Pam  00:01

Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  00:10

And I'm Kim Montague, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  00:18

Because we know that algorithms are amazing human achievements. But, ya'll, they are not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop. Even if you develop conceptual understanding first. 

Kim  00:37

In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships. 

Pam  00:43

We invite you to join us to make math more figure-out-able. Hey, Kim.

Kim  00:49

Hey. Tell you that when we got started, I hadn't adjusted my volume.

Pam  00:54

Oh. Was I yelling in your ear?

Kim  00:56

Real lout in my ear.

Pam  00:58

Sorry.

Kim  00:58

No, no. It's all me.

Pam  01:00

Hi.

Kim  01:01

Hi. I was excited about. Hey, fellow math. 

Pam  01:04

Hey! Yeah. Gosh. Yeah, I've been too many places lately where the microphone volume is not quite set quite at the moment. And, yeah. I think I need to when I test the microphone. It's like, "Hey, test, test, testing." I usually sit in kind of a normal like I'm talking to the person next to me voice, but then when I start talking, I do it in my "I'm presenting to a group", voice.

Kim  01:30

Yeah.

Pam  01:31

And yeah, duh. You'd think I'd be better at that by now. 

Kim  01:35

I mean...

Pam  01:36

Yeah. Hey, speaking of presenting. So, something that's been happening to me recently a little differently than in the past. In the past, you know, 20 years ago, when I was doing Problem Strings, nobody had heard of Problem Strings.

Kim  01:49

Mmhm. 

Pam  01:49

And then a lot of people started hearing about Cathy Fosnot's work, which is where I got Problem Strings. And then they would say, "Are these the same things as number strings?" Which I would giggle a little bit because I would say, "Well, it depends on how you define number strings because they're not really strings of numbers." They're strings of problems, right? And I get it. Like, there's a whole conversation about whether you call them Problem Strings or number strings. I called them Problem Strings because they're strings of problems, and we can do that in data and geometry. And pick another area of math. Like, functions, and graphs, and tables, and transformations. Like all that kinds of things we can do Problem Strings with those things. So, it's strings of problems. But now, interesting, I get kind of the same response to people or same request, question from people. Every once in a while where they'll say, "Hey, hey, hey. Yeah, like these Problem Strings you're doing. Those are the same thing as Peter Liljedahl's thin slicing, right?" So, Kim.

Kim  02:44

Yeah.

Pam  02:45

Let's talk about those. You know, one of the questions is people will say. "So, yeah. Those. That string we just did with you. I'll do that at a vertical, non-permanent surface, right?"

Kim  02:53

Mmhm. 

Pam  02:53

Alright, so let's chat. Are Problem Strings the same as what people are referring to as Building Thinking Classrooms thin slicing? 

Kim  03:02

Yeah, I think it's a great question. So, in one of his books, I think it's Mathematics Tasks for the Thinking Classroom.

Pam  03:08

Mmhm. 

Kim  03:08

Dr Peter Liljedahl says, "We can build a sequence of tasks that get incrementally more challenging as the ability of the student increases. This is called thin slicing." So, that is kind of their definition in this book for thin slicing.

Pam  03:25

Yeah. So, we also have sequences of tasks.

Kim  03:30

Mmhm. 

Pam  03:30

Do they get incrementally more challenging? Sometimes. Often. But not always. We definitely want the ability of students to increase. (unclear).

Kim  03:40

Yeah, so I can see why people might come to you with the question.

Pam  03:44

Absolutely. Especially because...

Kim  03:46

(unclear).

Pam  03:46

Yeah, series of problems. We're calling them Problem String, string or series of problems. 

Kim  03:51

Yeah.

Pam  03:51

Alright, so to have this conversation today, Kim, let's do a quick Problem String, not a thin slicing.

Kim  03:57

Mmhm. 

Pam  03:58

Together. And so, that we can refer to it as we kind of talk about how they're the same.

Kim  04:02

Okay.

Pam  04:02

And dun, dun, dun, how they are different, how Problem Strings and thin slicing are different. And maybe how we can take advantage. Knowing that can help us help more learners learn more.

Kim  04:16

Mmhm. 

Pam  04:16

I like that actually. Help more learners learn more. Huh.

Kim  04:19

Quote it.

Pam  04:21

We should think about saying that more often. No? Do you hate it? 

Kim  04:24

No, I don't hate it. 

