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  
We know that algorithms are amazing, historic achievements, but they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

Kim  0:30  
In this podcast, we hope you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:38  
We invite you to join us to make math more figure-out-able. 

Kim  0:42  
Hey there. 

Pam  0:44  
[Sighs]. How's it going?

Kim  0:47  
It's good. You sound tired.

Pam  0:50  
You know what? We're doing lots. My life is full of wonderful things.

Kim  0:55  
That's my favorite. Which means I'm tired.

Pam  0:58  
It means I'm really, really busy, but I'm not saying that. I'm saying my life is full of wonderful things. I'm always open to new opportunities. 

Kim  1:08  
They are wonderful things.

Pam  1:10  
They are, they are, they are.

Kim  1:12  
So, it is almost Fibonacci Day.

Pam  1:16  
That's a cool day, right? So, we were talking about the fact that Fibonacci Day was coming up. So, Fibonacci Day is November 23 because if you write November as the eleventh month, and then 23rd next to it, so you have 1-1-2-3, those are the first four terms of the Fibonacci sequence. 1-1-2-3. And I said, "Okay, but wait. When is really Fibonacci? When's the Fibonacci Day going to be where the next, where the year will pipe in, right? We started talking about that. So, the Fibonacci sequence is 1-1-2-3 because you add the... Once you start with the two ones, then you add the numbers together. So, 1 plus 1 is 2. And then that second 1 plus the 2 is 3. And then that 2 plus the 3 is 5. And then that 3 plus the 5 is 8. And then you could keep going, 5 plus 8 is 13. So, I said, "Really, the real Fibonacci Day is coming... Oh, okay. Not so soon." I was like, "It's soon! Oh, 2058. 11/23/2058, we will have like, ta-da! Fibonacci Day! That will be amazing... There you go. Fibonacci sequence appears in so many random kind of cool places. 

Kim  2:23  
Yeah. 

Pam  2:24  
It's a cool thing.

Pam  2:25  
And yeah, okay. 

Kim  2:26  
We should make a Fibonacci Day shirt. 

Pam  2:29  
Oh, okay.

Kim  2:30  
One year before. In '58.

Pam  2:32  
We'll tell Sue that we need a Fibonacci Day shirt. Oh, that's actually not a bad idea. Alright, listeners, would you? Anybody interested in a Fibonacci Day shirt? Have to let us know.

Kim  2:42  
Okay. So, we're going to do something a little bit different today, right? So, normally you will give me problems to solve, and then talk through them. And so, let's do something a little bit different. How about I give you some problems in a Problem String, and then you share how you're thinking

Kim  2:59  
about it. What do you think?

Pam  3:01  
Alright, cool. That

Pam  3:02  
sounds great. Let's do it. 

Kim  3:05  
You got some paper and a pencil?

Pam  3:05  
I have my iPad and my Apple Pencil. Yeah.

Kim  3:05  
Hmm, your fancy.

Pam  3:05  
Is that a pencil? Do you think an Apple pencil is a pencil?

Kim  3:11  
I say Apple pen. 

Pam  3:13  
You say Apple pen?

Kim  3:14  
I think I do (unclear). Yeah, I totally say (unclear). 

Kim  3:17  
I mean,

Pam  3:17  
on Apple Pencil, it literally has an apple, and then the words Pencil Pro. 

Kim  3:21  
Then it is a pencil. Look at you using a pencil!

Pam  3:24  
It has my name because they would engrave it for free, so I put Pam Harris on it. 

Kim  3:24  
What?!

Pam  3:24  
I

Pam  3:24  
know how cool is that. 

Kim  3:24  
You didn't get Math is Figure-Out-Able on

Kim  3:31  
it?

Pam  3:34  
I guess I could have.

Kim  3:35  
Still waiting on my Math is Figure-Out-Able pencils.

Pam  3:40  
Indeed, you are. 

Kim  3:41  
Alright, let's get started.

Pam  3:44  
You know why we haven't done that? Because everyone who sells pencils only sells mechanical pencils that they write on or they sell terrible pencils. They're

Pam  3:47  
not good pencils. 

