to spend the time on this, and it might mean that I have less time for a direct instruction lesson. But Kurt, I know for me when I did these kinds of problems that ensured that students were accessing their own understanding, the learning lasted longer. When I would feel that pressure of time and say, okay, I would love for them to do this exploratory activity on this topic, but gosh, there's only, you know, eight minutes left in class, so I'm gonna lecture instead. I would almost always would find the next day they didn't remember a word of what I told them from lecture anyway. So it's sort of that take the time now and it will save you time down the road because it actually shifts the learning experience and you know what the students get out of the activity and what they remember long term. So it's it's hard to put that mindset into practice, but every time I did, I never regretted it. It always worked out better in the end. How about you? What would you suggest for finding the time to do these kinds of things?

SPEAKER_01 14:40

So I love that you brought up that um we have this perceived, I'm giving up time to make time for this um kind of piece. And I know that I felt that that too until I started trying to play with some of these ideas. And I'm thinking of an example while you were talking there. I was thinking about this example that came to mind. Um, and it and it's it's loosely a modeling task. We didn't, I didn't have the students do it uh themselves directly, but one of the greatest ways I always played with my students trying to get uh them to understand this idea of uh p-value and significance in um in an in a statistics course, and and in particular, I tried to get them to get to this idea of what is a what's the significance level and why do we reject the null hypothesis, uh, if you will, whenever um this this probability um decreases below a certain value. Like how do I why did we choose 5% as this rejection level, as this alpha level that we always refer to? And I would I would do this, and I borrowed this from Penny Smelzer. Penny, if you're listening, this is awesome. I want to give you uh props for this. But um, this goes around and lots of statistics teachers do this. But um I would have my students, I would get a deck of cards. Um, in fact, I'd get two decks of cards, and the night before we would do this activity, I would take a deck of cards, um, and I'd I would get my Xacto knife out, and I would open up this the plastic on the bottom, and I would take out the cards from both decks, and I would mix them such that I had one deck of all red and one deck of all black cards. Okay. And I would um then I would get really careful. I mean, I got tricky. I got down to like having double sticky-sided tape, and I mean, I was really careful to refold that that plastic in there. And so then when I would come into class the next day and we would talk about um, you know, I would make a big deal with them that, you know, like whoever gets, you know, the next uh black card from this deck is gonna get this great, and I brought in a giant candy bar or something like that, and I would have that sitting up on it. And so whoever could draw, and so there was a little bit of skin in the game for them, right? And so then we would shuffle the deck, I'd make a big deal out of opening it up, you know, brand new deck, and shuffle it, and then I would draw cards and I would have them draw cards, and we would keep track of the number of cards that came out and the and the probability of such uh an occurrence, right? So each time they would draw, okay, you know, this is the first time they'd get a red card, okay, you know, one half. And then the next one, oh, they got a red card. Oh, that's one half times one half, one fourth. And then we'd keep this up. And invariably, at the end of this this experiment, um, at about five red cards in a row, between four and five red cards in a row, somebody in the room says, Wait a minute, wait a second.

SPEAKER_00 17:38

That's a stacked deck. I'm calling you out.

SPEAKER_01 17:42

Yeah, they would. And so this experience, right? And and that absolutely corresponds to that that whole idea of between six and three percent, right? And so this this like we just have a human innateness that like one out of twenty seems unusual to us whenever we actually experience it. And um, so then that you know that really helped drive home this idea of you know what is a significance level and why do we do this rejection at this particular um level. And so that that really helped us get.

SPEAKER_00 18:18

I just wanted to say I I love that example so much because what that connects to that, you know, we said before when we were talking about what modeling is exactly, it is about tapping into their own common sense and helping them understand, like, no, you like you have so much mathematical skill just within your brain. This isn't about what have you learned in this class, right? Or what procedure have you have you mastered as a math student. This is about what makes sense to you in terms of you know, your own brain and how you interact with the world. And I think that's so important.

SPEAKER_01 18:52

I bring up the card thing because um you were mentioning this whole idea of of it taking time and choosing to take time to bring that up. So I brought that up because I had students that remembered that demonstration years after the fact and and would comment on this this idea that man, I I don't remember lots of things, but I remember why we rejected a null hypothesis. I remember why this idea of five percent made sense to me, and it was that experience, right? That that real tangible uh experience of developing a real sense, and it was revealing even for me to feel that the first time of like, ah, that's a stacked deck, and and to be able to tie that directly to that probability was super super huge, and so that was helpful for me. So I got one more question for us to maybe talk about. And I think I'll I I would like to to let you get started on this question for sure. Um and that is uh so we've we've talked about we've talked about first of all context, we've talked about making sure that these things are open, we've talked about that that really we want to do questions that where where students um are addressing things that they know and um specifically we we want to be kind of thinking about um targeting standards that are close to the standards that we're supposed to be teaching at this time and making sure that that happens. But you know, I'll be frank, that feels like a lot to me. Um thinking back to my days of trying to plan for the classroom and trying to plan for what my next day was going to be, uh oftentimes I was pretty overwhelmed at just choosing what we should be talking about, um, much less thinking all the way through some kind of a modeling task or modeling uh scenario. So how do we even get started? Like, where should we go? What should we do to try to get this thing started?

