5 ms·

On Becoming a Math Whiz: My Advice to a New MIT Student

81 points by da5e 15y ago

anonlawstudent 15y ago
Sort of mirrors the 10,000 hours of practice theory.
- vecter 15y ago
  In some sense, yes. If you read this post of his though: http://calnewport.com/blog/2008/11/25/case-study-how-i-got-the-highest-grade-in-my-discrete-math-class/ 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.
  - nsfmc 15y ago
    the adage i learned in grad school for this was work smarter, not harder.
    - deleted 15y ago
      [deleted]
  - kenjackson 15y ago
    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.
    - Retric 15y ago
      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.
      - kenjackson 15y ago
        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.
        xiaoma 15y ago
        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.
- radicaldreamer 15y ago
  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!
benthumb 15y ago
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/
- orborde 15y ago
  This blog is amazing. Thanks for posting it.
  - benthumb 15y ago
    That's what I thought, too. Glad I'm not the only one who appreciates it...
davidtgoldblatt 15y ago
What I don't like about this sort of advice is that it's almost completely non-actionable. "Don’t just sit and stare at it: think hard; until you’re exhausted; then come back the next day and try again" is the piece of advice to the stuck student. The distinction between "think hard" and "look at the problem and wait for a flash of inspiration" isn't something that a beginning math student will be able to see. A better set of advice for a stuck math student might be: - 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-solving-strategies/ http://terrytao.wordpress.com/2010/10/21/245a-problem-solvin... are much more useful than posts that say "well, try hard".
- nsfmc 15y ago
  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.
- sliverstorm 15y ago
  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.
- greenyoda 15y ago
  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.)
bodski 15y ago
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!
- Entaroadun 15y ago
  How is that reassuring? If I'm learning something I don't want to just "get used" to it. I want a fundamental understanding.
  - spinchange 15y ago
    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.
    - Entaroadun 15y ago
      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.
      - bodski 15y ago
        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.
akdubya 15y ago
Sounds like the Feynman algorithm: 1. Write down the problem. 2. Think real hard. 3. Write down the solution.
- vinoct6 15y ago
  This works only for Feynman.
- lkozma 15y ago
  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!”
chegra84 15y ago
I just did all the questions in the Maths and Statistics book.(Guaranteed A+) Avoid books that don't have practice exercises.
- mg74 15y ago
  Which "Math and Statistics" book?
Maro 15y ago
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.
- jacobolus 15y ago
  (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.
xiaoma 15y ago
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.
VB6_Foreverr 15y ago
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.
astrec 15y ago
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."
atakan_gurkan 15y ago
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.
lkozma 15y ago
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?"
sireat 15y ago
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
da5e 15y ago
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.

81 points by da5e 15y ago

anonlawstudent 15y ago

Sort of mirrors the 10,000 hours of practice theory.

vecter 15y ago
In some sense, yes. If you read this post of his though: http://calnewport.com/blog/2008/11/25/case-study-how-i-got-the-highest-grade-in-my-discrete-math-class/ 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.
- nsfmc 15y ago
  the adage i learned in grad school for this was work smarter, not harder.
  - deleted 15y ago
    [deleted]
- kenjackson 15y ago
  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.
  - Retric 15y ago
    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.
    - kenjackson 15y ago
      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.
      - xiaoma 15y ago
        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.
radicaldreamer 15y ago
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!

benthumb 15y ago

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/

orborde 15y ago
This blog is amazing. Thanks for posting it.
- benthumb 15y ago
  That's what I thought, too. Glad I'm not the only one who appreciates it...

davidtgoldblatt 15y ago

What I don't like about this sort of advice is that it's almost completely non-actionable. "Don’t just sit and stare at it: think hard; until you’re exhausted; then come back the next day and try again" is the piece of advice to the stuck student. The distinction between "think hard" and "look at the problem and wait for a flash of inspiration" isn't something that a beginning math student will be able to see. A better set of advice for a stuck math student might be: - 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-solving-strategies/ http://terrytao.wordpress.com/2010/10/21/245a-problem-solvin... are much more useful than posts that say "well, try hard".

nsfmc 15y ago
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.
sliverstorm 15y ago
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.
greenyoda 15y ago
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.)

bodski 15y ago

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!

Entaroadun 15y ago
How is that reassuring? If I'm learning something I don't want to just "get used" to it. I want a fundamental understanding.
- spinchange 15y ago
  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.
  - Entaroadun 15y ago
    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.
    - bodski 15y ago
      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.

akdubya 15y ago

Sounds like the Feynman algorithm: 1. Write down the problem. 2. Think real hard. 3. Write down the solution.

vinoct6 15y ago
This works only for Feynman.
lkozma 15y ago
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!”

chegra84 15y ago

I just did all the questions in the Maths and Statistics book.(Guaranteed A+) Avoid books that don't have practice exercises.

mg74 15y ago
Which "Math and Statistics" book?

Maro 15y ago

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.

jacobolus 15y ago
(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.

xiaoma 15y ago

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.

VB6_Foreverr 15y ago

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.

astrec 15y ago

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."

atakan_gurkan 15y ago

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.

lkozma 15y ago

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?"

sireat 15y ago

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

da5e 15y ago

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.