Ep 182: Integer Addition

Pam  00:01

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

Kim  00:07

And I'm Kim.

Pam  00:08

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 mentor students to think and reason like mathematicians did when they were young students. 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  00:36

Hello. How are you doing? 

Pam  00:39

How am I doing? You know, I'm doing pretty well, pretty well. We have lots going on right now. Would you agree with that?

Kim  00:45

Yes. Lots of things.

Pam  00:48

We have a lot happening. But we're having fun. We're having fun with it. And we are excited about the work that we're doing. We love the work we do. Yeah. We do. 

Both Pam and Kim  00:57

(unclear).

Kim  00:58

It's funny you say that because I found a review for us.

Pam  01:00

Ooh! Nice! 

Kim  01:01

And it kind of actually mentions some of the work we do. So, I'm going to dive in because it's a little bit long. 

Pam  01:06

Okay. Alright, go. 

Kim  01:08

So beachbum2010. I like it. Says, "I simply cannot find the words to describe how I have changed as a mathematician, since beginning to follow Pam and her work. I am about to begin year 33 as a K-6 educator.

Pam  01:25

Whoo! Clap. Clap. Clap. Clap. Clap. 

Kim  01:28

"Over the past three years, I have listened to every podcast. Some several times. I've participated in every challenge, I think. I have immersed myself in the DMR twice. And I've done two other deep dive workshops. I am currently working on funding to become a JourneyLEADER, so that I can help my colleagues know better, so they can do better." Love that. 

Pam  01:47

Aw.

Kim  01:47

"Doing math with the confidence I now have is so much fun, as well as addictive. Do yourself a favor and join the Math is Figure-Out-Able movement. You will not be sorry." (unclear) endorsement!

Pam  01:57

Wow! That is wonderful. Thank you so much beachbum2010.

Kim  02:01

Yeah, I want to know who you are. Send me an email. 

Pam  02:03

Yeah! Like, tell Kim who you are. That's awesome. Some wonderful things that you can join in. We run challenges at least three times a year. We have the Developing Mathematical Reasoning workshop. That's the DMR. It's still free out there. You can take. We have deep dive content workshops, and then our Journey program. Kim, we're having a lot of fun. 

Kim  02:24

That's my favorite. 

Pam  02:25

That is. Yeah, yeah. And you do an excellent job in there. And we do some really, really cool things in Journey. And then, we have leader support in our JourneyLEADER program. So, if you're a coach, a leader. Anybody out there who leads mathematics teaching in any way, check out our JourneyLEADER Program. And we're so glad beachbum2010 that you're in there. Love it! Or that you're getting in there. Yeah.

Kim  02:46

Yeah. 

Pam  02:47

Let's make it work.

Kim  02:47

Okay.

Pam  02:47

Okay, cool.

Kim  02:48

Let's dive in. We have lots to say. I feel like our episode last week was a little long, but. 

Pam  02:52

It was a little long.

Kim  02:53

It's fine. If they hate it, they can turn it off. We'll just keep talking.

Both Pam and Kim  02:57

(laughs).

Kim  02:58

Okay, so last week, we did chat kind of big picture, major things to know, major things to consider when working with integers, kind of in general. So, today, let's dive in and get specific to the work of operations with integers. Which I think a lot of people will love. So, let's chat about integer addition today. 

Pam  03:18

Yeah. Integers. Negative, positive numbers. Let's really think about how we can operate. I told a little bit in the last episode about how I was working with just... It was kind of a random group of kids that I grabbed. And I was just like, "Hey. Hey, based on the work that you've done with whole numbers, just kind of tell me how you're thinking about it." And because they had already just been reasoning about whole number addition and subtraction, they just continued to reason about integer addition and subtraction. And I thought it was fantastic. I've done a lot of work since then to make sure that it works for all kids, and that it makes sense, you know, mathematically and everything. That was just kind of my early "Let's experiment." I've done that a lot, Kim. A lot of the work that we have put out in the Math is Figure-Out-Able movement is stuff that I bumped into, ran into, tried anecdotally, and then did the, "Let me make sure that mathematically this is true." And, you know, did all the reading and everything to make sure that it would fit, and then continue to experiment. Let's do that today. I'm going to actually do kind of a string that I did and do with people to get at integer addition. So, Kim, do you have any idea the elevation that we are at here? We're south of Austin.

Kim  04:31

It's four hours to the beach. I don't know how high we climb to get. 

Pam  04:36

Okay, when you run cap to coast? So, cap to coast is Austin to the coast, the Gulf Coast.

