Pam Harris  00:02

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

Kim Montague  00:09

And I'm Kim. 

Pam Harris  00:10

And we're here to suggest that mathematizing is not about mimicking, or rote memorizing. But it's about thinking and reasoning; about creating and using mental, mathematical relationships; empowering teachers and students. We answer the question, if not algorithms, then what?

Kim Montague  00:30

In today's episode, we are going to gear the conversation towards teachers and parents of younger learners, and talk all about counting. So we're going to call this episode 'The Language of Counting'. 

Pam Harris  00:42

And you might find it interesting that the first thing we're going to talk about is language. If we're talking about math and counting, we're going to talk about language. But we are because our language influences how kids learn to count and learn the counting sequence. In a huge way, the names, the words that we have given to numbers can either sort of clear up or make more clear the counting sequence, or it can actually sort of muddy the counting sequence. What do I mean by that? Well, for example, when kids learn to count you know, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, then 11...11, where did that name come from, 11? Because really, if you look at the way we write 11. 11 means a 10 and a 1, right? There's that one 10 and that one 1. And you might be like, "Well, Pam, of course that's what 11 means." Well, interestingly enough, in the language of English, we have made up a word, 'eleven'. In several languages, that's a new word when we get to 11. However, in many Asian languages, the word for 11 isn't a new word at all. It isn't a made up word. It's actually the word ten. And the word one put together. So if I was speaking many Asian languages, and I start to count, there's a tone or word for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And that has its own tone. But then when I get to 11, I actually say ten-one. And when I get to 12, actually say ten-two. Those are the words the names for those sort of teen numbers. And I'm actually not even in the teens yet. When I get to 13 it's not well, thirteen. That actually sounds kind of like it right? It almost sounds like three-teen. We almost kind of then follow the pattern. You're laughing at me, three-teen. Well, four-teen? I mean, it kinda like that makes sense in the pattern, right? In fact, we noticed that when kids get to 14, they're kind of confident again, four-teen, but then it doesn't sound like it should again, fifteen. Shouldn't be five-teen? It should, right, if it makes sense? You're laughing, you're like five-teen. Well, six-teen? Seven-teen? Those sound kind of like they should, right? And even if we were to help the students a little bit better, it would even be better if it was ten-one, ten-two, or even teen-one, teen-two, teen-three, but we could actually name the numbers, something that they mean. And my point is that there are languages out there that instead of sort of having these new words or words that kind of sound like what the numbers mean, they actually are what the number means: ten-one, ten-two, ten-three, ten-four, ten-five. ten-six, ten-seven, ten-eight, ten-nine. And so that can be helpful to kids when they're learning how to count. So in our language, since we've kind of made up these words, for whatever reason, we have to tell kids or at least I'm going to suggest to help kids learn to count, let's be clear with them what 11 means. Let's be clear with them what 12 means, what 13 means. That will help them feel the pattern. If they've already got the pattern of 1, 2, 3, 4, 5 down, then when we say, "Well, that's just 10 and 1, 10 and 2, 10 and 3, 10 and 4." Then they're like, "Oh yeah, well, I could do that pattern." "Oh, and we call it 11. So ten-one, this number means 10 and 1, but we call it eleven. This number right here, this means 10 and 4, but we call it fourteen. This number here, it means 10 and 7, and we call it seventeen. Oh, look, there's the 7 right in there seven-teen." So one of the things that we're gonna suggest in today's podcast is to help students learn to count, help them understand what the numbers actually mean, not just what they're called. So that number means 10 and 3, and we call it thirteen. 

Kim Montague  04:18

Yeah, the first time I ever heard you say this, it was mind blowing, because I didn't teach Kinder and first grade. So you know, for me, it was more about these numbers just are named with their names. And I remember hearing you kind of give this spiel about teen numbers, and it just rocked my world a little bit. And at the time, I had a really young learner -

Pam Harris  04:39

Tiny, tiny Cooper.

Kim Montague  04:40

Tiny Cooper and he was starting to sing song count, right? He had heard the sequence of numbers. And I obsessively came behind him when he was counting. And I would say, "Hey, let's count numbers together." Let's count the things and when he got past 10, and he would say "11?" I would whisper behind him, "ten and one." And he would say, "12." And I would say, "ten and two." And he would say, "13." And I would say, "ten and three." And I felt like it was my job at the time, to provide some meaning to what that word he was saying meant, and I was - 

Pam Harris  05:15

Brilliant! Totally you were telling me about that, and I was like, yes! That's awesome. It's amazing, because now Cooper has kind of heard the song of counting. He's able to say the next word for the number, but you're providing meaning kind of on top of that. And it just solidifies for him that he's got the counting sequence, right, and what those words mean, what the numbers actually mean as he gets into the teens. That was brilliant. When you told me about that, I was like, Yes! We need to get all parents and teachers of younger students to do that kind of thing. So well done. Well, then in our number system as kids are counting then they get to the decades or the multiples of 10. Right? So 16, 17, 18, 19... 20. What does 20 mean? 

