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 keeps 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 we have to fess up a little bit this week. And admit that recently, we realized that even though we've been doing Problem Strings of sorts.

Pam Harris  00:48

Lots of them.

Kim Montague  00:49

Yea, with listeners on the podcast. We've never actually taken the time to really define Problem Strings and talk a little more deeply about what they are and what they are not.

Pam Harris  01:00

Yeah. So it was actually kind of funny, right? Because there was something we were going to do. And I said, "Well, you know, like, we defined Problem Strings in the podcast, and you were like, "I don't know that we've ever done that." So we totally went back and looked and Oh, hmm. Maybe we haven't. Oops. Alright, so let's do that today. So here is a definition of sorts. In fact, Kim, I'll never forget the day we were doing a presentation together. And I had "definition of a problem string.' And you were like, "Of sorts". And I was like, "Of sorts? What does that even mean?" You're like, "Pam, really? That's 'the' definition?" Yeah, so a lot of what we're gonna talk about today is a little, we're still kind of creating a lot from our experience, a lot of what we know, but here is a definition of sorts, because it's kind of an art. Problem Strings are kind of an art. 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.

Kim Montague  02:08

Yeah. And like you said, we're kind of still working out how we know what a Problem String is, and what we know about them.

Pam Harris  02:17

Yeah, interesting. We've run across some examples of sequences of problems or lists of problems. 

Kim Montague  02:25

Yeah. 

Pam Harris  02:25

And people have said, "Hey, is that a Problem String?" Or "Here's the Problem Strings." And the two of us were like, "Nu-huh." And then when we've pressed each other on that line. Or we've seen another one, and we're like, "Ooh, nice. That's a great Problem String." And we've pressed each other on that like, "Yeah, how do you know? Why did we both initially go hmm, no, not that." That one is, and that one isn't? What makes a list of problems, what makes it a Problem String or not? Kim, a story that comes to mind is, so there is a curriculum out there called MVP, Mathematics Vision Project that now has a new name. Oh, new name. I should have looked that up. We will put in the show notes with a new name. A-a-a Nope. I thought it would come to me. Maybe I'll look it up in a minute. So it has a new name, but it was called the Mathematics Vision Project. It's a high school, curriculum. And I'm familiar with it. I've worked with teachers with it. I've looked at it. I know the authors of it. It's some good stuff. And I ran into some of the authors at a national conference not too long ago. As I ran into them, I said to them, "Hey, are you familiar? Are you aware that that you've written these, what they call investigations, that some of them are far better, they work better if you would facilitate them as a Problem String, not as an investigation." In other words, not as a rich task. So like, when you throw out a rich task, you kind of throw it out and students work on it. They work together in groups, it's longer, it's a rich task, versus a Problem String is a series of things that you're going to do. And you're going to talk in between and you're going to represent strategies in between each of the problems. And I said, "Several of the what you're calling 'investigations' are far better, they work better in class, if you facilitate them as a Problem String." Well, what was funny was they all laughed. And I was like, "Wait, what?" And that's when they started kidding. They were ribbing each other a little bit because they're like, "Oh, yeah, that's how Travis facilitates them." So Travis is one of their master facilitators. And they were recognizing that when he facilitates those investigations that he parses out which one is more like a rich task, and which one is more like a Problem String. If you can see my hands right now I'm like, waving in the air and like putting some of them over on this side, and some of them over on the other side. And that if you facilitate them differently, then they go better. And so they were aware that there's a difference between how you might facilitate some things. 

Kim Montague  04:50

Yeah, so I guess an important question that we want to tackle today is how do you know if you're looking at a Problem String? Right. So -

Pam Harris  04:59

Let me interrupt real quick, because also we looked at, I think we've mentioned on the podcast that you and I were writers in Bridges in Mathematics. And we wrote grades three through five. And when we looked at a lot of what was written in the kindergarten, first and second grade Bridges - we didn't have a part of that - we recognized, right, we were like, "Oh, there's a Problem String." Even though they didn't label them as Problem Strings, we recognize that there were series of purposely planned problems that facilitated like a Problem String would work really well as a Problem String. So yeah, how do you know, how do you recognize if you're looking at a Problem String?

