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  00:10

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. 

Pam  00:19

Let's be on a mission together. 

Kim  00:20

We're on a mission together to improve math teaching. 

Pam  00:23

Hey, how come you say a reasoner who now knows how to share her thinking not her reasoning with others? Is that because we don't want to (unclear). 

Kim  00:30

Probably because I don't want to say "reasoning" twice. 

Pam  00:32

Ah, okay. Alright.

Kim  00:33

Yeah.

Pam  00:33

Hey, ya'll, we know that algorithms are amazing human achievements, but let's be clear, they're terrible teaching tools because mimicking step by step procedures actually traps students into using a digit-oriented approach to everything and using less sophisticated reasoning than the problems are intended to develop. And they then question their math identity, and we get this pandemic of "I'm not a math person" all stuff.

Kim  01:00

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

Pam  01:08

We invite you to join us to make math more figure-out-able. You don't like it when I go off script so much, huh?

Kim  01:14

No, I think it's fine.

Pam  01:15

Ah, okay.

Kim  01:16

It can be boring to listen to the same thing to the start of an episode.

Pam  01:19

I mean, it's boring for me to say the same thing every time. 

Kim  01:22

Yeah.

Pam  01:23

Alright. We'll get to episode number something cool, and we'll change the intro again, but not today.

Kim  01:28

Yeah. 255 is not the time. 

Pam  01:30

Nope. 

Kim  01:31

Okay. So, yay! If you've been listening the last two episodes, then you know we have chosen a Problem String. We've unpacked all the things to think about when you're choosing. 

Pam  01:43

Mmhm.

Kim  01:43

And then we unpacked all the things that you could think about when you're planning the string. So, you have a string. You've planned the string. And you have facilitated the string. And so, now maybe you're wondering and you're thinking about what you've learned from your students. You still have some wonders. So, what happens after a Problem String.

Pam  02:03

Yeah.

Kim  02:04

Yeah. And before we get started to that, let's have a string in the beginning here, so that we have something to talk about. We can do some math together. 

Pam  02:12

Alright, so let's do a little math, and then we can talk about that as we talk about, if you've done this one, what are you thinking after that? Alright. So, Kim, I am thinking about the fact that I was in Canada recently. And in Canada, they have these M&M-like candies. 

Kim  02:27

They're not M&Ms. 

Pam  02:28

They're not, but they're like. They're like. And they're called Smarties. 

Kim  02:32

Yeah, confusing.

Pam  02:33

So, people in the States, yeah, if you think of a Smartie, you think of that chalky, kind of disgusting sweet. I don't like Smarties in the United States. They don't make me happy. They're (unclear). 

Kim  02:42

Can, I tell you?

Pam  02:44

It's like sugar. Yeah. 

Kim  02:45

My kid is in this dual credit history class right now.

Pam  02:49

Okay.

Kim  02:49

And apparently, right before a test, they have this, I don't know, quizzle wazoo, whatever. Like, whatever they do. Whatever this thing is.

Pam  02:57

Like, electronic quiz thing? Is that what you mean? 

Kim  02:59

Yeah. Yes, I guess. 

Pam  03:00

Okay, alright. 

Kim  03:01

And he comes home. And they've done two or three of them, and he, you know, is like the king, apparently of this whatever this thing is. And he comes home and he dumps a pocket of his backpack out and just like lands Smarties all over the kitchen counter. Like...

Pam  03:19

That's what he won?

Kim  03:19

...piles and piles and piles of them. I'm like, "Hey, can you share with your friends? We don't want those."

Pam  03:25

Okay, so are you talking American Smarties then? 

Kim  03:27

Yeah, the chalky ones. The chalky ones.

Pam  03:29

The chalky. Yeah. Bleh. 

Kim  03:31

I don't actually hate them. 

Pam  03:32

Oh, do you like them? 

Kim  03:34

I wouldn't say like.

Pam  03:34

That would make sense because I don't like them, so that would make sense that you would. Which is why we get along so well when we travel together, right?

Kim  03:40

(unclear) nostalgia. I had him as a kid, yeah.

Pam  03:42

Oh yeah, no. 

Kim  03:43

You know, I don't really eat candy all that much anymore, but I think so.  Yeah. 

Pam  03:43

Not worth. No. No. Yeah. Okay, anyway, when I was in Canada, a dear friend, Cathy, gave me several boxes of Smarties. Thanks, Cathy. And not watching my waistline there. So, these are like M&M like kind of candies. And the one particular box, when I counted the Smarties. Because I sort smart. Do you sort? Like, if you're eating M&M's or Skittles, or whatever, do you sort them by color? Okay, alright.

