Ep 149: Subtraction: The Operation That Will Surprise You

Pam: 0:00

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

Kim: 0:07

And I'm Kim.

Pam: 0:08

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

Kim: 0:36

It is so fun to hear from listeners. Ya'll, do more of that, please. A few weeks ago, we did a short series about fraction division, which was super fun and has been well received. After episode 142, we heard from Christina Stevens, and she said, "I love that this makes sense to me. I trusted the roles, but I admittedly didn't get the why. Hooray! I get it!" That is super exciting, right?! And it's exactly what we're hoping for our students. For adult mathematicians, it's kind of why we do the podcast. So, well done, Christina!

Pam: 1:09

Yeah. Nice. It was super good to get so much feedback, especially (unclear)...

Kim: 1:13

Yeah.

Pam: 1:13

...Christina and everybody about how much fun everybody was having. Somebody was telling me they were doing it in their car with their nephew, and they were like, "It's math in the car!" I'm like, "Alright, that's awesome." Yeah, nice.

Kim: 1:25

We did get a listener say don't listen to the podcast in the car because we often do Problem Strings and, you know, you might want to write something down. Anyway, today(unclear).

Pam: 1:34

Don't write while you're in your car. Don't do it. No writing while you're in your car. Not while you're driving anyway. Yeah.

Kim: 1:40

Yeah. Okay, so today, we're diving into what we think is the most undervalued but incredibly important operation. So, we hear a lot about multiplication, and facts, and division, but not as often do we hear about subtraction. And wow, it's so important in so many ways. So, let's kick off a few episodes. You know, because we have a ton to say about this. Few episodes about subtraction.

Pam: 2:07

Now, ya'll, don't stop listening just because we said"subtraction", you high school teachers. Hang on. Hang on. Or anybody who's like, "Oh, subtraction, you know, like that's... You just do the steps, and you're done." Hang on because there's some important things that we're going to talk about with subtraction that will impact lots and lots of mathematics. So, Kim, we often talk about the routine I Have, You Need.

Kim: 2:35

Mmhmm.

Pam: 2:35

And so, if I say something like, "I have 87, what do you need to make 100?" Go ahead.

Kim: 2:40

Thirteen.

Pam: 2:41

Thirteen. So, 87 and 13 is 100. What does that have to do with subtraction? (unclear)

Kim: 2:50

So I...

Pam: 2:50

Yeah, go go.

Kim: 2:51

Yeah, I know that we often talk about it. And because of the way that you say, "I Have..." and then the kids think about "You Need..." it feels a lot like addition. But this routine came about because subtraction across zeros. Early, early, early days. Subtraction across zeros. I was like, "That is awful. That is a horrible thing. And if we could just get kids to know their partners of 100, then that would never be something they ever had to deal with." So, it very much came out of a need for work on subtraction.

Pam: 3:26

Yeah. And so, I have to interrupt you really quickly because this is just reminding me of a story when... And I may have said this on the podcast before, but I'm going to tell it again. I was doing a lot of professional learning with secondary teachers. We're doing graphing calculators and, you know, kind of higher math and everything. And at one point... Well, when I got really interested in younger math because of my kid and everything, my kid's school...in fact, your teaching partner at the time, Danna Fincher...asked me if I could do some professional learning with the teachers. And I was like,"Yeah!" Like, "I would really like to do that. But what would we do?" Like, I knew so little about younger math. And, you know, I was really clear what I could help teachers do with graphing calculators. Wasn't going to do graphing calculators with, you know, K-5 educators. And I was like, "What would be a thing?" You know like, "What do you guys want to learn? What do you need help with?" And in all seriousness, the response was,"Subtraction across zeros!" And I said, "Okay. Absolutely, we will do that. What is that?" Like, I had no idea. I literally was like, "What does that mean?" So, what is it? Kim, tell us. What does it mean? When you say subtraction across zeros, what does that mean?

Kim: 4:32

Oh, gosh. So, when you have 100. So, if you are teaching with an algorithm, subtraction across zeros is always a beast because you would write 100 minus 87 below it. And then(unclear).

