Ep 33: Problem Strings vs Problem Talks

Pam Harris: 0:40

Hey, fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.

Kim Montague: 0:47

And I'm Kim.

Pam Harris: 0:48

And we're here to suggest that mathematizing is not about mimicking, or rote memorizing. But it's about thinking, reasoning about creating and using mental mathematical relationships. That mathematics class can be less like it has been for so many of us and more like mathematicians working together. We answer the question: if you're not teaching algorithms, then what?

Kim Montague: 1:12

So today, Pam, we're gonna dive into a hot topic out in the world.

Pam Harris: 1:17

Hot topic.

Kim Montague: 1:18

And one that you speak about quite a bit. And that is Problem Strings, and Problem or number Talks. So for this episode, I'm just going to lob some questions out to you and get your perspective on some key parts. Is that possible?

Pam Harris: 1:32

Absolutely. Yes, let's do it. Alright. So first of all, you call what we're going to talk about today, Problem Talks, where a lot of the world would call them number talks. Can you share a little bit about that? Yeah, sure. So number talks are great. I thoroughly respect the authors of the number talk books. I think they've done a lot of good work. I think, all of the blogs and the stuff on MTBos, and there's a lot of really good things out there, where people are talking about what they call number talks. And like I said, I really appreciate that work. So I just want to start with well done, great job, you have definitely added to our repertoire of mathematics teaching. However, I think it's a little short sighted to call them 'number talks' for a couple of reasons. I would, for a couple of reasons, I would prefer to call them 'Problem Talks'. I'm K-12. So I do work Pre K-12. I do work with really, teachers of really young students. Sometimes I will say, really young teachers, to me, teachers of really young students, to university people. We work together on teaching pre-service teachers, and then the gamut in between. So, you know, high school was where I started, and then I dove back in and sort of consider myself an expert now Pre K-12. So, because of that, when I learned about number talks to me, they're not talks about 'number'. They're talks about 'problems'. And those problems could be number oriented. They could be geometry oriented. They could be statistics oriented. Yeah, in fact, Jo Boaler's just put out a bunch of really nice statistic talks called 'data talks'. But I think we could have just called them Problem Talks, and their Problem Talks about statistical stuff like data. So if we have this more general, sort of, these are Problem Talks, then we can have specific Problem Talks about or we can have Problem Talks about specific topics in math, and still use the same sort of procedure, that they have the same sort of characteristics becomes the same instructional routine, this is a Problem Talk, oh, but we're gonna now specify that we're in these different sort of categories. There's another reason why I kind of push back against number talks, or the term number talks is because it kind of, in the same way that I pushed back a little bit against people calling them 'math talks', because then people say,"Oh, this is when we talk in math." As if that's the only time we talk in a math class. So not true, right? Like, I want kids talking all the time. It's just that there's some times where we're gonna have a particular instructional routine, and we have to call it something so I'm okay calling it a Problem Talk. Because we're gonna talk about problems. So therefore, it's a Problem Talk. And that's just it's a more general name that I think fits a little bit better. I don't have any problem calling them number talks, as long as we then differentiate the instructional routine from the other instructional routine Problems Strings, which we'll talk a little bit more about later.

Kim Montague: 4:36

Yeah, so you're gonna hear us talk today and call them Problem Talks. So share with us a little bit about your goals for a Problem Talk.

Pam Harris: 4:44

Yeah, totally cool. So again, I like Problem Talks. They're excellent, an excellent instructional routine. But I think they have different goals than other instructional routines. So I'm glad you asked me about goals right off the bat. So I think there's a couple short lived goals for Problem Talks, and then a much longer lived goal that lives and lives and lives. So the short lived goals. One of them is to sort of poke around, like, I kind of want to, it's formative assessment. I want to sort of feel where my students are, get a flavor for how they're thinking about something. So even before I start a unit on something, I might give them a Problem Talk, where I throw out a problem. It's a rich problem, you could use lots of strategies to solve it, and see what's happening in my classroom before I dive in with the unit, not knowing what they're already doing with the material. So if I throw out a Problem Talk, then I can get a really good sort of formative idea of what's happening out there. And then I can respond, my instruction can respond. But then I've done that, right? And so now it's time to move in and actually develop relationships. There are better things to use to then develop relationships. So it's good to kind of poke around, now move on to do other things. There's another short lived one, which is to prove to students that there's more than one way to solve the problem. Many of us grew up in circumstances where we were given the algorithm. It was one and only one way to solve problems. Maybe we did stuff in our heads, but it was never talked about or showed or whatever. So Problem Talks can be a great venue to prove to students and teachers, parents, everybody that there's more than one way to solve a problem that we could use relationships and connections to solve problems. But y'all that needs to be short lived, too. I fear that way too many people continue to have that purpose, that they're like,"No, I need to prove to these kids, there's more the way to solve it, look, see, see, this person did that way. And that kid did this way, that student did it this way. And yay, look how flexible we are." When in reality, the class isn't flexible at all. The first kid doesn't know the second student's strategy. And the second student doesn't know the third. And they didn't develop it right there just by seeing it. We have to do other more focused things to help students develop, most students develop those strategies. Every once in a while, Kim and I'll see somebody use a strategy and bam, we've got it, we own it. That's probably true for everybody that they can grab some things. It's right on their zone, or the edge of their zone of proximal development. So they grab it, they're ready to dive into it. Most students most of the time need more concerted, focused effort to help them develop the relationships necessary to learn a particular strategy. So it's not enough for us to just say,"How'd you do it? How did you do it? How'd you do it," continually over and over and over again and expect that students will pick up those strategies. So it's, Problem Talks can have two fine purposes

