[00:00:01.380] - Pam
Hey, fellow math-ers! Welcome to this podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned math-er.

[00:00:11.410] - Kim
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.

[00:00:19.170] - Pam
Y'all, we know that algorithms are super cool historic achievements, but they are 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.

[00:00:33.380] - Kim
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

[00:00:40.180] - Pam
Y'all, thanks for joining us to make math more figure-out-able. Kimberly.

[00:00:45.710] - Kim
Hello, Pamela. How are you?

[00:00:48.050] - Pam
My life is full of wonder, and I'm always ready for new opportunities. Why are you laughing?

[00:00:55.270] - Kim
Because I know. I know what that means.

[00:00:59.150] - Pam
Do you think that means I'm feeling a little full today?

[00:01:01.100] - Kim
You're full. Your brain is full.

[00:01:04.210] - Pam
You know, I took a week off and we went... Oh, have we told? Have we said this on the podcast?

[00:01:10.420] - Kim
I don't think so.

[00:01:10.450] - Pam
I recorded the audiobook for Developing Mathematical Reasoning.

[00:01:13.480] - Pam
Yeah.

[00:01:13.720] - Pam
Avoiding the Trap of Algorithms, K-12. Super, super fun. To do that, I took a few days off. Well, 2 weeks off. So, it really wasn't "off", right, because I was recording the audiobook the whole time. Why was I... What? Oh. Because, yes. Then when I got back, I had 2 weeks worth of meetings to catch up on. So... (unclear).

[00:01:29.760] - Kim
Yeah.

[00:01:30.880] - Pam
I've been in meetings upon meetings about, you know, it's just like the kind where you don't get a breath, you know. And then like a dork, I just schedule them so one ends on the hour and one begins on the hour. That's never so...

[00:01:42.410] - Kim
5-minute window, 5-minute window.

[00:01:44.030] - Pam
Need at least a 5-minute window. Okay, note to self. Come on. 5-minute window. Alright, Kim, today I am kind of excited to... This is going to be a little bit of a... What's a good word? Beat out conversation. Not in a bad way.

[00:01:56.700] - Kim
Are we going to wrestle?

[00:01:58.450] - Pam
Well, we're for sure going to grapple. Like, we're, yes. Grapple, wrestle. Like, you know, I'm really intrigued with language and...

[00:02:07.670] - Kim
Yeah.

[00:02:08.780] - Pam
...communication and actually understanding what each other means. And golly, you and I have known each other for a long time, so it's always fascinating to me when you go, "Wait, wait. What?" And then we kind of get under a word and we're like, "Oh, that's actually not. I didn't think that part of it," or whatever.

[00:02:21.710] - Kim
Yeah, yeah.

[00:02:22.450] - Pam
So, I've been in some situations lately where a kind of meaning has come up. And so, y'all, when you listen to this episode today, be aware that it's not like we're trying to force my definition on you or Kim's definition on you. We're trying to talk about a concept, a thing, like an idea, and put some words to it, so that we all kind of get more clarity around it, so that when we can talk about it in math education, and math teaching, and working with kids, and working with teachers that we can communicate. So, it's not about who's right or wrong or who's using the best word or anything like that. It's about let's communicate. Is that a fair way to start this?

[00:03:02.090] - Kim
I think so, yeah. Because if you're hearing somebody say something or you're listening to a talk and you think you know what they mean, but they actually mean something different, that not only could be confusing, but it could change the course of how you hear the rest of the information that they're sharing.

