Ep 212: Making Progress - Modeling Your Strategies

What's next on the landscape for teachers after they've built their own math understanding in a particular area? In this episode Pam and Kim discuss math for teaching, the next layer on the Success Map that teachers should consider as they are learning how to make math more figure-out-able for themselves and their students.

Talking Points:

• Math for teaching is different body of math than the math anyone does
• Understanding and naming the major strategies and models for each operation
• Judiciously using the major strategies and models
• Crafting rich problems
• Build understanding in this stage alongside your students

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Register for free Developing Mathematical Reasoning Workshop: https://www.mathisfigureoutable.com/freeworkshop
Download the free Major Strategies ebook: https://www.mathisfigureoutable.com/big 

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Transcript

Pam  00:00

Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam Harris, a former mimicker turned mather.

Kim  00:09

And I'm Kim Montague, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  00:17

We know that algorithms are amazing historic achievements, but they're not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop.

Kim  00:31

In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  00:38

We invite you to join us to make math more figure-out-able. Bam. Alright. Hey, Kim, how's it going? 

Kim  00:44

Hey. Good. How are you? 

Pam  00:46

Alright, so I think we're in step two, series two, part two of a series that we're going to do. As we kind of pull back the curtain on how we differentiate for teachers when we do workshops, and zooms, and online things. In our journey group and in our asynchronous workshops, if we are working with leaders or teachers in some way...like you just said in our in our opening that we're trying to improve math teaching...is we're working with those groups of teachers. We know that we're working with people who are all over the map experience wise. They're all over the map geographically, for sure. But they're also all over the map experience wise. Some people have more experience making math more figure-out-able for themselves. They've kind of maybe always thought about relationships. Though, it's interesting. You know, often I will meet those people. In fact, I just met one in a workshop that I just did. And Whoa, she was like, "Oh, yeah, my brain just does that." But then we got into some other math. And she's like, "Oh, I've never thought about this math." That was kind of interesting. As we sort of progressed through the Developing Mathematical Reasoning graphic, we went from additive to multiplicative, to proportional reasoning. And when we got to proportional reasoning... Before that, she would be like, "Oh, yeah. My brain just does this. No, my brain does that." And I was making her thinking visible. And when we got to proportional reasoning, I asked the question... Some of you if you've ever done a proportional reasoning Problem String with me, you know that there's a point where I'll often ask can you can this middling strategy, does it always work? And if it's a proportional relation, yes. If it's a linear function, yes. But any other function maybe not. Like it really has to do with the constant rate of change. And it was interesting in that moment because she was like, "Oh. No, yeah. It works for all functions." And I was like, "All functions?" "Oh, yeah. All functions." And then the more we thought about it, she was like, "Oh, wait. I don't know that I've ever figured out this part of math." So, my point is whether you have always in a specific part of math sort of figured it out or you haven't, we would suggest that everybody needs experience figuring out that part of math. But then after that, there are kind of other stages as we work with teachers that were aware of, that you might need more experience in this particular part of math or math for teaching. Or you might need more experience in this particular part of math teacher moves. Or you might need more experience in this particular part of sequencing and advancing tasks and advancing the math. So, we're aware that there's kind of this map of different landmarks where teachers all over the place. Ya'll, if you haven't listened to the last episode, we would encourage you to listen to the last episode where we kind of outlined and described the way we kind of think about this map and kind of the first stage or place that people might be gaining experience. We kind of described that in the last episode, so if you want to make progress, check that episode out, so you kind of hear about that, first stage where we talked about what does it mean to make a particular kind of math figure-out-able where you are solving problems using what you know, reasoning, with mathematical connections that you own. And in the process, making more connections. 

Kim  04:09

Yeah.

Both Pam and Kim  04:10

(unclear).