Kim Montague  05:59

Two tens!

Pam Harris  06:00

Well, yeah, they're exactly. And if you go look at those Asian languages, the word for 20 is two-ten. And literally, it's what it means. 20 is two tens. And you might be like, "Well, yeah, okay, that's what 20 means?" Well, so, sometimes I joke with teachers, "So really, we should have called 20, 'tooty'." Right? You're like, "What?" Tooty? Threety? Right? 30 should be Threety? And you're laughing you're like, "Really tooty, threety?" Well, fourty? I mean, fourty sounds like it should right? Then shouldn't it be fivedy? I know that sounds goofy. But then it would at least sort of follow the pattern. And as you laugh fivedy? Well, sixty, seventy, eighty. Why is it that we've had we have some of these multiples of 10 - these decades - why do some of them sound more like they should and others don't? I don't know. I'm not a linguists. I'm sure that there's some fine history about why the words sound the way they do. But because it's just funky history that turned it into that we can help bring meaning to it. So as kids are talking about 20, we can say, "Hey, that actually means two 10s." We can kind of do what Kim did and kind of come behind kids, kind of over their shoulder and kind of say, "Oh, yeah, 20 that's like two 10s." And so what does 21 mean? Well, it's like two 10s and 1. What does 22 mean? It's like two-ten-two. You're laughing. You're like, "Really Pam?" Like y'all in Asian languages, how do you say 25? You say two-ten-five. That's the way they name the numbers. That gives them a real advantage as they are learning the counting sequence. They don't get near as confused. All teachers of younger kids and many parents will be able to say, yeah, one of the places kids get stuck is as they're counting higher, right? We know we want kids to count to 100. That's a landmark we want kids to get to. Kids will do things like 36, 37, 38, 39.... And then they kind of look at you with this, like, "What is next? I don't know." You know, and the parents like, "Come on, you can do it, you can do it, you could do it. 39... " And then often we'll either supply it, or the kids will guess or whatever. Instead, we could say to them, "Well, what comes after three-ten-nine?" If you've got 3 10s and 9, what should be next? Oh, that's obvious. It's four 10s. Because they already have the 3, 4 sequence, they're going to have the three-ten, four-ten sequence. What they don't have is sort of these memorized kind of made up words that we made up, fourty. I mean, at least 'fourty' sort of sounds like it. Maybe I should have chosen fourty-nine. Because then the kid might say 'fivety' right? Because then it makes sense to follow the pattern. So when they say 'fivety' parents, teachers don't say to them, "No, wrong. No, it's fifty." No, in that case, you go, "Yeah, nicely done! And we call that fifty. That five-ten number. Yeah, you guessed it, right. It's like fivety, good job. You're better than our linguists. But we call that fifty. But you're right, it's five-ten. It's fivety." So we want to help kids understand the meaning of the numbers, and what we call them. I'm not suggesting that your kids going to go to school and as when the teacher says to them, "Hey, I want you to count." And the kid goes, literally gives just the the meaning of the numbers. Of course, I'm okay that the kid knows -  the student knows what the names we have given them in English are. I want... Am I saying that well, Kim? Help me out here. 

Kim Montague  09:11

Yeah, I feel like you know, to give kids meaning for the numbers first, and then supplying the name that we call it afterwards is a little bit better and easier, a lot better and easier than supplying the kind of fake name that we give it, the social name that we've decided it is and then try to come back with the meaning afterwards. 

Pam Harris  09:31

And so at least let's bring meaning. Let's make sure that the meanings there. Maybe you're teaching them first, then let's do the meaning first, and then tack on that social term. But if your kids already heard it, whatever, then at least bring the meaning into the situation. Cool. 

Kim Montague  09:45

You said 'advantage' earlier. And I feel like I've heard you say something about advantages for solving multi-digit addition problems as well. 