Pam  04:25

Okay, first problem of the Problem String today. What is 48 and 10?

Kim  04:30

58.

Pam  04:31

And how do you know that by the way?

Kim  04:34

I mean, there's some stuff that happens in our place value system. Like, it's just 1 more 10, so the 40 changes to the 50 and the 8 stays the same. 

Pam  04:43

Cool. And so, if I was really asking this with a real group of kids, that this is really a question that they would have access to. It's not going to be super challenging. I could have kids who need to count by ones, but not for this particular string. You'll see as the rest of the problems come out. Probably don't want have students still counting by ones. So, we've been talking about the place value patterns that are happening here.

Kim  05:05

Mmhm.

Pam  05:05

They should kind of fairly readily be able to do 48 plus 10. Cool. As they do it and somebody says exactly what you said, "Well, it's just 58. You know, it's just one more 10." Then I'm going to represent that for this particular string with an open number line starting at 48, jumping 10, landing on 58.

Kim  05:22

Yep.

Pam  05:22

Not going to spend very much time on this problem. This is whole group. Everybody's listening. I'm calling on students. Next problem I'm going to ask. What's 48 plus 9? I'm going to pause just a little bit. I'm going to wait till there's a lot of thumbs or maybe even smiles. I might not wait too long on this one. And I'll say, "Hey, what'd you get?" And I might ask for a choral response at this point because I'm pretty sure everybody's not struggling with this one too much. So, Kim, what is 48 plus 9?

Kim  05:48

57.

Pam  05:49

Cool. And then I'm going to say, "Did anybody use the problem before? You didn't have to. But anybody used the problem before to help you?" I'm looking to see who that might be. Kim, did you use the problem before?

Kim  05:58

Sure did. 

Pam  05:59

(unclear). Go ahead. 

Kim  06:00

If adding 10 was 58, then only adding 9 is going to be 57 because it's adding 1 less.

Pam  06:07

And as you were saying that I would on the board have drawn a number line right under the one that I had before, put the 48 right underneath the 48, draw a jump of 10, put that jump. Because you said just like the one we had before, so I'm going to draw the one before. And then you said but it would be 1 less, and so then I would back up 1. I would do a jump back minus 1. And then I may even say, "So, what was that again? What's 1 back from 58? And you said? 

Kim  06:32

57.

Pam  06:32

57. And then I would write down that landing spot at 57. I've already put the 57 next to the problem 48 plus 9 because that's what you said you got. And then I sort of now the two number lines are lined up. 

Kim  06:44

Mmhm.

Pam  06:44

So, you can see the 48s are vertically in the same place. The 58s are vertically in the same place. And the 57 is a little bit to the left. Okay, cool. Next problem. What about 37 and 20? Go ahead.

Kim  06:56

57.

Pam  06:57

And I might just say, "Go ahead" to the whole class. It might get like a choral response here. Because again, if I'm doing this Problem String, you'll see as the problems go on, that I don't want this to be a lot of work for kids. That we should have been doing some work, maybe some Count Arounds, to make sure that these are happening. When you said that, then I might say something like, "Hey, nobody needed to draw a picture. But I'm going to just kind of make visible  maybe what's happening in your head with the relationships." And so, then I'll draw a number line with starting at 37. By the way, this number line has started to the left of the number line that started at 48 because 37 is to the left of 48. So, even though it's a new number line, it's kind of in relation to the two that I had before. And then I'll do a bigger jump. And I'll actually say that as I'm drawing it. I'll say, "Ooh, a bigger jump of 20. And you guys are saying that that lands on 57." And I'm trying to get that 57 to be kind of where it was in the problem before. You're probably not writing, so you don't even know that we had a 57 in the problem before, but I'm kind of lining those up. Then, the next problem. What's 37 plus 19? Now, I'm going to pause a little bit. I'm going to let kids work. I'm probably going to have some kids that are getting to a friendly number. I might have kids that are doing the algorithm if they've come to me that way. I'm just going to kind of watch. And then I might call on a student. This might be an opportunity for me to call on a student who I haven't heard from for a while. This also might be an opportunity for me to call on a student who... How do I say this? Is really ready for a lot of challenges, and I don't necessarily want to call on that kid who explains really well during later in the string when I want an explanation. Because if that kid explains it too well, we won't get enough thinking going on. So, I might call on that kid now just so that kid gets a chance to contribute, but not in a place where his or her too well, too too clear explanation is going to... Anyway. So, alright, I'll call on somebody and say, "Hey, did you use the problem before?" If they did, then I'll say, "Yeah, tell us about that." So, Kim, did you use the problem before?

