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

[00:00:10.920] - Kim
And I'm Kim Montague, 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.

[00:00:18.710] - Pam
We know that algorithms are amazing historic achievements, but they are not good teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

[00:00:32.240] - Kim
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

[00:00:39.140] - Pam
We invite you to join us to make math more figure-out-able. Kimberly.

[00:00:45.250] - Kim
Hello.

[00:00:46.460] - Pam
Somebody called me Pamela the other day while I was on a trip.

[00:00:48.860] - Kim
Did you think I was there?

[00:00:50.790] - Pam
A little bit. And I kind of wanted to turn right back to him and go, "Kimberly!" I don't know. You might be the only one that... Yeah. Or my mom. Anyway.

[00:00:59.699] - Kim
Yeah.

[00:01:00.320] - Pam
Alright, so we have kind of an exciting episode to get into today. You and I have been talking about this topic for a long time, in and out, lots of different ways. We think we have a new slant on helping people kind of understand what we're thinking about. So, as we were talking about this episode, there were two things that you brought up that were kind of random and not related until you said a little bit more about them. And I was like, "Oh, my gosh. That's totally. It totally fits." Alright, so can we just dive right in?

[00:01:28.360] - Kim
Sure, let's do it.

[00:01:29.720] - Pam
Okay. So, Kim, there's kind of these two phrases that you and I might kid each other a little bit about. I think you said them both, right?

[00:01:38.570] - Kim
Yeah.

[00:01:39.110] - Pam
And yeah, I'll just dive right in. So, Kim, write that down. Now, nobody else is laughing right now.

[00:01:45.950] - Kim
No.

[00:01:46.160] - Pam
They're like, "Why is that funny?"

[00:01:47.220] - Kim
Well, unless they've heard this story before.

[00:01:49.400] - Pam
Yeah, maybe, maybe. So, we have a video of Kim with Mikayla, a 7th grade student. And Kim's doing some amazing mathematical... Well, really, Mikayla's doing. Kim's doing amazing mathematical teaching and Mikayla is doing amazing mathematics. And in the middle of it, you say... Mikayla says something amazing, and you say, "Write that down." And just kind of the way you say it and the timing of it.

[00:02:12.690] - Kim
Very bossy.

[00:02:12.780] - Pam
Or whatever. A little bossy and a little bit like super directed, like super like focused. Like, there's no, "I invite you to write that down." None of that, right? It's like, "Do it!" It's like in that moment, bam. And she does. She writes it down, and then continues to do amazing things. And we have a conversation often when we show that video about cognitive load and how you don't have to hold it all in your head. And that maybe especially Mikayla, but for me. Let's just say, especially for Pam. Often, when I am math-ing, I can't hold it all in the ether. I can't just like sort of have it all floating around, whatever. But if I can write something down, then I could do more math.

[00:02:52.210] - Kim
Yeah.

[00:02:53.180] - Pam
And so, in that, you'll say, "Hey, if I can get Mikayla at certain points just write down what she's thinking about, then she has a... I was about to say easier. Is that the right word I want to use? It's more fluid.

[00:03:04.360] - Kim
Freer, yeah.

[00:03:05.780] - Pam
Freer. Ooh, I like that. Like, almost like... I picture my hands are like lifting a weight off your shoulders. Like less weighty. It's all of a sudden you're not weighed down as much, if you write that down.

[00:03:19.110] - Kim
Yeah.

[00:03:19.520] - Pam
But maybe what's noteworthy about that story... Well, no. So, let me tell the other story, and then we'll say why it's noteworthy. Because then you said, "Sure wish I knew something about quarters." As we were planning, right? So, it's not in the same video with Mikayla. This is in a different video. And both of these videos I showed semester after semester when I was teaching at the University of Texas, when I was teaching at Texas State University. I would show these videos where my students were studying you and studying the students and their math-ing. And in the middle of a video where you're working with 4th grade students?

[00:03:52.460] - Kim
Mmhm.

[00:03:52.710] - Pam
Kids were playing around with multiplying 125 times 36, 32. One of the two. 125 times...

[00:03:59.410] - Kim
36.

