Pam  00:01

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

Kim  00:09

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

Pam  00:16

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

Kim  00:31

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

Pam  00:38

Because we grapple with each other. And you're going to hear maybe some of that today. We invite you to join us to make math more figure-out-able. And we like the word "grapple", yeah?

Kim  00:49

I think you (unclear) more than I do, but I like it, yeah

Pam  00:52

I love... Yeah, we did martial arts in our family for quite a while. Sort of reasons we stopped. But there were not good reasons. It was health reasons actually. But we really enjoyed that family thing, and I just loved grappling. I don't know.

Kim  01:08

Yeah.

Pam  01:08

Maybe that says something about me. But getting on the floor, you know, just... Yeah, it was good.

Kim  01:13

It's like rough housing. 

Pam  01:14

Yeah.

Kim  01:15

You had boys. 

Pam  01:16

It just... Yeah. Well, in fact, one of my favorite days of all was one time when Craig moved up into the adult class, and it was the first time I'd ever sparred with him.

Kim  01:26

Mmhm.

Pam  01:26

And I was like, "Oh, this is cool! Like, bring it on!" He was kind of a little kid. He's totally grown up now, but he was kind of a smaller kid. And I don't even know how old. Anyway, it was fun. It was a lot of fun. It was a little kid whipping me. He was quite good.

Kim  01:40

Yeah, I don't think I'd be good at martial arts. I think I would just like resort to tickling or something because I can not stand to lose, but (unclear).

Pam  01:51

What I struggled with was pulling punches. I would actually pull my punches.

Kim  01:56

What does that mean?

Pam  01:57

It means that I wouldn't hit full force.

Kim  01:59

Ah.

Pam  02:00

And my instructors are like, "No, you have to actually practice hitting full force or you're going to be, you know, out in self defense, and you're you're going to pull your punch." And I was like, "Okay, but I don't want to hurt anybody." They're like, "No. Like, hit!" Anyway, yeah. Alright, alright, alright.

Kim  02:14

Wow.

Pam  02:14

Hey, should we talk about math?

Kim  02:15

Sure. Okay, last week we spent some time talking about choosing a Problem String. And there are lots of things to think about when you're choosing a Problem String, but now you have your string you've chosen. Now what? 

Pam  02:29

Yeah.

Kim  02:30

Lots of preparation still.

Pam  02:32

Bam. Lots to do. Lots to think about. Don't let that keep you from doing Problem Strings. But we thought, since you're trying them, we'll give you some kind of insight into things that we (unclear).

Kim  02:41

Alright.

Pam  02:42

Alright, so Kim to have a Problem String to refer to, we're going to do a quick Problem String here. So, first problem. What is 314 plus 400?

Kim  02:52

714.

Pam  02:54

And I'm just going to quickly draw a number line with 300. a huge jump of 400. It's actually not that big on my paper, but relatively big. And then you landed on 714.

Kim  03:04

Mmhm.

Pam  03:04

Next problem. What is 314 plus 399.

Kim  03:10

713. 

Pam  03:12

Because? 

Kim  03:13

Because plus 400 was 714, but I'm going back 1 to get to 713 because 399 is 1 less than 400. 

Pam  03:23

Nice. And so, 314 plus 399 is 713. 1 less than the problem you had before. How about 428 plus 300.

Kim  03:34

728.

Pam  03:36

So, I've just sketched down really quickly 400. Oh, I didn't say what I did for the problem before. So, the problem before I wrote down the same problem that you'd had before, the 314 plus 400, and then I backed up 1 to one before the 714. Alright, so this time, I have scooted my number line to the right because the first number line I wrote down started at 300. This number line is going to start at 428, so I've scooted it to the right. Which is kind of why I said I drew a big number line but not so long that it covered the whole paper because I needed to be able to scoot over now.

Kim  04:10

Mmhm.

Pam  04:11

So, that 428 plus a big jump of 300. Not as big as the jump that I had before because the jump before was 400, so this is 300. And you said that it landed on 728. Okay, next problem. Do you want to guess what my next problem is?

