# Math is Figure-Out-Able!

Math teacher educator Pam Harris and her cohost Kim Montague answer the question: If not algorithms, then what? Join them for ~15-30 minutes every Tuesday as they cast their vision for mathematics education and give actionable items to help teachers teach math that is Figure-Out-Able. See www.MathisFigureOutAble.com for more great resources!

## Math is Figure-Out-Able!

# Ep 219: Find the Mean, Median, Mode

Statistics Problem Strings? Yes Please! In this episode Pam and Kim demonstrate a Problem String that helps students develop intuition for the meanings of mean, median, and mode.

Talking Points:

- Names are social knowledge, but the feel for what they mean in the data is logical mathematical knowledge
- Experience with data helps students develop real understanding of mean, median, mode
- Changing the data values and comparing how that affects the measures of central tendency deepen understanding

Check out our social media

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Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

Pam 00:00

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

Kim 00:10

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 00:19

We know that algorithms are amazing human 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 00:33

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

Pam 00:41

And we invite you to join us to make math more figure-out-able. Hey,

Kim 00:45

Hey, Pam.

Pam 00:46

Hey, Kim, How's it going?

Kim 00:47

Good. Hey, before we start today's episode, we just wanted to share that we heard from a friend of ours...not too long ago, actually...about the new intro and a particular portion of it where we have started, or you've started, saying "mather". "Hey, fellow mathers." Yeah. And so because we we usually have said "mathematicians", and this particular person reached out and said, "Hey, so did you know..." And... "Why haven't you been giving credit?" Because Deborah Parrott has been talking about "mather" as a word for quite some time. And so this person very kindly pointed that out to us and said, you know, "Why didn't you give acknowledgement there?" And, unfortunately, I had to admit that I didn't actually know Deborah's work. And I'm so happy that this person pointed it out because we would not want it all to seem that we were not crediting people with their work, and we are really excited that everybody is helping to move math forward for all students. So, thank you for pointing that out to us. And, Deborah, thank you for helping improve math for everyone.

Pam 01:57

Yeah, absolutely. So, thank you. Yeah, we definitely want to give credit where credit is due. And so... You know, I think I've actually met Deborah, but I had not heard that "mather" was her thing. And so, fantastic. Great minds, yeah. And let's give credit where credit's due.

Kim 02:14

Cool. Alright, so I wanted to share a review with you also, Pam. I know you haven't heard this one, (unclear).

Pam 02:20

I love reviews.

Kim 02:20

I know. The title is "Excellent!" And I wish I could give credit for this person, whoever it is, but it's a series of letters as the name. Which is probably what I would do if I were leaving a review. Coming up with a catchy name for the (unclear).

Pam 02:34

Okay, that's hilarious because I'm looking at it right now. It's B, F, H, G, T,

Kim 02:38

Yeah.

Pam 02:41

(unclear).

Kim 02:41

Hey, well, don't read. Don't read any of it.

Pam 02:43

Okay, I'm not reading. I'm not reading.

Kim 02:44

So, the review says, "One of the best podcasts I've listened to when it comes to teaching math and helping students truly strategize and think. Short and sweet and easy to listen to on the way to work."

Pam 02:56

Bam!

Kim 02:57

Yeah, it's fun.

Pam 02:58

That's awesome.

Kim 02:59

The danger about listening on the way to work, though, is in so many of the episodes we do math. And yeah, that could be tricky. You might want to write something down.

Pam 03:08

Yeah, that's fun.

Kim 03:09

Don't do it. Don't do it while you're driving.

Pam 03:11

Well, (unclear), thank you so much for your review.

Kim 03:18

I thought your microphone messed up for a second. I was like, "Oh no. Here we go."

Pam 03:24

Oh, no. I was just trying to pronounce that name. Hey, seriously, thank you for the review. I don't know, listeners, if you know, but if you review and give us a rating, that helps other people find the podcast because it does the algorithm thing where then it shows the podcast to more people, and then we can spread the word that Math is Figure-Out-Able more and more. So, we appreciate that. Thanks tons, yeah.

Kim 03:44

Okay, let's get on with it. So, the last episode, we started talking about Mean. And we actually were talking about making sense of what the average is, other than just telling kids, "Hey, here's these numbers, add them up and then divide by however many you have." So, we're going to continue that work today. (unclear).

