Ep 150: Memorizing Subtraction Facts

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 it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We believe we can mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but, ya'll, rotely repeating steps actually keeps students from being the mathematicians they can be.

Kim: 0:38

So, I'm laughing a little bit as you're talking because I'm listening to the things that you're saying, and I'm kind of nodding. I'm like, "Yes." And then, I look up a little bit, and I glance at the title, what we titled this episode. We call this episode Memorizing Subtraction Facts. (unclear)

Pam: 0:55

(unclear).

Kim: 0:55

Well, Pam. So, we probably need to get into it really quickly because I'm sure there's some people who are like, "What? What are they going to talk about on memorizing?"

Pam: 1:06

Yeah. So, Kim, I find this fascinating. You know, I'm a little bit of a wordsmith. I'm a little bit. I enjoy the exercise of debating. I enjoy, really getting under meaning. I enjoy, you know like, really understanding what people are talking about, and asking them to define their terms, and just being really super clear on things. And one of the things that has been fascinating to me, as I've been working with people to build their powerful numeracy over the last 20-25 years, is this confusion over the word"memorize". That we really think it means something different than like as we're communicating. So, I want to get really clear today about what we mean when we say, "We want kids to know their facts," and what we don't mean. So, the word"memorize", I think, is super tricky. Because I think people will use it to mean that "kids should rote memorize stuff", and I think other people will use it to mean "the kids should know stuff".

Kim: 2:09

Yep.

Pam: 2:10

And even that's a little fuzzy. And so, let's parse those two out. When I say that I want kids to own something deep down, then that means I want it at their fingertips. I want it to not cause a huge cognitive load for them to redo, refigure. Especially find it in some less sophisticated way than the operations calling for. I want it to be something that they own with connections to other things, so that they have multiple neural pathways that are involved in the knowing, the owning the really having that, so that it can ping in their intuition, so that when they own that fact...say we're talking about multiplication facts today...when they own that fact, that things ping for them, relationships and mathematical connections occur to them. They have intuition around it because it is deeply owned. Now, if that's what you mean, when you say "Yeah, so they've memorized it." You know, "They have it in their memory." If that's what you mean, okay, great. Then yay, let's do that. But I think the word "memorize" is tricky because I think it can also mean the way you get there, the way you own it.

Kim: 3:31

Yep.

Pam: 3:31

And it can mean... And so, I don't know. Several years ago. I stopped saying the word"memorize" when what I meant was"rote memorize". So, I think that "memorize" often can mean to people that we're "rote memorizing" something. And you're like, "What does that mean?" Well, that's like where I admit that it's not really understandable. It's not really figure-out-able. It's something that is rote memorizable, and I need to, therefore, have a rhyme, or a rap, or a pneumonic, or some kind of memory technique to be able to memorize that thing. That it could be a disconnected way, something that doesn't have anything to do with it. I don't know something about Mrs. Week sitting on a chair. Do I mention that here Kim? I don't know.

Kim: 4:22

No, that's a whole other podcast.