Kim  03:46

I did Skittles because they have different flavor, M&M's, I don't care about because they tastes the same. 

Pam  04:14

So, you don't want to be surprised. You don't want to just grab a Skittle and then pop in your mouth and be surprised. 

Kim  04:18

Well, I have favorite flavors.

Pam  04:20

See, okay.

Kim  04:20

But that's when it matters.

Pam  04:21

There you go. Unlike like M&M's and Smarties, you don't. They're not. They taste the same, so maybe it's not whatever. I still sort them. I don't know why. My daughter and I sort together. They used to talk about sorting chips. Anyway, here we go. So, (unclear) as I was sorting them, there were 26 Smarties in the box. So, there's 26 Smarties in the box. How many Smarties are in 10 boxes?

Kim  04:44

260.

Pam  04:45

Okay, so I've just drawn a quick table, and I've put "Box" on the left and "Smarties" on the right. 1 box to 26 Smarties. And then underneath that I wrote 10. And then you said it was 260, so now I have 1 to 26 and 10 to 260, yeah? 

Kim  04:59

Mmhm. 

Pam  05:00

Because if you have 10 times the number of boxes, you have 10 times the number of Smarties. Cool What about 5 boxes? If I had 5 boxes of Smarties? How many Smarties? 

Kim  05:08

130.

Pam  05:09

Because?

Kim  05:11

Because Five is Half a Ten, so 260 Halved is 130.

Pam  05:16

So, if you have half the number of boxes, then you would have half the number of Smarties. And you said that was 130. Cool. What if we had 15 boxes of Smarties that had 26 Smarties in them each? 

Kim  05:28

Then you would have 10 boxes and 5 boxes.  And so, 260 Smarties plus 130 Smarties is 390 Smarties.

Pam  05:31

Okay. Alright, so I've written in the table 15 and 390. And I have drawn a little bracket from the 10 and the 5. And I'm like 10 plus that 5 boxes is 15. And then a little bracket. 260 plus 130 Smarties is 390 Smarties. Okay, cool. What if we wanted 16 boxes of Smarties? 

Kim  05:57

Okay. So, it's going to be 26 more than the 15 boxes, which was 390, and that's going to be 416.

Pam  06:07

Smarties.

Kim  06:08

Smarties.

Pam  06:09

Because you added the 15 and the 1 box. So, again, I might draw brackets over there between the 1 and the 15 and the arrow down to the 16. Then, the corresponding Smarties, draw a bracket. And then, yeah. 416. Cool.

Kim  06:21

Mmhm. 

Pam  06:21

But what if I really like chocolate? Which I do. It's terrible. What if I wanted to have 50? 

Kim  06:28

Can I...

Pam  06:28

50? Yes?

Kim  06:29

This whole time, I know we had this conversation about the Smarties.

Pam  06:33

Oh, okay. 

Kim  06:33

But you kind of threw me when you said chocolate because I've been picturing Smarties, chalky Smarties. Okay. I was like, "Wait, new context." Okay, so how may did you say?

Pam  06:43

50. 

Kim  06:44

50.

Pam  06:44

They're colorful. They're colorful. I think they're actually bigger than

Kim  06:47

Yeah. 

Pam  06:47

M&Ms.  Oh, yeah. Okay.

Kim  06:50

Then you would have (unclear)... 

Pam  06:51

You're not excited because you're not a chocolate fiend. You don't... You can turn chocolate away. 

Kim  06:57

Yeah.

Pam  06:58

Yeah. Okay.

Kim  07:00

1,300.

Pam  07:01

Because? 

Kim  07:03

Because 5 boxes was 130, so 10 times as many boxes give me 10 times as many Smarties. And you

Pam  07:10

had 130. And 10 times 130 is 1,300.

Kim  07:14

Correct.

Pam  07:14

Okay, cool, but that's too many Smarties, too much chocolate, so only 49 boxes.

Kim  07:23

Then I have 1,274 Smarties. 

Pam  07:27

I have to ask, did you just play I Have, You Need in your head? 

Kim  07:29

I did indeed. 

Pam  07:31

So, it's like, if you're going to take... Well... Yeah.

Kim  07:36

So, I had 1300.

Pam  07:37

Thank you. 1300 Smarties. 

Kim  07:39

Mmhm. And I wanted to get rid of 26 Smarties. 

Pam  07:44

Because? 

Kim  07:44

Because I want to get rid of 1 box. 

Pam  07:46

Okay. 

