Pam  0:01  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned mather.

Kim  0:09  
And I'm Kim, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching. 

Pam  0:16  
Y'all, we know that algorithms are nice achievements, but they're terrible teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop. Hang on a second. The way I said that. "They're nice achievements." That could sound like it's nice for kids to achieve doing them. It's not what I mean. I mean that it was nice that we have historically created them. That's an achievement, not using it. Okay, sorry. Keep going.

Kim  0:45  
In this podcast, we help teach math-ing, building relationships with your students, and grappling with mathematical relationships. 

Pam  0:52  
We invite you to join us to make math more figure-out-able. And there's no ADD happening in this podcast today. 

Kim  0:58  
Well, my foot's asleep. I was going to stand up. 

Pam  1:01  
Well, do you want to stand up? 

Kim  1:04  
No. Well, I have to wait now for my foot to... 

Pam  1:06  
I am actually standing. I'll stand up for you. I'm standing up.

Kim  1:09  
I need a standing up desk. I'm jealous when I hear the button...rrrrrr.

Pam  1:13  
You can hear it go up and down.

Kim  1:15  
Yeah. 

Pam  1:16  
I don't know how I ever lived without it. I'm... Yeah. I'm definitely not 50/50. Honestly, when we record, I'm usually standing up. If I do a webinar or something, I'm usually standing up. There's just something about the energy is different. I don't know. I never taught sitting down. Do you know I had a teacher at the university level that taught the entire class period sitting at his desk?

Kim  1:37  
I had a high school math teacher that was like that.

Pam  1:39  
Gah, come on, people. It wasn't physical. It wasn't like he needed that physically. Like, I get it if someone's, you know, physically can't. But. Anyway, energy. Energy is a good thing. How did I get off on that? Alright, Kim.

Kim  1:52  
I don't know. 

Pam  1:52  
Here we go. So, I'm going to ask you a question. We're going to start this episode with a math question. Everybody, think about it. Make sure you pause the podcast. Think about it a little bit. And then we're going to kind of build off of that for the rest of today. So, our question is 39 times 25.

Kim  2:16  
Mmhm.

Pam  2:17  
Alright. So, pause the podcast. Everybody think about it. Kimberly.

Kim  2:17  
Pamela.

Pam  2:17  
Will you share? 

Kim  2:17  
Yeah, I was going to say. You want me to solve this?

Pam  2:18  
Yeah. Will you play a little bit? Like, is there any way I could get you to just start verbalizing what's happening in your head? 

Kim  2:24  
Yeah, yeah. Okay, so you said 39 which screams Over strategy to me. 

Pam  2:29  
Okay.

Kim  2:29  
Something with a nine. So, the first thing I thought about was I'm going to solve 40 times 25.

Pam  2:35  
Okay. 

Kim  2:36  
And then back up. So, I know 4 times 25 is 100. Which means 40 times 25 is 1,000. 

Pam  2:43  
Nice.

Kim  2:44  
But I have one too many 25s, so I just wrote 1,000 minus 25, which is 975.

Pam  2:50  
1975. Oh, 1,000. Wow. 

Kim  2:50  
1,000 minus 25.

Pam  2:50  
Yep, I have an extra 1 on my... Okay, 975. Got it. 

Kim  2:54  
Yeah. 

Pam and Kim  2:56  
Okay. 

Kim  2:58  
I also... I'm assuming I can play some more. I also like...

Pam  3:03  
Yep, keep playing.

Kim  3:04  
Okay, I also like 25. I think about a fourth.

Pam  3:09  
Okay.

Kim  3:10  
So, a fourth of 39 and then scale.

Pam  3:14  
Okay.

Kim  3:14  
So, half of 39. I kind of thought about Over two, a little bit. Half of 40, would be 20. But it's a little bit less than that. So, half of 39 would be 19 and a 1/2.

Pam  3:27  
Nice. 

