Pam  0:00  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  0:09  
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  0:17  
Because we know that algorithms are amazing human achievements. You know, I got to figure out the intonation on that sentence. It's like the same "We know that..."

Pam  0:27  
"amazing". 

Kim  0:27  
I know. Same thing for me. Let's see.

Pam  0:30  
Ya'll, we know that algorithms are amazing human achievements, but they're TERRIBLE TEACHING TOOLS because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop. I had to take a breath in the middle of

Pam  0:44  
that one. 

Kim  0:44  
You need breaking up sentences. You say that (unclear).

Pam  0:48  
Alright, yeah. 

Kim  0:48  
In this podcast, we help you teach mathing, building relationships with your students and grappling with mathematical relationships. 

Pam  1:00  
Do you take a breath in that one? 

Kim  1:02  
No, but "mathematical relationships" is a tongue tie for me. 

Pam  1:02  
Huh. I don't

Pam 1:00  
think I've ever heard you like stumble on that one. Mathematical relationships. I'll have to say that four times fast. Alright, ya'll, thanks for joining us to make math more figure-out-able. We're on one today, so hold on tight. This is going to be fantastic. Whoo! Alright, Kim.

Kim  1:10  
Right before we got on, I told Pam that I have the oldest eyes. Man, when I hit 40, something happened. But I found a lovely short review, which I'm excited about. MeeksMcD. Probably, McDonald's. 

Pam  1:25  
Oh, could be. Yeah. 

Kim  1:27  
I can see it, but it's a tough go. MeeksMcD said, "I am hooked! Five stars!" Yay!

Pam  1:33  
Aw, hooked. Yay!

Kim  1:35  
"This podcast is fantastic. Pam and Kim challenge me in the most delightful ways. 

Pam  1:40  
Oh, I love that you say that. That's awesome. Yay! 

Kim  1:42  
I mean, I think we challenged a lot of people, and they don't always love it. So, yay.

Pam  1:46  
Well, to be clear, I think we challenge each other, right? 

Kim  1:49  
We're

Kim  1:50  
mostly pleasant on the podcast. Our team, you know, is... The rest of our teammates are like.

Pam  1:54  
What, when we challenge each other in meetings?

Kim  1:57  
They probably don't like it. 

Pam  1:59  
I mean, I think we challenge each other on the podcast, just pleasantly.

Kim  2:02  
Yeah, yeah, yeah.

Pam  2:03  
Maybe. Are we always as pleasant when we're not on a... Hey, speaking of old eyes. I'm feeling like I have this old shoulder, and it's crazy. I'm doing these like exercise things. What is with my left shoulder? Ah!

Kim  2:16  
When we both had shoulder surgery, mine was left. Was just right or

Kim  2:19  
left? 

Pam  2:19  
No, it was left. It's the one I had the surgery on. Yeah.

Kim  2:22  
(unclear) get that looked at. 

Pam  2:23  
I don't have time. Not right now. 

Kim  2:26  
You never have time.

Pam  2:27  
It's on the list. I got to get my hormone levels straightened out. Ya'll, if  you mess with your thyroid, just join me in my because I'm not only hypothyroid, I'm hypo parathyroid. Which no one knows what that is because it's super rare. It's actually a rare condition. Anyway, I got to get that figured out, and then. But in the meanwhile, I'm trying to exercise, and it's just making me crazy that my left shoulder. Anyway. Alright, let's... Should we do some math? 

Pam  2:53  
Let's do it. 

Pam  2:53  
How did I get onto that?

Kim  2:55  
(unclear).

Pam  2:56  
Okay. Oh, wait. Oh, there you go. A while ago, not too long ago, we did a podcast series about Problem Strings. And one of the podcasts that we did was facilitating Problem Strings. 

Kim  3:07  
Yeah.

Pam  3:07  
And in that podcast, I guess we did something that we don't always do because people are telling us. They were like, "Hey, do more of that." And I was like, "Wait, don't we (unclear)..." 

Kim  3:13  
Do more what?

Pam  3:14  
Yeah, we do Problem Strings all the time on the podcast. And they said, "What was different was this time, we kind of painted a picture of the moves that we would make, when and why, and maybe alternatives. And really...

Kim  3:30  
Yeah.

Pam  3:30  
Yeah, really, kind of painted a picture of what it would look like in a classroom to do that particular Problem String. So... 

Kim  3:36  
We've

Kim  3:36  
heard that before, right? Sometimes I think that we try to communicate what's in our heads, but unless you're in our heads, you know.

Pam  3:45  
Which is a scary place to be. 

Kim  3:46  
Sue, who we love, has said you need to unpack more because maybe you're not saying enough sometimes to give people a good image. So, this is another realization that describing more might be helpful.

Speaker 1  4:00  
Yeah. And specifically, I think we've done a good job on the podcast as describing the string, and the relationships, and the math at hand. But less the classroom, and the look, and what would actually be happening with kids.

Kim  4:13  
For sure. 

