Pam Harris  00:01

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

Kim Montague  00:08

And I'm Kim. 

Pam Harris  00:09

And we make the case that mathematizing is not about mimicking steps, or rote memorizing facts. But it's about thinking and reasoning, about creating and using mental relationships. We take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keep students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what?

Kim Montague  00:39

So this is going to be really good episode, y'all. Last week, we talked in detail about what Problem Strings are and are not. We kind of defined them a little bit. We thought this week, we would share what we have over the years called some do's and don'ts of Problem String facilitation. People ask all the time, how they should make Problem Strings go smoothly, and what could go well, for both them and their students. So today, we're going to focus on some tips and things to think about whether you're starting Problem Strings, or you've done some for a while. Absolutely. So newbies, and veterans, we think we have some suggestions about ways that you can facilitate Problem Strings to get the most bang for your buck. So here we go. Let's start with just repeating the definition of sorts about what a Problem String is. And then we're going to kind of parse through some nuances. So a Problem String is a purposefully designed sequence of related problems, that helps students mentally construct mathematical relationships, and nudges students towards a major efficient strategy, model, or big idea. Alright, so with that in mind, some things about Problem Strings are they are meant to be mini lessons, about 10 to 15 minutes long. Now, Kim - Short, Pam, short!

Pam Harris  02:00

Kim's going to give me a hard time because I have a harder time making them short. But there is a reason for that, right?

Kim Montague  02:06

 Yeah.

Pam Harris  02:06

Like, in the classroom, when you know you have students over a period of time, we want them to be short, snappy, to the point, get it done. Because we know we get students back tomorrow. And we know we can build on those tomorrow. When I'm teaching my university classes, I do a better job of that. When I do workshops with teachers, and I only have them for that day. Or for heaven's sakes lately, that hour in a virtual workshop, then I try to get more done. Because I feel like I only have then for right then and so I tried to build it too much probably in that moment. So -

Kim Montague  02:38

You also have some teacher talk embedded, I will give you that. 

Pam Harris  02:42

Oh, yeah, that's true, too. Yeah, whether you want me to or not. Because it's hard for me not to because that's who the students are in front of you, you're teachers. There you go. Alright, so another thing, the problems are related problems, they're not unrelated. So sometimes we'll see things that people will call Problem Strings and they're really just sort of a list of problems that aren't really related. They're just kind of the problems that you can all solve using the same algorithm or something. That's not a Problem String. Alright, what else Kim?

Kim Montague  03:13

So Problem Strings are one at a time, right? They're not to be given to students all at once. I am going to admit that I have seen several times where it's a fine Problem String, a list of related problems, that could be a lovely string. But they have been typed out on a slide and thrown up as like morning work, or handed out on a piece of paper. And -

Pam Harris  03:39

Like a warm up? Like here's your bell ringer, like all at once. Uh huh. 

Kim Montague  03:42

And really, because the idea is to generate conversation, it's not about doing the same thing every time, then throwing all the problems out at once is not going to convince somebody to try something new or to open their mind to the strategy that's happening. They're just gonna do the same thing every time.

Pam Harris  04:00

Yep. And many of the Problem Strings that I've been writing lately, you won't even know what to do based on just handing you the problem. There's often other things to do, like graph this function, or talk to me about the end behavior, or how many sides are parallel. Like you can't even put it down on paper sufficiently that the students would even know what to do without those prompts. Yeah, so it's not a worksheet, not on a slide. Alright, so another difference is, the teacher chooses who shares. 

Kim Montague  04:33

Yes. 

Pam Harris  04:33

So it's not about everybody sharing. A teacher is going to circulate and look at what students are doing and then say, "Ooh, to move the math forward, I need this strategy shared next." It's not about choosing the student who has not necessarily the best strategy or the most clear explanation. Actually, sometimes we'll choose the student who's just on the cusp of making sense of things, that they have the idea kind, it's sort of there, it's a little muddy. And we ask that student to share, because that is going to give us the opportunity to have other students join in and help clarify and clear up and more learning can occur. So it's not about having the most clear explanation shared, it's about what will move the math forward here. Sometimes it's even having a student just share the beginning of an idea, and they get us started and then students could sort of jump in and add on. And so we're always looking with an eye toward equity as we choose who shares but it's absolutely not a free for all. It's a thoughtful sharing.

