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PROFESSOR: OK.

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Today, teaching equations and
how to teach equations in a

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way that promotes long lasting
learning and understanding.

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OK.

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So the first example--

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I'm going to give you
two choices for

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starting the example.

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So this is example one for
teaching equations.

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This is Hardy-Weinberg
equilibrium.

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So here's one way
you could start.

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So if a locust has n alleles
and the organism is

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polyploid--

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so that means it has, let's say,
C copies of chromosomes.

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OK.

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So there's n alleles.

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So the allele frequencies
are P1 through Pn.

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Then we're going to deduce
the genotype frequencies.

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So this is the multinomial
middle coefficient C. Choose

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K1, K2, K3, all the way to Kn.

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And K1 through Kn are the
number of copies of each

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allele with K1 plus blah, blah,
blah, to Kn, all equal

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to C.

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So you could actually state
that and then prove it.

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That's option A. So now I'll
give you option B for starting

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the same topic.

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So option B is from Hardy.

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So in Hardy's Mathematician's
Apology, he writes, I have

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never done anything useful.

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No discovery of mine has made or
is likely to make, directly

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or indirectly, for good or ill,
the least difference to

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the amenity of the world.

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That's a very interesting,
very bold statement.

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And amazingly, for
such a brilliant

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mathematician, it's wrong.

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And today, what we're going to
do is we're going to look at

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an example of the discovery
by Hardy--

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the Hardy-Weinberg
equilibrium--

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that does make a difference to
the amenity of the world and

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helps you actually understand
genetics.

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OK.

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So the option B is to quote
Hardy and then point out the

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contradiction between what
he said and what

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we're going to do.

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Now, also, in option B, you
can go a bit farther.

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You can say, well, why
would Hardy have said

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something like that?

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And indeed, why would he have
said something like that in

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his Apology?

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Well, it's actually probably
hard to appreciate on this

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side of the Atlantic, because
since the Civil War, we've

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never had a war that just
basically devastated entire

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countries on our own soil.

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Whereas in Europe, the memories
of World War I are

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very strong.

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So if anyone's from Europe,
you know that.

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In every English chapel, there's
names on the wall of

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all the people who died in
the, quote, Great War.

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So World War I had
a very strong

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effect on European society.

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And one of the effects was on
European science and people's

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attitude towards science.

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So poison gas was invented
partly by German chemists.

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Hopper was one of them.

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And in the end, he committed
suicide, partly maybe because

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of what he had done.

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No, sorry.

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His wife committed suicide.

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I forget if--

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I should check that.

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But basically, people in the
family were so unhappy about

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what had happened just there
that there was a suicide.

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And furthermore, science after
that was considered--

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World War I was considered
the chemists' war.

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So there was a wish to distance
oneself from those

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kinds of horrible effects.

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And the quote from Hardy is
actually a reflection of that.

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He himself was very
strongly anti-war.

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He left Cambridge because
Cambridge fired Bertrand

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Russell for protesting World War
I. So Hardy left Cambridge

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for a professorship in Oxford
and only came back basically

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12 or 13 years later
to Cambridge.

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So that statement of his partly
a wish that he's hoping

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nothing he's done has any effect
on the world because in

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his mind, maybe most
effects are bad.

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Well, but actually, he
did have an effect.

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The effect is Hardy-Weinberg.

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Even if you discount everything
you did in number

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theory, there's Hardy-Weinberg,
and it has an

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effect and we're going
to look at it.

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So let me ask you rhetorical
question.

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If you're a student, which would
you find more engaging,

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more inviting?

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This one or this one?

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00:06:55,890 --> 00:06:57,370
Who votes for that one?

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OK.

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Who votes for that one?

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Yeah.

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Now, why?

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So now think of-- so the
question I'm going to ask you

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to think about just for a minute
with one or two of your

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neighbors is why.

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So yeah, this is definitely
more engaging.

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In terms of the principles we
talked about last time, what

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makes this way of introducing it
more engaging, more likely

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to bring students in, more
likely to make them want to

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learn about Hardy-Weinberg.

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OK.

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So find a neighbor or two we
think about the principles

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behind

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option B. OK.

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So I'll rudely interrupt you.

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And good thing I did the voice
exercise so I can project all

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the way into the back.

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So what reasons for option B or
against option A, which is

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00:08:04,680 --> 00:08:06,530
the same thing?

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00:08:06,530 --> 00:08:06,840
Yes.

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Can you tell me your name?

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AUDIENCE: Brian.

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00:08:08,470 --> 00:08:08,730
PROFESSOR: Brian.

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00:08:08,730 --> 00:08:11,310
I'm going to try to learn
people's names.

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So go ahead.

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AUDIENCE: The reason
against option A--

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when you start out that way, you
are activating the ability

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that students have to transfer
material from the blackboard

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to their notes and then pass
it to their brains.

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PROFESSOR: Right--
through the wave.

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Yeah.

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I'm activating the ability
of the students to

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basically take dictation.

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Or as someone said, teaching
is an excellent way to

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00:08:35,000 --> 00:08:38,330
transfer material from the notes
of the teacher to the

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00:08:38,330 --> 00:08:40,100
notes of the student without
it passing through

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00:08:40,100 --> 00:08:42,559
the minds of either.

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So I'll call that not A. So
that's the not symbol.

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00:08:45,680 --> 00:08:46,930
Not A is--

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so I'm basically asking to do
dedication which doesn't

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necessarily lead to any
kind of learning.

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00:09:02,510 --> 00:09:05,290
They could, for example, copy
all of that down but not

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00:09:05,290 --> 00:09:07,110
really understand of it.

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00:09:07,110 --> 00:09:07,410
OK.

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00:09:07,410 --> 00:09:09,270
Other--

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00:09:09,270 --> 00:09:10,315
yes, can you tell
me your name?

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00:09:10,315 --> 00:09:11,100
AUDIENCE: Susannah.

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00:09:11,100 --> 00:09:12,540
PROFESSOR: Susannah.

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00:09:12,540 --> 00:09:14,940
AUDIENCE: This is right brain?

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00:09:14,940 --> 00:09:19,260
So it's easier to remember
and ingest it.

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And maybe even identify
the theorems.

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PROFESSOR: OK.

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Right.

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And so B is the story.

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I'll give you an example of
how stories can be so

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00:09:35,870 --> 00:09:39,330
powerful, and just
even the word.

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So a big commercial
publisher--

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this was several years ago--

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they wanted me to write a
freshman physics textbook.

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I had actually just put up a
proposal on the web saying, we

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should write a freshman physics
textbook that is based

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00:09:54,450 --> 00:09:56,210
on the history of science.

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And they saw it and they
thought, oh, that's great.

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So they flew me from England to
California to talk to them.

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00:10:02,050 --> 00:10:04,160
And they liked it mostly.

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00:10:04,160 --> 00:10:07,100
But they said, history--

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00:10:07,100 --> 00:10:08,410
so there was the word.

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They liked the idea as I talked
about it, but the word

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history really frightened
them.

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00:10:13,480 --> 00:10:15,790
And then, I'm still amazed--

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00:10:15,790 --> 00:10:20,010
I had this amazing insight
of just two letters.

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00:10:20,010 --> 00:10:20,900
I thought, oh, wait.

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00:10:20,900 --> 00:10:23,320
I can actually explain
to them what I mean.

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00:10:23,320 --> 00:10:27,130
I said, well, actually, if you
just take away that part, what

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00:10:27,130 --> 00:10:28,990
I'm really talking
about is that.

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00:10:28,990 --> 00:10:30,400
And it was interesting.

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As soon as they saw the word
story and they saw that

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00:10:34,360 --> 00:10:36,770
history doesn't have to be
names, dates, facts--

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00:10:36,770 --> 00:10:38,170
I think that's what they
were seeing it as--

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00:10:38,170 --> 00:10:41,780
they saw it as story, which
is actually its origin

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00:10:41,780 --> 00:10:44,110
in French or Latin--

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00:10:44,110 --> 00:10:46,070
[FRENCH].

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00:10:46,070 --> 00:10:47,565
They all of a sudden were
totally convinced.

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They said, oh, yeah.

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00:10:48,430 --> 00:10:50,740
That's exactly how we should
do the textbook.

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00:10:50,740 --> 00:10:55,600
For various reasons, basically
because I wanted to be a

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00:10:55,600 --> 00:10:56,800
freely licensed book, we didn't

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00:10:56,800 --> 00:10:58,690
actually sign a contract.

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But it's an example of how
powerful story is and how

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00:11:01,940 --> 00:11:05,270
people who actually spend their
lives thinking about

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00:11:05,270 --> 00:11:08,090
teaching and reaching students--
in other words,

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00:11:08,090 --> 00:11:11,630
this educational publisher knew
the importance of story.

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00:11:11,630 --> 00:11:15,300
But if you present it as just
dry history, it's not so

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00:11:15,300 --> 00:11:16,040
interesting.

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00:11:16,040 --> 00:11:17,200
So story--

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00:11:17,200 --> 00:11:18,050
yeah.

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00:11:18,050 --> 00:11:18,840
So let's see.

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00:11:18,840 --> 00:11:20,480
Who haven't I heard from?

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00:11:20,480 --> 00:11:22,280
I haven't heard from Adrian.

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00:11:22,280 --> 00:11:25,192
Yeah, and then you're next.

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00:11:25,192 --> 00:11:27,424
AUDIENCE: So the fact that it's
a contradiction creates

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00:11:27,424 --> 00:11:30,152
some sort of tension from
linear thought.

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00:11:30,152 --> 00:11:32,640
And it's more of
a exploratory.

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00:11:32,640 --> 00:11:33,110
PROFESSOR: OK.

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00:11:33,110 --> 00:11:34,280
So there's a contradiction.

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00:11:34,280 --> 00:11:37,050
And the contradiction creates
some kind of tension.

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00:11:37,050 --> 00:11:38,610
So the tension creates
interest.

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00:11:42,100 --> 00:11:45,620
So every good story needs
some kind of tension.

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00:11:45,620 --> 00:11:49,240
So the tension here, the
contradiction, is that Hardy

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00:11:49,240 --> 00:11:51,875
wanted to do nothing that could
harm people, basically.

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00:11:51,875 --> 00:11:56,980
And so he was like, OK, well,
anything could be used, even

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00:11:56,980 --> 00:11:58,090
if it's for good, for ill.

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00:11:58,090 --> 00:12:01,190
So let me just back off from
all of that and say I'm not

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00:12:01,190 --> 00:12:04,710
going to do anything that has
an effect, an application.

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00:12:04,710 --> 00:12:07,050
I'm a pure mathematician.

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00:12:07,050 --> 00:12:10,850
His most famous book is A Course
of Pure Mathematics.

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00:12:10,850 --> 00:12:13,030
Well, maybe his most famous
book is that book that I

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00:12:13,030 --> 00:12:15,100
quoted from, A Mathematician's
Apology.

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00:12:15,100 --> 00:12:17,740
His most famous math book is
probably A Course of Pure

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00:12:17,740 --> 00:12:19,430
Mathematics.

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00:12:19,430 --> 00:12:21,020
And that's what he wanted
to be known for.

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00:12:21,020 --> 00:12:23,530
And that's in contradiction
with the fact and no, even

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00:12:23,530 --> 00:12:26,320
Hardy couldn't help
an application--

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00:12:26,320 --> 00:12:28,940
an actually very,
very common one.

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00:12:28,940 --> 00:12:31,320
It's taught in every
single introductory

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00:12:31,320 --> 00:12:34,380
biology course, probably.

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00:12:34,380 --> 00:12:34,650
OK.

231
00:12:34,650 --> 00:12:36,000
So you need tension.

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00:12:36,000 --> 00:12:39,130
Now, that's a general principle
of learning there--

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00:12:39,130 --> 00:12:42,890
the idea of story and
tension and paradox.

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00:12:42,890 --> 00:12:48,270
So the one way that, as a
undergraduate in physics, I

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00:12:48,270 --> 00:12:51,310
learned a ton of physics, me
and my friends were doing

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00:12:51,310 --> 00:12:52,500
problem sets together.

237
00:12:52,500 --> 00:12:55,170
And a lot of problems were just
grinding through math.

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00:12:55,170 --> 00:12:56,920
So we didn't learn a hell
of a lot from that.

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00:12:56,920 --> 00:12:59,470
But we were in the library late
at night, the physics

240
00:12:59,470 --> 00:13:01,570
library, ordering pizza, and
trying to do a problem set,

241
00:13:01,570 --> 00:13:04,890
and we just got to making
up physics paradoxes--

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00:13:04,890 --> 00:13:06,080
perpetual motion machines.

243
00:13:06,080 --> 00:13:08,890
We'd invent perpetual motion
machines and try to get the

244
00:13:08,890 --> 00:13:10,620
other person to figure out
what was wrong with it.

245
00:13:10,620 --> 00:13:12,310
Sometimes, we didn't even know
what was wrong with it and

246
00:13:12,310 --> 00:13:13,990
we'd both try to
figure it out.

247
00:13:13,990 --> 00:13:16,710
So from that kind of tension--

248
00:13:16,710 --> 00:13:20,260
tension is almost, in a way,
self teaching, because as long

249
00:13:20,260 --> 00:13:22,250
as the tension's there,
you know you're not at

250
00:13:22,250 --> 00:13:23,650
the end of the story.

251
00:13:23,650 --> 00:13:25,300
You know there's more to do.

252
00:13:25,300 --> 00:13:27,550
So the same thing-- as long as
there was perpetual motion

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00:13:27,550 --> 00:13:30,120
going on and we hadn't found the
reason, we knew we weren't

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00:13:30,120 --> 00:13:30,930
done with the problem.

255
00:13:30,930 --> 00:13:32,660
We didn't have to ask a teacher
to say, well, is this

256
00:13:32,660 --> 00:13:35,070
right, because it was basically
self teaching.

257
00:13:35,070 --> 00:13:37,180
As long as the tension
was there, we knew

258
00:13:37,180 --> 00:13:39,740
that we weren't finished.

259
00:13:39,740 --> 00:13:40,730
So yeah--

260
00:13:40,730 --> 00:13:43,510
another general principle,
stories and tension.

261
00:13:43,510 --> 00:13:44,070
Let's see.

262
00:13:44,070 --> 00:13:46,850
There was a comment over--

263
00:13:46,850 --> 00:13:47,140
yes.

264
00:13:47,140 --> 00:13:49,600
Tell me your name?

265
00:13:49,600 --> 00:13:50,360
AUDIENCE: Wing Ho.

266
00:13:50,360 --> 00:13:51,240
PROFESSOR: Wing Ho?

267
00:13:51,240 --> 00:13:51,650
AUDIENCE: Yeah.

268
00:13:51,650 --> 00:13:54,278
So this along the line that
the teacher should make

269
00:13:54,278 --> 00:13:58,976
themselves more human, because
Hardy-Weinberg sounds like

270
00:13:58,976 --> 00:14:00,715
such a big name you
would tattoo.

271
00:14:00,715 --> 00:14:05,500
That student may be driven to
believe in the equation just

272
00:14:05,500 --> 00:14:08,745
by the matter of the fact
that it's authority.

273
00:14:08,745 --> 00:14:10,790
But then making a
story makes--

274
00:14:13,760 --> 00:14:14,270
PROFESSOR: OK.

275
00:14:14,270 --> 00:14:22,860
So the teacher makes himself
more human and makes one of

276
00:14:22,860 --> 00:14:25,600
the people who invented the
Hardy-Weinberg equilibrium

277
00:14:25,600 --> 00:14:26,590
more human.

278
00:14:26,590 --> 00:14:29,510
So it's actually something
much more easy for the

279
00:14:29,510 --> 00:14:32,460
students to connect to.

280
00:14:32,460 --> 00:14:34,070
Yes, can you tell
me your name?

281
00:14:34,070 --> 00:14:34,580
AUDIENCE: Gregor.

282
00:14:34,580 --> 00:14:35,245
PROFESSOR: Gregor.

283
00:14:35,245 --> 00:14:35,670
AUDIENCE: Yeah.

284
00:14:35,670 --> 00:14:40,738
So my problem is that two
different options are very,

285
00:14:40,738 --> 00:14:41,690
very different.

286
00:14:41,690 --> 00:14:48,370
And the amount of legwork it is,
it's maybe more tense to

287
00:14:48,370 --> 00:14:54,130
me, because you can discuss it
without legwork, but I think

288
00:14:54,130 --> 00:14:57,730
also that option A is pretty
interesting, because you can

289
00:14:57,730 --> 00:15:00,500
approach this biological
problem from more of a

290
00:15:00,500 --> 00:15:01,490
qualitative approach.

291
00:15:01,490 --> 00:15:04,559
So depending on what kind of
learner you have and what kind

292
00:15:04,559 --> 00:15:08,230
of question you choose,
A would be also very

293
00:15:08,230 --> 00:15:08,490
interesting.

294
00:15:08,490 --> 00:15:11,626
But at that point, with the
background I have right now,

295
00:15:11,626 --> 00:15:14,360
it's very difficult
to pursue that.

296
00:15:14,360 --> 00:15:14,740
PROFESSOR: OK.

297
00:15:14,740 --> 00:15:19,740
So Gregor's point, which is a
good one, is that B doesn't

298
00:15:19,740 --> 00:15:23,790
have the same quantitative depth
as A. And actually, what

299
00:15:23,790 --> 00:15:28,320
I'm going to show you is a way
to get to this starting from

300
00:15:28,320 --> 00:15:32,200
here and using the principles
that we're talking about so

301
00:15:32,200 --> 00:15:35,740
that by the time you get here,
or to something like this, it

302
00:15:35,740 --> 00:15:38,690
actually makes sense
to the students--

303
00:15:38,690 --> 00:15:42,070
so that you can have both.

304
00:15:42,070 --> 00:15:44,290
So that's a promise.

305
00:15:44,290 --> 00:15:48,100
And hopefully, I'll deliver
on it for you.

306
00:15:48,100 --> 00:15:50,015
But does that address what
you're talking about?

307
00:15:50,015 --> 00:15:50,940
AUDIENCE: Yeah.

308
00:15:50,940 --> 00:15:52,990
PROFESSOR: --that B doesn't have
the quantitative depth--

309
00:15:52,990 --> 00:15:54,142
and that's true.

310
00:15:54,142 --> 00:15:57,446
AUDIENCE: Actually, I mean, you
can talk about one hour on

311
00:15:57,446 --> 00:16:03,391
B. But if you really want to
deliver an idea or a concept,

312
00:16:03,391 --> 00:16:07,482
that is as quantitative as
option A, then you have to get

313
00:16:07,482 --> 00:16:10,390
to A at some point.

314
00:16:10,390 --> 00:16:13,200
PROFESSOR: So the point made
is that if you want to get

315
00:16:13,200 --> 00:16:16,360
here, you do have to say
this at some point.

316
00:16:16,360 --> 00:16:18,630
You can't just say the
story for now.

317
00:16:18,630 --> 00:16:20,450
Well, it depends on the
purpose of the class.

318
00:16:20,450 --> 00:16:23,600
If the purpose is the history
of biology, maybe you'd

319
00:16:23,600 --> 00:16:26,540
continue with the story or the
history of science and war.

320
00:16:26,540 --> 00:16:29,540
But if you want students to be
able to solve problems in

321
00:16:29,540 --> 00:16:33,240
genetics, maybe you
need to go here.

322
00:16:33,240 --> 00:16:38,440
Or what we'll find is I'll
continue with option B moving

323
00:16:38,440 --> 00:16:42,030
towards A. And I'm going to show
how you can get there and

324
00:16:42,030 --> 00:16:45,570
still have the opening
of B preserved.

325
00:16:45,570 --> 00:16:46,340
OK so let's see.

326
00:16:46,340 --> 00:16:47,170
There was--

327
00:16:47,170 --> 00:16:47,543
yes.

328
00:16:47,543 --> 00:16:48,640
Can you tell me your name?

329
00:16:48,640 --> 00:16:50,510
AUDIENCE: Roger.

330
00:16:50,510 --> 00:16:52,760
I think one of the reasons
we like B is because we

331
00:16:52,760 --> 00:16:54,505
see A all the time.

332
00:16:54,505 --> 00:16:58,790
So I don't know to what extent
that's [INAUDIBLE].

333
00:16:58,790 --> 00:16:59,900
PROFESSOR: OK.

334
00:16:59,900 --> 00:17:03,400
So B is new.

335
00:17:03,400 --> 00:17:05,829
Right.

336
00:17:05,829 --> 00:17:08,650
So against A, A is very
familiar and common.

337
00:17:08,650 --> 00:17:11,310
So in fact, yeah,
I chose this--

338
00:17:11,310 --> 00:17:13,349
I didn't really make
a straw man here.

339
00:17:13,349 --> 00:17:16,480
This is how a lot of things
are introduced.

340
00:17:16,480 --> 00:17:21,910
For example, in mathematics
class, theorem proof--

341
00:17:21,910 --> 00:17:25,480
so the theorem will be
introduced without any of the

342
00:17:25,480 --> 00:17:29,290
struggle, the wondering that
led to the theorem.

343
00:17:29,290 --> 00:17:31,560
Why would anyone even care
about such a thing?

344
00:17:31,560 --> 00:17:33,530
Or if they cared about it, why
would they come up with

345
00:17:33,530 --> 00:17:35,450
something like that?

346
00:17:35,450 --> 00:17:37,480
How can you see that?

347
00:17:37,480 --> 00:17:41,240
So yeah, A is seems very
familiar and therefore, maybe

348
00:17:41,240 --> 00:17:42,890
not as interesting.

349
00:17:42,890 --> 00:17:46,790
Although, it may also be
intrinsically less interesting

350
00:17:46,790 --> 00:17:47,840
for the other reasons.

351
00:17:47,840 --> 00:17:49,650
But I take your point, too,
that it could be just

352
00:17:49,650 --> 00:17:50,590
familiarity.

353
00:17:50,590 --> 00:17:50,860
Yes.

354
00:17:50,860 --> 00:17:51,990
Can you tell me your name?

355
00:17:51,990 --> 00:17:52,390
AUDIENCE: Yeah, I'm Meg.

356
00:17:52,390 --> 00:17:52,600
PROFESSOR: Meg.

357
00:17:52,600 --> 00:17:55,573
AUDIENCE: It seems that you
could really incorporate the

358
00:17:55,573 --> 00:17:59,085
quantitative side into B if you
wanted to by having the

359
00:17:59,085 --> 00:18:03,010
way the proof occurred
[INAUDIBLE].

360
00:18:03,010 --> 00:18:05,208
And then you wind up reversing
the traditional order, where

361
00:18:05,208 --> 00:18:07,653
you have a proof and then a
theorem, because the way it

362
00:18:07,653 --> 00:18:10,098
happened in reality and the
way it happens when you're

363
00:18:10,098 --> 00:18:12,176
going to do research, and the
way it happens to students

364
00:18:12,176 --> 00:18:13,032
when they're trying to work
it out themselves.

365
00:18:13,032 --> 00:18:15,477
And so I feel like that's much
more of a natural flow

366
00:18:15,477 --> 00:18:16,727
[INAUDIBLE].

367
00:18:20,000 --> 00:18:20,300
PROFESSOR: OK.

368
00:18:20,300 --> 00:18:24,890
So your point, which is an
excellent one as well, is that

369
00:18:24,890 --> 00:18:27,690
B doesn't preclude
the quantitative.

370
00:18:27,690 --> 00:18:33,180
What you could do is you could
talk about the history, start

371
00:18:33,180 --> 00:18:36,160
with the history of Hardy saying
that, which is not

372
00:18:36,160 --> 00:18:38,310
chronological, because
he said that in 1940.

373
00:18:38,310 --> 00:18:38,890
But that's OK.

374
00:18:38,890 --> 00:18:40,275
It doesn't have to chronological
just because

375
00:18:40,275 --> 00:18:42,060
it's history.

376
00:18:42,060 --> 00:18:45,980
Go to 1940 and then backtrack to
well, how did this problem

377
00:18:45,980 --> 00:18:48,320
actually come to Hardy's
attention, which actually

378
00:18:48,320 --> 00:18:50,700
turns out to be quite an
interesting story.

379
00:18:50,700 --> 00:18:53,440
And then talk about how they
solved it and what was the

380
00:18:53,440 --> 00:18:54,220
publication history.

381
00:18:54,220 --> 00:18:56,960
And all the quantitative ideas
would come out in that way.

382
00:18:56,960 --> 00:19:01,380
And it would be almost backwards
from the usual way.

383
00:19:04,510 --> 00:19:08,390
The usual way here is
completely general.

384
00:19:08,390 --> 00:19:11,120
And that's probably not the way
it was first figured out.

385
00:19:11,120 --> 00:19:15,950
And the advantage of telling
about it that way is that

386
00:19:15,950 --> 00:19:19,530
you're preparing students
themselves for a research

387
00:19:19,530 --> 00:19:24,000
career, because nobody comes
up with theorems full blown

388
00:19:24,000 --> 00:19:26,070
like Athena out of
the head of Zeus.

389
00:19:26,070 --> 00:19:28,300
You come to them from struggle
and wondering.

390
00:19:28,300 --> 00:19:29,460
Hm, I wonder.

391
00:19:29,460 --> 00:19:32,280
Everything has some historical
background to it.

392
00:19:32,280 --> 00:19:39,370
So it's called the, you could
say, the genetic approach.

393
00:19:39,370 --> 00:19:42,935
So in biology, there's a slight,
not misconception, but

394
00:19:42,935 --> 00:19:46,720
a saying that ontology
recapitulates phylogeny--

395
00:19:46,720 --> 00:19:50,630
in other words, that the
organism, as it develops, say,

396
00:19:50,630 --> 00:19:55,000
in the womb, goes through all
the evolutionary stages that

397
00:19:55,000 --> 00:19:57,670
it went through over the last
200 million years or whatever

398
00:19:57,670 --> 00:19:58,900
to become, say, a person.

399
00:19:58,900 --> 00:20:01,410
You're at first a fish and then
maybe you're a monkey and

400
00:20:01,410 --> 00:20:02,250
then you're a person.

401
00:20:02,250 --> 00:20:04,360
So roughly speaking-- it's
not exactly true.

402
00:20:04,360 --> 00:20:08,280
But there's a lot of truth to
that for learning ideas--

403
00:20:08,280 --> 00:20:12,230
that you have to recapitulate
the history of ideas to really

404
00:20:12,230 --> 00:20:15,480
understand where we are
now, the ideas today.

405
00:20:15,480 --> 00:20:22,360
So a great example of that is
Newton's Second Law of Motion,

406
00:20:22,360 --> 00:20:25,740
the idea that force and
acceleration are connected.

407
00:20:25,740 --> 00:20:27,980
So for thousands of years,
millions of years, people

408
00:20:27,980 --> 00:20:30,260
thought force and velocity
were connected.

409
00:20:30,260 --> 00:20:32,800
And it's actually force
and acceleration.

410
00:20:32,800 --> 00:20:35,790
So actually, you can guide
people to the understanding of

411
00:20:35,790 --> 00:20:38,550
force and acceleration being
connected by showing them the

412
00:20:38,550 --> 00:20:42,510
history of how people thought
it was force and velocity,

413
00:20:42,510 --> 00:20:45,280
because for those same reasons,
those are the reasons

414
00:20:45,280 --> 00:20:46,790
that students will
think it, too.

415
00:20:46,790 --> 00:20:48,260
So you're actually helping
them overcome their

416
00:20:48,260 --> 00:20:49,900
misconceptions.

417
00:20:49,900 --> 00:20:53,830
So the history actually,
generally speaking, helps

418
00:20:53,830 --> 00:20:55,130
overcome misconceptions.

419
00:20:55,130 --> 00:20:56,470
So I'll put that here.

420
00:21:00,700 --> 00:21:03,410
And the history is actually
quite interesting.

421
00:21:03,410 --> 00:21:07,520
I think it was Punnett
from Punnett squares.

422
00:21:07,520 --> 00:21:10,570
He played cricket with Hardy
and Hardy loved cricket.

423
00:21:10,570 --> 00:21:12,600
And he just asked him
about this problem.

424
00:21:12,600 --> 00:21:14,030
And Hardy said, oh.

425
00:21:14,030 --> 00:21:15,950
Yeah, no, it's just this,
and sent off a paper

426
00:21:15,950 --> 00:21:18,660
to Science or Nature.

427
00:21:18,660 --> 00:21:22,530
So basically, it's because
they were cricket playing

428
00:21:22,530 --> 00:21:26,090
colleagues in Cambridge, that's
how a mathematician

429
00:21:26,090 --> 00:21:28,800
ended up interested in
a biology problem.

430
00:21:28,800 --> 00:21:30,730
Punnett was a biologist.

431
00:21:30,730 --> 00:21:31,350
OK.

432
00:21:31,350 --> 00:21:32,760
So any other reasons?

433
00:21:32,760 --> 00:21:33,980
Yes.

434
00:21:33,980 --> 00:21:35,690
AUDIENCE: I'm going to
play devil's advocate

435
00:21:35,690 --> 00:21:36,180
for a second here.

436
00:21:36,180 --> 00:21:36,910
PROFESSOR: Sure.

437
00:21:36,910 --> 00:21:38,260
Can you tell me your name?

438
00:21:38,260 --> 00:21:38,715
AUDIENCE: Paul.

439
00:21:38,715 --> 00:21:39,810
PROFESSOR: Paul.

440
00:21:39,810 --> 00:21:43,500
AUDIENCE: So when I had
biochemistry the first time,

441
00:21:43,500 --> 00:21:49,120
the first month of the class,
the teacher would tell the

442
00:21:49,120 --> 00:21:53,220
history about the people
developing quantum mechanics.

443
00:21:53,220 --> 00:21:54,800
And that was the lectures.

444
00:21:54,800 --> 00:21:56,745
We didn't touch the
material one bit.

445
00:21:56,745 --> 00:21:59,500
And because his philosophy was
the material was in the book,

446
00:21:59,500 --> 00:22:00,492
you don't need to teach it.

447
00:22:00,492 --> 00:22:02,040
You just go home and read it.

448
00:22:02,040 --> 00:22:04,590
But I remember being in that
class-- and maybe that's

449
00:22:04,590 --> 00:22:08,825
because I normally don't see
things taught like that-- but

450
00:22:08,825 --> 00:22:12,993
everyone was outraged,
because we didn't

451
00:22:12,993 --> 00:22:14,420
feel we learned anything.

