Ep 120: Rounding

Pam  00:00

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

Kim  00:07

And I'm Kim.

Pam  00:08

And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But ya'll it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor mathematicians as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be.

Kim  00:37

In today's episode, we're going to discuss a common sticking point for many teachers and students.

Pam  00:44

Let me tell you. So, teachers, early on, when I very first... I dove into the research. I dove into my kids classrooms. I was doing a lot of experimenting. I was reading all this stuff, trying activities. I started doing professional development with teachers. And so, teachers were coming out of the woodwork. They're like, "Okay, so you're kind of entertaining. It's, you know... We're learning stuff. This is good." They started asking me for things, and one of the things they asked me for really early on was, "Okay, alright. Pam, next professional learning, we want you to teach us how to do rounding." And I was like, "Huh?" 

Kim  01:15

Yeah.

Pam  01:16

I was like, "How do you teach rounding?" I don't know. It was so like... I had no idea what that even. Because then they would say... I mean, it so floored me, in part, because then they would say, "Well, yeah. Like, we've tried..." And then, they would list these things. And I was like, "Oh, not that!" Because they would list these crazy rules, or colors, rhymes, directions. All the steps! Like, I just... In the intro, I just said rotely, repeating steps actually keep students from being the mathematicians they can be. This was such a. Because teachers would say, "Well, I've tried this one: Four or less, let it rest. Five or more, let it soar." Or one of them was, "Four or less, let it rest. Five or more, up the score." Because, you know, we're scoring things. One of them was, "Five and above, give it a shove." 

Kim  02:07

Oh, no. Oh no, no.

Pam  02:10

 They were showing me things about, "So, you underline this number, and you circle the one before..." They had these arrows to sore. I mean, I saw one with wings. I mean, Kim, it was all the. Now, I get it. I get it. If you are the teacher who learned math as fake math, and so it's all about steps and procedures. And you're going to help those kids learn those steps. Well, if it is, then you're going to try to make it fun, cute, and. Ya'll, if you look at something about teaching math, and it looks cute, may I just respectfully suggest it's an attempt to make fake math palatable. It's an attempt to make fake math rote memorizable. In other words, fake math! Fake math alert! That's a fake math alert. Can we actually reason about relationships? So, it's another time where I went up to you and I was like, "Kim, how do you teach rounding?" And you were like, "Duh." You're always so cute because you would just look at me like, "Well, you know, the only way that a sensible person would." And I'm like, "Okay, tell me. Tell me what that way is. Tell me what that sensible way is." So, Kim, I'm going to let you fly. Tell me about rounding?

Kim  03:20

Well, so just like we talked about in last week's podcast episode, I found such power in when you shared the open number line with me. And I'm super grateful because, while I was messing with numbers in my head, I didn't really have the best ways necessarily. Other than just talking at my students, I didn't have the best ways to help them visualize what was happening. I didn't have a great way of recording things that were happening. 

Pam  03:49

Really making that thinking visable.

Kim  03:51

Yes, absolutely. 

Pam  03:52

Yeah.

Kim  03:52

And so, you know, when you came to me saying like, "What about rounding?" It was a natural thing for me to say, "Oh, you would just put it on number line." Because that's kind of where numbers are, right? Like, I have this visual in my head of a number line now. And so, numbers have relationship to each other. And so, you know, if you were to ask me, like, say 53. If you were going to ask me, "Where does 53 fall in relation to other numbers?" Then, I would sketch a number line. I would put 53 down. And it was an opportunity for me to say, "Where does 53 fall? Oh, it's 3 away from 50." And I would.

Pam  04:31

You're asking students this, right?

Kim  04:33

Well, I'm just telling you what I would do in my head, and then.

Pam  04:36

Ah, okay. Sorry, sorry.

Kim  04:37

This is kind of the conversation I have with myself, but the same conversation I have with students. So, if we're trying to figure out where 53 is and which 10 is closest to, then the question is, "How far away is it?" So, the first thing that I would share is that I think about which 10s it falls between. And so, 53 falls between 50 and 60, and so whichever one it's closest to is what it would round to. And so, 53 is only 3 away from 50. So, I'd make a jump of 3 backwards to get to 50. And then, I would say, "How far away is it from 60?" And it's 7 away. And so, naturally, it's closer to 50, and so it would round to 50.

Pam  05:19

Nice. If you are rounding to the nearest 10.

Kim  05:22

Right.

Pam  05:22

So, what if we round to the nearest 100?

