October 03, 2023
Pam Harris

Math is Figure-Out-Able with Pam Harris

Ep 172: The Most Sophisticated Strategy for Addition

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Math is Figure-Out-Able with Pam Harris

Ep 172: The Most Sophisticated Strategy for Addition

Oct 03, 2023

Pam Harris

What makes a strategy sophisticated? In this episode Pam and Kim build the most sophisticated strategy and talk about how and why to develop it with students.

Talking Points:

- What are the most sophisticated strategies?
- Strategies that require more simultaneity require more time to develop
- Simultaneous versus sequential
- Representing Give and Take
- We need more than one strategy

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What makes a strategy sophisticated? In this episode Pam and Kim build the most sophisticated strategy and talk about how and why to develop it with students.

Talking Points:

- What are the most sophisticated strategies?
- Strategies that require more simultaneity require more time to develop
- Simultaneous versus sequential
- Representing Give and Take
- We need more than one strategy

**Pam **00:00

Hey fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam.

**Kim **00:06

And I'm Kim.

**Pam **00:07

And Kim's having a morning, ya'll. It's going to be fantastic. And you found a place where math is not about memorizing and mimicking, where you're waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can actually mentor students to think and reason like mathematicians do. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be. Alright, Kim.

**Kim **00:09

I'm having a morning. I mean, ya'll, it's 9 a.m., and I've been to school four times already.

**Pam **00:44

Oh!

**Kim **00:45

It's a thing. Oh my gosh. Alright, well, I'm happy to be here with you.

**Pam **00:49

The traffic near those schools are... That's intense.

**Kim **00:52

It is intense.

**Pam **00:53

Alright, so before we get going, let's just make sure. Kim, do you have a pencil.

**Kim **00:56

I do actually. I have four.

**Pam **00:58

Because I have a pen. What?!

**Kim **01:00

I have four. You know, it's been crazy the last couple of days. And so, I will get up with pencil in hand and wander, and then set them down like on a counter or, you know, wherever, and then I go get a new one. And so, this morning, I collected pencils because I knew we were going to do some math, and so I have four. Three with little pencil cap erasers, and one without, so I won't touch that one.

**Pam **01:00

Hey, Kim, I have one. I got one pen in my hand. One pen. Pen. Pen. Alright.

**Kim **01:27

The negative part about pencils, I will admit, is that I like mine super crazy sharp. And so...

**Pam **01:35

Doesn't last.

**Kim **01:36

Yeah. I do start the day with several. So. Okay, anyway.

**Pam **01:38

Hey, that's actually one of the reasons why do pen. Because I'll do pencil for the five strokes that it's super sharp. And then, I'm like, "Ah! I want it sharp! Come back!" And it's not, and I don't have a sharpener handy, and so then it just bugs. The texture.

**Kim **01:41

Yeah. Yeah.

**Pam **01:54

Is that the right word? It just bugs. But we should probably do the episode. Here we go.

**Kim **01:58

Right. So, hey, recently, we did episode 168. Do you remember that it was about the line y equals x?

**Pam **02:07

Oh, yeah, that was cool.

**Kim **02:08

Yeah, it was super fun. But hey, we got this email from Anna Mary. And she said, "Mind blown when you asked what would the graph look like if you were a perfect guesser?" She said, "The picture in my brain went to the graph of y equals x, and it was amazing!" She said, "I love this activity and cannot wait to share it with my students."

**Pam **02:25

Oh my gosh, that's so true! Anna Mary, I had the exact same experience. Right?

**Kim **02:29

Yeah, I think you described that.

**Pam **02:31

Oh, yeah. That is wonderful. Good. I'm glad we could share that with more people. That's awesome.

**Kim **02:35

Yeah, yeah. So, if you're a high school teacher, you might want to check out episode 168. But really, even if you just want to make sense of the math that you were probably taught early high school or eighth grade, you should check that out too. It could make sense for you. Alright, we are finally ready to get started. Oh my gosh. So, everyone probably already knows by the title of the episode that we're going to talk about addition. And that's why I have my pencil and you have your pen because we know we're going to do some math.