[00:03:59.660] - Pam
Let's just go with 36. Okay, 36. And the kid says something about quarters. And then a couple of the other kids in the group were kind of going off. And you said, "Hmm, sure wish I knew something about quarters." And it just focused the kids. It just helped them go, "Yeah, we could use quarters here. Like..." Because the one kid was kind of wanting to play with it. The others were going somewhere else. And you wanted to encourage them to go that direction. And so, it was definitely more of an invitation. "Hmm. Sure wish..." It was kind of you lobbing it out. "Sure wish I knew something about quarters." And man, they just grabbed it and decided to mess with quarters. And they came up with this brilliant strategy to think about 36 quarters and how that could play out with 36 times 25. And it was great. Okay, so why did we bring up? Why, Kim. You did. Why did you bring up these two random, "Write that down."

[00:04:50.580] - Kim
Well it's...

[00:04:50.700] - Pam
"Sure wish I knew something about quarters." Yeah, go.

[00:04:52.300] - Kim
It's funny because just now you didn't say to me that you thought it was like random that I was saying that. You must have just stayed quiet when I raised them because you didn't look at me like I was crazy. So, you know, we were talking about this idea of explicit teaching and inquiry-based. And like we were just raising up these words. And these are two phrases that you and I say to each other kind of out of nowhere sometimes. When we talk about the idea of lobbying something out, every once in a while one of us will say it to the other people, and I think the people in the room, you know, either on our team or around us, probably are like, "What are you talking about?" But it's like the phrase means something to us more than just it did in that moment. So.

[00:05:37.030] - Pam
Because we have a common experience around that phrase. And it represents... You just said lobbing out. Can I say a little bit more about that? I think I interrupted you.

[00:05:45.440] - Kim
Well, I was just going to say what the two phrases mean to us.

[00:05:48.800] - Pam
Oh yeah. Go. Yeah.

[00:05:49.120] - Kim
So, the first one, when we say, "Write that down," this is more about like, I know what you need or I know what I need. It is a little bit more directed to say, "I'm working with you. I'm telling you something to do or something to record." But it's not about how to do math. It's like social. Or it's a behavior that we're going to engage in. But it's something that I feel okay about telling because I know that student. And the other one. So, that's the "Write that down." Like, I'm giving you a direction. I'm very clear, this is the direction.

[00:06:23.960] - Pam
Very clear. It's direct. It's intentional. Well, intentional. That's tricky because both of these are intentional.

[00:06:29.160] - Kim
Yeah. But like, I knew Mikayla. I had been her teacher, and so I knew that if I could say to her, "Write that down." And it wasn't like I was, you know, shaking my finger over her. But I was like, "Hey, write that down." And I was telling her what she should do in that moment to get her brain being able to move forward in her math. The other though, the "Sure wish I knew something about quarters," like you said earlier, really is about lobbying something out. This was a math thing where the "Write that down" wasn't a math thing I was directing her about. This is a math thing. It's logical. We wouldn't want to tell these specific moves to make. But when I said, "Sure wish I knew something about quarters." I was lobbying out something to see if any of those students would grab hold. It was strategy. It was something that they could use, and I just lobbed it out to see if anyone would run with it.

[00:07:19.880] - Pam
It's a relationship, a mathematical relationship, a connection that you put in the ether, and you basically invited them if it's within their zone of proximal development, that it would flash for them, it would ping for them, it would ignite something in them. If it's within their zone of proximal development, you are asking them to consider. You're inviting them to consider it. And if it's within their zone of proximal development, then they can run with it.

[00:07:48.230] - Kim
Yeah.

[00:07:49.170] - Pam
And they can do something with it because that kind of math, math-ing, is logical. It's mathematical. It is internally consistent. It's a system that you can logic your way through. You can use what you know to then determine the next thing. You can run with it. You can notice the pattern, use the pattern, ask yourself if it's still making sense.

[00:08:13.910] - Kim
Well, so what I didn't do was say, "Hey, y'all. This has a 25 in one of the numbers. Everybody should be using quarters here."

[00:08:22.910] - Pam
Or, "Let me tell you how to use quarters."