Kim  3:28  
And then if I want a fourth of that, I actually need another. I need to halve the 19 and a 1/2 again. And so, I'm going to do 18. If I was halving 18, that would be 9. And then half of 1 and a 1/2 would be 75. So, I think a fourth of 39 is 9.75.

Pam  3:53  
Yep.

Kim  3:54  
But I'm going to scale up times 100, and so it's the same 975.

Pam  3:59  
Nice, nice. 

Kim  4:01  
Yeah. 

Pam  4:01  
I like it. Alright, do you want to play anymore?

Kim  4:05  
Well, I was going to say, I'm also thinking about...

Pam  4:08  
Never stop Kim from playing.

Kim  4:09  
You gave me that option. You know, it's, it's like some of the same things are going to come out. But I was also thinking about like Double Halve. So, like halving the 39 a couple of times and doubling the 25 to get to the same 9.75.

Pam  4:23  
100 x 9.75, yeah, that's nice. 

Kim  4:26  
I'm actually wondering right now if there's a major strategy I would not want to use. Like, that's what I sometimes think about. 

Pam  4:33  
Would you ever think about 25 times 4 is 100, and then 39 divided by 4? And then think about...

Kim  4:41  
Say that again... Wait, you said? I'm writing it down.

Pam  4:42  
Sorry, so quadruple, quarter. 

Kim  4:44  
Oh, yeah, yeah. Yeah.

Pam  4:46  
But then you're kind of thinking about quartering 39. I guess that's how you thought about it before you halved it, and then halved again. 

Kim  4:55  
Yeah.

Pam  4:55  
I was thinking about 40 divided by 4, subtracting 1 divided by 4.

Kim  4:57  
Oh, that's nice. Yeah. 

Pam  4:58  
Okay.

Kim  4:59  
Yeah.

Pam  4:59  
Just kind of threw me more into the 40 than the half, half. Half of...

Kim  5:04  
Yeah.

Pam  5:04  
Yeah. Okay, nice. Alright, so lots of nice things that you could play with that problem. Listeners, would you consider that if you grew up learning math the way I did, you might have seen the problem 39 times 25, and you might have said, "Okay, I've seen this problem before. I've been taught how to do this. I know how to do this. I'm going to line up." I might write the 39 down first, and then the 25 underneath it. Then, I draw a line, and I put a little x off to the side. And then I say to myself, "Now, I got to do a bunch of single-digit multiplication." So, now I'm thinking about the smallest numbers in the problem, the least consequential numbers. And I'm thinking about 5 times 9, and then I'm thinking about 5 times 30, or 3. Wow, I can't even do it out of context anymore. 5 times... Oh, that's funny. My daughter's calling. Hold on just a second. Okay.

Kim  5:59  
Hey, Abby! 

Pam  6:00  
I only have that set for certain people to be able to be able to break through when I have it on.

Kim  6:13  
Yeah, sure.

Pam  6:13  
I wonder if that means she's called twice. Anyway, okay. So, then I might have to do 5 times 3. But then I have to do the, you know like, write the numbers down next to each other for that first line. Then, they have the magic zero. Then, I'm thinking about 2 times 9. I'm carrying that 1 from 18. And then I'm thinking about 2 times 3. Or really 20 times 30. Whatever. So, there's a bunch of single-digit multiplication that I have to do. And then I have these two lines that I have to add together. And so, now I have a bunch of single-digit addition that I have to do to add those two lines together. And I would end up with 975. Yeah?

Kim  6:37  
Yeah, mmhm.

Pam  6:38  
Okay. If you're thinking that's what math is, you got to know a bunch of single-digit... Well, first of all, you got to know which algorithm you're going to do. 

Kim  6:51  
Mmhm.

Pam  6:51  
Then, you got to know a bunch of single-digit multiplication. 

Kim  7:00  
Mmhm.

Pam  7:00  
You might have to think about whether you're going to line them up or move the decimal, if you had decimals in there.