**Pam **03:03

Bam! And if you've listened to the podcast for very long at all, you know that Kim is the Over Girl. In fact, Kim, the other day I was in Oklahoma, and somebody said something like, "Kim, the Over Girl." I know it was this funny like, "Is that what Kim's going to get known by?"

**Kim **03:23

Maybe I need a t-shirt?

**Both Pam and Kim **03:24

Oh!

**Kim **03:26

Yeah. For when we go to conferences, I can just wear a t-shirt that says "The Over Girl."

**Pam **03:31

Oh, my gosh, we're going to do that. Hey, we could probably tell everybody. We're going to be at NCTM and NCSM. Which I should have just said those... No, that's the right order. Because it's (unclear) this year. In DC. Washington, DC

**Kim **03:44

Yeah, super soon.

**Pam **03:44

Soon. I think it's the end of October. So, if you're going to be there, we will let you know. And that would be fun to meet up.

**Kim **03:52

Yeah.

**Pam **03:52

Alright. So, Kim, the Over Girl, as you would share with me this idea of adding a bit too much and backing up, or subtracting a bit too much and then having to adjust, or multiplying by too much and having to adjust down, and dividing by a bit too. You know like, this idea of Over. I was simultaneously figuring out what are the major strategies? And sure enough, the Over strategy is one of them. And I was kind of weighing out. And I knew you used the Over strategy, so in my mind, I was wondering if there's a hierarchy of strategies, is the Over strategy the most sophisticated? Or at least more sophisticated than other ones? Because that's part of my work, right? Part of my work is identifying what are the major strategies and kind of their level of sophistication and narrowing that down, so that really we can not memorize one algorithm. It's not about rote memorizing. But really gain the relationships in our heads, so that these major strategies become natural outcomes. And what are the major ones we can focus on? Well, so I'll just sort of say that as a preface that I was kind of wondering if the Over strategy was the most sophisticated. Alright, before we actually do the Problem String for today, let's play a little I Have, You Need just quickly. We believe in I Have, You Need. We want to encourage teachers to do it quickly, often. And so, maybe we'll do that in the podcast. You know, we'll just do it a little bit, so you kind of are like, "Oh, that's a thing. I should just do a little bit often with kids."

**Kim **05:21

Yeah.

**Pam **05:22

Alright, So Kim, we're going to play total 1,000.

**Kim **05:27

Oh, okay.

**Pam **05:28

Now that might be... You know like, that's kind of a lot. But you know, total 1,000. And we'll have you share how you're thinking about it. If I have 783, what do we need to make 1,000? If I have 783, what do you need?

**Kim **05:33

Okay. You need 217.

**Pam **05:47

How do you know that? 783. 217. How do you know that? Or how did you figure that out?

**Kim **05:53

So, I knew I was at 700 something, so I knew the partner would be 200 something. And then...

**Pam **06:01

Don't you mean 300?

**Kim **06:04

Yeah, but I have some extra beyond that 700, right. So, I knew that the 83 and its partner, so,I kind of thought about like 700 and 83. And I knew I wanted the partner for 700 to be 200 because the 83 has a partner of 17. And that 83 and 17 total 100 would then be added to the 700 and the 200.

**Pam **06:30

Nice.

**Kim **06:30

So 217.

**Pam **06:31

Thanks. I appreciate you explaining that. Alright, cool. So, ya'll, that's I Have, You Need. you just throw out a number, have a person explain back. And you could do another one. We won't do another one today. But just do it often, you know. And it's not about rote memorizing those relationships. It's about figuring them often, so those kinds of become natural inclinations. Alright. Here is the Problem String for today. Ready, Kim?

**Kim **06:53

Oh, yeah. Am I doing? I'm doing some math? Okay.

**Pam **06:56

Yeah, you're doing the math. Okay. So, the first problem 46 plus 29. 46 plus 29. What do you got?

**Kim **07:03

I want to do 46 and 30.

**Pam **07:05

Alright.

**Kim **07:06

Which is 76.

**Pam **07:08

Okay.

**Kim **07:08

That's 1 too much, so it's 75.

**Pam **07:11

You're pretty confident on that?

**Kim **07:12

Yeah.