[00:03:20.020] - Pam
Totally. Yes, yes, yes. Okay, so I thought I'd start with just a brief experience that happened quite a while ago. I was doing some training in Texas for leaders around the state of Texas. It was a wonderful opportunity I had. I had just written Building Powerful Numeracy for Middle and High School Students. And I did eight days. Not altogether. But I had a group of leaders around the state that I got for I think four days in a row, and then maybe two and two. Or maybe it was two, four. I don't remember. It was a while ago. And we were doing... We were building powerful numeracy, so I was literally doing addition and subtraction, multiplication, division Problem Strings with leaders, so that then they could turn around, and they could work with their teachers to build powerful numeracy. Blah, blah, blah. In that, I was doing a subtraction Problem String for the Over strategy. So, like I was giving problems, and I was having people share strategies, so that they would think about a problem like, I don't know, something random like 63 minus 49. They might go, "Well, I know what 63 minus 50 is." A little too much, go a little Over. And if I know that 63 minus 50 is 13, then I can think about only backing up, not 50, but only backing up 49. You know, just don't go as far. So, you kind of have to, you know. If I've jumped back 50, but I was only supposed to jump back 49, I can jump up 1. And so, really the answer would be 14. So, I'm looking for that. I give the problem, 63 minus 49. And this gal raises her hand, she goes, "I mean, I'm just kind of weird, but I just see 7s." And I look at the problem, and I like, "There's not a 7 in that problem, sweet participant." And she's like, "It's just like I see nine 7s and seven 7s." And I kind of put my hand up. I was like, "What?" "Nine 7s minus seven 7s. So, it's just two 7s. I know two 7s is 14, so the answer is 14." Kim, in that moment, I was like, "Oh, that is clever. That is really clever. Ooh." Kind of the "ooh" factor where I was like, "Is that a thing? Can you like factor?" Now, I know we've done this strategy on the podcast before. If you haven't heard of it, go check out the factor to subtract or factor to add Problem Strings that we've done on the podcast. But in that moment, I remember just having this, "Ooh, that is really clever."

[00:05:36.060] - Kim
Yeah. Do you think that it was clever to you at that time because it was something you hadn't seen before?

[00:05:43.720] - Pam
Something I didn't do, I didn't play with, I didn't own those relationships?

[00:05:46.260] - Kim
Yeah, yeah. Because I don't know where you were at the time, like in your development.

[00:05:51.640] - Pam
I had never thought about factoring numbers to subtract, ever.

[00:05:56.300] - Kim
Yeah.

[00:05:56.550] - Pam
And in that moment, I had to ask myself is that generalizable? Like, will that ever? Is that just one problem? It'll never happen again.

[00:06:03.910] - Kim
Yeah.

[00:06:04.170] - Pam
Now, I write whole Problem Strings with it.

[00:06:05.690] - Kim
Yeah, yeah, yeah. So, you said it was clever, but what you didn't say was interesting. So, you might've also thought it was interesting, but the word you  just used is that you thought it was clever. So, do you think that you thought it was clever because it was new to you? Or that the strategy itself is clever?

[00:06:26.500] - Pam
I think that's a...

[00:06:26.850] - Kim
Like, even now as you've seen it and written strings.

[00:06:31.560] - Pam
I think it's a really good question. I think the strategy's clever. Now, we need to define "clever". And this whole episode's going to be kind of like what is clever, a little bit about that. But...

[00:06:42.160] - Kim
Yeah.

[00:06:42.400] - Pam
Because I want to kind of pull out what you think and what other people think.

[00:06:45.590] - Kim
Yeah.

[00:06:45.780] - Pam
It definitely was new. It definitely was new. And there are times where things are new. I'd like to talk about what makes it clever, even if it isn't new.

[00:06:55.640] - Kim
Okay.

[00:06:56.010] - Pam
Is that?

[00:06:56.330] - Kim
Okay.

[00:06:56.880] - Pam
Yeah, I feel like you're going to ask something. Go ahead.

[00:06:59.050] - Kim
Well, before we define "clever", can I tell you one thing that I'm thinking about right now? Because I... We... Maybe we should define "clever" first. So, tell me if we should. But listen, I, we have been in situations, and we've even... Gosh, when I was brand new to working with numeracy, I would say things like, "Solve this problem however you want, and then when you solve it, I'd like for you to solve it a second way."

[00:07:30.460] - Pam
Mmhm.

[00:07:30.510] - Kim
Or maybe like the piece of the directions on the assignment would say, "Solve the problem in two different ways." And you and I...

[00:07:40.390] - Pam
I remember papers that had two columns. Solution 1, solution 2. Mmhm.

