39 - Optional Interview with Keith Devlin

دوره: Coursera – Learning How to Learn / درس 39

Coursera – Learning How to Learn

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39 - Optional Interview with Keith Devlin

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Dr. Keith Devlin is a very big guy in the world of math education,

not to mention the world of math research.

I’m introducing you to him here because I love

how he thinks about learning in the world of mathematics,

and how his ideas about learning relate to learning in

all sorts of topics way outside of mathematics.

Among many other duties,

Keith is a Co-founder and executive director of

Stanford’s human sciences and technologies advanced research institute.

A co-founder and president of an educational technology company,

Brain Quake, that creates mathematical learning video games.

He’s also a World Economic Forum fellow,

a fellow of the American Association for the Advancement of Science,

and a fellow of the American Mathematical Society.

Last but definitely not least,

he is ‘‘The Math Guy’’ on national public radio.

Welcome Doctor Devlin.

Thanks for being with us here today Dr. Devlin,

let’s just dive right in.

What’s the biggest tip you can share with our viewers about how to

most effectively approach a math problem that they might be stuck on?

Well, the first thing of course is don’t panic.

Not-so-famous Douglas Adams motto from The Hitchhiker’s Guide to the Galaxy,

and it holds just as much for mathematics as it did for for the Hitchhiker’s Guide.

You have to begin by understanding what the problem is about,

what are you trying to do?

What do you want to achieve?

And what information do you have?

You’ve really got to avoid the thing that actually teachers try to

impress on you of trying to see if you can match it to some template,

and to act quickly.

One of the worst things that I think happens in the school system is

people come away thinking that solving

math problems is something that has to be done at speed.

It absolutely is not.

So, you have to sit back,

take a deep breath, begin by understanding the problem.

If the problem involves numbers,

put in a few simple numbers threes and fours,

and just play with it to get a sense of it.

If it is geometric problem,

do some little diagrams,

I often end up writing simple little graph diagrams with dots and lines connecting them,

just something to give me a sense of what that problem’s about.

At that point, you can ask yourself,

'’do I know any techniques that look as though they would work with this problem?


If they do and you mean look, the problem is solved.

Actually that’s usually very unsatisfying

because you’ve just applied somebody else’s thinking.

Mathematics gets interesting when you have to force

yourself to find out when a new method or some new approach.

So, let’s assume that none of the techniques you know work.

The next thing is, ‘‘can I adapt an existing technique?


Maybe you can maybe you can’t.

If you can usually leaves you with a nice feeling because you’ve done something yourself.

The really challenging ones and the ones where it gets actually exciting and fun,

is where it becomes clear,

at least you think that nothing you know is going to help you with that.

At that point, what you do is,

you bash away at it several times with just naive approaches.

Almost certainly they won’t work,

you just try this, you try that.

You you come up with ideas based on what the problem is asking you to do,

because solving a problem is a journey.

You begin with the information you’ve got.

Either the information you’re given with the problem,

or information you already know.

You may have to add some extra information,

you may do a Google search find some extra information about it.

At that point, the journey

takes place in that you have to get to the goal and solve the problem.

If none of the methods you try for the first half hour,

maybe I’m depending on the problem.

It could be half hour, couple of hours,

sometimes it’s a day or a few weeks.

But if nothing happens,

then the best thing to do is walk away from the problem.

Go and do something else.

In my case, for most of my career,

I was a long-distance runner.

If I verily came up against it,

I would put on my running gear, whatever the whether,

and I’d go out for three or four of hour run,

and when I came back, I felt good,

I felt relaxed, and quite often on the course of that ride,

on that mono, when I got back,

the problem seemed different,

and I had some new ideas.

As I got older, my knees gave away from 25 years of heavy duty running.

These days I’ve added bicycle.

But I go on my bike, and I go for three or four hour bike ride, and I come back.

First of all, I can approach the problem with a fresh eye.

Secondly, in some cases,

what seemed to be an impossible problem has melted away.

I look at it and think, ‘‘oh, gee,

that was obvious, why didn’t I see it before?


Well, I don’t quite think that,

because I know that the feeling you have when you solve a problem is always,

'’Ooh, dah, why didn’t I think of that before?


If that doesn’t work, you start the cycle again.

