There is so much that is good happening in the math teaching community! In this episode Pam and Kim discuss how we can determine how much of which good things to put into our classrooms.
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Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam. And you found a place where math is not about memorizing or mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. Ya'll, we can mentor students to think and reason like mathematicians. Not only are algorithms really not helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be. Alright, Kim, on today's episode, I thought I would tell you. We talk a little bit about... I had dinner at CAMT. So, CAMT is the Conference for the Advancement of Mathematics Teaching. It is the Texas State Conference. It's super good. If you ever get a chance to come to Texas, ya'll, come join us in our super state conference. And I sat down with a bunch of folks and Curtis Brown was there. We had a nice chat over dinner. And I didn't know that Curtis Brown and Joanie Funderburk have a podcast. And it is called The Room to Grow - A Math Podcast podcast. I put the word podcast in there twice. But A Room to Grow... No. sorry. Room to Grow - A Math Podcast by Curtis and Joanie. Yeah, it was great. I've known Curtis for a while. We had a nice chat and everything. And so on my drive home from Fort Worth...I live in Austin, so it's about a three and a half hour drive...I listened to some of their podcasts. And one of the episodes they did we thought we'd put our little spin on. So, we'll...
And I'm Kim. They have a great report.
Yeah, they do.
You sent me an episode to listen to, and it was super fun.
Yeah, it was nice. So, we are going to talk about their episode. I think it was 3.6 Don't Lose the Mathematics. And they talked about kind of nixing the tricks. Yeah. And so, we thought we'd put our spin a little bit on, and add to that conversation. We would wholeheartedly agree that we should nix the tricks. There's a fine booklet out there. It's online. You can get it for free. You can order the book, if you want to actually have the hardcopy. Called Nix the Tricks. And, Kim, I think you and I would both agree that it does a fine job of identifying some tricks that math teachers have kind of maybe found over time, that unfortunately they're promoting as doing math. They're like, "Hey, here's a here's a trick to do this. Here's a different trick to do that." And many of us are saying, "Nah, let's nix those tricks. Let's actually focus on what mathematicians do, the way they think, and not teach these tricks." And so, Curtis and Joanie, in that episode, mentioned a few of the the tricks and came up with some things to think about. Do you want to mention anything else about the episode? Or, then, I'll just dive into what I was thinking about.
Yeah. So, there were a couple of things that they talked about that prompted me to revisit the Nix the Tricks website. And I had forgotten that it was kind of curated, I can't remember who it was, but a couple of people. But a lot of it was submitted by the math community in general, which I absolutely love. And it's like a collection of things. And I got to tell you, Pam, when I got on the website, I was like, "Wow, there's even more tricks than..." I mean, like, this is not the kind of thing, listeners, where we're saying, "Go check it out, and like learn a bunch of new things to do in your classroom." We would absolutely recommend that you don't go learn some things. We won't learn them. It's just people telling you like, "Here's a thing, a trick you can do." But yeah, there were several on there that I was like, "Wow, I've never heard that before." How is that? There's like 70 different.
It's not math, yeah.
Like, can you imagine being a middle school kid, and you're like, "Here's one more."
"Let's try to memorize all of those."
Oh, my gosh. Oh, my gosh. Yeah.
Which is kind of what we get, right? The older the students get, if that's the kind of math class they've been in, it just gets harder and harder to say, "Woah, woah, woah. Yeah, you get answers with those tricks. But you can actually think about that stuff, actually reason through using what you know." Oh, let's see. We might have to back up and know some more things.
One of the things that I thought we could kind of add to the conversation a little bit is, one of the things they talked about is how do you know if something is a trick or not?
