[00:00:00.820] - Pam
Hey, fellow math-ers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned math-er.

[00:00:10.520] - Kim
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.

[00:00:18.540] - Pam
Y'all, we know that algorithms are amazing historic achievements, but they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

[00:00:31.620] - Kim
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

[00:00:38.260] - Pam
Y'all, thanks for joining us to make math more Figure-Out-Able.

[00:00:42.510] - Kim
Hi there.

[00:00:43.230] - Pam
Hey, how's it going?

[00:00:44.430] - Kim
It's great. I'm excited for this episode.

[00:00:47.240] - Pam
Oh, me too.

[00:00:47.880] - Kim
So, it's a bit of continuation from the last two. And in this episode, we are going to talk about ways that we are explicit in Problem Strings that we are delivering with our students and when we are lobbing things out. So, you know, sometimes people will say to us, "Well, I did a Problem String, but, you know, it didn't necessarily go the way that I thought or I wanted." And I think maybe this will be helpful because sometimes we are very explicit. And...

[00:01:20.740] - Pam
Some things we do are very explicit. Yeah, in the midst of actually kids grappling and looking for patterns.

[00:01:29.000] - Kim
Yeah.

[00:01:30.120] - Pam
Alright.

[00:01:30.530] - Kim
Let's do it.

[00:01:31.330] - Pam
Let's do it. So, unlike other times on the podcast where I kind of give a problem, and then Kim answers, and we sort of, I don't know, role play the way it would look in a Problem String.

[00:01:41.050] - Kim
Yeah.

[00:01:41.450] - Pam
Today, I'm going to just sort of tell you some things that I'm thinking about while I would facilitate a specific Problem String. So, if you've never heard of a difference versus removal Problem String, check out the show notes. We'll send you to a podcast episode or two where we've done a distance versus removal Problem String. The goal of that particular Problem String is to help kids go, "Huh. Subtraction can mean that I can remove the second number from the first number or it could also mean that I could find the distance between the numbers. And that's sort of two different ways I can interpret subtraction. And that's interesting. And it might make sense to let the numbers influence how I do that subtraction." That's kind of the goal of the string.

[00:02:26.330] - Kim
Yep.

[00:02:27.150] - Pam
So, when I do a distance versus removal string, the first thing that I'm really looking at is the board, the space that I have to write on. And I'm very careful to put the problems down the left-hand column. Which is kind of typical for Problem String. But then I'm going to keep two equal size columns in the middle. Or the rest of the board are going to... It's going to kind of be broken into two chunks. I'm going to ask the first question, but I have that spacing in mind to start off with. So, I give the kids the first question. I often do it in a context. Well, I'll tell you the first question. No, I'll do it in a context. Where the kids are going to... No, I'm going to tell you the question. "Pam, make up your mind!" So, I'm going to say to kids, "If you're reading a book, and the book is 312 pages long." I write down 312. So, I say it, then I write it. It's very intentional. "And you're on page 288. How many pages do you still need to read?" Then I'm going to write down minus 288. So, the book is 312 pages long. You're on page 288. How many pages do you still have to read? Now, very intentional. I have given a context that kids are sort of thinking about. I'm on 288, and I've got to read up to 312. So, I'm intentionally giving them that kind of context. Sometimes I'll give a context with smaller numbers where I'll say something like... The problem is 53 minus 48, and I'll say the score at halftime is 53, and you've got 48. Quick, how much do we need to score to catch up?" But again, I'm writing down 53 minus 48. So, those are two contexts that I'm very intentionally using. We don't always use context in Problem Strings. We love it when we can find a microcontext. Thanks, Kara M. for the term "microcontext". We love it when we can find a good one. So, in this case, it's very explicit. Then, there's nothing direct happening. Then, kids are solving the problem. Now, I'm letting them grapple with those numbers. With these numbers, I'm probably circulating a little bit. I'm looking at what kids are doing. They may or may not be writing a whole lot depending on how long I've been working with them.

