Pam  0:01  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned mather.

Kim  0:09  
And I'm Kim, 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  0:16  
Because

we know that algorithms are amazing human achievements, but 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  0:34  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:40  
Ya'll, thanks for joining us to make math more figure-out-able.

Kim  0:45  
Hey, I'm excited about today's episode. 

Pam  0:47  
Yeah, me too.

Kim  0:48  
Because we're finishing what we started last

week. 

Pam  0:50  
Yeah. So, last week, we dove into a question that I'm just getting asked more and more, which is are Problem Strings, our favorite Instructional Routine, are those the same as Dr Peter Liljedahl's thin slicing. And I think that's kind of been... I don't know. I'm not sure he started off knowing that that's what they were. He kind of talked about the theory of thin slicing and how it kind of fits with what he was doing. And now, people kind of call it that. I think it'd be worth mentioning what he says in his book. So, in his recent Mathematics Task for the Thinking Classroom, he said, "We can build a sequence of tasks that get incrementally more challenging as the ability of the students increases. This is called thin slicing."

Kim  1:34  
Mmhm. 

Pam  1:34  
So, it kind of looks like a list of problems, and he has you do them at vertical, non-permanent surfaces in randomly chosen groups. And you might look at a Problem String and go, "Oh, sweet. Here's a whole bank of thin slicing. Bam. We're going to go do that the way that Liljedahl talks about doing his thin slicing." And so, last episode, we kind of, with some detail, went through a Problem String, so that then we could kind of point out what that Problem String and the thin slicing from Dr. Liljedahl what they would have in common, and then how they are different. So, ya'll, if you have not listened to last week's episode, highly recommend that you go do that. Especially listen to the Problem String. You could actually maybe skip the rest of it if you want to. That's where we talk about what they have in common. Today, we're really going to talk about how they are different. Yeah?

Kim  2:24  
Yeah. And I think this is important because they serve different purposes. And so, I think that if you think that you're getting the same thing out of Problem Strings and thin slices, thin slicing, I think just creating awareness is important. So

one...

Pam  2:42  
Let me actually add on to that if you don't mind. Because if you take our Problem Strings and you facilitate them like you would like Dr. Liljedahl suggest that you facilitate a thin slicing, you won't get out of them for all students what you could if you facilitated it like a Problem String. Do you agree with that?

Kim  3:02  
Yeah, yeah. 

Pam  3:03  
Okay, alright. 

Kim  3:05  
So, an important part of thin slicing for Dr. Liljedahl is that they are at vertical, non-permanent surfaces. His research is that students are up at whiteboards. They have an opportunity to kind of erase and not make permanent their thinking. For us...

Pam  3:23  
And it's

vertical. 

Kim  3:24  
Mmhm. 

Yeah.

Pam  3:24  
I'm going to add to that, and... Can I do the group thing now? No. Keep going. Vertical, non-permanent surfaces. It's erasable. It's

vertical. Keep going.

Kim  3:32  
Yeah. And so, in a Problem String, we actually want a record of students thinking that is not erasable. We actually want a record that we can circulate and see students work, so that we see kind of where they started and where they're moving, and we can look back on their recording. That we can, even if they're in a different problem, we can see an evolution of their thinking over time. So, we would prefer that if students are recording. And sometimes we don't have them record. But if they're recording, we have them record in a notebook, on piece of paper. There are a few times where we have them record like on a on a whiteboard, but it is not necessarily up at vertical and non-permanent surfaces, like it's important to him that they are in thin slicing.

Pam  4:15  
We don't feel like it's necessary or even all that helpful. I do think...you and I've had the conversation...if we had all the room in the world to...

Kim  4:25  
Oh, yeah. 

Pam  4:25  
...have all of our students standing up individually with a marker, okay.

Kim  4:30  
Mmhm.

Pam  4:30  
 But we don't think it's necessary. We think we can still get the bang for the buck out of the Problem Strings without causing that disruption.