Pam: 4:25

I'll mention this one that was on my kids wall in third grade that said something like,"6 times 8 like the garden gate is made of 6, so it's 56." Wait... 6 times 8? And I do that wrong on purpose. 6 and 8 isn't 56. I do that wrong on purpose to exemplify the problem with rote memorizing facts because I could rote memorize them wrong. I could rote memorize that rhyme. It should be, "7 times 8, like the garden gate is made of 6, so it's 56." But if I rote memorize it wrong, then I'll get it wrong. I could rote memorize the 9's trick with my fingers wrong, and I could get the 9's wrong. I could rote memorize the fact 13 minus 8 wrong, and I could get it. (unclear) real consistent about 13 minus 8. But if I've memorized it wrong, if I've rote memorized it wrong, my rhyme, my rap, the pneumonic, the picture I've drawn, the story I've told. If I somehow get that wrong, then I'm going to get that fact wrong all the time. Or another alternative is, I'm going to forget it. I'm going to not be able to recall it, and then I'm stuck with nothing. "(unclear) I can't remember that rhyme that I had, or that story that I told, or whatever." In fact, I was just reading in Dr. Nikki Newton's Running Records this morning a little bit about some of the introductory stuff, and she very nicely parsed out where she was talking to students, and she said, "So, if this student has rote memorize the fact..." I think she's the word "memorize" not "rote memorize", but I'll say, "If the student had rote memorized it, when they get to it on somewhere where they need to have it there, and they can't recall it from rote memory, then they'll just skip it. Or they'll guess." Like, that's the only recourse that they have. And so, again, I want to just kind of at the top of this episode parse out the difference between what does it mean to have, to own the fact, so that, yes, I can pop it out when I need it. I can have it at my fingertips. But more than just have it as a disconnected fact that that's all I have. It doesn't have multiple mathematical connections that ping, that help me intuitively do other things with it. Help me. Kim, is there any that you want to kind of(unclear)?

Kim: 6:41

Yeah, I mean, I think it's so good because either... The word "memorize" could either mean something you've committed to memory, in whatever way you've committed it. Which is what you and I are interested in. We do want things to be automatic. We do want kids to own things. But the other version of "memorize" means an action of things to do to put it in your memory. And those are the things that we're not interested in rote memory. We're not interested in the rote memory, like the action of,"This is what we think you do over and over and over and over and over and over again to get it into your memory."

Pam: 7:17

Yeah, so what actions are we interested in? We're interested in actions that are related to relationships. We want kids to deal with the numbers a lot and get... Not just deal with the numbers, but deal with the kinds of relationships that are important, so that they're thinking about doubles. They're thinking about close to 10. They're thinking about using facts they know, and then being able to compensate from there. Those relationships are the super important things that we want to deal with a lot, so that those neural connections get really strong. So, that in the case that we ask a kid something like 13 minus 8. In that case when we ask them that, then if it doesn't just pop for them, if they don't intuitively just know, like say it, or have it right there. That they then intuitively have things ping for them that are those (unclear) relationships, that are additive relationships, so that they quickly refigure it in a way that isn't a big cognitive load. And they're not stuck with, "Oh, I can't remember. I don't know. Nothing." Or, "I'll guess." Because their only recourse, if it was rote memorized, if they can't remember the rap, the rhyme, the pneumonic, the whatever, then their only recourse is to guess or they don't know, right?

Kim: 8:32

Yeah. I also think that it's really important to note right at the beginning here that we recognize that people get really hung up on facts. It's a huge topic of conversation. And a lot of times, that's because if you're only going to do the algorithms, then you literally just work with the basic facts over and over and over and over and over again. And so, if that's your jam, if you are,"I'm full on leading towards an algorithm," then I understand why it would make sense that you would say, "Just memorize these 144 things. And we're just going to repeat them over and over and over and over again." Seems like it would be an easy job. You know, it's not. But that's not our goal, right? We are about place value, and magnitude, and thinking, and relationships. And so, since our goal is different, it's not just to get answers, we are less inclined to work with facts in the ways that other people do. But we are interested in making sense of the facts as ways to work with strategies and thinking with less sophisticated numbers, so that those strategies and thinking can happen with all size of numbers. That's our outcome. The ways that we work with the facts, then lend themselves towards the work we do as the numbers get bigger.

Pam: 9:46

Yeah, and maybe it will be helpful to illustrate that. But I've just been using the fact 13 minus 8. So, a way of addressing subtraction facts is, "Okay there's these 144 separate subtraction. If I look at the deck of cards, I'm going to drill kids on these. There's 144." 12 by 12. I don't know. I don't know how many subtraction facts there are. 87. "However many there are, I'm going to drill kids on those. And they're going to have these separate, disconnected flashcards rote memorized. And however, we accomplish that." But if what we're about is what you just said, what it's really about mathematical relationships, and connections, and patterns, and properties, that if we develop those with these less sophisticated numbers can extend to more sophisticated numbers. An example of that would be, if a kid's thinking about 13 minus 8 as the difference between 8 and 13. That's 5. Like, I can go from the 8 to the 10 is 2, and from the 10 to the 13 is 3, so that's a total of a distance of 5. If they're thinking about like that, then they can also think about a problem like 130 minus 80. In a similar fashion, they go, "Well, I can do the same thing. I can find the distance between 80 and 130." They can think about something like 1,300 and 800. 1,300 minus 800. Because those problems become drastically different problems if I'm lining them up in algorithms, right?

