conrad wolfram TED talk (transcript)
Joe Corneli
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 little 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 not 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 calculation 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, 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 really 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 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 when you're using a computer, you can just substitute. 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're 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 got
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 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 a
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 critical reform we have to do in computer-based math. We
have 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)
-- http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html