On Becoming a Math Whiz: My Advice to a New MIT Student(calnewport.com)
calnewport.com
On Becoming a Math Whiz: My Advice to a New MIT Student
http://calnewport.com/blog/2011/04/28/on-becoming-a-math-whiz-my-advice-to-a-new-mit-student/?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+StudyHacks+%28Study+Hacks%29&utm_content=Google+Feedfetcher
14 comments
I agree; "Don’t just sit and stare at it: think hard; until you’re exhausted; then come back the next day and try again." is bad advice. Don't just think about it, whatever that means to you; just thinking tends to get people's minds running in smaller and smaller circles-which is why going away and coming back the next day helps. But thinking, and working, differently will help you more and help you more quickly. Try diagramming the problem, try working a simplified or more specific example, try generalizing the problem to see if it fits something you learned in another context. There are many things you can do to work on a problem.
Polya's How to Solve It is mostly simpler types of math, it is intended for teacher training after all, but the general methods he demonstrates can often help with much harder and more complicated problems.
Wickelgren's How to Solve Mathematical Problems (originally titled, How to Solve Problems, the new title is more accurate) has both basic and more advanced tactics for problem solving.
The best way I have found to use these types of books, after reading them through quickly for an overview, is to stop and browse in one of them when you get badly stumped on a problem. Then go back to the problem; if you still can't make headway, stop and browse a bit more.
ADDED: Mathematics involves three distinct types of learning and work: learning the mathematical theory, which I generally find fairly easy. Learning and applying problem solving methods to apply theory to actual problems, which is much harder. And doing the calculations to solve the problems once you have worked out how to apply mathematics to the problem, which I find really, really hard, fortunately this is the easiest aspect to automate (calculators and Mathematica, for example).
Polya's How to Solve It is mostly simpler types of math, it is intended for teacher training after all, but the general methods he demonstrates can often help with much harder and more complicated problems.
Wickelgren's How to Solve Mathematical Problems (originally titled, How to Solve Problems, the new title is more accurate) has both basic and more advanced tactics for problem solving.
The best way I have found to use these types of books, after reading them through quickly for an overview, is to stop and browse in one of them when you get badly stumped on a problem. Then go back to the problem; if you still can't make headway, stop and browse a bit more.
ADDED: Mathematics involves three distinct types of learning and work: learning the mathematical theory, which I generally find fairly easy. Learning and applying problem solving methods to apply theory to actual problems, which is much harder. And doing the calculations to solve the problems once you have worked out how to apply mathematics to the problem, which I find really, really hard, fortunately this is the easiest aspect to automate (calculators and Mathematica, for example).
Good point.
I might not have been the best at math but I've sometimes been considered a "math whiz" - I took the undergraduate math seminar at UCLA when I was a High School senior.
I've always had the impression that what made people bad at math is the exactly the "grind" attitude - "focusing" on a problem only reduces your creativity. In fact, whenever I took this attitude, I became bad too.
Playing with a given problem every way you can is good. Enjoying a problem and finding it interesting is important. Grasping the concepts is good. Letting a given problem go whenever you can't solve it is good - I think there's a place below conscious awareness where problem solving can happen well.
But don't just grind on mathematics, that's poisonous.
I might not have been the best at math but I've sometimes been considered a "math whiz" - I took the undergraduate math seminar at UCLA when I was a High School senior.
I've always had the impression that what made people bad at math is the exactly the "grind" attitude - "focusing" on a problem only reduces your creativity. In fact, whenever I took this attitude, I became bad too.
Playing with a given problem every way you can is good. Enjoying a problem and finding it interesting is important. Grasping the concepts is good. Letting a given problem go whenever you can't solve it is good - I think there's a place below conscious awareness where problem solving can happen well.
But don't just grind on mathematics, that's poisonous.
Let me add (cynically):
1. When nothing else works, copy the examples and proofs from the text. Over and over again, thinking about what it all means, writing out the intermediate steps.
2. Hunt around the internet for classic and introductory texts on whatever, if you need it. Just like Cliff Notes, dont tell anyone, but use them anyway. Take your prof's recommendation of "introductory" of introductory with a grain of salt.
3. Brown nose the grad students and ask their advice about teachers and texts. Go with their consensus above the profs...