Pam  4:14  
So, let's do that today. 

Kim  4:15  
Okay. 

Pam  4:15  
Today, we're going to do a very specific kind of Problem String. So, there are different structures of Problem Strings. So, we thought we'd focus on a very specific structure of a Problem String called a Helper Clunker string because different structures of Problem Strings, you would facilitate differently. And...

Kim  4:33  
Yeah.

Pam  4:33  
...we've done that on the podcast, but I don't know that we've been as explicit about it, so let's do that. Alright. Kim, first question.

Kim  4:40  
Yep.

Pam  4:40  
Well, sorry. I'm painting the picture. So,I might say to students, "Hey guys, we're going to do a quick Problem String. For this particular Problem String, you're welcome to write. I don't know that you're going to need to. But don't make this about working memory. So, go ahead and have something handy." Which means either scrap paper and pencil. Paper, pen, whatever. Or they could have a...

Kim  5:06  
Dry erase?

Pam  5:06  
Miniature. Dry erase, yeah? Like the... What am I trying to say? Like, they could run on their desk if it's dry erase. They could have the dry erase board. Whatever.

Kim  5:06  
Yeah. 

Pam  5:06  
We don't want them but vertical in groups because Problem Strings are not done in groups. Problem Strings are done everybody's doing their own individual work. 

Kim  5:22  
Yeah. 

Pam  5:23  
Okay, so then I would say, "Hey..." You know, yeah. Here we go. First problem, I would probably say the problem. 99 plus 123. I might say the problem and write it at the same time on the board.

Kim  5:33  
Mmhm. 

Pam  5:34  
Either way, I'm going to write it before I expect them to solve it. So, I'm not expecting them to just hear it. Most of us don't have working memories where we can hear a lot of digits like that and be able to hang on to it, so I'm definitely going to write it.

Kim  5:48  
Yep. 

Pam  5:49  
Then I'm going to pause for a second. And it just occurred to me that I'm not doing... Kim?

Kim  5:57  
What?

Pam  5:57  
Can I change the... Ha! Today, we're going to talk about facilitating a an equivalence Problem String structure. Not a Helper Clunker. So close, and yet so far. 

Kim  6:11  
Okay, sounds good. 

Pam  6:13  
So, there you go. Alright an equivalence Problem String structure. See, when I just gave you that first problem, I was like, "Wait, that's not a helper. What am I doing?" Yeah, okay, I'm prepared. I'm alive. It's the shoulder. Blame the shoulder. Alright, so I'm going to give kids this problem. 

Kim  6:26  
Okay.

Pam  6:26  
And 99 plus 123, depending on what we've been doing.

Kim  6:29  
Yeah. 

Pam  6:30  
If I'm going to do this Problem String, we would have done other Problem Strings before. We would have developed other strategies than the strategy I'm about to do. 

Kim  6:39  
Okay.

Pam  6:40  
So, I would expect kids to do something. I wouldn't expect for them to take too long, to work too hard, because we would have built other strategies.

Kim  6:46  
Sure.

Pam  6:46  
But I'm going to let them do it. I might circulate briefly to just see what kids are doing, but I'm probably just going to say, "Hey, what did you get?" And then, Kim, what did you get? 

Kim  6:57  
Oh, do I solve it? 

Pam  6:58  
Yeah. 99 plus

Pam  6:59  
123. 

Kim  7:00  
I got 222.

Pam  7:03  
222. And I probably should have said, "Pause the podcast. Solve it, everybody." Then, I'm probably going to ask. "Did anybody think 99..."

Kim  7:11  
Mmhm. 

Pam  7:11  
And then, Kim, did you?

Kim  7:13  
Yeah.

Pam  7:14  
Okay, so (unclear). 

Kim  7:15  
So, start at 99, and if you add 1 more, that gets you to 100.

Pam  7:19  
Okay. As you say that, I just drew on the board a number line. I started at 99. So, I have tick mark, 99. When you said you're almost to 100 to add 1. I did a jump of 1 and landed on 100. Okay, keep going.

Kim  7:37  
And then since I've already added 1, I just have 122 left. So, I'm adding 122.

Pam  7:41  
So, then I would draw a big jump of 122, and I would say, "And here's the hard part. What is 100 plus 122." And then you would repeat, 222."

Kim  7:51  
Yep. 

Pam  7:51  
And then I would draw. So, I wouldn't just do it. I would draw the jump of 122, draw the tick mark, and I would say, "And then that's the hard part. What is 100 plus 122?" And then when you said the answer, then I would write down the 222.

Kim  8:05  
Yeah.

Pam  8:05  
And then I would say something probably like, "Hey, you could have solved this in a couple of different ways. I just wanted to get one way up here to kind of establish the relationship. Cool. Next Problem."

Kim  8:15  
Yeah. Can I interrupt you?

Pam  8:16  
Yeah.

