In case you haven't noticed, Kim has a favorite strategy. In this episode Pam and Kim discuss one of the most important strategies and how to use it for addition.
Pam Harris 00:00
Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.
Kim Montague 00:06
And I'm Kim.
Pam Harris 00:07
And you found a place where math is not about memorizing and mimicking, 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 mentor mathematicians as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.
Kim Montague 00:34
So before we jump into today's episode, I want to take a minute or two really quickly to share an email that we got that really struck me. It's from Brad Ballenger, and he was listening to an episode recently. I think it was episode 120 about rounding. And this is what he says, "One reason to learn to make sense of mathematics rather than learn it by rote rhyme whatnot, is that there are simply too many rules and relationships in math to carry them all around in your brain. It's much better to have a habit of thinking that allows you to nimbly take stock of a situation and figure out what makes sense. A rule is easily misremembered. But if you frequently check for things making sense, you catch your own errors. For me, at least it's easier, more reliable, and less stressful to have the habit of sensemaking. Perhaps this is a point you've made in one of your podcasts. I haven't heard them all. But I think it deserves to be reinforced." And this is so exactly we couldn't agree more. And it's one of the reasons why we take the time each week to make the podcasts. And listeners, we are so grateful for you for tuning in every week and sharing your thoughts with us. It's so fun, and we appreciate you.
Pam Harris 01:43
Totally, Brad, like right on. I mean, everything you said. I was like yes, like a habit of sensemaking. I mean, that's totally brilliant. When you said carry, let's see how did he say it? Carry around to have...where did it go? Hang on, I'm trying to find it. "Too many rules and relationships in math to carry them all around in your brain. It's much better to have a habit of thinking that allows you to (oh I love this part) nimbly take stock of a situation and figure out what makes sense." Love it, Brad. Well done.
Kim Montague 02:11
Yeah. So it's been a little while since we've done some math on the podcast. And we thought we would spend a few weeks developing a strategy that might possibly be one of my favorites.
Pam Harris 02:23
Kim Montague 02:23
It may not be the best. But it's definitely something that I use a lot in my daily life. And I think it's crucial that we develop in more people. And a bonus is that it's a strategy that spans across multiple grade levels, and all four of the operations.
Pam Harris 02:38
And when you say it might not be 'the best' maybe for a particular problem. For a particular problem it might not be like the pinnacle of strategy, the one that we sort of end on with all students, but it's a good one. It's one of our major, right? It's one of the top that we say for all operations. So it's definitely one that we like. Okay, you love and I like. I mean, I think I'm becoming, I'm getting there, I'm getting to the love place, pretty much. In fact, it's interesting, most of the work I've done lately has been, how do I say it? It's not my stock presentation, but I've actually had to, help me, what's the word? Personalize, I've had to personalize the presentation. So I've gone to some interesting places, and they're more advanced. So I don't give him the kind of the beginning thing. I actually have to sort of personalize things. And my go to lately has been this strategy. Alright, so let's get right, let's get to it. We thought we'd start today a little differently by posing a problem. And then, I'm gonna give you our lesson plan. Here's our lesson plan for the day. I'm gonna pose a problem, ask you to think about it, then we're going to do a Problem String and ask you again, how you might think about that problem, given the relationships that we create and ping for you in the Problem String. All right? All right. So here's the problem to think about. The problem to think about is 4257 plus 2989. Let me say that again: 4257 plus 2989. So if you think about that problem. So pause the podcast, think about it, do your best, give it a shot, solve it, then come on back and engage with us in a Problem String that then might get you thinking about that problem, might, maybe, not Kim, but maybe other people. People like me, at least how I used to be, might think about it differently. And then we'll see kind of how you think about it after we do the Problem String. Yeah?
Kim Montague 04:43
Pam Harris 04:43
All right. So Kim, you ready for my Problem String?
Kim Montague 04:45
Pam Harris 04:46
You're on the hot seat. Here we go.
