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TEDGlobal 2010

Conrad Wolfram: Teaching kids real math with computers

July 9, 2010

From rockets to stock markets, many of humanity's most thrilling creations are powered by math. So why do kids lose interest in it? Conrad Wolfram says the part of math we teach -- calculation by hand -- isn't just tedious, it's mostly irrelevant to real mathematics and the real world. He presents his radical idea: teaching kids math through computer programming.

Conrad Wolfram - Mathematician
Conrad Wolfram runs the worldwide arm of Wolfram Research, the mathematical lab behind the cutting-edge knowledge engine Wolfram Alpha. Full bio

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We've got a real problem with math education right now.
00:15
Basically, no one's very happy.
00:19
Those learning it
00:22
think it's disconnected,
00:24
uninteresting and hard.
00:26
Those trying to employ them
00:28
think they don't know enough.
00:30
Governments realize that it's a big deal for our economies,
00:32
but don't know how to fix it.
00:35
And teachers are also frustrated.
00:38
Yet math is more important to the world
00:40
than at any point in human history.
00:43
So at one end we've got falling interest
00:45
in education in math,
00:47
and at the other end we've got a more mathematical world,
00:49
a more quantitative world than we ever have had.
00:52
So what's the problem, why has this chasm opened up,
00:56
and what can we do to fix it?
00:58
Well actually, I think the answer
01:01
is staring us right in the face:
01:03
Use computers.
01:05
I believe
01:07
that correctly using computers
01:09
is the silver bullet
01:11
for making math education work.
01:13
So to explain that,
01:16
let me first talk a bit about what math looks like in the real world
01:18
and what it looks like in education.
01:21
See, in the real world
01:23
math isn't necessarily done by mathematicians.
01:25
It's done by geologists,
01:28
engineers, biologists,
01:30
all sorts of different people --
01:32
modeling and simulation.
01:34
It's actually very popular.
01:36
But in education it looks very different --
01:38
dumbed-down problems, lots of calculating,
01:41
mostly by hand.
01:43
Lots of things that seem simple
01:46
and not difficult like in the real world,
01:48
except if you're learning it.
01:50
And another thing about math:
01:53
math sometimes looks like math --
01:55
like in this example here --
01:57
and sometimes it doesn't --
02:00
like "Am I drunk?"
02:02
And then you get an answer that's quantitative in the modern world.
02:07
You wouldn't have expected that a few years back.
02:10
But now you can find out all about --
02:13
unfortunately, my weight is a little higher than that, but --
02:16
all about what happens.
02:19
So let's zoom out a bit and ask,
02:21
why are we teaching people math?
02:23
What's the point of teaching people math?
02:25
And in particular, why are we teaching them math in general?
02:28
Why is it such an important part of education
02:31
as a sort of compulsory subject?
02:34
Well, I think there are about three reasons:
02:36
technical jobs
02:39
so critical to the development of our economies,
02:41
what I call "everyday living" --
02:44
to function in the world today,
02:48
you've got to be pretty quantitative,
02:50
much more so than a few years ago:
02:52
figure out your mortgages,
02:54
being skeptical of government statistics, those kinds of things --
02:56
and thirdly, what I would call something like
03:00
logical mind training, logical thinking.
03:03
Over the years
03:06
we've put so much in society
03:08
into being able to process and think logically. It's part of human society.
03:10
It's very important to learn that
03:13
math is a great way to do that.
03:15
So let's ask another question.
03:17
What is math?
03:19
What do we mean when we say we're doing math,
03:21
or educating people to do math?
03:23
Well, I think it's about four steps, roughly speaking,
03:25
starting with posing the right question.
03:28
What is it that we want to ask? What is it we're trying to find out here?
03:30
And this is the thing most screwed up in the outside world,
03:33
beyond virtually any other part of doing math.
03:35
People ask the wrong question,
03:38
and surprisingly enough, they get the wrong answer,
03:40
for that reason, if not for others.
03:42
So the next thing is take that problem
03:44
and turn it from a real world problem
03:46
into a math problem.
03:48
That's stage two.
03:50
Once you've done that, then there's the computation step.
03:52
Turn it from that into some answer
03:55
in a mathematical form.
03:57
And of course, math is very powerful at doing that.
04:00
And then finally, turn it back to the real world.
04:02
Did it answer the question?
04:04
And also verify it -- crucial step.
04:06
Now here's the crazy thing right now.
04:10
In math education,
04:12
we're spending about perhaps 80 percent of the time
04:14
teaching people to do step three by hand.
04:17
Yet, that's the one step computers can do
04:20
better than any human after years of practice.
04:22
Instead, we ought to be using computers
04:25
to do step three
04:28
and using the students to spend much more effort
04:30
on learning how to do steps one, two and four --
