Pam Harris  00:01

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

Kim Montague  00:07

And I'm Kim.

Pam Harris  00:08

And we're here because mathematizing is not about mimicking or rote memorizing. It's about creating and using mental relationships, that math class can be less like it has been for so many of us and more like mathematicians working together. We answer the question, if not algorithms, then what?

Kim Montague  00:27

So we just had some time off from school, right? Big holidays, big breaks, time with family. I actually, Pam, got to do some algebra homework with my niece and nephew. 

Pam Harris  00:38

Fun!

Kim Montague  00:38

And so it got us thinking, Pam and I thinking about how we typically spend time talking with the littles in our lives when we see them after it's been a while. And of course, we do math with them. So we decided to call this episode Ten Minute Math.

Pam Harris  00:55

Because if I've got 10 minutes, I'm doing math, right? I mean, when we run into those littles, like you just said, people maybe we haven't seen for a minute, we like to play around and playing around with mathematics is fun. So we decided to title this episode, after a book that we both love that we used several years ago called 'Ten Minute Math'. It's got some great ideas for doing math in short bits of time. Today, we thought we'd talk about some of the things that we do when we have just a few minutes. So last Saturday, I was working with some gals in my church, and we were helping out a family and there were lots of jobs to do. And I got the best job because I got to play with the kids. So I had 4 kids. And I sort of just kind of kept them entertained for a couple of hours. We did some swings and stuff. But in the meanwhile, there were some times we kind of had some downtime. And I would grab that two year old, four year old, five year old, and seven year old. Yeah so they were littles right, and how do you deal with that range? That's a gap, right? I mean, that's quite the gap from two to seven and in between. What do you kind of do with those kids? So we thought we'd just share, like some of the things that we kind of do as we have kids. And not only entertain them, but ourselves, because it's so fascinating to hear how they think. So one of the things I did was I did the Finger Flash game. So I literally would hold up fingers. Like maybe I'd hold up three fingers on one hand, and I would say, "Hey, how many?" And I would kind of have to have some chats with the older kids about not to yell it out when I was letting the tiniest two year old - Oh, gosh, Kim, she was so cute. She totally could count to 10. She kind of messed up between six and eight a little bit. But for the most part, I'd throw up those three fingers and she would go, "1, 2, 3!" It was so exciting. And then I got to say back to her, "So how many?" 

Kim Montague  02:45

Yeah. 