that are short lived: 7:29

one to poke around, and two that there's more than one way to solve the problem. But once you've done that, then we need to do other things. So what's the major strategy? The major strategy that we want to use Problem Talks for is to compare already constructed strategies for efficiency. Now, there's a lot to unpack there. We want to compare already constructed strategies. That means we've already had to construct it, not just a kid has one and a different kid has another. But we need a whole class to be messing around with a few strategies, or at least two strategies. Now, let's compare. Now I give you a rich Problem Talk, a rich problem where we can now discuss, ooh, that strategy or this strategy and why. That's the major, overarching, ongoing, forever living purpose, for a Problem Talk to compare those already created already developed strategies towards efficiency. Because now we want students to be able to choose strategies. So there's the main reason.

Kim Montague: 8:29

So you do think that they're valuable?

Pam Harris: 8:31

Absolutely. In fact, let me prove it to you. They're so valuable. I created MathStratChat. Like MathStratChat is a global Problem Talk, where once a week I throw out a problem and the entire world throws in their strategies. And you might be like, "Oh, Pam, I learned so much from MathStratChat." Yeah, absolutely. But I bet you would have learned more quicker if we'd been doing more focused work towards actually developing particular strategies. Yeah, for most students need a bit more focused work. But I love them, right? We do MathStratChat once a week, I put a lot of time and effort and love the outcomes. I love the response. We are getting response from all around the world. It's so fun to read the different ways that people are solving problems. So the global Problem Talk is MathStratChat. So of course, I think they're wonderful. I love them.

Kim Montague: 9:17

So let's talk about this other focused thing that you are talking about. And this is kind of what you're known for, right? You emphasize Problem Strings. Tell us a little bit about those.

Pam Harris: 9:29

So Problem Strings are also an instructional routine and the brilliance of instructional routines is that they can become routine. So that as the what I'm supposed to do, and how it looks, and what happens here, as that becomes routine, then students can not put their mental and emotional energy into knowing what to do. They can put their mental and emotional energy into playing with the numbers, and developing the strategies, and developing the relationships, and comparing their strategies with others, and all the kinds of things we want to happen in an instructional routine. So Problem Strings are this really, really important instructional routine that then fill the purpose that I just said Problem Talks don't quite do. Problem Talks don't help students necessarily develop the relationships that they need in order for strategies to become natural outcomes. Problem Strings are the vehicle to develop. They are the vehicle for construction, for having kids' brains literally change into more and more sophisticated thinkers. So what is a Problem String? A Problem String is a string or series of related problems that are purposefully designed and purposely sequenced in order to construct mental relationships in the learner's head so that strategies, models, and big ideas become natural outcomes.

Kim Montague: 10:51

So what is the purpose of a Problem String?

