Let's do some guessing! In this episode Pam and Kim develop some functional reasoning by analyzing the relationships between their guesses and reality.
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Hey, fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able! I'm Pam. And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to... 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 our algorithms really not very helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be. Whoa!
And I'm Kim. Hi, Pam.
Kim, I'm sitting here with my pen in hand. I'm going to bet you have a pencil.
I do, but it has no eraser, and I'm super sad about that.
Actually, you know...
We should probably stop recording right now if you don't have an eraser.
Well, I'll be okay because I found in my kids like art table drawers. I forgot about these. But you can buy Ticonderoga erasers and they are not shaped like you know the standard, rubber pink eraser thing. You know what I'm talking about it's kind of...
The rectangular prism one?
Yeah, yeah, yeah. This one is super cute because it's shaped exactly like a fat pencil. It's got a pink eraser and a little. I'll have to show you sometime. They're super cute. But I found those, and I forgot how much I love them.
(unclear) Is it kind of like you're holding a pencil, but you're holding an eraser?
Yeah, it's completely shaped like a pencil.
So, you're actually smiling with your Ticonderoga eraser.
But it's nubby. It's like two inches long and really fat. So, yeah. It's not full. Anywho. Hey, we got a... We're a mess. We got the best review. I want to share it with you. Oh, cool. Okay. So, it's from KelseyAnderson68. And I love when it's like an actual real name instead of 06253.
Anyway, she said...
68. I'm not quite sure 68 is really part of Kelsey's name, but okay.
Well, maybe it's her favorite number.
Kelsey could be a he.
So, she... Oh, I didn't think about that. Sorry, Kelsey, if you're a man. Anyway, they said, "Obsessed" as their title. And it says, "I listened to every episode. Every math teacher, and anyone who has an opportunity to shape children's lives needs to listen to this podcast."
Oh, thanks, Kelsey. We appreciate that. Yeah. Ah, nice. Yeah, let's do it. Cool. Alright, Kim, for today's episode, let's talk about a super cool experience that I had and do a little bit of math. So, I was just in Oxford, England. Like, Oxford college. Oxford University. I was at the Queens College at Oxford University. An amazing, amazing place. I think my jaw is still open. I had such a good time. And one of the coolest people I know was there. So, Gail Burrill is a past NCTM president. She's a high school math and statistics teacher. And she has long been on my list of people. So, Kim, you know I have this list when I go to a conference. I don't actually look at session titles. Especially first. The first thing I do is go to the speaker index, and I look for speakers. And if Gail Burrill is at a conference, I will go hear what she says because she is so thoughtful. And I swear, every time I listen to her talk, she says something new that she's been thinking about. Or maybe I hear it for the first time. But yeah, she's super thoughtful. And I really appreciate listening to her. So, kind of funny. When I was a brand new teacher, I had had some really unique experiences early, early on. But one of the things that happened. I think it was my second year of teaching. No, I'm sorry. It was my third year of teaching. I got an opportunity to go to my first NCTM regional conference. And I hadn't been to one before. I was kind of in an isolated. My first couple of years, I worked with a remarkable teacher. I had this great collegial experience. Then, I moved to a school where I worked with very nice people, but who were not very interested in working together. And so, I was really looking forward to the conference because I was like, "I'm going to go, you know, learn all these new things," and, you know, have more of that collegial experience. And I really was hungry to learn. So, I was then actually kind of disappointed. Session after session. I was like, "Ah, this is like same old stuff. I'm not learning anything here." You know, I kind of wanted some new ideas. I had really been diving into some of the research by Frank Demana and Bert Waits about visualizing math and using technology with power visualization. But I was now teaching Algebra 1, and I didn't have... That was when I was teaching pre-calculus in my first position. And they had a new book out and all sorts of things, but there wasn't really a lot of things happening in Algebra 1 at that stage. At least that I was aware of. And so, you know, I was looking for that. And I went to a session by Gail Burrill. I didn't know who she was. I walked into the session, and she asked some questions. And in the middle of her series of questions, all of a sudden, I was like, "OH!! YES!!" Like, "This is what I was looking for!!" And it was so striking because I had been through the entire day of the conference, and I was so disappointed. And then, in that moment, it made it all worth it. I had this flash of inspiration of what I could do like I'd been doing in pre-calculus. I could do that same kind of thing in Algebra 1. And I was just thrilled. So, I thought we'd do a little bit of that today.