You bash away to it again, make no progress,

and then you got to have another bike ride,

or another swim, or whatever it is you’d like to do, a nice long walk.

But it’s important to keep oscillating between bashing away at it,

and then backing off and doing something else to let the brain do it’s own thing.

I think that was one of the hardest things for me to

realize was how important it is to back away.

Sometimes, especially when it’s something difficult,

it is important to be able to focus on it,

as it is to be able to back away from it and then come

back later with a fresh perspective and a fresh mind.

Oh yeah.

For difficult problems,

I have never ever solved a difficult problem

sitting at a desk or even sitting in my chair thinking about the problem.

I’ve never done that, and I’m not sure it’s possible to do that,

because if a problem solution requires something new,

applying all of your conscious thinking by definition can’t solve it,

because you’re looking for something new.

The weird thing about the brain is when you’ve done

that preparation of bashing away without success,

and you go and do something else.

I’ve mentioned physical activity because I’m a physical active guy,

although the biggest result in my PhD,

was actually when I was getting out of the bath.

I’d taken a long bath,

you don’t have to exert lots of energy.

You have to do something different.

Having a bath is

a famous one amongst mathematicians for solving problems.

But somehow, weird stuff happens inside the brain.

When you’re doing something else,

and what three hours ago seemed like an impossible problem,

just melts away and everything just falls into place.

The brain sorts it out when it’s left alone to do it.

That’s exactly the kinds of ideas that we’re talking about in in our course.

But it is important to do that initial work.

It’s no good just going out for a bike ride,

you’ve got to have gone through that process of

really coming up against it and seeing everything that doesn’t work,

because that seems to set up the mind in

order to do this mysterious stuff it does when you’re engaged in something else,

when your conscious mind is engaged in something else.

Absolutely, that’s so important.

One thing that you’ve described is that in higher mathematics,

there should be a lot less doing and a lot more thinking.

Can you tell our viewers what you mean by that,

and how does that relate to areas of learning that are outside mathematics?


This is really where you’ve got to

unlearn lots of the things that you’re taught at school.

Actually, teachers do that for very good reasons because

of the way we especially in the United States and parts of Europe,

the way we subject people to endless test-taking with timed tests.

The way to get through the school system is to learn to act fast on the pressure of time,

and that’s the last thing you can do when it comes to mathematics.

You’ve really got to let it take its course.

People have different speeds.

I’ve worked with some very fast mathematicians.

The young math professor from Stanford who won a Fields Medal recently,

she has gone on record as saying,

her teachers at school thought she wasn’t good at mathematics,

because she was so slow.

Well, she may be slow but she now has the equivalent of

a Nobel Prize in mathematics and congratulations to her for doing that.

You really have to let it take its time.

It’s a slow process.

You’ve got to remember that mathematics is relatively recent,

two or three thousand years.

So, you’re bringing what’s essentially a stone age brain to a domain,

there’s a couple of thousand years old.

In fact most mathematics is only a few decades, or so.

How can you take this brain that evolved to survive in

the wild and then more recently to survive in social environments,

how can you take that brain and apply it to solve this abstract problem in mathematics?

Well, the answer is, we don’t really know how we do that.

What we do know, is that if you familiarize yourself with

the problem to such an extent that that problem is like a member of your family,

then the act of solving that math problem is actually not

unlike solving a difficult problem you have at home.

We’re very good at solving social problems and problems about our environment,

because evolutions set us up for that.

So, somehow we have to figure out how to

let that brain solve that abstract problem in mathematics.

The act of getting inside the problem and thinking about it for a long period of time,

so that it’s so familiar.

Then the natural circuits in the brain that

serve us well all of the time actually come into play for a math problem.

There’s no proof of that, but that’s certainly my own experience of what goes on.

Yes, all I can say is three cheers for slow thinking and I love

the idea of thinking about math and

a difficult problem as being something that’s a member of your family.

Yes, in many ways,

mathematic’s problems are not

fundamentally different from other problems except in one respect,

they are about totally abstract things.

So someone solving a math problem has

an initial problem that someone doesn’t have if they’re solving a problem in real life,

or a coaching problem in sort of football.

Most of the time, and I used to spend a lot of time rock-climbing.

