Pam  00:01

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

Kim  00:10

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

Kim  00:30

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

Pam  00:37

We are so glad that you're on this journey to make math more figure-out-able! Bam! Alright, Kim, we have been doing a series of what we call our Success Map of how we look at working with teachers, how we look at differentiating as we work with teachers. We differentiate less by separating teachers out into low and high groups and giving them different things to do. And we differentiate more...

Kim  01:05

Did you air quote your fingers because we can't hear that.

Pam  01:07

Oh, totally, you can't. We don't do that. We do not separate teachers out in low and high and give them different things to do.

Kim  01:13

Okay.

Pam  01:14

What we do do is do the same kinds of tasks with teachers no matter experience level they are on recognizing that they can have different lenses through which they experience the learning that we're doing. And we help teachers recognize through which lens they could kind of focus on, so that as they focus there, they can make progress. And so this Success Map really is helpful for us. So, as we plan things to do with teachers, we really think about are we making sure that in that task it's open enough for teachers with all these different experience levels? And it's also helpful for teachers to be able to say, "As I do this task here, I'm going to focus in this way." So, a teacher might say, "I've never really thought about this area of mathematics. Bam, I'm going to dive into this problem." Say we're doing a Problem String with them. "I'm going to dive into this problem and try to solve it using what I know. I'm going to reason using what I know. And if I'm tempted, I'm inclined to use those memorized step-by-step procedure, I'm going to lean away from that. I'm going to try to lean into reasoning, using mathematical relationships."

Kim  02:26

Yeah.

Pam  02:26

A teacher might also say, "Nah, I can do that. In this area of mathematics, I can use what I know to solve problems. But do I know the major models and strategies for this operation?" I might, as a teacher, be focused in that area. Like, "Okay, I can solve it, but what are the other ways and what are the other models that I could use to make that thinking visible?" And then I might be a teacher who says, "Yeah, yeah. I got that. I got that. But now I'm going to start eliciting that from students. And as I listen to student thinking, with all that background we just talked about, now I'm going to represent and make that thinking visible. And I'm going to really work on listening purposefully to students, and then choosing the best model at the best time and making thinking visible, so that we can have conversations across strategies. And I'm really working sort of there." Or you might be a teacher who says, "No, no. I got that. I got that. What I'm really working on now is how I see you're like doing these Problem Strings. I see this private response signal, and I see this neutral response, and these teacher moves, and the different tasks you're using." Maybe you're in a place where you're really thinking about incorporating those major teacher moves in your practice. Or maybe you're like, "I don't know. Like, I think, Pam, I think I've kind of got those down," and you're now in what we would call the fifth stage, which is all about...

Kim  03:47

Sequencing or advancing the math. 

Pam  03:51

Yeah.

Kim  03:52

Yeah. And, you know, before we talk about that, I'm going to say one thing. That I'm glad that you say that it is... We're giving them a lens, giving teachers a lens on what they can be focusing on. Because it's not like you do all stage one, and you're perfect at stage one. And then you go to stage two, and you're perfect at stage two. But you don't ever do anything else in any other stage. It really is... I'm kind of doing teaching. Teachers do all those things all the time. But they're focusing their study, their craft. They're getting better at a particular stage. So, for like stage one, you might have never thought about numeracy before. You might have never thought about math being figure-out-able. So, that might be a place where it's all brand new. But if you're teaching, you're probably using some teacher moves, but you're sharpening better teacher moves and when to use those teacher moves, when not to use those teacher moves. So, it's not like you have to be perfect at a stage before you can go on to another stage. It's just what you're focusing on in that moment. (unclear).

Pam  04:55

Absolutely. And when the math changes. So, then you might shift and you might I think, "Okay, now I'm thinking about geometry." And now I'm like, "Okay, can I sell this problem thinking and reasoning? And what does it look like to? What are the major models and strategies happening here?" And so, as you shift your focus, we all kind of cycle back through these stages. You and I hate the word stage. We just can't think of a better word for it. But I got to tell you Kim. The other day, I was listening to a podcast my son turned me on to. Three Blue, One Brown. Three Brown, One Blue. Grant. His name's grant. Anyway, he has this great math YouTube channel. And I was listening specifically to the one where it was the big ideas of linear algebra. And he did something in the middle of this one video, and I was like, "Man, I am stage one right here, right now." He did this very visual way of explaining matrix multiplication. Well, he did this visual thing, and then he said, "Now, look at this." And I was like, "Oh! It's matrix multiplication!" Like, I had this huge "aha".

