Pam  00:01

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

Kim  00:07

And I'm Kim Montague. 

Pam  00:09

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 students to think and reason like mathematicians do. 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  00:36

So, we have some longtime listeners, but I want to have a special welcome to the parents who are tuning in this week. 

Pam  00:43

Welcome! 

Kim  00:43

Last week... Yeah! That's really special, right? Listening to something...

Pam  00:47

We appreciate that. 

Kim  00:49

Yeah...for your students. So, welcome, welcome to the parents. Last week, we started talking about some parent tips, and some activities and routines that you can do with your student in teeny tiny bits of time that we know make a huge impact. And so, if you didn't tune into the last episode, Episode 170, we'd encourage you to check that one out as well. And so, if you listened to that episode, and you thought, "Man, they talk fast, and they said a lot," we would highly encourage you to check out that free download that we shared last week. And we'll mention again today. You can find that at mathisfigureoutable.com/parenttips. One word, "parenttips". Today, we're going to share some more tips and activities that you can do to help build mathematicians in your home.

Pam  01:39

We love active parents, parents who care about their kids, parents who dive in and really want the best for their students. And we honor the fact that you are here. So, let's get right at some tips that we have to work with your students. So, a couple of general tips to start with. Kim. 

Kim  01:58

Yeah?

Pam  01:59

Let's talk about... One of the most interesting things you and I find is that when we are interacting with kids maybe at church, or on the ball field, or in a store, not our own, and we asked them a question, we often find that parents dive in super fast. 

Kim  02:17

Oh, yeah. Absolutely.

Pam  02:18

It's almost like they're like, they'll say things like, "You know that. You know that one. You know..." Like, "We've learned that. We practice that." And I want to just like, "It's okay. It's okay. It's alright. Let them think." Like, let them think. It's not about retrieving from rote memory. It's not about, "Can I find that in my memory banks?" It's really more, "How are you thinking about that?" So, if a kid has to think about it a little bit, I'm okay with that. I'm going to wait a little bit longer. So, parents, our first tip for the day, wait a little bit longer for responses to math questions. Maybe to any questions, but for sure math questions. Give them a chance to think. Yeah, go ahead. 

Kim  02:54

Yeah. I mean, it tells them that you value their thinking, right? You're not going to interrupt their thinking. And so, I will say, Pam, that when we are waiting, we're not talking into that space, either, right? We're not trying to direct it. We're not trying to give them too many hints or clues. We are big fans of, "Do you need time or help?" That's a huge question. And a lot of times, most kids, just need a little bit more time. And when we rush in, we're not giving them the actual help that they need. So, the next one that I want to mention.

Pam  03:28

Ooh, before you go on, could I also suggest that if I were looking at Kim's face right now, while she was working with a kid giving that kid that time, she would have a look on her face of intense interest. What it wouldn't be would be, "Come on. Come on, you can do it." Like, kind of this wavy thing, like I'm impatient. Like, it's about speed. I'm snapping. It's fast. It wouldn't be a... Well, how else do I describe?

Kim  03:53

It's not impatient.

Pam  03:54

Yeah. It's very patient. It's very interested. It's very, like... Sometimes we'll kind of crinkle our brow and we'll kind of look kind of like, "This is hard." 

Kim  04:04

"Why don't you know this?" Yeah. 

Pam  04:07

Yeah. Like, we give him the like, "You're thinking. Good. I'm thinking. Like, "We're both thinking here." It's kind of the thinking face. It's not the expectation face. It's also not... Yeah, just the impatient face. It's kind of just like, "Yeah. Hey, let's think about this." Let's put some weight on the fact that we're thinking. So, we're waiting and we're putting weight. We're allowing there to be a little time.. Ya'll we're not suggesting that we want your kids to be slow. We're suggesting that it's okay for them to think. There's a difference. Yeah.

Kim  04:38

Right. 

Pam  04:39

Cool.

Kim  04:40

Along with that, while you're waiting, we want to resist the urge to tell the right answer. And if you grew up in a world where the answer was everything, which I think a lot of us did, then you might feel the urge to supply the answer, assume that means they now own it, and move on. Which is not what we believe.