Pam: 4:47

Line it up, right? Make sure you line it up correctly. Mmhmm.

Kim: 4:50

Yep. Super abstract for kids because you're doing things like 0 minus 7. And you"can't..." I'm air quoting"can't do that". So, then, you have to go to the 10's to borrow. Oh, wait. There's none there, and so you keep going. And there's all this like crossing out and, "Is it a 10 or is it a 9." And it's so little to do with place value because kids are just following steps. They don't have a clue most of the time, and they're just trying to mimic what's happening. And so it always. If you are a third grade teacher starting to talk about subtraction across zeros, you know the pain of mimicking those steps (unclear).

Pam: 5:31

Lots and lots of pain, yeah.

Kim: 5:32

(unclear) and demonstrating 100 million times. And then, when you get to maybe later third grade and it's 1,000, you're like, "Oh, here we go again." So, yeah, that was. You know, in my very first year when I taught pretty traditionally, it was like, "Oh, gosh. I can't. This is not working."

Pam: 5:51

You know, and you just said that there's no place value as they're doing that.

Kim: 5:56

Not for the kids.

Pam: 5:57

Maybe I'll tweak it to say there's no place value understanding happening because there's a lot of value embedded in that algorithm.

Kim: 6:03

Oh, one hundred percent. Yeah.

Pam: 6:04

And that's why it works. But the understanding, we try. You know, "Okay, you're in the 10's column, so you're borrowing a 10, and you're putting the 10 over here, and so you're crossing that out. It's becoming a 9, but you're in the 10's.""What's a 10 in the 10's..." Like it's...

Kim: 6:16

Yeah.

Pam: 6:17

It's so abstract. Especially for those young learners. Yeah.

Kim: 6:20

Yeah.

Pam: 6:21

So, when I just said, "I have 87, what do you need?" It's kind of like that missing add in problem, right? 87 plus something is 100. And then, you said 13. But it's also the answer to 100 minus 87 is 13. Or we could turn it around 100 minus 13 is 87. So, subtraction across zeros kind of becomes a non issue if we get students working with these partners of 100, partners of 1,000. And you might be like, "Well, do they need to have partners of 200?" I mean, if they have partners of 100, then they're already thinking about that 100 in that 200 to use for subtraction. Maybe we'll parse that a little bit more later. But it can be super helpful to have kids own those partners, and we can use the routine I Have, You Need to build those. And like you said, the reason that we started to work on that, it was because subtraction was difficult. But not only just subtraction problems. The more that you and I built our numeracy for multiplication and division, we also started to see that the sticking point, the tricky part came not so much in thinking about the multiplication and division for kids, but in doing the subtraction. Like, let's give an example maybe. So, if I were to say something like, Kim, I want to find 99 times 87. We've been just dealing with 87. Let's stay with 87's.

Kim: 7:50

Yeah.

Pam: 7:51

So, if I said ninety-nine 87's, what are you thinking about?

Kim: 7:56

It's really close to 100, so I'm going to say one hundred 87's is 8,700. But then, because I have one too many 87's, then I have to subtract 99 from the 8,700.

Pam: 8:14

Wait.

Kim: 8:14

So, then...

Pam: 8:16

Wait, subtract (unclear).

Kim: 8:16

I'm sorry, subtract 87.

Pam: 8:19

You only want 99. You only want ninety-nine 87's.

Kim: 8:21

Yeah.

Pam: 8:21

So, you're going to subtract. Yeah, subtract 87 from 8,700. Mmhmm.

Kim: 8:25

So, again, if you're teaching pretty traditionally, even if you're starting to dabble, then your kids would be thinking, "Oh, I have 8,700 minus 87." And there's all those zeros again, right? So, that's not what I would do. I would subtract 100 from 8,700, which would be 8,600. And then, I subtracted too much. 100 was too much, so then I add back 13 because it was 13 too much.

Pam: 8:56

Because you know the partner of 87 to 100 is 13.