Kim  04:12

I was going to say you might be listening to this as a leader who wants to think about working with their teachers. But you might... Because you've mentioned leaders several times. But you might also be a teacher who doesn't work with other teachers, but you're like, "Where am I on my journey? Like, how would I describe where I am?" And as you're listening to this series, you might hear some things that you're like, "That's where I am." And it's important because then you know where you're going, right? So, if you know where you are and you know where you want to be, then you can identify where you are on your journey and have something to move towards. Nice. And have some ways to move towards it. Because we're going to give you some action items today. If you recognize, you're like, "Ooh, for this particular area of mathematics, bam. You guys are talking about. This resonates with me. This feels like the kind of work that I'm working on. Then we'll give you some action items to do that you can move forward and gain more experience in this particular stage, so that then you can work on the next stage. Yeah. Yeah, so thank you for bringing that out. You could be a leader listening to this working with teachers, but you could also be a teacher identifying through what lens might you be kind of participating in all the stuff that we're doing here?  Yeah. 

Pam  05:20

Cool. So, today, we thought we would talk about kind of the next stage. So, if the first stage is building your own mathematics, your own... We'll say numeracy, but that only applies to like numeracy. I wish there was a building your own mathacy, building your own... Like, there's got to be some way of saying your mathematics in a particular area. Once you've kind of done that, you can solve problems using relationships you know. Then, we think that there's now a new area that opens up for you. And I kind of call this mathematics for teaching. And before we dive into it too much more, I just want to give a shout out to Deborah Ball from University of Michigan. I think Deborah Ball might be one of the people that I listened to speak where I'm like, "Oh, man, you talk so fast. I can't keep up." Like, she's so well spoken, and I love listening to her speak. One of the things that she has given us is she's done a lot of really nice work in helping us realize that there is mathematics, but there's also this other body of mathematics for teaching. And that I could be an engineer or a computer scientist or a physicist. Pick somebody who uses math, like in their in their work. And I could do the math that I need for that work. But I might not, and probably don't necessarily know, the math for teaching, that there's a different body of mathematics that we actually need for teaching. And so, we're going to dive into what does it mean not to just be able to solve problems using what you know, but also, the.... Well, I'll let you go into that a little bit more.

Kim  06:58

So, have already come to the conclusion that Math is Figure-Out-Able and that you've done some work to just try to dive into problems, you might be ready for the next stage. And that stage is all about building your knowledge. Or if you're a leader, the knowledge of your teachers, about the major strategies and models for each operation. So, at that point, you're thinking, "Okay, I'm focusing on addition right now, and I want to learn tons about addition." Or, "I want to really focus on multiplication." And the major milestones for that stage, we've got got a bullet list for you. So, in that stage, you might be wrapping your head around naming and using the major strategies. So, you might not just do whatever you want for any kind of problem. But you might say to yourself, "Okay, I own the four major strategies for addition, and I can think about this problem in a variety of ways because I own the major strategies. And I'm going to choose which strategy that I want to do because I own them well enough. And I look at the numbers of the problem, and I'm going to pick. Not just whatever occurs to me in that moment because I only have one way to do it."

Pam  08:10

and so not only are you confident like, "Yes, I can solve that reasoning using what I know," but I can list off the four major proportional reasoning strategies that I might use to solve this proportion. Or I can name the three major strategies students might use to find the y-intercept, so they can write the equation of this line. Or I might be able to know the four main ways, four main... How do I say that? Four main aspects that kids need to know about rational functions as their graphing rational functions? So, like it's not that you can... How do I say this? You can do it. You can use what you know. But now you're like, "Oh, but what are all the major strategies? And can you list them off? Do you know what they what they are? And, yeah. Maybe I won't go further in that because we've got another bullet coming up.  Yeah,

Kim  08:11

Yeah. So, you also at this stage are working to name and use the major models for an operation. So, if we're working with addition again, you might say I know this major model is preferable and why. "Here, is maybe another model I might use and when I might give up that model. And why I might give up that model in favor of a different model. So, not only naming using major strategies but also the major models. And I'm going to...

