-
Not Synced
The following content is provided under a Creative Commons license.
-
Not Synced
Your support will help OpenCourseWare continue to offer high-quality
-
Not Synced
educational resources for free. To make a donation or to view additional material
-
Not Synced
from hundreds of courses,
-
Not Synced
visit MIT OpenCourseWare at ocw.mit.edu.
-
Not Synced
Ok, teaching equations, and how to teach equations in a way that promotes
-
Not Synced
long lasting learning and understanding.
-
Not Synced
Ok. So. The first example.
-
Not Synced
I'm going to give you two choices for starting the example.
-
Not Synced
So the first choice, the example is —
-
Not Synced
So this is example one for teaching equations.
-
Not Synced
And it's —
-
Not Synced
This is Hardy-Weinberg Equilibrium.
-
Not Synced
So, here's one way you could start.
-
Not Synced
So if a locus has n alleles
-
Not Synced
and the organism is polyploid
-
Not Synced
So that means it has say "c" copies of chromosomes
-
Not Synced
Okay so, and the, so there is "n" alleles
-
Not Synced
and the allele frequencies are.
-
Not Synced
So the allele frequencies are "p1" through "pn."
-
Not Synced
Then, we're going to deduce the genotype frequencies
-
Not Synced
so this is the multinomial coefficient
-
Not Synced
c choose k1, k2, k3 all the way to kn.
-
Not Synced
And k1 through kn are the number of copies of each allele.
-
Not Synced
With k1 plus, blah blah blah, kn all equal to c.
-
Not Synced
So, you could, actually state that,
-
Not Synced
and then prove it, that's option A.
-
Not Synced
So now I'll give you option B
-
Not Synced
for starting the same topic.
-
Not Synced
Options B,
-
Not Synced
So option B, is, from Hardy
-
Not Synced
so in Hardy's Mathematicians apology
-
Not Synced
he writes, I've never done anything useful.
-
Not Synced
No discovery of mine has made,
-
Not Synced
or is likely to make,
-
Not Synced
directly or indirectly,
-
Not Synced
for good or ill,
-
Not Synced
the least difference to the amenity of the world.
-
Not Synced
And that's a very interesting,
-
Not Synced
very bold statement.
-
Not Synced
And amazingly for such a brilliant mathematician, its wrong.
-
Not Synced
And today, what we're going to do
-
Not Synced
is we're going to look at an example
-
Not Synced
that. Of a discovery by Hardy
-
Not Synced
Hardy-Weingberg Equilibrium.
-
Not Synced
That does make a difference to the amenity of the world.
-
Not Synced
and helps you actually understand genetics.
-
Not Synced
Ok so the option B is to quote
-
Not Synced
Hardy and then show the,
-
Not Synced
point out
-
Not Synced
and then point out the contradiction
-
Not Synced
between what he said and what we're going to do
-
Not Synced
now also, in option B,
-
Not Synced
you can also go a bit farther,
-
Not Synced
you can say,
-
Not Synced
well why would Hardy have said something like that?
-
Not Synced
Well indeed why would he have said something like that, in his apollogy?
-
Not Synced
Well, it's, actually probably hard to appreciate
-
Not Synced
on this side of the Atlantic
-
Not Synced
because we've never actually had
-
Not Synced
Well since the Civil War, we've never had
-
Not Synced
a war that just basically devastated
-
Not Synced
entire countries, on our own soil.
-
Not Synced
Whereas in Europe,
-
Not Synced
the memories of World War One
-
Not Synced
are very strong are very strong.
-
Not Synced
So if anyones from Europe you know that.
-
Not Synced
In, you know in every English Chapel
-
Not Synced
there's names on the wall
-
Not Synced
of all the people who died
-
Not Synced
in the quote Great War.
-
Not Synced
So, it's World War One
-
Not Synced
had a very strong effect
-
Not Synced
on European society.
-
Not Synced
And one of the effects was on
-
Not Synced
European science.
-
Not Synced
And people's attitude toward science.
-
Not Synced
So, poison gas was invented
-
Not Synced
partly by a German Chemist, Haber,
-
Not Synced
was one of them.
-
Not Synced
And in the end
-
Not Synced
he committed suicide
-
Not Synced
partly maybe because of what he had done.
-
Not Synced
He was, so, did he?
-
Not Synced
Yeah, I think he.
