# Ep 100: Celebrating Place Value

May 17, 2022 Pam Harris Episode 100
Math is Figure-Out-Able with Pam Harris
Ep 100: Celebrating Place Value

We love that we've been able to make 100 episodes! We also love that 1 hundred is 10 tens, so let's talk place value! In this episode Pam and Kim discuss why students struggle with place value, and suggest some of their favorite place value activities.
Talking Points:

• Why 'place labeling' is ok, but not enough
• Count Arounds
• Multiple Towers
• Base 10 materials
• Using correct place value language

Pam Harris  00:01

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

Kim Montague  00:08

And I'm Kim.

Pam Harris  00:09

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. 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  00:38

I'll tell you what, Pam, it's the 100th episode, right. We're super excited. And since it's the 100th episode, we thought we would celebrate our base 10 system.

Pam Harris  00:49

Right, right. Yeah, let's bring it on. Let's talk tens.

Kim Montague  00:52

Fun fact. My kid is 10. Right. Yeah.

Pam Harris  00:56

Okay. Alright.

Kim Montague  00:57

And my grandmother is about to be 100.

Pam Harris  01:00

No.

Kim Montague  01:02

Yeah. Is that amazing?

Pam Harris  01:03

Seriously?

Kim Montague  01:04

Yeah. For real?

Pam Harris  01:05

Wow.

Kim Montague  01:06

It's pretty okay.

Pam Harris  01:07

So I'll just join right in their claim to fame. I'm half of your grandma-ish. I'm in my 50s. And then we won't tell you which 50 I am. But I'm in my 50s. So yeah, like five 10s. We're talking about 10 today, 100, 10 squared. Yeah. So lots of nice. Yeah, I can't believe your grandma's almost 100.

Kim Montague  01:25

I know. It's amazing.

Pam Harris  01:27

Nicely done Grandma. So we thought as we talked about our base 10 system today that we would emphasize something that we find really interesting to talk with teachers about, work with teachers, work with students, especially because often we might be talking past each other just a little bit. We might not know exactly what we mean. So we're gonna dive in today. When I was very first doing early professional learning with elementary teachers, I had been doing professional learning stuff with secondary teachers for a while, graphing calculators, how to make higher math visible. But when I began doing that early professional learning work with teachers, teachers asked me, "Hey, you know, Pam, what are the things we need to do? We talk about place value, like our kids place value." And I kind of dove in. And the more that I saw what was happening, the more that I, when I talked to teachers, and I understood what they were doing in class, I got really clear, the teachers were spending an awful lot of time on what I would call 'place labeling' work, and not really 'place value' work. Because teachers were asking me, "Pam, pam, come help us with place value." "Okay, okay, tell me what you need." And then I would see these six week units...

Kim Montague  02:46

Yes.

Pam Harris  02:47

...on 'place value'. I just put quotes, air quotes, place value in quotes.

Kim Montague  02:50

Yeah.

Pam Harris  02:50

And I would like, "When's the place value work gonna start?" And they're like, "Well, like right here." Then I'm like, "You're, you're having them label the places. Okay, good. That's necessary." Like they need to know the name of the places they need to be able to, like sort of write the number in expanded notation, all that's necessary. But it is not sufficient for kids to really understand and deal with and own place value. Yeah?

Kim Montague  03:13

Right. Absolutely. I remember, my very first teaching job was third grade. And I remember being handed binders like, "Hey, this is what we do." And you're not joking, six weeks, the first six weeks was very much about 'place value', and it was all...

Pam Harris  03:34

Are you quoting that? Is it place value really?

Kim Montague  03:37

Oh, I'm air quoting, sorry. Sorry, you can't see me? Yeah, absolutely. Because the idea was, first we get kids good at place value, and then they can actually do things with it. And so I remember feeling really awkward about that. And, like, just kind of going along with it. And I think you've probably stumbled into that conversation. On my campus as well.

Pam Harris  04:01

Yeah, absolutely. Yeah. Well, many teachers, right. I mean, you guys were doing your best. There was, like you said, it was all set up, you dive in, you're doing the thing. And then I would hear the frustration and teachers as they would discuss students' complete lack of reasonableness, you know, like the kids wouldn't estimate. They would get to the end of a problem. And the answer was so completely out of the ballpark. So of course, teachers were frustrated. And they could, y'all had a sense that there was a lack of place value or a lack of understanding that the value of the number depends on the place the digits in.

Kim Montague  04:39

Right.

