Ep 95: If Not the Addition Algorithm, Then What?

Pam Harris  00:01

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

Kim Montague  00:08

And I'm Kim Montague. 

Pam Harris  00:09

And we make the case that mathematizing is not about mimicking steps or rote memorizing facts, but y'all, it's about thinking and reasoning; about creating and using mental relationships. We take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keeps students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what? And boy howdy, are we going to answer that question, better than maybe ever before today.

Kim Montague  00:45

Absolutely. So last week, we talked a little bit about what we're going to be doing in the rest of this series, and we are going to answer the question: If not algorithms, then what? So we are going to dive in today to addition, and we're going to share what we believe are the major addition strategies that your students need to know.

Pam Harris  01:05

Students need to own these strategies. And if they do, we can throw any problem at them that's reasonable to solve without a calculator, and they will have power over solving it. They will have the ability to solve it without having to fuss or fumble or re-invent. They'll be like, "Yep, I got relationships to use, bam, and I can solve that problem confidently, fluidly." We've got students that are adders with the best of them. Can I say adders? Is that a thing? Adders? 

Kim Montague  01:35

Adders? Sure. 

Pam Harris  01:37

Anyway, all right. So before we start, well, we're just gonna say there are four, there are four major addition strategies that we've identified that we want to develop in students. However, there's also sort of a precursor strategy. And we don't usually lump that in. If you've ever heard me say five, it's probably because I lumped it in. Most of the time when we say four, we don't lump this precursor strategy in. Because research has proven that if we don't force a traditional digit oriented, column oriented strategy, or algorithm, on kids, if we don't force that on kids, then almost all kids will develop partial sums as a strategy all by themselves. In other words, if they're looking at a problem, like 48, plus 39, and we're just going to use that as an example today: 48 plus 39. And we don't force kids to line them up, and then think about the ones and then borrow and carry and all that stuff. You don't really borrow with addition, but carry and all that stuff. If we don't do that, then students will be thinking about 48 and 39. And as they think about 48, I totally just spit when I sit 39. Yuck! 48 and 39. Kids are thinking about 40 and 30. 

Kim Montague  02:49

Yeah. 

Pam Harris  02:50

And they'll pull those together and get 70. And then they'll say to themselves, "Okay, what was it? It was 48 and 39, okay." Then they'll think about the eight and the nine, they'll pull those together. And then they'll pull that 70 and that 17 together. So that's called partial sums. Or sometimes we call it Splitting by Place Value, you kinda split everything by place value, pull the place values together, and then pull all of it together. So that's kind of a precursor strategy, we would expect that you would do work early, or when you're helping students early in addition, you would do work with students to help them get confident in that sort of partial sum or Splitting by Place Value strategy. When I say early, early with whole numbers. When you start adding decimals, we would also expect that to show back up, students are going to be making sense of adding decimals together. They're going to split those decimals up by place value, pull the place value parts together, and then pull everything back together. So that's kind of a precursor strategy. But from there, we don't recommend that you go to a traditional addition algorithm. Instead, we want to develop these four strategies.

Kim Montague  03:56

So you're saying four. Right? So let's name them. Although they may not mean a whole lot to begin with.

Pam Harris  04:02

The names, yeah.

Kim Montague  04:02

Yeah, the names are not as important as building the idea behind them. And so let's - we'll give some examples. But let me name them first. The first is 'Add a Friendly Number'. And then we have 'Add a Friendly Number - Over'. We have 'Get to a Friendly Number'. And we have 'Give and Take'. So those are the four major addition strategies,

Pam Harris  04:23

And see how we just tell you those and instantly, if you don't have experience with those instantly, you're probably like, "What? How's that helpful?" Yeah, it's not. So let's give you experience, so you can build some relationships, so that then those words tagged on to these ideas, then you're like, "Oh, okay, that's what it means." Alright. 

Kim Montague  04:43

Okay, first problem. Or the problem. We're gonna give this example or this problem and give examples of each of those strategies. So your problem is 48 plus 39. 

Pam Harris  04:56

Yep, yep. 48 plus 39.