Kim  04:27

I'm going to go plus 299.

Pam  04:30

Good guess. It was plus 298. Excellent guess. What do you got? 

Kim  04:34

726.

Pam  04:36

So, how'd you do it?

Kim  04:38

Plus 300 would be 728.

Pam  04:41

Mmhm.

Kim  04:42

But I want two less than that, so I went back to to get to 726.

Pam  04:46

So, I redrew the number line starting at 428, making sure my 428 lineded... Lineded? Lined it it up. 

Kim  04:53

Lined it it up, mmhm. 

Pam  04:54

Wow, with the problem above it. With the number line above it. And then I jumped the same 300. So, that lined up with the 728 above it. And then I backed up 2, and so now the 726 is before, to the left of the 728. Cool. Next problem. How about 400 and... Where am I 457 plus 498. But I'm not going to give you a helper for that one. And I'm curious if you follow the same pattern as the sort of two sets of problems that we've done before, could you think of a helper for 457 plus 498?

Kim  05:34

Well, yes, I can. 

Pam  05:35

Okay.

Kim  05:35

I'm going to add plus 500.

Pam  05:38

So, 457, plus 500?

Kim  05:40

Mmhm.

Pam  05:41

Okay.

Kim  05:41

That's 957.

Pam  05:43

Okay.

Kim  05:43

And then I'm going to backup 2 to get to 955.

Pam  05:48

So, I'm going to admit, now that we've done this Problem String. Oh, maybe I should tell you what I did. So, I wrote down your first, your 457, the helper that you made up.

Kim  05:57

Mmhm.

Pam  05:58

And I added a huge jump. So, I've also gone just a little to the right with a 457. So, it's not underneath the 428. It's a little to the right. 457. And then I've done a the biggest jump I have on my paper now is plus 500. You said that that was 957. And then I didn't redraw it for the plus 498. I just backed up 2, and then wrote 957, 955. Is that right? Yeah. So, then I'm kind of stuck in where I wanted to go, which is why I wasn't really paying attention to what you got. I'm realizing, as I gave you that problem, I wish I would have actually said for that last problem not 457 plus 498. That's too much like the problem before where I added 298.

Kim  06:43

Mmhm.

Pam  06:44

I wish I would have done 457 plus, something like 495.

Kim  06:49

Mmhm. 

Pam  06:51

Instead of 98. Like a different. The first problem was 399. The second one was 298. For the third one, I shouldn't have done 498. I should have done something more like 495. That would have been a better, I don't know, ta da. Like, you might not have anticipated it as much.

Kim  07:07

Mmhm. 

Pam  07:08

And I wouldn't maybe have a kid just kind of looking, "Well, the first problem was back one. And the second problem was back 2. I bet the third problem is back 3." You know like some weird pattern that's not really holding true. I want them to actually be thinking about the problem. Anyway, that's probably more than you wanted to know. 

Kim  07:24

Well, it's good to know, to hear, as you, as the listeners, become Problem String writers. For now, you know, we're suggesting that it's great to choose a pre.

Pam  07:34

Pre-written, mmhm. 

Kim  07:35

But looking at the numbers in a pre-written string and wondering why they have those numbers. 

Pam  07:40

Yeah. 

Kim  07:40

Is really, really helpful,

Pam  07:42

Yeah, and as you do a lot of pre-written Problem Strings, you might get to a point where you're like, "Ah, that's almost too formulaic. I'm going to tweak that last one." And give it a go, you know like, see what happens. And I'll be honest, there's some times where I've written a string where later I look back on it, kind of like right now, and go, "Ah, it's a little too formulaic. I wish I would have done a little bit more of a change there." So, yeah, like as soon as you've kind of done a few, you get a feel for it, feel free to kind of start to play a little bit. It could be  kind of playful. Cool. 

Kim  08:12

Mmhm.

