Ep 116: Structuring Your Math Class Examples

September 06, 2022 Pam Harris Episode 116
Math is Figure-Out-Able with Pam Harris
Ep 116: Structuring Your Math Class Examples

What does teaching Real Math look like at your grade level? In the last episode Pam and Kim discussed the One Third/Two Thirds rule to structure math time. In this episode they give specific examples at various grade levels of what it might really look like.
Talking Points:

• 1st Grade - doubling and halving
• 3rd Grade - concepts around 100
• 5th Grade - multiplication and volume
• 7th Grade - rates (see Episode 112 for Unit Rate Problem String)
• Algebra I - exponential functions
• Pre-Calculus - functions

Our workshops are open for registration! Take advantage now to change your math classroom! https://www.mathisfigureoutable.com/workshops

Pam Harris:

Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.

Kim Montague:

And I'm Kim.

Pam Harris:

And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns and reasoning using mathematical relationships. We can mentor mathematicians as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be.

Kim Montague:

So last week, we talked about how Pam and I view what it could look like to fit in all kinds of great things into a math classroom. And we shared that we'd like to think of a two third/one third split of time as kind of our structure of a math class. This week, we want to dive into some specific examples at a few grade levels to give you a better idea of what we mean.

Pam Harris:

Kim Montague:

Post-it notes.

Pam Harris:

Are there a lot of post it notes in a set?

Kim Montague:

Yeah.

Pam Harris:

Okay.

Kim Montague:

You can buy little stacks, big stacks, yeah.

Pam Harris:

Okay, something where there's a lot so that they have to, or at least more. A number that I think that they can really fuss with. But then I might hand a different group of students, the markers, and I, you know, like, "Here's a marker set of eight. And how do you know if you have half?" Again, I'm being judicious to kind of make sure I'm giving students experiences that are right on the edge of their zone of proximal development to really move them forward. And then I'm going to continue to give them better and bigger collections. So that as they have them, they gain more experience, we would do a math Congress about what were some ways that you were thinking about halving. I would definitely want to pull out a strategy, like, "Ooh, Was anybody kind of dealing out? I give you one, and then I get one, and you get one, and I get one." Because that's a strategy of making sure that we have an, anyway, blah, blah, blah. I don't want to get too deep into any one of these. So this idea of an Investigation and a Congress around halving, then we might follow that up with a doubles game. We might have a game where you sort of roll a die and double that amount, and then move that forward on the board, so that kids are like, again, they're sort of messing with this idea of doubles. Do you see how we're kind of doing things that are all kind of around the same idea? I might follow that with a quick image problem string where I am flashing doubles. And we're looking at using the structure of the 10 frame or structure of a rekenrek to help students kind of glance at doubles and see something. So in that kind of two thirds time, I've just described a sequence of tasks that could happen to really help students thinking about doubles. Really, it's about doubles, it's less about halving, it's about doubles. We halve to give them kind of this nice context of game playing. I'm not trying to cinch halving. I'm trying to get that on their horizon to think about, but really, I'm giving them lots of experiences to count where the count matters.

Kim Montague:

Right?

Pam Harris:

So it's a big idea in first grade. Now, if that's the two thirds time, what am I doing in the 1/3 time? Well in the one third time I'm thinking about past games that we've already played that I want to keep going all year long. So I'm like, bring those back. We might be doing Count Arounds so that we're looking at patterns in the base 10 number system, like what are things that need to be happening all year long, really important things? And I'm going to continue to do those during that time. Yeah, Kim, how does that feel?

Kim Montague:

Yeah, it feels good. And I want to mention again, that I think a small piece of what you said that maybe people maybe didn't catch is that you said, this was not like, all in one day. You're not tackling all of the things that you've said in one day. It might be over the course of two days or three days or, you know. I think it can be really easy to say, like, I want to master doubles in a day. Right? And we know that that's not gonna happen.

Pam Harris:

It's one standard. That one standard in isolation. Okay, that's Tuesday.

Kim Montague:

Yeah, yeah. Yeah, I mean, it can feel like that is, you know, what you have to do. But we're taking this kind of like a wider approach to what can I do to give kids experiences over the course of several days or week or over time. And I love that 1/3 bit of time, because that's when you can cycle back to things that you know, you've approached, but you still want to, you know, doubles are going to be something that you make use of all year long in the future.

Pam Harris:

Absolutely.

Kim Montague:

So I was thinking about grade three, and I, you know, I kind of was thinking, in two days, what would a two day plan look like for me? And again, it's not that, you know, everything's gonna be mastered in two days. But I thought, okay, if I'm third grade, addition is kind of a big deal in third grade.

