Pam  00:01

Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam Harris, a former mimicker turned mather. 

Kim  00:09

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  00:17

We know that algorithms are amazing historic 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. I made it all the way through that sentence! I took a really deep breath. Could you hear me? I took a totally deep breath right as I said that. I did a singer's breath there.

Kim  00:42

In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  00:49

We invite you to join us to make math more figure-out-able. 

Kim  00:54

Oh, Hi, Pam.

Pam  00:55

Hey, Kim. So, listeners, you're going to have to tell us. Do you hate it when we totally mess up the beginning? Because I don't know.

Kim  01:00

It's not going to help. So. 

Pam  01:02

Yeah, what am I saying? If they say, "Yeah, we hate it," then we're not going to mess up again? I don't think that's a thing. Yeah.

Kim  01:09

It's May. May, Pam. End of the school (unclear).

Pam  01:12

Hallelujah!

Kim  01:13

No kidding. 

Pam  01:14

Whoo!

Kim  01:14

Exciting. Okay, so it is in the school year for me, but not for everybody. And calculators are a hot topic for teachers and students as they get older, aren't they? I know that's less of a thing for elementary. Kids have access to them. We let them use them. And at times, it might feel like building numeracy isn't as important if the kids are just going to reach for them. Or is it? So, today we're going to talk about something that we love to help students become judicious technology users. 

Pam  01:45

Yeah. Because 

Kim  01:46

(unclear). 

Pam  01:46

Yeah, absolutely, We now know that kids are forever going to have technology at their fingertips.

Kim  01:51

Sure.

Pam  01:51

And so, rather than slapping their hands, or "Put it away," or giving them free rein to use technology whenever they want, I would prefer to help them, for us as teachers to be purposeful in helping them become judicious users of technology. Yeah. When can they have power over something? When does it make sense to reach for technology? And you're right. We we found a way to make this happen. So, Kim, we found this early in our work together because the two of us were working with Investigations in Data, Number, and Space. 

Kim  02:28

Yep. Yeah. 

Pam  02:29

And it's called Beat the Calculator. And so.

Kim  02:33

Yes.

Pam  02:33

Is it a game? A routine? I don't know what you call it. is it a game? it's kind of a game.

02:37

Yeah, I don't remember. But I remember that it was in our fifth grade. Might have been in other grades too. But I remember as a fifth grade teacher. In Investigations, there was this thing, activity called Beat the Calculator. And I don't know that it matters as much the setup. But kids, you know, you have cards. You set kids up in partnerships. And there's a stack of cards. And one partner has a calculator, and the other person has paper and pencil or dry erase. However you want to record. And in that routine, activity, game, the students would flip a card over and on it would have a problem. And the game was called Beat the Calculator because the person who had paper pencil or, you know, dry erase, was asked to solve the problem while the other person was typing it in. (unclear).

03:32

Mmhm. And the idea is can that kid. Yeah, can you beat the calculator, right? 

Kim  03:38

Yeah.

Pam  03:39

And you might be listening, going, "That's a dumb game. The calculator is going to win every time." Or is it? Right? Like, yeah. So, the Investigation authors were super clever that they wrote problems that if you are just going left to right. Yeah, calculators probably going to beat you. But if you think, if you get outside the problem, look at it kind of from the 20,000 foot view, and like, "What do I know?" Ooh, bam, chances are super high, you could actually beat the calculator. And what a fantastic message to send to kids. That I mean, you can. You pull that out, and you type all that in. But bam, I just used what I know. And there, you know, I got an answer. And I think it's a brilliant message to send kids. 

Kim  04:23

Yeah. 

Pam  04:23

Yeah. So, Kim, let's play a little bit. 

Kim  04:25

Okay. 

Pam  04:26

Okay. So, you and I decided earlier, and I came up with a couple and you came up with a couple, and so I'm going to give you a first one. Okay. Now, we can't really play Beat the Calculator here because I don't... Like, one of us would type. Because we're sort of listening over the, you know, fairways, whatever. So, I'm going to read the question. Listeners, you might want to write it down because often it's like kind of just suggested that you might not want to do it left to right. You might want to like look at it. So, we're not going to really race, but we'll talk about, you know, how might you solve it. How long do you think it would take? Not like timing it, but do you think it would take longer to do in a calculator or just like reasoning. So, let's just start. So, Kim, here's the first one I made up. 17 plus 8 plus 3. 

