Ep 253: Choosing a Problem String

Which Problem Strings are appropriate for your students? In this episode Pam and Kim discuss what they consider when choosing a Problem String for a classroom.

Talking Points:

• A fraction addition example string
• Things to consider when choosing a Problem String:
  • Prewritten vs write your own
  • Time of year
  • Strategies students already own
  • Building a model vs building a strategy
  • Order of sophistication of strategies
  • Building a model vs strategy
  • New content versus review/reteach

See episodes 33, 71, 72, 163, 226, 241 for more on Problem Strings!

Check out our social media
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Facebook: Pam Harris, author, mathematics education
Linkedin: Pam Harris Consulting LLC 

Transcript

Pam  00:00

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

Kim  00:09

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  00:17

We know that algorithms are amazing achievements, but they're not good teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop. 

Kim  00:31

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

Pam  00:38

We invite you to make... We invite you to join us to make math more figure-out-able. Kim, this doesn't bode well for the episode. I don't know.

Kim  00:49

It's going to be great. 

Pam  00:50

Whoo! Alright, we can do it. I'm alive. I'm awake. 

Kim  00:54

Oh, gosh. Okay, well, at least somebody likes the things that we talk about here.

Pam  00:59

Bam.

Kim  00:59

We got a review from CJTheriot? Last names.

Pam  01:00

Harriet? Theriot? Theriot? Yeah, one of those two. Yeah. CJ. 

Kim  01:09

CJ says, "I'm pretty good at math, and over time, I sort of figured out a few of the strategies that Pam and Kim teach on my own." So, he's kind of like me, maybe.

Pam  01:18

Nice. 

Kim  01:18

"But at no point did I have any way to model what I was thinking until I found this podcast."

Pam  01:23

Woohoo! 

Kim  01:24

Yeah. So, "Had these models and strategies been taught when I was in elementary, middle, and high school, not only would it have made me better, but it would have helped so many of my friends who thought that math was only rote memorizable and not figure-out-able. And now that I teach math, I try to incorporate thinking and reasoning over algorithms and rote memory every opportunity I get."

Pam  01:44

Oh, that's amazing! 

Kim  01:46

Yeah, that really resonates with me and kind of the story of my growing up. Yeah, that's pretty cool. Thanks. (unclear).

Pam  01:52

Well, and I'm noticing you pasted it in here that it says "Game changing!!!!" with four exclamation points. Aww, nice.

Kim  01:58

Yeah. 

Pam  01:58

Yeah. Five stars, thanks. Yeah, that's so well said that math is not rote, memorizable . Wait, "...my friends who thought that math was only rote memorizable and not figure-out-able." Yeah, that's super well said. Yeah. Yeah. Cool. 

Kim  02:08

(unclear). I'm excited about the things we're going to talk about the next couple weeks because we get to probably share some stories about us going into classrooms, and we spend some time, part of the work, besides writing and producing some content. Part of the work that we do is going into classrooms. 

Pam  02:33

Mmhm.

Kim  02:33

And we film strings with other people's students. 

Pam  02:37

Mmhm.

Kim  02:37

Which is super fun, and we're so grateful for the teachers who've let us into their rooms.

Pam  02:42

Yes.

Kim  02:43

But it also can be tricky because problem sting... Problem stings. Ugh! 

Pam  02:51

We're doing great. 

Kim  02:52

I know, right? So, Problem Strings, right, you want to make careful choices in planning, and preparing for them, and what you do with students.

Pam  03:01

Mmhm.

Kim  03:01

And so, that work that we're doing going into other people's classrooms, we thought that we would spend some time unpacking a little bit. So, if you want to do a Problem String, what are the things that you need to think about.

Pam  03:13

And to start this, kick it off, let's do a quick Problem String that then we can refer to...

Kim  03:19

Sure.

Pam  03:19

...throughout the episode. Alright, so Kim, I got a few questions for you.

Kim  03:22

Okay.

Pam  03:23

I just heard when I was actually in California recently, a bunch of people said to me, "We really enjoy the fact that you give Kim these problems, and we get to kind of hear what you're thinking about." So, let's do that. (unclear).

Kim  03:33

Okay. We'll see if I'm sharp enough today.

Pam  03:37

Today, neither of us might be.

Kim  03:38

Right. 

Pam  03:39

It's going to be great. It's going to be great. So, I have my handy pen.

Kim  03:43

Got a pencil. I just sharpened a pencil right before we started today.