Pam 04:03

Yeah, rather than having... Excellent. Rather than having Mean be a procedure where kids go, "Oh, yeah. Is that the one where you add them all up and divide by the number? Or is that where you..." Instead of that being the feeling tone, the intuition, the perception that kids have for Mean, what could we do to make things more sensible, figure-out-able, so that kids actually have a feel and perception for Mean, and Median, and Mode? And Mean, Median, and Mode are things that we call "measures of central tendency". They measure the tendency to the center of data. And so, those measures of central tendency can be things that we can give kids a feel for. Now, there are parts of those measures of central tendency that are social, the conventional that we deem to be so. Somebody gave the mean, the name Mean. Somebody called the thing that happens in the middle, the middle number in the data set were called the Median. And then the thing that happens the most often, the data entry that happens the most often we call the Mode. Those names are social. It's conventional. But the feel for what those mean in the data, that's logical, mathematical. And so, we need both. We need to tell kids the name of it and give them some sense of why we gave them that name. But also help them develop the perception of what a Mean tells you about the data. And develop a perception about what the Median can tell you about the data. And also what the Mode can tell you about the data. But also, we can change up the data to help give that perception. So, today, Kim.

Kim 05:43

Yes.

Pam 05:44

We're going to do a Problem String. And I'm going to give you a data set. And when I give you that data set, I'm going to ask you for the Mean, Median, and Mode. And you can tell us in any order. If I were doing this Problem String with real kids, not fake kids like you, Kim. You're a fake kid. If I was doing this with students, on the board, I would write the data set and I would also have three columns to the right of it where I would have the word Mean, and the word Median, and the word Mode. And we're going to fill those in. But I would also tell kids. Be very upfront. You can fill them in in any order. So, whichever order you want, I'm happy to have you fill those in. So, Kim, on my paper right now, I have the word Mean, and the word Median, and the word...I'm actually writing it right now...Mode.

Kim 06:32

Okay.

Pam 06:33

(unclear) three columns. And then next to that (unclear) kind of on the right side of my paper. Next on the left side of the paper, I'm about to give you a data set. Are you ready?

Kim 06:40

Okay, yep.

Pam 06:40

Alright, if the data sets a five value data set, if that data set is 20, 20, 20, 20, and 20. If that's the data set five 20s.

Kim 06:53

Mmhm.

Pam 06:54

Pick one of those. Anyone that you want to pick first that you're just like, "Well, bam, It would have obviously be..."

Kim 07:01

Mode. The most often occurring, that's 20.

Pam 07:05

Alright.

Kim 07:06

The Median is the one in the middle. That's 20.

Pam 07:09

Okay.

Kim 07:11

And the Mean is also 20.

Pam 07:14

And how do you know? Is that too... I don't know. Can I ask you that? If the values are all the same...

Kim 07:23

Yeah.

Pam 07:23

...then they're all even, right? If we were to line up like... And if I were to have a column up on the board of like grid paper that was 20 high, and I had five columns of... Sorry rows. 20 rows, and I had 5 columns. And I was like, "Hey, if this grid paper right here, if I colored in this first column up to 20, and this column up to 20, and this column..." They were all up to 20. They're kind of even, right? It's like evened out. And which of... If I was describing this evening outness, which of the Mean, Median or Mode is that describing?

Kim 07:57

The Mean.

Pam 07:58

The Mean? It's like I sort of evened it out, and that's kind of the meaning of the Mean. We did that in the last episode. And so, yeah, okay. And then do you see the median in that? In that look, if you kind of are looking at these five columns of 20? I don't know if you see the Median. Do I see the Median?

Kim 08:13

Yeah, it's the one in the middle,

Pam 08:15

Well, and so we'd have to line them up, right? Like...

Kim 08:18

Yeah.

Pam 08:18

...I could put those values in a different order. So, we'd have to kind of line it up, and then the value that shows up in the middle. Okay, cool. I don't know if the Mode shows up in that picture.

Kim 08:28

So, if they were different levels, it's the one that is the most often occurring. Like, where they... Like, if you look horizontally, which one has the same value all the way across.