Kim: 9:00

Yeah.

Pam: 9:00

Absolutely. So, now, you've added back 13 to 8,600. And that's the hard question, right? What is 8,600 and 13. Ha, ha, ha.

Kim: 9:09

Yeah.

Pam: 9:09

It's 8,613.

Kim: 9:09

Yep.

Pam: 9:10

And so, that's a great strategy for multiplying. Use that Over strategy. What were you going to say?

Kim: 9:15

Yep. Yeah, I was going to say, you know I... You've mentioned more than one time that I'm a super big Over fan. And I like to think "more than" and backup in lots of operations. And so, subtraction, has become really important, particularly in that strategy.

Pam: 9:34

Yeah. Nice. Nice. And I might just... Another way to kind of think about that 8,700 minus 87 is... And maybe this is just kind of another way of envisioning. But I just wrote down on my paper a number line where I have 8,600 on the left and 8,700 on the right, and I'm kind of thinking about that 100 in between there. And I'm not really thinking about subtracting the 100, and then adding the 13. I'm kind of just thinking about there's that 100 in there. And I need to subtract 87, so I've kind of drawn that jump of 87 back from 8,700, and that leaves me with that little bit of space between 8,600, and that jump back.

Kim: 10:14

Yep.

Pam: 10:15

And then I can say, "Ooh, what is the partner of 87?" That's 13. And so, now, I have this. In that space between 8,600 and 8,700, I have a jump of 13 and a jump of 87, and I want to land in that where those two jumps meet. Or "meet"?

Kim: 10:33

Yeah.

Pam: 10:33

Is that the right word? And so, that is 8,613. That's sort of in there.

Kim: 10:40

Yeah, I think it's really important that you mentioned that because if you're an older grades teacher, and you hear about I Have, You Need, then you're like, "Cool, but that's just 100. That's super low numbers compared to what my kids are working with." And you mentioned the 200 earlier. Do you do I Have, You Need with 200. But really, once you've established that partnership with 100, then I do think that you bring in other amounts of hundreds and help kids understand that a 400 is a 300 and a 100. And then, it's always that partner of 100 work. And then, you come back and tack on that 300 at the end. So, there's always that 100 within any amount of hundreds that you can find the partner for.

Pam: 11:22

Yeah, so like in this problem that 100 is within that 8,600 and 8,700. And so, I deal with what I know about 100 in there, or the 13 and the 87. Then, I just tack on that 8,600 back on the beginning.

Kim: 11:34

Yeah, yep.

Pam: 11:34

Yeah, nicely said. So, subtraction is super helpful in things like this crazy multiplication problem we just did. 99 times 87. And we could do similar things with division where the problem actually becomes like, "Can I think about subtraction in the midst of this higher math problem?" So, let's parse out a little bit about two ways of thinking about this 87 plus what is 100? When I said,"I have 87, what do you need?" You said, "13." That's a way of thinking about kind of the distance between 87 and 100, right?

Kim: 12:10

Yep.

Pam: 12:11

I'm at 87, would I need to make 100? 87 plus something is 100. I can look at a number line. I'm at 87. I've got to jump up to 100. That's a way of thinking about distance or difference. But I can also think about 100 minus 87. I can think about removing 87. That looks less like starting at 87 and going up to 100, and that looks more like starting at 100 and going back 87. And I might think about going back 80 to 20, and then going back 7 more to 13. I might also think about going back 90 and landing on 10, and then I went too far, so I'd go back up 3 to get to 13. But in that way, I'm really thinking about removing. I'm thinking about removing 87 from 100, going back from 100 to land on 13. Those two types of thinking, one of them's sort of distance going from the 87 up. One of them's removing, going from the 100 back. Those are the two ways that we need to help kids think about subtraction. Maybe we could call it the two interpretations of subtraction that we can actually think about the problem 100 minus 87 as"What do I need to do to get from 87 up to 100?" Or as "What do I get when I remove 87 from 100." Now, if we're stuck in the traditional algorithm, we often haven't built that intuition about distance, about difference, that we haven't built that part of it. If we're stuck in the subtraction algorithm, we might only ever think about removing.