Pam  03:46

I believe you, yeah. Do you have an electric sharpener (unclear)? 

Kim  03:49

Oh, yeah, for sure. 

Pam  03:50

Do you ever do a hand or it's always electric? 

Kim  03:53

Oh. Oh, that would be be kind of fun. I like a good hand cranking pencil sharpener, but these are smooth. Electric pencil sharpeners are so smooth. And...

Pam  04:02

Oh, keep going.

Kim  04:03

I was going to say. Well, and I don't share it with a classroom of kids anymore, and so you know, they break up when there's a bunch of kids messing with it.

Pam  04:09

Yeah, totally. I can handle a good hand one, but it can't be the kind that like it's crooked and like and the pencil comes out.

Kim  04:20

You need a trash can right underneath it. 

Pam  04:22

Yes. Yes. If it's the kind that doesn't catch the shavings. No, can't do it. Can't. Can't have that stuff everywhere. It's like (unclear).

Kim  04:30

Alright, give me a problem. 

Pam  04:31

Alright, here we go. Kim, what is 10 and 1/2 plus 6? 

Kim  04:37

Oh, sweet. It's 16 and 1/2.

Pam  04:39

Okay, so that's not too bad, right?

Kim  04:40

Yeah. 

Pam  04:40

So, even though that's not too bad, I'm going to go ahead and just sketch on my paper. So, if I was doing this with students, I would say, "Hey, great. You know like 10." I might even say, "So, you thought about like 10 and 6, and then you added the..." Oh, I have to take my glasses off, so I can see. And then you might tack on the half at the end. Whatever. So, I might say like if 10 and 1/2. And I've just written a number line. And on a location at the left hand side, I've written 10 and 1/2.

Kim  05:06

Mmhm. 

Pam  05:06

So, that's where we are. And then you jumped 6, and we landed. And so, now I've done the jump of 6, and the landing place is 16 and 1/2. And I've just kind of written that number line. Cool. Next problem. How about if I asked you for 10 and 1/2 plus 5 and 3/4. So, I know this is audio, so let me just say we have 10 and 1/2 plus 6 is 16 and 1/2 already on the board. And then I've asked you 10 and 1/2 plus 5 and 3/4. Okay, what are you thinking about?

Kim  05:35

6. Adding 6 was a little bit too much. So, then I'm going to say 5 and 3/4 is one-quarter smaller than 6. So, if adding 6 landed me at 16 and 1/2, then I'm going to back up a fourth to land on 16 and 1/4.

Pam  05:57

Cool. And how do you know that 16 and 1/2 minus a fourth is 16 and a 1/4.

Kim  06:02

Because one-half is two-quarters. 

Pam  06:05

So, did you do a lot of like common denominator and like some rule to find a...

Kim  06:10

No. 

Pam  06:10

Or do you know something about halfs and fourths?

Kim  06:13

Mmhm, yeah. 

Pam  06:13

And ideally, would that be like a third grade thing?

Kim  06:17

Yeah.

Pam  06:17

If you've got a half, then you have two-fourths. If you have two-fourths, you have a half. And that relationship would be like kind of the definition of fractions.

Kim  06:26

Yeah.

Pam  06:26

Like, what does it (unclear).

Kim  06:27

I actually think it could be earlier because...

Pam  06:29

Okay.

Kim  06:30

...at least in the States when you mess with money.

Pam  06:33

So, for sure by third grade.

Kim  06:35

Yeah.

Pam  06:35

That should be something that kids could just own and not turn into some procedure that they... Okay, cool. What if... Next problem. 4 and 1/5. 4 and 1/5 plus 4.

Kim  06:48

8 and 1/5. Yeah, I wondered actually, if you were going to redraw a new number line or if you were going to add on to the one that

Pam  06:49

Okay, so I'm just going to sketch that out. 4 and 1/5. Oh, I didn't tell you what I drew on the 10 and a 1/2 plus 5 and 3/4. As you were talking, I added the 10. I started at 10 and a 1/2 and I added 6 just like the one I had done above. So, I created a new number line. Added 6. Landed in the same sort of location. I've kind of lined them up, landed at 16 and a 1/2. And then when you said it was too much, I backed up a quarter, and then just backed up a tiny bit. And before that 16 and a 1/2, I wrote 16 and a 1/4. Just so because we're audio. (unclear).

Kim  07:26

I. 

