Lec 2 | MIT 5.95J Teaching College-Level Science and Engineering, Spring 2009

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Ok, teaching equations, and how to teach equations in a way that promotes

long lasting learning and understanding.

Ok. So. The first example.

I'm going to give you two choices for starting the example.

So the first choice, the example is —

So this is example one for teaching equations.

And it's —

This is Hardy-Weinberg Equilibrium.

So, here's one way you could start.

So if a locus has n alleles

and the organism is polyploid

So that means it has say "c" copies of chromosomes

Okay so, and the, so there is "n" alleles

and the allele frequencies are.

So the allele frequencies are "p1" through "pn."

Then, we're going to deduce the genotype frequencies

so this is the multinomial coefficient

c choose k1, k2, k3 all the way to kn.

And k1 through kn are the number of copies of each allele.

With k1 plus, blah blah blah, kn all equal to c.

So, you could, actually state that,

and then prove it, that's option A.

So now I'll give you option B

for starting the same topic.

Options B,

So option B, is, from Hardy

so in Hardy's Mathematicians apology

he writes, I've never done anything useful.

No discovery of mine has made,

or is likely to make,

directly or indirectly,

for good or ill,

the least difference to the amenity of the world.

And that's a very interesting,

very bold statement.

And amazingly for such a brilliant mathematician, its wrong.

And today, what we're going to do

is we're going to look at an example

that. Of a discovery by Hardy

Hardy-Weingberg Equilibrium.

That does make a difference to the amenity of the world.

and helps you actually understand genetics.

Ok so the option B is to quote

Hardy and then show the,

point out

and then point out the contradiction

between what he said and what we're going to do

now also, in option B,

you can also go a bit farther,

you can say,

well why would Hardy have said something like that?

Well indeed why would he have said something like that, in his apollogy?

Well, it's, actually probably hard to appreciate

on this side of the Atlantic

because we've never actually had

Well since the Civil War, we've never had

a war that just basically devastated

entire countries, on our own soil.

Whereas in Europe,

the memories of World War One

are very strong are very strong.

So if anyones from Europe you know that.

In, you know in every English Chapel

there's names on the wall

of all the people who died

in the quote Great War.

So, it's World War One

had a very strong effect

on European society.

And one of the effects was on

European science.

And people's attitude toward science.

So, poison gas was invented

partly by a German Chemist, Haber,

was one of them.

And in the end

he committed suicide

partly maybe because of what he had done.

He was, so, did he?

Yeah, I think he.

No sorry, his wife committed suicide.

I forget now, I should check that.

But, basically people were, in the family were so unhappy

about what had happened

just there that there was a suicide

and furthermore science after that was

considered you know World War One was considered

The Chemist's War

So there was a wish to distance one's self

from those kinds of horrible effects

so the quote from Hardy is a reflection

of that. He himself was actually very anti-war

he left Cambridge because Cambridge

fired Bertrand Russell for protesting World War One.

So Hardy left Cambridge for professorship at Oxford

And only came back basically

12 or 13 years later

to Cambridge. So that statement

of his is partly a wish

that he's hoping that nothing he's has any effect on the world

becausein his mind, maybe most effects are bad.

Well, but actually he did have an effect,

the effect is Hardy Weinberg

even if you discount everything he did in number theory

there's Hardy-Weinberg

and it has an effect amd we're going to look at it.

So let me ask you

a rhetorical question,

if you're a student,

which would you find more engaging,

more inviting?

This one or this one?

Who votes for that one?

Okay, who votes for that one?

Yeah, now, why?

So now think of,

so the question i'm going to ask you

to think about in, just for a minute

in with one or two of your neighbors is

why?

So, yeah this is definitely

more engaging.

In terms of the principle we talked about last time,

what makes this way of introducing it

more engaging, more likely to bring students in,

more likely to want to learn

about Hardy Weinberg?

Okay so find a neighbor or two

and think about

the principles behind option B.

Okay so...

I'll rudely interupt you.

And good thing I did

the voice excercise so I

can project all the way to the back.

So, what reasons for option B?

Or again stops your name

which is the same thing.

Yes, could you tell me your name?

I'm going to, Bryan, I'm going to try to learn people's names.