[00:08:24.520] - Kim
"Write that down. Use the quarters. And this is the way you should use it." It really was for me like a wondering to see if maybe they weren't thinking about in that moment, but if the slight nudge, the slight inkling would then make them go, "Oh, yeah. Of course. We can do that." And I believe that particular group, they did run with it. But other groups may not have. And I didn't at that point say, "Come on, y'all. I lobbed it out, and you didn't catch hold, so now I'm going to be more direct and tell you, you have to do it this way."

[00:08:59.100] - Pam
You're saying

[00:08:59.660] - Pam and Kim
It's not that.

[00:09:01.150] - Pam
It literally is, is this something that you guys can grab onto and run with? Then do. Then do. And if not, okay. Then what are you thinking about? And let me see if I can nudge you along that road on the landscape of learning. So, I want to just parse out a little bit that one of them has much more to do with what we would call conventional or social knowledge. When you said, you know, Mikayla, and you know the math, and you know the landscape. And in this moment, as she's solving problems, since you know her, you're like, "Ooh, if she write..." She came up with a great idea. And you're like, "Yeah, write that down, and now you'll be able to use it and keep logic-ing your way through the problem." But in that moment you said... Part of it is I'm going to directly tell you what to do. I'm going to like, "Write that down." That there are parts of mathematics and there are parts of mathematics teaching where we are going to tell kids. We're going to be direct. But there are other, and I'm going to suggest it's a much bigger set of mathematics and teaching, that is logical-mathematical, that has everything to do with mathematics as a consistent system. It leads. One thing leads from another. That's how we created it, right? Like, humankind created mathematics. So, it's internally consistent. It's logical. You can use what you know to move from there. And so, then there are these other times where you might lob something out. You might put something in the ether. And then if they grab it a little bit, then you will higher dose them with it, give them a higher, more concentrated dose of the pattern, so that then they can potentially pick up the pattern and use it to continue to reason.

[00:10:50.190] - Kim
So, in the situation or the scenario that we gave earlier about Mikayla, that was more of like a "I know you, student." Like, behavioral type thing. But you're saying that in just mathematics in general, there are these two categories. There's social-conventional stuff that we would tell, and there's logical-mathematical stuff that is more of that lob out. Or, elicit.

[00:11:13.950] - Pam
An experience.

[00:11:14.760] - Kim
Yeah.

[00:11:15.300] - Pam
Yeah. Now, we want to have kids experience high dosing of patterns, so that, you know, human beings are wired to pick up on patterns and to use patterns. Most of us need a higher dose than we get just wandering through life. Or just even seeing a teacher show steps of an algorithm doesn't typically give most of us— pick me— enough of the relationships behind the algorithm for me to then use them to solve problems. Or I get stuck in the, you know, the teachers pretend... Well, how do I say this? If a teacher says to me addition is these steps, I just believed it. I believed that addition was social, that I had to wait until you would tell me what to do, and then I would mimic that. That means that the teacher was treating it as social, and I was eating it. I was eating it? I was buying it. I was ingesting it as social. And so, therefore I treated it as social. Whereas addition isn't social. Addition, the nature of adding stuff together, we need to experience that in relationships.

[00:12:19.250] - Kim
And before we go on, what's very tricky is that was true for you, but it is true for so many. We were probably all taught that all areas of mathematics was social. I mean, I certainly was taught that growing up, that this is the way that you do math. And, you know, here are the steps that you do. And so, it can be very tricky for us as adult teachers to parse out the difference between social-conventional and logical-mathematical because we were told all of it was social.

[00:12:52.850] - Pam
And let me add to the trickiness of that. That's if your perspective, the way that you viewed mathematics as a child, as a student was what we just described, that you sort of believe the teacher. But maybe you were the person who saw the shark. If you want to listen to other episodes, you can hear us talk more about that. That when the teacher showed you steps of an algorithm and said, "This is the definition of..." I don't know, pick one. Multiplying fractions. You said, "Ooh, but I can see the relationships that are happening here, and I'm making these connections, and I'm really thinking about the numbers." And now, you might potentially think that's the best way to teach kids because that's how you learned. We're suggesting you could have gone farther and faster if somebody actually helped you do what you were kind of doing on your own behind the scenes inside your own head. Whereas the teacher was really just showing you a bunch of steps. You were making the system logical and internally consistent and using the relationships to logic your way to the next thing, so you might think that's the way to do it.
And so when we say some of it's logical-mathematical, and some of it's social- conventional, you're like, "No, it's like all together. It's all the same thing," because none of us have had a lot of experience parsing those out. And, Kim, I think you and I would both agree that this comes up in this sort of weird controversy. Is weird the right word? (unclear).