Kim  4:37  
Yeah. And part of that is because we believe firmly that Problem Strings are an individual piece of work. And so, if you, you know, if you have 10 groups, you can get them all at a vertical, non-permanent surface. We just don't have the space for 30 kids standing around in a room. And because we want individual thinking, not group thinking, for a Problem String, it's just not feasible.

Pam  5:01  
And that's not to say that we don't ever want kids working together. We do.

Kim  5:04  
Sure.

Pam  5:05  
Just not during a Problem String.

Kim  5:07  
Right, right. 

Pam  5:08  
And a minute ago, I said something about disruption. Let me just be clear why I said that. One of the tenets of Peter's work is that he will say they want to sort of disrupt the norms of the way we've been doing things. 

Kim  5:19  
Right.

Pam  5:20  
That's the kind of disruption I meant. Not that putting kids up at vertical non-permanent surfaces was disruptive. I mean, unless it's disruptive of the sort of norms that have been happening in class. But yeah. Not in a negative way.

Kim  5:31  
Right, and we would agree that for Rich Tasks, that is a great change. 

Pam  5:36  
We

like vertical non-permanent surfaces.

Kim  5:38  
Yeah.

Pam  5:38  
Rich Tasks, But not for Problem Strings.

Kim  5:41  
Right,

right. 

Pam  5:42  
Cool. 

Kim  5:43  
Also, we'll talk for just a minute about the increasing difficulty, but we would believe that the series of problems is not always from easy to more complex. Because sometimes we're building ideas. We're building connections between problems. For example, in a Doubling and Halving Problem String, which is the equivalent structure, you might have a difficult problem like 25 times 36 as the very first problem. Which a lot of people would not say is an easy problem. But then you might follow it up with one-fourth of 36. And then 0.25 or 25/100 of 36. So, it doesn't... You know, in a thin slicing, often they move from the simpler problems to a new type which is more difficult than another new type which is more difficult. We often might have a helper clunker where it's an easier problem, a more challenging problem, an easier problem, a more challenging problem. So, it doesn't fall into the general idea of always easy to hard.

Pam  6:46  
Yeah. And in fact, it's interesting. When you gave those numbers 25 times 36, I actually thought the place you were going to go was then 50 times 18 or 100 times 9. That's a different equivalent structure. I did that kind of fast. 25 times 36, we could Double and Halve. You kind of used a quarter strategy.

Kim  7:05  
Yeah.

Pam  7:05  
...that you could think about. Yeah. So, those are a couple of different examples of Problem Strings that don't necessarily go, like you said, an easy problem to a harder problem to a harder problem. In an equivalent structure Problem String, we're looking for equivalence. And the idea is that then you ask students, what are you noticing? And why is that happening? Why do these problems have the same answer? What's happening here? Ooh, and I wonder if that could be helpful in a next series or set of problems.

Kim  7:35  
Mmhm. 

Pam  7:35  
Yeah. So, we do have Problem Strings that maybe you could look at that they go from easier to harder. But often not. In fact, because everyone dutifully listened to last week's episode, you heard the Problem String that we did, and in that particular Problem String, we had problems 48 plus 10, 48 plus 9. So, an easy problem to a harder problem. But then 37 plus 20, that's arguably easier. And then 37 plus 19, that's arguably harder. But notice that the 48 was bigger than the 37. So, is that easier or harder? And then the next problem, 55 plus 40 and 55 plus 38. That 55 plus 38, that's unarguably more difficult than all the problems that we've had so far. But then for the last problem, we end with 46 plus 37. Actually, the numbers got smaller. So, again, it's just in a Problem String, it's strategically planned sequence and it's not just always easier to harder. 

Kim  8:37  
Yeah.

Pam  8:38  
Yeah. That's a difference. 

