Pam  0:02  
Hey, fellow math-ers! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam Harris, a former mimicker turned math-er.

Kim  0:11  
And I'm Kim Montague, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching. 

Pam  0:20  
We know that algorithms are super cool achievements, but they are not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop. 

Kim  0:32  
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:40  
We invite you to join us to make math more figure-out-able. Kim!

Kim  0:45  
Yes, hi.

Pam  0:46  
Oh, my gosh. Can I tell you about a recent experience that I just had? You and I have chatted about a little bit, but I want to dive into it today. I have been thinking about this. I keep thinking about it. Okay, so I was recently... had the opportunity, wonderful opportunity, to go to South Korea, again. I've been there once before. This is now time two. Both times I lost my suitcase. Really, what is that?! It's fine. They ended up finding it in time both times. But anyway. Wonderful Chadwick International is an independent school there near Seoul, South Korea. And one of the things that they wanted me to do was some lab lessons. So, we did some pre-conferencing about what was going to happen, do the math and make sure everybody's ready, then go in. I did Pre-K through grade eight. Oh, look at me. See, I've been in South Korea. Eighth grade. I'll probably do that because I'm thinking about South Korea. So, each one of those, so nine different lab lessons. And then we would do a post thing to kind of talk about what happened. And like I call it a post mortem. You know, I sort of discuss all the things and everything. Okay, so let's just chat about one of them. In our last episode, we did the candy bar string, where I said something like if three-tenths of a candy bar weighs 12 ounces, what does one-tenth weigh? Right?

Kim  2:07  
Mmhm.

Pam  2:07  
Okay, so I did that particular Problem String with... I think it was grade five. Pretty sure it was grade five. And then the next day I was supposed to go in. So, one of the other things that I did were some full school PD sessions. And also, I went in and just observed some teachers, and then kind of gave some feedback. In the fifth grade lesson, Sam had seen me do this Problem String, and he was like, "Oh! Hey, when you come into my room, you can watch me a little bit, but can you do that Problem String with my students in his grade four classroom? And I was like, "Are you kind of trying to get out of me watching you?" You know, we joked around. He's a great guy. Wonderful, super guy. I really enjoyed watching him teach and interacting with him. So, we joked around a little bit. And I was like, "Yeah, I'm happy to do that." Kind of funny. So, I just said, If three-tenths of the candy bar weighs 40 ounces. Sorry. No, that's the whole candy bar. I just gave it away. If you haven't watched that Problem String or... Watched? If you haven't listened to that that episode last week, you might go check it out. Sorry, the problem is if three-tenths of the candy bar weighs, help me, 12 ounces. Right, okay. I'm trying to think here. I'm going to write that down. If three-tenths of the candy bar was 12 ounces. Well, when I was doing it with the fifth grade classroom, I hadn't thought about the fact that they don't deal in ounces. So, half the kids are Korean nationals and half the kids are not. They're expats, embassy kids, teacher kids, people that work for high-tech companies over there. And so, for sure, the Korean nationals. But all the kids, while they're living there, everything's in metric. And so, I just... I had thought about a lot of other cultural things, and I missed that one. So, I was a little chagrin standing in front of those kids. I'm like, "Okay, if three-tenths of a candy bar was 12..." Oh, crud. I'm like, "Kilograms?" And I looked around. Kim, how are your kilograms? Can you make sort of sense of...?

Kim  3:21  
Yeah.

Pam  3:44  
I pointed at a kid, and I was like, "Hey, how much do you weigh?" And then I thought, "Can I ask that?" I mean, they're grade five. Can you ask a grade five? I said, "Never mind. Don't tell me how much you weigh. How much does like a normal, like average fifth grade..." Normal? See, did I say that? I don't even know. Like, an average fifth grader? I'm trying to get a feel for kilograms because I know I weigh a whole lot less kilograms than I do pounds. And I'm trying to think what does a candy bar weigh? But, Kim, funny enough. I'm thinking if three-tenths of a candy bar was 12 ounces, but I was thinking about the whole candy bar. And I'm thinking, can a whole candy bar weigh 12 kilograms? That seems heavy to me. Does that seem like a big candy bar? The kids are all laughing. They're like, "Whoa! that's a really big candy bar!" But we were all thinking about the whole candy bar. Kim, if three-tenths of the candy bar weighs 12 kilograms, that means the whole candy bar weighs a whole lot more. Anyway, we're totally joking. It was great. The kids were so... I don't know. They were just in it. We're all trying to figure out kilograms and stuff. And like a dork, I said, "I don't think you can weigh 12 grams." Well, it totally could. In fact, after that string, when I did the first time in the fifth grade, somebody handed me a granola bar. Are you ready? Oh, so, again, go listen to the Problem String. But if three-tenths weighs 12 ounces, the whole candy bar weighs 40 ounces. So, they handed me a granola bar that, are you ready, weighed 40 grams. So, it totally could have been 12 grams, and it would have like made sense that the whole candy bar could have weighed 40 grams. Anyway, all the things. 

Kim  5:34  
What did you do at that point?

Pam  5:37  
In the class, we did kilograms like a dork because I just...

Kim  5:40  
Oh.

Pam  5:41  
I know. It's that moment in... Well, in fifth grade. In fifth grade, we did kilograms, because I was like, "Okay, everybody, picture the biggest candy bar you can," and then we went with it from there. And it still went great. But in between then, they showed me the granola bar, and I'm doing the... What do you call that when you put your hand to your forehead? You're like. You know like? 

Kim  6:00  
Yeah.

Pam  6:01  
Yeah, that thing. Like, "Duh, duh, duh." So, when I did it with grade four, I did do grams, and it worked. And we didn't even joke about it. Which almost kind of lost some of its fun a little bit. Anyway, it was great. But so I go in... Wow, I'm telling the story crazy. I go in to Sam's classroom to do this lesson that he's kind of finagled me into doing. Way to go, Sam. And he's chatting with his kids, so of course, I'm going to listen a little bit. He can already tell his great rapport with his students. And he says... As I walk in, he says, "Hey, am I right that you guys say the denominator first, but you also write it first? And I'm like doing that zoink thing where I'm like, "Say what?" And he goes, "You know, when you're talking in Korean..." Because half the kids are Korean nationals, so they're totally bilingual. In fact, it's amazing, Kim, I was talking these Pre-K kids who spoke no English in November, and they were full sentences. It was amazing. Anyway. So, he says, "Am I right? That, you know, when you guys say like a fraction," and he wrote four-fifths on the board. I said, "So, when you guys say a fraction like four-fifths, you say the denominator." And then he said the word for denominator in Korean. He said, "You say that first, and then you say the 4, the numerator," and he said it in Korean. And all those kids are like, "Yeah!" Like, not even like, "Yeah." They're like, "Yeah!" Like, of course. Like, normal. I'm like, my jaws on the floor. Now, I'd kind of heard a little bit that they said the numerator and the denominator in the order we don't right? Four-fifths, we say the 4 first, and then we say the 5. 4 for the fifths. Anyway, four-fifths. And we often say four 1/5s. I kind of knew there was this thing where maybe they would say 5, 4. But the way he explained it, it's kind of like we've cut the whole into 5. You're going to need 4 of them. It's kind of shorter than that, but that's the meaning that comes across in Korea. That's interesting. 

Kim  8:08  
Yeah, it is.

Pam  8:09  
Like, but what I hadn't realized is that not only do they say that, "I've cut the whole into 5, and I need 4 of them." Or "The 5's cut into whole." "I have fifths, and I need 4 of them. I have fifths, need 4." It's kind of like that shortened. Not only do they say it that way, Kim, they write it that way. 

Kim  8:30  
Yeah. 

Pam  8:30  
So, on the board, he goes, so like, "For three-fourths, you would say, I've got a thing cut into 4." He writes the 4. He draws the line. And then on top, on the numerator, he writes the 3, and I need 3 of them." And the kids are like, "Yeah!" And I'm like, "What?!" Okay, mind blown. If you could see my hands right now, I'm doing that thing above my head. You're like, boom. Ah. And, Kim, what are the ramifications for the fact that we say the numerator first? How do we think about fractions?

Kim  9:06  
Yeah, I have so many questions, honestly. Can I ask my question first before we do that? 

Pam  9:11  
Yeah, yeah, yeah. Totally. 

Kim  9:12  
So, Sam was talking to his own students and asking them how they do it. Did he mean like... Did he not... Like, had he not talked to fractions with them yet? Has he... Like, how did he not? Was he asking what they do in Korean, but that's not what he's teaching?

Pam  9:30  
Yes.

Kim  9:31  
Okay.

Pam  9:31  
Yeah, so the language of instruction in the whole school is English. 

Kim  9:35  
Ah, okay, okay. 

Pam  9:36  
Yeah, so everybody speaks English in the whole school, and the Korean nationals have been going to that school since Pre-K. 

Kim  9:44  
I see. Okay.