**Pam **07:14

Last week, when I was talking to some kids, they said something. So, I... Sorry. I would describe your strategy as an Over strategy. You added 30. That's a bit too much. It's more than 29. So, that's kind of the Over strategy we mentioned earlier. Not surprised that you did that. When I was talking to some kids last week, they did something more where they said they added 30, but they added it to 45. Do you have any thoughts on that?

**Kim **07:41

Yeah.

**Pam **07:42

Does that make sense to you?

**Kim **07:43

Yep.

**Pam **07:43

Can you tell me why?

**Kim **07:45

Yeah. So, it sounds like they were thinking about an equivalent problem. So, they took 1 from the 46 to give it to the 29 to make the new problem be 45 plus 30. So, took 1 from the 46, gave it to the 29.

**Pam **08:02

Okay, and 45 and 30 is also 75. That would be a different way of solving the problem.

**Kim **08:06

Yeah.

**Pam **08:06

Yeah, this was kind of interesting. A little different than what you did. Cool. Next problem. How about 99 plus 67?

**Kim **08:12

99 plus 67 is... That's 166.

**Pam **08:17

Okay, and how do you know?

**Kim **08:19

I borrowed your kids strategy and said, "I'm going to take 1 from the 67 and give it to the 99." So I wrote down 100 plus 66.

**Pam **08:29

Cool. And you kind of ended up with a problem that is almost the answer, right?

**Kim **08:32

Yeah.

**Pam **08:32

What is 100 plus 66? 166. Nice. Hey, is there anything about the order of the numbers in the problems that may have nudged you one way or the other? Yeah, that makes sense to me. So, like the first problem 46 plus 29.

**Kim **08:43

Yeah, absolutely. You can Over when 99 is the first number. But it's a little like you're thinking in your head maybe a little bit of the commutative property. And so, typically, I find that I Over more when I want to Over the second number, when I want to adjust the second number in the problem. Yeah.

**Pam **08:48

You're like, "It's almost 46 plus 30. I'm just going to do that, and back up." If I would have said 67 plus 99, you might have said, "Oh, it's so close to 67 plus 100. I'm Just going to Over." But I flipped it, and I gave you the 99 first, and so you're like, "Oh, that's almost 100. Let me grab 1 from the 67 to make it the 100. Now, I'm just left with kind of an equivalent..." Is that a way of just kind of describing?

**Kim **09:34

Yep. Absolutely.

**Pam **09:35

And so, kind of when you said commutative property. That would be like, you could have said 99 plus 67. I'd rather think about that as 67 plus 99. Oh, now that feels like Over. I just wanted to kind of explain a little bit more of the commutative property just briefly. Okay, cool. Next problem. How about 495 plus 378.

**Kim **09:57

495 plus 378. The 495 is really close to 500, so I'm going to grab 5 from the 378 to make that 373, and I'm going to give that 5 to the 495. So, my new problem is 500 plus 373, which is 873.

**Pam **10:21

Yeah, that makes a lot of sense. Grab 5 from one, give it to the other. It's almost like I said to you, "Hey, Kim. You've got 495 marbles in this pile, and you've got 378 marbles in that pile. How many marbles do you have?" What did you do? Describe that with the marbles.

**Kim **10:36

Yeah, I took 5 marbles from the second pile, and I just moved them into the first pile.

**Pam **10:42

And then said, "I will happily now solve. Because that equivalent problem of 500 plus 373. Bam, is just like..." So, yeah, cool.

**Kim **10:50

Yeah.

**Pam **10:51

I wonder if that influences how somebody might do a problem like... Kim, do me a favor and pause on this one. Let's let the podcast listeners think about this one.

**Kim **10:51

Okay.

**Pam **10:51

Would what Kim just did influence how you might solve a problem like 12,783 plus 4,219? So, I'm just going to say that again. 12,783 plus 4,219. So, pause, maybe podcast listeners. Is there anything that's percolating for you? Alright, Kim, go ahead.

**Kim **11:23

Yeah. I was just thinking when we were pausing for a second that we should probably give people a warning at the beginning. "Hey, we're going to do some math. Grab some paper, and be prepared to pause."

**Pam **11:32

Yeah.

**Kim **11:33

Okay, so 12,783 is pretty close to 13,000. So, I want to give it 217.

