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  00:09
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 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  00:31
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  00:39
And I'll just there pause for no reason at all.

Kim  00:44
You know what's horrible? I was like, "What's going on?" I have a Charlie horse in my foot (unclear).

Pam  00:48
Oh no!

Kim  00:49
I'm rubbing it while we're talking. I don't know what happened to me.

Pam  00:52
So, you're thinking, "I'm so glad Pam paused because (unclear). 

Kim  00:55
I thought you were going to stop (unclear).

Pam  00:56
Dude, eat more potassium. Go get a banana. Are you okay?

Kim  01:00
Yeah, I'm fine. 

Pam  01:01
Alright, we invite you to join us to make math more figure-out-able. Are you running in these days? I don't even... 

Kim  01:06
Yeah.

Pam  01:07
Yeah. Alright, Kim. Tell. How far do you run? What do you…

Kim  01:10
Oh, no. I'm just running with my… My entering eighth grader is running cross country, so I wake him up at 7:30 in the summer. Yay, mom! And we just go like two miles in the morning.

Pam  01:21
Just two miles. 

Kim  01:22
Well, yeah.

Pam  01:23
Just a little short two miles. Says the marathon runner. Alright.

Kim 01:26
Okay,

Pam  01:27
Does he like cross country?

Kim  01:29
Yeah, he's got some friends that do it, and so he likes it. And, you know, I think running is fantastic because it's the kind of thing that you could maybe do in your life. You know, I don't know a lot of people who are going to play football at 45. Or, you know, maybe pick up basketball. I don't know. But he enjoys it, and I'm super excited that he enjoys it.

Pam  01:49
Yeah. Yes you are.

Kim  01:51
Okay.

Pam  01:51
Alright, podcast.

Kim  01:52
Hey, I found us another review. You know it makes me happy. This one’s title is “Life Changing for Teachers and Kids”.

Pam  02:01
Aww. 

Kim  02:01
I know. 

Pam  02:02
Aww. 

Kim  02:03
And the handle, the name, is “Stuck in Fourth Grade.” So sweet. So, yeah, I don't know if you're stuck because you wanted another grade, or you never get out of fourth grade because you're amazing and you stay there forever and ever.

Pam  02:17
Yeah, there you go. I think “stuck” might be because you're like, “I'm here!” You’re purposely in fourth. Yeah, okay.

Kim  02:25
Well, okay. So, it says 30 years ago, so maybe. Yeah. “30 years ago. I hung an 8 by 2 sign in my first classroom that merely stated, ‘THINK,’ in all capital letters, 

Pam  02:36
Nice. 

Kim  02:36
“And then I proceeded to teach my kids to mimic.”

Pam  02:41
That sounds like my classroom.

Kim  02:43
I know, right? So, “Following this podcast has opened my eyes to the surface level of understanding that I have been chasing all these years. This past year, I have listened to Pam and Kim sometimes four times before recreating the same Problem String in my classroom. And my fourth graders have blown me away and literally outperformed me in their developing intuitive sense of numeracy. I'm so grateful for the power this gives my students, and even more grateful that the Problem Strings can now be purchased in a book. I will continue to listen for the insight and encouragement to teach counter than wrote algorithm culture, but can drive safely without jotting things down.”

Pam  03:25
That's unbelievable! I love it.

Kim  03:26
I know, I know, I know. And what's super exciting is the fourth grade Problem String book is out for pre-order, and we'll be shipping in the next couple of weeks. So, Stuck in Fourth Grade, I'm so excited that you're going to have that for this school year.

Pam  03:41
Oh, that is fantastic. 

Kim  03:43
Isn’t that fun? (unclear).

Pam  03:43
Drive safely without jotting things down because it's in a book now. That's fantastic. 

Kim  03:47
Yeah, yeah. 

Pam  03:47
And I also like the “counter rote algorithm culture”. I like that. Yeah, I like it a lot. And I tell you, Stuck in Fourth Grade, I am right there with you. As soon as I started turning over the thinking to my kids in not just like in the way that we're suggesting, right? Not just in some random. Like, I was always trying to get kids to think like you were. I just didn't know what that meant. But as soon as I actually knew what it meant, I am continually, continually blown away by kids. I'm still thinking about the kids I met in Australia. 

Kim  04:18
Yeah. (unclear).

Pam  04:20
(unclear). Oh. In fact, Kim. One of the things, most fun things that I had on that trip when I went to the Blue Mountains with that principal and his kids. At a couple of times, I was with either the seven year old or… Golly, how old was the old one? Nine years old? 10? Sorry, don't tell him how I didn't remember how they were. And we just played math games as we were trucking up and down these lots and lots of steps. Oh my gosh. I went up and down. I did more steps that day. Anyway, it was fantastic. But we just… The math. The mathing those kids were doing. It's just so… I just love it. It's one of my favorite things to do ever. Just ask kids questions. Cool. 

Kim  04:52
Very cool. 

Pam  04:53
Alright. Hey, Kim, people ask me sometimes questions that kind of give me the feeling that they think all I do is numbers and numeracy. Like, literally, they'll say, “Why do you only do numeracy? Why do you only do numbers?” So I don't. Like, I do all of mathematics. I was a high school math teacher. I like geometry. You know, yes. There are lots of different areas of mathematics. Sometimes you might see us just do numbers and do Problem Strings with numbers because it's a good inroad. It's a good entry point for people to kind of get on the Math is Figure-Out-Able track. But hey, we thought we'd change things up a little bit. So, Kim. Kim, I said, “Kim, let's do some geometry.” And then I said, “You pick.” So, it is your pick today. What are we going to do?

Kim  05:39
Okay, so when we were writing… I don't know. Can we say? Is that what we're doing? Have we already said that a million times?

Pam  05:47
Go ahead. Yeah. I believe so.

Kim  05:48
(unclear). Okay, so when we were writing for Math Learning Center, the Bridges in Mathematics curriculum, one of the things that we thought long and hard about was making sense of geometry and the hierarchy of the way shapes fit together, two dimensional shapes. And so, we wrote some lessons in there. And one of the things that we put in those lessons were these geometry riddles. I don't even know what we called them. They're riddles. And so, I grabbed a couple of those, and I'm going to give them to you today. 

Pam  06:20
Oh (unclear).

Kim  06:21
Give you some riddles. And you can think out those riddles. They're kind of like clues. I'm going to give you a clue. You're going to tell me what you know about it. Then I'll give you another one, another one, another one until we figure out what shape I'm thinking of. 

Pam  06:33
Nice. Alright, I'm excited. Let’s do it.

Kim  06:35
Are you thinking hard today? It's always me. I always have to think hard out loud.

Pam  06:39
So, this is good turnaround. Alright.

Kim  06:41
Okay, ready?

Pam  06:42
Bring it on. 

Kim  06:43
Riddle one. 

Pam  06:44
Okay.

Kim  06:45
I have four sides.

Pam  06:47
Golly, I think you are a quadrilateral.

Kim  06:50
Okay.

Pam  06:50
But I don't think I know... I don't know anything else about you. So, you're a polygon with four sides. Polygon with four sides, and we call that a quadrilateral. Okay.

Kim  06:57
Okay, I have two pairs of parallel sides.

Pam  07:01
So, two pairs of parallel sides means you're not a trapezoid. Well, depending on which definition you use. But you are a parallelogram,

Kim  07:09
Mmhm. Which means I'm also a quadrilateral and also a polygon.

Pam  07:16
Yes. 

Kim  07:17
Like, we're narrowing, right? So, you're everything you said, and what you just recently said.

Pam  07:21
So, that's kind of what you meant by a hierarchy. 

Kim  07:24
Yeah.

Pam  07:25
So, in the hierarchy, we're kind of narrowing down. 

Kim  07:27
Mmhm. 

Pam  07:27
Yeah, and you are a… Did I ever tell you that my high school math analysis teacher… So, it's kind of like pre-calculus. It's called math… I don't know why they called it math analysis. Anyway, what was her name? I think it was Mrs. Barton. Barton? Was that her name? Anyway, she was from the south, and so when she said parallelogram.

Kim  07:45
Oh.

Pam  07:45
It was a little different than I'd ever heard before. Parallelogram.

Kim  07:49
Sounds like power. Power-llelogram. 

Pam  07:51
Parallelogram. She was awesome. Anyway, okay. So, we have a parallelogram. What else?

Kim  07:56
Okay. I have four right angles.

Pam  08:00
Ooh. 

Kim  08:00
Tell me what you’re thinking right now. 

Pam  08:02
Yeah, so I pictured a parallelogram that's kind of skewed, you know? And I just like straightened it up. If you can see my hands, they just straightened up. You said four right angles, right? 

Kim  08:13
I did. 

Pam  08:13
So, if I have a parallelogram, opposite sides are parallelogram… Opposite sides are parallel. And I straighten up those sides, so that the angles are right angles, then I have a rectangle.

