Pam Harris  00:01

Hey fellow mathematicians, you're listening to the podcast where math is Figure-Out-Able. I'm Pam.

Kim Montague  00:08

And I'm Kim.

Pam Harris  00:10

And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But 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 keep students from being the mathematicians they can be.

Kim Montague  00:41

So in this episode, we want to talk about something that many of you love or hate, or love to hate, and that is decimals.

Pam Harris  00:50

Decimals. So we've done episode before on fractions-

Kim Montague  00:54

Decimals, and percents. Yeah.

Pam Harris  00:57

And let's dive into decimals. Like what about decimals? And specifically, we have gotten some questions and concerns, comments from listeners about three things. And we thought, ooh, these three things have to do with each other. Let's lump them in. Let's get at them in one episode, we can sort of pay attention. And so we'd ask you to consider today when we talked about these three things with decimals, kind of pause and consider the role that these three things play in your teaching of decimals. How do you handle them? What kind of role do they play? So let's kick this off. Go.

Kim Montague  01:37

Yeah. Okay, so we got a message from Kathy, who's one of our Journey members, your membership site, implementation support. 

Pam Harris  01:48

Love our Journey members. 

Kim Montague  01:49

Yeah, we do. She congratulated us on our 100th episode, where we talked about place value. And one of the things that she mentioned, she sent a message and she said, "Hey, when using base 10 blocks, we begin by asking, 'Today, what's the value of small cube?' kind of like a legend on a map. And then once we knew that, we could identify the value of the rod flat and large cube for our work. And then when I taught sixth grade, the small cube could have represented a 1, 10, 100, 1/10, etc. And it depended on the day." She said, "I tried not attaching the name 10 to the rod, in hopes that the kids would know that it didn't always represent a 10. In general, I wanted the kids to learn that the models could represent different amounts. And it was important to find out what they were. I thought this would support flexible student thinking to build relationships between the base 10 blocks." And I think that's absolutely a fantastic suggestion. Because what she said-

Pam Harris  02:46

Nicely done.

Kim Montague  02:47

-at the very end, right was build relationships between the base 10 blocks. And she asked the question, because in our place value episode, we kind of railed a little bit on don't call them flat, rod, cube, unit, whatever we want to call them, or people are calling them.

Pam Harris  03:05

They will give them these random names. Yeah. 

Kim Montague  03:07

And I think we at that time said, if you're going to call them something, at least call them 1, 10, 100. But Kathy is absolutely right to poke back a little bit and say, we need to consider the relationships between those models. And what is the unit of the day? Like, what does one represent? So I think that's something that is a consideration for all teachers. Hey, starting the day saying whether your first, second, your first introducing base 10 materials, if this represents one, then what might the value of this be? What might the value of this be? If holding something else up? What if this represents one, then -

Pam Harris  03:49

And just because we've done probably some high school teachers on Well, when you're saying 'this', so for example, if you held up like the little unit cube, there's one little tiny cube not broken into anything. You could hold that up, and you could say, "If this represents one," then you could hold up the thing that has 10 of those in a line, 

Kim Montague  04:10

Yes. 

Pam Harris  04:10

You could say, "What would this represent?" But conversely, you could say, "What if this little tiny guy, this one unit cube, what if it represents 10, then what would this sort of rod looking thing?" We wouldn't call it that, right? You would hold it up. 

Kim Montague  04:29

Hold it up, yes.

Pam Harris  04:31

"What do 10 of them, 10 of them stuck in a row here in a line, what would that represent? If if one of these was 10? Then what does this represent?" And then you could also do 10 of those 10s. So now that sometimes teachers will call that a flat, often, we sort of said at least call it a 100. But if the unit cube represented 10, then 10 of them would represent 100. So now I've got that sort of 10, that line of 10 cubes that sometimes people call a rod. That thing now represents 100. Well, if I've got 10 of those, that flat thing that's 10 of those rods, what would that represent? Now I'm using words to sort of help people kind of know what, we don't want to call them that. We just want to hold them up, then what would this represent?

Kim Montague  05:20

Because there's a lot of re-unitizing, that happens, and that really can be challenging, right? 

Pam Harris  05:25

Oh, you just threw that word out. We might want to define that. Sorry. Keep going.

Kim Montague  05:30

I was just gonna say that the point is the relationship that those models have with each other, it's a really nice manipulative, and you can do some nice questioning around it. But the questions can change. And really, if you're wanting kids to be flexible thinkers, they should change. We should, we shouldn't really be saying, this is a 'blank' for a number of years. And then surprise one day in fourth grade, we go no, now this is a 'blank'. We really should give kids the experience to wrap their heads around, if this, then this. Okay, well, now if this, then what?

