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  00:10

And we're here to suggest that mathematizing is not about mimicking, or rote memorizing. But it's about thinking and reasoning, about creating and using mental relationships, empowering teachers and students. We answer the question, if not algorithms then what?

Kim Montague  00:29

Thanks so much for joining us. We're having such a great time each week talking about math topics, and we love hearing from you about what you're interested in knowing more about. Feel free to share your ideas with us on your favorite social media. Or you can reach us directly at Pam@mathisFigureOutAble.com and Kim@mathisFigureOutAble.com. Send us your request. We'd love to hear from you. 

Pam Harris  00:51

In today's episode, we want to share with you some really important ideas that we think about when we work with teachers. 

Kim Montague  00:59

Yeah, we've been doing a lot of work around teaching teachers and working with teachers as they improve their craft. And one of the things that we keep coming back to is this. When you say the words math or mathematics, a lot of things come to mind. And when you say teaching, a lot of other different things come to mind. But when you say mathematics teaching, the number of things to consider all at the same time, it can be overwhelming. There's a lot of layers to consider. 

Pam Harris  01:27

Absolutely. So we want to address some of those layers in today's podcast that we are calling Mathematics for Teaching. And it's all about identifying and honoring what are some of the most important areas we're all working on as we grow and become the math teachers, we want to be.

Kim Montague  01:44

Sure and it might be possible that some of you haven't really stopped to consider all of these layers that we'll mention today, perhaps because some of them happen naturally, or it's not something you really have to think about. And there's probably others of you who recognize these layers. But have never verbalized them. We hope to put some words today to the act of math teaching. It's a lot. How do you tackle all these things at once? 

Pam Harris  02:09

Yeah, so Deborah Ball from the University of Michigan has helped us all think about mathematical knowledge for teaching, and how that's different from just knowing the mathematics, that there's this set of knowledge about mathematics that a mathematician doesn't necessarily need to know. But a teacher, a math teacher does need to know a different sort of set of stuff. It's not just maths, but math for teaching and teaching math. 

Kim Montague  02:36

We recognize that there's a lot of things that teachers juggle. And just because you can read doesn't mean you're a good reading teacher. And just because you eat does not make you a chef. We recognize that these layers that you have to be aware of what they are, and that maybe we can help you focus on what to work on right now. 

Pam Harris  02:56

So Kim, start us off. Tell us about one of these areas, we can focus on. 

Kim Montague  03:00

Okay, so one layer that we feel is really important is that we're always continuing to build our own numeracy. We want to use relationships to solve problems, and not just jump to a rote memorize algorithm every time we encounter a problem, right? For some of us, that means knowing that there are things other than the algorithms to even do and becoming more dense in our brain structure. 

Pam Harris  03:23

Yeah, that means that we have more relationships that we can call on. So we let the numbers and the structure of a problem influence the strategy that we use, the way we use relationships to solve a problem. Ya'll, I didn't realize how important it was for me to have such a vertical understanding. I as high school math teacher used to say, "I teach mathematics, that's just arithmetic" - with kind of a hand wave and like a like foofoo, that like that piddly stuff that you don't really need to know, which really meant I wasn't all that good at arithmatic. So if I would make an arithmetic error then it was like, "oh whatever that's just arithmetic." Now I'm very aware of the power of being able to manipulate numbers and relationships about how we can build a ton of algebraic reasoning on that sense of numeracy. And we can develop properties like the commutative, associative and distributive property by generalizing what students are doing with the numbers. So once I knew it was a thing to have numeracy and that it didn't just mean getting really good at the algorithms, like I was good at the algorithms. I could solve all the problems. But I didn't do any of that stuff mentally. I wasn't using relationships. I was just mimicking steps. Once I realized that there was this whole set of stuff that I could do these relationships I could use. Then I really began to sort of figure out what are the strategies that are worth working on? Is there some vast unknowable, unnumerable set of strategies that you could use to solve problems? Or is it really that there are specific strategies for specific numbers. It's not just any strategy for any problem at any time. 

Kim Montague  05:06

Right. So another important area for teachers of math to know is that there are major strategies that kids need to know for each given operation, and what the major models that can be used to build them. 

Pam Harris  05:20

Yeah, so most teachers I talked to, and most really good PD people out there are - see if this fits with kind of what you see, they're all about exploring doing some estimation work, kind of getting kids a sense for what's going on. And then you teach the algorithms. Like then kids need to get good at the step by step procedure. So we teach it to them, they practice it till kids get good at it. That's not what I'm suggesting. So if math is not about algorithms, if that's not our end goal, then what is it? Well, I spent a lot of time working on sort of generalizing classes of problems that would best fit for different strategies. So like pick an operation, it became really important for me to know that as I was working on my own numeracy, what were we working towards? Was it just about like, all these different relationships? Or was it really that there was a set of relationships that we could kind of focus on? And if you gave me sort of any number, let's pick multiplication. If you gave me any random number, any problem that's reasonable to solve without a calculator, could I solve it without the algorithm? That became a really important question for me. Because if we're not going to focus on algorithms on these rote memorize procedures, then what are we going to focus on? 