Kim  05:13

Oh, okay. 

Pam  05:14

Okay. So, if you were to see 17 plus 8 plus 3. 

Kim  05:18

28. 

Pam  05:19

Because?

Kim  05:20

17 and 3 is 20 plus 8 is 28.

Pam  05:24

So, like looking at it, you can almost see 28. You got the 17 and 3, just. So, not going left to right, right? You can choose the book ends, and.

Kim  05:32

Yeah.

Pam  05:33

Do you think that kids could see 17 plus 8 plus 3. Grab the 17 and 3 to make 28. Quicker than they could type those in? 

Kim  05:42

Oh, for sure. For sure. 

Pam  05:44

I think most of them absolutely. 

Kim  05:46

Yeah.

Pam  05:46

Yeah, cool. And then maybe for older kids, I might have said 17 plus 8 plus 3 plus 12. 

Kim  05:51

Mmm, nice. Yeah. 

Pam  05:53

And why did I choose 12. 

Kim  05:55

Because 12 and 8 go together. And that would be 20 and 20, 40. 

Pam  05:58

And you're just at 40. And surely, even though I added that last number on, I think you're going to beat the calculator by more with that one because it's going to take you longer to type plus 12 than it would be to just consider, "Oh, that 20. Bam," and then we're at 40. Cool.

Kim  06:13

Yeah. Cool. 

Pam  06:14

So listeners, hope you get the idea that it's kind of this like, you know, we're going to give you problems that you might be like, "Oh, you're cooking the numbers." Yes. Yes, we are. 

Both Pam and Kim  06:22

Yeah, yeah. 

Pam  06:23

We're absolutely cooking the numbers because we want kids to realize the numbers are often that figure-out-able.

Kim  06:29

Yeah. Okay. You want one? 

Pam  06:30

Yep, I want one. What you got for me?

Kim  06:32

Okay, I've got you 0.25 times 7 times 4.

06:38

Does that look like 25 divided by 100? Like over 100? Like as a fraction? 25/100? 

Kim  06:43

No.

Pam  06:43

No. Say it again.

Kim  06:45

0.25

Pam  06:46

Oh, 0.25 (unclear).

Kim  06:46

0.25 times 7 times 4.

Pam  06:49

Okay, so 0.25 times 7 times 4. Oh, nice. That's nice.

Kim  06:53

Okay.

Pam  06:53

Cool. Okay, so I'm the 0.25 times 4 is instantly 1. And then times 7 is 7. Bam.

Kim  06:59

Mmhm.

Pam  07:00

Yeah.

Kim  07:00

Nice.

Pam  07:00

Yeah. And like with 0.25, kids, even if they don't type the 0, there's a lot more characters they have to type in. But if I can just think of a fourth of 4. And then times 7. Yeah, I'm going to beat the calculator on that one. Alright. I got one for you. Ready? 

Kim  07:15

Yep. 

Pam  07:16

How about 123 minus 98?

Kim  07:19

Mmm. Yep. Got it. 

Pam  07:22

Okay. What are you doing? 

Kim  07:23

I got 25.

Pam  07:25

How? 

Kim  07:27

This time I just said 2, 23, 25. So, 98 plus 2 to get me to 100 plus 2?

Pam  07:36

(unclear). So, you just found the difference between the numbers. 

Kim  07:39

Mmhm, Yeah. 

Pam  07:39

Yeah. 98. up to 123. Bam, you're at 125. 

Kim  07:43

Yeah.

Pam  07:44

That one might be a little closer for some kids because they might have to think about it. But I don't know very. I mean, not by much. Okay. (unclear).

Kim  07:53

(unclear) I'll give you some middle school. What if I ask you to find the equation of a line? 

Pam  08:02

Okay. 

Kim  08:02

With the point 0, 0 and 1, 3?

