Pam Harris  00:01

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

Kim Montague  00:09

And I'm Kim.

Pam Harris  00:10

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 could 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 keeps students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what?

Kim Montague  00:51

Today, we're going to be wrapping up the series we've been talking about for a few weeks now: models and modeling. 

Pam Harris  00:57

Very important. We hope we have clearly made the point that for us modeling is not about CRA,  concrete representational abstract or concrete pictorial abstract, that sort of series of using models and modeling in math - that is not what we do. Because that, all too often, is just a way to get kids to do the algorithms, not our goal. Our goal for math is much more about giving students problems, helping them make sense of them, pulling out how students are reasoning, representing that thinking using important models, making that thinking visible. Then over time, and with lots of experience, those models become tools for thinking. That's how we think about modeling.

Kim Montague  01:49

And that there are particular models that we think are useful for students: the open number line, the open array, and the ratio table are some of the most important and most useful. Today, we're going to address a few common models that people really love. And we'd like to push back on exactly what they are, and are not useful for.

Pam Harris  02:09

Yeah, so let's get a little controversial today, Kim. Let's talk about a few things that we hear lots of teachers talk about and use. We see them all the time in classrooms. And like you said, let's get a little clarity around what exactly they're useful for or maybe not so useful for. So let's just list them out right off the bat. 

Kim Montague  02:32

Okay. 

Pam Harris  02:32

Hundreds charts. Base 10 blocks, base 10 materials. 

Kim Montague  02:37

Yep.

Pam Harris  02:38

And algebra tiles. 

Kim Montague  02:39

Some really popular ones, right? 

Pam Harris  02:41

Really popular, we see them a lot. Of course, depending on the grade levels, you might see different ones. They have some things in common, which is why we're talking about all three of them today at the same time. But let's get really clear just for listeners, we have lots of grade levels listening. So Kim, will you describe briefly hundreds charts? What is a hundreds chart?

Kim Montague  02:58

Sure. Okay. Yeah. So a 100 chart is a chart that goes from zero, I'm sorry, one to 100.

Pam Harris  03:04

Well, could be. Could be 0 to 99.  

Kim Montague  03:07

Yeah that's why I said that, I was picturing it. So one to 100. And it's arranged in a 10 by 10 grid. So on the first row would be one to 10. And then the next row would be 11 to 20. Continuing on till the last row is 91 to 100.

Pam Harris  03:23

And there's some pretty cool patterns that you can see with a hundreds chart. In fact, I remember, briefly - briefly meeting: Pam be brief. That's me, telling myself to be brief about the story. When I was working with my kids, second grade teacher, I would often go to my kids teachers at the beginning of the school year, and I would say, "Hey, I'll help. Give me anything. I'll make copies. I'll laminate or do whatever you need me to do for the first few days. And then after that, can I do some math with you?" And so one of the things she said to me was, "Oh, yeah, will you put the 100 chart back together?" So she had a pocket chart, and cards with those digits in it. And she said, "If you'll just put these" - all the cards were out and the pocket chart was empty - she said, "Will you just put them where they belong." I looked at the first card and it was 54. And I thought to myself, where does this go? Like as a high school math teacher looking at this hundreds chart, and to me, that was like a really cool thing, for me to have to figure out where 54 went. And kind of like you just said it could be zero to 99 or could be one to 100. And I needed to know that before I could put 54 on that chart. I needed to know, you know what was her range, the set of numbers that she was gonna put in. Anyway. So all that's cool. There's some really cool things we could do with 100 charts and there are less cool things we can do with 100 charts. Let's keep going. Okay, Kim, what are base 10 blocks or base 10 materials?

Kim Montague  04:39

Oh, base 10. Okay, so you're going to have a little tiny cube, that's going to represent one and then 10 of those little tiny cubes attached together makes what people would call a rod, so it represents 10. And then 10 of those rods attached together make what somebody would call a flat. Most common name, I think flat. That would represent a hundred. So 1, 10, 100. And then 10 flats, put together make a larger cube, which represents 1,000.

