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

Kim  0:10  
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  0:18  
Because we know that algorithms are amazing human achievements, but they are terrible teaching tools. Mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop. 

Kim  0:31  
In

Kim  0:31  
this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical

Kim  0:37  
relationships. 

Pam  0:38  
Y'all, thanks for joining us to make math more figure-out-able. Hello.

Kim  0:44  
Hi.

Pam  0:45  
Hey.

Kim  0:45  
You know how last week you said that we might do a video component?

Pam  0:51  
Yes. 

Kim  0:51  
I've been thinking about that because how many (unclear) drink water, or like scratch my head, or adjust my... Ugh.

Pam  1:00  
That's your thing? See, I don't mind drinking in front of

Pam  1:03  
people. 

Kim  1:04  
Well, it's not like I, you know... I just heard you.

Pam  1:07  
So, yeah. So, my big deal is I don't want to put makeup on. 

Kim  1:10  
Oh.

Pam  1:12  
Not even. Or I feel like there's this thing where either I either want to look the same every time, or I want to look different every time.

Kim  1:22  
Different feels like a lot of work. But, I mean, I...

Pam  1:25  
It's a lot

Pam  1:26  
of work either way. But listeners, we are awfully curious. Do you want us to go through the trouble of doing video? Or you're just like, "No, stay audio." I'm actually... We're both kind of hoping that you say that. But we're willing to think about video if really you guys think video would be better. 

Kim  1:42  
Yeah.

Pam  1:45  
Notice I said "think about". We're willing to consider.

Kim  1:47  
Um, yeah. I mean, we've been doing this a long time. You know, maybe a change-up would be good. Like five years.

Pam  1:54  
Alright, well let's see.

Kim  1:54  
Alright, alright. Okay. 

Pam  1:56  
Let us know, you guys. Let us know what you

Pam  1:57  
think. 

Kim  1:57  
Okay, so we have been doing some celebrating with the launch of your newest book. I'm going to see if I can... Developing Mathematical Reasoning: The Strategies, Models and Lessons to Teach the Big Ideas in K-2. If you are a K-2 teacher, you should be on the lookout. If you are a coach, this is for you. Everybody needs this book. Parents. Parents of K-2. I don't know.

Pam  2:23  
We're super excited. 

Kim  2:24  
It's super fun. 

Pam  2:24  
Yeah, one of these days, we'll write actually a parent book. But if you're a parent into it, I think you'll definitely like it. Yeah. So, just to be clear. We wrote Developing Mathematical Reasoning: Avoiding the Trap of Algorithms. That was the K-12 anchor book. It's really the book study book for entire districts, so that you can get the conversation going with everybody about what mathing, the mental actions of mathers actually are and how algorithms can trap students in lots of ways. We go through those traps. That's the anchor book. In the anchor book, K12 book study book, that's the one that really gets at the traps. What not to do. There's definitely a lot in there about what to do. But then we're now writing a series that comes after it. And the first one is out for kindergarten, first, and second grade teachers. All about what to do - The Strategies, Models, and Lessons to Teach the Big Ideas in K-2. So that you know, the 3-5 book will come out sometime in around February of 2026. So, that one's been written. We're now in the editing stage. In fact, Kim, I don't know if you know, I just got copy edits for the first couple chapters back yesterday. Which is...

Kim  3:30  
There's so many rounds. 

Pam  3:31  
It's excellent. But it's also like, "Oh, one more thing to do," because currently we're writing the 6-8 book, and that one will come out six months later. And then we will work on the 9-12, book, and it will come out six months after that. So, a year from now, I will be on vacation without a book hanging over my head because I've had this series of five that I've loved, and I'm super excited about, but it's been quite a journey. Anybody that's ever written anything knows it's a big deal. 

Kim  4:01  
Well, and

Kim  4:01  
it's been alongside some other projects

Kim  4:05  
as well.

Pam  4:06  
That as well, yeah. But it has been a lifelong dream to really get this work out there, and so we're super excited. And I love the fact that the K-2 book is out.

Kim  4:15  
Mmhm. 

