Rote Learning Fragments the World: Sanjoy Mahajan at TEDxCaltech

Sanjoy Mahajan obtained his PhD in theoretical physics from Caltech in 1998, after an undergraduate degree in mathematics from Oxford and in physics from Stanford. In March 2010, MIT Press published his textbook "Street-Fighting Mathematics: The Art of Educated Guessing" and "Opportunistic Problem Solving," available in print and online under a Creative Commons Noncommercial ShareAlike license.

In problem solving as in street-fighting:
Rules are for fools!

(Laughter)

(Applause)

Let's see how far we can go
by bending rules

as we estimate the fuel efficiency,

the miles per gallon of a 747.

The fuel is used to fight drag,

the force of air resistance,

what you would feel

if you stuck your hand
out of a moving car --

don't try this at home --

or try to run in a swimming pool.

There are at least two ways
that you can use

to figure out the drag.

You could spend
10 years learning physics

and you write down
the Navier–Stokes equations:

the differential equations
of fluid dynamics.

And then you spend another
10 years learning mathematics

to solve for the pressure.

And whereupon you find

that actually there's no exact solution

for the flow around a 747,

or, in fact, for most of the situations

which you want to know.

Rigor, the rigorous approach,

the exact approach
has produced paralysis,

rigor mortis.

(Laughter)

We need a different way.

The street-fighting way,

which starts with
a home experiment.

Chair please.

Props please.

(Laughter)

Small cone, big cone. Coffee filters.

They're the same shape,

but this one has
one-fourth the area.

This one has four times
the area, twice the diameter,

but otherwise the same shape.

When I drop them,

how fast do they fall
relative to one another?

Is the big one roughly twice as fast?

Are they comparable in speed?

Or is the small one
roughly twice as fast?

Take ten seconds and think.

What do you believe?
What does your gut tell you?

And then we'll take a vote.

Check with your neighbor.

(Laughter)

(Crowd murmuring)

OK, let's take a vote.

You don't have to agree
with your neighbor.

(Laughter)

That's the beauty of democracy.

So, cheer if you believe
that the big cone

will fall roughly twice as fast
as the small cone.

(Faint cheering)

OK, I hear a few.

Cheer if you believe
that they'll be roughly comparable.

(Louder cheer)

And cheer if you believe the small cone

will be roughly twice as fast.

(Loudest cheer)

A lot of cheering for that one.

OK, well, as Feynman said and believed,

in science we have a
supreme court: experiment.

So, let's do the experiment!

One, two, three.

(Cheering) (Applause)

They're almost the same.

Within experimental error.

So what does that mean?

What can we use
that experiment to tell us?

Well,

the cones fell at the same speed.

They fall in the same air.
It has the same density.

The same properties.
The same viscosity.

The only things different
between the two cones

is this one has four times the area,

the cross sectional area of this one,

and their drag force is different.

How different?

Well, the drag force
is equal to the weight.

Because they were falling
at a steady speed with no acceleration.

So the drag and the weight cancel.

So we have a very sensitive measure

of the drag force
without any force sensors.

All we do is measure the weight.

So this one has four times
as much paper as this one.

So it's four times heavier,
four times the drag.

Only change, four times the area.

The conclusion:
drag is proportional to area.

Not square root of area,
not the square of the area.

but just the area.

That's the result
of our home experiment

without the rigorous
rigor mortis method.

How can we use that?

Well, that one constraint,

along with the next street-fighting tool

of dimensional analysis,
solves the drag force.

We match their dimensions.

We match the dimensions of force,
drag force on one side

with what we have on the other,

which is area, density, speed
and viscosity.

But we already know how to put in
the area, just one of them.

That gives us length squared,
meters squared.

Now we look and we say,
"Oh, there's kilograms over here,

we have to get a kilogram over here."

The only place to get it from is density.

Speed and viscosity, the kinematic
viscosity, have no mass in them.

So we put in one density.

Now what we need still
is meter squared / second squared,

out of speed and viscosity.

The only way to make it is speed squared.

So there is our drag force.

One experiment for a constraint.

Dimensional analysis
for the rest of the constraints.

Drag Force =
Area x Density x Speed squared.

How can we use this?

Well, the fuel consumption
is proportional to the drag force.

So, let's compare the fuel consumption
of a plane with a car.

Rather than calculating the plane
from scratch, compare it to a car.

Another street-fighting technique.

So there're three factors in
the comparison, in the ratio:

the area, the air density
and the speed squared.

Do them one at a time.

So, the area. Well,
in the old days of plane travel,

you could lie down on three seats

and there were three sets of those seats.

So three people wide.
Plane is about three people high.

So it's nine square people.

A car: Well,
from nocturnal activities in cars

you know you can sort of lie down
in cars a bit uncomfortably.

(Laughter)

And you can stand up.
So it's one square person.

So it's roughly a ratio of ten,

maybe nine or ten.

So the plane is 10 times
less fuel efficient for that.

What about air density?

Well, the planes fly high,
about Mt. Everest.

So the density is about one third.

So that helps the plane.

But they fly about ten times faster,

600 miles an hour versus 60.

That means planes pay a factor
of a hundred, 10 squared.

The result is planes are 300 times

less fuel efficient than cars.

Oh, no. By flying here,
did I damage the environment

300 times compared to driving?
(Gasp)

What saves it?

300 people on my plane!

So the conclusion is planes and cars

are roughly equally fuel efficient.

(Laughter)

All from that.

(Applause)

So let's say the plane is
30 miles per gallon.

Crossing the country
back and forth 6,000 miles,

30 miles per gallon, 2 dollars a gallon.

That's 400 dollars of gasoline.

That's not that different than
the price of my plane ticket,

which may explain why
airline companies teeter on bankruptcy

and why they charge us for peanuts.

(Laughter)

So connection between
the 747 and the cones.

They increase our enjoyment of the world

and expand our perception.

And that, making connections here

was enabled by street-fighting reasoning,

by getting away from rigor mortis.

Making connections is so important

because it builds ideas and isolated facts

into a coherent story.

Imagine each dot is an idea

and the lines are the connections
between them.

As I increase the fraction of connections

from 40% to 50%, to 60%,

the big story, the red connection network,

grows to fill the whole space.

That's the long lasting learning.

That's what we want
to build in our thinking

and in our teaching.

The goal of teaching should be

to implant a way of thinking
that enables a student

to learn in one year
what the teacher learned in two years.

Only in that way
can we continue to advance

from one generation to the next.

In fifty years, all education

will, I believe and dream,
be based on this principle.

Richard Feynman, I think,
would have agreed.

Thank you.

(Applause)