Ep 17: Social vs Logical Knowledge

Pam Harris  00:01

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

Kim Montague  00:08

And I'm Kim. 

Pam Harris  00:09

And we're here to suggest that Real Mathematics is not about mimicking, or rote memorizing. But it's about thinking and reasoning about creating and using mental relationships. We answer the question, if not algorithms, then what are we doing in math class? Alright, so first, today, we're going to share a really cool message that we got the other day from a parent who's listening to the podcast. So thanks to Nikina.

Kim Montague  00:36

Yeah, she wrote, "Hey, Pam, I just want to let you know how your podcast is helping me. I don't remember exactly how I found you. But it was from the other podcasts that I was listening to, specifically for elementary math info. I'm not a teacher, just a parent who wants to get my math brain fired up so that I can engage my children with thought provoking games, when they are developmentally ready. My daughter's only four. But even already, she will get excited about the fun patterns we see in numbers. And it makes me very excited that I'm learning so much from you and Kim. I've been working with her on number recognition in the car by calling up the numbers we see on speed limit signs, when I introduced to her that the two numbers are actually 35 rather than a three and and five, she was engaged. Just a few days of me telling her the number after her calling up the single digits, she recognized the pattern and I had a proud parent moment when she yelled out sixty-zero. Obviously, I know she's wrong, but I didn't correct her right away because it showed me that she understood the concept rather than just memorizing what I was telling her. I love that I'm thinking more mathy by listening to you, and that thoughtfulness is coming through in my parenting. Thank you."

Pam Harris  01:49

So cool! Love that she shared that. We would love to hear from all of you guys about how you are using the podcast. Feel free to send us an email, Pam@mathisfigureoutable.com, or Kim@mathisfigureoutable.com, or even better share what you think and your rating at your favorite podcast place.

Kim Montague  02:06

That's so awesome. Okay, so last week, we talked about the language of counting. And we're aware that part of counting is about the social convention of what we call things. But there's also part of counting that's all about kids needing to experience counting, so that they really own their relationships. In today's episode, we're going to talk about this interconnectedness of the social-convention part of math, and the experiential-development part of math.

Pam Harris  02:33

And Kim, I think this is so important. I can't tell you the number of times that I've been on Twitter or Facebook, or we were having conversations with math people. And we get stuck here and we begin to talk past each other, and we're not really communicating. And so today's episode is really important that we're going to differentiate this idea of what we have to own in math versus the social convention part of math. So let's define a couple of terms and where they come from. So Jean Piaget was a Swiss psychologist - shout out to Swiss woohoo (my mom is from Switzerland). So he, a while ago, and he talked about types of knowledge, two of which pertain to this conversation. So one type of knowledge is called 'social knowledge'. And then the other type is called 'logical mathematical knowledge'. I kinda wish we had a better name for that, but whatever. So let's define social knowledge. Social knowledge, is "that which we deem to be so." It's by convention. It means that as a society, we sort of said, this is going to be the way it is. And in different countries, we might actually have different social conventions. If you look at something, and it's different in different language, that could very well be a sign that that's something that's social. It's not the other kind of knowledge that we have to build. But it's just that that language has named it that way. And so it's the kind of thing that we have to sort of tell kids.

Kim Montague  04:00

So you said 'something that is deemed to be so'. Who makes those decisions?

Pam Harris  04:05

Yeah, so that's actually less specific than you might think, um. Mathematicians sort of make the decision about what we deemed to be so, what's going to be true by convention. But kind of over time, it's like as a society, or we look at a body of work, and we kind of adopt certain conventions. Let me give you a historical example. So Rene Descartes, 'I think, therefore, I am', was also a mathematician. And one of the things that he's less known for in history is that he started using in his work - he was doing a lot of writing and a lot of people were reading his work. His work was very well known. And so because of that, people wanted to understand what he was doing, and they begin to take on some of his convention. For example, he used 'a, b and c' for constants, and 'x, y and z' for variables. And before him, that wasn't necessarily what everybody was doing but because so many people read his work and it was so widely known and people wanted to understand it, then it sort of became the convention. And so today, we typically use 'a, b, and c' for constants and 'x, y and z ' to represent variables. It's not wrong to use the variable y to represent a constant, we just don't typically do it. By convention, we've decided as a society that we just don't typically use that convention. We don't usually do it that way. And so one of the unfortunate things could be that a math teacher today could say, "No, no, that's wrong. You can't use that variable. You can't use the variable 'a' to represent - or in this case the letter 'a' to represent a variable." Well you can, we just don't by convention. It's not right or wrong, mathematically, but as a society, we've sort of deemed it to be one way or the other. So it's really important that we treat things differently if they are social and things that are logical mathematical.