Kim Montague 04:47
Pam Harris 04:48
What is oh...so listeners, so you might, it might be handy, it might be helpful, not if you're driving. But if you're not driving, if you have a hands free, it might be helpful for you to actually sketch these problems down. So that then you can kind of look at the Problem String, as we talk about it later. So we're gonna actually start to talk about the Problem String a little bit. So it might be helpful to, however, we'd rather have you solve the problems than get caught in just copying the problems down. We never want students to get their brain engaged in, "Oh, I must get these problems down." That's not interesting. So think about the problems, but you also might sort of sketch them down so that we can talk about the string itself. All right, first problem 27 plus 10.
Kim Montague 05:29
Pam Harris 05:30
You didn't pause on that or anything. If 27 plus 10, I wonder if that can help you think about 27 plus nine?
Kim Montague 05:37
Yeah, 27 plus 10 is 37. So 27 plus nine is 36.
Pam Harris 05:42
And so if I was doing this with students, I would have when you did 27 plus 10, I would have drawn an open number line and very quickly, started at 27, made a jump of 10 and ended at 37. And then when you said, "Yes, you can use it to help you with the next one," I would have jumped, well, yeah, when you said it's just plus nine, I think you said it's just one less. Then I would have hopped back one to say, "Oh, it's just one less? And one less than 37 is 36." So on the number line, I've drawn that jump of 10, I've jumped back one, a little tiny jump of one, and then labeled that 36. Cool. Next problem 56 plus 20?
Kim Montague 06:21
Pam Harris 06:22
And you didn't need to think about that one either. So again, quickly, I would have drawn a number line, I would have drawn 56, drawn a jump of 20, to get to 76. And again, I would have said something with students, like, "Nobody needed to draw that. I'm not asking you to draw. You don't need to. I'm just gonna put it up here just so we just sort of have a relationship just kind of up in front of us." And then I might say, "Anybody want to guess the next problem? Just for fun?" Fifty-six plus 19. What do you got?
Kim Montague 06:51
Seventy-five. So plus 20 is 76.
Pam Harris 06:54
If I could see you I know you're like cracking up right now because you're like, "Do I pause on that? Do I just say it really fast?" Am I right?
Kim Montague 07:01
Pam Harris 07:02
Like you don't need, you don't actually take that long to think about it. You're trying to give people time to think about it, which is very kind of you but maybe you don't have to do it that long.
Kim Montague 07:09
Pam Harris 07:10
So then where I have drawn that 56 plus 20 to be 76. I'm popping back one again, to hit that 75. And like maybe I should have asked you how you thought about it, because I didn't really give you a chance.
Kim Montague 07:22
Well, yeah, you would just ask me plus 20. So knowing that plus 20 was 76 plus 19 was 75.
Pam Harris 07:30
How about 347 plus 200, 347 plus 200? Go
Kim Montague 07:35
Five hundred forty-seven.
Pam Harris 07:37
You think that's 547. So I would write down 347, I would make a bigger jump, like make it bigger than the 20 or the, by the way, the 20 jump was twice as big as the 10 jumpish. And the 200 jump is just bigger. I don't have room on my paper to really make it as big as it should be. But it's definitely bigger. And you said that that plus 200 was 547. The next problem is 347 plus 198. Ha! Anybody that was thinking about 199, good try.
Kim Montague 08:09
That is 545.
Pam Harris 08:11
Because? Because 347 plus 200 is 547. But now you're asking me plus 198. That's just two less than 200. So that answer is gonna be two less. So it'll be two less. And I might at some point, draw that jump of 200 and backup two. But I also might just draw the jump of 198 in there. I might like, draw the jump like I might say, "But where is the 198?" And pull out of students that well, because like looking at my number line right now I've got 347. I have a jump of 200. I have jumped back up two. I have the 545, but nowhere do I see the 198. So might ask students, "Where is that 198?" That's an important question, especially if students didn't use the problem before to help them. And so to help them kind of anchor what are they doing? Oh, it's like they're making a smaller jump. And that smaller jump I just drew from 347 all the way to that 545. It's just like it's almost as big as the 200. It's just two, just two, two shorter, just little, just little tiny, just, just two shorter. And so then you can see that 198 and that two put together is that big jump of 200. That can be helpful for students to go. "Oh, that's what you guys are doing. All right." All right. So the next problem: 4257 plus 3000, 4037 plus 3000.