Kim  8:16  
I think it's really important that you said that you are asking and you're waiting for the answer because I've been watching some Problem Strings lately where the teacher assumes they know what the kids are doing, and they're drawing it as the kids are saying it. Which then gives the impression that you're not really doing what the kids are saying. So, recording after the kids are saying it is hugely important if you are making it about their strategy.

Pam  8:40  
Mmhm, mmhm. Yeah, in fact, I've gotten a little lazy lately, and I have... How do I even say this? That as they're saying their strategy, I will at least be a little behind them. But sometimes it's good to hear them say it, and then say, "Let's represent that. Can you repeat that?" Or even have somebody, "Hey, did somebody get... Somebody repeat that." And now, I'll represent your thinking. So, sometimes it's good to have them get it out.

Kim  9:09  
Mmhm.

Pam  9:09  
And then represent

Pam  9:10  
it. 

Kim  9:10  
That also helps if they aren't doing exactly what you think they're going to do.

Pam  9:15  
Yeah, because in that moment I might say, "Ah, I hear what you're doing. Actually, I was wondering was anybody..." blank. 

Kim  9:20  
Mmhm.

Pam  9:21  
And then you can get that strategy out. That's a way. Yes. Sometimes teachers will say, "Pam, what if they don't give you the strategy that you're looking for?" Well, that's one way to do it. Have them say what they did, and then you can go, "Oh, okay, nice strategy. Did anybody use this to do that? Okay, say that. Oh, yeah. Okay, let's put that up here."

Kim  9:38  
Yeah. 

Pam  9:38  
And again, you're trying really hard to. It's not like you're saying bad on your first strategy. You're saying, "That's not what we're doing today. That's not the one I'm going to represent today. Nice strategy. But anybody do this relationship? Okay, cool."

Kim  9:51  
Mmhm. 

Pam  9:51  
Alright. So, Kim, on the board right now, we have 99 plus 123 equals 222 with then next to it an open number line showing Get to a Friendly Number strategy. 

Kim  10:01  
Mmhm.

Pam  10:02  
Alright, so the next problem that I might ask is what is 100 plus 122? 

Kim  10:07  
Mmhm.

Pam  10:08  
I'm going to write that on the board pretty much as I say it, and then I'm going to smile, and I'm going to say, "I'm not going to wait very long on that one. What do you guys all get?" I'll probably get a choral response on that.

Kim  10:19  
Mmhm, mmhm. 

Pam  10:19  
And what is 100 plus 122? 

Kim  10:21  
222. 

Pam  10:22  
And then I'll say something like... I might not say anything at this point. I might smile. So, you guys aren't looking at the board. I mean, maybe you're writing it down while you're listening to the podcast. But I now have two problems that have the same answer. I'm not going to make a big deal of it. I might smile, but then I'll give the next problem.

Kim  10:41  
Do you think you would say, "Nobody wrote this down, but you..." No, I don't think you would. 

Pam  10:46  
Okay. What do you mean? 

Kim  10:48  
Sometimes I hear you say, "Nobody represented their thinking this way, but I could." But I think you actually do that with Over, so never mind. 

Pam  10:57  
Oh, okay, okay.

Pam  10:59  
I might... No, no. Then I'll probably give the next problem. And I might say something like, "Hey, what is 144 plus 100.

Kim  11:00  
Mmhm. 

Pam  11:00  
I'm not waiting long on that one. So, I've written down 144 plus 100. And then I'll say. I'll look at them smile. "I'm not waiting very long on that one. What does everybody get?" I'll probably choral response that one. Let everybody just yell out. Now, I might pause long enough to make sure everybody's actually thought about it before I say, "What is that everybody?"

Kim  11:22  
Mmhm.

Pam  11:22  
And then what is 144 plus 100, Kim? 

Kim  11:22  
244.

Pam  11:22  
244. I'm just going to write down 244. 

Kim  11:22  
Mmhm.

Pam  11:22  
And then I'll say, "Okay, what is 146 plus..." Where am I? "...plus 98?"

Kim  11:23  
Mmhm. 

Pam  11:47  
Then, I'm going to pause. I'm going to go, "Oh, okay, that one's not as quick. You guys go ahead and solve that any way you want. Use a strategy." Now, I'm watching.

Kim  12:06  
Mmhm. 

Pam  11:45  
I'm looking out. There's probably going to be I'm anticipating. If I see kids go, "Oh, yeah, this one's not as easy. I'm going to have to work at that." They put their head down. They start to write. That's good information for me. I'm going to let them think. I'm going to let them use a strategy. I'm going to let them go. I'm looking though. I'm watching, I'm looking. I'm going to see if there's any kids that kind of look at me knowingly with a grin.

Kim  12:08  
Mmhm. 