Kim Montague  06:40

Yeah, so can we help the listeners understand what you're talking about here with some examples? 

Pam Harris  06:44

Yeah, let's get kind of nitty gritty. So sticking with multiplication, for example, if I said 12 times anything, like pick any random number. I've got something for that. It's gonna be 10 of them plus two of them. Ten of them plus two of them. That gets me 12 of them. Let me give you another example. If I said 25 times anything, what do I know about quarters? Well, could I find a quarter of that number and then do some scaling to find 25 times that thing? You might need to think about that a little bit, because that was a little bit more of an advanced strategy. 

Kim Montague  07:19

Yeah. 

Pam Harris  07:20

What about 36, 22, 44 times anything? What do those numbers have in common? Well, let's start with 22. 22 times anything. Well, could I double the thing? Now I have two of them. Well, now that I have two of them, could I scale that times 10? Now I have 20 of them, right? Two of them times 10 is 20 of them. When I've got two of them, and 20 of them, add them together, you got 22 of them. Similarly for 36 or 44. If I can get to four of them, double to get 2 double to get 4. Now I've got 4 of that thing. Scale it up to get 40 of that thing, add it back to the four now I've got 44 of whatever that number is. I could do the same thing with 36. Once I've got three of them, scale times 10 to get or sorry, double to get six of them. Now I've got 30, and six of them and add those together. So you can give me any random number. Do I have a strategy to sort of solve that number times anything? Let me do a couple more. Because I get excited. What if I asked you for 49 times anything? Well, if you could find 50 of those things, could you just get rid of one of them to get 49 of them? Well, how do you get 50 of them? Well, that's just half of 100 of them. So if I get 100 of them, that's easy. Right? Scale it times 100, half that's 50, get rid of one that's 49 of them. In fact, nine times anything is just that 10 times it, back up one. So 69 times anything, 49 times 39, times anything. If I could find 40 times that thing that I can back up to get 39 times that thing. Now, sometimes people will go, "Oh, yeah, well, what about sevens? Like, what about 27 times anything, 47 times anything?" Well, one of my favorite ways is to switch it and do the other thing. So like if it's 47 times 22, do the 22. Right? That's 20 of them. And the two of them, don't focus on the 47. So give me any random number and I can sort of tell you how I'm thinking about finding that random number times anything. It might be some pattern with that number, it might be used the other factor, but I've kind of thought about what it means to multiply any number without using the algorithm. How can I use relationships I know? it's interesting because the curriculum out there called Everyday Math, really tried kind of in a way to do something like this. They said to teachers, "Hey, get kids to kind of mess around with these different strategies. Have the kids sort of investigate how these different algorithms work and that will help kids kind of gained some relationships. And then and then you can kind of focus." But unfortunately it kind of confused a lot of teachers. Teachers were in this bent of kids are supposed to memorize the steps. And so it actually kind of threw teachers off because teachers said, "Oh, I'm supposed to help kids memorize all these different strategies. Well, that felt weird, because all these different algorithms, why would we memorize all these different ways? Can we just have one way?" And then unfortunately, Everyday Math kind of had kids settle on the partials like partial sums, partial differences, partial products, partial quotients. And I'm here to suggest those aren't efficient enough. If we settle on those partials. Kids, when they start dealing with bigger numbers, or multiplying by decimals and fractions, all of a sudden, then kids are too inefficient too unsophisticated. And then the middle school teacher sort of hate us and because then they feel like they have to back up and teach kids how to get more efficient and sophisticated at those. So where Everyday Math kind of wanted kids to sort of play around with a bunch of algorithms and strategies and then default to the partials. I'm suggesting that the partials aren't efficient enough for decimals and fraction work. So what is? What could carry through to higher math? And it's actually kind of cool. Because the strategies that I came up with, y'all, they show up, like, if you just ask random mathy people remember, we believe everybody could be mathy. We just have to sort of help them develop those relationships. When I throw questions out on MathStratChat, people from all around the world use those main strategies, that those are the relationships that people use when they solve problems. I realized these major strategies exist. And so now I'm helping people realize what are those major strategies? They're a thing and how we can help students develop those strategies. So it's important to build your own mathematics, your own numeracy and know what the major models and strategies are for your grade level. 

Kim Montague  11:55

Right. And so I'm thinking as you were talking that, you know, I had some good numeracy. Right? And I messed around with some of these strategies. But one of the biggest things and one of the layers that I learned about so much from you was that I didn't model my thinking ever really, except with the equations. And while you were trying to solidify your numeracy and mess with strategies, you modeled a lot. And it became clear to me that it was something that I needed to learn to do and do a good job of modeling my students' thinking. So that's the third layer. I wanted to raise that math for teaching means recognizing and making sense of others use of relationships, and putting those ideas on paper to help them and others make sense of the ideas. 