Pam Harris  05:09

Because I've got 10 of those flats, oh, or hundreds and 10 hundreds is another name for 1,000. And so it nicely represents sort of place value, right? Because we've got sort of one 1 and then one 10 and then one 100, and then put those together and we get one 1,000. And so hey, look at all that place value that is embodied in those manipulatives... Oh, let's see, we talked about that in the last episode. So if that's a little unclear to you, what I'm trying to do, like jest a little bit here, is that we sometimes think that because we've made sense of manipulatives, and we've put those values in there that it naturally just occurs to kids. Just as all of a sudden is clear. Oh, it's so clear the place value in our number system - or not. And it can really get obscured when then we don't even call them 1, 10, 100, and 1000. We call them -  I don't know, what do they even call the one?

Kim Montague  06:00

Um, unit? Usually, yeah. 

Pam Harris  06:03

So unit and then rod? Rod, and then flat. 

Kim Montague  06:06

Stick, stick is another.

Pam Harris  06:09

Yeah. So you might be interested to know that when I was, again, same sort of year, I was working with my kids teacher, and after I did the first week, and so later, I'm like, "Hey, can I do some math?" One of the things I did was pull out some kids and like, enrich them. I was just like, "Give me some kids. And I'll do some things." Which really meant that I sort of played around with math with them as I tried to figure out what younger kids were doing with math because I was a high school math teacher. And I was just starting to dip my toes into what it meant to do younger math, like really think about how kids were thinking. Anyway, so I'm in the midst of that one day, they said, "Hey, we're not gonna let you have a group of kids. We're just gonna give you this one student. She's really struggling with addition. And so here, go teach her how to add with these base 10 stuff." I mean, they just really handed me like this bin. Like, go teach her with that. And I was like, what are these? As a high school math teacher, I hadn't really ever seen them. And I looked at them and I was like, "Oh, okay." And I could kind of see the place value; I had a lot of relationships already built in me. So therefore, I could kind of see some things happening in the manipulatives. And so I sat down with her, I said, "Hey, tell me about these." And she goes, "Well, that's a" - I honestly don't remember what she called the one. Maybe I don't - yeah, I don't remember. But then she said, "That's a rod. And that's a flat." And I said, "Okay, so like 42, like, make 42 for me." And she just like, looked at me. And I said, "Well, you know, like, what? Grab a ten." She goes, "There aren't any 10s." Like, "What? Okay, rod, but what does a rod mean?" She goes, "It's a rod." And I was like, "Yeah, but like, it's 10. Right?" She goes, "It's a rod." Bless her heart. I mean, in that moment, I was like, "Oh, like you really think this is a rod, not a 10." Like it didn't mean 10 to her. Now, maybe we could talk about how it could mean 10. But it was one of those moments where I was so clear that what was supposed to be clear to someone by looking at the manipulatives, or even messing with them, was not clear to a student who hadn't already constructed some ideas about what 10 meant, that one 10 is also 10 ones. And this young student who I think was in first or second grade had not constructed that. And so it didn't - it was a rod. And I was like, "Well, how many do you have in your hand?" And she's holding on to that rod. That 10. "How many do you have?" She looks at me, "One." I'm like, "You have one rod?" "Yeah, I've got one rod." "Well, how many does this represent?" - the flat. "One." And so anyway, it was just an interesting moment for me to go, "Ooh, there's more to this, we can't just hand these to kids." Yeah, and think that the math is gonna be apparent.

Kim Montague  08:47

And let me add in that, I don't think that just calling it 1, 10, 100, 1000 instead of unit rods, whatever - stick. That's not going to fix the problem that kids don't make sense of what it is. Right? So if that teacher maybe called it those names, calling it mathematical names isn't gonna fix the fact that the math is not embodied for them.

Pam Harris  09:10

That's not magical. Yeah. 

Kim Montague  09:12

It's not just they named it wrong. 