So, we're going to actually walk you through a similar experience of what I had there, and then we'll kind of do some math with it. So, Kim.
I asked you if you would go out and find 8 or 10...I don't know how many...8 to 10 celebrities, famous people-ish, and their ages. And then, you're going to tell me the famous person, and I'm going to guess their age. Which to be really clear, I'm bad at, so just like. I don't ever guess people's age because, well A, I'm bad at it, and B, I don't want to I don't mess with that (unclear)
Yeah, that's probably not a good situation.
Yeah, like it's important to people. And yeah. Anyway, okay. So, Kim, tell me a famous, relatively well-known person, and I will guess their age. Go.
Okay. The first person I found was Michael Phelps. Any idea who that is?
Okay. Yes, he's a swimmer.
And funny, we think he looks like my brother Michael.
Oh, is he very tall?
Very tall, long arms?
My brother's tall with long arms. My brother, he might swim a little bit, but he's a mountain biker. So, physique wise, they look similar.
How old do you think?
I'm trying to remember. Yeah, I'm trying to. I know how old my brother is. Does that count?
Both Pam and Kim 06:37
I'm going to guess that Michael Phelps is about 40.
Want me to tell you?
No, not yet. Yeah, thanks. So, let's go through all the people, I'll guess their ages, and then you're going to go back and tell me.
Is that okay?
Yeah, it's perfectly fine. Alright. Next, next, next. Michael Jordan. You know who that is.
Ooh, another Michael.
Okay. I do know who Michael Jordan is. And ooh, but let me think. I... Oh. I think Michael Jordan is 55. I'm not actually sure on that one. Oh, he might be a little older than that. But I'll stick with 55.
Wait, you don't like Richard Gere? Or is that a, "I have no idea."
Yeah, that was more of an umm. The only picture that comes in my mind right now is Pretty Woman.
Yes! And so many other great movies.
Which I liked, but shouldn't. So, Richard Gere. Richard Gere.
I wonder if anybody screaming ages in their cars.
Yeah probably, right? Yeah. 70? Is he 70? I don't know.
Okay. I'm so bad at this.
Okay. Prince George.
Prince George. Prince George. Like, the prince in England? That prince?
Prince. Should I?
Not like the musician that we used to? Prince who died? Not that one? No. Not Prince, not boy George. Prince. Ah, that's who. Yeah, I was putting those two together. Okay. Prince George. Like, literally like the son of...
The child of the Prince and Princess of Wales. Yeah.
Yes. Okay. I think he's 8.
Alright. What about...
I'm not sure I knew his name was George. Okay, go on.
What about Paul McCartney?
Well, funny. Our church youth group the other night, we were doing a karaoke. And I said, "Let's do a Beatles song!" And somebody said, "Who's the Beatles?" And I was like, "What?!"
Oh, I know. They're so young.
And then somebody said, "They're all dead." And I was like, "Wait, I don't think Paul McCartney is dead." Anyway.
Sorry. Paul McCartney. Is Paul McCartney 75? I'm so bad at this. That's my guess. Paul McCartney is 75.
What about Eminem?
Ugh. Now, that sound was, "Oh, I feel like a dork" because I know that he's a rapper, but that's about all I got. And I also am really clear. I just spelled his name wrong because I literally wrote the letter M and. I think it's E-M. Is that right? (unclear).
E-M. Yeah. E-M.
Yeah. Anyway, so Eminem. Eminem is 40. I have nothing. I have no idea. Okay.
Alright. Serena Williams.