Pam  4:15  
Yeah, super, super fun. So, we thought today we would give you a little insight into some of the magic that's in this K-2 Strategies, Models, and Lessons to Teach the Big Ideas book. And a way to do that, one of the  cool things that we do in the book is help kindergarten, first, and second grade teachers realize the things they do that grow up into more sophisticated strategies, pretty quickly actually, for students in a different way than I think many of us used to look at. "Oh, you got to know your facts because then you're going to repeat those facts over and over, over in a traditional algorithm. So, boy, we better get those memorized because those are the building blocks." Most of us have this visceral, embodied memory of doing 1-29. Not even odd. Like all 1-29.

Kim  5:07  
Yeah.

Pam  5:07  
Problem after problem after problem. Where, if you didn't have those facts down, it was just... What's a good word? Agony? Pain?

Kim  5:17  
Painful.

Pam  5:17  
Drudgery. That's the word I'm looking for. It was drudgery because you just over and over and over again, especially if you didn't know those facts. We're suggesting... Well, actually, let me say one other thing. Then when you memorize those facts to repeat those drudgery steps of doing like the addition algorithm, that didn't really help you, then you had to learn a completely new set of steps for the subtraction algorithm. Like, completely new. 

Kim  5:40  
Yeah.

Pam  5:40  
It didn't build on. That's different than strategy. If we build that small set of major relationships that lead to a small set of major strategies, that small set of young, less sophisticated strategies just grow up to be more sophisticated. 

Kim  6:01  
Yeah. 

Pam  6:01  
They build on each other. There's synergy in developing strategy, because now you just keep getting better at better and bigger and better at that thing.

Kim  6:10  
I mean,

Kim  6:11  
and how impactful does that have to feel as a K-2 teacher, knowing that what you're giving your students is directly related to what they're going to be doing older. Like, I think the connection between the grade levels is more apparent when you do work with strategies than anything else that you do.

Pam  6:32  
And it's not just in them getting more success answering questions. But it's literally in building their capacity. 

Kim  6:38  
Yeah.

Pam  6:39  
Their brains are stronger. They're thinking more sophisticatedly. Like you're literally creating more successful human beings, not just getting answers to math questions. Okay, cool. So, an example of that would be for teachers of young students that we work with them on Partners of Ten.

Kim  6:58  
Mmhm. 

Pam  6:59  
So, that they can use the strategy Get to Ten.

Kim  7:03  
Mmhm. 

Pam  7:03  
So, for example, we might help them learn that the partner of 10 for 7, 7 plus what is 10? 7 plus 3 is 10. We want them to get experience with ten-ness, so that they break it apart, and they realize that if I've got 7, I need 3 to make that 10, so that then they can use that. So, that's a kindergarten, first grade idea. So, that then in first grade, they can use that idea. Grows up, so that they can use it to Get to 10 for adding something like 7 plus 5.

Kim  7:33  
Mmhm.

Pam  7:33  
So, they can say to themselves, "Well, I know 7 plus 3 is 10 because I've done that ten-ness and that 7 and 3 partner-ness. So, if 7 and 3 is 10, but I was supposed to add 5, we've got 2 left over. Ah, then I could think about 10 and 2. So, that's the Get to Ten strategy. 

Kim  7:51  
Mmhm.

Pam  7:51  
Is there anything you want to say about that?

Kim  7:53  
About... 

Pam  7:53  
Oh, except for the fact that I just use a different example than I wrote down. Haha, that's funny.

Kim  7:57  
So, another example, right, is if students are given 8 plus 3. If they've done work with fluency for partners of 10, then they intuitively know that 8 and 2 is the portion that gets them to 10. And then for 8 and 3, they just have to add 1 more. And so, that fluency work that you're doing young helps with Get to Ten. And then later it grows up for something like Get to a Friendly Number, where they are given a problem like 28 plus 7, and they know that 28 and 2 will get them the 30. And since they...

Pam  8:31  
That 8 and 2 partner. Mmhm.

Kim  8:33  
Mmhm, mmhm. And if they're adding 7 altogether, and have used 2 to get from 28 to 2. 28 plus 2 is 30. Then they have just 5 left, so that they land on 35. And what's fantastic about the idea of this fluency work of partners of 10 is that it doesn't just stay in partners of 10 at a very young age. If they're working with partners of 10, then that helps them not only with 28 and 2 but with something like 280 and 20 to get to 300.

Pam  9:01  
Nice.

Kim  9:01  
Or 2800 plus 200 to get to 3,000. 

Pam  9:06  
Brilliant.