Pam  05:03

We'll see a lot of parents. You know, I'll say something like, "Hey, do you know 34 plus 10?" And the kid will kind of, you know, wait a little bit, and the parent will go, "44." Alright now, you know." And I'm like, "Umm..." Just because the kid heard 44 doesn't mean that the kid now owns. Like, if I'm messing with 34... I don't know if that was the best problem, Kim. But if I'm asking 34 plus 10. Or maybe a better example is if I said 34 plus 10, and the student said, "44. Haha, I know that." And then I said, "Well, what's 34 plus 9?" And the kid waits a little bit, and the parent goes, "Well, it's 43?" You know like.

Kim  05:37

"It's just one less." Yeah. 

Pam  05:38

Yeah. Oh okay, now the kid... Yeah, or "It's just one less." Now the kid owns it, right? In that moment, we don't want to steal the chance for the kid to grapple with that relationship. We don't want to take away the opportunity for the kid to consider, "Can I use plus 10 to help me with plus 9? Huh. What would that? Let me grind that. Let me create that path in my head, that then the next time it comes up, I had that that one time path. And boy, if I travel that again, it'll be a stronger path. And boy, then if I travel that again, it'll be an even stronger neural path. That then when I hit things like that, I have these paths that I've created. If you just pop in and answer, I don't get a chance to create that neural pathway."

Kim  06:22

Yeah, and that's what we want. (unclear).

Pam  06:24

That's what we want. Yeah, we want dense, rich, rich brains, dense brains. Lots of neural pathways in brains.

Kim  06:30

Yeah. Last week, we alluded to this a little bit. But when you're working with your student on any of these routines or game-like situations, important questions to ask would be, "What do you know?" When they say, "I don't know," you can say, "What do you know?" and relate to something that you know that they know. You could say, "How do you know?" Or, "How did you do that?" And then, intensely listen with curiosity. You could ask, "Does that always work?" Having kids verbalize their thinking can be super powerful to cementing that thinking in their own brain. And it's super fun to share about ways that we're learning and sharing strategies because then you're open to additional ways of thinking about it.

Pam  07:14

Yeah, super cool. Kim, do we have time for me to give a quick example? 

Kim  07:17

Sure. 

Pam  07:18

So, when you said, "What do you know?" I think this often comes... With parents, I'll see it often with facts. Do we want kids to know their facts? Absolutely. We want them at their fingertips. We want them to have facility with those facts. But how they get them is not really through rote memorization. So, I might ask a student or you might ask your student something like 8 times 7, that's an often missed fact. If the kid's going to miss a fact, they might miss 8 times 7. And when you ask a kid, you're like, "Okay, come on." So, first we're going to pause, right? We're going to give them some time. Wait a little bit longer. We're going to resist the urge to tell the right answer. And when a kid says, "I don't know," then we might say, "Well, what do you know?" Do you know some 8s? Do you know some 7s? We're trying to find 8 times 7. Do you know some 7s? Do you know some 8s? What do you know?" And then if a kid says, "Well, I know five 8s," then we might say, "Well, how do you know that?" "Well, I know my 5s." Or, "I know 10 times 8, so I can find 5 times 8. So, asking. Or, if a student says, "Let's see, 48, 56," then we might go, "Oh. Like, what did you? What were you just doing? How do you know that? How do you... What were you thinking about for the 48?" "Oh, well, I was thinking about 8 times 6, so I just need one more 7 to get..." Wait, I need one more 8. I don't usually use that one. So, if I know six 8s is 48, then I can do seven 8s would be one more 8. We want kids to have to verbalize. Like, I just did. I had to think about my own thinking there. We want kids to verbalize things like that when they are using relationships like that. And then, that last question, "Does it always work?" When a student says, "Well, yeah just add one more group," you could say, "Can you always do that?" Like, if you're looking for eight 9s, and you know seven 9s, can you just add one more 9? Or if you're looking for eleven 12s, and you know ten 12s, can you just add one more 12? Does that always work, that you just add one more group when you're only one group less?" That kind of generalizing is hugely important for students to kind of cement relationships. And it works towards algebraic reasoning, which they're going to do later in higher grades.

Kim  09:18

So, another suggestion that we would make is to play games with your students. So, Pam and I are both a big board game loving, kind of people. And there's some fantastic games out there that we highly recommend to build all kinds of things with students. But we're also going to dive in now to a couple of other routines that we would love. Pam, you want to say anything about games?"

Pam  09:41

I'll just say, it doesn't have to be a math game. It can be any game. But there's something about board games, something about (unclear).

Kim  09:48

Card games.

Pam  09:49

Card games. In those games, if there's a mathy thing that happens, say it out loud. Here's a real simple one. You're playing one of those you just move your pieces along the board, and you notice there's a pattern on the board. Like Monopoly. And so, you know, you're supposed to go so many spaces, and you just know how many they are, and so you skip ahead to it because you're like, "Well, you know, if I go to the..." I can't take now "...the railroad, then it's just going to be 1 past that." Or, I know that, you know, it was supposed to be 10, and 11 is just going to be one over the edge because I was right here." When you think about those moves in the game, think about it out loud. Just say out loud how you're thinking about that little mathy thing in the middle of the game. Could be a way to start the conversation about what's happening in your brain. You might say, "Pam, everybody knows that's what's happening in my brain." Oh, actually. You might be surprised. I didn't. As a kid. I would watch people do that as a kid, and I would like, "I don't even know what they're doing. Oh, well." And then I would just. Because it wasn't talk-about-able with whoever I was playing, so I didn't ask. But oh, I wish people would have said more of that out loud, so I could have started making those relationships more.

Kim  10:56

Yeah. Okay, so one thing we would recommend parents do with their students from a very early age is count forward and back from numbers other than 1, right? So, really young kids can sing the song 1, 2, 3, 4, 5 pretty early on. But the idea that you start with 16 or 24 and count on is a huge skill. But then also, even more difficult than that is counting backwards from numbers.

Pam  11:26

Like, choose random numbers, and say, "Hey, today, we're going to count on from 13. Go. 13, 14..." And just start counting forward. But then say, "Today, we're going to count backwards." Start from the nice 10 and count backwards and start from the nice 20 and count backwards, but someday say, We're going to start counting backwards from 13. Go. Bam! (unclear).

Kim  11:45

I mean, doesn't have to be by ones, right? (unclear)

Pam  11:47

Ah, yes, keep going. 

Kim  11:49

Yep. By 10s. 23, 43, 53, 63. Talking about the place value in that, what's actually happening when you count forwards by 10s? What happens to the tens place? What happens to the ones place? Once you've gotten some experience with counting forwards and backwards by 10s or by hundreds from strange numbers, what happens when you count forwards by 9? What does that do to the tens place and the ones place? So, you're talking about the value of numbers, as they're doing this small routine.

Pam  12:23

Yeah, super, super helpful. For younger kids, you can also count by 2s, but don't just count by 2 starting from zero.

Kim  12:30

Right.

Pam  12:31

Start with 3 and count by 2. Start with 11 and count by 2s. You can start with 16 and count by 2s. And then, start with 26 and count by 2s backwards. So, yeah. Lots of things that you can do. Kim, I used to do that going up or down stairs. So, somehow for me that was like a ping with my kids. That as we climb stairs, I'd be like, "Okay, say we're on 15. Ready, we're going to count by 5s. 15, 20..." And then, on the way down, I was like, "Okay, now we're on..." I don't even know "...73, and we're going to count back by 5s." Oh, that was (unclear).

Kim  13:00

Yowza!

Pam  13:01

That was a slower. Okay, maybe I didn't actually do that one. Usually stairs we did easier. 

Kim  13:05

But you could. Yeah.

Pam  13:06

But you could, you could. Oh, if I would have done it more, we could have done it even more. Yeah, absolutely. Cool.

Kim  13:12

Alright, what about Guess My Number"? 

Pam  13:16

Ooh, love it. Love it.

Kim  13:17

Oh, yeah. Okay, so Guess My Number in my family is where I think of a number, and my kids can ask me yes or no questions about that number. And so, I might be thinking of the number 68. I generally start with numbers under 100 for quite some time. So, I think about the number 68, and they might say to me, "Is your number odd? Yes or no?" Okay, well, if I know it's odd, yes or no, then I don't have to ask if it's even, but sometimes they do early on. "Is it less than 100? Does it have a 6 in the ones place? Is it a prime or composite number? Is it a multiple of a certain number?" And so they're gathering information about the number based on the questions they're asking until they determine what the number is.

Pam  14:04

Yeah, super cool. A slightly simplified version of that, that I like to play first with kids, is I might say, "I'm thinking of a number," and then they would just guess. So, I'll say, "I'm thinking of a number between 0 and 100," and they guess. It's funny how random they guess. "33." And I'll say, "My numbers higher than 33." And I want to play this game often enough to where kids finally start to go 50 is their first guess. Like, cut it in half. Because now, if they guess 50, they've cut out half the numbers, and they can... You know, it's for sure in that other half. So, don't tell them that though. Play with them often enough that along the line somebody goes, "Oh, I think 50 is a good guess." And so then I'll say, "My numbers higher than 50." And then, I hope they'll come back and say, "75." And then I'll say, "My number is lower than 75." And then, I hope they'll come back and try to go in between 50 and 75. Now, that takes a while for them to kind of get that, but boy when they do they own that idea that if I can sort of cut what's left, the span in half, then I've cut out half the numbers. I can guess between the numbers that are left. So, that's a little bit of a simplified version maybe for younger kids. 

Kim  15:13

Yeah.

Pam  15:13

You can make those numbers much bigger. I can have the range 0 to 1,000 and guess my number in between there. Go ahead, Kim. 

Kim  15:19

Yeah. Well, I was going to say both versions of that are very much about the richness of the number, and its approximation, and about what it comes between, its place. 

Pam  15:31

Neighborhood. 

Kim  15:32

Yep. Speaking of its place, we also are really big fans of talking about the value of number, rather than its digits, right? So, kids might be really used to in early years saying the number 391 is about... We'd love for them to know it's 300 and 90 and 1. But also talking about what it means in terms of other views. So, it could also be, if I said the number 391, it is 300 and 90 and 1, but it's also 200 and 191. It's also 39 tens and one.

Pam  16:18

And it's also 9 away from 400. It's 400 minus 9. Absolutely. And and let me just really point out that every time Kim just said the number 391, she never said 3-9-1. 

Kim  16:30

Right.

Pam  16:31

So, often we say numbers as digits, and we don't talk about the values that are happening. So, if you're saying the number 1,236, it's okay to say 1,236. It's okay to say 1,236. But try not to say 1-2-3-6. 

Kim  16:48

Yeah.

Pam  16:49

Like, just calling it digits. And then like Kim said, all the different kinds of ways that you could talk about that number flexibly. Like, that helps kids think about what it's near. It helps kids place that number and think about magnitudes, the size of numbers. Yeah, it's totally cool. Nice. 

Kim  17:05

Okay. Go ahead.

Pam  17:07

Another one I was going to mention, going a little higher math here, is that when you're talking about fractions. So, if you're talking about a fraction like four-fifths, the name that we give four-fifths in English is four-fifths, but it also means four 1/5s. Like, the fraction four-fifths if you only learn fractions and part-whole relationship the way we typically have taught, when I say four-fifths, you might be thinking, "Oh, that's 4 "line" 5," and you might only be thinking about 4 out of 5. To do that, you're only thinking about 4. You're thinking about 5. And maybe you're shading some pizza pieces or something. But we want kids to think about four-fifths as four 1/5s. But if I can think of something cut into five equal shares, and I call that a fifth. Then, four 1/5s, four-fifths is four of those 1/5s.

Kim  18:00

Right. 

Pam  18:00

So, similarly, if I had...I'm trying to think of another fraction...two-sevenths, then I want to think of that as two 1/7s. Or if I have had...

Kim  18:08

meaning a seventh and a seventh, right?

Pam  18:10

Exactly. I've got a seventh there and a seventh there, and so I've got two of those 1/7s, then I can call those two-sevenths. And similarly, if I had three-eighths, then I could say I've cut something into 8 equal shares. I've got 8 equal pieces. And I have three of those 1/8s. Three 1/8s is the same thing as three-eighths. You might be surprised how much clarity that can bring to some fractions when you call those mixed numbers, when you call those fractions by what we call the unit fraction, by the one piece, how many of that one piece you have. Cool.