Rock-climbing is partly physical,

but it’s also a lot of problem solving,

because you’re having to look for these moves,

you’re having to move your body in the right way.

So there’s an awful lot of problem solving that we

do all of the time and that’s what the brain evolved to do.

In the case of mathematics however,

the world in which you are solving those problems isn’t one we’re familiar with.

You initially have to create that world inside of you.

This I think, is why it’s essential to knock away at the problem for 10-15 minutes,

a day, two days before you let the brain do it’s own thing.

My guess and my perception of what’s going on is that

that process is making my mind familiar with that domain,

with that problem, to such an extent that that problem is just like

a problem I’d have with my family or my workplace or whatever.

I’ve got all of this apparatus for solving real-world problems and social problems.

Once that math problem is inside my mind,

it’s another problem just like that.

So, solving the math problem usually isn’t the hard part.

The hard part is getting that problem familiar inside your mind.

The only way to do that is just keep living in the problem,

get inside the problem,

really live inside that problem for 10 minutes,

half an hour, maybe even a few days.

In the case of Andrew Wiles,

solving Fermat’s last theorem, a few years ago,

it took him seven years to really get inside

that problem to the point where he could see the solution.

Well, part of what you do when you’re spending a lot of time with a problem is you’re in

some sense reifying and growing

the neural structures that are related to what you’re trying to think about.

One of the things that Dr. Sinalski and I emphasize quite strongly in

our course is how exercise helps actually build neural structures.

It allows you to learn better,

it allows you to remember better.

So how does exercise play a role in your own ability to do mathematics?

That’s a good one,

but it certainly is the case for me.

In fact when I was a PhD student,

my closest colleague was a mathematician from the United States,

came over to England when I was a student back in Bristol.

He was a rock climber, in fact I learned to rock climb with him.

We used to go on these three or four hour rock climbs

and we would almost certainly by the time we got to the top of the climb,

we’d done some mathematics.

Because you do a climb and you get to

a ledge and you’re tied up and then you chat for a while,

as you’re sort of belayed onto the rock face.

We’d talk about mathematics and then we’d forget the mathematics,

we don’ do the physical stuff of climbing.

When you’re climbing, as when you’re bicycling very fast,

your mind is totally focused on that physical activity.

So your entire world is physical,

so you’re focused on the climbing or you’re focused on riding your bike.

Stuff is going on in the background in your mind,

it’s working on those things that you’ve talked about.

We’d finish a pitch, we’d put ourselves back on belay,

we’d talk about the problem and we’ve made progress on the problem.

So as a graduate student,

I discovered this method of thinking,

doing, thinking, doing, was incredibly powerful.

I can make conjectures and sort of self

reflect as to why the physical activity is important.

I mean aspects clearly are you’re focusing on that activity, it’s very refreshing.

We tend to think these days that the mind is separate from the body.

I don’t buy that it certainly isn’t the case for me.

I find that if my body’s tired,

my mind is tired and vice versa they stimulate each other.

So the mind did evolve to help the body survive,

and I think it’s a mistake to try and think of those as separate.

If you nurture the body,

the mind follows this goes back

to the ideas of having physical education in schools and so forth.

This is hardly original,

but I think we need to take it seriously.

I’ve never ever, well I have solved problems in the bath,

that’s a somewhat more relaxing case.

But even in the bath, it’s an artificial environment in which you’re relaxed.

You’re in water, you’re doing things, you’re freeing up the mind,

the mind is absolutely not thinking about that problem,

it’s doing other things.

In my case definitely the bath example not withstanding,

physical activity plays a big role.

Exactly why?

I don’t know,

but I do know it works.

Well there is actually good research evidence.

They’re uncovering the bio physiological mechanisms by which it does take an effect.

But I have to ask,

with everything that you do,

do you ever catch yourself,

it’s hard to believe, but do you ever procrastinate?

If you do, what do you do to get yourself out of it?

Yeah, well first of all I do do many things,

I’ve got my own educational technology company,

I do lots of biking.

I run an institute at Stanford,

I do my teaching,

I’m actually on Twitter a lot.

I do a whole bunch of things.

You might think I’m a multi-tasker,

I absolutely do not multitask.