Kim  08:57

I did. 

Pam  08:58

And how?

Kim  08:59

Because I know that adding 20 to 37 is 57. But this time, instead of adding a whole 20, I'm only adding 19, so it's going to be 1 less.

Pam  09:08

So, very similar to what I did before. I've lined up the number lines. I've redrawn the 37 plus 20, and then I backed up 1 to land on 56. And I have that also in the equation. Cool. Next problem. How about 55 plus 40, Kim?

Kim  09:24

Mmhm. 95.

Pam  09:25

95. I'm going to go and draw that way to the right. It's kind of starting about where 55 was with the other number lines. I'm adding a huge jump of 40. Double actually, what we just had. And you said that that was 95. Next question. How about 55 plus 38? Pausing. Letting everybody think. Kim, did you use the problem before? 

Kim  09:45

I did, yeah. So...

Pam  09:46

I'm going to look for a kid like that. Go ahead. Share that, mmhm.

Kim  09:49

So this time instead of 40...

Pam  09:50

Oh, actually. Sorry, just a little facilitation move. At this point, I might call on a kid who I'm pretty sure was not using the Over strategy before this Problem String. So, I might look around for a kid who I think is just trying it for the first time and give that kid a chance to verbalize it because that's going to help cinch for that kid. And because that kid verbalizing it, chances are high it's going to be a little less succinct, a little less exactly correct. Then, that gives the rest of the class a chance to chew on that explanation. And we all kind of move forward because of that. Okay, sorry, Kim. 

Kim  10:28

That's okay.  So, this time I'm going to go back 2. And I'm going to purposely just say that much. So, you can (unclear).

Pam  10:36

(unclear). 

Kim  10:36

I'm just going back 2.

Pam  10:38

And so, then I might say back 2 from what? Like, so did you use the problem before? And oh, okay. So you started on 55 and you added 40. And that was we already said it was 95. And why are you going back 2? Kim, we've been going back 1. Have you been not watching the pattern?

Kim  10:53

Because I only need 38 not 40. 

Pam  10:56

Bam. And so, that's back 2. What is back 2 from 95? 

Kim  10:59

93. 

Pam  11:00

Alright, so you're thinking that we land on 93. I'm going to finish the equation. I've got the number lines all lined up. Then, I'm going to probably say something like, "Hey, can somebody put words to what?" And now, Kim, you're the only one answering the question. Let's say you were the new kid. You were the kid who was, you know, never really tried this strategy before. And so, that kid I called on for that last problem. I might say, "Can anybody? You know like, what was Kim doing there? Why? You know, what words would you put to that?" And then I'm going to again call on a kid who I'm pretty clear hasn't been using that strategy a lot, is developing it. Again, we get an opportunity to kind of hear some less. Kind of like you said "Back 2. You know, I subtracted 2." "Well from what? And what were you..." You know, and I'll kind of trying to pull out some language. But here, I might try to be a little bit more general.

Kim  11:44

Mmhm. 