Kim  05:25

So, if I. Same thing. I would put 53 down on a piece of paper, and I would say, "Oh, it's between 0 and 100."

Pam  05:33

"What 100s is it between?"

Kim  05:35

Yeah.

Pam  05:36

Yeah. 

Kim  05:36

And so, then,, I would say, "It's 53 away from 0. And it's 47 away from 100."

Pam  05:43

Mmhmm.

Kim  05:43

"And so, 53 is closer to 100 than it is to zero." It's all about where it is on that number line. And as I modelled, and as students got better with considering those numbers, I also found that their jumps on their number line got nicer, got more in line...

Pam  06:10

Precise.

Kim  06:10

...with. Proportional. Yeah, yeah, yeah. 

Pam  06:12

Yeah, yeah.

Kim  06:12

"Precise" is a good word. 

Pam  06:13

More reasonable. More...

Kim  06:15

Mmhmm.

Pam  06:15

Mmhmm, yeah. More realistic. 

Kim  06:17

Yeah.

Pam  06:17

All those words. You know what's interesting, Kim? As you're talking about this, I'm actually kind of acknowledging in my head that you're actually asking more of students in some ways than. 

Kim  06:27

Yeah.

Pam  06:28

Especially if you're just, like, repeating this rote memorized rhyme. But you're actually asking them, "How far was 53 from 50?" And, "How far was 53 from 60?" 

Kim  06:36

Yes. 

Pam  06:36

And correspondingly, "How far was 53 from zero?"

Kim  06:39

Yes.

Pam  06:40

How far is it from 100?" Well, bam! I Have, You Need. Like, those. 

Kim  06:43

Yeah.

Pam  06:43

Those partners are. You're actually asking students to figure out more than we typically do. You know, if you just say, "Hey, round to the nearest 10." Students could just, you know, think about which one it's closer to, but not actually find the differences. And so, you're actually asking more. That's interesting, I think. Sometimes, teachers will say to me, you know like, "Wait, you're really telling me not to spend six weeks on place value?" And I'll say, "Yeah, yeah, yeah. Do this. Do this work that we're suggesting." Because if you do like what Kim just suggested, ya'll. Notice the place value that's coming out. It's not just a bunch of rhymes about. What did I? I can't even remember. Shove them down? Give it a shot? Let it soar? What? It's not what rule, what rhyme to apply. No, no, no. It's like we're diving in. We're thinking about place value. "What 10s is this number closer to? What 100s is this number..." In fact,  we could get crazy and say, "Round to the nearest 1000." Would you similarly then say 53 is one of the 1000s? 

Kim  07:36

Yes.

Pam  07:37

On either side?

Kim  07:38

Yep.

Pam  07:38

So, can you run us through that? (unclear). 

Kim  07:39

Yeah. So, 53 comes between 0 and 1000. So, there's. it's only 53 away from 0. And it's going to be 947 away from 1,000.

Pam  07:55

From 1,000. Yeah.

Kim  07:56

Mmhmm.

Pam  07:57

Look at that extra work that she's doing. Isn't that brilliant, ya'll? So, we can get our students to start thinking? Go ahead. 

Kim  08:02

It's... There's such a richness to number that I think sometimes we miss opportunities to give kids those experiences. Right? We're constantly talking about experiences. Here's an opportunity for kids to get a really good feel for the richness of 1,000.

Pam  08:16

Absolutely.

Kim  08:17

 And where numbers fall.

Pam  08:19

Yeah. And like you said, they're placement on a number line gets better, gets more efficient, gets more precise. Not that... That's not what's important. But what's important is their sense, their perception, their intuition for number. And here's a brilliant way for us to save all that time that we're practicing the rhyme, and can they implement the rhyme, and the rule, and the color, and all the things. And instead, we're diving into actually the relationships. So, this is such a good example of, like you just said, experiences, the richness that we could just keep kids simmering in. I've just. My hands are kind of like I'm stirring a pot. Like, we're just keep them going in thinking about real math. Alright, so somebody out there is thinking about the fact that what I didn't give you is the number 50. 

Kim  09:09

Oh. 

Pam  09:09

Right? Or the number 55. 

Kim  09:12

Yeah.

Pam  09:12

So, like, tell me a little bit about... What do you do with, like, if it was. Instead of 53, what if it was 55? How do you know?

Kim  09:22

So, if it was 55, then it's going to be 45 away from one of the 10s and 55 away from the other 10. But what's really tricky for kids is when it is exactly in the middle.