[00:14:17.060] - Kim
Unfortunate. It's an unfortunate.

[00:14:18.590] - Pam
Unfortunate. That's a better, much better word. Where some people say, "Ooh, we should really use direct teaching." And other people say, "No, we should do inquiry teaching." And some people say, "Ooh, we should do discovery teaching." And some people say, "It should be problem-based learning" or "explicit teaching." And there's all these words that I think It's a super unfortunate argument because I think much of the time we're not communicating with each other. We don't even know the definitions that each other are using when we use those words. We think that a big part of this unfortunate, probably not needed controversy is really all about if we can parse out what parts of mathematics and mathematics teaching are more logical-mathematical and what parts are more social-conventional. Because when we can parse those out, if something is social-conventional, we made it up, we made it to be that, we deemed it to be so, then you can't figure that out on your own. Someone's going to have to tell you. That helps us as teachers realize we got to tell kids that stuff, "Write that down." But the rest of it is logical-mathematical. And in order for us to really own the logical-mathematical parts of math, we need to experience it. We need a high enough dose of the patterns that our brain can grab those patterns, and make sense of them, and then use them to logic our way using those patterns to create new math or to solve problems.

[00:15:46.260] - Kim
I'm going to... Yeah, I'm going to... Well, you said figure it out on your own. And I just want to...

[00:15:53.850] - Pam
Oh, yeah.

[00:15:54.090] - Kim
Because I don't think you mean like, "Go away, kid. Figure it out on your own."

[00:15:56.530] - Pam
No, I don't.

[00:15:56.600] - Kim
And that's part of the conversation I think that gets hard. You absolutely mean. Because then you followed up with like experiences and high-dose patterning. Like there's teaching involved in that.

[00:16:09.400] - Pam
A lot of teaching.

[00:16:10.080] - Kim
So much. So much important teaching.

[00:16:11.740] - Pam
A lot of very intentional teaching.

[00:16:14.390] - Kim
Yeah.

[00:16:14.880] - Pam
Yeah, yeah. So, let's do a little back and forth for a second. I'm going to give you a list of things, and we're going to just take a second and parse out is it logical-mathematical? Or is it more social-conventional? Or are there bits of both? Okay, so, Kim.

[00:16:31.880] - Kim
I feel like you're giving me a quiz.

[00:16:33.660] - Pam
I'm giving you a quiz. Here we go. Ready, ready? What do they call that? The hot seat? You only have so much time. You turn the timer on. No, okay. Alright. The look of numerals. So, super young. Kids are learning what the number 3 looks like.

[00:16:44.640] - Kim
Yeah, that's social. That's social.

[00:16:46.260] - Pam
Why?

[00:16:46.660] - Kim
Well, because we made up like how you would write it.

[00:16:51.100] - Pam
Like, 3 doesn't have to look that way? Someone could have made the symbol look differently.

[00:16:54.139] - Kim
Please don't change it. But yes. It could. And but like in other countries, right, somebody else's 3 isn't going to look like our 3 necessarily. So.

[00:17:02.410] - Pam
Different numeration. Different numeration systems. It actually looks different. Nice, nice.

[00:17:06.950] - Kim
But the value of the 3.

[00:17:09.730] - Pam
Ooh, value of 3.

[00:17:11.230] - Kim
That's logical-mathematical.

[00:17:12.270] - Pam
[Claps].

[00:17:13.230] - Kim
Mmhm.

[00:17:13.230] - Pam
That's logical-mathematical a little bit.

[00:17:13.670] - Kim
Were you clapping for me?

[00:17:15.250] - Pam
Uh, sure.

[00:17:16.190] - Kim
You were clapping 3 times.

[00:17:17.230] - Pam
Sure I was. Yeah. Yeah, I was clapping 3 times for you.

