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 and 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 the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what? Alright, so let's talk about this episode. Kim, a while ago, we've been working together for a little while, just a little bit. And a while ago, you said, "You know, it's like, if you know, your content, and you know your kids, then kind of everything falls under that umbrella. Like, that's kind of how we could sort of think about things." When you said that, and the more that you kind of talked about it, and we kind of talked about it, that resonated with me, really resonated, that we can kind of lump a lot of things together, we can sort of explain a lot of things, kind of help illuminate things in those two buckets. Like if you know your content, and you know your kids, then really we can do some amazing things as we are building these young mathematicians. 

Kim Montague  01:30

Yeah. I don't think that everything fits under those two categories. But I think a lot does. And those are kind of our charges, as teachers and educators, that those are the specific areas that when we know those two parts well, and how they interact with each other, gosh, then our answers are solved for so many things, right? So maybe we can explain a little bit. We've said this phrase before, you know, when we're talking with people, and sometimes people nod and I don't know that we've ever spent the time to really dive into what do we mean when we say know your content, know your kids. So maybe we do that today.

Pam Harris  02:09

So there's two parts: there's know your content, and know your students. Whoever your students might be. For me as a math teacher educator, it might be the participants in a workshop. At the university, it might be my students in university class. Whoever we're sort of teaching, know those people. Let's start with know your content. Kim, tell us more about that.

Kim Montague  02:33

So there's a lot of things that go into knowing your content. I think all educators know that, right? It's not a super simple thing. But what I mean by that is, you deeply understand the content that you're charged with helping your students understand. Both each individual, tiny little standard, and how they relate to the broader big ideas. There's like all these standards, all these expectations are under this umbrella, and these sometimes often shoved to the side process standards. 

Pam Harris  03:05

Or mathematical process standards. In Texas it's "Process Standards".

Kim Montague  03:09

Yah, that all fit together. So knowing individual standards and knowing how they relate with each other is one area.

Pam Harris  03:18

Yeah, absolutely. And while we're there, like you just said, specifically the ones that you're charged with helping your students understand. But not just, not just, the ones for your particular students. 

Kim Montague  03:33

Yeah, at a minimum, you've got to familiarize yourself with the content that comes before you, in the grade level before you. And also after you. To understand how they connect to your grade level. A lot of times we say, well, you know, this is what they can and can't do. But knowing what they are expected based on a list of standards is not only helpful to know where kids could be, but it's also the relationship between what's expected at that grade level. 

Pam Harris  04:04

Yeah, so you sort of know how it's been built, or should have been built or could have been built. And then where it leads to surely will affect how you teach the standard. It will surely influence the manner in which you attempt to make sure that your students are growing in that particular standard. Right, I was -

Kim Montague  04:22

Go ahead.

Pam Harris  04:25

Yeah, it's more than that. It's not just about knowing what standards come before and after. But it also has a huge thing to do with, oh, shoot, am I getting to know your kids? Once you know where your students are, then you can help them based on knowing that full band of content. Knowing really the landscape of the big landmarks that are involved in that particular sort of strand maybe that you're working on, how it's connected to other strands, how you can bring those together. And if you have a student that kind of already owns some of the content, they've had enough experience that they're able to really conceptualize and do well with the content that's required at your grade level, you now know how to, like nudge that student, how to lob some things that they can begin to tinker with that are, you know, maybe more horizontal standard, like getting really deeper into what your grade level content or moving them vertically, like nudging them a little bit higher. 

Kim Montague  05:23

I think it's important to know whether or not you're the first grade level to introduce something, whether you're continuing on a particular idea. Because I think you're gonna address those different ways. Right? Are you the last grade level that it's specifically mentioned? And that at that point, kids are supposed to - I'm air quoting - "own all the things", and then later are there grade levels that are skipped over in the standards that don't necessarily mention them again, but we're assuming that students will continue that knowledge, and then it's brought back up again, two years later.

Pam Harris  06:02

And that will be important to know, because if your grade level sort of skips a particular strand, say like, I don't know, geometry, and all of a sudden, maybe that's not the best, like data tips, we see this a lot where it's sort of mentioned in one grade level, and then it kind of skips a grade or two, and then it comes back in another grade level. You could, knowing that that's true, you could keep that going, with just little ways that you would tweak some of the rich tasks that you do and Problem Strings you do and discussions that you have. You just sort of lob some of that data in there to keep it fresh. You keep kids thinking about it and keep kids working on it.

Kim Montague  06:37

Yeah. So one of the things that I think about, and there are probably more if we had more than, you know, 15 minutes. 

Pam Harris  06:43

Hey, before you go on, sorry, I don't know that I want to leave. I think what you said was really important. And I don't know, I'm feeling like, I don't want to just repeat it. But I wonder if you can say any more about why do we need to know for the first grade level  that teaches something specific, or the last level.