Pam  9:45  
Once they start, they hang. They don't go in and out. But they also... This is just a Korean thing. They also go to these after school academies. Some of them go before school, and so they spend a lot of time in school. Some of the teachers were telling me last time I was there that the Korean government had to institute a new law a few years ago that those academies had to close by 10:30 p.m. Like, it was illegal for kids to go longer. So, these kids go to a lot of school. 

Kim  10:13  
Yeah.

Pam  10:14  
And one of the things that I was doing, a little aside, at the school is they said can you help us differentiate? Because we have these kids who go to these academies have a lot of experience. The academies are kind of rote-memory oriented. They're kind of algorithm. Kind of? They are algorithm focused. But it still gives kids a lot of experiences, more than say the sort of average expats kids or the teacher kids who, you know, are playing sports after school, or, you know, kind of doing regular kid things after school.

Kim  10:44  
Right.

Pam  10:44  
And so, because of that, you know, you might consider that what their challenge to differentiate is even more of a challenge with their students because they have so much experience versus kids who have just less experience with a topic, or, you know, kind of math. Anyway...

Kim  11:03  
So, okay. Yeah, so that clears it up for me because, you know, he's their teacher, and so I wondered why he was asking. So, thank you for that.

Pam  11:10  
What's he been doing?

Kim  11:11  
Well, no. I mean, yeah. I figured it was that he was using instruction kind of more similar to what we would say. So, we would say...

Pam  11:19  
Ooh, can I say one more quick thing? So, as I was doing the fifth grade. Remember, I did that the day before. The fifth grade lesson. The kids, every once in a while, when they were going to say three-fourths, they would say 4, 3. I mean three-fourths. Or when they were going to say four-fifths, they would say 5, 4. I mean four-fifths. And they would kind of catch themselves quick enough that I didn't, you know. And I knew this kind of thing about how they might say it backwards, and so it didn't bother me, and we just kind of moved on and everything. But I had no idea that they wrote it first. Anyway, sorry. I probably didn't need to say that again, but carry on. 

Kim  11:51  
Well, it's funny that you say they say it back, or they wrote it backwards. So, backwards for you. But they were trying to say it backwards to them. So, which is a huge indicator that what we're saying, the way that we're describing it. Like, you were talking about the same values. You were talking about a number of parts, and how many parts of the parts. This is a very part-whole way of thinking about a fraction. You're saying four 1/5s, right, because you're talking about the candy bar. But they were still thinking about... They were thinking about 5 parts. I have 4.

Pam  12:27  
Yeah, yeah.

Kim  12:28  
So, that's interesting to me because they're thinking about the whole amount, and then what is my share? Like, they're thinking what's the total first. But when we say three-fifths or four-fifths, or whatever fraction you used, we start by saying 4. Which, if you stop me at that point, I'm thinking 4 whole. Like, 4. And then I say, "Oh, wait. It's a portion of a whole. Four-fifths. Or four 1/5s." It's almost like they already have the... Yeah, they already have that it's a part of something. Or it's like...

Pam  13:04  
From the get-go.

Kim  13:05  
...the whole. Yeah.

Pam  13:06  
Yeah, yeah. Yeah, and you could argue that if you know you're thinking about fractions, and someone says 4 that you might not go to 4 whole. But you do kind of think about 4 pieces.

Kim  13:17  
Right.

Pam  13:17  
In that way, it's the unit. And then you have to picture the whole candy bar. It cut up into... Right? Yeah, and so neither is right or wrong. Well, I think the thing that I'm so fascinated about is how the social way that each language, each culture has decided, determined that they are going to... What's the word when you say that socially we've determined something? By convention.

Kim  13:45  
By convention.

Pam  13:46  
By convention, they've decided that we are going to name the numerator first, and therefore sort of thinking about the number of pieces that we've got first out of then the total number of pieces that we've cut the whole into. Versus they're going to think about how many they've cut the whole into. They're really focusing on the whole, and how many pieces they've equally divided into, and then how many of them they need. That social convention is impacting the logical, mathematical way that you're making sense of fractions. That's noteworthy. How many times, Kim, have we run into somebody that they'll say, "Well, that's the way it is. That's the math." And you're like, "I mean, how are you thinking about it? Like, the convention is influencing the way you're thinking about it." And even though those kids very quickly were like, you know, 5, 4. I mean, four-fifths, they're thinking, I've got 5 pieces cut out of the whole, and I need 4 of them. They're thinking that, and then they're translating it into English to say it the right way in English. They're not thinking about it, you know at least if they're saying it that way, they're not thinking about it the way we are. That's interesting. Yeah.

