What exactly is a Problem String? In this episode Pam and Kim dig into that question and give examples of what a problem string is and is not.
If you want to hear more about Problem Strings, check out Episodes 71 and 72.
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Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam Harris.
And I'm Kim Montague.
And you found a place where math is not about memorizing and mimicking, where you're waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians do. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be.
When you say that, sometimes I say it along in my head because there's certain phrases that I have down, but not the whole thing. One day I'll get it all.
You like memorized it? Is that what you're saying?
As you say it, I think it. Okay. So, Pam, we have had over 200 episodes so far. Can you believe that?
(unclear) That was really random.
I know. I know, I'm sorry.
200 episodes! Ow!
I know it's crazy. I can't believe we've been doing this so long. And over 550,000 downloads! Can you believe that out?
I know, that's pretty amazing. But here's the problem. There's only been 146 reviews. Where are the reviews people? Where are they? We did get one recently from TeachingWithAMission.
I know. So cute. And they said, "Best math teacher podcast I found!
So nice, right?
And there have been several others really, really thoughtful and kind. And we're so happy that you take the time to listen because we enjoy doing this, and we're so grateful for listeners. But we would love for you to put a review wherever you listen.
So, Kim, I have to tell you. The other day, I tried to review a podcast, and I couldn't figure out how. And I started to wonder if maybe that's why we don't have as many. So, hey, listeners, let me tell you what I figured out. Now, you might be like, "Pam, we've already figured that out long ago." But I had not. So, if I just can share with you. Now, I listen to Apple podcast, so if you're in some other podcast place, you know, obviously we're glad you're listening there. I'm not going to try to help you to review us there. But if you're on Apple podcasts, you have to click on the show. So, not just the episode. So, like when I listen, usually it's just the episode that's up. So, I go to the... On mine, it's purple. I'm assuming it's purple for everybody. I click on the show, and when I click on the show, then it still shows the episode that I'm listening to, and then it shows, I don't know, three or four or five more episodes. You have to go below that. So, scroll down, down down. Once you have the show up, you see a few episodes, scroll down, down. It doesn't show all the episodes. It will only show, I don't know, five or six of them. I could probably count better, but I'm not going to. And then, beneath that it says "ratings and reviews", and that's where you can go in, give a rating and leave us a review. We would love for you to do that. A, because it's fun to read. But B, because it helps more people find the podcast, and we can spread the word more that Math is Figure-Out-Able. So, that would be cool. Yeah. Nice. Hey, so, Kim, I wanted to tell you a story. So, you and I both met Howie Chua. We love Howie, right? Super cute. And he does a great job of explaining why some math works. He does a nice Mental Math Monday thing. He's been a guest on our challenge. We loved having him. Super guy. When I ran into him recently in Florida. I had an hour or two before I had to be the airport, so he and I just sat and chatted kind of in the little... What do you call it? What do you call? In the hotel, right there in front of the registration?
Oh, like the lobby?
Yeah, thank you. Lobby. That's a word. There's a nice little sitting area, and so we just sat and chatted for, I don't know, almost a couple of hours. I was almost late to the airport. But it was fantastic. He and I could talk forever. I like him. He goes, "Can I ask you a question?" And I was like, "Sure." He goes, "I'm a little confused about Problem Strings. Is this a Problem String?" And then, he proceeded to tell me how he would solve a problem, and I was like, "Well wait, say more about that." And he goes, "Well, you know, if I was going..." I don't actually remember what problem he was solving. But he kind of told me. And let's say it was a multiplication problem. I don't even know. Give me a multiplication problem, Kim.
15 times 18.
15 times 18. He might have said, "You know like, I would solve 20. I would take about 18 times 15, and I would find twenty 15s. So, the first problem in my string would be 20 times 15, and the next problem in my string would be 2 times 15. And then, the next problem in my string would be 18 times 15." Yeah, so what's going on there, Kim?
So, he's thinking that the chunks that he's using is a string. Like, he's describing his strategy, and he thinks that is the string. (unclear).
That that's what a Problem String is. Yeah. Because then he said, "Pam, I'm confused because I will look at some of your Problem Strings, and there's problems in the middle of it that don't help you get to the answer.
How do you hear that, Kim? Yeah, keep going.
Yeah. Well, so it sounds to me like he's thinking a Problem String are the steps that he... The steps. The pieces that he uses to solve a problem.
A particular problem, right? Yep.