Pam 08:43

Gotcha. And that one that shows up the most often, that would be the Mode. Cool. I like it. Alright, that was the first problem. That was really quick. Second problem. The first... So, again, it's a five value data set.

Kim 08:54

Okay.

Pam 08:54

The first value is 19, 20, 20, 20, 20, and 20. So, 19 and then four 20s.

Kim 09:04

4?

Pam 09:05

Yeah. What are you thinking about?

Kim 09:07

19, 20, 20, 20.

Pam 09:12

I'm sorry. Typo, typo, typo. The first number is actually 18. My bad.

Kim 09:18

Say them again. You want?

Pam 09:21

18.

Kim 09:22

Yep.

Pam 09:23

20, 20, 20, and the last one's 22. Holy cow! Did I sleep last night?! Like, I'm a mess.

Kim 09:30

Say them one more time.

Pam 09:31

Whew! Okay. 18, 20, 20, 20, 22.

Kim 09:38

18 20, 20, 20, 22

Pam 09:41

Yeah. Wow, sorry about that. I was just... That was intentional to fake everyone. Not really. Not in any way. I don't know what was wrong with me right there.

Kim 09:49

It's a good thing I have an eraser. Okay, alright. So, what do you want me to answer first. You don't care. Okay, I'm going to go look Mode is 20 because there are three 20s.

Pam 10:00

Okay.

Kim 10:01

I'm going to say that the median is also 20.

Pam 10:04

Because?

Kim 10:05

Because if I line them up in order from least to greatest, then 20 is in the middle.

Pam 10:09

Okay. And you're welcome because I kind of gave them to you lined up.

Kim 10:11

Thank you for that. And I'm going to say that the mean is also 20 because if I take 2 from the 22 and give it to the 18, then I have five 20s again.

Pam 10:23

And they're all kind of evened out.

Kim 10:25

Yep.

Pam 10:25

Okay, nice. Cool. Alright, next value set. I'll try to do this one correctly. Here we go. First number is 18, 18, 20, 22, 22.

Kim 10:38

Okay.

Pam 10:39

Alright.

Kim 10:40

I'm going to start with Mean this time, and I'm going to say that the mean is also 20 because just like I previously did, I'm going to take 2 from one of the 22s and give it to the 18. But I'm going to do that twice this time because I have two 22s and two 18s.

Pam 10:56

Nice. Nice. Mmhm. And when you do that, they're all 20.

Kim 11:00

Yeah, all 20. Mmhm.

Pam 11:01

Okay, okay.

Kim 11:02

And then I'm going to say that the Median is 20 again.

Pam 11:06

Okay.

Kim 11:07

And then, the Mode. I have 2 Modes.

Pam 11:11

Ah. Bimodal.

Kim 11:12

Bimodal. 18 and 22.

Pam 11:15

Nice. Alright, that makes a lot of sense. Cool. Next value set. Let's see, we've done three, right? So, I'm on the fourth one. Okay, next one. 18, 18, 18, 22, 22.

Kim 11:29

Mmm. Okay.

Pam 11:31

Oh, and you know what? Gah, Kim. There's a thing that I wish. So, you know, teachers, none of us are perfect. And here, I'll just admit I'm not perfect. So, I'm just realizing that when I gave you the first one, the first value set, data set, you said that all the Mean, Median, and Mode were all 20. And then, when I gave you the second data set, you said that the Mean, Median, and Mode were all 20. And I wish at that point I would have said, "What changed in the data set and what didn't change?" Because the Mean, Median, and Mode all stay the same.

Kim 12:01

Right.

Pam 12:01

But some things changed and didn't change.

Kim 12:03

Mmhm.

Pam 12:03

Do you mind? Can we back up? I know...

Kim 12:05

Sure.

Pam 12:05

...people in the car like, "What even were those?!" Yeah, so the first one was all 20s. And the second one was an 18, three 20s, and a 22.

Kim 12:13

Yeah. So, what changed is that I have two pieces of data, two values, that averaged to 20, but when left alone, they were not. So, 18 and 22, the average between those two is 20, but they are not 20 themselves.

Pam 12:31

Cool. And so how did that not change the Mode?

Kim 12:36

Because I also had a bunch of 20s. More of them than not 20s.