Pam  12:08  
And I'm going to kind of maybe look at them like, "Shh, wait. Hang on. Wait. You know like, pause, pause. Give everybody a chance to think. Give everybody a chance to think." You know, I'll kind of smile at them. Maybe wink a little bit. And then as kind of everybody's finishing, then I'm going to have somebody share a strategy. Now, let's kind of go to a different route. If nobody blinks. Everybody just looks at me like, "Duh." Because there's a relationship between it and the third problem. If everybody sees that relationship, I might not take as long there. So, that's why I said I'm really watching. And if anybody dives in to solve that problem, I'm giving them time to solve the problem. And then I'm probably calling on one of them to say, "Okay, it looks like you were, you know, you're thinking about that. Good. How are you doing that?" I'm going to give them lots of respect because they were diving in, probably using a good strategy. And at that point, I'm probably going to look for somebody. I might say something like, "Did anybody Over that one?" And I'm looking, I'm looking. And if nobody says so, I'm looking to see if anybody sparks at that to just say, oh, like they probably could. And when I see somebody spark, or if nobody does, I might say, "Could you? Could you?" I'm going to pause. "Could you?" And if somebody sparks or if somebody use the Over, then I might say, "Tell us about that." Then, I'll... Kim, can you describe an Over, and I'll say what I would write?

Kim  13:23  
Yeah. So, I would start at 146. And if I could add 100, that would be 246.

Pam  13:30  
So, I've written a number line with 146. And then you said you would add what?

Kim  13:34  
100.

Pam  13:35  
So, then I would have a big jump of 100. I could have maybe heard you out more, but keep

Pam  13:39  
going.

Kim  13:40  
And that gets me to 246.

Pam  13:43  
Often... Can I just... Sorry.

Kim  13:45  
Okay.

Pam  13:45  
You can tell me if you hate me right now. Often, when a kid does Over, they might say, "Well, I started at 146, and I added 100." Or they might just say, "I added 100." And then at that moment, I'll say, "Where'd you start?"

Kim  14:09  
Mmhm. 

Pam  14:09  
So, if I say, Hey did anybody do an Over?" and Kim goes, "Yeah, I added 100." Then I'll go, "What did you start with?" And they're like, "Oh, I started 146." And then I'll draw the 100. And then they'll go, "And then I subtracted 2." And I'll say, "Wait, wait, wait. When you added the 100, where did you land?" 

Kim  14:11  
Yeah. 

Pam  14:12  
So, an important teacher move is to really get kids to actually say where they started, where they land at each of the... So, they might just say, "Well, I added 100 and subtracted 2. That's okay. Don't say "Wrong! Bad!" But just go, "Where'd you start? And when you added the 100, where'd you land?" Okay, so now we're at 246.

Pam  14:34  
Go ahead, Kim. 

Kim  14:34  
And then I back up 2.

Pam  14:34  
And then what is back 2 from 246.

Kim  14:36  
244.

Pam  14:37  
And as you said that, I backed up two on the number line, wrote minus 2. And then where you landed, I wrote 244.

Kim  14:44  
Yeah. 

Pam  14:45  
So, then I would say, "Okay, cool. So, it sounds like you added 146 plus 98 by adding 100, backing up 2, and you landed at... I'll look at the number line kind of point. You landed at 244. Then I kind of step back and I just pause. And I'm looking for kids who now have seen that I have two problems that had the same answer.

Kim  15:03  
Yeah.

Pam  15:03  
And then I have another two problems that have the same answer. And I'm looking kind of curious right now.

Kim  15:08  
Yeah, I

Kim  15:08  
think this is important. I'm going to pause you here. 

Pam  15:11  
Yeah, please.

Kim  15:11  
Because when you say, "I step back, and I look."

Pam  15:14  
Mmhm. 

Kim  15:14  
Your body language of turning your body and looking at the board is causing people in the classroom to also direct their attention there.

Pam  15:23  
Mmm, mmhm. 

Kim  15:23  
So, like you're not looking at the kids, like, you know, "Fingers crossed. Hopefully you see it." You're looking at the board.

Pam  15:30  
Curiously. I have this curious look on my face. I'm looking. Then I am going to turn and look at the kids.

Kim  15:36  
Mmhm.

Pam  15:36  
Look back at the board. Look back at the kids. And then I might say, "Kim, I noticed that you have this kind of curious look on your face."

Kim  15:45  
Mmhm. 

Pam  15:45  
"What are you noticing? And now, that's a bit of a... You know, I could get some. I've actually had kids say things like... What have I had them say? "Same, same, same, same." You know like, same answer, same answer. And not have gone farther than that. And they'll say, "Oh, that's interesting. Is that coincidence? It's probably just coincidence. Yeah. Yeah, just just happenstance that that happened." Somebody's going to snicker a little bit or something. And I'll go, "Is there a connection? Is there a relationship between the problems?" And now, I'm going to try to get kind of a conversation to happen. And I anticipate. I don't know, Kim. Do you have something that you would anticipate a kid might say that's kind of half formed?

Kim  16:24  
They might say the pairs have the same answer. Or they're like almost the same problem.