Pam  08:13

So, you know, this idea of looking at a Problem String and really thinking about facilitation and the modeling and everything. We had an experience, Kim, where I was doing... We were filming a Problem String and a colleague of ours, who I love and respect and so grateful that we've been invited into her classroom. Abby Sanchez is a high school mathematics teacher here in the state of Texas, and she and I've worked together for several years. She's a great, great person. And super interesting. One day we were in filming in her classroom, and you were, especially at that point, hadn't done a whole lot of thinking about teaching high school math. Like, you took high school math.

Kim  08:52

Yeah, yeah, yeah.

Pam  08:52

You hadn't done a lot of thinking about high school math. And I had invited you that day to really kind of be a pedagogy check. Like, help me think about the teacher moves and everything. Not so much the math.

Kim  09:02

Mmhm. 

Pam  09:02

And Abby was doing a string that I had written, and in the middle of that string, you were like, "Uhhh." And you like grabbed me, "And I'm like, "Hey, the cameras are rolling. What are you doing? Like, don't bug me."

Kim  09:13

And I kept bugging you. Well, and I think the reason that I was talking to you about it is because we were going to do that string in other classes. 

Pam  09:14

You kept bugging me. And you're like, I don't even remember. I think you like pointed to something in the string, and you're like, "It would have been better this way." And I was like, "What?" And I couldn't. This is a little bit about me. I don't know if I can think about two things at once. And so, I kind of just like shooed you away. I was like, "Hey, give me a minute. Let her finish the..." Like, let's turn the cameras off, and then let me think about it. And you said, "I think this string would be better this way." I don't think that's quite the way you said it. What you said was, "What is she going for? If she's going for this, how come she didn't ask this?" Like, I could see your brain moving. Like, I think... It's almost like you said, "I think I know where she's going. Wouldn't this have worked better?" But you said it in a really inquisitive, curious, not a like, "Pam, you're a dork" kind of way. (unclear). Oh, you know what? I think I'm telling the story wrong. You're right. I think it was while Abby was practicing. I don't think the cameras were on yet. (unclear).

Kim  10:13

I think it was the first class was being filmed.

Pam  10:16

Okay.

Kim  10:16

And we had like, two, one or two more classes because she switched classes.

Pam  10:20

Right.

Kim  10:20

And I was like, "Hey, like, what about this?" But you were watching Abby because she was being filmed, and you were watching to make sure that it was going and you were running the shoot. 

Pam  10:31

Yeah. 

Kim  10:31

And so, I was probably should have saved my poking for a different time. 

Pam  10:38

So, interestingly, I said, "What..." You tried to tell me what you would change. And I said... And I don't even know if I was distracted or whatever. But for whatever reason, whether I said write it down or you just decided to, you wrote down a new string that was kind of like the the pre-written one, but with some tweaks, a few tweaks. But next to each problem, you wrote just a little bit of help for Abby. Like, I know you wrote, for one of them, you said something like, "Why is this happening?" And for one of them, you said, "Have we seen this before?" And for one of them, you said something like, "Wait, what?" Like, you literally wrote the word wait "comma" what. Question mark, exclamation point. And the idea was, in the middle of Abby facilitating it, Abby could put the problem up there and then go, "Wait, what?" Like, it was a moment for her to show curiosity and for her to go, "Is that like a typo I just..." So, the kids could look at the problem and go, "(unclear) problem." Like it was a moment, it was a way for you to kind of help her realize the feeling tone for that problem.

Kim  11:16

I remember that. Mmhm. 

Pam  11:18

And so... Oh, go ahead.

Kim  11:20

I was going to say, I remember writing enough on that paper that you would know what I meant. Because, you know, at times we have shared brains. And then I remember afterwards, when Abby was done, we went up to the board, and I kind of talked through it, and I asked her, "Do you want me to write more?" And she was like, "No, I totally got it." And so...

