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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;

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.)
- gjm11 15y ago
  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."
billswift 15y ago
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).
- joe_the_user 15y ago
  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.
forkandwait 15y ago
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
rojoca 15y ago
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.

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.)

gjm11 15y ago
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."

billswift 15y ago

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).

joe_the_user 15y ago
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.

forkandwait 15y ago

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

rojoca 15y ago

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.