Conrad Wolfram: Teaching kids real math with computers

We've got a real problem with math education right now.

Basically, no one's very happy.

Those learning it

think it's disconnected,

uninteresting and hard.

Those trying to employ them

think they don't know enough.

Governments realize that it's a big deal for our economies,

but don't know how to fix it.

And teachers are also frustrated.

Yet math is more important to the world

than at any point in human history.

So at one end we've got falling interest

in education in math,

and at the other end we've got a more mathematical world,

a more quantitative world than we ever have had.

So what's the problem, why has this chasm opened up,

and what can we do to fix it?

Well actually, I think the answer

is staring us right in the face:

Use computers.

I believe

that correctly using computers

is the silver bullet

for making math education work.

So to explain that,

let me first talk a bit about what math looks like in the real world

and what it looks like in education.

See, in the real world

math isn't necessarily done by mathematicians.

It's done by geologists,

engineers, biologists,

all sorts of different people --

modeling and simulation.

It's actually very popular.

But in education it looks very different --

dumbed-down problems, lots of calculating,

mostly by hand.

Lots of things that seem simple

and not difficult like in the real world,

except if you're learning it.

And another thing about math:

math sometimes looks like math --

like in this example here --

and sometimes it doesn't --

like "Am I drunk?"

And then you get an answer that's quantitative in the modern world.

You wouldn't have expected that a few years back.

But now you can find out all about --

unfortunately, my weight is a little higher than that, but --

all about what happens.

So let's zoom out a bit and ask,

why are we teaching people math?

What's the point of teaching people math?

And in particular, why are we teaching them math in general?

Why is it such an important part of education

as a sort of compulsory subject?

Well, I think there are about three reasons:

technical jobs

so critical to the development of our economies,

what I call "everyday living" --

to function in the world today,

you've got to be pretty quantitative,

much more so than a few years ago:

figure out your mortgages,

being skeptical of government statistics, those kinds of things --

and thirdly, what I would call something like

logical mind training, logical thinking.

Over the years

we've put so much in society

into being able to process and think logically. It's part of human society.

It's very important to learn that

math is a great way to do that.

So let's ask another question.

What is math?

What do we mean when we say we're doing math,

or educating people to do math?

Well, I think it's about four steps, roughly speaking,

starting with posing the right question.

What is it that we want to ask? What is it we're trying to find out here?

And this is the thing most screwed up in the outside world,

beyond virtually any other part of doing math.

People ask the wrong question,

and surprisingly enough, they get the wrong answer,

for that reason, if not for others.

So the next thing is take that problem

and turn it from a real world problem

into a math problem.

That's stage two.

Once you've done that, then there's the computation step.

Turn it from that into some answer

in a mathematical form.

And of course, math is very powerful at doing that.

And then finally, turn it back to the real world.

Did it answer the question?

And also verify it -- crucial step.

Now here's the crazy thing right now.

In math education,

we're spending about perhaps 80 percent of the time

teaching people to do step three by hand.

Yet, that's the one step computers can do

better than any human after years of practice.

Instead, we ought to be using computers

to do step three

and using the students to spend much more effort

on learning how to do steps one, two and four --

conceptualizing problems, applying them,

getting the teacher to run them through how to do that.

See, crucial point here:

math is not equal to calculating.

Math is a much broader subject than calculating.

Now it's understandable that this has all got intertwined

over hundreds of years.

There was only one way to do calculating and that was by hand.

But in the last few decades

that has totally changed.

We've had the biggest transformation of any ancient subject

that I could ever imagine with computers.

Calculating was typically the limiting step,

and now often it isn't.

So I think in terms of the fact that math

has been liberated from calculating.

But that math liberation didn't get into education yet.

See, I think of calculating, in a sense,

as the machinery of math.

It's the chore.

It's the thing you'd like to avoid if you can, like to get a machine to do.

It's a means to an end, not an end in itself,

and automation allows us

to have that machinery.

Computers allow us to do that --

and this is not a small problem by any means.

I estimated that, just today, across the world,

we spent about 106 average world lifetimes

teaching people how to calculate by hand.

That's an amazing amount of human endeavor.

So we better be damn sure --

and by the way, they didn't even have fun doing it, most of them --

so we better be damn sure

that we know why we're doing that

and it has a real purpose.