[00:04:31.450] - Pam
If I can see anything, I'm looking for kids who are thinking about starting at 288 and moving up. Now, often I won't be able to see anything. Especially if I'm beginning to work with kids. If you've been modeling kids' thinking, you might have kids now be drawing number lines, and they know how to represent their own thinking. If not, then I'm going to very intentionally have this question in my head that I'm going to say, "Hey, I'm kind of curious. Did anybody kind of picture the book? Anybody sort of think about you're on page 288, and you sort of started reading to get to the end of the book?" So, it's a very intentional question to see if I can get some... And then I'll say, "Tell me about that." Then, when a kid says, "Well, I'm on page 288," I very intentionally,draw a number line on the right next to the problem. Not right next to. But in kind of that first column. And then they'll in some way talk about how they read up. And often, they'll say, "Well, it took me 12 pages to get to 300." And so, then I'll draw a jump of 12 and land on 300. And they'll say, "And then I had another 12 pages." I'll draw another jump of 12 that is very intentionally the same size as the first jump of 12. And they'll say, "So, then I landed on 312." And then I like to, if I can, go, "So, I just landed on 312, is that your answer?" Just to get anybody who's kind of tuned out or whatever to kind of like perk back up. It's kind of a move where I'll try to make every once in a while where I know that what we're doing might seem a little bit easy to some of the kids. I want to maybe do something where I just get them to focus back in again. Like, "So is the answer 312? We landed on 312?" It's also a way for me to make sure that kids don't get in the habit of always thinking that the answer is where they land. Because, again, we're looking for the pages that they read. So, then they say, "No, no, no. It was the pages I read. It was the 12 plus the 12." Now, I'm going to write a plus between the two 12s and a 24.

[00:06:22.030] - Pam
Okay, so now I've got the problem on the left-hand side. Oh, 312 minus 288. I've now written equals 24 because we've got that answer. I have the strategy in that sort of first column. And then I'll maybe say something like, "Did anybody else do that? Cool. Nice. Next problem."

[00:06:38.950] - Kim
Before you move on to the next problem. There was a time that you said, "Did anyone do that?" And I want to name that question and what you're looking for as very explicit because you were circulating and you maybe didn't see, or maybe didn't have enough on people's papers to draw out that. But you were explicitly looking for a particular strategy. And when you caught the eye of somebody, you basically said to them, "Tell me about that." Not any strategy, that strategy.

[00:07:11.550] - Pam
Yeah. And sometimes if I'm looking for a specific strategy and I say, "Hey, did anybody do that?" And no one does. So, Kim, I was just in... Where was I? Utah. I was in southern Utah. Hey, Washington County. And I did several Problem Strings where I would say, "Did anybody do that?" You know, exactly what just happened. And nothing. And I would say, "Could you?" Bam! Like, half the room lights up. And they're like. And then I run with that. So, those are very intentional teacher moves that I have in mind. Because you're right, if I would have seen somebody sort of adding up from 288 to 312, I could have just asked that kid to share with everybody. It's when I don't see it, then I have to have that question in mind. Did anybody do that? And it is very intentional. Yeah, nice. Okay, cool. Then in this particular setup for a Problem String, then I'm going to give the next problem. I'll say, "Okay, next problem. Notice, as I'm going to the next problem, I did not ask for a different strategy. I didn't say, "Did anybody remove?" I just got the different strategy up there, and now I'm moving on.

[00:08:12.340] - Pam
Now, so 202 minus 13. So, if you're listening to the podcast, and you've never done a distance versus removal, you might think about that problem before you, and then pause or whatever, and then come back. But because we're kind of talking about where I am explicit. So, I'll write that one right underneath the first problem. And I'll say, "Go ahead and solve it any way you want." You know what? Sorry. I would probably stay in context. So, intentionally, I would say, "Hey, what if the book you're reading is 202 pages long and you're on page 13? Oof. And I might even give some sort of "oof" noise to be like, "Wow." And then if I catch a kid going, you know like, eyes widening or something, I might go, "A lot of pages, huh? Like, you still got a lot to read. You're not very far in the book." I might even say that out loud to just kind of feed into the notion that if kids are thinking about reading, that there would be a lot still to read. So, a very intentional move to kind of get that in the ether.