Kim Montague  05:54

Okay, so that's interesting that you share that because my oldest son is in Algebra right now. And he's always wondering about why things are done a certain way. And sometimes I don't have an answer for him. If it's not 'logical, mathematical'. So will you tell us a little bit about that one.

Pam Harris  06:08

Yeah, totally. So that's the other one, right? And I wish we had a better name, because it sounds like it's too much either logical OR mathematical. But it really has everything to do with experiential. It's the part of knowledge that we have to build. It's the sort of things that if we don't own, we can't really do something with it, we can't use it. So a way to kind of differentiate between the two, if you can look something up, and then you can use it, you can do something with it, It's probably social. If it's like, easy to, like, there's that fact. And now I can go just like, talk about that fact. Then, like, for example, a name in history, or a date or a place, or that's social convention. It sort of happened in history. And so we can kind of, like when somebody has to tell us that. We can't figure it out. We can't sit on our own, and use what we know to logically derive it, then it's probably social. If we look something up, and then we can't, like, use it right away, then it's probably logical mathematical, if we have to actually own it, if we need to work with it and understand it and understand the ramifications of what's connected to it, that's probably logical mathematical knowledge. Let me give you a couple of examples. So often, I use this one when I'm working with teachers. So I was working with a group of teachers one day, and I said, "Who was the eighth president of the United States?" And then I stare them down. And I'm looking to see like, what do they think about that? Are they figuring? Is there something we can do to, like, use logical relationships to decide who - no! I mean, that's a perfect example of social knowledge. It happened, we can look it up. And then we know the answer instantly. We can ask somebody and they can tell us who it is instantly. Okay. So for one time. There was one time where I asked that question, "Who's the eighth president of the United States?" And this gal starts to like, do this thing. She's kind of like ticking her hand off in front of her. It's almost like she's sort of counting and then she looks up. And she tells me. I don't even remember what it was. I still don't know who the eighth president is. And I looked at her, and I was like, "What were you just doing?" Now when I ask people that sometimes they'll say, "Oh, I bet she had a song. I bet she had a rap or a rhyme." Well,  that's interesting, because that's a really good sign that something's social knowledge. If you need a mnemonic, or a rap, or rhyme or story or some sort of memory technique to help you remember it, chances are, that's social knowledge, or at least you're treating it like it's social knowledge, because you're doing something to rote memorize it. So it was really interesting, I'm like, "What is she doing to like, figure that out?" And she said to me, "I'm driving home." I'm like, "What are you talking about?" Well, she lives in San Angelo, Texas and in San Angelo Texas the streets in her city are the presidents of the United States. And so, like Adams is twice so she kind of has to, like think about that when she kind of drives home to decide who the eighth president of the United States is. Again, that's very social, the names of those streets, they decided to be that way. Who became the president of the United States? We voted on it. It wasn't a logical mathematical, it wasn't something that's about relationships between logical understandings. It happened, that's social knowledge. Let me give you another example, more mathy example. Okay, so if I asked you what a polygon is. So polygon is a - often people will say it's a many sided figure. But if you actually think about the word 'polygon',  '-gon' is sort of the suffix that represents angle. So it's actually a many angled figure. And then I might ask you what's a 10 sided polygon. And you'll say it's a 'decagon', and a nine sided is a 'nonagon', and eight sided is an 'octagon', and seven sided is a 'septagon', and eight sided is a six sided is a 'sexa-gon'?

Kim Montague  09:51

Hexagon.