Pam Harris  12:40

Yeah, that's absolutely true. Because students can be good at modeling their own thinking, but not at first. 

Kim Montague  12:46

Right. 

Pam Harris  12:47

If we say to kids, "Hey, show us your thinking and words, numbers or pictures." Oftentimes, kids don't know how to do that until we kind of help them. So it's our job as teachers to help pull out their thinking, help them verbalize what they're thinking. And then the first step is for us to model their thinking, make their thinking visible, then students can say, "Oh, when my brain does that, it could look like that? Okay. Like when my brain uses those relationships, it could look like it could be visible like that? Oh, okay." Then we can expect them to model their own thinking. And then that kind of modeling, y'all, is a big equity piece. If we can make students' thoughts visible and put them out for the class to consider. Now all students have the opportunity to take up that thinking. We're celebrating students different ways of using relationships. So as we celebrate that, we're allowing more and more students to have access to that thinking. And then models can become tools to think with as tools for computation. So one of the areas we need to get good at as math teachers is modeling, representing student thinking. 

Kim Montague  13:53

Right. 

Pam Harris  13:53

Alright, Kim, what's another area? 

Kim Montague  13:55

So here's one I think that all math teachers recognize. Something that we have to consider are teacher moves that we make while working on our content. Right? It's not enough just to have numeracy and modeling strategies of our own and those of our students. There are so many other things to think about, like, let me name a few: Good questioning, appropriate responses to students, helping create disequilibrium, creating an atmosphere of intrigue, having students work together, creating and justifying arguments, critiquing the reasoning of others. It can be overwhelming, right? To consider it all at the same time. 

Pam Harris  14:35

Yeah, exactly. Kim, do you remember when we went to the RME, the Realistic Mathematics Education conference? And it was so fun because a colleague of ours Kara Imm, it was one of the first times we'd ever seen her present. And she did a Problem String with the participants in the session. And it was so fun because at every juncture that we had a chance to sort of turn and talk to each other You and I were like, "Oh my gosh, did you see that teacher move. Oh, right there! That was was amazing, I want to adopt that." And we were just all over her teachers moves were so amazing. And there were a few that you and I, they were kind of new to us. And so we were just loving how they fit within our system of what we think about teaching and learning. And that we were able to, like add, some important teacher moves to kind of our list of things that we try to do as we work with students. And one more area of work is the idea of how we sequence tasks. Like I see a ton of work out there, really good work by good people about types of tasks. Like I have sort of focused on Problem Strings, that's kind of an area of strength of mine. I like writing Problem Strings. I think developing Problem Strings takes a knack, really understanding how they they work. I've also seen people do really good work on Rich Tasks on three act tasks and how we can kind of get kids really investigating and exploring. But I see less work out there on sequencing tasks, on how to help students develop using tasks that build on each other that I can sort of use tasks to go from here and help kids just up the ante enough that I keep kids having access to the task, but we're also keeping kids challenged by the task. This is a huge area of interest for me. How do you decide what to do next? How do you decide what to do tomorrow next week, what influences the choices that you make? 

Kim Montague  16:21

That's a great thing to consider. So here's the areas that we've identified that we work with teachers on number one: building your own math. Number 2: knowing the major models, strategies and ideas for a domain. Number three: modeling student thinking to make it visible and take-up-able. Number four: high leverage teacher moves like questioning. And number five finally: advancing the mathematics by sequencing tasks. 

Pam Harris  16:48

And then interesting Kim, as you and I've sort of developed these sort of five major areas that we kind of work on, then it depends on the content because given new content, we actually sort of stuff cycle back through those five stages. Like, after I had developed all of that, we just talked about for the four major operations addition, subtraction, multiplication, division, then I went back through that cycle for fractions. And I went back through that cycle for Proportional Reasoning. And I'll be honest, I'm still working on that cycle with different areas of high school math. So listeners, if you haven't heard already, I've just launched my signature teacher implementation help. And I call it Journey. 

Kim Montague  17:28

Oh, we're so excited in Journey we've created a progress path with milestones and action items to help you figure out where you are and what you want to work on next. 

Pam Harris  17:38

So I invite you to check out Journey on the website. We'll put that link in the show notes where you go and read all about journey and how it can help us sort of decide where you are. If you're interested in implementation help, you like what you're hearing on the podcast, what you see on the website and you want to make it happen in your teaching. Check out Journey.

Kim Montague  17:59

Yes. As you listen today, you may have thought, "You know, I hadn't considered everything that goes into my daily work just in this one content area. I'm more aware now. I can put some words to my aspects of teaching." Maybe one of them pinged for you that you really want to focus on. Or maybe you're interested in checking out more in detail. Join us on Journey. 

Pam Harris  18:18

So thanks for listening today. Don't forget MathStratChat on your favorite social media on Wednesday evenings, and y'all thanks tons for the five star ratings on Apple podcasts. We love reading your comments and a-has that you're having. So we really appreciate you guys rating the podcast and giving us some comments and feedback. So if you're 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!