[00:07:47.070] - Kim
Yeah, yeah. So, the word "clever" has come up in lots of situations with you and I before because we have said things like we want people to seek for clever. And sometimes that happens when you've solved a problem the way that you are naturally kind of like... Naturally. The way that you... Kind of inclined. Mmhm. Yeah, the first way you think about it. And then you can like sit back, and then you can tinker. Kind of the way we did in the Problem String last week.

[00:08:16.570] - Pam
Yeah.

[00:08:16.900] - Kim
Like, I did what felt like my first inclination, but then we kind of sat back a little bit, and I looked for even more.

[00:08:23.390] - Pam
Other relationships that could come into play. And when you do that, you're kind of curious. I wonder if I can find one that feels clever.

[00:08:31.170] - Kim
Right. So, I think there's a difference here between, "Solve every problem two ways" and "Solve it however you want. But maybe then look for something also clever." So, let's define clever, but I want to come back to this idea of solving things in a couple of ways because I think there are people who are suggesting that and asking for that because we want kids to have lots of strategies.

[00:08:56.570] - Pam
Flexibility, right?

[00:08:57.450] - Kim
Maybe it's not getting them the results that they want. Yeah.

[00:08:59.830] - Pam
Yeah, yeah. So, one of the standards lists that we want kids to be fluent. And part of the definition of fluency is that we want them to be flexible problem solvers. And so, I think there is an idea permeating around that flexibility means thou shalt solve the problem in more than one way. And so, then it becomes a directive, a forced. And what we've seen is that when you force in order to get a grade. I'm going to mark it wrong if you get a correct answer, but you only did it one way. Then we get all these misunderstandings, misconceptions, miscommunications, really, where a parent might say, "You can't mark this wrong. The kid got the answer right." And the teacher then kind of has to say, "Yeah, but they have to solve it in two ways." And the parent's like, "Why? Like, they got the answer."

[00:09:46.270] - Kim
Yeah. And it just...

[00:09:47.720] - Pam
Yeah. And... Sorry, did you want to say something?

[00:09:50.080] - Kim
Well, I was just going to say at that point, it's just performative. It's...

[00:09:53.560] - Pam
Yeah.

[00:09:53.640] - Kim
You did it one way. Now, do for me another way.

[00:09:55.900] - Pam
Yeah. It doesn't feel very...natural, or helpful, or desirable. It's just like you're doing it just to do it. And then we find that when we're in that sort of performative, commanded, "I have to solve it two different ways to get a grade," especially then we find kids will solve it the best way they can because we're all lazy, so we do the best thing that we can think of. And then often kids will do something less sophisticated, less clever, less... Like they'll draw pictures that are super un— um.

[00:10:34.710] - Kim
600 tally marks plus 300 tally marks. Yeah.

[00:10:38.580] - Pam
There was a video that went viral shortly after the Common Core came out where the parent held up the paper, and it had a ton of circles. The problem was something like 600 divided by 7 or whatever. And there were 7 circles and then 600 tally marks, and it all divided out. And it could be as a result of a well-intentioned, well-intentioned sort of instruction to say, "You have to solve it more than one way," where a kid's like, "Okay, well, I've seen this way. I'll do that." That same thing can happen when we say, "You have to show your work." If the kid's never seen their thinking represented, actually their thinking, what they're really thinking about, and you say, "Can't use an algorithm. You got to show your thinking." And the kid's like, "I don't know how to show what's happening in my head," then they very often revert to something they've seen, which it could be a less sophisticated, counting one by one, not what we're... The whole thing when we're trying to mentor people to math, we don't want to force them to do less sophisticated math, right, just to get a grade, or get the nod, or get the okay. We want to actually be encouraging them to reason. I don't know. Do you want to add on any of that? Is that?

[00:11:52.610] - Kim
Well, when you said they'll revert to something less sophisticated. You know, if I've got a third grade student first 6 weeks of school, and I tell them to show me their thinking, but we're tinkering with something new, when I say model your thinking, they're probably going to be thinking about what they saw modeled a bunch of times in second grade because that's what they've had the most experience with. So, it's kind of this delicate balance where we want students to be able to represent and talk about their thinking.