[00:17:21.330] - Kim
Yay me.

[00:17:23.030] - Pam
So, there's part of three-ness that is logical-mathematical. But the look of the numeral. "You know like, shouldn't 3 have 3 bumps instead of 2? Just has, you know, 3. (unclear)."

[00:17:35.320] - Kim
3 points at the end. I don't know.

[00:17:37.390] - Pam
Oh, there are 3 points at the end, but there's... See, I focus on the...

[00:17:40.970] - Kim
Can we not though? Because then it gets into like touch math.

[00:17:45.300] - Pam
Oh. Well, that's not what I meant.

[00:17:46.890] - Kim
I know, I know, but...

[00:17:47.450] - Pam
I meant the round part of the... Anyway, okay. So, the look of the numeral is not logical. Somebody made it up, so that part's social. Okay. How about... And a whole new thing. 5 plus 6. Because Kim, let's be clear, this is a fact and kids just should memorize their facts.

[00:18:03.610] - Kim
Well, we want them to know what 5 plus 6 is, but it is logical-mathematical. Like, we can solve that problem using reasoning.

[00:18:14.630] - Pam
Do we want kids to have it at their fingertips?

[00:18:16.430] - Kim
Yeah, absolutely.

[00:18:17.040] - Pam
Yes, of course we do. But we really want kids to have it at their fingertips and know that 5 plus 6 is golly, 5 plus 5 and another one. It's 6 plus 6 and subtract 1.

[00:18:29.130] - Kim
Where it nestles between the 10 and 12 would be cool. Yeah.

[00:18:32.900] - Pam
Ooh, that's very nice. Yep, yep. And that it's related to 5 plus other things and it's related to 6 plus other things. And we really want this interconnected system of addition to be playing out in 5 plus 6. What we don't want is, "Okay, everybody. Picture a 5, and that's gonna be like the beehive. And 6, think of sticks. And so, if you add 5 plus 6, and you take the stick and you beat up the beehive..." I mean, I'm obviously not making up a rhyme that anyone has ever made up before. And then I'm supposed to remember 11 from that somehow. Like, we don't want kids to rote memorize 5 plus 6. Yes, we want it at their fingertips. But if we treat it like it's social, then we don't have all that meaning behind 5 plus 6. We don't... Kids aren't math-ing. That's like memorizing a river. Or the name of a river. Or the capital cities of Europe or something.

[00:19:25.570] - Kim
People right now might be thinking, "What are you talking about?" But I can assure you, our dear Kourtney on our team spent some time sending me some videos of just terrible rhymes. And we were like, Stop. Stop doing it.m It's going to be in your feed!

[00:19:39.580] - Pam
Yeah, there's whole... Yeah, because she was looking on YouTube, right? And there were YouTube videos. "Help your kids learn the facts!" And there were these horrible mnemonic devices to help kids rote memorize something that can be logically figured out. Now, once they've logically figured it out, and they've learned some strategies and relationships, we want to do that strategy relationship work enough that 5 plus 6 becomes pretty automatic for that reason. Alright, here's another one. How about pentagon?

[00:20:12.410] - Kim
Mmhm. The name of it? The name "pentagon" is social.

[00:20:15.480] - Pam
Well, yeah, what comes to mind?

[00:20:16.970] - Pam and Kim
The name.

[00:20:17.330] - Kim
The name "pentagon" is social.

[00:20:19.370] - Pam
And somebody might say, "I don't know, Pam. That's pretty logical because you could think about hexagon."

[00:20:23.850] - Kim
Yeah, but...

[00:20:24.600] - Pam
Septagon and octagon. You know, there's a pattern. But take that pattern down. Like, decagon, nonagon, octagon, septagon, sexagon... Oh, wait. Hexagon. Oopsie. Broke the pattern there. Pentagon. And then how about a four-sided? That's called a quadragon? A fouragon? Oh, quadrilateral. Social. The name pentagon is social. Is there part of that, that's logical, mathematical that... I don't know.

[00:20:49.750] - Kim
I mean, the way it relates with polygons maybe, but the name itself is the one that gets me. That we just memorize the name because we have to.

