# Ep 61: Ten Frames

August 17, 2021 Pam Harris Episode 61
Math is Figure-Out-Able with Pam Harris
Ep 61: Ten Frames

Continuing their series on modeling, Pam and Kim discuss the advantages and pitfalls of Ten Frames. They emphasize the power of models that can become tools for reasoning, and use Ten Frames as an example of how teachers can focus too much on models that may eventually get it in the way of reasoning.
Talking Points

• What are Ten Frames?
• The 3 limitations to Ten Frames
• It's not just the model, it's how you use the model
• Our preferred model for transitioning from counting to additive reasoning
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 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:

So thanks for joining us, if you missed the last few episodes 58 to 60, then you missed 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:

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, and kind of in a row, then I can also think about nine as five and those four and so I'm getting 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:

Yes, absolutely.

Pam Harris:

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

Kim Montague:

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

Pam Harris:

Or too much.

Kim Montague:

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

Pam Harris:

Kim Montague:

That's awesome.

Pam Harris:

Kim Montague:

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:

It's a procedure, right?

Kim Montague:

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

Pam Harris:

Before you move on. Stay with procedure for a second. So I give you a 10 frame, oh, I'll give you 2 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:

That belong to someone else, right? They made sense 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:

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, if 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:

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

Pam Harris:

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

Kim Montague:

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

Pam Harris:

Kim Montague:

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:

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 that you can have kids when they're learning to count. And so if they're using counting to solve problems, 10 frames are fine things for them to sort of organize their count that 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, muddy windows on a bus.

Kim Montague:

Yup.

Pam Harris:

Where they'll have a double decker bus where you can see, it's basically kind of a 10 frame kind of thing. And some of the windows have mud on them. And so you can't see, it's ike, How many? How many indows are there? Is it how m ny kid can you see? I can't rem mber.

Kim Montague:

Yeah.

Pam Harris:

Is it how many kids you can see?

Kim Montague:

I think so.

Pam Harris:

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:

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

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

Kim Montague:

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:

Those are some of our favorites. Maybe not all of them, but definitely some of our favorites. Okay, everybody. Next week. In the next episode, we are going to tackle some of the most controversial models. 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!