4. Put in more hours than anybody, but always get enough sleep and save three hours a week to excercise, and make sure you eat decently
1. When nothing else works, copy the examples and proofs from the text. Over and over again, thinking about what it all means, writing out the intermediate steps.
2. Hunt around the internet for classic and introductory texts on whatever, if you need it. Just like Cliff Notes, dont tell anyone, but use them anyway. Take your prof's recommendation of "introductory" of introductory with a grain of salt.
3. Brown nose the grad students and ask their advice about teachers and texts. Go with their consensus above the profs...
4. Put in more hours than anybody, but always get enough sleep and save three hours a week to excercise, and make sure you eat decently
You completely missed the kernel of wisdom:
come back the next day and try again
I personally grind away at something until I'm too tired to see straight because I don't know how to put something down until it's done, but several of my friends and at least one professor strongly advocate that approach as working quite well for anything requiring any amount of inspiration or insight.
come back the next day and try again
I personally grind away at something until I'm too tired to see straight because I don't know how to put something down until it's done, but several of my friends and at least one professor strongly advocate that approach as working quite well for anything requiring any amount of inspiration or insight.
another, which may be a hard pill to swallow if you've been 'good at math' all your life is:
- go to office hours and don't be afraid to ask for help from your TA and classmates.
- go to office hours and don't be afraid to ask for help from your TA and classmates.
This sounds very much like the advice that Polya gave in his famous book "How To Solve It" -- it's an old book, but still in print. I wish I'd known about that book when I was in college. (I first heard about it many years later when someone told me that at the time it was given to all new hires at Microsoft. Don't know if it still is.)
If so, it seems that Polya wasn't always as diligently persistent as he advised others to be. There's a famous story about G H Hardy seeing a bear in a zoo: it sniffed at the lock on its cage, hit the lock once with its paw, and then turned away to do something else. "He is like Polya: he has excellent ideas, but he does not carry them out."
One other tip I got was to make sure you understand your solutions. When you finally resolve a problem it is tempting (albeit sometimes necessary) to move straight on to the next problem without thoroughly understanding what you just did.
Forcing yourself to go over your own solutions, and in particular extracting the steps involved, can really consolidate your knowledge.
This is one of those hard work things.
Forcing yourself to go over your own solutions, and in particular extracting the steps involved, can really consolidate your knowledge.
This is one of those hard work things.
Aside from the learning the theory a lot of being good at math(s) is pattern matching type stuff and the intuition that goes with it. I suppose this is what the author is getting at, practice, practice....
"Young man, in mathematics you don't understand things. You just get used to them." - Jon von Neuman
One of the most reassuring quotes I know!
"Young man, in mathematics you don't understand things. You just get used to them." - Jon von Neuman
One of the most reassuring quotes I know!
How is that reassuring? If I'm learning something I don't want to just "get used" to it. I want a fundamental understanding.
It's reassuring precisely because it acknowledges that conundrum -- fundamental understanding often comes after a tough, rote (and uncertain) 'getting used to it.' -- so "stick with it," as it were.
I agree with that, because thats what it takes to internalize something, however the quote says understanding never comes. What I wanted to opine on is the idea that you shouldn't be satisfied with not understanding something just because someone tells you so.
You might be taking the quote too literally. I think what von Neumann was getting at is that the abstract world of mathematics is essentially artificial and alien to our natural sensibilities and that if you expect to 'understand' something to its very core you may be misguided. The deeper you go the more counter intuitive things can get.
It reassures me because it feels at times that the people around me who are so 'great' at mathematics are born naturals and that I may as well give up. For me is a great 'leveller' to hear that even the greatest minds struggle with these things, albeit to differing degrees.
It reassures me because it feels at times that the people around me who are so 'great' at mathematics are born naturals and that I may as well give up. For me is a great 'leveller' to hear that even the greatest minds struggle with these things, albeit to differing degrees.
There's quite a nice quote on the subject from Terence Tao (Livingston, Sir Ken; The Element; pp 100-101):
"I think the most important thing for developing an interest in mathematics is to have the ability and the freedom to play with mathematics -- to set little challenges for oneself, to devise little games, and so on. Having good mentors was very important for me, because it gave me the chance to discuss these sorts of mathematical recreations; the formal classroom environment is of course best for learning theory and applications, and for appreciating the subject as a whole, but it isn't a good place to learn how to experiment.