Kim: 11:14

Yep.

Pam: 11:15

In fact, maybe I'll just. Because I don't know that we were going to mention that today. But if I line up 13 minus 8, and do the algorithm. How many times, teachers? This is when you know the algorithm is failing your kids. When they look at a problem like 13 minus 8, and they line it up on. And so, now I have the 1, 3. I have the 8 below the 3. And the kids say, "Okay, what am I going to do first? I got to do 3 minus 8. Nope, can't do that. So, I got to borrow. Let's see. I'm going to bring a 10 over here. So, then that 10 over there. That 1 becomes a 0. And now, what do I end up with? I end up with 13 minus 8."

Kim: 11:52

All the time.

Pam: 11:53

That's a moment where you know they're not thinking about 13 minus 8 at all. They're performing a bunch of steps. And then, who knows what they do when they get to that 13 minus 8. Maybe count it by ones. Again, that's less sophisticated reasoning if they're counting by ones than what we want to have happening, which is additive reasoning, thinking in terms of bigger chunks than one at a time. Let me just carry that. So, if I'm thinking about 13 minus 8 by actually thinking about the difference between 13 and 8, then if I'm thinking about 130 minus 80, that becomes a completely different problem. Where now I've got 0 minus 0, and 3 minus 8. I mean, parts of it are similar and the same. But it's not... I'm not feeling the same relationships. I'm not letting intuition help me reason about those similar numbers. Yeah? So, also. I get why, like you just said, if you are algorithm... I don't want to say algorithm oriented. That's not quite. That's what I was going to say. But if that's your goal. Focused, yeah. If your goal has been... And we don't blame you because you grew up in an algorithm focus probably. That's the way you were taught. But if that's the way you've been teaching any of the four operations, then yeah it makes sense that you know when students get to big numbers, that they're going to have to do these little numbers over and over and over, so they might as well just memorize those little number combinations because you're going to use them over and over in these, you know, multistep algorithms. That totally makes sense. But if you're trying to build relationships. Well, let me stay there. So, it makes sense why you would want kids to do that. Which, then, also makes sense why you might want those quick. And you don't want kids in the midst of a long subtraction problem to be counting on their fingers, or to be grabbing manipulatives and removing the manipulatives and counting what's leftover. You might be saying to yourself, "I need you to just own all these individual little subtraction facts in the midst of this big subtraction problem, so that you can just do them, and you don't have to do this stuff that takes time in the middle of this." And so, that might lead you to say,"Ooh, I think a way to stop you from refiguring these facts all the time. Let's put a time limit on it." In fact, I just read a post the other day by Michael Pershan, who is a thoughtful guy, who puts out a lot of stuff. Some of which we agree with. Some of which we don't. But his bit was, "Let's give timed drills because we want to not allow kids the time to refigure, because we don't want to leave them there. We don't want to have kids doing all this refiguring all the time. We want to give them a time limit, so they're forced to memorize them." I think... Michael, you can tell me if I'm not representing your position accurately. So, I get why that would be a thing, if your goal is a bunch of disconnected set of facts that I can just pop out if I'm going to use them over and over, and so I want them to pop out. But that's not our goal.

Kim: 15:00

Right.

Pam: 15:01

If the goal is just that collection of rote memorize facts, fine. But not our goal. So, maybe we can agree that if our goal is to actually own the relationships, then I want to do a lot with those relationships because the more... If we give kids lots of experience with those relationships, and the properties, and the different strategies that we can use to figure out the facts, then kids will naturally... They will naturally gravitate towards owning those facts. But they will have so much more as well.