Yeah. The moves that he would make. And so, we would call that a strategy. Like, the moves you make, the way that you mess with the numbers. And I'm assuming that you shared with him that a string is focusing on relationships that build an idea. And it encompasses more than just one single problem.
Your leaning towards relationships, that then a strategy will be a natural outcome.
Yeah. Because I said, "Hey, that last problem in a Problem String is not necessarily the goal."
It's one of the problems to solve. We're going to solve lots of problems in the Problem String, like you said, in order to build relationships, to build connections. To quote, John Tapper. The other day, I was talking to John Tapper of All Learners Network. "To help students gain mathematical insight." So, he used that phrase, "mathematical insight". I thought that was really nice. And so, we want to create mathematical insight. We're not just listing out the relationships we would use to solve a particular problem. And let me be clear, he said I could totally quote him to clear it up. And so, hey, we thought we would do an episode on it. Does that make sense? Well, you were just going to say something. What were you going to say?
Well, I was going to say. So, in a classroom would Howie then say, "A kid..." You know, he throws a problem out there, and in a classroom, a student did one string when they describe their thinking, and then a different student was doing a different string. Like, is that?
Oh, yeah. I don't know. Yeah, probably. It sounds like it. It sounds like it, yeah. In other words, when we use the phrase, "Problem String", he heard it to mean, "That is the way, it is the string of thought, the string of steps, the string of relationships I'm using to solve a problem." And that's not what we mean. We mean a Problem String is an instructional routine that helps construct relationships and connections, so that strategies, models, big ideas, and concepts become natural outcomes.
Oh, I'm so glad he asked you that.
Yeah, right! I know, I know. And I was like, "Well, thank you for letting me clear that up." And he was totally cool about it. He's like, "Yeah, yeah, yeah. Totally use my name. It's great."
So, Howie we appreciate that. Thanks. I don't know if you're listening, but if you are, thanks for that. And again, thanks for the super conversation. Always gain, and it was great to see you in Florida. So, Kim, let's continue to parse out this idea of what a Problem String is. I'm going to ask you a problem, and will you tell us just how you would solve it. Just random problem.
So, what is 42 plus 39? Random problem. 42 plus 39? How would you solve it?
42 plus 39. I'm going to think about an equivalent problem of 41 plus 40. So, I'm going to take 1 from the 42, and give it to the 39 to make 41 plus 40. And that is 81.
Cool. So, under Howie's thought, we could say something like your series of relationships you used, your steps you used, the thoughts, the connections you used, was that you said to yourself, First, maybe, "I could create an equivalent problem that's easier to solve. And I'm going to grab 1 from the 42." So, that's maybe the first thing. I can use an equivalent problem. Second thing, "I can give 1 from the 42 to the 39." And then, your third thing is to create 41 plus 40." And then maybe your fourth step or thought that you had was, "Then, add the 41 plus 40 to get 81." So, that could be like a series or a list of things that you did in order to solve the problem. How is that different than a Problem String to help someone build a specific strategy to solve the same problem, 42 plus 39. So, let's do a Problem String toward that end. You ready?
Alright, Kim, first problem. What is 8 plus 10?
And that's not hard. And I would... If I was doing this with kids, I would actually write down an open number line. I would write the number 8. I would add 10, a jump of 10. And I would land on that 18. It wouldn't take very long.
Why are you laughing?
I'm laughing because as you were saying it, I had grabbed my pencil, and I was doing that. Because I know that we're going to explore some relationships, and so I want to be prepared.
I like it. And I had drawn that with my pen. And you had done that with your pencil. Nice. Okay, so then the next problem I would ask is, what is 8 plus 9? And I would say, "Solve that any way you want." And what do you get?
So, I know that one, but since we just did the plus 10, I was thinking plus 10, but then back 1. So, 8 plus 10 is 18, back 1 because I only want 9, so that would be 17.
And as you would say that, I would have drawn that. I would have drawn a new number line, and I would have drawn 8 plus 10 is 18. Then, I would have backed up one and land on the 17. So, I would have, and it would be right underneath, and it would be all lined up, the beginning numbers would be lined up, the 18s are lined up, and the jump of 10 is the same size to kind of make that strategy, now, a little more point-at-able and discussable. Yeah, cool. Next problem. What's 27 plus 10?
And I would just quickly kind of scoot over a number line more to the right because 27 is to the right of 18. And I would make that same sort of size jump of 10, and I would land on 37. And then, I would give you the next problem 27 plus 8.
So, 27 plus 10 was 37, but I don't want plus 10. I want plus 8. So, I'm going to back up from the 37. Back up minus 2 to get to 35.