452
00:22:14,420 --> 00:22:15,590
PROFESSOR: Right.

453
00:22:15,590 --> 00:22:15,890
OK.

454
00:22:15,890 --> 00:22:21,150
So the comment is that you were
in an actual class that

455
00:22:21,150 --> 00:22:23,830
was purely taught about
the history.

456
00:22:23,830 --> 00:22:24,710
People were outraged.

457
00:22:24,710 --> 00:22:27,860
They felt they weren't learning
actual content.

458
00:22:27,860 --> 00:22:29,460
And there's two answers
to that.

459
00:22:29,460 --> 00:22:33,390
One is that actually, I don't
recommend if, say, you're

460
00:22:33,390 --> 00:22:36,020
teaching a physics class,
that you teach it purely

461
00:22:36,020 --> 00:22:37,670
historically.

462
00:22:37,670 --> 00:22:41,600
But to the extent that you do
teach the history, you can

463
00:22:41,600 --> 00:22:44,080
actually teach the content
through the history.

464
00:22:44,080 --> 00:22:47,460
So it's not that the history and
the equations or the ideas

465
00:22:47,460 --> 00:22:48,690
are separate.

466
00:22:48,690 --> 00:22:52,470
It's that you can use the
history as a means of making

467
00:22:52,470 --> 00:22:55,720
the equations come alive with
a deeper understanding.

468
00:22:55,720 --> 00:22:57,890
So it's possible that the
teacher you're talking about

469
00:22:57,890 --> 00:23:01,540
actually didn't do that and just
talked about the people

470
00:23:01,540 --> 00:23:03,080
who did this and did that.

471
00:23:03,080 --> 00:23:05,350
But he didn't really know
what they did or why.

472
00:23:05,350 --> 00:23:08,980
So you're not forced
to do that.

473
00:23:08,980 --> 00:23:09,240
OK.

474
00:23:09,240 --> 00:23:10,095
So I'll give you an example.

475
00:23:10,095 --> 00:23:12,840
I'll continue with this in just
a moment after I take any

476
00:23:12,840 --> 00:23:15,120
other questions, showing you
how you can start with the

477
00:23:15,120 --> 00:23:17,610
history, and then you can
show the content.

478
00:23:17,610 --> 00:23:19,820
You can actually show
people exactly why

479
00:23:19,820 --> 00:23:20,690
that would be true.

480
00:23:20,690 --> 00:23:22,792
And I'll show you that
in one second.

481
00:23:22,792 --> 00:23:23,220
OK.

482
00:23:23,220 --> 00:23:23,960
There was a question there.

483
00:23:23,960 --> 00:23:24,350
Yes.

484
00:23:24,350 --> 00:23:25,165
Sharon.

485
00:23:25,165 --> 00:23:25,670
Oh, sorry.

486
00:23:25,670 --> 00:23:26,120
Not Sharon.

487
00:23:26,120 --> 00:23:27,600
Can you tell me your name?

488
00:23:27,600 --> 00:23:28,130
AUDIENCE: Me?

489
00:23:28,130 --> 00:23:28,880
PROFESSOR: No, in front.

490
00:23:28,880 --> 00:23:29,770
AUDIENCE: Cecilia.

491
00:23:29,770 --> 00:23:30,130
PROFESSOR: Cecilia.

492
00:23:30,130 --> 00:23:31,834
Yes.

493
00:23:31,834 --> 00:23:41,344
AUDIENCE: I was wondering that
Option B-- it's like targeting

494
00:23:41,344 --> 00:23:42,328
another audience.

495
00:23:42,328 --> 00:23:45,772
I come to a class and I'm not
very interested in the class.

496
00:23:45,772 --> 00:23:49,216
I need to be motivated here,
Genetics or something.

497
00:23:49,216 --> 00:23:54,910
If I already signed
up for genetics--

498
00:23:58,592 --> 00:24:01,790
because if I were being in a
class interested in that, I

499
00:24:01,790 --> 00:24:03,020
actually want to learn things.

500
00:24:03,020 --> 00:24:04,988
I want to know the rules.

501
00:24:04,988 --> 00:24:08,432
So I don't know if I want
to be distracted by

502
00:24:08,432 --> 00:24:13,330
some story that is--

503
00:24:13,330 --> 00:24:17,222
when you first said that, I was
like, oh my god, if this

504
00:24:17,222 --> 00:24:21,478
is going to be some other big
story, he's going to have the

505
00:24:21,478 --> 00:24:22,345
equation at the end.

506
00:24:22,345 --> 00:24:25,306
I'm not going to be
understanding what's going on

507
00:24:25,306 --> 00:24:29,460
or whatever you say.

508
00:24:29,460 --> 00:24:33,340
And I think that could be a
little bit of a turn off more,

509
00:24:33,340 --> 00:24:36,735
because-- try to make
it look easy.

510
00:24:36,735 --> 00:24:40,150
You have avoided writing the
equation on the board.

511
00:24:40,150 --> 00:24:41,320
PROFESSOR: Yeah, I'm going
to do that next.

512
00:24:41,320 --> 00:24:42,310
AUDIENCE: OK.

513
00:24:42,310 --> 00:24:46,765
And also, I guess there
are two kinds of

514
00:24:46,765 --> 00:24:48,745
stories you can tell.

515
00:24:52,854 --> 00:24:55,527
I don't know what story you're
going to tell exactly, but

516
00:24:55,527 --> 00:24:58,686
this seems to be removed from
the actual equation.

517
00:24:58,686 --> 00:25:04,040
This is more Hardy's
philosophy.

518
00:25:04,040 --> 00:25:04,870
It's not really--

519
00:25:04,870 --> 00:25:06,875
PROFESSOR: It's not directly
about that.

520
00:25:06,875 --> 00:25:11,470
AUDIENCE: Something you did a
second ago was more about the

521
00:25:11,470 --> 00:25:13,360
equation and more about
the concepts

522
00:25:13,360 --> 00:25:13,970
involved with the equation.

523
00:25:13,970 --> 00:25:15,360
PROFESSOR: That's true.

524
00:25:15,360 --> 00:25:19,350
So one question is how related
should the stories be?

525
00:25:19,350 --> 00:25:21,600
And there's a lot of
freedom in that.

526
00:25:21,600 --> 00:25:24,760
I would say the answer
is that you want it--

527
00:25:24,760 --> 00:25:27,260
if it does this, if it creates
interest, then it's already

528
00:25:27,260 --> 00:25:28,210
done something.

529
00:25:28,210 --> 00:25:31,280
Now, your other point was that
it depends on the audience.

530
00:25:31,280 --> 00:25:34,080
Suppose the audience is all
graduate students in genetics

531
00:25:34,080 --> 00:25:37,050
or people who want a crash
course in genetics.

532
00:25:37,050 --> 00:25:41,220
Maybe actually, they want to
just know the formula.

533
00:25:41,220 --> 00:25:44,480
And they would actually be
offended if you started

534
00:25:44,480 --> 00:25:46,800
telling them stuff that wasn't
the formula, because they feel

535
00:25:46,800 --> 00:25:48,570
like they're time's
being wasted--

536
00:25:48,570 --> 00:25:52,740
a little bit like what
you were saying.

537
00:25:52,740 --> 00:25:56,410
So now, did you have a
comment about that?

538
00:25:56,410 --> 00:25:56,980
AUDIENCE: No.

539
00:25:56,980 --> 00:25:57,350
PROFESSOR: OK.

540
00:25:57,350 --> 00:25:59,630
Then I'll come to
you one second.

541
00:25:59,630 --> 00:26:04,140
So one comment about that is
that actually, most of the

542
00:26:04,140 --> 00:26:07,030
students, at least in a big
class in genetics, are going

543
00:26:07,030 --> 00:26:08,520
to be students who are--

544
00:26:08,520 --> 00:26:09,630
for example, at MIT--

545
00:26:09,630 --> 00:26:11,690
are there not because they love
genetics and are going to

546
00:26:11,690 --> 00:26:12,550
continue in genetics.

547
00:26:12,550 --> 00:26:14,570
They're doing it because they're
required to take

548
00:26:14,570 --> 00:26:16,690
introductory biology.

549
00:26:16,690 --> 00:26:19,040
And your job as a teacher is
to actually show them that

550
00:26:19,040 --> 00:26:20,840
this is a really fascinating
subject.

551
00:26:20,840 --> 00:26:22,130
It's the same in physics.

552
00:26:22,130 --> 00:26:25,300
It's often taught as if the
only students in a physics

553
00:26:25,300 --> 00:26:27,200
class were the physics majors.

554
00:26:27,200 --> 00:26:30,000
Now, when I was an undergraduate
in Standford,

555
00:26:30,000 --> 00:26:34,180
there were 1,600 undergraduates
and 12 physics

556
00:26:34,180 --> 00:26:34,990
majors every year.

557
00:26:34,990 --> 00:26:37,930
So 1% of the student body
was physics majors.

558
00:26:37,930 --> 00:26:40,830
So they were actually teaching
in a way, generally speaking,

559
00:26:40,830 --> 00:26:42,950
for 1% of the student body.

560
00:26:42,950 --> 00:26:44,190
What about the other 99%?

561
00:26:44,190 --> 00:26:47,020
It was really important for them
to reach them as well--

562
00:26:47,020 --> 00:26:49,520
important, also, for the health
and growth of the

563
00:26:49,520 --> 00:26:55,240
field, because the field needed
people who, even if

564
00:26:55,240 --> 00:26:57,710
they weren't professional
physicists, saw the value in

565
00:26:57,710 --> 00:26:59,650
the field and the value
in physics.

566
00:26:59,650 --> 00:27:03,050
So generally speaking, it is
actually a wise way to reach

567
00:27:03,050 --> 00:27:03,620
all of them.

568
00:27:03,620 --> 00:27:07,240
Now, suppose you have people
who just want this.

569
00:27:07,240 --> 00:27:10,680
The next thing I'm going to
do will show you a way of

570
00:27:10,680 --> 00:27:13,750
teaching to them that they'll
actually learn this better,

571
00:27:13,750 --> 00:27:16,950
because I guarantee you, almost
everyone, if you just

572
00:27:16,950 --> 00:27:20,065
tell them this and they take
dictation, now they go away,

573
00:27:20,065 --> 00:27:24,070
and you say, OK, everybody,
what did I write down?

574
00:27:24,070 --> 00:27:25,540
Most people can't
reconstruct it.

575
00:27:25,540 --> 00:27:28,020
They'll say, well, was
this a C or an N?

576
00:27:28,020 --> 00:27:30,465
In fact, if you look in
Wikipedia, the Wikipedia entry

577
00:27:30,465 --> 00:27:31,260
is incorrect.

578
00:27:31,260 --> 00:27:34,940
Actually, I think it
has an N over here.

579
00:27:34,940 --> 00:27:36,720
It seems plausible, but if you
actually understand the

580
00:27:36,720 --> 00:27:40,560
equation, you think, that
can't possibly be right.

581
00:27:40,560 --> 00:27:42,670
And I think it has
an N over here.

582
00:27:42,670 --> 00:27:45,160
So they actually do
not really have a

583
00:27:45,160 --> 00:27:46,310
command of the equation.

584
00:27:46,310 --> 00:27:50,070
So even for them, you
want to not--

585
00:27:50,070 --> 00:27:52,690
the story isn't the proof
of it for them, but the

586
00:27:52,690 --> 00:27:54,360
continuation will be
valuable for them.

587
00:27:54,360 --> 00:27:56,400
So again, I'll promise that.

588
00:27:56,400 --> 00:27:59,340
Now, there was a question
over there.

589
00:27:59,340 --> 00:28:01,000
Yes.

590
00:28:01,000 --> 00:28:04,552
AUDIENCE: The story, you can
basically use to-- and if you

591
00:28:04,552 --> 00:28:07,370
see you lose your students
when you go on this

592
00:28:07,370 --> 00:28:11,290
quantitative approach, then it's
maybe helpful to loosen

593
00:28:11,290 --> 00:28:15,700
up a little bit and basically
bought you interest to see

594
00:28:15,700 --> 00:28:17,460
what's a big context.

595
00:28:17,460 --> 00:28:20,148
And the other point was--

596
00:28:20,148 --> 00:28:21,810
PROFESSOR: So you're first point
was that you can add

597
00:28:21,810 --> 00:28:22,910
stories as needed.

598
00:28:22,910 --> 00:28:23,848
AUDIENCE: Yeah.

599
00:28:23,848 --> 00:28:26,193
So depending on what
it's like.

600
00:28:26,193 --> 00:28:28,808
And the other thing is when I
make my comments related to

601
00:28:28,808 --> 00:28:32,792
the problem, I see it
more from a various

602
00:28:32,792 --> 00:28:34,290
student point of view.

603
00:28:34,290 --> 00:28:37,137
So you point out that, depending
on what the audience

604
00:28:37,137 --> 00:28:39,500
is, you teach differently.

605
00:28:39,500 --> 00:28:39,970
PROFESSOR: Yeah, that's true.

606
00:28:39,970 --> 00:28:40,250
Yeah.

607
00:28:40,250 --> 00:28:42,720
AUDIENCE: So when you
give your examples--

608
00:28:42,720 --> 00:28:44,490
PROFESSOR: Oh, I should tell
you who the audience is.

609
00:28:44,490 --> 00:28:47,200
AUDIENCE: --interesting to know
at a certain point what

610
00:28:47,200 --> 00:28:48,010
kind of audience it was.

611
00:28:48,010 --> 00:28:49,210
PROFESSOR: Yeah, OK.

612
00:28:49,210 --> 00:28:50,460
Good point.

613
00:28:52,654 --> 00:28:55,250
AUDIENCE: So I'm undergrad.

614
00:28:55,250 --> 00:28:58,824
So I don't know how it is to
teach first year university.

615
00:28:58,824 --> 00:29:00,318
I've never had students.

616
00:29:00,318 --> 00:29:05,595
So how much basics or how much
story you have to put in

617
00:29:05,595 --> 00:29:09,060
compared to how much math.

618
00:29:09,060 --> 00:29:12,040
So it would be very helpful
to see what our target is.

619
00:29:12,040 --> 00:29:12,880
PROFESSOR: OK.

620
00:29:12,880 --> 00:29:15,110
Your comment is that it would be
helpful when I'll give you

621
00:29:15,110 --> 00:29:18,180
an example like this to say,
OK, for the particular

622
00:29:18,180 --> 00:29:20,000
audience, what would you do?

623
00:29:20,000 --> 00:29:20,340
Yeah.

624
00:29:20,340 --> 00:29:21,620
And I'll try to do that.

625
00:29:21,620 --> 00:29:22,560
That's a good point.

626
00:29:22,560 --> 00:29:24,790
Now, there was another
question--

627
00:29:24,790 --> 00:29:26,040
behind Cecilia.

628
00:29:29,960 --> 00:29:31,836
Did you still have a comment?

629
00:29:31,836 --> 00:29:34,101
AUDIENCE: Oh, I just wanted to
mention that in choice A,

630
00:29:34,101 --> 00:29:35,460
there's a lot of jargon.

631
00:29:35,460 --> 00:29:36,010
PROFESSOR: OK.

632
00:29:36,010 --> 00:29:43,720
So there's a lot of jargon in
choice A, which makes it much

633
00:29:43,720 --> 00:29:45,220
harder to understand.

634
00:29:45,220 --> 00:29:47,220
You have almost--

635
00:29:47,220 --> 00:29:49,160
for example, the multinomial
coefficient--

636
00:29:49,160 --> 00:29:51,930
you have to keep that in your
head, as well as all these

637
00:29:51,930 --> 00:29:54,820
subscripts, keep it all in
your hand and try to

638
00:29:54,820 --> 00:29:56,710
manipulate that object--

639
00:29:56,710 --> 00:29:58,050
very difficult.

640
00:29:58,050 --> 00:30:01,830
One of the points about that
related is the chunking.

641
00:30:01,830 --> 00:30:03,540
So the chunking paper--

642
00:30:03,540 --> 00:30:07,660
the idea is that these, for a
student, each of these things

643
00:30:07,660 --> 00:30:10,310
is going to fill up one of the
chunk slots, almost, because

644
00:30:10,310 --> 00:30:12,710
it's all new to them.

645
00:30:12,710 --> 00:30:16,390
You've flooded the
chunking system.

646
00:30:16,390 --> 00:30:17,460
You've overflowed it.

647
00:30:17,460 --> 00:30:19,770
And they can't actually
manipulate this as one object

648
00:30:19,770 --> 00:30:22,140
anymore, because it's
far too many chunks.

649
00:30:22,140 --> 00:30:24,970
Now, you, as a professional
in the field, are

650
00:30:24,970 --> 00:30:25,630
like, well, of course.

651
00:30:25,630 --> 00:30:26,720
It's a multinomial
coefficient.

652
00:30:26,720 --> 00:30:28,060
What else could it be?

653
00:30:28,060 --> 00:30:29,290
For you, that's one chunk.

654
00:30:29,290 --> 00:30:32,400
For the student, that's
is it C or N?

655
00:30:32,400 --> 00:30:33,035
K1?

656
00:30:33,035 --> 00:30:33,470
K?

657
00:30:33,470 --> 00:30:33,940
Why K?

658
00:30:33,940 --> 00:30:35,470
What the hell is K here?

659
00:30:35,470 --> 00:30:37,750
Is it commas or spaces?

660
00:30:37,750 --> 00:30:38,550
Parentheses?

661
00:30:38,550 --> 00:30:39,340
What about brackets?

662
00:30:39,340 --> 00:30:40,910
I've never seen something
like that.

663
00:30:40,910 --> 00:30:42,810
Shouldn't there be as many
things up here as here?

664
00:30:42,810 --> 00:30:44,970
So the way they look at it
is completely different.

665
00:30:44,970 --> 00:30:47,420
Every symbol is almost
a chunk.

666
00:30:47,420 --> 00:30:51,136
So this is just massively
overflowing with chunking.

667
00:30:51,136 --> 00:30:57,110
So this is sort of related
to the jargon.

668
00:30:57,110 --> 00:30:58,043
Yes.

669
00:30:58,043 --> 00:30:58,536
AUDIENCE: Scott.

670
00:30:58,536 --> 00:30:59,029
PROFESSOR: Scott.

671
00:30:59,029 --> 00:31:01,987
AUDIENCE: I have
a meta comment.

672
00:31:01,987 --> 00:31:04,365
In this example, if you don't
know how you're actually going

673
00:31:04,365 --> 00:31:06,442
to complete these, it's created
a lot of tension.

674
00:31:06,442 --> 00:31:08,740
I'm dying to know how you're
going to do it.

675
00:31:08,740 --> 00:31:09,530
PROFESSOR: OK.

676
00:31:09,530 --> 00:31:11,960
Fair enough.

677
00:31:11,960 --> 00:31:15,730
Would you be very offended if I
gave everyone a break for a

678
00:31:15,730 --> 00:31:17,100
few minutes and then
finished it?

679
00:31:17,100 --> 00:31:17,970
AUDIENCE: OK--

680
00:31:17,970 --> 00:31:18,370
PROFESSOR: OK.

681
00:31:18,370 --> 00:31:19,780
So finish your comment.

682
00:31:19,780 --> 00:31:22,359
AUDIENCE: My question is in a
situation like this where you

683
00:31:22,359 --> 00:31:25,877
have people who are making so
many comments, and the other

684
00:31:25,877 --> 00:31:29,230
half's just dying to know, how
do you know when to stop?

685
00:31:29,230 --> 00:31:30,550
PROFESSOR: Good question.

686
00:31:30,550 --> 00:31:32,630
So how do I know when to stop?

687
00:31:32,630 --> 00:31:34,890
Because I don't want to exhaust
people's patience, but

688
00:31:34,890 --> 00:31:36,600
I do want to take questions.

689
00:31:36,600 --> 00:31:40,190
And in this case, there's the
American teaching phrase saved

690
00:31:40,190 --> 00:31:43,130
by the bell, which is that
I don't have to make the

691
00:31:43,130 --> 00:31:44,920
decision too hard, because
it's probably

692
00:31:44,920 --> 00:31:47,480
time for a break anyway.

693
00:31:47,480 --> 00:31:49,440
But how do I make
the decision?

694
00:31:49,440 --> 00:31:53,340
Partly, I listen to
people's comments.

695
00:31:53,340 --> 00:31:55,840
For example, suppose I got yet
another comment saying, well,

696
00:31:55,840 --> 00:31:57,430
we still don't know
the equation.

697
00:31:57,430 --> 00:31:59,040
And I'm saying for the
third time, oh, wait.

698
00:31:59,040 --> 00:32:00,760
I'll show you a way to
get to the equation.

699
00:32:00,760 --> 00:32:02,910
I think, you know, it's probably
time for me to do the

700
00:32:02,910 --> 00:32:05,380
equation and then I'll take
more questions later.

701
00:32:05,380 --> 00:32:09,720
So listen for the tension in
people's voice, which is also

702
00:32:09,720 --> 00:32:12,740
a good reason to have everyone
do the voice exercise first

703
00:32:12,740 --> 00:32:16,350
and free the tension so that if
the tension creeps back in,

704
00:32:16,350 --> 00:32:19,010
you know it's something
you've done.

705
00:32:19,010 --> 00:32:19,300
OK.

706
00:32:19,300 --> 00:32:22,010
So I'll take more questions
after the break.

707
00:32:22,010 --> 00:32:24,610
But it's 10:02 by that clock.

708
00:32:24,610 --> 00:32:26,630
10:05, we'll start again.

709
00:32:26,630 --> 00:32:29,030
You can jump up and down, do
jumping jacks or whatever it

710
00:32:29,030 --> 00:32:32,270
takes to get the
blood flowing.

711
00:32:32,270 --> 00:32:33,880
I'll take a couple more
questions and then I'll show

712
00:32:33,880 --> 00:32:36,760
you how to continue it to
get towards that in

713
00:32:36,760 --> 00:32:38,310
a perceptive way.

714
00:32:41,790 --> 00:32:42,150
OK.

715
00:32:42,150 --> 00:32:47,800
So let me, as promised, continue
along the lines--

716
00:32:47,800 --> 00:32:50,870
not continuing the story, but
continuing that approach, the

717
00:32:50,870 --> 00:32:56,160
alternative mirror image
approach of A, which is to

718
00:32:56,160 --> 00:33:00,740
say, well, what would I do
after telling the story?

719
00:33:00,740 --> 00:33:04,620
So first of all, I would try to
fix some of the problems in

720
00:33:04,620 --> 00:33:06,940
just basically blasting
people with that.

721
00:33:12,250 --> 00:33:17,750
So I try to make it as clear and
unjargony as possible-- so

722
00:33:17,750 --> 00:33:19,230
as you notice, jargon
up there.

723
00:33:25,710 --> 00:33:27,290
So imagine a gene with
two flavors--

724
00:33:35,160 --> 00:33:36,730
sickle cell or not.

725
00:33:48,920 --> 00:33:50,390
And two chromosomes--

726
00:33:50,390 --> 00:33:55,710
so just like people have two
copies of each chromosome.

727
00:33:55,710 --> 00:33:56,030
OK.

728
00:33:56,030 --> 00:33:58,820
So now, before I continue,
what have I done

729
00:33:58,820 --> 00:34:01,120
just by doing that?

730
00:34:01,120 --> 00:34:06,560
Well, first of all, I've made
something concrete.

731
00:34:06,560 --> 00:34:07,900
It's sickle cell or not.

732
00:34:07,900 --> 00:34:10,159
So right away, you
can imagine it.

733
00:34:10,159 --> 00:34:12,480
So that's continuing the
idea of a story.

734
00:34:12,480 --> 00:34:16,199
It's much easier to imagine
a concrete situation.

735
00:34:16,199 --> 00:34:18,630
Either you have sickle cell
anemia or you don't.

736
00:34:18,630 --> 00:34:22,679
Or you have a gene for it or a
gene that doesn't cause it.

737
00:34:22,679 --> 00:34:25,280
And the two chromosomes--

738
00:34:25,280 --> 00:34:29,050
so rather than having C
chromosomes, I'm just

739
00:34:29,050 --> 00:34:31,510
restricting it to two.

740
00:34:31,510 --> 00:34:34,679
Now, you might think, well,
that's a terrible restriction.

741
00:34:34,679 --> 00:34:35,889
That's a specialization.

742
00:34:35,889 --> 00:34:37,570
Therefore, it's bad.

743
00:34:37,570 --> 00:34:38,010
No.

744
00:34:38,010 --> 00:34:42,130
Actually, it's good for that
reason, because it makes it

745
00:34:42,130 --> 00:34:48,080
possible to understand, in the
next step, the idea behind

746
00:34:48,080 --> 00:34:49,409
that equation.

747
00:34:49,409 --> 00:34:52,940
And once you understand the
idea, then that equation is

748
00:34:52,940 --> 00:34:55,270
not so mysterious.

749
00:34:55,270 --> 00:34:59,440
And furthermore, what's nice
about two chromosomes?

750
00:34:59,440 --> 00:35:02,440
Yeah, it's a specialization,
but--

751
00:35:02,440 --> 00:35:02,835
AUDIENCE: Same two.

752
00:35:02,835 --> 00:35:04,880
PROFESSOR: Yeah, we
all have two.

753
00:35:04,880 --> 00:35:07,580
There's some plants that have
more and some plants have

754
00:35:07,580 --> 00:35:08,350
less, I think.

755
00:35:08,350 --> 00:35:10,940
But people have two.

756
00:35:10,940 --> 00:35:14,820
So it's already interesting to
us just for that reason.

757
00:35:14,820 --> 00:35:15,170
OK.

758
00:35:15,170 --> 00:35:29,780
So then the question is how
frequent are the combinations?

759
00:35:35,710 --> 00:35:37,410
So we want to answer how
frequent are the three

760
00:35:37,410 --> 00:35:40,330
combinations sickle--

761
00:35:40,330 --> 00:35:42,390
lowercase S is not sickle.

762
00:35:42,390 --> 00:35:54,000
So this is S. This
is lowercase S.

763
00:35:54,000 --> 00:35:55,330
What happened to the
fourth combination?

764
00:35:58,620 --> 00:35:59,870
Why are there only three?

765
00:36:02,080 --> 00:36:02,410
Yeah.

766
00:36:02,410 --> 00:36:03,000
It's the same.

767
00:36:03,000 --> 00:36:03,960
S, S--

768
00:36:03,960 --> 00:36:06,570
this and this are the same.

769
00:36:06,570 --> 00:36:08,540
OK So I would actually
ask that, too.

770
00:36:08,540 --> 00:36:11,490
Any time there's something
interesting--

771
00:36:11,490 --> 00:36:12,810
it should have been four.

772
00:36:12,810 --> 00:36:14,870
Plausibly, it could have been
four, but it's actually three.

773
00:36:14,870 --> 00:36:15,600
Why three?

774
00:36:15,600 --> 00:36:17,150
OK, it's because we're actually

775
00:36:17,150 --> 00:36:18,870
lumping these two together.

776
00:36:18,870 --> 00:36:20,530
OK.

777
00:36:20,530 --> 00:36:22,930
So how frequent are those
three combinations?

778
00:36:22,930 --> 00:36:31,290
Well, that's provided
by this formula.

779
00:36:35,250 --> 00:36:41,720
P squared, q squared, 2pq--

780
00:36:45,910 --> 00:36:50,520
so p and q are the frequencies
of S and little S.

781
00:36:50,520 --> 00:36:51,770
So this is the--

782
00:36:59,400 --> 00:36:59,720
OK.

783
00:36:59,720 --> 00:37:02,380
So now, because you've asked
them about the three versus

784
00:37:02,380 --> 00:37:06,550
four, this two makes sense,
already somehow.

785
00:37:06,550 --> 00:37:07,750
But how could make
sense of the p

786
00:37:07,750 --> 00:37:11,360
squared, pq, and q squared?

787
00:37:11,360 --> 00:37:14,730
And so that's where I would
continue with the following,

788
00:37:14,730 --> 00:37:15,980
which is this.

789
00:37:19,980 --> 00:37:23,405
So here is 0 to 1.

790
00:37:27,320 --> 00:37:29,910
And this is p.

791
00:37:29,910 --> 00:37:30,670
And that's q.

792
00:37:30,670 --> 00:37:33,830
So this is the frequency
of sickle cell.

793
00:37:33,830 --> 00:37:37,130
This is the frequency
of not sickle cell.

794
00:37:37,130 --> 00:37:39,250
So right away, you have
p plus q equals 1.

795
00:37:42,130 --> 00:37:43,405
I'll draw this up here too.

796
00:37:53,100 --> 00:37:54,350
So I make a square.

797
00:38:01,570 --> 00:38:04,720
And I have four regions.

798
00:38:04,720 --> 00:38:06,300
So it's not magic.

799
00:38:06,300 --> 00:38:09,080
Why am I making a square?

800
00:38:09,080 --> 00:38:10,755
What about the problem
tells me square?

801
00:38:14,530 --> 00:38:15,780
So I would ask the class that.

802
00:38:21,900 --> 00:38:22,780
I'm looking for area.

803
00:38:22,780 --> 00:38:23,860
And why an area?

804
00:38:23,860 --> 00:38:25,440
Why two dimensions?

805
00:38:25,440 --> 00:38:27,150
AUDIENCE: Because there
are two chromosomes?

806
00:38:27,150 --> 00:38:28,860
PROFESSOR: Because there
are two chromosomes.

807
00:38:28,860 --> 00:38:32,860
So this is your C. The number
of dimensions is eventually

808
00:38:32,860 --> 00:38:35,610
going to become this thing, so
you can see how we're going to

809
00:38:35,610 --> 00:38:37,660
get there eventually.

810
00:38:37,660 --> 00:38:40,420
So I'm looking for an area,
because I have two

811
00:38:40,420 --> 00:38:42,510
chromosomes.

812
00:38:42,510 --> 00:38:44,720
So now, let's just look at
these various areas.

813
00:38:44,720 --> 00:38:48,420
Here's a p squared.

814
00:38:48,420 --> 00:38:49,260
What's this area?

815
00:38:49,260 --> 00:38:51,170
There's q by p.