Kim  8:39  
So,

I think the most important differences are about the actual mathematics that's happening in thin slicing versus Problem Strings and the teacher's role. So, let's camp here for quite some time. The mathematics that's happening in thin slicing. I want to call attention to page 46 of Mathematics Tasks for the Thinking Classroom. Right around there, he gives tasks 1 through 10 and 11 through 20 that you would roll out to students. And he says, "Adding two-digit numbers can be approached through thin slicing. Adding two-digit numbers is not one type of task. It's made up of multiple types." And he goes on to talk about the three types of problems that are within these tasks. And he says, "Type one..." I'm paraphrasing a little bit. "Type One is where the sums of the ones is less than 10 and the hundreds are less than 100." So, essentially, there's no regrouping that's happening in type one.

Pam  9:37  
Like, you then kind of add the columns of numbers, and you don't have to... Nothing spills over.

Kim  9:41  
Right.

Pam  9:42  
Mmhm.

Kim  9:42  
In type two, within those sets, he says, "Sums of ones is more than 10 and the hundreds is less than 100." So, one grouping, one regrouping in the ones place is happening in those problems.

Pam  9:56  
In type two. 

Pam and Kim  9:57  
Mmhm. 

Kim  9:57  
And then, "In type three, the sums of the ones is more than 10, and the sons of the hundreds is more than 100." So, two regroupings. So, in these tasks, the change that's happening between type one, type two, and type three are all about the regrouping that quote "would be necessary to solve those problems".

Pam  10:18  
And it's not necessarily suggesting that he's demanding the algorithm. But it is kind of a perspective of that what's difficult for kids to think about adding numbers is that they are going to be kind of lining them up, and they're going to be adding by place value. And that's an assumption that we're making, that that's what becomes more difficult for kids. Would you agree with that? 

Kim  10:42  
Yeah. Yeah, I think so. 

Pam  10:43  
So, that's really interesting. When you and I saw that, we... I'll just speak for myself. I took a deep breath and I was like, "Wow, that's not how we look at adding multi-digit numbers at all."

Kim  10:55  
Yeah. 

Pam  10:55  
I think there's some different... How do I even say this? Some underlying things that we would want to build in students that don't have anything to do with whether the sums are more than 10 or more than 100 that have to do with some underlying understandings of what it means to add and what multi-digit numbers are. 

Kim  11:17  
Also,

we talked about... So, that the three types was interesting. Also, Peter says that we want students to be in the flow and have productive struggle. He talks about that on page 47. We would absolutely agree that that's important. And I loved a comment that he made about teachers give hints that either increase ability rather than decreasing challenge. So, he talks about two hints, and you can either increase ability or you can decrease challenge with your kinds of hints. We would also agree that we want to give hints that increase ability. But I also think that there's some stuff missing there, that when we talk about facilitating a Problem String, it is much more teacher facilitated and the role of the teacher is much more important than I read that the teacher's role is in their books.

Pam  12:10  
Yeah, let's dive into that. Before we go, what you just said is really interesting. I want to spend just another second on that teacher hint thing. When he says that there's sort of two ways that teachers give hints, he's not suggesting that decreasing challenge is a good thing. Right?

Kim  12:27  
Right.

Pam  12:27  
I think he's suggesting that if you give a hint that just makes it easier, it decreases the challenge, that that might look good in the moment, feel good. The kid's off the hook. He's got an answer. She can move on. But in reality, that it's not helpful for future learning. And so, like him, we also don't think that giving hints that decrease challenge is optimal. But we do want to give hints. He suggests increase ability. I don't know about increase ability. I want to maybe increase access or hints that scaffold, hints that help students use what they know, help students realize what they know in the problem, and so they can then logically deduce from there.

Kim  13:08  
I would think that that's what he means. You know, we often say do you need time or help, and that helps a student to make choice for themselves, gives them some agency. And then we would talk about... You know, you've talked before about focusing and funneling questions,

Pam  13:22  
Mmhm. 

Kim  13:23  
As a way. And I think that maybe what he's describing as "decreasing challenge" are these funneling questions where it's we're just helping you get the answer (unclear).

Pam  13:33  
Funnel a kid to an answer. 

Kim  13:34  
Yeah.