Pam  11:44

Instead of like talking about these numbers, like what did we do every time? You add a 10 to add 9? Well, now I'm kind of giving you an idea. Like, could somebody be general about kind of like? And, oh, just lately, Kim, I've been doing this thing. I don't know if you've even seen me do this. Where I'll stand in front of the board, and I'll kind of make my arm go from left to right, and then back up a little bit. So, I'll go like. And I'll just, you know like, my hands going that big jump to the right, and then just back up just a little bit. I kind of like, "Somebody put words to that. Why were we every time? You know like, what is it?" And some kid is going to say something like, "Well, we added a bit too much, and then we kind of went... You know, we took off the extra." And when we kind of get there, then I'll say, "Huh. Okay, well, that seems like an interesting way that we could solve some problems. Not everybody's doing that, but you're saying you could. Okay, cool. I wonder. I wonder if you were to follow that same pattern, could you come up with a helper? Like, you'll notice that I gave you 48 plus 10 to help you with 48. Or you could have used that to help with 48 plus 9. 37 plus 20. You could have used that to help you. Think about 37 plus 19. 55 plus 40. You could have used that to help you think about 55 plus 38 in this kind of a way. Could you come up with what that helper would be if I gave you a problem like 46 plus 37?" Now, I'm going to pause. And I may at this moment say, "When you're ready, when you've come up with a helper problem for that, will you turn to the person next to you and share your. But make sure they're ready. Don't talk over their thinking. Give everybody a chance to think." This might have been a time where I've purposely had kids sit next to each other, where I've got kids who like to think out loud, and I might have put them next to each other because they're going to turn and talk to each other no matter what, and they're going to kind of ignore each other. But it gives them a chance to like say out loud without interrupting someone else's thinking. But I might put kids who are real thoughtful thinkers next to each other, and they're going to be, what, courteous of each other, and they're going to watch. Now, that's not to say I'm not going to group kids differently other times, but in this moment, when I know I want kids to give each other some respect, I might say, "Like, let's take a moment and, you know, share with your partner when they're ready." And then after they've kind of shared that I'm... Well, as they're sharing it. I'm circulating and I'm listening for somebody that's come up with a helper problem that's got some  conversation around it. Again, I'm always thinking equity. I'm like, whose voice is it time to kind of pull forward who will help move the math forward? And so, Kim, did you? Could you come up with a helper problem that's kind of following that pattern? 

Kim  14:18

Mmhm. Yes.

Pam  14:19

Okay, what do you got?

Kim  14:20

So, if you're asking me to add 37.

Pam  14:22

Mmhm. 

Kim  14:23

Then I might instead decide to add 40. 

Pam  14:25

To what? 

Kim  14:26

To the 46.

Pam  14:27

Okay, so we started with 46, and you're going to add 40. And why 40, Kim? 

Kim  14:31

Because it's a nicer number. It's a little bit more than 37. It's easier to add. 

Pam  14:35

There's that kind of that "Wah" number. It's like it's too big. It's like... Yeah, cool. "What do you guys think? Did anybody else create a problem like Kim did?" A lot of people are going to. "Oh, I'm seeing a lot of nods. Okay, cool." We might have kids who followed a different pattern, and I might let them do that. "But if you follow sort of the same pattern you guys think that Kim did..." This time, I'm going to draw the 46 plus 40. Again, kind of trying to line up with the numbers above it. 46, that big jump of 40, landing on... What do you land on by the way? 46 plus 40?

Kim  15:05

86.

Pam  15:06

So, I'll land on 86 and then I'll say, "But Kim, why did you create that problem?" Because it was what? 

Kim  15:12

Yeah, 40 was easier. A little bit more than 37.

Pam  15:15

A little bit more. How much more? 

Kim  15:17

3 more. 

Pam  15:17

Oh, it was 3 more. So, now you're going to back up 3 to use that. Alright, here's the hard part. What is 86 back up 3, minus 3?

Kim  15:25

83. 

Pam  15:25

83 cool. And so, you're saying that 46 plus 37 might be 83. "Hey guys, let's put some words. Again, this kind of thing." Like, and again, have maybe one more kid put some words to the pattern that we're kind of finding. We'll probably do this kind of a Problem String one or two more times with what? A third grade class? I could probably do this in the second grade class near the end of the year.

Kim  15:46

Mmhm. 

Pam  15:47

If I'm starting in fourth grade, I could do this. I might have gone to bigger numbers in fourth grade if I was doing this with fourth grade kids. 

Kim  15:53

Yeah. 

Pam  15:54

And so, then when most of the kids are playing with this relationship, then I'll make an anchor chart, and we'll continue to help cinch the relationships. Alright, I did a lot of teacher talk to kind of let you into what would be happening if I were to do this Problem String. That's kind of important. And I don't know if anybody's like, "Pam, we've heard you do Problem Strings like this before." But now we want to lean back on some things that happened during that Problem String. Or at least that I talked about would happen in a class with real kids. So, Kim, here's a definition of a Problem String. "A Problem String is a purposefully designed sequence of related problems..." So, far we kind of sound like thin slicing. Let me keep going. "...help students mentally construct mathematical relationships and nudges them towards a major efficient strategy, model, or big idea." So that's a... We call it a definition of sorts.

Kim  16:48

Mmhm. 

Pam  16:49

Because there's more that goes into a Problem String. Let's kind of dive into what are some of the things that Problem Strings and thin slicing have in common.