Pam  09:35

Well, and so what if I said 55 to the nearest 10? 

Kim  09:38

Oh, I thought we were back on 100. Sorry about that. 

Pam  09:40

No, you're good, you're good.

Kim  09:41

So, yeah, that's definitely an interesting conversation to have, right? I love this conversation because we get to talk about the fact that it's both 5 away from 50 and 5 away from 60. And so, instead of your rhyme about giving it a shove, it's a really nice chance for me to go, "Oh, wow, you're right. It is exactly in the middle of 50 and 60." And like, it's an opportunity for conversation, and for kids to go, "But I think that it's...", "We should round down," and "I think we should round up." And like, context for when you would want it to be rounding down or rounding up. And then, I could just share that piece of knowledge that's social that I say, mathematicians have just decided that when it's right in the middle, then we round up. 

Pam  10:33

Yeah, when you said that, "we just decided," you didn't mean just now.

Kim  10:37

No, no, no. They have decided for us.

Pam  10:40

That is a thing. That is a thing that mathematicians. And we have to make decisions, sometimes, like that. And so, we call that "social knowledge", like you said. And they just decided. But brilliant to have the conversation. And oh my gosh, I loved when you just talked about context, that there are times when the context determines if you're going to round up or down. Right? 

Kim  11:00

Yeah.

Pam  11:01

Like, brilliantly. In fact, can you... I'm going to put you on the spot. Can you think of a context where you would not round 55 up? Or one that you (unclear).

Kim  11:11

Where I wanted to round down? 

Pam  11:12

Yeah.

Kim  11:13

When it's a bill I owe.

Pam  11:14

Haha! Yeah, but you can't. You're still going to have to pay. Can you think of one where you actually would round down? I can. I think so.

Kim  11:24

Yeah, go for it.

Pam  11:26

Help me fix it if this doesn't work because I'm. This is off the cuff here. So, if I have... If I'm looking at purchasing something, for $55, but I've only got $50. That's... I don't know. Help me fix it. Something like that, where I'm like, "Mmm, can't get that." Like, I'm not going to round up in that case. It's not $60. 

Kim  11:48

Yeah.

Pam  11:49

It's close to the $50 I've got, but I can't. It's not perfect. There's gotta be a better. Hmm. Well, ya'll, you can tell we record this podcast live.

Kim  12:00

I was going to say, we should have people send us in an opportunity (unclear).

Pam  12:02

Oh, let's do that. Yeah. You're like screaming at. You're all listening to this podcast. You're like, "Pam! Obviously, this one! Here's a good example!" Hey, send us your good examples of when you would naturally want to round down. 

Kim  12:12

Yeah (unclear).

Pam  12:13

Or need to round down. Yeah. Even though it's five or above. And so, the rule says to round up, but yeah, you wouldn't. Oh, I know there's a few. I just can't think of what it is right now. Alright, if you can think of one the rest of the episode, that would be cool. Hey, you've talked to me a little bit about clothesline math. And we've done some work with that. 

Kim  12:34

Yeah.

Pam  12:34

Tell us more about that.

Kim  12:35

Yeah, so in a variety of grade levels, I have been in classrooms...and have done a little work myself...with the idea that when you're trying to build intuition about number and relation to other numbers. There's some really great activities where you have a clothesline that spreads across the room, and you put. Like, say you give kids a card that has like 7, or 10, or 6, or 5, or 4. And you say, "I'm going to put the 0 and the 20 at the beginning and the end of the clothesline. And kids come up, and they place numbers where they think it would go on their clothesline number line. And then, they place numbers in relation to each other. So, like, the 10 kid might come and put the 10 right in the middle, and then other kids place their numbers around it. And it just. As we were talking about rounding, it reminds me that those ideas about placing numbers in relation to each other are absolutely related to rounding because numbers are in relation to each other. 

Pam  13:38

They're all relation. Yeah. 

Kim  13:39

Yeah. And so, you can then say like, "What if my number line went from 0 to 10?" Or whatever the numbers are that you're going to change. But you give kids experiences coming up and moving numbers around and shifting, so that they can see them proximity to each other?

Pam  13:55

Well, one of my favorite ways of doing that is, let's say that you did your 0 and 20. So, you might say, "Okay, so here's 0 and here's 20." And then, you might give kids the number 30. And so, if they had put 0 and 20 at the two ends of the thing, they're going to have to move that 20... 

Kim  14:09

Yeah.

