Pam  00:01

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

Kim  00:09

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

Pam  00:16

We know that algorithms are amazing human achievements, but they are 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. 

Kim  00:42

Hey, there. 

Pam  00:43

Kimberly.

Kim  00:44

Yeah. You know what... It's good. 

Pam  00:48

Yeah? 

Kim  00:48

It's good, yeah. Yeah.

Pam  00:53

Should we start? Should we get going? (unclear)

Kim  00:55

(unclear) Okay. Hey, so I'm really excited about today's episode because we get this question a lot, and, you know, we thought that we would spend some time in the podcast talking about it because then we can refer people to it. But the question is so good that I think it's worth diving into. So, in one of the Facebook groups, the Math is Figure-Out-Able teacher Facebook group, Amy left a message, and she said, "Hi, I'm absolutely loving everything Pam and Kim have to offer. Problem Strings have been a game changer for kids and teachers. I'm curious what others think about older students who need reasoning with all operations." Yes. 

Pam  01:34

Right. 

Kim  01:34

"Do you stick with one operation? Bounce between them? How do you plan Problem String sequences for these students?" And then she goes on to talk about what she does, and she says, "I get a lot of kids who are considered to be, quote, struggling, but I find they've been filled with procedures and respond amazingly to other practices." Yeah. 

Pam  01:52

Yeah, isn't that the truth?

Kim  01:53

Mmhm. "I've got many grade four through six students, and while I want to do multiplication, division Problem Strings, they've not had experience with reasoning with addition, subtraction. I often see them only a little bit of time." And then she's basically trying to figure out like how do I do all the things? What is a recommended sequence or plan for them? 

Pam  02:12

Yeah, I'm so glad you asked this question. 

Kim  02:15

Yeah. 

Pam  02:15

Let's dive in. Kim, as we do the very first thing that I would suggest. So, maybe I'll just start. Kim worked with younger students. I taught older students, so I'm... But you also taught older elementary students. So, both of us had students who had been at rote memorizing, mimicking for a hot minute, right? Like, they'd been memorizing and mimicking, and we both got them and had to decide where do we start with them? Would you agree with that? 

Kim  02:48

Yeah, for sure. 

Pam  02:49

Though, you had a few years at the kids, my kids school where your kids came up to you not rote memorizing and mimicking. 

Kim  02:58

Yeah. 

Pam  02:58

I remember when you moved to the new school.

Kim  03:01

Mmhm.

Pam  03:01

And you and I were chatting, and you were like, "Wow, I didn't realize how good I'd had it." 

Kim  03:05

Right, right.

Pam  03:06

Yeah, I remember you saying that, but we both... So, we've both been in this situation where, similar to Amy, where we have to kind of make this decision. Like, if we have students who have been rote memorizing and mimicking, whether they've been successful at it or not, what do we do with them now? Because we're recognizing if we think about developing mathematical reasoning from counting to additive to multiplicative to proportional to functional, we recognize that while our content might be here, we want to shore up the things that kind of support those, right? The hierarchical that they're built on and based on. So, where do you start? So. 

Kim  03:38

Yeah. 

Pam  03:39

A major thing that I want to suggest, whatever you do, know your students well enough. Don't start with something that will make them feel dumb. Don't start with something that will make them feel like you're going, "Alright, ya'll, you don't know anything, so we are going to back up to second grade because you don't..." Like, whatever you want to do, you want to do something that makes them feel cool. So, you want to (unclear).

Kim  04:06

And...

Pam  04:07

Yeah, go ahead. 

Kim  04:07

Well, I was going to say, you might not be saying those words, but you might be giving that impression by backing up too far and in certain ways. So, you know, people aren't going to say, "Oh, yeah. I say that." No. Nobody's saying that. (unclear).

Pam  04:21

Well, of course. And the way I just said that might... Listeners might be like, "Pam, I'm not going to make my students feel dumb. Who do you think I am?" So, yes. I'm absolutely not trying to like patronize anybody. I'm.. Yes. So, thank you, Kim. What I am trying to say is, if you back up too far in, without making it look, feel, sound like cool, like it's intriguing, then the chances are super high that they're going to like roll their eyes, sort of slump over and go, "Really? Really? You think we're really we can't do that?" Even though we as teachers know, yeah, you actually can't do that. You actually do need work here. So, here's (unclear).

Kim  04:58

And you run the my risk...

Pam  04:59

Oh, go ahead.

Kim  05:00

...that you're cutting off opportunities even when you get to the stuff that's in your grade level because you set the tone that, you know.

Pam  05:07

Yeah, yeah. 

Kim  05:08

It's this baby stuff.