-
Not Synced
No sorry, his wife committed suicide.
-
Not Synced
I forget now, I should check that.
-
Not Synced
But, basically people were, in the family were so unhappy
-
Not Synced
about what had happened
-
Not Synced
just there that there was a suicide
-
Not Synced
and furthermore science after that was
-
Not Synced
considered you know World War One was considered
-
Not Synced
The Chemist's War
-
Not Synced
So there was a wish to distance one's self
-
Not Synced
from those kinds of horrible effects
-
Not Synced
so the quote from Hardy is a reflection
-
Not Synced
of that. He himself was actually very anti-war
-
Not Synced
he left Cambridge because Cambridge
-
Not Synced
fired Bertrand Russell for protesting World War One.
-
Not Synced
So Hardy left Cambridge for professorship at Oxford
-
Not Synced
And only came back basically
-
Not Synced
12 or 13 years later
-
Not Synced
to Cambridge. So that statement
-
Not Synced
of his is partly a wish
-
Not Synced
that he's hoping that nothing he's has any effect on the world
-
Not Synced
becausein his mind, maybe most effects are bad.
-
Not Synced
Well, but actually he did have an effect,
-
Not Synced
the effect is Hardy Weinberg
-
Not Synced
even if you discount everything he did in number theory
-
Not Synced
there's Hardy-Weinberg
-
Not Synced
and it has an effect amd we're going to look at it.
-
Not Synced
So let me ask you
-
Not Synced
a rhetorical question,
-
Not Synced
if you're a student,
-
Not Synced
which would you find more engaging,
-
Not Synced
more inviting?
-
Not Synced
This one or this one?
-
Not Synced
Who votes for that one?
-
Not Synced
Okay, who votes for that one?
-
Not Synced
Yeah, now, why?
-
Not Synced
So now think of,
-
Not Synced
so the question i'm going to ask you
-
Not Synced
to think about in, just for a minute
-
Not Synced
in with one or two of your neighbors is
-
Not Synced
why?
-
Not Synced
So, yeah this is definitely
-
Not Synced
more engaging.
-
Not Synced
In terms of the principle we talked about last time,
-
Not Synced
what makes this way of introducing it
-
Not Synced
more engaging, more likely to bring students in,
-
Not Synced
more likely to want to learn
-
Not Synced
about Hardy Weinberg?
-
Not Synced
Okay so find a neighbor or two
-
Not Synced
and think about
-
Not Synced
the principles behind option B.
-
Not Synced
Okay so...
-
Not Synced
I'll rudely interupt you.
-
Not Synced
And good thing I did
-
Not Synced
the voice excercise so I
-
Not Synced
can project all the way to the back.
-
Not Synced
So, what reasons for option B?
-
Not Synced
Or again stops your name
-
Not Synced
which is the same thing.
-
Not Synced
Yes, could you tell me your name?
-
Not Synced
I'm going to, Bryan, I'm going to try to learn people's names.
-
Not Synced
So go ahead,
-
Not Synced
That's right, through the mount wave
-
Not Synced
or in fact maybe even
-
Not Synced
so yeah, I'm activating the ability of the students to
-
Not Synced
basically take dictation.
-
Not Synced
Or, as someone said,
-
Not Synced
teaching is an excellent way
-
Not Synced
to transfer material from
-
Not Synced
the notes of the teacher to
-
Not Synced
the notes of the student
-
Not Synced
without it passing through
-
Not Synced
the minds of either.
-
Not Synced
So I'll call that 'not a.'
-
Not Synced
So, this is the 'not' symbol.
-
Not Synced
'not a' is.
-
Not Synced
So I'm promoting dictation
-
Not Synced
which doesn't necessarily.
-
Not Synced
I'm, basically asking to do dictation
-
Not Synced
which doesn't necessarily
-
Not Synced
lead to any kind of learning.
-
Not Synced
It could for example
-
Not Synced
copy all of that down,
-
Not Synced
but not really understand any of it.
-
Not Synced
Okay, other...
-
Not Synced
Yes, could you tell me your name?
-
Not Synced
Susanna, yeah?
-
Not Synced
Yeah.
-
Not Synced
Okay so,
-
Not Synced
right and so B is a story.
-
Not Synced
Or, I'll
-
Not Synced
I'll. You know, so here's an
-
Not Synced
I'll give you an example
-
Not Synced
of how stories can be so powerful.