Kim: 13:55

Yeah.

Pam: 13:55

And when you were doing all that stuff with subtraction across zeros. 0 take away 7, you can't do that, so then we have to go to the next one. It's always "take away", right? It's always, "Can I remove this number from that one?" And so, we might have kids stuck in only that one interpretation of subtraction and never actually build intuition about the other interpretation of subtraction. And we need both.

Kim: 14:18

Yeah.

Pam: 14:18

One of the ways that we can do that is we can use the open number line to represent what students are doing, the way they're thinking about things. That's a great tool. We like to use it. And actually, as I was just talking about both of those ways of thinking about 100 minus 87, I was... If you could see my hands. I had my hands kind of spanning. So, when I was talking about 87 plus what is 100, I literally had my hands kind of spanning 100. My hands are going back and forth. Here's 0 to 100. I wonder if you can hear that. Can you hear that over the microphone? No, it's like air. Anyway, so 0 to 100. And then, I'm thinking to myself, "But In between that span of 0 to 100, here's 87." And I've just put my hand at 87. And I'm thinking,"If my hands here at 87, what's that distance to get up to 100?" I'm literally have this number line in front of me that's either in my head, or it's between my hands. Or on the other way of starting at 100, and then I'm backing up. And now, my hands kind of backing up from 100 to the 87, and I can kind of think about... Or sorry. Backing up. Ha. Not to the 87. That would still be difference. But backing up 80 and 7. Like, removing that 87. So, both of those are kind of happening on an open number line, whether I'm writing it or it's kind of what's happening in the space in my head. So, ya'll the open number line is not a picture that we're dumbing down to help your kids, I don't know, do fuzzier math. No, no, no. It's literally what's been happening in mathematicians' minds this whole time. We're just making that thinking visible. Go ahead.

Kim: 15:55

Yeah. And I think this is really important that you're talking about it because especially with younger grade students, there is intuition in kids to find the distance between numbers. I think you'd be hard pressed to find a kid who when you said 100 minus 99, they couldn't come up with. I mean, depending on their age. But I think many kids would just say 1. They're not thinking about remove 99 until we put a problem on the page in a stacking format and say,"Subtract."

Pam: 16:26

"Thou shalt..." Yeah.

Kim: 16:27

Absolutely. I think we schooled them out of some of that intuition. And so, if we can help them recognize that that's a thing and say, "Oh, here's a representation of this distance, this gap, that space that you just called out as 1." They could take that representation and do so many more things with it, but.

Pam: 16:50

And you can almost picture that ornery boy that's like,"I'm not going to do that. I'm not going to write down your steps, teacher. I just know 100 minus 99 is 1." And then, I can picture that well meaning teacher that goes, "Yeah, but you're going to need this for harder problems, so you got to show me all these steps."

Kim: 17:04

I mean, we have some of those boys.

Pam: 17:06

You and I. You and I birthed some of those boys. That would look at that teacher and go, "No, I'm not doing all those steps for that crazy problem." But that will meaning teacher that says, "No, no. No, really. You're going to find harder subtraction problems. So, we gotta get these steps down with these easier..." And so, what we are respectfully saying is,"Let's not try to beat that intuition out of kids." Let's actually say, "Yeah, yeah, yeah. Let's think about the fact that 100 minus 99 is just the difference between 99 and 100. I mean, Pam. I'm sorry, go ahead. Let's use that intuition.

Kim: 17:26

I'm having a hard time keeping my brain stopping.

Pam: 17:28

Let's use that intuition whenever we can. Now, what were you going to say Go, go, go.