Pam  08:06

Nice points. So, instantly, I can see the y-intercept is 1.

Kim  08:11

Mmhm.

Pam  08:11

And that the rate of change is 3. So, that's just 1 plus 3x. Nice. So, what would kids do in a calculator? I've often...too often because it shouldn't ever happen...have teachers say, "Alright, when you see this item on the high stakes test, you're going to go into the List Editor, and you're going to put those points in as a scatterplot. Like, stick them in List Editor. And then you're going to do the whatever the buttons are for your particular calculator to get the line of best fit. And there. Now, you've got the equation of a line between those." I can definitely beat the calculator here. Definitely. Hands down.

Kim  08:47

Did you say 1 plus 3x?

Pam  08:51

Yeah. 

Kim  08:52

Okay. 

Pam  08:54

Oh, no, I didn't. I mean, I did, but I'm wrong. Sorry. It should just be 0 plus 3x or 3x. 

Kim  09:00

Okay. I was replaying your words, and I was like, "Did she say that or (unclear) y equals 3x?" I don't know what you said.

Pam  09:07

I meant to say y equals 3x. 

Kim  09:09

Okay.

Pam  09:10

0 plus 3x. Yeah. 

Kim  09:11

Okay. 

Pam  09:11

Good call. Good catch.

Kim  09:12

Alright, alright. 

Pam  09:13

Yeah. Okay. I've got one for you. Ready? Okay. The fraction 1/5 times the fraction 5/42 So, (unclear).

Kim  09:23

Oh, yeah, yeah, yeah. Yep.

Pam  09:25

Okay. What do you got? 

Kim  09:27

1/5 of 5 is 1. But we're talking about 42. So, 1/42.

Pam  09:36

Bam. Kids could type that in, but when they type it in, unless they're in a cast calculator, they're going to get the decimal equivalent to 1/42, right? 

Kim  09:46

That's funky. Yeah, yeah, yeah. 

Pam  09:47

Yeah, but we can just reason through that guy. Yeah. Alright. Nice, nice.

Kim  09:52

Okay. So, Pam, like or dislike, Beat the Calculator.

Pam  09:58

I like it for certain periods of time to give kids... I think you can switch content, which makes it nice, and it gives kids this idea that they can think and reason and can be just as powerful or more powerful as the calculator. One other quick thing I'll say about calculator use. I've done a lot of work with TI and their Teachers Teaching With Technology group. And one of the things that one of the instructors. I can't remember who it was. But it was a t-cubed instructor. Said once was that the calculator... That we often say to kids, "Check your work with a calculator," which gives this impression that the calculator is the be it end all because you're going to check your work. It has the right answer. It's the thing. Instead, they said, "Strike that from your vocabulary." Don't say "Check your work on the calculator." Say things like, "Okay, do you know enough to put that in your calculator? Like, this calculator is this limited machine. Do you have enough to know how to enter that in your calculator? If you do, go ahead,. You can do that. And then see if if you knew enough to put that in your calculator. You can see if you guys agree." So, it's a different vantage point of it's not the calculator is like, "Go check. Go check your limited work." No, no, no. It's like, "Alright, you're powerful enough. Like go. You use what you know to see." Like, put that in the calculator. Bam. Because it's this limited machine. You have to know how to put it in there, so that it can recognize what you're doing. Yeah, I think that's a decent way to do that. How about you, Kim? Like or dislike?

Kim  11:28

Oh, strong like. Strong, strong like. I think there's so much that can be done with this routine at different times of the year for different purposes. Maybe we'll talk about that in a minute. But I'm a huge, huge fan. But what I do think probably is happening in some people's heads right now is they're like, "Wait a second. It's about beating the calculator, and so that means there's got to be some sort of speed component. Feels like competition. That doesn't seem right coming from you guys."

Pam  11:56

Let's parse those out a little. 

Kim  11:58

Yeah.

Pam  11:58

Let's talk about speed on one hand and competition on the other. Can I start with competition real quick?

Kim  12:02

Sure. 