Pam  09:31

Go ahead.

Kim  09:31

I was going to roll right into that next bullet.

Pam  09:33

Yeah.

Kim  09:34

That because you own the major strategies and major models, you are judiciously using the major strategies and models for our specific problems. So, I might see a problem and say, "Ah, I would represent it this way, and I'm going to choose to use this strategy based on what I'm thinking about today. And based on the numbers of the problem, like there's no other strategy that makes sense." If I'm going to add 27 and 99, then there's really one major strategy. Maybe two. Yeah, I was going to say two. Might be two. But there are strategies that are possible that I'm not going to use because the numbers in that problem are just so good for a different strategy.  Yeah. And you're mentioning teacher, but I believe that in this stage, you're still really thinking about your own learning. So, even beyond the work of a teacher, you can be at this stage and still growing yourself, right? You are thinking, "Gosh, I have this problem in front of me. And what do I know?" And you kind of have a Rolodex in your head of the major strategies and the models, and you're choosing based on all the work that you've put in to own them. A really nice milestone for this stage is also the ability to create a rich problem for a Problem Talk. You know, we do Problem Talks. Which we've kind of turned into a Problem String in MathStratChat. But sometimes, you know, people will say, "Oh, my gosh. Like, I didn't know that there were so many different varieties of strategies that you could use for this one problem. And it's because it's the problems are purposefully written, so that lots of good strategies will come out. And so, being able to create a rich problem means that you have an ability to understand lots of strategies, so that you can kind of put some numbers together that will draw out a variety of strategies.

Pam  10:14

Yeah, so we would hope that in this stage, you would do work to say what's a rich problem? So, for example, for this proportion, wow, I could use any of the major strategies. And I'm going to model that solution with, say, a particular ratio table because if it's rich and I can like really feel all those relationships. But at the same point, you might say, "Ooh, but for this percent problem, it's not super rich. It's actually very pointed. Kind of like your 99 plus 27. For this particular percent problem, I'm finding 49% of something. Bam, I'm going to use Five is Half a Ten or a little Over/Under. And I'm going to model that on a percent bar. Like, I'm choosing the model that will A, either help students really feel the relationships right now. Or then I might, like you said, I might give up that model, and we might move to a model that's better as a tool for solving that problem. So, sometimes we use models to build spatial sense and relationship. And sometimes we use models, then later we kind of slide into them to be the tool to solve the problem. They become a better tool. An example that I would give is sometimes with solving fraction problems, we might use an area model to really feel what's going on. But then we might move from that area model when... Not when we're going to some procedure that we've like memorized on the area model. But when we actually feeling the relationships with the fractions, then we might move to an equation model where we're like Doubling and Halving because we can feel those relationships happening. Yeah, so it's about judiciously choosing the strategy and judiciously choosing the model. And maybe I'll get even a little bit more pointed. As a solver, you choose the model and strategy that fits the problem. And then separately as a teacher, you choose the model to represent to help students grow in a way or use that model as a tool for solving. So, it's kind of two different levels that we might think about as judiciously choosing one as a solver yourself. But the other, like you said, math for teaching, as a teacher which model and strategy are you going to choose to work on today, to use today? And it's all about judiciously choosing those. Mmhm. And I completely agree with everything you just said. And I would just add and you can be building your math for teaching as you are working with students at the same time. 

Kim  13:55

Yeah. 

Pam  13:55

Like, please listeners don't hear us what we are not saying is, "Hey, if you feel like you're in this stage, stop the presses, really build yourself here before you ever do any work with students. Nah. 

Kim  14:06

Yeah.

Pam  14:07

You know like, this is concurrent work that you can be doing along with your students. 