Pam  16:31  
And so, as they say the pairs have the same answer. And maybe they haven't said "It's almost the same problem," yet. I might say. Direct them. So, good. Let's make sure everybody's seeing that. "What are the addends have to do with each other?" So, if they only are focusing on the answers, kind of half formed, then I'm going to say, "Do the  addends in the problems have anything to do with each other?"

Kim  16:32  
Mmhm. 

Pam  16:32  
And what? You know like, "Where? What are you seeing?" And then I might expect a kid to say something like, "Well, the 99 is almost 100. And the 23 is almost the 122. And the 144 is almost that 146. And the 100 is almost at 98." So, they're noticing that they're close. So, then I might direct. Well, I might just try to keep calling on kids. You know, because I'm hoping, but I might have to ask the question, "How  close are they?" And then if a kid says, "Well, the 100 is just one away from 99. And the 123 is 1 away from 122. Same with 140 and 446 is there just 2 away, 2 way." And now I'm trying to get direction in there. So, again, I'm going to try to let it go and hope kids come up with that. But if they don't, I'm going to nudge like, "Well, 2 away, 2 away. 2 up? 2 down? So, oh, okay. The 146 was 2 more than 144 and the 100 was 2..." Well, now I'm doing two directions. "The 146 was 2 more the 144 and the 98 was 2 less than the 100. Oh, well, that's interesting. If the addends go up and down by the same amount, does that make sense that we'd have the same answer?" And then I might just... If we've gotten, if we've raised the question, I'm not going to solidify anything (unclear). 

Kim  18:08  
Right, right.

Pam  18:09  
I'm going to raise it and kind of go, "Do you think that'll always be true?" And now, I probably might give the next question.

Kim  18:15  
Yeah. So, in this though.

Pam  18:17  
Mmhm. 

Kim  18:17  
You are letting what the students say help you be informed about how far you're going to carry the conversation. This is not you hoping that by the third pair these kids have, you know, all these great words about it because especially if this is your first string of this type, you're like raising some curiosity. You're raising some wonder, and you're watching the students, and you're getting feedback from them. And if not enough students are participating in the conversation, then you know you're going to come back to this in another string, in another string, and so it's just enough to raise them some wonders.

Pam  18:56  
And to be clear, when you just said, "If not enough students are participating in the conversation," what you didn't mean was if not enough students are being engaged, if they're being willing. What you meant was that if not enough students have any idea of what's going on and seeing the relationships, making connections...

Kim  19:13  
Right. 

Pam  19:14  
...then you're going to put we're going to continue this string to have those relationships and connections ping more before we try to tie too much together.

Kim  19:24  
Right. 

Pam  19:24  
If I'm in a high school class right now, I'm expecting, you know, the older the kids are, the more... Maybe I should say the more experience I think the kids have had.

Kim  19:33  
Right. 

Pam  19:33  
I'm expecting more kids to be making more connections. But I'm taking the temperature of the class, and I'm trying to decide how many connections are being made. Which will help me then decide how fast I'm going to try to pull things together.

Kim  19:47  
Mmhm. 

Pam  19:47  
Okay, so then, without pulling too much together at this point, I'm going to say, "Hey, here's another problem. 236 plus 198. Go ahead and solve it any way you want." Now, I'm anticipating... Well, go ahead and solve that, guys. So, podcast listeners, solve that. 236 plus 198. Come on back. I am anticipating at this point because I've put the 198 as a second addend right under a problem where we Overed plus 98, I'm anticipating that somebody's going to think Overing again. And so, I might say at this point, "Hey, did anybody Over for this one?" But I probably should have said, Kim, I'm going to let kids work on this. 

Kim  20:26  
Sure. 

Pam  20:27  
And as I'm letting kids work, I'm looking to see if somebody Overed. I'm also looking to see if anybody's trying to use the pattern that they might have seen in the first two sets of problems.

Kim  20:37  
Mmhm. 

Pam  20:37  
And I'm noting those kids.

Kim  20:39  
Yeah.

Pam  20:39  
But I'm probably going to call on a kid who Overed. Alright, so, Kim, can you walk us through that over strategy?

Kim  20:45  
Yeah, so 236 plus 200 is 436.

Pam  20:49  
So, I've written a number line with 236. I've slid it over to the right of the two number lines I already have on the board. So, the first problem started at 99, ended at 222. The second one started at 146. So, kind of in the middle of that first number line, I've started the second number line. But this one starts at 236, so it's almost to the right of the second number line because the second number line ended at 246, so this 236 is starting just before that a little bit. So, these number lines are one of them started right where I was. The next one started to the right. The next one started to the right. And this is all because I've looked at these numbers ahead of time, and I've planned out on the board that I have room to do this. 

Kim  21:27  
Yeah. 

Pam  21:28  
Okay, so I started at 236. You said added 200, so a longer jump than I have on the board so far. Big, big jump at 200.

Pam  21:35  
Okay.

Kim  21:36  
That's 436, and then I backed up 2.

Pam  21:38  
Make sure they tell me where they landed. Backed up 2 lands you at?