And the way you're describing is very nice. Often, when I do this with kids or teachers, they'll just say, "Well, it's just back 2, 35."
And then, I will have to provide kind of like the language. I'll kind of nudge kids to say more about, "Oh, so you used the problem before. And then, why did you backup 2? Oh, because 8 is to less than 10." But you did... Oh, go ahead.
Which is modeling for them what you... It's thinking out loud, right? And it's modeling for them what you would hope that they would communicate to others.
And it's often the relationship they used. I'm just like helping put words to that and kind of pulling it out. Cool. Next problem. How about 33 plus 20?
33 plus 20 is 53.
Okay, and so then I would scoot over a little bit because 33 is just to the right of 27. And I would start at 33. I would do a big jump. So, this jump now should be twice as big as the jumps of 10 because this is a jump of 20. So, 33 plus 20 is, and then it would land on 53. Next problem. What is 33 plus 19?
Plus 20 was 53, so plus 19 is just back 1 from the 53, which is 52.
Cool. And depending on if I had done this strategy with kids before, a string like this, I may or may not redraw the 33 plus 20. I probably would, especially the
It's so funny because I totally, for the first time, did not redraw.
So, the first two problems you redrew the helper problem. And for this one, you're like, "Eh, I'll just use that helper problem." Yeah. So, it depends. But you and I both kind of have this gut instinct of when we're going to redraw the helper problem, and when we're just going to kind of use the helper problem. Now, if this was the first string of this type that I'd ever done with students, I would probably redraw the 33 plus 20, and then back up the 1 to land on 52.
But if I done a few of these, then I might just like, "Did anybody use this problem? You did? You did? Tell us about that." And then, I would just draw it on that first number line that I've drawn for the 33 plus 20. Cool. Alright, next problem. How about 42 plus 39. And I'm kind of curious. I'm kind of curious. For for this string of problems that we've just done, I kind of gave you 8 plus 10, and you used that to help you with 8 plus 9. And I gave you 27 plus 10, and you used that to help you for 27 plus 8. And I gave you 33 plus 20, and you use that to help you with 33 plus 19. But this time, I didn't really give you a helper. So, I wonder, before you solve 42 plus 39, could you create a helper that's like the pattern that we've kind of been using to solve this one? What might be yours? So, don't even solve it. But what might be the helper that would kind of follow the pattern?
I'm going to go with 42 plus 40 because 39 is really close to 40.
Nice. And so, go ahead and do that now? What is 42 plus 40?
So, I went ahead and wrote down 42. Big jump of 40 and landing on 82. And then, how is that going to help you solve 42 plus 39?
So, we did a jump that was too big by one, so I'm going to backup 1 from 82 to get to 81.
To get to 81. And that is a Problem String, a string or series of problems to help kids build the strategy of adding a bit too much. We call that the Over strategy. So, like 8 plus 10 helped you with 8 plus 9. 27 plus 10 helped you 27 plus 8. 33 plus 20 helped you with 33 plus 19. And then, you came up with your own helper problem, 42 plus 40, to help you with 42 plus 39 by going a little too much. 42 plus 40, a little too much, too high. And so, then, you backed up the one. Yeah. So, I'll just... Probably people don't have a pen or pencil in their hands and might not remember. But the first problem that we talked about was that same last problem. 42 plus 39. And you actually solved it a different way.
So, yeah, go ahead.
Well, I was going to say. So, I'm thinking about Howie's comment, and what he was saying was that given a particular problem, the thought that you went through the series of thinking to solve that problem really only gives you the answer to that one single problem. And in a Problem String, you're doing a bunch of problems, but it's less about the answers to those individual problems and more about constructing that big idea that then can be transferable to a ton of problems.
Nice. Yeah, I like it. So, a Problem String is an instructional routine. It's a way of teaching. It's a thing to do with students to help them construct mathematical relationships.
Yeah, nice. And there are different structures for Problem Strings.
The one that we just did we call a "helper clunker structure". And shout out right now to Rachel Lambert and Kara Imm, and... Oh, that third gal. Oh, crumb. I can't remember her name. Sorry. Oh, is it... Jennifer! Jennifer... Oh, I've never said your last name. DiBrienza? I'm slaughtering your last name. I'm so sorry. Jennifer. There we go. That they helped me. They were the first ones to go, "There are different structures of Problem Strings." And I was like, "Oh, my gosh. There are!" And we've worked more on that to kind of get solid on what some of those different structures are. And we just did a structure called a helper clunker structure. So, Kim, let's do a different structure of Problem String to get at a different strategy. So...