Pam Harris: 10:54

The purpose of a Problem String is to construct. It is to develop, it is to literally help students' brains get more and more sophisticated as they think. Which means that we use Problem Strings to help students construct relationships, which turn into strategies. So that's the place where we 'teach' strategies, we don't 'direct teach' strategies. It's not about 'I do, we do, you do." We use Problem Strings to help students learn those strategies and models and big ideas as well. But it's the huge place where we learn strategies. So let me tell you some things that are particular about Problem Strings. So like I said, it's a series of related problems. So if you just got one problem, that's not a Problem String. If you've got a bunch of problems, but it's kind of like a menu that you can choose from, that's also not a Problem String. Problem Strings are in particular order, the order matters, they are designed to help develop. So it's not about just picking and choosing between problems. It's not sort of a random, unrelated group of problems, or even related that you pick and choose from. But it's the sequence matters. So if the sequence matters, and we're helping students develop those relationships, then that's a Problem String. They're also focused. So a Problem Talk is much more, it's less focused. It's like, however you do it is a great. We're going to compare those strategies. In a Problem String, it's more focused. It's not laser focused. We're still letting students solve the problem any way they can. So when we give students a problem in a string, we don't say, "Use this strategy." We say, "Solve it any way you want to." But then we share the strategies that will help towards that focus. We know what the focus of the string is, as teachers if we're giving, if we're doing that Problem String. And so we craft the conversations around where we're trying to go. Now, the first time that you do a Problem String, you will be less focused. You'll entertain more strategies. And then you sort of, I kind of picture like helping students focus on the relationships that we want to focus on. Kind of like I'm picturing, like a spotlight kind of focuses by crafting that conversation to help them kind of narrow it down so that the conversation in the class is more about whatever it is you're trying to develop in that particular Problem String. So Problem String, systematically nudged towards more efficient and sophisticated strategies. It's not about leaving kids wallowing in their own caveman-like strategies. We want kids to start there, they have to start there, that's a necessary starting place. But then we help them get more sophisticated and efficient by using Problem Strings.

Kim Montague: 13:26

What do you see as the commonalities between those two routines?

Pam Harris: 13:31

Absolutely. So they're both instructional routines. That's important.

Kim Montague: 13:33

Yeah.

Pam Harris: 13:34

So they will take a little bit longer at the beginning to get kids used to. Once, Kim and I know that, but once kids, once it becomes routine, then you should be able to do them as a mini lesson. So they're meant to be mini lessons, not they shouldn't necessarily take up the whole class period. We think 10 to 20 minutes is ideal. Like I said, it'll take you longer the first time you do a Problem String, or a Problem Talk. And the first time you do one of its kind, of its type with the material that you're doing, it's going to take a little bit longer. But let's say I do a Problem String to develop, hey, last week we talked about the Doubling and Halving strategy, that with a given class, we might do a Problem String that's focused towards developing Doubling and Halving multiplication strategy and get a lot of kids sort of thinking about it. But then we got to come in again, up the ante, do a string, like it get more kids sort of thinking about it, and kids kind of solidifying the ideas. Do it again, get all kids kind of like at least playing with the ideas. And then we kind of bring all the way along, we sort of are bringing words into it. But we don't really generalize it until most students are really kind of playing with the relationships, and they're getting it down. Then we put words to it to help that continue to solidify as we then generalize the relationship that's happening. So as we go, as we do the second, third or fourth Problem String towards a big idea, model, a strategy, a particular one, then it becomes more like a 10 to 20 minute kind of thing that can happen in your class. It's a mini lesson. What else do they have in common? They also in a Problem Talk or Problem String, we believe that the teacher chooses who shares purposefully. Now it's not about calling on a teacher's pet. It's not about calling on the kid who has it right all the time. It's not about the kid who's fastest, not that either. It is more about what is the direction we're trying to go right now. So for example, in a Problem Talk, I'm going to choose two chairs purposely because I want to get a variety of strategies. So I don't necessarily want to choose students who've all done the same strategy and then put that on the board 14 times. That's not going to help move the mathematics forward. And we only have so much time, right? We have precious time. So in order to move the mathematics forward, we want to get several. Now I might have a student say, "Yeah, I just think different." And you know we're trying to get several different ones, and then not be really clear about the difference between models and strategies. So they might give us the same strategy on a different model. And that's a fine time for me to then clarify that. "Oh, so use the same relationships that so and so did. Well , we already have that one on the board. So let's get some more up here." Because I'm going to choose the specific model as well, so that I can help compare. If the teacher's thoughtful about the model they use to model the relationships, the strategies that kids are using, that becomes an easier task of comparing to the point where even sometimes we're purposeful, to choose to have kids share different models with the same strategy, because it's about being purposeful. Now that I have these different models with the same strategy. Now kids can actually really focus on the strategy. So it's about being purposeful. It's about what can I do here to help move the mathematics forward? I'm putting big asterisk in the air caution, caution, caution, with always an eye towards equity. So let's say today, we're going to focus the conversation towards moving a particular strategy forward. And I'm walking around the room and seeing what kids are doing or I'm asking specific questions to draw specific strategies. And I see several students using the thing that I want to move the math. I know it will help move the math forward, then I think equity within that group of kids. I'm thinking to myself,"Okay, who have I not call them for a while? Who do I need to position as a sense maker? Who do I need to help position as someone that is, that does have good ideas in this class?" So I'm always thinking towards equity. Now, I'm also at the same time thinking bigger than that, because I know just this set of students have a strategy that's going to move the math forward. So I've got to also be aware of equity of who didn't have that strategy. And I need to make sure I've pulled their voice in at some other, later time. Not maybe in that moment, because it's not going to move the math for that moment. But I absolutely, am thinking about equity all the time to make sure that that's happening as well. So because of that the two routines, Problem Strings and Problem Talks also have in common that the teacher is modeling student thinking with a model. At all times, we are trying to make student thinking visible, in order to construct the relationships in a Problem Strings, or compare the strategies in a Problem Talk. It's easier, we can facilitate that better when the relationships are visible when they're up in front of the class so that we can point to them, we can talk about them, we can pull them apart, we can compare. So we really, really like the idea of having, it's necessary to have visual models, whenever is possible, like open number lines, open arrays, ratio, tables, graphs, all those kinds of things will be really helpful. So teachers choosing models, and they're modeling student making with a model, a visual model when possible. And then the last thing that I'll mention that they have in common is that all strategies are not equal. So that unfortunately can kind of become a side effect of just doing number talks. No one ever intends that. But you might notice this, teachers, that if you've ever tried number talks, and you're like, "I don't know, I'm often getting the bang for my buck." It might have been, may not necessarily true, but it might have been that because you're just sort of letting all strategies fly, that you didn't really have a purpose towards your Problem Talk or even your Problem String that it kind of gave the feeling tone, but like all strategies are equal. Everything is sort of game, I shouldn't say it that way because everything is game, but that we want the feeling, tone in the class to be, "Oh, cool. Nicely done. You did that. Now, can we get even more efficient? Can we get to be more clever? Can we use those relationships and even more sophisticated way? It's not that you're not good enough. It's that oh, we're always striving for the cool, clever strategy, that that's kind of the atmosphere in this class, ooh, I want my brain to do that next time, real growth mindset, kind of an idea." And so one of the reasons that I do this comparison is to sort of help teachers realize these different goals and that we need to be cognizant of the reason we're doing the routines, so that we can get the bang for the buck. Because I'm a little bit concerned that if the only thing teachers do is they hear Jo Boaler, or somebody else talk about number talks, and they're like, "Oh, I'm going to go do it." And so they do it with a purpose of there's more than one way to solve a problem. They're not going to get the bang for their buck that they would if they do both Problem Strings and Problem Talks. Okay, I've gone on for quite a while. Kim, what are their questions? Yeah.