Pam  14:10

...to be able to get the 30 on the end. And then, you might give them 100. And then, they're going to have to shift that 20 and that 30 down in order to put the 100 on the end. And then, you might give them 50. And then... And that might help you, like, kind of judge how the 20 and 30 are fitting with a 50. And then, you could start talking about rounding in the midst of that conversation. So, again, another really rich place where we get place value, and relationship, and you can also drop rounding. I was kind of. I wanted to ask you about that because as we were talking about 53. I wonder if you put 53 on. And it doesn't have to really be a clothesline. It can be any kind of like rope or string or something. We've seen teachers with really strong magnets and they sort of put the magnets, and then the string, and then use index cards.

Kim  14:53

Yes.

Pam  14:54

And fold the index cards in half and that's what you. Right? It be really nice to laminate some of those index cards because then you can change the number. You don't have to, like, waste index cards. But if you did that with the 53, we can say, "Alright, ya'll, here's 53. Put the 10s that are closest to it." So, then some kid goes up and puts 50 and somebody puts 60. And then, we talk about how they do. And then, what you said, the differences. Right? And then, you can say, "Okay, so now I want to put the closest 100s, Where are those?" Well, Whoa. Like, depending on where they put that 50 and 60, they might have to squish those way in, in order to get the 0 and the 100 up there. And then, you know, "Should 53 be closer to the 0 or the 100?" And like you said. And then, if you are putting those magnets against a whiteboard, you can be actually drawing in those differences that you were talking about. 

Kim  15:44

Mmhmm.

Pam  15:44

And then, you could say, "Okay, cool. Now, we want to put it... Where are the 1,000s? What 1,000s is it close to it?" Well, now you're, like, sucking all those in. So, in order to get them in. And you might have to even take some off, if they're not going to fit, to get them into those closest 1,000s. And we sort of get this collapsing number line a little bit where you're sort of zooming in and zooming out. And again, brilliant ways to help kids really think and reason, gain relationships. Yeah? 

Kim  16:14

Yeah, I like it. 

Pam  16:16

Hey, when you were telling me about rounding, there was a point you said something about "easier".

Kim  16:22

Yeah. 

Pam  16:22

Can you say a little bit more about that?

Kim  16:24

Sure. So when you have a number like 53 that you're going to put on the number line. I think it's a whole lot easier for kids to say, "Well, I know that it's 53 away from 0. So, going to the lesser 10, the lesser 100. It's easier for them to say, "It's only 53 away." But finding the greater 100 or the greater 10 requires a little bit of addition and subtraction, right? It's a little bit more thinking involved to find that greater number, which I think is why I did it. Anytime we get a chance to do a little bit extra in there.

Pam  17:01

And maybe, let's get specific. So, if I said, "Round to the nearest 100."

Kim  17:04

Okay.

Pam  17:05

With 53. You're like, "Well duh. 53 is 53 away from 0." 

Kim  17:10

Yep. 

Pam  17:10

But in order to tell how far away 53 was from 100. 

Kim  17:13

Yeah, you have to do a little work.

Pam  17:14

You had to use. Yeah, you had to do a little work. And if you know. If you've played I Have, You Need, you can be like, "Well, that's got to be 47." 

Kim  17:20

Yeah. 

Pam  17:20

There's that partner. And so, you're saying it's the. When you said the lesser is easier. So, like if I said, "53 but round to the nearest 1,000." 

Kim  17:29

Yep. 

Pam  17:30

(unclear).

Kim  17:31

(unclear).

Pam  17:31

Yeah. Go ahead. Keep going.

Kim  17:32

0 is only 53. away. There's not a lot of work there. There's not a lot of... It's just it's that far away from 0. But to get to the 1,000. The number, the 10, or 100, or 1,000 that's greater, Then, there's definitely some thinking involved. You have to figure out that it's 947 away.

Pam  17:54

I wonder if that's when I Have, You Need was born. 

Kim  17:58

I don't know...

Pam  17:59

I wonder if you were doing rounding and you said to yourself, "Man, these guys need that. They don't own these. They need those. Let's see. Hmm, what can I do?" 

Kim  18:06

That's a great question. I don't know. 

Pam  18:08

We might have just discovered the birthplace of I Have, You Need. 

Kim  18:13

I'm not sure. 

Pam  18:14

So, ya'll, when you're rounding, don't just round. Like, do it as a part of the rich experience of helping students really learn all of these numerical relationships that will keep them in good stead, that will serve them so well with everything else that we're doing. So, 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!