Serena Williams is totally cool and has a sister, and they play tennis.
I don't know, that might... She might not be that old. I don't know. Okay.
Alright. See you've done great. You've known who all these people are. What about Clint Eastwood?
He's still alive. Okay. That guy's gotta be ancient. 85. I'm thinking 85. (unclear). Yeah, I don't know.
Both Pam and Kim 09:58
Yeah. Now, my parents who are not quite 85 are going to be mad that I said that was ancient.
Both Pam and Kim 10:02
I actually was talking about age with a friend the other day. And she's 51. And I said 51's not old at all because I'm, you know, about to be 45. And I said to her son, "Do you think that's old?" And he was like, "Yes." Okay, kid.
Yeah, age is kind of was relative.
How close you are things. Mmhmm.
Alright, last one.
Kylie Jenner, who used to be Bruce Jenner, I believe.
No, no, no.
Do I have the right?
No, not the right person.
Oh, I don't know who that is then. Kylie Jenner.
Okay, well you're still going to have to guess.
Kylie Jenner. Okay, I'm going to have to guess. I don't know who this person is.
So, that's going to be hard. Are they related to? Caitlyn. (unclear).
They are. Caitlyn is who you're thinking yeah.
(unclear). Okay. Kylie Jenner.
No hints, huh. No hints?
Okay, I'm saying 26. I have nothing. I don't know.
Alright. How old is Michael Phelps really?
You said 40. He's actually 38. (unclear).
(unclear). Alright, alright. Cool. Michael Jordan. How old is he?
You said 55. He's actually 60.
Ooh, a little older. Okay. Okay. How about Richard Gere?
Hey! Okay, alright.
I know, you're doing so well.
70. I said 70. 73 Alright. How about Prince George, little guy?
Little guy. You said 8. He's actually 10.
10. That's funny because I actually thought he might have been more like 6. But okay. What's last time I paid attention to? Paul McCartney I said was 75, but he's actually?
81. Alright, not too bad, not too bad. Eminem, I thought was 40.
Oh, okay. (unclear).
I was surprised by that one. Yeah.
Alright, alright. Serena Williams I thought was 36.
Man, those guys hold their age well. Both sisters. That's cool. Those guys are athletes.
Oh, best. Yeah. Best tennis ever.
Yeah. Alright, Clint Eastwood.
Woah! So, I said 85 was ancient. What does that mean 93?
Way to go, Clint! Alright.
That's pretty cool. Okay, cool. And then, the person I don't know. And I can't even read my own writing. Is it Kylie Jenner? Yeah. Okay, who is that?
One of the daughters. One of the Jenner girls.
Ah, okay, okay.
I couldn't name them all. There is one named Kim. Because you know, I know that.
You know that because you're name is Kim.
Courtney. There's a Courtney. Anyway. She is 25.
Your best guess was somebody you didn't even know. Look at you. You are a good guesser.
Both Pam and Kim 12:49
That's terrible. Well, so that then becomes the question. So, in this moment, Gail Burrill paused. She gave us a bunch of famous people...different famous people obviously...and asked us to guess their age. And then, she told us their correct ages. And then she said, I'll never forget, "How do you know if you're a good guesser?" And I started. I was like. Now, I'm not going to give away what was going on my head quite yet. But like, Kim, you said I was the best guesser for the one I didn't know, and you said that because I was only one off. Right? Like, her age is 26, and I said 25. Some of them I was way far off. Eminem I was 10 off. Was I any any further off?
Nope, just that one.
Eminem was my worst. I mean, I was 8 off for Clint Eastwood. So, are you a good guesser? Like, what does it mean to be a good guesser? So, a couple of things that we could do right now. Let's say that I was working with middle school kids? Could I find the differences between those guesses? Because I just kind of did. One of them, I said I was 10 off, so that would be like 10. And since I'm in middle school, could I even do... Maybe I was in fifth grade. I could just find how far off I was either direction, and we could represent it with a positive. It's that 10. You wouldn't know if I was 10 over, 10 under. You just know I was 10 years off. But if I was in middle school, could we actually represent like... I don't know. Do you want over to be bad? So, like if I was over, then it would be negative 10. I think I was under. I was under, so maybe that would be a 10. I was under 10 years that would be 10. But if I was over 10... Was I over on any of them? Did I always guess under? Oh, Michael Phelps I was over.