Kim Montague  04:58

How about you pick a strategy?

Pam Harris  05:00

Alright, I'm going to Add a Friendly Number. And we would often think that this might be the first one that you start to develop with kids. So we would say, "Well, if you're going to keep one of those addends whole. You're going to keep 48 whole, and you're going to add 39, well like what's friendly?" And so I might say, "Well, I'm gonna start with 48. And I'm supposed to add 39, I'm gonna add 30, I can think about that." Now young learners might add 10, 10 and 10. We want to encourage them to do bigger, fewer jumps. So I'm gonna think about that 48 and 30. And that's 78. But I still have that nine leftover. And so now that I have that 78, then I'm gonna think about adding that 78 and nine, and however I do that is less important. But I might add 2 to get to 80. And then I have seven leftover. And so we're at that 87. So it's all about that kind of first move. And the first move was starting with 48. And I'm going to add that friendly 30. And then I'm going to kind of adjust from there. 

Kim Montague  06:00

I was actually wondering what you were going to say about the nine because there's a couple of other strategies within that that you could do.

Pam Harris  06:07

Yeah, for sure. Let's stay with this 'Add a Friendly Number'. Kim, what if I said 39 plus 48?

Kim Montague  06:14

Oh so just the commutative property there? 

Pam Harris  06:18

Yeah. 

Kim Montague  06:19

Oh, okay. So in that one, then I would start with 39. And instead of, so if I want to Add a Friendly Number, then I would start with 39. And I would add 40 to get to 79. And then I would add the eight to get 87.

Pam Harris  06:34

And you sort of adjust from there. Yeah. Cool. So we gave you a problem that you kind of think about either adding a friendly numbers starting from either number. Cool, nice. All right ya'll, so that's the 'Add a Friendly Number' strategy.

Kim Montague  06:49

Let's reiterate that the name comes from that first move. I know you said that. It's the first move that you make. The reason why we call it that is because you added a friendly number first.

Pam Harris  06:58

Yeah, and the reason we do that is it's your plan of attack. It's as you hit the problem, do you do that very first thing that we talked about that Splitting by Place Value? Do you say to yourself, "I'm splitting both these numbers up." And then go from there? Well, then you're Splitting by Place Value, using partial sums. But if you don't do that, and you keep one added whole, and you decide to from that 48, I'm going to add that really friendly 30. Then that was your your plan of attack. Your plan of attack was keeping one addend whole, and adding that next number, but adding a friendly part of it, and then adjusting from there. Adjusting up from there adding the rest of it on. Cool. Alright. So next one is related to it and I'm so glad I'm going to ask you to do this one Kim. I think this might be your fave, 'Add a Friendly Number - Over' what does that mean? So same problem: 48 plus 39. How are you going to Add a Friendly Number but specifically the over version.

Kim Montague  07:55

So Add a Friendly Number - Over you just described Add a Friendly Number and stayed underneath the total. And added smaller chunks. Add a Friendly Number - Over is when you would start with the first addend or start with an addend and then add a little bit too much. So if I was adding 48 plus 39, I'd start with 48. And instead of adding 39, I'm going to add 40. Because it's really close to that 40. So 48 plus 40, to get to 88. And then I asked myself, or tell myself, "I went too far. How much too much?" I added too much. How much too much? It was one too much. So then I go back one to get to 87. So 48 plus 40, and then backup one to get 87.

Pam Harris  08:41

And then that 40 and backup one is the 39. So that's your way of adding the 39 was to add 40 and backup one. Yeah. And the reason I sort of kidded you a little bit about it being your favorite is because I think it is your favorite. 

Kim Montague  08:55

Yeah. I love Over.

Pam Harris  08:57

When I met you, I had never conceived of the Over strategy. Alright, all you listeners who just like looked down on me, be nice, be nice. There are lots of us out there that never kind of got outside of the rules. And when I would listen to Kim, I was like, "How did you know to do that?" She goes, "Well 39 is so close to 40. Like it's so pleasant, we can just do that." And similarly, if I were just to give you the other example, you started with 48 and added 39 by adding 40 and subtracting one. You could also say, "I'm going to start at 39 and I'm supposed to add 48. So instead I'm going to add 50. 39 and 50 is what 89? But I was only supposed to add 48. I've added 50, So I've added two too many, so then I would back up 2 from 89." And that would be another way of getting to 87. 