Kim Montague  05:33

Well, and even as you choose who shares, you choose who shares in what order. I've seen you say, "You're number one. You're number two." Because as you're circulating, you are telling kids in what order you want them to share, right?

Pam Harris  05:48

Yeah, in a really good facilitation, you have thought through the kinds of things that you expect students to do. And you're thinking about sharing them in an order that will help everybody have access and help move that math forward. Help create more and more connections. 

Kim Montague  06:04

Yep. Okay. So also, in a Problem String, the teacher models the student thinking. We're very clear that we believe that kids can do more than they say, and they can absolutely do more in their heads than they can record. So we want them to focus on relationships, not necessarily trying to record things so perfectly that it can be thrown up on the board. There are absolutely times where we have kids come up to the board and record their thinking, but not in a Problem String, the teacher is purposely choosing the model that can best represent the ideas for the students. 

Pam Harris  06:39

Yeah, and you and I have talked at length before we do a Problem String about what the final board will look like. We plan what that final display will look like, because that look on the board, what feels very organic to students, that final look can really help create relationships and help students make connections and see patterns based on how we put it up there. And so we don't have anything against students modeling their own thinking, but they don't know what that final look is supposed to be. And so they can't, "Hey, come to the board, no that's wrong." It sort of won't work because they don't know what the final plan is. And so we want it to feel very organic to students. "How did you do that? Ooh, can I model your thinking?" But I have a final goal in mind. And so I'm going to put it up on the board in such a way that helps students make connections easier and find those patterns. Yeah. So another difference is that as students are explaining their thinking, and the teacher is modeling that thinking, the teacher's job is to draw out relationships, it's to ask questions to help students make those connections, and to listen for those glimmers and draw out those connections or relationships.

Kim Montague  07:53

However, it is not a time for a teacher to directly teach the strategy. Right? If they're drawing out relationships that the kids are sharing, there's not an opportunity or a reason for a student to directly teach, and then use those problems to practice. And I think you actually have a story about that.

Pam Harris  08:14

Oh, it was so painful. So we were working with teachers early on, and it was probably - I'll take all the blame. I probably wasn't very clear about what Problem Strings are and facilitation. I was still learning, you know, I was still doing them and learning about them and watching videos about them and getting better at them. But, at one point, I went into a teacher's classroom, I was observing teachers and kind of giving feedback and everything and the teacher was all proud and excited. And  said, "Okay, we're going to do a Problem String. First problem. Here's how you do it. Step one, step two, step three, step four. Okay, everybody got it? Now, use the rest of the problems in the string and go practice what I just taught you." I was in the back of the room going, "What? No!" That's an example of not a Problem String. It's not about just like using the first problem to directly teach the strategy and the rest of the problems are practice. 

Kim Montague  09:04

Nope. 

Pam Harris  09:04

It is discussion between each problem, modeling, making thinking visible in between each problem. Yep. Cool. So Kim, you have a story about something else a Problem String is not. It's also not students using all using the same strategy.

Kim Montague  09:22

Right. So we, Pam and I, were in a classroom once and we were excited because there was some really good thinking going on. And Pam and I walked in the room. We might have already told the story, but I'm going to tell it again. We walked in the room, and I went one direction and Pam went the other direction. And -

Pam Harris  09:39

Oh my gosh, I was so excited. I was like, I'm thrilled.

Kim Montague  09:42

Pam is across the room staring at me kind of giving me thumbs up and I'm shaking my head at her like kind of -

Pam Harris  09:48

I mean, I'm like high fiving I'm almost woo-hoo! I'm like ready to 'whoop' out loud. I'm so excited and you're like, "No." And I'm like, "What?" And you're like, "No."