Pam Harris  09:52

Hexagon. Okay, but wouldn't it - come on, it'd be more fun if it was a 'sexa-gon'. Anyway, so we've got these sort of names for these polygons, right? They're kind of following a pattern and then it breaks down a little bit when we get to hexagon like it should be sex-agon. Or then what's a five sided? Should be a fifth-...? It's a Pentagon. Okay, cool. What's a four sided? Well, then all of a sudden it's a four sided, it's not a quad-agon, it's not a four-agon. It's a quadrilateral. Ooh, look at that word quadrilateral. Now we're actually to a sided figure. Now, it's a four sided figure. You guys, this is all social. Somebody deemed it to be so. We gave these figures those names. That's social knowledge. There is kind of a pattern happening and that we break it. When that happens, when there's kind of this logical pattern, and then it breaks, often that's a hint that we're dealing with social knowledge. Somebody decided to name polygons that way. Okay, so let's keep going. We've got a four sided quadrilateral, what's a three sided polygon? A 'tri-agon'? Well, at least it's a 'triangle', you know, angles still in there. So again, social knowledge: we've deemed it to be so. As a society of mathematicians, we sort of given things names. Often vocabulary are an example of social knowledge.

Kim Montague  11:10

So this sounds really important. But why do we care? Like, what are we supposed to do about that?

Pam Harris  11:17

Yeah, so why are we making a big deal about social versus logical mathematical knowledge? It has everything to do with how we teach the different things. So if something is social, if it's by convention we've deemed it to be so, tell kids that. That's the kind of stuff that we just need to explain, we just need to give, we just need to hand out, we just need to make sure that it's clear. And we don't wait on when we ask kids. So if I'm asking kids, something that's social, I'm not going to stare them down, I'm not going to use my wait time in that case. I'm not gonna be like, okay, figure it out, like math is Figure-Out-Able. Well it is,  but that part of it, if that part of it is not Figure-Out-Able, well, then don't wait. That's like, rude, it's mean. If they can't pull it from rote memory, waiting longer is not gonna help them pull it from rote memory, even better. If, however, the thing is logical mathematical, then don't just tell kids that. It won't work. Kids won't own it. Kids need to experience logical mathematical. They need to build the mental relationships. They need to change their brain so that they actually can handle more of the simultaneity so that your brain can deal with things more simultaneously. If you haven't built your brain to be able to deal with these concepts, then they're not gonna be able to handle it. Let me give you a quick example. Craig and I, he's our third. He's home right now. BYU has remote classes. And so he decided to take remote classes because his fiance's here. And so he stayed home. to be near his fiance. And he's taking a computer science class. And he came downstairs last night, and I said, "Hey, how's your how's your current program going?" And he said, "We're dealing with this logic stuff." And I was like, "Oh, tell me more about that." And he started talking about syllogisms, and different logic gates and all these, he started throwing out some terms. And I said, "Well, what does that mean? And what does this mean?" And I knew some of it. I've taken Logic, I've dealt with some logic. And there were some things that he could tell me the name, and he could tell me the term, but boy until we like, talked about it and had some examples and had some non examples. Like just knowing the name and the term didn't really help me understand the logic of what's happening. It's funny, because I'm using the word logic. But that's an example of a mathematical kind of logical mathematical knowledge, that it wasn't just enough for him to just tell me the sort of terms, we had to dive into it and dig into it enable to use it and able to do something with it.

Kim Montague  13:43

So that actually makes me think of a really important, like, elementary example. The names of the shapes, like you said earlier, are social; kids can guess those. 

Pam Harris  13:52

Yup.

Kim Montague  13:52

But the relationships and the classifications that we do based on attributes, those are logical mathematical, they should be experienced.

Pam Harris  14:01

That's a really nice way of parsing that out.

Kim Montague  14:03

And rounding. Somebody asked recently, on Twitter, I saw about rounding. Someone decided that halfway between numbers, rounds up. Like when the number ends in a five, we decided it rounds up. But the logical mathematical part says that we need to consider the situation and make sense of when we would round up or down given a situation. Can you give me an example in higher math, though?

Pam Harris  14:26

That's a really good example in rounding. In fact, one of the things that came out on Twitter, is that they were talking about how in Canada, they don't use the penny anymore. And so if you're sort of at that closer to like 10 cents, if you're at 10 cents, 11 cents, 12 cents, then you're going to round down, and you're just going to give him back a dime in change. But if you're closer to the nickel, then you're going to sort of use the nickel. If you're at 13 or 14 or 15 cents, and you're gonna round up to the nickel. How interesting is that? So it really depends on context. And you have a really good point that that is all about logical mathematical. We have to think about the situation in the context to make sense of it. Yeah. Very cool. But the part of it that's social is where we decided sort of to round up or down, kind of in the middle. What do you do in the middle? We've decided to round up, five rounds up. But it might depend on the situation where we bring the logical mathematical understanding in. Alright, so you asked for an example from higher math. You were telling me the other day that your son was dealing with the Pythagorean theorem. Wait, was that the parallel, parallelogram theorem, Pythagorean - those are, like long mathy sort of names, right? Well, Pythagoras was a guy, and we named the theorem after him. That name, that's social. But to be able to deal with the Pythagorean theorem, how to derive it, and how the relationships and how to understand, for example, if I'm trying to find the distance between two points, could I use the Pythagorean theorem and not even memorize a separate new formula? Can I just like, use what I know about the Pythagorean theorem? To help me find the distance between two points? Yeah, then we don't have to have anybody rote-memorize the distance formula. They could just use what they know based on the Pythagorean theorem that they know. What part of that is social? The name, the name is social, but how I use it, how I understand how it's related to the distance between two points: that's the logical mathematical part of it.