Kim  3:47  
Yeah, I need Ticonderoga to step up their game and do engraving.

Pam  3:54  
We need to

Pam  3:55  
find somebody who will engrave on Ticonderoga. That's what we haven't found yet. You find that. You find it, I'll print them.

Kim  3:59  
I'm sure there's an Etsy person out there who would do it. Oh. Well, that's true. 

Pam  4:03  
That would be a pricey pack. 

Kim  4:05  
Alright, let's do some math. You ready for your first problem? 

Pam  4:05  
Yes, go. Problems, go.

Kim  4:05  
Okay, first problem is 59 plus 94. 59 plus 94.

Pam  4:08  
Okay, 59 plus 94.

Kim  4:13  
Yep.

Pam  4:15  
I am thinking about 100. So, the 94, I'm making that into 100. So, I added 6, so I'm going to take away 6 from 59 to get 53. And 53 and 100 is 153. Oh, but I just saw something else I could do. 

Kim  4:31  
Okay,

Kim  4:31  
well, take your pick. It sounds to me like you did a little Give and Take. You took...

Pam  4:36  
I did.

Kim  4:37  
...6 with a 59 to give it to 94 to make 100. 

Pam  4:41  
Yep. 

Kim  4:42  
Okay, fantastic. So, I actually put on my paper minus 6 and plus 6 using an equation and wrote underneath that, 53 plus 100 equals 153.

Kim  4:55  
Okay. That makes sense, yeah. 

Pam  4:55  
Okay. Ready for the next problem? That would correctly represent my thinking. Yeah, go ahead.

Kim  4:59  
Great. You're next problem is 54 plus 99. 

Pam  5:03  
So,

Pam  5:04  
that was actually the thing I thought about after I did my first strategy.

Kim  5:08  
Mmhm. 

Pam  5:08  
So, 54 plus 99, I'm thinking Over, so I'm going to do 54 plus 100 is 154. But 54 plus 99 would be 1 less than that, so it's 153. Which magically is the same as the problem before. So, I'm noticing there's a connection. Can I talk about that? 

Kim  5:29  
Yeah. 

Pam  5:30  
So, the first problem, 59 plus 94, we got 153. And the second problem, 54 plus 99, we got 153. I noticed that the ones digits swapped places. The 9 and the 59 went to the 90. And the 4 in the 94 went to the 50. 

Kim  5:52  
Nice noticing. 

Pam  5:52  
Yeah, it's almost like you had piles of things.

Kim  5:53  
Mmhm.

Pam  5:53  
And you just sort of moved piles between the two numbers.

Kim  5:56  
Nice.

Pam  5:57  
Cool. 

Kim  5:57  
Alright, ready for the next one? 

Pam  5:08  
Yep. 

Kim  6:12  
139. 

Pam  5:08  
Okay. 

Kim  5:08  
Plus 491

Kim  5:08  
139 plus...

Pam  5:08  
I'm probably going to Give and Take again. 

Kim  6:12  
Okay.

Pam  6:12  
I'm going to make the 139 into 140. 

Kim  6:15  
Okay. 

Pam  6:16  
By taking the 1 away from the 491. 

Kim  6:18  
Mmhm. 

Pam  6:19  
So, then I have 140 plus 490. Ugh. Maybe I shouldn't have done that. No, I take it back. I want to give... I want to take 9 away from the 139 and give it to the 491. So, the 491 became 500, and the 139 became 130, and so I have 130 plus 500 is 630. Yeah, that's embarrassingly better.

Kim  6:47  
I think it shows mathematical behavior that you tried something. And your strategy was the same, you just made a different choice about which number you wanted to make nice.

Pam  6:55  
Mmm, sure enough.

Kim  6:58  
You did a nice job there. Okay. And so what did you get

Kim  7:01  
for your total? 

Pam  7:02  
630. 6 hundred thirty.

Kim  7:02  
630, nice.

Kim  7:03  
Okay, next problem.

Pam  7:05  
Okay.

Kim  7:06  
131 plus 499.