You were over for Kylie Jenner and Michael Phelps. Yeah. So, like Kylie Jenner I'd be negative 1. Michael Phelps I was 2 over, so it would be negative 2. And then, we could kind of maybe find the mean of those. Could we average those differences? And we can kind of get a sense for maybe who in the class was the best guesser. What would we be looking for? A high mean? A low mean? Any ideas, Kim?
Would it be be a low Mean?
Because they're the closest to age.
To the actual. Yeah. Like, if they were all close, then I would have a low mean. In fact, maybe I would even want a 0 mean, if possible. Except. It's interesting. Like, what could happen? What if I had a 0 mean? What could be true?
I mean, what if you guessed off by 20 one time and over by 20 another time.
Like, you're a horrible guesser, but either way.
Yeah. So, there's some nice conversation that we could have about what "mean" means. What is an average? And then, we could also have conversations about integers and how a negative 20 could sort of cancel out a positive 20. And that idea of 0 pairs could come up. And so, lots of nice things. Not actually where Gail Burrill was going. So, Gail Burrill in that moment said, "What if we graphed these as ordered pairs?" What if we put the actuals on the x axis. Sorry, the guesses. The guesses because that's the first one we did. So, we'll just put those on the x axis. The guesses on the x axis and the actuals on the y axis as an ordered pair. So, for example, Michael Phelps I would go over 40 because I guessed 40, and I go up 38, and I'd make a point. So, over 40, up 38 I'd put a point there on the graph. And for Clint Eastwood, I'd go way over 85 because I'd guessed 85, and I'd go up higher. I'd go up higher 93. So, my Michael Phelps guess of 40, 38 would be further to the right. I'd be over 40, but I'd only go up 38. I wouldn't go up quite as high. But my...help me...Clint Eastwood guess, I'd go way over, 85. But I'd go even farther up 93. And so then, when she said, "What would a perfect guesser look like?" That's when I really dropped my jaw. I said, "Oh, my gosh. She's going for the line y equals x." What does that mean? What does it mean where the actuals would equal the guesses? Like, I don't think I got any spot on did I? I didn't have any.
One really close. A few actually pretty close.
Like, what's one that's close?
Kylie Jenner. 25, 26.
Alright, so for Kylie Jenner, I would have gone over 26, but only up 25. But that's almost like if she was 26 years old. And I got over. I guessed 26, then I would go up 26. If I did that for everyone. Say Michael Phelps. If I had guessed 38, and he's actually 38, I would go over the same amount I would go up. And if I did that for every age, for every x value, I went up the same amount.
Kim, can you picture what that line would look like? Like, if you were to describe the line in space? What would it would it be? Horizontal? Would it be vertical?
It would be diagonal.
Because for every... It would be 0,0. 1,1. 2,2. 3,3. All the way up to, you know, however old you can get.
If I'd guessed 100. and the person was actually 100, then I would go over 100, up 100. And if I connected all those dots. If I guessed they were a half a year old, and they were actually a half a year old. I could fill in, and I would have that diagonal line, and that 45 degree line (unclear). And that is actually what the line y equals x looks like. And often, Kim, when I taught algebra, the line y equals x was this anomaly. It was this weird line that didn't match the pattern. Because they were used to graphing things, at least the way I was taught, and the way I saw most of my colleagues teaching. They were used to graphing y equals mx plus b. And so, there was always some sort of slope. It wasn't usually 1. Because in this case, the slope was 1 when I go over, I go up the same amount. The rate of change is always go over and up one at a time, or it could just simplify to one at a time. So, the slope, in this case, is 1. They were used to having slopes that weren't 1. And then, plus B was the y intercept. In this case, the y intercept, like you just said. If I guessed 0, they were 0 years old, the y intercept is 0. That was weird. It was weird to have a line that had a slope of 1 and a y intercept of 0. So, that was almost one they would miss if they ever find the slope between these two points.