Pam  12:04

It's like she was sharing our brain then as well. Yeah, yeah. Once you talked through it, she's like, "No, your notes make sense." And so, then she did that kind of tweaked string in the next class. It totally worked better. Those were excellent tweaks. So, that has become a thing. Before that, we would just write down the string, and we would go to the board and facilitate the string. After that, from that moment on, we began to... We call it Kim's Notes for a while. We began to make little notes to ourselves next to each of the problems. And especially like if you and I were facilitating a workshop together. When you would do the string, I would be like, "Oh, I wouldn't have thought of that." And I would take my note card out and make a little note, so that I would remember the move you just made that wasn't natural to me, but I loved it.

Kim  12:06

Yeah, yeah, yeah. Mmhm. 

Pam  12:53

And we started this tradition of making this sort of facilitation notes page where that we had the string written, but then next to it we had, you know like, say this or do that. And then we also started the tradition of planning what the board would look like. What's that display that we were going to have at the end of the Problem String and kind of including those together. And that's been a great tradition.

Kim  13:19

Mmhm. 

Pam  13:20

For us to begin and keep going, 

Kim  13:22

Yeah, and it's been really nice because we can share those with the world, right? So, it's a way to when we write strings, we can share with the world what we would focus on.

Pam  13:33

Yeah.

Kim  13:33

And when we would focus on something. So, the string itself is just a series of problem. And if you're a Problem String writer, you know that is very carefully constructed, one problem after the next. But also very carefully planned is the facilitation. And so, in a note card, let's talk about some of the things that you would put on it. 

Pam  13:33

Mmhm. 

Kim  13:33

One of the things that you would do when you're preparing a note card is say, "Okay, I need to think through the math. What is the strategy that's happening? What is the model?" So, we will take a note card. If you've seen any video of either of us, you've seen us hold a note card in our hand or post it or something. And we will sketch out exactly what we think the final board is going to look like.

Pam  14:16

Yep. 

Kim  14:16

So, we're going to have the exact replica of a miniature model. So, if it's a number line, you're going to see the number line for each problem on our note card and it's going to be like the layout in a miniature version. And I got to tell you, Pam, my note cards look amazing. They are 100% fantastically laid out, proportional. And the work that I continue to have to do is the blown up version on the board because I am the spastic when I'm doing Problem Strings. And I'm kind of like all over the place. And I'm so into the conversation with the kids that sometimes I look back and I go, "Wow, that board is slanted." Or, you know, it's like towards the end of the string, I get sloppy with my stuff. 

Pam  15:01

Well, and you're left handed, so I think sometimes you tend to write up the board.

Kim  15:05

I do.

Pam  15:06

Which is interesting. I tend to go down. Like, my lines are all a little bit down. Yours are little bit up. Yeah.

Kim  15:10

I just watched a video of me in a fifth grade classroom, and at one point I said on camera to the kids, "How could you let me do that?" Because I asked them. I was like, "Hey, make sure I'm doing a good job." So, the layout of the board is there. But what's also super important is the note that we make beside some of the problems. Now, every single problem might not have a note. But like Pam just mentioned, some of them might have a place that you pause with little asterisk. Or "Wait, what?" Sometimes, they might have a "pause here". Like, let it sit. Some might have... Sometimes next to mine, teacher moves that I don't use as often, but I want to in this particular string. So, I might say, "Did anyone hear what so and so said?" Or, "Can anyone restate? Can anyone add on? (unclear).

Pam  15:37

Or partner talk. Like, let's turn and talk here. Give everybody a chance to kind of get some ideas out before you do any of what you just said.

Kim  16:08

Yeah, I might...

Pam  16:09

(unclear) start us off.

Kim  16:10

Yeah. I might also make a note about when I want to choose more than one strategy. Maybe at the beginning of the string, I might say take two. Or I might write, take Over and Give and Take. Because in the facilitation of the string, you're thinking so much about circulating, and about talking to the kids, and listening carefully. Then, if you can free your brain up because you've already planned out the choices that you're going to make.

Pam  16:11

Mmhm. 

Kim  16:13

It can be really helpful.