Perhaps one character trait which does help is the ability to focus, and perhaps to be a little stubborn. If I learned something in class that I only partly understood, I wasn't satisfied until I was able to work the whole thing out; it would bother me that the explanation wasn't clicking together like it should. So I'd often spend a lot of time on very simple things until I could understand them backwards and forwards, which really helps when one then moves on to more advanced parts of the subject.
I don't have any magical ability, I look at a problem, and it looks something like one I've already done; I think maybe the idea that worked before will work here. When nothing's working out then I think of a small trick that makes it a little better, but still is not quite right. I play with the problem, and after a while, I figure out what is going on. If I experiment enough, I get a deeper understanding. It's not about being smart or even fast. It's like climbing a cliff -- if you're very strong and quick and have a lot of rope, it helps, but you need to devise a good route to get up there. Doing calculations quickly and knowing a lot of facts are like a rock climber with strength, quickness, and good tools; you still need a plan -- that's the hard part -- and you have to see the bigger picture."
"I think the most important thing for developing an interest in mathematics is to have the ability and the freedom to play with mathematics -- to set little challenges for oneself, to devise little games, and so on. Having good mentors was very important for me, because it gave me the chance to discuss these sorts of mathematical recreations; the formal classroom environment is of course best for learning theory and applications, and for appreciating the subject as a whole, but it isn't a good place to learn how to experiment.
Perhaps one character trait which does help is the ability to focus, and perhaps to be a little stubborn. If I learned something in class that I only partly understood, I wasn't satisfied until I was able to work the whole thing out; it would bother me that the explanation wasn't clicking together like it should. So I'd often spend a lot of time on very simple things until I could understand them backwards and forwards, which really helps when one then moves on to more advanced parts of the subject.
I don't have any magical ability, I look at a problem, and it looks something like one I've already done; I think maybe the idea that worked before will work here. When nothing's working out then I think of a small trick that makes it a little better, but still is not quite right. I play with the problem, and after a while, I figure out what is going on. If I experiment enough, I get a deeper understanding. It's not about being smart or even fast. It's like climbing a cliff -- if you're very strong and quick and have a lot of rope, it helps, but you need to devise a good route to get up there. Doing calculations quickly and knowing a lot of facts are like a rock climber with strength, quickness, and good tools; you still need a plan -- that's the hard part -- and you have to see the bigger picture."
One of the most entertaining things I've read period, and it's by a mathematician talking about his coming of age:
http://www.xamuel.com/homeless-by-choice/
http://www.xamuel.com/homeless-by-choice/
Sounds like the Feynman algorithm:
1. Write down the problem.
2. Think real hard.
3. Write down the solution.
1. Write down the problem.
2. Think real hard.
3. Write down the solution.
He had a more practical "algorithm" as well:
"You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say, “How did he do it? He must be a genius!”
This works only for Feynman.
Similar advice by Paul Halmos on reading maths:
"Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"
To be impressive in a sea of impressive people like MIT, I don't think just hard focused work is enough. It's also important to consider nurturing your general cognitive abilities. Primarily, don't drink too much in college! For an extra edge, get cardiovascular exercise regularly. It leads to higher levels of neurogenesis.
from the article:
"Junior graduate students think senior graduate students are smarter, but they’re not: they simply have more practice.
Senior graduate students think junior professors are smarter, but they’re not: they simply have more practice."
My M.S advisor is a really hard worker, so he has a lot of experience doing research and is very good at it. For a while as an undergrad, I thought he was not particularly smart, just hardworking. However, after I started working on my first real problem as part of my thesis, I changed my mind. I realized that to work hard you actually need to be smart (of course there are many other factors related to time and money), otherwise it is very frustrating.
Senior graduate students think junior professors are smarter, but they’re not: they simply have more practice."
My M.S advisor is a really hard worker, so he has a lot of experience doing research and is very good at it. For a while as an undergrad, I thought he was not particularly smart, just hardworking. However, after I started working on my first real problem as part of my thesis, I changed my mind. I realized that to work hard you actually need to be smart (of course there are many other factors related to time and money), otherwise it is very frustrating.