Pam Harris:

Very much so.

Kim Montague:

And we expect kids to come to us as third grade teachers, having some solid addition. But we know that that's not always the case. And then we panic, because, like, we have to tackle multiplication. So I was thinking over the course of a couple of days, for a two thirds time, I might start with I Have, You Need. A whole class, just a routine to kind of get some things started moving in their brains, or they might do it with partners on day one. And I might do it with partners of five and multiples of 10. So fives and 10s. Just to start warming up and saying like, "We're going to start doing some thinking within 100."

Pam Harris:

And if you've never heard of I Have, You Need that's just where you say, if it's partner of 10, you're saying, "I have eight, what do you need to make 10?" And then students are finding those partners of 10.

Kim Montague:

Yeah, but I'm, but I'm thinking, I really am thinking that I'm going to do partners 100 by 10s, to just bring out the idea of 100, and then just see what's happening.

Pam Harris:

Kim Montague:

Yeah, yeah. Well, I mean, if they've never played ever before, then that might be a nice entry point. So then I think I'm going to do a Rich Task, or maybe an Open Middle activity, where students are working with those same partnerships. If I'm moving towards total of 100, or total of 200. It's really important for me, in third grade to have kids recognize that if I know 42, and 58, then I know that 142 and 58. Is that next 100. I think that's a really essential thing for kids to make sense of. That there's just these partnerships within hundreds. And so I would absolutely spend some time doing some work around that.

Pam Harris:

Let me give another example. So you're saying like, if I know if 32 and 68, partner to 100. I also know 332 and 68, partner to 400. Yeah, I think it's a really big idea. And so I might

Kim Montague:

Yeah. I Have, You Need, and then I might do an Investigation where kids are drawing out possible combinations. And it could be like an Open Middle and we treat it like a really rich time where kids are messing with what can they find? And are they looking for any patterns? So that might be like the end of day one for me. And then in my 1/3 time, on that day, I might be doing something where they're playing Close to 20. Or they might be doing a, you know it might be something that I'm I know that I'm eventually going to play Close to 100, some might be introducing some games in a small group type setting. So the next day, I might come back together. And I might say, let's do a Get to a Friendly Number string with some numbers where I'm having them look at 27 and saying how much more to the next friendly number? So we're still looking for partnerships of 10s and hundreds. And then we're going to have a Congress to pull out the patterns that the kids noticed in the thing that they did the day before. Or maybe I can see playing Close to 100, which is one of our favorite games. Maybe during the one third time, like I said, I'm playing close to 20 or 100, even 1000 with different students, because I know they need something different. Which is brilliant, right? Because all the kids are playing, quote unquote, the same game. Yeah.

Pam Harris:

But you've differentiated the total that allows them to gain more experience where they need more

Kim Montague:

Yeah, yeah. Or in my 1/3 time, I might be, you experience. know, like I mentioned in the last episode, sometimes looking back at work, or sometimes doing a problem string with slightly different numbers. You know, if I haven't heard the voice of some of my students in whole group, then I might pull half the class and do a problem string and so that I can, like more focus in on what some of my kids are thinking. So you know, it might all be centered around 100 for a little bit, a couple of days, but we're like you said the first grade doubles is kind of the big idea, and how they can use doubles. You're tackling it in a couple of different ways and I think 100 is crucial in third grade. So,

Pam Harris:

Yeah, hey, really quick when you said at the end of the, golly, I think it was the 1/3 time. Okay, I'm not even sure when it was. But when you were talking about that you might pull a small group in prep for the next day.

Kim Montague:

Oh, yeah.

Pam Harris:

Can you say more about that?

Kim Montague:

Sure. So if I know that I'm going to play, I'm going to introduce Close to 100, which is, you know, one of our favorites. And I know that I'm going to play that game. And I'm going to introduce it maybe whole class. Because we like to do that. Play against the class. So they really understand how to play. But it's a game that I'm going to play all year long. It's not like a one shot game.

Pam Harris:

You start playing, sorry, start playing against the class. So they really understand how to play, so that then you

Kim Montague:

You can send them in partners. Yeah, but there can see can- might be times, there might be classes, there might be kids who I know, maybe there's a lot of instruction to the game, or maybe notating their thinking might be something that they're not really sure how to do. It might be that there's a whole host of reasons why I might pull a small group and say, "Hey, tomorrow, we're going to play this game. And I want you guys to be my experts. Like, I really want you to know how to play this game." And it just boosts a little bit of confidence. Maybe? Or maybe, you know, it's not the kid who's always the expert in the thing. So there are times where I might pull a group the week before or a couple of days before and give them some experience before we play as a class.