Pam 12:43

Okay. And then how did that not change the Median?

Kim 12:47

Because one was lower than the average and one was higher than the average.

Pam 12:52

How does that affect the Median?

Kim 12:55

Because the average is in the middle.

Pam 12:58

That's the Mean, right? The average in the middle?

Kim 13:01

What did you ask me?

Pam 13:02

(unclear). Oh, do you mean the middle value when you say the middle?

Kim 13:04

What did you ask me?

Pam 13:05

I don't even know. So, I'm asking about Median.

Kim 13:09

Oh, okay. So, you asked me, why is that? How does that affect the Median?

Pam 13:14

The Median, yeah. And so, when you said one is lower, one is higher, is that...

Kim 13:17

One is below the Median and one is above the Median.

Pam 13:21

Okay, and so does that affect the middle value?

Kim 13:26

Yes, because one's less than the middle and one's more than the middle.

Pam 13:30

And so, therefore, the middle didn't change.

Kim 13:32

Right.

Pam 13:33

Okay, okay. Alright, cool. I think so. Alright, from the second data set where it's 18, 20, 20, 20, 22

Kim 13:41

Yep.

Pam 13:42

To the third one, which was 18, 18, 20, 22, 22.

Kim 13:46

Mmhm.

Pam 13:47

The only thing that changed was the mode. So, I don't know if you can put any words to why did the Mean not change?

Kim 13:56

Because when you evened them out, they still evened out to all 20s.

Pam 14:01

Okay, that makes sense.

Kim 14:03

But this time there were two pairs that had to adjust. So, there were two less than the median and two more than the Median.

Pam 14:14

Okay, so even though those, the four numbers on the sides, of the Median changed, it's still in the middle.

Kim 14:25

Right.

Pam 14:26

Okay, okay. Alright, cool. Alright, so for the last data set. Now, I don't even remember. I gave you 18, 18, 18, 22, 22, right? Did I say those out loud? I have them written down. I think I said them. (unclear)

Kim 14:39

Say them again. 18, 18, 18, 22, 22. Yeah,

Pam 14:43

three 18s, two 22s. And then we haven't talked about that one yet, right?

Kim 14:48

Um... No, I haven't said anything about that.

Pam 14:50

Okay, what do you got for that one? Sorry, I'm a mess.

Kim 14:53

(unclear) 18s, 20s, and 22s (unclear).

Pam 14:54

I know I'm just a mess. Okay, so let me just say it one more time for everybody. This one that we're now currently talking about. 18, 18, 18, 22, 22.

Kim 15:03

Okay.

Pam 15:04

Okay. Which one you want to do first?

Kim 15:06

So, the first thing I notice is that there are... We've been living in the land of 20 is the Mean and the Median, but now I have more values below that. So, I think it's going to shift everything down a little bit.

Pam 15:21

Interesting. Okay.

Kim 15:23

So, I'm going to start with the Mean again.

Pam 15:26

Okay.

Kim 15:26

Well, no I'm going to start with Mode because 18 is the Mode. Just that pops out pretty (unclear).

Pam 15:30

Bam. You got three of them. (unclear).

Kim 15:31

Yeah.

Pam 15:32

Yeah, okay. Nice, that makes sense. Mmhm.

Kim 15:33

And same with Median. If I line them up, 18 is the middle.

Pam 15:38

Okay.

Kim 15:39

And then I'm going to take two from both of the 22s...

Pam 15:44

Mmhm

Kim 15:44

...and give them to the 18s.

Pam 15:46

Mmhm.

Kim 15:47

So, now my paper, I have 20, 20, 18, 20, 20.

Pam 15:51

Mmhm.

Kim 15:53

So, I have one number that's two less than the average.

Pam 15:59

Mmhm.

Kim 16:01

So... How do I explain? If I want to even out... So, I said two less, but I also have... Oh, I got to write the numbers again. 20, 20, 20, 20. I guess I could re-even it out and make it be some 19s.

Pam 16:22

Mmm, okay.

Kim 16:24

I don't really want to mess with that, though. So, I basically have 8 extra above 18... If everybody had 18, then I would have 8 extra to share among 5 numbers.

Pam 16:39

Ooh, okay.