Pam Harris  04:40

And that real sense of value was missing. And you guys had a sense of that. You all had a sense of that it was missing. But you knew you'd spent all his time on it. All his time and effort. And so you wanted to see more bang for your buck. And you were really interested in what were some things you were missing or some little tips and tricks that you could to make the was placed labeling units better? And my answer was, "Stop doing all that."

Kim Montague  05:04

Yeah.

Pam Harris  05:05

Like we're in a huge way we are treading water here, we're sort of wasting time. Kids that can do it, do it and don't really get a lot out of it and kids that can't do it. Keep not can't do it. You know, it's like a lot of memory work and everything, and we're not really developing value. So Kim, what are some of our favorite ways to actually develop place value with students so that they really understand the magnitude, the sense of size of numbers, and how all of that depends on which place they're in, what place that digit is in, those digits in that number. Because we have thIS super, super cool place for the system.

Kim Montague  05:43

So one of my very favorite activities or routines is a Count Around by tens.

Pam Harris  05:49

Nice,

Kim Montague  05:49

Yeah, so if you're not familiar with Count Arounds, you know, people have different images of what that might look like. But for me a Count Around is where we start with a number. Sometimes I'll ask my students, "What number do you want to start with?" Sometimes I have a predetermined number. But let's say I'm starting with second or third graders, and I might say, "Give me a digit between or a number between one and nine." And so a kid might say: seven. And so I say, "Okay, so we're going to start with seven, and we're going to count by 10s." And so I will record as we count. I'm going to record the numbers. I'm kind of doing this with my hands, but vertically, where each number is recorded one beneath the next so that the place values line up. And so somebody might say seven, and then the next person might say 17.

Pam Harris  06:38

And you just wrote that below. So you have seven, and then below that you have 17 and then below 27.

Kim Montague  06:43

37, 47. And we'll go around. And then the most important part of that is to step back and take a look and say, "What do you notice? Like what patterns do you notice? What do you see?" And so that conversation is bringing out the place value-ness. The things like, "I noticed that all the ones place continues to be seven and seven and seven, and seven and seven." And so we'll have a conversation about why is that? if I start with seven, why does every subsequent 10 that I add on still going to add in and seven?

Pam Harris  07:20

Yeah.

Kim Montague  07:20

And if I started with nine, what would those numbers end in. Also very early, I found that students were able to pick out the one 10, two 10s, three 10s, four 10s. And then as you get further down the line, they notice nine 10s, ten 10s, eleven 10s, twelve 10s. And I remember that being one of the most difficult conversations to have with kids when I was just doing placed labeling work. Because we would have to say things like in the number 123, how many 10s? And you know what kids would always say was two 10s, there was only two 10s.

Pam Harris  07:57

In 123 here's only two 10s. Yeah, total of two 10s.

Kim Montague  08:00

There was no way-

Pam Harris  08:02

Kim, that's correct. In the number of 123 there are only two 10s. Right? Not 'only'. Well, it sort of depends. It depends on if you've already stripped out the 100.

Kim Montague  08:12

Right.

Pam Harris  08:12

If you've already stripped out those ten 10s, then how many 10s are leftover? Okay, there's just two. Or there's two 10s in the 10 slot? Can I say it that way?

Kim Montague  08:22

Yeah.

Pam Harris  08:23

But only if you've already stripped out that 100, those ten 10s. So if you want the total number of 10s and 123. That has to be 12. There's twelve 10s. Yeah.

Kim Montague  08:33

Absolutely. And I just remember that being such a difficult conversation, because I didn't have an experience at that time to give them so that they could really feel the twelve 10-ness of the number 123. So Count Arounds are one of my favorites for a variety of ages.

Pam Harris  08:49

I feel like one of the things I want to emphasize is the fact that you were recording those values on the board. You made a really specific point to say that you were recording them vertically. But that's not always true. Right? It depends on how you're counting around. And the pattern that you want to bring out that.

Kim Montague  09:05

Sure. Sure. It just depends on the pattern that we want to bring out. But I feel like because we're talking about the base 10 system, that's the one that I have a convention, but um, yeah. And so-

Pam Harris  09:17

Can I give an example of another one? Just for grins.

Kim Montague  09:19

Sure. Yeah.

