Pam Harris  00:02

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

Kim Montague  00:10

And I'm Kim. 

Pam Harris  00:11

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; that math class can be less like it has been for so many of us and more like mathematicians working together to learn more math. We take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keep students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what?

Kim Montague  00:52

So thanks for joining us, if you have missed the last few episodes 58 to 60, then you'll miss out on the fact that we've been discussing models, modeling and all the things. We're going to continue that conversation today. In today's episode, we're going to dive into a specific model that can be a little bit tricky. So let's parse out 10 frames. So for older grade listeners, what's a 10 frame, Pam.

Pam Harris  01:19

Okay, so a 10 frame is a structure that helps kids think about our five and 10 structure. So picture a rectangle that is cut horizontally in half so it's got two rows, and then vertically into five sections. So it's got 10 squares in this rectangle, five on the top, five on the bottom. And in this 10 frame, we can do lots of really nice things. And it's nicely set up to be 10 because it's our base 10 number system. And to have that 10, cut into five and five, because we want kids to be able to think about numbers like six, as five and one more. So when we have kids think about 10 frames, we have them fill in - I might have that 10 frame, it's called a frame because it's literally just those lines, that is that rectangle cut into those 10 squares, five in the top five in the bottom. And we might have kids fill that in. And we might say, show me the number six, and we encourage them to sort of fill in the top five, and then one more so that they could kind of look at six as five and one more, or they can look at the number nine, fill that in almost any way you want to as 10 minus one, right? Because we're gonna fill up the whole 10 frame, and we're just gonna have one missing. And so we want to be able to think about that as 10 minus one. But also, if I filled in the top five, and then bottom four, in kind of in a row, then I can also think about nine as five and those four and so I'm getting a lot sort of the structure of 10 and five-and. Like how do I think about numbers? Oh, don't let me forget under five. So if I filled in the first four then I can think about that as one less than five, because I sort of get used to the fact that that top row is going to have five and when I have one less than that, then that's four. I can also do things like fill in three. And now I see three, but I also see the unfilled two. And so five is made up of that three and that two, and so I can keep going. Lots of relationships can kind of sort of start to ping at kids as they study and work with 10 frames. So do we like 10 frames, Kim?

Kim Montague  03:38

Yes, absolutely. 

Pam Harris  03:40

But Kim, you said they were tricky. What do you mean? 

Kim Montague  03:42

They are tricky. So we don't want to use them for too long, right? Not too long.

Pam Harris  03:47

 Or too much.

Kim Montague  03:48

Or too much, right? There are some good things about them, but they're not our go to model.

Pam Harris  03:53

Yeah, they are a fine model, but not a be it end all model. So in our modeling sequence of podcasts that we've been doing, we've been advocating a paradigm about modeling. That is, we start with a model of the situation, then we model students' thinking, and then through lots of time and experience, that model becomes a tool for thinking. But there are only certain models that we advocate because they go through that whole paradigm, because there are only certain models that are really good models to be tools for thinking, to be tools in order to think and sorry 10 frame, you don't make the cut. But that doesn't mean that you're bad, you're ugly, and that we can never use them. But let's get picky. So Kim, I'll tell you a story. 

Kim Montague  04:41

Okay. 

Pam Harris  04:42

I was doing a workshop, and I had a principal come to the workshop, which is a little unusual. So I want to give this principal some kudos, he knows who he is, for coming. He had a teacher who had come to workshops before and was really advocating for change in the school. And this principle was like, "I don't know, I mean, Pam sounds a little crazy." He had some confidence in mathematics. And he said, "I'm gonna go to the workshop. I'm gonna check this out. I'm going to push back on what she's saying, I want to make sure that this is really good stuff." So I love that. I love a couple things about this principle. So he shows up at the workshop, we do a lot of work. He's like, this is amazing. You're going to revolutionize the way we teach. We're going to do this. And he went back to school. Another thing that I really liked about this principle is he was working with some students. So he made time out of his week to work with a couple of students who were having some struggles. And he particularly said, "I'm gonna work with these students. "

Kim Montague  05:40

That's awesome.

Pam Harris  05:40

It is awesome. So we respect him for that, right? So he came back to me, and he said, "Pam, it didn't work." And I'm, like, "Help me like, what didn't work?" We had this very interesting conversation, where he said, "Okay, like, I went to your workshop, I love all things, you're gonna revolutionize the way we teach math. I went back and I took these kids, they cannot add single digits." So young kids, he is a principal of elementary school. And he said, "So I grabbed those kids. And I said, 'alright, guys, this is what you do. So here's this 10 frame. Now notice, you're going to add seven plus eight.'" Let's just use that for an example. "We're gonna add 7 plus 8. So here's the 10 frame, and make seven. Do it." And the kids could do that, the kids totally made seven. He said, "Okay, now here's another 10 frame, make eight." The kids could totally make it and he goes, "Okay, guys. Now, here's what you do. Like there's seven, there's eight, you just you just clearly move some from one of these 10 frames into the other one, so that it's now a full 10. Right? So pick whichever one you want. Let's say that we take the one with seven, it has how many leftover, it has three empty, right? And so then we have one with eight. And so we grab three from the one with eight, and we put it into the one with seven. Now the one that had seven is now full, so that's a full 10. And now the leftover one, the one that had eight, we've taken three away, and so it has five left, right?" And he's like, "Okay, read off the answer, like there it is." Like you've got the full 10. You got the five, you gotta count those the kids clearly did that. No problem. They were able to do it, they got the answer. And he said, "But it didn't transfer." He's like, "Then I would give them problems to solve. And they couldn't do it." He's like, "It didn't work." And I was like, "Oh, okay, ah, let's parse this out." So Kim, help me parse this out, like what's going on? All the things that were not quite what we meant. 