Kim Montague: 20:02

Last one. So you're talking about these two routines and time is precious. You mentioned that. So what is the interplay between the two? If you had to do one or the other more? Where would you suggest?

Pam Harris: 20:14

So you're hearing me clearly that I like both of them, right? They're both really, really good. However, I think that Problem Strings give us more bang for our buck as far as construction, as far as actually helping the relationships happen. So I think we need to do them more often. Like 80% of the time that we're going to do these two routines, I think we should be doing Problem Strings. And then 20% of the time, we can do Problem Talks. So in 80% we're structuring strategy, models and big ideas. And then in the 20% of the time we're comparing those strategies towards efficiency. Does that make sense? Like we can't compare already constructed strategies if we haven't already constructed them. So Problem Strings construct. 80% of the time we're doing these two, you're doing Problem Strings. And then once you've got a couple of constructed, then we do a Problem Talk to sort of compare those already constructed strategies. Back to Problem Strings, construct, construct, construct. Then get through another Problem Talk to compare. Back to Problem String, construct, construct construct. Then we throw in a Problem Talk to compare.

Kim Montague: 21:14

That was a lot to digest. So if you've heard of number talks, this may be a little bit of a different perspective. And if it resonates with you, and you want to remember some of the things that Pam said, we have a download for you. It's a one page doc that outlines some of the things that Pam talked about today.

Pam Harris: 21:33

Yeah, so if you'd like that download, you can go to mathisFigure-Out-Able.com/strings V talks. Math is readable.com/stringsVtalks, download that handout, the handy dandy handout, or you're welcome to check out the show notes.

Kim Montague: 21:49

Perfect. So remember to join us on MathStratChat on Facebook, Twitter, or Instagram on Wednesday evenings, where we explore problems with the world.

Pam Harris: 21:57

Please share the podcast with your friends and colleagues. Give us a rating and review so that more people can find the podcast so that more people can learn that Math is Figure-Out-Able. So if you're interested to learn more math and you want to help yourself and your students develop as mathematicians, then don't miss the Math is Figure-Out-Able podcast because Math is Figure-Out-Able.