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Annotated captions of Conrad Wolfram: Teaching kids real math with computers in English

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

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Basically, no one's very happy.

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Those learning it

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think it's disconnected,

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uninteresting and hard.

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Those trying to employ them

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think they don't know enough.

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

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but don't know how to fix it.

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And teachers are also frustrated.

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Yet math is more important to the world

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than at any point in human history.

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So at one end we've got falling interest

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in education in math,

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

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a more quantitative world than we ever have had.

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

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and what can we do to fix it?

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Well actually, I think the answer

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is staring us right in the face:

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Use computers.

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I believe

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that correctly using computers

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is the silver bullet

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for making math education work.

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So to explain that,

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

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and what it looks like in education.

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See, in the real world

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math isn't necessarily done by mathematicians.

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It's done by geologists,

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engineers, biologists,

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all sorts of different people --

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modeling and simulation.

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It's actually very popular.

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But in education it looks very different --

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dumbed-down problems, lots of calculating,

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mostly by hand.

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Lots of things that seem simple

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and not difficult like in the real world,

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except if you're learning it.

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And another thing about math:

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math sometimes looks like math --

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like in this example here --

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and sometimes it doesn't --

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like "Am I drunk?"

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

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

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But now you can find out all about --

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

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all about what happens.

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So let's zoom out a bit and ask,

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why are we teaching people math?

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What's the point of teaching people math?

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

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Why is it such an important part of education

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as a sort of compulsory subject?

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Well, I think there are about three reasons:

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technical jobs

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so critical to the development of our economies,

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what I call "everyday living" --

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to function in the world today,

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you've got to be pretty quantitative,

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much more so than a few years ago:

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figure out your mortgages,

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

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and thirdly, what I would call something like

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logical mind training, logical thinking.

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Over the years

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we've put so much in society

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

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It's very important to learn that

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math is a great way to do that.

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So let's ask another question.

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What is math?

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

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or educating people to do math?

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

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starting with posing the right question.

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

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

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beyond virtually any other part of doing math.

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People ask the wrong question,

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and surprisingly enough, they get the wrong answer,

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for that reason, if not for others.

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So the next thing is take that problem

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and turn it from a real world problem

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into a math problem.

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That's stage two.

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

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Turn it from that into some answer

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in a mathematical form.

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

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

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Did it answer the question?

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And also verify it -- crucial step.

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Now here's the crazy thing right now.

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In math education,

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we're spending about perhaps 80 percent of the time

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teaching people to do step three by hand.

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Yet, that's the one step computers can do

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better than any human after years of practice.

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Instead, we ought to be using computers

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to do step three

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and using the students to spend much more effort

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

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conceptualizing problems, applying them,

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

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See, crucial point here:

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math is not equal to calculating.

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Math is a much broader subject than calculating.

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Now it's understandable that this has all got intertwined

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over hundreds of years.

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

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But in the last few decades

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that has totally changed.

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We've had the biggest transformation of any ancient subject

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that I could ever imagine with computers.

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Calculating was typically the limiting step,

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and now often it isn't.

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So I think in terms of the fact that math

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has been liberated from calculating.

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

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See, I think of calculating, in a sense,

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as the machinery of math.

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It's the chore.

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

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

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and automation allows us

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to have that machinery.

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Computers allow us to do that --

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and this is not a small problem by any means.

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I estimated that, just today, across the world,

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we spent about 106 average world lifetimes

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teaching people how to calculate by hand.

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That's an amazing amount of human endeavor.

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So we better be damn sure --

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

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so we better be damn sure

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that we know why we're doing that

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and it has a real purpose.

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I think we should be assuming computers

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for doing the calculating

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

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And I think there are some cases.

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For example: mental arithmetic.

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

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People say, "Is such and such true?"

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

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

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So I think practicality is one case

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where it's worth teaching people by hand.

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And then there are certain conceptual things

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that can also benefit from hand calculating,

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but I think they're relatively small in number.

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One thing I often ask about

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is ancient Greek and how this relates.