Kim  7:01  
Yeah. 

Pam  7:01  
Then you have to do a bunch of single-digit addition. 

Kim  7:01  
Yeah. 

Pam  7:01  
Okay. What if you're thinking more like the way Kim is?

Kim  7:08  
Mmhm.

Pam  7:09  
What kind of mental actions did Kim need to know?

Kim  7:14  
Mmhm.

Pam  7:15  
Kim, what are some of the things you wrote?

Kim  7:17  
Yeah, it's interesting. As you were talking, I didn't think about a single-digit fact other than 4 times 25 helps me get 40 times 25. So, really, what was more important there was the connection between 4 times 25 and 40. So, like, I think I was thinking about some scaling, and I was thinking about, like the first thing I thought about was when I saw the problem, it was like I could make a different, related problem. So, that was like kind of underlying that as some properties, like some distributive property. Like I can, I can use what I want to know. And I was looking for patterns and relationships. Like, I was listening to describe the things that you would have been told, "This is what you think about," and it was so much less of that and more like, "What do I notice? And what do I want?" More choice and stuff. So.

Pam  8:10  
Well, and maybe I can mention that if it's that first way, the way I was taught, then I was thinking about things like, "Okay, since I'm going to have to do all these single-digit facts. Oof, I really don't want to get bogged down in skip counting, so maybe I'll just memorize all these single-digit facts."

Kim  8:26  
Yeah.

Pam  8:26  
Like, as soon as you've done 1-29, Odd, 29 times, you get this feel of like, "I just better know these multiplication facts because this is just really a monotonous." But it's... What's the word I want? It's I'm getting bogged down because I don't know them. And I don't want to do that, so I better memorize them all. And also, I don't want to be the kid who doesn't remember all the steps. So, I wonder how many of us kind of have this feeling tone ingrained in us that math is about mimicking these steps. And if I'm going to mimic those steps a bunch of times, then I want to know which steps to do. And I also want to know all the pieces in there. I want to have those memorized, so I don't get bogged down. I don't want to be the last one done with 1-29, Odd. You know like, I want to be able to like whip through them. And maybe that's why we hand kids the multiplication tables because they're just getting so bogged down they can't do enough. So, here, let's just help them retrieve them. 

Kim  9:23  
Yeah.

Pam  9:24  
But if... Kim, I didn't hear any of that in what you were saying. I almost felt more from you curiosity and hmm.

Kim  9:30  
Yeah, for sure. 

Pam  9:31  
How do I  use what I know to...

Kim  9:35  
It's more of like what do I want to do with this particular problem? 

Pam  9:38  
How do these numbers speak to me?

Kim  9:40  
Yeah.

Pam  9:41  
What's occurring to me right now that I could mess with well. Yeah.

Kim  9:45  
And, you know, this, this extension of like what would I not want to do? Like, I find myself asking myself questions like, "What would not be good here? What would...?" It's, so, yeah. Curiosity. And I think this is bringing up for me the idea, the conversations that we sometimes hear from people who go, "I'm not opposed to what you're suggesting. I'm not opposed to this idea."

Pam  10:11  
That's cool.

Kim  10:11  
Yeah. "It's cool." Like, "Yeah, I'm not opposed to it." But I think there's this bit where
maybe teachers, or leaders, or parents, or pick a whoever, kids, say that's cool for them. Like, yes, I have students where it's really good for them. It's good to extend them to let them think about strategies. But I do have some kids who don't know the basics, and so they're not. They can't. I'm not going to talk to them. And it's this interesting thing when you talk about if you grew up thinking math was this or if you had an experience where you picture math like this that maybe there's something to the viewpoint that teachers, or kids, or parents, or leaders have where they go like what really is math? And who can have the math that we describe? 

Pam  11:12  
And I think one of the things that you said to me when we were talking about briefly before this podcast was, is it possible that we've got a whole generation or generations of people who say, "Yeah, I see what you're doing. That was kind of cool. I had to think about that."