[00:09:11.030] - Pam
Now, again, I'm going to circulate and kind of see what I can see. If I see somebody removing 13 from 202, I might call on that kid. Making sure everybody's ready. So, I'm intentionally waiting till everybody's ready. Then when somebody, I'm pretty sure... Well, let's see. Let's say I don't see anything. Then I'm going to have this question, this very intentional question. I'm going to explicitly ask, "Did anybody not do the same thing that you did last time? Did anybody think about this one slightly differently? Like, you're reading and instead of kind of reading up..." And I'm watching. Nobody sparks. I'm like, "Did you do anything different?" I'm still watching. If nobody sparks, like, "Did anybody like start on the 202?" Now, somebody's going to be like, "Yeah, it's like, I don't want to read that far up. I just subtracted the 13 from the 202." As soon as I can pull that out of somebody, I'm going to now go into the right-hand column. And, in the right hand... So, the rest of the string, every time somebody does a distance or difference strategy, I'm putting it in that middle column. And as soon as anybody starts discussing a removal strategy, I'm going to put it in the right column. Those are very intentional moves, plan that I'm making ahead of time. So, in that right-hand column, I'm going to draw a number line. I'm going to write down 202. And then the kid's going to say, "Well, I subtracted 2 to get to 200." Kid or teacher. "And then I've already subtracted 2. I needed to subtract 13, so I'm going to subtract 11 more." And at that point, I might go, "Do you know 200 minus 11? I'm just kind of curious." That's intentional because in that moment, I want to just raise the idea that you could think right now? Do you know 200 minus 11? Because a lot of kids might just start doing smaller chunks. They might go, "Okay, 10 and 1." But I kind of want to raise the idea that you could actually think about the, you know. I might even say something like, I Have, You Need, depending on if kids have played I Have, You Need. If I have 11, what do you need to help them think about the fact that that's 189? The partner of 11 is 89. So, intentionally, right now. And I've been drawing as I've been talking. I have the two problems on the left-hand side with the answers because we landed on the...

[00:11:15.010] - Pam
Oh, sorry. Let me end. As we've drawn that number line for the 202 minus 2 to land on 200 minus 11 to land on 189, I might then say, "So, is this your answer? You landed on the 189. Or is the answer also on the top like it was in the other one?" Like, we might play that out. I might also wait on that. If this is the first of this kind of string, I might not point out those differences the first distance versus removal string that I ever do. Somewhere along the line I will. Okay, so then I've written down the 189 because now we have that answer. Then I'm going to very intentionally step back, and I'm going to say, "Huh. Weird, I gave you two subtraction problems. They both had to do with reading. And you guys did them. Many of your brains did them differently. That's interesting." Pause. Now, depending on the ether, I might right now, if there doesn't seem to be a lot of sparks going on, I might just do the next problem. If there's some sparks happening, like I can see kids are kind of ruminating on it, they're like, "Huh, that's interesting," I might have them turn and talk and get some words out. Not going to spend too, too much time here. Then I'm going to give them the next problem. Now, the next problem is going to be in the first of a set. And it's going to be something like 1,606 minus 1598. 1,598.

[00:12:33.570] - Pam
And then I'm going to do the same kind of thing. Now, this time I'm not going to be in a context. Though, I could. I don't know. What are we reading? War and Peace here or something? Like super long? I think my kid's favorite novel might be that long. The Way of Kings by Brandon Sanderson. I'll just throw that out because my kids are now smiling if they're listening to the podcast. So, I may keep it in reading. But I might not. I might just say, "What is your brain thinking about doing for this problem?" I'm going to look then intentionally for a kid who is finding the difference between those numbers. I'm going to intentionally represent that strategy in that distance or difference column. Then I'm going to give the next problem. 3,041 minus 46. And then I'm going to ask, "I wonder what your brain wants to do with these numbers? Is your brain trying to be more like Reading up? Or is your brain just sort of subtracting that 13 from the 202? Hmm." I'm going to let kids solve. I'm going to find somebody either by circulating or by asking a question. "Did anybody look at that 3,041 and just decide to remove, just jump back?" Like, I'm going to say as little as I can until somebody sparks. I'm going to grab that kid and I'm going to represent that strategy in the right-hand column. So, now, at this point, we've asked four questions. In the left-hand column, I've got two finding the distance, difference strategies. In the right-hand column, I've got two removal strategies. Now, I'm going to create a conversation. "What's happening? Why? What is it about? I wonder if there's something about the numbers." This is me lobbing out, right? I'm lobbing out an idea. "I wonder if there's something about the numbers that's kind of given you the inclination that you just wanted to like find out how far apart they were, just sort of read up and find out how many pages there are. Or is there something about the numbers over here in the right-hand column where you're like, nah, I'm not going to do that. I'm just going to remove the second number from the first number. What's going on?" Now, depending on how many thumbs-up I see, I may start a class conversation.

[00:14:32.880] - Pam
I don't see very many thumbs up. I might then have them partner talk and have a conversation. Either way, I'm going to get some conversation going. If they partner talk, then I'm going to have a whole group conversation. I might say, "Did anybody's partner say something? Or does somebody want to start us off?" Those are intentional moves I'm going to make to create a conversation around the idea that, hey, if those numbers are close together, just find that little distance. But if the numbers are really far apart compared to each other, then that the second number is probably smaller, tiny compared to the first one, then just remove it from the first one. You're looking to garner that kind of conversation out of kids.

[00:15:09.870] - Kim
Yeah.