Pam Harris  02:45

And she would count again, "1, 2, 3!" And I would say, "And so how many?" I was totally testing for cardinality. Right? And the four year old that second time that I said it just couldn't keep it anymore. And she would go, "Three, there's three!" And the two year old would look at her like, "Oh, is that what I'm supposed to say?" And I was like, "Yay, there's three!" And then I would flash some more fingers. And when the two year old kind of got a little bit tired of it, which honestly doesn't take all that long. Then I turn to the older kids. And then I started flashing like five, and the seven year old instantly, "Five." And then I would flash six. And interestingly enough, the seven year old had to recount. So five, I could flash and she knew five. But when I flashed six, then she recounted the five to get to six, and then I would say, "So how many?" And she would say, "Six." And so I'm like, Ah, so you can recognize numbers under five and she could. Any any numbers I flashed under  five, she and the five year old, they could both recognize and sort of say what they were. But as soon as I got over five, that was less, and they would kind of count them all. So I started doing a little bit of work with them. So I would instead of just flashing seven, I would flash five, they did five, then I'd flash seven. They'd recount and I would go, "Oh that's interesting. So once you got to five then you just said six, seven Hmm." And then I would maybe do the six after that. I'd flash five then I'd flash six. "Well that's interesting. After you get to five, then you just said six." Y'all then when I did seven again or eight and I would flash the five and then I would flash the eight - pretty soon that five year old and that seven year old started starting with the 5, "5, 6, 7, 8." I was like sweet! Learning is occurring as we're talking. Then I would flash 10. "Oh 10! We know ten!"  Then I would flash nine, those sweet little dears would start over and count them all. And I would go, "That's interesting. You said this was how many?" "10!" "And you just said this was how many?" Oh my gosh, that first time they recounted to nine. And I was like, "Yep, sure enough that's nine. So wait, this was 10 and this is nine? 9, 10 interesting." And then I would work with some other numbers. Let me mention: I flashed 10 again, then I flashed 10 and one more and they had to count and then we got to 11 and - anyway so that was fun. That was one thing to kind of do, kind of have some fun. As soon as that got a little boring for all of them, which again, doesn't take too long, remember, these are short bits of time. We don't want to kill it. We're not trying to bore anybody. We're just trying to add a little bit of fun and then move on. So then I started to do I Have, You Need. So I said, "Hey, guys, if I want 10 fingers," and I hold up both hands, you know, flash all 10 fingers, "if I want 10, and I only have eight," and I would put down two fingers, "if I only have eight." So I'm calling it now. Now I'm calling it eight. They can see my two fingers are down. I'm like, "How many more do I need to have 10?" Well, sure enough, those older kids were just looking at the two fingers I had that down. And they're like, "All you just need are those two." And I was like, sweet. "Well, what if I have five? How many more?" "Oh, you just need five." So we just played I Have, You Need with them kind of looking at how many fingers I have down. Now if I'd had some more time with them, and they had a little more stamina then I might have then put my fingers down and tried it, "Hey, so if I have nine, how many do you need if I have 10?" Notice that I'm using numbers that are close to 10. I don't say things like, "If I have one, how many do you need?" not without fingers first. Because those numbers that are farther from 10, if kids have not conceived of the partner say between seven and three, then if I say," I have seven, how many do you need?" Well, they can count those three, right? They might even be able to not even see my fingers and say, "7, 8, 9, 10." They might be able to do that without seeing my fingers. But y'all if I start with three they've got to count seven to get to 10 that's much harder for them if they haven't kind of conceived already of the partnership for 10. So we played a little I Have, You Need. That worked for a little while. We also did the Alligator song which I'm learning not everybody knows. I learned this one growing up where you're like, it's kind of like the five Monkeys Jumping On The Bed where along comes an alligator quiet as could be and snapped that monkey right out of that tree. So if we started with five monkeys in the tree or five monkeys jumping on the bed, and then you snap that one, then how many are left? How many are left in the tree? Or how many on the bed or whatever? Oh so four little monkeys hanging in the tree teasing Mr. Alligator, "Can't catch me." Along comes Mr. Alligator quiet as can be and snap that monkey right out of the tree. And then, of course, they're all jumping with the snap, you know, and that's exciting. And then they get to sort of you know, sort of count down. Then you might be like, "Pam, if they can count up why are you messing with counting down?" Oh, y'all, it's a whole different animal. We need kids to have practice both counting up and counting back. Counting back is very important for kids to sort of get that progression of number words, forwards and backwards. So that could be a way to do that. Whatever little ditty you've got, whatever song that's kind of fun to kind of count backwards. So one more thing that I did with them. We were on the swings a little bit and we had a little bit more downtime. And so I was like racking my brain and like what else can we do that would be kind of fun? Because it's fun, right? And I wanted to have a lot more fun. I decided to try something I've heard Kim do. I think I was a little less successful than Kim. But it was fun. It was still fine. I kinda wish she was there. Because I really kinda wanted to poke your brain a little bit. You'll have to tell us a bit more about how you do it with your kids. But I'll start off with what I did. So I said, "Hey, how old are you?" And she said, "I'm five." And I said, "Oh, and how old are you?" "I'm seven." I was like, "Cool. Cool. So you're five and you're seven? How old will you be next year?" And the five year old kinda looked at me? And then she's like, "Six!" And I was like, "Yay you're gonna be six. That'd be great. And you're seven. How old will you be next year?" "I'll be eight." "Okay, cool. So if you're five and you're seven, next year, you'll be six and eight. How old would you guys be the next year?" And we did that a little bit, right? That didn't last very long. That wasn't really exciting. I tried to do, "Well so if your four and you're five, then if next year you'll be five. How old will you" - pointing at the five year old. That was a little bit too much. They couldn't kind of think about how far apart they were. 