And we thought we'd add a little bit of the conversation from Piaget. So, John Piaget was kind of a renaissance guy, did a lot of things. But one of the things he did was talk about education and learning psychology. And he talks about different types of knowledge, and if we could concentrate on two different types of knowledge that pertain to kind of what we're talking about. One of them is social knowledge, and one of them he calls "logical, mathematical". And I'm just going to call it logical knowledge. The difference between social knowledge and logical knowledge is social knowledge is by convention. It is the stuff that we have deemed to be so. We have defined it that way. We've put a tag on it. One way to think of that is if you change languages, someone would have to tell you the new word. So, it is socially created. It's by convention. It's the kind of stuff we have to tell kids. That's the stuff we need to tell kids. Those are things that I'm kind of okay if you have a pneumonic for because otherwise... To remember them, it could be helpful to have a pneumonic to remember that stuff. One of the ones that is true, that Joanie and Curtis mentioned in their podcast, was SOHCAHTOA. So, SOHCAHTOA is a bunch of letters to help kids remember the way that we've designed trig functions. So, we have defined trig functions that the Sine is the opposite over the hypotenuse, and that's the "SOH" part of it. So, we've got an S... How do I even do that? See, I haven't done this for so long. Sine, opposite over hypotenuse, right? So, Sine OH. So, S-O-H is the "SOH" of SOHCAHTOA. That's funny because see I don't even actually use SOHCAHTOA to help me do because I just own these so much. So, then the definition of Cosine and Tangent would come in there as well. Well, somebody made up the word "Sine" to represent that we could have the ratio of opposite over hypotenuse. The ratio of the opposite to the hypotenuse is the Sine. We just made that word up for Sine. We could have called it the "Kim", like Kim could have been the ratio of the opposite to the hypotenuse. So, I'm okay that... I mean, I want you to use it a lot, so that it just becomes a part of you. But because somebody made up the word "Sine", and they decided that that one is connected to the ratio of the opposite to the hypotenuse, I'm okay that we sort of. Well, we have to tell kids that because we decided it was going to be that way. So, that's an example of something that's social. So, yeah. Go ahead and tell kids. But boy, then to actually use Sine, we need a lot of experience. That's a logical relationship. So, first of all, since it's a ratio, we need to give kids experience reasoning proportionally. Kids need to be able to reason proportionally about ratios or all they see it as number over a number. That's positional. They're not going to be able to actually reason about what's happening with that ratio, if they don't have some sort of ratio proportional relationships. And then, they can't use Sine, the Sine function, to do all sorts of other things. If all they do is memorize SOHCAHTOA, if they just use that pneumonic to memorize it, then we're sort of stuck in this kind of trick area where we're pretending that math is all social. There's some parts of math that are social, but most of it, the vast majority of the things that we want kids to learn, are not social at all. They are that logical type of knowledge. And for that, we want to give kids experience.
So, that we can add to the conversation a little bit, that if you can look at the thing you're trying to teach, and say to yourself, "Hey, is this... Is part of this social?" Well, then I'm going to tell kids that, and I might help kids rote memorize that. But the vast majority of whatever it is, is probably logical, mathematical, then bam, we're jumping in, and we're going to actually have experience learning that thing, experience gaining relationships and making the mental neural connections stronger and stronger.
Yeah. I think more and more, thankfully, we're hearing people say, "These are some tricks. Don't teach kids the tricks. It's not about cutesy. It's not about a bunch of things to memorize." I mean, I think we're, hopefully, trending towards that becoming less and less of a thing. But what I am seeing, I think you're seeing this too, is that instead of those tricks than what people are doing is just explaining more clearly, or maybe explaining things with a little bit more mathematics. But it's still a little bit too much of explaining. Right, like a little too much of "Teacher has all this knowledge. Let me just tell you."
"Let me just dump it in your head."
Yeah. Which is also not something that we advocate, right? We want to build mathematicians by giving them experiences, by posing important questions, letting them rustle and (unclear) some stuff, having deep conversations, letting them be a part of the process, rather than just standing at the front telling them with a little bit more meat than some of the tricks we might have done in the past.
Yeah, because we actually believe that's actual learning.
We don't believe that you can unzip a kid's head, and pour some stuff in, and then they own it. That's like saying, "Go ask GPT how to, and then..." Pick something that's... In fact, here's here's a way that sometimes I determine whether something's logical, mathematical, or social. If you can ask Google or chat GPT a question, and their answer actually you can own it, you can do something with it, you can run with it, then it's probably social. Like, if it's enough. If you're like, "Hey, when was..." Pick something. "When was Mount Rushmore created?" Some random social thing. That's a social thing. It happened. There's no way you can figure that out.
"Who were the signers of the Declaration of Independence?" Bam, like I'm going to go, I can ask that, when I get the answer, it's satisfying. I'm done. But if I say, ask chat GPT all the connections and relationships to build a nuclear bomb, which I don't even think it will answer that question. Or to everything about quadratic equations. It might tell me a lot of stuff about quadratic equations. But that telling, if I don't have enough to experience to own that stuff, then then I'm not going to own it. I can't do anything with the answer. So, chat GPT might give me a great answer. But if I don't have experience grappling with those relationships, then I'm not going to own them enough to do something with them.