Kim  17:00

I think they actually have quite a few things in common. And I can understand why people would see Problem Strings and think, "Oh, that's a great. I've captured a good thin slicing." So, what they have in common are they are not all at once. So, when we deliver problem strings, we would say do not hand on a piece of paper all the problems, send students off to solve the problems.

Pam  17:22

Mmm, like don't put them on the board. "Here's your warm up. I'm going to take roll while you do all of these problems."

Kim  17:27

Yep.

Pam  17:27

Yeah, we don't. They're not all at once. Mmhm. 

Kim  17:29

Yeah. And I think that Liljedahl would say the same thing. Like, he purposefully has the problems cut up and students come get a problem one at a time.

Pam  17:38

Yeah, and I know lots of people have come up with some inventive ways to do that. I'll also mention, when I was writing Discovering Advanced Algebra with my wonderful co-authors, one of the things that was kind of suggested along the ways. I think, one of the editors said, "We should put your Problem Strings in the problem sets of the textbook."

Kim  17:55

Mmhm. 

Pam  17:56

And I was like, "No. Like, you can't give them to students all at once." And the question was, "Well, why not?" And and the answer, I think... And I'm not going to speak for Dr. Liljedahl, but I think that the answer is some kids might pick up on the patterns that we're trying to pull out of a Problem String or maybe thin slicing. But most kids won't. Most kids, they need a higher dose of patterning. And also, often you don't even know what to do in a Problem String. The one that we just did is really clear. You add the numbers. But there's a lot of Problem Strings we've written that kids might not even know what the question is really unless the teacher facilitates that question as they're facilitating the Problem String one problem at a time.

Kim  18:41

Mmhm.

Pam  18:41

Okay.

Kim  18:42

So, also there is some form of increasing challenge in the Problem String that you and I just talked about. We started with 48 plus 10, and then 48 plus 9, and we ended with 46 plus 37. So, the numbers increased a bit. The number that, you know, you also increase the difficulty by asking me to provide a helper problem.

Pam  19:02

Mmm, mmhm. 

Kim  19:03

So, two different ways that you up the ante. There is some form of increasing challenge in thin slicing We'll talk a little bit later about the type of challenges that are increased. So, if you glance at a thin slicing, you might see, you know, the numbers getting harder or change in some way. 

Pam  19:21

Mmhm.

Kim  19:21

So, that would be a bit similar.

Pam  19:23

Hey, before you go on, I'll just point out in our particular Problem String, it could be argued that the third problem was actually more difficult than the last problem.

Kim  19:32

Mmhm.

Pam  19:32

Because the third problem, at least the clunker, was 58 plus 30... 55 plus 38. Those are higher numbers than the last problem, which was 46 plus 37.

Kim  19:42

Mmhm. 

Pam  19:43

So... 

Kim  19:43

Yeah, that's true that. 

Pam  19:44

It might be noteworthy that in a Problem String, it's not necessarily true that you could look at any one problem in a sequence and say the next one's going to be, quote, unquote, "more difficult". 

Kim  19:57

Yeah, for sure. 

Pam  19:58

Yeah. Okay.

Kim  19:59

Super thing they have in common is about teacher planning. You know, we would say that Problem Strings... People ask all the time, "Man, how did you get it to look like that? When I facilitate them, you know, it doesn't. It's a little clunkier. Or, you know, it doesn't come across that way." And we would say, first of all, that we've been delivering Problem Strings for a few decades. And that sometimes people don't consider the planning that goes into it. We wouldn't say you have to be, you know, perfect before you do them. But as you facilitate more, you'll realize how much planning goes into the facilitation, the choices that are being made. And I think that that Liljedahl and his co-authors would agree that teacher planning is really important. Planning the order of the problems, planning what the thin slicing is really matters.

Pam  20:48

Mmhm. 

Kim  20:49

Also, we both would agree that there needs to be an entry for all. They describe low floor, high ceiling. We would suggest that there needs to be some open access. You mentioned that in the Problem String that you would want 48 plus 10 to be readily accessible for the most of the students, so that they have an entry point into the Problem String. We want just enough challenge.

Pam  21:11

Yeah, thank you. Just enough challenge. What that means is if 48 plus 10 or even 55 plus 40 is not a problem that most kids can pretty readily figure out, then this is not a great Problem String for that class yet.

Kim  21:25

Right.

Pam  21:25

We'd want to do some other work first. Yeah.