Yeah. (unclear) starting point. Like, this wasn't established, and then?
It is now the way I teach.
But that's not common in high school?
Not that I've ever seen. No.
No, not at all. No, in fact, this was a special, weird case that we just didn't even talk about very often. I know. But we could. Yeah, we absolutely could have this be kind of the starting point that kids could conceptualize, "Well, if my guesses equaled the actual, then bam, I would have this line, this diagonal line." And then, every other line could be a translation or transformation of that line. We could rotate it some. We could make the slope higher. We can make the slope lower. If you could see my body right now, I've got my arms are like at this 45 degree angle. As I said "the slope higher", I kind of increased the slope. I made it more steep. And "the slope lower", I decreased my arms and made that. And my pivot point is my two hands are coming together. I don't know if you can picture that. But almost like... How would I even say that? Like, my two hands are coming together in front of me in the middle, and that's kind of where I'm considering the 0,0. So, I'm sort of pivoting around that point, and I'm making the line either steeper, the line more shallow, depending on if we made the line steeper or more shallow. And then, I could move it up and down, depending on if I had a different y intercept. But for this, this task really helped me conceive of that line y equals x as a thing to start with, as a thing that we could have kids like kind of focus on. Then, a thing that we could do with kids is we could say, "Okay, if we've got that perfect guess line, where do all your points lie? Are your points above that line? Oh, then you must be an over guesser. Are your points below that line? Oh, then you must be an under guesser." And then, I pause. Because Is that true? Is it true that if my points are above the line, that I'm an over guesser. Kim, can you pick out of that series of points a point that would be above that perfect guest line?
Yeah. When you were talking earlier, I sketched it out. So, Clint Eastwood.
You guessed 85. No, you guessed 93. So, you go over,
I guessed 85.
Yeah, you did. You guessed 85. So, did I plot that properly, then? Yeah. I think I went over 85.
And then, because he's actually 93, then I went up the actual line 93. And that put it over the line. I put it under the under the line.
Under the line? Over the line?
Did I plot it right?
I don't know. Let's say that I guessed correctly. That my guess... Let's say he was actually 85. Where's that point? Over 85.
Oh, actually no. I put it over the line. I put the points backwards. I can't listen and do at the same time. So, it would be above the line. Clint Eastwood would be above the line.
For me. For me.
Yeah, because my guess was 85, but I went way higher than and plotted the point.
And if I'd guessed 85 and he was actually 85, that point would be over 85, up 85. But my point is over 85, up 93. Way higher, and so it'd be above the line. But in that point above line, I was under... Yeah. Under guessing, right? So, if I was under guessing, why was my point above the line? Shouldn't an under guess be under the line? Let's do one more just really quick. So, where was a point... Did I ever over guess?
Yes, Michael Phelps.
Yes, Michael Phelps. Michael Phelps I over guessed,. It's a good thing. So, for Michael Phelps, I would go over 40 and up not as high, up only 38, and plot a point. So, a perfect guess would have been over 40 and up 40. But I only went up 38. So, that point is under the perfect guess line, even though I over guessed. I guessed he was 40, but he's only 38. I thought he was older than he is because my brother's a little bit older than that. So, an over guess is under the line. And to be clear, on purpose. So, there might be some algebra teachers out there right now going, "Pam, just do the thing, and it will..." Yeah, I know, I know. So, hang on a second. We'll talk about that. But in this moment, I didn't want it to be too obvious when I worked with students. I wanted, when I did this task with students, for students to have to think about a point under the line, under that perfect guess line, what did it mean? And a point over the line, what did it mean? I didn't want them just go, "Duh, it's under, so therefore it's an under guess" and "Duh, it's over, therefore it's an over guess." I wanted them have to actually reason through that. And then, over the next few days, we would do more tasks like this, or extensions of it, and I would warn them. I would say, "Hey, I'm going to switch it on you one of these days." Well, what do I mean by that? Well, those algebra teachers that are smiling right now, some of you might have said, "Pam, why did you graph that your guess on the x axis and the actual on the y axis." Which I think Kim is what you did. You switched them.