Yeah, so some people are saying, "Don't teach the tricks." We're saying, "Don't just explain more. Actually build mathematicians by giving them experience."
Yeah. So, the other thing that I think is fun to bring it today is that there are lots of educational leaders out there. And we're so lucky to have a bunch of people doing really good work to move away from tricks and to do a better job of giving kids these experiences. And so, you think that there's a lot of richness to bring to a classroom. But the one that you probably focus on a lot more, because you love them and because they're so beneficial, are Problem Strings. Right, like that's a...
That's kind of my schtick, yeah.
It's kind of an important thing. Maybe because once it becomes routine, it's a quick routine. That there's so much meat. Like, there's there's so much bang for your buck in a Problem String. So, you do Problem Strings. Some people do number talks. Some people do, you know, all these other things. And let's talk for just a second about the idea that when you go all in on a thing, like a Problem String or a number talk or whatever, there are some really good things about it, but there's also some things to be on...
(unclear) about, be aware of. Because we've both had conversations where people say, "Well, I do this thing." But when they describe... Let's say number talks. They do number talks. But when they describe what they're doing with number talks, or how they're sharing number talks in the classroom, or what they think they're getting out of number talks, we've heard some scenarios where we feel like people were just kind of like missing the boat a tad bit. And then, let's talk about the kind of the negative to that. Like, what can happen when people go all in on one thing, and that's the only thing that they they focus on?
Yeah, and especially if they maybe misunderstand the thing or the purpose of the thing. Yeah. So, you mentioned number talks. Let's start there. So, are number talks great? Absolutely.
But I got to tell you, the first time that I heard the phrase "number talks", or the title "number talks", I pushed back on that a lot. Because I said to myself... Or maybe even math talks. People are like, "Oh, you know. Yeah, do a math talk," as if that's the only time you talk in math class. And I was like, "I want to be talking a lot in math class, not just during this one sort of thing that we do." So, a thing to consider is, if you've heard math talk or number talk, and you're like, "Oh, yeah, that's when we talk in math," I'm going to push back on that, and say, "No, no, no. No, we should be talking a lot in math class." Sure, there are times where students are quietly doing their own work. I think that is an appropriate use of math learning time a little bit. But mostly we're having conversations, were grappling, we're having kids wrestle with. We're giving students good problems to tackle, so that they can then wrestle and grapple with the relationships. And then, we're making that thinking visible, and we're pointing at it. But we're doing that not just during a number talk or math talk time. We're doing that all the time. That should be happening during almost all the work that we're doing. So, that would be an example of something that maybe we'll bring up. A tweak that we would suggest is, sure you can do number talks sometimes in your class. But that doesn't mean that you're not talking in other times of class. Cool. So, I'll mention another one, Kim, that I've been thinking about. I hear sometimes people say, "Oh, Exploding Dots is the best thing ever!" So I think Exploding Dots, depending on how you define it, is kind of cool. I like to listen to...
Thank you. James Tanton. I can't believe. Whoa! I got sleep last night. James Tanton is an enjoyable presenter to listen to. He tells good stories. He's pretty funny. However, I would mention that I think Exploding Dots does its best work when we use it to think about place value.
So, how does our base 10 place value system work? And so, if I have this dot in this place. Like, say I have one dot in the tens, place, bam, it can explode into these 10 ones in the ones place, or I can gather 10 tens in the tens place. And I can grab those 10 tens, and I can stick them into one dot in the hundreds place. Or I can explode it back and have 10 tens in the... You know like, whatever. That is a great and it's a fine way to kind of think about our base 10 place value system. But then, if you'll consider that other times where he talks about Exploding Dots, or other ways that people have used Exploding Dots, really then is a way to understand the traditional algorithms. Well, if your goal is to understand the traditional algorithms, okay, then go ahead. Yeah, use Exploding Dots to help you understand them. But if you've listened to the podcast very long at all, you know that mimicking those traditional algorithms, or even understanding those traditional algorithms, is not really our goal. We want kids to solve problems fluently, and do that as they build their brains to reason mathematically. That's our goal in a math classroom. So, we're not really... It's not a big goal to understand the algorithm, so we kind of de-emphasize Exploding Dots in the work that we do. So, some people have asked me, "Why don't you Exploding Dots more?" I'm like, I mean, they're fine for understanding place value, but for doing anything else, not my goal. So, I'm not denigrating. I'm not saying that they're bad in any way. It's just, I wouldn't go all in on them. What were you going to say?