Pam  13:34  
Mmhm. 

Kim  13:34  
But I think that maybe what he means with increasing ability is a lot of what you're saying. We want to focus them on what's happening, remind them of the things that they know, give that support, so that they recognize that they have ability. And we're kind of like nudging them there.

Pam  13:50  
Yeah, I'm not sure why "increase ability" bugs me. Why does it bug? Not horribly. Just, I don't know, it almost sounds like you're increasing...

Kim  13:58  
We'll

beat that out sometime because I really love it actually.

Pam  14:01  
Oh, you

love "increase ability". Okay, not today. Alright.

Kim  14:04  
Not today.

Pam  14:04  
Not today. Okay.

Kim  14:05  
That's funny.

Pam  14:06  
Yeah, so we would definitely agree that the teacher's role. I think Peter would say that because the kids are up at vertical, non-permanent surfaces in these randomly chosen groups, the teacher's role is to circulate and to be helping groups move along to give those hints that don't... Not the hints that decrease challenge but the hints of the increase ability. And he also talks about the the role of the teacher and how important that is. And we would agree with that. And we would really suggest that the math is all important in maybe a slightly different way. When, Kim, you and I looked at these types that he talked about where... And again, this is only one example. So, this is we're giving one example here. But the types being kind of where kids are really dependent on kind of lining those numbers up. They don't have to be lining, but they're really thinking about the place values, and they're adding the ones together and they're adding the tens together. And so, the double-digit numbers get more difficult just based on if when you add the ones together, it's more than 10. And if you add the tens together, it's more than 100. We think there's a lot more math involved for kids to really get good at multi-digit addition. And I don't know that we even parse out double-digit edition so much. It's really more about the relationships involved in multi-digit addition. So, for example, kids need to know any number plus 10, any number plus multiples of 10. That's an important pattern in our number system, and we can, you know like, help kids develop that. They need to know what friendly numbers are. Which are definitely multiples of 10, but also quarters are friendly. And it depends on the problem what numbers might be friendly in that problem. For example, if I get to decimals, a whole one might be friendly. Or decimals and fractions, a whole number might be friendly. Kids need to really understand place value, not just the place labeling, that the value of the place depends on the place that the digits in. Doubles are important. We need kids to have experience sort of owning doubles, being able to adjust from a friendly number. If I add a bit too much, can I adjust. If I add not enough, I add a friendly number that's not big enough, can I adjust more than that? The commutative property, kids need to like feel the commutative property. Not just in naked numbers, but hugely in context. The associative property is really what we're basing all of this adjusting on, that it's not just enough to have partial sums. Which is kind of what these types make me feel like, that partial sums is all about place value partial sums, where I'm always adding things by place value. But really, if I can use the associative property well, then that's where I'm doing things like adding 55 plus 38 by thinking about adding 55 plus 30, and then adding the 8. Or in the example that we did in the Problem String last week, adding 58 plus 40, and then adjusting back by 2. And that's all about using the associative property. All of those. It's not about random ways of chunking numbers. It's very specific ways about chunking numbers. I was just looking at a different progression the other day, and they particularly had lumped together addition problems where you keep one of the addends whole and you compensate... Not compensate. That wasn't the word they used. Basically, you use the associative property with the other number. And I said, "Well, does it matter how?" And they were like, "No. You can chunk it any way you want." I was like, "Ooh." We would suggest there's two really important ways that kids need to develop. Sure, they could chunk it any way they want, but we don't really want them doing that. We want them chunking it either to get to a friendly number and then add the rest or to add a friendly part first and then adjust. 

Kim  17:46  
And

it because those are based on different relationships that are being used. It's not that one is better than the other necessarily. But one is about getting to a friendly number, so combinations of 10 or combinations of 100. And the other is based on adding a friendly 10, so the place value that happens when you add more tens. Those are those totally different thinking.

Pam  18:06  
Yeah,

those are different aspects of place value. And so, we want kids to develop both of those.