Pam  12:03

So there is a bit of a social movement right now, has been maybe for a hot minute, about no competition. Competition is bad. Everybody gets a trophy. I don't actually adhere to that. I think competition is fine. I think kids should lose sometimes. I think kids need to learn how to lose, and learn how to win gracefully. And if they are a rotten winner, that's going to play out, and it's not going to work well for them socially. And so, I think that's a necessary thing to help our kids learn resilience. And I'm going to suggest gently that maybe we have a little bit of resilientless... Not less because that sounds like we don't have any. Kids who struggle with resilience because of some social things that happen to say, "Every kid should get a trophy. And there shouldn't be competition. And we're all winners." So, I'm not at blood curdling, ruthless competition. But I do think a little competition is a fine thing. And I think what we really should strive for is varying in our classes the kinds of competitions we have, so that kids can rise to the top of the things they are good at. And we give everybody a chance to be good at things they're good at, honestly good at, so that the pride they have in winning is honestly won, honestly earned. Not actually sure how you think about that, Kim. What do you think?

Kim  13:46

Well, you know, the phrase humble winner and gracious loser has been said on more than one occasion in my house because we do like a good competition in our house. I was the parent who never let my kids win at candy land or at whatever if I could help it. Well, that's a bad example because it's all luck there. But yeah, I'm not about letting people win. But I think there's, you know, (unclear).

Pam  14:08

I'm sorry. I have to interrupt you because I'm going to forget, and I want to say it. So, Candyland? Candyland is the easiest game to cheat out when your kids are young enough because you just find the card that gets that dumb game over soon.

Kim  14:17

I know. But I'm also not a cheater, Pam. 

Pam  14:20

Well, I just found the card. I didn't care who won. I wanted it over. I'm not a big fan of games where there's nothing I can think about. 

Kim  14:28

Yeah, I hear you.

Pam  14:28

Okay, sorry. I interrupted you. (unclear).

Kim  14:29

No, that's okay. I forgot what I was going to say. 

Pam  14:33

Oh! 

Kim  14:34

Oh!

Pam  14:34

Sorry.

Kim  14:34

So, you know, I'm like a highly competitive person. But I also (unclear).

Pam  14:37

You are a little bit.

Kim  14:38

I am. But I also recognize that that is not where everybody's coming from. So, if some kids in your class like a little competition and some don't, I love that you said we want varied experiences, right? I don't think that we could never have competition and that doesn't... The people that are motivated by that don't ever get an opportunity to be motivated. And if we, you know... We need both. So, I am okay with having some experiences with competition. What I love about this one in particular is that you can get better. This is not something that it's like you're either good at it or you're not. This is about something that you have the ability to learn. You have the ability to learn to be judicious. And as you gain more strategy, then it becomes more readily available. I could see having this experience at the beginning of the year. And kids don't maybe own a lot of strategies, and so they're like, "Ugh, man. The calculator was winning more. So now we are going to grow. We are going to grow. We're going to learn more strategy, and then check in." I could see doing it with some operations and not others. I just think there's so much that can be done here.

Pam  15:48

But that's really a nice way for kids to kind of be like, "Okay, yeah, calculator beat here. I was close in some of them."

Kim  15:56

Yeah. 

Pam  15:56

And then develop their strategies, and they come back and their whopping the calculator. (unclear).

Kim  16:01

Yeah, I think (unclear) so much to thinking and reasoning. 

Pam  16:04

Yeah, that's nice. I like it. (unclear).

Kim  16:06

Okay, so let's talk speed because speed is also involved in some ways.