Pam Harris:

Just sort of front load their experience.

Kim Montague:

Yeah.

Pam Harris:

Yeah. And then when you're doing it whole class, because of that they're able to join in. What a wonderful way to give them that experience that just a little extra experience that they need to then be able to join in. That's nice.

Kim Montague:

Yeah.

Pam Harris:

And I loved it how you couched it as you guys are gonna be my experts.

Kim Montague:

Yeah.

Pam Harris:

That's cool.

Kim Montague:

Everybody wants to be an expert at some point, right?

Pam Harris:

Yeah, absolutely. All right. I really liked that third grade plan centered around a really big idea. Yeah, nicely done.

Kim Montague:

Okay. So let me give you a little bit about fifth grade, too.

Pam Harris:

Kim Montague:

So a lot of this depends on the time of the year, because, you know, both of us talked about Problem Strings and games and some Rich Tasks. And those are, I think, essential Jey, before we go to one third. for the whole year. But there are times where we're getting towards test taking I mentioned or we're getting towards, I need to, I you know, I've introduced a topic. And I feel like I need to spend more time with you on it. So fifth grade, depending on the time of the year, I might start with a multiplication string to just get gears turning. And then we might do an Investigation about, say, like, how many boxes could they make with 24 chocolates, which of course leads to conversation about volume and later on surface area. And we have a rich Investigation. I might follow up that with a Problem String about prime factorization, or another one of our favorites, the Product Game. In the 1/3 time, I might be- Yeah.

Pam Harris:

So that sounds like you're really maybe focusing around either multiplication or volume. Like that sounds like a really fifth grade thing to do that we're really getting more sophisticated in multiplication. So let's use volume to do that. Oh, and volume is this thing we need to do anyway.

Kim Montague:

Yeah.

Pam Harris:

So we're just sort of kind of everything, you're doing different things, but they're all building toward facility with multiplication and volume.

Kim Montague:

Yeah, since they're so connected, right? There's so many strategies for multiplication that are going to help so that it's not, here's this formula for volume.

Pam Harris:

Yeah.

Kim Montague:

So in that third time, which is kind of extra, could be set aside from the two thirds or could roll right together, I might be setting up a game or again, maybe an Open Middle puzzle for the class to work on. I might be saying, "Okay, today, you guys are gonna play the Product Game or play," or whatever, pick a game that I've already introduced. And that's, you know, we've gone over the expectations for how that's played. And, you know, here's your partners, or pick your partners or whatever. And I'm going to pull small groups of students during that time. So you know, for 20 minutes you guys are going to play, they know how to play because we've gone over it together. I'm pulling small groups of kids, or maybe I'm free, and they're coming to me one on one with, "Hey, I had a chance to look at this assignment that you passed back and I want to talk about whatever this this thing that I wasn't sure about," or "I want to talk to you about this thing that I thought I understood and maybe I need some more support." or "Hey, look at me, I did so fabulous on this and, and I just want to celebrate it." So it might be that I'm freer, and they're, they just have some kind of back and forth flow between the game and meeting with me.

Pam Harris:

And that game or that task, whatever they're doing at that point is something or centered around a thing you know, it's important all year long. And so that's going to be one that you're going to continue to bring back all year.

Kim Montague:

Yeah.

Pam Harris:

So the kids were like, oh, yeah, like this is a good engaging thing. While, it frees you up. I'm going to mention one other kind of thing that you might do well, it frees you up. It frees you up to circulate and listen in.

Kim Montague:

Yeah.

Pam Harris:

Right? So you get a chance to kind of hear where students are. And that gives you really good information about what you do next, because now you know, "Oh, everybody's really mastered this game. There's not a lot of thinking going on anymore. Bam, it's time to move up."

Kim Montague:

Yep, for sure.

Pam Harris:

Yeah. Cool. Hey, it seems like we're kind of in an odd grade habit here, because I'm gonna go to grade seven.

Kim Montague:

Okay. All right.