Kim Montague  05:38

Well, if it's just a sequence of harder and more complicated problems, then that could be like what you find in any textbook, right? You could just open up the book, and you could look at one through 29 odd and it's likely that one through five is going to be kind of your less complicated problems. And then as you go through the series, they add on and, so the question really is what makes Problem Strings intriguing and strategy building?

Pam Harris  06:06

Not just a series of more complicated problems? Yeah, that's a really good question. So you and I have talked about this just recently, like, how do we parse out what makes a series of problems a good Problem String, and we think we have two ways. So today, today, we're talking about two ways. One is the 'why and the purpose'. And two is the 'how you facilitate it'. And I want to break those down a little bit. So let's back up. So the one way that we can recognize if just a list of problems is a Problem String, or if it's just a list of problems is the why and the purpose. So we would suggest that the 'why and the purpose' of a Problem String is development. And often it's developing strategy. Though it might be developing a model or a big idea, or a model and a big idea. But it's about development, it's about helping students get more sophisticated in their reasoning, it's about helping them understand things more deeply and clearly. It's not just about getting answers. And so all too often a sequence of one through 29 odd is about, "Well, if you can do these easy problems. Now we make it slightly harder. Can you do these medium problems? And then they get harder? Can you do these harder?" But "do these problems" means "can you get answers to these kinds of problems"? We're not about just getting answers to kinds of problems. We're about building mental relationships and connections literally making brains be able to think more sophisticatedly so that you can answer those kinds of problems. And that's a big difference that we talked about, somewhere in the podcast about answer getting versus really building brains. And that's that big why. So the why of a Problem String is about developing. We want to help develop brains to be more sophisticated thinkers. Number two, the how. How you facilitate that series or that list of problems. Well, if it's just a textbook series, the how is you know, "Here, guys, I've given you an example problem, we've worked through the steps, maybe I've done a second one. We've worked through as a worked example. Now go. You go do that series of problems."

Kim Montague  08:26

Off on your own.

Pam Harris  08:27

Off on your own. Good luck. And then you come back and we give you maybe feedback tomorrow on your homework. Versus a Problem String is much more about the sequence in the sharing, and the modeling, making thinking visible. That all creates connections and mental relationships. So the sequence is very important. But it's not just about getting harder. In fact, sequences in a Problem String often goes back and forth. You might have a helper problem to help you do a clunker, and a helper problem to help you to do a clunker. Or you might have a building string, but the increasing complexity isn't just about getting harder, it might might go back and forth between different relationships, not just harder relationships. It also might be where I have a problem, and then an equivalent problem and then another equivalent problem. And then another equivalent problem. We call that structure sort of the equivalent structure, where it's all about the fact that you're getting the same answer. And so how are those problems related? It's about developing the relationship between those problems, not just solving problems, not just about getting answers to increasingly harder problems. But also then it's about the sharing. So in between each problem, we're sharing strategies, we're sharing the thinking, and then we're making that thinking visible, we're using modeling and models to represent that thinking so that that thinking can be shared, it can be discussed, it can be compared. That's hugely different than just sitting down and cranking out a bunch of answers to increasingly sophisticated problems. Also in that "how", how you facilitate the problem, the teacher is intentionally causing disequilibrium. Because we intentionally want students to have to rethink and grapple with relationships we want the students to have a chance to grapple, to be off balance, because as they work out that confusion, that's learning occurring. I hope that was just a snap, I have no idea how a snap will come across on a podcast. Yeah, but that's learning occurring as students grapple with that disequilibrium, and how to make sense of things. In sort of Piaget language, they have to re-schematize, they have to figure out how to make their schema fit with what's happening. That's learning occurring. And that's just different than the sort of other just lists of problems that get increasingly more difficult. So, Kim, in the podcast, we have done some Problem Strings. Let's use an example of maybe one or two of those to kind of walk through what we mean. 

Kim Montague  11:03

We've done several, right. 

Pam Harris  11:05

Yeah, several.

Kim Montague  11:07

So let's, let's talk about one that we've done that I think is like easy accessibility. So we have done a Problem String, where we gave problems. And it was like a something plus 10. And then the next problem would be something plus nine. And then maybe the next problem was something plus 20. And then the next problem was something plus 19. All in the attempt to get this idea of like a little bit too much. And then a little bit over, right, a little bit too much over and then back up. 

