Often people misunderstand what this game we call mathematics is and feel they aren't good enough to play. In this episode Pam and Kim discuss how students need the opportunity to experience and play in the real game of mathematics.
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 make the case that mathematizing is not about mimicking steps, or rote memorizing facts. But it's about thinking and reasoning; about creating and using mental relationships. Y'all, we take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keep students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what? Alright, in this episode, we are going to have a little bit of fun, because I've been thinking about something and I'm gonna live ask Kim what she thinks. You're gonna hear maybe more than you ever wanted to about what Kim and Pam do in their free spare time when they chat. So often, when Kim and I are on the phone or on a zoom, or we are going to a workshop together or film shoot together. And we're just chatting. It's funny, because we didn't talk about some personal stuff. But often it will be you know, like, we'll beat something out. And we both of us really enjoy kind of getting to the heart of the matter and helping each other understand what we really think and believe about some things.
Kim Montague 01:16
Pam Harris 01:17
All right. So, Kim, I have been playing with an idea. We are really clear, you and I are really clear that doing mathematics, that being a mathematician, mathematizing is not about speed.
Kim Montague 01:30
Pam Harris 01:30
It's not about being fast. And we both have some experiences in our life, with people that are close to us, who've had very negative experiences with math because they, because of time pressure. My husband was one of them. My daughter didn't like time pressure. I think you have some people in your life.
Kim Montague 01:49
My kids have had that.
Pam Harris 01:51
I wasn't gonna volunteer that it was your kids.
Kim Montague 01:52
Yeah, I definitely have.
Pam Harris 01:53
And so I've been thinking a lot about the fact and I mentioned it every once in a while, but I kind of want to dive into it a little bit more about how that's similar to the fact that I played basketball. So if you ever watch me on a zoom, you'll notice I have a basketball behind me kind of in my background. Basketball was fun for me. I played a lot as a youth. I've played as an adult a little bit. And interesting, for me, I was not particularly quick. So I could last the game. Like I had pretty good stamina. I don't know, I could, you know, like, I could run the whole game. But it wasn't the quickest kid on the court. But I was allowed to play. And I wonder sometimes, if I were to try to compare this to math, I wonder sometimes if we have misunderstood the game, now not maybe everybody, but I think a lot of people. If you've ever taken our quiz, if you haven't taken our quiz yet, we'll put that in the show notes to go take our perspectives quiz on what math is. But I think there's a lot of people who, like me, thought that math was a disconnected set of facts to memorize and rules and procedures to mimic. And that's what we did. And if you were fast at recalling those disconnected sets of facts, and you were fast at repeating and mimicking those procedures and steps, then you were sort of considered good at math. And I'm suggesting that is not the game of math. And basketball when I was allowed to play the real game of basketball. And I wonder if by misunderstanding the game, we have not allowed kids to play the real game of mathematics. And even if, Kim, if they have a different perspective, or had a different perspective, growing up more like you did, where you did a lot of things, using relationships and connections. You played a lot with numbers. But am I correct that even in your life, there was a little bit of misunderstanding about the game?
Kim Montague 03:53
Yeah, I mean, I definitely had teachers who,
Pam Harris 03:58
Like did you show what was going on in your head?
Kim Montague 04:00
No, no, no.
Pam Harris 04:02
Say more about that.
Kim Montague 04:03
Because it wasn't a thing, right? It was, I always, you know, I loved my teachers. But I was a teacher pleaser, to the degree that I never spoke about what I did in my head to anyone. It wasn't what math was in quotes. And, you know, I definitely mess with numbers a ton, but it was about following their rules and recording on paper. You know, I can remember my dad being kind of irritated with me when I would do homework, because I would have a conversation with him about whatever. And then I would dutifully write on my paper all the you know, carry the ones and whatever. And he would be like, "What are you doing?"
Pam Harris 04:43
Because you just had a conversation about what's happening in your brain.
Kim Montague 04:45
Pam Harris 04:46
But you would record what you were supposed to record. You
Kim Montague 04:48
You know, I want to see something before you get away with it.
Pam Harris 04:51
Kim Montague 04:51
And I don't know where you're going because we haven't talked about this. But when you just said people weren't allowed to play the game. Instantly when you said you were allowed to play basketball. And even though you weren't necessarily fast, it makes me think of Around the World and how the kids that don't continue to play, like they're just shut out because they weren't fast. They weren't-
Pam Harris 05:13
Let's maybe describe it. Maybe everybody hasn't, so Around the World. Two kids stand up. The other teacher flashes of fact. Whoever says it first stays up.
Kim Montague 05:22
Pam Harris 05:22
So then a new kid stands up. Teacher flashes effect. Who ever gets it right stays up.
Kim Montague 05:27
Pam Harris 05:27
Okay, so now what were you saying?