Yeah, I did. Do you ever ask students what would make it...
What would make that fit in their minds maybe?
Yeah. And I guess I could have asked listeners, like what could we do with this data, so that under guesses will be under the line and over guesses will be over the line? And it's exactly what Kim did. If we switch the axis. So, instead of graphing... Like, I had asked for guesses first, so that was kind of my first variable, and so I let that that'd be x. But instead, if we put the actuals on the x axis, and the guesses on the y axis, then under guesses would be under the line and over guesses would be over the line. And then, it would be kind of like you would expect. Again, I don't do that at first with students because I want them to have to think through it. But then, I would warn them, "Hey, you know, I'm going to switch it, and you'll never know, so you've always got to be paying attention." Kim, the next three times we did an extension of this task, I would still will keep guesses on the x and actuals on the y, so it would still be not the way they're thinking. And then, I would surprise them like on the fifth iteration. Then, I would be like, "Oh, here it is." Yeah, and I tweaked assignments and everything to make it look like it, so that kids would keep thinking. Because the goal in my class is thinking and reasoning, making sense of things. Not about memorizing, and not about just being the obvious thing that you don't have to think about. I wanted to build that thinking. Yeah, totally cool. Hey, one of the things that then came up for me. And Doug Smelts, who is a fantastic t-cubed instructor, teachers teaching technology instructor that I met out of Ohio. Helped me think about the fact that once we have that line y equals x, then we can use that as the parent function. He said, "Could you consider that once kids really own y equals x, it's not only that then we could mess with lines from there, where we could up the slope, or down the slope, or move the y intercept. But we could actually think, "Well, if I know what y equals x looks like, then y equals x^2, I could think about y equals x. But every time I go over, instead of going up the same amount, now I'm going to go up that amount squared." And so, I almost take all those y values, and I bend them into their squares, and I get a parabola. So, I'll just kind of drop that as an interesting thing. If we can really conceptualize the line y equals x, it doesn't only get us more lines, it also gets us all polynomials. And in fact, maybe, all functions. And we'll just sort of mention that and kind of have fun with it. And then, could we then have all transformations be built on and based on that unique parent function? Kind of cool. I will never forget that moment in that NCTM regional conference, where I had this flash of mathematical insight. And I'm quoting a colleague, John Tapper, who the other day was talking to me, and he goes, "Pam, don't you think it's all about helping students create and have mathematical insight?" And I was like, "Oh, that's a nice phrase." So, I'll just label that moment that I had with Gail Burrill as a flash of mathematical insight, that then I was able... For me. Not just about teaching.
But then, I was able to take it into teaching. Yeah (unclear).
Nice. Well, Kim, thanks for getting some ages for me, so that we can play a little bit (unclear).
I'm super excited about this. I've got, you know, kids. One moving into working with lines, so I'm have to go find some ages that he will be able to guess.
Oh, some people that he'll be able to guess, and then yeah.
Yeah. For sure.
(unclear). Tell us how it goes. Tell us if he's a good guesser. But maybe less important if he's a good guesser or not, and more importantly what does he do with like thinking about whether a good guesser or not? Yeah. Nice. Cool. Ya'll, we do a task and tasks like this in my Building Powerful Linear Functions workshop, so if that interests you, check out the "All Workshops" page at mathisfigureoutable.com/workshops, where we do tasks like this to build all higher math. Because, ya'll, everyone is telling you that higher math is rote memorizable, and we're telling you that all math is actually figure-out-able. So, 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!