I think we should be assuming computers

for doing the calculating

and only doing hand calculations where it really makes sense to teach people that.

And I think there are some cases.

For example: mental arithmetic.

I still do a lot of that, mainly for estimating.

People say, "Is such and such true?"

And I'll say, "Hmm, not sure." I'll think about it roughly.

It's still quicker to do that and more practical.

So I think practicality is one case

where it's worth teaching people by hand.

And then there are certain conceptual things

that can also benefit from hand calculating,

but I think they're relatively small in number.

One thing I often ask about

is ancient Greek and how this relates.

See, the thing we're doing right now

is we're forcing people to learn mathematics.

It's a major subject.

I'm not for one minute suggesting that, if people are interested in hand calculating

or in following their own interests

in any subject however bizarre --

they should do that.

That's absolutely the right thing,

for people to follow their self-interest.

I was somewhat interested in ancient Greek,

but I don't think that we should force the entire population

to learn a subject like ancient Greek.

I don't think it's warranted.

So I have this distinction between what we're making people do

and the subject that's sort of mainstream

and the subject that, in a sense, people might follow with their own interest

and perhaps even be spiked into doing that.

So what are the issues people bring up with this?

Well one of them is, they say, you need to get the basics first.

You shouldn't use the machine

until you get the basics of the subject.

So my usual question is, what do you mean by "basics?"

Basics of what?

Are the basics of driving a car

learning how to service it, or design it for that matter?

Are the basics of writing learning how to sharpen a quill?

I don't think so.

I think you need to separate the basics of what you're trying to do

from how it gets done

and the machinery of how it gets done

and automation allows you to make that separation.

A hundred years ago, it's certainly true that to drive a car

you kind of needed to know a lot about the mechanics of the car

and how the ignition timing worked and all sorts of things.

But automation in cars

allowed that to separate,

so driving is now a quite separate subject, so to speak,

from engineering of the car

or learning how to service it.

So automation allows this separation

and also allows -- in the case of driving,

and I believe also in the future case of maths --

a democratized way of doing that.

It can be spread across a much larger number of people

who can really work with that.

So there's another thing that comes up with basics.

People confuse, in my view,

the order of the invention of the tools

with the order in which they should use them for teaching.

So just because paper was invented before computers,

it doesn't necessarily mean you get more to the basics of the subject

by using paper instead of a computer

to teach mathematics.

My daughter gave me a rather nice anecdote on this.

She enjoys making what she calls "paper laptops."

(Laughter)

So I asked her one day, "You know, when I was your age,

I didn't make these.

Why do you think that was?"

And after a second or two, carefully reflecting,

she said, "No paper?"

(Laughter)

If you were born after computers and paper,

it doesn't really matter which order you're taught with them in,

you just want to have the best tool.

So another one that comes up is "Computers dumb math down."

That somehow, if you use a computer,

it's all mindless button-pushing,

but if you do it by hand,

it's all intellectual.

This one kind of annoys me, I must say.

Do we really believe

that the math that most people are doing in school

practically today

is more than applying procedures

to problems they don't really understand, for reasons they don't get?

I don't think so.

And what's worse, what they're learning there isn't even practically useful anymore.

Might have been 50 years ago, but it isn't anymore.

When they're out of education, they do it on a computer.

Just to be clear, I think computers can really help with this problem,

actually make it more conceptual.

Now, of course, like any great tool,

they can be used completely mindlessly,

like turning everything into a multimedia show,

like the example I was shown of solving an equation by hand,

where the computer was the teacher --

show the student how to manipulate and solve it by hand.

This is just nuts.

Why are we using computers to show a student how to solve a problem by hand

that the computer should be doing anyway?

All backwards.

Let me show you

that you can also make problems harder to calculate.

See, normally in school,

you do things like solve quadratic equations.

But you see, when you're using a computer,

you can just substitute.

You can make it a quartic equation. Make it kind of harder, calculating-wise.

Same principles applied --

calculations, harder.

And problems in the real world

look nutty and horrible like this.

They've got hair all over them.

They're not just simple, dumbed-down things that we see in school math.

And think of the outside world.

Do we really believe that engineering and biology

and all of these other things

that have so benefited from computers and maths

have somehow conceptually gotten reduced by using computers?