Pam Harris  11:41

And then adjust back. 

Kim Montague  11:42

Yeah, right. Mhmm. 

Pam Harris  11:43

Developing the Over Strategy. A number plus something that's really nice, too big. And then that same number plus just a little bit less, can you use that helper problem to help you develop this idea? It's not about getting answers to those problems. It's developing this strategy, these relationships that oh, there are these friendly numbers that I can add that are too big. And since I can do that easily, I'll just do that. And then I'll just adjust back to get the answer to this problem that's really related to that. So that goes well, we build those relationships well, when we discuss that in between as we do that sort of easy problem, and we model it, we make that visible. And then we say, "Alright, what about this next one?" And then when students describe their thinking, and they gain clarity as they put words to what they're doing. And we represent that. So they describe that. And then we say, "Could you use that same kind of thinking in a problem like this?" And then we give them the next sort of series of problems. And maybe we end that Problem String without a helper problem. And we ask them, "Could you create a helper using the same kind of relationships you've just been developing to answer this question?" It's about again, it's about there is a specific sequence. But there is sharing, there's modeling, making the thinking visible, we're comparing, we're verbalizing those relationships. We're potentially causing disequilibrium by making each of the paired problems a little bit different. And so they have to, it's not exactly using the same relationship because they have to sort of adapt a little bit. And then again, verbalizing and making visible and comparing and discussing and all of that is helping students develop their brains to be able to use that relationship.

Kim Montague  13:23

Right. 

Pam Harris  13:23

Yeah. 

Kim Montague  13:24

And really, the takeaway is the generalization that students can make. It is not about the celebration of the hardest last problem.

Pam Harris  13:32

Oh, that's well said. 

Kim Montague  13:33

Right. 

Pam Harris  13:33

Yeah. Yes. That is a big difference. Yeah. 

Kim Montague  13:35

Yeah. It's the strategy that the students leave with not a generalized procedure. It's not just about the problems one through 29. But which problems with which strategy, right? We want them to leave with with an overall strategy, not just here's my page of problems that I can solve today. 

Pam Harris  13:52

Yeah. And really important, it's a strategy that works for certain numbers. It's not like you said a generalized procedure. Okay. Now, guys do this for every problem, every single problem that you ever see, you're going to do these steps. No, it's about like, what did you build? What relationships did you build, so that now when you see new problems, what sparks for you, what pings for you? Because you've built those mental relationships now that those relationships can spark, they can ping, they can activate in your brain, you go, "Oh, yeah, I could use this strategy for these problems, because of these numbers, this structure. But I wouldn't use it for those numbers, or for that structure. Something else pings for me for those." It's not this generalized procedure I use every time but it is kind of a generalized set of relationships that I could then sort of use depending on the numbers in the structure that show up.

Kim Montague  14:44

Well, I'm going to jump in to say that I feel like sometimes we've given the impression maybe that you do one Problem String. And then the generalization is there, which is not what we would suggest at all.

Pam Harris  14:57

Especially the first time that you're trying to build those relationships. 

Kim Montague  15:01

Absolutely.

Pam Harris  15:01

One Problem String is probably not going to build those relationships for every student - well enough that then you're done. So we suggest you're going to have to do some more work. You do another Problem String to continue to do that. 

Kim Montague  15:13

Right.

Pam Harris  15:14

Yeah, absolutely. And also just to differentiate, it's not about showing students the strategy. It's about pulling out the glimmers that students have in their brains. It's about, like listening for that glimmer, and pulling out those relationships and saying back to them, "Is this what you mean?" And modeling it, making it visible. It's not about demonstrating, it's about modeling their thinking and helping connect that thinking. Taking those not fully baked ideas, maybe not even half baked ideas, try to put words to that tiny spark that's happening and putting that in a visual model in a way that students can see the relationships and helping build those relationships over time.

Kim Montague  15:59

Yeah. So if you're interested in checking out a problem string with students that Pam and I have done, you can check out those on the website at www.mathisFigureOutAble.com/ps for problem strings.

Pam Harris  16:16

And you can go and you can see some example Problem Strings, both just written out and also videos of us doing Problem Strings with real students. 

Kim Montague  16:24

Yeah.

Pam Harris  16:24

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