Kim  21:42  
434. 

Pam  21:43  
434. And so, you're saying the answer to this problem is 434. I've now put the 434 on the number line. I've now walked back over to where I have the 236 plus 198, and I'm also writing the 434 there.

Kim  21:55  
Mmhm. 

Pam  21:55  
Cool. Then I'm I might, at this point, if I have enough kids that dove in the conversation, enough kids looking like they're using the connection, I might say, "I wonder if anybody has a guess what my next problem is." Now, I'm not going to wait too long there because I think there's going to be some kids that are not looking at patterns at all. They were just diving in using an Over strategy. That's a good place for them to be, so I'm not making this too much of an emphasis of, "You should be here," just kind of like a wonder. This is a way for me to kind of keep kids challenged that are ready for a challenge, while not making anybody else feel like they're less than for not going there.

Kim  22:35  
Mmhm. 

Pam  22:35  
And then I might say, "My next problem actually is 234 plus 200."

Kim  22:41  
Mmhm.

Pam  22:42  
Pause. I'm looking at the board. I'm looking at the kids. Pausing. Like, I'm not really pausing on this because that 234 plus 200 is hard. But I'm giving weight to that there's a relationship happening here, that there's a pattern I'm hoping that they're starting to look at. "Ya'll, what is 234 plus 200?" They'll say, "434." And then, again, I'm pausing. I'm stepping back. I'm looking at the board. I'm looking at them. Now, depending on kind of how the feeling tone is this point, I might joke around a little bit. "Again? Like, what?" I'm just kind of in this like, "I don't know. There's this pattern. Same answers. Like, what are you guys... What's happening in the problems that's impacting why I'm putting..." You know like, "What patterns are you seeing?" So, Kim, what patterns are you seeing?

Kim  23:36  
I'm seeing. It's close to the problem that we had before, that the 236 went down 2 the 234. And the 198 went up 2 to be 200.

Pam  23:49  
So, at this point, I might say, It's almost like you're telling me that if I have a pile of marbles, and I had 236 in this pile, and I had 198 in that pile. How did the piles change from the 236 plus 198 to the next problem 234 plus 200?

Kim  24:06  
Took 2 marbles from the 236, and I gave it to the 198. 

Pam  24:10  
"It's

Pam  24:11  
almost like you like took..." And now, I'm physically like taking my hand and I'm putting it near the first addends. 

Kim  24:18  
Yeah. 

Pam  24:18  
And I'm like then moving it to the second added. And I'm like, "So, you're telling me if I just move these 2 marbles over here, did I change the total number of marbles? Huh. Looks like I didn't." Now, I'm going to go back to the other two sets of problems, and I'm going to go, "I wonder if we can make sense of these with piles of marbles. Will you turn to the person next to you? Have a brief partner conversation. Could you talk about piles of marbles with these problems?" I'm going to dive in and listen to what kids are saying. Now, at this point, ya'll, I'm probably not going to go to kids who are seeing the pattern and they're clearly being able to articulate it. I might go to kids that I'm hoping to kind of keep engaged in the Problem String, and I might kind of listen in on them. "Like, hat are you guys thinking about?" And they're like, "I don't even know what you're talking about. Where are the marbles?" So, then I might be like. You know, I might help them sort of visualize. "I'm thinking of a pile of 200." You know, maybe you just kind of repeat. They might just need to hear it one more time.

Kim  25:15  
Mmhm. 

Pam  25:15  
Kim, is that true for you? When I said that, did you have a different instinct of who you might have made a beeline for? I'm just curious. I don't expect an answer here. I'm actually curious.

Kim  25:25  
No, I think I'm on board with what you're saying. 

Pam  25:27  
Okay, okay.

Pam  25:30  
I think there is a world where I might have gone to my kids who I know are ready to be challenged, and I might have said, "How are you guys verbalizing this?" Or I might have said, "Can you guys come up with a problem that you could use this?" So, I might start getting them generalizing at a higher plane.

Kim  25:51  
I think it might depend on the age of the students. So, like, if you're picturing high school students, for sure.

Pam  25:55  
Okay.

Kim  25:57  
You know, I'm picturing...

Pam  25:59  
Younger? 

Kim  25:59  
3, 4, 5. You know, fifth grade probably there's going to be some who like really understood the magnitude of the numbers in these problems. But if they're younger, there might be still tinkering with. You know, third graders still tinkering with hundreds, size of numbers and hundreds. I'm wondering when at what point. When you said, "It's like moving marbles."

Pam  26:19  
Mmhm. 

Kim  26:19  
And it was the 236 to the 198. Would you record for this first string the plus 2, minus 2. Like, are you recording? You haven't said you're recording anything at that moment. Like, I picture you moving your

Kim  26:32  
hand.

Pam  26:32  
Yeah. Uh, I did not record. But you know what? I might have. 

Kim  26:36  
Yeah.