I was going to say, do you remember one time we were interviewing some students, and we had a young lady who we gave her a problem. And it was a not super complicated problem for maybe her age. And we had seen her do some work, where we knew this problem was not likely to be super challenging problem for her. She had done some other things that were maybe a little bit more complicated. But because she had been using Exploding Dots so much, she did that for every problem. And, you know, I don't pretend to know James Tanton.
Well, did that. Let me describe what she did. So, it was like a problem like 7 plus 8 or something.
And she dutifully wrote down 7 dots and dutifully wrote down 8 dots, and then moved her paper over, and then like circled 7 plus 3 and said, "Okay, now that's one 10. And then, what's leftover?" And then, she's like, "Okay, so that's 15." And I looked at her, and I was like, "Do you know 7 plus 8?" Were you interviewing her? Both of us there I remember. And she's like, "Oh, yeah, it's 15." And we're like, "But did you have to do all that work?" "No, no, no. But that's what I'm supposed to do."
Right. So, I think that's the point, right? That's the point is that I don't know what James Tanton's take would be on that. You know, I don't. But there are things. And so, her experience with either her teacher, or previous teachers, or whatever was that "This is what you do." And so, a thing that is maybe a really good idea, maybe Exploding Dots is fantastic for a place value or whatever, but what we see happening is these really good things turn into "Do the same thing all the time," or "This is how the one way that you use them." And so, I think that's challenge, right, is hearing something and going like, "In what way do I want to use this idea that somebody's sharing about and (unclear)..."
"...and have it not go awry."
Right. And when would I not use it? And how is it not helping my students in these particular instances? Can I bring up another one?
Or do you want to say more about that?
No, go ahead.
You get asked a lot, "What about manipulatives?" And it's kind of an interesting question because people will say, "What about manipulatives," question mark. And It's like, Wait, what's the question there? What? Manipulatives?"
What about manipulatives?" Yeah. Like, say more.
How? When? Which one? Mmhmm. So...
You'll need a long answer, but in brief.
Mmhmm. So, I have to tell you. The first thing that comes to mind is Gail Burrill was a former NCTM president. And when I was a young teacher, she was the NCTM president, and she did a president's message. And it was when the Wendy's commercials... I'm totally dating myself here. The Wendy's commercials had come out. "Where's the beef? Where's the beef?" And their whole thing was, you know, "We have we have big hamburgers," I guess. But "Where's the beef in everywhere else?" But her thing was, "Where's the math?"
And she said, as she was traveling around as the NCTM president, she saw a lot of manipulatives out in classrooms, and her concern was, "Where's the math?"
And so, I share that concern. And here's where I would add to that conversation. We do a lot of talking about developing mathematical reasoning, in that we need kids to learn to count and solve problems using counting strategies, but then we want to advocate additive reasoning and thinking in terms of bigger chunks of numbers. If we give kids one to one manipulatives all the time, kids will not necessarily be nudged, be encouraged to build their brains to think in terms of chunks of numbers because they're counting one by ones. If we give them one to one manipulatives, they will continue to count one by one.
So, the biggest caution that I would give. It's not the only one. But the biggest caution I would give about manipulatives is to think about what is my goal here, and is the manipulatives supporting answer getting only or is it supporting the goal of building reasoning? And what kind of reasoning? So, then, we actually have to identify what kind of reasoning we're trying to build. I'll give you a quick example. Early, early, when I dove into elementary. So, I was secondary teacher. I got super interested in elementary. I started diving into research. One of the things I did was volunteer at my kids school. And one day, they said, "Hey, today, we're not going to give you that group of kids you've been kind of extending." I had a group of kids, and I was just experimenting and trying some things with them. And they said, "Today, we don't have time for you to do that. Sorry, we didn't, you know, tell you ahead of time. But today, can you just help this one student. She's really struggling." And I was like, "Oh, I don't even know what I will do with this second grade student, but sure." And so, I started chatting with this student, and they said, "Help her add." And it was like add two digit numbers. And they gave me base 10 materials.