04:33
conceptualizing problems, applying them,
04:35
getting the teacher to run them through how to do that.
04:38
See, crucial point here:
04:41
math is not equal to calculating.
04:43
Math is a much broader subject than calculating.
04:45
Now it's understandable that this has all got intertwined
04:48
over hundreds of years.
04:51
There was only one way to do calculating and that was by hand.
04:53
But in the last few decades
04:56
that has totally changed.
04:58
We've had the biggest transformation of any ancient subject
05:00
that I could ever imagine with computers.
05:03
Calculating was typically the limiting step,
05:07
and now often it isn't.
05:09
So I think in terms of the fact that math
05:11
has been liberated from calculating.
05:13
But that math liberation didn't get into education yet.
05:16
See, I think of calculating, in a sense,
05:19
as the machinery of math.
05:21
It's the chore.
05:23
It's the thing you'd like to avoid if you can, like to get a machine to do.
05:25
It's a means to an end, not an end in itself,
05:29
and automation allows us
05:34
to have that machinery.
05:36
Computers allow us to do that --
05:38
and this is not a small problem by any means.
05:40
I estimated that, just today, across the world,
05:43
we spent about 106 average world lifetimes
05:46
teaching people how to calculate by hand.
05:49
That's an amazing amount of human endeavor.
05:52
So we better be damn sure --
05:55
and by the way, they didn't even have fun doing it, most of them --
05:57
so we better be damn sure
06:00
that we know why we're doing that
06:02
and it has a real purpose.
06:04
I think we should be assuming computers
06:06
for doing the calculating
06:08
and only doing hand calculations where it really makes sense to teach people that.
06:10
And I think there are some cases.
06:13
For example: mental arithmetic.
06:15
I still do a lot of that, mainly for estimating.
06:17
People say, "Is such and such true?"
06:20
And I'll say, "Hmm, not sure." I'll think about it roughly.
06:22
It's still quicker to do that and more practical.
06:24
So I think practicality is one case
06:26
where it's worth teaching people by hand.
06:28
And then there are certain conceptual things
06:30
that can also benefit from hand calculating,
06:32
but I think they're relatively small in number.
06:34
One thing I often ask about
06:36
is ancient Greek and how this relates.
06:38
See, the thing we're doing right now
06:41
is we're forcing people to learn mathematics.
06:43
It's a major subject.
06:45
I'm not for one minute suggesting that, if people are interested in hand calculating
06:47
or in following their own interests
06:50
in any subject however bizarre --
06:52
they should do that.
06:54
That's absolutely the right thing,
06:56
for people to follow their self-interest.
06:58
I was somewhat interested in ancient Greek,
07:00
but I don't think that we should force the entire population
07:02
to learn a subject like ancient Greek.
07:05
I don't think it's warranted.
07:07
So I have this distinction between what we're making people do
07:09
and the subject that's sort of mainstream
07:12
and the subject that, in a sense, people might follow with their own interest
07:14
and perhaps even be spiked into doing that.
07:17
So what are the issues people bring up with this?
07:19
Well one of them is, they say, you need to get the basics first.
07:22
You shouldn't use the machine
07:25
until you get the basics of the subject.
07:27
So my usual question is, what do you mean by "basics?"
07:29
Basics of what?
07:32
Are the basics of driving a car
07:34
learning how to service it, or design it for that matter?
07:36
Are the basics of writing learning how to sharpen a quill?
07:39
I don't think so.
07:43
I think you need to separate the basics of what you're trying to do
07:45
from how it gets done
07:48
and the machinery of how it gets done
07:50
and automation allows you to make that separation.
07:54
A hundred years ago, it's certainly true that to drive a car
07:57
you kind of needed to know a lot about the mechanics of the car
08:00
and how the ignition timing worked and all sorts of things.
08:02
But automation in cars
08:06
allowed that to separate,
08:08
so driving is now a quite separate subject, so to speak,
08:10
from engineering of the car
08:13
or learning how to service it.
08:16
So automation allows this separation
08:20
and also allows -- in the case of driving,
08:22
and I believe also in the future case of maths --
08:24
a democratized way of doing that.
08:26
It can be spread across a much larger number of people
08:28
who can really work with that.
08:30
So there's another thing that comes up with basics.
08:33
People confuse, in my view,
08:35
the order of the invention of the tools
08:37
with the order in which they should use them for teaching.
08:40
So just because paper was invented before computers,
08:43
it doesn't necessarily mean you get more to the basics of the subject
08:46
by using paper instead of a computer
08:49
to teach mathematics.
08:51
My daughter gave me a rather nice anecdote on this.
08:55
She enjoys making what she calls "paper laptops."
08:58
(Laughter)
09:01
So I asked her one day, "You know, when I was your age,
09:03
I didn't make these.
09:05