Kim  08:25
So, you have a polygon that is a quadrilateral, that is a parallelogram, that is a rectangle.

Pam  08:32
Yeah, a special parallelogram a rectangle. 

Both Pam and Kim  08:33
Mmhm.

Kim  08:34
And I have four congruent sides.

Pam  08:38
Whoa! And I just took that elongated rectangle I was picturing, and I squished it, so that the sides are all congruent, and now I have a square.

Kim  08:46
Mmhm (unclear).

Pam  08:47
Alright, do you want me to name them all? I have a polygon that's a quadrilateral, that's a parallelogram, that's a rectangle, that's a square. 

Kim  08:55
Nice. 

Pam  08:55
Or a square… Let me say it a different way. A square is a special rectangle, which is a special parallelogram, which is a special quadrilateral, which is a special polygon.

Kim  09:04
Nice. And I think going both directions is super helpful. How many times do we have kids who say, “Is it a parallelogram or is it a rectangle? Is it a rectangle or a (unclear).”

Pam  09:16
Or a square. 

Kim  09:16
Mmhm. 

Pam  09:16
Rectangle or a square. Mmhm, yeah.

Both Pam and Kim  09:18
Yeah. Okay. 

Pam  09:19
Cool.

Kim  09:19
You want another one?

Pam  09:21
I do. That was fun.

Kim  09:22
Okay. Alright, here we go. 

Pam  09:23
Okay.

Kim  09:24
I am a closed figure with three or more sides.

Pam  09:28
I'm a polygon.

Kim  09:29
I have exactly three sides.

Pam  09:32
Alright, so now I'm a triangle. I'm a triangle that is a polygon.

Kim  09:36
Mmhm. 

Pam  09:36
Okay.

Kim  09:37
I have one right angle.

Pam  09:41
Only one right angle. Which is good if you're a triangle. So, that's… You’re a right triangle.

Kim  09:46
Okay. And none of my sides are congruent.

Pam  09:50
Ah, so you're a scalene right triangle.

Kim  09:53
Yeah.

Pam  09:54
Okay, so you're a scalene right triangle, that is a special right triangle, that is a special triangle, that is a special polygon.

Kim  10:01
Mmhm.

Pam  10:01
Okay. Nice.

Kim  10:03
Okay, I have two more for you. 

Pam  10:04
Okay, all right.

Kim  10:06
Riddle four. Or riddle three, sorry. I am a closed figure with three or more sides.

Pam  10:12
Okay, so I'm a polygon.

Kim  10:15
Mmhm. All of my sides are congruent.

Pam  10:18
A regular polygon.

Kim  10:20
Nice.
 
Pam  10:21
I had to think about that for a second. Okay, sides congruent, regular polygon. Mmhm.

Kim  10:25
All of my angles are congruent.

Pam  10:28
So, I'm, again, a regular polygon. 

Kim  10:31
Mmhm.

Pam  10:32
Okay,

Kim  10:33
All three of my angles are acute.

Pam  10:38
Oh, three of your angles. So, now I know that you're a triangle. And if all three angles are acute, you are an acute triangle.

Kim  10:47
Mmhm.

Pam  10:48
Okay.

Kim  10:49
Nice. Riddle four. 

Pam  10:52
Bam. Wait, do you want me to do the hierarchy there? 

Kim  10:55
Oh, yeah, yeah, that's good. 

Pam  10:56
Okay, so I am an acute triangle that is a triangle, that is a regular polygon, that is a polygon. Or I'm a polygon, and a special polygon because all my angles and sides are congruent, so I'm regular. And I have three sides, so I'm a triangle, but I'm a special triangle because the angles are all acute. Which kind of falls from a regular triangle. Because as soon as I'm a regular triangle, I have to be an equilateral triangle. Which also means that the angles are acute.

Kim  11:26
Yeah.

Pam  11:26
Yeah. Okay, cool. Alright, I'm ready for four. Bring it on.

Kim  11:29
I am a closed figure with three or more sides.

Pam  11:32
I'm a polygon.

Kim  11:34
All of my sides are congruent.

Pam  11:37
So I’m regular.

Kim  11:38
All of my angles are also congruent

Pam  11:41
Still regular. 

Kim  11:43
All six of my…

Pam  11:44
I guess I could also say I'm equiangular. So, I'm equilateral and I'm equiangular. I could throw those words in there. Sorry. The next one.

Kim  11:52
All six of my angles are obtuse.

Pam  11:56
Six angles. So, I'm a hexagon. I don't know why hex. Why not sexagon? Okay, hexagon. 

