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

Kim  0:09  
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  0:16  
Y'all, algorithms are super cool 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 and use. 

Kim  0:29  
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships. 

Pam  0:35  
We invite you to join us to make math more figure-out-able. Da Da Dum

Kim  0:41  
In this episode, we're taking a look back at some of the episodes where we've chatted about important topics in K-2 classrooms. We're highlighting names of teen numbers, nudging students off of counting by ones, most important early strategies for first grade, and how strategies we build with young students grow into strategies we want to develop in older students. Now on to the show!

Pam  1:03  
You might find it interesting that the first thing we're going to talk about is language. If we're talking about math and counting, we're going to talk about language. But we are. Because our language influences how kids learn to count and learn the counting sequence. In a huge way, the names, the words that we have given to numbers can either sort of clear up or make more clear the counting sequence. Or it can actually sort of muddy the counting sequence. What do I mean by that? Well, for example, when kids are to count, you know, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Then 11. 11? Where'd that name come from? Eleven? Because really, if you look at the way we write 11, eleven means a 10 and a 1, right? There's that one 10 and that one 1. And you might be like, "Well, Pam, of course, that's what 11 means." Well, interestingly enough, in the language of English, we have made up a word "eleven". In several languages, that's a new word when we get to 11. However, in many Asian languages, the word for 11 isn't a new word at all. It isn't a made-up word. It's actually the word "ten" and the word "one" put together. So, if I was speaking many Asian languages, and I start to count, there's a tone or word for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And that has its own tone. And then, when I get to 11, I actually say 10-1, and when I get to 12, I actually say 10-2. Like, those are the words, the names for those sort of teen numbers. And I'm actually not even in the teens yet. When I get to 13, it's not... Well, 13. That actually sounds kind of like it, right? It almost sounds like 3-teen. We almost kind of then follow the pattern. You're laughing at me. 3-teen. Well, 4-teen. I mean, it kind makes sense in the pattern, right? In fact, we noticed that when kids get to 14, they're kind of confident again, 14. But then it doesn't sound like it should again. Fifteen? Shouldn't it be 5-teen? It should, right if it made sense. You're laughing. You're like 5-teen. Well, sixteen, seventeen. Those sound kind of like they should, right? And even if we were to help students a little bit better, it would even be better if it was 10-1, 10-2. Or even teen-1, teen-2, teen-3. But we could actually name the numbers something that they mean. And my point is that there are languages out there that instead of sort of having these new words or words that kind of sound like what the numbers mean, they actually are what the number means. 10-1, 10-2, 10-3, 10-4. 10-4, buddy. 10-5, 10-6, 10-7, 10-8, 10-9. And so, that can be helpful to kids when they're learning how to count. So, in our language, since we've kind of made up these words for whatever reason, we have to tell kids. Or at least I'm going to suggest. To help kids learn to count, let's be clear with them what 11 means.

Kim  3:53  
Yeah.

Pam  3:54  
Let's be clear with them what 12 means, what 13 means. That will help them feel the pattern. If they've already got the pattern of 1, 2, 3, 4, 5 down, then when we say, "Well, that's just 10-1, 10 and 2, 10-3, 10 and 4," then they're like, "Oh, yeah. Well, I could do that pattern." Oh, and we call it eleven. So, 10-1, this number means 10 and 1, but we call it eleven. This number right here, this means 10 and 4, but we call it fourteen. This number here, it means 10 and 7, and we call it seventeen. Oh, look, there's a 7 right in there, seventeen. So, one of the things that we suggest in today's podcast is to help students learn to count, help them understand what the numbers actually mean, not just what they're called. So, the number means 10 and 3, and we call it thirteen. 

Kim  4:39  
Yeah, the first time I ever heard you say this, it was mind blowing because I didn't teach kinder and first grade, so you know for me it was more about these numbers just are named what they're named. And I remember hearing you kind of give this spiel about teen numbers, and it just rocked my world a little bit. And at the time, I had a really young learner, who was starting to count.

Pam  5:00  
Tiny, tiny Cooper.

Kim  5:01  
Tiny Cooper. And he was starting to sing song count, right? Just, he had heard the sequence of numbers. And I obsessively came behind him when he was counting, and I would say, "Hey, let's count numbers together," or "Let's count the things." And when he got past 10, and he would say "eleven," I would whisper behind him, "10 and 1," and he would say, "12," and I would say, "10 and 2," and he would say "13," and I would say, "10 and 3." And I felt like it was my job at the time to provide some meaning to what that word he was saying meant. And I was (unclear) about it. 

