What comes to mind when someone says the word "strategy"? There are at least four different meanings for the word "strategy" in mathematics teaching. In this episode Pam and Kim parse out the different meanings and discuss how those meanings shape the outcomes of our teaching.
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, reasoning, about creating and using mental relationships. We take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keeps students from being dumb mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what?
Kim Montague 00:37
So in this episode, we're going to talk about strategies. So Pam, when I say that word, what pops into your mind?
Pam Harris 00:45
I think about how you solve a problem.
Kim Montague 00:48
Like problem solving strategies? Somebody actually asked me about that. And we've really already done an episode, it was Episode 15. If you're interested in what we think about problem solving strategies,
Pam Harris 01:00
So check out Episode 15 if you really want to hone in on problem solving strategies. What do we mean by that? Because I don't mean that. I don't mean those, not problem solving strategies. When you said strategy and I said, how you solve a problem? I mean, how you mess with the numbers or the structure to solve problems. You know, that big ebook that we put out not too long ago. We had a whole series, what is it, episodes 90 something through 90 something, where we talked all about: If Not Algorithms, Then What? We had If Not the Addition Algorithms, Then What? We had: If Not the Subtraction Algorithms, Then What? We had: If Not the Multiplication/Division Algorithms, Then What? And in those we talked all about those strategies. Come on, Kim, you do that. So interesting, we've been having conversations lately, where it has become apparent to me that while we think we are talking to each other about strategies, we might actually have four distinct meanings of the word strategies just in math teaching. And so we thought we'd take just a minute today, in this episode to parse out the meanings. What could be the four different meanings of strategies? So Kim, I'm gonna hone in on the one I was just talking about. In that whole series, If Not Algorithms, Then What?, where we literally worked out, or we worked with people, what are the top five addition strategies? What are the top five subtraction strategies, top five multiplication - strategies being if I'm going to solve the division problem, if I want to build a kid who's really thinking and reasoning about division, what are the top five relationships they should own? So that those five strategies become natural outcomes. They can hit a division problem. And they're like, "Yeah, I've got I've got some relationships I could use and I can choose between these two. I'm gonna choose that strategy for this problem. That strategy. Oh, maybe actually, that, yeah, that one, because that's really efficient, it was sort of where my brain went today." That's a meaning, a sort of a numeracy strategy, how you mess with the numbers or structure. So give us a little bit more of a sense of the ones that earlier would when I said, "Problem solving strategy." You said, "Problem solving strategy?"
Kim Montague 03:12
Yeah, right. So those are the ones that, like I said, we mentioned in a different episode, where somebody might say, one problem solving strategy might be to solve a simpler problem, or a problem solving strategy might be to look for patterns, or to look back to make sure that it makes sense. And we're actually, although some of those things are fantastic problem solving strategies, we also don't mean acronyms, where it says underline this, and circle this, and check for this, like some of those acronyms where it's like CUBES or STEPS And yeah,
Pam Harris 03:47
All the different, we can't even remember them. Right? They're acronyms, right? But they're kind of step by step procedures that kids have to follow all these steps every time they... Every student for every problem. Yeah. But that could be a meaning of the word 'strategy', that kind of problem solving strategies. We don't advocate those, memorizing a bunch of steps to solve problems. If you want to hear more about that, check out Episode 15, where we really dive into problem solving strategies, what to and not to do for those. So what's the third meaning that we might have as we're talking to each other, we're math teachers, we're interested in teaching math? What might be a third meaning about strategies?
Kim Montague 04:24
Yes. So we have problem solving and numerical strategies, but there's also teaching strategies, right? Teaching Strategies. And when we say teaching strategies, we also, I think, kind of get twisted a little bit because we might be talking about teaching like an adjective, or like a noun. Are we talking about teaching students some strategies, or are they teacher strategies?
Pam Harris 04:50
Yeah, parsing that out is tricky. So if we're talking about, doesn't seem like, you said adjective or noun and I'm, like stuck. You have to tell me which one this is. Okay. So if I'm talking about strategies like: good questioning, and wait time, and using a private signal so that all students have enough time to respond, we don't have this sort of speed thing going on. Or if I talk about partnering kids strategically and appropriately. Or if I talk about, like all the discourse moves to create discourse in my class, things that the teacher does. Teacher moves, we call them high leverage teacher moves. Those are some teaching strategies, right? Yeah. Those are teaching strategies the teacher does. Or teacher moves, teaching strategies.
Kim Montague 05:35
I guess, I kind of in my mind, I think it's less important that it's an adjective or a noun. But I think of those as their teaching. Like they're, how do I say this?
Pam Harris 05:45
Happening as you teach.
Kim Montague 05:49
I don't know how to say what I'm thinking in my head. Yeah, I don't know.