Pam  16:11

Absolutely. And one of the things I think that is super tricky right now and maybe has been for a while in math education is this idea of speed that we culturally have built kind of an expectation that we don't agree with. That being good at mathematics is equivalent to being the fastest at, I don't know, retrieving things from rote memory. And we don't think that. We don't think that those are equal. That just because you're fast means that you're good. For example, you can have a kid that you could say, 7 times 8, and they quickly in their head go like, "The garden gate is made of 6, so it's 56. Bam, 56!" And they say that fast enough, and they spit it out. And we're like, "Oh, good. Look how good you are at multiplicative reasoning." When we might have a kid who goes, 7 times 8. I know 7 times 7 is 49, I need one more 7, that's 56." And that kid is already built more multiplicative reasoning just by knowing they can chunk the fact into chunks they know. And then they're thinking about those chunks, and and they're putting them together. I would much rather have a kid do that kind of multiplicative reasoning to figure 7 times 8 than repeat a rhyme because they're not building multiplicative reasoning at all. They're just dealing with a memory pneumonic. So, it's not... We clearly are going to say they're not equivalent. We also would clearly say that we don't want a kid finding 7 times 8 by going 1, 2, 3, 4, 5, 6, 7, 8. 1. 9, 10, 11, 12, 13, 14, 15, 16. 2. Like we don't want him counting by ones and keeping track of those groups. If we see that kind of slow speed happening, we want to get under it. We want to figure out why kids are taking a while. If they're taking a while because they're using a less sophisticated strategy, well, then we want to help build their brains to be able to think more sophisticatedly. If they are taking a little bit longer because they're really grappling with sophisticated relationships, bam, we want to support them and celebrate that thinking. But at the same time, on the other hand, if they're being super quick, and they're doing it in a memorized kind of way, then we want to be like. I want to get under that too. I want to really encourage you to think and reason. Alright, so all that little background on speed. Thanks for listening. What about this game and speed? Kim, what would you say? Like, so why are we good with this game and speed? Or do you (unclear).

Kim  18:32

No, that's a good question. So, for me, it's about are you thinking at all before you dive in? And I feel like it's acknowledging that it's worth thinking first. Yeah, you know, I (unclear) put words to it.

Pam  18:51

Yeah, for me, it's a lot about acknowledging that there is something about efficiency. There is something about we don't really want you bogged down and taking forever when that's not necessary. 

Kim  19:06

Yeah. 

Pam  19:06

So, I think we can say, Yeah, we do want you to be efficient, and look how efficient you can be when you are powerful, when you own these powerful relationships. 

Kim  19:14

Yeah. 

Pam  19:15

And put the calculator in its place that it is only as efficient, and smart, and able to do with what you can put in it.

Kim  19:24

I think that might be it for me. And I am absolutely loving where this conversation is going because I have to tell you, I've had a post it note on my monitor for the last month, and I can't wait to share this with you. But it's knowing that there are some problems that you would never touch a calculator for. And as we build your brains to be more and more sophisticated, then you identify more of those. But there are some problems that are funky, and you could think through them, but maybe given what you're trying to do in that moment, it's not necessarily worth the time.

Pam  19:56

Absolutely.

Kim  19:56

If I'm stringing along a ton of problems, then (unclear).

Pam  20:00

Well, if you're in the midst of a large problem. This is my best example. You're in the midst of a large problem. You're really messing with something say proportional or even multiplicative. And in the middle of it, you got to add some crazy numbers together. In that moment, I might just type those in. Or I might just type in, because my focus is on this bigger math, big thing over here. And I just need this little calculation. Yeah, I might not take the time to go over there and do that. Another example. We were in a meeting the other day, and we were trying to do some business planning for our small team, and some numbers came up whatever. And somebody snarkily was like, "Pam, what do you got for? You know, what's your strategy for that?" And I was like, "I typed it in." Because in the moment, we were thinking about these big things, and it wasn't a moment where I was trying to build my brain. It was a moment where I was trying to do something with the number. 

Kim  20:47

Yeah.

Pam  20:47

And I think I think we want kids to be able to feel that tension. 

Kim  20:51

Yeah. 

Pam  20:51

And not just... We all know what it means when a high school kid reaches for a calculator to add 2 plus 3. I would submit that kid hasn't dealt with that tension. We need to put that tension up in front of the kids. (unclear) 

Kim  21:03

Hey, let me tell you this post-it not real fast. 

Pam  21:05

Oh, okay. Go,

Kim  21:06

I got to tell you. 

Pam  21:07

I wrote down what I'm going to say, so I don't forget (unclear)

Kim  21:08

(unclear) Hang on to it.

Pam  21:09

Go, go, go.