Pam  7:11  
I see what you're doing there. So, I'm going to Over this one. 131 plus 500 would be 631. But I added 500 instead of 499, so then we're going to back that up to 630.

Kim  7:25  
So, when you said, "I see what you did there," what was it that you saw?

Pam  7:31  
That you moved some piles of things around again. So, we had 139 in the problem before, and this one was 131. And we had 491 in the problem before, and this was 499. So, the ones place shifted. The 9 went where the 1 was, and the 1 went where the 9 was. And then I was able to do a very nice Over strategy.

Kim  7:53  
And I'm going to wonder if you already knew what the sum was going to be. 

Pam  7:58  
Hmm. When I

Pam  7:59  
said, "I see what you did there," I could have just used the problem before? 

Kim  8:02  
Mmhm.

Pam  8:03  
That actually didn't occur to me in the moment. Even though I recognized that you'd swap things, it didn't occur to me that therefore it would have the same sum.

Kim  8:11  
Okay, and so you added 500, so I would draw plus 500, and then back a little 1, and you got 630. 

Pam  8:19  
Yeah.

Kim  8:20  
Okay. 

Pam  8:20  
Now that you've played for that, played with that. You can tell me I can't do this. But I could have turned... I could have swapped the 10s places from that original problem. 

Kim  8:29  
Mmhm. 

Pam  8:30  
So, that 139 plus 491 could become 199. And the 491 would become 431. And then I could have over that way. Just a different Over. 

Kim  8:40  
Mmhm.

Pam  8:40  
And then it's a different swap from the second problem. 

Kim  8:43  
Mmhm.

Pam  8:43  
The second problem was 131 plus 499.

Kim  8:46  
Mmhm.

Pam  8:47  
To get to my third problem I just created, I could take the 10s and the ones and swap them. 

Kim  8:52  
Mmhm.

Kim  8:53  
Yeah. 

Pam  8:53  
Anyway, whatever. 

Kim  8:54  
Yeah. Alright, you ready for the next one? 

Pam  8:58  
Yep. 

Kim  8:58  
938 plus 299. Curious what you're thinking when you see that problem.

Pam  9:05  
The very first thing I thought of is do I want to over by 300? And then I said, "But we've been doing this swapping thing," and I don't really want to have to add over the 1,000. So, I was like, "What if I swapped the 99 from the 299 and put it in the 900? 

Kim  9:25  
Mmhm.

Pam  9:25  
So, that would be 999 plus. And then the 38 that was with the 900 is now going to go with the 200, so 238.

Kim  9:31  
Mmhm. 

Pam  9:31  
999 plus 238. Little Give and Take is going to be 1,000 plus 237. 

Kim  9:38  
Mmhm.

Pam  9:39  
Which is just 1,237.

Kim  9:40  
Nice.

Pam  9:41  
That's a nice problem. 

Kim  9:43  
Yeah. And I like that you noticed that you could. The 299 is not a bad, a bad thing to... 

Pam  9:50  
Add 300, yeah.

Kim  9:51  
...add 300, right? 

Pam  9:52  
But 938 plus 300 just makes me think more than I want to.

Kim  9:57  
Mmhm.

Pam  9:57  
I mean, I can.

Kim  9:58  
It's good, it's good.

Pam  9:59  
I can. 

Kim  9:59  
Mathematicians often are a little bit lazy, clever. Is that what you say? Alright, can you put some words? I know you've been talking about like some things moving. Look back at the problems.

Pam  10:13  
Yeah.

Kim  10:13  
Can you put some words to what you're noticing?

Pam  10:18  
I'm noticing that you gave me a lot of problems with nines.

Kim  10:21  
Mmhm.

Pam  10:22  
That were not necessarily in the most friendly place to start with. 

Kim  10:26  
Okay.

Pam  10:27  
So, when I see problems with nines, I could consider breaking. I could consider thinking about the number as the hundreds, the tens, and the ones and wondering if I just move that pile of tens to the other number or that pile of ones. or maybe the tens and the ones. Or I could also think about moving the hundred to the other one, swapping the hundreds. 

Kim  10:44  
Yeah.