Kim Montague 05:28
So we're rewarding fast, instantly. And then even, it's so contradictory because even in those teachers minds, the way that it's supposed to work, is it supposed to help everyone get better, but it still only rewards the kids who already know or are the quickest at knowing, right?
Pam Harris 05:48
And who gets practice during that?
Kim Montague 05:50
The kid who is the fastest! It's ridiculous.
Pam Harris 05:53
Everybody who doesn't know a fact, sit down. And then they get no more practice. Yeah. So like, whose universe, is that a good activity? It's so funny.
Kim Montague 06:03
Pam Harris 06:04
Oh. Yeah, good. Yeah. Excellent point. I want to bring up a little bit about something that kind of comes up in this. This is all not very well formed in my head. So thanks, everybody, for letting me sort of beat it out. But I'm aware that there's this sense of, in me about what the game really is, and allowing kids to play. So even though I was a little slow, and I'm short. So I wasn't quick. I'm short. I was allowed to actually play the game, here's what didn't happen: I wasn't put over in a like, go over on the sidelines and until you can dribble for so much time, and until you can pass like, let me let me separate these little tiny skills into, and until you can do that I'm not letting you on the court. Like, in fifth grade, they put us on the court, like threw the ball at us and said go and we just had a blast, like just playing the game. Now I'm not saying we didn't ever pull things apart and look at some specific skills and get better at things. But let me take one in particular. So as a high school player, I learned what a triple threat was in basketball. And I may get way too, some people are like, Pam, really, you're talking about basketball? Well, bear with me for just a second. So I learned this thing called a triple threat. And if you hold the ball just right, and you get your body in the right stance, then it's called the triple threat. Because in that moment, I can shoot from that place, I can shoot immediately quick, bam, that the shot is up and I'm scoring. I can immediately go and dribble from that, if I'm in the right place, my body and the ball, go and dribble, or I can pass. Bam! So I could do three things right from there, I have no wasted motion. I have no, so it's really important. Because if you can get that down, then when you catch the ball, you're instantly in that triple threat. Now you are a threat in three ways. And the defense now has to sort of watch for all those things bubbled up. My point in bringing that up is, even though I spent time: A getting that stance down and having the ball the right place; and B practicing all three of those options. In that game, when I caught the ball, I then had those options. I was free to choose how to play the game. I would submit that in math, if we have misunderstood the game of mathematics, we have said to kids, "Thou shalt dribble." Or in this case, when you catch the ball, in this case on this place in the floor, you have to shoot and we've taken away. We haven't even taught them the other things. We've given them something to do that may or may not make sense to them. It might be the series of steps that they, it's not their work. And then they have no freedom, no agency, no I don't know what's another word, to choose. They're not empowered at that point. It's just like, go do the thing. So if I can sort of compare the two activities, I think even worse than that. We've taken some kids and if they couldn't do one of those things, then we haven't even let them on the court.
Kim Montague 09:12
Pam Harris 09:13
And I think that that's interesting. Okay, so what do I mean by that in math. We have said, "If you don't have your facts memorized, then you just keep working over here. You just keep doing your facts." And we just keep giving them things to do, steps to do it over and over every year. We just keep drilling the same things. And kids never get a chance to actually play the real game. I don't know. Does that work for you?
Kim Montague 09:37
Yeah, it does. And I'm thinking specifically about in classrooms where you have opportunities to have kids shine in other areas, and they don't even get to participate in math because they're sitting on a table somewhere working on their facts. Like you have a rich task going on. And they don't even get to participate in the conversation or the puzzles or the, because they have to get out of, you know, fact practice or they go to a small group somewhere. And so they miss something that's happening in the classroom that's richer, because they don't quote, 'get the basics yet.' It's like a precursor.
Pam Harris 10:15
I remember you as a teacher telling me, "Quit taking my kids out for their," I don't know what you call it that point, but like RTI or special where, kids will be pulled out of your math class for math help. You were like, "No, no, no, no, thank you. Like, leave them in here: they A they need this experience; and B we need them here."
Kim Montague 10:35
Pam Harris 10:36
So we might have a creative thinker. And we might have a spatial thinker. We might have a kid who is a really good argue-or like creates really good arguments and justifying thinking. We might have students who are more adept at deductive reasoning and students that are more adept at inductive reasoning. And if those kids don't happen to have this one little like, bit of something done, then yeah, like you said, we sort of set them aside. All right you go over there do that thing while, and not only, I'll say it again, not only are they missing out, but we are missing out on what they would bring to the table.
Kim Montague 11:10
Yeah, don't you think also that it has something to do with, like somebody who has a very linear approach to learning, this idea that, like, there are all these skills in order and you can't move on until you get all these skills? And that's just not where you and I are coming from.