Pam  05:10

Yeah. And so, there's this whole like line to walk, this kind of balance beam of how do you get a bit of review in without making students go, "Oh, we're just going to review." Because, as we've said before, if you just start reviewing, you're going to have one of two things happen. Either kids are going to say, "Oh yeah. That's the stuff that we did before. I know how to do that. Sweet. I'm not going to have to work in here," and you've set the precedence for that terrible work ethic. They're not going to have to work in your class. Or they go, "Oh, my gosh. I was never good at that. I'm not going to be good. This is going to be a terrible experience," and now you've set that precedence for your class. Those are sort of the only two kind of things that happen. So, what can you do? We suggest that you start with something that feels intriguing to your students, something where they can go. "Oh, wait a minute. What? Well, that was... Oh. Okay. Well, I'm tentatively interested. I'm listening. Okay, do more of that." 

Kim  06:05

Yeah.

Pam  06:05

That's the sweet spot that you're kind of interested in. 

Kim  06:08

And, you know, there are some strategies that are really slick, and fun, and intriguing, so you don't have to make stuff up. You just got to start with the ones where kids are like, "That is really useful. I want to know more about that." So, the Problem String books... Go ahead. 

Pam  06:25

Maybe I'll just say the Over strategy. For students who have never reasoned, Over strategy where you do some problems that they can kind of access, and you move into high dose pattern them, and move into some problems where they're totally. You know, you give the helper, and they can use that helper to solve it. And you're like, "Hey, how did you just solve that problem?" And they're like, "Whoa! I just did that. Like, that's..." So, that's a place for you to look, to think about. Over strategy with any of the operations could be a place where almost everybody is super intrigued. I often start with a version of the Over strategy with adults, and they're plenty intrigued. 

Kim  07:04

Yeah.

Pam  07:04

Yeah. Alright, so sorry, Kim for interrupting.

Kim  07:06

No, it's okay. I was going to say that, you know, there's a lot of language in the Problem String books that we worked on about guidance for strategies to start with, kind of a sequence and some order, and maybe had a flow back and forth between the two of them, and so that that support is there in the Problem String books. So, we'll mention a few of the things here. It's a really important question that Amy is asking because the answer depends kind of where you are in your class. We've talked about different kinds of time in your classroom before, and it'll depend based on which section you're talking about.

Pam  07:41

Yeah, and so I think what you're referring to is we've kind of talked about how in our ideal situation, setup, scenario of how we teach math, we like to think of kind of a two-third, one-third track.

Kim  07:54

Yeah.

Pam  07:55

For me with older students, it's a bit more of a three-fourth, one-fourth track. Either way that there's kind of this separate track that's a little shorter, and then there's this kind of main track that's kind of the bulk of your instruction. And so, Kim, I'm just going to ask you. When you think of that one-third or that not main track but the kind of other track, what do you think about happens during that time period in your class?

Kim  08:24

Yeah, so that's the kind of catch all time. That would be maybe playing I Have, You Need, or Count Arounds or some other routine that I want to spiral through some things that we've either previously worked on. Or it's something that I know has a lot of social convention to it, so I want kids to hear that all through the year. You know, I taught in tested grades for our state, and so I would have done some sort of test formatted practice during that time just to help them be aware of what is going to be coming. I pulled small groups during that time, played games during that time. That was the time where we really strengthened different kinds of things and reviewed. 

Pam  09:06

Mmhm.

Kim  09:06

Yeah, and it was kind of a hodgepodge. It was very purposeful, and it was very... You know, I added things to our repertoire as we went along, so that then I could have my kids really own some things, so that I was freed up during that time if I needed to be. So, that's kind of where we set the tone for like here's some choice and some options.

Pam  09:24

When you say you added things, you could hear added content, but I think you meant added like a new routine. Like, you might say, "Hey, at the beginning of the year, I'll start with this major routine, and then I'll add this one, and then I'll add..." and you got kids good at them, like you said, so then you could kind of would free you up to pull a small group. (unclear).

Kim  09:40

Right, right. So, it might have been all together or it might have been in pairs or small groups. So, that's where I set the tone for you know how to play these games. It might have been, like I said, whole class and we're doing this thing together to get you prepared for something else. But it was all this kind of separate review, spiral, fluency kind of work. 

Pam  10:02

So, high school teachers, an example of that might be where you use a factor puzzle. A factor puzzle is where you give kids the product and the sum, and then you say, "What could the two numbers be?" without all of the binomial, you know, the parentheses and the x's and stuff. You just literally mess with those number relationships, so that when you get later in the year to factoring quadratics, kids are already have those number relationships happening, and now you just have to bring the algebra in. So, that's kind of an example of a thing that you might do kind of all year long, so that you're building up to when you're going to need it.

Kim  10:39

Mmhm. 

Pam  10:39

Yeah. Yeah. Cool. So, if that's that kind of smaller or less time filled, short. I'm trying to say shorter. It's the (unclear).

Kim  10:49

Well, a third is smaller. (unclear).