816
00:38:51,170 --> 00:38:53,220
So that's a pq.

817
00:38:53,220 --> 00:38:55,260
That's also pq.

818
00:38:55,260 --> 00:38:58,140
And that's q squared.

819
00:38:58,140 --> 00:38:58,520
OK.

820
00:38:58,520 --> 00:38:59,530
Let's look at all our pieces.

821
00:38:59,530 --> 00:39:02,710
Oh, we have a p squared there.

822
00:39:02,710 --> 00:39:04,650
We have a q squared there.

823
00:39:04,650 --> 00:39:06,810
Oh, and pq, pq--

824
00:39:06,810 --> 00:39:09,540
2pq.

825
00:39:09,540 --> 00:39:15,470
So in fact, this picture
explains the entire set of

826
00:39:15,470 --> 00:39:16,720
frequencies here.

827
00:39:18,820 --> 00:39:22,900
And if you see this picture,
there's nothing really new to

828
00:39:22,900 --> 00:39:24,610
understand here.

829
00:39:24,610 --> 00:39:27,500
In fact, all this picture
is showing is that--

830
00:39:32,030 --> 00:39:35,010
it's a picture of p plus
q squared equals 1.

831
00:39:35,010 --> 00:39:38,410
So if p plus q is 1, its square
is also equal to 1.

832
00:39:38,410 --> 00:39:42,150
And you break this up into the
four terms, group two of them

833
00:39:42,150 --> 00:39:45,230
together, and you get
three terms, three

834
00:39:45,230 --> 00:39:46,590
different kinds of terms.

835
00:39:46,590 --> 00:39:50,110
You get three different
frequencies.

836
00:39:50,110 --> 00:39:53,210
So now, this square is actually
very easy to use.

837
00:39:53,210 --> 00:40:00,610
Suppose someone tells you that
sickle cell anemia, people who

838
00:40:00,610 --> 00:40:05,460
have both genes, are 1%
in the population.

839
00:40:05,460 --> 00:40:10,090
Well, what fraction of the
population has no sickle cell

840
00:40:10,090 --> 00:40:11,770
gene that all?

841
00:40:11,770 --> 00:40:12,100
OK.

842
00:40:12,100 --> 00:40:15,290
Well, we can just look
at the square.

843
00:40:15,290 --> 00:40:18,620
The information was
that this is 1%.

844
00:40:18,620 --> 00:40:22,010
So that means this 0.1.

845
00:40:22,010 --> 00:40:26,326
And that's 0.9.

846
00:40:26,326 --> 00:40:27,680
That's 0.9.

847
00:40:27,680 --> 00:40:30,710
So this here is people with
no sickle cell at all.

848
00:40:30,710 --> 00:40:33,750
It is 0.81.

849
00:40:33,750 --> 00:40:35,870
And so the square actually makes
everything really easy

850
00:40:35,870 --> 00:40:37,180
to understand.

851
00:40:37,180 --> 00:40:40,550
So now, the question is how
do you generalize it?

852
00:40:40,550 --> 00:40:43,040
Well, what are some of the-- the
first generalization I'll

853
00:40:43,040 --> 00:40:47,410
do is I'll say, suppose
that there are--

854
00:40:47,410 --> 00:40:49,050
well, I'll ask you.

855
00:40:49,050 --> 00:40:52,760
Should I generalize it to more
dimensions or more different

856
00:40:52,760 --> 00:40:54,020
copies of the gene?

857
00:40:54,020 --> 00:40:56,670
What's easier to draw?

858
00:40:56,670 --> 00:40:59,740
More copies, because
more dimensions--

859
00:40:59,740 --> 00:41:01,630
I don't know how to draw
in three dimensions.

860
00:41:01,630 --> 00:41:03,360
But I can draw three copies.

861
00:41:03,360 --> 00:41:04,610
That's pretty easy.

862
00:41:08,700 --> 00:41:12,800
So now, take a piece of paper
and just on your own, draw

863
00:41:12,800 --> 00:41:17,870
this same figure for three
copies of the gene with

864
00:41:17,870 --> 00:41:21,258
frequencies p, q, and r.

865
00:41:26,040 --> 00:41:29,965
So people have a picture?

866
00:41:32,540 --> 00:41:34,370
Does anyone want to describe
their picture to me?

867
00:41:48,940 --> 00:41:49,566
Yeah.

868
00:41:49,566 --> 00:41:52,725
AUDIENCE: I did a three by three
square with p squared

869
00:41:52,725 --> 00:41:54,912
and q squared and r squared
diagonal [INAUDIBLE].

870
00:41:58,330 --> 00:41:58,760
PROFESSOR: OK.

871
00:41:58,760 --> 00:41:59,790
So I do the same.

872
00:41:59,790 --> 00:42:03,640
I make a p, a q, and an r.

873
00:42:03,640 --> 00:42:04,890
And then I do the same here--

874
00:42:07,670 --> 00:42:08,920
qr.

875
00:42:16,550 --> 00:42:16,960
OK.

876
00:42:16,960 --> 00:42:23,730
So there's a p squared,
q squared, r squared.

877
00:42:23,730 --> 00:42:26,930
Now, how many guys-- so we have
nine square, sub squares.

878
00:42:26,930 --> 00:42:28,170
Three of them are this.

879
00:42:28,170 --> 00:42:30,620
There's six more.

880
00:42:30,620 --> 00:42:32,370
Now, which of them
look similar?

881
00:42:32,370 --> 00:42:35,620
So this is a pq.

882
00:42:35,620 --> 00:42:38,370
Are there any other pq's?

883
00:42:38,370 --> 00:42:40,950
Yeah, one other, right?
pq over here. .

884
00:42:40,950 --> 00:42:43,370
And here is a pr.

885
00:42:43,370 --> 00:42:44,990
Is there a pr?

886
00:42:44,990 --> 00:42:47,030
Yea, another pr over here.

887
00:42:47,030 --> 00:42:50,360
And here's a qr and a qr.

888
00:42:50,360 --> 00:42:53,880
So we have p squared.

889
00:42:53,880 --> 00:42:55,970
And that's that guy.

890
00:42:55,970 --> 00:43:09,516
We have q squared, r squared,
and then 2pr, 2qr, and 2pq.

891
00:43:14,960 --> 00:43:19,640
So that's the generalization to
three copies of the gene.

892
00:43:19,640 --> 00:43:23,320
So it turns out that all
these coefficients--

893
00:43:23,320 --> 00:43:25,530
1, 1, 1, 2, 2, 2--

894
00:43:25,530 --> 00:43:30,206
those are binomial, multinomial
coefficients.

895
00:43:35,580 --> 00:43:37,810
So now, let's generalize
one more.

896
00:43:37,810 --> 00:43:38,730
So what have we done?

897
00:43:38,730 --> 00:43:46,440
There, before, we've
written out this.

898
00:43:46,440 --> 00:43:50,000
And those are the nine terms
grouped into six.

899
00:43:50,000 --> 00:43:56,130
So this was our C. So now, we
know how to generalize.

900
00:43:56,130 --> 00:44:00,600
p plus q plus r to
the C equals 1.

901
00:44:00,600 --> 00:44:02,550
If this is two chromosomes--

902
00:44:06,170 --> 00:44:07,595
this is C chromosomes.

903
00:44:14,560 --> 00:44:17,280
So this is C chromosomes
with three copies.

904
00:44:17,280 --> 00:44:21,050
Well, what happens if
we have n copies?

905
00:44:21,050 --> 00:44:27,810
p1 plus p2 plus one of those
plus pn to the C equals 1.

906
00:44:27,810 --> 00:44:29,780
And all you have to do
is expand that out.

907
00:44:32,390 --> 00:44:34,620
You can use a square in
higher dimensions.

908
00:44:34,620 --> 00:44:36,930
Or you can actually use
formulas for math.

909
00:44:36,930 --> 00:44:40,510
And that's where these
come from--

910
00:44:40,510 --> 00:44:43,930
p1 to the 1 power, p2 to the
next power, all the way to pn

911
00:44:43,930 --> 00:44:44,620
to the next power.

912
00:44:44,620 --> 00:44:50,400
And these guys are the
coefficients that count the

913
00:44:50,400 --> 00:44:51,760
multiplicity.

914
00:44:51,760 --> 00:44:53,586
Yes.

915
00:44:53,586 --> 00:44:56,480
AUDIENCE: Why do you have
to add up to 1?

916
00:44:56,480 --> 00:44:57,000
PROFESSOR: Oh.

917
00:44:57,000 --> 00:45:00,720
They have to add up to 1 because
p plus q plus r, if

918
00:45:00,720 --> 00:45:05,550
you only have three flavors of
the gene-- either you have

919
00:45:05,550 --> 00:45:08,120
sickle cell A, B, or
C, let's say--

920
00:45:08,120 --> 00:45:11,070
then these are probabilities,
probability p.

921
00:45:11,070 --> 00:45:13,880
AUDIENCE: Oh, so parentheses
says how often it's done?

922
00:45:13,880 --> 00:45:14,430
PROFESSOR: Yes.

923
00:45:14,430 --> 00:45:16,710
Frequency and probability--

924
00:45:16,710 --> 00:45:19,050
so p plus q plus r equals 1.

925
00:45:19,050 --> 00:45:21,660
So if you square it,
you still get 1.

926
00:45:21,660 --> 00:45:25,050
So you start from the idea that
p plus q equals 1, which

927
00:45:25,050 --> 00:45:27,440
is what we did over here.

928
00:45:27,440 --> 00:45:29,440
So p plus q is equal to 1.

929
00:45:29,440 --> 00:45:30,330
Then you square it.

930
00:45:30,330 --> 00:45:31,190
It's still equal to 1.

931
00:45:31,190 --> 00:45:33,920
So now, you just another
way of writing 1.

932
00:45:33,920 --> 00:45:37,330
So it turns out that
Hardy-Weinberg is all just

933
00:45:37,330 --> 00:45:40,080
fancy ways of writing 1.

934
00:45:40,080 --> 00:45:41,840
Starting from this
picture, we've

935
00:45:41,840 --> 00:45:44,170
successively complicated it.

936
00:45:44,170 --> 00:45:47,330
This is C equals
2, n equals 2.

937
00:45:47,330 --> 00:45:52,890
There's C equals
2, n equals 3.

938
00:45:56,160 --> 00:45:59,080
Here is the same thing again.

939
00:45:59,080 --> 00:46:05,130
Here is n equals 3, general C.
Here's general n, general C.

940
00:46:05,130 --> 00:46:11,660
OK so what we've done is we've
basically got here--

941
00:46:11,660 --> 00:46:17,720
by stages of successive
approach, one step at a time.

942
00:46:17,720 --> 00:46:22,680
So at every stage, it's clear
what is going on.

943
00:46:22,680 --> 00:46:24,160
And what's the core idea?

944
00:46:24,160 --> 00:46:27,300
The core idea is the one you
just asked a question about,

945
00:46:27,300 --> 00:46:29,730
which is that the frequencies
add up to 1--

946
00:46:29,730 --> 00:46:32,340
p plus q equals 1-- so if you
square the frequencies, you

947
00:46:32,340 --> 00:46:33,710
still get 1.

948
00:46:33,710 --> 00:46:37,210
And there's nothing more to
Hardy-Weinberg than that.

949
00:46:37,210 --> 00:46:38,434
Yes.

950
00:46:38,434 --> 00:46:39,300
AUDIENCE: I'm Julie.

951
00:46:39,300 --> 00:46:40,165
PROFESSOR: Julie.

952
00:46:40,165 --> 00:46:42,640
AUDIENCE: My question has to do
with the original way you

953
00:46:42,640 --> 00:46:44,372
presented the problem--
using the word

954
00:46:44,372 --> 00:46:45,610
flavors instead of alleles.

955
00:46:45,610 --> 00:46:49,075
And I've always been fond of
teaching that you should

956
00:46:49,075 --> 00:46:52,050
always use the probable
vocabulary of your students.

957
00:46:52,050 --> 00:46:53,302
PROFESSOR: To switch.

958
00:46:53,302 --> 00:46:55,900
AUDIENCE: So you got the
same thing as the--

959
00:46:55,900 --> 00:46:56,800
PROFESSOR: Yeah,
good question.

960
00:46:56,800 --> 00:46:59,050
So I used flavors instead
of alleles--

961
00:46:59,050 --> 00:47:03,580
so right here, imagine a
gene with two flavors.

962
00:47:03,580 --> 00:47:06,100
So actually, probably the
best way to do it is

963
00:47:06,100 --> 00:47:07,240
to combine the two.

964
00:47:07,240 --> 00:47:10,310
So you say flavors, because--

965
00:47:10,310 --> 00:47:15,140
so this is a question of
transmitting information to

966
00:47:15,140 --> 00:47:17,300
the student without noise
on the channel.

967
00:47:17,300 --> 00:47:19,780
So if you say the
word allele--

968
00:47:19,780 --> 00:47:21,220
so this is again related
to chunking.

969
00:47:21,220 --> 00:47:24,070
If you say the word allele,
the problem is that now,

970
00:47:24,070 --> 00:47:26,840
you're expecting them to try to
understand this new idea as

971
00:47:26,840 --> 00:47:30,740
well as this new item for taking
up one of their chunks

972
00:47:30,740 --> 00:47:32,130
that they have available.

973
00:47:32,130 --> 00:47:35,430
So when I initially presented
it, I would use flavors.

974
00:47:35,430 --> 00:47:38,020
And then I'd say, OK, now we
understand the whole idea.

975
00:47:38,020 --> 00:47:40,410
The thing has kind of seeped--
it's not really part of short

976
00:47:40,410 --> 00:47:41,470
term memory anymore.

977
00:47:41,470 --> 00:47:44,230
It's connected to something
else that they know-- for

978
00:47:44,230 --> 00:47:46,570
example, this. p plus q
equals 1, so p plus q

979
00:47:46,570 --> 00:47:48,100
squared equals 1.

980
00:47:48,100 --> 00:47:50,500
Now, it's not taking up so
many chunks anymore.

981
00:47:50,500 --> 00:47:55,090
Now, they're ready to hear
the word allele.

982
00:47:55,090 --> 00:47:57,120
So I'll say, OK.

983
00:47:57,120 --> 00:47:59,180
Colloquially, we could say
flavors, but actually, the

984
00:47:59,180 --> 00:48:01,690
word in the literature
is alleles.

985
00:48:01,690 --> 00:48:05,560
So at no point, you're
overloading the system.

986
00:48:05,560 --> 00:48:08,940
So again, it's philosophy
based on,

987
00:48:08,940 --> 00:48:13,050
say, history and chunking.

988
00:48:13,050 --> 00:48:14,195
Yes.

989
00:48:14,195 --> 00:48:17,930
AUDIENCE: When you presented
this whole second way, you

990
00:48:17,930 --> 00:48:19,896
made, actually, use of
the equation that you

991
00:48:19,896 --> 00:48:21,870
wrote down in the--

992
00:48:21,870 --> 00:48:22,735
PROFESSOR: Oh, over there?

993
00:48:22,735 --> 00:48:23,150
AUDIENCE: Yeah.

994
00:48:23,150 --> 00:48:23,640
PROFESSOR: Oh, no.

995
00:48:23,640 --> 00:48:25,890
I was just saying that
for your benefit.

996
00:48:25,890 --> 00:48:27,410
AUDIENCE: But I thought,
actually, for me, it was very

997
00:48:27,410 --> 00:48:29,875
useful to see what the
kind of equation was.

998
00:48:29,875 --> 00:48:32,650
And then as you were going
through the argument, I could

999
00:48:32,650 --> 00:48:33,170
reason out myself--

1000
00:48:33,170 --> 00:48:34,200
PROFESSOR: Good question,
good point.

1001
00:48:34,200 --> 00:48:35,470
AUDIENCE: --where everything
was coming from.

1002
00:48:35,470 --> 00:48:35,890
PROFESSOR: Yeah OK.

1003
00:48:35,890 --> 00:48:36,590
That's a good point.

1004
00:48:36,590 --> 00:48:42,080
So it actually was helpful to
actually see the final goal to

1005
00:48:42,080 --> 00:48:43,040
know where you're going.

1006
00:48:43,040 --> 00:48:45,250
Actually, that's a
good suggestion.

1007
00:48:45,250 --> 00:48:48,400
It's actually not bad to say,
OK, this is what we're going

1008
00:48:48,400 --> 00:48:49,750
to try to understand.

1009
00:48:49,750 --> 00:48:51,420
I don't expect to understand
it now.

1010
00:48:51,420 --> 00:48:53,140
It's full of all kinds
of squigglies.

1011
00:48:56,330 --> 00:48:59,380
So for example, maybe a really
good way to do the whole thing

1012
00:48:59,380 --> 00:49:01,705
would be to start with the Hardy
story and say, we're

1013
00:49:01,705 --> 00:49:03,070
going to talk about
Hardy-Weinberg.

1014
00:49:03,070 --> 00:49:06,280
Say, what does Hardy-Weinberg
say?

1015
00:49:06,280 --> 00:49:09,210
Well, in its full generality,
it says, blah.

1016
00:49:09,210 --> 00:49:11,160
I don't expect you to make any
sense of that right now.

1017
00:49:11,160 --> 00:49:14,680
Let's talk about what the
core ideas of it are.

1018
00:49:14,680 --> 00:49:19,870
With this, you approach it,
creep up on it bit by bit,

1019
00:49:19,870 --> 00:49:23,130
successive generalization,
and then you get there.

1020
00:49:23,130 --> 00:49:25,500
So actually, it's a good idea.

1021
00:49:25,500 --> 00:49:27,480
Leave that on the board the
whole time so you have a

1022
00:49:27,480 --> 00:49:29,760
context and a goal, like
a mountain peak

1023
00:49:29,760 --> 00:49:31,310
you're trying to scale.

1024
00:49:31,310 --> 00:49:33,960
Thanks for the suggestion.

1025
00:49:33,960 --> 00:49:34,800
Yes.

1026
00:49:34,800 --> 00:49:35,740
Tell me your name?

1027
00:49:35,740 --> 00:49:36,140
AUDIENCE: Scott.

1028
00:49:36,140 --> 00:49:36,915
PROFESSOR: Scott.

1029
00:49:36,915 --> 00:49:38,986
AUDIENCE: Last week, you did
present an example where you

1030
00:49:38,986 --> 00:49:41,140
wrote down this complicated
physics--

1031
00:49:41,140 --> 00:49:42,225
PROFESSOR: The Navier-Stokes
equation.

1032
00:49:42,225 --> 00:49:43,978
AUDIENCE: Yeah, you wrote down
the Navier-Stokes equation,

1033
00:49:43,978 --> 00:49:47,700
and then you said, yeah, I
sometimes do that just to

1034
00:49:47,700 --> 00:49:52,475
create this tension in the
students [INAUDIBLE].

1035
00:49:52,475 --> 00:49:55,774
So given that when you complete
this method-- you

1036
00:49:55,774 --> 00:49:57,600
actually didn't use
the story at all.

1037
00:49:57,600 --> 00:50:03,012
It seems to me that if you had
presented that and then you

1038
00:50:03,012 --> 00:50:06,323
stood back and made a joke about
[INAUDIBLE], it would

1039
00:50:06,323 --> 00:50:08,370
have been perfectly
satisfactory.

1040
00:50:08,370 --> 00:50:09,620
PROFESSOR: I think
that's true, too.

1041
00:50:12,660 --> 00:50:16,100
So a lot of teaching is
preference of the instructor.

1042
00:50:16,100 --> 00:50:18,780
I, for example, happen to really
like that story because

1043
00:50:18,780 --> 00:50:21,420
I taught in Cambridge for a long
time and I can imagine

1044
00:50:21,420 --> 00:50:24,270
Hardy running into Punnett
at the cricket ground.

1045
00:50:24,270 --> 00:50:28,260
So it's interesting to me, so I
can tell it with enthusiasm.

1046
00:50:28,260 --> 00:50:30,950
And I think it says something
about the relation of science

1047
00:50:30,950 --> 00:50:32,510
on society, which I
think is important

1048
00:50:32,510 --> 00:50:33,600
for students to learn.

1049
00:50:33,600 --> 00:50:34,620
So I might use that story.

1050
00:50:34,620 --> 00:50:38,240
But an instructor with different
purpose might do it,

1051
00:50:38,240 --> 00:50:42,160
as you say, which is make a joke
about how impenetrable

1052
00:50:42,160 --> 00:50:45,100
this is and say, it seems really
cryptic, but we'll

1053
00:50:45,100 --> 00:50:46,300
actually understand
this by the end.

1054
00:50:46,300 --> 00:50:47,700
Don't worry.

1055
00:50:47,700 --> 00:50:48,870
And that's a good way
of-- actually, I

1056
00:50:48,870 --> 00:50:49,320
think you're right.

1057
00:50:49,320 --> 00:50:52,620
It's a good way of creating
and releasing tension.

1058
00:50:52,620 --> 00:50:53,360
Yes.

1059
00:50:53,360 --> 00:50:54,000
AUDIENCE: Eric.

1060
00:50:54,000 --> 00:50:54,340
PROFESSOR: Eric.

1061
00:50:54,340 --> 00:50:57,202
AUDIENCE: So what level of
background do we assume the

1062
00:50:57,202 --> 00:50:57,942
audience has?

1063
00:50:57,942 --> 00:51:00,900
Because you use sickle cell as
an example, and it's obviously

1064
00:51:00,900 --> 00:51:03,858
not critical to the example
that they know

1065
00:51:03,858 --> 00:51:05,943
what sickle cell is.

1066
00:51:05,943 --> 00:51:07,430
Do you assume that
they know that?

1067
00:51:07,430 --> 00:51:07,700
PROFESSOR: Right.

1068
00:51:07,700 --> 00:51:09,010
So what level of background?

1069
00:51:09,010 --> 00:51:12,110
So if they don't know what
sickle cell is, or maybe I'd

1070
00:51:12,110 --> 00:51:13,750
use cystic fibrosis, which
they're maybe even

1071
00:51:13,750 --> 00:51:14,860
less likely to know.

1072
00:51:14,860 --> 00:51:15,680
AUDIENCE: --presentation.

1073
00:51:15,680 --> 00:51:15,840
PROFESSOR: Yeah.

1074
00:51:15,840 --> 00:51:17,090
I could be adding noise.

1075
00:51:17,090 --> 00:51:18,700
So it's a flip side.

1076
00:51:18,700 --> 00:51:20,300
It's a two edged sword.

1077
00:51:20,300 --> 00:51:22,440
So are you adding noise by
saying sickle cell to someone

1078
00:51:22,440 --> 00:51:24,160
who doesn't know what it is?

1079
00:51:24,160 --> 00:51:26,190
Yeah.

1080
00:51:26,190 --> 00:51:27,270
It would add noise.

1081
00:51:27,270 --> 00:51:29,800
So maybe it's actually worth
saying just one sentence.

1082
00:51:29,800 --> 00:51:30,940
What is sickle cell anemia?

1083
00:51:30,940 --> 00:51:33,840
It's a mutation of the red blood
cells that makes them

1084
00:51:33,840 --> 00:51:37,640
take a different shape and makes
you unable to transfer

1085
00:51:37,640 --> 00:51:40,200
oxygen as effectively
and makes you more

1086
00:51:40,200 --> 00:51:41,990
resistant to malaria.

1087
00:51:41,990 --> 00:51:44,410
So actually, that part, I
wouldn't say at the beginning.

1088
00:51:44,410 --> 00:51:46,440
I'd say, well, if
it's so bad--

1089
00:51:46,440 --> 00:51:48,370
can't transport oxygen
so well--

1090
00:51:48,370 --> 00:51:50,330
how come it's still around?

1091
00:51:50,330 --> 00:51:52,860
Just let people think about that
for the day and come back

1092
00:51:52,860 --> 00:51:55,130
the next time and tell
them the answer.

1093
00:51:55,130 --> 00:51:55,650
But yeah.

1094
00:51:55,650 --> 00:51:58,180
So I'd probably say one sentence
if the people haven't

1095
00:51:58,180 --> 00:51:58,980
heard of sickle cell.

1096
00:51:58,980 --> 00:52:00,870
When I was saying it, I was
assuming mentally that they

1097
00:52:00,870 --> 00:52:01,740
know what it is.

1098
00:52:01,740 --> 00:52:03,470
But I think you're right.

1099
00:52:03,470 --> 00:52:05,590
Many people wouldn't know what
sickle cell anemia is, and

1100
00:52:05,590 --> 00:52:07,270
it's worth saying just
one sentence.

1101
00:52:10,380 --> 00:52:13,450
So let me just see if there's
anyone who hasn't made a

1102
00:52:13,450 --> 00:52:14,340
comment or question.

1103
00:52:14,340 --> 00:52:14,820
Yes.

1104
00:52:14,820 --> 00:52:15,590
Can you tell me your name?

1105
00:52:15,590 --> 00:52:16,600
AUDIENCE: [INAUDIBLE].

1106
00:52:16,600 --> 00:52:18,264
PROFESSOR: [INAUDIBLE].

1107
00:52:18,264 --> 00:52:20,460
AUDIENCE: Could you point to the
last little bit, because

1108
00:52:20,460 --> 00:52:21,710
I'm curious to see
[INAUDIBLE].

1109
00:52:26,570 --> 00:52:30,190
PROFESSOR: Oh, how do
you get this piece?

1110
00:52:30,190 --> 00:52:30,440
AUDIENCE: Yeah.

1111
00:52:30,440 --> 00:52:30,790
PROFESSOR: OK.

1112
00:52:30,790 --> 00:52:32,690
So how do you get this
whole thing?

1113
00:52:32,690 --> 00:52:32,990
OK.

1114
00:52:32,990 --> 00:52:34,480
So let me do that by analogy.

1115
00:52:38,080 --> 00:52:41,100
So for example, suppose--

1116
00:52:41,100 --> 00:52:42,590
let me do C equals 3.

1117
00:52:46,850 --> 00:52:48,290
I'm running out of board.

1118
00:52:48,290 --> 00:52:49,540
Let me erase this one.

1119
00:52:59,490 --> 00:53:02,770
And in fact, let me just use n
equals 2, because it doesn't

1120
00:53:02,770 --> 00:53:05,860
illustrate any new ideas
to crank n up.

1121
00:53:05,860 --> 00:53:08,910
But the cranking the c up, the
number of copies, actually

1122
00:53:08,910 --> 00:53:11,290
makes it important to see why
you need to add them all up.

1123
00:53:11,290 --> 00:53:15,740
So let's do p plus q cubed.

1124
00:53:15,740 --> 00:53:18,380
So this is an organism with
three chromosomes.

1125
00:53:18,380 --> 00:53:21,590
But it's either sickle cell
or not at each spot.

1126
00:53:21,590 --> 00:53:21,850
OK.

1127
00:53:21,850 --> 00:53:24,410
Well, let's actually just
write this whole thing.

1128
00:53:24,410 --> 00:53:27,010
You could do it by looking at
a cube and seeing all the

1129
00:53:27,010 --> 00:53:28,260
chunks, because you could
actually just do it with

1130
00:53:28,260 --> 00:53:29,220
algebra, too.

1131
00:53:29,220 --> 00:53:32,390
So there's a p cubed.

1132
00:53:32,390 --> 00:53:34,727
There's a ppq.

1133
00:53:44,350 --> 00:53:51,060
So there's a ppq,
pqp, and qpp.

1134
00:53:51,060 --> 00:53:53,230
So those are all contributing
to p squared q, and there's

1135
00:53:53,230 --> 00:53:55,590
three of them.

1136
00:53:55,590 --> 00:53:55,850
OK.

1137
00:53:55,850 --> 00:53:57,870
Now, the three isn't what
we're talking about.

1138
00:53:57,870 --> 00:54:00,580
We're talking about this
exponent here.

1139
00:54:00,580 --> 00:54:04,560
Why is K1 all the way
up to Kn equal to C?

1140
00:54:04,560 --> 00:54:08,780
Well, let me just put
that 1 in there.

1141
00:54:08,780 --> 00:54:10,040
What does this equal?

1142
00:54:10,040 --> 00:54:13,790
The sum of those guys is 3.

1143
00:54:13,790 --> 00:54:17,710
That's p to the cubed
q to the 0.

1144
00:54:17,710 --> 00:54:19,100
That's also 3.

1145
00:54:19,100 --> 00:54:19,870
What's the next term?

1146
00:54:19,870 --> 00:54:22,830
Well, there's a 3
p cubed squared,

1147
00:54:22,830 --> 00:54:26,090
and there's a q cubed.

1148
00:54:26,090 --> 00:54:28,786
Well, that's q cubed
p to the 0.

1149
00:54:28,786 --> 00:54:29,900
That's 0 plus 3.

1150
00:54:29,900 --> 00:54:31,800
This is 1 plus 2.

1151
00:54:31,800 --> 00:54:33,360
It's always three.

1152
00:54:33,360 --> 00:54:35,380
And it's always the
C. Why is that?

1153
00:54:35,380 --> 00:54:37,840
Well, you only have
three products--

1154
00:54:37,840 --> 00:54:39,090
p plus q.

1155
00:54:43,130 --> 00:54:45,010
So you have three factors.

1156
00:54:45,010 --> 00:54:47,100
And you get to choose, when
you're writing out all the

1157
00:54:47,100 --> 00:54:49,460
terms-- there are eight of them
and we've combined them

1158
00:54:49,460 --> 00:54:50,850
into four groups.

1159
00:54:50,850 --> 00:54:54,700
There's three here, three here,
one here, and one there.

1160
00:54:54,700 --> 00:54:57,290
You get one from each
of the factors.

1161
00:54:57,290 --> 00:54:59,090
So you get one exponent
from each factor.

1162
00:54:59,090 --> 00:55:02,020
So the total of all the
exponents has to be 3.

1163
00:55:02,020 --> 00:55:04,780
So once you understand it
for 3, then this is

1164
00:55:04,780 --> 00:55:06,960
just for C in general.

1165
00:55:06,960 --> 00:55:09,000
Does that help?

1166
00:55:09,000 --> 00:55:10,810
So that's a good example,
actually.

1167
00:55:10,810 --> 00:55:13,310
Someone probably would ask that
question or should ask

1168
00:55:13,310 --> 00:55:14,290
that question.