Kim: 15:35

Yep.

Pam: 15:35

I wonder...

Kim: 15:35

(unclear) I'm going to dive in real quick.

Pam: 15:37

Okay, go, go, go.

Kim: 15:37

Because I think that there are lots of people in the memorization, rote memorization camp, who think that people like you and I mean, "You shouldn't do mad minutes. You shouldn't do time drills. You shouldn't do whatever."

Pam: 15:53

Flash cards.

Kim: 15:53

"You just gotta let kids think." So, then there's no work to be done. And that is absolutely not true. We are not like, "Let kids do whatever they want." Like, "Get (unclear) 10, 5. You know, 5 minutes to think about it is fine." Like, we're saying, "No we're actively doing a lot of work to own them. It's just not the rote memorization work." And I think that's really important for us to say often because it's not rote memorization or nothing. And I think that we have experienced teachers who say, "I hear you. I buy it. I see that it's not helping my kids. They stress about the time thing. It's not improving them when I do it day after day after day with no work." Of course, it's not going to help them. So, they're like,"Okay, I'm Team 'I'm not going to do that stuff'." And then, they just do nothing. And the next year teacher's like,"Umm, your kids got nothing."

Pam: 16:47

Or they come to us and say,"So, what do I do?"

Kim: 16:50

Yes.

Pam: 16:50

Right? Like, let's give them that. Like, there are tons and tons of teachers that are like, "Okay, I'm buying in. I'm on Team 'let's figure these out'. But now, I don't know what to do because I have this arsenal of stuff to do to help them rote memorize. What is the arsenal now?" Because it does take work. It takes work for kids to own them. We're just suggesting that, that work is not time based and it's not disconnected drill based. So, Kim, what is the work? And, Kim, I don't want to forget. I guess I could write this down somewhere. I don't want to forget that maybe we want to bring up the Trinity first grade thing. What happens when we do all... You know, we're about to talk about the arsenal of things to do. What happens when kids just kind of don't? Do you know what I mean?

Kim: 17:35

No, but you could tell me again when it's time to talk about it.

Pam: 17:37

Okay, I've written it down. I won't forget.

Kim: 17:38

Okay, sounds good.

Pam: 17:39

She's like, "Pam, I can't read your mind." Alright, so, Kim, what is some of the work that we can do to help kids? Go.

Kim: 17:46

Yeah. So, we recognize that there are some major relationships for single digit facts, right? We have talked about those. We're going to bring them up again here. So, if you didn't listen to episode 149, which is the last episode, you'll definitely want to listen to that again because we talked at length about distance or difference versus removal as the two interpretations of subtraction. So, that work is super important, having kids experience both of those. But there are a couple of major relationships with basic facts. And so, let's talk about a couple of those.

Pam: 18:21

And specifically this episode with subtraction basic facts. Right?

Kim: 18:25

Yes.

Pam: 18:26

Yeah. Okay.

Kim: 18:27

So, if there was a problem 15 minus 8, it's...

Pam: 18:32

Often missed

Kim: 18:32

...probably missed. Yeah, it's a problem.

Pam: 18:34

Yep.

Kim: 18:34

So, a major relationship is this idea of remove to 10. So, if I'm thinking about 15 minus 8, and I'm thinking about removing to 10, then I might think about 15 minus 5, which gets me to the 10. But then, since I've removed 5, I still have to remove 3 more to make the 8. So, then 10 minus that last 3 would be 7. So, 15 minus 8 would be 7. That's a student who's thinking about removing to 10. And it's super cool because we can anchor everything on that 10. Which is the basis of teen numbers, right? If you've done any work with young grade students, you know that teen numbers are an important place to hang for a while because every teen number is 10 and some. 10 and some more. So, if we can think subtraction, go back to that 10, that's an important and helpful strategy. So, another strategy is thinking about doubles. We are huge fans of Doubling and Halving numbers at a very young age. We want there to be a lot of work with Doubling basic facts when you're thinking about addition. And then, that comes in really handy when you're thinking about subtraction. A kid who is thinking about 15 minus 8 might say, "I know 8 plus 8 is 16, but that's too much. I need to just be back 1 to get to 15. So, then I have to adjust one of those add-ins and call it 8 plus 7. So, 8 and 7 is 15. So, 15 minus 8 would be 7.