Kim  05:59

Oh, fun. 

Pam  06:00

Oh, it was super, super fun. 

Kim  06:01

Yeah.

Pam  06:02

And so, we all in different areas of mathematics will sort of go through these stages if we want to teach those (unclear) mathematics. And like you said, we're not always... We're not ever just in one. But we can focus. We can look through that lens and kind of focus our work in a particular place. Today...

Kim  06:20

And to be clear, every single one of us is at probably every single stage for something different, and we're moving forward based on our focus of the day. It's kind of like, you know, you're filling out your buckets, and you're filling different ones at different times based on the need that you have in that moment.

Pam  06:38

Mmhm.

Kim  06:39

So, today, let's talk about stage five.

Pam  06:41

Yeah. And it can be super helpful to recognize, "Ooh, for this, I'm in this particular place. Bam. Let me work on those things." It helps you focus. Yeah, cool. Alright, so stage five. Sequencing, advancing the math. Yeah. What do you have to say about that?

Kim  06:56

I'm going to talk to leaders for just a second because as leaders, we can feel pressure to jump to stage five quickly. We, as leaders, might think we need to know everything in stage one through four before we can actually lead. Or we can say to our teachers, "Hey, we got to hurry up and get to stage five because this is where you build a curriculum, and I need your comment on the curriculum that we've been handed. Or we're going to set that aside and build something together." And we can jump there too quickly or feel like we have to have all the answers. This stage is all about progression. It's all about the flow of lessons to move students and mathematics forward. And it's understanding that one or two experiences doesn't help students or teachers get "it". You just air quoted. I did. The "it". They don't get "it".

Pam  07:47

Yeah, "it". What's the "it"? Like, multiple. We want multiple access tasks, so that everybody learns and everybody grows.

Kim  07:53

Yeah. So, it's thinking about the varying needs in your class and the sequence of things in your classroom, the sequence of lessons that you're going to do to move the math forward. So, there are quite a few major milestones because it's a very complex stage. Lots of things to consider, to have in your mind in this stage. So. 

Pam  08:14

We're going to talk about several things a little bit. A little bit about several things.

Kim  08:18

Yeah. Okay. So, the first one is creating a similar or an echo Problem String based on a pre-written string. So, you have a string in front of you. In this stage, you're able to create a echo string.

Pam  08:32

Or at least you're working to create. Yeah, your desire is to really know the string well. And so to really know the string, that instructional routine well, you're diving in and you're saying to yourself, "How can I write like an echo or a similar? Use different numbers but accomplish the same things?" And that is not trivial.

Kim  08:50

No.

Pam  08:51

Yeah, to create a good one. Let me tell you. I have tried to create echo strings from some, and I'm like, "Wow, these numbers are just so tight. But it helps you really own the relationships better to do that.

Kim  09:03

We should, on a podcast sometime, write a string together. 

Pam  09:07

Oh, that would be good. 

Kim  09:08

So, we can talk out. Yeah, some are some are easier than others.

Pam  09:12

Sometimes I think people hear me say Problem String. I was thinking about this a lot. I've done a lot of presentations lately. And I wonder if people hear... Listeners, you'll have to tell us. I wonder if I say, "Hey, do a Problem String," if they hear, "Oh, do a bunch of problems in a row. Okay, I'll go find a bunch of problems that are in a row," and they just go ahead and grab them from anywhere. So, we almost had to come up with a different name for Problem String because it's a very specific, purposefully written series of problems that are related and in a specific order.

Kim  09:42

That's too long of a title.

Pam  09:43

Yeah, I know. So, anyway, it's not... So, in order to get to know specific relationships better, a way to do that is to write echo strings. (unclear). Next.

Kim  09:52

The next is creating a "next step" or what we call an up the ante Problem String based on a pre-written Problem String. So, you have the Problem String. You can create an echo string, but also are working to create a "next step". 

Pam  10:07

You're going to up the ante with those kids. What does it look like to up the ante? And I know, Kim, you're really good at this because you'll say don't change everything at once. 

Kim  10:14

Yeah. 

Pam  10:15

So, if you're going to... Maybe up the numbers. Make the numbers look more complex. That's one thing we could change. What else? 

Kim  10:21

You might change the model.

Pam  10:23

Mmhm, mmhm. So, same strategy, same relationships, but now on a different model, different spatial look. 

Both Pam and Kim  10:28

Mmhm. 

Pam  10:29

Yeah.

Kim  10:30

You might have a more high level starting point. So, the numbers might still be in three digits, but you start right away into three digits rather than work your way up into that.