Pam  07:27

And you know what? I asked myself that question as you were solving it. And I decided if I was really with students, I would have redrawn the number line. At least for this first problem. Okay, second problem. 4 and a 1/5 plus a fourth. What did you say? Yep.

Kim  07:42

8 and a 1/5. 

Pam  07:43

Okay, so I've put 4 and a 1/5 in a place, and I have jumped 4, and I've landed at your 8 and a 1/5. Next problem. What is 4 and 1/5 plus 3 and 9/10?

Kim  07:57

Mmhm.

Pam  07:59

Okay.

Kim  07:59

So, if I add 4, that's 8 and a 1/5. 

Pam  08:04

Because we did that before, mmhm.

Kim  08:05

Mmhm. But that's one-tenth too much because 3 and 9/10 is one-tenth less than 4. So, I want to back up a tenth. 

Pam  08:16

From? 

Kim  08:17

From 8 and a 1/5. 

Pam  08:19

Okay, so 8 and a 1/5. Back up a tenth. 

Kim  08:21

Mmhm.

Pam  08:22

Yeah.

Kim  08:22

So, one-fifth is the same as it's equivalent to two-tenths.

Pam  08:27

Mmhm. 

Kim  08:28

So, if I back up a tenth, then I'm at 8 and 1/10.

Pam  08:33

And so, I've drawn the same thing I had drawn before. I, again, drew two number lines. So, I had your first 4 and a 1/5 plus 4 is 8 and a 1/5. And then I redrew that. And then I backed up from the 8 and 1/5. Backed up a tenth. Then, I wrote 8 and a 1/10 as the landing spot. Cool. Next problem. How about if I ask you 6 and 1/3 plus 5. 6 and a 1/3 plus 5.

Kim  08:56

Mmhm.  Mmhm. 11, and 1/3. 

Pam  08:57

Cool. So, I've just kind of sketched over here 6 and 1/3. I've been a jump of 5, and I've landed on 11 and 1/3. Next problem. How about 6 and 1/3 plus? And I'm just going to pause for a second. So, I realize, if you're driving in the car, wherever you're listening to us, you haven't been writing all this down. But I might at this point say, "Hey, if you kind of looked at the board up here, or looked at paper or whatever, anybody want to guess my next problem?" Because I had 10 and a 1/5 plus 6, 10 and a 1/2 plus 5 and 3/4. 4 and a 1/5 plus 4, 4 and a 1/5 plus 3 and 9/10. 6 and a 1/3 plus 5, 6 and a 1/3 plus? And then I'll let kids guess. And I'm going to say it is 6 and a 1/3 plus 4 and 5/6. Alright, what do you got?

Kim  09:20

So, 4 and 5/6 is one-sixth less than the 5 I've already added.

Pam  09:54

Okay.

Kim  09:54

Or could have added. So, I'm going to back up one-sixth from 11 and a 1/3. Which is what I got when I added 5.

Pam  10:02

Mmhm.

Kim  10:03

And back 1/6 is 11 and 1/6.

Pam  10:07

11 and 1/6. And how do you know that?

Kim  10:11

Because when I had one-third, that was two-sixths. And so, if I back up a sixth, then I'm at 11 and 1/6. 

Pam  10:19

11 and 1/6. And how do you know that two-sixths is one-third.

Kim  10:26

That's a great question. How do I do that? I mean, I guess I've just messed with. I think about like the ratio of 2 to 6 is the same as 1 to 3.

Pam  10:36

Okay, okay. And if we had something we were sharing equally, three of us were sharing equally, you'd get a third, I'd get a third, the other person would get a third. But if I said, "Yeah, but we're actually going to share 6 people, 6 of us are going to share equally." How would those sixths relate to those thirds? And we would want to make that connection, right, early, early with kids. That's like the definition of unit fractions. What does it mean to be one-sixth? How does that relate to one-third? If you actually have kids take a strip of something and say, "Hey, fold that into sixths, often, they'll do it two ways, and one of the ways is the way you just did it. They'll fold it into thirds, and then they'll cut those thirds in half to get sixths.

Kim  10:38

Mmhm.

Pam  10:39

Can you think of another way kids might do that?

Kim  10:48

Can I admit that I'm not listening?

Pam  11:05

Sure.

Kim  11:07

Sorry.

Pam  11:13

Another way the kids might might find 1/6 is they might fold the thing into half, and then cut the half into 3 chunks to get sixths.