-
Not Synced
And just even the word.
-
Not Synced
So a, a big commercial publisher,
-
Not Synced
this was several years ago,
-
Not Synced
they wanted me
-
Not Synced
to write a freshman physics textbook.
-
Not Synced
I'd actually, I'd just put up a proposal on the web saying
-
Not Synced
you we shoudl write up a freshman physics textbook
-
Not Synced
that is based of history and science
-
Not Synced
and they saw it and they thought,
-
Not Synced
"ohh that's great"
-
Not Synced
so they flew me from
-
Not Synced
England to California to talk to them.
-
Not Synced
And, they liked it mostly
-
Not Synced
but they said, "hmm, history."
-
Not Synced
So there was the word.
-
Not Synced
They liked the idea as I talked about it
-
Not Synced
but the word history really frightened them.
-
Not Synced
And then, I'm still amazed as how
-
Not Synced
I had this amazing insight
-
Not Synced
of just two letters.
-
Not Synced
I thought, "ohh wait, I can actually explain to them what i mean."
-
Not Synced
I said, "well actually, if you just take away that part,
-
Not Synced
what I'm really talking about is that."
-
Not Synced
And it was interesting
-
Not Synced
as soon as they saw the word story,
-
Not Synced
everybody.
-
Not Synced
And they saw that, well history doesn't have to be
-
Not Synced
names, dates, facts.
-
Not Synced
I think that's what they were seeing it as.
-
Not Synced
They saw it as 'story' which is actually
-
Not Synced
its origin in French or Latin, 'histoire.'
-
Not Synced
They all of a sudden were totally convinced.
-
Not Synced
They said, "ohh yeah that's exactly how we should do the textbook."
-
Not Synced
And for various reasons, basically because
-
Not Synced
I want it to be a freely licensed,
-
Not Synced
want it to be a freely licensed book,
-
Not Synced
we didn't actually sign a contract.
-
Not Synced
But, it's an example of how powerful 'story' is
-
Not Synced
and how people who actually spend their lives thinking about
-
Not Synced
teaching and reaching students.
-
Not Synced
This educational publisher
-
Not Synced
knew the importance of story.
-
Not Synced
But if you present it as just dry history,
-
Not Synced
it's not so interesting.
-
Not Synced
So, 'story.'
-
Not Synced
Yeah, so it's, let's see.
-
Not Synced
Who haven't I heard from?
-
Not Synced
Adrian, yeah.
-
Not Synced
Okay, so there's a contradiction.
-
Not Synced
The contradiction creates some kind of tension.
-
Not Synced
Right so that tension creates interest.
-
Not Synced
So. Right so every good story needs some kind of tension.
-
Not Synced
You know, so the tension here,
-
Not Synced
the contradiction is that, you know,
-
Not Synced
Hardy wanted to do nothing that could harm people basically.
-
Not Synced
So he was like, "Okay well anything can be used, could be used, even if it's for good for ill.
-
Not Synced
So let me just back off from all that
-
Not Synced
say I'm not going to do anything that has an effect, an application.
-
Not Synced
I'm a pure mathematician."
-
Not Synced
His most famous book is of course, Pure Mathematics.
-
Not Synced
Well maybe his most book famous is that book, which I quoted from,
-
Not Synced
A Mathematician's Apollogy.
-
Not Synced
But his most famous math book is,
-
Not Synced
probably the Course of Pure Mathematics.
-
Not Synced
And that's what he wanted to be known for.
-
Not Synced
And that's in contradiction with the fact
-
Not Synced
that, no, even Hardy couldn't help an application.
-
Not Synced
An actually very very common one,
-
Not Synced
it's taught in every single Introductory Biology course probably.
-
Not Synced
Okay so you need tension.
-
Not Synced
Now, that's a general principle of learning there.
-
Not Synced
You know the idea of story and tension
-
Not Synced
and paradox.
-
Not Synced
So you.
-
Not Synced
The one way that as a undergraduate in physics
-
Not Synced
that I learned a ton of physics,
-
Not Synced
me and my friends were doing problem sets together
-
Not Synced
and you know lots of the problems were,
-
Not Synced
just grinding through math.
-
Not Synced
So we didn't learn a hell of a lot from that.
-
Not Synced
But, we were in the library late at night,
-
Not Synced
you know, the physics library,
-
Not Synced
and ordering pizza and trying to do our problem set
-
Not Synced
and we just got to making up physics paradoxes.