1169
00:55:14,290 --> 00:55:15,968
And that's how I would
answer it.

1170
00:55:19,320 --> 00:55:19,430
OK.

1171
00:55:19,430 --> 00:55:22,390
Now, Cecilia, you had
another question.

1172
00:55:22,390 --> 00:55:23,180
Yes.

1173
00:55:23,180 --> 00:55:24,600
Did you have another question?

1174
00:55:24,600 --> 00:55:29,310
AUDIENCE: What are we proving?

1175
00:55:29,310 --> 00:55:31,665
For example--

1176
00:55:31,665 --> 00:55:32,150
PROFESSOR: Yeah.

1177
00:55:32,150 --> 00:55:34,020
You'd think, on one hand,
what are we proving?

1178
00:55:34,020 --> 00:55:37,160
Well, we just proved p plus
q plus r squared equals 1.

1179
00:55:39,690 --> 00:55:41,120
So it looks like there's
no content.

1180
00:55:41,120 --> 00:55:42,370
AUDIENCE: [INAUDIBLE].

1181
00:55:48,160 --> 00:55:49,640
PROFESSOR: How has that
done anything?

1182
00:55:49,640 --> 00:55:52,170
AUDIENCE: The key fact is how
you prove that the 1, the 3,

1183
00:55:52,170 --> 00:55:56,712
the 3, and the 1 in
each [INAUDIBLE].

1184
00:55:56,712 --> 00:56:00,664
For example, are we supposed
to prove that

1185
00:56:00,664 --> 00:56:02,156
today or break even?

1186
00:56:05,280 --> 00:56:08,920
PROFESSOR: Well, I would say
there's no less information--

1187
00:56:08,920 --> 00:56:11,320
well, what we've learned is
that, for example, we've

1188
00:56:11,320 --> 00:56:13,030
learned this.

1189
00:56:13,030 --> 00:56:15,300
Why are there various
products here?

1190
00:56:15,300 --> 00:56:18,530
And then what this thing is--
we've learned intuitively what

1191
00:56:18,530 --> 00:56:20,280
this thing does.

1192
00:56:20,280 --> 00:56:22,890
This thing counts for the
number of copies.

1193
00:56:22,890 --> 00:56:25,590
So in the original--

1194
00:56:25,590 --> 00:56:27,780
n equals 2, C equals 2--

1195
00:56:27,780 --> 00:56:30,400
it was either there's two copies
or one copy of each of

1196
00:56:30,400 --> 00:56:31,060
these guys.

1197
00:56:31,060 --> 00:56:33,940
So it adjusted for the
number of copies.

1198
00:56:33,940 --> 00:56:37,390
When you have p plus q plus
r, there's one copy,

1199
00:56:37,390 --> 00:56:38,470
one copy, one copy.

1200
00:56:38,470 --> 00:56:40,210
There's two copies of that
one, two of that

1201
00:56:40,210 --> 00:56:41,490
one, two of that one.

1202
00:56:41,490 --> 00:56:42,660
So it's that factor.

1203
00:56:42,660 --> 00:56:45,630
It's the number of
copies factor.

1204
00:56:45,630 --> 00:56:48,590
And then in probability course,
you learn how to count

1205
00:56:48,590 --> 00:56:51,130
those factors in terms
of factorials.

1206
00:56:51,130 --> 00:56:53,220
And that has a definition
in terms of factorials.

1207
00:56:53,220 --> 00:56:56,090
But I wouldn't focus, for
example, in a biology course

1208
00:56:56,090 --> 00:56:59,030
on why it's factorials,
necessarily.

1209
00:56:59,030 --> 00:57:02,250
I'd want them to understand that
this thing counts for the

1210
00:57:02,250 --> 00:57:03,380
number of factors.

1211
00:57:03,380 --> 00:57:06,220
But actually calculating the
number of factors for general

1212
00:57:06,220 --> 00:57:12,646
C and n wouldn't be my first
goal in the course.

1213
00:57:12,646 --> 00:57:22,426
AUDIENCE: So if I didn't have
that in mind, why would I--

1214
00:57:22,426 --> 00:57:25,849
so the general purpose is
teaching the terms in the

1215
00:57:25,849 --> 00:57:27,830
multinomial expression.

1216
00:57:27,830 --> 00:57:28,100
PROFESSOR: Right.

1217
00:57:28,100 --> 00:57:30,030
So it's terms in a multinomial
expansion--

1218
00:57:30,030 --> 00:57:35,010
so basically, it's from writing
out 1 equals 1.

1219
00:57:35,010 --> 00:57:37,140
So it seems like, oh my god,
we've done nothing.

1220
00:57:37,140 --> 00:57:38,080
What have we learned?

1221
00:57:38,080 --> 00:57:40,630
We've just learned 1 equals
1, which we already knew.

1222
00:57:40,630 --> 00:57:43,440
But actually, by splitting up
1 on the other side, you've

1223
00:57:43,440 --> 00:57:45,360
actually given meaning
to each of the terms

1224
00:57:45,360 --> 00:57:46,140
on the other side.

1225
00:57:46,140 --> 00:57:51,100
So this is the frequency of
probability of having three

1226
00:57:51,100 --> 00:57:55,990
copies of gene type A and
none of the other type.

1227
00:57:55,990 --> 00:57:59,400
This is the probability of
having two copies of gene A

1228
00:57:59,400 --> 00:58:03,635
and one of gene B. And this is
the probability of having one

1229
00:58:03,635 --> 00:58:07,460
of gene A and two of gene B.

1230
00:58:07,460 --> 00:58:09,970
So there are three, because
there's different

1231
00:58:09,970 --> 00:58:10,730
ways of doing it.

1232
00:58:10,730 --> 00:58:12,886
AUDIENCE: So do you have these
chromosomes with different

1233
00:58:12,886 --> 00:58:14,136
flavors [INAUDIBLE]?

1234
00:58:18,100 --> 00:58:20,310
PROFESSOR: That's right.

1235
00:58:20,310 --> 00:58:23,340
Because these organisms we're
talking about come with

1236
00:58:23,340 --> 00:58:24,590
multiple copies of
chromosomes.

1237
00:58:29,040 --> 00:58:29,990
Yes.

1238
00:58:29,990 --> 00:58:31,240
AUDIENCE: [INAUDIBLE].

1239
00:58:36,290 --> 00:58:38,070
PROFESSOR: What is that?

1240
00:58:38,070 --> 00:58:38,340
Yeah.

1241
00:58:38,340 --> 00:58:40,170
So what's the phenotype
in the end?

1242
00:58:40,170 --> 00:58:42,200
And you're right.

1243
00:58:42,200 --> 00:58:43,370
So with the sickle cell-- so

1244
00:58:43,370 --> 00:58:44,560
actually, that is an advantage.

1245
00:58:44,560 --> 00:58:47,400
So the point is you
need to link this

1246
00:58:47,400 --> 00:58:48,970
phenotype and the genotype.

1247
00:58:48,970 --> 00:58:50,260
And that's true, actually.

1248
00:58:50,260 --> 00:58:54,200
I've erased the sickle cell, but
actually, the sickle cell

1249
00:58:54,200 --> 00:58:55,970
is a good example
for doing that.

1250
00:58:55,970 --> 00:58:59,410
If you have no copies of the
sickle cell gene, then you're

1251
00:58:59,410 --> 00:59:00,600
perfectly healthy.

1252
00:59:00,600 --> 00:59:02,920
If you have two copies of the
sickle cell gene, you have

1253
00:59:02,920 --> 00:59:04,430
sickle cell anemia.

1254
00:59:04,430 --> 00:59:06,810
What happens if you have one
copy of the sickle cell gene?

1255
00:59:06,810 --> 00:59:11,050
Well, you do have some
symptoms of it.

1256
00:59:11,050 --> 00:59:14,010
So the question is why does
that gene survive?

1257
00:59:14,010 --> 00:59:14,920
Well, it's because
you're actually

1258
00:59:14,920 --> 00:59:17,300
more malaria resistant.

1259
00:59:17,300 --> 00:59:19,520
At least, that's one
of the theories.

1260
00:59:19,520 --> 00:59:23,200
So somehow, I guess the malaria
parasites can't eat

1261
00:59:23,200 --> 00:59:24,980
those red blood cells as well
because they have a different

1262
00:59:24,980 --> 00:59:26,820
shape or can't invade it.

1263
00:59:26,820 --> 00:59:29,300
So it actually gives you
some advantage and some

1264
00:59:29,300 --> 00:59:30,970
disadvantage, but they're
sort of balanced.

1265
00:59:30,970 --> 00:59:34,490
And yeah, if you have full blown
sickle cell with two

1266
00:59:34,490 --> 00:59:37,010
sickle cell genes, then
you're in trouble.

1267
00:59:37,010 --> 00:59:41,060
But that's much rarer than
the 2pq, as long

1268
00:59:41,060 --> 00:59:44,390
as q is small enough.

1269
00:59:44,390 --> 00:59:46,670
So then you can actually
continue that example.

1270
00:59:46,670 --> 00:59:49,190
You can look at the frequency
of sickle cell gene in

1271
00:59:49,190 --> 00:59:51,460
different populations and say,
OK, well, is it higher in

1272
00:59:51,460 --> 00:59:52,990
areas where there's malaria?

1273
00:59:52,990 --> 00:59:55,650
And test that theory.

1274
00:59:55,650 --> 00:59:57,772
Yes.

1275
00:59:57,772 --> 01:00:00,748
AUDIENCE: I noticed that the
first one that you did was to

1276
01:00:00,748 --> 01:00:04,220
stay mathematical with
the number line.

1277
01:00:04,220 --> 01:00:07,940
When I've, in the past, taught
[INAUDIBLE], I always stick

1278
01:00:07,940 --> 01:00:09,190
with conceptual--

1279
01:00:12,652 --> 01:00:15,380
p is this allele, and
then you've got a

1280
01:00:15,380 --> 01:00:16,915
phenotype in the end.

1281
01:00:16,915 --> 01:00:20,026
I'm just wondering if you chose
this because you know

1282
01:00:20,026 --> 01:00:23,380
that you're teaching mostly to
people with math rather than

1283
01:00:23,380 --> 01:00:24,739
scientists.

1284
01:00:24,739 --> 01:00:27,570
And if you were teaching to,
say, English majors--

1285
01:00:27,570 --> 01:00:28,310
PROFESSOR: Good question.

1286
01:00:28,310 --> 01:00:30,430
Yeah, so I think you're right.

1287
01:00:30,430 --> 01:00:32,480
I did this kind of
mathematically, compared to

1288
01:00:32,480 --> 01:00:35,730
maybe how you've taught it
to biology students.

1289
01:00:35,730 --> 01:00:38,240
And yeah, I guess I'm implicitly
assuming-- but I

1290
01:00:38,240 --> 01:00:39,750
didn't say, and I should say--

1291
01:00:39,750 --> 01:00:41,400
that this is MIT students.

1292
01:00:41,400 --> 01:00:44,490
I'm just, in the back of mind,
thinking MIT students.

1293
01:00:44,490 --> 01:00:47,640
But I didn't make that
assumption explicit.

1294
01:00:47,640 --> 01:00:51,730
So MIT students are perfectly
happy with squaring binomials

1295
01:00:51,730 --> 01:00:52,820
and trinomials--

1296
01:00:52,820 --> 01:00:55,040
trinomials, usually.

1297
01:00:55,040 --> 01:00:56,270
And to the C power--

1298
01:00:56,270 --> 01:00:58,550
sort of stretching
it, but it's OK.

1299
01:00:58,550 --> 01:01:02,490
But generally, that way of doing
things for them is good.

1300
01:01:02,490 --> 01:01:05,010
If it's people, English majors,
yeah, you're right.

1301
01:01:05,010 --> 01:01:07,920
I would try to actually do
even more conceptually.

1302
01:01:07,920 --> 01:01:12,390
But one thing that's good for
all of them is the picture,

1303
01:01:12,390 --> 01:01:15,890
because once you see the
picture, then you actually

1304
01:01:15,890 --> 01:01:18,620
understand the idea
in one grasp.

1305
01:01:18,620 --> 01:01:20,460
It's really just
one chunk now.

1306
01:01:20,460 --> 01:01:20,980
Oh.

1307
01:01:20,980 --> 01:01:23,060
It's just a square
in four pieces.

1308
01:01:23,060 --> 01:01:23,725
Oh, OK.

1309
01:01:23,725 --> 01:01:25,640
It's just a different
way of writing 1.

1310
01:01:25,640 --> 01:01:28,680
This is one area 1 and
there's four pieces.

1311
01:01:28,680 --> 01:01:32,170
So that I would keep, no matter
who I would talk to.

1312
01:01:32,170 --> 01:01:34,880
And the question is how much
would lead up to it?

1313
01:01:34,880 --> 01:01:35,320
Yes

1314
01:01:35,320 --> 01:01:43,718
AUDIENCE: So I think I just had
an idea here that I think

1315
01:01:43,718 --> 01:01:48,164
highlights why you can't just
go in with this, because

1316
01:01:48,164 --> 01:01:50,387
there's a disconnect
between what these

1317
01:01:50,387 --> 01:01:52,900
terms mean at the end.

1318
01:01:52,900 --> 01:01:54,045
PROFESSOR: Why you can't
go in with A.

1319
01:01:54,045 --> 01:01:56,190
AUDIENCE: Right.

1320
01:01:56,190 --> 01:02:00,505
So I think you have to draw a
connection between, say, the

1321
01:02:00,505 --> 01:02:05,038
areas of the boxes in
the diagram and

1322
01:02:05,038 --> 01:02:06,288
the phenotype frequency.

1323
01:02:09,760 --> 01:02:10,620
PROFESSOR: You're right.

1324
01:02:10,620 --> 01:02:13,290
So what you're saying is that
you can't just launch in with

1325
01:02:13,290 --> 01:02:16,110
this, because even though
you could give--

1326
01:02:16,110 --> 01:02:18,050
say you gave really exact
definitions of what all these

1327
01:02:18,050 --> 01:02:18,920
things are.

1328
01:02:18,920 --> 01:02:21,620
It's not clear that exact
definitions are

1329
01:02:21,620 --> 01:02:24,860
computationally productive
for a student.

1330
01:02:24,860 --> 01:02:27,260
Yeah, they may have the exact
definitions, but they can't

1331
01:02:27,260 --> 01:02:28,090
actually use it.

1332
01:02:28,090 --> 01:02:30,910
For example, let's do an exact
definition of chess.

1333
01:02:30,910 --> 01:02:33,020
I'll tell you all the
rules of chess.

1334
01:02:33,020 --> 01:02:36,910
And that's enough of an exact
definition to be able to

1335
01:02:36,910 --> 01:02:39,210
decide what the best first
move in chess it.

1336
01:02:39,210 --> 01:02:40,650
But it's computationally
useless.

1337
01:02:40,650 --> 01:02:42,420
I still don't know what the best
first move is, even if I

1338
01:02:42,420 --> 01:02:43,670
know the rules of chess.

1339
01:02:43,670 --> 01:02:46,670
So just telling the student
all the rules that, say,

1340
01:02:46,670 --> 01:02:50,350
define what a genotype is and
what's polyploidy, doesn't

1341
01:02:50,350 --> 01:02:52,370
mean they can actually use
it in any problem.

1342
01:02:52,370 --> 01:02:55,290
So if you want to transfer, all
these things have to have

1343
01:02:55,290 --> 01:02:56,360
meaning for them.

1344
01:02:56,360 --> 01:03:00,030
And that's what the goal
of this approach is.

1345
01:03:00,030 --> 01:03:03,210
Now, I think the approach has
been improved from your

1346
01:03:03,210 --> 01:03:04,680
suggestions.

1347
01:03:04,680 --> 01:03:07,040
I just wanted to show you
a direction to go.

1348
01:03:07,040 --> 01:03:09,215
But I think you've actually
taken it farther and extended

1349
01:03:09,215 --> 01:03:11,700
it and improved upon it.

1350
01:03:11,700 --> 01:03:14,020
And the general principles
among all of them are

1351
01:03:14,020 --> 01:03:17,855
that you want to--

1352
01:03:17,855 --> 01:03:20,370
I would say one of the
key ones is chunking.

1353
01:03:20,370 --> 01:03:23,410
You want to not overload
the chunk system.

1354
01:03:23,410 --> 01:03:25,230
You want to somehow bring people
in so they'll even

1355
01:03:25,230 --> 01:03:25,990
listen to you.

1356
01:03:25,990 --> 01:03:27,920
If they don't listen to you,
if they don't care, the

1357
01:03:27,920 --> 01:03:29,360
learning is going to
be so much less.

1358
01:03:29,360 --> 01:03:31,990
So all those are for that.

1359
01:03:31,990 --> 01:03:36,410
So the dictation and jargon
oppose chunking.

1360
01:03:36,410 --> 01:03:40,240
And these all go together.

1361
01:03:40,240 --> 01:03:42,290
OK.

1362
01:03:42,290 --> 01:03:49,900
So now, what I want to do is
give you a short answer to one

1363
01:03:49,900 --> 01:03:51,820
of the questions that
was raised earlier.

1364
01:03:51,820 --> 01:03:55,030
How you become a good teacher?

1365
01:03:55,030 --> 01:03:56,410
The reason I want to
do that is that is

1366
01:03:56,410 --> 01:03:58,420
exactly the same thing--

1367
01:03:58,420 --> 01:04:01,060
if you understand how you become
a good teacher, you

1368
01:04:01,060 --> 01:04:03,780
understand how you become
good at anything--

1369
01:04:03,780 --> 01:04:06,130
how you become good at chess,
how you become good at

1370
01:04:06,130 --> 01:04:08,550
biology, how you become good at
solving physics problems,

1371
01:04:08,550 --> 01:04:11,350
how you become good at playing
concert piano.

1372
01:04:11,350 --> 01:04:15,330
So if that's what you want to
teach your students-- to be

1373
01:04:15,330 --> 01:04:18,260
good at those things-- well, you
want to understand that in

1374
01:04:18,260 --> 01:04:19,965
a context, say, that
you're working on--

1375
01:04:19,965 --> 01:04:21,810
say, being a good teacher.

1376
01:04:21,810 --> 01:04:29,480
And to do that, there is the
following set of experiments.

1377
01:04:29,480 --> 01:04:30,730
So projectors--

1378
01:04:33,370 --> 01:04:33,680
OK.

1379
01:04:33,680 --> 01:04:35,340
So the way I'm going to
illustrate this is I'm going

1380
01:04:35,340 --> 01:04:39,340
to show you a chess position.

1381
01:04:39,340 --> 01:04:42,640
And the goal is to try to
remember the chess position.

1382
01:04:53,620 --> 01:04:54,870
I'm going to give you
two more seconds.

1383
01:04:58,720 --> 01:04:59,850
OK.

1384
01:04:59,850 --> 01:05:03,440
So everyone got the position?

1385
01:05:03,440 --> 01:05:06,700
Now, I'm going to ask you
instead of to remember

1386
01:05:06,700 --> 01:05:08,320
exactly, to reconstruct it.

1387
01:05:16,010 --> 01:05:17,390
How many pawns were there?

1388
01:05:22,680 --> 01:05:25,010
That would be E.

1389
01:05:25,010 --> 01:05:27,670
So who votes for A?

1390
01:05:30,560 --> 01:05:30,810
OK.

1391
01:05:30,810 --> 01:05:33,430
Who votes for B, nine pawns?

1392
01:05:33,430 --> 01:05:37,900
Who votes for C, 11 pawns?

1393
01:05:37,900 --> 01:05:41,850
Who votes for D, 13 pawns?

1394
01:05:41,850 --> 01:05:45,590
Who votes for E, none
of the above?

1395
01:05:45,590 --> 01:05:46,470
OK.

1396
01:05:46,470 --> 01:05:50,250
So let me show you.

1397
01:05:50,250 --> 01:05:52,970
Then I'll explain why I asked
you this particular question.

1398
01:05:52,970 --> 01:05:55,450
So there are actually 11.

1399
01:05:55,450 --> 01:05:59,830
Now, it's a very hard task.

1400
01:05:59,830 --> 01:06:02,020
So this actually, this
very tasks--

1401
01:06:02,020 --> 01:06:03,240
not counting the number
of pawns--

1402
01:06:03,240 --> 01:06:04,990
I made that slight variation.

1403
01:06:04,990 --> 01:06:09,460
But the so-called reconstruction
task was given

1404
01:06:09,460 --> 01:06:13,260
to chess players of
various abilities.

1405
01:06:13,260 --> 01:06:16,240
Grandmaster slash master
is one group.

1406
01:06:16,240 --> 01:06:17,450
What are called experts--

1407
01:06:17,450 --> 01:06:20,180
experts are people who are not
quite chess masters, but

1408
01:06:20,180 --> 01:06:22,150
close, in chess lingo.

1409
01:06:22,150 --> 01:06:24,660
And then class A players,
which is pretty strong

1410
01:06:24,660 --> 01:06:25,670
tournament players.

1411
01:06:25,670 --> 01:06:30,255
So they were given the task of
looking at a position for four

1412
01:06:30,255 --> 01:06:31,255
to five seconds.

1413
01:06:31,255 --> 01:06:34,440
The position was knocked down,
and they were asked to

1414
01:06:34,440 --> 01:06:37,680
reconstruct it as accurately
as they could.

1415
01:06:37,680 --> 01:06:37,930
OK.

1416
01:06:37,930 --> 01:06:40,330
So it's even harder than
the task of counting

1417
01:06:40,330 --> 01:06:41,580
the number of pawns.

1418
01:06:44,400 --> 01:06:52,310
So the results are
very striking.

1419
01:07:04,300 --> 01:07:05,665
So by level of chess player--

1420
01:07:08,790 --> 01:07:20,080
so class A is the strong
tournament players, experts or

1421
01:07:20,080 --> 01:07:21,330
grand master or master.

1422
01:07:25,440 --> 01:07:26,690
So the percent correct--

1423
01:07:29,990 --> 01:07:39,410
51% of the pieces correct,
72% or 93%.

1424
01:07:39,410 --> 01:07:42,140
So the grand masters and the
masters were amazing.

1425
01:07:42,140 --> 01:07:44,810
And in fact, for the number of
pawns, I don't think they ever

1426
01:07:44,810 --> 01:07:47,890
make mistakes on that, because
pawns are one of things that

1427
01:07:47,890 --> 01:07:50,650
you just know as a really
strong chess player.

1428
01:07:50,650 --> 01:07:53,270
So 93% percent correct--

1429
01:07:53,270 --> 01:07:54,520
and that's amazing.

1430
01:07:59,860 --> 01:08:04,460
So there's an related story,
which is about the memory of

1431
01:08:04,460 --> 01:08:07,260
chess players, which is the
Bobby Fischer-- yes.

1432
01:08:07,260 --> 01:08:10,062
AUDIENCE: So I'm thinking the
positions, the positions

1433
01:08:10,062 --> 01:08:10,372
wouldn't work.

1434
01:08:10,372 --> 01:08:11,463
It wasn't possible.

1435
01:08:11,463 --> 01:08:13,650
They wouldn't do it as well.

1436
01:08:13,650 --> 01:08:14,900
PROFESSOR: I'll come to
that in one second.

1437
01:08:17,550 --> 01:08:23,189
So Bobby Fischer was in a
tournament and some strong

1438
01:08:23,189 --> 01:08:25,170
master was playing in it.

1439
01:08:25,170 --> 01:08:27,520
And Bobby Fischer went
to the bathroom.

1440
01:08:27,520 --> 01:08:29,410
And as he went to the bathroom,
he happened to see

1441
01:08:29,410 --> 01:08:33,520
that master playing a game and
just continued walking.

1442
01:08:33,520 --> 01:08:35,460
And then about six months later,
he ran into him at

1443
01:08:35,460 --> 01:08:37,130
another tournament.

1444
01:08:37,130 --> 01:08:40,180
Fischer said to him, oh, in
that position in that

1445
01:08:40,180 --> 01:08:42,529
tournament, did you play blah?

1446
01:08:42,529 --> 01:08:44,310
And the guy said, well,
actually, I had no idea what

1447
01:08:44,310 --> 01:08:45,520
the position was, even.

1448
01:08:45,520 --> 01:08:47,949
Fischer said, oh, and he set up
the board and said, well,

1449
01:08:47,949 --> 01:08:50,520
see, this was the thing that
you really needed to do.

1450
01:08:50,520 --> 01:08:51,910
And by then, he sort
of remembered.

1451
01:08:51,910 --> 01:08:55,080
But Bobby Fischer remembered
at one

1452
01:08:55,080 --> 01:08:58,130
glance six months later.

1453
01:08:58,130 --> 01:09:03,979
So what's a natural conclusion
from this data?

1454
01:09:03,979 --> 01:09:09,270
You'd say, well, naturally,
grand masters and masters--

1455
01:09:09,270 --> 01:09:12,649
they were born with better
visual memory.

1456
01:09:12,649 --> 01:09:17,140
But in fact, the crucial
experiment was

1457
01:09:17,140 --> 01:09:20,439
then done in 1973.

1458
01:09:20,439 --> 01:09:30,510
So this experiment was done
in 1948 by de Groot.

1459
01:09:30,510 --> 01:09:34,560
What Chase and Simon did in
1973 was that they showed

1460
01:09:34,560 --> 01:09:38,100
positions that were random.

1461
01:09:38,100 --> 01:09:40,490
So they redid the experiment--

1462
01:09:40,490 --> 01:09:42,380
they confirmed these results.

1463
01:09:42,380 --> 01:09:45,529
And then, as you suggested,
they showed just positions

1464
01:09:45,529 --> 01:09:46,890
where the pieces were scattered

1465
01:09:46,890 --> 01:09:48,600
arbitrarily over the board.

1466
01:09:48,600 --> 01:09:54,520
And then everyone was basically
at 12%, give or take

1467
01:09:54,520 --> 01:09:57,495
1% or 2%, just random
variations.

1468
01:10:02,980 --> 01:10:04,120
So what does that show?

1469
01:10:04,120 --> 01:10:05,560
It's not that the--

1470
01:10:05,560 --> 01:10:07,550
maybe Bobby Fischer
was an exception.

1471
01:10:07,550 --> 01:10:10,960
But for almost everybody else,
even the very strongest

1472
01:10:10,960 --> 01:10:13,230
players, it's not that
they're born with a

1473
01:10:13,230 --> 01:10:15,080
better visual memory.

1474
01:10:15,080 --> 01:10:18,990
It's that they've learned
somehow a way of looking at

1475
01:10:18,990 --> 01:10:24,310
chess positions that there's
less to remember for them.

1476
01:10:24,310 --> 01:10:25,300
OK.

1477
01:10:25,300 --> 01:10:28,050
So now let's compare.

1478
01:10:28,050 --> 01:10:32,600
Here, when the student sees
this, there's a ton for the

1479
01:10:32,600 --> 01:10:36,200
student to remember, just like
when we look at a chess

1480
01:10:36,200 --> 01:10:39,050
position, every piece
is separate.

1481
01:10:39,050 --> 01:10:40,840
But what does a chess
player see?

1482
01:10:40,840 --> 01:10:43,070
When a chess master looks
at a chess position--

1483
01:10:46,460 --> 01:10:48,190
so I'll put the position back
and I'll show you what the

1484
01:10:48,190 --> 01:10:49,685
chess master sees.

1485
01:10:49,685 --> 01:10:52,420
The chess masters see something
very different.

1486
01:10:52,420 --> 01:10:57,910
They see groups of related
pieces together.

1487
01:10:57,910 --> 01:11:04,550
So for example, here, the
chess master sees--

1488
01:11:04,550 --> 01:11:07,350
this king here is not a
surprising thing for this

1489
01:11:07,350 --> 01:11:08,240
chess master.

1490
01:11:08,240 --> 01:11:09,710
That's when you castle
your king.

1491
01:11:09,710 --> 01:11:10,910
That's where it goes.

1492
01:11:10,910 --> 01:11:13,150
Then the rook goes next to it,
and then you usually move your

1493
01:11:13,150 --> 01:11:14,110
rook into the middle.

1494
01:11:14,110 --> 01:11:15,560
So that's not surprising.

1495
01:11:15,560 --> 01:11:17,670
This rook is also not
surprising, the second one,

1496
01:11:17,670 --> 01:11:19,950
because it usually comes from
this corner into the middle.

1497
01:11:19,950 --> 01:11:22,060
So all of this almost contains
no information

1498
01:11:22,060 --> 01:11:24,080
for the chess master.

1499
01:11:24,080 --> 01:11:31,200
Here, these three pawns are very
common, with the castled

1500
01:11:31,200 --> 01:11:32,130
king on that side.

1501
01:11:32,130 --> 01:11:33,470
But then it looks
kind of strange.

1502
01:11:33,470 --> 01:11:36,610
There's some new information
there, because maybe this king

1503
01:11:36,610 --> 01:11:38,120
castled, but then the
rook went all

1504
01:11:38,120 --> 01:11:39,000
the way to the corner.

1505
01:11:39,000 --> 01:11:41,670
So actually, maybe black
never castled.

1506
01:11:41,670 --> 01:11:45,290
And his kind just sort of
wandered into this area.

1507
01:11:45,290 --> 01:11:46,180
What does that suggest?

1508
01:11:46,180 --> 01:11:48,480
This suggests that the black
king is really vulnerable.

1509
01:11:48,480 --> 01:11:50,440
Maybe it's time for an attack.

1510
01:11:50,440 --> 01:11:53,510
And in fact, I'm pretty sure
this is a position from one of

1511
01:11:53,510 --> 01:11:55,170
Garry Kasparov's games.

1512
01:11:55,170 --> 01:11:57,270
And in fact, that is the
right conclusion.

1513
01:11:57,270 --> 01:11:59,370
The right conclusion is that
it's now time to sacrifice

1514
01:11:59,370 --> 01:12:02,990
your knight and take this pawn
and draw the king out.

1515
01:12:02,990 --> 01:12:05,270
And he actually won
using that--

1516
01:12:05,270 --> 01:12:06,680
by sacrificing his knight.

1517
01:12:06,680 --> 01:12:10,240
So the chess master looks at it
completely differently than

1518
01:12:10,240 --> 01:12:11,810
the novice.

1519
01:12:11,810 --> 01:12:13,590
I'm a novice when
I play chess.