Not least because I know from the research that’s been done here at Stanford,

a lot of it by my late colleague Cliff Nass.

That shows that a, human beings cannot multitask,

and b, the more you think you’re good at multitasking the worse you are at doing it.

So, I absolutely don’t try to use this serial processor to multitask.

What I do is I serial task,

I will work exclusively flat out for periods of time.

Now they may only be 10 minutes,

they may be half an hour or they may be.

I mean recently I had to develop a whole set of 75 puzzles,

a new set of puzzles for my educational video game that we launched last fall.

We wanted to bring out a different version.

I worked nonstop on that for four days.

I didn’t do anything else other than eat,

sleep and bicycle ride and in

my entire focus everything else went to one side was to get that thing done.

Four days later I had the 75 puzzles

worked and tested and everything and then I moved to something else.

So what I do is I work very intensely,

and very focused for periods of time.

I mean it’s amazing that friends and family are still by me,

because this is very focused and it excludes everything else.

I must have been really difficult to live with at those times.

But everything gets excluded, I focus on it,

and then comes an end either because I finished the job or I just get

tired and want to switch and I’m really getting frustrated with it,

then I’ll switch and do something else.

What I’ve learned to do very efficiently is switch from one to the other,

it doesn’t take me more than a few minutes,

at most, to switch from one to the other.

Procrastination comes from recognizing that I need to make that switch.

I’ll keep going and keep going I think

I want to do this other thing, I want to do this other thing.

I do find myself doing lots of other little things I’ll go and start off

read my Twitter feed and things for a while before making the switch.

So the one moment where I will

procrastinate is starting something new, for obvious reasons.

You’ve been in the groove,

you’ve been making progress and it’s slowed down and you’ve got to get in another groove.

I know that that takes willpower and effort and

like everybody else I do have a procrastination streak.

But the only time it affects me is making that switch.

Once I’ve done it, I’m away in that domain until I’ve

either solved that issue or done it or completed it or got tired of it and need a break.

That I found works extremely well.


That’s actually something that we talk about in this course is

just that idea of you don’t need to use willpower all the time,

you just need to put it one little place wherever you need,

where you’re getting that cue,

that’s where you need to be applying willpower.

That seems to be exactly what you do.

It is true that I’m fortunate to have been able to carve out a career for myself,

where most of the time I’m doing stuff I really want to do.

So even though I might not want to write that blog post,

I know at the back of my mind that

the moment I get into it and I’ve written the first sentences,

writing that blog post will be the most important,

and most wonderful and rewarding thing in the world until it’s finished.

But then I have to sort of stop that and do something else,

and that’s when there’s these hiatus moments when I procrastinate.

It’s the switching gears to something else that I find,

I won’t say I find it difficult because I’ve been doing it all my life.

But you do need willpower to do that.

At least I find I need willpower to do that.

If I’m not careful it will just fritter away with social media or whatever.

I feel the same way in how I handle things.

One thing I like to do is I think of my life as like this enormous funnel where there’s

all these things that are demanding to be done

but then only one thing can go through that funnel at a time.

So, you just got to give it up and realize one thing is all you can be focusing on.

If you have that approach it makes it a

little easier to realize that you just can’t do everything,

you just do as much as you reasonably can.


I mean one thing I do do is when I actually

just upgraded my computer system to a to a new Macintosh.

The first thing I did was went into

the system controls and switched off every single alert.

I don’t want visual alerts,

I don’t want noise alerts.

So I don’t want to know if an email has come,

I don’t want to know if something is going on.

I literally switch off all the alerts because I do not want things pushed to me,

I want to go and find out things when I need them.

Because the last thing I want if I’m in the middle of working on an article say

is to have some little beep going off to tell me some emails has arrived.

I don’t want to know that,

because that’s a distraction I want to be in the world of writing that article.

Because if you take the human brain and you put it into a problem,

into a domain and you set it free,

it loves what it’s doing,

it doesn’t want to give up and that sure doesn’t want to be distracted.

So, I don’t distract it I just let it do what it does.

Well, I thank you so much Dr. Devlin for sharing your wisdom with us here today.

It’s just been a great pleasure talking with you and I

know our viewers will really learn a lot from what you had to say.

Okay the pleasure is been all mine, absolutely.

Thanks so much.

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