Pam  9:07  
So, lots of ways that those relationships grew up. First partners of 10, which grows up into Get to 10, which grows up into Get to any Friendly Number. And those partners grow up in place value and magnitude. And we can then use that Get to a Friendly Number for problems like 2,898. And I can think about well that 8, I just add that 2 right there. Bam, I'm at 3,000, and then I can just add whatever is leftover to whatever I was adding to begin with. Cool.

Kim  9:41  
So, that's one strategy. Get to Ten growing up to Get to a Friendly Number. What about Add Ten and how it grows up to Add a Friendly Number? 

Pam  9:52  
Nice.

Pam  9:52  
So, for young students, we encourage teachers to work with them to think about any digit, any single-digit number plus 10.

Kim  9:59  
Mmhm.

Pam  10:00  
So, if I've got 2 plus 10, what does that mean? Well, that's like 10 and 2. Oh, yeah. That's a teen number. What do we call that? We call that 12.

Kim  10:06  
Mmhm.

Pam  10:06  
Or 8 plus 10. That's like 10 and 8. Oh, that's a teen number. What do we call that? 18.

Kim  10:11  
Mmhm. 

Pam  10:12  
And so, we want kids to realize that if they have a single-digit number plus 10, that's a teen. We want to have them have lots of experience with that. Then that Add Ten strategy can grow up to Adding a Friendly Number.

Kim  10:25  
Mmhm.

Pam  10:25  
So, now if I've got things like 28 plus 14, they can think about 28 plus 10. If they've thought about, you know, plus 10, it grows up, and now we want plus 10 for larger numbers. 28 plus 10 gets you to 38, and then I can add whatever's left over. And I can continue to do that with larger numbers. So, if I'm at 48 plus 20, I can think about... Or 48 plus 26, I can think about, "Well, what's 48 plus 20? That's a friendly number. So, if I've got plus 10 down, now I can add multiples of 10. So, 48 plus 20 is 68. Then I can add that leftover part in whatever way that I want to." So, that also grows up. Now, I can do large numbers where I can be thinking about 3,486 plus 2,247, whatever it is. And I can say to myself, "Well, I know what 3,486 plus 2000 is."

Kim  11:20  
Mmhm.

Pam  11:20  
And once I get that, then I can work on adding the rest of them.

Kim  11:23  
Mmhm.

Pam  11:23  
Both of the strategies that we've just talked about, starting really young with either Partners of Ten or just Adding Ten, grow up to these larger strategies of Getting to a Friendly Number or Adding a Friendly Number don't just get us those strategies. Notice that we're talking about place value and reasonableness. And kids have a sense of magnitude and how big numbers are, and they're literally thinking about. Like, I just said 3,486 plus 2,000. Like, what is that? And then what would I do with the leftovers? All of that is necessary work for most students to get a high enough dose of those relationships so that they've built those connections in their

Pam  11:59  
head. 

Kim  12:00  
What a drastically different feeling for students and what they're developing than 3,487 is 3-4-8-7. And I know you talked about that before. But when we're doing this kind of work, you...

Pam  12:12  
It's not a collection of digits. 

Kim  12:14  
Right. It's it's magnitude. It's nearness to other numbers. Yeah.

Pam  12:19  
Yeah, nice. Alright, Kim, what about a strategy that we think about with young students about adding 10, adding too much, 10, and adjusting back?

Kim  12:29  
Well,

Kim  12:30  
I'm going to back up for a second.

Pam  12:31  
Oh, okay. 

Kim  12:31  
Because you just said... Well, it's connected to what you just said. You were talking about the meaning of teens. 

Pam  12:37  
Mmm.

Kim  12:37  
And I loved... I feel like I heard you say 8 and 10 are the teen numbers. 

Pam  12:44  
Mmhm.

Kim  12:44  
And I think that's really important because I think a lot of teachers focus on 10 and 8, 10 and 7, 10 and 4. But as you're developing the meaning of teens, also thinking about 4 and 10, 8 and 10. Because this also leads into the idea that when you add 10 to a number, then you can go further with it to say, "Well, I know 8 and 10 is 18, but what about 8 and 9?" So, I've just added a little too much. And if they know 8 and 10 gives me that fluency with that teen number, then if I'm adding 8 and 9, then I can just back up 1 to get to 17. And we could do that with small numbers. But we can also use this Over strategy for larger numbers. So, where they might say, "Okay, the problem is 6 and 9."