Kim Montague  08:57

Yeah. 

Pam Harris  08:57

So that we could go on because Kim, I think that's what you've done. Am I right?

Kim Montague  09:00

Yes, that's my favorite. That was my favorite thing to do. So my boys are three years apart almost exactly. They're three years and like 10 days apart. And so we explored that a lot, particularly because one would turn older and then they'd be four years apart for 10 days, supposedly. And so when Cooper was old enough, I can't remember exactly how old he was around this age, maybe a little bit older than the kids you're talking about. But I would say to him, "Hey, Coop. So you're seven. How old is Luke again?" And he would just know and he would say, "Oh, he's 10." And I would say, "So when you're 10 how old will Luke be?"

Pam Harris  09:40

So nice.

Kim Montague  09:41

Right? And so I would ask him all the time, like, "Hey, how far apart are you guys?" And he would say, "We're three years apart." Then I would say, "When you're - oh my gosh, when you're driving and you're 16, how old will Luke be then?" And he for a while would count up 16, 17, 18, 19. And then it was crazy because I asked him a lot. And especially if I was at a benchmark year like 20 or 15, he could just tell me. He just knew three more. And so to introduce the idea, or the game with my older one, who was three years older, and needed to be a little bit more challenging, and I am 30 years older, almost exactly than my oldest son. I had him when I was 30. So, we would do the exact same questions, I would say, "Hey, Luke, you're nine right now. How much older am I?" "30." "Okay, so how old am I right now?" "You're 39." And we would scale up and down, basically a number line asking different questions. When mom's 49 how old will you be? How old will Cooper be? And we played that game a ton, just to get that comparison.

Pam Harris  10:52

So I'm curious, did you ever say, "So how old was I when you were born?"

Kim Montague  10:56

They already knew that. But I did a lot of like, older age stuff. When you're my age, how old will I be? 

Pam Harris  11:02

Nice. 

Kim Montague  11:03

So like not including the numbers, specifically, but what they already knew.

Pam Harris  11:07

Yeah. And so brilliant. Because you're exactly 30 years, and 30 years isn't that hard to handle. Right? Of all the numbers that could feel harder. So I'm curious. Did you ever bring your husband's age into that? 

Kim Montague  11:17

Oh, yeah. Because he's two years older than me. So there's a lot of age conversation happening in our house for quite a while. Now they're older. And so they're not as interested in that particular activity. But I have others that I do with them. In fact, one of them -

Pam Harris  11:37

Tell us!

Kim Montague  11:37

- is  MathStratChat. My younger son -

Pam Harris  11:41

Kim, I gotta tell you, I did not conceive that when we came up with MathStratChat, the number talk for the world, I did not conceive that your kids were the ones that were playing with it. Because how old are your kids now?

Kim Montague  11:53

They are 9 and 12.

Pam Harris  11:54

Yep. Oh, yeah. Three years, right.

Kim Montague  11:58

They are.

Pam Harris  11:59

Hey, they're still three years apart! Oh that's interesting.

Kim Montague  12:04

So the other night, actually, I think it was last week Cooper came running in. And he said, "Mommy, what day is it?" And I said, "Tuesday." It was either Tuesday or Thursday. I don't know. No, must have been Tuesday. And I said, "Hey, don't worry, we haven't missed anything." Because I knew exactly why he was asking. There's only one night that he asked me what day is it? And it is Wednesday night, around bedtime, he knows that MathStratChat is coming out. And he wants to know, is it going to be something he could do? And I said, "Don't worry, we haven't missed it." And he was so disappointed. 