Kim  07:47

And so, then I was looking at the 26 and the 1300 and I said, "What amount is going to give me the partner to one of the hundreds?" So, I know it's 1200. And the partner to 26 to make that last 100.

Pam  08:02

Nice. And so, if I have 26. You need 74. Yep. Nice. Cool. So, you had... For 49 boxes, you got 1,274 Smarties, whether they're chocolate or chalky, either one. Brilliant. Okay, so there's a Problem String that we can kind of refer back to where we kind of ended with the problem 49 times 26. But to be clear, the goal of the Problem String wasn't to solve the last problem. The goal of the Problem String is to develop Five is Half of Ten in this case. The strategy. And get lots of answers and build relationships in kids heads with like times 10 and Five is Half of Ten and 50 is half of 100. Or in this case, 50 is 10 times 5.

Kim  08:32

Mmhm. Mmhm. 

Pam  08:45

So, all those and some nice subtraction relationships. The goal of the Problem String is to develop all those relationships, not just to get one answer. Alright, so now we have a string to talk about. If we do that string in class, what are some things you're thinking about now?

Kim  09:02

So, if I'm working with a group of students then, and I facilitate a string like this.

Pam  09:08

Mmhm. 

Kim  09:08

When as soon as that's over, I'm kind of taking in all the information about the students and what do I know about each of the students in the class that I heard from? It's very unlikely that I heard from every one of the students in class or saw every single one of the students record something on paper, so I have to make a choice. How much evidence do I have from the majority of the class?

Pam  09:32

Mmhm. 

Kim  09:32

If I don't have enough evidence from the majority of the class, I have to think about do I want to do another Problem String like this, so I can get around to more students? So, things to think about. Would I want to do another Problem String like this, same size numbers but do it on a different model? So, on this one, you mentioned that you wrote it on a ratio table. 

Pam  09:55

Mmhm. 

Kim  09:55

So, do I also want to build Five is Half of Ten with about the same size of numbers but maybe do it on an array, a ratio, an area model? 

Pam  10:04

Yeah, yeah. 

Kim  10:05

Do I want to change the size of numbers or the type of numbers? Would I maybe want to continue to work with a ratio table with Five is Half a Ten but maybe increase the size of the numbers, so that I can get more into deeper content? If I'm a fourth grade teacher, and I have students and I start with, like a two-digit number, when do I want to switch to maybe a three-digit number? Is it... Do I have enough evidence that the majority of my kids were hanging with the strategy  as I've done this string that I can bump up a little bit?

Pam  10:42

Yeah. And let me get... If we could be a little specific with just this particular string. I used 1 to 26. So, one of the things you said is do I want to keep about the same numbers? So, if I was going to do that, I would probably choose another even number.

Kim  10:54

Mmhm. 

Pam  10:55

Because that's going to impact how things play a little bit.

Kim  11:01

Mmhm, mmhm. 

Pam  11:01

I might choose... But if I wanted to up the numbers, then I might choose an odd number or I might a higher number.

Kim  11:09

Mmhm.

Pam  11:09

Because that's going to impact when I start adding the 10 boxes...

Kim  11:13

Yep.

Pam  11:13

...5 boxes together. It also impact the subtraction when I'm going from the 50 box to the 40 box. Go ahead.

Kim  11:19

Well, so you might change the number of Smarties, but you might keep the relationships the same. 1, 10, 5, 15, 16, 50, 49.

Pam  11:27

Yeah, exactly. 

Kim  11:28

You might also choose to do another. So, we call these an echo string, where you're keeping kind of the same relationships, the same size of members. You might change it to like a similar number of Smarties like 24. Change a few of the relationships that are playing out. So, not 1, 10, 5. 

Pam  11:47

Yeah. 

Kim  11:48

15, 16. You might choose different, little bit Over and Unders of the Five is Half of Ten. 

Pam  11:52

Yeah, so I might do 150. I might do 100 boxes, and then 50 boxes, and then 150 boxes. That would be another. And then 149 boxes or 151 boxes. That would be a way to sort of up the numbers. But then, if I have maybe fifth grade, sixth grade, and up students, I might find a half of a box.

Kim  12:12

Mmhm. 

Pam  12:13

And then find 15.5 boxes, or 49.5 boxes. So, other ways of, kind of making. Well, in fact, 1.5 boxes to get to the 150 boxes.

Kim  12:26

Yeah.

Pam  12:26

So, other ways of making the numbers not just bigger but more complex.

Kim  12:30

Mmhm. 

Pam  12:31

But still getting at the same strategy as Five is Half of Ten. So, kind of asking myself what are the numbers that my students are expected to mess with and upping the ante in one of those ways, either the number of Smarties in the box or changing the numbers of boxes that we're finding. 