Kim Montague  09:56

She said, "What's going on?" So what we were seeing was that every single student for every single problem was using the exact same targeted strategy. And let's be real, we've been in classrooms, there's no way that every student is using the same strategy for every single problem. Because the problems were good. These were rich problems, where lots of strategies can be used.

Pam Harris  10:18

And what was funny was I didn't recognize that. So all I saw was that we had some good strategies happening. And I wasn't really like noticing that the problems were rich, and that the students, if we really said solve the problem any way you want, some students should have been having some pings spark to solve the problem in different ways using different relationships, and they weren't. And so they were all too lockstep.

Kim Montague  10:41

Yeah.

Pam Harris  10:42

It's not about being lockstep. It's not about forcing strategies for every problem. The whole idea of numeracy and strategies and mathematizing is that you let the numbers and the structure influence the strategy that you choose. That's mathematizing not just lockstep doing the same thing every time. So it's also not about taking the time to put up a student's crazy strategy on the board, or as I like to joke around a crazy student's strategy. But what do I mean by that? Of course, we're going to investigate and explore students' strategies. Of course, we're going to take things that students do, and we're going to look into them and like, will this work every time? And that we really want to honor students' thinking. But probably not during a Problem String, if you don't recognize what's happening quick enough. What do I mean by that? I mean, as you're circulating, and a student's doing something, and you're like, I'm not really clear on that. Well, you can take a little bit of time to see if you can figure out what the student's doing, to see if you can generalize what will work every time. But we don't want you to get bogged down in that moment, in that moment meaning: during a Problem String. We absolutely want you to dive into it and figure out what's going on. But we need to keep Problem Strings snappy. Problem Strings are instructional routines. Like we said earlier, they're short-ish, right? We don't want kids to get bored. We also don't want to get super sidetracked during a Problem String. Now, we might take that strategy, that crazy student, I mean, that student's crazy strategy, and say, "Hey, this is really interesting. I'm gonna look at it later." Ask your colleagues, phone a friend, phone me and Kim, let us know like what you're thinking about. And work it out. You might take some time later, when you have planned some time where you can kind of go down that rabbit hole and we're you've had a chance to sort of figure out what's happening, because then you don't derail the learning that was going to happen. So it's not about dissing student thinking. We absolutely want you to go there, just not during a problem string, if you can't sort of figure it out quick enough. Because it's about the sequence, the sharing, about making the thinking visible by modeling and representing students' thinking. That's what creates the connections and mental relationships, causing disequilibrium to give students a chance to grapple with what's happening. We want to keep that package tight, so that kids know this is what happens during a Problem String.

Kim Montague  13:00

Right. And so people ask us about writing problem strings. And I gotta tell you, Pam, and I really believe strongly that even before you need to worry about being a string writer, becoming a master facilitator of strings is crucial. We used pre written strings for a very long time before we made the attempt to write strings, because they are so carefully crafted.

Pam Harris  13:22

Yeah, absolutely. So if you want to write Problem Strings, go for it! Like we will support you the best we can. However, we think that that will be an easier thing to do, a more...Not even just easier, but a thing that you can do at all, if you sort of get a feel for what Problem Strings are. What does it mean to facilitate a problem string? What are the different structures of Problem Strings? What are the different types? What are the different goals? Like all the things you can get better at as you facilitate pre written strings, and then you'll have a better sense of writing strings. We will talk more a little bit about writing strings a bit later in the podcast. But for now, we'd really encourage you grab some really well written Problem Strings and give them a go. Like, grab your neighbor, grab your spouse, grab your kids, grab like whoever you can, your class, your colleagues, facilitate Problem Strings. Get used to those Problem Strings. What does it mean to have that sequence? What's embedded and what are the relationships happening? How do you make that thinking visible? Then you'll be more ready to sort of write Problem Strings. Yeah?

Kim Montague  14:23

Yep. 

Pam Harris  14:24

So if you want to learn more mathematics and refine your math teaching, so that you and your students are mathematizing more and more, then join us and the Math is Figure-Out-Able Movement and help us spread the word that Math is Figure-Out-Able!