Kim 16:40

So, if I'm sharing 8 among 5 number. So, 8 divided by 5.

Pam 16:48

Mmhm.

Kim 16:48

Then everybody would get 1 and 3/5, which is 1.6.

Pam 16:55

Nice.

Kim 16:56

So, then I'm going to say that the average is that 18 where they're at plus the 1.6. So, it's 19.6. Super cool. Is the Median.

Pam 17:08

Yeah. Yeah, I like it.

Kim 17:09

And I suppose I could have just subtracted. That was kind of lame. (unclear).

Pam 17:12

What do you mean?

Kim 17:14

I could have started. Which is I think what I wanted to do first, but I switched. I could have subtracted 0.4 from the 20 they were originally at.

Pam 17:28

Okay. Wait, so when you had 20, 20, 18, 20, 20, you could have? Where did you get the 0.4 from?

Kim 17:35

Well, in the previous problem, the average was 20, the mean was 20. Oh, okay. So, I could have said, "Okay, I have two less total to share among the 5 numbers." And that's 0.4

Pam 17:50

Two shared among 5. Two-fifths is 0.4. That's nice. So, I think you actually did two really nice things. One, was kind of based on looking at the new value set and really sharing out. You were kind of dealing out among the five values and kind of making sense of that. And the other was comparing to the one before and kind of using that sense of if you add it all together, the total was only two less. But it had that 2. That 2 from the total had to be shared among the five values. So, there's kind of these two ideas that I think we want to build both in students. We want to build this idea of kind of evening out the values, but we also want them to get this sense of I could add everything up together and share it among them. Not that we tell them that that's a procedure, that that's what it means. But they they can actually start developing that sense. Exactly like you just did. Kind of, ooh, from this one, it was only 2 different. What is 2 different mean over all of them? Ah, so that's a way to kind of develop that sense. Go ahead.

Kim 18:56

And it feels very different than 18 plus 18 plus 18 plus 22 plus 22 divided by 5.

Pam 19:03

Especially if you're going to long divide that total. Mmhm.

Kim 19:06

Mmhm.

Pam 19:07

Yeah, nice.

Kim 19:07

Yeah.

Pam 19:08

Yeah, super. Okay. Are you ready for one more?

Kim 19:10

Am I... I was going to say am I done?

Pam 19:13

You want to be done? No, not quite.

Kim 19:15

Okay.

Pam 19:15

Alright, the next one. 18... I know you're like, "Really?! Change the numbers!" Okay, so the first one's 18, 18, 22, 22, 22. Now, before you do any work, can I just point out that the one before it was 18, 18, 18, 22, 22. This one is 18, 18, 22, 22, 22.

Kim 19:36

Yeah, mmhm.

Pam 19:37

So. Okay. Just so the listeners can kind of hear that.

Kim 19:39

Yeah. So, I'm going to go back to... I have no idea which problem this was. But when we had 18, 18, 20, 22, 22 that that Mean was 20. Then the problem I just did was four-tenths less than that one, so it was 19.6 is the average. Mmhm. Was

Pam 20:05

the Mean. Mmhm.

Kim 20:07

Mmhm. This time, I'm noticing that... I'm kind of ignoring the 18s and 22s on the sides. Like...

Pam 20:14

Okay.

Kim 20:15

...we know those are going to work out to 20. So, I'm really just looking at the Median, and I'm saying that the Median when it was... When 20 was the fifth number and the Mean was 20.

Pam 20:27

Mmhm.

Kim 20:27

Then we went to 18 was the Median, and the average was 19.6. This time I'm going above. So, I'm circling on my paper. I'm not saying enough out loud. But in the previous problem, we were four-tenths under 20, so 19.6. This time, we're four-tenths above 20, so the average is going to be 20.4.

Pam 20:52

Yeah, that makes sense. I like it.

Kim 20:54

I don't know if it makes sense if you can't see anything on my paper, but.

Pam 20:56

No, yeah. I can totally follow that. Nice, nice. I like it. If I forced you to do something just with these numbers. I don't know if you have something. Because you kind of said that you were going to take 2 from the 22s to give to the two 18s.

Kim 21:12

Yeah. So, if I did that, then I would have 20, 20, 22, 20, 20.

Pam 21:17

Mmhm.