Kim Montague 09:40
Seven thousand, two hundred fifty-seven.
Pam Harris 09:43
Next problem: 4257 plus 2989.
Kim Montague 09:53
So plus 3000 was 7257. That's too much. It's 11 too much. So I'm going to backup 11 and get 7246.
Pam Harris 10:07
Because the problem before it was 11 too much.
Kim Montague 10:10
Pam Harris 10:11
So listeners, you might recognize that problem as the one that we started the podcast with: 4257 plus 2989, could be solved by saying to yourself, "Hmm, what do I know?" Oh, sounds like Kim knows what 4257 plus 3000, a little bit too much more than, 11 more than 2989. Why 11, Kim? That's weird. Why would you choose 11?
Kim Montague 10:39
Because 11 and 89 are partners of a 100.
Pam Harris 10:42
Sounds like you play I Have, You Need a little bit.
Kim Montague 10:44
I have played a few times.
Pam Harris 10:45
So when I said 4257 plus 2989, what pinged in your head? What did you think of?
Kim Montague 10:52
Oh, totally thought of 11. And that 11 partner 89. So when I saw 2989, I didn't want to mess with it. And so I thought of what was around there. What was close to that number, then I just decided I could back up a little bit.
Pam Harris 11:10
Cool. Next problem. How about 4.7 or four and seven tenths plus two?
Kim Montague 11:16
Six point seven.
Pam Harris 11:18
How about 4.7 plus 1.9, or one and nine tenths?
Kim Montague 11:24
Six point 6, six and six tents?
Pam Harris 11:26
So if 4.7 plus two is 6.7, did you use that? Sorry, I was putting words in your mouth.
Kim Montague 11:33
Sorry. Yeah, yes, yes, yes, when you asked me four point seven plus two, I wrote down 6.7. And so then when you asked me one and nine tenths, then I compared it to the previous problem and said, "This is 1/10 less."
Pam Harris 11:49
Nice. Last problem, six and one half, plus 10.
Kim Montague 11:56
Sixteen and a half.
Pam Harris 11:59
Six and one half plus nine and three quarters.
Kim Montague 12:04
You said last problem on the last problem. I thought I was done.
Pam Harris 12:08
Sorry, last of problem set. I didn't do that on purpose, I promise.
Kim Montague 12:13
So, plus 10 will be 16 and a half. That's too much. And it is 1/4 too much. So if I'm at 16 and a half, and I want to back up 1/4, then that's going to be 16 and a fourth.
Pam Harris 12:31
So it's almost like for many of those problems, you said to yourself, "Hmm, I could use the problem before." But how was the problem before related to the problem? Why use the problem before? How were those all related?
Kim Montague 12:44
Yeah, yeah. So the problem before was really close to the kind of yucky problem. It was just a little bit more than, it was a little bit over. It was a friendlier number, a little bit over what you were asking for.
Pam Harris 12:58
And so we typically call the strategy that you were using the Over strategy. So you can look at a problem like 4257 plus 2989. And you can say to yourself, "2989 is really close to 3000. So I'm just gonna go a little over, add that, ah, nice, friendly. And that, but it was 11 too much that I added so backup, just take off that extra 11." And then you've added what could have been a crankier problem. Might be used a nicer problem that was a bit too much, or why we call it the Over strategy. Yeah.
Kim Montague 13:33
Yeah. And I think the reason why I love Over so much is that I tend to use it a lot when I'm thinking about things like 99, or 98, or 95. And if you think about, like daily real life, you're going to the grocery store lots of things into 99 cents and 97 cents and 95 cents. And so as you're throwing items in your cart, and you want to know about how much you're putting in there, you can easily round up to the next dollar and then kind of backup a little bit, if you ever needed to know the exact amount.
Pam Harris 14:04
Kim Montague 14:05
And so, and frankly, in you know, school math, where people want to give harder problems, when people are thinking about like, "Oh, I, you really want them to have to use a traditional algorithm and like borrow or whatever." A lot of times, they throw in lots of sevens, eights, and nines, just to see if they can make it a little bit more challenging. It's brilliant for Over problems, in fact,
Pam Harris 14:27
Kim Montague 14:28
What did I say?