Kim: 17:42

Well, I was going to say I would love to be in a third grade classroom right now saying, "At what point?" Like,"Where's the point where you say, 'You know, if it's 99 and 100,' they're going to say 1." Right, it's just 1. And you model the number line? And I would love to say, "What's the threshold?" Like, "At what point do you switch from finding the distance between them to..." It's like if you said, 100 minus 1, then they're going to remove. And I think it's different for every kid based on their familiarity and like different strategies. But I almost want to like plunk numbers on a number line and say, or ask them questions and say, "Okay, what do you want to do when it's 100 minus 20? What about 23? 76?" Like, there's... At some point, before we mess with kids too much, they have this gut reaction to where they want to start removing, and where they want to find the distance. I just think it would be a fun exercise and like, "What are you thinking about? And what numbers do it for you in either way?"

Pam: 18:48

Mmhmm. Yeah, absolutely. Yeah, that's super cool. I want to hang with the open number line for just a second, and then I want to go back to what you were just saying. Let's see if we can do both of those things.

Kim: 18:57

Stay focused. Okay.

Pam: 18:58

Hanging with the open number line for a minute. You know, teachers of higher math, you might be like, "Okay, Pam. You told us to listen to this episode because it's important some way." Yeah, let me just take that a little bit. Because as we think about the difference. Say, I've got something like 105 minus 97. Do we really want to remove 97 from 105? Or would it make more sense to kind of think in your mind,"Where's 97? Where's 105? How far apart are those?" As we do that kind of action, we're building this sense of, "I can use subtraction to represent that." So, from that 97 up 3, and then up another 5. So, I can think about 105 minus 97 as that distance of 8, that 3 and then that 5 in between there. But I can write that as 105 minus 97. That subtraction symbol can mean"Find out how far apart they are." That so relates to stuff in higher math, that we can see places in higher math where there is a subtraction symbol and it doesn't mean "remove". It means "Find the difference between the numbers." And I'm just going to say a couple of examples, and we'll do some episodes later where I get into these more deeply. But if you think about finding the slope of a line between two points. If I'm thinking about y2 minus y1, does that subtraction symbol mean "subtract y1 from y2." Or does it actually mean "How far apart are those y values?" So, I'm going to let teachers of higher math think about that a little bit. X2 minus x1. Does that mean, "remove x1 from x2." Or can it actually mean, "How far apart are those x values?" Because is that rate of change formula talking about the distance, the difference between the y values and the ratio of that, ratio of the distance to the y values to the distance of the x values, the difference between those. I would suggest it's far more about that and not at all about removing things. And we could go even higher than that. We could talk about calculus, and the way that we represent intervals when we're trying to find the area under a curve. When we represent intervals, we often use the subtraction symbol to represent an interval. An interval is the distance between two points. It's not about removing one from the other, but we're using subtraction. If I was finding the area between two curves, I'm going to subtract f of x minus g of x, and then take the integral from a to b of that difference. But it's... I just said"difference". The difference between those two functions, between the two y values of those curves. It's the difference between the y's of those curves. It's not about removing a function from another function. Anyway, there's a couple of examples of where we're actually building intuition towards that, so that when we get them in higher math, we have both interpretations of subtractions. And, Kim, I got to mention one more. Not to mention integers like integers.

Kim: 21:49

Oh, yeah, I was going to say, "Wait a second, you missed a big one!" Because there's all these rules, right, for when do you add, when do you subtract. And I have been living in that with my sixth grader because he's doing things like 37 minus negative 42. And I'm like, "Oh, you're not removing." He's like, "Oh. Ohhh, those are 79 apart on the number line." I'm like, "There you go." There's your other interpretation of subtraction.

Pam: 22:17

Yeah. And what's interesting. And we'll do a whole episode on just integer subtraction soon, but what's interesting about integer subtraction is we actually need both interpretations to make sense of integer subtraction problems when we subtract a negative. We have to think about the difference, the distance between the two numbers, and then we have to consider from that first number, "Am I subtracting something smaller than it?" That's what we usually do. And if you subtract something smaller than that first number, then you end up with a positive answer. But if you subtract something bigger than it, then you're going to have a negative answer, even though the distance was positive. So, I know I just said that. We'll have to do an episode where we totally build that at some point.

Kim: 22:56

Yeah.