Pam Harris  09:55

Yeah, absolutely. So one of the first strategies that we expect students to develop as they are adding multi-digit numbers, is what we call Split by Place Value. So let's talk about that for just a second. If I have a problem like 27 plus 38. And if we haven't forced the traditional algorithm on kids, like we're recommending. Do not force the traditional algorithm on students. If we haven't, then students will think about 27 and 38. And they'll think, "Okay, how do I add those together?" And as they're thinking about it, typically, the first strategy kids will develop is they'll split those numbers by place value. So 27 and 38, they'll think about 20 and 7, and 30 and 8. And they'll bring that 20 and 30, together to make 50. And they'll bring the 7 and the 8 together to make 15. And then they'll think about the 50 and the 15. And they'll add those together to get 65. So they sort of split the 27 into its place value parts: 20 and 7. And they split 38 into its place value parts: 30 and 8. They pull the place values together: 20 and 30, 7 and 8. And then they pull it all together in the end. We call that Splitting by Place Value. Research has shown that most kids will develop that kind of on their own, if we haven't forced the traditional algorithm. It's a great strategy. But interestingly enough, can you imagine if we were actually calling those numbers, what they mean? If we were an Asian language, and 27 was two-ten-seven, and then we were going to add three-ten-eight. So just think about that problem for a second two-ten-seven plus three-ten-eight. It almost begs splitting by place value: two-ten-seven plus three-ten-eight. I'm thinking about two 10s and three 10s. Well, that's just five 10s. And then you can add the 7 and 8 together. Those students almost can't help themselves, just thinking about how many 10s there are, and then adding the 1s together and then pulling them together. It's a very natural strategy for students to use, especially if the language kind of helps them understand what the numbers actually mean. So you know, sometimes I look at kind of the test results of the USA versus some of the Asian language countries. And of course, those younger students are doing better earlier because their language is naturally helping them sort of develop that strategy. Well, we can do the same thing. We can help students understand what the words, actually, excuse me, what the numbers actually mean, what the words translate into, so that they're actually thinking about 27 as two 10s, and 7, and they're thinking about 38 as three 10s, and 8. And so that Splitting by Place Value strategy will become all the more natural for our students as well, because they're understanding the numbers. 

Kim Montague  12:25

So all this talk about counting, I want to take it back to Kinder for just a second. So it reminds me of a question in our Journey Facebook group from our friend Holly, and she was working with Kinder teachers and students. And I think her question was something like: should we always use the same finger patterns when talking about a specific number? Like when we show the number 3, should it always be the 3 middle fingers with your thumb and pinky down? And I think you answered with something specific. 

Pam Harris  12:53

Yeah, yeah, I definitely have an opinion on this. So if you think about showing the number three, often in the States, I see what you just said, where you put the thumb and the pinky down, and the three middle fingers are sort of up. And that's a 3. I can't tell you the number of kids that I've messed around, messed around with, can I say that? I can't tell you the number of kids that I've sort of experimented with when I'm in the grocery store, and they're in the cart in front of me. Or I'm at church and the kids are hanging around or wherever I'm around, my own personal kids. I'm at schools, where I'll show to a young learner, I'll show them 3. But I'll do it in a non-traditional way. So maybe I'll put my thumb up, and my first two fingers, and my fourth and fifth finger are down. And I'll say, "How many?" And they'll look at me like, "I don't know." Or they'll have to count them. When they can sort of see the kind of more traditional, where the first three fingers or the middle three fingers are up where they can just instantly say three. And so then I'll put just my thumb and my pinky and one of the other fingers up. And I'll say, "How many?" And then the kids are like a little stymied. And the parents kind of look at me, like, "Why are you doing it wrong?" Ya'll, we want to have different iterations for fingers. We want to have kids to be able to recognize the three-ness of a number that they can see three, um, in any way. Like, I might want to put my thumb up on my left hand and 2 random fingers on my right hand and still have that be 3. It's not about kids memorizing a specific way your fingers are up. In fact, I think the resource that the question was being asked about was suggesting that you should always have your fingers up in a certain way so that it kind of represents an open number line. That if you're going to represent 3, then it should be your thumb and first 2 fingers, because those are kind of together on the number line. And if you're gonna do 4, then it would be your thumb and your first 3 fingers and your pinky would be down. And then that way you're kind of always representing it as sort of as it would look on a number line. I mean, sometimes that's okay, but I don't want to get kids locked into that either. I don't want it to be a rote memory about what your fingers look like. I want kids thinking about three-ness. What does it mean to have 3? Or what does it mean to have 2 fingers? I can have 2 fingers, one on each hand. And I still want kids to know that that's 2. They don't have to be next to each other because they're next to each other on the number line. I think that's artificial, if they - Go ahead. 