Kim  18:11  
Yeah, yeah. 

Pam  18:12  
Yeah, not so that they find their favorite, and they, "Okay, kids got this one. We're going to leave them in that strategy." No.

Kim  18:17  
Right.

Pam  18:17  
Because then they're missing a whole... I was going to say "chunk", but that's not the word I want. A whole relationship. A whole set of relationships that we need to build in kids. Not so that they're just adept at adding multi-digit numbers, so that they are reasoning additively. Like, our goal is additive reasoning, and so all this stuff comes in. When you just mentioned that one of those was all about getting to or using the friendly partner, so partners of 10, partners of other friendly numbers, partners of 100, those are super important relationships and inherent. And that's going to be Over. And then also the notion of equivalence is super important. I feel like you took a breath. Did you want to say something? 

Kim  18:55  
No, no. (unclear). 

Pam  18:56  
Okay, cool. So, if all of that mathematics is super important for kids to develop, so that they are reasoning additively, just compare that just a little bit with for students. I can't quote him exactly. But something in the book about if we want students... Adding two-digit numbers can be approached through thin slicing because we have these multiple types of tasks, and the types of tasks are all about if we have to kind of kick over the place value. I mean, that's maybe a way that you can get kids adding two-digit numbers. Not our goal. Our goal is that kids are adding two-digit numbers because they're building additive reasoning. We want additive reasoners. And as a byproduct of that, they're super successful in adding two-digit numbers. That's a shift. That's a different way of sort of looking at the purpose of math class. So, how do we do that? Well, "do that". What you're "that", Pam. How do we get kids reasoning additively? Well, we would advocate that you're going to do Problem Strings, facilitate Problem Strings for adding 10 plus any number, and you're also going to do Problem Strings for Adding a Friendly Number where it's not just adding 10 but you're adding multiples of 10. Then, or in addition to, at same time, you're going to do Problem Strings for getting to a Friendly Number, and then adding what's left over. You're also going to do Problem Strings where you add too much, and then adjust back. That's like the Problem String that we did in last week's episode. We also think that you should facilitate Problem Strings where it's more of an equivalence perspective where kids kind of get good at simultaneously Giving and Taking. Sometimes people call that compensation, but it's a really specific way of we're kind of compensating all along. All of any of the major strategies use a form of compensation.

Kim  20:44  
Mmhm. 

Pam  20:44  
Give and Take is a very specific form where you're kind of simultaneously giving to one addend, taking that to make it super friendly, while you're simultaneously taking it from the other to end up with an equivalent problem that's easier to solve. All of that builds senses of magnitude, and size, and place value, and a sense of what it means to reason additively. If we do that kind of work, then subtraction... Excuse me, students won't see subtraction as a whole new thing. Consider that if we're really building additive reasoning, we're already moving around this number line map that we've created in their minds. Again, listen to last week's episode where I painstakingly described how I was representing the thinking. If I'm in a thin slicing, I don't have that opportunity. I might at some students boards, and I could pull kids together at the end and sort of show some of the modeling I was doing. But if we're doing a Problem String whole group, then that modeling, that representing of the thinking, creating that mental map, having the discussion about that mental map is happening in between each problem to therefore high dose kids with that pattern, with that mental map, so that then they have the opportunity to use that thinking in the next problem. 

Kim  22:03  
Mmhm.