Pam and Kim  12:50

Yeah.

Pam  12:50

Nice.

Kim  12:50

So, we tend to call those echo strings or next step strings. So, when I've done the Problem String, I've gotten information, and I have to decide do I do an echo string? My kids need some more experience. 

Pam  13:04

It's like a very alike string where not much has changed, right?

Kim  13:07

Mmhm. You call them sometimes sister strings.

Pam  13:09

Yeah.

Kim  13:09

So, they're within the same range of numbers, same model, not a lot changed. 

Pam  13:14

(unclear).

Pam and Kim  13:14

Mmhm, mmhm.

Kim  13:16

Or am I ready to bump it up a little bit to do a next step? And we are big fans of saying that you don't want to change too much at once. So, you might switch models but keep the numbers and the strategy the same. 

Pam  13:28

Yep. 

Kim  13:28

You might switch size of numbers but keep the rest of the same. You might switch type of numbers but keep the rest of the same. You might say, "Wow, my kids really owned that strategy, and I could see them pretty quickly using this strategy, so now I'm ready to introduce the next kind of strategy." Maybe you've done a few of this particular Five is Half a Ten string on a ratio table. You've done the gamut of numbers. You feel really good and confident that your kids are owning it. You see it in the rest of their work. And now, you're ready to switch to a different strategy. 

Pam  14:00

Yeah, and let me maybe mention one other kind of next step that you might consider or upping the ante for this particular string is you could begin to ask, give kids the number of Smarties and ask for the number of boxes.

Kim  14:14

Mmhm. 

Pam  14:14

So, that's like division. Like, the problems that you and I just did were kind of multiplication problems. But if I switch that out and say, hey, what if I've got 1274 Smarties? How many boxes? Now, the kids have to kind of do division problems. And I would want to help them realize they're doing division. So, that's another way of kind of upping the ante.

Kim  14:33

Mmhm. 

Pam  14:33

Another way of upping the ante, depending on your grade level. If I'm a middle school teacher, I definitely want to get into decimals at some point. But then I also want to graph these as ordered pairs. So, the number of boxes are going to be x, and the number of Smarties is going to be y. And, you know, is there any pattern to that graph? And then once we've graphed that pattern, we're going to see it's pretty linear. Why is it linear? Because for every 1 box, we have the same number of Smarties. Another box over. If you could see my hands, I'm like going over a box and up the number of Smarties. And over a box and up the number of Smarties. That's a constant rate of change. If you have 0 boxes, you have 0 Smarties, so it's also a proportional relationship. Since it is, we could early, pretty early. Often, it says seventh grade we could write the equation for that proportional relationship. Around eighth, ninth grade, we could shift that and actually talk about transformations. So. Well, a way of upping the ante in the string itself for the students I have is starting with content that kind of helps them fill their relationships, and then going to the content that I teach in my grade level. Or, like we said, you know, upping the ante with the numbers, and then maybe going further. Maybe the first time you only do the numbers, and second time you graph it and write the equation, and the third time you transform the equations.

Kim  15:50

Mmhm. 

Pam  15:50

Yeah. So, things to think about like the next kind of thing that you might do. 

Kim  15:54

You also might make different choices for different students in your class. So, you might do a Problem String on Monday, and then another same strategy on on Tuesday. You might be ready to do a next step or up the ante string on Thursday or Friday. But you might say, "Man,  I've got a group of kids in my class who I feel like could really use one more experience before we up the ante." So, you might pull a small group and say, "Okay, let's do one more echo," Or like a half step before you make, you know, give an opportunity for the whole class to do a bigger next step. Just little. A little bit of intervention there.

Pam  16:30

So, you can make a different choice for a small group. 

Kim  16:33

Yep. 

Pam  16:33

Yeah. Nice. 

Kim  16:34

You also might consider that once a strategy has been developed, then you could take that strategy to a game or some sort of other practice where they are applying the strategy outside of a Problem String, and you're seeing what happens there before you make a decision about what to do with the next group of strings that you're going to do. 

Pam  16:51

Yeah, nice.

Kim  16:52

Yeah.

Pam  16:52

Nice, yeah. 

Kim  16:53

You also might have to think about when is it time to anchor? And I'm going to suggest that it's not appropriate to anchor the very first time. But you have to consider, when do I want to anchor?

Pam  17:04

Yeah, so if you've done a Problem String, like the one that we just did.

Kim  17:09

Mmhm.