Kim  14:11

Yeah, this is a really rich stage to be in. And there's a ton of learning that happens here. And so, we would encourage that you don't feel rushed through this stage. If you're a leader, you're allowing your teachers time to experience an operation and letting the strategies become intuitive. If you are the learner yourself, you are not rushing through this stage. It takes time for you to really own well and deeply all the major strategies and models, and you'll be coming out the other side with so much more knowledge to be able to help your students. and you can grow with them. We would encourage you practicing modeling your own thinking with various models and to share those models and your thinking with others because future stages build off of the work that you'll do here.

Pam  14:56

Yes, super, super nice. So, what are some things that you could do if you're like, "Man, I feel like I'm in this stage. I can solve problems. I can solve division problems using what I know. I can solve fraction operation problems using what I know. But the major strategies? I don't know those are (unclear)." 

Kim  15:14

You might be the person who does the same strategy over and over again. "I'm not using an algorithm, but I really like this one strategy, and I'm going to use it all the time." If that's you, you might be in this stage.

Pam  15:26

Yeah. So, what are some things that you could do? Well, we would invite you to take one of our online workshops where we dive in. It's one of the major things that we accomplish in our asynchronous online workshops is identifying and helping you build mathematics for teaching the major models and strategies for the different areas or for the specific area that that workshop covers. So, take one of our workshops, including.... We only open workshop registration a few times a year, and so if workshop registration isn't open right now, you can take our Developing Mathematical Reasoning free online workshop. That's a fantastic place to start. You can register for that at mathisfigureoutable.com/freeworkshop. We'd highly encourage you to do that. We'd also invite you to participate in MathStratChat every Wednesday evenings. And when you do, participate in it not only to solve the problem the way that you can  but noticed that many people participate by putting two or three strategies. So, like work on that. Solve it, throw that strategy in there, but maybe before you hit enter, or you can put in multiple entries, try to solve it using a different major strategy. And then importantly, look at others what they've put in and try to identify do you think this was an example of a major strategy? Do you think it was just kind of, "Oh, that might work for this problem, but it's not really a major strategy." Comment on other people's strategy, ask questions about how they're thinking about it. As often as you can, participate in Problem Strings. And when I say that, that could be you as a teacher. Grabbing a problem string from our website, from our Problem String books, and in front of your class, ask the first question. Now, planning is better, but at a minimum, you could throw the first question out, and see what people are doing, and represent their thinking, and throw the next question out, and see what kids are doing, and represent their thinking. Like, that would be a great way for you to build in yourself what the major models and strategies are. Compare that work then with our major strategies ebook. Oh, I didn't even think about having that handy. That would have been a really good one to have for today, Kim. We'll put that link in the show notes. What is it? Mathisfigureoutable.com/....

Kim  17:37

(unclear). 

Pam  17:40

Man, it'll be in the slow notes. Show, slow, show. I can't say that anymore. 

Kim  17:46

We'll have it for next time. 

Pam  17:47

You want to talk for a minute, Kim? Let's see. I can find it.

Kim  17:51

I think slash big. Somebody on our team is dying right now because we don't know.

Pam  17:58

Try that. mathisfigureoutable.com/big. And if it's not, we'll put it in the show notes for sure. But that ebook has in it what the major models and strategies are for the four operations. 

Kim  18:08

Yeah.

Pam  18:09

And totally free. You're welcome to sign up for that. And lastly, I'll suggest that coming out soon. I don't know when you're listening to this podcast episode. But coming out soon, we're about to announce the major book that I am writing with Corwin called Developing Mathematical Reasoning - Avoiding the Trap of Algorithms. And one of the big things that's going to happen in that book series is outlining what the major models and strategies are. So, highly encourage you to check that out. I think it's going to be available for preorder at NCTM, NCSM in September of 2024. So, check that out. Super, super excited about being able to get that resource out to the world. 

Kim  18:49

Excellent. Okay, so we have tackled two stages, and have lots of lots of ideas for people to participate in. And so next week (unclear).

Pam  18:59

You are going to want to hear. You are going to want to hear (unclear)

Kim  19:02

What's next after you know the major models and strategies? 

Pam  19:05

Yeah, bam. Alright, 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!