Kim  04:42

I have no idea.

Pam  04:42

Well, let's ask you. When you run it, do you run uphill or downhill? Is it flat? Is it totally flat the whole time? 

Kim  04:49

No. And you probably shouldn't ask me because I tend to like the hill-y ones. I think there's some uphill and downhill. So, overall, I would assume it's downhill. But...

Pam  05:03

Because you? 

Kim  05:03

Because we're heading towards the beach. 

Pam  05:06

Because you're heading (unclear). Okay.

Kim  05:07

I mean, I am. But I don't know. Are we? 

Pam  05:11

Well, we are 425 feet above sea level. 

Kim  05:14

Yeah, I don't feel like it's... 

Pam  05:16

(unclear) Yeah. 

Kim  05:17

I feel like we're pretty flat. No mountains here. Sorry.

Pam  05:19

So, when you run, you run a little downhill, but it's not like big downhills and stuff?

Kim  05:24

No.

Pam  05:25

Yeah. Okay. I know that our Scouts. When my kids were in Scouts, they did a canoe trip from here to the coast. And so, the water runs that way, so that's got to be downhill a little bit, right? If the water runs that direction. Okay, there's the geography for Pam today. 

Kim  05:42

Geography is not my thing. 

Pam  05:45

Okay, so elevation. If we're talking about above and below sea level, could we picture ourselves on the coast. There's these hills. Maybe there's sand dunes. Maybe there's like the hill by the. You know like, there's some above and below the kind of ocean level here. So, we're on the beach. And, Kim, you decide to walk up the sand dune, and it is a 25 foot tall sand dune. So, that's where you're starting. You're 25 feet above sea level. And then, you saw a higher sand dune. And you're like, "Ooh, that's 34 more feet." So, you started at 25 feet. You walked up 34 feet. Where are you?

Kim  06:23

I'm at 59 feet.

Pam  06:25

Above sea level. Okay, cool. 

Kim  06:26

Yeah. 

Pam  06:26

And I could represent that on a number line. And if I've drawn a number line on my paper, I'm actually going to throw 0, kind of in the middle of my number line here. And then, I'm going to go over to 25 to the right, and I'm going to make a jump of 34. And you said when you did that, you're at 59. I could have done a vertical number line. I don't know why, but I chose horizontal. I think you can do either. (unclear). 

Kim  06:49

You should do both. (unclear).

Pam  06:50

It's good to have both. Absolutely. Yep. Good to have both. Okay, first problem. 25 plus 34 is 59. Don't have to think about that very hard. On purpose. Next problem. You swam down, and you're in this underwater sea cave that is 25 feet below the sea level. And in that sea cave, you can go up and down and everything, and you walked down. And you've now walked down. That it says, "Hey, if you want to take this trail, it's going to go 34 feet further down." So, you are adding a walk of 34 feet down. So, I've now written on my paper minus 25, negative 25, plus negative 34. What do you got?

Kim  07:31

I am at negative 59.

Pam  07:34

How do you know?

Kim  07:35

Feet. I drew a vertical number line, and I put negative 25. But then I went ahead and put 0 also, just to kind of like orient my head. And I just was at negative 25, and I jumped down negative 34. And that was negative 59.

Pam  07:51

Yeah, nice. Let me tell you what my board would look like if I was doing this Problem String. So, I have that first number line where I have 0 kind of in the middle, where we did 25 plus 34. Where you started 25, and then you walked up 34 more. The 0 is in the middle. And then, I have started 25 on the right, to the right of it. Jump 34 and end on 59. So, I kind of have this jump of 34 over on the right from 25 to 59. But then, the second number line for the second problem, which was negative 25, starting at negative 25, and then walking down another 34. I have a number line underneath it. But the 0s are lined up. 

Kim  08:29

Yeah. 

Pam  08:29

So, the 0s are lined up where you can see. Yeah, they're lined up. And then, I went the same distance back that I went forward 25. So, the 25s are the same distance away from 0, but the negative 25 is to the left of 0, where the positive 25 is to the right. And then, I'm jumping 34 back. Kind of like you jumped down. I'm jumping 34 back, and I'm landing at negative 59. So, if you were to see my paper right now, I actually kind of have the same thing on two sides of the 0. I have the same jump of 34. One is going to the right because we were adding 34. And one is going to the left because we were going further down in the cave 34. One is landing on positive 59. One is landing on negative 59. So, I wished you could see my paper, but that jump is to the right and the other one is to the left. And 0 is kind of that constant, kind of in the middle. Next problem. How about if you started in that sea cave. You're down negative 25 feet. But this time, you walk up in that sea cave, and you walked up 34 feet? Where are you? Are you still below? You know, because you start at negative 25. Are you still below? Are you above the water? Are you... Like, where are you as far as the water level goes, the sea level?