Pam: 22:57

So, what's important to know about integers is we have to have both interpretations in order to make sense of integer subtraction. And that's huge because then, if we hadn't made sense of that, what are we left with? Well, we're left with all those rules, right? And then, kids are using Keep, Change, Flip no matter whether they're dividing fractions or they're operating with integers. And neither of those are a good rule because then we don't know when to use them, and we don't know why they work, and we're not reasoning. We're pretending that math isn't figure-out-able, pretending math is rote memorize. Okay, so there was one other thing I wanted to do. What was the other thing, Kim? I said there were two things. Oh, oh. I think I wanted to talk a little bit about like. Kim, I'm going to give you a problem to do.

Kim: 23:42

Okay.

Pam: 23:42

And so, if I were to say... Well, actually don't do the problem, please. If I were to say 328 subtract 96. Will you line those up as if you were going to do the algorithm? So, 328 minus 96.

Kim: 23:57

Okay.

Pam: 24:00

(unclear).

Kim: 24:00

Oh, I'm laughing because the last time I told you I attempted to do an algorithm, I totally did it wrong. So, let's see if I can do it. Okay. 328. let me write down. Minus 96.

Pam: 24:10

Three hundred twenty-eight minus 96.

Kim: 24:11

Okay.

Pam: 24:12

If you were doing the algorithm. If you were doing the algorithm. Which, I know it's been a hot minute, since you've done that.

Kim: 24:17

Yeah. So, I'm starting in the ones place. So, 8 minus 6 is 2.

Pam: 24:21

Because? Did you just think about the difference between 8 and 6? Or did you subtract? Do you even know? I mean, I don't know if you can think about that. 8 minus 6.

Kim: 24:28

I think I probably found the distance between them.

Pam: 24:32

Because? Like, why? What about 8 and 6 made you think about distance between them?

Kim: 24:36

Yeah, they're really close.(unclear).

Pam: 24:39

So, the distance is small, right? The distance between them is small. So, yeah. Might as well just find that distance. I didn't mean to put words in your mouth. Sorry.

Kim: 24:46

No, that's okay. So, then, I'm in the 10's place.

Pam: 24:49

Ooh, but before you move on. Sorry. But could you imagine that we might have kids that were. If we're teaching them. If we're saying, "Thou shalt do this algorithm." Could you imagine that the kid is saying,"Okay, 8 minus x. Minus means take away. So, I've got to take away 6. 8, 7, 6, 5..." And they're counting on their fingers until they tell you. 6 fingers up, and they land on a 2. And they're like, "Okay, so it's 2." Alright.

Kim: 25:11

Yep.

Pam: 25:12

So, I've written down the 2. Now, carry on with the algorithm

Kim: 25:14

And, then, maybe if you're lucky, they'll look at the whole problem again, but probably not because then they're just moving into the next column.

Pam: 25:20

Yeah, the trying to remember the next step. Yeah. Okay, next column. Mmhmm.

Kim: 25:22

Yep. So, 2 minus 9. Which they will say, "Can't do that." So, you got to go to the 3, cross out the 3, make the 2 a 12. And then, 12 minus 9.

Pam: 25:38

And as you think about that now 12 because you brought that 10 over there.

Both Pam and Kim: 25:42

Yeah.

Pam: 25:43

Again, minus 9. And so, often, now kids are going to say, "Alright. 12, 11, 10..." And they count backwards. The 9, right. Even though 12 and 9, if I think about those on a number line, it'd be so much easier to think about those as the distance between 9 and 12, right, to get that 3. So, it's just interesting that if we pigeonhole kids into, "Here are the steps that thou shalt do," even in those digit oriented problems, where we've got the 8 minus 6, and then 12 minus 9. If we're saying minus, and we're in that algorithm, kids are only thinking about removal. They're not thinking about distance or difference, which would make so much more sense for the exact numbers that we just did, right?

Kim: 26:33

Mmhmm.

Pam: 26:34

Okay. Now, I'm going to let you think. How would you actually do 328 minus 96?