Kim  11:38

There's a red Cardinal out my window. So sorry.

Pam  11:42

Hey, do they do sound pretty? Because we were out on a walk, and I was like, "What is that sound?" Like, it was so... My son pointed out. He's like, "I think it's that Cardinal right there."

Kim  11:52

Maybe. We have a ridiculous amount of birds, and so we hear them all the time, but I have not distinguished one from another kind. 

Pam  11:59

Do you do bird feeders? Is that what you have birds? 

Kim  12:01

We have one in the front of our house. It probably doesn't have any stuff in it because that takes regular maintenance.

Pam  12:07

(unclear) regular maintenance (unclear).

Kim  12:10

Oh, geez. 

Pam  12:11

Let me guess, you didn't hang it up.

Kim  12:13

I did not.

Pam  12:14

Whoever hung it up, you were like, "Okay, it's on you, sucker."

Kim  12:18

My youngest just recently said, "Hey, we should probably put some stuff on that." Nice.

Pam  12:22

Alright. So, Kim, we've done a string where I gave you a problem where we kind of added a nice number.

Kim  12:29

Mmhm.

Pam  12:30

And then I gave you a problem where we added something that wasn't as nice that was a little bit smaller than that.

Kim  12:35

Yep. 

Pam  12:35

And you chose every time to kind of say, "Well, if I know what that adding that nice number is, then I can just back up a little bit to kind of adjust." And so a little Over with fractions. Okay, cool. So, we did that Problem String. 

Kim  12:47

Mmhm.

Pam  12:48

If we were going to do that Problem String, or if we're in a place where we're trying to choose Problem String to do with kids, what are some things that you would ask yourself as you're thinking about, hmm, how would you choose this string? Or like what in general kind of, "What do you ask yourself about...

Kim  13:07

Yeah

Pam  13:08

...choosing a Problem String.

Kim  13:09

So, one of the first things that I think we could think about is, am I going to choose a pre-written string? Or am I going to try to write my own? And I don't think that that's a trivial question. I think there's a lot of people who are attempting. You know, they're seeing great strategies out there, and they're like messing with numbers, and they're thinking, you know, what do I want to do with my students? And so, they're attempting to write strings. And, you know, I think up front, we should just say it was a lot of time had passed before you and I were writing Problem Strings. 

Pam  13:43

And not just time, right? It's not just like...

Kim  13:45

Oh, yeah. An experience. Yeah.

Pam  13:47

Experience.

Kim  13:47

Yeah. 

Pam  13:48

We were facilitating pre-written Problem Strings that someone else wrote, who'd thought carefully about the relationships, knew the structures of strings. And we would be like, "Huh, I wonder how this one's going to play out?" 

Kim  14:00

Yeah. 

Pam  14:00

And then we give it a go. And also one of the cool things, I think, that was super, what, helpful for me was I would do a Problem String a few times in a workshop or with students, and then you and I would do one together, like a workshop together, and you would facilitate it and I'd be like, "Huh, why did you do that?" Like, you do a move I hadn't thought about or you wouldn't do a move I did. And then the two of us we would like scurry together. You know like, "Hey, hey, tell me more." Oh, my gosh. I totally just dropped stuff on the floor. Our editor's like, "What are you doing, Pam!" Sounds go everywhere. That was really impactful for me to realize that I was making certain choices to facilitate the strong string in a certain way and you were making other choices. And, man, we learned from each other. Like, there were times where I would adopt what you were doing, and there were times, I think, that you were adopted what I was doing. And there were times where we'd both stop something and choose, you know, to wonder about something else. So, that was, yeah.

Kim  14:01

And some of those moves were specifically about facilitation, but some of those moves that we made were based on the numbers and the string. And so, at that point, we really looked hard at the types of numbers and really hone skills on writing strings. But the good news is you can still facilitate Problem Strings even before you feel ready to write your own. It's great. It's a great practice. Start writing your own and compare with other people. But you can still facilitate well written strings even before you feel ready to write your own. We have tons of Problem Strings on the Math is Figure-Out-Able website. And great news, we have grade level Problem String books that you can purchase to get a years full of strings. So, first decision.

Pam  15:53

Well, let me pause. So, that's kindergarten through fifth grade. And if you're a middle school, then I've got Lessons & Activities for Building Powerful Numeracy. And if you're high school, Algebra Problem Strings Advanced Algebra Problem Strings. Great pre-written strings. And, Kim, I run into a lot of people who say, "Okay, I know I'm supposed to be writing strings." And I'm like, "Whoa!"