-
Not Synced
You know perpetual motion machines.
-
Not Synced
We'd invent perpetual motion machines
-
Not Synced
and try to get the other person
-
Not Synced
to figure out what was wrong with it.
-
Not Synced
Sometimes we didn't even know what was wrong with it,
-
Not Synced
we'd both try to figure it out.
-
Not Synced
So, from that kind of tension,
-
Not Synced
tension is almost in a way self-teaching.
-
Not Synced
Beacuse, as long as the tension's there you know you're not at the end of the story.
-
Not Synced
Right so you know there's more to do.
-
Not Synced
So the same thing,
-
Not Synced
as long as there was perpetual motion going on,
-
Not Synced
and we hadn't found the reason,
-
Not Synced
we knew weren't done with the problem.
-
Not Synced
We didn't have to ask the teacher,
-
Not Synced
"Well, is this right?"
-
Not Synced
Because it was basically self-teaching
-
Not Synced
as long as the tension was there
-
Not Synced
we knew that we weren't finished.
-
Not Synced
Okay so, yeah,
-
Not Synced
another general principle stories of tension.
-
Not Synced
Let's see there was a comment over...
-
Not Synced
Yes, tell me your name.
-
Not Synced
Wing, Wing-ho.
-
Not Synced
Right.
-
Not Synced
Okay so, yeah the, it, the teacher makes himself more human
-
Not Synced
and makes the person who invented,
-
Not Synced
or one of the people who invented the Hardy-Weinberg Equilibrium more human.
-
Not Synced
So it's actually something much more easy,
-
Not Synced
easy for the students to connect to.
-
Not Synced
Yes, could you tell me your name?
-
Not Synced
Gregory.
-
Not Synced
mhm.
-
Not Synced
Okay so, so Gregor's point which is a good one
-
Not Synced
is that B doesn't have the same quantitative depth as A.
-
Not Synced
And, well actually what I'm going to show you
-
Not Synced
is a way to get to get to this
-
Not Synced
starting from here,
-
Not Synced
and using to the principles that were talking about
-
Not Synced
so that by the time you ger here or to something like this
-
Not Synced
it actually makes sense to the students.
-
Not Synced
Okay so that you can have both.
-
Not Synced
So, that's a promise and,
-
Not Synced
hopefully I'll deliver on it for you.
-
Not Synced
Okay does that?
-
Not Synced
But does that address what you're talking about?
-
Not Synced
That B doesn't have the quantitative depth.
-
Not Synced
And that's true.
-
Not Synced
You, well so you so the,
-
Not Synced
the point made is that
-
Not Synced
if you want to get here you do have to
-
Not Synced
say this at some point.
-
Not Synced
And you know you can't just say the story for an hour.
-
Not Synced
Well, it depends on the purpose of the class.
-
Not Synced
If the purpose is the history of Biology,
-
Not Synced
maybe you would continue with the story
-
Not Synced
or the history of science and war.
-
Not Synced
But if you know you want the students
-
Not Synced
to be able to solve problems in gentics
-
Not Synced
you know maybe you need to go here.
-
Not Synced
Or, what we'll find is the next.
-
Not Synced
I'll continue with option B
-
Not Synced
moving towards A
-
Not Synced
and show you how you can get there
-
Not Synced
without and still have
-
Not Synced
the opening of B preserved.
-
Not Synced
Okay so let's see,
-
Not Synced
there was yes.
-
Not Synced
Yes, can you tell me your name?
-
Not Synced
Okay.
-
Not Synced
Yeah okay, okay so B is, new.
-
Not Synced
Right, A is actually.
-
Not Synced
So against A, A is very familiar and common.
-
Not Synced
So in fact yeah I chose this,
-
Not Synced
I mean i didn't really make a strong ???man??? here.
-
Not Synced
This is how alot of things are introduced
-
Not Synced
for example, in mathematics class
-
Not Synced
theorem proof.
-
Not Synced
Right so the theorem will be introduced
-
Not Synced
without any of the struggle, the wondering,
-
Not Synced
that lead to the theorem.
-
Not Synced
Like, why would anyone care about such a thing?
-
Not Synced
Or if they cared about it,
-
Not Synced
why would they come up with something like that?
-
Not Synced
You know, how can you see that?