Kim Montague  07:22

Yeah. Well, so to begin with, there's a kind of a mentality of I do, we do, you do. A little bit of, I'm going to tell you what to do with this 10 frame. And that's problematic. So it's because - 

Pam Harris  07:41

It's a procedure, right? 

Kim Montague  07:42

Yes it's a procedure. There's also this idea -

Pam Harris  07:45

Before you move on. Stay with procedure for a second. So I give you a 10 frame, oh, I'll give you two 10 frames. So I provide them and now do the thing, do the thing. So they make the seven, make the eight. Now move the -  it's the steps, right. Now, even if you're trying to like, Okay, what are we trying to find? Even if you're questioning through the steps, it's this idea of steps - 

Kim Montague  08:06

That belong to someone else, right? They belonged to him. And he was trying to tell the steps to somebody who had not yet experienced what it meant to break apart an eight and give it, right like a Give and Take Strategy, to give it to the seven to create an equivalent problem, which is a sophisticated idea.

Pam Harris  08:27

And who also hadn't, like maybe thought about the fact that if I make this friendly 10, then it will be easy for me to see the 10 and that five leftover and call that 15. If I haven't created that idea myself, then who cares? Oh, maybe next time, because I'm following your procedure, so maybe next time, I'm just going to bring a couple over. And now I have nine and I have a bunch leftover. And what do I do now? What he saw is exactly what I'm describing, then the kids would count them all. And he's like, I'm trying to get him out of counting. And I'm like, "But they don't understand why you created the full 10." He's like, "How can you not understand that, it's clear, I told them." In a huge way, the background behind that is a belief that teaching math is about unzipping kids' heads and pouring previously constructed ideas that I have into your head and you will be able to latch on to them and make sense of them and know why to do them. And so it's clear. It's like the math is embodied in this model. Wallah, do it. Okay, so that's one. What was the other thing you were gonna say?

Kim Montague  09:31

Um, so the idea that I'm reading off an answer, right, is problematic.

Pam Harris  09:37

I mean, but Kim, we want kids to get answers. Why is that problematic?

Kim Montague  09:40

Sure, it's problematic because they're not cognitively involved in the thinking of what's happening in the situation. 

Pam Harris  09:46

Oh, this is so important. I think the first time I heard about this, I think, pretty sure it was Cathy Fosnot talking about that if you do some stuff, and then read off the answer that that is a, these are my words, but that's a ping. That should be a, "Oh, wait, maybe I wasn't just mathematizing." That if I get to a place where I'm almost surprised by the answer, where I've done a bunch of things, and it's like, "Oh, hey, look, there's the answer," means I wasn't using the relationships. Like you said, I wasn't cognitively involved in the relationships and doing something with them to make that happen. I'm sort of like, "Oh, yeah, I did the thing I'm supposed to do, I moved," maybe the kid understands I'm supposed to make one of those 10s whole, and now I see a 10 and a five. And so I read off that 15. But I wasn't thinking about how eight is related to 10. And how seven is related to 10. And how eight and seven were related to each other, and how those related to 15. I was just like doing the thing, getting an answer. Now, if you're new to our podcast, you might be like, I do not understand what's wrong. We got an answer.

Kim Montague  10:49

Yeah. 

Pam Harris  10:50

You might want to go back - I don't have it handy. We'll put it in the show notes about the episode where we talk about it's not about answer getting. It's about building reasoning. Which leads me to one other thing I want to talk about. So sort of three things that kind of what just went awry. One: It was too procedural. Two: Your're reading off the answer at the end, which could be a real clue that you were just too procedural. And then the third thing I'd like to mention is, that notice that for kids to do that, in order to fill that 10 frame, they had to count one by one. 

Kim Montague  11:22

Yeah. 

Pam Harris  11:23

They had to count 1, 2, 3, 4, 5, 6, 7, to fill that 10 frame. 1, 2, 3, 4, 5, 6, 7, 8, to fill that eight 10 frame with the eight. Then I don't know if they counted to move over the three, probably not. I didn't give you the best example for now the problem that I'm going to have at the end, because I wish that now they were going to have to count more. The 10 and the five is almost too easy to go 15. Let's say that I had a problem like nine and seven. And so nine, I've moved one over, I made a 10. And I've left over six. Now I've got this full 10 frame of 10. And I have this other 10 frame with six in it. But do I know that? And all too often what we have is what his kids were doing, they didn't really understand why they were doing what they're doing. And so they move things around, and then they just ended up counting one by one, all of the counters, not using the structure of the 10 frame to help them see what the final result was. 