Kim: 26:39

So, 328 minus 100 is 228, but I took off too much by 4, so I'm going to add 4 back to get 232.

Pam: 26:51

Because 4 to that 228 you landed on is 232. Nice.

Kim: 26:56

Yep.

Pam: 26:56

So, we could think about that. Could you also think about... And that's a great strategy. Could you also think about if I had sort of that 96 on a number line, and I had the 328 on a number line, and I wanted to find the difference between them? Could you talk through that? What would that look like?

Kim: 27:11

Yeah, so if I was at 96, then I'd probably add that 4 first, instead of... When I did Over, I added it at the end. But I'd add that first to get to 100. Because then once I'm at a friendly number, I know the space between 100 and 328 is 228. So, 228 and the 4 is 232.

Pam: 27:36

Which is then a really fun conversation you can have with students if you compare those two strategies. Because in both cases, you ended up adding 228 plus 4. You just did it in a different order. Now, your thought process was completely different, but that's a fascinating conversation to have with students, right?

Kim: 27:54

Yep.

Pam: 27:54

Because in the one case, you subtracted 100, and you landed on 228. So, then, you had to add the 4 back because you subtracted too much. So, 220 plus 4. But in the other case, she added the 4 first to get up to the 100, and then you just like, "Well, and now I still have 228 left to get up to the 328." In both cases, you ended up adding 4 and 228. But in a different order and for different reasons. But, ya'll, that's making sense of what's happening with place value, and the order of numbers, and the magnitude of numbers, and you're having kids actually think about numbers like 328 and 96, and what they're close to, neighborhood and nearness. Not 3-2-8 and 9-6. And 8 minus 6. And 2 minus 9. "Oh, can't do that. Kill the 3, make it a 12." You know like, all of those super small numbers that can get(unclear). Alright, cool. So, some things, Kim, that we could think about as we leave this podcast. Super young kids, we need to help them count backwards.

Kim: 28:56

Absolutely.

Pam: 28:57

They need to be able to think about minus 1 repeatedly is sort of counting backwards. So, you can. Now, don't only, teachers of young kids, don't only go, "Okay. Everybody ready? 10, 9, 8..."

Kim: 29:11

Yeah.

Pam: 29:11

That's a fine one. You can do that.

Kim: 29:13

Yep.

Pam: 29:13

But also say, "Hey, we're going to count back today. Ready?" Start from 8. "8, 7..." Or, "Let's start from 12. 12..." Or start from 20. Start from 18. Like, pick a (unclear) number, and let them count backwards. Did I hear you say the other day that you actually had some kids counting backwards by 2's?

Kim: 29:30

Yeah. So, I think we were talking about a kindergarten classroom that I was in. And we talked about some Count Arounds, and starting with numbers and counting just back by 1's and writing kind of off to the side"minus 1, minus 1, minus 1, minus 1" and looking at patterns. But then, also counting back by 2's to recognize the repeating digits that happened.

Pam: 29:58

Yep, nice. Nice.

Kim: 30:00

And, you know, you and I saw a Problem String not too long ago where some second grade students were having some... I don't know if it's difficult. They're grappling with the idea of if you're at 80, and you subtract 1 is that 81? Is that 89? Is that 79? So, that's a really. It's a real situation. And so, giving kids more experience to deal with like that decade before. You know, we do a lot of counting up and a lot of moving forward with addition, but (unclear).

Pam: 30:34

Well, and kids get used to that pattern, right? "76, 77, 78, 79..." And they have to think about what that next decade is. "80." And then, they kind of continue (unclear) "81, 82..." And so, they get kind of used to that. But we need them to get used to "82, 81, 80... Is that 89 that comes next? Is that 71 that comes next?" Like, that's a tricky... What's the word I want? Tricky landmark to bump over.

Kim: 31:01

Yeah.