Pam Harris:

Let's stay odd a little bit here, 1, 3, 5, 7. All right. So in grade seven, what could it look like? I'm thinking that rates are really important. Often non unit rates are really important in grade seven. And so I might think about doing a rate string. We did a rate string on the podcast not too long ago, I don't know a couple months ago, maybe we'll put that in the show notes of when I did that rate string. But, you know, if you're covering 20 feet in four seconds, how fast are you going? And then 19 feet in four seconds, a string like that, where kids are kind of starting from non unit rate to a unit rate, and back and forth, and forth and back. They're also getting some really nice division experience as they do that they're getting the sense of, of a speed. And what that means. And all of that is sort of front loading for what I know, they're going to do later with slope of a line. So this rate stuff. So I might start with that rate string. And then to continue to do that kind of an idea, I might have them do a motion detector Investigations. I love motion detectors.

Kim Montague:

Yay you do.

Pam Harris:

Kim Montague:

Yeah.

Pam Harris:

All right. So there's a great seven. A look at what might happen. Oh, and I'm gonna

Kim Montague:

Okay. mention again, just like you said, that's not all happening in one day. That's over a couple of days. You can't do all that in one day. I don't think so. Even that Investigation, the Investigation itself, should take most of the class period, especially if you do the Congress after it.

Pam Harris:

Then the strings are coming on either ends of those.

Kim Montague:

Pam Harris:

Absolutely. And so if you're listening to this podcast, and you're like, "Whoa, I mean, that sounds interesting. I'd like, hand that to me, and I'll go do it." And you don't want to create it all. We don't think that teachers should be expected to be master teachers and master curriculum writers at the same time.

Kim Montague:

Yeah.

Pam Harris:

So to where might you start? What might you, if you want to dip your toes in and give something a try? Just consider carving out some time. Find some time once a week that

you can commit:

I'm going to do a Problem String. Start there. Start with, "Hey, y'all" like Fun Friday, Marvelous Monday, Terrific Tuesday, whatever it is, a time of the week where you feel like, "I can get 15 minutes, I'm going to commit, I'm going to do this thing." A place to start is to start doing Problem Strings. Just quickly, the reason to carve out that

time is A:

so you'll actually do it. But B: it's also, by carving it out, I also mean, not just like the time itself, but actually the space that you set up this time and space with students so that they know during that time and space, you're doing something different, you're doing this thing, and they know their ex, they know what their role is, they know the expectations during that time. And they're able to dive in and do thinking and reasoning. That allows them to feel safe during that time that they're like, "Yes, I can, I don't have to worry about anything. I just like dive in to actually think. I know my job here is to use what I know." But it also protects you a little bit, because kids aren't going to be going, "Hey, miss, I thought we were supposed to like do that thing." It's not you saying, "Alright, forever after we're going to do this from now on." And then you slip, you forget, you, you know, go back into I do,we do, you do just naturally, because that's sort of what you're used to. And now the kids are giving you this hard time or whatever. It gives you this carved out time where you're like, "During this time, this is going to happen." You get better at it, they get used to it, then it can start to permeate more and more of what you do. You know, it's interesting, one of the fascinating comments that I keep getting in our message boards. So you know, we have these online workshops that go on. And I'm in the message board, interacting with people, answering their questions. I keep getting this interesting comment that teachers and leaders are saying that they finally get to experience all the things, not just hear people talk about them. They say like, "Pam, people talk about growth mindset, they talk about equity, and they talk about discourse," and you know, blah, blah, all the things. And they're like, "But Pam, we actually experience it in your workshop." Well, I really appreciate that. Because they say that it actually is happening naturally that what they don't see is me saying, "Y'all equity is important and discourse is important. And growth mindset is important." But they experienced that the way I'm teaching is growth mindset, that it is equitable, that you are seeing discourse happen. And then we have this really cool module in our workshops, where we sort of step outside, and we rewatch some of the clips from what they've just experienced. And we point out, "Hey, did you notice this was actually a purposeful move I did to have equity happen. This, this was actually a purposeful move I did that created that discourse that allowed the sense and a feel of what was happening so that this discourse happened. Or that I just did that right now see how that encourage the growth mindset that then right here you see a teacher having, because of what we just set up." So I think it's a really unique part of our online workshops, where you do the online workshop, and you learn the math. And then we take a look back at certain specific things. And we call out and identify really important things that happen kind of naturally in the workshop. And now you can make it happen because you're like more, you've experienced it.

Kim Montague:

You're aware.

Pam Harris:

Now you're aware of the things that you have control over as a teacher to instigate so these things can actually happen in your classroom. And I'm particularly happy about that.

Kim Montague:

And can I just add that it's an experience that we don't often get to have in our classrooms, because time is passing by as we're making the moment to moment decisions. And so you don't get often a chance to look back. But in the workshops, you participate in the math, and then you get to look back and you get to like internalize what's happening a little bit and go, "Ah, like, I believe that I want to do that."

Pam Harris: