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 achievements, but they're not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop.

Kim  00:30

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

Pam  00:37

And we invite you to join us to make math more figure-out-able. Not rote memorizable but figure-out-able. Bam. Hey, Kim.

Kim  00:46

Hi.

Pam  00:46

Alright, we're going to dive in today. Kim was like, "Surprise me." And I was like.

Kim  00:51

Uh, yeah. It's a little scary to say that to you.

Pam  00:54

Well, like I said, you're the one who did it. You're like, "Surprise me." I'm like, "Ah, sweet!" So, Kim has no idea what we're doing today.

Kim  01:00

None. No idea.

Pam  01:02

Super fun. So, I was recently in an eighth grade classroom. We were filming. Filming and videoing at the same time. Fideoing. We were... What is the right word filming? Videoing? You could say either. We were shooting? Shooting. Shooting is probably not a good thing to say with schools.

Kim  01:17

We were taking video of you...

Pam  01:19

There you go.

Kim  01:19

...and students.

Pam  01:20

I was teaching a Problem String to eighth grade students. We filmed all day long. It went super well. I had a really good time. The teacher was amazing. The instructional coach was in the room. We had such a good time. We put these videos in our membership, so teachers can watch Problem Strings with real kids. And, you know like, not fake kids. How we pull out the... I know you hate it when I say that.

Kim  01:40

I hate when you say real kids. 

Pam  01:42

Which is why said not fake kids. 

Kim  01:43

I know.

Pam  01:43

Real classrooms. They're not staged. That's what I mean. What I mean is it's not a stage. Like, we don't cut stuff. Whatever happens, happens. Sometimes I like, "Oh, my..." It's like, yeah. You know, every once in a while. And then we actually talk over them for our membership to like give kind of a voiceover. Anyway, blah, blah, blah. When I was doing it, I did this particular string. It went super well. And it was kind of fun to watch kids kind of come alive a little bit because you could tell they had done some of this kind of math, but that we were actually kind of figuring it out. And yeah, okay. So, here we go, Kim, ready? 

Kim  02:15

Yeah. 

Pam  02:16

Kim. 

Kim  02:16

I guess. 

Pam  02:17

If I were to give you a bunch of numbers and ask you to find the Mean of those numbers.

Kim  02:22

Yep. 

Pam  02:22

Do you remember which? Like, there's Mean, Median, Mode. That's kind of social knowledge to know which word goes with the thing? 

Kim  02:29

Yeah. The Mean is the average. 

Pam  02:31

And what do you mean by "average"? Maybe I'll bive me some numbers. Can I give you numbers. 

Kim  02:36

Yeah. 

Pam  02:36

Okay. So, if I gave you these numbers. 10, 10, 10, 10, and 10. 

Kim  02:44

Pretty easy. 

Pam  02:45

So, I think you have five 10s. Do you have five 10s?

Kim  02:47

I do.

Pam  02:47

Let's make sure the same. Okay, you have five 10s. So, what's the Mean of that set of 5? It's 10. Okay, but maybe that's the Mode. Do you have that wrong? Like, what's the...

Kim  02:58

I mean, it's that too.

Pam  03:00

What's the Median then? 

Kim  03:01

Still 10. 

Pam  03:02

Okay, and how do you know? So, let's just briefly like, what are. Do you remember what Mean, Median, and Mode are?

Kim  03:06

Mode is the most often occurring.

Pam  03:09

Okay.

Kim  03:10

What were the other ones?

Pam  03:12

Median. 

Kim  03:13

Median is if you consider the lowest amount to the highest amount, which one is in the middle?

Pam  03:19

You're going to line them up, and then one right in the middle is the Median. Okay. And then the Mean, like you said, was the average. Yeah. And so, one way to think about that is if I kind of on my paper here... Say I had a bunch of like grid paper, and I kind of drew a column of 10, and a column of 10, and another column of 10. Like, five columns of 10. 

Kim  03:21

Yep. 

Pam  03:23

And I was like, "You know like, where do they even out?" You're like, "Well, they're pretty even already."

Kim  03:42

Yep.

Pam  03:42

And so, a way to kind of think about the Mean is like kind of where those columns even out.