More actionable advice would be to read George Polya's seminal work How to Solve it.
http://en.wikipedia.org/wiki/How_to_Solve_It
http://en.wikipedia.org/wiki/How_to_Solve_It
Sort of mirrors the 10,000 hours of practice theory.
That should really just be called the hours of practice theory- the 10,000 doesn't hold up to much scrutiny except for a few select cases. It's great if you want to write popular non-fiction though!
In some sense, yes. If you read this post of his though: http://calnewport.com/blog/2008/11/25/case-study-how-i-got-t... you get the sense that it's not only about brute forcing hours, but also spending those hours wisely. In the case of math, that was making sure he could understand and recreate every proof perfectly. This was probably a lot more effective than spending those hours, say, studying only problem sets or lecture notes.
The 10,000 hours theory though requires something called Deliberate Practice. The theory notes the difference between simply doing something a lot, versus doing something with the intent to get better, presumably with some form of feedback.
The 10,000 hours theory relates more to the human lifespan than any specific training method. If you can be really good at something at 25, then at most you had ~15 years * 52 weeks per year * ~40 hours a week of practice = ~30,000 hours. Now, change that to deliberate practice and your looking at around 10k hours. However, some things like go blow those numbers out of the water. You can focus your life on the game a 8 and still be improving at 50.
Well the Dan Plan is looking to show that it is about training. But I should also note that the theory isn't that you stop improving at 10,000 hours, but that's how many hours it takes to be an expert.
Maybe Go is an outlier. I don't know enough about the game. But I'd be surprised that someone who did 10k hours of deliberate practice wouldn't be pretty good by most metrics of the Go community.
Maybe Go is an outlier. I don't know enough about the game. But I'd be surprised that someone who did 10k hours of deliberate practice wouldn't be pretty good by most metrics of the Go community.
Go is likely similar to certain parts of language acquisition and musical talent in that there's a critical period involved. If you wait until you're an adult to start, not even 20k hours under a great teacher will give you real mastery. AFIK, every top player started training as a child.
I think what Newport is saying is that you have to work smart harder in order to get smarter and more hard-working. It's after the hard work and learning that the inspiration comes. The paradox is that if you want something to "come to you" you have to go after it first.
I am no math whiz but I did have one road to Damascus moment that helped me feel less daunted.
Proofs of theorems can be very intimidating. With a lot of effort I might eventually be able to understand and reproduce a proof. However I'd think that the mind that could come up with such a proof in the first place must be orders of magnitude smarter as some of intermediate steps would seem so unintuitive. As in what possessed him to try that route?
But what you have to remember that what you're seeing can be the result of years of effort, trial and error that eventually gets tidied up into a narrative that's analagous to sticking your arm into a haystack and picking out the needle in one smooth action.
It would help people a lot if this was pointed out more by teachers I think.
But what you have to remember that what you're seeing can be the result of years of effort, trial and error that eventually gets tidied up into a narrative that's analagous to sticking your arm into a haystack and picking out the needle in one smooth action.
It would help people a lot if this was pointed out more by teachers I think.
Being a math wiz really only pays off if you become a mathematician. Even a physicist doesn't really have to be a math wiz. Eg. being good with people is a much better thing to get good at and has a much wider payoff horizon.
(a) Why are these mutually exclusive? (b) “Math wizardry” in the context it’s being used here means something like “able to think and focus in a deep and prolonged way about the connections and patterns in numbers and structures”. This is about as useful a skill as you can develop in a huge number of fields, including most sciences and engineering disciplines, finance, architecture, mechanical jobs like auto repair, many kinds of art and music projects, some of the trickiest aspects of law and government, and so forth.
- Take a concrete case of the abstract (e.g. instead of trying to prove cauchy-schwarz for inner product spaces, prove it for R^n, and generalize).
- Strengthen your assumptions, then weaken them (prove something for finite dimensional vector spaces, then weaken your conditions to include all vector spaces).
- Try to find a counter-example of what you're trying to prove, and figure out why it's so difficult.
To me, posts like this (themed toward real analysis but parts of it generally informative): http://terrytao.wordpress.com/2010/10/21/245a-problem-solvin... are much more useful than posts that say "well, try hard".