Pam  16:17

Yeah, there's a particular string I love to do in high school about function notation and writing the equation of a line. And there's a strategy that I really want to pull out, but it's not natural to me. I don't, in the moment, my brain kind of shuts off. And so, I'll just make a little note on that card, and I'll be like, "Oh, that's right. I want to go find that." Because somebody in the room has either done it. Or I could nudge it a little bit, and someone will do it. And then it's super helpful for that one to come out for the second to last problem because it will influence how people will try to solve the last problem, and so I want to make sure that one comes out. And in the moment... I tell you, before I wrote it on my card, I would be in the middle of the Problem String and be like, "Oh, there's something I'm supposed to be looking for now. And I can never remember what it is in that moment. But until I've written it down. And so, I've gotten better over time at realizing this is what I need to write down for this string. It's also true that if you look at my card today, if I'm doing a Problem String I've done several times before, you might not see any notes.

Kim  17:40

Right.

Pam  17:40

Any of the problems. Because I've done that particular string a lot of times. But then you'll on this in the exact same card, you'll see the second string, and it will have several notes. You know, it just depends on kind of how well you planned, how many times you've done it. Go ahead.

Kim  17:56

I'll tell you where I have the most filled out note card. 

Pam  17:59

Oh, yeah? 

Kim  18:00

Is when I'm going to go film.

Pam  18:02

Yeah, for sure, for sure.

Kim  18:04

Just because I want to make sure that we we put in maybe some examples of different teacher moves that, you know, I have to think through.

Pam  18:12

Sure.

Kim  18:12

The other thing is that, you know, you might emphasize something in one string. In a same string in one 4th grade  classroom, you might emphasize one thing, but in a different year or with a different group of kids, you might not emphasize that one thing because the strategy might be the emphasis. But there might be that you're trying to get kids to talk to each other more. Maybe you're trying to build community with one class that already meshed well with another, and so you might be looking for a purposeful time for them to partner talk. 

Pam  18:47

Sure.

Kim  18:47

You might be trying to start to solidify, so you might have a note that says, "look back". And so, towards the end of the string, you might look back as a class of what's been happening here?  Mmhm.

Pam  19:01

For example, you might... Like, in the string that we did today, you might say, "Hey, before I give you this 457 plus 495. Or I wish it would have been... Yeah, I wish it was 495. That last problem without a helper. I might have said, "Hey, Kim. Let's take a look at the two pairs of problems that we have. What kind of pattern are you seeing that that at least it's on the board? Maybe you didn't use, but you know what have we heard the class kind of using?" Put some words to that and then say, "Huh. Well, if that's kind of that pattern that you just put words to that you could keep the first number whole and add something that's a bit too big, and then, oh, you could just adjust back, what might be a good helper problem to follow that pattern for this problem?" Yeah. That's what we mean by "look back". Like, look up the string. 

Kim  19:03

Mmhm.

Pam  19:03

Talk about the patterns. Pull out of kids what they're thinking about the pattern they're seeing. It might be the first time a kid starts to think about that pattern. Because they're just solving problems. 

Kim  19:57

Yeah. So, when you are preparing a Problem String, you know, there's a lot of things that you can think about.  Yeah. 

Pam  19:57

Maybe they've paid attention to kind of the strategy that you're modeling, maybe they haven't, but that could be a moment for them to start to put words to it. I was just watching a Problem String this morning that Holly did. Holly's a colleague of ours, a fantastic fourth grade teacher at that point. And there were a couple of times where she did exactly that. She's like, "Let's put some words to this." And part of the the reason that it's coming to mind today is the words they used were so just natural and un... Help me. unprecise. Like, it wasn't precise mathematical vocabulary. And to watch her just massage it into, "What about the friendly number? Oh, you were getting to a friendly." Like, she just took what they said, and then would like pop back just a little bit, and then they would kind of refine. And then she would just kind of push a little, and they'd refine. And by the end of it, the kids were using fantastically precise mathematical vocabulary, but in a very natural. Yeah, it was super fun to watch. Mmhm.

Kim  21:03

But also offer Problem Strings and and suggest some half sheets, some Kim's Notes, some... What do we call them now? Problem string facilitation?

Pam  21:12

I think we call them facilitation notes. 

Kim  21:13

Yeah. 