Pam  10:52

So, that's like the third of your time. If it's just two-third, one-third split, and that was the one-third split of your time, Kim, what's happening in the two-thirds split?

Kim  11:02

Yeah, so the two thirds time is that bit of time where you might typically have like a scope and sequence from your school or district or, you know. That's kind of like all the big standards kind of marched in order the way that, you know, you're expected to do or that you've chosen to do. You've clustered them in certain ways. That's going to be like your typical daily lesson. It might be like the Rich Task if you're, you know, doing those. But it's generally maybe chosen for you where you might have more freedom to make some of the choices in the one-third time.

Pam  11:39

That makes sense. Yeah, yeah. So... 

Kim  11:42

Kind of the meat the core, right? The core of what your lesson is.

Pam  11:47

Yeah, of what your lesson typically is, yeah. Like, this is what we're learning in these two weeks or three weeks, this sort of unit. That's what's kind of happening in there. Yeah.

Kim  11:55

Yep. Yeah. 

Pam  11:56

So, if you're thinking K-2.

Kim  11:59

Well, so we... You know, it depends on where you're talking about strings. So, Amy's question was what strings do you choose and how do you decide? If it's in that two-third of the time, then it's very likely going to be a string that is around the topic that you're working on. So, if you're working on multiplication, then it's going to be a multiplication string. If it's something that's, you know, division, whatever, fractions, it's going to be something within that time. But if it's in the one-third time, you might have some more choices. So, in in K-2 I'm thinking about these are the major, global things that I want to accomplish this year, and I'm thinking about the kind of the big strategies that need to happen for that particular grade. In 3-5, you know, you've got multiple operations, and so you're going to think about I need to span numbers. So, if I'm doing subtraction in grade three, I'm thinking about mostly whole numbers, right? But if I'm a fifth grade teacher, I'm talking about, I've got whole numbers. I've got decimals. And if I'm thinking about Over, I'm doing it with whole numbers, decimals, fractions, maybe even later. And it means that I've probably done some fraction stuff before. So, within all of those ranges, we're going to do lots of different things for a particular strategy.

Pam  13:15

Yeah, so that makes sense. So, you're kind of thinking like how do I? If I'm a particular teacher and I've got this thing going on, how do I span that over the year? Can I build it so that it progresses, it gets more, like kind of like you just said, from whole numbers to decimals to fractions? What I'm hearing you say is it's less that I'm going to accomplish that all right now.

Kim  13:34

Right.

Pam  13:35

And more that I'm going to think about how do I kind of span that over the year, so that by the end of the year kids are really good at the Over strategy with all of those numbers. 

Kim  13:43

Yeah, so like in fourth grade. Let me be a fourth grade teacher for a minute. I'm thinking, "Okay, I've got these four operations. And these are the major strategies within each operation." And then for each of those strategies, I'm also going to be thinking about the span of number from whole number to decimal to fraction. So, there's a lot of layers to consider for me to fit it in throughout the year. And I'm going to maybe do a couple of strategies in whole number early on, then cycle back around to do those same strategies with decimals a bit later. And then I might do some of those same strategies with fractions even later in the year. 

Pam  14:26

Yeah. But if I'm a middle school teacher, I might do, I might think to myself, really, I'm focused on rational numbers, and so I'm going to pick a strategy, and I'm going to do a Problem String that starts with whole numbers, moves into decimals, and ends with fractions.

Kim  14:42

Right. 

Pam  14:43

But then I might do tomorrow another string toward that same strategy with all of those numbers because I'm a middle school teacher, and so I'm going to kind of lump those things together a little bit more, and then I'm going to switch strategies. Again, starting with whole numbers, moving into decimals, and then maybe ending with fractions. Maybe that very first time I do that strategy, maybe I don't end with fractions. Maybe I just really get the whole number and decimal thing down, then pop in the fractions the next time that I do it.

Kim  15:11

Yeah, and I don't know that we say this often enough that we don't do one string for a particular strategy. So, also to layer into that is, okay, I'm going to be working on this particular strategy, and I'm going to do it with whole numbers. And I might need to do two or three strings within whole numbers. Or moving from whole numbers to decimals whatever my grade is. So, we're not saying this is a one hit. You did it once. Great. Checkbox. Move on to the next strategy. 

Pam  15:41

They got it. They got it. They got it. We're done. 

Kim  15:43

Yeah, they don't have it after one string. 

Pam  15:46

Or do they need more experience? We're going to high dose them over time. 

Kim  15:50

Yeah.

Pam  15:51

Yeah. Yeah, nice. And so, if I'm a high school teacher, I might be thinking about like what are the major content areas that I need to hit this year? But I also want to shore up some numeracy. Well, I might shore up that numeracy in that one-third time, some. But I also might say to myself, "Hey, we're about to do exponential functions, so why don't I do a quick numeracy Problem String today that starts with whole numbers and does a little bit of decimal and fraction stuff that gets into multiplying with the base multiple times. You know, where it's like some kind of a where I'm I'm Overing a little bit, but I'm heading toward this sort of multiplicative relationship that I know is going to be sort of necessary for me to build exponentiation on that. Yeah. Nice. Cool. 