Kim  03:47

Yep.

Pam  03:47

In this case, duh, it's just like 10. Okay.

Kim  03:49

Yep. 

Pam  03:50

Next problem. Here is your set of numbers. Ready? 9, 10, 10, 10, and 11? 

Kim  04:00

Yep. 

Pam  04:01

Now, before you go too far. Can I just... On the board, I would have a column of 9, a column of 10, 10, 10. So, three columns of 10. And then a column of 11. And then the question is, what is the Mean? 

Kim  04:14

Are you ready for me to answer?

Pam  04:15

I am ready. 

Kim  04:16

Okay.

Pam  04:17

Sorry.

Kim  04:18

It's 10.

Pam  04:18

And how do you know it's 10? 

Kim  04:20

I'm going to give 1 from the 11 to the 9 and now everybody has 10. Because

Pam  04:25

You just sort of evened it out. And bam. And literally I could kind of circle that 1 on the 11, that 1 that was kind of above 10, and kind of scoot it over to the 9, and now they're all 10. Cool. Okay.

Kim  04:37

Yep.

Pam  04:37

Next problem. This time there's only four values. 

Kim  04:40

Okay.

Pam  04:40

5.

Kim  04:41

Yep. 

Pam  04:42

5.

Kim  04:43

Yep.

Pam  04:44

15. 15 

Kim  04:47

Okay.

Pam  04:47

Okay, so listeners maybe pause. Ask yourself what's the Mean? Kim, go ahead when you're ready.

Kim  04:53

It is 10.

Pam  04:55

Okay, now that was the problem before. You're on the right problem? 

Kim  04:58

I am.

Pam  04:59

Tell me what. Or tell us why. Ooh, look at me. I just caught it. Tell us why. 

Kim  05:03

Well, there's only one of you here, so that's fair. 

Pam  05:05

Well, I mean, there's hopefully more than one person listening to the podcast. Please. Thank you, listeners, for making this worth it. We appreciate your listening. And hey, maybe I'll be actually really serious there. We appreciate your time and attention. Like, that's not a trivial thing. Thank you. Thank you. Sit up tall. Were like grateful. Okay, go ahead. 

Kim  05:28

Okay, so I'm going to take 5 from the 15 and give it to the first 5. 

Pam  05:34

Okay.

Kim  05:34

And I'm going to take 5 from the other 15 and give it to the second 5, so now everybody has 10. 

Pam  05:39

Bam. You evened it out, and so the mean is still 10.

Kim  05:42

I'm not a mom of four kids, but that's how I would do it at my house.

Pam  05:46

You'd make sure it's all even?

Kim  05:48

Yeah.

Pam  05:48

Nice. Okay, next problem. 6, 7, 9, 10. And we want the Mean. This time we want the Mean. 

Kim  06:00

Yeah. This time.

Pam  06:04

This time.

Kim  06:05

Okay. Do you want me to tell you where? You just want the number? The amount?

Pam  06:08

Well, tell me the Mean first, tell us the Mean first, and then tell us how.

Kim  06:15

Well, I have to think for a second. I'm going to take two (unclear)...

Pam  06:18

Thinking is allowed. Thinking is encouraged.

Kim  06:21

Encouraged even. I'm going to take 2 from the 10 and give it to the 6. Now, those are both 8. And I've got a 7 and a 9. Okay. So, I'm going to give that 1 from the 9 to the 7, and now everybody's got 8. So, that's just 8.

Pam  06:38

(unclear). So, you're saying the Mean is 8. 

Kim  06:39

It is.

Pam  06:40

We just kind of move things around a little bit, and bam, once are all kind of even you're good. Okay, next problem.

Kim  06:45

Yep. 

Pam  06:45

16 (unclear).

Kim  06:46

Wait, new paper, new paper. (unclear).

Pam  06:47

Oh, sorry. New paper.

Kim  06:48

16. Mmhm. 14

Pam  06:49

Do you had a pencil or pen?

Kim  06:50

I do. And a tiny little post-it. I didn't know I was going to write all these numbers.

Pam  06:53

Oh, sorry. Yeah.

Kim  06:54

That's okay. (unclear)

Pam  06:54

You're going to need a few post-its. 14.