Pam and Kim  21:14

Yeah.

Pam  21:14

Yeah. That include the Problem String with the little notes next to them and that sample final display. 

Kim  21:21

Yeah.

Pam  21:23

Mmhm.

Kim  21:23

Mmhm. But we suggest that you make them your own. You know your kids, and so you have to mesh the idea of what's happening in this Problem String and what do I know about my students?

Pam  21:33

Yeah, nice, nice. So, Kim, one of the things that we've noticed with teachers of younger students is that they'll look at a rekenrek Problem String, a Problem String that's intended to use a number rack or a rekenrek, and they will assume that they can... Well, maybe, I'll just say it's for me for sure. I'll just get up there and start moving beads around, and then all of a sudden realize wait a minute. Which side should the beads start on? 

Kim  22:00

Mmhm.

Pam  22:00

Where should they end? Am I covering up some of the beads? Those are some things that you need to think about ahead of time. And we've tried to put those kind of notes in our pre-written strings, so that you're really clear what it should look like. And, in fact, often we'll show here's what the number rack shows.

Kim  22:17

Start.

Pam  22:18

Yeah, when you start, here's what it would look like. And then as kids talk about their strategy, here's the movement that you would make.

Kim  22:25

Mmhm. 

Pam  22:26

And then here's kind of the end result that it would look like for a problem. And so, try to give teachers kind of that help because we had to really think through what that would look like.

Kim  22:34

I mean, yeah. Much respect to kinder and first grade teachers who do it daily because I actually think it's more for me to think about than recording on a number line than a ratio table

Pam  22:45

Some of the other models. 

Pam and Kim  22:47

Yeah.

Pam  22:48

Yeah, so we just encourage you to read through those notes, and then practice a little bit. Like, actually put the number rack up. Be aware of which beads you have to the right and to the left. And then which ones you're going to cover when. Just a little bit, you know. Be super good work to do in a PLC. Leaders, teachers, if you can grab a teaching partner and say, "Hey, just like does this make sense for me to do this in this? Let me just practice there. Okay." Yeah, that would be really good kind of pre work to do. 

Kim  23:14

Yeah.

Pam  23:14

Yeah. 

Kim  23:15

Another thing to consider when you're planning for a Problem String when you're preparing, is what you need to consider when you're facilitating. Are you going to use color in any way to highlight anything on the board? 

Pam  23:26

Nice.

Kim  23:27

What are your labels going to look like for your ratio tables? Where are you going to land numbers? What are you going to circle? What are you not going to circle? When are you going to circle? 

Pam  23:36

When are you going to erase on a ratio table. Yeah. 

Kim  23:39

When are you going to write equations or not? Yeah.

Pam  23:42

Mmhm, yeah. 

Kim  23:42

What are times that you're going to circulate? When are you going to narrow the amount of strategies that you choose? 

Pam  23:49

Are you going to have kids write or is it all without paper and pencil for this particular? Mmhm, that's a good one. Yeah.

Kim  23:56

Lots to think about.

Pam  23:57

And as you think about those things, consider noting those on your half sheet, Kim's Notes, facilitation notes, whatever you're calling it. Note those things as you're planning. Because if you're like me, you stand up and you're like, "Wait, what was I going to..."

Kim  24:10

Mmhm.

Pam  24:11

And then as it as you do the string or at the end, take a little note to yourself and keep those for next year, so that you have that note then for you to use next year. Yeah. 

Kim  24:21

Mmhm.