Pam  26:36  
Today during the podcast, I did not. But you're pinging for me that I actually might have in a class. So, between the 236 plus 198 and the 234 plus 200, I actually left a space.

Kim  26:48  
Yeah.

Pam  26:48  
Maybe. Maybe subconsciously I was.

Kim  26:50  
I think you probably would have.

Pam  26:51  
Yeah, so I might have said, "So, if I move..." So, under the 236, I might have put minus 2 in between the 236 and the 234." And then with the 198, I might have put plus 2. So, now you kind of see this almost looks like equation solving steps, but it's not because it's on the same side of the equal sign.

Kim  26:52  
It's

Kim  26:52  
in context to what they're saying. 

Pam  26:58  
Yeah.

Kim  27:00  
So, right? Like that is so valuable that...

Pam  27:07  
I'm modeling what they were saying. Mmhm.

Kim  27:13  
Yeah. Kids are talking about it.

Pam  27:17  
Exactly, yeah. And so, as they say, "So if it took 2 marbles from the 396," then I would have said, "Oh, so like minus 2, and then we'd give it to the 198. Oh, so like there's the plus 2?" Yes, so it's in context. Yeah, nice, nice. So, then I might say, "What did you just talk about with your partner?" and have them give some. You know, have one or two kids talk about what they said with their partners. If, in that moment, somebody is too clear. If somebody goes, "Yeah, you just plus 2, minus 2, and you get the same answer." I don't know if that's clear or not. But if what they say, in that moment, resist the urge to go, "Yeah, you got it! Good! We're done!" 

Kim  27:57  
Right, right, right. 

Pam  27:58  
Really, in that moment, look at them like, "Really?" Like, you want to look at them and go, "Hmm. Does that make sense to anybody? Can somebody add on to that or say that in your own word?

Pam  28:07  
Is that..."

Kim  28:07  
Yeah. 

Pam  28:07  
Like, you're really...

Kim  28:09  
It's not playing dumb, though. It's not like, "Hmm, I'm confused. What do you mean?"

Pam  28:13  
It's

Pam  28:13  
not playing dumb.

Kim  28:14  
Right. 

Pam  28:14  
it's being intensely interested... 

Kim  28:17  
Right.

Pam  28:17  
...in what they're thinking. It's really wanting to hear how everybody's making sense of that. It's putting weight by waiting. You're putting weight on the fact that this is not trivial and making sense of what's happening right now. We're going to spend a minute here. We're going to like. I'm putting some weight on this. Yeah, it's not fake. It's you're intensely interested in making sure that we're sort of giving everybody a chance to make sense of it. Yeah. Okay, so at that point, now we put some words to it, and kids are like, "Yeah, I think if  move some marbles, then I didn't change the total amount, so I could just..." I might steer it towards, "So, we made one of these numbers nice? It's almost like you decided. Because we didn't randomly move marbles. Like you made one of those. That 236 to 234, that's pretty random. Oh, no, it's because we made the 198 nice to make it 200. Okay, so there were some. We made the 98 nice to make it 100. Or we made the 99 nice to make it 100. So, really, you were making one of the numbers Nice. Those were the marbles you were moving around. Well, that seems kind of interesting. I wonder which of these numbers might you want to make nice?" And then I'll give the next problem, and I'm going to write up 395 plus 357. "I wonder what if you make one of the... Now, go and solve it any way you want, but I wonder if you can make one of those numbers really nice by moving some marbles around. Where would those marbles come from?" Now, I'm letting kids solve. I'm walking around. I'm listening. Kim, what might we expect to see kids

Pam  29:47  
do? 

Kim  29:48  
Oh, we would expect them to make 395 nice

Kim  29:53  
by taking 5.

Pam  29:54  
As a kid says that, I'm going to say 300. You said take 5. So, go ahead. Keep going.

Kim  29:59  
Take 5 from the three. 357 and give it to the 395.

Pam  30:03  
And when you say take the 5 from the 357, I might say, "That seems really random. Why are you taking 5 from the 357."

Kim  30:09  
Mmhm. 

Pam  30:10  
"Oh, it's because you're trying to make the 395 nice by giving the 5 marbles to it. And what is 395 and 5? That's that nice 400." So, I've just written plus 5 under the 395. Drawing a line. That's 400. And what do you get when you take the 5 from the 357? I'm going to let the kids say that. Which is?

Kim  30:29  
352. 

Pam  30:29  
352. "Hey you guys, what is 400 and 352? Oh, that's nice. That's just 752. 

Pam  30:36  
Kim, I'm not drawing a number line now.

Kim  30:38  
Right.

Pam  30:39  
So, on the board right now I have 395 plus 357. Underneath that, I have plus 5 minus 5, respectively. Underneath that, I have drawn a line. And then 400 plus 352 equals 752.

Kim  30:52  
Right.