Kim  18:48

And a small tweak to that is that you would hear both of us say, "Four-fifths" about as much as we would say "four 1/5s". So, when we're describing the fraction four-fifths, we also will say, "Oh, you have four 1/5s.

Pam  19:02

Yeah, so it's not like we give up one name for the other because they're going to hear four-fifths in common usage and everything. And four-fifths is fine. We just need the meaning of four 1/5s as well, so we just often say both of them. Yeah?

Kim  19:13

Yep.

Pam  19:13

What did you say? Yep, totally cool. 

Kim  19:15

While you're talking about fractions, what about comparing fractions?

Pam  19:18

Oh, yeah. So, often when we get to comparing fractions, if kids haven't been taught a lot of rules up to this point, almost always there comes these funny like butterfly. These things are not mathematical. These weird. So, when you're comparing fractions, consider that you can compare them to landmarks. So, when I just said four-fifths, if I got four 1/5s, five 1/5s would make the whole, right? I'd have the whole candy bar, if I had four of those 1/5s. I'm almost to the whole. So, I can sort of use that four 1/5s, that idea of being so close to that landmark 1 whole. I can use that to help me think about comparing it to something like six-fifths. So, if I'm thinking about six 1/5s. Six 1/5s? Wait, five-fifths is the whole thing. Five 1/5s, I've got the whole thing. So, I've got 6 of them. Woah, I've got more than the candy bar. Well, right there if I'm comparing four-fifths and six-fifths, I can just think about how those both compare to five-fifths. I don't have to do any fancy butterfly, cross multiply, cross cancel, like whatever all these rules are. I can just compare to a landmark. Kim, what if I was comparing something like four 1/5s to two 1/5s? How do I have some sense of? Oh, maybe that one's almost too easy. Let me do four-fifths and... Oh, Pam thinking live here. And... No, that's going to work.

Kim  20:39

Two-sevenths? 

Pam  20:40

Thank you. You knew where I was going.

Kim  20:42

So, I knew what you were going to ask. So, if I know, I understand that four-fifths means four 1/5s, then I can think about, "Where's halfway?" Like, four-fifths is more than half. More than half of the fifths. And two-sevenths, I only have 2 of the sevenths. That's less than half of the sevenths. So, if I have one fraction that's more than a half and one that's less than a half, that's usually very helpful to to compare them. So, four-fifths would be bigger. Yep.

Pam  21:12

Yeah, nice. And I could have even done something like four-seventeenths. So, now the 4s are the same. I've got four 1/17s. And I've got four 1/5s. And so, now I can... Well, that's probably... I just left the one-half didn't I? That's not what I meant to do.

Kim  21:30

But there's lots of ways to compare with meaning. I think is the point, right? There's lots of ways to compare...

Pam  21:35

Absolutely. Yes. We'll stay there.

Kim  21:36

...meaning rather than some rote memorized things. 

Pam  21:39

Yeah. So, when you're thinking about fractions, see if you can use that unit fraction, that one share, the one-fifth, the one-seventh, the one-fifteenth to help you reason about more of that one share. The five-sevenths and the six-fifteenths. And that could be helpful in reasoning about how close that fraction is to a landmark fraction. Cool. Alright, ya'll, one of the most frequent questions that we get from parents is, "Do you have all this stuff written down that we can refer to it? We just listened to your podcast, but how are we supposed to have all this in our back pocket?" Well, the answer is, yes. We have a tip sheet for parents. It's a free download. It lists the things that we've talked about and more. And it has links to learn more about many of them. You can get it all at mathisfigureoutable.com/parenttips. To get our free download, click on the link... Click. You can't click. Well, you could click on the link in the show notes. Or go to mathisfigureoutable.com/parenttips. And don't worry, it's totally free. Parents, picture a day when you have more confidence talking math with your child. Picture a day when you are sharing your ideas and wonders, and your student is intrigued and sharing with you too. Picture a day when your child says, "It's cool that Math is Figure-Out-Able! And we talk about it in our family." For help to make that happen, download our parents tip sheet at mathisfigureoutable.com/parenttips. Ya'll, thanks 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!