Kim  06:56

14. Okay.

Pam  06:58

10.

Kim  06:58

10.

Pam  06:59

8. 

Kim  07:00

8. Okay. 

Pam  07:02

And the Mean, please. This time we're going to find the mean.

Kim  07:06

All the Mean. Alright, I'm moving 4 from the 16 to the 8. Now, those are both 12. I'm taking 2 from the 14, give them to the 10, so now they're 12. So, the mean is 12 all the way around. 

Pam  07:17

(unclear). So, you're just being able to just really move these guys around, and you're comfortable that then you kind of everybody has the same, and that's the mean. Okay, that seems kind of helpful. This next series has 10 numbers. Excuse me, five numbers 

Both Pam and Kim  07:30

Five numbers.

Kim  07:30

Okay.

Pam  07:31

Sorry, five. One of them is 10. That's what I saw. Okay. So, 12.

Kim  07:35

Yep. 

Pam  07:35

8.

Kim  07:36

Mmhm.

Pam  07:36

10. That's the 10 I saw.

Kim  07:38

Mmhm.

Pam  07:38

12. 

Kim  07:39

Mmhm

Pam  07:40

8.

Kim  07:41

Yeah, so that Mean's 10. 

Pam  07:43

Ooh. You didn't even give people a chance to think (unclear).

Kim  07:46

Oh, I'm sorry, people. I'm sorry, one listener.

Pam  07:49

What was happening that you were able to... Like, it almost seems like you were kind of doing something as I went.

Kim  07:54

Yeah. When I saw the 12 and 8, I was kind of hoping that it would be like 10. 10 was right between them. And then I got excited when you said 10. And then the next 12 and 8, 10 was right between them. So, I was kind of thinking about what's between the two numbers you gave me.

Pam  08:12

Nice. So, you're welcome for giving you problems that kind of made you excited. Okay, next set. 24. 

Kim  08:21

Yep. 

Pam  08:21

24. 

Kim  08:22

Yep.

Pam  08:23

20.

Kim  08:24

Okay.

Pam  08:25

16. 16. 

Kim  08:28

Okay. 20 It is. 

Pam  08:30

Whoa!

Kim  08:31

Dang it! I'm sorry, listeners. Shoot!

Pam  08:34

Tell us about that.

Kim  08:35

I need to put my thumb up.

Pam  08:37

Oh, yeah. That would be good. So, listeners, get ready. Get your thumb on your podcast and push pause.

Kim  08:43

Right. 

Pam  08:43

So, that you get a chance to think before Kim just blurts out the answer.

Kim  08:46

I will stop doing that. I apologize. 

Pam  08:48

Because again, that kind of seemed like you were thinking something as we were going. Which, to be clear, I didn't really catch that with those eighth graders. I don't know that there was a whole lot going on as a... Maybe, maybe. Maybe there was, maybe. I bet there was somebody in the room. 

Kim  09:00

So, the 24 and 24...

Pam  09:02

Yeah.

Kim  09:05

...are both 4 up. They're connected to the 16 and 16. So, like, there 4 above the 20. And the 16 is 4 below the 20. 

Pam  09:13

Mmm, mmhm, mmhm. It's almost like you can kind of feel this like balance thing kind of happening. Like, you've got these extras over here. You've got less over here. If you kind of throw those... What are you throwing? 4? Are you throwing 4 onto the 16? From the 24 to 16.

Kim  09:28

From each 24, yeah. 

Pam  09:30

Do you know what you were thinking when I said 24, 24, 20?

Kim  09:33

I was thinking about how much above the 24 was from the 20. Like, I was up 8 at that point from the 20. 

Pam  09:41

Yeah. 

Kim  09:41

But also I didn't know that 20 was going to be the...

Pam  09:46

The middle?

Kim  09:46

The Median until you said the 16. 

Pam  09:50

It is the Median, isn't it? 

Kim  09:51

Yeah. 

Pam  09:52

Is it the Mode?

Kim  09:55

No. Once I average them, 20 would be the mode but it's not actually.

Pam  10:02

Which values are happening the most often.

Kim  10:05

20 is not the Mode. 24 and 16 are the Mode. 