Pam Harris  12:41

Because it wasn't out yet.

Kim Montague  12:44

It was the night before. And I said we've got that to look forward to tomorrow, because he thinks of math as fun and playfulness and exploring. And so I said  - 

Pam Harris  12:54

To be clear. That's because you have made it this fun, exploring, cool thing to think about and to do. I just wanted to give credit for that. 

Kim Montague  13:02

And he's had some great teachers who have done the same. They've both had some really nice experiences. Anyway, so I said to him, "Let me see what else I've got up my sleeve. Sorry about this kid." And so I said, "Oh, have we ever done, Which One Doesn't Belong?" And my other son came wandering up around the same time. And they were like, "I don't know what you're talking about." So I went to Twitter, and I typed in WODB.

Pam Harris  13:26

For Which One Doesn't Belong.

Kim Montague  13:27

And I found lots of Which One Doesn't Belong. And that is another thing that they love now. And what's amazing about both of those routines, MathStratChat and Which One Doesn't Belong, is that there are so many different entry points, right? There's Which One Doesn't Belong shows four images. It could be four numbers, four pictures, four whatever. Problems. 

Pam Harris  13:52

Or graphs.

Kim Montague  13:54

And the idea, right, you know, is which one of the four doesn't belong.

Pam Harris  14:00

One of these things is not like the others. 

Kim Montague  14:02

But the trick of it is, or the thing about it is, you can find something that doesn't belong for all four of them. 

Pam Harris  14:10

That's an extension, right? That helps kids keep thinking, yeah.

Kim Montague  14:14

Yeah. And so both my boys were over my shoulder. We were scrolling and we had a great time. And, you know, now I've got to find another thing.

Pam Harris  14:24

So let's land on Which One Doesn't Belong for just a minute, because I've actually had some teachers push back a little bit. I've been throwing out Which One Doesn't Belong on Facebook and Twitter and Instagram, and I had a couple teachers push back when they said, like, they gave a for instance, "I think it's C, C doesn't belong." And I said, "Okay, cool. I like your reason for C not belonging. But do you have reasons for A, B and D?" And they said, "Ah, is that a thing?" And a couple of teachers are like, "Oh no! I did it wrong?" And I'm like, "It's not wrong or right, it's just more fun to come up with reasons why each one of the choices doesn't belong." And then they were like, "Okay." But then a couple of teachers pushed back and they said, "Well, I wouldn't require kids to do that for all four." And I said, "Yah but let them."

Kim Montague  15:06

What if they can? Yeah.

Pam Harris  15:07

It's not about 'requiring'. Like then they were saying, "Are you going to grade them wrong if they don't, you know, you gonna take points off if they can't find one?" To me, it's so far from grading, it's not about taking a grade on Which One Doesn't Belong. It's like having fun, 

Kim Montague  15:21

Right.

Pam Harris  15:21

So it's about stretching. Can you find a reason for each one of them not belonging. And then once the class or the group has sort of shared, "Ooo I could think of a reason for each one of these not belonging," then can you think of a cooler one, a more clever one. We're looking for the 'Ooh factor", like, "Ooh, nice! Oh I like that reason!" Like, we're sort of, that's what we're striving for. That's why your kids find it so fun to talk math with you. Because then you have those moments where you're like, high fiving. And like, "Oh, sweet thinking." Kind of fun. So another thing that we like to do when we have a few minutes with maybe some older kids is: who knew how important it was that I can find anything plus 10? Like pick a number, and we need kids to know that number plus 10. In fact, I was at our church the other day, we were helping clean the church. And there was a few moments where I was standing in the foyer. A 10 year old was hanging around. And, you know, what are we gonna do? We got 10 minutes. We're gonna chat math. So I said, "Hey, do you know what's" - and I just kind of randomly chose a two digit number - "what's 28 plus 10?" And she looked to be like, this is no problem, "It's 38." And I said, "What's 28 plus nine?" Oh, then there's that look on her face, like -  and so I said, "Well, you said 28 plus 10." And I don't finish the sentence right away. 20 plus 10 was 38. She says, "Yeah, 38." "So 28 plus nine would be..." Kim.