Kim Montague  09:46

You know what I'm doing right now as you're talking? 

Pam Harris  09:48

Probably the same thing I am. 

Kim Montague  09:50

Oh?

Pam Harris  09:50

We should take a picture. I'm drawing number lines on my paper.

Kim Montague  09:54

I am too! I'm sketching what you're saying. I was just thinking that if somebody owns some of these strategies and they're like, "Yeah, it totally makes sense." They could totally practice modeling.

Pam Harris  10:04

Oh, nice. Yeah. 

Kim Montague  10:07

I'm just drawing while you are talking.

Pam Harris  10:08

That would be really good. So let's just be really clear, we're writing it down because - or I am, I'm not gonna speak for you. I'm writing down the number lines to follow what we're saying, because that helps me follow what we're saying. Like I'm able to keep track of the relationships better by putting him down on paper, as we're saying them. I could probably do it in my head. But it's easier for me. I can not have my mental space being taken up by holding on to that, because I'm holding on to it on the paper. And then I could be thinking about what we're talking about, and actually investing in the conversation rather than being so worried about the numbers, you know, like just holding the numbers in my head. Yeah. So hey, if you guys want, listeners, if you're interested, you could get some modeling practice in as we're talking about these strategies. Nice. Nice. Alright. So we've done 'Add a Friendly Number' we've done 'Add a Friendly Number - Over'. How about 'Get to a Friendly Number'? So Kim, same problem: 48 plus 39. How would you do the strategy 'Get to a Friendly Number'?

Kim Montague  11:06

Okay, so if I start at 48, and then if I want to get to the next friendly number, then I would add two from the 39, to get to 50. And since I've already added two of the 39, then I would just make one big jump of 37 to get to 87. So I made a jump of two and a jump of 37. And that's where my 39 comes from.

Pam Harris  11:31

Nice. So instead of Adding a Friendly Number, like we were just talking about, you still kept one of the numbers whole. But your first move wasn't to add a big friendly chunk, your first move was to get to a nice, friendly number, like in this case, to get to that nice, friendly 50.

Kim Montague  11:48

Which is really nice, because then I can add whatever's left in one big chunk.

Pam Harris  11:55

Yeah, now young learners might not, right off the bat. 

Kim Montague  11:57

Right. 

Pam Harris  11:57

But we want to help them add that leftover part in one big chunk if they can. In fact, y'all I just did some work just yesterday with a group of teachers and a third grade teacher was really frustrated. She's like, "I don't think this way, I have never been good at math." All the things. And in the middle of her - we were doing a 'Give and Take' string, which we're about to do. And this 'Get to a Friendly Number' is the precursor strategy for 'Give and Take'. Because in order to 'Give and Take', you have to sort of know I'm going to get to that friendly number first. That's kind of a precursor thing. And this particular teacher had gotten to the friendly number. And then I said, "Well, what's leftover?" And it was fascinating to me that she was able to make that big jump. She didn't have to make little jumps to add the rest of what was leftover. And I looked at her and I was like, "Oh, don't you tell me that you're not good at math. You're absolutely good at Real Math. Like you just skipped a step, I would have expected you to have to take it your progression. Bam!" Like it was a very rewarding day to watch this teacher just sit up taller, and taller through the day and just be so excited. Just to be really clear. Perhaps she was never good at fake math. But she's absolutely brilliant at Real Math. It was really fun. 

Kim Montague  13:07

So fun.

Pam Harris  13:08

Hey, do we need to do 'Get to Friendly Number' by flipping that problem around? And y'all, the reason we're doing this isn't that we necessarily want to kill this with students. It's not like, "Okay, now you better do it another way." I don't want to give that misimpression. It's just, if we can kind of do a couple of examples that we hope that that will help you get a little bit better of an idea of the relationships we're using. Do you mind doing it?