Pam  10:41

Yeah. And then whatever you do, don't change all of those and the strategy at the same time. 

Kim  10:46

Oh, yeah. 

Pam  10:47

Yeah, so like within a strategy, change like one or two of those things. Up the ante enough that everybody can still enter and everybody still grows.

Kim  10:55

Okay, another one is creating a sequence of Problem Strings to build a major model, strategy, or concept. So, not just one single string, but a sequence.

Pam  11:07

Yeah, because it's all about progression, right? We're progressing. So, it's not... Often, teachers will say, "Hey, I did your thing. I did a string. They didn't get it. "It" in quotes. And I'll say, "They didn't get 'it'?" And they're like, "Yeah, they weren't all doing your thing at the end." And I'm like, "What were they? Did they all learn and grow?" "Well, yeah. Yeah, they all learned and grew, but they weren't all doing 'it'?" "Well, yeah. That's why you have to do a sequence of strings." The idea is that multiple access, everybody learns and grows, then you up the ante. And I'm doing my fingers. You can't see it. But like, where everybody has an access point, everybody learns something, and you keep doing that over and over again. And so, it's about sequencing. And that's... Let's be clear. That's not trivial. I think these other two are a little bit easier to create an echo string, to create a next step string, you're kind of basing it on something. To create a sequence of strings, and that's... Kim, we know that. We've just been working really, really hard to write books of sequences of Problem Strings. And it's not trivial. We have a lot of experience behind us. We've done a lot of experimenting, a lot of work with students and with teachers to create those sequences. But that is a thing then that we need to all get better at. And maybe I'll just say you don't have to necessarily create them. You could use ours. 

Kim  12:19

Yeah. Well, and we would say that if you thought that you had to create them before you could start using them, then you're missing a lot of opportunity to build into yourself and your students understanding. Do what we did. We used somebody else's.

Pam  12:37

Yeah. So, for sure, when you're in stages one through four. And when you get to stage five, then you're kind of in a place where you're like, "No, like I'm in stage five. I want to build some sequences." Or at least choose good sequences for the time. (unclear).

Kim  12:50

Yeah, and it might be true for an operation or two, but not all. 

Pam  12:54

Yep. 

Kim  12:55

Right? Okay, we got to move on. 

Pam  12:56

Yep. 

Kim  12:57

So, you can create a sequence or working on creating a sequence of Problem Strings, but you're also creating and facilitating a sequence that might be a Problem String, then a Rich Task, then a congress, then a Problem string. So, a sequence of different types of lesson types. 

Pam  13:13

That actually work into each other. 

Kim  13:15

Yeah. 

Pam  13:15

Yeah, they flow from each other. 

Kim  13:17

Yeah. 

Pam  13:18

And we really liked that small... There's a word I want. That chunk of a string that leads into a Rich Task, followed by that discussion at the end, and then lead out of that with a string to kind of cinch something that happened in the Rich Task. So, we like that chunk, that... We're all about sequences here. We're all about progression. And that's a really nice chunk that you could iterate. So, working on that chunk is a part of this stage that you're in.

Kim  13:49

Mmhm. I feel like our list is going from like smaller pieces to bigger pieces because the next one is focusing on moving the math forward by sequencing tasks. So, you have that chunk that you just... Maybe I described. That was the string, task, congress, string. But then also within a major unit, sequencing what are the big tasks that you're going to do?

Pam  14:14

That then you'll... Yeah, and you'll have Problem Strings on either side of them, right?

Kim  14:18

Yeah.

Pam  14:18

So, yeah, I kind of think of a unit. At least this is the way we think of units is like these are the main things that we want to accomplish, need to accomplish in this unit, and so we have these major tasks that are kind of the anchors. And we get in and out of those tasks using Problem Strings and other instructional routines peppered in as we go. So, yeah. Creating that sequence is work to do in stage five. 

Kim  14:40

Yeah.

Pam  14:41

Yeah, cool. 

Kim  14:41

So, I've mentioned this in stage two. Knowing the progression of models for your chosen operation. So, in stage two, you're learning all about the major strategies and models, and you'll start your work of thinking about which models when. But in stage five, you're really understanding and thinking carefully. about when to leave them behind, when to introduce a particular model, where to, how to identify where your students needs are, so that you can give them the model that they need at the right time.

Pam  15:13

Yeah, and I would get even more vertical than that. Like, having a sense and feel for how that model was used in the grade before. 

Kim  15:20

Yeah.