Pam  24:22

Hey, so, Kim, one of the other things that we've noticed is that we seem to be in an era. PowerPoints and electronic displays. And so, I used to go into classrooms and there was a whiteboard. Sometimes there was a document camera. Now, I go into classrooms and there's often still some whiteboards, but there's often some sort of electronic display where it's either a big TV screen or... Yeah, basically a big computer screen. And sometimes teachers write on that big computer screen with, you know, like electronic marker or whatever. Sometimes teachers are using a tablet to write on and it's sort of showing up. I'm okay with all of that. Again, I want you to sort of think. I invite you to think through what that means for your facilitation. But here's the thing that I would maybe ask you to consider is that often teachers are getting used to the idea that the lesson is sort of prepared, pre-prepared because they have this electronic delivery system, and so they're writing the problem in like a PowerPoint kind of fashion. So, the Problem String that we did today, you might be like, "Oh. Well, you know, of course, over the podcast. We can't see that. So, you know, if you were doing it in a classroom, you would have created a PowerPoint where the first problem 314, plus 400 would show up." And maybe it would show up off to the left. And so, it would give me plenty of space to draw a number line. But I've even seen teachers put the problem on the left, and then click forward, and a drawn number line, a pre-created number line, shows up to the right. So, if I could put a big red X over that. Like, so not that. In fact, not even having the problem pre show up. We think it's pretty important that this feels organic. It feels like kids are coming up with answers on the spot and that you're actually representing their thinking. So, even though we're... You're like, "But Pam, Kim, didn't you just clearly tell us to plan what the end display is going to look like? And Kim just said that often it doesn't quite look like her carefully prepared one? Wouldn't it be better if we created this perfect display and we just revealed it a little bit at a time?" And everyone shake your head and say no. We are not suggesting that. We are actually suggesting recreating that display every single time. And I'm saying that as a high school teacher who taught seven periods in a day. Every single time, I would erase the board and recreate the whole thing. Now, there are a few times where I'll create pre-create something to show up. But that's pretty rare. But let me give you an example. Sometimes, I will... I've done some geometry strings with dot paper. I'm not going to draw dots on my board every time because then I'm trying to erase. I'll project somehow dot paper, and then I'll draw stuff on there. So, that. Sure. I might even provide the dot paper for kids, so that they can recreate the shapes that are sort of happening. There are times where I'll project a grid on an axis. A coordinate axis. But many times I won't. And it depends. But the thing that's always at the top of my head is if I put this grid up here is it going to inadvertently have kids counting by ones? And I don't want to do that. So, I'm very careful to think about whether I'm putting a grid up or not. I don't want to promote that you could solve this problem just counting. That's counting strategies. I want to really promote Relational Thinking. So, teachers, when you're doing like area models, if you're like, "Ooh, I'm just going to create all these exact to the dimensions and have them appear," highly recommend that you don't do that, that you actually redraw those arrays, even if they're not perfect. But you say out loud, "Hey, I'm trying to draw this. Does it really look like a 30 by 42?" Or whatever it is. So, the kids kind of help you make it look and they get a sense and a feel for the proportionality because they watch you try to make things proportional. Yeah, there you go.

Kim  26:11

Mmhm. 

Pam  26:12

Yeah, nice examples. Yeah. 

Kim  26:19

So, we like to create sample final displays of what the board could look like. We also like to create half sheets, so that we are what we figure out is important, that we want to emphasize. And yours might differ from ours, but we have lots of examples of what both of those look like for you. 

Pam  28:43

Yeah. And you can find example of those in all of our Building Powerful Mathematics online workshops. And you can also find them for every Problem String in our Problem String books. The ebook has them as well where you literally can just like copy this half sheet, this facilitation notes that you can take with you as you facilitate your Problem Strings. You can also see examples of those in our major strategies ebook that you can get at mathisfigureoutable.com/big. B-I-G because it's big news, big thing. So, if you have not downloaded yet our free major strategies ebook, check that out. It's got sample Problem Strings where you literally can see what we're talking about. This example, facilitation notes.

Kim  29:28

And if you are a lucky participant who's ever been in one of your Building Powerful Numeracy workshops, they've had a chance to see these half sheets as well.

Pam  29:37

Yeah, totally.

Kim  29:38

Yeah. 

Pam  29:38

If you'd like to see the half sheet or the facilitation notes for the exact string that we did today in today's podcast, you can download that www.mathisfigureoutable.com/big, and this exact Problem String is there with the facilitation notes and that sample final display that you can see there. 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. And keep spreading the word that Math is Figure-Out-Able.