Kim  09:46

So, I just went back to the number line that I had just done, and I went back to the negative 25. And I know I was going up 34, so I went up 25 to get back to the 0 that I had. So, I moved up 25. I'm at 0. I haven't gotten all 34 yet. So, then I knew I that I had 9 more to go. And so, then, I went up a jump of 9 to land on 9.

Pam  10:12

Because from 0 a jump of 9 would land you on 9.

Kim  10:15

Mmhm.

Pam  10:15

Cool. So, if you started 25 feet below and you walked up 34 feet, you're telling me you're now above the sea level by 9 feet. You're 9 feet above sea level. Yes?

Kim  10:25

Mmhm.

Pam  10:25

Cool. So, on my paper, and what my board would look like if I was doing this with students is, again, I have these 0s lined up. So, on that first problem, I kind of have this jump of 34 on the right. On the second problem, I have this jump of 34 back on the left. But on the third problem. This one we just did. Like you, I started at the same negative 25. So, the negative 25s are lined up. But this time, I'm kind of doing the jump of 34 to the right over 0. Kind of like you said. So, up 25. And then, up 9 more. And landing on the 9. Okay, cool. Next problem. How about if I said, hey you're above sea level. You're on that sand dune. The 25 feet above sea level. And you are... You don't really dive off a sand dune. You're not on a sand dune. You're on a cliff above the sea level. And this is cliff diving. And you decide that you are going to dive. And how did we measure this? This might not have been my best context ever. Because you are going to dive 34 feet. And I'm kind of curious. Where are you if you dive 34 feet? Are you above the sea level still? Like, are we catching you mid dive and you haven't hit the water yet? Where are you?

Kim  11:33

I'm in the water. So, when I jumped 25 down, then I was at 0. But I hadn't completed the 34 feet.

Pam  11:42

Because you started at 25, right? 

Kim  11:44

Yeah.

Pam  11:44

You start at 25. Jump 25 down. You're at 0. Okay.

Kim  11:46

Mmhm. And then, I still had 9 more to go. So, one more jump of negative 9 is at negative 9.

Pam  11:54

So, on my paper, I've got those number lines where all these 0s are lined up. And the positive. There's a positive 25 on that first problem. And so, this last problem, two places where you started, 25 feet above sea level. Those 25s are lined up. And this time I drew that back 25 to 0. And like you said, another 9 is back at negative 9. So, this one kind of looks different than all the other ones, except it also has a jump of 34. It also has a jump of 25 and 9. Whereas the problem before it had a jump forward of 25 and 9. This one had a jump backwards of 25 and backwards 9. Interesting that we kind of have the same numbers that are kind of all falling through here. One more thing I'll say is that, then I also had the equation for this problem. Where I wrote 25. You started at 25. And then, I wrote plus negative 34 because I kind of said something about how then you decided to dive, so decided was like this positive thing. And the dive was like negative 34. I could have written minus 34. You could have a conversation about whether you mean that you're removing 24, falling down, diving, or whether you decide to dive. That can be like add negative. It's a little bit of word play there, but you can get kind of either equation out of that. So, I've got 25 plus negative 34 equals. And you said you would land 9 feet below sea level. So, negative 9. Okay. So, let me just kind of review a little bit how this Problem String was a little different than the one we did last week. So, the one we did in last episode was a lot of contexts all at once, with a lot of different numbers. Positive, negatives, all kind of floating around. And we're just really trying to get what the difference between positive and negative numbers and what's happening. In this one, I'm much more interested in what the equations are looking. I'm still staying in context. We're still really reasoning about what's happening. But I'm kind of focused a little bit more on what the equations look like, and how they compare to the number lines, and how that number lines compare to each other. So, we're really kind of looking at what's the same and what's different. So, at the end of this string. Which ya'll can't see because it's a podcast. If I was with students, and we'd be looking at the board, we'd be looking for where are the 25s showing up? Where are the 34s showing up? And why? Like, what's happening? Why do we keep seeing 9? Why do we keep seeing 34? Why do we see 59 and negative 59? What's happening here? So, Kim, maybe I'll ask you a couple those questions. Why do we see 59 and negative 59?