Kim  16:09

Yeah, yeah.

Pam  16:10

You can just facilitate pre-written strings. I don't think teachers have to be content creators and expert teachers at the same time.

Kim  16:18

Right.

Pam  16:19

I think they can find good content and just get really good at facilitating. Yeah.

Kim  16:24

Yeah, because there's so much to consider, even in a bank of problems. So, it's really about knowing your content and knowing your kids. So, things to think about. Where are you in your year?

Pam  16:35

Mmhm. 

Kim  16:35

Is it the beginning of the year? Middle of year? End of the year? What precursor knowledge do students already have or do they need to have in order to do a string that you thinking about choosing? 

Pam  16:46

Yeah. 

Kim  16:46

So, like the string that we just did.

Pam  16:50

Yeah. Can I tell you what I was thinking about?

Kim  16:51

Yeah.

Pam  16:52

So, one of the things that I considered was what would be numbers that I was pretty sure podcast listeners could kind of hang on to? Some of you are like, "I had a hard time hanging out with those fractions." And I'm aware of that. Like, I'm aware that when I say 10 and a 1/2 plus 6 that probably most people were able to kind of hang on to that, adults listening to the podcast. And so, that, on purpose, was like the first problem because I was pretty sure you'd be able to hang on to that. And then, I don't know if you noticed, but I tried to say often, "So, if 10 and a 1/2 plus 6 is 16 and a 1/2." You know like, say that over again, so that then maybe you have a prayer of hanging on to 10 and a 1/2 plus 5 and 3/4. Because 10 and a 1/2 and 5 and 3/4 quarters, many people will be like, "Whoa! Like, I got to write that down to hang on to those." Totally that makes sense. But I was pretty sure that the fractions I chose, especially adding a whole number, at least people could hang on to that first bit, you know, the helper problem of the string. But I was also aware there's probably going to be some people listening to the podcast who are like, "Wait a minute, you put a fraction on a number line?" And that might be a new idea. I often work with teachers where I'll start putting fractions as a location on a number line, and they go, "Whoa, I've I've never thought about that. I've always thought about, you know like..." We did one of the problems was three-fourths. "I've always thought about that as three "over" 4, and I've counted 3, and I've counted 4, and I've shaded the 3 out of the 4." So, they've only ever thought of three-fourths as 3 and 4, and not really as a location on a number line, that I could go over 5 and then 3/4 more of a unit and mark that as a location.

Kim  18:34

Mmhm. 

Pam  18:35

So, I was aware of that tension, but I thought the way we did it would help people kind of come along. And one of the reasons that I kept talking about what I was drawing was to help people realize you can put a fraction as a location on number line. 

Kim  18:53

Yeah. Yeah, because the idea is that they can enter into the Problem String, that right off the bat it's not like, you know, smack in the face. I'm not even sure how to get started. So, where you are in the year and what pre-cursor knowledge is super important to think about. The other thing that is important to think about, I think we all will address, is what content do you need to address right now or that you're trying to get at? While it's fine to be working on, you know, an addition string while you're also working on division, there are some times where staying in a particular lane is super helpful for kids. And we'll talk later in another episode about how you choose what to do at what time. But if you're working on division, and that's the meat of your class, you might want to stick with a multiplication or division string, so that kids are tinkering with strategies that you're going to want them to use in the rest of their math time.

Pam  18:55

And they're building multiplicative reasoning, which is necessary for them to reason through division.

Kim  19:38

Yeah.

Pam  19:39

Yeah, nice. 

Kim  19:40

Another really important consideration is what size numbers are your students ready for?

Pam  19:47

Yeah. Yeah, that's a big one.

Kim  19:48

Yeah. So, if it's beginning of the year, third grade, clearly, I'm not going to start with three-digit multiplication times two-digit multiplication. We might get there, but we want to think about what can my kids handle, and what's the ramp up of this string look like. Even strings that start with smaller numbers in the beginning could ramp up pretty quickly inside of a string. So, just looking at the numbers, kind of analyzing what's happening throughout the string. Are the kids going to be able to handle what's going on? We want a little bit of a push but not so much that that right off the bat they're struggling with the numbers and they can't own. Yeah, they can't mess with the strategy that's happening. Because when the numbers are too big, often we see kids revert back to less sophisticated strategies.