[00:12:18.690] - Pam
Absolutely.

[00:12:19.230] - Kim
But it's this very delicate balance of we need to give them the tools to be able to do that and experience, you know, under a careful eye of an educator who can...

[00:12:30.160] - Pam
Pull out.

[00:12:30.670] - Kim
...be there just in the moment. Right.

[00:12:32.150] - Pam
Yeah, pull out their thinking, represent it for them. "When your brain does that, it can look like this." And they get that experience. I love that you said experience, so they're like, "Oh, when my brain does this, it could look like that." Now, then we start to say, "Hey, your brain's doing that thing. Can you represent it? Can you show me what it's doing?" Because you can tell me right now. I love you used to say this. I'll quote you right now. You used to say, "You could tell me right now, but this paper, I'm taking that home. I'm grading that at home. You won't be there to tell me, so what could you put on the paper, so I know what you're thinking?" It's a great way of sort of honestly helping kids realize that what we put on paper can be about communication. It can be, you know, "How can I understand what you are thinking about when you're not here to tell me or draw, you know, talk it out loud."

[00:13:15.450] - Kim
Well, and I think we're kind of getting off topic, but this is fun anyway. But in that moment, then when I take that paper home, and I look and see what they put, and I know that kid, then I go, "Oh, they're putting something that looks like this, and I need to help them put what actually is happening in their mind." I've talked to them enough. I know the way that they're thinking about relationships. And I know that this is not necessarily representing their thinking. I mean, we have this beautiful picture of my Luke when he was like 5. Has all this... This piece of paper from first grade, I think it was. Sarah Hempel, amazing. She took Luke's paper, which had all these drawings.

[00:13:58.630] - Pam
One by one.

[00:13:59.200] - Kim
She was like, "Hello, sir, I know that that's not what you're thinking about." And she represented for him and helped him see what he could do because she knew he wasn't, you know drawing and counting by ones.

[00:14:11.640] - Pam
He wasn't drawing the 14 squirrels and the 6 squirrels that were running down the tree and crossing them out, whatever. He did that after he'd figured out the answer because he didn't know another way to represent the fact that he was thinking about 8 plus 6 to help him think about 14 minus 6.

[00:14:25.980] - Kim
Yeah.

[00:14:26.480] - Pam
Yeah, nice. Something like that. I don't know the exact numbers. I think this all points to...

[00:14:31.700] - Kim
Let's define clever.

[00:14:31.960] - Pam
This points to why you and I are encouraging teachers to help students seek for clever, for clever solutions, clever strategies, clever ways of thinking, clever ways of using relationships. But what do we mean by clever? So, I could have this wrong. I'm going to say upfront. I've traveled a little. I have some friends in different countries. So, this is totally my take. And then we looked it up on the internet, and the internet disagreed with me. So, we'll give you both takes. When I use "clever", I use it different than the way I read it in Harry Potter. So, when I've spoken to a couple of British people, it felt like British people were saying a person is clever. They're smart. They're brilliant. They're intelligent. I don't use it... When I say, "Seek for clever," that's not go find the smart kid in your room. That's not what we mean. We mean encourage to be on the... To try to find, what, innovative, elegant, creative, slick strategies. It's less of a definition of who you are. Like, "You're clever. You're not clever." Like, Kim, would you call me clever?

[00:15:47.380] - Kim
I feel like it's a trick question, Pam, because I would say that you have clever ideas, and I think that you look for clever strategies. But, you know, we're talking about here that people... It's interesting because we're going to talk about how people have behaviors and we're talking about there's a behavior of seeking for clever.

[00:16:07.360] - Pam
Yeah. Like, trying to have ingenuity, trying to adapt, trying to be resourceful. Ooh, I just was talking to Canadians. Resourceful. That's funny. I think flexible fits in there. Earlier when I was talking about the really cool factor to subtract strategy, and I thought it was clever, and I was like, "Ooh. It's kind of that ooh factor." Like, one of the things that we seek for. We're doing a Problem String. We're working with kids. I'm even just chatting with a kid and they say something and I'm like, "Well, what about..." Or, you know, I just like, "Does that apply here?" I'm seeking for a kid to go, "Ooh, that's..." You know, or a kid to say something and the rest of the class to go, "Nice, ooh." When they do that, that kind of describes this thing that we're calling "clever".