1520
01:12:13,590 --> 01:12:16,640
To me, every piece is a new
bit of information.

1521
01:12:16,640 --> 01:12:18,980
I'm way overloaded past
my chunk threshold.

1522
01:12:18,980 --> 01:12:21,560
I can't hardly remember
the board at all.

1523
01:12:21,560 --> 01:12:25,310
So your students are in exactly
the same position when

1524
01:12:25,310 --> 01:12:27,660
they're learning material that,
for you, you're the

1525
01:12:27,660 --> 01:12:28,910
chess master.

1526
01:12:31,020 --> 01:12:33,130
So you're teaching
Hardy-Weinberg.

1527
01:12:33,130 --> 01:12:34,820
Well, clearly, you've been
appointed to teach

1528
01:12:34,820 --> 01:12:37,110
Hardy-Weinberg because you
have a Ph.D. In biology.

1529
01:12:37,110 --> 01:12:38,710
You know a lot of biology.

1530
01:12:38,710 --> 01:12:39,930
You're the biology master.

1531
01:12:39,930 --> 01:12:42,590
So this doesn't surprise
you that much.

1532
01:12:42,590 --> 01:12:46,370
But for the student, every
single almost letter in there

1533
01:12:46,370 --> 01:12:48,010
is news to them.

1534
01:12:48,010 --> 01:12:51,240
So what you want to do is you
want to find ways of thinking

1535
01:12:51,240 --> 01:12:55,150
about it that you can group
the ideas into chunks.

1536
01:12:55,150 --> 01:12:56,950
So here is almost one chunk--

1537
01:12:56,950 --> 01:12:58,520
for example, the idea
that really, it's

1538
01:12:58,520 --> 01:13:02,990
just p plus q squared.

1539
01:13:02,990 --> 01:13:05,090
And there's a picture for it.

1540
01:13:05,090 --> 01:13:08,040
And once you understand that,
there's another chunk.

1541
01:13:08,040 --> 01:13:11,790
There's another idea which is
well, you could actually have

1542
01:13:11,790 --> 01:13:14,230
three kinds of flavors
instead of two.

1543
01:13:14,230 --> 01:13:15,880
OK. p plus q plus 4--

1544
01:13:15,880 --> 01:13:17,550
oh, I can transfer it there.

1545
01:13:17,550 --> 01:13:18,900
Oh, and then once you understand
that, you can

1546
01:13:18,900 --> 01:13:24,260
transfer it to n flavors.

1547
01:13:24,260 --> 01:13:26,000
Now, you can increase the
number of copies of the

1548
01:13:26,000 --> 01:13:27,040
chromosome--

1549
01:13:27,040 --> 01:13:29,070
so into two dimensions,
three dimensions,

1550
01:13:29,070 --> 01:13:30,550
four dimensional picture.

1551
01:13:30,550 --> 01:13:34,450
So then, you can actually
make sense of all this.

1552
01:13:34,450 --> 01:13:38,010
You can have a way of
understanding the position.

1553
01:13:38,010 --> 01:13:42,160
And not too long ago, there's,
I think, a not well enough

1554
01:13:42,160 --> 01:13:42,770
known paper--

1555
01:13:42,770 --> 01:13:44,590
I'll put the reference
on the website--

1556
01:13:44,590 --> 01:13:47,700
which shows the relative
importance of symbolic

1557
01:13:47,700 --> 01:13:50,120
calculation versus perception.

1558
01:13:50,120 --> 01:13:53,980
So this is a perceptual mode.

1559
01:13:53,980 --> 01:13:56,720
So much of our teaching is,
let's say, left brain-- very,

1560
01:13:56,720 --> 01:13:58,630
very symbolic.

1561
01:13:58,630 --> 01:14:01,310
Well, there was this really
interesting study done of

1562
01:14:01,310 --> 01:14:02,330
chess grandmasters--

1563
01:14:02,330 --> 01:14:05,070
in fact, of the strongest
chess grandmaster

1564
01:14:05,070 --> 01:14:06,500
today, Garry Kasparov.

1565
01:14:06,500 --> 01:14:12,270
What's the relative importance
of perception versus analysis

1566
01:14:12,270 --> 01:14:15,180
in his really strong
chess playing?

1567
01:14:15,180 --> 01:14:17,210
So the way they tested that--
there was a really good

1568
01:14:17,210 --> 01:14:18,240
experiment--

1569
01:14:18,240 --> 01:14:21,490
is he would play simultaneous
exhibitions all the time.

1570
01:14:21,490 --> 01:14:23,360
So the way you do a simultaneous
exhibition is

1571
01:14:23,360 --> 01:14:27,220
there's, for example, let's
say, 10 opponents.

1572
01:14:27,220 --> 01:14:31,130
And you would just go around one
component after another.

1573
01:14:31,130 --> 01:14:33,930
They have the full time, say,
three minutes, to think until

1574
01:14:33,930 --> 01:14:35,160
you come back.

1575
01:14:35,160 --> 01:14:37,450
But as soon as you get to a
board, you just think for

1576
01:14:37,450 --> 01:14:40,000
about five or six seconds, maybe
10, and make a move, and

1577
01:14:40,000 --> 01:14:42,510
go to the next board, so that
by the time you come around

1578
01:14:42,510 --> 01:14:45,090
back to that same opponent,
they've had their couple

1579
01:14:45,090 --> 01:14:46,280
minutes to think.

1580
01:14:46,280 --> 01:14:49,880
So now, he played simultaneous
exhibitions against very

1581
01:14:49,880 --> 01:14:52,850
strong grandmasters.

1582
01:14:52,850 --> 01:14:55,830
And you can then measure
his performance there.

1583
01:14:55,830 --> 01:14:57,680
So why is that a good
experiment?

1584
01:14:57,680 --> 01:15:00,850
Well, he's now not able to do
all of his calculation that he

1585
01:15:00,850 --> 01:15:02,370
does normally.

1586
01:15:02,370 --> 01:15:04,960
He's normally able to think for
three minutes, maybe five

1587
01:15:04,960 --> 01:15:07,500
minutes, and do a whole bunch
of analysis, symbolic

1588
01:15:07,500 --> 01:15:08,600
computation.

1589
01:15:08,600 --> 01:15:12,190
But when he has five seconds,
10 seconds to think, mostly

1590
01:15:12,190 --> 01:15:13,890
it's perception.

1591
01:15:13,890 --> 01:15:17,810
Well, his chess rating
effectively dropped by maybe

1592
01:15:17,810 --> 01:15:19,750
50 or 60 points.

1593
01:15:19,750 --> 01:15:22,480
So 50 or 60 points, to
give you an idea--

1594
01:15:22,480 --> 01:15:25,040
his chest rating is the
highest in the world.

1595
01:15:25,040 --> 01:15:28,080
It dropped to a level which only
five people in the world

1596
01:15:28,080 --> 01:15:29,100
are higher.

1597
01:15:29,100 --> 01:15:31,110
So it's a very small drop.

1598
01:15:31,110 --> 01:15:34,470
So he still plays incredibly
strong chess, better than

1599
01:15:34,470 --> 01:15:37,570
almost every other grandmaster
on the planet.

1600
01:15:37,570 --> 01:15:41,090
So almost purely with
perception--

1601
01:15:41,090 --> 01:15:44,640
so what that shows is that the
way Kasparov has become so

1602
01:15:44,640 --> 01:15:47,710
good and in general, experts
have become so good is it they

1603
01:15:47,710 --> 01:15:49,310
look at the world different.

1604
01:15:49,310 --> 01:15:52,810
Their perception is different.

1605
01:15:52,810 --> 01:15:56,720
So how do you do that
as a teacher?

1606
01:15:56,720 --> 01:15:57,700
That's one question.

1607
01:15:57,700 --> 01:15:59,510
How do you promote that
in your students?

1608
01:15:59,510 --> 01:16:01,990
Well, you want to give the ways
of looking at the world

1609
01:16:01,990 --> 01:16:03,650
that change their perception.

1610
01:16:03,650 --> 01:16:08,230
That's why I'm so focused on
the story, the tension, the

1611
01:16:08,230 --> 01:16:12,360
human, the right brain, the
pictures, the chunking,

1612
01:16:12,360 --> 01:16:14,780
because it's those that are
actually producing long term

1613
01:16:14,780 --> 01:16:19,120
expertise, whereas this is
producing what would be the

1614
01:16:19,120 --> 01:16:21,000
equivalent of programming
a chess computer.

1615
01:16:21,000 --> 01:16:22,230
But that doesn't work.

1616
01:16:22,230 --> 01:16:24,410
That may work for chess
computers to play good chess.

1617
01:16:24,410 --> 01:16:27,010
But it doesn't actually work for
people to be able to use

1618
01:16:27,010 --> 01:16:29,250
the knowledge later.

1619
01:16:29,250 --> 01:16:31,560
So now, what produces that?

1620
01:16:31,560 --> 01:16:35,690
So there's one short answer
which is that for teaching,

1621
01:16:35,690 --> 01:16:38,100
you want to change your
perception of

1622
01:16:38,100 --> 01:16:39,720
how students think.

1623
01:16:39,720 --> 01:16:43,110
If you have a correct, new,
good perception of how

1624
01:16:43,110 --> 01:16:47,000
students are thinking, then
you're actually able to make

1625
01:16:47,000 --> 01:16:48,590
teaching judgments on the fly.

1626
01:16:48,590 --> 01:16:54,490
You can plan chess move, your
lecture, like a chess game.

1627
01:16:54,490 --> 01:16:55,850
Your intuition is right.

1628
01:16:55,850 --> 01:16:57,500
So you want to tune
your intuition.

1629
01:16:57,500 --> 01:16:59,900
Well, that is why I do this.

1630
01:16:59,900 --> 01:17:02,870
I've found the single most
important thing that has

1631
01:17:02,870 --> 01:17:05,200
improved my teaching, and I
highly recommend, is the

1632
01:17:05,200 --> 01:17:09,710
feedback sheet, because for
example, I learned what was

1633
01:17:09,710 --> 01:17:12,140
confusing in question one.

1634
01:17:12,140 --> 01:17:14,040
And in question two, I learned
what works and

1635
01:17:14,040 --> 01:17:15,190
what doesn't work.

1636
01:17:15,190 --> 01:17:17,900
So as I see what works and
doesn't work, I start to build

1637
01:17:17,900 --> 01:17:20,730
up a more and more accurate
model of you.

1638
01:17:20,730 --> 01:17:25,360
And I start to be able to plan
and reason about how to teach

1639
01:17:25,360 --> 01:17:27,990
you and, in general, how
to teach students.

1640
01:17:27,990 --> 01:17:30,140
So I'll talk about that more.

1641
01:17:30,140 --> 01:17:32,240
That's the idea of deliberate
practice and expertise.

1642
01:17:32,240 --> 01:17:34,740
I'm going to talk about that
more in the subsequent

1643
01:17:34,740 --> 01:17:37,820
sections in more detail and show
you some of the studies

1644
01:17:37,820 --> 01:17:38,620
around that.

1645
01:17:38,620 --> 01:17:42,740
But the general rules is you
want reflective, quick

1646
01:17:42,740 --> 01:17:46,790
feedback on what you're doing in
order to become an expert.

1647
01:17:46,790 --> 01:17:50,130
And that's true whether you're
in teaching, concert piano,

1648
01:17:50,130 --> 01:17:52,370
physics problems, whatever
it may be.

1649
01:17:52,370 --> 01:17:56,010
So with that said, if you can
fill out the sheets so that I

1650
01:17:56,010 --> 01:18:00,210
can become a better teacher,
that will be very helpful.

1651
01:18:00,210 --> 01:18:06,470
And one announcement, which is
that next week is a Tuesday.

1652
01:18:06,470 --> 01:18:08,140
MIT is open on Tuesday,
except it's

1653
01:18:08,140 --> 01:18:09,860
Monday's schedule of classes.

1654
01:18:09,860 --> 01:18:12,410
So we don't have a
class next week.

1655
01:18:12,410 --> 01:18:15,310
The week after that,
I'm a witness in an

1656
01:18:15,310 --> 01:18:16,530
administrative law trial.

1657
01:18:16,530 --> 01:18:17,210
So I'm not here.

1658
01:18:17,210 --> 01:18:19,550
So there's no class for
the next two weeks.

1659
01:18:19,550 --> 01:18:21,990
So we'll meet again
in three weeks.

1660
01:18:21,990 --> 01:18:26,340
And I'll post some readings and
a short problem set for

1661
01:18:26,340 --> 01:18:27,390
you to work on--

1662
01:18:27,390 --> 01:18:30,790
some readings growing out of
what we've done today.

1663
01:18:30,790 --> 01:18:31,450
OK.

1664
01:18:31,450 --> 01:18:38,720
So if you could bring up your
sheet and your index card to

1665
01:18:38,720 --> 01:18:40,880
separate piles, that would
be very helpful.

1666
01:18:40,880 --> 01:18:44,890
And there's going to be another
class coming in--

1667
01:18:44,890 --> 01:18:46,160
a big class, I think.

1668
01:18:46,160 --> 01:18:49,940
So I'll just go outside and
answer any questions that

1669
01:18:49,940 --> 01:18:52,640
people I have right outside
so that the new

1670
01:18:52,640 --> 01:18:56,270
class can come in.

1671
01:18:56,270 --> 01:18:58,610
SPEAKER: Answers from Lecture
3 to questions

1672
01:18:58,610 --> 01:19:00,090
generated in Lecture 2.

1673
01:19:02,880 --> 01:19:06,060
PROFESSOR: I'm going to first
answer questions from before,

1674
01:19:06,060 --> 01:19:08,120
since there are lots of
questions, and all

1675
01:19:08,120 --> 01:19:08,880
interesting.

1676
01:19:08,880 --> 01:19:12,820
And I'm also going to do another
equation example.

1677
01:19:12,820 --> 01:19:15,950
There was lots of requests for
another equation example to

1678
01:19:15,950 --> 01:19:19,550
see how it plays out in a
different field and a

1679
01:19:19,550 --> 01:19:23,470
different way of approaching
equations, not just from the

1680
01:19:23,470 --> 01:19:25,320
entry point of a story.

1681
01:19:25,320 --> 01:19:26,550
So I'll show you that.

1682
01:19:26,550 --> 01:19:29,410
And then we're going to look at
misconceptions in various

1683
01:19:29,410 --> 01:19:33,150
fields and the fundamental
importance of understanding

1684
01:19:33,150 --> 01:19:36,030
that so that you can understand
how to change your

1685
01:19:36,030 --> 01:19:37,730
teaching and how to reach
the students.

1686
01:19:37,730 --> 01:19:40,580
Basically, if you can't
understand where they are, you

1687
01:19:40,580 --> 01:19:42,350
can't come to them.

1688
01:19:42,350 --> 01:19:46,650
So questions from before--

1689
01:19:46,650 --> 01:19:50,830
one comment was that I don't
often enough summarize the end

1690
01:19:50,830 --> 01:19:54,390
result with the transferable
lessons for later.

1691
01:19:54,390 --> 01:19:56,700
So thanks for that suggestion.

1692
01:19:56,700 --> 01:19:59,720
I'll make sure to do that.

1693
01:19:59,720 --> 01:20:03,810
Another question was graduate
versus undergraduate classes.

1694
01:20:03,810 --> 01:20:07,910
We talked a fair amount about
audience a bit last time in

1695
01:20:07,910 --> 01:20:09,220
response to questions.

1696
01:20:09,220 --> 01:20:11,910
So how do you change your
teaching in response to

1697
01:20:11,910 --> 01:20:15,160
questions, in response to
the change in audience?

1698
01:20:15,160 --> 01:20:17,940
So the particular question
here is graduate verses

1699
01:20:17,940 --> 01:20:19,420
undergraduate classes.

1700
01:20:19,420 --> 01:20:22,730
And the sense I got from some
of the questions was that

1701
01:20:22,730 --> 01:20:26,210
somehow, it's harder to do what
we were talking about

1702
01:20:26,210 --> 01:20:30,300
last time, which is teaching
equations in an intuitive way,

1703
01:20:30,300 --> 01:20:33,450
in a graduate class than
a undergraduate class.

1704
01:20:33,450 --> 01:20:36,760
Actually, in some ways,
it's the opposite.

1705
01:20:36,760 --> 01:20:40,470
It's true that generally, in
graduate classes, people just

1706
01:20:40,470 --> 01:20:41,790
put up a ton of equations.

1707
01:20:41,790 --> 01:20:44,430
For example, in quantum field
theory, you just get a

1708
01:20:44,430 --> 01:20:47,320
gazillion integrals with
epsilons floating all over the

1709
01:20:47,320 --> 01:20:49,520
place and then path integrals.

1710
01:20:49,520 --> 01:20:50,370
You integrate this.

1711
01:20:50,370 --> 01:20:52,540
And there are some 2 pis and
you take a bunch of limits.

1712
01:20:52,540 --> 01:20:55,000
And it seems like a whole bunch
of methods gymnastics.

1713
01:20:55,000 --> 01:20:58,140
But A, it doesn't have
to be that way.

1714
01:20:58,140 --> 01:21:00,520
And also there's another
characteristic of graduate

1715
01:21:00,520 --> 01:21:03,390
students which you don't have
so much with undergraduates,

1716
01:21:03,390 --> 01:21:06,560
which is that graduate students
know how to read.

1717
01:21:06,560 --> 01:21:08,930
Now, this may seem like a
bizarre statement, because

1718
01:21:08,930 --> 01:21:12,290
surely, everyone knows how to
read by, say, age three or

1719
01:21:12,290 --> 01:21:14,150
four or five or whenever
they teach reading in

1720
01:21:14,150 --> 01:21:15,730
school these days.

1721
01:21:15,730 --> 01:21:18,200
But what I mean is that
undergraduates--

1722
01:21:18,200 --> 01:21:21,450
generally, they have no
experience on how to read a

1723
01:21:21,450 --> 01:21:24,980
textbook because they've had
so much experience with us

1724
01:21:24,980 --> 01:21:26,980
telling them stuff everything
on the board.

1725
01:21:26,980 --> 01:21:29,630
So they have no incentive to
actually read the textbook.

1726
01:21:29,630 --> 01:21:30,730
And they don't learn
how to do that.

1727
01:21:30,730 --> 01:21:32,750
They think textbooks are
read the way you

1728
01:21:32,750 --> 01:21:34,250
read Jane Austen novels.

1729
01:21:34,250 --> 01:21:37,790
You just read for plot.

1730
01:21:37,790 --> 01:21:39,830
Something happened to some
equation, then something else

1731
01:21:39,830 --> 01:21:41,090
happened to some equation.

1732
01:21:41,090 --> 01:21:44,860
And you just carry on, paragraph
by paragraph, as you

1733
01:21:44,860 --> 01:21:46,330
would a novel.

1734
01:21:46,330 --> 01:21:49,190
That way of reading is
completely hopeless for

1735
01:21:49,190 --> 01:21:51,150
technical material.

1736
01:21:51,150 --> 01:21:52,660
But graduate students--

1737
01:21:52,660 --> 01:21:54,290
not always, but generally--
have much more

1738
01:21:54,290 --> 01:21:55,380
maturity about this.

1739
01:21:55,380 --> 01:21:59,070
So graduate students actually
can or often are closer to

1740
01:21:59,070 --> 01:22:01,290
being able to read technical
material with skill.

1741
01:22:01,290 --> 01:22:03,510
For example, graduate students
often read papers

1742
01:22:03,510 --> 01:22:04,580
in their own field.

1743
01:22:04,580 --> 01:22:07,390
And you know you have to read
a paper differently than you

1744
01:22:07,390 --> 01:22:08,880
would a Jane Austen novel.

1745
01:22:08,880 --> 01:22:12,050
So because of that, you can
actually teach equations very

1746
01:22:12,050 --> 01:22:13,080
differently.

1747
01:22:13,080 --> 01:22:17,360
What you do is you give all the
long, messy, yucky parts--

1748
01:22:17,360 --> 01:22:20,940
you leave that for the notes,
for the book, somewhere where

1749
01:22:20,940 --> 01:22:24,940
everything is printed in a
very easy to read format,

1750
01:22:24,940 --> 01:22:27,880
rather than copying long, long,
long, long strings of

1751
01:22:27,880 --> 01:22:29,240
symbols off the board.

1752
01:22:29,240 --> 01:22:32,210
So that connects back so what
we talked about last time,

1753
01:22:32,210 --> 01:22:33,830
which is chunks.

1754
01:22:33,830 --> 01:22:36,920
So if you put long equations
up on the board, generally,

1755
01:22:36,920 --> 01:22:39,150
you overflow the chunk system.

1756
01:22:39,150 --> 01:22:42,020
And once that happens, people
start making mistakes.

1757
01:22:42,020 --> 01:22:44,190
So you want to avoid doing
that as much as possible.

1758
01:22:44,190 --> 01:22:46,740
And with graduate students,
it's even easier to do,

1759
01:22:46,740 --> 01:22:50,990
because you leave all that for
a type set, professionally

1760
01:22:50,990 --> 01:22:54,620
published book or type set by
yourself, but somehow printed

1761
01:22:54,620 --> 01:22:56,880
in a clean way with
no mistakes.

1762
01:22:56,880 --> 01:23:00,350
And you can then, in
class, discuss the

1763
01:23:00,350 --> 01:23:02,120
meaning of the terms.

1764
01:23:02,120 --> 01:23:03,430
What are the terms?

1765
01:23:03,430 --> 01:23:04,390
Where do they come from?

1766
01:23:04,390 --> 01:23:07,590
Why would you expect
that kind of term?

1767
01:23:07,590 --> 01:23:09,490
So I'll give you an example
of doing that

1768
01:23:09,490 --> 01:23:10,800
with an equation today.

1769
01:23:10,800 --> 01:23:14,070
But generally speaking, all of
what I was talking about last

1770
01:23:14,070 --> 01:23:17,470
time applies perfectly well to
graduate classes, even if

1771
01:23:17,470 --> 01:23:21,090
people at first think
that it doesn't.

1772
01:23:21,090 --> 01:23:23,790
So I should say if any questions
occur to you as I'm

1773
01:23:23,790 --> 01:23:25,040
answering questions--

1774
01:23:25,040 --> 01:23:28,330
basically, questioning beginning
questioning--

1775
01:23:28,330 --> 01:23:31,730
please raise your hand and
ask them right now.

1776
01:23:31,730 --> 01:23:32,980
OK.

1777
01:23:35,410 --> 01:23:37,810
I found that the square
diagram muddied the

1778
01:23:37,810 --> 01:23:39,850
development of the
Hardy-Weinberg equation.

1779
01:23:39,850 --> 01:23:42,370
So the square diagram
was this one.

1780
01:24:01,080 --> 01:24:04,720
So the conclusion from that was
the question-- which is

1781
01:24:04,720 --> 01:24:08,300
shouldn't college students be
comfortable with expanding p

1782
01:24:08,300 --> 01:24:09,880
plus q squared?

1783
01:24:09,880 --> 01:24:11,130
Why do they need a diagram?

1784
01:24:21,190 --> 01:24:25,460
And the answer isn't that people
aren't comfortable with

1785
01:24:25,460 --> 01:24:28,580
expanding p plus q squared,
although you will find people

1786
01:24:28,580 --> 01:24:33,850
for whom this and this is just a
symbol replacement strategy.

1787
01:24:33,850 --> 01:24:36,200
In other words, it's something
like you program in a computer

1788
01:24:36,200 --> 01:24:38,680
whenever you see this
pattern do this.

1789
01:24:38,680 --> 01:24:41,410
But the terms don't actually
have meaning for people.

1790
01:24:41,410 --> 01:24:43,640
They don't know why those
terms are that way.

1791
01:24:43,640 --> 01:24:46,750
And if they'd mismemorize and
put cubes here, they would

1792
01:24:46,750 --> 01:24:48,080
write that down too.

1793
01:24:48,080 --> 01:24:51,150
So the picture actually makes
it clear what the meaning of

1794
01:24:51,150 --> 01:24:51,730
the term is.

1795
01:24:51,730 --> 01:24:54,650
So it gives, actually, meaning
to what people might be

1796
01:24:54,650 --> 01:24:56,700
otherwise comfortable
with or having done

1797
01:24:56,700 --> 01:24:59,030
from lots of practice.

1798
01:24:59,030 --> 01:25:03,500
So I should say there's a
difference between just rote

1799
01:25:03,500 --> 01:25:06,510
matching between here and
here versus this kind of

1800
01:25:06,510 --> 01:25:07,470
understanding.

1801
01:25:07,470 --> 01:25:10,690
And that's illustrated by the
following research, which is

1802
01:25:10,690 --> 01:25:14,960
about people's ability to
multiply and add when people

1803
01:25:14,960 --> 01:25:16,040
have brain damage.

1804
01:25:16,040 --> 01:25:18,460
So the technical term
is brain lesions.

1805
01:25:18,460 --> 01:25:21,850
The people with brain
lesions in the--

1806
01:25:21,850 --> 01:25:22,720
let's see if I can
say this right.

1807
01:25:22,720 --> 01:25:27,380
Some people with brain lesions
in the arithmetic areas lose

1808
01:25:27,380 --> 01:25:32,030
their ability to add, but
they can multiply fine.

1809
01:25:32,030 --> 01:25:36,940
And people with brain lesions in
the verbal areas, they lose

1810
01:25:36,940 --> 01:25:40,420
their ability to multiply,
but they can add fine.

1811
01:25:40,420 --> 01:25:44,660
Now, this seems kind
of strange, right?

1812
01:25:44,660 --> 01:25:48,600
So why would multiplication
and addition to not go

1813
01:25:48,600 --> 01:25:51,860
together when you get brain
damage in the arithmetic area?

1814
01:26:01,100 --> 01:26:03,870
So if the damage is either an
arithmetic or verbal, here's

1815
01:26:03,870 --> 01:26:05,120
what you lose.

1816
01:26:07,600 --> 01:26:15,760
So arithmetic area damage,
you lose addition.

1817
01:26:15,760 --> 01:26:18,000
Here, you lose multiplication.

1818
01:26:21,920 --> 01:26:25,160
And that is very bizarre,
because you'd think you should

1819
01:26:25,160 --> 01:26:27,440
lose multiplication here, too.

1820
01:26:27,440 --> 01:26:31,040
And the reason that it doesn't
work that way is because of

1821
01:26:31,040 --> 01:26:33,810
the way multiplication is
generally learned and taught.

1822
01:26:33,810 --> 01:26:38,430
So for example, in England
the way it's taught is--

1823
01:26:38,430 --> 01:26:41,450
this is research from Brian
Butterworth in England.

1824
01:26:41,450 --> 01:26:45,060
The way multiplication is taught
in England is in the

1825
01:26:45,060 --> 01:26:46,840
multiplication table,
same is here.

1826
01:26:46,840 --> 01:26:50,660
And people memorize it
as six 9's is 54.

1827
01:26:50,660 --> 01:26:52,590
So six 9's is 54.

1828
01:26:52,590 --> 01:26:54,970
Here, I think I learned
multiplication both places.

1829
01:26:54,970 --> 01:26:58,390
So 6 times 9 is 54.

1830
01:26:58,390 --> 01:27:02,010
Either way, that's a purely
linguistic string.

1831
01:27:02,010 --> 01:27:07,070
It's no surprise that when you
lose verbal ability, you lose

1832
01:27:07,070 --> 01:27:09,180
the ability to memorize
linguistic strings.

1833
01:27:09,180 --> 01:27:11,570
So the multiplication
table went.

1834
01:27:11,570 --> 01:27:14,050
So what that tells you is that
most people-- and I don't know

1835
01:27:14,050 --> 01:27:16,560
if this is true for everyone,
but my guess is it depends on

1836
01:27:16,560 --> 01:27:18,540
how you learned the
multiplication table.

1837
01:27:18,540 --> 01:27:21,400
For most people, they learned
the multiplication table in a

1838
01:27:21,400 --> 01:27:25,250
way that is not meaningful.

1839
01:27:25,250 --> 01:27:27,050
They've learned the
multiplication table purely

1840
01:27:27,050 --> 01:27:28,530
linguistically.

1841
01:27:28,530 --> 01:27:34,540
And that is the problem with
just going down this path

1842
01:27:34,540 --> 01:27:36,600
only, which is that--

1843
01:27:36,600 --> 01:27:38,400
again, this is where it's
important to know where the

1844
01:27:38,400 --> 01:27:39,460
students come from.

1845
01:27:39,460 --> 01:27:42,110
If you know students basically
have just linguistically

1846
01:27:42,110 --> 01:27:47,270
memorized this transformation
to that transformation, like

1847
01:27:47,270 --> 01:27:48,830
the rock bands in
other countries

1848
01:27:48,830 --> 01:27:50,120
that sing in English.

1849
01:27:50,120 --> 01:27:51,710
They know when they
see English words.

1850
01:27:51,710 --> 01:27:52,780
They know what to say.

1851
01:27:52,780 --> 01:27:55,840
But they don't necessarily
understand any of the words.

1852
01:27:55,840 --> 01:27:58,300
That is actually very common.

1853
01:27:58,300 --> 01:28:01,400
And this is another linguistic
transformation.

1854
01:28:01,400 --> 01:28:03,060
There's no meaning
underneath it.

1855
01:28:03,060 --> 01:28:05,810
So here to give it some meaning
underneath it, some

1856
01:28:05,810 --> 01:28:08,670
picture, it actually
incorporates another brain

1857
01:28:08,670 --> 01:28:13,750
area into the understanding,
which is where addition

1858
01:28:13,750 --> 01:28:15,460
actually has some meaning
to people.

1859
01:28:15,460 --> 01:28:19,130
So that brings up a related
question, which is if people

1860
01:28:19,130 --> 01:28:21,580
have learned multiplication in
this linguistic way, how could

1861
01:28:21,580 --> 01:28:24,250
you teach it in a non linguistic
way so that people

1862
01:28:24,250 --> 01:28:26,200
actually understand it?

1863
01:28:26,200 --> 01:28:28,570
There are several ways.

1864
01:28:28,570 --> 01:28:31,560
I don't know of any studies that
show that after doing it

1865
01:28:31,560 --> 01:28:35,040
this way and then brain damage,
it doesn't get lost.

