Ep 18: This, Not That

Pam Harris  00:01

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

Kim Montague  00:08

And I'm Kim Montague.

Pam Harris  00:09

And we're here to suggest that Real Mathematics is not about mimicking or rote memorizing. It's about thinking and reasoning about creating and using mental relationships. We answer the question, if we're not using algorithms, then what are we doing in mathematics? Alright, I really like the title of today's episode. This, Not That,

Kim Montague  00:33

Yeah, we're gonna share some of the things that we hear our fellow educators or parents say, that make us cringe just a little bit and offer some alternatives that are much more mathematical. So Pam, we're going to start right off, and I know that one of your personal favorites is 'add a zero'.

Pam Harris  00:52

Oh we have to start with that one. Okay. Yeah. So when we multiply by 10, in our really cool base 10 system, you know that there's this zero thing that happens, right? That when we multiply by 10, it's sort of that many 10s. And so we have zero ones. And so there's this zero that shows up. But it makes me slightly crazy when people say, "Oh, yeah, just add a zero." Because when I was in kindergarten, if I had five puppies, and zero puppies crawled in the room that I didn't have 50 puppies, right? You're not adding a zero, it's really more about multiplying by 10 and what happens in our base 10 system when you do that. So when we multiply by 10, or divide by 10, it's not about adding a zero. In fact, really quickly, if we're multiplying decimals by 10, you can't add a 0. 3.2 times 10 isn't 3.20. So we don't want rules that expire,

Kim Montague  01:46

Right. 

Pam Harris  01:47

So we don't want to talk about adding zero.

Kim Montague  01:50

So also, I think that what I should have mentioned earlier is that sometimes it's less that people don't know the right thing to say. I think sometimes we get a little bit sloppy in our language.

Pam Harris  02:02

Little bit lazy.

Kim Montague  02:04

Maybe this is a sloppy one. But what we want kids to be thinking about multiplying 10, 10 times greater or 10 times a number. Sometimes we say 10 times more than a number. But when we divide by 10, we don't want to say 10 times less, we want to say one 10th of.

Pam Harris  02:22

So 10 times more could be like adding 10. Sorry, 10 more would be like adding 10. 10 times a number is a great way to say it. A 10th of a number is a great way to say it. Here's another not a great way to say it. Don't 'butt-cheek'. When I mentioned that we were multiplying by decimals, some teachers were like, "Oh, now you don't add a zero. Now you just butt cheek." The first time that came up, Kim was actually working with a seventh grade kid, and she called me and she's like, "Hey, this kid is talking about butt-cheeking." 

Kim Montague  02:53

What is that?

Pam Harris  02:55

That's a that's an unfortunate thing that sometimes, again, in our sort of sloppy language, we kind of treat this idea of multiplying by 10 is just like moving the decimal and our attempt to kind of make math cute or help kids remember, like, if you move the decimal, you sort of draw these butt cheeks. I mean, Kim, I'm from a good little Mormon family. We don't even say the word 'butt'. But since you brought that recording to me. Now it's all over the place. Butt cheeks. 

Kim Montague  03:21

Sorry about that.

Pam Harris  03:21

Unfortunate. Butt cheeks do not belong in mathematics. Let's be clear.

Kim Montague  03:25

Right. And so actually, that's part of a whole place value conversation, right? And so we're gonna have a whole podcast series about place value, because it's definitely a big topic.

Pam Harris  03:35

Yeah. So just briefly, when you're talking about times 10, and talk about the pattern in our number system, where you have that many 10s. And so there's a zero that shows up, because it's sort of times how many of the ones are in that one. It's not about 'butt cheeks' or 'adding a zero'. Okay, cool. So here's another fave. I'll shoot back at you, Kim, how about the word 'cancel'? Well, I like that one, right?

Kim Montague  04:02

Yeah, that's important.

Pam Harris  04:03

Okay, so what's wrong with the word cancel? Cancel is so helpful because it comes up in so many places. But y'all that's part of what's bad about using the word cancel or difficult about using it because it shows up in so many different places to actually mean different things. And if we're not really explicit about what's happening, kids just start to cancel everywhere. So you can appreciate sometimes we might have something like -2x + 2x. And sometimes teachers will say, "Oh, they just canceled out." Well, could we talk about that -2x + 2x as adding to zero? Or we might have just -2 + 2 and those add to zero? And if we have zero in this equation, then we can sort of deal with the fact that we have, now it's zero there. Because if it just sort of cancels then what's left there? Well, it's kind of important that they sort of added to zero. So now we know we have a zero left that we still have to deal with. Why is that important? Well, because another place where we often use the word 'cancel' is when a thing is divided by itself. So we might have five divided by five. And especially if it's in kind of like a rational thing where we have 5a squared divided by 5b squared, then people might say, 'Oh, the fives cancel.' Or if we're simplifying fractions, those factors cancel. But do those canceled to zero? No, they cancel to one, right? See, I just used the word cancel. They divide out to one. A thing divided by itself is one. Now in that multiplicative relationship, I have a one hanging around. Oh, well, that's hugely different than having a zero hanging around in a multiplicative situation. So it's important that kids actually talk about what's happening, and not use this word 'cancel', that can get used to mean way too many things. Let me tell you, as a high school teacher, I start to get them that if it looks like a fraction, you know, like irrational things, things divided by things, they just start to cancel everywhere. And all of a sudden, they look at a say fraction addition question, and they say, "Um, let's see, this is where I find a common denominator. No, this is where I invert, multiply. No, this is where I multiply straight across," or my favorite, "this is where I cross cancel," which isn't even a thing. So we want to be careful about how we use this word, 'cancel'. I would prefer that we actually talk about what's happening in that situation. So kids are mathematically aware of what's going on and it doesn't become this crazy 'Cancel everywhere!'.