Pam Harris 11:27
Yes. And I also think there's a little bit, boy, I was gonna go somewhere. Can you hang on to that?
Kim Montague 11:31
Pam Harris 11:32
Where I thought you were going was we might have a learner who sees all connections, and we want to have, we want to keep that learner in the discussion next to the learner who's a little bit more of like, "Wait, I can't think about that till I think about this. And I've got to like, wait, I have to connect this to that before." Like we have different strengths.
Kim Montague 11:52
Pam Harris 11:52
Those two learners will be better for interacting with each other.
Kim Montague 11:57
Yeah, well, and on a personal example is, I've got a son who does think deeply about stuff and makes choices about stuff. And so you know, what it does, it slows him down. But it has absolutely nothing to do with his ability. In fact, I happen to think that he has more connections than many people. But he has to wade through which ones are going to be useful in that moment.
Pam Harris 12:20
And that takes him a little bit longer. So he doesn't maybe appear to be, I don't know-
Kim Montague 12:26
He's not the fastest one in the group, for sure.
Pam Harris 12:30
He's definitely not appearing to be the fastest one, which then could translate to. "Oh, I guess you're not as mathematically adept, or you don't understand as much." When in reality, he understands far more. And it's the fact that he's actually parsing through what he knows, because it's so cool, because he's so connected. But if you're not asking questions, and you're not giving students experience in rich tasks, you're not going to see that. You're not going to be able to acknowledge that. And then kids will be the ones that are not allowed on the court. Then they, "Okay, you're over here, all you can do are the sort of these drills over here, and we're not going to let you actually play the real game of math." So there's another part of this, it's connected in my head, we'll see that if it actually sort of connected. Where sometimes people will talk to me about talent. And they'll say, "But Pam, some kids just have the math gene. Some kids are just better at math. And there's kids that are better at-" So this is tricky for me and, Kim, feel free to like, help me do this. You know me well enough.
Kim Montague 13:26
Pam Harris 13:26
So I'm gonna cop to the fact, I'm gonna I'm gonna own up that. Yeah. Like, I think there is some talent involved. Just like when I play basketball. I was not the most talented kid on the court. I was not the quickest. I was definitely not the tallest. In other words, I didn't have some inherent things. You're like, "Pam, tall is not talent." Yeah. But it's like an inherent thing, right? My parents are normal. Normal, I just said normal. My parents are average size, not normal. So I you know, I'm five, six and a half, which is always funny because my basketball coach always listed me as five, seven. Sorry, I have it backwards. The basketball coach listed us short. The volleyball coach listed us tall. It was all this mind game, it was kind of, anyway. So I didn't have sort of some natural things that you might be like, "Pam, that's such a game. Don't play." But I was allowed to play.
Kim Montague 14:16
Pam Harris 14:16
Like, not just here you gotta go do all this crazy, these weird, disconnected drills and stuff until you're sort of good enough to get on the real court. No, no, no like, I was allowed to get on the court and actually play the game. And because I worked so hard, then I was good enough that I made the varsity team. I played, we always took at least third, okay, fourth at the state championship game. And I had the opportunity to play semi pro basketball in Switzerland. I mean, I was allowed to play the game to whatever extent that I wanted to play because it was the real game. And I worry sometimes that we look at sort of this natural talent thing. And we say, "Oh, well maybe because I don't see you quickly, doing something that I consider to be Oh, well, you better be able to do that at math," I have no idea what my snap sounds like on a podcast, snapping when I do that. You don't care. I worry that we're like, "Well, if they don't have this natural talent at math, then then there's no way that you're going to be able to, to do the real math. So we're gonna, we're gonna put you over here until you look like you can. I don't know, repeat your facts over and over quick enough." Okay. If I may-
Kim Montague 15:26
Can I jump in.
Pam Harris 15:27
Kim Montague 15:28
Because I just had a short conversation with one of our Journey members who said that some of her co-workers think that what we're talking about Problem Strings and rich math, tasks, and like Real Math is for some kids only. And what they've identified is it's for, quote, unquote, 'advanced kids'. That it's that it's not for every kid.