**Pam **11:45

Whoa! That number just sounds familiar. How does that?

**Kim **11:51

I like how you gave me the I Have, You Need before that.

**Pam **11:55

So, just anybody who just popped in the podcast, you didn't listen to beginning. At the beginning, I asked, "Hey, if I have 783, what do you need to make 1,000?" What?! Like we planned this. And so, Kim had already thought about that partner of 1,000. So, when I say 12,783, listeners, I wonder how many of you were like, "I've heard that before. It almost feels like 13,000. It feels like it's 217 from 13,000." Okay, sorry, Kim, I'm interrupting.

**Kim **12:20

No, it's all good. So, then, if I want to give the first add-in 217, then I'm going to take that from the second add-in. So, when I take 217, then I'm I'm left with 4,002. And the resulting problem is super, super nice. 13,000 plus 4,002, which is 17,002.

**Pam **12:42

Bam!

**Kim **12:43

Yeah.

**Pam **12:43

Alright. So, hopefully, everybody could follow that one. You can just think about like getting to that next friendly 1,000. And then, so you got to grab it from the other one. And Whoa! Nice equivalent problem that's super easier to solve. So, Kim, that only works with whole numbers, right? You can't like grab one from one number and give it to the other one. Could you do something similar with like, 2.8 plus 1.9? 2 and 8/10 plus 1 and 9/10?

**Kim **13:14

Yes. And I'm pausing for a second because I have to decide.

**Pam **13:19

Decide what?

**Kim **13:20

They're both really, really close to a friendly number.

**Pam **13:26

Ah.

**Kim **13:26

So, I'm going to go with taking... I actually thought $0.10. I'm going to take $0.10 from the $2.80, from the 2.8. I'm going to take a 0.1 from that and give it to the 1.9. So, I've made 2.7, or 2 and 7/10, plus 2. Which is 4 and 7/10.

**Pam **13:45

4.7. Nice. Why did you decide to do that? Why the dime?

**Kim **13:51

Um...

**Pam **13:53

Why not?

**Kim **13:54

Yeah. Yeah, I guess maybe it was less to Give and Take.

**Pam **14:00

Okay. Alright. So, you could have also said, "Hey, 2.8 is so close to 3. I'm going to give that 0.2, 2/10." Made that 3 plus. And then, 1.7, or 1 and 7/10. And which would also get the 4.7. Cool, nice. But that doesn't work with fractions, right? Like, you can't Give and Take with fractions. So, if I had something like 9 and 3/4 plus 1 and a 1/2. Any thoughts on that?

**Kim **14:27

9 and 3/4 plus 1 and a 1/2. Yeah, I actually want to take a 1/2 from the 9 and 3/4, and give it to 1 and a 1/2.

**Pam **14:39

Okay.

**Kim **14:40

I'm going to call that 9 and a 1/4 plus 2, which is 11 and a 1/4.

**Pam **14:45

Nice, cool. I giggled a little bit there because that's not what I was anticipating you would do.

**Kim **14:52

Yeah.

**Pam **14:53

Yeah? Do you want to guess what I was anticipating?

**Kim **14:55

Yeah, I assume that you thought maybe I would make it be 10.

**Pam **14:58

You just wanted to like mess with me.

**Kim **15:02

Well, it's a nice problem, right? Because you could adjust either way.

**Pam **15:07

Sure.

**Kim **15:08

And that would be fun conversation.

**Pam **15:10

Can you do the second adjusting?

**Kim **15:12

Sure. So, I would take a fourth from the 1 and a 1/2, to give it to the 9 and 3/4, so then that way it would be 10 plus 1 and a 1/4.

**Pam **15:23

Which is also 11 and 1/4.

**Kim **15:25

Mmhm.

**Pam **15:26

Super cool. So, if you were in my late middle school, early high school, or any high school class, I might then at this point say something like... So, Kim, if you have a number... Like, let's just call it "a". Plus another number. Let's just call it "b". So, I've got a plus b. We could say that's the sum, a plus b. Could we kind of generalize what we just did? Could we say something like... Every time in this problem when I gave you a number "a" plus a number "b"... And you don't have to use a and b. In fact, maybe I shouldn't have said a and b at first. Let me stay general. Kim, I gave you two numbers. Could you just put some words to what you were doing?