Kim  14:32

Because when you're adding the positive 25 and 34, then you have the positive 59. When you're adding the negatives 25 and the negative 34, then you have the negative 59.

Pam  14:46

Totally makes sense. Yeah. Why did we see 9 show up? Like, for one problem the answer was 9. For the other problem, the answer was negative 9. Like, why 9?

Kim  14:57

Well because the distance between 25 and 34 is 9. 

Pam  15:01

Wow. Sure enough. 

Kim  15:03

So, it depends on if one of the numbers is negative. Like, when you had negative 25 and you increase 34, then that gave you 9.

Pam  15:17

Because your kind of looking at the difference between those two. Yeah. So, notice in this Problem String, we kept the numbers very consistent, so that kids didn't really have to mess around with addition and subtraction. Once they had figured the 59, they could just use that in the next problem. Once they'd found the difference of 9, they could just use that in the next problem. So, we're not getting kind of hung up in diving into the relationships between the numbers. We're like, "Okay, we got the relationships." Well, what's happening here? Where are we? Are we negative? Are we positive? Are we looking at what's in between them or not? Are we adding together? And so, we can really kind of focus there. So, that's just a different kind of structure of a Problem String. So, we kind of relieve some of the cognitive load a little bit, so we can focus on the part that we wanted to.

Kim  16:00

Right. 

Pam  16:01

I would suggest you can do strings just like this with temperature, Sort of stay in temperature and have the storm comes. The temperature drops. There's a heatwave. You walked into a sauna and the temperature rose. You started in the... Where's a place where you'd start that's really cold? You started outside, and you... Alaska. You started outside in Alaska, and you walked into the house. Like, there's ways that you can kind of sort of play with temperature and different contexts. Also use debt. Especially when you're with addition. You can add a debt. That makes sense. I can add a debt. So, how much money do I have now? I start in debt, and I earn money. I start in debt. I add more debt. So, a fine thing to do. And then, again, you can use American football. But always, always, always from the line of scrimmage. Not from the 0 line on the football field. Cool. Alright.

Kim  16:53

Actually, we talked a little bit, or at least mentioned the teacher Facebook group last week. And I decided to go in there and see what questions had come up. And I answered them at the time, but I'm curious if you give your answer to a couple of the questions. (unclear).

Pam  17:09

Yeah, yeah. Because integers has been a hot topic. 

Kim  17:12

(unclear).

Pam  17:12

People have been asking questions. Absolutely. Alright, what do you got?

Kim  17:15

So, one of them was about the minus or subtract symbol versus the negative sign.

Pam  17:19

Yeah, yeah, yeah that's interesting. So, it's a notation thing. But it's also really connected to meaning. And it's tricky. I don't know, teachers, if you remember the first time that you looked at a problem that had a subtraction sign and a negative sign in it, and you're like, "Are those the same thing?" Like, what does it mean? Are we subtracting 25? Does adding a negative 25? Is that equivalent to subtracting 25? Does the negative sign... Is it little, and it kind of goes up? You know how you see sometimes typesetting, where I'll write negative 34 with a little tiny negative in front of the 34. But sometimes we write it as the same symbol as the subtraction minus sign. So, that can be super confusing. So, what I usually do students is do what we've just done in last episode, in this episode, where we are in context, and I'm representing as they're thinking about it. And I will be incredibly inconsistent. Now, somebody just was like, "No!" In other words, sometimes I will write the negative 34 as a little tiny negative by 34. And sometimes I will write it as a big minus in front of the 34. And sometimes I'll put parentheses around negative 34. So, I might have written... I didn't today. Interesting. But sometimes I will. I'll right, 25 plus parentheses negative 34 end the parentheses, so that you can kind of really tell that it's negative 34, and I'm not writing plus minus. I'm writing plus negative 34. I think often that's when we put the negative up high and kind of small is when there's an operation in front of it. So, if it's like plus negative 34, then we'll write the addition symbol, and then we'll write that tiny, little negative. But sometimes we don't. Sometimes, we'll write plus, minus 30. You know, the subtraction symbol 34. So, I will as students are reasoning about contexts, and we're not trying to get them good at operations, but we're really reasoning about the meaning of integers. I'll be super inconsistent, so that kids kind of get used to lots of different notation. Then, as we start to get into operations, then I'll say, "Wait a minute. Is this negative? This symbol right here. Does that mean minus? Does that mean plus?" And we start to reason that it can mean both. And it depends on what you're thinking about. And we can bring both meanings to bear sometimes. Especially when we're doing subtraction. That it can mean both. So, I might have something like 25 minus 34. And I can say, "Is that 25 subtract 34? Is that 25 plus a negative 34? And how can we make sense of both of those? And that's actually a good place to be. Where we want students to sort of realize that we can think about subtraction as adding the negative number. 25 minus 34, we can think about as 25 plus negative 34. But so, that's complicated. Part of it is notation, where we just have to tell kids this is what we do. Part of it is, let's actually work through kind of the meanings of both of those.