Pam  13:33  
Mmhm. 

Kim  13:34  
"I can think about 6 and 10 is 16, so 6 and 9 must be 15." But even larger than that. 36 plus 19 might be a problem. And I can think about going a little over. 36 plus 20. And because they've done some work with adding friendly numbers, 36 plus 20 is 56. But it's a little more than I need, so I can back up 1 to get to 55.

Kim  13:58  
Nice.

Kim  14:00  
And I love this idea because for the first time, you know, when the two problems that we already shared, when they're Adding Ten, that grows up to Add a Friendly Number. And when they're Getting to Ten, that grows up to Get to a Friendly Number.

Pam  14:13  
Mmhm.

Kim  14:14  
Students are breaking apart numbers to add some amount at one time. But for the first time ever, when they're thinking about the Over strategy, we're also building capacity for thinking about a different problem entirely. They're thinking about what new problem might I want to do that helps me? And instead of kicking them into just break it apart and do some things, they're thinking about what new problem might help me solve the problem that I've been given.

Pam  14:44  
So, like an example would be if I had 3,486 plus 1,995.

Kim  14:49  
Mmhm.

Pam  14:50  
A student using one of the two first strategies, Add a Friendly Number or Get to a Friendly Number might just start diving in and adding pieces of 1,995.

Kim  14:57  
Yeah, yeah. 

Pam  14:58  
But you're suggesting that helping students develop the Over strategy helps develop the capacity for them to say, "Well, 3,486 plus 1,995. What's a problem that's near that, that's friendly, that I could use that's not even here?" 

Kim  15:14  
Right. 

Pam  15:14  
Could I use 3,486 plus 2,000? Ah. Okay, that. That I can have a handle on. But that problem wasn't even on the page, right?

Kim  15:22  
Right, right. 

Pam  15:23  
They've created something new that's really nice. It's a bit too much. And then they can just sort of back up from there. 

Kim  15:29  
Yeah, yeah.

Pam  15:30  
If I could describe that as sort of getting outside the problem a little bit. It's almost like you're kind of looking up. If you'd seen my hands, I'm kind of like getting the 20,000 foot view a little bit and saying what do I know? Ooh, I could create this, like you said, a problem that's not even there. We want to help that develop that in students. And one way to do that is helping them develop this strategy. Nicely

Pam  15:51  
said. Cool. 

Kim  15:52  
At one point, I heard you say something about another strategy that we love for young students. And it's Using Doubles. And... 

Pam  16:02  
Oh, Kim.

Kim  16:03  
...you said something about how it's connected to Give and Take. So, why don't you talk about that for a minute?

Pam  16:09  
So, you and I have had this conversation a couple times, but I don't know that we've had it... I think we're going to have it more today. 

Kim  16:15  
Out loud?

Pam  16:16  
Yeah, so one of the major strategies that we suggest in Developing Mathematical Reasoning: The Strategies, Models, and Lessons to Teach the Big Ideas in K-2. I had to look at the title.

Kim  16:18  
I was going to say did you read that? Have it memorized yet?

Pam  16:27  
I was totally looking at the book on my desk. I don't have it memorized yet. One of the strategies that we promote in young learners is for them to use doubles. So, let me just sort of... I'll just describe that strategy, and then we'll kind of talk about what you and I... I don't know if we disagree, but maybe a different viewpoint. We'll see.

Kim  16:45  
It was a nice conversation. We're beating out what we know and believe, yeah.

Pam  16:48  
We're

Pam  16:49  
going to have it together on the podcast. Okay, so if I ask a kid an often missed question like 7 plus 8, then I might say... Or a student might say what do I know? And often kids know doubles. And if they don't, we can work with them. Doubles as a thing that brains kind of latch on

Pam  17:07  
to. Yes? 

Kim  17:07  
Can I interrupt you for just a second?

Pam  17:09  
You're going to anyway.

Kim  17:10  
Sorry. 

Pam  17:10  
No, go, go. 

Kim  17:11  
You can tell me no. I was talking to my 17-year-old not too long ago. And he... We were talking about some math stuff, and he was like, he's like, "It feels like doubles just kind of happen for people. Why is that? Like, why do we just know doubles?" And I was like, "Yeah, they do. Good job." 