Pam: 20:09

Nice. Nice. (unclear)

Kim: 20:12

(unclear) using doubles.

Both Pam and Kim: 20:13

Yep.

Pam: 20:14

I was going to just do that. Yep. Removing ten, doubles.

Kim: 20:17

Sorry, sorry.

Pam: 20:17

Okay, what's the? Nah, I just want to make sure we kind of helping listeners sort of focus or? Yeah.

Kim: 20:21

Yeah.

Pam: 20:21

Those two. Is that it?

Kim: 20:23

So, finding the difference. If we're thinking about not just talking about removing. So, the first one, remove to 10 is a removal version of subtraction. If you're thinking about finding the difference, a kid might say,"Ah, I'm thinking about how far is it from 8 to 15. Let me think about that gap between them." So, somebody might say, "Mmm, I know 8 and 2 is 10. And then, from 10 to 15 is 5. And so, that 2 and that 5 was the gap, so that's 7."

Pam: 20:55

Nice.

Kim: 20:56

And then, one of my favorites is Remove a Friendly Number, Over. And this is super helpful when you're thinking about 7's, and 8's, and 9's. So, if you are Removing a Friendly Number, Over with 15 minus 8, you might say, "Mmm, I don't want to subtract 8. I'm at 15. I want to remove 10 because that's easy. And so, then I'm at 5. But I've removed too much." And I would say, "How much too much?""I've removed 2 too much because I did 10 instead of 8, so then I had to tack that 2 back on. So, I was at 5, tack the 2 back on, and that would put me at 7."

Pam: 21:34

Brilliant. Nice.

Kim: 21:34

So, 4 major relationships that are super important.

Pam: 21:38

That we want to help build with students. We want to help them develop those relationships, so they own those relationships. It's not about rote memorizing those relationships either, right?

Kim: 21:49

Yeah.

Pam: 21:49

It's not about saying,"Okay, kids. Today, you're going to rote memorize the steps for removing to 10." It's not about that. It's about giving them experiences where we...and often it's Problem Strings...where we give them strings of problems, where we do these relationships, and we put them in front of kids, and we let them see the patterns, and we talk about the patterns, and with that verbalizing them, the patterns get stronger, and kids own them, and now they have intuition about how they want to attack a problem if... Well, let me say. Yeah. They attack problems a lot, and they use these strategies over and over and over and over and over so often that they own a lot of the the smaller subtraction facts. And then, the ones they don't own, the few that they don't own, they quickly use one of these sophisticated strategies. They can often with a low cognitive load, because the strategies are... They own the strategy. They own the relationship. So, it's a low cognitive load to just quickly do one of these because they've done it. They own them. Just for those few that they don't maybe have at their fingertips yet. And the more that they do that, then they might have all of them at their fingertips by the(unclear).

Kim: 22:59

Yeah.

Pam: 22:59

But again, if they don't, then they have a recourse. So, Kim, if you don't mind. Quickly, I'm just going to give you one other problem, and let's just run through those four strategies one more time.

Kim: 23:07

Okay, cool.

Pam: 23:07

So, if I gave you 17 minus 9. Do the remove to 10 for me.

Kim: 23:14

Okay. So, 17 minus 9, and we're removing to 10. So, I'm removing 7 first to get to 10, and then I still have 2 more to remove, so that would be 8.

Pam: 23:25

Cool.

Kim: 23:25

So, I removed 7, removed 2.