Pam  30:53  
"Wow, it really seemed like moving marbles around for this problem seemed pretty nice. Made one of those numbers nice. Got the marbles from the other one. Huh. I wonder if you could do something similar..." Okay, depending on how the kids are handling that, if enough kids were doing it, I might just move on to the next problem. If not enough kids were, I might say, "Huh. Well, interesting. That was the next problem in my string. Now, if nobody came up with that helper problem, if this is the very first time we've done this and there's... I shouldn't say nobody. But if a lot of students were like, "I don't know what you're doing," then I might have just given 400 plus 352.

Kim  31:31  
Mmhm.

Pam  31:32  
And then we'd probably be done with this string.

Kim  31:35  
Right. 

Pam  31:35  
I'd probably come back to it later. But if enough kids, most of the kids, were kind of like, "Yeah, I'm tracking with you to move those marbles around, then I could end this Problem String by saying, Alright, it looks like we made that first number nice. That seemed really helpful. I wonder if we want to make this first number nice. So, how about 468 plus 292. Want to make that first number Nice? Now, I'm looking for kids to smile. Kim, you want to make that first number nice?

Kim  32:03  
No.

Pam  32:04  
No? Why? Because you're contrary? Go ahead. What would you do?

Kim  32:09  
I would make the second number nice.

Pam  32:11  
By?

Kim  32:11  
By taking 8 from the 468 and giving it to the 292. 

Pam  32:15  
So, if I

Pam  32:15  
give 8 to the 292, that's 300. And I wrote that first even though you said take it from the other one.

Kim  32:22  
Mmhm. 

Pam  32:22  
So, where'd those 8 marbles come from? Now, I'm going to go back over to the 468. Write minus 8. And what is 468 minus 8? Hey, that's just 460. And then 460 plus 300. Is that just 760? 

Kim  32:35  
Yep.

Pam  32:35  
Huh. Now, I'm going to say something general like, "It really seemed helpful to think about marbles. Can somebody put into words what you're thinking about right now?" And then, Kim, can you just? I don't know. What do you think a third or fourth grader might say right now?

Kim  32:51  
I think they say something about making one of the numbers nice. Nice and friendly. 

Pam  32:56  
Mmhm. And then

Pam  32:58  
I might say, "Can somebody add on to that?"

Kim  32:59  
Yeah.

Pam  32:59  
And I'm looking for somebody to say, "Yeah, it's like we're moving marbles around." And I might say, "Can somebody add on to that?" Maybe we might get to the point where somebody says we're making an easier problem to solve. 

Kim  33:10  
Mmhm.

Pam  33:11  
And I might then say the word "equivalent". "Huh. So it seems like if we can move some marbles around to make an equivalent problem, that might be a strategy worth thinking about. Great work today,

Pam  33:19  
guys."

Kim  33:20  
You might have a kid who's like, "Miss, miss, miss, I made the other number friendly."

Pam  33:24  
Mmm, mmhm.

Kim  33:26  
Because moving 32 from the 292 to the 468 also is a really nice equivalent strategy.

Pam  33:35  
And if you've been playing I Have, You Need, that's not inconceivable.

Kim  33:40  
Right.

Pam  33:40  
Yeah, absolutely. Kid might have. You know, Play I Have, You Need a little bit. If I have 68, you need 32. Bam.

Kim  33:46  
Yeah. 

Pam  33:46  
And like you said, taking that 32 from the 292, sweet!

Kim  33:50  
So, that's really nice because this is an opportunity for students who have made sense of the strategy. Because as other ones are thinking of one way to make it nice, you have a problem where you could say to the other guys, "Is that the only way you can make it nice?"

Pam  34:05  
Yes.

Pam  34:06  
And here we are naturally differentiating.

Kim  34:08  
Right.

Pam  34:09  
Yeah. We've written problems that naturally differentiate. And so, yeah, we could keep all kids being supported and all kids being challenged at just the appropriate level. 

Kim  34:19  
Yeah.

Pam  34:19  
Nicely done. Thanks for bringing that out. I don't know that I would have thought of that, Kim. I like it. Alright, ya'll, hope you enjoy a podcast episode where we kind of walk through what it would look like to facilitate an equivalence Problem String structure. Alright, ya'll, if you like that Problem String, here's the thing to know. That particular Problem String comes from our grade four Numeracy Problem String book that came out not too long ago. 

Kim  34:45  
Yeah.

Pam  34:45  
We're super excited about our Problem String books. It is one of the equivalence structured Problem Strings that is meant to build the Give and Take structure. In that particular book, we actually have four lessons that are devoted to this particular strategy. And it is one of, golly, about 48 lessons in the book. For this particular string, we have this string and two, what we call, echo strings that could be what you could do tomorrow and the next day if kids were they need a lot more experience. We also have two next step strings that are kind of baby steps before you get to then that next lesson that's on the exact same structure. So, lots and lots of helps and guidance for facilitating Problem Strings in our numeracy grade level books. Check those out. We'll put those links in the show notes. Thanks for being my student today, Kim.

Kim  35:40  
Yeah, no problem. 

Pam  35:41  
Students. Alright, ya'll, thanks for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!