[00:16:54.150] - Kim
Yeah, I think it's important that you said what you just said right now, that there is a kid who might say something and the rest of the kids go, "Ooh," because everyone in the class can provide that opportunity for the rest of the kids to say, "Ooh." Like, there are plenty of times where I do something that I go, "Oh, that was clever." And there are plenty of times where I go, "That was not very slick." You know like, we...

[00:17:22.020] - Pam
Like, you trudge through. You trudge through a problem. You do the best you can. You look back and you're like, "Wow." Oh, especially then if when you look back you go, "Oh, man! I could have just...!" Yeah, yeah.

[00:17:33.250] - Kim
Yeah.

[00:17:33.510] - Pam
Then, the that feels clever in that moment. Like, when you see the connection that you're like, "Oh, wow, look at that. That's..." Or in the midst of it, you're like, "Ooh! Ooh! Ooh!" And you want to share that "Ooh! Ooh! Ooh!" It's that "ooh" moment. That's the innovative. I'm going to say some words again. Elegant, slick, creative. The "ooh" factor. Ingenuity, adapting, using relationships, connections, being resourceful, flexible. Those are the things that we're sort of hoping for. We don't want to kind of wreck when we just go, "Okay, you have to solve it two different ways. You have to. You're not going to get a grade if you don't." Yeah.

[00:18:13.500] - Kim
Yeah, and so in classrooms if we talk about this idea of there being a reason to seek for clever, if there's a reason to take a second look at your work, if there's... Now that you've... You know, you just said, "I have trudged through problems before." I have. Absolutely. And then I take a breath, I pause, I look back again. Because there is still learning to happen. There is still a reason for me to look back and go, "Okay, I have made it through one pathway. What now could I think about?" And maybe there's not something in that moment, or maybe there's not at all, but often I have found that I look back, and I go, "Oh, haha, that would have been more clever. That would have been slicker. That would have been, you know, more flexible." But I didn't see it in that moment.

[00:19:04.670] - Pam
And we're seeking to build the habit of mind you just described.

[00:19:08.440] - Kim
Yes.

[00:19:09.230] - Pam
We're seeking to build that in ourselves and in our students. Now, you might be thinking, "Really? When you're like on a test, you want them to look..." I mean, maybe. Maybe. You know, the only way I learned to check answers on a test was to undo. Or, you know like, do the other operation or whatever. But instead, I could look for other relationships to use. Math, math-ing involves, includes the mindset that Kim just described of solving problems.

[00:19:36.850] - Kim
I mean, I think life.

[00:19:38.310] - Pam
Well.

[00:19:38.550] - Kim
I think life does.

[00:19:39.500] - Pam
One would hope.

[00:19:39.520] - Kim
I mean, I think there are plenty of things I do the hard way, and then go, "Oh, that, okay. Noted for next time." I mean, I think that's a mathematical behavior, but it is also a life skill that we're building in students to say, "This is not the one and only way. Can I look back, and what next could I do the next time?"

[00:19:59.640] - Pam
And math...

[00:19:59.710] - Kim
And that's...

[00:20:00.140] - Pam
Keep going.

[00:20:00.690] - Kim
Go ahead. I was going to say that's drastically different than the always two ways. Solve every time on every assignment, always solve it two ways to show flexibility. I would much rather have students give it their best go and then instill this behavior in them, and then sometimes they share with me something that they've come up with that they thought would be more clever than the next time.

[00:20:23.610] - Pam
Yeah, this sort of mindset that math-ing is looking for relationships, and looking for connections, and trying to seek for clever And and that's only possible, I think, when Math is Figure-Out-Able. Like, if you're approaching math as rote-memorizable, then you're like, "I have to solve this in two ways. Golly, I have to memorize twice as much as I did before?" Like, that's not the perspective. And so, I think part of the reason we wanted to talk about "clever" here today was just to kind of infuse just one more angle of what it means for math to be figure-out-able. And I think it might be helpful to maybe just have an example. Kim, can you?