Kim Montague  16:55

Oh, sorry.

Pam Harris  16:58

So Kim, I didn't warn you? You okay? Alright. So if I choose some random number, like 83 plus 10, that is?

Kim Montague  17:06

93.

Pam Harris  17:06

So then what's 83 plus 9?

Kim Montague  17:09

92.

Pam Harris  17:10

What if I have 151 plus 10?

Kim Montague  17:14

 161. 

Pam Harris  17:16

So then what's 151 plus 9?

Kim Montague  17:20

160. 

Pam Harris  17:22

Oh, right, right. Now kids are like thinking about the pattern, even though one plus nine is just 10. They're thinking about the pattern. So thinking about that 161 minus one, we get some really nice sort of patterning going on. Now, if that's like too easy, you know, maybe I might throw something out like, what is 747 plus 10? 

Kim Montague  17:42

757.

Pam Harris  17:44

So 747 plus nine?

Kim Montague  17:47

756.

Pam Harris  17:49

Bam. Then I might throw something out, like so now we've got anything plus 10, anything plus nine, could I do - hey, let's go back to that. 48. Hey, what's 48 plus 100? 

Kim Montague  17:59

148.

Pam Harris  18:00

So what's 48 plus 99? 

Kim Montague  18:03

147. 

Pam Harris  18:05

Right? So then could I say something like, What's 202 plus 100? 

Kim Montague  18:11

302.

Pam Harris  18:12

So then what's 202 plus 99? 

Kim Montague  18:15

301.

Pam Harris  18:16

Right? It's just one less than that 100. And I'm starting to get kids to just kind of think about this idea that if I know what any number plus 10 is, I can think about any number plus nine, any number plus 100 I can think about that plus 99. So Kim, if I said to you, what is 2,466? Plus 100. 

Kim Montague  18:36

2,566.

Pam Harris  18:39

So then what's 2,466 plus 99?

Kim Montague  18:44

 2,466 plus 99?

Pam Harris  18:47

Yep.

Kim Montague  18:47

Yep, 2,565.

Pam Harris  18:51

Bam. And do we let kids keep track? Absolutely. In fact, I'll be honest, I totally just wrote down the number 2,466, in order for me to check you because I have a harder time keeping all those numbers in my head. 

Kim Montague  19:03

Right. And the goal is not that. Right?

Pam Harris  19:05

Right. So that's what I'm saying. If you need to keep track - now, if you're driving, maybe don't keep track while you're listening to the podcast. But we don't limit kids from being able to sort of keep track. What we don't want kids to do is write down 2, 4, 6, 6. And then 99 underneath that then draw the line to do all the - we want them to think about plus 100 and then how that can relate to plus 99. Y'all we could be crazy. We can ask things like what is 240... What have I not done? 249 plus 1,000. 

Kim Montague  19:35

1,249. 

Pam Harris  19:38

That's not too bad. Right? So then what's 249 plus 999?

Kim Montague  19:45

1,248.

Pam Harris  19:48

Right? Yeah. And then I could keep going and we could keep at it. The bigger the numbers get, absolutely we would want to have kids be able to sort of keep track. So big idea is to ask sort a random number plus 10, then follow it right up with plus nine. Then if you can ask plus 100, then follow that with plus 99. And if you can ask plus 1,000, and then follow that with plus 999. It's a great ways for kids to start thinking about, "Oh, those are friendly numbers." Those are some relationships that I can use to solve problems, even when we're not just playing around with things. Alright, so Kim, um, I kind of only a little bit warned you that I was gonna do this. I want you to tell us, how do I even explain? Y'all Kim is this master of assessment. What do I mean by that? What I mean is that I can give her a kid and I can say, "Kim, tell me what this kid knows." And Kim does this bizarre, kind of crazy cool thing. I've learned so much from watching you work with kids. Where literally, I've just said, "Hey, can you tell me-" you know, like, in fact, my niece, one day. We sat down with my niece. And I was like, "Help me figure out kind of what's going on with my niece." And Kim will do - one day, Kim, I saw you literally pull out the coins out of your purse, and you threw them on the table. And you were like, "Hey, yeah, Pam wants me to talk to you about math. But before we do that, will you just help me real quick? I need to know how much money I have." 