SPEAKER_00 21:04

I want to throw in this idea about in in terms of getting started, like maybe even it's thinking about the time of within the studying of a unit or a particular concept that you bring this task in, right? Like it, I I think I know, especially my early career, had this perception that like I have to give students all of the skills they need before I can give them this authentic, open-ended problem that they would use those skills to solve. And actually, there's some research that says the opposite works much more effectively, uh, and that is to pose the problem and create the need for the skills, right? And I think it also does a great job as being as serving as a formative assessment tool. If I started the unit with what I think of as a modeling task or an open ended contextual problem involving the math of the unit, it gives me a good sense about what it. Kids already know and and where what are they where where are they like not able to progress past on this problem? That helps me drive my instruction a little bit better as well. So I love this idea of throw it at the beginning of a unit. And again, it's a great way to send the message to students that every problem I give you, I don't expect a solution, right? Like it's not like here, take this problem, step, step, step, solution. I want, I want to think about how you're thinking about the mathematics. I want to see what you know and understand and how you reason about this. Or throwing in a task in the middle of the unit to sort of assess where, you know, what have you already covered that, gosh, actually nobody really gets that yet? Or hey, we're really ready to move on. So I'm not just thinking about how do I um how do I find tasks and what do I use, but when do I use it is another thing to think about when we're getting started.

SPEAKER_01 22:54

For sure. That brings to mind another thing that I used to do with my stats classes and thinking about starting with simulation, right? So instead of going in and and I know I'm talking specifically to the stats community here, but thinking about, you know, when I started um hypothesis testing and and confidence intervals and that whole inference space where we really jump in and we use these uh probability density functions and they're these these big um these big mathy things, and we're really actually applying some calculus, even though my students weren't doing calculus to get it done. We're applying these these results from calculus. And what I wanted to do um was flip everything on its head, and let's just get started by simulating and figuring out probabilities based on on you know simulation, either by uh some sort of by hand sort of simulation, which I love to start units with, um, similar to the to what we did with the cards and coming up with p-value, but also then looking towards some sort of a technology simulation to help us kind of do larger samples and do those kinds of things. I think I feel like that sort of, hey, we don't really know the math part yet, but we can at least experience this to get us in the concept range, and then we can kind of begin to develop the mathematics uh around it, and I think that was helpful to get us started. So I love putting a modeling task at the beginning of these things. One thing we haven't talked about, and I would like to try to have us bring up is okay, so we might think about um choosing a modeling task and putting it at the beginning or putting it in the middle, or even putting it at the end where it sometimes might feel natural to us uh of a unit. What what do we need to be thinking about from the complexity of the task, maybe the complexity of the thinking involved in that? Like what should we be thinking about in that regard when we consider a modeling task?

SPEAKER_00 24:55

Yeah, I I love this question. And I think thinking just about how, again, students are gonna engage with this, right? So if we're giving uh your simulation idea is the perfect example. I know when I taught statistics, simulations were so amazingly powerful, but it took my students a little bit to really get what we were doing. Like, why am I pulling cards if this is a problem about drunk driving and reaction times? Like to make that connection between this thing we're doing to simulate the probabilities and the actual context of the problem. So there was a lot of cognitive complexity for them in connecting what we were doing to the scenario itself. So, or if I if you're you know, we're working with younger students and giving a scenario that they might not be have a lot of familiarity with, right? There's cognitive complexity in in working with the problem and what it is we're actually trying to do. And then there's the cognitive complexity of the mathematics that we're gonna do. So if this is, you know, an end-of-unit task and we're asking students to apply mathematics that they're they're not really solid on yet, like they've just learned this in the last few weeks. Um, that creates some cognitive complexity as well. And I really think setting ourselves and our students up for success in managing a balance between those two. So when there's a lot of cognitive complexity around the task itself, lower the cognitive complexity of the mathematical demands, make the math a little bit simpler so that their mental energy can go towards understanding the task, the situation, the problem, the scenario. On the other hand, when it's a really familiar scenario and they don't have to put a lot of cognitive energy into that, then upping the cognitive complexity of the math is a lot more reasonable. So I think it's important to manage both of those, both so that your students are set up for success and you're set up for success as a teacher. For me, that's like my general advice. When you get started with anything new that you haven't been doing, set yourself up for a quick win. Make it something that the kids enjoy, make it something that you feel good about. Don't try to tackle the hardest problem you can find or say you're gonna do this every week. Like that, you're just setting yourself up for struggle and failure, and then it's much harder to try again later. So start small, set yourself up for success.

SPEAKER_01 27:27

Joni, I echo those thoughts. Um, especially the start small and the do it not real often.

unknown 27:36

Yeah.

SPEAKER_01 27:36

I think that's one of the biggest, I think that's one of the biggest things. Anytime you're starting something new, especially in the classroom, you know, the first year even, um, it's so advisable to sit down and and consider, okay, I'm gonna do this once this semester. I'm gonna pick, I'm gonna pick one unit where I can insert a modeling task, and I'm gonna spend some time um looking for that modeling task and thinking about it, and and it's gonna be one thing that my students are really gonna love and jump into um and dive into, and that's the that's gonna be our big modeling task. And then, you know, as you as you do one, then you'll get ideas for ones that that um you could do again later, and you go, Oh, yeah, back when we had that three units ago, I could have done this. And you know what? You could do it now if you want to. But the idea would be just to start small and and don't try to do them too often. I agree with that wholeheartedly. Well, Joni, this is uh this has been a good like eye-opening conversation for me. Um, thinking about several things here. Uh, I really appreciate um your insight and the thoughts that you've had today. So um really enjoyed chatting with you about this, and I'm looking forward to our next conversation.

SPEAKER_00 28:52

Yeah, me too, Kurt. I always love our conversations, and I'm I'm looking forward to hearing uh thoughts back from our audience as well. So I hope that the folks listening, you know, take the time to check out resources in our show notes and tell us what they thought about this.

SPEAKER_01 29:06

That's awesome. Well, I've sure to enjoy it, and we're gonna continue growing.

SPEAKER_00 29:12

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'd 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.