How to Improve Mathematical Thinking and General Problem Solving Skills?

Question

I'm a sophomore in university and seriously feel that I'm bad at solving mathematical and algorithmic problems (be it discrete math, calculus or just puzzles). I noticed that I'm only good at solving questions that are similar to the ones that have been taught to us.

Here's how I generally approach it:

• What is the problem? What do I need to do here?
• Does it look like I've encountered this before?
• Can I think of a smaller problem to solve instead?

If the answer is no to all the above then I sort of blank out. I stare at it and force my brain to run through a wide variety of stuff, almost like a brute force attempt of solving it. Obviously that leads me to nowhere everytime. I simply can't think "outside the box."

What can I do to improve my situation?

Answer 1

I think proving theorems really develops your thinking. Try to prove a few important theorems from calculus as well as discrete math, or try to understand someone's proof. Of course, the more you know the better, so that is why we say math is not a spectator sport. You need to do more than just the homework if you want to improve. Sometimes many results that you learn in, say discrete math, might seem confusing, but once you see why they are important in a different context, for example in number theory or algebra, you should remember them. To be honest, I think understanding and being able to prove theorems is actually relevant to math, whereas puzzles are just for fun. The best advice I can give is: Do not try to memorize math and simply remember everything for an exam because that way you might get a good grade, but you will forget everything a few days after the exam, instead try to understand why something is true. This way you will remember something practically forever, because you will be able to derive it when you forget.

Answer 2

You might want to read Thinking Mathematically. (I read it and it's excellent. It will teach you exactly what you're looking for.)

Answer 3

I belonged to a school education system where we were made to do lots of different problems, but we were never told to try and understand the underlying theory behind the problems. This made me scared of math. What I basically had was a cookbook of a variety of wonderful recipes without realizing why I needed to add salt or sugar to a dish. May be you are facing the same problem? May be you are learning all these different techniques to solve problems without really understanding the theory behind why the problems can be solved using those techniques? Hence, because you don't understand the theory behind the techniques, once you get a problem that cannot be solved using the techniques you are familiar with, you get stuck.

While I agree with glebovg that trying to develop an intuition for how to write proofs is essential, I feel that you should make the effort to start reading proofs first. For instance, a book that really helped me understand Calculus was Spivak's Calculus. Try going through the proofs there, and learn the underlying theory. This is coming from someone who was in your position not too long ago.

I encourage you to read books that emphasize problem solving, but at some point you will just have muster the courage to open a book with proofs, and read through it.

Also, the issue of memorization is kind of a slippery slope. You will find that often even when you are trying to understand the theory, you will just have to memorize some computational techniques here and there. I think Terry Tao has a good post where he addresses the issue of memorization. I agree with him that certain basic things have to be memorized. For instance, you will have to memorize what the axioms of a group or a field are. I think memorization and understanding go hand in hand. Certainly your goal should not be to only memorize techniques to solve problems.

Here is more advice from a master:

http://terrytao.wordpress.com/career-advice/solving-mathematical-problems/

http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/

http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-grades-and-exams-and-methods/

http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/

All the best!

Answer 4

I don't know about puzzles, so I write only about solving mathematics problems. In my experiences in this site, I find it far easier to solve problems in a field(like abstract algebra) I know well than in a field(like analysis) I know less. I think it's like walking in a town. If the town is where you live, you know every corner and you think you can almost walk with blindfold. On the other hand, if you are new in the town, you lose your way easily.

So the question is how we know a field well. Read textbooks, understand proofs, try to prove a theorem before reading the proof of a textbook. Reconstruct a proof without seeing a textbook. do exercises, try to find examples and counterexamples, try to find problems by yourself and solve them, etc.