Kim Montague  06:58

Well I think it actually connects to the next thing I was gonna say. And we can hang here for a while. But I think it's really important to make sense of appropriate models and strategies within a particular idea. So I might spend more time building some ideas with some maybe more introductory models or introductory strategies. If I'm the first person to introduce something. If I'm the first person to talk about multiplication, for instance, I might need to spend a little bit more time there thinking about what is multiplication? What does that even mean? And making sense of the whole idea of multiplication, connecting it to addition.

Pam Harris  07:36

Bridging from additive reasoning.

Kim Montague  07:39

Versus if I'm a fifth grade teacher, and I feel pretty certain that kids have had some experience with multiplication over the last couple of years, I see evidence of that in their thinking, then I'm going to spend less time introducing that and building, bridging towards strategies that are appropriate to move that math forward.

Pam Harris  07:59

So less time on the intro and more time on the bridging and the moving forward. Yeah, that makes sense. And then what if you're the last teacher that's really going to work on that? 

Kim Montague  08:08

Yeah, I think at that point, then we need to make some generalizations, right? We're going a little bit abstract, if you will, we're talking a little bit more about generalizations that I want kids to walk away with, with some big ideas of multiplication, connecting multiplication to division, making sure that kids are really solid on that connection.

Pam Harris  08:27

And it's not like we haven't been generalizing the whole time. Like we're generalizing little bits as we go. We're helping kids abstract out as we go. But there's this sort of emphasis difference, like if you're starting a topic - I was going to try to think of another one - if I'm solving equations in middle grades, I'm going to do a lot more introductory things with scenarios that makes sense. If I'm later and we're solving equations, we're going to be quite a bit more abstract. Now, I need to know the content well enough that if I have students who haven't had the experience that I wish they had in the middle grades of solving equations, then I might need to provide some of that experience for students. Because that's what we believe, right? We believe that students just need experience to construct relationships. That's why we need to know the content, we need to know sort of the gamut and know where we fit in that. So that we can be really concentrating on the beginning, introductory kinds of things, focusing on oh, we're just gonna keep going with that and get you more sophisticated. Maybe, and here's know your kids, figure out what strategies your kids already own. And so which ones you still need to work on with that particular class if you're sort of that middle teacher. And then if you're kind of the be it end all teacher, this is kind of where the rubber hits the road - I shouldn't say where the rubber hits the road, it hits the road everywhere. You're the teacher where it's the last time that particular content shows up in the standards, then you're the teacher who's going to really maybe emphasize that standard a little bit more, maybe spend a little bit more time there, definitely talking in more general terms, you're going to be more abstract - not maybe from the beginning. But you're going to get there quicker, you realize the onus is on you to do the best you can to get everybody to sort of master that content this year, because it's the last time that it sort of shows up in the standards. Again, knowing your kids, knowing your kids so that you're not going to push kids who aren't ready for that, you know, you're going to get them as far as you can from where they were. Go ahead.

Kim Montague  10:26

Yeah. And, I was listening to you talk, and it made me think, you know, it could sound very much like we're saying, if you know, your content, then you know, that's fantastic. But the reason to know your content is because you're going to have live human interaction with students where you're wanting to question, right, you're wanting to nudge. And you cannot question and nudge well, if you don't know to what end. And that's the point of knowing your content.

Pam Harris  10:53

Or even the beginning, right, like, you have to sort of know the whole landscape. You're asking good questions, because you know the content, and then you're able to hear the answer and go, "Okay, you own that. Okay. Okay, you own that. Okay. And oh, here's an edge, I can see an edge. I can see the edge where I can lob some stuff out, we could do some work here, I can ask this question that will nudge you forward." In other words, another way of explaining that could be we know the zone of proximal development for our students. We know, "Ooh, if I ask this question that's gonna be right in the center, that's boring, that's not gonna intrigue you. But if I get on the edge of your zone of proximal development, ooh, now it's gonna spark, like it has the potential to cause some disequilibrium and help you re-schematize and structure and look for patterns." If it's outside the zone of proximal development. Then we get like frustration and confusion and kids that give up and kids that are -. So we can see that frustration and confusion, two places, right, we can see it outside the zone. But we can also see it in the middle of the zone, like really, is it supposed to be hard? Like why she's spending time on this? I don't think - this is so obvious. Like it's, we're aiming for that sweet spot. If you could see my hands right now, I'm like, forming a sphere. I'm like, right on the edge of the zone of proximal development, where we could just really ping where kids are. Did I just go into know your kid? You had already gone into know your kid so that's fine.

Kim Montague  12:23

Yeah, I think we're all over the place, and that's typical. So I think maybe in summary: What do they already know, and can currently make sense of, that you can leverage to make other mathematical connections? Right, you're talking about being on their zone. And so I think it's all about leveraging what they do know, to make connections to what comes next. What's on the horizon that needs further cementing?

