توضیح مختصر: 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, the way to get through the school system is to learn to act fast under pressure of time, and that's the last thing you can do when it comes to mathematics. 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.
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فایل ویدئویی:ویدئوی آموزشی درس « Optional Interview with Keith Devlin from Stanford, the NPR "Math Guy" »
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Doctor 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, BrainQuake, 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, Dr. Devlin. Thanks for being with us here today, Doctor 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. That’s a 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’s about. What are you trying to do? What do you want to achieve? And, what information do you have? You must have, you, I mean you’ve really gotta 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’s a geometric problem, do some little diagrams or you, I often end up writing simple little graph diagrams with dots and lines connecting them, just some things 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’re in luck, the problem’s solved. Actually that’s usually very unsatisfying because you’ve just applied someone else’s thinking. Mathematics gets interesting when your, you have to force yourself to find a new, a new method or some new approach. So, lets assume that none of the techniques you know work. The next thing is can I adapt an existing technique. Well, maybe you can, maybe you can’t. If you can that usually leaves you with a nice feeling because you’ve done something yourself. The really challenging ones are the ones where it gets actually exciting and fun is where you, you 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, you, you basically, you bash away at it several times which just naive approaches almost certainly they won’t work. You just try this and you try that. You, you come up with ideas based on what the problem’s asking you to do because solving a problem is a journey. You begin with the information you’ve got, either the information your 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 and you have to get to the goal and solve the problem. If none of the methods you try for the first oh, half hour, maybe 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 really came up against it. I would put on my running gear, whatever the weather and I’d go out for a three or four hour run, and when I came back, A, I felt good, I felt relaxed, and quite often on the course of ride or that, that, that run or 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 ride a bicycle. But I go on my bike and I go for a three or four hour bike ride and I come back, and first of all I can approach the problem with a fresh eye and secondly in some cases what seemed to be an impassible problem just 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 oh, duh, why didn’t I think of that before. And if that doesn’t work you start the cycle again. You bash your way at it again, make no progress and then you go, you go and have another bike ride or a swim or whatever it is you 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. That sometimes, especially when it’s something difficult, it’s as 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. Okay, I, it, it, for, 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 you think about it, 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’ll mention physical activity because I’m a sort of 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 sort of exert lots of energy, you have to do something different. Having a bath is a famous one among 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. And that’s exactly the kinds of ideas that we’re talking about in in our course. 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. It, that’s so important. Well 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, how does that relate to areas of learning that are outside mathematics? Yeah, and, and this is really where you’ve got unlearn lots of the things that you’re taught at school. And actually you’re taught, 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 in, with timed tests. The, the way to get through the school system is to learn to act fast under 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. It’s not, I mean people have different speeds, I, 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, his teachers at school thought she wasn’t good at mathematics because she was so slow. Well, she may be slow, but she now has a, the equivalent of a Nobel prize in mathematics and, 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, you know 2 or 3,000 years, so your bringing what’s essentially a stone age brain to a domain that is a couple thousand years old, In fact, most mathematics is only a few decades old. So, how can you take this brain, that evolved to survive in the wilds 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’s 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. You know, we’re very good at solving social problems and problems about our environment, because evolution set us up for that. So, somehow we have to figure out how to let that brain solve that abstract problem in mathematics. And 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, that, that serve us well all of the time actually come into play for a math problem. Now it, there’s no proof of that, but that’s certainly my own experience of what goes on. Yes. Well, all I can say is, three cheers for slow thinking. And I love the idea of thinking about math and, and a difficult problem as being something that’s a member of your family. Yeah, it’s a, you know, it. Math in many ways mathematics problems are not, it’s not, they’re not fundamentally different from other problems except in one respect they’re about totally abstract things. So, someone solving their math problems has an initial problem that someone doesn’t have if they’re solving their problem in real life or a coaching problem in, in sort of football. Or, or, or, in a, you know, most of, I used to spend a lot of time rock climbing, and rock climbing is, is partly physically, 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. And in the case of mathematics however, the world in which you’re solving those problems isn’t one we are familiar with. You initially have to create that, that world inside of you, and 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 demand, with that problem to such an extent that, that problem is