Kim  20:31

Yeah. And the more we have kids wrestle with the meanings and think about things in context, then it doesn't just look like a symbol, and they don't. They're not really sure what to do with it. Here's how you know that you didn't look at my questions ahead of time. The next question I was going to ask you was do you ever put parentheses around negative numbers? Do you have anything you want to add to that? You kind of mentioned it.

Pam  20:53

So, here's kind of what I'll add is I'm sure there are some middle school teachers right now that are a little bit like steam is coming out of your head. Or what's that picture where you're like, "Wait, Pam, what?!" Because you're clear. Let me just give you the homage to say, I am with you that this is complicated for kids. And I'm going to guess that some of you out there right now are like, "No, no, no Pam. We have to be super consistent or kids get really confused and mixed up." To which I'm going to say, "They're going to reach this inconsistency."

Kim  21:23

Yeah. What about next year?

Pam  21:25

Even if you don't do it, it's going to show up because we, as a culture, are not consistent. And because there is this ambiguity about are we subtracting or are we adding a negative? And because that ambiguity exists, I would rather face it upfront, make sense of it, give students the space to grapple with it while I'm there as that helpful other, more knowledgeable other. Where I can see, "Ah this is right on the edge of your zone of proximal development. Let me kind of scaffold help for you to to grapple here." So, we really want kids to grapple. I know it is so much easier to just tell kids "plus, plus, minus, minus" or "minus, minus, plus, plus". Like I just said, the roll backwards. Like, it is so much easier to just give kids a rule. They do it. They're getting answers. Whoo! They're less stressed. The anxiety level goes down.

Kim  22:26

On the day that you tell them that, and then what about five days later? 

Pam  22:29

Well, to be clear, as that stress goes down and sort of the grappling goes down, the thinking goes down, the reasoning goes down. So, it's not about having kids guess what's in your head. It's not about having them like figure it out on their own. It's about giving them the space to grapple in contexts where you're helping them make sense of it, where you are there to help scaffold and not let it get to where it's unproductive struggle. And super, super important. I hear you clearly, middle school teachers, where you're like, "That will take so much longer." Um, "so much"? Notice, we just built in a lot of stuff that most textbooks don't do. Most textbooks have maybe a section where they're like "Negative, positive numbers. Done! Bam!" And then, we move on. I'm really suggesting that you spend some time getting kids really to ground themselves in what negative and positive numbers mean. You do that using Problem Strings like we did in this episode, in the episode before. And in that space, they're making sense of it. I know that's harder. I know that takes longer. But the alternative is what we then see, which is mixed up, messed up. Kids memorizing. Kids mimicking. And they're not doing real math. So, do I want to get kids to where they can just look at integer operations and just sort of be fluent? Absolutely. But I want fluency with understanding, not because they are just mimicking rules, and then mimicking them incorrectly. Because, let's be clear, that's going to happen all too often.

Kim  24:02

Mmhm. Do we have time for one more question? (unclear) 

Pam  24:03

Yeah, except, let me just say... Well, let me just say one more quick thing.

Kim  24:06

Okay. 

Pam  24:06

Because you might be like, "No, Pam. But I got kids that get it. You know, they get the rules. They move on." So did I. I was the student that got the real fast, applied it correctly every time. And I was robbed from real math. So, don't rob the me's of the world, either from really reasoning about what's happening? I don't know. Do I? Tell me the question. I'll see how deep it is. 

Kim  24:30

Oh. Well, it's more... It might be a topic. What are your thoughts on integer chips? 

Pam  24:36

Oh, dear.

Kim  24:36

Maybe we do it next week.

Pam  24:38

Yeah, I think we better put that one off. Let's do that next week.

Kim  24:40

Sorry, listeners. Listen to next week.

Pam  24:42

Right?! Listen next week, and we'll talk about integer chips. What are the pros and cons? And when to and not to? Alright, cool Bam! Thank you. Alright, 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 mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able! Hey and chances are high we'll also start integer subtraction next week. Alright, bye, ya'll!