Kim  14:54  
It's so funny that you say that, you know, the "right way" we say in English because you and I...

Pam  14:58  
Did I say "right"? Did I say "right way"?

Kim  14:58  
Maybe you didn't. Maybe you say the way that we say in English. The funny thing... But that's making me think because, you know, growing up, I would have said three-fourths. Like, that was the way to say it. But you and I also say three 1/4s.

Pam  15:16  
Yeah. Now, we do.

Kim  15:17  
And people kind of go, "Say what?" And kind of like raise an eyebrow sometimes. And it's not that we say one over the other. We say them both. We say both three-fourths and three 1/4s because three one fourths is a way to help students understand that three fourths means three 1/4 pieces. So.

Pam  15:39  
A fourth, and a fourth, and fourth.

Kim  15:40  
Yeah, it's like that nudge is helpful.

Pam  15:44  
Yeah, yeah. So, then I took a deep dive. A deep dive? I took a shallow dive and into Google, and I just kind of like. I thought what about quarters? Like, what about other languages? Are there other languages? And for whatever reason, I chose the fraction three-fourths. And so I just said, "Hey, what are other languages doing with three-fourths?" And I was fascinated to find that a lot of languages will say three-fourths parts or three 1/4s. So, like three quarters in English or tres cuatros in Spanish. Forgive my accent there. Hindi has a way to kind of say. A lot of languages say that. And then I realized, Kim, I wonder if quarters was actually not a good denominator to look up. And sure enough, it's like specific for quarters. A lot of languages have a word like we do. So, like we have three-fourths, right? One-fourths, three 1/4s. But we also have this quarters thing. A lot of languages has this quarters thing. Fifths. Do we have a word for fifths besides fifths? Not so much. So, then I looked up fifths. Oh, then it all switches for fifths. There's really only a couple of languages that then say... And noteworthy, Korean, Chinese, say 5 parts 3. So, for three-fifths, the way they say three-fifths is 5 parts 3. I've got 5. The whole is cut into 5 parts. I need three of them. Japanese says 3 of 5 parts. Everybody else is more like a part whole. 3 out of 5 parts. Or three-fifths. That's interesting that both Korea and Chinese have this... Or Korean and China. Can I say that? Korea and China or Korean and Chinese have this kind of different way of saying it. But everybody else kind of has this part-whole way of saying fractions, and Korean and Chinese have this whole-part way of saying fractions. But almost nobody has the helpful way of saying four 1/5s. Or three 1/5s.

Kim  17:51  
Yeah. 

Pam  17:51  
But we can pull that in to help with the meaning.

Kim  17:54  
I like how you just said that, that it's like part-whole, but Korean and Chinese say whole-part.

Pam  18:00  
Yeah, yeah. 

Kim  18:01  
Because they both are attacking that particular meaning of fractions, that part-whole relationship, but in a slightly different way. But what's also really interesting is that what's changing is just the social part of it. And we're still talking about the same mathematics. And so, you know, like, we, we say three 1/4s. This also happens when we talk about decimals. Like, people get really like maybe a little concerned that we say all the different. We say point. We say money versions. We say tenths and hundredths because they're real, right? That's like real life. We say 3 point 2 billion.

Pam  18:43  
No one says 3 and 2/10 billion. Like, no one says that.

Kim  18:47  
Yeah.

Pam  18:47  
Yeah, and so people sometimes will harp on us like, "You shouldn't say 3 point 2." And we're like, "Well, how are you going to talk about our national debt?" Really? 

Kim  18:54  
Yeah. 

Pam  18:54  
So, yeah. We like to say all three of them, so that they... I'm sorry, I'm finishing your sentence. 

Kim  19:00  
No, it's okay. I was just to say, we say all those because it's real life, and because kids are going to encounter that, and we want to help create a situation where the social part of it, the social, the language part of it, not only doesn't get in the way, but it supports their understanding. 

Pam  19:17  
Oh, nicely said.

Kim  19:18  
Yeah.

Pam  19:19  
I like how you said that. Yeah, so social and logical. Understanding the different kinds of... What am I trying to say? Different kinds of ways of...

Kim  19:29  
Mentions?

Pam  19:30  
Knowing things or the few different kinds of knowing... That. It's important. It is helpful to parse those out. And in this case, wow, it was just like screaming at me, and I had such a good time. And thanks for letting me tell you about that. 

Kim  19:44  
Yeah, no problem.

Pam  19:44  
Yeah. Alright, 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 mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able.