1866
01:28:35,040 --> 01:28:38,190
But my speculation is that if
you learn it, actually, pretty

1867
01:28:38,190 --> 01:28:40,780
much the way I learned it, you
wouldn't actually lose it.

1868
01:28:40,780 --> 01:28:42,750
And the way is, for example,
suppose you have to

1869
01:28:42,750 --> 01:28:46,010
multiply 6 times 9.

1870
01:28:46,010 --> 01:28:49,230
Rather than memorizing that
as 54, you think about

1871
01:28:49,230 --> 01:28:50,120
what it should be.

1872
01:28:50,120 --> 01:28:51,570
And you reason your way to it.

1873
01:28:51,570 --> 01:28:52,485
And we think, oh, well,
that's slower.

1874
01:28:52,485 --> 01:28:54,190
Yeah, it's slower in
the beginning.

1875
01:28:54,190 --> 01:28:56,820
But in the end, you get
to the same place.

1876
01:28:56,820 --> 01:28:58,510
But you've put it in a different
part of the brain.

1877
01:28:58,510 --> 01:29:00,440
So the way you could do
this one, for example,

1878
01:29:00,440 --> 01:29:02,040
you say 6 times 9.

1879
01:29:02,040 --> 01:29:08,710
That's 6 times 10, which is
really easy, minus 6.

1880
01:29:08,710 --> 01:29:12,240
So 6 times 10 you know not
by memorizing but from

1881
01:29:12,240 --> 01:29:14,110
understanding the
number system.

1882
01:29:14,110 --> 01:29:16,560
So you know that that's 60
because of the way the number

1883
01:29:16,560 --> 01:29:17,180
system works.

1884
01:29:17,180 --> 01:29:18,770
You don't have to
memorize that.

1885
01:29:18,770 --> 01:29:21,420
That's the kind of example
where I think calculators

1886
01:29:21,420 --> 01:29:25,380
should be programmed to self
destruct or at least not work

1887
01:29:25,380 --> 01:29:28,680
for about a week if you type
in a problem like that.

1888
01:29:28,680 --> 01:29:32,650
They should just freeze as an
incentive to actually think

1889
01:29:32,650 --> 01:29:35,030
about these things before you
put them into the calculator.

1890
01:29:35,030 --> 01:29:38,780
So minus 6, and you get 54.

1891
01:29:38,780 --> 01:29:40,870
So now, that's a bit longer
the first time you do it.

1892
01:29:40,870 --> 01:29:42,950
It's longer than
memorizing it.

1893
01:29:42,950 --> 01:29:45,310
But after you do multiplication
a bunch of

1894
01:29:45,310 --> 01:29:50,060
times like this, you actually
reinforce the meaning of the

1895
01:29:50,060 --> 01:29:52,310
number system, and you come
up with the same answer.

1896
01:29:52,310 --> 01:29:54,140
And eventually, you will
memorize it, but you've

1897
01:29:54,140 --> 01:29:55,980
memorized it in a
different way.

1898
01:29:55,980 --> 01:29:57,230
Here's another example.

1899
01:30:03,760 --> 01:30:05,100
From the 12 times tables--

1900
01:30:05,100 --> 01:30:06,640
8 times 12.

1901
01:30:06,640 --> 01:30:09,050
Now, should you memorize
that as 96?

1902
01:30:09,050 --> 01:30:09,730
No.

1903
01:30:09,730 --> 01:30:17,730
You should write that as 10
minus 2 times 10 plus 2.

1904
01:30:17,730 --> 01:30:24,060
So that's equal to 10 squared
minus 2 squared equals 96.

1905
01:30:24,060 --> 01:30:26,310
And you can even draw a picture
for this and show what

1906
01:30:26,310 --> 01:30:27,760
happens to squares and areas.

1907
01:30:27,760 --> 01:30:30,260
So again, you've come up with
the same answer, but you've

1908
01:30:30,260 --> 01:30:33,450
done it in a meaningful way.

1909
01:30:33,450 --> 01:30:37,620
So that's one answer to why it's
worth showing pictures,

1910
01:30:37,620 --> 01:30:40,840
even if people can
do the algebra.

1911
01:30:40,840 --> 01:30:42,090
It reinforces the algebra.

1912
01:30:46,210 --> 01:30:48,670
Is determining what constitutes
a chunk simply a

1913
01:30:48,670 --> 01:30:51,000
matter of my intuition about
students' level?

1914
01:30:51,000 --> 01:30:52,820
Well, to some extent, it is.

1915
01:30:52,820 --> 01:30:55,550
But in the chess playing
research I talked about, they

1916
01:30:55,550 --> 01:30:59,180
actually had more objective
ways of determining

1917
01:30:59,180 --> 01:31:00,140
what a chunk was.

1918
01:31:00,140 --> 01:31:02,530
And what they did is they put
eye trackers on people.

1919
01:31:02,530 --> 01:31:05,615
And they had the chess masters
and the non chess masters and

1920
01:31:05,615 --> 01:31:08,390
the experts look at the board,
and they tracked their eyes to

1921
01:31:08,390 --> 01:31:09,560
see what they did.

1922
01:31:09,560 --> 01:31:13,820
So the non chess masters, their
eyes wandered all over.

1923
01:31:13,820 --> 01:31:16,930
But the chess masters looked
at pieces in groups.

1924
01:31:16,930 --> 01:31:18,940
They would go here, here, here,
here, and then here,

1925
01:31:18,940 --> 01:31:20,060
here, here, here.

1926
01:31:20,060 --> 01:31:21,920
And the assumption was that
that's a chunk and

1927
01:31:21,920 --> 01:31:22,810
then that's a chunk.

1928
01:31:22,810 --> 01:31:25,200
Or if they went here then to
there, that somehow, these two

1929
01:31:25,200 --> 01:31:26,540
chunks were related.

1930
01:31:26,540 --> 01:31:28,390
So there was ways in the
chess playing problem

1931
01:31:28,390 --> 01:31:30,140
of measuring chunks.

1932
01:31:30,140 --> 01:31:33,830
And then how do you apply that
to, say, teaching physics?

1933
01:31:33,830 --> 01:31:36,850
Well, that's a matter,
partly, of intuition.

1934
01:31:36,850 --> 01:31:39,180
And the idea is to help the
students build up the chunks.

1935
01:31:39,180 --> 01:31:41,880
So you have to be on the watch
for what chunks you use.

1936
01:31:41,880 --> 01:31:44,950
So you have to introspect.

1937
01:31:44,950 --> 01:31:46,200
OK.

1938
01:32:00,530 --> 01:32:03,810
How do you keep things
interesting and reveal

1939
01:32:03,810 --> 01:32:05,500
material in a time
appropriate way?

1940
01:32:05,500 --> 01:32:08,010
So the comment was that it
took a while to do the

1941
01:32:08,010 --> 01:32:11,760
Hardy-Weinberg with the story
and then building up to it.

1942
01:32:11,760 --> 01:32:14,940
So this is a question of how
do you plan lecture time.

1943
01:32:14,940 --> 01:32:18,090
What's worth doing in lecture?

1944
01:32:18,090 --> 01:32:20,420
Is it worth spending
a bunch of time on

1945
01:32:20,420 --> 01:32:22,290
understanding the concepts?

1946
01:32:22,290 --> 01:32:26,400
And there's two extremes
to this.

1947
01:32:26,400 --> 01:32:29,650
I'm basically towards one
extreme, which is that if you

1948
01:32:29,650 --> 01:32:32,530
don't understand the concepts,
it's not even worth learning

1949
01:32:32,530 --> 01:32:36,180
the thing, because as a
student, if you don't

1950
01:32:36,180 --> 01:32:38,610
understand the concept, you
might as well just forget the

1951
01:32:38,610 --> 01:32:40,140
material right now, because
you're going to

1952
01:32:40,140 --> 01:32:41,470
forget it soon anyway.

1953
01:32:41,470 --> 01:32:44,920
So the only benefit from taking
the course if you don't

1954
01:32:44,920 --> 01:32:46,930
understand the material, is
you just get a great on a

1955
01:32:46,930 --> 01:32:48,990
final exam and you pass
a requirement.

1956
01:32:48,990 --> 01:32:50,990
But in terms of actually
changing how you see the

1957
01:32:50,990 --> 01:32:52,940
world, it ha no value.

1958
01:32:52,940 --> 01:32:56,220
So there was a study done, I
think, at Carnegie Mellon.

1959
01:32:56,220 --> 01:32:57,250
Yeah, this was at
Carnegie Mellon.

1960
01:32:57,250 --> 01:32:59,480
They studied freshman
physics students.

1961
01:32:59,480 --> 01:33:02,100
So they had students who took
freshman physics and students

1962
01:33:02,100 --> 01:33:03,480
who didn't take freshman
physics.

1963
01:33:03,480 --> 01:33:06,710
And then a year later, they gave
them the freshman physics

1964
01:33:06,710 --> 01:33:08,070
final exam--

1965
01:33:08,070 --> 01:33:10,640
I think it was the same one,
pretty much, as the students

1966
01:33:10,640 --> 01:33:13,240
who took freshman physics took,
except with numbers

1967
01:33:13,240 --> 01:33:15,360
changed, but otherwise,
it was same problems--

1968
01:33:15,360 --> 01:33:17,870
to see whether taking freshman
physics had any effect on

1969
01:33:17,870 --> 01:33:19,510
whether you did well on
a freshman physics

1970
01:33:19,510 --> 01:33:20,690
final a year later.

1971
01:33:20,690 --> 01:33:24,610
And the conclusion was that it
had no effect a year later.

1972
01:33:24,610 --> 01:33:26,960
So yeah, sure, if you're give
them the final the next day

1973
01:33:26,960 --> 01:33:29,580
after the final exam that they
took, they would have done

1974
01:33:29,580 --> 01:33:31,550
pretty well, maybe 50%.

1975
01:33:31,550 --> 01:33:33,140
Who knows what the
time constant is.

1976
01:33:33,140 --> 01:33:36,190
But certainly, by a
year, it was gone.

1977
01:33:36,190 --> 01:33:40,010
So what that tells me is that
that regular way of teaching

1978
01:33:40,010 --> 01:33:42,220
material actually did no good
to the students except for

1979
01:33:42,220 --> 01:33:45,470
passing a requirement, but no
intrinsic good for their way

1980
01:33:45,470 --> 01:33:47,140
of analyzing the world.

1981
01:33:47,140 --> 01:33:49,970
So what that says is then
you really need to

1982
01:33:49,970 --> 01:33:51,220
find something different.

1983
01:33:51,220 --> 01:33:53,890
And yeah, if it means it takes
extra time in lecture, a bunch

1984
01:33:53,890 --> 01:33:55,300
of time in lecture, so be it.

1985
01:33:55,300 --> 01:33:58,000
At least people will understand
something and they

1986
01:33:58,000 --> 01:34:01,620
will change how they
see the world.

1987
01:34:01,620 --> 01:34:04,360
And that's the motive for today,
which is to really

1988
01:34:04,360 --> 01:34:06,210
understand misconceptions,
because if you don't

1989
01:34:06,210 --> 01:34:08,400
understand the misconceptions,
you're not going to be able to

1990
01:34:08,400 --> 01:34:12,420
teach in a way that produces
long lasting learning.

1991
01:34:12,420 --> 01:34:12,750
OK.

1992
01:34:12,750 --> 01:34:15,790
So another one was how
do I apply this to

1993
01:34:15,790 --> 01:34:17,240
something really abstract?

1994
01:34:17,240 --> 01:34:20,220
The way of approaching
equations, like the proof of--

1995
01:34:20,220 --> 01:34:24,460
oh, this is actually one of my
favorite infinite series.

1996
01:34:24,460 --> 01:34:26,920
So the question is how do you
apply it to something really

1997
01:34:26,920 --> 01:34:28,170
abstract like this?

1998
01:34:36,290 --> 01:34:38,730
So this is a famous
infinite series.

1999
01:34:38,730 --> 01:34:39,980
What's the sum of that?

2000
01:34:42,440 --> 01:34:45,442
And it turns out to be
pi squared over 6.

2001
01:34:45,442 --> 01:34:47,435
Well, even that, you think,
well, how can

2002
01:34:47,435 --> 01:34:48,290
I apply it to that?

2003
01:34:48,290 --> 01:34:54,440
And it turns out there's great
stories about that one, too.

2004
01:34:54,440 --> 01:34:58,050
I think this is a story about
this problem, which is that no

2005
01:34:58,050 --> 01:35:00,450
one knew how to do this sum
from 1 to infinity.

2006
01:35:00,450 --> 01:35:02,280
It's quite a hard sum.

2007
01:35:02,280 --> 01:35:04,590
And so it was set as a problem,
basically, for the

2008
01:35:04,590 --> 01:35:06,480
mathematicians and physicists
of Europe.

2009
01:35:06,480 --> 01:35:09,510
And then someone produced
a solution.

2010
01:35:09,510 --> 01:35:12,210
I think someone produced
a solution anonymously.

2011
01:35:12,210 --> 01:35:14,820
And everyone basically figured
out who had done it, because

2012
01:35:14,820 --> 01:35:16,630
it had their handy mark.

2013
01:35:16,630 --> 01:35:19,780
So the solution was by Euler.

2014
01:35:19,780 --> 01:35:23,420
And it involved a whole bunch
of trickery with polynomials

2015
01:35:23,420 --> 01:35:24,750
and infinite degree
polynomials.

2016
01:35:24,750 --> 01:35:27,640
And it was a really
sly method.

2017
01:35:27,640 --> 01:35:30,540
So if I was going to teach
this equation, I would

2018
01:35:30,540 --> 01:35:34,440
actually teach the history of
it-- how was really hard, how

2019
01:35:34,440 --> 01:35:36,130
you could actually guess this.

2020
01:35:36,130 --> 01:35:37,540
What are ways you could
guess this?

2021
01:35:37,540 --> 01:35:40,190
Well, you could actually
approximate the sum, get a

2022
01:35:40,190 --> 01:35:43,670
number that's one or two or
three digits accurate.

2023
01:35:43,670 --> 01:35:46,360
And then you feed it into--

2024
01:35:46,360 --> 01:35:47,610
does everyone know this guy?

2025
01:35:50,040 --> 01:35:52,040
If you Google for that, you
should probably find it.

2026
01:35:52,040 --> 01:35:54,880
It's called the inverse
symbolic calculator.

2027
01:35:54,880 --> 01:35:55,900
It's a fantastic thing.

2028
01:35:55,900 --> 01:35:57,260
I do not know how it works.

2029
01:35:57,260 --> 01:35:58,530
And I would love to
know how it works.

2030
01:35:58,530 --> 01:36:01,990
But what it is is you feed in a
number, and it will tell you

2031
01:36:01,990 --> 01:36:04,000
all the ways of producing
something

2032
01:36:04,000 --> 01:36:05,490
really close to that.

2033
01:36:05,490 --> 01:36:11,550
So for example, if you put in
3.141, it'll say a bunch of

2034
01:36:11,550 --> 01:36:12,890
numbers that get near here.

2035
01:36:12,890 --> 01:36:14,660
And one of them is pi.

2036
01:36:14,660 --> 01:36:17,840
If you put in this to one or
two or three digits, it'll

2037
01:36:17,840 --> 01:36:20,650
probably guess for you
pi squared over 6.

2038
01:36:20,650 --> 01:36:22,620
So that's one way of getting
at an answer.

2039
01:36:22,620 --> 01:36:24,450
So part of the way of teaching
it is to say, well, let's

2040
01:36:24,450 --> 01:36:26,610
somehow get an answer
with ways we can do

2041
01:36:26,610 --> 01:36:27,920
that aren't too abstract.

2042
01:36:27,920 --> 01:36:30,030
And then let's see if we can
justify that answer.

2043
01:36:30,030 --> 01:36:32,280
So even there, you're
not lost.

2044
01:36:32,280 --> 01:36:33,600
There's always stuff
you can do.

2045
01:36:38,570 --> 01:36:40,930
Oh, another question, which
was readings-- how do you

2046
01:36:40,930 --> 01:36:44,180
incorporate readings into a
course so that students do it?

2047
01:36:44,180 --> 01:36:46,460
So that was asked twice,
actually, because I didn't

2048
01:36:46,460 --> 01:36:47,980
answer it the first time.

2049
01:36:47,980 --> 01:36:51,350
So one way to do readings
is something

2050
01:36:51,350 --> 01:36:52,700
called reading memos.

2051
01:37:05,480 --> 01:37:10,640
And it's an MIT invention by
Edwin Taylor, who's recently

2052
01:37:10,640 --> 01:37:12,470
retired from the physics
department.

2053
01:37:12,470 --> 01:37:14,670
And what a reading memo is--

2054
01:37:14,670 --> 01:37:15,680
I'll put up the handouts.

2055
01:37:15,680 --> 01:37:17,800
So I have often done
this in my classes.

2056
01:37:17,800 --> 01:37:20,690
I'll put up the handout for you
to use that I give out and

2057
01:37:20,690 --> 01:37:22,860
you can just copy it or
do whatever with it.

2058
01:37:22,860 --> 01:37:27,500
So a reading memo is a request
to the students to write you a

2059
01:37:27,500 --> 01:37:29,670
short memo about something
that you've

2060
01:37:29,670 --> 01:37:30,600
asked them to read.

2061
01:37:30,600 --> 01:37:33,000
It could be the draft notes for
your textbook that you're

2062
01:37:33,000 --> 01:37:35,300
working on, which is what
I often do it with.

2063
01:37:35,300 --> 01:37:38,170
Or it could be the textbook
someone else wrote and you ask

2064
01:37:38,170 --> 01:37:39,660
the students to read chapter.

2065
01:37:39,660 --> 01:37:43,400
And what it's not is it's not a
summary of the text, because

2066
01:37:43,400 --> 01:37:44,420
you already know what
the text says.

2067
01:37:44,420 --> 01:37:45,730
There's no point in
getting a summary.

2068
01:37:45,730 --> 01:37:48,300
What it is is students'
reactions to it.

2069
01:37:48,300 --> 01:37:52,590
So anything that questions,
things that puzzle them--

2070
01:37:52,590 --> 01:37:55,910
oh, I didn't understand
why you did this or

2071
01:37:55,910 --> 01:37:56,800
the author did this.

2072
01:37:56,800 --> 01:37:59,790
And then maybe three pages
later, oh, now, I see--

2073
01:37:59,790 --> 01:38:01,940
which, if you're the author of
those notes, you know that you

2074
01:38:01,940 --> 01:38:03,510
explain the two things
out of order and you

2075
01:38:03,510 --> 01:38:05,190
should connect them.

2076
01:38:05,190 --> 01:38:09,960
But what that does is it teaches
students how to read

2077
01:38:09,960 --> 01:38:12,950
actively, because again, I like
I talked about, people

2078
01:38:12,950 --> 01:38:16,910
just do Jane Austen's approach
to reading technical material.

2079
01:38:16,910 --> 01:38:19,640
And by getting students to read
actively and formulate

2080
01:38:19,640 --> 01:38:24,380
questions, by doing that,
students learn a different way

2081
01:38:24,380 --> 01:38:26,390
of reading, a way necessary for

2082
01:38:26,390 --> 01:38:28,850
reading technical material.

2083
01:38:28,850 --> 01:38:32,410
And also by writing their
questions down and you seeing

2084
01:38:32,410 --> 01:38:35,070
the questions, you actually
get a view into how the

2085
01:38:35,070 --> 01:38:36,430
students are thinking.

2086
01:38:36,430 --> 01:38:39,690
So it's actually a way of
understanding what their

2087
01:38:39,690 --> 01:38:42,310
misconceptions are, their
conceptions of the field are,

2088
01:38:42,310 --> 01:38:43,710
and tuning your teaching.

2089
01:38:43,710 --> 01:38:45,880
Just automatically, you'll
find your teaching will

2090
01:38:45,880 --> 01:38:48,610
impedance match to where the
students are just by reading

2091
01:38:48,610 --> 01:38:49,750
the reading memos.

2092
01:38:49,750 --> 01:38:53,040
It has a further benefit, which
is that it inverts the

2093
01:38:53,040 --> 01:38:56,660
normal power relationship
between teacher and student.

2094
01:38:56,660 --> 01:39:00,310
So for example, most
assignments, problem sets,

2095
01:39:00,310 --> 01:39:02,505
there's the correct answer,
which you know.

2096
01:39:02,505 --> 01:39:05,150
And you're seeing whether they
know the correct answer.

2097
01:39:05,150 --> 01:39:07,690
So they're now writing an
answer, worried whether

2098
01:39:07,690 --> 01:39:09,160
they're correct or not.

2099
01:39:09,160 --> 01:39:10,380
And then you're judging them.

2100
01:39:10,380 --> 01:39:19,070
So normally, p set, the power
hierarchy is you and then the

2101
01:39:19,070 --> 01:39:21,910
student down here.

2102
01:39:21,910 --> 01:39:24,860
And the student is looking
up to you for validation.

2103
01:39:24,860 --> 01:39:27,930
So now, this is not a
good thing to teach.

2104
01:39:27,930 --> 01:39:31,140
And maybe it's hard to avoid
with problem sets.

2105
01:39:31,140 --> 01:39:33,420
You have to do problems sets
somehow when you teach people

2106
01:39:33,420 --> 01:39:33,990
to do problems.

2107
01:39:33,990 --> 01:39:35,940
But you want to minimize this
as much as possible, because

2108
01:39:35,940 --> 01:39:38,820
it's not a transferable way
of dealing with the world.

2109
01:39:38,820 --> 01:39:41,250
They can't use that when
they go elsewhere.

2110
01:39:41,250 --> 01:39:44,040
And it teaches bad habits of
deference to authority.

2111
01:39:44,040 --> 01:39:45,660
So that's normal.

2112
01:39:45,660 --> 01:39:46,920
How does a reading memo work?

2113
01:39:46,920 --> 01:39:48,190
Well, it's the other
way around.

2114
01:39:48,190 --> 01:39:52,570
If the student says, this is
confusing, by definition, they

2115
01:39:52,570 --> 01:39:53,150
are correct.

2116
01:39:53,150 --> 01:39:54,240
It's confusing.

2117
01:39:54,240 --> 01:39:56,680
They are the expert on what's
confusing or not.

2118
01:39:56,680 --> 01:40:01,370
So it inverts the hierarchy
to this.

2119
01:40:01,370 --> 01:40:04,240
And you become very interested
in what the

2120
01:40:04,240 --> 01:40:05,560
students are saying.

2121
01:40:05,560 --> 01:40:07,130
They are the experts now.

2122
01:40:07,130 --> 01:40:09,960
And I've had very good results
with doing reading memos.

2123
01:40:09,960 --> 01:40:12,890
And my explanation is that it's
because of this inversion

2124
01:40:12,890 --> 01:40:14,920
of power hierarchy.

2125
01:40:14,920 --> 01:40:17,190
Now, what I mean also
by good results--

2126
01:40:17,190 --> 01:40:17,820
two things.

2127
01:40:17,820 --> 01:40:20,360
One is that I get fantastic
feedback on

2128
01:40:20,360 --> 01:40:22,350
the things I'm writing.

2129
01:40:22,350 --> 01:40:24,960
The other is that I find
students actually want to do

2130
01:40:24,960 --> 01:40:27,830
reading memos after the
class finishes.

2131
01:40:27,830 --> 01:40:30,740
They say, oh, if you have more
notes, can we just do some

2132
01:40:30,740 --> 01:40:32,610
more reading memos?

2133
01:40:32,610 --> 01:40:34,060
Great.

2134
01:40:34,060 --> 01:40:35,270
Let's do that.

2135
01:40:35,270 --> 01:40:37,570
And it's because
it's actually--

2136
01:40:37,570 --> 01:40:39,770
if you write the problem sets,
they'll often say, can we do

2137
01:40:39,770 --> 01:40:41,130
more problem sets?

2138
01:40:41,130 --> 01:40:42,780
But that requires a fair
amount of work to

2139
01:40:42,780 --> 01:40:43,920
minimize the hierarchy.

2140
01:40:43,920 --> 01:40:47,115
But it's automatically here
in the correct hierarchy.

2141
01:40:47,115 --> 01:40:48,750
So students actually
enjoy doing

2142
01:40:48,750 --> 01:40:50,880
that and want to continue.

2143
01:40:50,880 --> 01:40:53,120
So that's one excellent
way of incorporating

2144
01:40:53,120 --> 01:40:55,110
reading into class.

2145
01:40:55,110 --> 01:40:58,110
So now, the problem is, what you
do when you have 50 people

2146
01:40:58,110 --> 01:41:01,190
in a class and you
get 50 memos?

2147
01:41:01,190 --> 01:41:02,490
So I've had this problem.

2148
01:41:02,490 --> 01:41:05,470
And one thing I do is I just
feel overwhelmed and I just

2149
01:41:05,470 --> 01:41:07,550
flip through them but I
don't know what to do.

2150
01:41:07,550 --> 01:41:10,280
But another is I revise
my notes based on it.

2151
01:41:10,280 --> 01:41:14,280
But the I think correct solution
is an online system.

2152
01:41:14,280 --> 01:41:16,510
So what you want is an online
system where you can post a

2153
01:41:16,510 --> 01:41:21,460
PDF file, and then people make
comments on the PDF file.

2154
01:41:21,460 --> 01:41:24,760
So everyone gets to see an image
of the page, and they

2155
01:41:24,760 --> 01:41:27,910
can just click and
make a comment.

2156
01:41:27,910 --> 01:41:30,640
And everyone gets to see their
own comments, and then when

2157
01:41:30,640 --> 01:41:32,870
they submit them, they get to
see everyone else's comments.

2158
01:41:32,870 --> 01:41:35,990
So I actually wrote half
of that system.

2159
01:41:35,990 --> 01:41:39,640
And there's a graduate student
in EECS who I think has now

2160
01:41:39,640 --> 01:41:42,780
written a whole system
independently of me.

2161
01:41:42,780 --> 01:41:45,460
So I'm going to try it out and
see how it works and try it in

2162
01:41:45,460 --> 01:41:47,240
some of my classes
this semester.

2163
01:41:47,240 --> 01:41:50,040
So the benefit of that is that
you can then see all the

2164
01:41:50,040 --> 01:41:53,610
comments at once, rather than
flipping through 50 sets of

2165
01:41:53,610 --> 01:41:57,580
reading memos with page
numbers on them.

2166
01:41:57,580 --> 01:41:58,830
OK.

2167
01:42:04,990 --> 01:42:07,390
How do I come up with
intuition examples?

2168
01:42:07,390 --> 01:42:10,030
How do I know if what builds
intuition for me will also

2169
01:42:10,030 --> 01:42:12,420
build intuition for
the students?

2170
01:42:12,420 --> 01:42:14,150
It's a very good question.

2171
01:42:14,150 --> 01:42:16,980
Is it just my personal opinion
or is it just the teacher's

2172
01:42:16,980 --> 01:42:18,080
personal opinion?

2173
01:42:18,080 --> 01:42:21,240
One of the whole themes about
this class is yes, teaching

2174
01:42:21,240 --> 01:42:24,020
does have a fair amount of art
and there is a fair amount of

2175
01:42:24,020 --> 01:42:25,270
personal opinion in it.

2176
01:42:25,270 --> 01:42:28,370
But there's also a fair amount
of science and things you can

2177
01:42:28,370 --> 01:42:30,730
do to make it more objective.

2178
01:42:30,730 --> 01:42:34,790
And one of them is actually
to do reading memos.

2179
01:42:34,790 --> 01:42:38,550
Any way you can to learn how
students think will make it so

2180
01:42:38,550 --> 01:42:41,330
that your intuition about the
students actually matches how

2181
01:42:41,330 --> 01:42:42,720
the students really think.

2182
01:42:42,720 --> 01:42:45,230
That's the whole purpose of
today about talking about

2183
01:42:45,230 --> 01:42:46,440
misconceptions.

2184
01:42:46,440 --> 01:42:48,620
Reading memos are way
of understanding

2185
01:42:48,620 --> 01:42:51,370
what students think.

2186
01:42:51,370 --> 01:42:53,210
Oh, there they are--
the reading memos.

2187
01:42:53,210 --> 01:42:55,260
So once you understand what
students think, it's much

2188
01:42:55,260 --> 01:42:58,470
easier to realize, just
intuitively choose things that

2189
01:42:58,470 --> 01:42:59,650
you know are going
to work for them.

2190
01:42:59,650 --> 01:43:02,750
The other way is the
feedback sheet.

2191
01:43:02,750 --> 01:43:04,870
So every time the students
tell you, oh, this really

2192
01:43:04,870 --> 01:43:08,290
helps me or this really didn't
help me at all, you now have

2193
01:43:08,290 --> 01:43:11,350
one more piece of feedback about
what works for them and

2194
01:43:11,350 --> 01:43:13,130
what doesn't work for them.

2195
01:43:13,130 --> 01:43:16,690
So then, you can actually choose
intuition examples.

2196
01:43:16,690 --> 01:43:18,530
How do you invent them
from scratch?

2197
01:43:18,530 --> 01:43:20,650
Well, there are some
general principles.

2198
01:43:20,650 --> 01:43:23,570
One is use pictures
whenever you can.

2199
01:43:23,570 --> 01:43:25,760
Generally, that speaks to
people's intuition, just

2200
01:43:25,760 --> 01:43:28,670
because people have much more
hardware for pictures than

2201
01:43:28,670 --> 01:43:30,620
they have for equations.

2202
01:43:30,620 --> 01:43:35,570
So I try to put myself in the
position of the student, and I

2203
01:43:35,570 --> 01:43:38,020
say, well, for example, here.

2204
01:43:40,520 --> 01:43:44,130
I say, yeah, all these equations
may well be true,

2205
01:43:44,130 --> 01:43:47,810
but I want a way that makes
me see it instantly.

2206
01:43:47,810 --> 01:43:50,530
And that just forces me to start
looking for pictures.

2207
01:43:50,530 --> 01:43:54,590
And that tunes me, actually,
towards what students need.

2208
01:43:54,590 --> 01:43:55,840
So you can do the same.

2209
01:44:00,660 --> 01:44:04,710
I think that was most
of the questions.

2210
01:44:04,710 --> 01:44:08,550
The other questions were
similar to that.