Kim Montague  13:32

No, not at all. So then I would think 39 plus 48. And I would start at 39. And to get to the next friendly number I only need one. So one more would get me to 40. And that one is from the 48. So now I just need to add 47. And 40 and 47 is 87.

Pam Harris  13:54

Nice. And an example of what I just meant about having to add smaller chunks is once you get to that one, you got to the 40, you said to yourself, "I have 47 left." If a kid says, "Okay, 40 and 47. Let's see I can add 40 and 40 to get to 80, how much do I have left? Seven. Okay, now I'm going to add the seven." That would be an example of kind of needing to make some more jumps in there. And what I was saying is that teacher yesterday didn't have to do that. She made the whole jump of 47 all at once. Very nice. Okay, so so far we've Added a Friendly Number, we've Added a Friendly Number - Over now we've just done Get to a Friendly Number. Let's pause here for just a second. We would consider that these three strategies, 'Added a Friendly Number', 'Added a Friendly Number - Over' and 'Get to a Friendly Number' need to be developed sort of simultaneously. Not in the exact moment. But kind of all throughout the time period that you're developing strategies with students, sometimes you would work on 'Added a Friendly Number', sometimes you would work on 'Added a Friendly Number - Over' sometimes you would work on 'Get to a Friendly Number'. As I say that you'd probably do 'Added a Friendly Number - Over' a little bit later. You would start with 'Added a Friendly Number'. But 'Added a Friendly Number' and 'Get to a Friendly Number' are about on the same level of sophistication. They're about the same difficulty. And so we kind of want to be developing both of those. But then we want to move to the most sophisticated addition strategy. And that is 'Give and Take'. So what does Give and Take look like? Alright, who's doing this one? 

Kim Montague  15:24

You go. 

Pam Harris  15:24

Okay, so same problem: 40 plus 39. If I was gonna Give and Take, then I'm thinking to myself, "What can I grab? What can I take from one of the numbers to make one of the other numbers really nice." That kind of feels like Get to a Friendly Number. But I simultaneously consider if I need 2 to make 48 really nice. Like, it's 48 plus 39, I need 2  to make 48 really nice. Can I grab it from the 39? Ooh, yes. And then can I consider that resulting 50 and 37, kind of all at the same time. It's sort of the simultaneous look. It might feel like Get to a Friendly Number, because  Get to a Friendly Number is a bit more sequential. I start at 48. I get to a friendly number. I take a breath. I think about what's left, then I add what's left. It's much more sort of sequential, I kind of do something and then I take stock and I do something else. And I take stock. Give and Take is much more about: Ooh, I can make that 48 into a 50. And look I could do it by getting that two from that 37 or from the 35 to make 37. And bam, I'm just at 40...50. I should stop trying to do this fast. Let me try that again. 48 and 39, I can make that 48 really nice. 50. Oh, and I can get it from that 39. Bam, it will be 37. And I sort of end up with this 50 and 37, almost simultaneously. And I can add that to get the 87.

Kim Montague  16:56

Yeah, it's like you're looking for a different problem.

Pam Harris  16:59

Ooh, nicely said. In fact, that is a hallmark of some of the most sophisticated strategies that we're going to outline for each of the operations is that sometimes a strategy is all about creating an easier problem to solve. Say more about that, Kim.

Kim Montague  17:18

So I actually was thinking about a different problem, because I was gonna Give and Take the other way. But as you were talking, I was thinking about how this idea of creating an equivalent problem that looks a little bit different, that's just nicer to solve at the end. So that when you landed with the 50 and 37, they just kind of fell together a little bit easier. Right? At that point. I don't know if we've mentioned this yet. But at that point, then I can kind of just like read the numbers left to right. So a little bit of 50, 87. And so there's no in your head having to do a lot of work at that point, because you created a nicer problem.

Pam Harris  17:57

Yeah, like if we would have started this with, "Okay, everybody, we're really gonna focus on strategies. And here's the example problem. 50 plus 37." Everybody would be like, what? Like, it's just 87. It's so just like sitting there as 87. That's part of our goal is to use strategies that can create equivalent problems that are easier to solve. That just take less like, oh, yeah, just see 50 plus 37. Well, so Kim we better have you Give and Take the other way. What would that look like?