Kim  17:25  
I mean, God gave us two hands, two arms, two feet, two eyes, two ears.

Kim  17:25  
Yeah. And he said, he like kind of run his hands up and down. He was like, "You think it's because of the symmetry in our body?"

Pam  17:36  
Ooh.

Kim  17:36  
I was like, "I have no idea."

Pam  17:39  
Who knows? Even-ness in the world. Yeah. 

Kim  17:41  
Yeah.

Pam  17:41  
Good question. So, if kids... A, we want to build doubles for kids, and then we could use them. So, if I say 7 plus 8, a student might say, "Well, I know 7 plus 7 is 14, so then 7 plus 8 would be 1 more, 15. Or I might know 8 plus 8 is 16, so 7 plus 8 is 1 less. It must be then not 16, but 15." So, they could use doubles. And then I have said... So, that that's the Using Double strategy. We want kids to learn that. It's a good one. It's good young one. I have said that then that grows up to the Give and Take strategy. And you've pushed back on that a little

Pam  18:17  
bit.

Kim  18:17  
I did. I did because... And I think it was hung up on the "grows up to". 

Pam  18:22  
Can I describe the

Pam  18:23  
Give and Take strategy first?

Kim  18:25  
Yeah. 

Pam  18:25  
And then we could do... Sorry, I probably should have done that first. Okay, so if you had a problem like 49 plus 53.

Kim  18:31  
Mmhm.

Pam  18:32  
You could say to yourself, "Ooh, that's close to the double of 50 and 50." But I'm going to do a little kind of Give and Take here. I'm going to do a little compensation. So, like 49. If I gave that 1, that would be 50. And if I took 3 away from the 53, that would be 50. So, it's kind of like you're down 1, up 3. So, down 1, up 3, you're kind of up 2. So, instead of double 50, you've got double 50 plus 2. It's like 102.

Kim  19:01  
Okay, so in this example.

Pam  19:03  
Yeah? 

Kim  19:04  
I would agree with you that it grows up.

Pam  19:08  
To the way I just use that example?

Kim  19:09  
In this example, yeah. And I think that when I was pushing back, we were talking about the single-digit facts where there is some work within doubles that you have some anticipatory thinking in the way that you do with Give and Take because you're thinking about what double is within another number, and like what am I going to do with the extra? So, we hadn't talked about this example. So, if you are trying to find a double and moving parts of numbers around, so that you can find a double that you like, then I would agree with you that it will grow up to Give and Take. I think we were just talking about different examples.

Pam  19:54  
Interesting.

Kim  19:55  
Yeah.

Pam  19:55  
So, like if I had something like 356.

Kim  19:58  
Mmhm.

Pam  19:58  
Plus 348. 

Kim  20:00  
Yeah.

Pam  20:00  
And I said to myself, "Those are so close to 350. I know the double of 350 is 700. But 356 is up 6. 348 is down 2. So, up 6 down 2, I'm really up 4. So, it's not 700, it's 704? You're like, yeah, that kind of. That's using doubles and Give and Take?

Kim  20:22  
Oh, so the way that you described that, I don't know that I would call that Give and Take. I would think that you found a double, and then dealt with what was left. What I wrote on my paper when you said those numbers was 354 plus 350. 

Pam  20:37  
Ah. 

Kim  20:38  
So, I did a little like Give and Take to get as close to a double as I can, and then deal with the extra 4.

Pam  20:45  
And that seems...

Kim  20:46  
This is all like such nuanced ways to make use of relationships.

Pam  20:53  
Mmhm.

Kim  20:53  
So, in a way, if you are building the idea of doubles, then that is something that you can make use of when you are using Give and Take. So, in that way, I would say they're connected. There's parts of it that grows up.

Kim  21:10  
And so, (unclear).

Pam  21:11  
Would you say that there's a part that grows up that if you are... If we encourage kids, help them to develop Using Doubles with single-digit facts. And so, they think about 7 plus 8 as either 7 plus 7 or 8 plus 8. That, that capacity to say, "What do I know that's near that?" and then adjust from there. What's a problem that's sort of inside of it or a little bit more than it that I can kind of adjust?" That, that we use in Give and Take?

Kim  21:38  
Yes. I think in single-digit facts, it's a little bit tricky because the adjustment is 1 more, 1 less because... You know, I think there's a chance that if you were working with something like 6 plus 8, and you were adjusting to 7 and 7, that feels more like I'm trying to find a double, and I'm using some Give and Take.