Pam Harris  12:50

That's really nice. I like that. I bet we can quote that. That was really good. Yeah, I mean, like leveraging what they know, oh, that was really sweet. Yeah, to make connections to what you're trying to teach. It's not about okay, blank slate, "Here we go. I'm gonna unzip your head and pour in this information and zip it back up, hang on to it. No, no, don't unzip. No, don't let that leak out." That's not the way we believe that human beings learn. We believe human beings attach meaning to current schema, or they have to re-schematize and make sense of the disequilibrium that's happening. And they have to sort of, like how do I balance that so that -  help me Kim? How do I merge those together? How do I make what's happening in my head and what I'm experiencing and seeing how do I make those fit together? Do I need to re-schematized? Does it fit nicely in the file cabinets that I have in my head? Or do I have to create some new file cabinets because whoa, what's happening here is not - I'm in a little disequilibrium, a little off balance by what's happening. And bam, as I make those fit, that's learning occurring, as I grapple with and then make that stuff sort of oh, okay, this is the way I can sort of structure it and schematize what's happening. That's learning occurring now I've created the new neural connections. What are some other things that you look for that you think about when somebody says know your kids?

Kim Montague  14:22

Um, so you know, this might not be necessarily mathematical but I think a lot of times about how best do students work? Right? It's not necessarily just about preferences, but how am I going to get the best version of them could be sometimes solo, could be sometimes with somebody, could be sometimes that they need support in something that maybe is this particular day, this particular time. this particular thing they need a little bit more. And I think that's true for all students, that they all need support in something. So taking in that, and I think a lot of teachers do this well, looking at the human in front of them, the person and saying, "What do I know about you as a person? And how can I help you?"

Pam Harris  15:13

Yeah, and I'm thinking of some specific examples, I can think of a very bright young man that we both know who's on the spectrum. And communication is tricky for this particular student. And so while the math is not an issue at all, like we see brilliant things happening all the time, we don't always necessarily hear it, you know, it takes some pulling out, for that student. It takes some modeling, and some - let me use modeling in a different way - and model demonstrating about how to communicate about the kinds of ideas that that particular student might be having.

Kim Montague  15:49

And since part of mathematics is communicating your thinking, and interacting with other mathematicians, that is a thing to work on.

Pam Harris  15:56

Nice. Yeah, absolutely. I'm reminded, when we were filming for the algebra workshop, the linear function workshop that we're creating, there was a particular student that I watched. Our model teacher, our exemplar teacher, Abby Sanchez is amazing. As I watched her work with this particular student, I was like, "Oh, nice", because the particular student kind of had a bit of a chip on their shoulder. This is a bit more of a sort of personality, kind of maybe life circumstance thing. You could kind of see some hard edges. Maybe it had been a hard day, I don't know. Everybody gets a hard day, right? Everybody can have sort of - and this student was definitely having that. And as I watched Abby treat the student with compassion, but with firm boundaries, and you know, like, this is what's gonna happen today. And these are the boundaries that we have to work in. But I'm also giving you kind of a - I'm gonna let you take a minute, you take a minute, collect yourself, I'll be back in a second. And then the student, I watched, the student just like take a deep breath, and kind of calm down a little bit. And then when Abby came back and said, "Alright, what are you working on?" Then the student was like, "I don't understand." And Abby's like, "Look, what do you not understand?" "Well, what are you even asking here?" And so then they were able to kind of work through what that student needed in that moment. And it could have been lots of things. But it was brilliant that Abby like knows that student well enough to know, this student right now needs some time. I'm going to walk away, give that student some time, then I'm going to come back and give the students some help. So knowing your students is going to really help you be able to answer questions like, "What do they need? What does this particular student need support with?"

Kim Montague  17:29

It's all about how can you best pull out their current thinking so that it can be built on. What's the best circumstances for them?

Pam Harris  17:38

Yeah. And you have to know your content, and the kid to be able to do that, right?Because to be able to pull up their current thinking, you have to know the lay of the land of the content in order to ask the questions so that you can then pull out their current thinking. Hey, I'm a little nervous. It just occurred to me in my head, that somebody might be listening to our podcast going, "Well, Pam, what if I don't know the content? Like okay, so I just don't do any of this until I like, have it all down?" No, no. So we're not saying that at all, right?

Kim Montague  18:05

Like, it's a goal, right? It's something to work toward. 

Pam Harris  18:07

Keep learning. Keep learning. 

Kim Montague  18:08

Yeah. And get better at, all the time.

Pam Harris  18:11

Which is why we do things like we have a success map where we talk about different stages of teachers learning, so that teachers can help themselves identify where am I sort of on my landscape of teaching with this content? Like, what else could I be working on? Where are some places I could go to continue to learn that content so that we can learn and know that content better and better as we go. So it's important to know your content and know your kids. It's a great goal, something to work towards and get better at each year. Kim it was super fun talking to you all about our mantra of know your content, know your kids.

Kim Montague  18:54

But before we go I want to mention something we're super excited about, right? We have some Math is Figure-Out-Able merchandise coming. Finally! People have been asking for so long. We finally are ready to announce November 26. We have products coming, Black Friday, do your shopping for your favorite mathies in your life.

Pam Harris  19:15

Bam! So we're gonna have some fun products that we're gonna release. We're super excited about our store. We'll give you more information as it comes. You'll be able to wear some Math is Figure-Out-Able merch around and help us spread the word that Math is Figure-Out-Able. Super exciting. So if you want to learn more math and refine your mathematics 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!