Pam Harris  09:13

That's really good point. Because you could have heard me mean that. Yeah, thanks for clarifying. Yeah. Okay, so let's talk about that third model. So algebra tiles. Algebra tiles, we typically use once we have kids mess around symbolically with x's and variables, and we want kids to kind of have some sense of what they mean. And so a typical thing that we might do with algebra tiles - oh, I'm supposed to describe them. Don't do yet Pam, describe. So I might have a rectangle. It's kind of usually a rounded rectangle that we call an x. And then I have a square that's the same height as the rectangle, but not as big of a width. To me, it looks like it's, I don't know about a fifth or a sixth of the rectangle that represents x. And that's a unit So we've got sort of a block that represents - they're flat, so they're not real, I mean, they are 3d, but they look more 2d-ish. So this flat rectangle sort of represents x, and then we have a smaller square that represents a unit. And then if we take that x, and we make it x by x, we make a square out of that x, then that's x squared. So they're actually sort of similar to base 10 blocks in that we kind of have these physical squares and rectangles that are representing kind of unknowns. And so like a one by one is that unit and then a one by x is that rectangle that represents x. And then an x by x is the rectangle that represents x squared. And then similar to two color counters, there are two sides, and one color represents positive ones x's and x squares and the other color represents negative of the opposite of those. And so we kind of have positive sides and - what are you laughing?

Kim Montague  11:05

I'm laughing because, thank goodness for Google. Because if people can't make sense of what you're saying, you can just look up what an algebra tile looks like, right? 

Pam Harris  11:13

There you go. That would probably be wise. It's one of those things. Don't listen to the podcast while driving. Maybe listen to the podcasts while googling: What do 100s charts, base 10 blocks and algebra tiles look like? Okay, so what's kind of funny for me Kim, is that you're laughing as I'm trying to describe algebra tiles, to like maybe elementary teachers. I'm laughing a little bit while you're describing 100 charts and base 10 blocks to high school teachers. Okay, so if that's what the materials are, let's talk about maybe some limitations to those manipulatives. 

Kim Montague  11:46

Okay. 

Pam Harris  11:46

So in the last episode, we talked about three limitations to 10 frames, and I'm going to mention those same three limitations, and we're going to apply them to these manipulatives. Okay, so the three limitations. Procedural: is the thing that you're doing really more about procedures and steps? That's number one. Number two, do you find yourself reading off the answer almost like, surprise, what's the answer at the end? That should be a ping, that you're doing less mathematizing and more mimicking. And three, is there a bunch of one to one, one by one counting? Remember the Development of Mathematical Reasoning, counting is this less sophisticated thing. And so if we're supposed to be building Additive Reasoning, or Multiplicative Reasoning, or Proportional Reasoning, or Functional Reasoning, I don't want to demand that in the midst of that kids have to count. That they can solve the problems getting away with less sophisticated thinking, when I'm trying to build the more sophisticated thinking. Okay, so those three things, Kim, if I were to give you a problem, like 48 plus 54. What are some ways that we're using 100 charts to solve 48 plus 54. And let's pick out those three things.

Kim Montague  13:02

Okay. So 48, and 54, I could start with either number, right? But I'm just going to start with 48, because it's the first one. 48 plus 54, I'm going to find the marker on 48. I might put my finger on it, put a, whatever, a cube on it, start at 48. And then I might move forward four. And I would count 1, 2, 3, 4. And I might have to, you know, I would have to roll over to the next row. And then -

Pam Harris  13:35

You're doing this on that chart?

Kim Montague  13:36

Right, you're probably moving your finger, mhmm. Or your manipulative, whatever your counter or your cube. And then to move 50 I'm gonna go down a row, five times. Down ten, down ten, down ten, down ten, down ten. Except, you just gave me 48 and 54. But on a 100s chart is called a 100. And so 48 and 54, I'm not going to actually find on the hundreds chart, right? You could also start at 48 and go -

Pam Harris  14:08

 Well maybe slow down a little bit, because if anybody's not doing the math, so 48 -  maybe just go by the 10s? 48, 58. 