Pam  10:44  
If I can swap some of those piles around, I might be able to turn it into a really nice Over or Give and Take

Pam  10:50  
problem. 

Kim  10:50  
Yeah, as long as you maintain some equivalence. 

Pam  10:54  
Yeah, I have to actually move them. I can't change them. The piles have to move, so that I still have the same total. It just sort of shifted where they were. 

Kim  11:01  
Okay, you ready for the final problem? 

Pam  11:03  
Yes.

Kim  11:05  
What about 1,949? Did you write that down? 1,949 plus 9,285. It's a little funky.

Kim  11:11  
Let me make sure. 1,949. 9,285.

Kim  11:12  
Yep. What you want to do? 

Pam  11:18  
So, my very first thought is the 9,000. I'm like, "That's not very close to 10,000. Sure wish it was." The 1,949 is... It's only 51 away from 2,000, so I could get it to 2,000, and it wouldn't be too bad to take that from the 85 over in the 9,285. But I'm noticing nines. 

Kim  11:42  
Yeah. 

Pam  11:43  
So, I really wanted that 9,000 to be closer to 10,000, so I'm super tempted to take the 900 from the 1949 over, swap it with the 200 in the 9,000. 

Kim  11:54  
Mmhm. Yeah.

Pam  11:54  
So, that now becomes 9,900.

Kim  11:56  
Mmhm. 

Pam  11:56  
I'm going to keep the 80 there with the 9,285. I'm going to keep that 80 there. But I'm going to swap the 9 from the 1,949 with the 5 from the 9,285. So, I think I've correctly got 9,989 as the second addend. I got to figure out what the first addend is. I left the 1,000 there. I brought the 200. I swapped the 200 over. I left the 4 there. And I swapped the 5 over.

Kim  12:21  
Yep. 

Pam  12:21  
So, I now have 1, 245 plus 1980... Sorry, 9,989. And that sounded like it took a long time. It didn't really. I just put the biggest numbers together. 

Kim  12:35  
Yep. 

Pam  12:35  
9,989 is just 11 away from 10,000.

Kim  12:40  
Yep. 

Pam  12:40  
So, I'm going to make that. I'm going to add 11 to get to 10,000. I'm going to subtract 11 from the 1,245 to get 1,234. Now, I have 10,000. Well, I have 1,234 plus 10,000.

Kim  12:54  
Yeah. 

Pam  12:54  
Which is 10,100... 1,002... Oh, my gosh. Which is 11,000. Sorry. I was going to say 10,000 1,000. It's 11,234.

Kim  13:03  
Nice. 

Pam  13:05  
That's a nice problem.

Kim  13:06  
Yeah.

Pam  13:07  
Especially because that 8 might throw you off, the 80. The 80. But it's so close, really.

Kim  13:12  
Yeah.

Pam  13:12  
Nice.

Kim  13:12  
Yeah. I think this is one of my favorite not major strategies because it's one of the things that I used early on, this idea of accumulating the largest amount I could before I then did something with

Kim  13:31  
the problem.

Pam  13:31  
By like swapping those piles around? 

Kim  13:31  
Yeah.

Pam  13:31  
That's interesting, yeah. 

Kim  13:33  
And

Kim  13:34  
you notice that there are a lot of nines. Having a lot of nines and eights in the problem makes it really nice if you're going to then use Over. But it's certainly not the only like requirement. They don't have to have nines and eights in order to swap it. Just sure is nice when they are. 

Pam  13:51  
Yeah,

Pam  13:51  
because then it turns into either a nice Over or a nice Give and Take problem that's just ah. 

Kim  13:55  
Yeah.

Kim  13:56  
So, I just want to ask you. So, I know that when I was recording what you were saying, I kept the majority of it with equations. What do you mostly have on your paper?

Pam  14:07  
Oh, gosh. I only have equations. 

Kim  14:08  
Yeah. 

Pam  14:08  
Even when I Overed, I just wrote down the result. 

Kim  14:13  
Yeah.

Pam  14:14  
Yeah.

Kim  14:14  
So, a lot of times people ask us about models and strategies and which is important and how do you know which model when? So, let's really quickly before we wrap too far up. 