Pam Harris 14:29
You said "over problems."
Kim Montague 14:31
Over strategies, sorry.
Pam Harris 14:32
We all have problems. It's alright.
Kim Montague 14:35
So one of my sons recently came home and he's in a PE class, like an athletics class. And he said they've been working on planks and sit ups and you lifting weight and you know, whatever athletic type stuff they do. And he said that one of the things they're doing is keeping a rep count for the group. And if the goal is 100, you know, these sweet kids are like missing their goals. Some are going a little over, some are hitting right on the goal and some are going a little under. And so it's like I get to 96, or I get to 97, I get to 99. And they want to collectively find out their total. And so many of the kids are like, "Oh, just write them all down and add them all up." And Cooper's like, "No, like, you have 97 and 99, and 98, that's like 300 back a little bit." And they just, you know, some of his sweet friends are blown away by that idea. So real life everywhere.
Pam Harris 15:28
That's awesome. Cool. So we mentioned a little bit earlier in the podcast, where that this strategy might not be the pinnacle strategy. Like we are looking towards developing at least one more strategy in students. But this is a great one. And in the hierarchy of strategies, it is a more sophisticated strategy. It's one that we want kids to develop. So it might not be the most sophisticated, but it's definitely one of them. I would challenge you to look at the major strategies free ebook that we've given out. We'll give that link in just a minute. That you can look at kind of where this strategy fits in the hierarchy. But it's a super one. It's super important for a lot of reasons, but one of them is place value. I asked you earlier, listeners, so you didn't have to. But if you recorded the problems that we did in the string, you might notice that we got students thinking about nine and how it's compared to 10. Or 19, how it's compared to 20; 198 how it compares to 20; 2,989, how it compares to 3000; 1.9, how it compares to 2; nine and three quarters, how that... nine and three quarters, is that like Harry Potter? Anyway, how that compares to 10? Well, all of those are placed value relationships. And what we find is that when we do more and more work with students to build this sense of the Over strategy, is kids actually get better at place value, because they're dealing with the place values, especially if we would have ended that string with something like 387 plus, and then whatever. And we want that, and I say to kids, "Ooh, is 387 close to something? Like could you add something nice?" Well, for those kids to have to then come up with the helper problem, like what would be clear and they say, "Well, that's, that's really close to?" Well, they could say 390. And often they do. And I'll go, "I mean, is that really, you know, that's maybe a little more helpful to add 390, but go for the gusto. Like, what's it really?" "Oh, it's really close to 400." "So could you just add 400 and then backup one?" That's dealing with place value. That's kids really messing with that the place determines the value, and how then I can look at numbers that are friendly around it. It's a super way for kids to really build that. We also talked about some models, while we were doing it. Modeling is important for kids. It's really super important to make that bigger jump and then help them kind of see where the smaller jump kind of backs up a little bit and where the cranky jump is kind of inside that bigger nice jump. So modeling for addition on an open number line, using jumps, using that nice problem that can be really important to help students see that. Y'all if you want to see what it might look like to model someone's thinking for the Over strategy, you can check out a free download that has a variety of examples on appropriate models. You can find that at mathisFigure-Out-Able.com/over. That's a specific download, I don't know that we've given this one out on the podcast before. That's a specific download that just gets at this Over strategy that we're talking about today. I also mentioned the hierarchy of strategies. If you have not downloaded this free ebook yet, y'all get on that! That is at mathisFigureOutAble.com/big, B-I-G because it is big. So thanks for listening in. Pam has been traveling all over and it's so fun for her to meet and then share with me how people tune in, listen with their kids and do math right along with us. If you'd like to leave us a comment or rating on your favorite podcast platform, we would love that. So thanks for tuning in and teaching more and more Real Math. To find out more about the Math is Figure-Out-Able movement visit mathisFigure-Out-Able.com. Let's keep spreading the word that Math is Figure-Out-Able.