And so, I wasn't even sure what to do with base 10. And so, I said to her, "You know, we got this problem. 28 plus 37. Like, what are you going to do?" And she goes, "Well, I think I'm supposed to grab these rods." And I said, "What's a rod?" Because like, this is me early, right? I'm like, I don't even know what a rod is. She goes, "Well, you know, it's this thing right here." And I said, "Well, what is this?" expecting her to say "It's 10." But she goes, "It's a rod." And I said, "Right, but you know like, if this is 1, what's this?" And she goes, "It's a rod." And now, you might be like, "Well, Pam, that student misunderstood," or whatever. But when I pulled out the 100, and she said, "That's a flat." And she goes, "I think it's called a flat." And I was like, "Um... But like, how many of these little guys are in that?" She goes, "I don't know." I'm like, "Well, how many of these little ones are in this 10." And I think I literally said, "How many of these ones are in this 10?" And she goes, "It's a rod." And I'm like, "Right, but how..." So, then, she took the little one, and she lined it up against the rod to see. And she goes, "Well, I guess in this one there's 10."
"In this one." Yeah.
Yeah, so my point is that I'm not sure what the teacher had tried to do. Obviously, you could do a better job of that. But in manipulatives, we have supposed, we've... How do I say this? We've created manipulatives to represent the mathematics that we have created in our minds and supposed that students can see the mathematics in that manipulative. And that is not true. Students cannot just look at this pre-constructed rod that we've stuck together and all of a sudden go, "Oh, yeah, that's a 10. And see, there's 10 of those in that flat,. And there's 10 of those in that cube." And so, we can't... I'm going maybe longer than you wanted me to. But we can't just assume that that because we've created the relationships, and we now know the mathematics that are involved in that manipulative, that then therefore it Woah magically appears to the students in the manipulative. So, I don't... Yeah. There's four other things I can think about manipulatives, but we won't make this too too long.
Yeah. Well, I don't know how long we want to go. But I think the big recap, the point here is that are manipulatives, or number talks, or Exploding Dots, are they bad? Absolutely not. There's value in all of those and many more other things.
What's important to keep in mind is that if you think only doing that thing is going to get you very far.
We have to do some (unclear) for everything. They have always be out, and we have to always have kids using. "Kids don't have manipulatives in their hand, you're doing a bad job teaching." We're saying no, not with that.
Sorry to interrupt.
That's okay. And also, you know, there are ways that these things are being used that miss the mark of even the people who are sharing about them. Right? So, like just just recently we talked about Problem Strings and how, you know, we think we're being super clear about them, and yet there's still questions. And so, I love when people continue to ask, right? Like, "What's the value in this?" And, "How often should I be using it?" And, "In what way should I be using?" And, "How do we know if they're being effective?"
Yeah, and you know that you're on the best journey possible when you continue to ask those questions. And we all continue to get better at refining what is the best way to use each of these individual things, what are ways that we want to de-emphasize, and how are we using them, so that we can get the best mileage, not just getting answers, but actually creating mathematicians in our classrooms? Yeah, nice. Cool. So, we can't possibly talk about all of that here. We are going to mention. Not mention, we're going to dive in and do some really good work with these kinds of things and more of them. Like, it's a super good thing, but let's really get more. Let's clarify how best to use and maybe how not to use. We're going to do more of that in the free challenge that's coming up very soon. We have a free challenge coming up. It is August... Oh, golly, Kim, I don't have the date handy. It's August something. I've almost got it. I've almost got it. It's 2023. So, if you're listening right now, we have a challenge coming up August 23rd through the 25th of 2023. But, ya'll, if you're listening to the podcast at some other time, and it's not that time for the challenge, we run challenges. And so, the best thing you can do is make sure that you get on our email list, so you'll know what we are doing, coming up free to continue to clarify. One thing that's really awesome about the challenge is, unlike this podcast, they're visual, so you can see what we're doing. We get to interact. It's super fun. We love to have people join us in the challenge. So, definitely join our email list. Go to mathisfigureoutable.com There'll be a pop up. You can get on the email list, make sure. That way you can hear what is coming up as we continue to all help each other refine what we're doing in our math classrooms. So, ya'll, 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!