So go ahead,

That's right, through the mount wave

or in fact maybe even

so yeah, I'm activating the ability of the students to

basically take dictation.

Or, as someone said,

teaching is an excellent way

to transfer material from

the notes of the teacher to

the notes of the student

without it passing through

the minds of either.

So I'll call that 'not a.'

So, this is the 'not' symbol.

'not a' is.

So I'm promoting dictation

which doesn't necessarily.

I'm, basically asking to do dictation

which doesn't necessarily

lead to any kind of learning.

It could for example

copy all of that down,

but not really understand any of it.

Okay, other...

Yes, could you tell me your name?

Susanna, yeah?

Yeah.

Okay so,

right and so B is a story.

Or, I'll

I'll. You know, so here's an

I'll give you an example

of how stories can be so powerful.

And just even the word.

So a, a big commercial publisher,

this was several years ago,

they wanted me

to write a freshman physics textbook.

I'd actually, I'd just put up a proposal on the web saying

you we shoudl write up a freshman physics textbook

that is based of history and science

and they saw it and they thought,

"ohh that's great"

so they flew me from

England to California to talk to them.

And, they liked it mostly

but they said, "hmm, history."

So there was the word.

They liked the idea as I talked about it

but the word history really frightened them.

And then, I'm still amazed as how

I had this amazing insight

of just two letters.

I thought, "ohh wait, I can actually explain to them what i mean."

I said, "well actually, if you just take away that part,

what I'm really talking about is that."

And it was interesting

as soon as they saw the word story,

everybody.

And they saw that, well history doesn't have to be

names, dates, facts.

I think that's what they were seeing it as.

They saw it as 'story' which is actually

its origin in French or Latin, 'histoire.'

They all of a sudden were totally convinced.

They said, "ohh yeah that's exactly how we should do the textbook."

And for various reasons, basically because

I want it to be a freely licensed,

want it to be a freely licensed book,

we didn't actually sign a contract.

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

and how people who actually spend their lives thinking about

teaching and reaching students.

This educational publisher

knew the importance of story.

But if you present it as just dry history,

it's not so interesting.

So, 'story.'

Yeah, so it's, let's see.

Who haven't I heard from?

Adrian, yeah.

Okay, so there's a contradiction.

The contradiction creates some kind of tension.

Right so that tension creates interest.

So. Right so every good story needs some kind of tension.

You know, so the tension here,

the contradiction is that, you know,

Hardy wanted to do nothing that could harm people basically.

So he was like, "Okay well anything can be used, could be used, even if it's for good for ill.

So let me just back off from all that

say I'm not going to do anything that has an effect, an application.

I'm a pure mathematician."

His most famous book is of course, Pure Mathematics.

Well maybe his most book famous is that book, which I quoted from,

A Mathematician's Apollogy.

But his most famous math book is,

probably the Course of Pure Mathematics.

And that's what he wanted to be known for.

And that's in contradiction with the fact

that, no, even Hardy couldn't help an application.

An actually very very common one,

it's taught in every single Introductory Biology course probably.

Okay so you need tension.

Now, that's a general principle of learning there.

You know the idea of story and tension

and paradox.

So you.

The one way that as a undergraduate in physics

that I learned a ton of physics,

me and my friends were doing problem sets together

and you know lots of the problems were,

just grinding through math.

So we didn't learn a hell of a lot from that.

But, we were in the library late at night,

you know, the physics library,

and ordering pizza and trying to do our problem set

and we just got to making up physics paradoxes.

You know perpetual motion machines.

We'd invent perpetual motion machines

and try to get the other person

to figure out what was wrong with it.

Sometimes we didn't even know what was wrong with it,

we'd both try to figure it out.

So, from that kind of tension,

tension is almost in a way self-teaching.

Beacuse, as long as the tension's there you know you're not at the end of the story.

Right so you know there's more to do.

So the same thing,

as long as there was perpetual motion going on,

and we hadn't found the reason,

we knew weren't done with the problem.

We didn't have to ask the teacher,

"Well, is this right?"

Because it was basically self-teaching

as long as the tension was there

we knew that we weren't finished.

Okay so, yeah,

another general principle stories of tension.

Let's see there was a comment over...

Yes, tell me your name.

Wing, Wing-ho.

Right.