Kim Montague  14:14

Okay, well that's what I was doing. 48 plus 54. So I might say 48 plus 50. And I'm going to go down five rows, down 5 10s. And that would put me at 98. 48 plus 50 is 98. And then I'm going to go over one, two to get to 100. But then I'm off the chart at that point, because it's 102.

Pam Harris  14:36

There is no answer.

Kim Montague  14:37

Right. Right. Or I mean, theoretically, you could have a 200s charts which then is a whole other thing.

Pam Harris  14:47

Or you could stack a 100s chart on top of a 100s chart or next to or whatever.

Kim Montague  14:52

So anyway, the only options you have are to move by ones or move by 10s one at a time.

Pam Harris  15:01

Yeah, so procedural, we see kids often just - they go to the first number. And like you just said, I'll just go to the first number, they don't even think necessarily about starting with the higher number so that they can then add less on. And then they either, like you said, add the ones, and then kind of go down by rows for the 10s or reverse it. And so that feels very procedural. This is our number one. That's procedural. And then when they get to that 102, tada, it's almost like tada. And when you said you're moving your finger, or you're moving a marker or something, they sort of move their finger with their marker, and then they almost have to look underneath it. Right? They almost have to -they've put their finger on it to get there. And then they have to like, "What is under there? I don't know, because I just did the thing. I did the procedure. And now it's like, whoa, where am I?" Yeah, they're reading off the answer. So that's our number two thing that should - Oh, yeah, keep going. 

Kim Montague  15:58

You know why? Because really, if once you started at 48, the only thing you're considering is: I got to move 54.  I have to move 54. And as long as I've moved 54, I don't have to consider where I've landed, I don't have to consider anything else, because I'm only going to move by 10s or ones.

Pam Harris  16:19

And you also didn't have to consider really where you started. 48 is close to what? Or 48 plus 50, do I know that? Like you're just doing, every time. Adding ones first and then 10s or adding tens first and then ones. Procedural. So you're not considering what kinds of jumps you're making. Or  like you said,  I just repeated what you said, Sorry, it's okay. Sometimes I repeat what you said. Because I'm making sense of what Kim is saying. And that's me making sense of it out loud. So it's not me saying Kim didn't make sense. It's just that- 

Kim Montague  16:54

And sometimes I don't, it's okay. 

Pam Harris  16:56

It's just that I'm slow. It's all about me being slow. And I'm okay with that. I'm okay with the fact that I'm slow when I do Real Math, because I'm doing Real Math. I would prefer to do Real Math slowly than fake math quickly. 

Kim Montague  17:06

Yep. 

Pam Harris  17:07

Okay, so let's see. So we got the fact that it's procedural. Yeah, the fact that we're reading off the answer and then noteworthily, in order to add those four, we had to count by ones, right. In order to add the five 10s, you had to count by 10, one at a time. All this one by one counting, those are our three, it hits all three of our markers, bam, that we don't, we don't love that for 100 charts. Okay, cool. So we're not a big fan of using 100 sharps for computing. Similarly, let's do the same kind of thing for base 10 blocks. So, me or your? Who's gonna describe this problem?

Kim Montague  17:45

I will. 

Pam Harris  17:46

Okay. 

Kim Montague  17:47

So base ten blocks, right. We just talked about what they look like, so if I'm adding, let's do the same problem, 48 and 54. So what I would be doing in that case is I would grab, hopefully not 48 cubes, I would grab four rods to represent my 40. And I would grab eight units to represent the eight and now I've built 48, on my desk, or wherever. And then I would grab 54. So that would look like five rods and four units. And I would lay them all out, and then it's time to add. So at that point, I would collect my eight units and my four units. And I would exchange those because I have 12 units, and I would exchange those because somebody told me that I can't have 12 units. So I would take 10 of my units, and exchange that for the next rod. And then I would have four rods, and now six rods. And then I would take those 10 rods and then exchange them again, for a flat. So I would be left with a flat and two units. And then I look at what I have on my desk. And I would say 102.