Pam  22:01  
And

Pam  22:02  
you gave and took to do that. Give and took.

Kim  22:04  
Mmhm, mmhm. 

Pam  22:04  
Yeah. Okay, okay. So, then Give and Take is more than just the two examples I did. Like I did 49 plus 53 and 356 plus 348. And then it's kind of giving and taking, but they were based on doubles. But we should also use Give and Take to do something like 39 and 140... Tell me. 6. I don't know if that was the best example.

Kim  22:27  
That's an interesting problem. 

Pam  22:28  
Yeah. I could change that. How about 126? It's not bad, but 126.

Kim  22:32  
Yeah, okay. 

Pam  22:33  
So, then I might say to myself, "Well, 39. That's almost 40, so I'm going to make that 40. And I got the 1, and I'm going to make that 125. Now, I have 40 plus 125."

Kim  22:41  
Right. 

Pam  22:42  
That those aren't... I'm not using doubles in any way, but I'm giving and taking.

Kim  22:46  
Yeah.

Pam  22:47  
Okay.

Kim  22:48  
Yeah, I like. So, I think some of the work that we're talking about here in this last bit of the conversation with the doubles and how it's connected to changing the problem is developing a level of sophistication that will help you with Over and help you with Give and Take. And so, if that work is beginning to happen in a K-2 classroom, then we're developing a part of your brain that goes looking for it. We're trying to notice how you can change the problem, so that it's easier for you. 

Pam  23:19  
Yeah, because

Pam  23:19  
like you said, those first two strategies that we did, Add a Friendly Number, Get to a Friendly Number, it's kind of attack the problem using relationships.

Kim  23:26  
Right.

Pam  23:26  
I keep one number whole, and then I add on parts of the other one, but it's the problem as it sits. 

Kim  23:32  
Right.

Pam  23:32  
Both in Over and in Give and Take we want students to look for the idea, notice the idea that they could kind of get outside the problem.

Kim  23:41  
Yeah, right. 

Pam  23:41  
Use something different to solve it. Go a little bit over, and then have to adjust. Or take one from one number to give to the other to make an equivalent problem that's easier to solve.

Kim  23:53  
Yeah.

Pam  23:53  
Both of those are more sophisticated. 

Kim  23:56  
Mmhm.

Pam  23:56  
And if we can help kids develop those, like you just said, noticing and looking for it younger, then when the numbers get bigger and more complicated, their brains are stronger already with a smaller, less complicated numbers, that they have a better opportunity to do that with larger numbers.

Kim  24:16  
Yeah. And so, when you just said if we can help kids look for it earlier, look for it younger, this is a really nice example where we've got kids who maybe are ready for more than maybe some of their peers at a younger age, and often we say, "Okay, you know, people are encouraged to like make the bigger numbers and move them into, you know, the grade above." We think this is a better way to help kids in the content that they're doing just become stronger, just become more dense in their own understanding of number. Help them think about strategies that are then going to grow up and be helpful when they're messing with numbers that are in the next grade

Kim  25:01  
level. 

Pam  25:02  
So, in other words, differentiate in the way that all your kids are working on single-digit facts.

Kim  25:07  
Mmhm. 

Pam  25:07  
But because you're developing strategy, you are differentiating because some students are attacking the problem and using what they know. And other students are being encouraged because they're ready, it's on the edge of their zone of approximate development, to think outside the problem and go a bit Over, or to Give and Take.

Kim  25:22  
Yeah.

Pam  25:22  
Yeah, that's a great. I like how you brought that out. Nice, nice, nice. Alright, it's not about learning steps to do, y'all. It's about building relationships. Strategies are synergistic. They build on and grow up as we go. And then students are using the mental actions of mathematicians, which is what we are aiming for. If you are interested to learn more about Developing Mathematical Reasoning: The Strategies, Models and Lessons to Teach the Big Ideas in K-2, go check out that book that is on bookshelves now. In bookstores? How do you say that now? At Amazon. 

Kim  25:54  
I don't know.

Pam  25:54  
It's everywhere.

Kim  25:55  
It's online. 

Pam  25:57  
Get a book wherever 

Pam  25:58  
you get a book. Alright, y'all, 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!