[00:20:59.860] - Kim
Oh. Oh, good. Yes. Because I grabbed an example. You shared the factor to subtract. You know, one of the fun things about MathStratChat. MathStratChat is on every social media on Wednesday evenings. And there are problems that you put out that really get people excited. And often, multiplication ones are really people chat a ton. We put out a problem, 48 times 52, one time. And this is a strategy that I think is incredibly clever. And again, for me, it's not because I didn't own it before or that it was new. Although, I did have that same reaction that you had in your story at the beginning. I went, "What?! You can do that?!" But I think it's a clever idea. For 48 times 52, people will sometimes think about the difference of squares. And so they'll think about, uh, 50 minus 2. Times 50 plus 2. And so that, for me, is a really clever idea that you would think about a multiplication problem as difference of squares. And it's pretty slick.

[00:22:07.810] - Pam
It's nice.

[00:22:08.290] - Kim
It's pretty slick. Pretty inventive. Definitely people are thinking flexibly because they look at and they go, "Oh, okay. How far apart are those numbers?"

[00:22:18.230] - Pam
Yeah.

[00:22:18.710] - Kim
It's one that I want to do a better job of seeking for. And so when people do it I go, "AHG!"

[00:22:25.360] - Pam
Having it occur to you. You want it to occur to you. When I say, "I want my brain to do that next time. I want my brain to have that inclination, that spark," to "Ooh, I wonder if I could look for difference of squares." Just to be clear. So, that 48 times 52. You said 50 minus 2 times the quantity 50 plus 2. So, 50 times 50, that would be 2,500. And then the minus 2 times 50 and the plus 2 times 50 add to 0. So, then you have minus 2 times minus 2 is minus 4. So, really you end up with 2,500 minus 4. Bam! And you've got 48 times 52. That's pretty clever. You know, in a way, both of the examples we used, I don't think that... I want to parse out that... I'm going to name a commonality between the two problems that we used as examples, and then suggest that doesn't... That you're clever doesn't have to include that commonality. One of the commonalities in our two examples is that you kind of use something you learn later to solve something you learned earlier. So, like I was solving a subtraction problem using multiplication facts. Like, factoring. Usually you subtract before you, you know, factor multiplication facts by thinking about nine 7s minus seven 7s. You were solving a multiplication problem using something we usually do in Algebra where we have a binomial times a binomial, and we distribute all the multiplication all the way out, and then we get a trinomial. These two examples we used have that properties, that description, that they fit. That you're using something you learn later to solve an earlier problem in a very clever way. I think you don't have to do that to be clever. I think there are lots of other ways to use relationships and connections that just seem to kind of, I don't know, be the straightest journey between two points, and you don't have to do all the extraneous. I mean, a very simple example would be 1,000 minus 998. If the only thing you've ever learned is an algorithm, when you learn that subtraction can be the difference between numbers and you're like, "Well, how far apart are 998 and 1,000?" You're like, "Whoa!" Again, not because it's new, but because it's... You don't have to trudge through all those steps. You can just reason using a connection that you know.
That can...  That's... Yeah, nice.

[00:24:39.960] - Kim
Yeah, agreed.

[00:24:40.930] - Pam
Cool. So, always two ways, Kim?

[00:24:45.420] - Kim
No. Not always two ways. I mean, if that's a goal you want to set for yourself, absolutely. Be on the hunt for thinking about a way to solve a problem because it's only going to help you to like, you know, build. If I see a multiplication problem, then I want to say to myself, is this a good one for difference of squares? But if I am assigning students work, I am not asking them to solve it two ways every time for a grade.

[00:25:10.110] - Pam
And you're not grading it if they don't. Like, that's not a part of the grade. Nice.

[00:25:14.770] - Kim
Yeah.

[00:25:15.320] - Pam
Alright, y'all, let's all seek for clever strategies. 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. Let's keep spreading the word that Math is Figure-Out-Able!