I don't think so -- quite the opposite.

So the problem we've really got in math education

is not that computers might dumb it down,

but that we have dumbed-down problems right now.

Well, another issue people bring up

is somehow that hand calculating procedures

teach understanding.

So if you go through lots of examples,

you can get the answer,

you can understand how the basics of the system work better.

I think there is one thing that I think very valid here,

which is that I think understanding procedures and processes is important.

But there's a fantastic way to do that in the modern world.

It's called programming.

Programming is how most procedures and processes

get written down these days,

and it's also a great way

to engage students much more

and to check they really understand.

If you really want to check you understand math

then write a program to do it.

So programming is the way I think we should be doing that.

So to be clear, what I really am suggesting here

is we have a unique opportunity

to make maths both more practical

and more conceptual, simultaneously.

I can't think of any other subject where that's recently been possible.

It's usually some kind of choice

between the vocational and the intellectual.

But I think we can do both at the same time here.

And we open up so many more possibilities.

You can do so many more problems.

What I really think we gain from this

is students getting intuition and experience

in far greater quantities than they've ever got before.

And experience of harder problems --

being able to play with the math, interact with it,

feel it.

We want people who can feel the math instinctively.

That's what computers allow us to do.

Another thing it allows us to do is reorder the curriculum.

Traditionally it's been by how difficult it is to calculate,

but now we can reorder it

by how difficult it is to understand the concepts,

however hard the calculating.

So calculus has traditionally been taught very late.

Why is this?

Well, it's damn hard doing the calculations, that's the problem.

But actually many of the concepts

are amenable to a much younger age group.

This was an example I built for my daughter.

And very, very simple.

We were talking about what happens

when you increase the number of sides of a polygon

to a very large number.

And of course, it turns into a circle.

And by the way, she was also very insistent

on being able to change the color,

an important feature for this demonstration.

You can see that this is a very early step

into limits and differential calculus

and what happens when you take things to an extreme --

and very small sides and a very large number of sides.

Very simple example.

That's a view of the world

that we don't usually give people for many, many years after this.

And yet, that's a really important practical view of the world.

So one of the roadblocks we have

in moving this agenda forward

is exams.

In the end, if we test everyone by hand in exams,

it's kind of hard to get the curricula changed

to a point where they can use computers

during the semesters.

And one of the reasons it's so important --

so it's very important to get computers in exams.

And then we can ask questions, real questions,

questions like, what's the best life insurance policy to get? --

real questions that people have in their everyday lives.

And you see, this isn't some dumbed-down model here.

This is an actual model where we can be asked to optimize what happens.

How many years of protection do I need?

What does that do to the payments

and to the interest rates and so forth?

Now I'm not for one minute suggesting it's the only kind of question

that should be asked in exams,

but I think it's a very important type

that right now just gets completely ignored

and is critical for people's real understanding.

So I believe [there is] critical reform

we have to do in computer-based math.

We have got to make sure

that we can move our economies forward,

and also our societies,

based on the idea that people can really feel mathematics.

This isn't some optional extra.

And the country that does this first

will, in my view, leapfrog others

in achieving a new economy even,

an improved economy,

an improved outlook.

In fact, I even talk about us moving

from what we often call now the "knowledge economy"

to what we might call a "computational knowledge economy,"

where high-level math is integral to what everyone does

in the way that knowledge currently is.

We can engage so many more students with this,

and they can have a better time doing it.

And let's understand:

this is not an incremental sort of change.

We're trying to cross the chasm here

between school math and the real-world math.

And you know if you walk across a chasm,

you end up making it worse than if you didn't start at all --

bigger disaster.

No, what I'm suggesting

is that we should leap off,

we should increase our velocity

so it's high,

and we should leap off one side and go the other --

of course, having calculated our differential equation very carefully.

(Laughter)

So I want to see

a completely renewed, changed math curriculum

built from the ground up,

based on computers being there,

computers that are now ubiquitous almost.

Calculating machines are everywhere

and will be completely everywhere in a small number of years.

Now I'm not even sure if we should brand the subject as math,

but what I am sure is

it's the mainstream subject of the future.

Let's go for it,

and while we're about it,

let's have a bit of fun,

for us, for the students and for TED here.

Thanks.

(Applause)