Pam  10:08

There you go. Okay. And that's (unclear).

Kim  10:09

Oh, I was saying if you let me give you the readjusted numbers of 20, 20, 20, 20, 20.

Pam  10:15

Oh, gotcha. 

Kim  10:15

That's not how it works, Pam.

Pam  10:16

So, in this particular... Nah, you're good. In this particular one, we would call that bimodal. There's two modes.

Kim  10:21

Okay.

Pam  10:22

So, these numbers are happening (unclear)

Kim  10:23

Look at you give me words as they occur.

Pam  10:26

we would call that just in time vocabulary, right? Okay, just to time. Cool. Alright, Kim.  And

Kim  10:31

Yeah?

Pam  10:32

We only have two problems left. You're doing fantastic. This time, maybe take a breath.

Kim  10:36

I feel like the clunker is coming.

Pam  10:38

Oh, I don't know. I was just saying maybe take a breath before you do this one. 

Kim  10:41

Okay. 

Pam  10:42

Okay, 12, 12, 12, 10, 10. Don't say it. Don't say it. Don't say it. Everybody thinking. Everybody thinking.

Kim  10:53

Mmhm.

Pam  10:54

So, three 12s and two 10s, yeah?

Kim  10:56

Yeah. 

Pam  10:56

Okay. Alright. I have no idea what you're going to do. I'm super curious.

Kim  11:02

What do I want to do? I'm seeing the 12s are 2 above the 10s. I'm thinking 11 is going to be somewhere in there. So, if I gave 1 from the 12, the two 12s that I first wrote down, to the 10s, then I would have 11, 11, 12, 11, 11.

Pam  11:23

Yeah, I'm just going to say that again. Just so podcast listeners who are driving can hear that. 

Kim  11:27

Okay.

Pam  11:27

So, now you have 11, 11, 12, 11, 11. 

Kim  11:30

Mmhm.

Pam  11:31

So, that could have been a different set that I gave you that would have the same Mean. 

Kim  11:35

Yeah, mmhm.

Pam  11:35

Okay, cool. That seems kind of interesting. Okay. 

Kim  11:38

So, I'm going to make that middle number 11 by taking away 1. 

Pam  11:42

Okay.

Kim  11:43

I'm going to kind of like hold it to the side. So, now I have a whole bunch of 11s. 

Pam  11:46

All 11s, mmhm. 

Kim  11:47

Mmhm. But I have this extra 1. 

Pam  11:50

Uh-oh. What are you going to do with (unclear).

Kim  11:51

I know. I need to share it among all five. So, everybody gets a 0.2

Pam  11:56

Haha, I knew you were going...

Kim  11:57

11.2

Pam  11:57

Did you think percent? I was super curious if...

Kim  12:00

I didn't. I thought about $0.20.

Pam  12:02

(unclear) Alright, well, I mean, that totally works, right? It's like you have $1.00, and you have to spread it among those five. That was the hardest thing for the eighth graders. They got the 11s like you did, and then there like, "But there's this extra 1."  Yeah.  "What did we do with that 1?" And I was like, "I don't know. I don't know. What can you do with that $1.00 and those five people?" And then they're like, "Ah!" And then some of them still really had to think about $1.00 divided by 5.

Kim  12:23

Sure. 

Pam  12:23

Which is interesting. You know, one of the things that we talk about is there are some fractions that we really ought to spend more time on, so kids kind of own more. Like, there are special fractions that matter more. And fifths, I think, would be one of them. Okay, so once you spread that 1 among those 11s, and you got that 0.2, the question was what's the Mean?

Kim  12:45

11.2 

Pam  12:46

11.2. So, they're all at 11 and that extra 0.2. And I had written down 11 and 1/5. Oh, okay. So, should I mark you wrong?

Kim  12:55

Absolutely. NOT!

Pam  12:57

NOT! Alright, cool. Alright, Kim, last problem of the string.

Kim  13:02

Okay.

Pam  13:02

I hope this has been kind of fun for you. 

Kim  13:03

Yeah, I like it.

Pam  13:04

Okay, here we go. Ready? 12, 10, 12, 10, 12. 

Kim  13:12

Okay. 