Kim Montague  16:24

That's kind of like when we ask kids to make sense of and use the distributive, associative and commutative properties. So using those properties, that's logical mathematical. But sometimes I see teachers requiring kids to name the properties without giving them a lot of opportunity to learn the names and practice the names. That's the social part, right?

Pam Harris  16:46

Yeah, that's an excellent example. So I'm okay if you use mnemonics to help them remember the name of the property. But boy, they need to experience those properties and the ins and outs and how they relate to what's happening in the strategies. That's far more important than the names. And you bring up a really good point, because the standards name the strategies, students should use these properties. But we think that most of the standards we've seen, that doesn't mean that the kids need to name the properties. They just need to be able to use them. If kids are learning really good strategies for numbers, they're actually using the properties, well. They don't need to name them. That's not an important part. It's not going to be on a high stakes test where it says: what is this property? And if it is, then it's a lame item on the test, and we shouldn't be asking that. Okay, cool. So it's really important that we don't treat social things as logical mathematical, and we don't treat logical mathematical things as social. Or this can happen. Like we might have teachers that say, "Oh, let's see, how am I gonna help kids learn" - pick a thing - multiplication of fractions? Well, it's the multiplication fractions. Is that logical mathematical or social? Well, if we say to kids, "Multiplication, fractions, multiply fractions straight across." I heard somebody sing that one time, then what we're treating multiplication of fractions as social knowledge. We're giving them this little ditty to sort of memorize. And then if they memorize the ditty wrong, then they're not going to multiply fractions correctly. But I would argue, if they've just memorized that they're not building their brain to be able to multiply fractions. They're just doing a thing. They're not understanding what multiplication of fractions is, understanding how multiplication of fractions relates to the area model. But also how we can think about the relationships, think about a half of something to help me consider what a fourth of it is. And if I can think about a half of something to think about a fourth of it, then I can think about three fourths of that thing. Like that kind of understanding is logical mathematical. That's the part of it that we really want kids to own deep down inside, not treat it like it's something social. Another huge one that I think we talk past each other are the multiplication facts. We're gonna have a whole series of episodes on the multiplication facts, but I'll just say briefly, multiplication facts are Figure-Out-Able. So if you're treating the multiplication facts, like that social knowledge, if you're doing rhymes and raps and songs and poems and stories, whatever, to rote memorize the multiplication facts, you're treating them like they're social, and they're not. They're logical mathematical. We want kids to know that the multiplication facts are Figure-Out-Able. So y'all I hope you've enjoyed this conversation about social knowledge versus logical mathematical knowledge and how important it is.

Kim Montague  19:37

So you don't treat logical things as social. And don't wait on things that are social. Wait time is important, right? But we want to honor kids' thinking by giving them time for things that they can figure out. We don't want to give them an extended amount of wait time for things that they cannot work through.

Pam Harris  19:49

Right. So just consider when you're waiting for kids, are you asking them to think about something that's logical mathematical? Let's wait on that. Give them a chance to use what they know to figure it out. But if you're asking them something social, don't wait on that. Just like give it back to them or call on somebody else because social, it's either there or it's not. Fabulous conversation about the difference between logical mathematical understanding and social knowledge.

Kim Montague  20:16

Thanks for joining us today. Don't forget MathStratChat on your favorite social media on Wednesday evening, and please head over and give us a rating and a comment because it helps more people find the podcast.

Pam Harris  20:27

Alright, so if you are interested to learn more math and you want to help students develop as mathematicians then the Math is Figure-Out-Able Podcast is for you. Because math is Figure-Out-Able!