[00:20:59.930] - Pam
Because somebody decided we were going to call a 5-sided polygon a pentagon, and so we can't expect kids to figure that out on their own. We got to tell them that. And then give them lots of experience with pentagons, so they have some sort of feel for what a polygon even is. And this is a 5-sided one. Okay. How about 7 times 8? Kids just got to memorize the facts, Kim.

[00:21:18.650] - Kim
Listen, I need you to stop saying "like the garden gate" because now I cannot. Like, it's ridiculous.

[00:21:24.550] - Pam
You've heard me say that too many times?

[00:21:25.650] - Kim
It doesn't help anybody like learn what the fact itself is.

[00:21:29.270] - Pam
7 times 8 like the garden gate is made of sticks, so it's 56. That was on my kids' third grade wall.

[00:21:33.970] - Kim
I believe you, but all I remember is "like the garden gate," and I'm like, "What's that fact mean?" Yeah, it's social. I mean, the song you're singing is social. It's logical, mathematical. There's so many different relationships you can use for 7 times 8.

[00:21:49.200] - Pam
So many.

[00:21:49.720] - Kim
So many.

[00:21:51.780] - Pam
2 and double, double, double, right?

[00:21:53.300] - Kim
7 times 7. Which is a square number. That's cool.

[00:21:56.440] - Pam
8 times 8, and then you can get rid of it a little bit extra. So many. 5 times 8. Because a lot of kids know 5s. You just need two more 8s. Now, you're adding 40 and 16.

[00:22:03.860] - Kim
You know what?

[00:22:04.560] - Pam
All of the...

[00:22:05.290] - Kim
Sorry, I'm just kind of random. You know what I don't think we've ever said is when I'm using 5 times 8 to help me with 7 times 8, I'm also getting more fluency with 5x8. So, like when I'm using facts to help me with other facts, then those facts that are known or like getting more known become even more known when I use them for the more unfamiliar facts. Like we keep reinforcing it.

[00:22:30.720] - Pam
That's a nice example of parsing. Yeah, yeah. Parsing out a place where you are high dosing those... Oh, I had a good word, and it flew out of my head. The facts you're putting together. The component facts? The component facts. Like, if you're saying, I can think about five 7s and two 7s to... Oh, that was seven 7s. Sorry. If I can think about five 8s and two 8s to get seven 8s, what you're saying is you've just high-dosed yourself again with five 8s and two 8s. Yeah, yeah. You're using them. That's an example of how we want. People will say, "No, it takes too long for them to figure it out." And we're saying, "No, we're giving kids experience getting better at all the facts because they're logically figuring them out and getting a higher dose of them as we go." Alright, how about one... No, a couple last. A couple more. Parentheses. Why do I even say that? In math, right? Math.

[00:23:23.980] - Kim
There are a lot of different ways parenthesis are used.

[00:23:26.480] - Pam
What's one thing parentheses means in math?

[00:23:29.220] - Kim
Multiplication.

[00:23:30.840] - Pam
Okay, so we definitely have, you know. We use it as a symbol to mean multiplication. But that's not the only way. How about ordered pairs? Like a point, right? How about interval notation, high school teachers? How about f(x) uses... Yeah, Grouping symbol. Nice. f(x), f prime of x, f inverse of x. Like, all of those use parentheses to mean different things. That's super interesting and confusing, and it's very social. Now, it's social, that somebody decided we were going to use parentheses in lots of different ways, and it means this here, and it means that here. But then in each of those cases, we want to have kids have enough experience that they then logically can use parentheses to help them with as a grouping symbol, or to help them represent transformations when they're using f(x). Say, like, you know, what is g of x if it's equal to f of the quantity x plus 2 all squared, and then that f of that quantity all plus 5. All of that, we can use that as a tool. We want to give kids lots of experience there. That's more logical, mathematical. Alright, here's another one. How about you got to tell kids how to solve equations? Like, what are the steps? Solving equations. Is that logical math? I just wrote it down. I should have just... Could I just have said how to solve equations? Solve equations. There, I'll just do it that way. Solve equations.

[00:24:53.730] - Kim
I think a student would have to be told when you see something recorded in this way, then here's what's being asked of you. Like, this is the way it's notated. But then the way that you solve equations is social. I mean, it's logical.