Okay so, yeah the, it, the teacher makes himself more human

and makes the person who invented,

or one of the people who invented the Hardy-Weinberg Equilibrium more human.

So it's actually something much more easy,

easy for the students to connect to.

Yes, could you tell me your name?

Gregory.

mhm.

Okay so, so Gregor's point which is a good one

is that B doesn't have the same quantitative depth as A.

And, well actually what I'm going to show you

is a way to get to get to this

starting from here,

and using to the principles that were talking about

so that by the time you ger here or to something like this

it actually makes sense to the students.

Okay so that you can have both.

So, that's a promise and,

hopefully I'll deliver on it for you.

Okay does that?

But does that address what you're talking about?

That B doesn't have the quantitative depth.

And that's true.

You, well so you so the,

the point made is that

if you want to get here you do have to

say this at some point.

And you know you can't just say the story for an hour.

Well, it depends on the purpose of the class.

If the purpose is the history of Biology,

maybe you would continue with the story

or the history of science and war.

But if you know you want the students

to be able to solve problems in gentics

you know maybe you need to go here.

Or, what we'll find is the next.

I'll continue with option B

moving towards A

and show you how you can get there

without and still have

the opening of B preserved.

Okay so let's see,

there was yes.

Yes, can you tell me your name?

Okay.

Yeah okay, okay so B is, new.

Right, A is actually.

So against A, A is very familiar and common.

So in fact yeah I chose this,

I mean i didn't really make a strong ???man??? here.

This is how alot of things are introduced

for example, in mathematics class

theorem proof.

Right so the theorem will be introduced

without any of the struggle, the wondering,

that lead to the theorem.

Like, why would anyone care about such a thing?

Or if they cared about it,

why would they come up with something like that?

You know, how can you see that?

The, so yeah A is seen very familiar

and therefore maybe not as interesting.

Although it may also be intrinsically less interesting.

For the other reasons, but I take your point too

that it could just be familiarity.

Yes, can you tell me your name?

Meg, yeah.

mhm.

Okay.

Okay so your, your point which is an excellent one aswell

is that, you, B doesn't preclude the quatitative.

What you could do is you could talk about

the history.

Start with the history of Hardy saying that

which is not chronological because he said that in 1940

but that's okay, it doesn't have to be chronological

just because it's history.

Go to 1940 and then backtrack,

to well how did this problem

actually come to Hardy's attention

which is actually, turns out to be quite

an interesting story.

And then talk about how they solved it

and what was the ???public, date in history???

All the quantitative ideas would come out

in that way

and it would be

an almost backwards from the usual way.

In one way is that the usual way well,

the usual way here is completely general

and that's probably not the way

it was first figured out.

And, the advantage of

telling about it that way

is that you're preparing students themselves

for a research career.

Because, they're not,

nobody comes up with theorems

full blown you know like seen

out of the head of Zeus.

You come through it from struggle

and wondering, "hmm I wonder."

Everything comes, has some historical background to it.

So the, it's called the, you could say it,

the Genetic approach.

So in, in Biology, there's a slightness,

not misconception, but a saying that

???oncology recapitulates via logically.???

In other words, that the organism,

as it develops in the womb

goes through all the evolutionary stages

that it went through

over the last 200 millions years,

or whatever it took to become,

say, a person.

You're first a fish

and then maybe you're a monkey

and then a person.

So roughly speaking,

it's not exactly true.

But there's alot of truth to that

for learning ideas.

That you have to recapitulate the history

of ideas to really understand where we are now,

the ideas today.

So a great example of that

is the Newton's Second Law of Motion.

The idea that force and acceleration,

are connected.

So for thousands of years,

millions of years, people thought

that force and velocity were connected.

And it's actually force and acceleration.

So actually you can guide people

to the understanding of force and acceleration being connected

by showing them the history of how people

thought it was force and velocity,

because, for those same reasons,

those are the reasons the student will think it too.

So you're actually helping them overcome their misconceptions.

So the history actually

generally speaking,

helps overcome misconceptions.

So I'll put that here.

And the history is actually quite interesting.

I think it was Punnett, from Punnett squares.

He played cricket with Hardy,

and Hardy loved cricket

and he just asked him about this problem

and Hardy said, "Ohh yeah I know, it's just this."

And sent off a paper to science or nature.