Transcriber: Yifat Adler Reviewer: Denise RQ
Karl Schaffer: Good morning.
Erik Stern: How do you do?
ES: How do you do? KS: Good morning.
ES: Hey, hey. KS: My good man.
KS: Hi. ES: Here we go.
(Laughter)
KS: Don't I know you?
ES & KS: Ha!
(Laughter)
ES & KS: Ha!
ES & KS: Yow! ES: Forget it.
ES: Oh, hi. KS: Hi.
KS: Hi, nice meeting you. ES: Nice to see you.
KS: Well. ES: Well.
KS: Will be seeing you. ES: Will be seeing ... you.
KS: Ah.
KS: Well. It was nice meeting you.
ES: OK. I got you.
KS: So long. Take care.
ES: Hi. I'm Erik Stern. KS: And I'm Karl Schaffer.
ES: And for a very long time we've been fascinated
with the connections between ideas and movement.
KS: Our creative experiments in the studio have taken us
both to the stage and to the classroom.
ES: This almost vaudevillian routine was the first of our dances
that got us talking about the mathematics of dance.
KS: We created it by collecting ways we can connect and ...
misconnect.
And then arranging them theatrically.
ES: We begin the take apart of our process from the point of view of combinatorics
which is the area of math concerned with counting combinations.
KS: So, this dance became the opening sequence
in our first show about dance and mathematics.
But it also inspired our first classroom activity.
ES: In that activity students work in groups
to create three to five unusual handshakes
and then arrange them in an intriguing order.
KS: Then we switched to thinking.
We say, let's define a connection such as right hand to right hand
to be the same connection or handshake no matter what form it takes:
right hand to right hand, that's one handshake.
And we ask, given this definition:
how many ways are there for two people to shake hands,
if each person may only use one hand at a time?
ES: This activity and the many others we've taught for over 20 years,
led us to discover a few key things about how we learn.
KS: First, embodying the problem is memorable.
It's social, it's creative, it makes the mathematical ideas accessible.
ES: Second, physical energy in the classroom,
far from being a distraction,
can be an opportunity for all ages and all disciplines.
KS: Third, choreographic and mathematical thinking
are composed of similar building blocks.
Noting changes.
ES: Remembering sequences.
KS: Asking if things are bigger of smaller.
ES: Checking your work to see if it's consistent and so on.
KS: Here are some of the answers we've got to that handshake counting problem.
[4 handshakes - R/R R/L L/R L/L] [3 handshakes - R+R, L+L, L+R]
ES: The more we taught this, the more we saw
that the mathematical discourse about what counts as a handshake
is as challenging and valuable as getting the right answer.
KS: It illustrates for us how complex "simple" counting questions can be.
We've been told countless times by teachers and students,
"I wish I could have studied math this way."
ES: What if teachers were to say,
"Put the books away, push the desks aside.
Let's warm up. It's time to do mathematics."
(Music)
ES: hey, hey, Sorry about this. Stop, stop. Dr. Schaffer...
KS: Yeah? ES: You littered.
KS: Oh. ES: Oh.
ES & KS: Well, I'll get it. You get it.
ES & KS: Stop.
KS: Hmmm.
ES: I've been looking for that sock for quite a while.
What just happened there was interesting,
when Dr. Schaffer was imitating everything I did,
it was almost as if there were a mirror in between us.
KS: You know, when Mr. Stern was copying my every move ...
ES: Copying what? KS: There's a name for that.
ES: Oh, yeh? KS: It's called symmetry.
ES: Oh, that name.
Symmetry is much easier to see than to do.
You are all going to be my mirror image right now.
All of you are going to be my mirror image right now.
Now look how... Well, you can't look.
I'm looking at how well you do this, and I'm assuming that's because ...
you see yourself in the mirror every morning.
So, you are rather used to this. So, let's try something harder.
KS: Raise your right hand and repeat after me:
"If I lean to my left, you lean to your left."
So, now, if Mr. Stern were to turn half way around,
he and I would be in the same shape.
ES: Good luck.
(Laughter)
KS: You know, dancers have to learn how to do this very quickly,
but maybe not that quickly.
(Laughter)
ES: The mathematician William Thurston wrote,
"We tend to think more effectively with spatial imagery on a larger scale.
It's as if our brains take larger things more seriously
and can devote more resources to them."
What we were just doing when we were thinking and moving together,
is how we often begin our classes.
Students are guided through movement problems
and we introduce dance and math concepts as we go.
KS: Even though symmetry is part of the core curriculum
we are not using these things to sugarcoat the mathematics
so it goes down more easily.
It's the connections between that are the heart of the matter.
And we're just as passionate about the dance ideas.
As choreographers, we find concepts like combinatorics and symmetry
to be powerful tools
in the science of dance as well as the art of mathematics.
ES: Hey, here's a symmetry exercise. I'd like you all to follow me.
(Clapping)
We're going to clap at four.
KS: Now, how is clapping and slapping symmetry?
ES: A pattern that repeats.
KS: Ah, let's clap at three. Two, three.
ES: Hey, woh, woh, woh ...
Nice clapping, but you're forgetting the most important thing
and that's that fours are better.
KS: You know, I think I like threes much better.
ES: Yeah, but fours are so much more fortuitous. Wouldn't you say?
KS: No, no, I don't think so.
But, I'm going to come over here,
and I'm going to clap at three with this half of the room.
ES: Fine. My half, we're going to clap at four at the same time.
Watch. We start by clapping up.
Ready, ES & KS: Set, go.
ES: And ... (Clapping)
(Laughter)
KS: You know, I like how it sounded when it got fast.
ES: Yeah?
KS: Do you think you could clap at four as fast as I could clap at three?
ES: Absolutely. KS: Just watch.
KS: You know, I think there's a great math puzzle here.
If the two of us start in the center at the same time,
and then one of us, he, returns to the center every four beats
and I return to the center every three beats,
then after how many total beats
will we both be back in the center at the same time?
Twelve. It's the smallest number that's an even number of threes
and an even number of fours, the least common multiple.
ES: Hey, look what I found in the wing.
(Laughter)
You see, we had threes and then fours,
and then when we played it together we made up a new rhythm.
It sounded like this: Taah-Ta-Taah-Taah-Ta-Taah
Taah-Ta-Taah-Taah-Ta-Taah.
KS: Look what I found. ES: How nice.
So, it's just like what you do when you subtract a fourth from a third,
convert to twelfths... We won't worry about that today.
ES: This is how I did it.
(Laughter)
KS: We are not saying that all classes must necessarily be movement-based,
just as all classes need not exclusively involve sitting at desks.
ES: But we do get very excited when artists, educators, and scientists
figure out ways to give students and teachers room to move,
especially when they're drawing on deeper underlying connections.
KS: And we will continue to play with ideas in the studio.
(Applause)