Kim 21:18

If I did some evening out. Then I would have 2 extra in the 22 that I would share among the 5 numbers. (unclear).

Pam 21:26

Nice. And 2 shared among 5 numbers is four-tenths. Yeah. Cool. I like it. I really like your your comparison one from the one before. Cool. So, you've given us the Mean for this set. If you don't mind (unclear).

Kim 21:38

Oh, I kind of ignored the other ones. Median is 22 and mode is 22.

Pam 21:43

Alright, cool. Last one. You have done a fantastic job. Here we go. 18, 18, 18, 18. So, five 18s. Last one is 22. Sorry, four 18s. Oh, my gosh. Four 18s. Blah! Four 18s, and the last one's 22.

Kim 22:02

Okay. I don't even know what my... I want to go back and compare, but I got to tell you, I've crossed out so much I don't even know if I know what any of the previous problems were.

Pam 22:10

Which, can we pause here for a second?

Kim 22:12

Yeah.

Pam 22:12

Teachers, this is why I make such a big deal about that it's less about what kids have on their papers, and it's super important that I have on the board a display that kids can refer to.

Kim 22:24

Yeah, that's really important.

Pam 22:25

(unclear) Yeah, so it wouldn't matter that you have... I love what you've been doing on your paper. But you would be able to look up at my paper and compare. So, it's a little unfortunate that we're on the airways, but (unclear).

Kim 22:38

Yeah, honestly, I want so badly to look at what I have for previous problems. I have everything crossed out and new numbers written above them.

Pam 22:45

Mmm, yeah.

Kim 22:45

So, ugh. Okay, but I'm trying to look above that, and I'm seeing that there was a problem a few back...

Pam 22:53

Okay.

Kim 22:53

...where we had 18, 18, 18, 22, 22.

Pam 22:57

Which is very close to this, 18, 18, 18, 18, 22.

Kim 23:01

Mmhm.

Pam 23:01

Mmhm.

Kim 23:02

Really, the difference between this problem you're giving me and that problem I just mentioned is 4 (unclear).

Pam 23:13

Because one of the 22s is now an 18.

Kim 23:15

Yeah.

Pam 23:16

Mmhm.

Kim 23:18

Oh, but I didn't even line this up right on my paper. I don't remember what the average was there. (unclear).

Pam 23:22

That one was 19.6

Kim 23:25

Okay.

Pam 23:25

Okay.

Kim 23:26

So, if this is 4 less shared among 5 numbers, then that would be 0.8 less. So, I think this one's going to be 18.8, but I want to confirm.

Pam 23:41

Okay.

Kim 23:42

Yeah, yeah, yeah. Because if there was... If the last number was 18 instead of 22...

Pam 23:48

Mmhm.

Kim 23:48

...then I would have 4 extra to share among 5 numbers, and that's 0.8. So, yeah, 18.8.

Pam 23:53

Nice, nice. How did you get the 0.8 again?

Both Pam and Kim 23:57

(unclear).

Kim 23:59

Either way. 4 divided by 5 is that what you did? Oh, four-fifths. Yeah.

Pam 24:03

Four-fifths is 0.8.

Kim 24:04

80%, 0.8.

Pam 24:05

Yeah. And you're sort of happiness with fifths. So, Kim has had a good relationship with fifths in decimals and percents, and so I just wanted to slow that down just a little bit.

Kim 24:17

Gotcha. Okay,

Pam 24:17

4 divided by 5 is four-fifths, which is also equivalent to 0.8, and that's where you're sort of getting that 0.8 from (unclear).

Kim 24:23

I like this string.

Pam 24:24

Oh, I'm so glad. cool.

Kim 24:26

I'm going to make my kids do some math tonight.

Pam 24:29

I love that you do that. Hey, if you didn't have the other ones and you were just going to use the five value data set, the 18, 18, 18, 18, 22, is there a different?

Kim 24:40

I think I would have done what I did the second time to check my thinking. Is I would have said, "Okay, they all have an 18 within them. What's extra?" And re-even that. So, the evening out that extra 4 that's in the 22.