Pam Harris 05:54
For me, it's the kinds of things that we - in our deep dive workshops, we have a whole module set aside in the deep dive workshops, where we really look at everything we've worked on and the math, and we relook at how we did it. And we talk about Ooh, like was was it on purpose that I asked that question at that point? Was it on purpose that I asked every person in the room a question? Was it on purpose? Why did I say could you restate what that person said? Or does everybody understand this person strategy strategy? There are numeracy strategy. I was about to say numeracy problem solving strategy. That's tricky. Did everybody understand what could have been either? Right? Did you understand how this person went about solving that problem, numeracy wise, the relationships wise? Or did you understand how they used a simpler problem? Or how they went and asked a colleague for a hint. Like there could be problem solving strategies. There could be numeracy sort of mathematical relationship. As we sort of point those out, what teachers are doing to teach, those I call 'teaching strategy' teacher moves. We usually call them high leverage teacher moves. What are the ones that we can really get a lot of leverage out of, if we just change this little thing, tweak this to create discourse in class? Those are sort of teacher moves. Because Can I think we've got one more kind of thing that could be, we could call a teaching strategy.
Kim Montague 07:16
Pam Harris 07:17
But it's less about, kind of what a teacher does as the teacher's teaching.
Kim Montague 07:22
And it's not about the moves they make.
Pam Harris 07:24
Yeah, it's not a high leverage teacher move. It's more of a routine. Yeah. Like, sometimes we'll have teachers talk about, "Well, I use numberless word problems," which by the way we love. Or "I leave the answer off at the end of word problems, to get students to really think about what's happening in the problem. And then I'll say-", when I say leave the answer off, I mean, sorry, I actually meant leave the question off. So I might, I might have the whole word problem, but then not ask a question. Just set up the scenario. And then say, "What what questions could we ask if this was the scenario?" What questions could be asked and let the kids sort of generate a bunch, and they can answer them. And then you can then when they look at a word problem later, and all they have to do is answer the one question, like, "Just this one? We don't have to come up with a bunch of just-?" That's an, all of those kinds of things. Another one that comes to mind, Peter Liljedaul and his work with Building Thinking Classrooms has some ways of teaching like using vertical non permanent surfaces. So you put kids up at these vertical non permanent, so like a whiteboard, or something that you can erase. That's a way of teaching, that's a teaching strategy to get kids up. And if you add to that, that there's only one marker and you add to that, that you're visibly randomly grouping them. All those are teaching strategies to help you teach it a different way. And from his perspective to create a thinking classroom. Those are sort of ways of arranging your teaching, ways of kind of building set routines in classroom. So we could be talking about that kind of strategy. So, I don't know there's like four different kinds of strategies that we sort of talked about today. We could have a numeracy strategy: how you mess with the numbers of structures solve the problem. we could have a problem solving strategy: how do you attack a problem? What's the way that you kind of help yourself understand what the problem is asking? And then how do you kind of decide how you're going to employ a numeracy strategy to solve it? Maybe? That's number two. Number three, a teaching strategy, a high leverage teacher move, the the sort of discourse moves, the things that the teacher does, in the act of teaching, to help students learn. And then also the fourth one, kind of these ways of teaching, setting up more routines, where we might, like we said, numberless word problems, leaving the question off at the end of a word problem, vertical non purpose surfaces, randomly grouping those kinds of ways of setting up teaching.
Kim Montague 10:04
Pam Harris 10:05
Kim Montague 10:06
Well, and I think you and I were talking about this, because we wondered, like, in the conversations that we're having with people, how clear are we being? And how clear are other people being? And like, why does it even matter that we parse out the differences between these strategies? And it boils down to, there's a lot of really cool routines and strategies out there of all kinds. But people want to adopt them as ideas in the room, because they're interesting, and they get kids thinking. However, if you don't have the mathematical strategies to support the other three, then they all fall flat. Like there's nothing we're talking about. And there's nothing worth pairing kids for. There's nothing worth problem solving for. And so we center a lot of our conversation around the numerical strategies, because then you bring meat to the conversation.
Pam Harris 10:54
Yeah, I almost just, like want to have you repeat exactly what you said. Of those four strategies -
Kim Montague 11:01
Pam Harris 11:01
We spend so much time talking about and building with teachers, the mathematical strategies, the numerical relationships, the spatial relationships, the modeling relationships. We spend a lot of time building those strategies. That's why we did all those episodes in the 90s all about If Not Algorithms, Then What talking about those mathematical strategies, because if we don't have those, the rest of them, I mean, they might make your class a little more engaging, or a little less boring than if you just just teach traditionally. All of those things, you could do any of those other three kinds of strategies and they kind of fall flat. Because you're not actually mathematizing. You're not actually mentoring students to be mathematicians, if it's still, if the content of your classroom is still about mimicking step by step procedures, then have at, go at the other three kinds of strategies. It will, you know, probably perked things up a little bit. But if you really want a real thinking classroom and a really engaged mentoring set of students where they are in classroom community of learners where it's shared, created, understanding, then we need to have all four happening. And the most important, we think, is to build that mathematics. So we really have those mathematical relationships going, nicely said. Alright. 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