Pam  17:09

It's, what, most of the time, if not all the time, you wouldn't want to say, "Great. You guys all own the Five is Half of Ten strategy. We're done with that. Let's put it on a poster," and never revisit it. For all the reasons we just said, you're going to want to up the numbers, make the numbers more complex. You're going to want to change the number of Smarties versus changing the number of boxes. Get further away from the nice numbers and change the model. Like, there's more to do to really cinch that strategy. But I'll also mention, because I'm high school remember, that there are times that I might be trying to develop something in my high school students where we really don't have an opportunity to go back to that, but I'm going to raise it here quickly as I do a Problem String into content that I teach. The kids have now been at least had some exposure to Five is Half of Ten, but really I'm talking about writing the equation of that proportional relation. Or, like transforming it. So. 

Kim  18:10

So, what would you anchor in that situation?

Pam  18:13

Like in the situation where I...

Kim  18:16

Yeah, you've done it once, and (unclear).

Pam  18:18

Yeah, it's a good question. I don't know that I would anchor the Five is half a Ten Problem String. I might mention it. I might go, "Hey, that's a really helpful." Sorry, Problem String, I said. Meant strategy.

Kim  18:27

Mmhm. 

Pam  18:28

I might say, "That's a really helpful strategy when you guys are multiplying. But hey, let's anchor this idea. How do I know if it's going to be a proportional relation?" Or, "Hey, let's anchor the idea of how do I write the equation for that proportional relation?" Or, "Hey, let's anchor the idea of why could we transform this function this way, this time? If my goal that day is transformations. But let me be clear, if my goal that day is transformations, I did not just do enough to solidify transformation, so I'm not anchoring that on that first day.

Kim  19:03

Yeah, I was going to say are you anchoring because of time or is it because you think the kids are actually ready to anchor after day one?

Pam  19:10

I think there's a little bit of both, but I think it has everything to do with what's my real goal for that group of students and do they own it enough that anchoring it will... I haven't waited so long that they're all like, "Duh. We've got that."

Kim  19:26

Mmhm. 

Pam  19:26

I'm actually using it as a tool to help them gain clarity, to help them get more clear on their relationships.

Kim  19:33

Mmhm.

Pam  19:34

In other words, if we anchor, then that means I'm pulling from them. How are you guys thinking about this? And what kind of language are you using or, you know, could we use to describe the pattern that you're finding? And how could we make some sort of statements and pictures that would represent what you guys are thinking right now? As they put those kinds of relationships and patterns into words, that helps everybody gain more clarity. Especially if we bounce back and forth and then we kind of decide what it's going too look like.

Kim  20:01

Mmhm. 

Pam  20:02

That's a moment where, oh, now everybody is more clear, and now we can continue to cinch it, use it in our other work. So, anchoring doesn't mean we're done and we're putting a bow on it. Anchoring is an opportunity for continual refinement. And I'll specifically mention, high school teachers, this is not trivial. Kind of like Kim is pushing me to like think about, so when do you anchor? One the things that we tried really hard in our just current High School Problem Strings workshop. Which I'm so thrilled that we've gotten that done and it's out. One of the things that we've done is help you think about what would you anchor after this Problem String? In fact, we also help you think about how would you prepare for this Problem String? And then what's your goals during the Problem String? What are the teacher moves as you're facilitating this Problem String? And also, what are you thinking about after this Problem String? And again, high school content. Not numbers, not... What am I trying to say? Not numerical strategies only. But also, you know like, real high school content. Problem Strings, with graphs, and tables, and transformations, and functions, and all the things. Yeah, so it's one of the things that we help you do in our Building Powerful Mathematics workshops. But we also have books with options in our Problem String books. We've got kindergarten through fifth grade Problem String books. We give you not only a main string that we write about. Every lesson has a main string we write about. And then we give you two echo strings that are basically the same relationships that are just another opportunity to do that same thing two more times. But then we recognize that you might want to up the ante just enough, and so then we give you what we call next step strings, two next step strings. So, if there's 48 lessons in one of our grade level Problem String books, that's actually 48 times 5 because we have that main string, the two echoes, and the two next steps. Lots of resources to help you as you are planning Problem Strings, facilitating them, and then thinking about what am I doing now after this

Kim  20:15

Yeah. 

Pam  21:58

particular Problem String? 

Kim  22:00

Yeah. So, we just wrapped up the series on planning, choosing a Problem String, planning for a Problem String, and what you could do afterwards. We hope this has been really helpful. Thank you always for tuning in and spreading the word that Math is Figure-Out-Able. 

Pam  22:13

Alright, fantastic conversation about Problem Strings. Ya'll, thank you 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!