Pam Harris  18:58

And you sort of read off the answer. So bam, instantly, right there. We can sort of say I've done this trading thing, I've collected - well, first you built. And as you were building, I'll just mention you had to count one by one in order to build. Even if somebody is listening right now they're like, but I would draw it on paper, even if you're drawing on paper, you need to count one by one to create those eight and to create those four, or to create the four 10s. And to create the five, you counted one by one to do that. So lots of counting, when we really we want kids thinking about 48 and 54. We don't want them thinking about like counting 1, 2, 3, 4 that should have been done before. If we're adding 48 and 54, we're beyond that. And we want to be thinking about the numbers themselves. So counting one by one, we hit that. Reading off the answer, we just hit that, sort of like there they are. It's like, "Oh, hey, what did we end up with?" And then you just read that off. And then does that feel procedural? 

Kim Montague  19:50

Absolutely.

Pam Harris  19:51

Quick story. I was in a classroom and all of a sudden I would hear this smack. And I was like, "What is happening?" The entire class was smacking their hand on their desks, and then the teacher would say something and smack. And I was like, "I'm intrigued, what's happening?" And she would literally go, "Okay, everybody take all of your units and count 10 of them out and ready, ready, everybody trade." And when she would say trade, everybody in the room would smack their hand on the desk, and they would take those 10 units, and they would pull out that 10 or that rod that you're calling it. Or they would take the 10 rods, the ten 10s. And they would do "Okay, everybody ready? Got 10 10s everybody? And trade!" Once the teacher said that they'd smack their hands on the desk. To me that like smacks of procedural. It's, these are steps, this is what we're going to do. Can I just mention that when you were gathering all of those units too, and then you're like, "Okay, I've got 10 of them. So I'm gonna trade that for rod or for 10." I was like, in my head going, "Whoa, why are you gathering the units first?" Like I would have looked at the bigger numbers first. I've been doing Real Math long enough that my gut instinct was to think about the 40 and the 50 first, not the eight and the four and then trade up. And I get it. I know why you're doing that. Because when you trade up, then you don't have to go - how do you how do you describe that? If you go from small to big than it just sort of trades nicely. If you go from big to small, then you have to kind of trade as a second thought, right? 

Kim Montague  21:22

Yeah. There's a lot of trading.

Pam Harris  21:24

Yeah, there's more trading because you sort of add the 40 and the 50 together. And if I had to trade there, I would, which I don't for that problem, but if I would, then when I go over to the units, then if I get another 10 then I might have to retrade, anyway. So, yeah. So I get why you're doing it. Because if you're doing the procedure, then you would of course do it from small to big. But research is clear that if kids are thinking about numbers, and we have not superimposed the algorithm, kids will think about the big numbers first. So there's an extra thing that is going wrong with base materials, as we're working against kids' intuition by working with the small numbers first. 

Kim Montague  22:00

Alright, let's talk about algebra tiles.