Pam Harris  09:20

So I did a lot of Count Arounds with a group of second and third grade teachers a few weeks ago, and one of the ones that we did was we counted by twos. And I don't even remember what number we started with, but we just literally wrote, you know, maybe we started to zero even 2, 4, 6, 8, 10. And I wrote those horizontally. So 0, 2, 4, 6, 8, 10 and then I wrapped around. And then underneath each of those, oh, sorry, I wrote the 10 under zero. And then 12 was under two and then 14 was four. And then I said, "Hey, hey," and and before I could say much one of the teachers was like, "That, that pattern we just looked at it's happening again." And I was like, "Where?" She said, "Over there in the 10s. Look, there's 10, 20,30. We're seeing the tens count up again. And the zero staying-" You know, it was like the patterns that you just kind of talked about the number of 10s. And this was an adult, you know, like, "Hey, look, it's happening again." Different patterns can come out. If you're counting by fives, there's a fine reason to go 0, 5 and then wrap back around 10, 15 and then wrap back around 20, 25. But lots of different ways that you can kind of wrap around vertically, horizontally, where different patterns can come out. And because it's our base template, system, base 10 patterns are gonna pop. That's going to be one of the things is going to happen. Alright, sorry to interrupt.

Kim Montague  10:36

It's just an added layer, the recording and pattern noticing is an added layer that's the most important part to add on to what people are kind of sing-song counting 10, 20, 30, 40. Yeah, love it.

Pam Harris  10:50

And then also notice that I started with zero in the several counts I just did. But then you don't only want to do that. Like sometimes you want to count by twos, but you want to start with the number seven, and count by twos. You want to start with 13 and count by twos. Or a lot of kids can count by 10s, but only from zero 10, 20, like you just sing-song but starting with seven and then counting 10s, or starting with eight and then adding 10, whole different game. And then I have to mention one of the favorite things I've ever seen you do is then count by nines. So start with a different number and count by nines and talk a little bit about how that pattern pops out.

Kim Montague  11:25

Yeah, so especially fun when I'm working with adults. Adults are interesting learners because they maybe have a little bit of a different worry than kids do. And so it's so interesting to me. But anyway, counting by nines, you know, the pattern that can come out really nicely is that nine is made up of a positive 10 and a negative one. So a lot of times when I'm counting by nine, somebody will say, "Well, I added 10. And then I took away one." So nine to get from nine or maybe start with seven again. So seven, if you're adding nine, you can add 10 and then backup one to get to 16. And then you can add another 10 and backup one to get 26.

Pam Harris  12:12

So I could think, "16,26, 25."

Kim Montague  12:14

Yeah. And what I found interesting is that sometimes people will do the backup one and then add the 10. So it's like two different ways of thinking about it. But still plus 10 minus one or minus one plus 10 still is equivalent to nine. So that's a nice conversation that comes out with, it can come out with both kids and adults.

Pam Harris  12:35

And sometimes adults will have this funny, or even kids will have this funny kind of a trick or rule or something. They're like, "Oh yeah, adding nines, you just it's one less." And they'll kind of have the sense like they've done it a lot. But getting them to verbalize what you just said can be really powerful and kind of abstract that generalization can be a powerful thing to do.

Kim Montague  12:54

Yeah, yeah.

Pam Harris  12:55

What about the 10s pattern that happens when you're counting by nines? Can we do that in a podcast?

Kim Montague  13:02

I don't know what you mean?

Pam Harris  13:03

So you know, like a lot of teachers will follow that pattern that you say where they'll add 10 and the backup one. And so as we're going along, if I were to record 7, 16, and the 25, and then 34. And then what 40, helped me 43. And then so sometimes the teachers as they're going, so say we're at 43, they'll just start realizing that they look back at the numbers. So y'all feel is the podcast, you're gonna write these down. 7, 16, 25, 34, 43 then they start realizing, "Oh, look, the 10s are just counting up by one."

Kim Montague  13:38

Oh, yeah. And the ones are going down by one, right?

Pam Harris  13:41

Yeah, because it's so, if I look at the 10s right now I had 16. So that's, that's one 10, and 25 two 10, 34 three 10s. So 1, 2, 3, 4. But the ones like the ones are counting down six, five or seven, we start with 7, 6, 5, 4, 3. So then I'll go, "Oh, the next one is going to have to be count up by one 5, 50 and then count down 52. And then the next one should be count up 61. And the next one should be seven, the zero or 70. And then and then they stare. Then they look at it. They're like, "Okay, this sevens got to go up to eight. So 80..." And then they look at that 0, because they were at 70. And look at the zero and then they look back at the ones and it's counting down there like 7, 6, 5, 4, 3, 2, 1, 0. What in the world? What happens next? And then someone in the group will snicker a little bit and go, "What's what's the next one, 70 and nine more? 70 and nine because we weren't 70 nine more? But if they're stuck in that, so just 79 Right? It's not 80, like the 10s don't count up in that case. So the pattern kind of breaks between 70 and 79.