Pam Harris 15:55
I'm screaming in my mind, right, yeah. So to be clear, I think that that is a typical, I definitely have had colleagues who kind of had that mindset. Okay, I had colleagues who even had more of a, I literally had colleagues at one of the schools that I taught at, we had a math Hall and all the math teachers were in the same hall. And my colleague said to me, "If they don't want to be here, let's just don't have them in this hallway. Like we, if they, if they're not the advanced student, we don't even want them here." And I'm like, Ah! Because what they were judging on was sort of this natural talent thing that they sort of looked at. If you don't have the math gene, we don't want to deal with you. And what Kim and I are saying is, "Come and play with us. Come and play the real game." It is not just for the quote unquote, 'advanced learners', whatever that means for you. It's not just for the quick kids, or the high flyers, or the- Real Math is for all kids. And if I may, I just want to sort of stay with basketball for just a second. So part of what made me a good basketball player, like that allowed me to play in Switzerland is I knew the game. I was the one who could, like tell you intricacies about what the other team was doing, the plays they were running, what would, how we could counter them well. As an example, one of the players on my high school team who played ahead of me, so she was on the court before I was. She was in other words, like my coach considered her a better player than me, sat down one day in the middle of a game. And I was trying to figure out what kind of zone they were playing. So zone as a kind of defense. I was getting really particular about like, what kind of zone so that I would, you know, I could when I got out there, I was going to know better how it gets sort of counter that. And she turned to me and she said, "Are they playing zone or man?" Now I know we're in a math podcast. So you might be like, "Pam, why are you talking about basketball?" So that's like a really broad general question. That's like anyone in my mind, I was like, "Really, you don't even know if they're - okay." Like, and I was, I knew the game. So that kind of could counter some of her natural athleticism. Because I knew the game, that I had some skills that could sort of weigh up. I'm not saying that well, that could measure up. That's the word I want. I could measure up. I could get on the court with her, because she had some natural athleticism and I had sort of some natural sense about the game. Is that true in our mathematics class? And we're suggesting that that could be true. That is true.
Kim Montague 18:24
Can I add one more?
Pam Harris 18:24
Yeah, go ahead.
Kim Montague 18:25
Yeah. So I don't know a ton about basketball. But I do know that Michael Jordan was pretty average. Right? And what he did was give himself a lot of experience. So he was, like, notoriously known for his childhood growing up being every day on the court after school on this his home, you know.
Pam Harris 18:51
Time, time, time, experience, experience, experience.
Kim Montague 18:53
Right. And so the more experience we can give all kids, we've continued to say is going to help all kids like, experience matters so much.
Pam Harris 19:05
Experience matters. And so if we set them aside. Go do these drills over here until we're going to allow you to play the real game. It's not the way to help everybody advanced because we need their voice in as well. Absolutely. So one more kind of unrelated thing that I wanted to bring in along this talent, kind of a thing. I just want to give one more illustration. So weird, I was in the jazz choir in high school. And in order to get in the jazz choir as a sophomore, because no, there were only juniors to seniors. I tried out and became the bass player. So I played the bass guitar. I was in the rhythm section. And just quickly, I wasn't all that good, but I could read music. What I wasn't, I wasn't the kid that could pick up the guitar and jam. I wasn't the kid that could just like, start playing. Like sometimes people were like, "Hey, let's just jam." And they would all start playing and I would sit there. I was like, "Hand me some music." Like I don't know what that means to just jam. That's my talents lie, to be able to read the music and play what's supposed to be. I can actually create a part, but I had to kind of know the key, and all the things. But the drummer in our jazz choir, he was phenomenal. That guy could play anything. And not only could play, he could sing. Oh, that guy could wail. I swear he was like Billy Joel. Like, every time we would do anything, he was, everybody loved John, and he was just, had this natural flair and talent. But when we went to competition, and we had to do the sight reading part. Very few people knew this, but John couldn't read music. He just faked it so well. And so, I hope John doesn't mind me like telling everybody that. I don't think he would mind. Hey, John out there. He's not a math teacher, it's fine. Um, so when we went to competition and we had to do the sight reading competition, they would give us a brand new piece of music that we'd never seen. And they would give us a certain amount of time and all this protocol and everything. And we would have to learn the music as fast as we could. And then they would come in, and they would judge us on how well we were able to perform that piece of music in a quick, you know, we only had so much time to quickly learn it and then play it. Well, as soon as they would give us the music and leave, the conductor would work with the singers. And I would turn to John and I would go, "Okay, dude, it's like this." And I would walk him through the music because I could read music. That was a talent I had. And then he would go, "Okay, this and then that?" And I'm like, "Yeah." And then they would walk back in and they would say, "Okay, everybody, here we go." And the conductor would lead us through it. And as he, as we would do it and everybody's singing, I would go, "John to me is two measures, one measure, go. And now it's gonna be the swing section. And now it's gonna be the jazz section. And that was gonna be the-" and I would like, talk him, because he didn't read music. And then we were phenomenal. Our rhythm section was amazing. But we had different talents. We brought different talents to the mix. And that made the world better. I would suggest, we asked you math teachers to consider, that all students bring a brilliance to our math classes when we play the real game of mathematics, when we're actually mathematizing the way mathematicians mathematize everybody can join in and everybody has a place on the court of a Real Math. So if you want to learn more mathematics and refine your math teaching so that you and students are mathematizing more and more then join the Math is Figure-Out-Able movement and help us spread the word that math is Figure-Out-Able.