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See, the thing we're doing right now

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is we're forcing people to learn mathematics.

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It's a major subject.

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

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or in following their own interests

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in any subject however bizarre --

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they should do that.

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That's absolutely the right thing,

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for people to follow their self-interest.

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I was somewhat interested in ancient Greek,

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

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to learn a subject like ancient Greek.

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I don't think it's warranted.

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

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and the subject that's sort of mainstream

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

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and perhaps even be spiked into doing that.

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So what are the issues people bring up with this?

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

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You shouldn't use the machine

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until you get the basics of the subject.

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

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Basics of what?

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Are the basics of driving a car

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

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

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I don't think so.

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

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from how it gets done

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and the machinery of how it gets done

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and automation allows you to make that separation.

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

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

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

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But automation in cars

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allowed that to separate,

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

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from engineering of the car

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or learning how to service it.

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So automation allows this separation

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and also allows -- in the case of driving,

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

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a democratized way of doing that.

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It can be spread across a much larger number of people

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who can really work with that.

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

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People confuse, in my view,

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the order of the invention of the tools

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

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So just because paper was invented before computers,

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

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by using paper instead of a computer

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to teach mathematics.

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My daughter gave me a rather nice anecdote on this.

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She enjoys making what she calls "paper laptops."

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(Laughter)

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

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I didn't make these.

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Why do you think that was?"

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And after a second or two, carefully reflecting,

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she said, "No paper?"

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(Laughter)

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If you were born after computers and paper,

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

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you just want to have the best tool.

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

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That somehow, if you use a computer,

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it's all mindless button-pushing,

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but if you do it by hand,

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it's all intellectual.

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This one kind of annoys me, I must say.

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Do we really believe

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that the math that most people are doing in school

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practically today

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is more than applying procedures

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

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I don't think so.

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

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

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

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

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actually make it more conceptual.

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Now, of course, like any great tool,

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they can be used completely mindlessly,

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like turning everything into a multimedia show,

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

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where the computer was the teacher --

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

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This is just nuts.

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

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that the computer should be doing anyway?

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All backwards.

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Let me show you

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that you can also make problems harder to calculate.

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See, normally in school,

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you do things like solve quadratic equations.

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

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you can just substitute.

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

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Same principles applied --

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calculations, harder.

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And problems in the real world

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look nutty and horrible like this.

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They've got hair all over them.

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

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And think of the outside world.

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Do we really believe that engineering and biology

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and all of these other things

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that have so benefited from computers and maths

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have somehow conceptually gotten reduced by using computers?

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

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So the problem we've really got in math education

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is not that computers might dumb it down,

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but that we have dumbed-down problems right now.

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Well, another issue people bring up

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is somehow that hand calculating procedures

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teach understanding.

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So if you go through lots of examples,

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you can get the answer,

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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,

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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.

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It's called programming.

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Programming is how most procedures and processes

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get written down these days,

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and it's also a great way

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to engage students much more

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and to check they really understand.

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If you really want to check you understand math

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then write a program to do it.

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

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So to be clear, what I really am suggesting here

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is we have a unique opportunity

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to make maths both more practical

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and more conceptual, simultaneously.

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

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It's usually some kind of choice

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between the vocational and the intellectual.

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

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And we open up so many more possibilities.

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You can do so many more problems.

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What I really think we gain from this

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is students getting intuition and experience

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

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And experience of harder problems --

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

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feel it.

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We want people who can feel the math instinctively.

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That's what computers allow us to do.

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

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

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but now we can reorder it

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by how difficult it is to understand the concepts,

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however hard the calculating.

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So calculus has traditionally been taught very late.

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Why is this?

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

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But actually many of the concepts

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are amenable to a much younger age group.

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This was an example I built for my daughter.

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And very, very simple.

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We were talking about what happens

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when you increase the number of sides of a polygon

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to a very large number.

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And of course, it turns into a circle.

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And by the way, she was also very insistent

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on being able to change the color,

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an important feature for this demonstration.