Kim Montague  14:51

And you can tell in that moment that people are, they found a pattern that's new and interesting to them. And they're so focused on the pattern that they stopped really considering the amount. They stop considering the value. And so it's a really nice opportunity to pause and have a conversation about like, patterns are cool. But you can't stop thinking about what's happening with number as well.

Pam Harris  15:14

Never stop thinking, never, never, never stop thinking. Yeah, so that's like one of my favorite things that come out. And then I have seen you masterfully, then count around by 100. So adding 100 to something starting with, like the number 34 and add 100. And then after they look at place value patterns, and they watch the sort of 10s now we can talk about the number of hundreds and the number of 10s. And that's a whole big deal, right? And, then then you'll say, "Okay, super job, everybody. Now, let's count by 99." I've also seen you do count back by 10. Count back by nine. Yeah, count back by 100. Count back by 99. And super fun, wonderful place value patterns can come out. Totally agree that Count Arounds are wonderful. A wonderful, super good way to develop place value with students. Do you want to add anything else about Count Arounds?

Kim Montague  16:05

I think we have a recording actually. Right? On your website?

Pam Harris  16:09

Kim Montague  18:44

I think we collected data about the 10th count. Maybe I'm remembering incorrectly but I almost feel like we said, "Hey, you guys, what did you get for the 10th one of yours? And what did you get for the 10th one of yours?" And then we collected that information to make on a chart as well.

Pam Harris  19:02

Absolutely. And so all these wonderful ways of getting, now so there's other neat patterns that can come from that. But since we're focused on our base 10 places some today, that that idea of what the 10th count was, and whoa, like every time ten 3s was 30, and ten 13s was 130 and ten 17s was 170, ten 25s was 250. And that that zero pattern, we could bring it out sooner, maybe then kids kind of stumbled on it and on their own. Like everybody's kind of stumbled on that pattern that when you multiply by 10, there's sort of this place where you shift when there's zero at the end that kind of represents the number of ones leftover when you multiply by 10. Everybody stumbles on that pattern. But we don't, maybe necessarily do it on purpose. Well, let's do it on purpose sooner. Like let's put it up in front of kids and get them thinking about why it was always the 10th one? And what's happening when we have ten 13s does that have anything to do with thirteen 10s? If we have thirteen 10s is that another way of thinking about 130? And again, we needed to have a little bit of place labeling happening. So we can sort of talk about what thirteen 10s would look like if I sort of put 13 in the 10s slot. Oh yeah 13 (unclear). If I counted by 10, that's a little easier sometimes for kids to feel 10, 20, 30, 40, what would it be if I get up to 13 of them?  Yeah. It'd be 130. Because Ten of them was 100. Ten 10s was 100. So  eleven 10s would be 110. Twelve 10s would be 120. They could kind of feel those 10s. And so by using the commutative property, we can get students to start thinking about those ten 13s as thirteen 10s, or those ten 27s as twenty-seven 10s. And just making more and more sense of our place value system. And we thought that Multiple Towers is a great way to do that.

Kim Montague  20:53

Can we take just a little bit of time to talk about everyone's favorite? Or maybe not favorite? Base 10 blocks.

Pam Harris  21:03

Yeah. Because obviously, I mean, Kim, we should have mentioned those first. They are the be it, end all. Right? If you really want to teach kids place value, we should have base 10 blocks from like in the cradle. Like kids should have them. They should be the things they play, because if kids play with those early, they will understand our place value system, like hands, like no problem. Why did we mention those first, Kim? So we're joking a little bit here. We're poking fun, just a little bit. Base 10 materials are a little tricky. There's a subtle thing happening in that often as adults, we have done some work, whether it be to understand the algorithms, whether it be to look at patterns, like we've just talked about. And we start to get a sense of what it means to have 10 ones and call that a 10. And to have ten 10s together and call that a 100. And so when we see base 10 materials, we can almost like recognize kind of some of the patterns that we've kind of built, especially if somebody kind of helps us work with those base 10 materials, we're gonna go, "Oh, wow, look at those patterns that I've kind of been fussing around with in my brain a little bit. And now I can kind of like, touch them and feel them. And oh, oh, I bet if we handed these to kids, we could bypass that step of having them have to notice patterns and fuss with them and struggle with them a little bit to make sense of them. I bet we could bypass that step and just kind of unzip their heads and pour base 10 materials in. Bam, they're just going to understand, they'll just like, get a feel for -

Kim Montague  22:42

It's so nicely organized, right?