Kim Montague  06:28

Yes. And so speaking of canceling and fractions, sometimes we ask kids to do my favorite thing, which is called 'reducing', and it gets me every time because the number doesn't get smaller, right? We're not reducing the number if I move from six eighths to three fourths, the numbers not getting smaller. In fact,

Pam Harris  06:49

Hey, Kim, you can have six eighths to the pie, but I'm gonna have three fourths of it. Wait, that would be smaller if I reduced it. Wait, I want the six eighths. If 3/4s is reduced, it's smaller. I want the bigger piece...

Kim Montague  07:04

Right. It's just silly. And actually what I was thinking about the other day was that, in fact, the size of the pieces in a part-whole scenario are actually getting bigger. So when we say reduce, the size of the pieces are getting bigger. 

Pam Harris  07:19

That's interesting.

Kim Montague  07:19

Right? So when we all agree from this point on-

Pam Harris  07:22

Wait, wait, wait, get specific, not everybody can picture that. So you're saying if I go from 6/8s, to 3/4s, some textbooks - and you know, it's horrible that even textbooks will say reduce - but if I go from 6/8s  and I quote unquote, 'reduce' it to three fourths, you're saying that the size of the pieces were 1/8s, but now they're bigger. They're actually 1/4ths. So we're reducing the fraction, but the pieces are getting bigger. So how about if we just decide not to use the word 'reduce' when we're talking about fractions at all?

Kim Montague  07:52

Yes, so let's use 'simplify'.

Pam Harris  07:55

Yeah, which I actually don't love, but at least it's a little bit better. Um, you know, it's funny staying on reduce for just a second, I actually have some colleagues who I respect, they've got great work, who said to me one day, "No, no, no, we could use reduce, because it's reducing the number of factors in the numerator and the denominator." So okay, that was a great attempt to make sense of a really bad name, like a really bad term that we use. So good job trying to make sense of it. How about if we just don't use it? Neither you or I really like simplify a ton. That's not our favorite, but it's so much better than reduce, that if you have to use one of the two. Let's use 'simplify'. Okay, cool. Let's do another example. So Kim, I'll shoot another one back at you. Often, when I work with middle school teachers, middle school teachers will get on me a little bit when I will say something like 'three-point-two'. So earlier, I talked about 3.2 multiplying by 10. You don't want to add the zero. You also don't want to butt cheek. But I said 'three-point-two'. What I didn't say was 'three and two tenths'. 

Kim Montague  08:52

Right. 

Pam Harris  08:52

Now I could have, right? The name for 3.2 is three and two tenths, but it's also $3 and 20 cents. 

Kim Montague  08:59

Yeah. 

Pam Harris  08:59

And I think we need all three of those. So I know middle school teachers get kind of excited, "Hey, if we were to use correct language, kids would understand better." So I have a case in point. I can give you a specific instance where it didn't help to use the language. Are you ready? 

Kim Montague  09:13

Yep. 

Pam Harris  09:13

I had a personal teacher who forced me to whenever I talked about a fraction I had to give it its fractional name. Didn't help me one wit. I memorized what the fraction names were. It didn't ever occur to me how they influenced what those like, three-point-two, three and two tenths. I could say 'three and two tenths' till the cows came home. It didn't help me think about that number any differently. So just using the vocabulary doesn't necessarily mean that we're also helping to have the kids think differently. However, I do think we should use all three. So if I'm talking about three-point-two, I think sometimes I say three-point-two. Sometimes I say $3 and 20 cents, and sometimes I say three and two tenths and I think we need all three of those, but not just using their names. We also need to help kids understand why we're using them; what their names actually mean. So I think we need all three of those descriptions.

Kim Montague  10:08

Right. And I feel like that's part of the place value conversation that we're going to have a little bit more in our multi part series. But I've also heard you mentioned that, um, we want to have language that is part of the masses, right? So we talk about our national debt.

Pam Harris  10:25

Oh, yeah. Yeah. How often have you heard a politician say, "Oh, let's see right now our national debt is 4.2 trillion, or four and two tenths trillion? 

Kim Montague  10:35

Yeah. 

Pam Harris  10:36

I don't even know where we're at right now. I'm just throwing out a random number. No one ever says 'four and two tenths trillion dollars'. That's not a thing. Thanks for bringing that up. We do need to have the popular sort of nomenclature also happening. So kids are sort of clear that all of that's legal to use to name those numbers. 