Pam Harris  06:07

Absolutely. And this idea of unitizing goes all the way back really young. The very first time we asked kids to think about representing three, like as soon as we say, (claps 3 times) three, and I hope you can hear my clapping, but that three claps represent, or can be represented by one word 'three', or one symbol, the numeral three. We're asking them to unitize taking a multiple of things and representing it with one thing. Then what you're describing is that we've got these materials that we might assume, "Oh, of course, that little one unit cube of course, that's one." And you put 10 of them together, "Of course, that's 10." By the way, so as you get to 10, you're re-unitizing in a different way, because now you've got 10 objects (claps 10 times), think that was 10, represented by two numerals, right? 

Kim Montague  07:00

Yeah.

Pam Harris  07:01

We had numbers that were represented by one now they're represented by two. And so now, what we don't, what we're suggesting is that as Cathy Fosnot would say, the mathematics is not embodied in the manipulative. It's not just apparent and obvious, that we want to help students be flexible with those materials. So I say that well?

Kim Montague  07:21

Yeah, you did. And really, we said that this is going to be an episode about decimals, right? And so I think a lot of times that early grades, we don't see kids struggle quite as much with, you know, base 10 materials, because we've only asked them to consider it 1, 10, 100, maybe 1000, right? We've only said this is this is the thing that they are, it's when we hit decimals, that all of a sudden, we've re-unitized. We've shifted what we asked kids to be considering. And it happens to be with decimals. And so we go, "Oh, they don't understand decimals." So it's not that the model is bad. It's probably some of the telling that we do with relation to base 10 materials. And we can just shift that a bit. And I think that's absolutely necessary.

Pam Harris  08:13

So let's shift from telling, "Here's what they are," to asking, "If this cube?" And so if you think of the large cube that has 1000 of those little unit cubes in it. If this cube represents one, then what is this flat of 100 of those unit cubes? Or what is this rod of 10 of those unit cubes? How do those relate?

Kim Montague  08:39

And we might have to supply the name tenth, hundredth, thousandth. But they certainly can tell us how many of them are needed to create one whole.

Pam Harris  08:49

We could certainly figure it out and in the figuring in the deciding what those then how many of them are. And we can supply the name. As kids wrestle with those relationships, that's the power of the base 10 materials. 

Kim Montague  09:02

Right. 

Pam Harris  09:02

We're not just then handing it to them and saying, telling, "Oh, now today, this is what they are." No, no. It's like in the in the questioning and the having the kids grapple with those relationships. Alright, that's totally cool. One other thing I'd like to bring up about base 10 materials. What we've just said is again, if you question well, they're good, if you just tell it's not so good. But also, that's all about building relationships. We've said before in other podcast episodes, that base 10 materials are not good for computing. They're not good tools for computation. We don't want to give kids base 10 materials and say now go build the number, build the number, add them together, collect all the things and then read off the answer. That's not helping students develop this sense of size and magnitude and facility with strategy. It's literally like telling them do this thing. Do this thing. Now gather them all together, and then read off the answer. And, again, I'll quote Cathy Fosnot, whenever you read off the answer that should be a clue that less thinking is happening and more just sort of doing and then you're kind of like almost, Oh surprised at the answer. So in that way, we're fine with using base 10 materials by questioning and helping students grapple to build relationships. But we're not great with those as tools for computation. The reason I bring that up, is the second thing we wanted to talk about in today's podcast, is Exploding Dots. So James Tantan, great guy, mathematician, has some nice stuff out there, has this thing where he talks about Exploding Dots. And people will often ask me, "Pam, what do you think about exploding dots? It's so good." Whenever, I would suggest that Exploding Dots is like, dots is like, dots are like, the thing of this. How do you say that? The idea of using Exploiting Dots helped from what I've heard. So when I heard James Tantan introduces for quite a while ago, actually. When I heard his story about kind of how Exploiting Dots came to him, it was all about him understanding our base ten place system, but most of it was sort of, at least when I heard it, most of it was kind of geared around understanding the algorithms. He said, "Oh, I get what's happening, I get that there, it's sort of in this dot (exploding noise), it could explode into 10 of those things that are sort of in it. And in this dot (exploding noise), it sort of explode into 10 of those things that are in it." That in a huge way he was understanding our place value system, that 100 has (exploding noise) 10 tens in it. Ten dots would pop out of there, that would represent tens. And each of those tens could explode into 10 ones. And that I'm fine with Exploiting Dots. For the same thing, the same role that I'm fine with base 10 materials in that they can help students build relationships, the relationships between the different values and our base ten place system. I am not fine with Exploding Dots being so emphasized as a way to understand the algorithms. I mean, if that's your goal, okay. So I've sat in sessions, and I've talked to other leaders, and they were so excited. They're like, "Oh, my gosh, this is gonna revolutionize all the things." And I said, "To what end?" And they're like, "Because now we finally understand the algorithms." Which I chuckled just a little bit. Because I'm like, "Yeah, like it took, like, how long and how much and all the things before you understand. But how is that helpful?" Okay, if your goal is understanding the algorithms, you bet. Not my goal, not our goal, our goal is not getting more and more proficient at the algorithms or understanding them any better. Our goal is something completely different. So hear me clearly I'm not saying James Tantan bad, Exploding Dots bad nope, nope. He had fantastic insight into our base ten place system and how the algorithms work. And if that interests you, go for it. I don't find it all that particularly helpful in computation. So just parsing those out again, both base 10 materials, and Exploding Dots are fine to help students understand, to help question them through, have them grapple about the relationships. I'm not interested in using them as tools for computation.