Pam  20:51

Mmhm, mmhm.

Kim  20:52

And so, we want to keep it within their zone of proximal development.  For sure. 

Pam  20:56

Would you say that when we're starting a strategy, we actually kind of keep the numbers really manageable, so that kids aren't... They're not fussing with the numbers so much. They're noticing the pattern of the strategy. But after they kind of have an inkling of the strategy and they've started using the relationships, then we make the numbers harder, so they're also building kind of place value and sense of numbers because they kind of have a sense of the strategy. Yeah, nice. 

Kim  21:06

Yeah. 

Pam  21:07

What else?

Kim  21:14

Also something to consider is what strategies do students already own? So, you know, we have talked a lot about the different strategies for each operation. And you know, in your your big ebook, we've outlined what those strategies are. And there's kind of a sophistication order to them. A lot of people see that most sophisticated strategy and they say, "Oh, sweet, that's the one I'm going to teach my kids."

Pam  21:48

Let's go there. Let's skip all that unsophisticated stuff. 

Pam and Kim  21:52

Yeah.

Kim  21:52

But in learning and developing the precursor strategies, the other strategies, kids are building their brains. Let's say multiplicatively. If it's multiplication strategies, they're building their brain multiplicatively as they learn more strategies, so that they can be equipped to handle the most sophisticated ones. So, what strategies do kids already own comes into play. You're not going to start a year teaching the most sophisticated strategy.

Pam  22:18

Mmhm. Depending on the age of the kids. So, I might take high school kids.

Kim  22:23

Oh, that's true. Yeah, older kids you might. 

Pam  22:24

Yeah. I might take high school kids and just do Give and Take for addition 

Kim  22:28

Yeah.

Pam  22:29

And not mess with the precursor ones because ideally at that... Now, it depends on the kids. But if I have sort of a what's a mainstream kind of Gen Ed population, I think I probably could start with some down the numbers again, but like getting them kind of thinking about if I got pile of marbles. Yeah, so depending on the age of the kids, you can.

Kim  22:50

Yeah, I agree because... 

Pam  22:51

Skip some of the precursor.

Kim  22:53

Their work is not... While you need to build additive reasoning with them, that's not the work of their grade level neccessarily,

Pam  23:00

Right, so we're going to do a little bit of that work. Yep, yep. Nice. 

Kim  23:03

And people will notice that we did that in the K-5 Problem String books. Oh, yeah. That's true. I don't know if we've ever said that out loud, but as the books get older grade wise, we made some choices about leaving off some of the lesser, sophisticated strategies, so that we can deepen, you know, broaden the size of the numbers (unclear). 

Pam  23:22

And as the grades go down, we even built precursor strategies with smaller numbers, like single-digit numbers, before then it kind of grew up into, kind of the major strategy with larger numbers. Yeah.

Kim  23:34

Yep.

Pam  23:34

Yep. 

Kim  23:34

So, another thing to consider is am I trying to build a model or am I trying to build a strategy?

Pam  23:39

Mmm, mmhm. 

Kim  23:40

Because there are strings that do both. And if it's a time where my kids have not messed with a model before, they haven't experienced that, we haven't developed it, then I'm going to maybe make some choices about the numbers, so that they can make sense of the model.

Pam  23:55

Yeah, down the numbers, so we can just make sense of the model. 

Kim  23:58

And when we say "down the numbers", that means we're just going to be looking for pre-written strings that have smaller numbers. That doesn't... Don't hear that as us saying, "Okay, now write one that has smaller numbers." We're still living in the land of there's lots of pre-written stuff to choose from, but those are the kinds of things you're thinking about. 

Pam  24:14

Yeah. And "down" also can mean less complex. 

Kim  24:19

Mmhm.

Pam  24:19

It doesn't necessarily have to mean smaller. So, I might deal with like the Over strategy that we just did. I might deal with, you know, add 200, add 199.

Kim  24:29

Mmhm. 

Pam  24:29

That could be less complex than add 80, add 77. Because it's just one down versus 3 down.  I don't know if that's the best example, but. Or add 600 to help with add 599 is going to be easier than add 200 to add 188. 188 to 200 is a bigger gap. Yeah, okay. So, less complex numbers. Mmhm. 

Kim  24:37

Yep. Yep.

Pam  24:54

Okay. 

Kim  24:54

Also thinking about what models have they experienced? So, are you building the model? Are you extending the model?