Kim  11:30  
Yeah.

Pam  11:30  
"I really had to like put some mental energy into like seeing the connections you were using. And sure, that was kind of hard for me. If that was hard for me, and I already memorized all those basics."

Kim  11:43  
Mmhm, for sure. 

Pam  11:44  
Then there's an order. That they're presupposing, an order that we've got to do this rote memorize stuff, that one way of doing the problem first, before kids can access the kinds of things that you were just playing with. 

Kim  11:58  
100%. We hear that a lot in training. Mmhm.

Pam  12:00  
And we're suggesting that that order that you think is true isn't actually necessary. 

Kim  12:08  
Mmhm.

Pam  12:08  
That we get why you would think that. I thought that. I was the one that was like, "Okay, I got all this down, and now you're showing me this cool stuff. So, then, surely, if that's kind of hard for me, surely it comes after what I've done."

Kim  12:20  
Yeah.

Pam  12:21  
So, listeners, we're asking you to consider it doesn't have to. In fact, that's not the best order because many kids will never make it there if you have to slog them through the stuff that we're suggesting isn't even math-ing. It's rote memorizing and mimicking. Yeah, go.

Kim  12:37  
It's a tricky thing. When that is what you've grown up in, and maybe that's what you've taught for a really long time, it is a tricky thing to wrap your head around the idea that maybe what we've experienced wasn't not only the only way, it was maybe not a preferred way, a like more helpful way. But that's not to say that we don't think that there are some basics. I mean, we're not saying, you know, you just raise this idea and magically kids are going to know strategies. We do think that there are some foundational relationships that we want to develop with kids. They're just probably not the foundational relationships that have been considered the basics, you know, forever and ever and ever.

Pam  13:28  
Yeah, and I really just kind of want to land on that for just a second if you don't mind. We hear a lot of people say, "No, they got to get the basics down first before you can do all this hoity, toity, this higher flyer stuff." What we're suggesting is, sure, but we vastly disagree on what those basics are. 

Kim  13:45  
100%.

Pam  13:47  
It's not the basics that I just walked through and I was like, "Okay, what do I need to know in order to do this algorithm?" You're like, "If they don't need to know the times tables..." Not the way you're thinking. So, what are the basics of math? If the goal is to rote-memorize and mimic, it's one set of things.

Kim  14:05  
Mmhm.

Pam  14:05  
But if it's what Kim was just doing, if it's what we're proposing here that math is actually figure-out-able. 

Kim  14:11  
Yeah.

Pam  14:12  
There are underlying relationships that are the goals to develop, so that we can get kids to develop these major strategies, and they're actually reasoning the way mathematicians actually reason. 

Kim  14:22  
Yeah. Yeah, because we have taught them about getting answers. Like, we in classrooms have spent a lot of time teaching kids how to get answers. And what's unfortunate, an unfortunate side effect is that we have given them maybe a cheap and easy way to get answers, and then... Like, I was just saying this yesterday, if you gave me a quick-ity quick thing that could make me be super, super strong, I'm not going to the gym on a regular basis. Like, I'm going to like give me the quick-ity quick that's going to build my muscle. But we've given them cheap ways to get answers, and then they don't want to think.

Pam  15:05  
Ooh, ooh. Can I see if I can do this? 

Kim  15:07  
Yeah.

Pam  15:07  
It's almost like you said if you give me a way to move that barbell from  the rack up there, then you wouldn't take that. You wouldn't use the, gol, forklift or whatever to move the barbell because all it did was move the barbell.

Kim  15:24  
Right.

Pam  15:24  
Like, you didn't gain any muscle. And what we've done is we've had generations now of math teaching where kids just got answers, but they never move the barbell themselves. They didn't ever get stronger. They got stronger at dumb things.

Kim  15:38  
But we have to help them understand. Parents, teachers, kids, we have to help them understand that the goal is not moving the barbell. The goal is to strengthen you, so that you are capable of moving the barbell. 