2211
01:44:08,550 --> 01:44:12,130
And I'll answer any that were
new that I haven't answered--

2212
01:44:12,130 --> 01:44:14,290
but I think most of them
I have answered--

2213
01:44:14,290 --> 01:44:16,650
at the beginning of
the next lecture.

2214
01:44:16,650 --> 01:44:19,015
So any questions that were
generated by the questions?

2215
01:44:24,350 --> 01:44:24,805
Yes, question.

2216
01:44:24,805 --> 01:44:29,064
AUDIENCE: So your story about
the physics retension issue

2217
01:44:29,064 --> 01:44:32,550
made me question the idea of the
survey courses or topics

2218
01:44:32,550 --> 01:44:33,843
in class, where it's like,
I don't really

2219
01:44:33,843 --> 01:44:35,040
need to learn this.

2220
01:44:35,040 --> 01:44:37,530
I just want you to-- been
exposed to it so that they

2221
01:44:37,530 --> 01:44:39,321
remember enough to know
to go back to it

2222
01:44:39,321 --> 01:44:40,518
if you need it someday.

2223
01:44:40,518 --> 01:44:41,768
That research, that that has
any prayer of working.

2224
01:44:43,780 --> 01:44:44,030
PROFESSOR: Right.

2225
01:44:44,030 --> 01:44:47,490
So the comment was what I'd
said about even a year of

2226
01:44:47,490 --> 01:44:49,720
intense freshman physics--

2227
01:44:49,720 --> 01:44:50,590
maybe it was a semester.

2228
01:44:50,590 --> 01:44:51,720
I forget if it was
just mechanics.

2229
01:44:51,720 --> 01:44:54,100
But it was either one semester
or a whole year freshman

2230
01:44:54,100 --> 01:44:58,420
physics did no good towards long
term understanding and

2231
01:44:58,420 --> 01:45:00,390
change of understanding
of freshman physics.

2232
01:45:00,390 --> 01:45:03,410
Well, what does that say about
these big survey classes,

2233
01:45:03,410 --> 01:45:06,190
where you're not expected
to understand?

2234
01:45:06,190 --> 01:45:07,850
So in freshman physics,
at least they have the

2235
01:45:07,850 --> 01:45:09,520
expectation that you're
supposed to understand

2236
01:45:09,520 --> 01:45:10,240
everything.

2237
01:45:10,240 --> 01:45:12,710
Now, what about the courses
where they start with the

2238
01:45:12,710 --> 01:45:13,960
expectation that you're
not going to

2239
01:45:13,960 --> 01:45:15,430
understand most of it?

2240
01:45:15,430 --> 01:45:17,160
That one is going to be
totally hopeless.

2241
01:45:17,160 --> 01:45:19,370
Then I think that is
basically true.

2242
01:45:19,370 --> 01:45:24,180
And maybe you could justify
back in the day--

2243
01:45:24,180 --> 01:45:29,760
let's say, 400 years ago, even
when books were around but

2244
01:45:29,760 --> 01:45:31,470
there was no web.

2245
01:45:31,470 --> 01:45:33,600
People need to know what
books are out there.

2246
01:45:33,600 --> 01:45:35,700
So this is a traditional thing
in Cambridge, for example.

2247
01:45:35,700 --> 01:45:37,640
They just give people big
huge reading lists.

2248
01:45:37,640 --> 01:45:39,130
So then you go back.

2249
01:45:39,130 --> 01:45:42,180
Later, you're like OK, these are
the key books in the area.

2250
01:45:42,180 --> 01:45:45,190
So you know one place to
go for a reading list.

2251
01:45:45,190 --> 01:45:48,700
That's kind of obsolete
now with the web.

2252
01:45:48,700 --> 01:45:50,380
If you want to find
out something--

2253
01:45:50,380 --> 01:45:52,320
for example, suppose there
was an equation

2254
01:45:52,320 --> 01:45:53,450
I didn't know about.

2255
01:45:53,450 --> 01:45:58,810
Let's say the Black-Scholes
equation.

2256
01:45:58,810 --> 01:46:01,670
Would I say I wonder if I
took any classes about

2257
01:46:01,670 --> 01:46:04,380
Black-Sholes--

2258
01:46:04,380 --> 01:46:06,550
maybe, let me go flip through
all my course notes?

2259
01:46:06,550 --> 01:46:07,380
No.

2260
01:46:07,380 --> 01:46:10,750
You just type it into some web
search and see what shows up.

2261
01:46:10,750 --> 01:46:15,640
And that is much more likely to
be more relevant than some

2262
01:46:15,640 --> 01:46:17,800
notes that you might have
had or not had.

2263
01:46:17,800 --> 01:46:19,520
So yeah, I think the
survey courses

2264
01:46:19,520 --> 01:46:21,500
are completely pointless.

2265
01:46:21,500 --> 01:46:23,290
Now, that doesn't mean
that introduction

2266
01:46:23,290 --> 01:46:24,700
to a field is pointless.

2267
01:46:24,700 --> 01:46:27,090
But it means that the way to do
the introduction has to be

2268
01:46:27,090 --> 01:46:27,970
very different.

2269
01:46:27,970 --> 01:46:31,690
You can't just scatter a bunch
of topics at people.

2270
01:46:31,690 --> 01:46:33,670
What you have to do is figure
out-- and we're going to talk

2271
01:46:33,670 --> 01:46:35,190
about this when we talk
about course design.

2272
01:46:35,190 --> 01:46:39,130
You have to figure out what are
the core reasoning ideas

2273
01:46:39,130 --> 01:46:41,940
special that that field has
to offer to the world?

2274
01:46:41,940 --> 01:46:43,210
For example, history--

2275
01:46:43,210 --> 01:46:44,570
what's special about history?

2276
01:46:44,570 --> 01:46:48,630
Well, historians have a sense
of how to evaluate the

2277
01:46:48,630 --> 01:46:51,120
validity and reliability
of evidence and

2278
01:46:51,120 --> 01:46:52,540
contradictory evidence.

2279
01:46:52,540 --> 01:46:54,800
That's something you don't
get in many other fields.

2280
01:46:54,800 --> 01:46:58,820
For example, it's in between
a science and a

2281
01:46:58,820 --> 01:47:00,830
pure literature field.

2282
01:47:00,830 --> 01:47:04,180
In just straight literature,
reading novels, there's

2283
01:47:04,180 --> 01:47:06,030
historical evidence and things,
but generally, you're

2284
01:47:06,030 --> 01:47:07,750
reading in a different way.

2285
01:47:07,750 --> 01:47:11,800
In math, it's hard to know where
contradictory evidence

2286
01:47:11,800 --> 01:47:13,970
comes in, although there are
ways of teaching math which I

2287
01:47:13,970 --> 01:47:15,060
like which do that.

2288
01:47:15,060 --> 01:47:18,730
But generally, history has
something new to offer, which

2289
01:47:18,730 --> 01:47:21,150
is it's a messy world.

2290
01:47:21,150 --> 01:47:22,430
You have noisy evidence.

2291
01:47:22,430 --> 01:47:23,270
What do you do?

2292
01:47:23,270 --> 01:47:26,380
Well, that's something that an
intro survey course could

2293
01:47:26,380 --> 01:47:27,710
actually teach.

2294
01:47:27,710 --> 01:47:29,600
And that, somebody could
transfer, even if they don't

2295
01:47:29,600 --> 01:47:33,240
remember when did the Magyars
invade Europe and all the

2296
01:47:33,240 --> 01:47:35,400
random stuff that would
be in a survey course.

2297
01:47:35,400 --> 01:47:39,510
So those would be grist for the
mill hung off big ideas.

2298
01:47:39,510 --> 01:47:41,220
So I'm going to talk about
that when we talk

2299
01:47:41,220 --> 01:47:41,900
about course design.

2300
01:47:41,900 --> 01:47:42,850
But yeah, you're right.

2301
01:47:42,850 --> 01:47:45,370
That course design is completely
hopeless the

2302
01:47:45,370 --> 01:47:48,340
general big survey.

2303
01:47:48,340 --> 01:47:49,840
Other questions?

2304
01:47:49,840 --> 01:47:50,820
Yes.

2305
01:47:50,820 --> 01:47:51,170
Yes.

2306
01:47:51,170 --> 01:47:52,260
Could you tell me your name?

2307
01:47:52,260 --> 01:47:52,550
AUDIENCE: Meg.

2308
01:47:52,550 --> 01:47:52,850
PROFESSOR: Meg.

2309
01:47:52,850 --> 01:47:53,570
And what was your name?

2310
01:47:53,570 --> 01:47:53,990
AUDIENCE: Amy.

2311
01:47:53,990 --> 01:47:54,250
PROFESSOR: Amy.

2312
01:47:54,250 --> 01:47:55,140
Thank you.

2313
01:47:55,140 --> 01:47:57,515
AUDIENCE: I was [INAUDIBLE].

2314
01:47:57,515 --> 01:48:00,840
I was just thinking about how
for me, even if I understand

2315
01:48:00,840 --> 01:48:02,619
something really well at the
time, and I know that I'll use

2316
01:48:02,619 --> 01:48:04,907
it again, it takes me--

2317
01:48:04,907 --> 01:48:07,039
having a test in front of me
will not actually reflect

2318
01:48:07,039 --> 01:48:09,173
whether or not I'm going to
understand it, given the short

2319
01:48:09,173 --> 01:48:11,610
period of time to
remind myself.

2320
01:48:11,610 --> 01:48:15,346
And so I'm wondering, if testing
people out of the blue

2321
01:48:15,346 --> 01:48:17,392
a year later is actually
capturing whether the people

2322
01:48:17,392 --> 01:48:21,680
who take this and perform
faster [INAUDIBLE].

2323
01:48:21,680 --> 01:48:23,100
PROFESSOR: Yeah.

2324
01:48:23,100 --> 01:48:23,930
That's a good question.

2325
01:48:23,930 --> 01:48:27,520
So maybe it was a slightly
unfair test, because they were

2326
01:48:27,520 --> 01:48:29,540
just tested out of the blue.

2327
01:48:29,540 --> 01:48:31,539
AUDIENCE: If they had been using
it all along-- if, in

2328
01:48:31,539 --> 01:48:33,990
that period of time, they had
taken courses that built on

2329
01:48:33,990 --> 01:48:35,020
that material--

2330
01:48:35,020 --> 01:48:36,717
then, they would be reinforcing
it all the

2331
01:48:36,717 --> 01:48:37,580
time if they can.

2332
01:48:37,580 --> 01:48:37,860
PROFESSOR: Right.

2333
01:48:37,860 --> 01:48:40,130
So if they had taken courses
that used freshman physics

2334
01:48:40,130 --> 01:48:42,000
throughout, maybe they would
have remembered the freshman

2335
01:48:42,000 --> 01:48:42,560
physics better.

2336
01:48:42,560 --> 01:48:45,010
And I'm sure that that's true.

2337
01:48:45,010 --> 01:48:49,600
So I can tell you one story
from my graduate time.

2338
01:48:49,600 --> 01:48:53,140
So I did a Ph.D. In physics.

2339
01:48:53,140 --> 01:48:55,050
And I had to do the
qualifying exam.

2340
01:48:55,050 --> 01:48:57,330
And to do the qualifying exam,
you have to study a whole

2341
01:48:57,330 --> 01:48:59,550
bunch of undergraduate physics
and then take the exams on it.

2342
01:49:04,260 --> 01:49:06,700
And then you take a bunch of
courses in various fields.

2343
01:49:06,700 --> 01:49:10,000
Now, the only thing, basically,
I remember from all

2344
01:49:10,000 --> 01:49:14,670
the electromagnetism is one
thing, which is I understand

2345
01:49:14,670 --> 01:49:16,810
pretty well the index
of refraction.

2346
01:49:16,810 --> 01:49:17,880
And why is that?

2347
01:49:17,880 --> 01:49:20,300
That's because I was
really pissed off--

2348
01:49:20,300 --> 01:49:22,510
sorry for the camera.

2349
01:49:22,510 --> 01:49:26,570
I was really annoyed about
the following thing.

2350
01:49:26,570 --> 01:49:29,070
So this actually goes back to
what I was talking about about

2351
01:49:29,070 --> 01:49:32,190
contradictions, which is that
you're always told in

2352
01:49:32,190 --> 01:49:41,030
relativity that speed of light
equals C. And that's the great

2353
01:49:41,030 --> 01:49:42,280
postulate of relativity--

2354
01:49:42,280 --> 01:49:45,380
that the speed of light does
not change, dammit.

2355
01:49:45,380 --> 01:49:46,650
And that's what Einstein said.

2356
01:49:46,650 --> 01:49:50,000
And there's all these thought
experiments with trains and

2357
01:49:50,000 --> 01:49:52,550
lightning bolts and people
throwing rocks from the train

2358
01:49:52,550 --> 01:49:54,140
at different speeds.

2359
01:49:54,140 --> 01:49:56,410
And it's C. It's C. It's C.

2360
01:49:56,410 --> 01:49:59,820
And then, somewhere later in
an electromagnetism course,

2361
01:49:59,820 --> 01:50:03,510
they say, the speed of light in
a medium with an index of

2362
01:50:03,510 --> 01:50:09,550
refraction N is C over N, where
N is typically around 1,

2363
01:50:09,550 --> 01:50:13,780
maybe a little bigger
like 1.001 for air

2364
01:50:13,780 --> 01:50:16,570
maybe, 1.33 for class.

2365
01:50:16,570 --> 01:50:22,370
And how do those fit together?

2366
01:50:22,370 --> 01:50:24,320
So that really annoyed me.

2367
01:50:24,320 --> 01:50:27,570
And I wanted to get to the root
of it and say, well, how

2368
01:50:27,570 --> 01:50:29,920
could it be that you could have
a speed of light that's

2369
01:50:29,920 --> 01:50:33,280
always C, yet it looks like
the speed is some

2370
01:50:33,280 --> 01:50:34,880
lower number, V?

2371
01:50:34,880 --> 01:50:38,050
So I worked out a whole bunch
of stuff about how electrons

2372
01:50:38,050 --> 01:50:40,620
scatter radiation and all the
scattered radiation adds up

2373
01:50:40,620 --> 01:50:43,085
and makes it seem like it's
slowing down the light.

2374
01:50:43,085 --> 01:50:46,300
And because of that, I actually
understand this.

2375
01:50:46,300 --> 01:50:48,970
And also because of that, which
may be not so good,

2376
01:50:48,970 --> 01:50:50,710
every time there's an
electromagnetism problem I

2377
01:50:50,710 --> 01:50:53,830
have to do, I always try to fit
into a scattering problem.

2378
01:50:53,830 --> 01:50:56,260
And if I can't do that, then
I just can't do it at all.

2379
01:50:56,260 --> 01:51:00,130
And that's despite taking two
years of electromagnetism--

2380
01:51:00,130 --> 01:51:03,100
two years as an undergrad and
one year as reviewing as a

2381
01:51:03,100 --> 01:51:03,820
grad student.

2382
01:51:03,820 --> 01:51:09,370
So the point I'm trying to
make by that is that most

2383
01:51:09,370 --> 01:51:10,500
stuff disappears.

2384
01:51:10,500 --> 01:51:13,240
And the way to really make stuff
stay is you really have

2385
01:51:13,240 --> 01:51:14,490
to struggle with something.

2386
01:51:16,850 --> 01:51:18,910
And that's the most efficient
way to make something stay.

2387
01:51:18,910 --> 01:51:21,410
And that's not what happens in
your traditional class, and

2388
01:51:21,410 --> 01:51:22,795
even less in a survey class.

2389
01:51:22,795 --> 01:51:25,320
AUDIENCE: I'm just
also asking--

2390
01:51:25,320 --> 01:51:28,004
it's almost impossible sometimes
when you see

2391
01:51:28,004 --> 01:51:30,932
material for the first time--
you might have to see it three

2392
01:51:30,932 --> 01:51:34,348
or four times before anything
comes, you're really going to

2393
01:51:34,348 --> 01:51:34,592
understand it.

2394
01:51:34,592 --> 01:51:37,727
And if the class is set up so
that you will only see it one

2395
01:51:37,727 --> 01:51:39,472
time because that's all the time
you have, then you have

2396
01:51:39,472 --> 01:51:40,950
to take more classes on it.

2397
01:51:40,950 --> 01:51:41,310
PROFESSOR: Right.

2398
01:51:41,310 --> 01:51:43,420
So maybe you need to see
something a few times to

2399
01:51:43,420 --> 01:51:44,670
really understand it.

2400
01:51:47,850 --> 01:51:49,785
So if a class doesn't give you
that chance and then you have

2401
01:51:49,785 --> 01:51:52,170
to take Thermo Two and
then Thermo Three.

2402
01:51:52,170 --> 01:51:53,074
AUDIENCE: Hopefully the class
gives you a chance.

2403
01:51:53,074 --> 01:51:55,340
Say that it's building
on this.

2404
01:51:55,340 --> 01:51:55,680
PROFESSOR: Right.

2405
01:51:55,680 --> 01:51:57,440
So you should try to build
that into the class.

2406
01:51:57,440 --> 01:52:00,120
So there's a name for that,
which is called the spiral

2407
01:52:00,120 --> 01:52:01,390
curriculum.

2408
01:52:01,390 --> 01:52:02,980
And there's a lot of
sense to that.

2409
01:52:06,740 --> 01:52:08,370
So I'll just put a quick--

2410
01:52:08,370 --> 01:52:11,530
so you show the idea
in its crude form.

2411
01:52:11,530 --> 01:52:14,930
And then you spiral back to it
in a more sophisticated way.

2412
01:52:14,930 --> 01:52:19,280
But you'd like to do that soon,
before the connection is

2413
01:52:19,280 --> 01:52:22,980
gone, before this is actually
wafted away.

2414
01:52:22,980 --> 01:52:24,870
You want to spiral back to it.

2415
01:52:24,870 --> 01:52:26,340
And so you'd like your
class to do that.

2416
01:52:26,340 --> 01:52:28,810
So what that shows is that you
should start with the big

2417
01:52:28,810 --> 01:52:31,770
ideas and then you should refine
them, because if you

2418
01:52:31,770 --> 01:52:33,750
start with all the little
details here, you'll just

2419
01:52:33,750 --> 01:52:35,140
flood the chunking system.

2420
01:52:35,140 --> 01:52:36,920
And actually, there will be no
memory of it here and then you

2421
01:52:36,920 --> 01:52:39,382
have to start over from
scratch here.

2422
01:52:39,382 --> 01:52:41,320
You had a question.

2423
01:52:41,320 --> 01:52:42,520
Could you tell me your name?

2424
01:52:42,520 --> 01:52:43,220
AUDIENCE: Roderigo.

2425
01:52:43,220 --> 01:52:45,048
PROFESSOR: Rodrigo, yeah.

2426
01:52:45,048 --> 01:52:48,450
AUDIENCE: Regarding the reading
memos, do you think

2427
01:52:48,450 --> 01:52:53,310
good to also ask the students
questions about the readings?

2428
01:52:53,310 --> 01:52:56,874
The reason why I am asking is
because I know of a class

2429
01:52:56,874 --> 01:52:59,170
that's actually implemented that
PDF annotation system.

2430
01:52:59,170 --> 01:53:03,972
And a lot of the comments
that they get are too

2431
01:53:03,972 --> 01:53:05,908
trivial, so to speak.

2432
01:53:05,908 --> 01:53:09,296
They don't ask those
type of questions.

2433
01:53:09,296 --> 01:53:10,546
[INAUDIBLE].

2434
01:53:15,080 --> 01:53:18,530
PROFESSOR: So the comment is
that a class that actually

2435
01:53:18,530 --> 01:53:21,070
implemented the online
annotation system, they found

2436
01:53:21,070 --> 01:53:24,190
that the comments are too micro
level and not broad

2437
01:53:24,190 --> 01:53:28,500
enough about what's really
confusing or interesting.

2438
01:53:28,500 --> 01:53:32,740
And so I haven't tried the
online system myself yet.

2439
01:53:32,740 --> 01:53:36,210
What I do know is that on the
paper, if you do it on paper,

2440
01:53:36,210 --> 01:53:39,310
you get really insightful,
detailed comments.

2441
01:53:39,310 --> 01:53:41,600
Now, I don't know what variables
are different.

2442
01:53:41,600 --> 01:53:45,760
One might be that the online
system just encourages--

2443
01:53:45,760 --> 01:53:48,190
because everyone does things
quick online-- it encourages

2444
01:53:48,190 --> 01:53:49,620
quick clicking.

2445
01:53:49,620 --> 01:53:51,900
So it may be that it encourages
less depth of

2446
01:53:51,900 --> 01:53:55,050
thinking, whereas writing it
on paper, I actually find I

2447
01:53:55,050 --> 01:53:57,710
get a mix of a whole bunch of
not trivial comments, but

2448
01:53:57,710 --> 01:54:00,950
small comments like, some typos,
that equation isn't

2449
01:54:00,950 --> 01:54:01,640
quite right.

2450
01:54:01,640 --> 01:54:05,765
But then I get things like, I
don't see the picture here or

2451
01:54:05,765 --> 01:54:07,490
I don't understand why
you did this now.

2452
01:54:07,490 --> 01:54:10,960
And that, to my mind, is it
useful comment-- or a question

2453
01:54:10,960 --> 01:54:13,320
like that can't be right because
of the following

2454
01:54:13,320 --> 01:54:14,570
counter argument.

2455
01:54:14,570 --> 01:54:16,920
So on paper, I get a lot
of interesting things.

2456
01:54:16,920 --> 01:54:19,550
So it may be that online
isn't as good.

2457
01:54:19,550 --> 01:54:22,230
And that's one reason I want
to try it and see.

2458
01:54:22,230 --> 01:54:25,720
So it may be that they need to
go back to paper despite it

2459
01:54:25,720 --> 01:54:27,420
being harder.

2460
01:54:27,420 --> 01:54:30,140
Another is also what
is the material?

2461
01:54:30,140 --> 01:54:32,470
If the material is really
boring, you're going to get

2462
01:54:32,470 --> 01:54:34,130
really micro comments.

2463
01:54:34,130 --> 01:54:36,700
So it helps to actually have
written interesting material

2464
01:54:36,700 --> 01:54:39,570
or give people interesting stuff
to read that people are

2465
01:54:39,570 --> 01:54:40,925
likely to think about
and make comments.

2466
01:54:43,720 --> 01:54:44,846
Question.

2467
01:54:44,846 --> 01:54:45,840
Can you tell me your name?

2468
01:54:45,840 --> 01:54:46,670
AUDIENCE: Brian.

2469
01:54:46,670 --> 01:54:47,000
PROFESSOR: Brian.

2470
01:54:47,000 --> 01:54:48,053
Yes.

2471
01:54:48,053 --> 01:54:51,524
AUDIENCE: I notice when you talk
about equations, you like

2472
01:54:51,524 --> 01:54:55,030
to give them the name that
they're most commonly

2473
01:54:55,030 --> 01:54:56,280
[INAUDIBLE].

2474
01:55:00,028 --> 01:55:02,992
Do you find that students get
more understanding of concepts

2475
01:55:02,992 --> 01:55:09,840
like this when they're given a
name based on discovery versus

2476
01:55:09,840 --> 01:55:13,139
terminology based on use
of the equation?

2477
01:55:13,139 --> 01:55:16,489
I think, from my mind, if
you want to describe the

2478
01:55:16,489 --> 01:55:18,808
relationship between stress and
strain in material, do you

2479
01:55:18,808 --> 01:55:20,041
want to call it Hooke's law?

2480
01:55:20,041 --> 01:55:23,985
Or do you just want to call it
the constitutive equation for

2481
01:55:23,985 --> 01:55:26,450
solid material?

2482
01:55:26,450 --> 01:55:27,010
PROFESSOR: Right.

2483
01:55:27,010 --> 01:55:27,330
OK.

2484
01:55:27,330 --> 01:55:31,960
So the question is what about
naming equations?

2485
01:55:31,960 --> 01:55:35,860
Should you name them by who made
them or by how they're

2486
01:55:35,860 --> 01:55:37,670
used or what they are?

2487
01:55:37,670 --> 01:55:40,150
So another example of that is,
for example, the fluid

2488
01:55:40,150 --> 01:55:40,910
mechanics equation.

2489
01:55:40,910 --> 01:55:42,610
Should you call them
Navier-Stokes or should you

2490
01:55:42,610 --> 01:55:45,100
call them fundamental
fluid equations?

2491
01:55:45,100 --> 01:55:46,620
And there's a tension there.

2492
01:55:46,620 --> 01:55:50,040
First of all, I'll say that you
do want to give a name.

2493
01:55:50,040 --> 01:55:52,570
The big win is giving the thing
a name, because that

2494
01:55:52,570 --> 01:55:55,760
makes a unit of thought
for the students.

2495
01:55:55,760 --> 01:56:00,140
So that's the first order bit,
the first order term.

2496
01:56:00,140 --> 01:56:03,240
The second order term is
how should you name it?

2497
01:56:03,240 --> 01:56:06,600
And there, there's something
which is that you want a name

2498
01:56:06,600 --> 01:56:07,260
that's common.

2499
01:56:07,260 --> 01:56:09,390
So if, for example, people look
it up elsewhere, they're

2500
01:56:09,390 --> 01:56:11,720
likely to find more
stuff about it.

2501
01:56:11,720 --> 01:56:13,620
On the other hand, you want
a name that's intuitively

2502
01:56:13,620 --> 01:56:14,130
meaningful.

2503
01:56:14,130 --> 01:56:15,260
So there's a tension.

2504
01:56:15,260 --> 01:56:17,670
There's not often a right
answer to that.

2505
01:56:17,670 --> 01:56:18,690
And you can go either way.

2506
01:56:18,690 --> 01:56:21,580
So for example, I wouldn't
probably use constitutive

2507
01:56:21,580 --> 01:56:24,490
equation for the solid, because
I have to think, what

2508
01:56:24,490 --> 01:56:26,000
the hell does constitutive
mean?

2509
01:56:26,000 --> 01:56:28,940
So myself, it doesn't
mean anything to me.

2510
01:56:28,940 --> 01:56:29,600
There's another word.

2511
01:56:29,600 --> 01:56:30,930
I'm trying to remember
what it is.

2512
01:56:30,930 --> 01:56:32,180
Epistemology--

2513
01:56:34,010 --> 01:56:36,790
people just use it like it's
just a plain, obvious word.

2514
01:56:36,790 --> 01:56:38,530
But every time I hear it,
I have think, what the

2515
01:56:38,530 --> 01:56:39,460
hell does that mean?

2516
01:56:39,460 --> 01:56:41,380
And then I put the translation
into the

2517
01:56:41,380 --> 01:56:42,410
sentence they're saying.

2518
01:56:42,410 --> 01:56:44,290
And then I can sort of parse
what they're saying.

2519
01:56:44,290 --> 01:56:48,130
So constitutive equation, to my
mind, that's a word that is

2520
01:56:48,130 --> 01:56:50,780
meaningful to the experts and
not so much to the students.

2521
01:56:50,780 --> 01:56:54,490
So I would maybe call it the
ideal spring equation, because

2522
01:56:54,490 --> 01:56:55,940
it is the ideal spring
equation.

2523
01:56:55,940 --> 01:56:59,030
It's just the proportionality
is slightly general because

2524
01:56:59,030 --> 01:56:59,880
you have tensors.

2525
01:56:59,880 --> 01:57:02,440
But otherwise, it is the
ideal spring equation.

2526
01:57:02,440 --> 01:57:04,400
So it connects to something.

2527
01:57:04,400 --> 01:57:05,860
And you can say, OK,
who discovered?

2528
01:57:05,860 --> 01:57:06,210
Hooke.

2529
01:57:06,210 --> 01:57:08,030
So we often called
it Hooke's Law.

2530
01:57:08,030 --> 01:57:09,940
So then they have both And
there's no harm in doing that.

2531
01:57:12,550 --> 01:57:13,715
Question.

2532
01:57:13,715 --> 01:57:16,140
AUDIENCE: Because you mentioned
a little bit before

2533
01:57:16,140 --> 01:57:17,390
[INAUDIBLE].

2534
01:57:20,990 --> 01:57:24,870
If you're trying to teach
something many times, then

2535
01:57:24,870 --> 01:57:28,770
maybe it's better to tell
the end, and then--

2536
01:57:28,770 --> 01:57:29,510
PROFESSOR: Yeah, the spiral.

2537
01:57:29,510 --> 01:57:30,002
AUDIENCE: Yeah.

2538
01:57:30,002 --> 01:57:34,922
And I was just thinking because
of the whole work--

2539
01:57:34,922 --> 01:57:40,630
so there are these really topics
like the ideal gas law.

2540
01:57:40,630 --> 01:57:51,136
And is it OK if I were to lie
or say to students just for

2541
01:57:51,136 --> 01:57:55,772
the sake of simplicity, for
example, do you want to say

2542
01:57:55,772 --> 01:57:59,024
whether you can actually use
that equation in some cases or

2543
01:57:59,024 --> 01:58:00,274
is it not logical always,
and then you go--

2544
01:58:03,580 --> 01:58:04,716
PROFESSOR: Could you
tell me your name?

2545
01:58:04,716 --> 01:58:05,112
AUDIENCE: Cecilia.

2546
01:58:05,112 --> 01:58:05,760
PROFESSOR: Cecilia.

2547
01:58:05,760 --> 01:58:07,250
Yeah, thank you.

2548
01:58:07,250 --> 01:58:08,820
That's an excellent
question as well.

2549
01:58:08,820 --> 01:58:12,270
So the question is, how much
should you lie, if at all?

2550
01:58:12,270 --> 01:58:15,240
For example, if I'm recommending
teaching the big

2551
01:58:15,240 --> 01:58:17,660
ideas in the overall approach
first, that's almost

2552
01:58:17,660 --> 01:58:22,040
necessarily going to involve
some amount of lying, because

2553
01:58:22,040 --> 01:58:26,740
the truth is complex and
messy, the full truth.

2554
01:58:26,740 --> 01:58:28,250
And basically, you
do want to lie.

2555
01:58:31,830 --> 01:58:32,770
Some people hate it.

2556
01:58:32,770 --> 01:58:35,220
But there's actually a
really good book that

2557
01:58:35,220 --> 01:58:36,410
follows this principle.