Pam  5:39  
Totally, you were telling me about that, and I was like, "Yes!" That's amazing, because now Cooper has kind of heard the song of counting. He's able to say the next word for the number. But you're providing meaning kind of on top of that. And it just solidifies for him that he's got the counting sequence right and what those words mean, what the numbers actually mean as he gets into the teens. Ah, that was brilliant. When you told me about that, I was like, "Yes!" We need to get all parents and teachers of younger students to do that kind of thing. So, well done. 

Yeah. So, Kim, you told me once about a group of students who were not using relationships, the strategies, but they were counting 1 by 1. And I was like, "What? Like, how is that a thing?" Because I was really convinced once we sort of opened the vision, opened their minds to the fact that they could just use them, I was like, "Then, that's magic, it's just like going to happen automatically." So, how is that a thing? And tell us all about that. What's the scenario? What was going on? 

Kim  6:41  
Yeah.

Pam  6:42  
Go.

Kim  6:42  
So, I spent some time on a campus for quite some time actually that was really into numeracy and really into relationships. And because I was there quite a while, I got to know the teachers well enough that we had a relationship. One day I was planning with the first grade team, and a few of the teachers said to me, "Kim, we are working on that stuff. We're working on the doubles. We're working on like partners of 10. They're still counting. What do we do?" So, I said, "Okay, let's take a look at what's happening." And so, I scheduled some time in their classes, and I sat down with the kids as they were working on problems, right? Some problems that they were supposed to be solving, problem solving whatever. 

Pam  7:21  
There were some word problems. There were some naked problems.

Kim  7:24  
Yeah.

Pam  7:24  
All different kinds of problems. 

Kim  7:25  
Yep.

Pam  7:26  
You sat down with them. So, that's a noteworthy thing I'm just going to point out right now. That what you didn't do was just kind of like glance or whatever, but you sat down and interacted with kids. 

Kim  7:36  
Yeah. Didn't look at the work afterwards, right, because then it's too late to know what they did. So, I sat down with the kids. And listen, they were just so quick at counting on their fingers or to 8, 9, 10, 11, 12. Right? That kind of like say it really fast. And we (unclear)...

Pam  7:51  
You said that so fast, I didn't even understand you. Say that again. Do that again, I dare you.

Kim  7:56  
8, 9, 10, 11, 12. And so, the kids were quick, right, at what they had been doing for quite some time, and so we needed to find out first who owned their doubles and who owned their partners of 10 before we could kind of interrupt that flow, before we could kind of say, "Wait, don't do that." And so, I talked with the teachers specifically about doubles and near doubles and partners of 10. And they kind of thought they knew who owned them, but they couldn't really name specific names, and so that was kind of our first. That's where we started. That was our first job. And so...

Pam  8:32  
Let me just be clear. You said to teachers, "Which of your students own doubles, near doubles? Which of your students own partners of 10? We need to sort of know that." 

Kim  8:42  
Right.

Pam  8:43  
And your teachers might have some ideas, but they weren't actually sure. They didn't know which students owned those relationships and which didn't. Okay.

Kim  8:52  
Right. So that was kind of the first thing we needed to figure out is who could we say own them, and who do we still need to work on and give more experience? And so that's where we started. We... I say we. I didn't. They pulled kids. So, quickly, here's what we did. We flashed. And I say "flash". It's probably not the right word. We flashed doubles. So, we held up for a couple of seconds each of the doubles up to 10. So, we show them a card with 9 plus 9, give them a few seconds, slide it to the side. Show them a card with 5 plus 5, and we just sorted them into ones I know right now, ones I don't know yet, to get a feel for which of the students owned their doubles. And then we did the same thing with partners of 10. And it literally took about 30 to 45 seconds per kid because there's only so many partners of 10 and only so many doubles. And so, we just had these cards ready. And it was really just about kind of sorting the kids. Can I rely on the fact that they own them in isolation at this point? And so what I said to them was, "We're going to make a list of these kids, and we're going to commit those names to memory. We're going to write them down somewhere. We're going to know in our heads which kids we can identify as ones that know these two categories or these groups of facts." And then we called those kids out. Right. When we saw them doing (unclear).

Pam  10:18  
What do you mean by that? You didn't embarrass them to be clear.