Pam 24:54

Gotcha, gotcha. That makes sense. Cool. And just so listeners can hear. Often, I will have kids say kind of like what Kim said, "Well, I can see 18s. There's four 18s and one 22. if I pretend that 22 is an 18, there's that 4 extra. Often, they'll say, "Well, I'm just going to dump that 4." And sometimes they'll dump that 4 into the four 18s, and then not realize that that last 18, that they've just sort of took that 4 away from the 22, and so now it's 18.

Kim 25:22

Yeah.

Pam 25:22

Well, now it's kind of lonely. But now they'll have 19, 19, 19, 19, and

Both Pam and Kim 25:27

18.

Kim 25:27

Yeah.

Pam 25:28

And so, now they have to kind of think about well what do we do to make that 19, 19, 19... The four 19s and 18. How do they even that out?

Kim 25:35

Yeah.

Pam 25:36

Ooh, can I just grab... What? A fifth from each of those?

Kim 25:40

Mmhm.

Pam 25:40

And now I've grabbed four 1/5s to give to the 18, so that's 18 and four-fifths. And I've left... I've grabbed one-fifth from each of those 19s, which leaves it with 18 and 1/5.

Kim 25:51

And so, that sounds like more than one shuffling, more than one evening out. And that that feels really important to let kids experience and not just tell them, "Oh, wait. You didn't see? You should have done this all at once." Like, that constant re-evening out is the learning that they need to have.

Pam 26:10

Well, and often, teachers will say, "Pam, how do we give kids more experience with fractions?"

Kim 26:15

Yeah.

Pam 26:16

Well, notice that here we are in a seemingly unfraction kind of task where actually we could get a lot of fraction understanding happening. Not only just with sort of adding that one-fifth or subtracting that one-fifth, but also the quotient meaning of that if I have 4 extra that I need to divide among these 5 data values, then I kind of have that quotient meaning of 4 divided by 5 is four-fifths. So, lots of nice things. Teachers, you might be thinking right now, "Pam, my kids don't own all that stuff. We can't do this kind of thing with that." But could this be an opportunity to actually develop those meanings of fractions, along with the sense of what's a Mean, and what's a Median, and what's a Mode, and changing the data values, how does that affect each of those measures of central tendency? And we can kind of have an open access task, where kids no matter where they are, everybody gets a little bit challenged to think and reason a little more sophisticatedly.

Kim 27:12

Yeah, I'm also thinking right now about in this last one I was tempted to make them all 18, and then deal with the extra 4 that I had. But there was one time, and I remember where it was, where for some reason, I wanted to go the other direction, and I was subtracting some amount,

Pam 27:37

Mmm, mmhm. Yeah.

Kim 27:39

And I don't remember which problem that was. But that was interesting. That was more challenging for me to think about going underneath instead of dealing with the extra on top.

Pam 27:49

Sure.

Kim 27:49

I wish I can remember what that was.

Pam 27:51

Well, and that's interesting too because you're such an Over girl that in this case you actually... Hey, maybe you have a little empathy for me when all the time you would go Over, and I'd be like, "Do that again. Let me see if I could track with you." Hey, the last thing I want to end this one with is just this idea that there is a difference between social, conventional knowledge and logical, mathematical knowledge. And just reiterate, like I said at the beginning, that the terms are social. That's the stuff that we tell kids. But the ideas, the ideas that spreading it out till it's even, that one value that shows up the most, or the one that's in the middle of the data set. Hammering down those concepts? That's logical, mathematical, and that takes experience. What kind of experience? Well, the kind of experience that we just had. Not just finding those, but discussing what's affected by what when you change the data. Why did that change and that not change? And giving kids experience of feeling those things change when the data change. This isn't the only kind of thing that you can do to help kids get a feel for those. I think it would be super helpful to have these data values mean something, so I also like to have a context involved. In fact, maybe even before I do what we just did today, we would have a context involved where it matters. Whether we're talking about like Median income, or free throw averages, or something like that where changing...

Kim 29:11

Ages.

Pam 29:12

...one of the... Ages. Lots of different things where changing a little bit of it, Whoa, that really makes a difference. And then, another idea is how does that change which measure of central tendency and so which one would you report out based on that context? So, maybe we'll do a string like that someday on the podcast.

Kim 29:29

Yeah, let's do it.

Pam 29:31

Alright. 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 mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!