Pam Harris  22:45

It's so nicely organized. It's just, it's right there in front of you, it's so clear. Like you should just be able to clearly pick it up. And we're suggesting that's not how learning occurs. Learning doesn't occur, as we and Cathy Fosnot said that we're pretending that the manipulative embodies the mathematics. We're pretending that the mathematics is so like, it's so - what's the word -transparently there, that kids should be able to just pick it up. And it's not, it's not what happens. In fact, a lot of teachers have to do, have to have some experience and mess around with even those base 10 materials before it makes sense. Another pushback that I have on base 10 materials is too often I see them used solely and only or for the main purpose of helping students understand the steps of a traditional algorithm. We're trying to get kids to understand why we're regrouping, why we're borrowing or carrying, what's happening. And so we represent the numbers and then we pull all the 10s together, ones together. And oh, hey, look, look we have more than 10. So we can exchange these 10 for a 10. Oh, look, now we have a bunch of 10s. And we can count those. Oh, look, we have enough 10s. So we could exchange 10 of those for 100. And oh, now we can count the number of hundreds, count the number of 10s, count the number of ones. And oh, we kind of understand now that carry. It's like we carried that one over there. It's kind of like we carried that 10 over there. And I would submit that may be one of the reasons why teachers are so enamored with base 10 materials is because it might be the first time you understood the traditional algorithm. Oh, that's what that little one is. That's what it means when we cross out that three and make it a 13. Or we cross out that nine and make it an eight so that we can then make that three over there a 13. That's what, like we're actually, like you, Oh, that's what's happening. That's brilliant if your goal is to understand the traditional algorithm. Not our goal. And if you don't remember that, check back a few previous episodes where we talked all about the different algorithms and what our goals actually are, I should say all the algorithms, we talked about the operations and the strategies that are our goals for those operations. And it's not really to build the traditional algorithm to build understanding of that. Do we want the place value understanding in the traditional algorithms? Yes, but we don't need the algorithms to do that. Right. Can I poke on a couple of things? And it's really not even the base 10 materials themselves, right? Like they're their objects. So it's a lot of times what we see happen with them as if doing a couple of activities is going to help kids understand. But if that were true, could we stop calling them unit, flat, rod? Like or unit, rod, flat? Like, can we at least call them the value that they're supposed to represent? So that's one thing. The other thing - Yes, wait,  wait. I have to tell you a story.

Kim Montague  25:37

Yeah.

Pam Harris  25:38

So early, like this is so early that my kid was in second grade. It's one of the very first things I ever did was I said, "Hey, can I just grab some of the, two kids from each of your classrooms and I'll just enrich them once a week?" One of the weeks I went, the teacher said, "Oh, sorry, we didn't tell you, we can't actually have you do that today. But hey, would you work with this student? This student need some help. Would you help the student? Here are base 10 materials. Help this student understand" - I think how to add with them or something. And I was like, "How do you what?" Like, whatever. So I just sat down, and I was like, "Hey, tell me about these." And this delightful students said, "Well, this is a -" I don't even know what she called the unit. I don't know what, I don't remember what that was. And she goes, "This is a rod." And I was like, "It's a what?" She goes, "It's a rod." And I was like, "Okay, but what is it?" And she goes, "It's a rod." And I said, "Well, I know. But like, how many things are like, there's 10 of them in there. Right?" She goes, "I don't know." I was like, "Wait, you don't know." And she goes, "You think there's 10 in there?" I was like, "I think there might be. Let's check that out." So like we counted, she goes, "There's 10." And I was like, "Okay, what's this thing?" She goes, "That's a flat." And I said, "Well, yeah, how many thicker there?" "I don't know." "Well do you think there might be 10 of these rods?" "Maybe." I mean it was unreal. So the idea that we sort of, that the names are could be important. Now, that doesn't mean that if we named them correctly, that it will magically make everything better. Certainly, it will make it less horrible.

Kim Montague  26:59

The other thing that I think, and I know we're going long, but the other thing I have to mention is that we will at like fourth grade teachers, right? We'll use base 10 blocks, as ones, tens, hundreds, the cube might be 1000. If you have one for your class set. Then we re-unitize those because it makes sense to us as adult learners or maybe to some of our students, we re-unitize those when we talk about decimals. And we say okay, now this is one. And we're gonna call this a 10th. And these hundredths.

Pam Harris  27:32

Yeah.

Kim Montague  27:33

Without giving kids any, we just the, next it's like, new unit, next day, bam, we've re-unitized. And we leave kids going, "I don't know what you just did. Because for years I've been calling this ones, tens, hundreds."

Pam Harris  27:47

Or even worse, one, rod, flat.

Kim Montague  27:50

Unit, flat, rod.

Pam Harris  27:51