Pam: 31:02

Not so much forward. I mean they do that a little more readily. So, we've got to give them experience doing that backward as well. Yeah. Nice. Super. So, another thing that we have to help kids when they're super young wrap their brains around is that when they first run into addition, subtraction problems, they're with action. And so, we often have problems with addition, where we'll say things like, "I've got 5 puppies. 3 puppies crawl in the room." We've got puppies joining the other puppies. But we also might have 7 balloons, and I popped 2 balloons. And we sort of have that removal action. And so, that removal action is a first thing that happens. And we do want kids to think about subtraction as removing, as take away. Those balloons are popping, or I'm eating the M&M's. Or, I'm... Help me. Popping, eating... Give me another action, Kim. I'm...

Kim: 31:56

[Kim laughs] I can't tell you what I just was thinking.

Pam: 32:00

Oh, okay. I'm grabbing marbles from my friend's pile of marbles. Like, whatever it is. So, that is a first thing. But then, we want to help them make those connections between the addition relationships, so they can start to think about more of a part, part, whole relationship. So, if I've got 3 red marbles. Are marbles red? I don't even know. I've got 3 red marbles, and you give me 2 blue marbles, how many total marbles do I have? I want to start connecting that with, "Well, what if I had those 5 total marbles, and I got rid of those 3 marbles? Ooh." So, there's this relationship between those removal and those combining problems.

Kim: 32:35

Yeah.

Pam: 32:35

And at first. Oh, go ahead.

Kim: 32:37

Oh, well, I was just going to say. And that is super helpful where you could do the work with a number line(unclear).

Pam: 32:43

Because we can see both of those parts.

Kim: 32:44

Yeah, and it's not just writing and rewriting fact families over and over again.

Pam: 32:50

Yeah.

Kim: 32:51

(unclear).

Pam: 32:51

Well, and you could do the same kind of work with a number rack because I've got those 5 beads, right? And I can see the 3, and I can see the 2 beads. So, I can have the 5, and I could remove 2. But I can have the 3, and I can bring in 2. I think what you were going to say is it's not just saying to kids,"This is a fact family." And,"Here, write it these 4 ways. No, not that way. That's the dumb way. That doesn't..." It's not about just memorizing ways of writing equations. It's really about understanding the relationship between addition and subtraction. Cool. Yep. And what I was going to say is, that's where CGI research comes in, and it's so helpful to help us recognize that we want to give kids different kinds of problems as they are ready for them, as it's on the edge of their zone of proximal development, that's when we want to give them. We want to start with those action oriented, combining or separating problems. But then, we want to give them those part, part, whole problems where there's not so much action. And then, we want to give them those comparison problems because in those comparison problems, that's really where that difference, the distance meaning really comes in. Now, I've got 5 marbles. You've got 3 more marbles than I do. How many marbles do you have? Or, "I've got 5 marbles, and you've got 8 marbles. How many more marbles do you have than I have?" Those comparison problems really lend themselves to that distance or that difference meaning of subtraction. And so, we really, really want to emphasize that those hardest kinds of problems... No, let me say this. We really want to emphasize that subtraction, we need to have both of those meanings happening. We need to build the connection between finding the difference between numbers and removing, and that both of those have everything to do with subtraction. So, Kim, I think we wanted to end with some things that teachers can do with their students.

Kim: 34:42

Yeah, so we mentioned quite a few within, so I'll just bring them back out again. Counting backwards. Super important particularly with really young students. Play I Have, You Need. Talk about both types of subtraction. The distance, difference representation, and removal. So, this is both interpretations.

Pam: 35:00

Mmhmm.

Kim: 35:00

Represent subtraction on a number line and rekenrek to give kids a feeling about subtraction not just the fact families.

Pam: 35:08

Mmhmm.

Kim: 35:08

And consider where subtraction impacts other operations in your grade because those might be the places where your kids get hung up. Not in the multiplication, not in the division, but that little bit of subtraction at the end. And then, they get a wrong answer. And then, we think they don't know what's going on. When really they just need more subtraction work.

Pam: 35:28

Yeah, nicely said. Alright, ya'll, this was a little bit of a long one. 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!