Why do you think that was?"
09:07
And after a second or two, carefully reflecting,
09:09
she said, "No paper?"
09:11
(Laughter)
09:13
If you were born after computers and paper,
09:19
it doesn't really matter which order you're taught with them in,
09:21
you just want to have the best tool.
09:24
So another one that comes up is "Computers dumb math down."
09:26
That somehow, if you use a computer,
09:29
it's all mindless button-pushing,
09:31
but if you do it by hand,
09:33
it's all intellectual.
09:35
This one kind of annoys me, I must say.
09:37
Do we really believe
09:40
that the math that most people are doing in school
09:42
practically today
09:44
is more than applying procedures
09:46
to problems they don't really understand, for reasons they don't get?
09:48
I don't think so.
09:51
And what's worse, what they're learning there isn't even practically useful anymore.
09:53
Might have been 50 years ago, but it isn't anymore.
09:56
When they're out of education, they do it on a computer.
09:59
Just to be clear, I think computers can really help with this problem,
10:02
actually make it more conceptual.
10:05
Now, of course, like any great tool,
10:07
they can be used completely mindlessly,
10:09
like turning everything into a multimedia show,
10:11
like the example I was shown of solving an equation by hand,
10:14
where the computer was the teacher --
10:17
show the student how to manipulate and solve it by hand.
10:19
This is just nuts.
10:22
Why are we using computers to show a student how to solve a problem by hand
10:24
that the computer should be doing anyway?
10:27
All backwards.
10:29
Let me show you
10:31
that you can also make problems harder to calculate.
10:33
See, normally in school,
10:36
you do things like solve quadratic equations.
10:38
But you see, when you're using a computer,
10:41
you can just substitute.
10:44
You can make it a quartic equation. Make it kind of harder, calculating-wise.
10:48
Same principles applied --
10:50
calculations, harder.
10:52
And problems in the real world
10:54
look nutty and horrible like this.
10:56
They've got hair all over them.
10:58
They're not just simple, dumbed-down things that we see in school math.
11:00
And think of the outside world.
11:04
Do we really believe that engineering and biology
11:06
and all of these other things
11:08
that have so benefited from computers and maths
11:10
have somehow conceptually gotten reduced by using computers?
11:12
I don't think so -- quite the opposite.
11:15
So the problem we've really got in math education
11:18
is not that computers might dumb it down,
11:21
but that we have dumbed-down problems right now.
11:24
Well, another issue people bring up
11:27
is somehow that hand calculating procedures
11:29
teach understanding.
11:31
So if you go through lots of examples,
11:33
you can get the answer,
11:35
you can understand how the basics of the system work better.
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I think there is one thing that I think very valid here,
11:40
which is that I think understanding procedures and processes is important.
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But there's a fantastic way to do that in the modern world.
11:47
It's called programming.
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Programming is how most procedures and processes
11:53
get written down these days,
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and it's also a great way
11:57
to engage students much more
11:59
and to check they really understand.
12:01
If you really want to check you understand math
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then write a program to do it.
12:05
So programming is the way I think we should be doing that.
12:08
So to be clear, what I really am suggesting here
12:11
is we have a unique opportunity
12:13
to make maths both more practical
12:15
and more conceptual, simultaneously.
12:17
I can't think of any other subject where that's recently been possible.
12:20
It's usually some kind of choice
12:23
between the vocational and the intellectual.
12:25
But I think we can do both at the same time here.
12:27
And we open up so many more possibilities.
12:32
You can do so many more problems.
12:35
What I really think we gain from this
12:37
is students getting intuition and experience
12:39
in far greater quantities than they've ever got before.
12:42
And experience of harder problems --
12:45
being able to play with the math, interact with it,
12:47
feel it.
12:49
We want people who can feel the math instinctively.
12:51
That's what computers allow us to do.
12:54
Another thing it allows us to do is reorder the curriculum.
12:57
Traditionally it's been by how difficult it is to calculate,
13:00
but now we can reorder it
13:02
by how difficult it is to understand the concepts,
13:04
however hard the calculating.
13:06
So calculus has traditionally been taught very late.
13:08
Why is this?
13:11
Well, it's damn hard doing the calculations, that's the problem.
13:13
But actually many of the concepts
13:17
are amenable to a much younger age group.
13:19
This was an example I built for my daughter.
13:22
And very, very simple.
13:25
We were talking about what happens
13:28
when you increase the number of sides of a polygon