just like a problem I’d have with my family. or, or, or my work place or whatever. I’ve got all of this apparatus for solving real world problems and social problems, and 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, and the only way to do that is just keep living in the problem, get inside the problem. Really live inside that problem for ten 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. And one of the things that Dr. Sejnowski and I emphasize, quite strongly in our course is how exercise helps actually build neural structures. It, it allows you to learn better, it allows you to remember better. So, how, how does exercise play a role in your own ability to do mathematics? That’s that’s a good one. But it certainly is the case for me. In fact, when I was a PhD student and my closest colleague was a, was a, was a, a mathematician from from the United States, came over to England when I was a student back in Bristol. And he was a rock climber, in fact I learned to rock climb with him. And we used to go on these three or four hour rock climbs. And we would almost certainly, by the time we got to the, 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, you’re tied up and then you chat for a while as you sort of belayed onto the rock face. And we’d talk about mathematics and then we’d forget the mathematics and we go and do the physical stuff of climbing. And when you’re climbing, as when, as when you’re bicycling very fast your mind is totally focused on that physical activity, that’s your your entire world is physical. So, you’re focused on the climbing or you’re focused on riding your bike, but stuff is going on in the background in your mind. It’s working on those things that you talked about. And we’d finish your pitch, we’d get ourselves back on belay we 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, 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 you know, 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, you know, the mind did evolve to help the body survive, and I think it’s a mistake to try to think of those as separate. If you nurture the body the mind follows. I, you know, this, 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, that’s a, a somewhat more relaxing case. But even in the bath it’s a really, you know, it’s a, it’s a, it’s an artificial environment in which you are relaxed. You’re in water, you’re doing things, you’re fresh, 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, the bath example not withstanding, physical activity plays a big role. Exactly why I don’t know, but I do know it works. [LAUGH] Well there is actually good research evidence. They’re uncovering the, the biophysiological mechanisms by which it does take an effect. but, I, I have to ask, with everything that you do. Do you ever catch yourself, it’s hard to believe but do you ever procrastinate, and if you do, what do you do to get yourself out of it? Yeah well first of all I do do many things you know, I’ve got my own educational technology company, I do lots of biking I, I run an institute at Stanford, I do my teaching, I do, I’m actually on Twitter a lot, I do a whole bunch of things. You might think I’m a multitasker. 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 ten 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, for my educational video game that we launched last fall. We wanted to bring out a different version. I worked nonstop on that for 4 days. I didn’t do anything else other than eat, sleep, and bicycle ride. And my entire focus, everything else went to one side, was to get that thing done, and 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, that, that friends and family are still by me, because this is very focused and it excludes everything else. You know, I must be really difficult to live with at those times, but everything gets excluded. I focus on it and then I, there comes an end. Now, either because I finished the job or I just get tired and want to switch and, and I’m really getting, getting frustrated with it, then I’ll switch and do something else. And 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. The procrastination comes from recognizing that I need to make that switch. I’ll keep going and keep going and think, I want to this other thing, I want to do this other thing. And I do find myself doing lots of other little things. I’ll go and start and 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 process and it’s slowed down and you’ve got to get in another groove. And I know that that takes will power 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. And that I found works extremely well. That’s, 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 will power, and that seems to be exactly what you do. And it’s 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 may not want to write that blog post, I know in the back of my mind that the moment I get into it and have 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 these hiatus moments when, when, I procrastinate. It’s the, the, the switching gears to something else that’s, that’s, 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. And 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, I like to do is I think of it, 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 the funnel at a time. And so you just gotta give it up and realize one thing is all you can be focusing on. And 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. Yeah, I mean, one thing I do do is when I actually just upgraded my computer system. to a new Macintosh. The first thing I did is went into the into the system controls and switched off every single alert. I don’t want visual alerts, I don’t want noise alerts, I don’t want know if an email has come, I don’t want to know if something’s going on. I literally switch off all the alerts because I don’t 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, little beep going off to tell me some email has arrived. I don’t want to know then because that’s a distraction. I want to be in the world of writing that article because if you take this human brain, and you put it into a problem, into a domain, and you set it free, it loves what it’s doing, and it doesn’t want to give up, and it sure doesn’t want to be distracted. And so, I don’t distract it. I just let it do what it does. Well, I, I, I thank you so much, Doctor Devlin for sharing your wisdom with us, here today. It’s just been a great pleasure talking to with you, and I know our viewers will really, really learn a lot from what you had to say. Okay, the pleasure’s been all mine, absolutely. Thanks, so much.