2558
01:58:36,410 --> 01:58:38,380
And even if you're not
interested in the typesetting

2559
01:58:38,380 --> 01:58:41,040
system, you can see how it's
played out in this book.

2560
01:58:41,040 --> 01:58:42,290
It's called The Tech Book.

2561
01:58:48,880 --> 01:58:52,030
So that's the manual for the
tech typesetting system, which

2562
01:58:52,030 --> 01:58:54,550
I use and many people in
math and physics use.

2563
01:58:54,550 --> 01:58:56,760
Now, the reason it's interesting
is that Knuth

2564
01:58:56,760 --> 01:58:59,360
actually tells you in
the preface, I'm

2565
01:58:59,360 --> 01:59:02,150
going to lie to you.

2566
01:59:02,150 --> 01:59:05,840
So what he does is he has three
levels of statements.

2567
01:59:05,840 --> 01:59:10,310
There's statements that
aren't marked with a--

2568
01:59:10,310 --> 01:59:12,780
so there's that sign.

2569
01:59:12,780 --> 01:59:17,450
On the road, it means slippery,
icy, something like

2570
01:59:17,450 --> 01:59:20,480
your car might fish tail
basically, danger.

2571
01:59:20,480 --> 01:59:23,570
So there are statements without
one of these, which

2572
01:59:23,570 --> 01:59:26,410
may have some lies in it, not
the full truth so that you

2573
01:59:26,410 --> 01:59:27,450
just get the idea--

2574
01:59:27,450 --> 01:59:30,350
what are the fundamental
concepts that tech uses.

2575
01:59:30,350 --> 01:59:33,320
When he starts to get into some
gory details, but not

2576
01:59:33,320 --> 01:59:33,900
super gory.

2577
01:59:33,900 --> 01:59:35,350
He puts one of those bends.

2578
01:59:35,350 --> 01:59:38,080
And when he has a super
gory details, there's

2579
01:59:38,080 --> 01:59:39,190
two of those bends.

2580
01:59:39,190 --> 01:59:42,160
And then he says, look, don't
read any of these things

2581
01:59:42,160 --> 01:59:44,990
unless you've been working with
tech for a year and are

2582
01:59:44,990 --> 01:59:45,920
pretty competent with it.

2583
01:59:45,920 --> 01:59:47,020
Don't worry about that.

2584
01:59:47,020 --> 01:59:49,100
You'll be able to do what you
need to do just by reading

2585
01:59:49,100 --> 01:59:51,670
this and maybe the single
bend sections.

2586
01:59:51,670 --> 01:59:54,770
So that's an example where
the lying was put

2587
01:59:54,770 --> 01:59:56,490
to really good use.

2588
01:59:56,490 --> 01:59:57,710
And yeah, you should lie.

2589
01:59:57,710 --> 01:59:58,660
In fact, you have to lie.

2590
01:59:58,660 --> 02:00:00,720
There's no way to avoid it.

2591
02:00:00,720 --> 02:00:04,230
And in fact, everything
is a lie.

2592
02:00:04,230 --> 02:00:08,540
It has to be, because to
understand the universe, our

2593
02:00:08,540 --> 02:00:10,340
brains are a constituent
of the universe.

2594
02:00:10,340 --> 02:00:13,960
So there's no way to understand
the full universe,

2595
02:00:13,960 --> 02:00:16,780
because that would involved
packing more than our brain

2596
02:00:16,780 --> 02:00:18,160
capacity into our brain.

2597
02:00:18,160 --> 02:00:21,410
So just there is a pigeon hole
principle proof that you have

2598
02:00:21,410 --> 02:00:22,910
to lie to understand
the universe.

2599
02:00:22,910 --> 02:00:25,660
So you have to say some stuff
that isn't quite true.

2600
02:00:25,660 --> 02:00:30,630
And where the art is is in
choosing what is a useful lie.

2601
02:00:30,630 --> 02:00:32,470
So what you want to
do is develop the

2602
02:00:32,470 --> 02:00:38,430
art of skillful lying.

2603
02:00:38,430 --> 02:00:40,740
And that's a mark of a really,
really good teacher--

2604
02:00:40,740 --> 02:00:43,140
skillful lying.

2605
02:00:43,140 --> 02:00:43,550
Yes.

2606
02:00:43,550 --> 02:00:47,280
AUDIENCE: How do we know that
you're not lying to us now?

2607
02:00:47,280 --> 02:00:49,190
PROFESSOR: I probably am.

2608
02:00:49,190 --> 02:00:51,250
The question is how do
you know that I'm not

2609
02:00:51,250 --> 02:00:52,430
lying to you now?

2610
02:00:52,430 --> 02:00:56,060
And I probably am, the reason
being that I've now practiced

2611
02:00:56,060 --> 02:00:58,830
lying so much, I don't even
know when I'm lying

2612
02:00:58,830 --> 02:00:59,690
and when I'm not.

2613
02:00:59,690 --> 02:01:00,970
So no, I'll give
you an example.

2614
02:01:00,970 --> 02:01:03,310
I am definitely lying
to some extent.

2615
02:01:03,310 --> 02:01:08,590
For example, there probably are
situations where you don't

2616
02:01:08,590 --> 02:01:11,790
want to any lies at all-- for
example, teaching people how

2617
02:01:11,790 --> 02:01:14,700
to manipulate the machines in
the intensive care unit.

2618
02:01:14,700 --> 02:01:17,880
Maybe the first thing you need--
if you have a half an

2619
02:01:17,880 --> 02:01:19,990
hour to teach them, you'd better
teach them, memorize

2620
02:01:19,990 --> 02:01:21,305
these damn things and
don't mess it up

2621
02:01:21,305 --> 02:01:22,720
or you'll kill somebody.

2622
02:01:22,720 --> 02:01:26,160
Maybe there are situations where
lying is less important

2623
02:01:26,160 --> 02:01:27,290
and lying is more important.

2624
02:01:27,290 --> 02:01:28,540
And I haven't talked
about those.

2625
02:01:28,540 --> 02:01:31,770
So right away, I have lied to
you, just because that.

2626
02:01:31,770 --> 02:01:34,130
And I've skipped details because
I wanted to get the

2627
02:01:34,130 --> 02:01:35,600
big idea across.

2628
02:01:35,600 --> 02:01:40,280
So lying actually comes very
naturally to me because I

2629
02:01:40,280 --> 02:01:42,530
think the most important thing
is the big details.

2630
02:01:42,530 --> 02:01:44,510
So just by saying big details
first, you're

2631
02:01:44,510 --> 02:01:46,320
automatically lying.

2632
02:01:46,320 --> 02:01:48,090
And I'm recommending it
highly to you, too.

2633
02:01:51,150 --> 02:01:51,450
Rodrigo.

2634
02:01:51,450 --> 02:01:53,090
AUDIENCE: I have a comment
on the lying.

2635
02:01:53,090 --> 02:01:59,461
As a student, I have students,
a lot of friends, who told me

2636
02:01:59,461 --> 02:02:01,556
that they don't like it
when teachers lie.

2637
02:02:01,556 --> 02:02:05,961
But what I think the
differentiating factor is

2638
02:02:05,961 --> 02:02:08,246
whether they tell you that
they're actually lying.

2639
02:02:08,246 --> 02:02:11,000
And if they do, then
it's totally fine.

2640
02:02:11,000 --> 02:02:15,040
And if they don't, and a couple
months from there, they

2641
02:02:15,040 --> 02:02:19,706
just told you that everything
was not really true, the

2642
02:02:19,706 --> 02:02:21,190
students might get upset.

2643
02:02:21,190 --> 02:02:21,920
PROFESSOR: Right.

2644
02:02:21,920 --> 02:02:24,640
So the comment is that students
often don't like when

2645
02:02:24,640 --> 02:02:26,270
they find out that they've
been lied to.

2646
02:02:26,270 --> 02:02:28,780
But it's OK if you tell them
that you're lying to them.

2647
02:02:28,780 --> 02:02:32,820
And there's a general principle
there which you can

2648
02:02:32,820 --> 02:02:34,570
use in all your teaching, which
is that whenever you do

2649
02:02:34,570 --> 02:02:37,850
anything slightly
nontraditional--

2650
02:02:37,850 --> 02:02:39,790
whenever you do anything like
that-- it's really important

2651
02:02:39,790 --> 02:02:43,390
to tell the students what you're
doing and why, because

2652
02:02:43,390 --> 02:02:44,560
they will go with you.

2653
02:02:44,560 --> 02:02:46,380
They'll go along with you
if you explain to them

2654
02:02:46,380 --> 02:02:47,490
the motive for it.

2655
02:02:47,490 --> 02:02:50,220
So you tell them, look, this is
a really complex subject.

2656
02:02:50,220 --> 02:02:52,650
There is no way to understand
the whole subject

2657
02:02:52,650 --> 02:02:53,840
at its first glance.

2658
02:02:53,840 --> 02:02:56,550
You need to develop high
level structures first.

2659
02:02:56,550 --> 02:02:58,890
And then you can put in the
details underneath that.

2660
02:02:58,890 --> 02:03:00,640
So I'm going to tell
you just the high

2661
02:03:00,640 --> 02:03:02,100
level structures first.

2662
02:03:02,100 --> 02:03:05,960
And there will be some untruths
in that, but I'm not

2663
02:03:05,960 --> 02:03:07,810
going to tell you things that
are completely false that you

2664
02:03:07,810 --> 02:03:08,910
have unlearn.

2665
02:03:08,910 --> 02:03:10,670
I'm going to tell you things
that you have to refine your

2666
02:03:10,670 --> 02:03:13,050
understanding of or that
aren't the whole truth.

2667
02:03:13,050 --> 02:03:16,750
So together with the idea of
lying, you want to minimize

2668
02:03:16,750 --> 02:03:19,920
stuff that you tell them that
they have to unlearn and make

2669
02:03:19,920 --> 02:03:21,430
it so that you're telling
them stuff that

2670
02:03:21,430 --> 02:03:22,360
they can keep using.

2671
02:03:22,360 --> 02:03:25,570
It's just not the full story.

2672
02:03:25,570 --> 02:03:26,970
OK.

2673
02:03:26,970 --> 02:03:32,440
So what I'm going to do is I'm
going to give you just a quick

2674
02:03:32,440 --> 02:03:37,050
example of another equation and
how you could teach it.

2675
02:03:37,050 --> 02:03:39,630
Then we'll take a short break
and then we'll do

2676
02:03:39,630 --> 02:03:40,830
misconceptions.

2677
02:03:40,830 --> 02:03:41,070
OK.

2678
02:03:41,070 --> 02:03:46,650
So the other equation that I
want to explain is this one.

2679
02:03:51,200 --> 02:03:54,870
So I did a biology example
before, so I chose a physics

2680
02:03:54,870 --> 02:04:00,460
equation this time, which
is the wave equation.

2681
02:04:09,030 --> 02:04:09,290
OK.

2682
02:04:09,290 --> 02:04:12,750
Now, also to vary it, with the
biology example, I introduced

2683
02:04:12,750 --> 02:04:14,240
it with a bit of history.

2684
02:04:14,240 --> 02:04:18,170
With the wave equation, I'm
going to actually introduce it

2685
02:04:18,170 --> 02:04:21,010
with a different approach, which
is not to talk about the

2686
02:04:21,010 --> 02:04:23,930
history but actually to get
the students to try to

2687
02:04:23,930 --> 02:04:25,930
construct the equation.

2688
02:04:25,930 --> 02:04:26,180
OK.

2689
02:04:26,180 --> 02:04:28,560
So the question is what the
hell is the wave equation?

2690
02:04:28,560 --> 02:04:30,015
Well, the wave equation
describes--

2691
02:04:33,165 --> 02:04:36,480
so here's some string.

2692
02:04:36,480 --> 02:04:39,740
And here's your coordinate x.

2693
02:04:39,740 --> 02:04:45,070
And you want to know how does
the height of this piece here,

2694
02:04:45,070 --> 02:04:50,120
the height being f of x as a
function of time as well,

2695
02:04:50,120 --> 02:04:52,370
change with position
and with time?

2696
02:04:52,370 --> 02:04:55,540
So you want to figure out an
equation for that behavior of

2697
02:04:55,540 --> 02:04:58,600
that string, that stretch
between two points.

2698
02:04:58,600 --> 02:05:00,890
For example, this might be a
guitar string, and these are

2699
02:05:00,890 --> 02:05:02,430
the two ends of the
guitar string.

2700
02:05:02,430 --> 02:05:03,960
And it' under tension.

2701
02:05:03,960 --> 02:05:07,150
And you want to know,
how does it move?

2702
02:05:07,150 --> 02:05:07,540
OK.

2703
02:05:07,540 --> 02:05:09,735
So we're going to construct
the equation.

2704
02:05:23,200 --> 02:05:27,690
So let's say this is a
differential equations class

2705
02:05:27,690 --> 02:05:31,870
for people in physics who are
learning mathematical methods.

2706
02:05:31,870 --> 02:05:33,020
And they want to learn
how to construct

2707
02:05:33,020 --> 02:05:33,680
differential equations.

2708
02:05:33,680 --> 02:05:36,620
So maybe they're engineers,
physicists.

2709
02:05:36,620 --> 02:05:38,700
And they're, say, juniors.

2710
02:05:38,700 --> 02:05:41,330
So they have some mathematical
sophistication and some

2711
02:05:41,330 --> 02:05:44,890
knowledge of forces in physics
and some knowledge of

2712
02:05:44,890 --> 02:05:46,050
differential equations.

2713
02:05:46,050 --> 02:05:49,375
So I'm going to write down the
rough form of the equation,

2714
02:05:49,375 --> 02:05:52,770
and we're going to try to figure
out all the pieces and

2715
02:05:52,770 --> 02:05:54,090
fill in the missing pieces.

2716
02:06:08,960 --> 02:06:11,020
So I'm going to give the general
form of the thing.

2717
02:06:20,000 --> 02:06:24,960
So there's some derivative of
f with respect to time--

2718
02:06:24,960 --> 02:06:26,800
one or two derivatives,
we're not sure--

2719
02:06:26,800 --> 02:06:28,750
is equal to something here.

2720
02:06:28,750 --> 02:06:32,120
And then there's some constant
here or maybe here.

2721
02:06:32,120 --> 02:06:34,680
So there's a whole bunch of
question marks to fill in.

2722
02:06:34,680 --> 02:06:40,090
And what we're going to do is
reason about what they are.

2723
02:06:40,090 --> 02:06:40,610
OK.

2724
02:06:40,610 --> 02:06:45,570
So the first question is
to figure out this guy.

2725
02:06:45,570 --> 02:06:49,820
How many time derivatives
do we need?

2726
02:06:49,820 --> 02:06:53,470
So what this equation describes
is the motion of

2727
02:06:53,470 --> 02:06:55,400
this point.

2728
02:06:55,400 --> 02:06:57,560
Now, why does the point move?

2729
02:06:57,560 --> 02:06:59,740
Well, you could ask the class,
and eventually, they'll come

2730
02:06:59,740 --> 02:07:02,170
up with, well, there's some
forces on it, because the

2731
02:07:02,170 --> 02:07:05,690
string's under tension, so
there's forces on the point.

2732
02:07:05,690 --> 02:07:07,920
So here's our point.

2733
02:07:07,920 --> 02:07:10,830
And here's, say, the string
going through it.

2734
02:07:10,830 --> 02:07:12,530
This is a blow up now
of this region.

2735
02:07:17,640 --> 02:07:18,080
OK.

2736
02:07:18,080 --> 02:07:20,360
So now, what are the
forces on this guy?

2737
02:07:20,360 --> 02:07:23,400
Well, there's a force from that
piece of the string and a

2738
02:07:23,400 --> 02:07:26,470
force from that piece
of the string.

2739
02:07:26,470 --> 02:07:26,760
OK.

2740
02:07:26,760 --> 02:07:30,530
Because there's force, what's
going to happen to the thing?

2741
02:07:30,530 --> 02:07:33,610
Is it going to have a velocity
or an acceleration?

2742
02:07:36,270 --> 02:07:37,010
Anyone?

2743
02:07:37,010 --> 02:07:37,790
AUDIENCE: Acceleration.

2744
02:07:37,790 --> 02:07:38,580
PROFESSOR: Acceleration.

2745
02:07:38,580 --> 02:07:38,870
OK.

2746
02:07:38,870 --> 02:07:41,490
How many derivatives does
acceleration have?

2747
02:07:41,490 --> 02:07:42,300
Two.

2748
02:07:42,300 --> 02:07:42,730
OK.

2749
02:07:42,730 --> 02:07:46,930
So we're going to put two
derivatives here.

2750
02:07:46,930 --> 02:07:48,080
So you need two derivatives.

2751
02:07:48,080 --> 02:07:51,620
So this, and now, we're going
to say this is some kind of

2752
02:07:51,620 --> 02:07:52,870
acceleration.

2753
02:07:56,910 --> 02:07:59,035
And maybe there's masses in
there and stuff, which would

2754
02:07:59,035 --> 02:08:00,790
all be slurped into
these constants.

2755
02:08:00,790 --> 02:08:02,410
So we have an acceleration.

2756
02:08:02,410 --> 02:08:06,520
And now, we have to decide what
generates acceleration.

2757
02:08:06,520 --> 02:08:10,310
This side is generating
acceleration or the force.

2758
02:08:10,310 --> 02:08:14,500
So we want to decide, for
example, one or two

2759
02:08:14,500 --> 02:08:16,310
derivatives.

2760
02:08:16,310 --> 02:08:18,270
Generally, most equations
either have one

2761
02:08:18,270 --> 02:08:19,290
derivative or two.

2762
02:08:19,290 --> 02:08:20,680
Some really nasty
ones have four.

2763
02:08:20,680 --> 02:08:22,040
But generally, it's
one or two.

2764
02:08:22,040 --> 02:08:22,880
So we'll just choose.

2765
02:08:22,880 --> 02:08:24,400
Is this one or two?

2766
02:08:24,400 --> 02:08:25,040
OK.

2767
02:08:25,040 --> 02:08:27,570
And then the last thing we're
going to do is to figure out

2768
02:08:27,570 --> 02:08:30,420
the constant.

2769
02:08:30,420 --> 02:08:30,710
OK.

2770
02:08:30,710 --> 02:08:32,670
So now, one or two derivatives--
one way to

2771
02:08:32,670 --> 02:08:35,990
decide that is to make something
that has just one

2772
02:08:35,990 --> 02:08:38,610
derivative in it.

2773
02:08:38,610 --> 02:08:43,390
So if here is my string and
here is the point--

2774
02:08:43,390 --> 02:08:50,030
so f has a non-zero--

2775
02:08:55,700 --> 02:08:57,363
so it has a non-zero df/dx.

2776
02:09:03,560 --> 02:09:05,975
But the second derivative
is 0, because

2777
02:09:05,975 --> 02:09:07,450
it's a straight line.

2778
02:09:07,450 --> 02:09:07,750
OK.

2779
02:09:07,750 --> 02:09:09,560
So we know the second
derivative is 0.

2780
02:09:09,560 --> 02:09:11,410
Let's see what we can figure
out what the force or the

2781
02:09:11,410 --> 02:09:12,790
acceleration should be.

2782
02:09:12,790 --> 02:09:13,510
OK.

2783
02:09:13,510 --> 02:09:15,830
Well, here is a point.

2784
02:09:15,830 --> 02:09:20,532
There's going to be a force on
it from that end and that end.

2785
02:09:20,532 --> 02:09:24,020
And what's the net result
of these two forces?

2786
02:09:24,020 --> 02:09:25,420
0.

2787
02:09:25,420 --> 02:09:29,170
So when the second derivative
is 0, in this case at least,

2788
02:09:29,170 --> 02:09:33,480
we would like the force to be 0,
which means that force has

2789
02:09:33,480 --> 02:09:35,875
to be connected to the second
derivative of position.

2790
02:09:39,650 --> 02:09:41,040
So we got that.

2791
02:09:41,040 --> 02:09:46,020
Now, the next problem is to
work out what goes here.

2792
02:09:46,020 --> 02:09:48,360
The first thing is
what's the sine.

2793
02:09:48,360 --> 02:09:49,880
Should it be plus or minus?

2794
02:09:56,840 --> 02:09:57,140
OK.

2795
02:09:57,140 --> 02:09:58,890
So I'm going to ask you that.

2796
02:09:58,890 --> 02:10:04,190
So find a reason whether it
should be plus or minus here

2797
02:10:04,190 --> 02:10:05,750
and a reason.

2798
02:10:05,750 --> 02:10:10,910
So find a neighbor or two,
and we'll take a vote--

2799
02:10:10,910 --> 02:10:13,180
plus or minus, intuitive
reason for it.

2800
02:10:24,230 --> 02:10:26,140
I can't use this example
to decide.

2801
02:10:37,700 --> 02:10:40,050
Let's vote.

2802
02:10:40,050 --> 02:10:41,340
Everyone have their
votes ready?

2803
02:10:41,340 --> 02:10:44,800
Who votes for plus?

2804
02:10:44,800 --> 02:10:47,380
So about--

2805
02:10:47,380 --> 02:10:50,580
who votes for minus?

2806
02:10:50,580 --> 02:10:50,730
OK.

2807
02:10:50,730 --> 02:10:51,260
That's great.

2808
02:10:51,260 --> 02:10:53,370
So we have a diversity
of opinion.

2809
02:10:53,370 --> 02:10:57,680
So right away, what that shows
you is that it's worth

2810
02:10:57,680 --> 02:11:01,590
actually discussing that point
in class, because if you just

2811
02:11:01,590 --> 02:11:04,840
tell people, they'll write down
something you tell them,

2812
02:11:04,840 --> 02:11:06,380
but they won't have actually
internalized it.

2813
02:11:06,380 --> 02:11:09,060
It'll just be something that
maybe contradicted what they

2814
02:11:09,060 --> 02:11:10,590
said, thought, or not.

2815
02:11:10,590 --> 02:11:12,210
And then they have to remember,
was it what I

2816
02:11:12,210 --> 02:11:13,800
thought or was what
I not thought?

2817
02:11:13,800 --> 02:11:15,850
Or was it what I not thought
or what I thought?

2818
02:11:15,850 --> 02:11:17,140
So they don't actually
understand why.

2819
02:11:17,140 --> 02:11:19,590
So it's actually worth going
through the discussion.

2820
02:11:19,590 --> 02:11:22,960
So what example could
you use to decide?

2821
02:11:22,960 --> 02:11:23,590
Who haven't I heard from?

2822
02:11:23,590 --> 02:11:24,135
Yeah.

2823
02:11:24,135 --> 02:11:25,010
Could you tell me your name?

2824
02:11:25,010 --> 02:11:25,530
AUDIENCE: Mike.

2825
02:11:25,530 --> 02:11:27,002
PROFESSOR: Mike.

2826
02:11:27,002 --> 02:11:29,046
AUDIENCE: Well, if you're at
the top of the arc, and you

2827
02:11:29,046 --> 02:11:30,510
don't it's going to want
to be pulled down.

2828
02:11:30,510 --> 02:11:31,980
It's coming down.

2829
02:11:31,980 --> 02:11:32,360
PROFESSOR: OK.

2830
02:11:32,360 --> 02:11:33,850
So let's use an arc like this.

2831
02:11:33,850 --> 02:11:35,940
We can't use this one,
because it's 0, and 0

2832
02:11:35,940 --> 02:11:36,740
doesn't have a sign.

2833
02:11:36,740 --> 02:11:37,440
So it doesn't help us.

2834
02:11:37,440 --> 02:11:39,730
So the next most complicated
thing is this--

2835
02:11:39,730 --> 02:11:41,380
so an arc like that.

2836
02:11:41,380 --> 02:11:42,490
Here's my point.

2837
02:11:42,490 --> 02:11:45,960
So the force is downwards, or
the acceleration is downwards.

2838
02:11:45,960 --> 02:11:47,680
So we should have negative
acceleration.

2839
02:11:47,680 --> 02:11:49,970
And what's the second derivative
with respect to

2840
02:11:49,970 --> 02:11:51,160
position here?

2841
02:11:51,160 --> 02:11:53,910
That's also downwards, because
the arc is like that.

2842
02:11:53,910 --> 02:11:56,870
So the derivative has
the same direction--

2843
02:11:56,870 --> 02:11:59,020
the space derivative and
the acceleration.

2844
02:11:59,020 --> 02:12:00,270
So it should be plus.

2845
02:12:02,530 --> 02:12:04,510
So we got the sign right.

2846
02:12:04,510 --> 02:12:06,760
And to emphasize the importance
of getting the

2847
02:12:06,760 --> 02:12:10,490
signs, there's an interesting
comment from Feynman back in

2848
02:12:10,490 --> 02:12:13,890
the '60s at some conference
about quantum electrodynamics.

2849
02:12:13,890 --> 02:12:19,120
He was commenting about how
crazy the whole process that

2850
02:12:19,120 --> 02:12:21,850
he invented for solving quantum
electrodynamics is.

2851
02:12:21,850 --> 02:12:24,020
And his reason that
is so crazy--

2852
02:12:24,020 --> 02:12:27,830
he said, well, we do the first
order term, and then we

2853
02:12:27,830 --> 02:12:29,720
calculate the second order term
and add it to the first

2854
02:12:29,720 --> 02:12:30,390
order term.

2855
02:12:30,390 --> 02:12:33,380
But it's very worrying that when
we calculate the second

2856
02:12:33,380 --> 02:12:35,440
order term after the first
order term, we don't know

2857
02:12:35,440 --> 02:12:37,710
whether the second order term
is positive or negative.

2858
02:12:37,710 --> 02:12:40,140
We just have to calculate it and
see what it's going to be.

2859
02:12:40,140 --> 02:12:41,940
But we can't predict ahead
of time whether

2860
02:12:41,940 --> 02:12:43,390
it's positive or negative.

2861
02:12:43,390 --> 02:12:46,620
And what that speaks to is the
importance that physicists

2862
02:12:46,620 --> 02:12:48,810
attach to knowing the
sign of an effect.

2863
02:12:48,810 --> 02:12:50,920
Is a plus or is it a minus?

2864
02:12:50,920 --> 02:12:52,210
And so that's fundamentally
important.

2865
02:12:52,210 --> 02:12:53,810
You want to make sure
you get that right.

2866
02:12:53,810 --> 02:12:58,020
Now, why did I do the sign of
the effect after this, after

2867
02:12:58,020 --> 02:12:59,580
the order of the derivatives?

2868
02:12:59,580 --> 02:13:01,210
Because the most important
thing is what

2869
02:13:01,210 --> 02:13:02,310
do the terms mean.

2870
02:13:02,310 --> 02:13:05,340
So this is a curvature.

2871
02:13:05,340 --> 02:13:07,815
This is an acceleration.

2872
02:13:07,815 --> 02:13:09,330
So here, let me write that--

2873
02:13:09,330 --> 02:13:10,580
curvature.

2874
02:13:12,800 --> 02:13:14,860
Until you know what the terms
mean, there's no hope of

2875
02:13:14,860 --> 02:13:17,130
figuring out what the sign
that connects them is.

2876
02:13:17,130 --> 02:13:19,160
But the sign is the very
next thing you do.

2877
02:13:19,160 --> 02:13:19,990
And it's really important.

2878
02:13:19,990 --> 02:13:20,740
It's a plus.

2879
02:13:20,740 --> 02:13:24,300
And now, we need to put in one
more thing, because the

2880
02:13:24,300 --> 02:13:26,040
dimensions are totally bogus.

2881
02:13:26,040 --> 02:13:30,910
This is a position divided
by a times squared.

2882
02:13:30,910 --> 02:13:34,240
This is a position divided
by a length squared.

2883
02:13:34,240 --> 02:13:37,980
So we need to actually multiply
by something to make

2884
02:13:37,980 --> 02:13:40,810
the units come out correct.

2885
02:13:40,810 --> 02:13:45,020
So we need something that puts
a time squared here and a

2886
02:13:45,020 --> 02:13:46,670
position squared over here.

2887
02:13:46,670 --> 02:13:49,310
So then the units will work
out-- or a position squared

2888
02:13:49,310 --> 02:13:55,250
over time squared, which
is C squared, which

2889
02:13:55,250 --> 02:13:56,380
is some speed squared.

2890
02:13:56,380 --> 02:13:59,590
So speed squared is position
squared over time squared.

2891
02:13:59,590 --> 02:14:02,010
And that makes all the
units work out.

2892
02:14:02,010 --> 02:14:02,560
So there you go.

2893
02:14:02,560 --> 02:14:05,680
There you have the
wave equation.

2894
02:14:05,680 --> 02:14:07,570
So now, if you're going to
actually formally derive it,

2895
02:14:07,570 --> 02:14:08,490
that's all fine.

2896
02:14:08,490 --> 02:14:12,590
But I would do this first so
that people know where every

2897
02:14:12,590 --> 02:14:14,530
single term in the equation
comes from.

2898
02:14:14,530 --> 02:14:15,780
And now, I've been
a bit sloppy.

2899
02:14:15,780 --> 02:14:17,500
These are really partial
derivatives.

2900
02:14:17,500 --> 02:14:20,750
But that's, again, an
example of lying.

2901
02:14:20,750 --> 02:14:22,930
I wouldn't worry about whether
it's a partial or total

2902
02:14:22,930 --> 02:14:24,700
derivative at the beginning,
because that's not the

2903
02:14:24,700 --> 02:14:25,490
fundamental idea.

2904
02:14:25,490 --> 02:14:28,770
The fundamental idea is that
it's curvature on this side

2905
02:14:28,770 --> 02:14:31,900
connecting to acceleration
on this side, and they're

2906
02:14:31,900 --> 02:14:36,370
connected by a positive sign
and by something that has

2907
02:14:36,370 --> 02:14:38,420
dimensions of speed squared,
which turns out to be the

2908
02:14:38,420 --> 02:14:40,190
speed at which the wave moves.

2909
02:14:40,190 --> 02:14:43,590
So that's a way to introduce
yet another equation, not

2910
02:14:43,590 --> 02:14:46,310
related to the history, but
actually connecting to the

2911
02:14:46,310 --> 02:14:48,360
intuitive approach.

2912
02:14:48,360 --> 02:14:49,610
Any questions about that?