Pam  14:26  
Okay.

Kim  14:27  
Define models and strategies.

Pam  14:31  
So, the strategy that we were just working on we call the Swapping strategy.

Kim  14:35  
Mmhm.

Pam  14:35  
Where we're sort of swapping those places. Like you said, it's not a major one. What we mean by that is kids can get by without it.

Kim  14:43  
Mmhm. 

Pam  14:43  
Meaning they can solve all the major problems that are reasonable to solve without a calculator, thinking and reasoning, using what they know. But it's super handy to start to develop in kids because of the place value that you're using.

Kim  14:55  
Right.

Pam  14:56  
And then you you get better at the sense of reasonableness and magnitude. Like, the fact that I had to think about what numbers were close to what numbers. And the Over strategy comes in or Give and Take. And all of that just kind of comes together. 

Kim  15:08  
Yeah.

Pam  15:09  
So, you could use strings like this to differentiate learning because you could have kids working on just using the strategies they've learned so far. 

Kim  15:17  
Yep.

Pam  15:18  
While you've got... And you might be modeling those like the Over thing that I did earlier.

Kim  15:21  
Yep.

Pam  15:21  
While you've got kids that are playing with the swapping idea. 

Kim  15:24  
Yep.

Pam  15:24  
It's a great differentiator.

Kim  15:25  
Right.

Pam  15:25  
So, the strategies are how you're messing with or playing with the numbers. What's your urge? You see the numbers and you're like, "I'm tempted to..." Like, what pings for you?

Kim  15:37  
Yep. 

Pam  15:37  
And how do you then use those relationships to solve? That's a strategy. You just said that the models that you have on your paper, my iPad screen, were mostly equations.

Kim  15:48  
Yeah.

Pam  15:48  
And for this string, as soon as you start swapping, really, you're just creating an equivalent problem.

Kim  15:54  
Right.

Pam  15:55  
So, that's why it becomes an equation. You got an equation sign with the equivalent problem next to it. I do think I did an Over which is probably why you have a number line.

Kim  16:02  
Right.

Pam  16:03  
That might be a model.

Kim  16:03  
Mmhm. 

Pam  16:04  
So, we wouldn't say I did a number line strategy. 

Kim  16:07  
Yeah, yeah. 

Pam  16:08  
It's like what I did on the number line was an Over strategy, and you use a number line to represent

Pam  16:15  
it.

Kim  16:15  
So, often, one of the questions people will ask is aren't you supposed to represent a lot of addition on number lines or multiplication, division on ratio tables or arrays. Is it ever okay to just have equations? And I think we're saying in this moment, that the best model for this particular strategy is the equation because of the structure of the numbers, because of the strategy that's being pulled out.

Pam  16:37  
Mmhm.

Kim  16:37  
Another question that people will sometimes ask is which is more important? The strategy or the model? Like, what should I be focusing on?

Pam  16:38  
So, before I answer that, can I just say a little bit more about when to use equations? 

Kim  16:56  
Yeah.

Pam  16:57  
So, we find that equations are the appropriate model to use when you're representing strategies when those strategies are equivalent strategies.

Kim  17:09  
Yeah.

Pam  17:09  
An equivalent strategy is when a student looks at the problem and then says, "How could I create an equivalent problem using relationships that's easier to solve?"

Kim  17:19  
I think that's true for addition.

Pam  17:22  
Okay. 

Kim  17:22  
But an equivalent strategy like Constant Difference, we would often put on a number line. Or with multiplication, we would do like maybe I'm an array. But since we're talking about addition, yeah, I just wanted to clarify that for this particular.

Pam  17:37  
But we could also use equations to represent the two equivalent strategies that you just mentioned.

Kim  17:43  
We could also use them, yeah. But in this particular case, we would only, for addition, use the equation. 

Pam  17:50  
Gotcha,

Pam  17:51  
yeah. I would say for flexible factoring in multiplication, you would only use.

Kim  17:56  
Yeah.

Pam  17:56  
So, that's a fruitful path to go down. Like, which model are you going to use to represent which strategy and why? 