**Kim **16:07

Yes. So, I have two numbers, and I'm taking some amount from one of the numbers and giving it to the other number.

**Pam **16:16

Sure enough. So, it's almost like if I had some number "a" plus some number "b", I could think about taking that "a". And you just said I took from that number, so I'll just go and take from that number, say "c". So, I've just written a plus b equals a minus c because I just took some number from it. I'll just use your words. And then, you gave that "c" to the other number. So, like b plus c. So, on my paper right now I have a plus b equals a minus c in parentheses, plus b plus c in parentheses. So, a minus c. I took an amount, "c", from the first number, and I gave it to the other one, b plus c. So, I've got a minus c plus b plus c, which simplifies to a plus b minus c plus c. Oh, and what's minus c plus c? Sure enough, that's 0. And so, we kind of have proved an identity. And "identity" means a sentence that is true no matter what we put in the variables. And so, sure enough, if we add something to one number and subtract it from the other one, in an addition problem, we can use the associative and commutative properties to prove that. Sure, we can do that, and we can have an equivalent problem. Kim, I'd like to focus. We've actually done an episode or two I think before, where we've kind of done this Give and Take strategy a little bit. One of the things I'd like to stress today that makes this episode a little different is this idea of creating an equivalent problem. And I think you even said it as we were kind of going through these problems in the string. You said something like, "Well, yeah. Like, this problem is easier to solve. Like, ended up with..." So, one of the problems you ended up with was 45 plus 30. Another one was 100 plus 66. Another one was 500 plus 373. Instead of the nasty problem...or gnarly at least...gnarly problem I gave you, you ended up with a problem, an equivalent problem that's easier to solve.

**Kim **18:14

Right.

**Pam **18:14

Yeah. Did you want to say something?

**Kim **18:16

Well, I was just going to say, I feel like it's a thing to look for ways to manipulate number, so that maybe it's less taxing. Yeah. Making something equivalent, so that then you're able to handle it maybe a little bit easier is something that I seek for.

**Pam **18:35

Yeah. And that makes a lot of sense. And I would suggest that that requires some anticipatory thinking.

**Kim **18:41

Yeah.

**Pam **18:41

So, thank you, Debbie Junk Plowman, who is a colleague of ours, who said to us one day, "Hey, there's some research out there that if you have to use anticipatory thinking, that's a more sophisticated strategy." So, as we talk about sophistication, I mentioned earlier, I thought maybe that the Over strategy was maybe the most sophisticated. I would suggest it is a more sophisticated strategy than some of the other ones because I think even Over requires a little bit of anticipatory thinking. You have to say to yourself, "Ooh..." I'm not just going to start doing stuff. I'm actually going to anticipate that if I add too much, I can adjust then. So, it requires a little bit of anticipatory thinking. But I would suggest this Give and Take strategy requires quite a bit more.

**Kim **19:31

Right.

**Pam **19:32

That you have to consider, "Hey, I can take this problem, and if I do some stuff, I'm going to end up with an equivalent problem, bam, that will be easier to solve. What can I do to mess around, so that I get an equivalent problem?" That requires kind of getting outside the problem, anticipating thinking, thinking about the future, and that it might be easier to solve an equivalent problem. That anticipatory thinking is a hallmark. Ya'll, that's a ping. When you realize that you kind of have to get outside the problem and anticipate what will come next or why you might do something, that could be a clue for you that sure enough this is more sophisticated. Another clue is the fact that in this strategy, you kind of Give and Take simultaneously. Would you agree with that?

**Kim **20:24

Yeah, absolutely. I think you're considering both numbers at the same time and they're nearness to something friendly.