Pam  15:20

And how it would be used after, so that you know... That you help students not get trapped in less sophisticated models but you bring up models. Every time I do work with middle school teachers, they just say, "Please tell every third, fourth, and fifth grade teacher that you work with to please use ratio tables for multiplication and division because oh then we could just soar with them." So, it's this idea of having the sense of the progression of models.  Yep. 

Kim  15:47

Yep. Yeah, for sure. Another milestone is knowing the connection between strategies. All the strategies that you learned in stage two, knowing the connections between them and making use of the properties to explain and compare. So, little bit of compare work happens on, but here is where you kind of get outside of the strategies and you're understanding the properties well enough that you can see the connections.

Pam  16:11

A bit more global.

Both Pam and Kim  16:12

Yeah.

Kim  16:12

Mmhm.

Pam  16:12

Yep. And lastly, we can actually help... In this stage, we can really focus on the different structures of Problem Strings. Problem Strings are so amazing. And one of the things that's amazing is there's different kinds of Problem Strings. Wnd we really can dive into learning that here.

Kim  16:27

Yeah. And so, you're probably never going to really exit stage five. There's always work at something new to learn and ways to grow. And let's say you're the most masterful stage five person ever, then that's in one area of mathematics. (unclear), like you've said before. In this stage, collaboration and reflection is key. And then you'll focus on a new area of mathematics.

Pam  16:50

Yeah, and zip back around, and be super cyclical through all of those stages. 

Kim  16:55

Alright, what can people work on?

Pam  16:56

Yeah, let's give you some action items. So, ya'll, if you have not taken the Developing Mathematical Reasoning free workshop, what is happening? So, mathisfigureoutable.com/freeworkshop where you can dive into the Developing Mathematical Reasoning workshop. And notice how we sequence tasks in that workshop. So, that would be a lens through which you could look if you're in the stage five. You might have noticed we've given you that as an action item in I think in almost all the stages.

Kim  17:20

Mmhm.

Pam  17:21

Because again, it's the lens through which you are looking as you take that. You might be like, "Pam, I already took your workshop when I was in different lens. Darn! I wish I could take it again." Do you know that you can register for it again? No harm, no foul. You can totally take it as many times as you want to. In fact, I'll just mention. If we have any university folks listening to the podcast, contact me at pam@mathisfigureoutable.com, and there is a high probability that I'll give you access to that workshop for your students in your methods courses. 

Kim  17:48

Yep. 

Pam  17:49

And then you get it for a whole semester. Yeah, go ahead. 

Kim  17:51

You know, every once in a while, I have a reason to look up somebody. Just a name in our system. And you can see that many people have taken DMR like three, four times. You can see each time they've registered. And somebody might say, "Oh, my gosh. Why are they watching it again? Didn't they watch it once?" Yeah. But maybe it's six months between or a year between because they're looking at it through a different lens.

Pam  18:15

Through a different lens. You're in a different place in your experience level. Yeah, absolutely. Ya'll, another action item is to check out our Problem String books because those are all about progressions of the problems in the string. We have "next step". We have echo strings. The similar strings. And we have next step strings. So, it would be a way for... You as a leader can have everybody look at that, just that progression of strings. That can be a conversation that you have with your advanced teachers about how, why this progression? How did they make these grow from each other? Along with that, if you're a high school teacher, the Discovery Math series, Discovering Algebra and Discovering Advanced Algebra have amazing tasks. Like, we talked about the sequence of a string, into a task, into a string. They have amazing tasks in the student books. And then I wrote Problem Strings to go along to exactly what we just said, to book end those tasks. So, Discovering Algebra and Discovering Advanced Algebra then are accompanied with the Algebra Problem String books and the Advanced Algebra Problem String books. Gives you a really good idea of those chunks of progressions. And, ya'll, you could just look at your scope and sequence and you could say to yourself, "How am I sequencing tasks?" And the kinds of things that we talked about today, how are you sequencing tasks and move the math forward? 

Kim  19:32

The good news is, is that everyone is on their own journey. And we get to celebrate that we're on this journey together. Some people's journey is faster, some is slower. And sometimes you'll meet people who are in different place on this journey.

Pam  19:47

Mmhm.

Kim  19:47

But it's all very exciting. It's good place to be.

Pam  19:50

Let's all celebrate that we're on this journey together. It's fantastic. Making math more figure-out-able. Ya'll, thanks for joining us for this series. We hope that you've found a place where you can focus on the learning that you're doing from a particular lens, through a particular lens as we all continue to progress. Okay, thanks, Kim. This was fun. 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 And keep spreading the word that Math is Figure-Out-Able!