Kim Montague  13:30

So what can we relate kids to?

Pam Harris  13:33

Oh, yeah, what can we do with decimals? So a third thing that people will ask us about is, "Pam, what do you think about using money to help kids understand decimals?" And Kim, what's our answer to that?

Kim Montague  13:48

Well, honestly, it's like the most relatable thing right? But people will say to us, "But kids don't know money." And to which we say, but they need to, right? 

Pam Harris  14:01

All the more reason. We must.

Kim Montague  14:03

Yeah, we have to give kids experience with money for so many reasons other than just life. Life in general, they need some experience with money. But it is by far the most important relationship that they can have to decimals is thinking about money. We love having kids think about tenths as dimes, and hundredths as pennies and we can not let go- 

Pam Harris  14:31

Thousandths can be parts of pennies. Yeah, absolutely. Parts of pennies. Not a problem at all. Yeah, sorry to interrupt you.

Kim Montague  14:39

No, it's okay. I was just gonna say that it gives us an anchor to recording decimals and recognizing decimals in written form.

Pam Harris  14:46

Absolutely. Sometimes people will push back on me, especially higher math people will say, you know, like high school teachers, maybe even middle school teachers. You know, honestly, it's probably more middle school teachers than high school teachers will say, "No, no, no, it's too limited, money is is too limited, we have more places, more places in our system, than just tenths and hundredths. And so you know, it's too limited, money. Money will will limit kids in understanding decimals." To which I say, "But that doesn't mean that it's a bad starting place. And it doesn't mean that it's not a great comeback to place to help me think about the magnitudes." That as I get and dive and delve into decimals, money can be a very helpful way for me to go, "Wait, let me make sure this is reasonable." Like I'm thinking about, I might be thinking about liters or I might be thinking about parts of something that doesn't have anything to do with money, and I might actually transfer into money in my head to go, "Oh, yeah, okay, that's reasonable." Or whoa, that seems, it shouldn't be, it should be dimes on dollars, or it should be pennies, not dollars, or pennies on dimes. Like oh, like, ah, that sense of magnitude that we can build with money can be helpful in having that sense of reasonableness. So money. Yeah, bam. Yeah, sorry. Go ahead.

Kim Montague  16:07

No, I was gonna say, I'm glad you just said that. Because there's plenty of times where I'm working with decimals. And it has nothing to do with money as the context. But I will step out of it and think about the problem that I've solved, just computation and go, "Would that be a reasonable amount of money? Yeah. Okay. Cool." And I know that I have made sense and thought through kind of the decimal placement well.

Pam Harris  16:31

Yeah, which is really the magnitude, right? You have been looking at the numerals and saying to yourself, "Is it 154? Is it 1.54? Is it one 5.4?" Which is 15.4. I was just trying to move the decimal around. Like what makes sense, is it .154? What makes sense here and money might be a helpful anchor to making sense of those magnitudes? Absolutely.

Kim Montague  16:54

So we tackled three separate things that people have asked us about. And I want to just talk a little bit about where this leads. So we got this email from Willow. And Willow said, "I was hoping that you ladies could do a podcast on multiplying decimal numbers less than one and a way to model or help students figure out what's happening when you multiply 0.3 times 0.7." And she says, "Halves are easier to explain, but the concept does not always transfer because 0.3 is not 1/3." Absolutely not the same.

Pam Harris  17:29

Sure enough. Good catch there. Good call. Point three is not 1/3, not equivalent to 1/3. Amen.

Kim Montague  17:35

So stay tuned for next week when we fold in what we talked about this week into decimal multiplication.

Pam Harris  17:44

So if you're excited for some decimal implications, stay tuned because that is next on the podcast docket. Y'all, 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 mathisFigure-Out-Able.com. Let's keep spreading the word that math is Figure-Out-Able