Kim Montague  12:16

Right.

Pam Harris  12:17

So if you, again, are new to the podcast, or you haven't remembered for a minute that we have this thing called the Development of Mathematical Reasoning, the Development of Mathematical Reasoning is a graphic where we talk about how we want to move kids from Counting Strategies to Additive Reasoning. Well, if I'm doing a problem, like seven plus eight, or what did I just say nine plus seven, if I'm doing those problems, ideally, if I've already learned to count, ideally, at this point, I'm building Additive Reasoning, not counting one by one. 

Kim Montague  12:47

Right. 

Pam Harris  12:47

So if I demand that's how kids solve problems, fill in the 10 frame, fill in another 10 frame, move them and then see what's leftover, kids could be doing an awful lot of counting one by one. In fact, they could be getting correct answers every time, very procedurally counting one by one. Not what we want. We want kids reasoning about friendly numbers, reasoning about landmarks and reasoning about the relationships between the numbers, so that then we can use that reasoning for bigger numbers and bigger problems and all the things because we want to build from Additive Reasoning to Multiplicative Reasoning. We can't have kids stuck in Counting Strategies when they should be thinking additively.

Kim Montague  13:26

Yeah. So I want to consider though, for some of our really young learners, right, some of our preK and kinder, there are good ways to use a 10 frame. And it's all about learning how to count, right. Making sense of the five structure and the 10 structure that's kind of embedded in this 10 frame. And it does allow kids to start using the structure, it kind of naturally pushes towards that five and 10 structure instead of counting one by one, which is the very beginning of additive thinking.

Pam Harris  13:56

Yeah, so some great things that you can do is do some Quick Images, where you show kids and you're like, "Whoa, how did you do that?" If it's quick enough that kids don't have time to count one by one. And so they start to be nudged a little bit towards, "Oh, like, how can I use the 10 structure, five or 10 structure to help me see those numbers?" You can even do that with multiple 10 frames. So those would be fine things. You can have kids when they're learning to count, and so they're using counting to solve problems 10 frames are fine things for them to sort of organize their count. It helps with one to one correspondence. It's all about when kids are learning to count, we are okay with using 10 frames. When kids are learning more additive thinking, then we don't want you to fill the frame. Don't have kids fill the frame, then really use Quick Images. And that's kind of - you'll have to show me another way to use it - ah, I can think of one other way. Bridges in Math uses 10 frames one other way where kids are learning Additive Reasoning, where they will show kids, not necessarily in a Quick Image, but they'll show them, I think they have Muddy Windows On A Bus. 

Kim Montague  15:04

Yup.

Pam Harris  15:05

Where they'll have a double decker bus where you can see, it's basically kind of a 10 framy kind of thing. And some of the windows have mud on them. And so you can't see, it's like, How many windows are there? Is it how many kids can you see? I can't remember. 

Kim Montague  15:22

Yeah.

Pam Harris  15:22

Is it how many kids you can see?

Kim Montague  15:23

I think so.

Pam Harris  15:23

And so then you're sort of using the structure and kids could count, but because some of the windows are muddy, then you're sort of like, "Ooh, can I count behind there?" And they mud out, not just the windows, but like the whole area. So you can't go 1, 2, 3 because it's like just this blob of mud. And so now you have to go, "Ooh, well, if I'm looking at this 10 frame, I have to sort of know what's there and then extrapolate what's not there." Those are fine ways to use 10 frames where kids are thinking about building Additive Reasoning. You don't want to force kids to have to fill the 10 frame counting one by one. 

Kim Montague  16:02

Well, I'm just gonna say that in that context of the muddy windows that the kids given, it's given to them. The kids are not filling something.

Pam Harris  16:09

 Good point.

Kim Montague  16:10

I think that's a good distinction. So it's all about representing a number and not doing stuff with the number, right when we would want to use a 10 frame. Once we need them to compute with that number, then we need a different tool. And we're going to suggest that it's a number rack and then an open number line.

Pam Harris  16:26

So we've talked about number racks a little before, number racks are also called rekenreks. Like you just said, those are our preferred tools: number racks and open number lines once kids need to compute.

Kim Montague  16:38

Yeah, so, so far in this series, we've talked about some models that we think are really good for using because they become really good tools to reason with and that's going to be a number rack and open number line and ratio tables.

Pam Harris  16:52

Those are some of our favorites. Maybe not all of them, but definitely some of our favorites. 

Kim Montague  16:56

Sure.

Pam Harris  16:57

Okay, everybody. Next week, in the next episode, we are going to tackle some of the most controversial models. Tada! So stay tuned, you are gonna love or hate maybe that one. It's gonna be really, really good. So we want to encourage you guys to pay attention to the next one because controversy is coming. So if you want to learn more mathematics and refine your math teaching so that you and your students on mathematizing more and more, then join the Math is Figure-Out-Able movement. Help us spread the word that Math is Figure-Out-Able!