Kim  10:20  
No, no, no. So I'm going to explain. So, when the teachers were sitting down next to them, and they saw them pull out their fingers, we gently covered their hands, and we reminded them, "You know these doubles." And when we heard them, "8, 9, 10, 11, 12," then we tapped them on the arm. We started speaking to them in the middle of the count. We did some things to interrupt the method that they had been doing for so long that it was a natural habit, right? So, if you have a habit, something has to interrupt the habit that you have to replace it with something new.

Pam  10:56  
So, you literally gently stop them mid count. 

Kim  10:59  
Yes, absolutely. Physically, by, you know, covering their fingers a little bit, softly, of course. 

Pam  11:07  
Just kind of tapping them on the hand.

Kim  11:09  
Or verbally if they were a verbal out loud counter or mouthing it, then we would speak to them in the middle of the count. So, sometimes we realized the kids weren't even aware that they were doing it, and so we just reminded them, "Hey, double. Oh, partner of 10." And they would go, "Oh, I know this one." Because we needed to correct the habit that they had gotten into for quite some time.

Pam  11:32  
Alright, so... Yes, we're all getting better at being more clear about all the things. Kim, what are the major relationships that we would want to build in, say, first graders as we look at single-digit addition facts?

Kim  11:44  
Yeah, so a really, really important one is partners of 10. Which I think then, you know, that basic understanding of partners also extends to partners of 20, partners of other things. But within each of those other groups of larger partners of 20, partners of 100, there's this basic partners of 10 relationship that we really want kids to own.

Pam  12:07  
You know, I remember the first time you said... When you think about partners of 20, I, in my head, thought about, like... How do I even say this? Kind of like all the partners of 20. And you said, "It's really 20 plus the partners of 10."

Kim  12:23  
10, and the partners are 10, yeah. 

Pam  12:26  
Oh, sorry. Thank you. Wait. Oh, yeah. Partners of 20. I was thinking about partners of 30. Ha. I don't know why. So, for partners of 20, yes. Wow. It's it's thinking about 10. And then like right now you have the 10. So, now you're in the teens, and so it's then you're just using partners of 10. So, let me go partners of 30. If you're thinking about partners of 30, that's probably when I thought of it actually. That's why it was in my head. It was the moment when you said, "Well, then you could work with kids for a short period of time in partners of 30," and I pushed back and said, "No, no, no. Don't do that." And you said, "Yeah, Pam. It's not any random number to get to 30. It's once you have 20 established, then you're really just dealing with partners of 10." If you were dealing with partners of 60, you really want kids to think about I got 50, and then partners of 10. That's the work you're doing is to realize that inside these bigger numbers, there's always kind of this big multiple of 10 and just partners of 10. 

Kim  13:21  
Yeah.

Pam  13:22  
Yeah.

Kim  13:23  
Yeah, and I think people can sometimes hear that when you know when we talk about I Have, You Need, they might hear it as like let me step up and do partners of 10 and partners of 20 and partners of 30, and once you get outside of the 10, then it's really what you're saying. The understanding of the larger multiples of 10 and the partnership that you're trying to develop. It's not, now, let me just step up and do... This week, we do partners of 30. And then the next week partners of 40.

Pam  13:51  
Or partners of 35. No, no, no. Yeah. No. Yeah, yeah. Cool.

Kim  13:54  
Yeah. Okay, so partnership 10 is one major relationship. Another major relationship is doubles. You know, kids early on have some intuitive sense about early doubles. But we can do some work to capitalize on that relationship because it extends far, far beyond just first grade. 

Pam  14:14  
Yeah, mathematicians play with doubles.

Kim  14:16  
Mmhm.

Pam  14:17  
And then once they kind of have the sense of doubling, they play with halving. Doubles and halves are super important. I did not know that as a non-real-mathy person for very long. And I remember the day when I started realizing doubles are super important. And yeah.

Kim  14:35  
Would you call yourself a non... I don't know what you just said. I would say you were a traditional.

Pam  14:41  
I said I was a non-real...

Kim  14:43  
Non-real math-er.

Pam  14:45  
Non-real math-er. I was not real math-ing. 

Kim  14:47  
Yeah.

Pam  14:48  
Not on purpose. 

Kim  14:49  
Yeah, no, no, no. Yeah, for sure.

Pam  14:51  
Yeah.