Pam  25:01

Mmhm.

Kim  25:02

Are you trying to take a strategy they know and they've experienced on one type of model, but now you want to experience that strategy on a different model?

Pam  25:12

Ooh, so a really good example of that would be the string that we did today.

Kim  25:15

Mmhm. 

Pam  25:15

I don't think I would do an Over strategy with fractions without having done an Over strategy with whole numbers first.

Kim  25:23

Right.

Pam  25:23

Which could mean not, not an entire. So, if I'm doing fractions like these, I'm probably in at least fifth grade up. Maybe fourth grade. I could probably do this in fourth grade. But not, for sure, until I had done work building whole number, the Over strategy with whole numbers. But if I was in like middle school or high school, and I wanted to do these fractions, I could have done these exact six problems that we did, but I probably would have done a helper, clunker pair to start this string with whole numbers.

Kim  25:49

Mmhm.

Pam  25:50

Or maybe even two with, you know like, more complex whole numbers, and then go into fractions. 

Pam and Kim  25:55

Yeah. 

Pam  25:56

What else?

Kim  25:57

Another thing. I kind of mentioned this earlier. But is this a in my content, right now, in my curriculum, that I'm supposed to be working on in my core group? Is this in my content string? Or is this a review string? Like, am I going to cycle back to a strategy maybe my kids have already experienced, but I don't really see them using anymore, and so but I want them to? It's sophisticated, and maybe they just aren't really using it. I want to circle back to it? 

Pam  26:23

Mmhm.

Kim  26:23

Or is it something we're just seeing for the first time? 

Pam  26:27

So, you want to consider both of those. Those should both be happening in your classes. Mmhm. What else?

Kim  26:31

When I go see students for the first time, one of the first questions I ask their teacher is have they ever done Problem Strings?

Pam  26:38

Mmhm. 

Kim  26:38

I know that about my own students. But is this string going to be routine? Or am I introducing procedures? That might affect the length of the string that I'm going to do. (unclear).

Pam  26:52

The intro, right?

Kim  26:54

Mmhm, longer strings.

Pam  26:56

My intro differently. If kids are starting to copy down everything, you might have to have a conversation about this isn't about copy mode. 

Kim  27:03

Yeah.

Pam  27:04

Yeah. Yeah, that's super important. You and I were just preparing to do some videoing, and both of us were asking that question. Like, have kids ever? And it was drastically different in the room that one of us was going in the other one. And so, one of us could just like dive in and do a new strategy with new numbers, and the other one was like, "Let's talk about what your role is in this." Because one of the things that we often get people missing, that they'll misunderstand, is that we're advocating some loosey goosey, go have kids wander around discovering math on their own. No. Not at all. And they they think that we're advocating like an unstructured, unclear, kids don't know what their job is. And very to the contrary, when we're doing a Problem String, we're going to be help kids be extremely secure in what their role is during a Problem String.

Kim  27:52

Mmhm. 

Pam  27:52

So much so that that's when kids go, "It's almost... It's almost like you want us to use what we know to solve the problem." Yes. Yes, that is your role right now. Use what you know. Let's explain each other's thinking. Yeah, yeah. What else, Kim?

Kim  28:05

I think the last one that that I was thinking was time. How long do you have to do a Problem String? Again, that will affect the number of problems that you want to do.

Pam  28:15

Mmhm, mmhm.

Kim  28:16

If you have just... You need a quick Problem string, you might do just a few problems. If you have a little bit more time to explore, then you might do a longer string. Again, that also has something to do with if it's routine yet or if they've used a particular strategy before.

Pam  28:33

Yeah, nice, nice. So, all really good things to think about when you're choosing a Problem String. 

Kim  28:40

Yeah.

Pam  28:40

Thanks for letting us in your head. 

Kim  28:42

So, we have other Problem String focused podcasts.

Pam  28:46

Yeah, so if you're interested to listen to us talk more about Problem Strings, check out episode 33, 71, 72, 163, 226, and 241. Good gravy, Kim, have we really done that many podcasts? That's awesome.

Kim  29:02

I mean, we do them all the time, but those are specific to learning more about Problem Strings. Listeners, we thank you for listening and for leaving us a review and for sharing the Math is Figure-Out-Able movement alongside of us. We love doing this, and we want as many people to know, so we are grateful for your sharing.

Pam  29:17

Yeah, thank you for tuning in 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!