Kim  18:04  
Yep. 

Pam  18:05  
And for this particular one, I would agree with you that it's more of a simultaneous equivalence thing that's happening, and so we're going to use equations. 

Pam  18:12  
Yeah. And then what was your question? 

Pam and Kim  18:14  
Which is more

Pam and Kim  18:15  
important? 

Pam  18:15  
The model?

Kim  18:16  
Do I need to focus on strategies? Do I need to focus on models? What's the most important thing to be worried about?

Pam  18:24  
I'm pausing a little bit. The most important thing for you as a teacher is to be clear on the model, so that you can help pull out the student's strategy. Use the model to make that thinking visible, point-at-able, discussable, so that students get better at both. Did I say that well?

Kim  18:45  
Yeah,

Kim  18:46  
I think the model is a vehicle for the conversation about strategies. 

Pam  18:51  
Yes, but...

Kim  18:52  
We're building... using relationships to build strategies in students.

Pam  18:57  
Yes, but we also want kids to gain spatial sense.

Kim  19:01  
Yeah.

Pam  19:01  
We also want them to gain sense with notation. And so, the models are important, but I would agree with your statement that they are a vehicle. It's kind of like we want area models because we need kids to have a sense of area. 

Kim  19:15  
Right.

Pam  19:16  
But they're also helpful to build strategy.

Kim  19:18  
Yeah.

Pam  19:19  
And as a vehicle to build strategy. So, it's kind of a combination there. 

Kim  19:23  
Yeah.

Pam  19:24  
From a teaching perspective, I would focus on pulling out students' thinking, which means their strategy, teacher represents it on a model, so that it becomes point-at-able, discussable, comparable, so that the student gets better at the strategy and also has now a sense of that model.

Kim  19:42  
Yeah, absolutely.

Pam  19:43  
Cool.

Kim  19:43  
Totally agree. Alright.

Pam  19:45  
Did you add anything on to that? 

Kim  19:47  
I don't think so. Swapping, not a major strategy, but a fun one for sure.

Pam  19:52  
So, let me just maybe end with that idea that there seems to be this misconception with a lot of people out there that as soon as you get into the algorithm is not the only way, then there's this vast unknowable.

Kim  20:04  
Yeah. 

Pam  20:05  
There's all these strategies. So, there are. There are a lot of kind of different ways to approach different problems. We are unique here at Math is Figure-Out-Able because we are suggesting to teachers that not anything goes. There is a small set of major strategies that we need to develop in kids, so they can solve any problem that's reasonable to solve without a calculator, thinking and reasoning, building relationships. But also, maybe more importantly, building their brain to math. So, that doesn't mean that we don't let kids use other strategies, and it also doesn't mean that we don't use strings to develop those strategies, the non-major ones. We just don't test kids on them. We don't emphasize them. We use them to help build kids' brains.

Kim  20:50  
It's funny that you're saying this because I was thinking this morning about an event that we had last night for the people in our coaching program called Journey. And one of the things that we talked about was where am I at in my stage of development? And how do I know where teachers are in their stage of development? And I was literally thinking about when teachers say there are so many strategies, it could be a really clear indicator that they're still building their own numeracy because it feels like there's all these. But when you have strengthened relationships, that means strategies are natural outcomes for you, and you are flexibly moving between strategies, and you're pulling out the one that's useful in that moment, it doesn't feel like there's this laundry list of things that you're trying to turn around in the classroom. You have a picture of the relationships that you build, and you just call on them naturally. So, if it feels like there's just big lists and how do I make it happen, it might be that as you develop your own numeracy and you get a handle on how to flexibly move around with numbers, you will find that you can settle into developing that with your students.

Pam  20:50  
And I'll add to that, settle into sort of the categories of what are the major models and strategies and big ideas. And bring that level of like... I don't know if "comfort" is the right word. But kind of like solidity. You feel more solid. Like, "Okay, I'm grounded in these major relationships. Bam, now I know how to begin to elicit and represent student thinking." Alright, Kim, nice. Thanks for tuning in, everybody, 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!