-
Not Synced
The, so yeah A is seen very familiar
-
Not Synced
and therefore maybe not as interesting.
-
Not Synced
Although it may also be intrinsically less interesting.
-
Not Synced
For the other reasons, but I take your point too
-
Not Synced
that it could just be familiarity.
-
Not Synced
Yes, can you tell me your name?
-
Not Synced
Meg, yeah.
-
Not Synced
mhm.
-
Not Synced
Okay.
-
Not Synced
Okay so your, your point which is an excellent one aswell
-
Not Synced
is that, you, B doesn't preclude the quatitative.
-
Not Synced
What you could do is you could talk about
-
Not Synced
the history.
-
Not Synced
Start with the history of Hardy saying that
-
Not Synced
which is not chronological because he said that in 1940
-
Not Synced
but that's okay, it doesn't have to be chronological
-
Not Synced
just because it's history.
-
Not Synced
Go to 1940 and then backtrack,
-
Not Synced
to well how did this problem
-
Not Synced
actually come to Hardy's attention
-
Not Synced
which is actually, turns out to be quite
-
Not Synced
an interesting story.
-
Not Synced
And then talk about how they solved it
-
Not Synced
and what was the ???public, date in history???
-
Not Synced
All the quantitative ideas would come out
-
Not Synced
in that way
-
Not Synced
and it would be
-
Not Synced
an almost backwards from the usual way.
-
Not Synced
In one way is that the usual way well,
-
Not Synced
the usual way here is completely general
-
Not Synced
and that's probably not the way
-
Not Synced
it was first figured out.
-
Not Synced
And, the advantage of
-
Not Synced
telling about it that way
-
Not Synced
is that you're preparing students themselves
-
Not Synced
for a research career.
-
Not Synced
Because, they're not,
-
Not Synced
nobody comes up with theorems
-
Not Synced
full blown you know like seen
-
Not Synced
out of the head of Zeus.
-
Not Synced
You come through it from struggle
-
Not Synced
and wondering, "hmm I wonder."
-
Not Synced
Everything comes, has some historical background to it.
-
Not Synced
So the, it's called the, you could say it,
-
Not Synced
the Genetic approach.
-
Not Synced
So in, in Biology, there's a slightness,
-
Not Synced
not misconception, but a saying that
-
Not Synced
???oncology recapitulates via logically.???
-
Not Synced
In other words, that the organism,
-
Not Synced
as it develops in the womb
-
Not Synced
goes through all the evolutionary stages
-
Not Synced
that it went through
-
Not Synced
over the last 200 millions years,
-
Not Synced
or whatever it took to become,
-
Not Synced
say, a person.
-
Not Synced
You're first a fish
-
Not Synced
and then maybe you're a monkey
-
Not Synced
and then a person.
-
Not Synced
So roughly speaking,
-
Not Synced
it's not exactly true.
-
Not Synced
But there's alot of truth to that
-
Not Synced
for learning ideas.
-
Not Synced
That you have to recapitulate the history
-
Not Synced
of ideas to really understand where we are now,
-
Not Synced
the ideas today.
-
Not Synced
So a great example of that
-
Not Synced
is the Newton's Second Law of Motion.
-
Not Synced
The idea that force and acceleration,
-
Not Synced
are connected.
-
Not Synced
So for thousands of years,
-
Not Synced
millions of years, people thought
-
Not Synced
that force and velocity were connected.
-
Not Synced
And it's actually force and acceleration.
-
Not Synced
So actually you can guide people
-
Not Synced
to the understanding of force and acceleration being connected
-
Not Synced
by showing them the history of how people
-
Not Synced
thought it was force and velocity,
-
Not Synced
because, for those same reasons,
-
Not Synced
those are the reasons the student will think it too.
-
Not Synced
So you're actually helping them overcome their misconceptions.
-
Not Synced
So the history actually
-
Not Synced
generally speaking,
-
Not Synced
helps overcome misconceptions.
-
Not Synced
So I'll put that here.
-
Not Synced
And the history is actually quite interesting.
-
Not Synced
I think it was Punnett, from Punnett squares.
-
Not Synced
He played cricket with Hardy,
-
Not Synced
and Hardy loved cricket
-
Not Synced
and he just asked him about this problem
-
Not Synced
and Hardy said, "Ohh yeah I know, it's just this."
-
Not Synced
And sent off a paper to science or nature.