**Pam **20:31

Yeah, yeah. And so, that simultaneity is also another huge clue, a ping, math teachers, that when you recognize that we're asking students to consider more things simultaneously than they were before, to grapple with them both at the same time, that that raises the level of rigor. That should be a clue for you, "Ooh, this is going to be something we need to give students time to grapple with, that they need experiences, multiple experiences, so their brains get used to that grappling, get used to traveling that path, and so it can become stronger." It's kind of like working out in the workout room. You've got to actually struggle and fuss and grapple, so that the muscles grow stronger. This is not a time to save students and go, "Oh, I can see you're like really grappling with that. Ooh, here, let me tell you what to do. Just Give and Take." No, like, give them the space, and the time, and the experience to grapple with why that Give and Take works because that sophistication is super interesting. Hey, can I just tell on myself a little bit?

**Kim **21:41

Sure.

**Pam **21:42

So, there was a time when I was, you know early, early, when I was learning. I was watching you do the Over strategy, and I was representing people's thinking, and so I would do Problem Strings with whoever I could. I would represent their thinking. And you said to me one day, "I disagree with how you're representing those problems."

**Kim **21:59

Oh, yeah, I remember. It's been a while. That's a good memory

**Pam **22:02

That was a while ago, eh?

**Both Pam and Kim **22:03

Yeah.

**Pam **22:03

And I said, "What are you talking about?" And you said, "Well, you've got problems that are Get to a Friendly Number." So, let me just give an example of that. In that... Let's see. Let's do the 99 plus 67. I would say, "Hey, everybody, 99 plus 67." And somebody would say, "I'm going to give 1 to the 99 and take 1 from the 67." And I would represent that Give and Take strategy as... I draw and open number line. I would write the 99 down. I would do a jump of 1 to 100. And then I would say, "What did you have leftover?" And they would say, "Well, I took 1 from the 67, so that's 66." And I would draw the jump of 66. And then you would land on what is 100 and 66? You land on 166. So, I'll just draw that. The number line looks like a number line, starting at 99, jump to 100 by 1. Jump 66 to 166. And you said, "That's not what the person did." And I said, "What do you mean?" And you said, Well, they didn't really give 1 to the 99, and then alright now I'm at 100. And then, think about..."

**Kim **23:03

What's left.

**Pam **23:05

"What's left. And then, add that." You're like, "They did it more simultaneous." And I said, "What?"

**Kim **23:10

I remember. I think we might have gone back and forth a little bit about it.

**Pam **23:16

Oh, like maybe longer than a minute. Yeah. And here's the upshot, everybody. It's because I wasn't thinking simultaneously. I was not giving and taking. And maybe let me just say out loud what I finally heard you say. I heard you say, "Pam, I'm considering both numbers at the same time. When I Give and Take..." I'm putting words in your mouth. But, "When I Give and Take, I look at the 46 and the 29, and I say to myself, 'Oh, if I just move 1 marble, I'm going to end up with a super nice problem.'" In other words, you were already considering that if you just move 1 marble, you would have an equivalent problem that would be easier to solve. I wasn't doing that. I was being much more, "Let's see. What could I do? Oh, for 495 plus 278. 495, I could get to 500. Okay. I like that a lot better. Now, what does that do to the other number? Well, what would I be left with?" Like, it was much more sequential. I was doing something, and then reconsidering. And you are really considering both numbers kind of simultaneously.

**Kim **24:17

At the same time. Yeah.

**Pam **24:18

Yeah.

**Kim **24:18

Yeah.

**Pam **24:19

So, that simultaneity is super important as a ping. That should... I'm snapping. I don't know if you can tell. That is the... "Ping" is the word that's hitting me today. To say, "Ooh, we have raised a level of rigor. This is a more sophisticated strategy." And that's important because we need to realize kids need more time and experience to build those relationships in their heads to actually be able to use that kind of thing.

**Kim **24:45

Yeah. And I think part of the difficulty, maybe, that you ran into in the beginning was that there's something to do with that 1 in the 99 plus 67. Both Getting to a Friendly Number and Give and Take are making a move or adjusting by a 1. And I think what we discovered is that you were trying to model that on an open number line, and because that's a sequential model, that creates maybe a little additional difficulty. So, we model give and take with equations because you can represent the sequential moves with equations. You're not able to do that on an open number line.

**Pam **25:27

Yeah, yeah. If you looked at my paper right now, all I have on my paper are equations.

**Kim **25:32

Yeah, me too.