Pam: 23:27

Yep. So, you removed to 10. Nice. Alright, let's do the doubles. How would doubles work with that?

Kim: 23:32

Okay, so I might say, "I know 9 plus 9 is 18, but I only need 17, so then it's going to be 9 plus 8." (unclear).

Pam: 23:41

Cool. And that's similar to the way that you just did. Oh, yeah. Sorry, I didn't mean interrupt. I got excited.

Kim: 23:44

That's okay.

Pam: 23:46

That's similar to the way you did the 15 minus 8. If I could do maybe a different version of the doubles.

Kim: 23:50

Yeah.

Pam: 23:51

I might think to myself, "I know 18 minus 9 is 9, but now I'm doing 17 minus 9, so it's only going to be 8." But I also might think to myself, "I know..."

Kim: 24:04

8 and 8 is 16.

Pam: 24:05

"...8 and 8 is 16."

Kim: 24:07

Yeah.

Pam: 24:07

Keep going. Yeah, keep going.

Kim: 24:09

And then, but I needed to get to 17, so then it's 8 and 9.

Pam: 24:12

Nice. And so, you kind of think about that partner of 8 and 9. Since we're subtracting 9, then the answer has got to be 8. Cool. Okay, what about finding the difference. If you were finding the difference 17 minus 9.

Kim: 24:22

So, I'm thinking about the gap between 9 and 17. So, from 9 to 10 is 1, and then 10 to 17 to 7, so that 1 and that 7 is the 8.

Pam: 24:35

Nice. And then, lastly, if you wanted to Remove a Friendly Number, Over. Your favorite.

Kim: 24:40

It is my favorite. So, 17 minus 10 would be 7. But I removed too much by 1, so then I put that 1 back on and make it 8.

Pam: 24:49

Brilliant, brilliant, brilliant, brilliant. Okay, cool.

Kim: 24:51

Yeah.

Pam: 24:52

So, we have these major relationships for single digit subtraction facts that if we work on those four, we work on developing those four major relationships, it can be super, super helpful for students.

Kim: 25:03

Yeah. And I think what's really even more important to us is not just that the kids have that work with basic facts, and they have something, so they don't just say, "Oh, I don't know that one." That's a huge bit, right? That's really important. But what I love more than anything is that these strategies are extending to higher numbers. It's not just like, "Let's figure out eight new strategies for large numbers. That get to 10, that remove to 10 is the same work that kids will do when they remove to 100 or remove to 1,000. (unclear).

Pam: 25:35

Or remove to any friendly number. (unclear).

Kim: 25:37

Yeah.

Pam: 25:38

Yeah.

Kim: 25:38

Yep, it's that general Remove to a Friendly Number strategy. Same with Removing a Friendly Number, Over. That is one of the major strategies that we use for larger numbers. Finding the difference, huge for kids in older grades, all the way up through high school.

Pam: 25:53

Absolutely.

Kim: 25:54

And using doubles. You know, doubles show up a lot more. Once you're aware of doubles, I think you realize that they show up a lot more often than we realize.

Pam: 26:03

They're so handy. Yeah, they're so handy.

Kim: 26:04

Yeah.

Pam: 26:04

Nice. Yeah. So, all of the these. I'll just restate it. That these four relationships that we just talked about, these four strategies, extend to larger numbers, and that's our biggest goal. We want to learn them here with these less sophisticated numbers, develop them in students so that they ping then with larger and larger, and then when we go to decimals smaller and smaller numbers. Absolutely.

Kim: 26:26

Yep.

Pam: 26:26

Yeah, super.

Kim: 26:27

So, we're going to wrap up because I think we talked for a while. But in our next episode, we're going to dive into what you can do to build things, tangible things that you can do with individual kids and with classroom activities. So, you're not going to want to miss that one. Join us in the next episode.

Pam: 26:43

Yeah, so in this episode we really talked about what the relationships are, the strategies are to build. Let's actually do it in the next episode. Ya'll, join us there. It's going to be great. 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!