[00:15:10.100] - Pam
Then I might say, "Okay, those are interesting ideas." And then I'm going to give the last set of problems. So, last set of problems might be something like 12.02 or 12 and 2/100 minus eight-hundredths or 0.08. "Do you feel like removing? Or do you feel like finding the distance?" Going to let them solve it, model. I'll let you listeners decide which one that is nudging towards, and then which column it would go under. Then I might give them a problem like 22.04 or 22 and 4/100 minus 21.97 or 21 and 97/100. Let them solve it. Ask them where it might go under. "Which column do you think this one might go under? Hmm." Depending on the kids, and if we've done this Problem String before, I might during this string start putting some words up. But I also might wait till the next time I do the string. I want to have enough kids grappling with the relationships that it would make sense to say, "So, what's kind of happening in the left-hand column?" And then I might write the words distance/difference. Like, you're sort of looking at how far apart the numbers are. Distance/difference. "What's happening in the right-hand column?" And then I might write the words removal or minus or subtract and write those numbers in either on top of, or in the bottom, or in the middle. Somewhere where I've left space for those two columns to kind of bring that language together now that they either talked about the partners or we talked about as a whole group. That explicitly, my board now at the end of this string has these three columns where I've got these six questions. Three of them were in the distance/difference column. Three of them were in the removal column. I've got some words down by it. Those are all very intentional moves. But notice, never during the whole string did I ever tell kids how to subtract.

[00:17:03.830] - Kim
Yeah. Yeah, I wrote down some things that what was explicit was the series of problems, the order of the problems. You went from some numbers in the 2 and 3 digit to 4 digit down to decimals. So, it was all intentional because you know the content goal. You're very clear about the content goal and where you're headed. That is 2 types of subtraction. You had some conversation throughout about diving in on the problem, that's a mathematical behavior, and using what you know. I could hear you being very explicit. Like, "What do we know about this? Could we use something we know?" You said, "Could we think about I Have, You Need?" So, all of that is very explicit. But when you got to the, "How would you solve this problem?" then you're lobbying and they're tackling the mathematics in a way that they want to solve it. But then when you pull back into the conversation, you're back to naming specific things and the layout on the board and focusing the conversation. I hear it kind of over time being... It's not... You're not narrowing it. But you're pulling their words in, sharpening where you're headed. Yeah.

[00:18:24.110] - Pam
Yeah, yeah. Focusing and sharpening. Absolutely. And never once am I saying, "Write this down, step one. Step 2. Step 3. Nope, borrow the sugar from next door. Step 4. Step 5." The math is kids, people, using the logically, internally consistent system to say, "If I know this, what else do I know?"

[00:18:46.420] - Kim
Yeah.

[00:18:47.060] - Pam
Yeah.

[00:18:48.240] - Kim
And you might have some students, very likely that you have some students who are listening and they're solving however they want. And then you're going to come back again with another Problem String. You're going to pick up from some of the conversation that happened in this Problem String, and you're going to have a similar Problem String where you're highlighting distance versus removal again. And you're going to remind them of some of the words that came out from students. And you're going to take the conversation a little bit further. And then there's this groundswell of students who are catching on to the things that you're lobbing out because you're lobbing them out in very specific times during out the Problem String.

[00:19:32.070] - Pam
And that is the dosing. We're giving them another dose, and another dose, and another dose until they get a high enough dose for that pattern to stick. Which really means they've traveled the mental path in their brain, and they've created this, now this schema in their brain, that subtraction has these two interpretations, these two meanings. I can let the numbers influence the way I think about it. And we just need to give kids a high enough dose, and they need to know that that's what math-ing is. Those are mathematical behaviors. What do I know? And how can I use what I know to know something else?

[00:20:07.620] - Kim
And how cool is it at the end of a series of Problem Strings for students to explicitly say, "There are two types of subtraction, and here's an example of when you would use this removal, and here's an example of when you would use the distance meaning."

[00:20:24.620] - Pam
And maybe we often do this, Kim, somewhere in that series of strings, what would be a problem that you could use either? And now we get those nice middling problems where either strategy is great. Alright. So, y'all, Kim and I are pretty explicit about parts of teaching mathematics. We have explicit goals. We have content goals. We have mathematical behavior goals. We have community goals that are very intentional and explicit. But the math-ing is the true mathematical behavior of really building kids' brains to reason mathematically. Y'all, tune into next week's episode because we're going to do it again with higher-level content. So, don't miss the next few episodes where we continue to work on when and how we are explicit in our teaching. Y'all, to find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Thanks for being here and spreading the word that Math is Figure-Out-Able!