Kim Montague  15:10

Sorry, I was gonna say it's not a number, question or thought, but it reminds me of we show a picture of a triangle. And kids think that one specific look of a triangle is a triangle and the other ones are not triangles. So giving them a variety of looks as well to help broaden and deepen their understanding. 

Pam Harris  15:28

Yeah, totally. We give them that equilateral green triangle, any teacher out there is going to recognize the green pattern block that "all sides are equal" triangle. And then when we show kids a right triangle where it has a right angle in it, kids will call it a wrong triangle. Because it doesn't look like that sort of stereotypical, yeah, equilateral where all the sides are equal triangles. So yeah, we want to show them right triangles and isosceles triangles and scalene triangles. We want to have all sorts of different examples of triangles to be "triangle" because they are. We don't want to have just only one look. An excellent connection, Kim. So we also have a task that we like to do with students that we call, Finger Flash, where we'll say to students, "How many? How many do you see?" And we'll flash a number of fingers at them. And when we do that, we want students to, like literally count the number of fingers or recognize the number of fingers that they can subitize the number that they're seeing. And so when we do that Finger Flash, again, we're not giving the same look for that number every time. One of the interesting places that this has actually come into question, or we've had an interesting conversation about, are my ASL friends. So I have several friends whose kids are deaf. I have a friend whose husband's deaf. And particularly the friend whose husband's deaf kind of got after me a little bit. She's like, "No, no, no, you must show them the ASL version of the numbers." I'm gonna disagree with that a little bit. I think when they're tiny - and I get it, I get it. She wants her kids to recognize the ASL symbols. And I think that's really important. I think in order for them to speak to their dad, it's hugely important that they can communicate with their father who's deaf, that they're learning ASL as young kids, and I love it. I love the fact that they're teaching their kids and it's a great family. But for number, I need kids to recognize various iterations of three-ness. So she and I came to an agreement that when we do fingers, we're going to do the ASL sign. But when we do any other way that the kids are going to recognize 3, we're going to have all sorts of different variations and permutations of the way 3 could look. So it's kind of funny that it kind of came up in the American Sign Language venue. So by the way, when kids are recognizing the numbers, and we sort of flash a finger at them, and we ask them the number, if kids then count, then that's an example of kids needing to count to find the total. If they don't say the end number, that means they don't quite have cardinality. 

Kim Montague  17:56

Yeah, let me explain that for just a second. 

Pam Harris  17:58

Thank you. Take it. Go. 

Kim Montague  18:00

Cardinality is a landmark, right? It's something that we want young learners to develop. And it just means that knowing the last number in the count represents the set. So if they don't own cardinality, they think "how many" means to sing that song of counting. So they count, and they say each of the counting words. And so we often test really young learners, and we'll say, "How many?" And if they go, "1, 2, 3, 4, 5, 6." And you respond with, "So how many?" If they recount and say, "1, 2, 3, 4, 5, 6." They don't have cardinality. But if they say, "1, 2, 3, 4, 5, 6." And you say, "How many?" And they say, "6." They have it. And you help them, right? So if they're singing the song, and you ask the question, "how many?" and they don't, you can supply, "Oh, so you have 6." 

Pam Harris  18:50

Yeah, that's totally a way that kids get that when I asked them "how many", their job isn't to sing the song of counting. Your job is to tell me how many. Like to give me that last counting word that represents the total number of the set. Exactly. Yeah. So my kids laugh at me. Because often in the grocery store, when there's a kid in front of me - though, honestly, with the mask's, it's a little harder. When there's a kid in front of me in the grocery cart, I'll just, you know, throw hold up three fingers. And I'll say, "How many?" And when the kid goes, "1, 2, 3." Then I'll say, "So how many?" And the parent looks at me like, "What are you?" And I'm like, "Don't worry, I'm teaching your kid carnality." The kid goes, "1, 2, 3." I'm like, "So how many?" And the kid goes, "1, 2, 3." You know, like the second or third time then I just go, "Oh, so there's 3." And then I'll kind of wink at the parent go, "Your kid's learning cardinality." And then we sort of move on. So that's totally a way that you can help kids learn the last number in the counting sequence represents the total and the count. So y'all, language is helpful and necessary, and it can get in the way of kids learning to count. So if we're purposeful in our use of language, we can help students develop important ideas in counting that go beyond just singing the song and the counting word. So if you're 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!