Kim  21:27

Yeah. Hugely important to us is that the problems are done in a very specific order. You mentioned in our definition of sorts. It's purposely designed sequence to draw out the specific relationships. I think that in thin slicing there there is an order. Although, Peter would say that some out of order is okay. And we would say in a Problem String, it is not okay to switch up the order.

Pam  21:53

Mmhm. Yeah, I think he's got groups often in his thin slicing, and it's okay to switch up the order in the group of questions.

Kim  22:02

Yep. 

Pam  22:03

And we don't. Problem strings do not have that. Problem String... The order in a Problem String is very purposeful, and you don't mess with the order. 

Pam and Kim  22:11

Yeah.

Kim  22:12

One more thing that they have in common is in the green Mathematics Tasks for Thinking Classrooms, they talk about variation theory, which is basically the idea that if other things are the same and one thing is being varied, you're drawn to the variation. I smiled when I read that because one of the structures that we use for Problem Strings is a variation structure. So, similar idea that when something is varying and you can notice that, but if too many things are varying, it's just too much change and it's harder to pull that out. So, we both like the idea of variation.

Pam  22:52

Yeah. And maybe I'll point out in the Problem String that we did today where we kind of used some of that. So, we had... All the problems had a two-digit number plus a multiple of 10. That was kind of the helper. So, 48 plus 10. 37 plus 20. 55 plus 40. But then we varied just a little bit by that it wasn't all the same decade. 48. 37. 55. And it wasn't all the same multiple of 10. We added 10. We added 20. We added 40. You might be like, "Pam, should the next one have been 30?" No, because then we don't want to be too predictable. Like, there's kind of a bit of an art to creating a Problem String that's a little bit unpredictable, so kids are on their toes. But then, you know, what do we vary? Well, we added 9, and then we added 19, then we added 38. At that point, a lot of kids are going to be like, "Haha, but this is going to be at 30. If we added 40, I bet it's going to be at 39." Well, that's a great moment where we get just a little bit about add 38. And then, you know, kids get a chance to, again, be kind of on their toes. And there's an opportunity for some almost some fun to be had with like. Kids get used to looking for those patterns because we haven't varied the pattern. But then we vary just a little bit more than they were expecting. Bam. That's a desirable outcome. We want to have just enough variation, but keeping some things the same, keeping enough similar. Yeah. Yeah, we like that. So, Kim, a lot of things that the Problem Strings and thin slicing have in common we've just talked about. Another thing that they have in common is that we would agree that we want kids to be in productive struggle.

Kim  24:40

Yeah.

Pam  24:41

And that we want to have all kids be challenged during a Problem String. 

Kim  24:45

Yeah.

Pam  24:45

Actually just got off of... We called a gig meeting. I don't know. Why do we call it that? With some leaders in schools around the world. And one of the things that this particular leader said is, "We're really looking for help with our..." and it was particularly middle school "...with our middle school teachers to differentiate. Like, we need to be able to scaffold for students who need it. But we also have a lot of students are ready for more challenge." And that the teachers were finding that challenging. She said, I'm quoting, "They teach to the middle."

Kim  25:20

Mmhm. Yeah, (unclear).

Pam  25:21

Yeah, go ahead. 

Kim  25:22

I was going to say I think that's common for lots of places. 

Pam  25:24

Well, we all taught to the middle.

Kim  25:26

Yeah.

Pam  25:27

Like, I think in a huge way, that's the experience that we had to draw from. So, yeah, I think Peter Liljedahl talks very eloquently about keeping kids in flow, and so we want the struggle to be just enough, but we also want them to be challenged. And we would agree that that is definitely a desirable outcome. 

Kim  25:45

Hey, Pam, I'm noticing the time, and we have so much more to say, and I'm going to suggest that maybe we leave this episode with some wonderful things that Problem Strings and thin slicing do have in common, but I want to make sure that we get to talk really nitty gritty about how they're not the same. So, what do you think about ending today and moving on to another episode? 

Pam  26:08

Alright, so we're going to finish the recording today at Kim's wonderful suggestion. Everybody's firmly it's (unclear). Alright, that's what they have in common. What? How are they different? And how can we take advantage of being really clear on those differences to get more bang for our buck? We will do that in our next episode. You are not going to want to miss that one. Ya'll, thank you 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. And keep spreading the word that Math is Figure-Out-Able.