**Pam **25:33

So, for example, on 495 plus 378, I have 495 plus 378 horizontally. Under the 495, I have plus 5. Under the 378, I have minus 5. And then, next to that equation, 495 plus (unclear). I'll stop trying to talk to fast., 495 plus 378, I have equals 500 plus 373, equals 873. I can't read that far without my glasses.

**Kim **25:59

Mine looks exactly the same. And I was just thinking it might be super fun after we do a string some time to take a picture of our papers.

**Pam **26:05

And then, compare. Yeah, cool.

**Kim **26:07

(unclear).

**Pam **26:08

I do have one number line, and that was when I was talking about what I was doing, the sequential Get to a Friendly Number. So, when you said it caused some difficulties, I think what you meant was, as I was trying to get people to develop the Give and Take strategy.

**Kim **26:21

Oh, yes. Mmhm.

**Pam **26:22

If I was modeling their strategies on a number line, people stayed thinking about Get to a Friendly Number. It didn't help nudge them into more simultaneous thinking. Yeah. Super interesting that the model can play in. I will say one small caveat, if somebody's thought about this. The reason that open number line, I'm going to suggest, is more of a sequential model is as I draw it. As I draw it, I kind of have to like do one thing, and then do the next. If I were to just look at an open number line, it's a little less sequential. If it's already built, it's already constructed, somebody could argue that that one's a little more simultaneous. But I still think the equations gives much more of a simultaneous feel and punch for strategies, where what the brain is doing is actually very simultaneous. Did I say that well?

**Kim **27:18

You did, but I'm not sure that I agree with you. We should talk about this a little more.

**Pam **27:22

Kim's going to push back. She's going to say, "No, no, if I look at..." So, when you look at an open number line, it's still just really looks sequential?

**Kim **27:28

It feels to me like you start somewhere, and then there's the next move. Yeah.

**Pam **27:32

I guess I'm kind of thinking about like...

**Kim **27:34

Now, what I will say is that when you look at a number line, if there's no signs, then (unclear).

**Pam **27:42

Plus. No plus this, plus that.

**Both Pam and Kim **27:44

Yeah.

**Kim **27:45

Then, I think that's a nice conversation too about what problem does this represent?

**Pam **27:51

Because it could be?

**Kim **27:51

Because it could be addition or subtraction.

**Pam **27:53

Sure. And it could be multiple.

**Kim **27:55

Yeah.

**Pam **27:56

Of each of those. Alright, we could keep talking about that. Hey, I think it might be noteworthy to mention that just because it's the most sophisticated strategy... So, I am going to suggest Give and Take, this idea of being able to move marbles around, is the most sophisticated addition strategy that we ought to help develop in students brains. But it's not necessarily the fastest one.

**Kim **28:20

Oh, yeah.

**Pam **28:20

Or maybe even more importantly, the one that you would use on the street. What do I mean by that? I mean, if I was on the street, and I didn't have a pen, Kim didn't have a pencil, and we had to solve a problem like 12,783 plus 4,219, we might consider using the strategy that the human calculator uses, where.... So, I always forget his name. Nice guy. Super nice guy. He calls himself "the human calculator". He has a strategy that you might use if you're walking down the street, and you don't have something that you can write down because it's kind of easier to hang on to the numbers sometimes, where he just kind of adds the left to right.

**Kim **29:04

Oh, yeah. That's what I was about to say.

**Pam **29:06

Yeah, and you just kind of keep. And that way, you have a running total.

**Kim **29:10

Right.

**Pam **29:10

And so, you don't have to keep track of anything because you just kind of keep a running total. Maybe we'll do a whole episode on that strategy some other time. But what I'm going to suggest with is... I guess what I'm saying is, the most sophisticated strategy isn't necessarily the fastest or the one that you would use on the street, but we do need to develop it in kids. Why? Because we want to develop their brains to be able to think more simultaneous, to be able to grapple with these relationships simultaneously and finding equivalent problems. I mean, any high school teacher right now is like, "Yes! Find equivalent problems!" Like, that's so important to do that. So important to help build, develop brains to be able to think more sophisticatedly.

**Kim **29:53

So, even though this is the most sophisticated strategy, it's not the one that we would say, "Go straight there. Just do this one. It's the most sophisticated, so this is all you need."