Pam  22:03  
If I don't do that till the end of a thin slicing where we consolidate, then I've lost out on that. Kids could solve every one of those problems using the same strategy the whole time. They could do it on a calculator. They may never see those patterns because they're solving any way they want, but then... We do that in a Problem String. We solve it any way you want. But then we highly influence students because we're high dosing them with patterns because of the modeling and the strategies that are happening in between each problem. Because we then have created that number line map in their minds, when they get to subtraction, it's not a whole new thing. If you think about subtraction, sort of talk traditionally, or even maybe now that we've learned to add, now we're going to learn to subtract. And kids start just kind of playing around with subtracting the ones digits and the tens digits. Sure, they're going to run into the same kind of when you have to regroup kind of places if you do it that way. But man, if we've done it with Problem Strings and creating these number line mental maps, students have already been moving around on that mental map. When we've been adding Over, adding a bit too much  like we did in last week's Problem String, they've already been moving forward and backward on the number line. Their subtraction isn't now this whole new procedure, this whole new thing to think about. It's now we're kind of building on what we've already been doing. Also, dump students into word problems, Kim. Kim, you and I have talked about this ad nauseam. That teachers see word problems as completely different than 1-29, Odd because we've kind of done this like, "Alright, like mess with these numbers in this way. Do what we've been doing over and over and over." But if we haven't build additive reasoning. Or maybe I'll do positive. When we build additive reasoning, students have been reasoning additively. When they get to a quote, unquote "word problem" or contextual problem, they've been reasoning additively. Now, they continue to reason additively. When they read the word problem, they go, "Huh. What's happening here? How can I use this mental map that I've been creating and these relationships that I've been creating. They're not... What they're not doing is saying, "Oh, what do I do here? I'm going to like flip a coin. Which operation am I doing?" They're really much more just reasoning through the problem. And if we've built their additive reasoning, they're able to reason additively and they're not just grabbing for formulas. Word problems become a whole... Golly, it's a natural extension of what we're doing as we're reasoning additively in a Problem String. Students won't just reach for that thing they were doing on a particular day. They'll reach in their minds for the connections they've built. 

Kim  24:50  
Yeah, and

I think all that was really good stuff. I think maybe what you're suggesting is that if we are building these things in the midst of a Problem String and immediately giving kids an opportunity to grab onto it and make use of it with the very next problem that there's much more likelihood that they will try something rather than consolidate at the end, and then maybe in another day or two hope that they will grab onto something that happened kind of at the end of a thin slicing.

Pam  25:19  
Mmhm. 

Kim  25:20  
That we're giving them a chance right away to to make use of what's going on.

Pam  25:25  
Make use of the connections, the relationships, the patterns that they're seeing. Yeah. We're highly suggesting those all the way through the Problem String.

Kim  25:33  
Yeah. 

Pam  25:34  
Some kids can look at thin slicing, and they can see the patterns, and they can get from the patterns. And even before the consolidation, they're like, bam. Other kids can get stuff from consolidating, and then hopefully use it again coming up. Most kids need more, a higher dosing of those patterns. And that comes from the facilitation in between the problems when you facilitate a Problem, String. 

Kim  25:57  
Yeah, I think... 

Pam  25:58  
Oh, go ahead. 

Kim  25:58  
I

was going to say I just think it's important to summarize, to say that for us it's not about solving problems. It's about building students brains to think more multiplicatively, to build more proportional reasoning, to build more functional reasoning. We really feel strongly that it's about building their brains to be able to handle more complex thinking.

Pam  26:20  
Building

those mathematical, mental connections. Yeah. So, then... 

Pam and Kim  26:24  
Yeah.

Pam  26:24  
Yeah, well said. Nice. Y'all, we have a lot in sync, pedagogically with Peter Liljedahl. There's great stuff there. Problem Strings are not the same as thin slicing. 

Kim  26:34  
And

hey, I'm going to interrupt here because we have been a bit remiss, because we've mentioned Dr. Peter Liljedahl this entire time and not his co-author. Which apologies to Maegan. I think... Her last name, I'm going to butcher.

Pam  26:49  
Go for it. 

Kim  26:50  
Giroux. Anyway, massive apology.

Pam  26:53  
Maegan, let us know how to say your last name, and we'll fix it. 

Kim  26:55  
Please, please, please. Yeah.

Pam  26:57  
Ya'll, there are some amazing tasks in that book. Already we've had a lot of fun looking at some of the amazing tasks. Problem Strings can be a way that you can help give kids a higher dose of the patterns. 

Kim  27:12  
Mmhm.

Pam  27:13  
Cool. 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!