Kim  14:52  
And then the other major relationship is the idea of plus 10 to help you with things like plus 9 or plus 8. So, a little bit over, little bit more than you need. But a lot of it is plus 10 or a multiple of 10. So, you're not going 18 to get to 17. 18 more to get you to get 17 more. It's really about capitalizing on the tens or the hundreds.

Pam  15:16  
Because there's this nice pattern that if you can think about adding 10. Especially, in first grade, if I have some single-digit number and I add 10, that's the definition of a teen.

Kim  15:26  
Mmhm.

Pam  15:26  
So, you know I've got 6 and I add 10. Bam, I need to know what 6 and 10 is. I need to know what 10 plus a single-digit number is, that that's a definition of those teen numbers. That's super important. And so, we can almost practice that while we kind of try to develop an Over strategy kind of thinking. Like, we can say, "Hey, what's 6 plus 10?" And the kid's like, "I know that. It's 16 because I know what the teens are." And you go, "Cool, what's 6 plus 9? If 6 plus 10 is 16. What's 6 plus 9?" Oh, it's one of my favorite, favorite questions of all time to ask a kindergarten, first-grade kid that's kind of, you know, so right on the cusp of what they're thinking about. Second grade if they've never really thought about stuff. I could even ask that question of third grade, fourth grade kids. Where, you know like, "Hey, what's 7 plus 10?" "That's 17. I know the teens. You're not going to get me here." "Okay, what's 7 plus 9?" "Let me think about that. Ooh, those are related. Hey, that's just 1 less!" And their eyes light up, and it's so exciting.

Kim  16:22  
Yeah.

Pam  16:22  
Cool.

Kim  16:23  
So, when I responded to our Journey member, and I shared those three major relationships, then I think part of the conversation was, so I wouldn't say let's do partners of 10, let's make sure all the kids own them really well. Now, let's move on to the new relationship. So, if my goal is that I want kids to get funky problems, and they have these major relationships, and they choose which of these relationships that you want to use, I'm going to start pretty early to develop kind of all of them, just, you know, with Problem Strings, and with routines, and with activities that we can do. And then highlight those relationships all throughout the year.

Pam  17:04  
Mmhm. Yep. And in order to choose, they have to have choice. Their brains are stronger. They're thinking more sophisticatedly. Like, you're literally creating more successful human beings. Not just at 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 10.

Kim  17:29  
Mmhm.

Pam  17:30  
So, that they can use the strategy, Get to Ten. 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 Ten for adding something like 7 plus 5. 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 three partner-ness. So, if 7 and 3 is 10, but I was supposed to add 5, I've got 2 left over. Ah, then I could think about 10 and 2. So, that's the Get to Ten strategy. Is there anything you want to say about that? Oh, except for the fact that I just used a different example than I wrote down. Haha, that's funny. 

Kim  18:27  
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 to 30. And since...

Pam  19:01  
That 8 and 2 partner.

Pam and Kim  19:03  
Mmhm.

Kim  19:03  
Mmhm. And if they're adding 7 altogether and have used 2 to get from 28 to... 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 with 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. Or 2800 plus 200 to get to 3,000.

Pam  19:36  
Brilliant. So, lots of ways that those relationships grew up. First, partners of 10, which grows up into Get to Ten, 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 left over to whatever I was adding to begin with. Cool. Alright. 

Kim  20:12  
So, that's one strategy, Get to Ten, growing up to Get to a Friendly Number. What about Add 10 and how it grows up to Add a Friendly Number?

Pam  20:22  
Nice. So, for young students, we encourage teachers to work with them to think about any digit, any single-digit number plus 10.

Kim  20:30  
Mmhm.

Pam  20:30  
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. Or 8 plus 10. That's like 10 and 8. Oh, that's a teen number. What do we call that? 18. 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 10 strategy can grow up to Adding a Friendly Number. 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 continue to do that with larger numbers, so that 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 2,000 is.

Kim  21:50  
Mmhm.

Pam  21:51  
And once I get that, then I can work on adding the rest of them.

Kim  21:53  
Mmhm.

Pam  21:54  
Both of the strategies that we've just talked about, starting really young with either Partners of 10 or just Adding 10, 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 head.

Kim  22:34  
K-2 is where we help students make sense of number and transition from counting to additive thinking. Such important work happens in these grades, and we are happy that you are helping to make math figure-out-able for your students. 

Pam  22:46  
Hey, 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!