Kim Montague  21:09

Yeah. 

Pam Harris  21:09

And then you just interacted with this kid for two or three minutes with those coins on the table. And then you looked up at me, you said, "This is what they need to work on next." So interesting.

Kim Montague  21:19

Well, in the land of number. Right? So, I kinda take some stuff with other areas. But it's less for me - first of all I absolutely my very favorite thing, above all, is interacting with students. I find such joy in that. And part of that probably is because I find what they have to say so interesting. And I find it so interesting to figure out kind of what are you thinking about right now? And what do you already know? And kind of what's next for you on the horizon? And so a lot of times, if you can have a conversation with a kid, not going into it with some pre-written like set of questions, and just genuinely asking things, and listening to what they're saying. So, you know, you say this about me all the time. And I wish that I had some magical answer. But really, I feel like I engage in a conversation. A lot of times you and I will say and other people will say, ask "How do you know?". But it's not... 

Pam Harris  22:24

It's not the only question, right? 

Kim Montague  22:25

It's not ask kids a question and say, "How do you know?" And they respond, and you say, "How do you know?" And they respond. It's less about that and more about, they say something and you're intentionally listening to what they're saying, to see if you can continue the conversation. You're not listening just to respond, you're listening to listen and to hear them. So you might - 

Pam Harris  22:50

To actually figure out what they're thinking about. 

Kim Montague  22:52

Right. So when you have a conversation with kids, you might say something like, "How did you think about that?" And then listen to what they say. And you might say, "I thought about that way too." And engage in that. Or you might say, "Actually, I thought about that a little bit differently." And respond in such a way that lets them know that you were also doing some thinking. For me, it's about trying to figure out what's happening in their minds. Sometimes I hear really good teachers that are so poised with the next question, that they're not actually listening. And if they would, what the children are saying, what the students are saying, would actually inform the next question better than a pre-written set of questions. So you know, specifically, I'm talking to parents at this moment, or aunts and uncles or whoever, try not to be as rushed. Maybe sometimes as we feel like we need to be when we're having conversations with kids, because what they have to say is so important.

Pam Harris  23:58

And it's not about the answer right? It's not about what you get. It's how do you, how are you thinking about that? Even if they get like a wrong answer. And that can be even helpful to go, "What were you doing to come up with that? That's interesting, I got something else. Let me tell you what I was thinking about." And parents, teachers have, you know, 20 to 30 kids in their classroom. So they might be a little more pressed for time. You're driving in the car, you're chatting with the relatives at whatever gathering, take some time, relax a little bit, kind of like put that personal interest into really kind of digging into the relationships that kids might be using in their head and share kind of maybe what you're thinking about. We'd like to give a quick shout out to Christopher Danielson who has both the Talking Math With Your Kid hashtag and also the Which One Doesn't Belong blog and examples in his book is great. We think he's a great example about how to have these kinds of conversations with kids. And also John Orr who has his Math Before Bedtime. Y'all, we'll put links to those guys and their work in the show notes because we think they have some great ideas about talking with kids when we have some time to really dive into it. Remember to join us on MathStratChat Facebook, Twitter, Instagram Wednesday eves where we explore problems with the world. Y'all if you're interested to learn more math and you want to help yourself and your students develop as mathematicians then don't miss the Math is Figure-Out-Able Podcast because Math is Figure-Out-Able!