[00:25:15.410] - Pam
In other words, you could look at an equation, and if there's a common factor, you could decide to factor out the common factor on both sides, divide out that common factor. But you could also decide to distribute that. You know, if there was a parenthesis, you distribute it through. Like, you have choices about how you're going to solve the equation. That's logical-mathematical. But we might have to help kids realize that the notation, the way it's set up, that means solve an equation. But we don't then tell kids how to do it. So, social part of it, the look of it, the sort of setup, what to do. Logical-mathematical, go for it. Use relationships and connections. That doesn't mean that we don't give kids lots of experiences with different ways of solving equations, so they see the patterns and they can then use those patterns. But that's all logical-mathematical. In other words, what we don't say to kids is, "Write that down." We don't say, subtract 2 from both sides. Now, multiply both sides by 3. Now, like... If you do that, you're treating solving equations like it's all social. We have to tell kids.  Okay, cool. Last one. How to divide. And I could even be more specific. How to divide fractions.

[00:26:22.310] - Kim
I see this similar to the solving equations one. If I've never seen any division notation young, someone's going to have to tell me this symbol means division.

[00:26:34.950] - Pam
I mean, how many different division notations can you like... So, I could what? Write the line with the two dots? That's one. What's another one?

[00:26:43.400] - Kim
Like a side slash. If they see it on a piece of paper.

[00:26:47.530] - Pam
Yep. And then I could make that side slash, if it's kind of a fraction symbol, I could make it horizontal, so now it's like a thing over a thing. Okay. How about the housetop division symbol?

[00:26:57.750] - Kim
Oh my gosh. That's so funny because I was like, "I think those are the only three."

[00:27:03.240] - Pam
Look at you been out of.... So, all of those notations are very social. In fact, when my mom does division, none of those symbols show up. She writes the divisor and then she writes the... No, she writes the dividend on the left. Then she writes a colon and then she writes the divisor on the right and then she does a bunch of steps. That's what her division looks like. Yeah. Interesting, huh?

[00:27:26.240] - Kim
Yeah.

[00:27:26.800] - Pam
Yeah. Some people tried to tell me that that colon, if you put a line between it, that became our division symbol. You know, the line with the two dots. Maybe. But it's still social, right? That look of the division symbol is social. But, Kim, if I say, "Ours is not the reason why, just invert and multiply," how am I treating division of fractions?

[00:27:46.480] - Kim
Social for sure.

[00:27:47.450] - Pam
Treating it like it's social. But is division of fractions social?

[00:27:50.610] - Kim
No, for sure not. There's... Yeah.

[00:27:52.830] - Pam
For sure logical-mathematical. Alright. So, Kim, well done. That was fun. I hope that gives everybody a little bit of a sense of some differences in mathematics and mathematics teaching about things that are social-conventional, we just got to tell kids. They can't decide on their own that parentheses can mean all those different things because we've just made it up as humankind. However, there are... Most of mathematics is a logical, consistent, internally consistent system that we can logic our way through. And that is logical-mathematical. And that needs to be experienced with high doses of patterns, and very intentional teaching to make those high doses happen and make those specific particular experiences happen, that's logical-mathematical. So, Kim, what's our definition of excellent math teaching?

[00:28:46.170] - Kim
So, we want to have explicit goals about math and math behavior. We want to have intentional moves. And we want to have a clear vision on different types of knowledge. Both the logical-mathematical and the social-conventional. So.

[00:29:03.190] - Pam
Because then we know what moves to do when and where.

[00:29:05.710] - Kim
Yeah. So, next week we're going to unpack what that looks like in a classroom. So, we're going to talk about times and instances where we want to be explicit. So, I think we have a challenge for listeners to take the time to think through when they think it's appropriate to be explicit, to be direct, to be, I'm going to say clear because you're always clear. That's not a good word.

[00:29:29.750] - Pam
Always be clear.

[00:29:29.940] - Kim
Yeah. When in your classroom do you think....

[00:29:33.010] - Pam
Do you tell kids? When do you tell? And when do you help kids experience? Nice. We'll see you next week, everybody. Right, 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. Let's keep spreading the word that Math is Figure-Out-Able.