Kim  12:05
No!

Pam  12:06
No? Sixagon? Sixagon. Should be a sixagon. Anyway, I'm a hexagon. Six-sided. What did you say about the angles? All my six angles (unclear). 

Kim  12:15
They're all obtuse. 

Pam  12:16
Obtuse. 

Kim  12:17
Yeah. 

Pam  12:18
Would that be true? As soon as we knew it was regular, did we know that the angles are obtuse? I think we did.

Kim  12:23
We just didn't know there was six at that time.

Pam  12:26
Gotcha. Yeah. Okay, so I'm a hexagon. Yep, a regular hexagon. 

Kim  12:30
(unclear) Okay, so we're taking what you currently know, right? You knew all of these words. You knew some of the relationships. But these riddles are a fantastic way to spiral some of the learning that you've already done. It's a fantastic way to continue the vocabulary, right? So many times in geometry classes, there's a chunk that's like here's the two weeks that we're going to talk geometry. And they are not… Those words leave the kids voices, they leave their heads. And so, once you've done some work with geometry, this is a fantastic way to continue that learning. They're super short, right? Just a couple of minutes each. It's great warm up. It's great thing that you can do in bits and pieces. I loved having riddles in an area of my classroom. Kids could quiz each other and just put some notes on the back about what the subcategories were. And it continues to build the relationships between the hierarchies

Pam  13:27
Nice. And I think that's a really nice way to take something that is social conventional knowledge, the vocabulary, and give it an opportunity to be kind of practiced in a meaningful, kind of fun, kind of purposeful way. Like, we could just say, “Take this vocabulary. Write it down. Put the definition in your notebook. Put it on a note card. Quiz each other.” Eh. Like, that's not only boring, it's not very effective. 

Kim  13:56
Right, 

Pam  13:56
But the vocabulary itself, the names. Like, I was sort of joking around about the hexagon and sexagon kind of thing. We want kids to know those names. And the best way to do that is to practice in context, to have those words come up along with the description, but in a in a purposeful way where like what you were just doing. We're sort of narrowing in on this kind of hierarchy in geometry where kids can get even better at the hierarchy. And they can be like, “Oh, yeah. Like, if I've got a special quadrilateral where the angles are congruent, then I know this about it. Wait, wait. But what's also true? Like, maybe if I…” And you didn't give me this one, but if I had a quadrilateral and I said the sides were congruent. Bam, then instantly you're sort of over on a rhombus.

Kim  14:46
Yeah. 

Pam  14:47
And then I could say now you have a rhombus, and I make the angles congruent. Oh, bam. Now, I've straightened up that rhombus into a square. That gives kind of this sense of the hierarchy of how those... The shapes, like you said before. We're kind of building that hierarchy. And the vocabulary at the same time. We like to do vocabulary when there's a… When you're sort of begging for it.

Kim  15:13
Right.

Pam  15:14
Like, when the kids are… They have the sort of sense of the thing in their head, and now there's a reason to. You know, they're trying to describe it. Kind of like… I can't remember exactly what you said. “I'm a…” How did you start three and four? “I'm a close sided...” 

Kim  15:27
I'm a closed figure with three or more sides.

Pam  15:30
Yeah, like you're giving sort of the definition. And we want kids to picture that thing. And then if they can't really come up with the word “polygon”, then we supply the word “polygon”. (unclear). 

Kim  15:39
Right.

Pam  15:39
“Remember, it's polygon.” Now, it's not like we don't ever want them to come up with come up with it, but this meaningful, sort of purposeful practice can be a great way to do that.

Kim  15:47
Yeah, and really, the relationships between the shapes and how they connect to each other is kind of what we want kids to know. How are they characterized, so that they're grouped similarly? And what makes them stand out? And why we have a need for a more specialized name? And I think that's the part that's sometimes tricky for kids is how can it be this and that? How come it's this but not that? And so, these riddles kind of sort out some of those misconceptions and some of the things that are maybe more challenging just because it gives them more repetition in a meaningful way.

Pam  16:23
Yeah. And maybe even in a little bit of a fun way. 

Kim  16:27
Yeah.

Pam  16:27
Cool. Alright, that was great! I'm going to put you in charge more often. Whoo! Super fun. 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... Did I just say Move? I don't even. Did I say movement? Didn’t sound like “movement” to me. To find out more about the Math is Figure-Out-Able movement, visit mathisfiguratable.com. And keep spreading the word that Math is Figure-Out-Able!