13:30
to a very large number.
13:32
And of course, it turns into a circle.
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And by the way, she was also very insistent
13:38
on being able to change the color,
13:40
an important feature for this demonstration.
13:42
You can see that this is a very early step
13:46
into limits and differential calculus
13:49
and what happens when you take things to an extreme --
13:51
and very small sides and a very large number of sides.
13:54
Very simple example.
13:56
That's a view of the world
13:58
that we don't usually give people for many, many years after this.
14:00
And yet, that's a really important practical view of the world.
14:03
So one of the roadblocks we have
14:06
in moving this agenda forward
14:09
is exams.
14:12
In the end, if we test everyone by hand in exams,
14:14
it's kind of hard to get the curricula changed
14:17
to a point where they can use computers
14:20
during the semesters.
14:22
And one of the reasons it's so important --
14:25
so it's very important to get computers in exams.
14:27
And then we can ask questions, real questions,
14:30
questions like, what's the best life insurance policy to get? --
14:33
real questions that people have in their everyday lives.
14:36
And you see, this isn't some dumbed-down model here.
14:40
This is an actual model where we can be asked to optimize what happens.
14:42
How many years of protection do I need?
14:45
What does that do to the payments
14:47
and to the interest rates and so forth?
14:49
Now I'm not for one minute suggesting it's the only kind of question
14:52
that should be asked in exams,
14:55
but I think it's a very important type
14:57
that right now just gets completely ignored
14:59
and is critical for people's real understanding.
15:02
So I believe [there is] critical reform
15:05
we have to do in computer-based math.
15:08
We have got to make sure
15:10
that we can move our economies forward,
15:12
and also our societies,
15:15
based on the idea that people can really feel mathematics.
15:17
This isn't some optional extra.
15:22
And the country that does this first
15:25
will, in my view, leapfrog others
15:27
in achieving a new economy even,
15:30
an improved economy,
15:33
an improved outlook.
15:35
In fact, I even talk about us moving
15:37
from what we often call now the "knowledge economy"
15:39
to what we might call a "computational knowledge economy,"
15:42
where high-level math is integral to what everyone does
15:45
in the way that knowledge currently is.
15:48
We can engage so many more students with this,
15:50
and they can have a better time doing it.
15:53
And let's understand:
15:56
this is not an incremental sort of change.
15:58
We're trying to cross the chasm here
16:02
between school math and the real-world math.
16:04
And you know if you walk across a chasm,
16:06
you end up making it worse than if you didn't start at all --
16:08
bigger disaster.
16:11
No, what I'm suggesting
16:13
is that we should leap off,
16:15
we should increase our velocity
16:17
so it's high,
16:19
and we should leap off one side and go the other --
16:21
of course, having calculated our differential equation very carefully.
16:24
(Laughter)
16:27
So I want to see
16:29
a completely renewed, changed math curriculum
16:31
built from the ground up,
16:33
based on computers being there,
16:35
computers that are now ubiquitous almost.
16:37
Calculating machines are everywhere
16:39
and will be completely everywhere in a small number of years.
16:41
Now I'm not even sure if we should brand the subject as math,
16:44
but what I am sure is
16:48
it's the mainstream subject of the future.
16:50
Let's go for it,
16:53
and while we're about it,
16:56
let's have a bit of fun,
16:58
for us, for the students and for TED here.
17:00
Thanks.
17:03
(Applause)
17:05

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Conrad Wolfram - Mathematician
Conrad Wolfram runs the worldwide arm of Wolfram Research, the mathematical lab behind the cutting-edge knowledge engine Wolfram Alpha.

Why you should listen

Conrad Wolfram is the strategic director of Wolfram Research, where his job, in a nutshell, is understanding and finding new uses for the Mathematica technology. Wolfram is especially passionate about finding uses for Mathematica outside of pure computation, using it as a development platform for products that help communicate big ideas. The Demonstrations tool, for instance, makes a compelling case for never writing out another equation -- instead displaying data in interactive, graphical form.

Wolfram's work points up the changing nature of math in the past 30 years, as we've moved from adding machines to calculators to sophisticated math software, allowing us to achieve ever more complex computational feats. But, Wolfram says, many schools are still focused on hand-calculating; using automation, such as a piece of software, to do math is sometimes seen as cheating. This keeps schools from spending the time they need on the new tools of science and mathematics. As they gain significance for everyday living, he suggests, we need to learn to take advantage of these tools and learn to use them young. Learn more at computerbasedmath.org.

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