Kim Montague  18:25

Well, you took two from the 39 to give it to the 48 to make 50 and 37. I actually wanted to make the 39 nice. And so I took one from the 48 and gave it to the 39 to make 47 and 40. So that was 87 as well.

Pam Harris  18:45

Again, an equivalent problem, that's easier to solve. Yeah, instead of solving 48 and 39, you decided to solve 47 and 40. That is the brilliance and beauty of 'Give and Take'. Yeah, super cool. Alright. We are suggesting that if students own these four major strategies that we can give them any problem to solve, that's reasonable to solve without a calculator, and they should be able to solve even cranky, decimal addition problems using one of those strategies. Kim, I'm a little aware that we're a little long on this episode already. But I did kind of want to ask you another question.  Sure.  Let's see if we can do it. So if I were to say to you, "Which of these strategies if I give you a problem, which of these strategies would you rather use?" Like... I don't know if I said anything plus 99, like 55 plus 99. Let's do 56. 56 plus 99. Like if I said that, what strategie is you're fav for that problem?

Kim Montague  19:45

Oh, I think a lot of times I go 56 plus 100 is 156 and then backup one to 155. 

Pam Harris  19:56

That's the Over strategy right? That's like Kim's fav. What if I would have said 99 plus 56?

Kim Montague  20:03

Oh, that's brilliant. Yeah. Because then at that point, then I feel a little less like going over. And I actually take one from the 56, to give it to the 99 to make 100 plus 55, which is 155.

Pam Harris  20:17

Yeah. So Kim, and I've discussed that we actually think that for the most part, when we hit an addition problem, we think of a couple things. The first thing that we kind of think about is, can we just Add by Place Value. So if I give you a problem, like 42, and 54, I might think to myself, "Well, that's just 40 to 54. That's just 90, what's leftover six." So we sort of by place value, notice that I started with the higher the 10s, I didn't start with the digits, nobody thinks about the tiny numbers, those aren't important. So I think about the 40 and 50. And I was like, that's 90 and six. So if we can just add by place value, we sort of do that. That's kind of a first shot when we hit an addition problem. It's when we can't, it's when you can't just Add by Place Value, then you say to yourself, well, alright, which of these four strategies, but honestly, after you've built all four, don't we really kind of lean on either an Over strategy or a Give and Take? I think, for the most part, maybe every once in a while, there might be a reason that something else kind of pings. But for the most part, I think we would then lean towards either an Over strategy, or a Give and Take strategy, those would be sort of - is that fair to say?

Kim Montague  21:29

Yeah, I think so. 

Pam Harris  21:30

Yeah. Cool. 

Kim Montague  21:31

Cool. Alright. So we last week shared that there is the probably most important download that you can get. It's fabulous. It is huge. It has all these strategies with examples and models. You can find that download at mathisFigureOutAble.com/big, and you're gonna want to get that today.

Pam Harris  21:53

B-I-G! BIG! Because this is a big thing. We've taken quite a bit of time to create this download, where we lay out the strategies, you're going to love it. Teachers, this is for you to learn what the strategies are, so you can help develop them and students. Leaders, you could use this with your teachers. This could completely be your professional learning with teachers that you do, where you help teachers develop these strategies. So they can then help them develop with students. This is not a handout that you are to download and print off and give to students. That is not the intent. This is for teacher learning. Teacher learning, not a student resource. We do have other student things that we do sometimes but this is not that. 

Kim Montague  22:32

Yep. 

Pam Harris  22:33

Alright. I hope you guys are liking this. If you like learning what the major strategies are for addition, wait till we get into subtraction next week. You are not going to want to miss next week's episode or the whole series where we attack each operation to talk about the major strategies for that operation. Alright, if you want to learn more mathematics and refine your math teaching so that you and students are mathematizing more and more, then join the Math is Figure-Out-Able movement and help us spread the word that Math is Figure-Out-Able!