Kim  21:10

So, one of my kids was doing volume and surface area, lateral area, like lateral surface area problems. And there was like a million, right? Like, a whole bunch of them. And he eally got down to, "Which of these three have the same?" Which have x and y, y x. Whatever. So, he wrote down 2 pi times 3 times 6 for one of them. 2 pi times 4.5 times 12 for one of them. And 2 pi times 6 times 9. And the question was, "Which are the two have the same...I guess, volume?" And he started to pick up his calculator or pull out desmos. And I went, "I'm sorry. Wait." And he looked at it, and he goes, "Oh, those are the same because double, half."

Pam  21:54

Nice.

Kim  21:54

And then he goes, "And the other one is a third as much." And I was like...

Pam  21:57

Bam!

Kim  21:57

..."There you go." Thinking. Like, as teachers, we have to intercede some time and say, "Think. Think."

Pam  22:03

"I think you think about this."

Kim  22:06

Yeah.

Pam  22:06

"I think you can. I think you can." Yeah, yeah. Or at least did you at least consider whether you can think about it or not. Hey, the one thing I was going to say that I wrote down, so I wouldn't forget is. You know, in the advent of Chat GPT I'm finding it fascinating listening to kids, and maybe specifically a couple of my own kids, talk about how much fun they're having coming up with prompts. And that's like becoming a whole science about how to come up with the best prompts. And it really speaks to in order to use Chat GPT well, you have to think. Like, you can't just say something, and then take what it spits out. Not to use it well. Like, really, there is an art, and it takes thinking to come up with the correct prompts to put in there.

Kim  22:52

Yeah.

Pam  22:52

I've actually used it a couple times recently, when I was trying to write some things, and I couldn't figure out a way to say something. And what's been fun for me to notice is I'll go, and I'll say, "Hey, what's another way of saying blank," and it will give me you know, a couple of ways. And then I'll say, "Try again," and whatever. And I don't know that yet I've chosen exactly what it spit out. But it does help spark some ideas.

Kim  23:11

Sure.

Pam  23:12

That especially when I give it a better prompt. When I go, "Oh, I see what you heard me say. Let me try that again. And so, yeah, it's all about helping us all become judicious users of technology. 

Kim  23:24

Yeah. So, we've mentioned a couple of... You know, I think I've brainstormed a little bit here about doing something at the beginning of a time where kids are going to learn some strategy or like maybe midway to see if they can see their own growth. So, I think there's a couple of different times that you could use this routine. But what's really important and valuable is the discussion that comes along with it. I think kids sharing how they solve the problems. Like, "I did find a way that was really efficient." But what I especially love, though, is when kids write their own.

Pam  24:00

Oh! That is a fantastic idea!

Kim  24:04

Yeah. When kids are able to see outside of strategy and pick numbers for which strategies would work well. That's really, really (unclear).

Pam  24:12

So, you're suggesting that you could actually say to kids, "Alright, we've played this game, and you now like..." "Okay, I beat the calculator. Oh, well that one you typed it in just as fast as... Okay, I wonder what strategy." That you're actually going to say to them, "Hey, you know strategies. Write some problems that you could figure using what you know quicker, faster, better than. You would never type that in the calculator because you could just..." That's a fantastic task. (unclear).

Kim  24:35

Well, and aren't we always looking for something that we can extend? (unclear). 

Pam  24:38

Oh, nice extension. 

Kim  24:39

Yeah. 

Pam  24:40

Nice extension. That was well done, Kim. Whoo! I like it. I like it. And this is also an example of a routine that is repeatable because, like you said, we can bring it up in different topics. And so once they kind of have the routine down, then we can kind of pick it back up and use it. And so, that's a wonderful thing that we like. We don't like to spend time on stuff that we can't bring back up in different ways and tweak. Yeah, nice. 

Kim  25:04

So, one more routine that we both very much enjoy. Got some favorites, but this one might be raising to the top again for me. 

Pam  25:12

Might be coming to the top depending on what we're talking about. 

Kim  25:14

Yeah.

Pam  25:14

Sweet. Alright. 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!