Pam  15:51  
Oh! We got to quote that one! Put that one out on social media. I think there's something really, really right about that metaphor. And I hope everybody's sort of getting... Like, I get asked a lot from leaders where they're like, "Hey, there's this novice expert argument where people are saying in order to become an expert, you've got to know the basics." And you and I both take a deep breath and go, "Yes, but define the basics."

Kim  16:15  
Right, right.

Pam  16:16  
So, Kim, could we take a minute here. And what would it look like to start developing the basics that kids would need to be able to do some of the playing that you were doing?

Kim  16:26  
Yeah, I feel like we've, we had a whole episode about this. Maybe I'm remembering wrong, but we can link to some. But quickly a few. Like, we need kids to understand like where numbers are in space and time. Like, how are they related to the numbers around them, like one more, one less, and 10 more, 10 less. Halving. Doubling Yeah, like what is a one place value shift up or down? Like, 10 more or 10 less, times 10, divided by 10? 

Pam  16:56  
Can I go with fractions for a sec?

Kim  16:58  
Yeah, for sure. 

Pam  16:58  
So, not just the number of parts shaded out of the total number of equal parts. But also so like three-fourths is not just 3 out of 4 equal parts, but it's also three 1/4s. 

Kim  17:09  
Yeah. 

Pam  17:10  
And we could also think about it as three-fourths of something.

Kim  17:14  
Mmhm.

Pam  17:14  
In the filming that we just recently did, we really worked with students who kind of had this basic idea of 3 out of 4. Kind of. And then they would do a bunch of rules with it. But we really worked hard on getting to think about it as 3 out of 4 and three 1/4s. And that was... Like, working on that? Boy, we saw some huge growth in just that one little idea kind of being emphasized in the work that we were doing. Yeah.

Kim  17:39  
Yeah.

Pam  17:41  
Yeah, and we could go on about like some of those other things. How could we develop? So, let's do a couple. Let's just talk through a couple of Problem Strings that could help develop the basics that kids would need, so that they could do some of the strategies that you used to begin when we...

Kim  18:00  
Yeah, when we started. Yeah.

Pam  18:01  
Yep, yep. 

Kim  18:01  
So, one of the strategies that I thought about was the Over strategy.

Pam  18:06  
Mmhm.

Kim  18:07  
And in that, I did some scaling as well. I was going to say I did some. So, a Problem String could be what is 2 times 25? Kids think about that, then they think about what is 4 times 25. So, the idea of doubling. And then using that scaling relationship to get to 40 times 25, so that you know that early relationship to the scale. And then the next problem could be 39 times 25. So 2, 4, 40, 39 to get Over. 

Pam  18:40  
Lots of place value. Like, you said, doubling, scaling times 10, the idea of what really happens with times 10. And then thinking about things like how can I think about 39 times 25 based on 40 times 25?

Kim  18:53  
Yeah.

Pam  18:53  
I know the Universal Screeners for Number Sense, one of the questions they asked that I just love. It's something like this where they give the kids the question. They say to them, "If we know that 4 times 30 is 120 how can you use that to help you think about 4 times..." What did I say? 4 times 30? "...4 times 29."

Kim  19:08  
Yeah.

Pam  19:09  
And the number of kids who just redo the problem. 

Kim  19:13  
Yeah, yeah.

Pam  19:13  
They can't think about how those two problems are related. Like those are the basics. That's part of the basics that we're trying to get at. How is 6 times 7 related to 7 times 7? Like, if you got six 7s, how's it related to seven 7s? That leads into this. If you got thirty-nine 25s, how does it relate to forty 25s? 

Kim  19:30  
Yeah.

Pam  19:31  
Those are part of the basics that we're trying to help kids really own, and they don't need that algorithm and all that stuff to get there. Go ahead, Kim. 