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You can see that this is a very early step

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into limits and differential calculus

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

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

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Very simple example.

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That's a view of the world

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

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

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So one of the roadblocks we have

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in moving this agenda forward

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is exams.

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

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it's kind of hard to get the curricula changed

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to a point where they can use computers

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during the semesters.

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14:12

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

tedtalks 14:12
14:15

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

tedtalks 14:15
14:18

And then we can ask questions, real questions,

tedtalks 14:18
14:21

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

tedtalks 14:21
14:24

real questions that people have in their everyday lives.

tedtalks 14:25
14:27

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

tedtalks 14:27
14:30

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

tedtalks 14:30
14:32

How many years of protection do I need?

tedtalks 14:32
14:34

What does that do to the payments

tedtalks 14:34
14:37

and to the interest rates and so forth?

tedtalks 14:37
14:40

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

tedtalks 14:40
14:42

that should be asked in exams,

tedtalks 14:42
14:44

but I think it's a very important type

tedtalks 14:44
14:47

that right now just gets completely ignored

tedtalks 14:47
14:50

and is critical for people's real understanding.

tedtalks 14:50
14:53

So I believe [there is] critical reform

tedtalks 14:53
14:55

we have to do in computer-based math.

tedtalks 14:55
14:57

We have got to make sure

tedtalks 14:57
15:00

that we can move our economies forward,

tedtalks 15:00
15:02

and also our societies,

tedtalks 15:02
15:05

based on the idea that people can really feel mathematics.

tedtalks 15:07
15:10

This isn't some optional extra.

tedtalks 15:10
15:12

And the country that does this first

tedtalks 15:12
15:15

will, in my view, leapfrog others

tedtalks 15:15
15:18

in achieving a new economy even,

tedtalks 15:18
15:20

an improved economy,

tedtalks 15:20
15:22

an improved outlook.

tedtalks 15:22
15:24

In fact, I even talk about us moving

tedtalks 15:24
15:27

from what we often call now the "knowledge economy"

tedtalks 15:27
15:30

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

tedtalks 15:30
15:33

where high-level math is integral to what everyone does

tedtalks 15:33
15:35

in the way that knowledge currently is.

tedtalks 15:35
15:38

We can engage so many more students with this,

tedtalks 15:38
15:41

and they can have a better time doing it.

tedtalks 15:41
15:43

And let's understand:

tedtalks 15:43
15:46

this is not an incremental sort of change.

tedtalks 15:47
15:49

We're trying to cross the chasm here

tedtalks 15:49
15:51

between school math and the real-world math.

tedtalks 15:51
15:53

And you know if you walk across a chasm,

tedtalks 15:53
15:56

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

tedtalks 15:56
15:58

bigger disaster.

tedtalks 15:58
16:00

No, what I'm suggesting

tedtalks 16:00
16:02

is that we should leap off,

tedtalks 16:02
16:04

we should increase our velocity

tedtalks 16:04
16:06

so it's high,

tedtalks 16:06
16:09

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

tedtalks 16:09
16:12

of course, having calculated our differential equation very carefully.

tedtalks 16:12
16:14

(Laughter)

tedtalks 16:14
16:16

So I want to see

tedtalks 16:16
16:18

a completely renewed, changed math curriculum

tedtalks 16:18
16:20

built from the ground up,

tedtalks 16:20
16:22

based on computers being there,

tedtalks 16:22
16:24

computers that are now ubiquitous almost.

tedtalks 16:24
16:26

Calculating machines are everywhere

tedtalks 16:26
16:29

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

tedtalks 16:29
16:33

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

tedtalks 16:33
16:35

but what I am sure is

tedtalks 16:35
16:37

it's the mainstream subject of the future.

tedtalks 16:38
16:41

Let's go for it,

tedtalks 16:41
16:43

and while we're about it,

tedtalks 16:43
16:45

let's have a bit of fun,

tedtalks 16:45
16:48

for us, for the students and for TED here.

tedtalks 16:48
16:50

Thanks.

tedtalks 16:50
16:57

(Applause)