Kim Montague  10:56

I've got an admission. The thing that I continue to need to work on, is I say 'over' a lot. I say over a lot. And I'm working on it. 

Pam Harris  11:09

Well, so let's be picky. When do we not want you to use 'over'? 

Kim Montague  11:15

So if I'm writing a fraction, and I'm describing it to somebody else, I will say, "two over three."

Pam Harris  11:23

For two-thirds. Yeah. So that's tricky, right? Because if you're trying to, like help a young learner, like figure out how to write two-thirds, you might say, "Ah, that the two goes on top, we call that the numerator, and then the three goes on the bottom." I mean, you might use positional words, to sort of help that youngest learner kind of figure out how to represent - that's social knowledge. We have another podcast on social knowledge if you want to listen to that one. But it's social knowledge about how we are gonna write that fraction. And you might use positional words, but over is a positional word. It's not a mathematical word. It doesn't describe a mathematical relationship. So when you are talking about two thirds, and you say 'two over three', then you're describing a position, not a mathematical relationship. So Kim, as long as we're admitting that is probably the hardest one for me as well. Not honestly so much with fractions, but with rational expressions. So if I'm talking like the parent function for rational functions is one over x. Oh, I mean, one divided by x. Like that is so difficult because of tradition, like I've just heard it that way so often, that yeah, that one has been a difficult one for me to switch. So what can we say? So instead of saying two over three, how can we name that fraction? 

Kim Montague  12:47

Yeah, so I definitely like 2 divided by 3 better, because it's mathematical.

Pam Harris  12:52

 Two one-thirds is a way of talking about it. When I'm talking about one over x, I can talk about one divided by x. I can talk about the ratio of one to x. I can talk about the rational function of one divided by x. So those are going to be better ways that then accurately reflect the mathematical relationships that are happening. But again, we recognize, you hear Kim and me talking about how 'over' is hard for us. 'Reduce' might be the one that's hard for you. I think that there might be some of these sort of non-mathy terms that kind of get in the way that we're used to. They've been tradition, we've sort of heard them a lot. It might take you a hot minute to sort of take them out of your practice. But we would just encourage you to work on that. We would encourage you to work on making your mathematical vocabulary as precise as possible. That will help your students understand. It will help them not - I'm about to do lots of negatives - it will help them not misunderstand as much. Hey, and I've got one for you Kim. So here's one that I didn't really understand when I first sort of ran into it in the state of Texas. I would hear teachers get really picky about the word 'and' when they were talking about numbers. And I was like, "Why are you guys being so picky?" Can you help me out with that? Like where does 'and' belong and not belong? And what are they getting picky about?

Kim Montague  14:13

Okay, so I feel like this is a conversation from testing situations that happened years and years and years ago. But a lot of teachers will say that you cannot say the word 'and' in a written form or when you're saying a larger number. Like, for instance, you cannot say 'one-hundred-and-seven'. It would be 'one-hundred-seven'. That it's mathematically incorrect to say 'one-hundred-and-seven'.

Pam Harris  14:40

So mathematically incorrect. I mean, maybe. But the reason - I think I agree with you - the reason teachers are making this big deal out of what is not a mathematically important thing, this word 'and', is because like you said it was on a high stakes test. So in the state of Texas, when TAAS came out, kids had to write out numbers in words. And if they put the word 'and' in 107, or 356, if they said 'three-hundred-and-fifty-six', then they were marked wrong. And so it became kind of this hyper vigilance of teachers to don't put 'and' in the wrong place. Well ya'll, that was a stupid test question. Like we shouldn't have asked that. That's not important mathematically. To make a big deal out of it is sort of nonsensical. Where should we say 'and '? I think it's what, three and two tenths. So 3.2 is three and two tenths. That's where we can say 'and' like, technically, and so teachers won't mark it off. If I say 'three and two tenths', but they'll mark it off if I say' three-hundred-and-two'. That's not important. So let's not make a big deal about using the word 'and' within numbers, especially big numbers. Not a big deal. Let's focus on the mathematical relationships, not on picky things that were once tested long, long time ago. Alright, you guys, today, we tried to map out just a few of our favorites 'This, Not That"' kind of things in mathematical vocabulary and terminology. We'd like to share some more in the future. And we've got more to share. But we're also interested in some of yours. So if you guys want to send us your faves 'This, Not That' then that'd be great. We'd love to hear from you. And we'll add them to the list. Awesome.

Kim Montague  16:23

Don't forget to join us on MathStratChat on your favorite social media on Wednesday evenings. And if you haven't yet, go watch the Problem Strings from the website where you'll see us work with students to pull out their thinking and reasoning. 

Pam Harris  16:35

So mathisfigureoutable.com/ps for Problem Strings and you can watch us work with real kids, pull out their thinking and reasoning. Lots of fun. So if you are interested to learn more math, and you want to help students develop as mathematicians then the Math is Figure-Out-Able Podcast is for you. Because math is Figure-Out-Able!