Kim  19:38  
Yeah, this is probably going to (unclear) a little bit, so give me a minute. I am thinking about an experience where my son was writing. We're saying "basics", but it's almost like it's basic, but it's also generalized. It's like what we're talking about here is this idea of teaching them Over, which is more general. I had my son help write a game for one of his former teachers who wanted a different version of something that I had. And he was messing for a little bit, and then he said, "You know what? Like, instead of just writing a particular one, I'm going to give her a generalized thing that she can manipulate in this way, so she can make as many of them as she wants."

Pam  20:22  
Make versions of the game board. Mmhm.

Kim  20:23  
Right. And I was like, "Well, aren't you smart?" Like that's like what we're wanting to develop is underlying properties and underlying strategies, and so that they're not... The thing isn't about 39 times 25. It's about solving problems. It's about categories of problems. 

Pam  20:42  
Nice. And it reminds me of when we very first started, and I said 39 times 25, and you were like, "Oh, 39. It's just begging for 40. It's like begging." That's this generalization that we want to help. It's not about 39. Yeah. It's about like any 9, kind of developing really neighborhood and nearness. Like, how close are you to something that, "Ooh, that's... I know I can work with that, and this is close."

Kim  21:06  
Yeah.

Pam  21:07  
Yeah, I like... I'm going to put a little bit deeper on your... It's not about... I think what you're trying to say, in one way maybe, is it's not about learning, memorizing, rote-memorizing, a bunch of discrete, non-connected, non-related like multiplication facts so that then I can do them in this algorithm, over, and over, and over. That's the old. That's the way I described. You're saying we really want to develop the sense of how can I use what I know? How are these numbers speaking to me? And specifically, in this case, Over. Like, what's... that 39 speaks to you as Over, can I do something with that? Ooh, yeah. With this problem, I can think about forty 25s, because I can think about four 25s. Or if I have to, I can think about two 25s. Let me give you an example of another. You used a different strategy where you thought about quarters. 

Kim  21:59  
Mmhm.

Pam  21:59  
So, yeah. Ideally, we would do work with kids, so that they get to a point where when they see 25, a thing that occurs to them to wonder if would be helpful here is to think about quarters. So, can I think about one-quarter of 40? Because the problem is 39 times 40, can I think about a quarter of 40? Then, can I use that to help me think about a quarter of 1? And if I've got a quarter of 40 and a quarter of 1, how does that help me think about a quarter of 39?

Kim  22:24  
Yeah. 

Pam  22:24  
Now, I'm sort of melding over 41, 39. But this idea of finding a quarter of each of those.

Kim  22:31  
Yeah. 

Pam  22:31  
And if we do that, we high dose kids with these important relationships, then kids are reasoning like mathematicians. They're not stuck just in the drudgery of rote memorizing a bunch of nonsense. It's not nonsense. A bunch of discrete facts that don't have to be rote memorized because we can reason to find them. 

Kim  22:51  
Yeah, I think, you know, we started talking about this because we have heard so many times, "I'm not opposed to what you're saying, but these kids aren't ready for it. These kids can't." And I think what we're kind of saying here is that might be an assumption on your part.

Pam  23:10  
Is it possible you're making that assumption because you were taught to memorize all this stuff and repeat all these algorithms, and then you see what we're doing, and you say, "Oof, I had to think about that a little bit. Okay. So, obviously you got to sledge through all this stuff before you get to this."

Kim  23:27  
Yeah.

Pam  23:27  
We're asking you to consider that's not actually the way it has to be. Consider that you can actually get kids to build mathematical relationships and mathematical behaviors where they can reason like Kim just did, never having rote memorized and mimicked all that stuff that you had to slog through.

Kim  23:47  
Yeah.

Pam  23:48  
And then more kids can do more math more successfully for longer. Whoo! What a good plan. Y'all, thanks 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. Thanks for spreading the word that Math is Figure-Out-Able. And, hey, Gerard, thanks for the question that prompted us to do this episode!