Hey, can you can you hang on that for just one more minute?
Because I don't know that we've ever. We've talked a ton about strategies, and strategies versus models, and kind of tried to clarify that a little bit over a couple of different episodes. This is the first time I think you've ever mentioned structures. And I don't know if you could say a little bit more about that, or you want to talk about that kind of at the end. But the idea of a structure versus a strategy might be muddy for some listeners.
Okay, yeah. Alright, so "strategy" is kind of what Howie was talking about. It's the relationships you use to solve a problem. It's how you mess with the numbers. When I'm talking about "structure", I'm back to the instructional routine of a Problem String. And so, a Problem String is a string or series of problems intended to create mental mathematical relationships in the learners heads. So, we could do that in a couple of different ways, in a few different ways. One way we can do that is to give students a helper problem, and then a problem that we call a clunker that they can use the helper problem to help them solve it. And that's the format of the string we just did, where I gave you helper, and then it could help you solve the next one. And then the very last one, we don't give you the helper, often, and we say, "Hey, based on the structure, or based on the pattern that you just used, what helper could you create to help you with this last kind of clunkerish problem?" That's one way of creating a Problem String, one way of the cadence, the way the Problems are related. But there are other ways in a string or series of problems that the problems are related that still help students build relationships, construct those mathematical relationships, but it's a different pattern in the string, it's a different way that the problems are related. That help?
Cool. So, here's a different structure.
And we call it... No, I'm going to tell you what we call it after. So, we'll just do it. Alright, so, Kim, first problem. What is... I just drew a big line on my paper to separate it from the last set of things. What is 48 plus 57? And you can solve it any way you want.
Okay, well, what's currently on my brain is the Over strategy because we just messed with that.
So, I'm going to go 48 plus 60, which is 108. And then, 60 is 3 too many, so I'm going to backup 3, and get to 105.
So, 105 is your answer to 48 plus 57.
And that makes a lot of sense. Cool. Next problem. What about 50 plus 55?
Also 105. I know that 50 and 50 is 100, and then 5 more is 105.
Cool. When you said "also 105", if you hadn't noticed that I would have been like, "Huh, same answer. That's..."
"Are those related? Is 48 plus 57 related to 50 plus? I mean, probably not. They just have the same answer. But other than that, they're not. There's no connection." Let me say them again. Between 48 plus 57 and 50 plus 55. Or is there? Can you see a connection? Is there anything that? What strikes you?
So, if you take 2 from the 57 and give it to the 48, then you've created an equivalent problem, which is 50 and 55.
So, on my paper, I have written 48 plus 57 equals 105. And below it, I wrote 50 plus 55 equals 105. And in between them, as you were talking, I just wrote underneath the 57 of the first problem. I wrote minus 2 because you said, "If you take 2 from the 57," and so then right underneath that is already the 55 for the next problem. And next to that, then I wrote plus 2 under the 48, and underneath that is the 50. So, it kind of looks like on my paper 48 plus 2, 50. And 57 minus 2, 55. And you can kind of clearly see these two problems are related, that you could kind of... It's almost like you took 2 marbles from one of the piles, from the 57 pile, and stuck it in the pile that had 48 marbles. Because if I had said, "Hey, Kim, I got 48 marbles in this pile, 57 marbles in that pile, how many total marbles do we have?" You decided to find that total of marbles after you move 2 marbles. You just like plop 2 marbles over there, and you're like, "How about if I just make it 50 and 55?" I have these two piles. You didn't lose any marbles. You didn't gain the marbles. And 50 and 55. Which one was easier to add the 40 and 57 or the 50 and 55?
Oh, at first glance, for sure the 50 and 55.
I mean, either one of them didn't take you too much mental effort. But at first glance, yeah, not too bad. Cool. Next problem. How about if I asked you 96 plus 148? Can you be a kid? And do the Over again because like your brain was still stuck in Over a little bit?
Sure, sure. I, actually, if it's legal for you today. And I'm not sure what you want. But I'm going to go with 148 plus 100. So, I'm going to make use of the commutative property, and I'm going to go 148 plus 100, which is 248. And then, I'm going to backup 4 because I needed 96 instead of 100. So, I'm going to go 244.
Cool. So, I heard you say, "I'm going to think about 96 plus 148 as 148 plus 96, and that 96 was 100 minus 4. So, you kind of used the Over strategy. Add 100, backup 4. Nice. Next problem. How about 100 plus 144?