Pam  13:13

Alright, five numbers. 

Kim  13:14

Yep. 

Pam  13:15

12, 10, 12, 10, 12. Alright, I'd love to hear your thinking out loud after. Pause the podcast listeners and figure it out. Okay, come on back, Kim.

Kim  13:22

I love when you say pause, and then you keep talking.

Pam  13:27

You hate it when I do that.

Kim  13:28

"Hey, everybody go think. I'm going to keep talking while you do that."

Pam  13:31

You're back there making faces at me.

Kim  13:33

Oh, I shake my head every time.

Pam  13:35

Kim's in the back of a workshop. And I do that. It's terrible. I will get better one day. I don't know when. I'm not going to promise when because it hasn't happened yet. But I'll be like, "Alright, everybody think," and then I keep talking. And Kim is in the back just shaking her head. Just looks at me like, "You dork." I don't know if dork is... Is dork the word you're thinking? 

Kim  13:54

No, I don't ever say dork. (unclear).

Pam  13:56

We don't swear on this podcast, Kim, so don't say what you're actually thinking.

Kim  14:01

I literally am like, "Oh, no. No. Stop talking!"

Pam  14:05

That's what you say. Your back there going, "Pamela." 

Kim  14:07

I am. 

Pam  14:08

"Pamela, you big dork." You don't say the word "dork"? What's wrong with the word "dork"? I'm kidding. I'm kidding.

Kim  14:12

I do not say "dork". Okay.

Pam  14:13

Okay.

Kim  14:14

So, they would all be 10 except for there's 4 extra. The two 12s have 2 extra, so there's 4 extra that then needs to be redistributed. 

Pam  14:25

Okay, so hang on a second. Let's make sure we have the same numbers. 

Kim  14:28

12, 10, 10, 10, 12.

Pam  14:29

No, sorry. 

Kim  14:31

Oh.

Pam  14:31

So, you were about to give me a great answer for a different problem.

Kim  14:34

Did you not say that? Did I hear (unclear).

Pam  14:35

I might have. I don't know. 

Kim  14:37

Okay. 

Pam  14:38

But here's what I (unclear).

Kim  14:39

We'll try again.

Pam  14:39

Here's what I meant.

Kim  14:40

Okay.

Pam  14:41

12, 10, 12, 10, 12.

Kim  14:42

12, 10, 12, 10, 12.

Pam  14:46

Yeah. 

Kim  14:48

Okay, so I have 6 extra. 

Pam  14:50

Okay. 

Kim  14:50

From the 10. 

Pam  14:51

Okay. 

Kim  14:52

And I don't know why I just thought about it that way. 

Pam  14:54

Yeah, I don't either. 

Kim  14:55

Why did you think about it that way? Because I'm thinking of a different way. 

Pam  14:58

Yeah. Okay. 

Kim  14:59

Which is Super fun. You want to hear (unclear)? 

Pam  15:00

I do. 

Kim  15:01

Okay. Maybe because there were a whole bunch of 10s. Ooh, I wonder if that's why. There were a whole bunch of 10s in the first thing, so I was kind of thinking like here's a base level everybody has and what's the excess?

Pam  15:14

Mmhm, mmhm.

Kim  15:14

So, that's like a way I could think about it. 

Pam  15:16

Yeah. 

Kim  15:16

But what I've been doing is redistributing kind of like to your neighbor as the other way of thinking about it. 

Pam  15:22

Yeah. 

Kim  15:23

So, because those less 10s this time, I kind of want to just have the 12s give 1 to the 10. 

Pam  15:33

Okay.

Kim  15:33

So, then I have 11, 11, 11, 11, 12. 

Pam  15:38

Okay. 

Kim  15:39

But I actually like that less. So, then I would do what I did before, and I just have 1 extra. Everybody would be 11, but I have 1 extra 1 because that last one was a 12.

Pam  15:50

Mmhm.

Kim  15:51

So, then I could deal out that 1, but I kind of actually like the first way I was thinking about it where everybody's got a base of 10. 

Pam  16:00

Okay. 

Kim  16:01

And if that were true, then I have 6 extra because there were three 12s. 

Pam  16:06

Yeah.