**Pam **30:03

Yeah.

**Kim **30:03

We definitely want to develop the other strategies. Because like you said, it's developing relationships. It's kind of precursor. It's prepping your brain to be able to reason. Get to a Friendly Number is a fantastic strategy that we would absolutely recommend and kids need. And also that work is necessary for them to really develop Give and Take.

**Pam **30:24

Yeah, you've told me that you've had lots of teachers, especially lately, mention in the Facebook group and other places where we interact with participants, that they've looked at the major strategies ebook, and they've seen. You know, we've said, "These strategies are in order, and so Give and Take is the most sophisticated." And they're like, "Oh. Well, we'll just skip all those other ones. We'll just go straight there." And what you're saying is, "No, no, no, no. Like, we have to develop those precursor things or they'll just be memorizing something again." It's not about rote memorizing Give and Take. It's actually owning all the relationships. Yeah. We want students to develop their brains to think more sophisticatedly. And it's about the choice. Once you own all of the strategies, then if Kim's solving a problem, and she does the Over and I'm like, "Kim, you should be using the most sophisticated Give and Take." No, no, no. No, no. Not at all. Because she owns them all, then she has the choice. That's empowering. It's empowering when you have choices. What we don't want to do is leave kids. So, another thing, another note. Don't leave kids in the one strategy they've got. "Oh, you don't need to learn these other ones. You've got that one." No, it's so important not to leave them there. We need to have, we want to have students have choice. So, develop the major relationships, so they can own the major strategies. And now, students are empowered to choose.

**Kim **31:42

Yeah. You joke a little bit about me being the Over Girl and kind of how I mess with number kind of all throughout my life. But there were strategies that I kind of naturally figured out, and some that I didn't use as much until later. So, I probably could have learned more if someone had helped me along the way, and I might have known this strategy as a young student and could have strengthened it earlier than I did. And I'm really glad that it's not too late. Right? It's not too late for us, for our listeners to learn, and develop, and strengthen new strategies.

**Pam **32:17

Absolutely. Hey, one last thing, since it's an equivalent strategy. And so, I'm going to say, we've kind of emphasized the fact that you're using anticipatory thinking to say, "Ooh, could I create an equivalent problem that's easier to solve?" Therefore, we call this an equivalent strategy. Since it's an equivalent strategy, it's fantastic to strengthen it. Once you've sort of done some work to build the Give and Take strategy, you can strengthen that with the instructional routine we call Relational Thinking. So, Kim, I'm going to throw one out at you. If I were...

**Kim **32:49

(unclear).

**Pam **32:49

One last problem. If I were to say to you, 5.9, or 5 and 9/10, plus 3.7, or 3 and 7/10, equals 6 plus blank, how would you fill in the blank?

**Kim **32:50

Yeah. So, I would need 0.1 more to make the 6. From 5.9 to 6, it's a 0.1 addition.

**Pam **33:15

Mmhm.

**Kim **33:16

So, then to maintain that equivalence, then I'm going to need a 0.1 less on the 3.7. So, then the blank, I think, is 3.6, 3 and 6/10.

**Pam **33:27

So, you're saying 5.9 plus 3.7 is equivalent to 6 plus 3.6?

**Kim **33:32

Yeah.

**Pam **33:33

Nice. And that is what we call Relational Thinking. And it's a super great way to help students kind of solidify this idea that we're creating equivalent problems that are easier to solve, which creates an equation, which then helps all sorts of understanding about equations. Meaning equivalence, not about things to do. Which is a thing we run into in higher math for sure.

**Kim **33:59

Yeah. And it's super cool because you have some Relational Thinking problems on your website. We'll make sure that that link is in the show notes, so that people can check those out. So, often, when participants, when learners get excited about Give and Take for addition, it's something that's pinging in their head for the first time ever, they say, "Can we do Give and Take for subtraction?" Right, that's always a big question. "Oh, what do we do about subtraction?" So, don't miss next week. We're going to be talking about the most sophisticated strategy for subtraction. It's coming up.

**Pam **34:29

Bam! Thank you for tuning in and teaching more and more Real Math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutble.com. Let's keep spreading the word that Math is Figure-Out-Able!

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