Kim  16:40

So, when I was in the classroom, one of the easiest ways for me to kind of make sense of what needed to happen was thinking about that two-thirds portion of the time. And I might say, "Okay, I know I'm going to teach these big ideas, and so I'm going to plug in maybe some strategies within the time period that's my two-thirds." And so, if I'm working on multiplication, I know that I'm going to have multiplication strings around that chunk in my two-thirds time. But I might also... And that might be like, say, on a Monday, Wednesday, Friday. I don't know. But then also, I think it's fair game to say in the work that I've done with my students, I see that many of them are not using the variety of choices that they have available to them in subtraction. They're doing maybe the same thing over and over again. And so, I want to strengthen subtraction strategies at the same time. But I'm doing multiplication in my two-thirds, so over in my one-third time, I might plug in a subtraction strategy just to kind of cycle back and remind them that they have that at their disposal as well.

Pam  17:45

Oh, gosh, especially if you're doing multiplication work like with the Over strategy, and so they're going to be doing subtraction in that multiplication work. That would be a really nice way to set them up for success there. Yeah. And, Kim, I've heard you say over and over again, and I think this is so brilliant. The first time you said it, I really had to think about it. And the more I think about it, the more brilliant I know it is. Don't change more than one thing at a time. As you think about progression, you've said don't change more than one thing at a time. Could you talk more about that a little bit? 

Kim  18:19

Yeah, so when we talked about having two or three strings, maybe in a row that you're focusing on, you can make shifts one at a time, generally, and kids can hang on to that. So, you might change the size of the numbers going from two digit to three digit or three digit to four digit. Or the complexity of the numbers, whole numbers to decimals. So, size of number, the number choice is an important one. The other one is important is the modeling that you use. We might say, "Okay, we want kids to think about multiplication on an area model or an open array, but also have facility on a ratio table.

Pam  18:58

Mmhm. 

Kim  18:58

So, what I would not do is say, "Okay, today we're going to do whole numbers on on an array, and tomorrow we're going to do decimals on a ratio table. There's... That's two shifts at the same time right back to back. (unclear)

Pam  19:13

You made the numbers harder, and you changed the model.

Kim  19:16

Yeah.

Pam  19:17

Yeah, (unclear). 

Kim  19:17

Yeah. And then we also don't shift strategy too quickly. So, I'm not going to switch the model, and the strategy, and the size of the numbers. Generally, I try to think about one change at a time. And you have lots of days of school, so you can have a string every single day, and there's lots that you can do. Especially if you have our Problem String books.

Pam  19:40

Yeah, so higher grade teachers like high school teachers, I'm thinking that you might do something in a table with students one day. And then you want to connect that to a graph. But maybe don't shift function types at the same time. Like, maybe it's like let's hang in this function yype for a while and connect tables and graphs before we then shift a different function type. Yeah, that might be a way of thinking about that. Yeah, cool.

Kim  20:11

I also think it's really important that we mention again, because I know we've said it, that early on in the year when you are facilitating strings in your classroom, it's going to take a little bit longer. They just... Kids are just kids aren't necessarily used to the routine or it's been a minute since they've done it. They will get faster. Until you switch a bunch of things again. So, you switch a strategy, all of a sudden, it's going to maybe take a little bit longer. Especially if you use the same number of problems that you've been doing and you switch strategy, you know, it's a different way of thinking maybe. So, when you make switches, it might take a little bit longer, but then the more you do, things speed up, and so you can get to those five to seven minute strings that we talk about. People are like, "Wait, I didn't. How does that happen?" You might not be doing them often enough, or switching too many times.

Pam  21:01

And switching too many things at a time. 

Kim  21:02

Yeah, yeah (unclear).

Pam  21:03

So, as you up the ante, take heart that it might take a little bit longer today, but you're going to gain that. We're going to go slow to go fast. We're going to gain that over time. If it ends up... If it takes a ton more time than you're anticipating, then maybe kind of consider did you perhaps switch up too many things all at once? Could be thing to consider. Alright, Kim, so as we think about sort of where we, how we are sequencing thing, one of the things that we can think about is this two-third, one-third idea, what you're doing at both of those times, and also how you are sequencing, how much you're changing all at once, and the idea that you're going to kind of cycle through things, upping the ante as you go throughout the year to get kids to learn from what they do know and can do to new and exciting, important things. 

Kim  21:52

Yeah. 

Pam  21:52

Nice. Cool. Alright, ya'll thank you for tuning in, teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfiguratable.com. Let's keep spreading the word that Math is Figure-Out-Able.