Yeah, that's really nice. That's 244.
And at first glance, that one's like "Duh" almost, right? 100 plus 44. Is that related to the problem before?
Mmhmm. If I had just taken 4 from the 148, and given it to the 96, then that's an equivalent problem.
And so, I kind of did the same thing on my paper. I've got the 96 and 148. And underneath the 96, I put plus 4. And underneath that is the new problem. T he first number was 100. And then, I have the 148 minus 4. And underneath that was the number from the second problem 144. That might have been real confusing how I just said that. Basically on my paper, I've got 96 plus 148 plus 4, minus 4, and then 100 plus 144. Kind of the two problems in a row with the kind of connection in between them. Does that make sense?
Cool. So, it's almost like you're saying you kind of move marbles around, and you could find an equivalent problem that's easier to solve. I wonder if you could do that for a problem like 197 plus 38? Could you move some marbles around for that one?
Sure. So, I like 197. I would love it to be 200. So, I'm going to take 3 from the 38 and give it to the 197 to create 200 plus 35.
And here's the hard part. What is 200 and 35?
And 35? Yeah, 235.
Bam, I mean, that's one of my favorite places to be when the question is the answer.
Cool. So, Kim, in this Problem String, let me just repeat it for everybody. The Problem String we just did was 48 plus 57, 50 plus 55, 96 plus 148, 100 plus 144, 197 plus 38. Yeah. In that string of problems, that Problem String, that instructional routine, what are students learning?
They're learning about equivalence, and it's heading towards Give and Take
The strategy of Give and Take.
Was that Problem String all about helping students solve the last problem, 197 plus 38? Is that the reason we did it?
No, no. I mean, it's really by the time you get to that clunker, that last problem, you're kind of checking to see if they're making use of the relationships that you've been focusing on in the string. But it's not about the answer to that final problem.
Yeah. And it's not also just about checking, though that's a huge part of what can happen in a Problem String, right? We can kind of get a feel and sense for where students are, the kinds of things they're using, so we know what to do next. But it literally is helping them construct those relationships. They're literally creating mental connections in their brain that they can then use in new and future problems. And the more that we do things like this, as strings like this, the more that we mention these relationships, and bring them out, and make them visible, and point at them, and discuss them, the more solid those connections get in kids heads. Cool.
Yeah, I'm really glad that you said that just now because we have run into a lot of people who say, "I did a string about Give and Take or equivalents, and then like three days later, it's like we hadn't done it."
Yeah. "Students aren't using it. Why aren't they using it in other other places?" Yeah, sure. We need to strengthen those relationships, and they need to come up again and again. And we advocate that you're almost never going to do one Problem String and call something done.
Right? Like, in the Problem String books that we're writing, kindergarten through fifth grade Problem String books that we're writing that will come out sometime in 2024, we have series of Problem Strings for every strategy, right?
So, it's funny that you picked these particular Problem Strings because, you know, you've mentioned many, many times that I love the Over strategy. And I...
I even call you the Over girl.
I know, right?
You're the Over girl.
I do, and I love all the strategies, and I love that I have those choices. But Over, for whatever reason, came early and came naturally to me, and I had to look for, like seek out, Give or Take a little bit more. Like, it made sense to me, and I could totally do it, but I tended towards Over strategy. And knowing that it was important that I had all these strategies, I had to work a little bit more towards Give and Take. And, you know, recently in one of the message boards for the workshop, a participant said, "If we only do the easiest things, will our brains ever grow and become more sophisticated thinkers? I'm here to say no." And I know that you completely agree and so do I. That we have to give ourselves the opportunities to explore the most important relationships and not just continue to do the same thing over and over again.
Yeah. Bam! One of the funnest things that you and I do is interact with participants in our online workshop.
Yeah, it's fun
That is a great quote about... I love that person was here to say no, and we are too. Like, it's not about the easiest thing. It's about helping brains grow. That's super cool. Alright, so to recap, Problem Strings are an instructional routine. It is a thing you do to help students learn. It's not just a series of what you did to solve a particular problem. It is about exploring relationships that are broad and transferable.
Yeah, and we've... Go ahead.
I was just going to say we've done some more work and had more conversations about Problem Strings and what they are, but they're not in Episode 71 and 72. So, if you want to hear more about Problem Strings, you should check out those two episodes.
Excellent. Thanks, Kim. Great conversation today. Alright, ya'll, thanks for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!