Pam Harris  22:03

Okay. So algebra tiles, a typical thing that we might do with algebra tiles is something like x plus three, times the quantity x minus two. And so if I've got these two linear binomials, and I'm multiplying them together, x plus three times x minus two, then I might think about that - see, now I'm trying to put all this meaning into it, not be more procedural. What we see kids do is they'll put that x, that rectangle, that represents the x and then three units, kind of down the side. So vertically, I've got a rectangle, and then a unit unit unit, and that sort of forming a vertical line. And then next to that, going horizontally, I would put an X and then I would put two units, but flipped over. So they're negative. So two units, but they're negative. Okay now that's x plus three by x minus two, and I'm sort of creating a rectangle where we're gonna look at the area, I had to put some meaning just to help you sort of like visualize it, like that's the visual, and then I'm going to fill in the area of that rectangle, I've have two dimensions: x plus three on one dimension, x minus two and the other dimension, and I'm gonna fill in the area of that rectangle, that's missing. And so then kids would, where I have the x by x, they would put an x squared and that x squared is an is a big square. So it fills that whole area of x by x. And then where I had the three units, now I have a one unit, one unit, one unit, well, a one unit by an x is x. So now I would put a horizontal x, another horizontal x, another horizontal x. Now I'm over on the other side, where I'm thinking about those two units by x. So I would have two vertical x's. And now I've got just this space in the sort of right hand lower corner that's empty, I need to fill it in. Well, that's going to be three units by two units. And so I'd fill that in with six units. And it's funny if you could see my hands, I'm literally like 1, 2, 3, 4, 5, 6. You can do that, which should scream at you. I'm counting by ones like all of those counting by ones that I was doing, when really, we want kids thinking algebraically not all that counting. So counting by ones, that's happening. After I've put all of those manipulatives down there, those cubes or rectangles. I guess they're all rectangles. If I put all those rectangles down there, all those - what's the word I'm looking for? Blocks? They're not blocks? Tiles? Yeah, algebra tiles, how about tiles? That's the thing. After I put all those tiles down there, and then I have to look at them and I have to go, "Okay, I've got an x squared. I've got a bunch of x's. How many X's do I have?" Look, I have to count them to see how many X's I have. And then I have a bunch of units. How many units did I end up with? Oh, I'll  count those. How many units did I end up with? And then I can say, "Okay, I've got x squared plus I've got five x's," I didn't actually do this problem. "I've got five x's, and then I've got six units, but they're all flipped over. So minus six and I've got  - plus negative six - negative six or minus six." So again, I'm sort of reading off the answer. I've done the work. I've done the procedure. So of our three things: I've done the procedure, I've done what you've told me to do, I put the ones on the left, I put the ones on the right, I filled in all the stuff in the middle. And then I've read off the answer after I've done the procedure. I've read off the answer. And I was counting one by one. All three of our things that we don't love. Doing with algebra tiles. Not my fav.

Kim Montague  25:24

Which people love, right? This is kind of why we're talking about all three of these manipulatives, is because they are so popular.

Pam Harris  25:32

Yeah. So, let me just mention, we're running out of time, we're really going long today. Thanks, everybody, for listening to this one. We actually like using these three models, but only to build relationships, not for what we just did. So what Kim and I just did with these three models was compute, we used them for computing. And that's when we have those three problems. So in a later episode, we will talk more about models for building relationships, versus models for computing for doing the math. And those are different. One is a subset of the other and we want to be careful about not using some models to compute when we get those three things that we don't want: where they're counting one by one, where it's all about procedure and when you're reading off the answer. And let me just mention one last thing about why we don't like these three particular models for computing, because kids get stuck in them. We have literally seen kids on some sort of assessment where they're supposed to add numbers together, redraw a hundreds chart in order to use it to add. That is not ideal, and we don't want it. Alright, so models and modeling and mathematics. Who knew there was all this stuff to talk about? We hope we've given you some things to think about and consider. When in doubt, remember the Development of Mathematical Reasoning and ask yourself, does this model help move students forward to more complex thought? Or might it inadvertently be trapping them in less sophisticated thought? Do you see a lot of one to one counting that's necessary they have to do when students should be building more sophisticated thinking rather than just counting? Does it feel like it's more about imposing someone else's thinking? Or does that model lend itself to representing the student's own thinking? Does that model have the potential to be a tool for working out the relationships, a tool for computing with relationships? When using that model do you tend to read off the answer at the end? Or were you cognitively involved in deciding how to use relationships? So Kim. 

Kim Montague  27:49

Yeah.

Pam Harris  27:50

How would you wrap up this? Anything else to add?

Kim Montague  27:54

I'm going to be succinct. In short, does the model that you're choosing help kids mimic or does it help them mathematize?

Pam Harris  28:03

Bam! Oh, that's brilliant. Alright. That's the big question to ask as you look at models and modeling and manipulatives in your class. So if you want to learn more mathematics and refine your math teaching so that you and students are mathematizing more and more, then join the Math is Figure-Out-Able movement and help us spread the word Math is Figure-Out-Able!