Kim  16:06

Man, I hope people are writing stuff down.

Pam  16:07

Haha.

Kim  16:09

So, then I want to redistribute that 6.

Pam  16:12

Over the five (unclear). Over the five terms, mmhm. And so, how does that work 6 divided among those? (unclear).

Kim  16:13

Over the five. 6 divided by 5?

Pam  16:21

Oh, is that what you're doing?

Kim  16:23

I have 6 to give out to the 5 numbers, 5 people, 5 numbers.

Pam  16:28

Okay. And they were all at 10. 

Kim  16:31

Yeah. So, they get they get another 1.2 because 6 divided by 5 is 1.2. So, now they... 

Pam  16:37

That is not what I thought you were going to do. 

Kim  16:38

Oh! 

Pam  16:39

Once you had all those 11s, and you had that... No, sorry, 10s. You had all those 10s, and you had that extra 6. I thought... Or at least this is what my brain did. My brain said, "Ooh, of that 6, I'm going to give out 5. 

Kim  16:51

Okay, yeah.

Pam  16:52

1 each. And then I'm back to the 1, the 0.2..  The 1 divided among 5. 

Kim  16:55

Yeah. Mmhm.

Pam  16:57

But I like your 1.2. So, you thought about six-fifths as 1.2. Oh, that's nice.

Kim  17:02

That's super fun. I liked this one. You were right. I would like.

Pam  17:06

So, Kim, there might be teachers out there thinking, "Well, alright, Pam. So, you had this fun with a little bit. You know like, do kids really own Mean?" Well, I would want to have a conversation at this point about what does it mean? (unclear) What were you doing in those last couple? What were we doing all over the place? Like, what does Mean mean? And this is a good example of social versus logical knowledge where the social knowledge is, is it Mean, Median, Mode. Which one? Okay, Mean. But now, let's get the logical knowledge. Let's get the experience of balancing. Like, nowhere in here where we lining them up (unclear).

Kim  17:37

Add them up, and then divide.

Pam  17:39

Well, that. We weren't doing that either. But we also weren't lining them up and finding the middle. That would be the median. And we also weren't counting the ones that shows up the most. We weren't finding the mode. And so, we really had this experience was all about Mean. And I don't know if you notice, but a couple times, I was like, "And this time, we're finding the Mean."

Kim  17:43

Right. Mmhm.

Pam  17:56

And that was a purposeful teacher move on my part to emphasize that this experience is this averaging thing is Mean, and it's called Mean. We're doing this thing, and this is what it's called. We're doing this thing to build the logical mathematical sense of the balancing and the redistributing until they're all even. And that thing's called the Mean. So, that was a sort of purposeful way of bringing in logical mathematical and social. Go ahead. 

Kim  18:18

Yeah. And I would argue that students using this experience, having this experience, understand what average or Mean means far more than, "Here's what mean is. Add a stack of numbers. Divide it with your calculator." But they have no clue that it's about balancing, about redistributing to see what everyone could have if it were equal. 

Pam  18:43

Yeah. Yeah, (unclear)

Kim  18:44

Which is what Mean is. Yeah.

Pam  18:46

And we want to build that sense. Now, I could ask them those last two problems, and I could say, "Tell me more about those last few problems. So, once you guys kind of found that base, and you had that extra, what were you doing?" And they're like, "Yeah, it's kind of like we had to like divide it among all of them." And I'm like, "Huh. So, mean has something with like taking all of it and dividing it evenly among. That seems kind of important." Now, we're kind of starting to connect. It's not like I don't ever want kids to think about the ways of finding the mean that they could add the total and divide it evenly among. But like they are...

Kim  19:21

Now, they know what it means. Yeah.

Pam  19:22

They know what it... It's funny how we use the word Mean. Hey, maybe sometime we'll do the next string that I followed up with that was kind of fun. Hey, thanks, Kim. This was fun. 

Kim  19:31

Yeah, thanks. 

Pam  19:32

Alright, ya'll. Ya'll, statistics is figure-out-able! Who knew? Bam, measures of center. Kind of fun. Ya'll, thank you for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit math is figure-out-able.com And keep spreading the word that Math is Figure-Out-Able!